**A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems**

M.H. Fazel Zarandi, Fereidoon Moghadas Nejad and H. Zakeri

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/47798

## **1. Introduction**

84 Fuzzy Controllers – Recent Advances in Theory and Applications

PhD. Thesis, Texas A&M University, Texas, USA.

[17] B. Sing, K. Jain (2003) Implementation of DSP based digital speed controller for

permanent magnet brushless dc motor. IE(I) Journal-EL., Vol. 84, pp. 16-21. [18] J.R. Rice (1983) Numerical Methods, Software, and Analysis. McGraw-Hill, New York. [19] T. Kim (2003) Sensorless Control of the BLDC Motors from Near-Zero to Full Speed.

> This chapter focuses on the basic concepts of novel fuzzy sets, three dimensional (3D) memberships and how they are applied in the design of type-1 and type-2 fuzzy thresholding in control systems. Automatic fuzzification and membership functions shape selection play a crucial role in fuzzy thresholding design and finally determination of outputs via defuzzification. The related methodology and theoretical base will be discussed in depth, using real examples in automatic control (Pavement distress detection and classification). In spatial domain, selection of membership functions is a difficult task. It should be noted that selection of a supper membership function is a golden key. This is one of the major aims of this chapter to introduce a robust method to consider the uncertainty of membership values by using flexible thresholding for controller problems.

> In direct approach to fuzzy modeling, deep knowledge of expert plays a key role for membership functions generation. In application, ambiguity of membership function assignment is the main problem with fuzzy sets and systems. So, different fuzzy membership functions may have various impacts on the systems and, then, different thresholds in control problems.

> To solve this problem, type II fuzzy thresholding is recommended. The upper and lower membership functions promote this dilemma; however the figure of uncertainty (FOU) has a fixed value that is equal to one, in upper and lower membership function. So, Type-2 fuzzy logic can effectively improve the control characteristics using FOU of the membership functions.

> In this chapter, a smart thresholding technique with its application will be presented, which processes threshold as flexible type-2 fuzzy sets. The concept of ultra-fuzziness aims at

© 2012 Zarandi et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

capturing/eliminating the uncertainties within fuzzy systems using regular (type I) fuzzy sets. A measure of 3D ultra-fuzziness is also presented. Several Experimental results are provided in order to demonstrate the usefulness of the proposed approach.

A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems 87

computations stand in front of extension T2 FSs in vast scale applications. Interval type-2 fuzzy sets (IT2 FSs) are proposed to reduce the complexity (Choi and Rhee, 2009). Many algorithms based on the T2 FMF have been proposed. (Choi and Rhee, 2009), (Hagras,2004), (Hwang. Rhee, 2004), (Hwang. Rhee, 2007), (John, 2000), (Karnik, J. Mendel, 1999), (Liang et al. 2000), (Liang, J. Mendel, 2001), (Makrehchi, et al. 2003), (Rhee, 2007), (Rhee, Choi, 2007),

In this chapter, we focus on the generation of 3D Polar fuzzy Memberships functions to use in hybrid expert system for systematic pavement distress detection and classification. In particular, we consider 3D polar type-1 fuzzy membership functions (3D T1 PMFs) that are generated from sample images and then developed to 3D polar type-2 fuzzy membership functions (3D T2 PMFs). First, we review three methods based on heuristics, histograms, and interval type-1 fuzzy C-means (IT1 FCM). For each method, the footprint of uncertainty (FOU) is only required to be obtained, since the FOU can completely describe a T1 PMF. We proposed two methods based on 3D domain and then 3D polar under the theory of type 2

In Section 2, we briefly review basic concepts and existing methods and background. In Section 3, we managed the IT2 FMF generation methods. In Section 4, concepts of polar fuzzy are discussed and we explain how our proposed IT2 PMF generation methods can be implemented. Section 5 approximate reasoning and fuzzy inference discussed. Finally,

The extension of T1 FSs to T2 FSs can be used to effectively describe uncertainties in situations where the available information is uncertain. T2 FSs consider as a blurred membership function. The blurring used to model the uncertainty of crisp T1 FSs. A T2 FS

( )

*x*

 

(Mendel, 2001). Footprint of uncertainty (FOU) is a region between the blurred membership

FOU constructed form upper membership function (UMF) and lower membership function

*f u*

*<sup>x</sup> u* is the blurred membership function and *xJ* is the original membership

, 0,1

 *FOU A and FOU A* (2)

(1)

*x*

(Rhee, Hwang, 2001), (Rhee, Hwang, 2002) and (Rhee, Hwang, 2003).

fuzzy sets.

**2. Background** 

where *f*

can be formulated as follow:

This paper is organized as follows.

Section 6 gives the summary and conclusions.

function. The FOU of ܣሙ can be expressed by as

(LMF). (Choi and Rhee, 2009)

 ( )

*x X A x*

*<sup>u</sup> <sup>A</sup> XJ x XJ*

*xX xX x*

, : [0,1] by *x X x x A A FOU A J x u u J*

We start with a real problem in control. The simplest method is to visually inspect the pavements and evaluate them by subjective human experts. This approach, however, involves high labor costs and produces unreliable and inconsistent results. Furthermore, it exposes the inspectors to dangerous working conditions on highways. Destructive Testing (DT) and Non Destructive Testing (NDT) are both costly and time consuming. To overcome the limitations of the subjective visual evaluation process; several attempts have been made to develop an automatic procedure (Moghadas Nejad and Zakeri, 2011,a,b,c) and (Daqi et al, 2009).

Most current systems use computer vision and image processing technologies to automate the process. However, due to the irregularities of pavement surfaces, there has been a limited success inaccurately detecting cracks and classifying crack types. In addition, most systems require complex algorithm with high levels of computing power. While many attempts have been made to automatically collect pavement crack data, better approaches are needed to evaluate these automated crack measurement systems (Moghadas Nejad and Zakeri, 2011,a,b,c) and (Daqi et al,2009)

A Hybrid Automatic Expert System (HAES) for automatic distress detection developed, based on complex AI methods (Expert system, Polar Fuzzy Logic) and image processing methods (Wavelet Transform, Inverse Wavelet Transform, 3D Radon Transform, Fast Fourier transform, EH, etc). Fuzzy logic methods are one among favorite and overwhelming architect that used for uncertainty simulations. Type-1 fuzzy sets (T1 FSs) have been successfully used many area such as image processing, pattern recognition, machin learning. (Choi and Rhee, 2009), (Hagras,2004), (Hwang. Rhee, 2004), (Hwang. Rhee, 2007), (John, 2000), (Karnik, J. Mendel, 1999), (Liang et al. 2000), (Liang, J. Mendel, 2001), (Makrehchi, et al. 2003), (Rhee, 2007), (Rhee, Choi, 2007), (Rhee, Hwang, 2001), (Rhee, Hwang, 2002) and (Rhee, Hwang, 2003). Automatic generation of T1 FMFs classified as a interesting and hot research area. many T1 FMF generation models have been tested and various degree of successes achieved (Choi and Rhee, 2009), (Makrehchi, et al.2003), (Medasani et al,1998), (Rhee, and Krishnapuram, 1993), (Wang, 1994) and (Yang and Bose, 2006). Heuristics, histograms, probability, and entropy are good tools to automate the T1 FMFs generation. Several methods under title of AI have been implemented to data sets to generate T1 FMFs. A good classification proposed for T1 FMFs by Choi and Rhee, (2009). Based on this classification, algorithms based on the fuzzy nearest neighbor, back-propagation neural network, fuzzy C-means (FCM), robust agglomerative Gaussian mixture decomposition (RAGMD), and self-organizing feature map (SOFM) were used to generate T1 FMFs must be a considered as FMFs generator. (Choi and Rhee, 2009).

Uncertain meaning, uncertain measurement and noisy data are main causes that we cannot obtain satisfactory results using T1 FSs, therefore in this mode employment of type-2 fuzzy sets (T2 FSs) for managing uncertainty solved the problems (Ensafi & Tizhoosh, 2005), (Choi and Rhee, 2009). Choi and Rhee (2009) stated that, because of the extra degree of freedom (DOF), T2 FSs can control the blurring better than T1 FSs. However, undesirable amount of computations stand in front of extension T2 FSs in vast scale applications. Interval type-2 fuzzy sets (IT2 FSs) are proposed to reduce the complexity (Choi and Rhee, 2009). Many algorithms based on the T2 FMF have been proposed. (Choi and Rhee, 2009), (Hagras,2004), (Hwang. Rhee, 2004), (Hwang. Rhee, 2007), (John, 2000), (Karnik, J. Mendel, 1999), (Liang et al. 2000), (Liang, J. Mendel, 2001), (Makrehchi, et al. 2003), (Rhee, 2007), (Rhee, Choi, 2007), (Rhee, Hwang, 2001), (Rhee, Hwang, 2002) and (Rhee, Hwang, 2003).

In this chapter, we focus on the generation of 3D Polar fuzzy Memberships functions to use in hybrid expert system for systematic pavement distress detection and classification. In particular, we consider 3D polar type-1 fuzzy membership functions (3D T1 PMFs) that are generated from sample images and then developed to 3D polar type-2 fuzzy membership functions (3D T2 PMFs). First, we review three methods based on heuristics, histograms, and interval type-1 fuzzy C-means (IT1 FCM). For each method, the footprint of uncertainty (FOU) is only required to be obtained, since the FOU can completely describe a T1 PMF. We proposed two methods based on 3D domain and then 3D polar under the theory of type 2 fuzzy sets.

This paper is organized as follows.

In Section 2, we briefly review basic concepts and existing methods and background. In Section 3, we managed the IT2 FMF generation methods. In Section 4, concepts of polar fuzzy are discussed and we explain how our proposed IT2 PMF generation methods can be implemented. Section 5 approximate reasoning and fuzzy inference discussed. Finally, Section 6 gives the summary and conclusions.

## **2. Background**

86 Fuzzy Controllers – Recent Advances in Theory and Applications

Zakeri, 2011,a,b,c) and (Daqi et al,2009)

capturing/eliminating the uncertainties within fuzzy systems using regular (type I) fuzzy sets. A measure of 3D ultra-fuzziness is also presented. Several Experimental results are

We start with a real problem in control. The simplest method is to visually inspect the pavements and evaluate them by subjective human experts. This approach, however, involves high labor costs and produces unreliable and inconsistent results. Furthermore, it exposes the inspectors to dangerous working conditions on highways. Destructive Testing (DT) and Non Destructive Testing (NDT) are both costly and time consuming. To overcome the limitations of the subjective visual evaluation process; several attempts have been made to develop an

Most current systems use computer vision and image processing technologies to automate the process. However, due to the irregularities of pavement surfaces, there has been a limited success inaccurately detecting cracks and classifying crack types. In addition, most systems require complex algorithm with high levels of computing power. While many attempts have been made to automatically collect pavement crack data, better approaches are needed to evaluate these automated crack measurement systems (Moghadas Nejad and

A Hybrid Automatic Expert System (HAES) for automatic distress detection developed, based on complex AI methods (Expert system, Polar Fuzzy Logic) and image processing methods (Wavelet Transform, Inverse Wavelet Transform, 3D Radon Transform, Fast Fourier transform, EH, etc). Fuzzy logic methods are one among favorite and overwhelming architect that used for uncertainty simulations. Type-1 fuzzy sets (T1 FSs) have been successfully used many area such as image processing, pattern recognition, machin learning. (Choi and Rhee, 2009), (Hagras,2004), (Hwang. Rhee, 2004), (Hwang. Rhee, 2007), (John, 2000), (Karnik, J. Mendel, 1999), (Liang et al. 2000), (Liang, J. Mendel, 2001), (Makrehchi, et al. 2003), (Rhee, 2007), (Rhee, Choi, 2007), (Rhee, Hwang, 2001), (Rhee, Hwang, 2002) and (Rhee, Hwang, 2003). Automatic generation of T1 FMFs classified as a interesting and hot research area. many T1 FMF generation models have been tested and various degree of successes achieved (Choi and Rhee, 2009), (Makrehchi, et al.2003), (Medasani et al,1998), (Rhee, and Krishnapuram, 1993), (Wang, 1994) and (Yang and Bose, 2006). Heuristics, histograms, probability, and entropy are good tools to automate the T1 FMFs generation. Several methods under title of AI have been implemented to data sets to generate T1 FMFs. A good classification proposed for T1 FMFs by Choi and Rhee, (2009). Based on this classification, algorithms based on the fuzzy nearest neighbor, back-propagation neural network, fuzzy C-means (FCM), robust agglomerative Gaussian mixture decomposition (RAGMD), and self-organizing feature map (SOFM) were used to generate T1 FMFs must be a considered as FMFs generator. (Choi and Rhee, 2009).

Uncertain meaning, uncertain measurement and noisy data are main causes that we cannot obtain satisfactory results using T1 FSs, therefore in this mode employment of type-2 fuzzy sets (T2 FSs) for managing uncertainty solved the problems (Ensafi & Tizhoosh, 2005), (Choi and Rhee, 2009). Choi and Rhee (2009) stated that, because of the extra degree of freedom (DOF), T2 FSs can control the blurring better than T1 FSs. However, undesirable amount of

provided in order to demonstrate the usefulness of the proposed approach.

automatic procedure (Moghadas Nejad and Zakeri, 2011,a,b,c) and (Daqi et al, 2009).

The extension of T1 FSs to T2 FSs can be used to effectively describe uncertainties in situations where the available information is uncertain. T2 FSs consider as a blurred membership function. The blurring used to model the uncertainty of crisp T1 FSs. A T2 FS can be formulated as follow:

$$\stackrel{\circ}{A} = \int \frac{\mu\_{\text{v}}}{\text{x} \, ^{A(\text{x})}\text{x}} = \int \underbrace{\left[ \int\_{\text{x} \in \text{X}} \frac{f\_{\text{x}}(\text{u})}{\text{u}} \right]}\_{\text{X} \, ^{\text{U}}\text{X}}, \text{X} \mathbf{J}\_{\text{x}} \subseteq \left[ \mathbf{0}, \mathbf{1} \right] \tag{1}$$

where *f <sup>x</sup> u* is the blurred membership function and *xJ* is the original membership (Mendel, 2001). Footprint of uncertainty (FOU) is a region between the blurred membership function. The FOU of ܣሙ can be expressed by as

$$FOL\left(^{=}\stackrel{=}{A}\right) = \bigcup\_{\forall x \in X} J\_x = \left\{ \left(x, u\right) \colon u\epsilon J\_x \subseteq [0, 1] \right\}
\overline{\mu\_{\tilde{A}}} = \overline{FOL}\left(^{=}\stackrel{=}{A}\right)
and \text{ by } \underline{\mu\_{\tilde{A}}} = \underline{FOL}\left(^{=}\stackrel{=}{A}\right) \tag{2}$$

FOU constructed form upper membership function (UMF) and lower membership function (LMF). (Choi and Rhee, 2009)

**Figure 1.** A possible way to construct type II fuzzy sets. The interval between lower and upper membership values (shaded region) should capture the footprint of uncertainty (FOU).

Although T2 FSs may be useful in modeling uncertainty, where T1 FSs cannot, the operations of T2 FSs involve numerous embedded T2 FSs which consider all possible combinations of secondary membership values. Therefore, undesirably large amount of computations may be required. An effectively method to reduce the computational complexity is interval type-2 fuzzy sets (IT2 FSs).

In General, FOU ( *A* ) can be expressed as: (Choi and Rhee, 2009)

$$\text{FOU}\left(\stackrel{\leftarrow}{A}\right) = \bigcup \left[ \underline{FOL}\left(\stackrel{\leftarrow}{A}\right), \overline{FOL}\left(\stackrel{\leftarrow}{A}\right) \right]\_{\forall \text{x} \in X} \tag{3}$$

A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems 89

management system this new generation of MF play a powerful link between several tools such as multi-resolution methods (wavelet and beyond the wavelet methods), image

A possible membership function can be defined for every category by expert with any tools. For example using image processing techniques and Radon transform, several membership

function generated and shows in Fig.3 for pavement cracking distress.

processing, NN and expert system.

**Figure 2.** Possible membership functions

**Figure 3.** Variety of membership functions

As a result, IT2 FSs requires only simple interval arithmetic for computing.

### **3. Automatic MF generators (AMFG)**

In this section, we introduce a method for effectively crating IPT-1 FMF automatically from images data. Several methods such as heuristics, histograms, and interval type-2 fuzzy Cmeans (IT2 FCM) are proposed by (Choi & Rhee, 2009) for generating IT2 FMF automatically from pattern data. Using scaling factor and heuristic T1 FMFs, IT2 FMF simply can be generating. The histogram based method uses suitable parameterized functions chosen to model the smoothed histogram for each class and feature extracted from sample data (Choi and Rhee, 2009),(Hagras,2004), (Hwang and Rhee, 2004), (Hwang and Rhee, 2007), (John, 2000), (Karnik, J. Mendel, 1999), (Liang et al. 2000), (Liang, J. Mendel, 2001), (Makrehchi, et al. 2003), (Rhee, 2007), (Rhee and Choi, 2007), (Hwang and Rhee, 2001), (Rhee, Hwang, 2002) and (Rhee and Hwang, 2003). The IT2 FCM based method uses the derived formulas of the IT2 FMFs in the IT2 FCM algorithm (Hwang and Rhee, 2002). A detailed description of each method is discussed. The heuristic method simply uses an appropriate predefined T1 FMF function, such as triangular, trapezoidal, Gaussian, S, or p function, to name a few, to initially represent the distribution of the pattern data. The following are some frequently used heuristic membership functions. (Choi and Rhee, 2009) Membership functions for fuzzy sets can be constructed by any method exact, heuristic and Meta heuristic, such as triangular, trapezoidal, Gaussian, S, or p function in the domain. Two most important constraints must be considered for selecting a membership functions first, A membership function must be restricted between [0 1] and the next *μA*(*x*) must be unique. Four possible membership functions are presented in Fig.2. Where type III and polar are new generation of fuzzy membership function that can be used in several application in the control and classification domains. In the field of pavement management system this new generation of MF play a powerful link between several tools such as multi-resolution methods (wavelet and beyond the wavelet methods), image processing, NN and expert system.

**Figure 2.** Possible membership functions

88 Fuzzy Controllers – Recent Advances in Theory and Applications

complexity is interval type-2 fuzzy sets (IT2 FSs).

**3. Automatic MF generators (AMFG)** 

In General, FOU ( *A*

**Figure 1.** A possible way to construct type II fuzzy sets. The interval between lower and upper membership values (shaded region) should capture the footprint of uncertainty (FOU).

) can be expressed as: (Choi and Rhee, 2009)

FOU ,

As a result, IT2 FSs requires only simple interval arithmetic for computing.

Although T2 FSs may be useful in modeling uncertainty, where T1 FSs cannot, the operations of T2 FSs involve numerous embedded T2 FSs which consider all possible combinations of secondary membership values. Therefore, undesirably large amount of computations may be required. An effectively method to reduce the computational

> *A FOU A FOU A*

In this section, we introduce a method for effectively crating IPT-1 FMF automatically from images data. Several methods such as heuristics, histograms, and interval type-2 fuzzy Cmeans (IT2 FCM) are proposed by (Choi & Rhee, 2009) for generating IT2 FMF automatically from pattern data. Using scaling factor and heuristic T1 FMFs, IT2 FMF simply can be generating. The histogram based method uses suitable parameterized functions chosen to model the smoothed histogram for each class and feature extracted from sample data (Choi and Rhee, 2009),(Hagras,2004), (Hwang and Rhee, 2004), (Hwang and Rhee, 2007), (John, 2000), (Karnik, J. Mendel, 1999), (Liang et al. 2000), (Liang, J. Mendel, 2001), (Makrehchi, et al. 2003), (Rhee, 2007), (Rhee and Choi, 2007), (Hwang and Rhee, 2001), (Rhee, Hwang, 2002) and (Rhee and Hwang, 2003). The IT2 FCM based method uses the derived formulas of the IT2 FMFs in the IT2 FCM algorithm (Hwang and Rhee, 2002). A detailed description of each method is discussed. The heuristic method simply uses an appropriate predefined T1 FMF function, such as triangular, trapezoidal, Gaussian, S, or p function, to name a few, to initially represent the distribution of the pattern data. The following are some frequently used heuristic membership functions. (Choi and Rhee, 2009) Membership functions for fuzzy sets can be constructed by any method exact, heuristic and Meta heuristic, such as triangular, trapezoidal, Gaussian, S, or p function in the domain. Two most important constraints must be considered for selecting a membership functions first, A membership function must be restricted between [0 1] and the next *μA*(*x*) must be unique. Four possible membership functions are presented in Fig.2. Where type III and polar are new generation of fuzzy membership function that can be used in several application in the control and classification domains. In the field of pavement

 

*x X*

(3)

A possible membership function can be defined for every category by expert with any tools. For example using image processing techniques and Radon transform, several membership function generated and shows in Fig.3 for pavement cracking distress.

**Figure 3.** Variety of membership functions

More simple and complex functions can be used under the form of discrete and continues. Generally the ordinary functions categorized in Triangular, Trapezoidal, Γ-membership, Smembership, Logistic, Exponential-like and Gaussian function. Additionally several more advanced membership function which generate by automatic generator introduced. More applications in image processing frequently used heuristic membership functions that can be generally categorized in Table.1.

A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems 91

*x X x X*

and *FOU A*

 (5)

are the minimum UMF and LMF among all UMFs and

0 , 1 

to generalization, Choi & Rhee, (2009) choose the min operation as intersection for obtain the

*FOU A FOU A or min FOU A FOU A*

LMFs, respectively. Heuristic method which proposed Choi & Rhee (2009) is summarized in

Histogram based method (HBM) for membership function generation is another method which is more flexible than heuristic methods. In HBM, distribution of the feature values, have a crucial role in T1 FMF determination elements. Choi & Rhee, (2009) clearly stated that,"membership functions generated from HBM may be considered more suitable for arbitrary distributed data than from heuristics". Based on this theory Choi & Rhee (2009) propose a new method for generation IT2 FMFs. Using smoothed histograms which generated by hyper-cube or triangular window and then normalized, the upper and lower membership function flourished and mapped to real data. Selection a well trained parameters function to model the smoothed histograms has a tangible ramification on performances of MF generator system. To avoid over fitting lowest, the suitable degree of the polynomial function (PF) is stood out as the knee point of error. As a result, HBM FSs

In our case, as a real example in control, Type, severity and extents of cracking in pavement surface transform in a transform realm to generate a simple features. Simple features can use for generation of T1 FMF. Approximate parameter values such as the number, height, and location of peaks which related to cracking used to determine the optimal parameter values

Choi & Rhee (2009) considered Gaussian functions as suitable to model the IT2 FMFs (Rhee and Krishnapuram, 1993). They used a heuristic approach (Choi & Rhee, 2009) to obtain the initial parameters. Choi & Rhee (2009) ignored the ones that have small peaks. This means

are obtained by again fitting PFs to the smoothed histograms. New again histograms crystallized upper and lower MFs fitted to PF. As dimensional parameters or overall size of problem increase, undesirably become more and more. These complexities arise due to the high process in smoothing and fitting. This is a challenging point that set in motion to

that we have a threshold that it considers as a crisp threshold. The *FOU A*

overall FOU by taking intersections of all upper and lower memberships.

min ,min α. . ,min

*x X*

FOU ,

requires good estimation of PF.

of the function.

where *FOU A*

Fig.4.

*A FOU A FOU A*

 

and *FOU A*


**Table 1.** Heuristic membership functions, (Choi & Rhee,2009)

By control parameters, one can select a various interval pattern. Theses parameters usually trained and learned by experts. Under the title of Control Parameter (ߙሻǡthe UMF of the IT2 FMF and LMF can be designed. The LMF and UMF determined by scaling ߙܽ݊݀ߚbetween 0 and 1, which can be also tuned in supervised and unsupervised manner or provided by an expert. Choi & Rhee, (2009) proposed a simple definition for FOU, which categorized in heuristic methods. For feature i

$$\begin{aligned} \mathbf{FOU}\left(\stackrel{\circ}{A}\right) &= \bigcup \left[ \underline{FOL}\left(\stackrel{\circ}{A}\right) \overline{FOL}\left(\stackrel{\circ}{A}\right) \right]\_{\forall \mathbf{x} \in X} \\ &= \bigcup \overline{FOL}\left(\stackrel{\circ}{A}\right) \alpha . \underline{FOL}\left(\stackrel{\circ}{A}\right) \Big|\_{\forall \mathbf{x} \in X} \operatorname{or} \bigcup \left[ \beta . \overline{FOL}\left(\stackrel{\circ}{A}\right) \overline{FOL}\left(\stackrel{\circ}{A}\right) \right]\_{\forall \mathbf{x} \in X} \end{aligned} \tag{4}$$

0 , 1 

90 Fuzzy Controllers – Recent Advances in Theory and Applications

be generally categorized in Table.1.

Triangular function

Trapezoidal function

S-function

heuristic methods. For feature i

More simple and complex functions can be used under the form of discrete and continues. Generally the ordinary functions categorized in Triangular, Trapezoidal, Γ-membership, Smembership, Logistic, Exponential-like and Gaussian function. Additionally several more advanced membership function which generate by automatic generator introduced. More applications in image processing frequently used heuristic membership functions that can

Gaussian function 2 2 ( ) /2 ( ) *x c x e*

P-function

**Table 1.** Heuristic membership functions, (Choi & Rhee,2009)

FOU ,

*A FOU A FOU A*

 

ܾ ൌ ሺܽ ܾሻȀʹ

By control parameters, one can select a various interval pattern. Theses parameters usually trained and learned by experts. Under the title of Control Parameter (ߙሻǡthe UMF of the IT2 FMF and LMF can be designed. The LMF and UMF determined by scaling ߙܽ݊݀ߚbetween 0 and 1, which can be also tuned in supervised and unsupervised manner or provided by an expert. Choi & Rhee, (2009) proposed a simple definition for FOU, which categorized in

,α. .,

*FOU A FOU A or FOU A FOU A*

*x X*

 

*x X x X*

 

  2

2

(4)

, 0

 

, (; , ,) <sup>2</sup>

, (; , ,) <sup>2</sup>

, 0

, 0

 

, 1

, 0

, 0

, 1

 , / , /

, /

*IF x a then IF a x b Then x a b a IF b x c Then IF c x d Then c x d c IF x d then*

, /

, 2 /

*IF a x b Then x a b a*

 

*<sup>b</sup> IF x c then x s x c b c c*

*<sup>b</sup> IF x c then x s x c b c c*

*IF a x b Then xa ca IF x c then*

*IF x a then*

, 12 /

*IF x a then IF a x b Then x a b a IF a x b Then c x c b IF x c then*

to generalization, Choi & Rhee, (2009) choose the min operation as intersection for obtain the overall FOU by taking intersections of all upper and lower memberships.

$$\begin{aligned} & \text{FOU} \left( \stackrel{\circ}{A} \right) = \bigcup \left[ \underline{FOL} \left( \stackrel{\circ}{A} \right) , \overline{FOL} \left( \stackrel{\circ}{A} \right) \right]\_{\forall x \ge X} \\ &= \bigcup \left[ \min \left\{ \underline{FOL} \left( \stackrel{\circ}{A} \right) \right\} , \min \left\{ \alpha , \underline{FOL} \left( \stackrel{\circ}{A} \right) \right\} \right]\_{\forall x \ge X} \text{or} \bigcup \left[ \min \left\{ \beta , \overline{FOL} \left( \stackrel{\circ}{A} \right) \right\} , \min \left\{ \overline{FOL} \left( \stackrel{\circ}{A} \right) \right\} \right]\_{\forall x \ge X} \end{aligned} (5)$$

where *FOU A* and *FOU A* are the minimum UMF and LMF among all UMFs and

LMFs, respectively. Heuristic method which proposed Choi & Rhee (2009) is summarized in Fig.4.

Histogram based method (HBM) for membership function generation is another method which is more flexible than heuristic methods. In HBM, distribution of the feature values, have a crucial role in T1 FMF determination elements. Choi & Rhee, (2009) clearly stated that,"membership functions generated from HBM may be considered more suitable for arbitrary distributed data than from heuristics". Based on this theory Choi & Rhee (2009) propose a new method for generation IT2 FMFs. Using smoothed histograms which generated by hyper-cube or triangular window and then normalized, the upper and lower membership function flourished and mapped to real data. Selection a well trained parameters function to model the smoothed histograms has a tangible ramification on performances of MF generator system. To avoid over fitting lowest, the suitable degree of the polynomial function (PF) is stood out as the knee point of error. As a result, HBM FSs requires good estimation of PF.

In our case, as a real example in control, Type, severity and extents of cracking in pavement surface transform in a transform realm to generate a simple features. Simple features can use for generation of T1 FMF. Approximate parameter values such as the number, height, and location of peaks which related to cracking used to determine the optimal parameter values of the function.

Choi & Rhee (2009) considered Gaussian functions as suitable to model the IT2 FMFs (Rhee and Krishnapuram, 1993). They used a heuristic approach (Choi & Rhee, 2009) to obtain the initial parameters. Choi & Rhee (2009) ignored the ones that have small peaks. This means

that we have a threshold that it considers as a crisp threshold. The *FOU A* and *FOU A* 

are obtained by again fitting PFs to the smoothed histograms. New again histograms crystallized upper and lower MFs fitted to PF. As dimensional parameters or overall size of problem increase, undesirably become more and more. These complexities arise due to the high process in smoothing and fitting. This is a challenging point that set in motion to

A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems 93

(7)

, (8)

is the LMF, and i is the feature's number. From our

where *FOU A*

and

is the UMF, *FOU A*

IT2 FMF in IT2 FCM expressed as (Choi & Rhee, 2009).

points of view, the main contributions of Choi & Rhee's methods are developing in membership's generation. These methods enable them to transfer the knowledge when expert facing with N dimensional features. These methods are applicable for images realm. We assert that this contribution is valuable. Nevertheless we would like to highlight that high process in discrete smoothing and fitting(first 1DUMF and 1DLMF calculation and then aggregation) faced us to problem to products an effective MF generator. Heuristic method to generate T2 FMF's, which proposed Choi & Rhee (2009) is summarized in Fig.5.

Choi & Rhee (2009) considered **fuzzy C-means (FCM)** functions to model the IT2 FMFs (Hwang and Rhee, 2007) (Choi & Rhee, 2009). The fuzzifier m in FCM, can be fired as a membership generator. IT2 FCM based method proposed by Choi & Rhee (2009). They stated that, "*Due to the constraint on the memberships we cannot design this region with any particular single value of fuzzifier m to be used in the FCM*". IT2 FCM algorithm was proposed to solving this problem (Hwang and Rhee, 2007). Indeed they products a simple dynamic fuzzifuyer AMFG to generating the Membership function. According to IT2 FCM, two fuzzifier m1, m2 are employed to control the blurring area in fuzzy domain. The proposed

> , :[ , ] *xJ x u u FOU A FOU A*

2 2 1 1

1 2

1

*K*

*<sup>m</sup> <sup>C</sup> ij*

*d d*

*<sup>m</sup> <sup>C</sup> ij*

*d d*

*ik*

*ik*

1

*K*

1 1

*K K*

1

1

1 1

*m m C C ij ij*

*d d d d*

*ik ik*

2 1

1

 

2 1

2

*ELSE FOU A*

*IF*

*FOU A THEN FOU A*

**Figure 4.** Heuristic based IT2 FMF generation method

product a new heuristic to handle computational load. Choi & Rhee (2009) proposed two steep methods **1)** calculate one-dimensional 1DUMF and 1DLMF for HBM. **2)** Obtain the overall *FOU A* and *FOU A* by Intersections operation. Intersections operation which

proposed for this aggregation expressed as

$$\text{FOU}\left(\stackrel{=}{A}\right) = \bigcup\left[\underline{FOLI}\left(\stackrel{=}{A}\right), \overline{FOLI}\left(\stackrel{=}{A}\right)\right]\_{\forall \mathbf{x} \in X} = \left\lfloor \min\_{\mathbf{y}} \left\{ \underline{FOLI}\left(\stackrel{=}{A}\right) \right\}, \min\_{\mathbf{y}} \left\{ \overline{FOLI}\left(\stackrel{=}{A}\right) \right\} \right\rfloor\_{\forall \mathbf{x} \in X} \tag{6}$$

where *FOU A* is the UMF, *FOU A* is the LMF, and i is the feature's number. From our

points of view, the main contributions of Choi & Rhee's methods are developing in membership's generation. These methods enable them to transfer the knowledge when expert facing with N dimensional features. These methods are applicable for images realm. We assert that this contribution is valuable. Nevertheless we would like to highlight that high process in discrete smoothing and fitting(first 1DUMF and 1DLMF calculation and then aggregation) faced us to problem to products an effective MF generator. Heuristic method to generate T2 FMF's, which proposed Choi & Rhee (2009) is summarized in Fig.5.

Choi & Rhee (2009) considered **fuzzy C-means (FCM)** functions to model the IT2 FMFs (Hwang and Rhee, 2007) (Choi & Rhee, 2009). The fuzzifier m in FCM, can be fired as a membership generator. IT2 FCM based method proposed by Choi & Rhee (2009). They stated that, "*Due to the constraint on the memberships we cannot design this region with any particular single value of fuzzifier m to be used in the FCM*". IT2 FCM algorithm was proposed to solving this problem (Hwang and Rhee, 2007). Indeed they products a simple dynamic fuzzifuyer AMFG to generating the Membership function. According to IT2 FCM, two fuzzifier m1, m2 are employed to control the blurring area in fuzzy domain. The proposed IT2 FMF in IT2 FCM expressed as (Choi & Rhee, 2009).

$$f\_x = \begin{cases} \left(x, u\right); uef\overline{FOL\left(\stackrel{-1}{A}\right)} \overline{FOL\left(\stackrel{-1}{A}\right)} \left\|\overline{FOL\left(\stackrel{-1}{A}\right)}\right\| & \tag{7} \\\\ \left(\begin{array}{c} 1\\ \text{IF}\frac{1}{\sum\_{k=1}^{c}\left(\frac{d\_{ij}}{d\_{ik}}\right)^{\frac{2}{\left(m\_1-1\right)}}} > \frac{1}{\sum\_{k=1}^{c}\left(\frac{d\_{ij}}{d\_{ik}}\right)^{\frac{2}{\left(m\_2-1\right)}}}\\ \sum\_{k=1}^{c}\overline{FOL\left(\stackrel{-1}{A}\right)} = \frac{1}{\sum\_{k=1}^{c}\left(\frac{d\_{ij}}{d\_{ik}}\right)^{\frac{2}{\left(m\_1-1\right)}}} \end{array} \right) \\\\ \left(\begin{array}{c} 1\\ \text{ELSE}\end{array}\overline{FOL\left(\stackrel{-1}{A}\right)} = \frac{1}{\sum\_{k=1}^{c}\left(\frac{d\_{ij}}{d\_{ik}}\right)^{\frac{2}{\left(m\_1-1\right)}}}\\ \sum\_{k=1}^{c}\overline{FOL\left(\stackrel{-1}{A}\right)} = \frac{1}{\sum\_{k=1}^{c}\left(\frac{d\_{ij}}{d\_{ik}}\right)^{\frac{2}{\left(m\_1-1\right)}}} \end{cases} \tag{8}$$

and

92 Fuzzy Controllers – Recent Advances in Theory and Applications

**Figure 4.** Heuristic based IT2 FMF generation method

and *FOU A*

proposed for this aggregation expressed as

 

FOU , , *i i*

overall *FOU A*

 

product a new heuristic to handle computational load. Choi & Rhee (2009) proposed two steep methods **1)** calculate one-dimensional 1DUMF and 1DLMF for HBM. **2)** Obtain the

> *A FOU A FOU A min FOU A min FOU A*

by Intersections operation. Intersections operation which

*x X x X*

(6)

$$\underline{FOL}\left(\stackrel{\scriptstyle\exists}{\bar{A}}\right) = \left\{ \begin{array}{c} 1 \\ \underline{I} \\ \sum\_{K=1}^{c} \left(\frac{d\_{ij}}{d\_{ik}}\right)^{\frac{2}{\binom{(m\_1-1)}{k}}} \end{array} \frac{1}{\sum\_{K=1}^{c} \left(\frac{d\_{ij}}{d\_{ik}}\right)^{\frac{2}{\binom{(m\_2-1)}{k}}}} \\ \begin{array}{c} \underline{THIEN}\ \overline{FOL}\left(\stackrel{\scriptstyle\exists}{\bar{A}}\right) = \frac{1}{\sum\_{K=1}^{c} \left(\frac{d\_{ij}}{d\_{ik}}\right)^{\frac{2}{\binom{(m\_1-1)}{k}}}} \\ \sum\_{K=1}^{c} \overline{FOL}\left(\stackrel{\scriptstyle\exists}{\bar{A}}\right) = \frac{1}{\sum\_{K=1}^{c} \left(\frac{d\_{ij}}{d\_{ik}}\right)^{\frac{2}{\binom{(m\_2-1)}{k}}}} \end{array} \right\} \tag{9}$$

A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems 95

**Figure 5.** Heuristic based IT2 FMF generation method

However IT2 FCM for updating cluster prototypes requires type-reduction. Using type-2 fuzzy operations therefore is essential. The crisp center obtained mean of centers of defuzzification as the centroid obtained by the type-reduction according Eq.10

$$V\_{\vec{x}} = \left[\underline{\underline{C}}, \overline{\underline{C}}\right] = \frac{\sum\_{\overline{FOL}\left(\stackrel{\circ}{A}\right) = I\_{x1}} \cdots \sum\_{\overline{FOL}\left(\stackrel{\circ}{A}\right) = I\_{x1}} \mathbf{1}}{\sum\_{i=1}^{N} \mathbf{x}\_{i} \overline{FOL}\left(\stackrel{\circ}{A}\right)\_{i}^{m}} \tag{10}$$

$$\overline{\sum\_{i=1}^{N} \overline{FOL}\left(\stackrel{\circ}{A}\right)\_{i}^{m}}$$

The UMF and LMF for class k and input pattern xj can be expressed by modifying

$$d\_{k\bar{\jmath}}^{\*} = \min\_{p} \left\langle d(\mathfrak{x}\_{\jmath'} V\_{\bar{\bf x}}{}^{k}) \right\rangle, p\epsilon \left\{ n\_{k} \right\} \tag{11}$$

Based on Choi & Rhee's (2009) method the membership values for the UMFs and LMFs are based on *m1* and *m2* and they are highly dependent on value selection of threshold which is itself considered crisp. Choi & Rhee's (2009) stated that IT2 FCM can desirably control the uncertainty that is quite simple handle all features of high dimensional problems. Their heuristic method summarized in Fig 6.

The accuracy of IT2 FCM highly dependent on fuzzifiers selection. These parameters have significant role in designing the FOU for a data set. In general, select unsuitable fuzzifier worth poor clustering. (Choi & Rhee's, 2009)

**Figure 5.** Heuristic based IT2 FMF generation method

*IF*

*FOU A THEN FOU A*

2 2 1 1

1 2

1

*K*

*<sup>C</sup> <sup>m</sup> ij*

*d d*

*<sup>C</sup> <sup>m</sup> ij*

*d d*

*ik*

*ik*

1

1 1

 

*i m*

*i*

*x x FOU A J FOU A J*

*x FOU A*

*FOU A*

*K*

1 1

*K K*

1

1

However IT2 FCM for updating cluster prototypes requires type-reduction. Using type-2 fuzzy operations therefore is essential. The crisp center obtained mean of centers of

1

*N i*

 \* ( , ), *<sup>k</sup> kj p j x <sup>k</sup> d min d x V p n*

Based on Choi & Rhee's (2009) method the membership values for the UMFs and LMFs are based on *m1* and *m2* and they are highly dependent on value selection of threshold which is itself considered crisp. Choi & Rhee's (2009) stated that IT2 FCM can desirably control the uncertainty that is quite simple handle all features of high dimensional problems. Their

The accuracy of IT2 FCM highly dependent on fuzzifiers selection. These parameters have significant role in designing the FOU for a data set. In general, select unsuitable fuzzifier

The UMF and LMF for class k and input pattern xj can be expressed by modifying

*x m N i i*

1

*ELSE FOU A*

defuzzification as the centroid obtained by the type-reduction according Eq.10

,

*V CC*

heuristic method summarized in Fig 6.

worth poor clustering. (Choi & Rhee's, 2009)

1 1

*m m C C ij ij*

*d d d d*

*ik ik*

2 1

1

1

  (9)

(10)

(11)

2 1

2

A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems 97

generator agent. First, the IT2 FMF algorithm introduced, and then our IT2 FPM based method are described. We selected Cubic Smoothing Spline (CSS) for generate the upper and lower membership functions because of non-uniform illumination of the Three Dimensional Memebership Functions (3DMFs). In the Type-2 domain, the estimation of the 3DMFu and 3DMFL are exanimate from the fitting of a cubic smoothing Spline,( Mora et al.,2011) to the 3DMF*(x,y)*. The select CSS is a special class of Spline that can capture the low 3DMF value that limited the non-uniformity of the 3DMF (Culpin, 1986). The fitting

<sup>2</sup> 2 2

∬ (12)

. ( , , ) (1 ) ( , ) ,

*M P f x y s x y p D S x y dxdy*

*Compactness:* measures how close the spline is to the data that reflect to the summation

*Smoothness:* measures the spline smoothness using its second derivative that reflect to

The smoothing factor p, controls the balance between being an interpolating spline crossing all data points (with p = 1) and being a strictly smooth Spline (with p = 0). The smoothing

<sup>2</sup> <sup>2</sup> <sup>2</sup>

where, *|z|2* represent for the sum of the squares of all the entries of ݊, N and M is the number of entries of x and y, and the integral is over the smallest interval containing all the entries of x and y. The default value for the weight vector w in the error measure is ones (size(x)). The default value for the piecewise constant weight function λ in the roughness measure is the constant function 1. Further, *D2f* denotes the second derivative of the function f. The default value for the smoothing parameter, p, is chosen in dependence on the given data sites x and y (Pal and Bezdek, 1994). The smoothing parameter determines the relative weight to place on the contradictory demands of having f be smooth vs having f be close to the data. For p = 0, f is the least-squares straight line fit to the data, while, at the other extreme, i.e., for p = 1, f is the variational, or 'natural' cubic spline interpolant. As p moves

from 0 to 1, the smoothing spline changes from one extreme to the other. (See Fig. 7)

spacing of the data sites, and it is in this range that the default value for *p* is chosen. For uniformly spaced data, one would expect a close following of the data for *p = 1/(1 +* 

:, ( ) (1 ) ( ) .

 

(13)

*p t D f t dt*

1 / 1 , / 600 , *min N M* with *h* the average

objective is to minimize the equation.

where, this equation include two parts:

spline *f* minimizes when

the integral term weighed by (1 - p).

1

*j*

*n*

1 1

term which weighed by the smoothing factor p,

*P j y j fxj*

The interesting range for p is often near <sup>3</sup>

*(min(N,M))3/6000)* and some satisfactory smoothing for

*m n*

*y x*

**Figure 6.** Heuristic based IT2 FCM generation method

## **4. Interval type-2 Polar Fuzzy Method (IT2 PFM)**

## **4.1. Type III-MF**

The Interval type-2 Polar Fuzzy Method (IT2 PFM) algorithm was proposed to automatically control the uncertainty. In this section, we proposed an intelligent IT2 FMF generator agent. First, the IT2 FMF algorithm introduced, and then our IT2 FPM based method are described. We selected Cubic Smoothing Spline (CSS) for generate the upper and lower membership functions because of non-uniform illumination of the Three Dimensional Memebership Functions (3DMFs). In the Type-2 domain, the estimation of the 3DMFu and 3DMFL are exanimate from the fitting of a cubic smoothing Spline,( Mora et al.,2011) to the 3DMF*(x,y)*. The select CSS is a special class of Spline that can capture the low 3DMF value that limited the non-uniformity of the 3DMF (Culpin, 1986). The fitting objective is to minimize the equation.

$$M = P.\sum\_{y=1:x=1}^{m} \sum\_{x=1}^{n} (f\left(\mathbf{x}, y\right) - s\left(\mathbf{x}, y\right))^2 + (1 - p)\iint (D^2 \mathbf{S}\left(\mathbf{x}, y\right))^2 dx dy,\tag{12}$$

where, this equation include two parts:

96 Fuzzy Controllers – Recent Advances in Theory and Applications

**Figure 6.** Heuristic based IT2 FCM generation method

**4.1. Type III-MF** 

**4. Interval type-2 Polar Fuzzy Method (IT2 PFM)** 

The Interval type-2 Polar Fuzzy Method (IT2 PFM) algorithm was proposed to automatically control the uncertainty. In this section, we proposed an intelligent IT2 FMF


The smoothing factor p, controls the balance between being an interpolating spline crossing all data points (with p = 1) and being a strictly smooth Spline (with p = 0). The smoothing spline *f* minimizes when

$$\mathbb{E}\left[\left|P\sum\_{j=1}^{n}\left(o(j)\left|y(\cdot,j)-f(\mathbf{x}(j))\right|^{2}\right)\right|+\left[(1-p)\left|\lambda(t)\right|D^{2}f(t)\right]^{2}dt.\right]\tag{13}$$

where, *|z|2* represent for the sum of the squares of all the entries of ݊, N and M is the number of entries of x and y, and the integral is over the smallest interval containing all the entries of x and y. The default value for the weight vector w in the error measure is ones (size(x)). The default value for the piecewise constant weight function λ in the roughness measure is the constant function 1. Further, *D2f* denotes the second derivative of the function f. The default value for the smoothing parameter, p, is chosen in dependence on the given data sites x and y (Pal and Bezdek, 1994). The smoothing parameter determines the relative weight to place on the contradictory demands of having f be smooth vs having f be close to the data. For p = 0, f is the least-squares straight line fit to the data, while, at the other extreme, i.e., for p = 1, f is the variational, or 'natural' cubic spline interpolant. As p moves from 0 to 1, the smoothing spline changes from one extreme to the other. (See Fig. 7)

The interesting range for p is often near <sup>3</sup> 1 / 1 , / 600 , *min N M* with *h* the average spacing of the data sites, and it is in this range that the default value for *p* is chosen. For

uniformly spaced data, one would expect a close following of the data for *p = 1/(1 + (min(N,M))3/6000)* and some satisfactory smoothing for

**Figure 7.** As p moves from 0 to 1, the smoothing spline changes from one extreme to the other.

$$p = 1/\left(1 + \left(\min\left(N, M\right)\right)^3 / 60\right) . p > 1\tag{14}$$

A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems 99

(16)

(17)

(19)

(1, ) and

1

1

(18)

,

*H*

1

*h*

(3 )( , )

(3 )( , )

*DRT i j*

*DRT i j*

*<sup>h</sup> <sup>h</sup>*

*max v*

*<sup>h</sup> <sup>h</sup>*

,

*max v*

(3 )( , )

*DRT i j v* is 3DRT value in the position *i* and *j*. Select a bigger h is worth a more enhanced distress for example in pavement distress detection and classification problem and smoother noisy background (see Fig.8). In order to define a type II fuzzy set, one can define a type I fuzzy set and assign upper and lower membership degrees to each element to (re)construct the footprint of uncertainty (Fig. 9) (Tizhoosh,2005). For example, when Radon Transform is applied to wavelet modulus, a distress (crack) is transformed into a peak in radon domain. Originally, every distress reflects to RT and has different intensity in 3DMF histograms. For example mean of 3DRT have variety range. According to the above Eq. 18 the max GR must be equal 1. To extend the fuzzy membership to type II fuzzy sets, ultrafuzziness should be zero, if the MF can be selected without any ambiguous such as type I. The amount of

The extreme case of maximal ultrafuzziness, equal 1, is worth to completely vagueness. pal and bezdek (1994) had extensive reviewed well known fuzziness index, two general classes proposed by them was additive and multiplicative class (Pal and Bezdek,1994). Based on

> <sup>2</sup> (, ) *near ka <sup>k</sup> H A dAA n*

Where, *k R* , d is a metric, and ����� is the crisp set close to the A. generally, based on *d*, weigh of k determined. The ���� �����) and linear or quadratic ( ) *H A ka* cab be determined

*DRT i j i j <sup>h</sup> <sup>h</sup> j i DRT i j*

*N M h*

*v*

where, M and N denotes the size of 3DMF platform,H is high platform, *h*

*MN max v*

1 1 (3 )( , ) <sup>1</sup> ,

(3 )( , )

(3 )( , )

*DRT i j i j <sup>h</sup>*

*v*

*v*

*DRT i j i j <sup>h</sup>*

(,)

(,)

ultrafuzziness will increase by rising uncertainly bound.

kufmann's Index of fuzziness for a set , *A <sup>n</sup> r x*

(,)

*GR*

and

(3 )( , ) *h*

by q-norms,

*RTMF*

can be input, but this leads to a smoothing spline even rougher than the variational cubic spline interpolate (Pal and Bezdek,1994).

$$p = \left(\frac{1}{1 + \frac{\left(\min\{\text{N}, M\}\right)^3}{600}}\right), p\_{\text{II}} = \left(\frac{1}{1 + \frac{\left(\min\{\text{N}, M\}\right)^3}{\alpha \times 600}}\right)\_{\text{II}}, p\_{\text{L}} = \left(\frac{1}{1 + \frac{\left(\min\{\text{N}, M\}\right)^3}{\beta \times 600}}\right)\_{\text{L}}\tag{15}$$

A reference smoothing factor *p e* 14 was obtained empirically for constructed MF in upper bound and *p e* 0.93 5 for constructed MF in lower bound. For example, in the case of image thresholding, After testing several thresholds, the general rule can be extract from 3DRT thresholds for upper and lower bounds by good selection݂Ƚ and ߚ.

#### *4.1.1. A measure of ultrafuzziness*

Using a simple method, we turned ultrafuzzy to the 3DRT fuzzy set. According a type II membership function, MF must be in [0,1]. One can be taking out the normalization form 3DMF using division every point by max 3DRT.

A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems 99

$$\text{RTMF}\_{\{i,j\}} = \left( \left[ \upsilon\_{\text{(3DRT)}(i,j)} \right] \left( \underset{\text{max}\left[ \upsilon\_{\text{(3DRT)}(i,j)} \right]}{\text{max}\left[ \upsilon\_{\text{(3DRT)}(i,j)} \right]} \right]^{\frac{1}{h}} \right) \tag{16}$$

$$\mu\_{\{i,j\}} = \left( \left[ \upsilon\_{\{3DRT\}(i,j)} \right] \left\{ \sum\_{\substack{h\\n \text{max}\left[ \upsilon\_{\{3DRT\}(i,j)} \right] }}^h \right\}^{\frac{1}{h}} + H\_{\text{\textquotedblleft}} \tag{17}$$

and

98 Fuzzy Controllers – Recent Advances in Theory and Applications

spline interpolate (Pal and Bezdek,1994).

*4.1.1. A measure of ultrafuzziness* 

3DMF using division every point by max 3DRT.

**Figure 7.** As p moves from 0 to 1, the smoothing spline changes from one extreme to the other.

 <sup>3</sup> *<sup>p</sup>* 1 / 1 , / 60 . 1 *min N M p* 

can be input, but this leads to a smoothing spline even rougher than the variational cubic

11 1 , , ( ( , )) ( ( , )) ( ( , )) 11 1

A reference smoothing factor *p e* 14 was obtained empirically for constructed MF in upper bound and *p e* 0.93 5 for constructed MF in lower bound. For example, in the case of image thresholding, After testing several thresholds, the general rule can be extract from 3DRT thresholds for upper and lower bounds by good selection݂Ƚ and ߚ.

Using a simple method, we turned ultrafuzzy to the 3DRT fuzzy set. According a type II membership function, MF must be in [0,1]. One can be taking out the normalization form

*pp p min N M min N M min N M*

*U L*

 

600 α 600 600

(14)

*U L*

(15)

33 3

$$\text{GR}\_{(i,j)} = \frac{1}{\binom{\text{MN}}{\text{MN}}^h} \sum\_{j=1}^N \Biggl[ \left( \text{ ${}^{\text{(3DRT)}} (i,j)$ } \right) \Bigg\}\_{\text{max}\left[ \text{ ${}^{\text{(3DRT)}} (i,j)$ } \right]^{\frac{1}{h}} \Bigg\} \tag{18}$$

where, M and N denotes the size of 3DMF platform,H is high platform, *h* (1, ) and (3 )( , ) *h DRT i j v* is 3DRT value in the position *i* and *j*. Select a bigger h is worth a more enhanced distress for example in pavement distress detection and classification problem and smoother noisy background (see Fig.8). In order to define a type II fuzzy set, one can define a type I fuzzy set and assign upper and lower membership degrees to each element to (re)construct the footprint of uncertainty (Fig. 9) (Tizhoosh,2005). For example, when Radon Transform is applied to wavelet modulus, a distress (crack) is transformed into a peak in radon domain. Originally, every distress reflects to RT and has different intensity in 3DMF histograms. For example mean of 3DRT have variety range. According to the above Eq. 18 the max GR must be equal 1. To extend the fuzzy membership to type II fuzzy sets, ultrafuzziness should be zero, if the MF can be selected without any ambiguous such as type I. The amount of ultrafuzziness will increase by rising uncertainly bound.

The extreme case of maximal ultrafuzziness, equal 1, is worth to completely vagueness. pal and bezdek (1994) had extensive reviewed well known fuzziness index, two general classes proposed by them was additive and multiplicative class (Pal and Bezdek,1994). Based on kufmann's Index of fuzziness for a set , *A <sup>n</sup> r x*

$$H\_{ka}\left(A\right) = \left(\frac{2}{n^k}\right) d(A\_\prime A^{near})\tag{19}$$

Where, *k R* , d is a metric, and ����� is the crisp set close to the A. generally, based on *d*, weigh of k determined. The ���� �����) and linear or quadratic ( ) *H A ka* cab be determined by q-norms,

$$d(A, A^{\text{near}}) = \left(\sum\_{i=1}^{n} \left| \mu\_i - \mu\_{A^{\text{mer}}, i} \right|^q \right)^{\bigvee\_q} H\_{ka}(q, A) = \left(\frac{2}{\frac{1}{n^q}} \left| \left(\sum\_{i=1}^{n} \left| \mu\_i - \mu\_{A^{\text{mer}}, i} \right|^q \right) \right|^q \right)^{\bigvee\_q} \tag{20}$$

Where � � �1, ∞). On the other side, Tizhoosh developed a simple ultrafuzziness index for the special case as fallow (Tizhoosh, 2005),

$$\tilde{\varphi}\left(\tilde{A}\right) = \frac{1}{\text{MN}} \sum\_{i=1}^{M-1} \sum\_{j=1}^{N-1} \left[ \mu\_{\text{ll}}(\mathcal{g}\_{i\bar{j}}) - \mu\_{\text{L}}(\mathcal{g}\_{i\bar{j}}) \right] \tag{21}$$

A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems 101

**Figure 8.** Basic rules for construction 3D fuzzy type II memebership function.

**Figure 9.** Three dimensional domain of FOU for 3D fuzzy sets, (3DFOU)

**Figure 10.** Example of Three FOU for 3D fuzzy sets, (3DFOU) using proposed algorithm

where 1/ *U A g g* and 1/ *U A g g* , (1,2]. and in general term it present as follow,

$$\tilde{\mathcal{T}}\left(\tilde{A}\right) = \frac{1}{MN} \sum\_{\mathcal{G}=0}^{L-1} \left[\mu\_{\mathcal{U}}(\mathcal{g}\_{ij}) - \mu\_{\mathcal{L}}(\mathcal{g}\_{ij})\right] \times h(\mathcal{g}) \tag{22}$$

Based on these theory and with respect to Tizhoosh's method (Tizhoosh, 2005),for developing ultrafuzziness on 2D data, a measure of ultrafuzziness �� for a platform 3DMF with M\*N sets, surf 3DMF and the membership function *<sup>A</sup>*(,) *i j* can be developed as follows:

$$
\tilde{\gamma}\{A\} = \left(\frac{1}{\left(\text{MN}\right)\_{\tilde{\gamma}}^{1}}\right) \left[\sum\_{j=1}^{M-1N-1} \sum\_{i=1}^{} \left|\mu\_{\mathbf{u}\{i,j\}} - \mu\_{\mathbf{L}\{i,j\}}\right|^{q}\right]^{\tilde{\lambda}\_{q}^{q}}
$$

$$
\frac{\partial}{\partial T}\tilde{\gamma}\{RT\_{\{i,j\}}\} = \frac{\partial}{\partial T} \left(\frac{1}{\left(\text{MN}\right)\_{\tilde{\gamma}}^{q}}\right) \left[\sum\_{j=1}^{M-1N-1} \sum\_{i=1}^{} \left|\mu\_{\mathbf{u}\{(i,j),T\}} - \mu\_{\mathbf{L}\{(i,j),T\}}\right|^{q}\right]^{\tilde{\lambda}\_{q}^{q}} = 0\tag{23}
$$

This basic definition relies on the assumption that the singletons sitting on the FOU are all equal in height (which is the reason why the interval-based type II is used), (Tizhoosh, 2005). The variation in the space can be measured by this method, therefore the new Index introduced in three dimensional domain of FOU for 3D fuzzy sets, (3DFOU). This method can resolve the problems about the ultrafuzziness index -"uncertainty (FOU) has a constant value, that equals one, in all the intervals of the universe of discourse" (Ioannis et al., 2008) using introducing flexible membership function across the intervals path (see Fig 9, 10).

Similarly, We are evaluated, proposed method, based four conditions *Minimum ultrafuzziness, Maximum ultrafuzziness, Equal ultrafuzziness* and *Reduced ultrafuzziness* that every measure of fuzziness should satisfy, which introduced by Kaufmann (Kaufmann, 1975). In a similar way, we established that the new index is qualified for measure of ultrafuzziness in 3D domain with these conditions.

**Figure 8.** Basic rules for construction 3D fuzzy type II memebership function.

1 (, ) *near*

*i*

the special case as fallow (Tizhoosh, 2005),

*dAA*

where 1/ *U A g g*

 

as follow,

*<sup>n</sup> <sup>q</sup> <sup>q</sup> near i A i*

and 1/

 *A*

surf 3DMF and the membership function *<sup>A</sup>*(,) *i j*

ultrafuzziness in 3D domain with these conditions.

*RT T T*

  

  1

*H qA*

<sup>1</sup> ( ) ( )

<sup>1</sup> ( ) ( ) ()

*U ij L ij*

*<sup>A</sup> g g hg MN*

Based on these theory and with respect to Tizhoosh's method (Tizhoosh, 2005),for developing ultrafuzziness on 2D data, a measure of ultrafuzziness �� for a platform 3DMF with M\*N sets,

1 1

*j i <sup>q</sup>*

(,) <sup>1</sup> ,, ,, 1 1

*i j u ij T L ij T j i <sup>q</sup>*

This basic definition relies on the assumption that the singletons sitting on the FOU are all equal in height (which is the reason why the interval-based type II is used), (Tizhoosh, 2005). The variation in the space can be measured by this method, therefore the new Index introduced in three dimensional domain of FOU for 3D fuzzy sets, (3DFOU). This method can resolve the problems about the ultrafuzziness index -"uncertainty (FOU) has a constant value, that equals one, in all the intervals of the universe of discourse" (Ioannis et al., 2008) using introducing flexible membership function across the intervals path (see Fig 9, 10).

Similarly, We are evaluated, proposed method, based four conditions *Minimum ultrafuzziness, Maximum ultrafuzziness, Equal ultrafuzziness* and *Reduced ultrafuzziness* that every measure of fuzziness should satisfy, which introduced by Kaufmann (Kaufmann, 1975). In a similar way, we established that the new index is qualified for measure of

 

*MN*

*MN*

1 1 1 , , 1 1 1 –

*U ij L ij*

 

(21)

(22)

can be developed as follows:

 

1 –0 *M N q q*

(23)

 

*M N q q uij L ij*

1

1

Where � � �1, ∞). On the other side, Tizhoosh developed a simple ultrafuzziness index for

1 1

*M N*

*i j <sup>A</sup> g g MN*

*U A g g*

 ,

1

*L*

*g*

0

1 1

1

(20)

1 , 1

 

 

*n q q*

(1,2]. and in general term it present

<sup>2</sup> ( , ) *near*

*ka i A i <sup>i</sup> <sup>q</sup>*

*n*

,

**Figure 9.** Three dimensional domain of FOU for 3D fuzzy sets, (3DFOU)

**Figure 10.** Example of Three FOU for 3D fuzzy sets, (3DFOU) using proposed algorithm

1. *IF i j*, consider as a type I fuzzy set *Then uij Lij* , , *AND A* 0

$$\tilde{\mathcal{Y}}\{A\} = \left[\frac{1}{\left(MN\right)\_q^{\frac{1}{q}}}\right] \left[\sum\_{j=1}^{M-1N-1} \left|\mu\_{\mathfrak{a}(i,j)} - \mu\_{\mathfrak{U}(i,j)}\right|^q\right]^{\frac{1}{q}} = 0 \text{ for } q \in [1, \infty). \tag{24}$$

A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems 103

is the radon transform of *f x y*, , θ represents

()

()

 (, , 00 0 , (32)

, (33)

 

(34)

 

> 

   

*,* (30)

*<sup>A</sup> r* should be

 

, (29)

· is the Dirac function, r[−∞,∞] is the perpendicular distance of a

*i j i j*

 

where *f x y*, represents an image, *P r* ,

> 

(Miao et al., 2012) Where

follows (see Fig.6):

is to minimize the equation.

such as 3DMF, the fuzziness can be calculated as follows (Tizhoosh, 2005);

*A min MN*

the line direction, and r is the distance away from the origin of coordinates. (Radon, 1919),

line from the origin and *θ*[0,*π*] is the angle formed by the distance vector. For the spatial case

1 1

*M N l A r A r j i*

1 1 <sup>2</sup> () ,1

*l A r A r*

determined. Tizhoosh present different functions, such as the standard S-function, the Huang and Wang function, LR-type fuzzy number (Tizhoosh et al, 1998; Huang and Wang, 1995; Pal and Bezdek, 1994; Pal and Murthy, 1990). Similar 3DMF presented in section 4.1, to generation of polar MF, CSS is used. The estimation of the MF also exanimate from the fitting of a cubic smoothing Spline, (Mora et al.,2011) to the 3DPMF*(r,*ߠ*(*. The fitting objective

<sup>2</sup> 2 2

∬ (31)

. ( , , ) (1 ) ( , ) ,

*M P f r s r p D S r drd*

where, this equation include two parts: *Compactness* and *Smoothness*. The smoothing factor p, controls the balance between being an interpolating spline crossing all data points (with p = 1) and being a strictly smooth Spline (with p = 0). In the polar transform, as p moves from 0

Using Radon transform for MF generation have several benefits such as Translation,

 

, , , , 0 *<sup>y</sup> Rf P*

*R fx x y y Pr r*

*R f xcos ysin xcos ysin P*

 

*x r*

1

0 <sup>2</sup> ( ) () ,1 *L*

*r A h r min*

*MN*

1 1

*m n*

*y x*

Rotation and Scaling in IT2 FPM. (Miao et al., 2012).

,

 

,( ,

 

To quantify the object fuzziness, a suitable membership function

to 1, the smoothing spline changes from one extreme to the other. (See Fig. 11)

, ,

where *M N* is subset *A X* with *L* radon transform value, 0, 1 *r L* , the histogram *h*(*RT*) and the membership function *μX*(*RT*), the linear index of fuzziness *γl* can be defined as

2. *IF* , , 1 *uij Lij* (high ambiguity) *Then A* 1.

$$\tilde{\mathcal{V}}\{A\} = \left[\frac{1}{\left(M\mathcal{N}\right)\_q^{\frac{1}{q}}}\right] \left[\sum\_{j=1}^{M-1N-1} \left|\mu\_{\mathfrak{a}\{i,j\}} - \mu\_{\mathfrak{U}\{i,j\}}\right|^q\right]^{\frac{1}{q}} = 1 \text{ for } q \in [1, \infty). \tag{25}$$

3. *<sup>A</sup> <sup>A</sup>* Where ���̅ �is type II fuzzy set and it's complement set can be determined by 1- *uij* , and 1- *Lij* , , therefore complement set defined as follow

For the complement set, the ultrafuzziness �� is equal:

$$\widetilde{\mathcal{T}}\left(\widetilde{\boldsymbol{A}}\right) = \left[\frac{1}{\left(\mathop{\mathrm{MN}}\nolimits\right)^{\mathrm{I}}}\right] \left[\sum\_{j=1}^{M-1} \sum\_{i=1}^{N-1} \left|1 - \mu\_{\mathrm{a}\left(i,j\right)} - 1 - \mu\_{\mathrm{L}\left(i,j\right)}\right|^{\boldsymbol{\theta}}\right]^{\mathrm{J}} = \widetilde{\mathcal{T}}\left(\widetilde{\boldsymbol{A}}\right) for \,\boldsymbol{q} \in \left[1, \infty\right) \tag{26}$$

$$\text{4.} \quad \text{IF 3DFOL}\_{\{i,j\}} < \text{3DFOL}\_{\{d,c\}} \text{ Then } \tilde{\gamma}\left(\tilde{A}\_{\{i,j\}}\right) < \tilde{\gamma}\left(\tilde{A}\_{\{d,c\}}\right).$$

#### *4.1.2. Finding the optimum interval 3DMF*

The general approach for 3DMF based on upper and lower MF is equal:

$$\xi = \left[1 - \frac{\min \tilde{\gamma}(i, j)}{\max \tilde{\gamma}(i, j)}\right], \text{SLIRF}\left(i, j\right) = \mu\_L\left(\mathcal{g}\_{\vec{\eta}}\right) \left[\xi + 1\right], \text{OR}\,\mu\_L\left(\mathcal{g}\_{\vec{\eta}}\right) \left[1 - \xi\right] \tag{27}$$

Where is ultra fuzzy coefficient and *i j* , �� ultra fuzzy value for *uij* , and *Lij* , , in upper and lower threshold.

#### **4.2. Interval type-2 polar based method**

Image processing is one among interesting applications of 3DMF. Instead of type reduce from Type-2 to type-1, we used a polar transform to make uniformity by same scale in [0,2��� The RT of a two-dimensional function *f x y*, in *r*,plane is defined as:

$$R\left(r,\theta\right) = R\left(r,\theta\right)\left[f\left(\mathbf{x},\mathbf{y}\right)\right] = \int\int f\left(\mathbf{x},\mathbf{y}\right)\delta\left(r-\mathbf{x}\cos\theta-y\sin\theta\right)d\mathbf{x}\,dy,\tag{28}$$

where *f x y*, represents an image, *P r* , is the radon transform of *f x y*, , θ represents the line direction, and r is the distance away from the origin of coordinates. (Radon, 1919), (Miao et al., 2012) Where · is the Dirac function, r[−∞,∞] is the perpendicular distance of a line from the origin and *θ*[0,*π*] is the angle formed by the distance vector. For the spatial case such as 3DMF, the fuzziness can be calculated as follows (Tizhoosh, 2005);

102 Fuzzy Controllers – Recent Advances in Theory and Applications

*MN*

*MN*

(high ambiguity) *Then*

*<sup>A</sup> <sup>A</sup>* Where ���̅

*MN*

*4.1.2. Finding the optimum interval 3DMF* 

determined by 1- *uij* ,

 

upper and lower threshold.

Where 

  

2. *IF* , , 1 *uij Lij* 

> 

3. 

consider as a type I fuzzy set *Then uij Lij* , ,

*j i <sup>q</sup>*

*j i <sup>q</sup>*

 and 1- *Lij* , 

1 1

1 1

The general approach for 3DMF based on upper and lower MF is equal:

For the complement set, the ultrafuzziness �� is equal:

*j i <sup>q</sup>*

4. *IF* (,) ( ,) 3 3 *DFOU DFOU i j d c Then A A* (,) ( ,) *i j d c*

[0,2��� The RT of a two-dimensional function *f x y*, in *r*,

 

is ultra fuzzy coefficient and

**4.2. Interval type-2 polar based method** 

1 , ,

*A for q*

1 1 1 , , 1 1

*A for q*

1 1 1 , , 1 1

1 – 0 [1, ). *M N q q uij U ij*

(24)

 

*A* 1.

 

1 1 1 1,

(26)

*M N q q uij L ij*

 

> .

 min , <sup>1</sup> , , 1 , ( ) 1 max , *L ij U ij i j SURF i j g OR g i j*

Image processing is one among interesting applications of 3DMF. Instead of type reduce from Type-2 to type-1, we used a polar transform to make uniformity by same scale in

*P r R r f x y f x y r xcos ysin dxdy* ,

 

,, ,

 

*A A for q*

*M N q q uij U ij*

1 – 1 [1, ).

(25)

 *AND*

1

1

�is type II fuzzy set and it's complement set can be

, therefore complement set defined as follow

1

 

> 

(27)

*i j* , �� ultra fuzzy value for *uij* ,

 

(28)

 

> and *Lij* , , in

plane is defined as:

,

*A* 0

1. *IF i j*, 

$$\mathcal{I}\_l \mathcal{I}\_l(A) = \left(\frac{2}{\left(M\mathcal{N}\right)}\right) \left[\sum\_{j=1}^{M-1N-1} \min\left[\mu\_{A\left(\tau\_{i,j}\right)}, 1-\mu\_{A\left(\tau\_{i,j}\right)}\right]\right],\tag{29}$$

where *M N* is subset *A X* with *L* radon transform value, 0, 1 *r L* , the histogram *h*(*RT*) and the membership function *μX*(*RT*), the linear index of fuzziness *γl* can be defined as follows (see Fig.6):

$$\mathcal{I}\_l \mathcal{I}\_l(A) = \left(\frac{2}{\binom{\text{MN}}{\text{MN}}}\right) \left[\sum\_{r=0}^{L-1} h(r) \times \min\left[\mu\_{A(r)}, 1-\mu\_{A(r)}\right]\right],\tag{30}$$

To quantify the object fuzziness, a suitable membership function *<sup>A</sup> r* should be determined. Tizhoosh present different functions, such as the standard S-function, the Huang and Wang function, LR-type fuzzy number (Tizhoosh et al, 1998; Huang and Wang, 1995; Pal and Bezdek, 1994; Pal and Murthy, 1990). Similar 3DMF presented in section 4.1, to generation of polar MF, CSS is used. The estimation of the MF also exanimate from the fitting of a cubic smoothing Spline, (Mora et al.,2011) to the 3DPMF*(r,*ߠ*(*. The fitting objective is to minimize the equation.

$$M = P.\sum\_{y=1:x=1}^{m} \sum\_{r=1}^{n} (f\left(r, \theta\right) - s\left(r, \theta\right))^2 + (1-p)\iint (D^2S\left(r, \theta\right))^2 dr d\theta,\tag{31}$$

where, this equation include two parts: *Compactness* and *Smoothness*. The smoothing factor p, controls the balance between being an interpolating spline crossing all data points (with p = 1) and being a strictly smooth Spline (with p = 0). In the polar transform, as p moves from 0 to 1, the smoothing spline changes from one extreme to the other. (See Fig. 11)

Using Radon transform for MF generation have several benefits such as Translation, Rotation and Scaling in IT2 FPM. (Miao et al., 2012).

$$R\left(\rho,\theta\right)\Big|\left.f(\mathbf{x}-\mathbf{x}\_{0'},\mathbf{y}-\mathbf{y}\_{0})\right|=P\left(r-r\_{0'},\theta\right),\tag{32}$$

$$R\left(\rho,\theta\right)\left\{f\left(\mathbf{x}\cos\varphi+y\sin\varphi,-\mathbf{x}\cos\varphi+y\sin\varphi\right)=P\left(\rho,\theta+\varphi\right)\tag{33}$$

$$\mathbb{P}\left(\rho,\theta\right)\left|f\left(\bigvee\gamma',\bigvee\gamma'\right)\right|=\gamma\mathbb{P}\left(\bigvee\gamma',\theta\right),\gamma\neq 0\tag{34}$$

A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems 105

On advantages of T2 PFM method is decrease on computational load in comparison with histogram based IT2 FMF. According our proposed method computational load can decrease, due to the stimulatory dimension in muli-scale surface and decrease computational load because of modified histogram smoothing process and fitting. Instead finding the one-dimensional UMF and LMF for each class label and feature which used by histogram based method, we fired all points in polar system with a cubic-spline. Next, we obtain the overall UMF and LMF Simultaneously. To obtain the generated IT2 PMF, it essential the three polar FOU (3D PFOU) be calculated. The UMF and LMF are designed by refitting cubic-spline. According proposed method the smoothed histograms have values that are above or below the mother fitted surface. Fig. 12a shows the one example

**Figure 12.** A)Three dimensional domain polar upper bound and lower bound, b) FOU for 3D polar

PFOU , min ,min

is the UMF, *PFOU A*

Using the upper and lower mother surface, to obtain the 3D PFOU, the PUMFs and PLMFs surface are designed. The UMF and LMFs surface normalized. Fig. 12,b. shows the IT2 PMF obtained by our proposed method. The shaded region between the 3D UMF and 3D LMF indicates the 3D PFOU. As shown in the Fig.13, our proposed method can effectively design IT2 FMFs based on the distribution of the input 3D data. The 3D PFOU can be expressed as

> *A PFOU A PFOU A PFOU A PFOU A*

*x X x X*

(35)

s the LMF, and *i* is the feature number.

constructed by polar upper and lower cubic-spline functions.

fuzzy sets, (3D PFOU)

where *PFOU A*

**Figure 11.** A sample of polar memberships function, As p moves from 0 to 1, the smoothing spline changes from one extreme to the other.

Where 00 0 *r x cos y sin* θ, ߛis the scaling factor and ߮ is the rotation angle. A rotation of *f x y*, by angle ߮ leads to a translation of *P*, in the variable θ. A scaling of *f x y*, results in a scaling in the ߩ coordinate, as well as an intensity scaling of *P*, . (Miao et al., 2012). For the Fuzzy Polar based Method, we proposed use the following heuristic approach. This method consists of seven steps to obtain the 3D membership function in the polar domain.


On advantages of T2 PFM method is decrease on computational load in comparison with histogram based IT2 FMF. According our proposed method computational load can decrease, due to the stimulatory dimension in muli-scale surface and decrease computational load because of modified histogram smoothing process and fitting. Instead finding the one-dimensional UMF and LMF for each class label and feature which used by histogram based method, we fired all points in polar system with a cubic-spline. Next, we obtain the overall UMF and LMF Simultaneously. To obtain the generated IT2 PMF, it essential the three polar FOU (3D PFOU) be calculated. The UMF and LMF are designed by refitting cubic-spline. According proposed method the smoothed histograms have values that are above or below the mother fitted surface. Fig. 12a shows the one example constructed by polar upper and lower cubic-spline functions.

104 Fuzzy Controllers – Recent Advances in Theory and Applications

changes from one extreme to the other.

*r x cos y sin* 

*f x y*, by angle ߮ leads to a translation of *P*

uniform data in multi-scale.

polar histogram generator.

lower PSG.

approximate parameter value (p).

Where 00 0

polar domain.

**Figure 11.** A sample of polar memberships function, As p moves from 0 to 1, the smoothing spline

results in a scaling in the ߩ coordinate, as well as an intensity scaling of *P*

surface from image and construct 3D data surface.

**Step 6.** Perform **PSG** fitting for the upper and lower histogram values.

,

2012). For the Fuzzy Polar based Method, we proposed use the following heuristic approach. This method consists of seven steps to obtain the 3D membership function in the

**Step 1.** Three Dimensional Surface (**3D Data)**, Using Radon transform generate the 3D

**Step 2.** Three Dimensional Polar Surface (**3D Polar)**, Transfer data to the polar domain and

**Step 3.** Polar Histogram Generator (**PHG)**, generates polar histogram in all direction using

**Step 4.** Approximate Smoother fitting parameters (**SF)**, Perform SF parameter to obtain the

**Step 5.** Polar Smooth Generator (**PSG)** smooths the histogram of the overall polar surface.

**Step 7.** Determine PFMF, by normalizing the height of the upper PSG and LMF by the

θ, ߛis the scaling factor and ߮ is the rotation angle. A rotation of

in the variable θ. A scaling of *f x y*,

,

. (Miao et al.,

**Figure 12.** A)Three dimensional domain polar upper bound and lower bound, b) FOU for 3D polar fuzzy sets, (3D PFOU)

Using the upper and lower mother surface, to obtain the 3D PFOU, the PUMFs and PLMFs surface are designed. The UMF and LMFs surface normalized. Fig. 12,b. shows the IT2 PMF obtained by our proposed method. The shaded region between the 3D UMF and 3D LMF indicates the 3D PFOU. As shown in the Fig.13, our proposed method can effectively design IT2 FMFs based on the distribution of the input 3D data. The 3D PFOU can be expressed as

$$\text{PFOU}\left(\stackrel{\text{\\_}}{A}\right) = \bigcup\left[\underline{\text{PFOU}}\left(\stackrel{\text{\\_}}{A}\right), \overline{\text{PFOU}}\left(\stackrel{\text{\\_}}{A}\right)\right]\_{\forall x \in X} = \bigcup\left[\text{min}\left\{\underline{\text{PFOU}}\left(\stackrel{\text{\\_}}{A}\right)\right\}, \text{min}\left\{\underline{\text{PFOU}}\left(\stackrel{\text{\\_}}{A}\right)\right\}\right]\_{\forall x \in X} \tag{35}$$
 
$$\text{where } \overline{\text{PFOU}}\left(\stackrel{\text{\\_}}{A}\right) \text{ is the UMF, } \underline{\text{PFOU}}\left(\stackrel{\text{\\_}}{A}\right) \text{s the LMF, and } i \text{ is the feature number.}$$

A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems 107

1

,

*H*

1

 

> [0,1] , *h*

(36)

(37)

(38)

(1, )

(3 )( , )

(3 )( , )

*h DRT*

 

*DRT*

 

*h D PMF*

 

1

*<sup>h</sup> <sup>h</sup>*

*max v*

*<sup>h</sup> <sup>h</sup>*

*max v*

0 0 (3 )( , ) <sup>1</sup> ,

*<sup>h</sup> <sup>h</sup> <sup>r</sup> DMF*

is 3DMF value in the position � ��� �. In thresholding, selection H

(3 )( , )

*GR d d r max v*

 

controller can use for select an optimum threshold based on type II fuzzy. Select a bigger H is worth a more enhanced maximum value. In order to define a type II fuzzy set in polar domain, first we develop a type I fuzzy set and assign upper and lower membership degrees to each element to (re)construct the footprint of uncertainty in polar system (Fig. 14). Hear we select H=0 to calculate the real 3D PMF. In polar system the definition for uncertainty is slightly deferent 3D FMF. Uncertainty can present in ring and height which reflect to polar memberships function. (Fig. 16) For example, when Radon Transform is applied to wavelet modulus, a distress (crack) is transformed into a peak in radon domain. Originally, every distress reflects to RT and has different intensity in 3DMF histograms. For example mean of 3D PRT have variety range. According to the above Eq. 38 the max GR must be equal 1. Similaty 3D FMF method, in 3D PMF, the amount of ultrafuzziness will

The extreme case of maximal ultrafuzziness in polar system, equal 1, is worth to completely vagueness. Based on Pal and Bezdek (1994) research on several fuzziness index, two general classes proposed by them was additive and multiplicative class (Pal and Bezdek, 1994).

*i j <sup>h</sup> <sup>h</sup>*

*v*

where, ��� denotes the size of 3DPMF platform,H is high polar platform *H*

*4.2.1. A measure of polar ultrafuzziness* 

 

( ,)

( ,)

Based on Kufmann's Index of fuzziness for a set ����(�), 

2 2

 

(,)

increase by rising uncertainly bound.

 

*PN*

division every point at ( ,)

and

and (3 )( , )

*h DRT v*

 

Polar ultrafuzzy can be calculated based on 3DMF fuzzy set. Such as defuzzifcation method proposed in measure of surface ultrafuzziness in section 4.1.1., type II membership function must be in [0,1]. Similaraway, normalization must be used for 3D PMF generation by

by max 3D PRT.

(3 )( , )

(3 )( , )

 

*D MF*

 

*D PMF*

*v*

*v*

2

 

**Figure 13.** Heuristic based IT2 FPM generation method

#### *4.2.1. A measure of polar ultrafuzziness*

Polar ultrafuzzy can be calculated based on 3DMF fuzzy set. Such as defuzzifcation method proposed in measure of surface ultrafuzziness in section 4.1.1., type II membership function must be in [0,1]. Similaraway, normalization must be used for 3D PMF generation by division every point at ( ,) by max 3D PRT.

$$PN\_{\left(\rho,\theta\right)} = \left( \left[ \upsilon\_{\left(3\,\text{PMF}\right)\left(\rho,\theta\right)} \right] \Bigg\times \max\limits\_{\max\left[ \upsilon\_{\left(3\,\text{PMF}\right)\left(\rho,\theta\right)} \right]} \right)^{\frac{1}{h}} \tag{36}$$

$$\mu\_{\left(\rho,\theta\right)} = \left( \left[ \upsilon\_{\left(3\,\text{D}\,\text{MF}\right)\left(\rho,\theta\right)} \right] \left\langle \rho,\theta \right\rangle \right)\_{\max\left[\upsilon\_{\left(3\,\text{D}\,\text{RT}\right)\left(\rho,\theta\right)}\right]}^{\frac{1}{h}} + H \tag{37}$$

and

106 Fuzzy Controllers – Recent Advances in Theory and Applications

**Figure 13.** Heuristic based IT2 FPM generation method

$$\boldsymbol{GR}\_{\left(i,j\right)} = \frac{1}{2\left(2\pi r\right)^{h}} \prod\_{0}^{2\pi} \left[ \left[ \boldsymbol{\upsilon}\_{\left(\text{3DMF}\right)\left(\rho,\theta\right)} \right] \left\langle \boldsymbol{\upsilon}\_{\left(\text{3DRT}\right)\left(\rho,\theta\right)} \right\rangle^{\frac{1}{h}} \right. \tag{38}$$

where, ��� denotes the size of 3DPMF platform,H is high polar platform *H* [0,1] , *h* (1, ) and (3 )( , ) *h DRT v* is 3DMF value in the position � ��� �. In thresholding, selection H controller can use for select an optimum threshold based on type II fuzzy. Select a bigger H is worth a more enhanced maximum value. In order to define a type II fuzzy set in polar domain, first we develop a type I fuzzy set and assign upper and lower membership degrees to each element to (re)construct the footprint of uncertainty in polar system (Fig. 14). Hear we select H=0 to calculate the real 3D PMF. In polar system the definition for uncertainty is slightly deferent 3D FMF. Uncertainty can present in ring and height which reflect to polar memberships function. (Fig. 16) For example, when Radon Transform is applied to wavelet modulus, a distress (crack) is transformed into a peak in radon domain. Originally, every distress reflects to RT and has different intensity in 3DMF histograms. For example mean of 3D PRT have variety range. According to the above Eq. 38 the max GR must be equal 1. Similaty 3D FMF method, in 3D PMF, the amount of ultrafuzziness will increase by rising uncertainly bound.

The extreme case of maximal ultrafuzziness in polar system, equal 1, is worth to completely vagueness. Based on Pal and Bezdek (1994) research on several fuzziness index, two general classes proposed by them was additive and multiplicative class (Pal and Bezdek, 1994). Based on Kufmann's Index of fuzziness for a set ����(�), 

$$dH\_{ka}\left(A\right) = \left(\frac{2}{n^k}\right) d(P\_\prime P^{near})\tag{39}$$

A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems 109

**Figure 14.** Basic rules for construction 3D polar fuzzy type II memebership function.

**Figure 15.** Three dimensional polar domain of FOU for 3D fuzzy sets, (3D PFOU)

Where, *k R* ,d is a metric, and *near A* is the crisp set close to the P. generally, based on *<sup>d</sup>*,weigh of k determined. The (, ) *near dPP* and linear or quadratic ( ) *H P ka* can be determined by q-norms such as 3D FMF,

$$d(P, P^{\text{near}}) = \left(\sum\_{i=1}^{n} \left| \mu\_{\left(\rho, \theta\right)} - \mu\_{A^{\text{now}}, \left(\rho, \theta\right)} \right|^q \right)^{\bigvee\_q}, \\ H\_{ku}(q, P) = \left(\frac{2}{\frac{1}{n^q}} \left| \left(\sum\_{i=1}^{n} \left| \mu\_i - \mu\_{p^{\text{now}}, i} \right|^q \right)^{\bigvee\_q} \right| \right)^{\bigvee\_q} \tag{40}$$

Where *q* 1, . Based on Tizhoosh ultrafuzziness index, we developed a new index in continues polar domain (Tizhoosh, 2005),

$$\tilde{\gamma}\left(\tilde{P}\right) = \frac{1}{4\pi\rho^2} \int\_0^{2\pi} \left[ \left( \mu\_\mathcal{U}(\mathcal{g}\_{\left(\rho,\theta\right)}) - \mu\_\mathcal{L}(\mathcal{g}\_{\left(\rho,\theta\right)}) \right) d\rho d\theta\right.\tag{41}$$

Where 1/ *U A g g* and 1/ , *<sup>L</sup> <sup>A</sup> g g* and in general term it present as follow,

$$\tilde{\gamma}\left(\tilde{P}\right) = \frac{1}{4\pi\rho^2} \int\_0^2 \prod\_{\ell=0}^{r-1} \left(\mu\_\mathcal{U}\left(\mathbb{g}\_{\left(\rho,\theta\right)}\right) - \mu\_\mathcal{L}\left(\mathbb{g}\_{\left(\rho,\theta\right)}\right)\right) \times h(\mathcal{g}) \,d\rho \,d\theta \,d\theta \,d\theta \tag{42}$$

A measure of ultrafuzziness �� for a polar 3D FMF in <sup>2</sup> <sup>4</sup> , polar 3D FMF and the membership function *<sup>P</sup>*( ,) can be developed as follows:

$$
\tilde{\varphi}(P) = \left(\frac{1}{\left(\operatorname{Area}\right)^{\frac{1}{q}}}\right) \left[\bigcap\_{0=0}^{2\pi r} \left|\mu\_{\mathcal{U}}(\operatorname{g}\_{(\rho,\theta)}) - \mu\_{\mathcal{L}}(\operatorname{g}\_{(\rho,\theta)})\right|^{q}\right]^{\frac{1}{q}},
$$

$$
\frac{\partial}{\partial T} \tilde{\varphi}\left(PMF\_{(\rho,\theta)}\right) = \frac{\partial}{\partial T} \left[\frac{1}{\left(\operatorname{Area}\right)^{\frac{1}{q}}}\right] \left[\frac{\gamma}{\gamma} \left|\mu\_{\mathcal{U}}(\operatorname{g}\_{(\rho,\theta)}) - \mu\_{\mathcal{L}}(\operatorname{g}\_{(\rho,\theta)})\right|^{q}\right]^{\frac{1}{q}} d\rho d\theta = 0 \tag{43}
$$

The variation in the polar space can be measured by this method, therefore the new Index introduced in polar dimensional domain of FOU for 3D polar fuzzy sets, (3D PFOU). This method can resolve the problems about the discontinues domain and in a same time reduce on dimension by using polar transform. (see Fig 15, 16).

Similarly, Kaufmann conditions consists of *Minimum ultrafuzziness, Maximum ultrafuzziness, Equal ultrafuzziness* and *Reduced ultrafuzziness* evaluated for polar method (Kaufmann, 1975). Polar Index is qualified for measure of ultrafuzziness in 3D polar domain with these conditions.

by q-norms such as 3D FMF,

Where 1/ *U A g g*

membership function *<sup>P</sup>*( ,)

  

follow,

 <sup>2</sup> (, ) *near ka <sup>k</sup> H A dPP n* 

1 1 ( ,) ,( , ) 1 ,

*A ka i P i*

(40)

 

(41)

*n*

 

and in general term it present as

 

1

1

 

, polar 3D FMF and the

 

> 

1 ( ) ( ) 0 *<sup>r</sup> <sup>q</sup> <sup>q</sup>*

   

Where, *k R* ,d is a metric, and *near A* is the crisp set close to the P. generally, based on *<sup>d</sup>*,weigh of k determined. The (, ) *near dPP* and linear or quadratic ( ) *H P ka* can be determined

> 1 1 <sup>2</sup> ( , ) *near* , , ) ( *near*

Where *q* 1, . Based on Tizhoosh ultrafuzziness index, we developed a new index in

2 ( ,) ( ,)

*U L P g g dd*

 

, *<sup>L</sup> <sup>A</sup> g g*

 

2 ( ,) ( ,)

 

<sup>1</sup> ( )( ) , <sup>4</sup>

<sup>1</sup> ( ) ( ) ( ) . . 4

1 ( ,) ( ,)

 

1 ( )( ) , *<sup>r</sup> <sup>q</sup> <sup>q</sup>*

*U L*

 

*U L*

 

(43)

 

*U L P g g h g d d dh*

 

 

(42)

*i i <sup>q</sup>*

 

2

and 1/

2 1

 

 

on dimension by using polar transform. (see Fig 15, 16).

*r*

0 00

A measure of ultrafuzziness �� for a polar 3D FMF in <sup>2</sup> <sup>4</sup>

*Area*

*q*

*Area*

*q*

0 0

can be developed as follows:

2

0 0

2 ( ,) 1 ( ,) ( ,) 0 0

*PMF g g dd T T*

The variation in the polar space can be measured by this method, therefore the new Index introduced in polar dimensional domain of FOU for 3D polar fuzzy sets, (3D PFOU). This method can resolve the problems about the discontinues domain and in a same time reduce

Similarly, Kaufmann conditions consists of *Minimum ultrafuzziness, Maximum ultrafuzziness, Equal ultrafuzziness* and *Reduced ultrafuzziness* evaluated for polar method (Kaufmann, 1975). Polar Index is qualified for measure of ultrafuzziness in 3D polar domain with these conditions.

*P g g*

 

*r*

*dPP H qP*

 

 

continues polar domain (Tizhoosh, 2005),

*n n q q <sup>q</sup> <sup>q</sup> near*

 

(39)

**Figure 14.** Basic rules for construction 3D polar fuzzy type II memebership function.

**Figure 15.** Three dimensional polar domain of FOU for 3D fuzzy sets, (3D PFOU)

A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems 111

1

*h*

*U*

(46)

*H*

*H*

 

> ,.

 

> ,

and

 

, �� ultra fuzzy value for *u*

**Then** Y is 1 ( ) *PC*

 

 

1

*<sup>L</sup> <sup>h</sup>*

*h DPRT*

 

(3 )( , )

 

> 

*DPRT*

1

1

*max v*

 

 *P P* ( , ) ( ,) *d c* .

*max v*

*<sup>h</sup> <sup>h</sup>*

3 ,

 

Where

*L* , 

Single Output (MISO):

*Rule 1*: **If** *X*1 is 11 ( ) *PA*

*Rule 2*: **If** *X*1 is 21 ( ) *PA*

 *P x*

 

*u u*

4. *IF* ( ,) ( ,) 3 3 *DPFOU DPFOU* 

*Area*

*q*

*4.2.2. Finding the optimum interval 3D PMF* 

 

, ,

 

2

0 0

*d c Then*

*<sup>P</sup>* is polar ultra fuzzy coefficient and

**5. Polar fuzzy type-2 approximate reasoning** 

 & *X*2 is 12 ( ) *PA* 

 & *X*2 is 22 ( ) *PA* 

1

, , | ,

Where 0 *H H U L* , for the complement set, the ultrafuzziness �� is equal:

The general approach for 3DMF based on upper and lower MF is equal:

memberships function of the interval type 2 polar fuzzy sets in position

1 ( ,) ( ,)

 

*P g g P for q*

*v*

*v*

*h DPRT*

 

(3 )( , )

 

1 1 ( ) 1 ( ) 1, *<sup>r</sup> <sup>q</sup> <sup>q</sup> U L*

 (47)

 , , , , 2 , , 2 *<sup>P</sup> L PU P IT PMF g OR g*

> 

 

, in upper and lower bound. For example, Fig.16 presents the principle polar

In this part, basic theory of fuzzy polar rule interpolation in fuzzy rule based will be presented for polar membership's function of type -2 fuzzy sets. Lets us show polar fuzzy rules with multiple antecedent and single consequent based on T2 PMF rules, Multi Input

& … & *Xm* is 1 ( ) *<sup>m</sup> PA*

& … & �� is �������)��**Then** Y is 2 ( ) *PC*

   

(48)

*DPRT*

3 ,

**Figure 16.** Example of Three polar FOU for 3D fuzzy sets, (3D PFOU) using proposed algorithm

1. **IF** �(���) consider as a type I polar fuzzy set **Then** ��(���) = ��(���) **AND** ��(�) = 0

$$\tilde{\gamma}\left(P\right) = \left[\frac{1}{\left(Area\right)\_{q}^{1}}\right] \left[\frac{2\pi}{\int\_{0}^{2} \left|\int\_{0}^{2} \left(\mu\_{L}\left(\mathcal{S}\_{\left(\rho,\theta\right)}\right) - \mu\_{L}\left(\mathcal{S}\_{\left(\rho,\theta\right)}\right)\right|^{q}\right]^{\frac{1}{q}}}{}{}\_{0}^{q} = 0 \text{ for } q \in \left[1,\infty\right). \tag{44}$$

2. *IF* , , 1 *u L* (high ambiguity) *Then P* 1

$$\tilde{\gamma}\begin{pmatrix} P \\ \end{pmatrix} = \left[ \frac{1}{\left( \begin{array}{c} 1 \\ \text{Area} \end{array} \right) \left| \begin{array}{c} 2\pi r \\ 1 \\ 0 \end{array} \right|} \left| \begin{array}{c} \mu\_{\mathcal{U}} \{ \mathcal{S}\_{\{\rho,\theta\}} \}-\mu\_{\mathcal{L}} \{ \mathcal{S}\_{\{\rho,\theta\}} \} \right| \\\\ \end{array} \right| \right]^{\mathsf{H}} = \mathbf{1} \operatorname{ for } q \in [1,\infty). \tag{45}$$

3. *P P* . Where *P* is type II fuzzy set and it's complement set can be determined by 1- *<sup>u</sup>* , and 1- *<sup>L</sup>* , , therefore complement set defined as follow

A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems 111

$$\tilde{\vec{P}} = \begin{pmatrix} 1 - \left( \left[ \upsilon\_{\text{(3DPRT)}} \right]\_{\left(\rho,\theta\right)} \right) \\\\ \left[ \upsilon\_{\text{r}} \mu\_{\text{u}\left(\rho,\theta\right)}, \mu\_{\text{u}\left(\rho,\theta\right)} \right] \\\\ 1 - \left( \upsilon\_{\text{(3DPRT)}\left(\rho,\theta\right)} \right) \\\\ \end{pmatrix} \pmod{\tau\_{\text{(3DPRT)}\left(\rho,\theta\right)}}^{h} \tag{46}$$

Where 0 *H H U L* , for the complement set, the ultrafuzziness �� is equal:

$$\tilde{\gamma}\left(\tilde{\overline{P}}\right) = \left[\frac{1}{\left(Area\right)^{\frac{1}{q}}}\right] \left[\frac{2\pi r}{\int\limits\_{0}^{2\pi r} \left|1 - \mu\_{L}\{\mathcal{g}\_{\left(\rho,\rho\right)}\} - 1 + \mu\_{L}\{\mathcal{g}\_{\left(\rho,\rho\right)}\}\right|^{q}}\right]^{\frac{1}{q}} = \tilde{\gamma}\left(\tilde{P}\right)for\,q \in \left[1,\infty\right) \tag{47}$$

4. *IF* ( ,) ( ,) 3 3 *DPFOU DPFOU d c Then P P* ( , ) ( ,) *d c* .

#### *4.2.2. Finding the optimum interval 3D PMF*

110 Fuzzy Controllers – Recent Advances in Theory and Applications

 

  

2. *IF* , , 1 *u L* 

3. 

 *P P* . Where

by 1- *<sup>u</sup>* , 

*Area*

 and 1- *<sup>L</sup>* , 

*Area*

*q*

*q*

2

(high ambiguity) *Then*

2

0 0

0 0

**Figure 16.** Example of Three polar FOU for 3D fuzzy sets, (3D PFOU) using proposed algorithm

1. **IF** �(���) consider as a type I polar fuzzy set **Then** ��(���) = ��(���) **AND** ��(�) = 0

1 ( ,) ( ,)

 

1 ( ,) ( ,)

 

*P g g for q*

*U L*

*P g g for q*

*U L*

1

1

*P* is type II fuzzy set and it's complement set can be determined

1 ( ) ( ) 0 [1, ). *<sup>r</sup> <sup>q</sup> <sup>q</sup>*

> *P* 1

1 ( ) ( ) 1 [1, ). *<sup>r</sup> <sup>q</sup> <sup>q</sup>*

 

, therefore complement set defined as follow

 

(45)

 

 

(44)

The general approach for 3DMF based on upper and lower MF is equal:

$$\tilde{\boldsymbol{\xi}}\_{P} = \left[ -\frac{\tilde{\boldsymbol{\gamma}}\left(\boldsymbol{\rho}, \boldsymbol{\theta}\right)}{2} \right] \boldsymbol{\iota} \text{T} \mathbf{2} \text{ PMF}\left(\boldsymbol{\rho}, \boldsymbol{\theta}\right) = \left[ \boldsymbol{\mu}\_{\mathcal{L}}\left(\boldsymbol{\mathcal{g}}\_{\boldsymbol{\rho}, \boldsymbol{\theta}}\right) - \boldsymbol{\xi}\_{P}\right] \boldsymbol{\iota} \text{OR}\left[ \boldsymbol{\mu}\_{\mathcal{U}}\left(\boldsymbol{\mathcal{g}}\_{\boldsymbol{\rho}, \boldsymbol{\theta}}\right) + \boldsymbol{\xi}\_{P} \right] \tag{48}$$

Where *<sup>P</sup>* is polar ultra fuzzy coefficient and , �� ultra fuzzy value for *u* , and *L* , , in upper and lower bound. For example, Fig.16 presents the principle polar memberships function of the interval type 2 polar fuzzy sets in position ,.

#### **5. Polar fuzzy type-2 approximate reasoning**

In this part, basic theory of fuzzy polar rule interpolation in fuzzy rule based will be presented for polar membership's function of type -2 fuzzy sets. Lets us show polar fuzzy rules with multiple antecedent and single consequent based on T2 PMF rules, Multi Input Single Output (MISO):

$$\begin{array}{ccccc} \text{Rule 1: If } X\_1 \text{ is (PA)}\_{11} & \& \text{ X}\_2 \text{ is (PA)}\_{12} & \& \dots & \& \text{ X}\_m \text{ is (PA)}\_{1m} \\\\ \stackrel{\stackrel{\equiv}{}}{\text{Rule 2: If } X\_1 \text{ is (PA)}\_{21} & \& \text{ X}\_2 \text{ is (PA)}\_{22} & \& \dots & \& \text{ X}\_m \text{ is (\overline{PA})}\_{2m} \\\\ \end{array}$$

$$\text{Rule } m\text{: If } X\_1 \text{ is } (\stackrel{\circ}{PA})\_{n1} \text{ \& \; X\_2 \text{ is } (\stackrel{\circ}{PA})\_{n2} \text{ \& \; ... \& \; X\_m \text{ is } (\stackrel{\circ}{PA})\_{nm} \text{ Then Y is } (\stackrel{\circ}{PC})\_n$$

Where �� denotes the pth T2-PFM antecedent and Y denote the T2- PFM consequence. ( )*nm PA* is the *n,m* th consequence of T2-PFM fuzzy set of Rule n. According polar MISO method, fuzzy interpolative polar reasoning result which denoted by *P* can be extracted based on observation polar fuzzy set *O* .

( )*nm PA* 

*Mamdani*:

**If** ܺଵ is

*Logical*:

**If** ܺଵ is

Where

Where *<sup>y</sup> P*

logical model, *PM*

constructed as three steps,

A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems 113

> *r*

can be extracted

*nr PC* 

*nr PC* 

*nr PC* 

is the

is the *n,m* th consequence of T2-PFM fuzzy set of Rule n. According polar MIMO

Logical and Mamdani, in the linguistic models considered as prepositions. The general form

**Then** Y is

**Then** Y is

is the T2 PFM as an antecedent variable in polar system

consequent variable, n denotes to number of rule and r denoted to *r* denote the number of

polar system. Concrete effective model proposed by Yager, is a complex method for

from 0 to 1. (Fazel et al. 2009). The defuzzication method for every ߠmust be computed based on theory which presented in section 4.2. Defuzzification agent in polar system

**Step 1.** Calculate the center of area inߠ, in which ߠdefined from 0, to ʹߨ)Gold Veins Root)

**Step 3.** Calculate the center of Gold veins and consider it as defuzzify conclusion result.

 

is consequents of mamdani model and ߚis control factor which move

in window r for conscience *r*.

*n*1

&

*n*1 *PC* 

&

is complement of the T2 PFM as an antecedent variable in

(49)

is result of polar type-2 fuzzy under

*n*2

*PC* 

*n*2

& …&

& …&

*PC* 

*PC* 

method, fuzzy interpolative polar reasoning result which denoted by *<sup>P</sup>*

*nm*

*PA* 

*nm*

*PA*  based on observation polar fuzzy set *O*

& … & *Xm* is

& … & ܺ is

*PA* 

*n*1

combination of these two models, presented as follow:

( ) (1 )( ) *PP P yL M*

is the Yager result(complex model), ( ) *LP*

**Step 2.** Plot the fuzzy Gold veins determined from first step.

of these linguistic rules shows as:

*n*1 *PA* 

*PA* 

*PA* 

consequences and

*n*1

*nm*

*Observation*: **If** *X*1 is 1 ( ) *O* & �� is 2 ( ) *O* & … & *Xm* is ( ) *O <sup>m</sup>* 

*Conclusions*: *O* is *P* .

This method at the first glance, is similar to method which proposed by Chen & Chang (2011) under the title of "fuzzy rule interpolation based on principle membership functions and uncertainty grad function of interval type-2 fuzzy sets"; however, the main difference between our proposed method with their method relay in type decreasing (Chen & Chang, 2011). Based on Chen & Chang (2011) method first type -2 reduced by type-1, then MISO applied, but in our method first MISO applied separately on upper and lower. After that the

type-2 polar reasoning result which denoted by *P* extracted and interval 3D PMF calculated based on section 4.2 theories. This method can be expanded for Multi Input Multi Output (MIMO) systems based on polar T2 PMF's. For example lets us show polar fuzzy rules with multiple antecedents and multiple consequent based on T2 PMF rules, Multi

Input Multi Output (MIMO):

*Rule 1*: **If** �� is 11 ( ) *PA* & … & *Xm* is 1 ( ) *<sup>m</sup> PA* **Then** Y is 11 ( ) *PC* & 12 ( ) *PC* & …& 1 ( ) *<sup>r</sup> PC Rule 2*: **If** *X*1 is 21 ( ) *PA* & … & *Xm* is 2 ( ) *<sup>m</sup> PA* **Then** Y is 21 ( ) *PC* & 22 ( ) *PC* & …& 2 ( ) *<sup>r</sup> PC Rule n*: **If** �� is 1 ( )*<sup>n</sup> PA* & … & �� is ( )*nm PA* **Then** Y is 1 ( )*<sup>n</sup> PC* & 2 ( )*<sup>n</sup> PC* & …& ( )*nr PC* 

*Observation*: **If** *X*1 is 1 ( ) *O* & … & *Xm* is ( ) *O <sup>r</sup>*

*Conclusions*: *O* is 1*P* & 2*P* & …& *<sup>r</sup> P* .

where, such as polar MISO, �� denotes the pth T2-PFM antecedent and Y denote the T2- PFM consequence which expanded in ( )*nr PC* , *r* denote the number of consequence in rule *n*.

( )*nm PA* is the *n,m* th consequence of T2-PFM fuzzy set of Rule n. According polar MIMO method, fuzzy interpolative polar reasoning result which denoted by *<sup>P</sup> r* can be extracted based on observation polar fuzzy set *O* in window r for conscience *r*.

Logical and Mamdani, in the linguistic models considered as prepositions. The general form of these linguistic rules shows as:

#### *Mamdani*:

112 Fuzzy Controllers – Recent Advances in Theory and Applications

type-2 polar reasoning result which denoted by *P*

& … & *Xm* is 1 ( ) *<sup>m</sup> PA*

& … & *Xm* is 2 ( ) *<sup>m</sup> PA*

& … & �� is ( )*nm PA*

 & …& *<sup>r</sup> P* .

 

where, such as polar MISO, �� denotes the pth T2-PFM antecedent and Y denote the

& … & *Xm* is ( ) *O <sup>r</sup>*

& *X*2 is 2 ( )*<sup>n</sup> PA*

can be extracted based on observation polar fuzzy set *O*

 & �� is 2 ( ) *O* 

& … & *Xm* is ( )*nm PA*

is the *n,m* th consequence of T2-PFM fuzzy set of Rule n. According polar

& … & *Xm* is ( ) *O <sup>m</sup>*

Where �� denotes the pth T2-PFM antecedent and Y denote the T2- PFM consequence.

MISO method, fuzzy interpolative polar reasoning result which denoted by

This method at the first glance, is similar to method which proposed by Chen & Chang (2011) under the title of "fuzzy rule interpolation based on principle membership functions and uncertainty grad function of interval type-2 fuzzy sets"; however, the main difference between our proposed method with their method relay in type decreasing (Chen & Chang, 2011). Based on Chen & Chang (2011) method first type -2 reduced by type-1, then MISO applied, but in our method first MISO applied separately on upper and lower. After that the

calculated based on section 4.2 theories. This method can be expanded for Multi Input Multi Output (MIMO) systems based on polar T2 PMF's. For example lets us show polar fuzzy rules with multiple antecedents and multiple consequent based on T2 PMF rules, Multi

 .

 

 

**Then** Y is 11 ( ) *PC*

**Then** Y is 21 ( ) *PC*

**Then** Y is 1 ( )*<sup>n</sup> PC*

 & 12 ( ) *PC* 

 & 22 ( ) *PC* 

, *r* denote the number of consequence in

 & 2 ( )*<sup>n</sup> PC* 

**Then** Y is ( )*<sup>n</sup> PC*

 

extracted and interval 3D PMF

 & …& 1 ( ) *<sup>r</sup> PC* 

 & …& 2 ( ) *<sup>r</sup> PC* 

 & …& ( )*nr PC* 

*Rule n*: **If** *X*1 is 1 ( )*<sup>n</sup> PA*

*Observation*: **If** *X*1 is 1 ( ) *O*

 is *P* .

Input Multi Output (MIMO):

 & 2*P* 

T2- PFM consequence which expanded in ( )*nr PC*

*Rule 1*: **If** �� is 11 ( ) *PA*

*Rule 2*: **If** *X*1 is 21 ( ) *PA*

*Rule n*: **If** �� is 1 ( )*<sup>n</sup> PA*

*Conclusions*: *O*

rule *n*.

*Observation*: **If** *X*1 is 1 ( ) *O*

 is 1*P* 

*Conclusions*: *O*

( )*nm PA* 

*P* 

$$\text{If } X\_1 \text{ is } \begin{pmatrix} \stackrel{\circ}{PA} \\ \stackrel{\circ}{PA} \end{pmatrix}\_{n1} \text{ & } \dots \text{ & } X\_m \text{ is } \begin{pmatrix} \stackrel{\circ}{PA} \\ \stackrel{\circ}{PA} \end{pmatrix}\_{nm} \text{ Then Y is } \begin{pmatrix} \stackrel{\circ}{PC} \\ \stackrel{\circ}{PA} \end{pmatrix}\_{n2} \text{ & } \begin{pmatrix} \stackrel{\circ}{PC} \\ \stackrel{\circ}{PA} \end{pmatrix}\_{nm} \text{ & } \dots \text{ & } X\_m \text{ is } \begin{pmatrix} \stackrel{\circ}{PA} \\ \stackrel{\circ}{PA} \end{pmatrix}\_{nm}$$

*Logical*:

$$\text{If } X\_1 \text{ is } \left[ \overbrace{\left( \overleftarrow{PA} \right)\_{n1}}^{\dots} \right] \text{ & ... & \& \newline X\_m \text{ is } \left[ \overleftarrow{\left( \overleftarrow{PA} \right)\_{nm}}^{\dots} \right] \text{Then Y is } \left( \overleftarrow{PC} \right)\_{n1} \text{ & \newline \left( \overleftarrow{PC} \right)\_{n2}}^{\dots} \text{ & ... & \& \left( \overleftarrow{PC} \right)\_{nm} \text{ & ...} \right]$$

Where *nm PA* is the T2 PFM as an antecedent variable in polar system *nr PC* is the consequent variable, n denotes to number of rule and r denoted to *r* denote the number of

consequences and *n*1 *PA* is complement of the T2 PFM as an antecedent variable in

polar system. Concrete effective model proposed by Yager, is a complex method for combination of these two models, presented as follow:

$$
\stackrel{\circ}{P}\_y = \beta(\stackrel{\circ}{P}\_L) + (1 - \beta)(\stackrel{\circ}{P}\_M) \tag{49}
$$

Where *<sup>y</sup> P* is the Yager result(complex model), ( ) *LP* is result of polar type-2 fuzzy under logical model, *PM* is consequents of mamdani model and ߚis control factor which move from 0 to 1. (Fazel et al. 2009). The defuzzication method for every ߠmust be computed based on theory which presented in section 4.2. Defuzzification agent in polar system constructed as three steps,


In step 1, center of area or center of gravity can be calculated in polar system using eq.38, which is a most common model used :

$$\mathcal{L}\left(\boldsymbol{\theta}\right) = \left(\frac{1}{Area\left(\boldsymbol{\theta}\right)}\right) \left[ \left.\left[\boldsymbol{\mu}\_{\mathrm{l}\mathbb{I}}\left(\mathbb{g}\_{\left(\boldsymbol{\rho},\boldsymbol{\theta}\right)}\right), \boldsymbol{\rho} - \boldsymbol{\mu}\_{\mathrm{l}}\left(\mathbb{g}\_{\left(\boldsymbol{\rho},\boldsymbol{\theta}\right)}\right), \boldsymbol{\rho}\right] \right] d\boldsymbol{\rho}.\tag{50}$$

A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems 115

**Figure 18.** a) Input T2 PMF longitudinal cracking, b)Final difuzzification points longitudinal Cracking, c) Input T2 PMF transverse cracking, d)Final difuzzification points transverse Cracking, e) Input T2PMF diagonal cracking, f)Final difuzzification points diagonal Cracking, g) Input T2 PMF block cracking, h)Final difuzzification points block Cracking, i) Input T2 PMF aligator cracking, j)Final difuzzification

points alligator Cracking.

and

$$\mathcal{C}\left(\boldsymbol{\theta}\right)\_{m} = \left(\frac{1}{Area\left(\boldsymbol{\theta}\right)^{q}}\right) \left[\left.\left[\boldsymbol{\mu}\_{\mathcal{U}}\left(\boldsymbol{g}\_{\left(\boldsymbol{\rho},\boldsymbol{\theta}\right)}\right)\right]^{q}\cdot\boldsymbol{\rho} - \left(\boldsymbol{\mu}\_{\mathcal{L}}\left(\boldsymbol{g}\_{\left(\boldsymbol{\rho},\boldsymbol{\theta}\right)}\right)\right)^{q}\cdot\boldsymbol{\rho}\right|\right]d\boldsymbol{\rho}\tag{51}$$

Gold Veins Root is a vector consist of paired , *C* , which is shows the direction center of gravity and it present a useful information from membership function variance and crisp result without type reduction. *<sup>m</sup> <sup>C</sup>* is a modified Gold Veins Root that can be control the final defuzzication result, in which q play a defuzzifer role. An example of Gold Veins Root extracted from polar T2 PMF is depicted in Fig.17. Two result from original and modified center of gravity by q=3, present in Fig 17.

**Figure 17.** Gold Veins Root in T2 PMF

In order to effectively implementation and test proposed model in some applications such as prediction problem and pattern recognition, such as the other models in fuzzy, we need set in motion to product a crisp result. We chose a heuristic function to generate a best crisp defuuzy value from Gold Veins Root, based on Eq.51

$$C = \left[\frac{\max\left(GVR\right) - \min\left(GVR\right)}{2}\right] \tag{52}$$

For the case Fig.18, C=96.33 worth in blurred section from 200,400 , provided good prediction orientation and radios for extension of T2 PFM. Now we present a logical method for type -2 in 3D techniques. Remove type reduction; turn T2 PMF as faster than existing techniques and to be more accurate model for type-2 inference techniques because of combination Logical & Mamdani models.

Gold Veins Root is a vector consist of paired

*Area*

which is a most common model used :

result without type reduction. *<sup>m</sup> <sup>C</sup>*

**Figure 17.** Gold Veins Root in T2 PMF

defuuzy value from Gold Veins Root, based on Eq.51

combination Logical & Mamdani models.

*C*

For the case Fig.18, C=96.33 worth in blurred section from 200,400

center of gravity by q=3, present in Fig 17.

and

In step 1, center of area or center of gravity can be calculated in polar system using eq.38,

1 . ..

 

   

 

(51)

is a modified Gold Veins Root that can be control the

 

, which is shows the direction center

(52)

, provided good

(50)

 , , 0

*<sup>C</sup> U L g gd Area* 

*<sup>C</sup> <sup>m</sup> <sup>q</sup> U L g gd*

 

 

, ,

 , *C*

of gravity and it present a useful information from membership function variance and crisp

final defuzzication result, in which q play a defuzzifer role. An example of Gold Veins Root extracted from polar T2 PMF is depicted in Fig.17. Two result from original and modified

In order to effectively implementation and test proposed model in some applications such as prediction problem and pattern recognition, such as the other models in fuzzy, we need set in motion to product a crisp result. We chose a heuristic function to generate a best crisp

> max min( ) 2 *GVR GVR*

prediction orientation and radios for extension of T2 PFM. Now we present a logical method for type -2 in 3D techniques. Remove type reduction; turn T2 PMF as faster than existing techniques and to be more accurate model for type-2 inference techniques because of

1 . . *r q q*

 

*r*

0

**Figure 18.** a) Input T2 PMF longitudinal cracking, b)Final difuzzification points longitudinal Cracking, c) Input T2 PMF transverse cracking, d)Final difuzzification points transverse Cracking, e) Input T2PMF diagonal cracking, f)Final difuzzification points diagonal Cracking, g) Input T2 PMF block cracking, h)Final difuzzification points block Cracking, i) Input T2 PMF aligator cracking, j)Final difuzzification points alligator Cracking.

## **6. Conclusions**

In this chapter the basic concepts of new fuzzy sets, three dimensional (3D) memberships and how they are applied in the design of type-1 and type-2 fuzzy thresholding in control systems are presented. The robustness of a system highly depends on automatic fuzzification and membership functions shape and defuzzification. The related methodology and theoretical base are discussed, using real examples in automatic control in civil engineering. Selection of a supper membership function is a golden key in fuzzy controls. A robust method to consider the uncertainty of membership values by using flexible thresholding for controller problems proposed in the special a polar domain presented in this chapter. Different fuzzy membership functions may have various impacts on the systems and, then, different thresholds in control problems. To solve this problem, type II fuzzy thresholding is recommended. The upper and lower membership functions promote this dilemma; however the figure of uncertainty (FOU) has a fixed value that is equal to one, in all the upper and lower membership function. Type-2 fuzzy logic can effectively improve the control characteristic by using FOU of the membership functions.

A Type-2 Fuzzy Model Based on Three Dimensional Membership Functions for Smart Thresholding in Control Systems 117

Fereidoon Moghadas Nejad and Hamzeh Zakeri

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A new fuzzy thresholding (flexible thresholding) technique developed, which processes threshold as a flexible type-2 fuzzy sets. Experimental results are provided in order to demonstrate the usefulness of the proposed approach. A review of types of fuzzy threshold methods in control problems provided and their algorithms presented. In type-2 thresholding method, measurement of fuzziness gives a quantitative index to vagueness. To quantify the object fuzziness, a suitable membership function based on thresholding for control problems introduced. A measure for ultra-fuzziness in 3D fuzzy model is proposed. A new method for thresholding algorithm based on 3D type-2 fuzzy and selection the optimum thresholding in 3D surface are addressed. By an example the validity of novel fuzzy algorithm in control systems, based on three dimensional membership functions demonstrated.

This paper presents a new type of fuzzy membership functions and uncertainly grade in the frame of polar systems. The proposed method can be used and generalized for several problems; however in this paper we present implementation of polar fuzzy type-2(PFT2) as a part of Hybrid expert system for pavement distress detection and classification.

Vast applications are predicted this fuzzy reasoning. The central idea of this work was to introduce the application of polar type II fuzzy sets.

The most important aspect of the proposed model is the ability of self-organization of the membership function and initial height platform without requiring programming.

Additional experiments reinforced this conclusion. More extensive investigations on other measures of ultrafuzziness and the effect of parameters influencing the width/length of FOU should certainly be conducted.

## **Author details**

M.H. Fazel Zarandi *Department of Industrial Engineering, Amirkabir University of Technology, Tehran, Iran*  Fereidoon Moghadas Nejad and Hamzeh Zakeri

*Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran* 

## **7. References**

116 Fuzzy Controllers – Recent Advances in Theory and Applications

In this chapter the basic concepts of new fuzzy sets, three dimensional (3D) memberships and how they are applied in the design of type-1 and type-2 fuzzy thresholding in control systems are presented. The robustness of a system highly depends on automatic fuzzification and membership functions shape and defuzzification. The related methodology and theoretical base are discussed, using real examples in automatic control in civil engineering. Selection of a supper membership function is a golden key in fuzzy controls. A robust method to consider the uncertainty of membership values by using flexible thresholding for controller problems proposed in the special a polar domain presented in this chapter. Different fuzzy membership functions may have various impacts on the systems and, then, different thresholds in control problems. To solve this problem, type II fuzzy thresholding is recommended. The upper and lower membership functions promote this dilemma; however the figure of uncertainty (FOU) has a fixed value that is equal to one, in all the upper and lower membership function. Type-2 fuzzy logic can effectively improve the control characteristic by using FOU of the membership functions.

A new fuzzy thresholding (flexible thresholding) technique developed, which processes threshold as a flexible type-2 fuzzy sets. Experimental results are provided in order to demonstrate the usefulness of the proposed approach. A review of types of fuzzy threshold methods in control problems provided and their algorithms presented. In type-2 thresholding method, measurement of fuzziness gives a quantitative index to vagueness. To quantify the object fuzziness, a suitable membership function based on thresholding for control problems introduced. A measure for ultra-fuzziness in 3D fuzzy model is proposed. A new method for thresholding algorithm based on 3D type-2 fuzzy and selection the optimum thresholding in 3D surface are addressed. By an example the validity of novel fuzzy algorithm in control

This paper presents a new type of fuzzy membership functions and uncertainly grade in the frame of polar systems. The proposed method can be used and generalized for several problems; however in this paper we present implementation of polar fuzzy type-2(PFT2) as

Vast applications are predicted this fuzzy reasoning. The central idea of this work was to

The most important aspect of the proposed model is the ability of self-organization of the

Additional experiments reinforced this conclusion. More extensive investigations on other measures of ultrafuzziness and the effect of parameters influencing the width/length of FOU

systems, based on three dimensional membership functions demonstrated.

introduce the application of polar type II fuzzy sets.

should certainly be conducted.

**Author details** 

M.H. Fazel Zarandi

a part of Hybrid expert system for pavement distress detection and classification.

membership function and initial height platform without requiring programming.

*Department of Industrial Engineering, Amirkabir University of Technology, Tehran, Iran* 

**6. Conclusions** 

	- H.R Tizhoosh., previous Image thresholdingnext using type-2 fuzzy sets, Pattern Recognition 38 (2005), pp. 2363–2372.

**Chapter 0**

**Chapter 5**

**Fuzzy Control of Nonlinear Systems**

Research on control of non-linear systems over the years has produced many results: control based on linearization, global feedback linearization, non-linear *H*∞ control, sliding mode control, variable structure control, state dependent Riccati equation control, etc [5]. This chapter will focus on fuzzy control techniques. Fuzzy control systems have recently shown growing popularity in non-linear system control applications. A fuzzy control system is essentially an effective way to decompose the task of non-linear system control into a group of local linear controls based on a set of design-specific model rules. Fuzzy control also provides a mechanism to blend these local linear control problems all together to achieve overall control of the original non-linear system. In this regard, fuzzy control technique has its unique advantage over other kinds of non-linear control techniques. Latest research on fuzzy control systems design is aimed to improve the optimality and robustness of the controller performance by combining the advantage of modern control theory with the Takagi-Sugeno

In this chapter, we address the non-linear state feedback control design of both continuous-time and discrete-time non-linear fuzzy control systems using the Linear Matrix Inequality (LMI) approach. We characterize the solution of the non-linear control problem with the LMI, which provides a sufficient condition for satisfying various performance criteria. A preliminary investigation into the LMI approach to non-linear fuzzy control systems can be found in [7, 8, 13]. The purpose behind this novel approach is to convert a non-linear system control problem into a convex optimization problem which is solved by a LMI at each time. The recent development in convex optimization provides efficient algorithms for solving LMIs. If a solution can be expressed in a LMI form, then there exist optimization algorithms providing efficient global numerical solutions [3]. Therefore if the LMI is feasible, then LMI control technique provides globally stable solutions satisfying the corresponding mixed performance criteria [4, 6, 15–20]. We further propose to employ mixed performance criteria to design the controller guaranteeing the quadratic sub-optimality with inherent stability property in combination with dissipative type of disturbance attenuation.

> ©2012 Wang et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**with General Performance Criteria**

Xin Wang, Edwin E. Yaz, James Long and Tim Miller

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48298

**1. Introduction**

fuzzy model [7–10, 13, 14].

cited.


## **Fuzzy Control of Nonlinear Systems with General Performance Criteria**

Xin Wang, Edwin E. Yaz, James Long and Tim Miller

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48298

## **1. Introduction**

118 Fuzzy Controllers – Recent Advances in Theory and Applications

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Research on control of non-linear systems over the years has produced many results: control based on linearization, global feedback linearization, non-linear *H*∞ control, sliding mode control, variable structure control, state dependent Riccati equation control, etc [5]. This chapter will focus on fuzzy control techniques. Fuzzy control systems have recently shown growing popularity in non-linear system control applications. A fuzzy control system is essentially an effective way to decompose the task of non-linear system control into a group of local linear controls based on a set of design-specific model rules. Fuzzy control also provides a mechanism to blend these local linear control problems all together to achieve overall control of the original non-linear system. In this regard, fuzzy control technique has its unique advantage over other kinds of non-linear control techniques. Latest research on fuzzy control systems design is aimed to improve the optimality and robustness of the controller performance by combining the advantage of modern control theory with the Takagi-Sugeno fuzzy model [7–10, 13, 14].

In this chapter, we address the non-linear state feedback control design of both continuous-time and discrete-time non-linear fuzzy control systems using the Linear Matrix Inequality (LMI) approach. We characterize the solution of the non-linear control problem with the LMI, which provides a sufficient condition for satisfying various performance criteria. A preliminary investigation into the LMI approach to non-linear fuzzy control systems can be found in [7, 8, 13]. The purpose behind this novel approach is to convert a non-linear system control problem into a convex optimization problem which is solved by a LMI at each time. The recent development in convex optimization provides efficient algorithms for solving LMIs. If a solution can be expressed in a LMI form, then there exist optimization algorithms providing efficient global numerical solutions [3]. Therefore if the LMI is feasible, then LMI control technique provides globally stable solutions satisfying the corresponding mixed performance criteria [4, 6, 15–20]. We further propose to employ mixed performance criteria to design the controller guaranteeing the quadratic sub-optimality with inherent stability property in combination with dissipative type of disturbance attenuation.

©2012 Wang et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the following sections, we first introduce the Takagi-Sugeno fuzzy modelling for non-linear systems in both continuous time and discrete time. We then propose the general performance criteria in section 3. Then, the LMI control solutions are derived to characterize the optimal and robust fuzzy control of continuous time and discrete time non-linear systems, respectively. The inverted pendulum system control is used as an illustrative example to demonstrate the effectiveness and robustness of our proposed approaches.

where

We have

for all time *t*.

input is given by

Using the notation

*x*˙(*t*) =

*y*(*t*) =

then the system equation becomes

*x*˙(*t*) =

*y*(*t*) =

*r* ∑ *i*=1

*r* ∑ *i*=1

*r* ∑ *j*=1

*r* ∑ *j*=1

*r* ∑ *i*=1

*r* ∑ *i*=1

*r* ∑ *j*=1

*r* ∑ *j*=1

Since, the following properties hold

*ϕ*(*t*)=[*ϕ*1(*t*), *ϕ*2(*t*), ..., *ϕp*(*t*)] (3)

Fuzzy Control of Nonlinear Systems with General Performance Criteria 121

*gi*(*ϕ*(*t*)) ≥ 0, *i* = 1, 2, 3, ...,*r* (6)

*hi*(*ϕ*(*t*)) ≥ 0, *i* = 1, 2, 3, ...,*r* (7)

*r* ∑ *i*=1

*r* ∑ *i*=1

*Hij* = *Ci* − *DiKj* (10)

*hi*(*ϕ*(*t*))*Fiw*(*t*)

*r* ∑ *i*=1

*r* ∑ *i*=1

*hi*(*ϕ*(*t*))*Kix*(*t*) (8)

*hi*(*ϕ*(*t*))*Fiw*(*t*)

*hi*(*ϕ*(*t*))*Ziw*(*t*) (9)

*hi*(*ϕ*(*t*))*Ziw*(*t*) (11)

*Mij*(*ϕj*(*t*)) (4)

*<sup>i</sup>*=<sup>1</sup> *gi*(*ϕ*(*t*)) (5)

*gi*(*ϕ*(*t*)) =

for all time *t*. The term *Mij*(*ϕj*(*t*)) is the grade membership function of *φj*(*t*) in *Mij*.

*p* ∏ *j*=1

*hi*(*ϕ*(*t*)) = *gi*(*ϕ*(*t*)) ∑*r*

> *r* ∑ *i*=1

> *r* ∑ *i*=1

It is assumed that the state feedback is available and the non-linear state feedback control

*r* ∑ *i*=1

*hi*(*ϕ*(*t*))*hj*(*ϕ*(*t*))(*Ai* − *BiKj*)*x*(*t*) +

*hi*(*ϕ*(*t*))*hj*(*ϕ*(*t*))(*Ci* − *DiKj*)*x*(*k*) +

*Gij* = *Ai* − *BiKj*

*hi*(*ϕ*(*t*))*hj*(*ϕ*(*t*))*Gijx*(*t*) +

*hi*(*ϕ*(*t*))*hj*(*ϕ*(*t*))*Hijx*(*t*) +

*u*(*t*) = −

Substituting this into the system and performance output equation, we have

*gi*(*ϕ*(*t*)) > 0

*hi*(*ϕ*(*t*)) = 1

The following notation is used in this work: *<sup>x</sup>* ∈ R*<sup>n</sup>* denotes n-dimensional real vector with norm �*x*� = (*xTx*)1/2 where (.)*<sup>T</sup>* indicates transpose. *<sup>A</sup>* <sup>≥</sup> 0 for a symmetric matrix denotes a positive semi-definite matrix. L<sup>2</sup> and *l*<sup>2</sup> denotes the space of infinite sequences of finite dimensional random vectors with finite energy, i.e. <sup>∞</sup> <sup>0</sup> �*xt*�<sup>2</sup> <sup>&</sup>lt; <sup>∞</sup> in continuous-time, and Σ<sup>∞</sup> *<sup>k</sup>*=0�*xk*�<sup>2</sup> <sup>&</sup>lt; <sup>∞</sup> in discrete-time, respectively.

## **2. Takagi-Sugeno system model**

The importance of the Takagi-Sugeno fuzzy system model is that it provides an effective way to decompose a complicated non-linear system into local dynamical relations and express those local dynamics of each fuzzy implication rule by a linear system model. The overall fuzzy non-linear system model is achieved by fuzzy "blending" of the linear system models, so that the overall non-linear control performance is achieved. Both of the continuous-time and the discrete-time system models are summarized below.

### **2.1. Continuous-time Takagi-Sugeno system model**

The *i th* rule of the Takagi-Sugeno fuzzy model can be expressed by the following forms:

#### **Model Rule** *i***:**

**If** *ϕ*1(*t*) is *Mi*1,*ϕ*2(*t*) is *Mi*2,... and *ϕp*(*t*) is *Mip*, **Then** the input-affine continuous-time fuzzy system equation is:

$$\begin{aligned} \dot{x}(t) &= A\_i x(t) + B\_i u(t) + F\_i w(t) \\ y(t) &= \mathbf{C}\_i x(t) + D\_i u(t) + Z\_i w(t) \\ &\quad i = 1, 2, 3, \dots, r \end{aligned} \tag{1}$$

where *<sup>x</sup>*(*t*) ∈ R*<sup>n</sup>* is the state vector, *<sup>u</sup>*(*t*) ∈ R*<sup>m</sup>* is the control input vector, *<sup>y</sup>*(*t*) ∈ R*<sup>q</sup>* is the performance output vector, *<sup>w</sup>*(*t*) ∈ R*<sup>s</sup>* is <sup>L</sup><sup>2</sup> type of disturbance, *<sup>r</sup>* is the total number of model rules, *Mij* is the fuzzy set. The coefficient matrices are *Ai* ∈ R*n*×*n*, *Bi* ∈ R*n*×*m*, *Fi* <sup>∈</sup> <sup>R</sup>*n*×*s*, *Ci* ∈ R*q*×*n*, *Di* ∈ R*q*×*m*, *Zi* ∈ R*q*×*s*. And *<sup>ϕ</sup>*1, ..., *<sup>ϕ</sup><sup>p</sup>* are known premise variables, which can be functions of state variables, external disturbance and time.

It is assumed that the premises are not the function of the input vector *u*(*t*), which is needed to avoid the defuzzification process of fuzzy controller. If we use *ϕ*(*t*) to denote the vector containing all the individual elements *ϕ*1(*t*), *ϕ*2(*t*), ..., *ϕp*(*t*), then the overall fuzzy system is

$$\dot{\mathbf{x}}(t) = \frac{\sum\_{i=1}^{r} g\_i(\boldsymbol{\varrho}(t)) \left[ A\_i \mathbf{x}(t) + B\_i \boldsymbol{u}(t) + F\_i \mathbf{w}(t) \right]}{\sum\_{i=1}^{r} g\_i(\boldsymbol{\varrho}(t))} = \sum\_{i=1}^{r} h\_i(\boldsymbol{\varrho}(t)) \left[ A\_i \mathbf{x}(t) + B\_i \boldsymbol{u}(t) + F\_i \mathbf{w}(t) \right]$$

$$\mathbf{y}(t) = \frac{\sum\_{i=1}^{r} g\_i(\boldsymbol{\varrho}(t)) \left[ \mathbf{C}\_i \mathbf{x}(t) + D\_i \boldsymbol{u}(t) + Z\_i \mathbf{w}(t) \right]}{\sum\_{i=1}^{r} g\_i(\boldsymbol{\varrho}(t))} = \sum\_{i=1}^{r} h\_i(\boldsymbol{\varrho}(t)) \left[ \mathbf{C}\_i \mathbf{x}(t) + D\_i \boldsymbol{u}(t) + Z\_i \mathbf{w}(t) \right] \tag{2}$$

where

2 Fuzzy Controllers

In the following sections, we first introduce the Takagi-Sugeno fuzzy modelling for non-linear systems in both continuous time and discrete time. We then propose the general performance criteria in section 3. Then, the LMI control solutions are derived to characterize the optimal and robust fuzzy control of continuous time and discrete time non-linear systems, respectively. The inverted pendulum system control is used as an illustrative example to

The following notation is used in this work: *<sup>x</sup>* ∈ R*<sup>n</sup>* denotes n-dimensional real vector with norm �*x*� = (*xTx*)1/2 where (.)*<sup>T</sup>* indicates transpose. *<sup>A</sup>* <sup>≥</sup> 0 for a symmetric matrix denotes a positive semi-definite matrix. L<sup>2</sup> and *l*<sup>2</sup> denotes the space of infinite sequences of finite

The importance of the Takagi-Sugeno fuzzy system model is that it provides an effective way to decompose a complicated non-linear system into local dynamical relations and express those local dynamics of each fuzzy implication rule by a linear system model. The overall fuzzy non-linear system model is achieved by fuzzy "blending" of the linear system models, so that the overall non-linear control performance is achieved. Both of the continuous-time

*th* rule of the Takagi-Sugeno fuzzy model can be expressed by the following forms:

*x*˙(*t*) = *Aix*(*t*) + *Biu*(*t*) + *Fiw*(*t*) *y*(*t*) = *Cix*(*t*) + *Diu*(*t*) + *Ziw*(*t*)

where *<sup>x</sup>*(*t*) ∈ R*<sup>n</sup>* is the state vector, *<sup>u</sup>*(*t*) ∈ R*<sup>m</sup>* is the control input vector, *<sup>y</sup>*(*t*) ∈ R*<sup>q</sup>* is the performance output vector, *<sup>w</sup>*(*t*) ∈ R*<sup>s</sup>* is <sup>L</sup><sup>2</sup> type of disturbance, *<sup>r</sup>* is the total number of model rules, *Mij* is the fuzzy set. The coefficient matrices are *Ai* ∈ R*n*×*n*, *Bi* ∈ R*n*×*m*, *Fi* <sup>∈</sup> <sup>R</sup>*n*×*s*, *Ci* ∈ R*q*×*n*, *Di* ∈ R*q*×*m*, *Zi* ∈ R*q*×*s*. And *<sup>ϕ</sup>*1, ..., *<sup>ϕ</sup><sup>p</sup>* are known premise variables, which

It is assumed that the premises are not the function of the input vector *u*(*t*), which is needed to avoid the defuzzification process of fuzzy controller. If we use *ϕ*(*t*) to denote the vector containing all the individual elements *ϕ*1(*t*), *ϕ*2(*t*), ..., *ϕp*(*t*), then the overall fuzzy system is

> *r* ∑ *i*=1

*r* ∑ *i*=1

<sup>0</sup> �*xt*�<sup>2</sup> <sup>&</sup>lt; <sup>∞</sup> in continuous-time, and

*i* = 1, 2, 3, ...,*r* (1)

*hi*(*ϕ*(*t*))[*Aix*(*t*) + *Biu*(*t*) + *Fiw*(*t*)]

*hi*(*ϕ*(*t*))[*Cix*(*t*) + *Diu*(*t*) + *Ziw*(*t*)] (2)

demonstrate the effectiveness and robustness of our proposed approaches.

dimensional random vectors with finite energy, i.e. <sup>∞</sup>

and the discrete-time system models are summarized below.

**2.1. Continuous-time Takagi-Sugeno system model**

**Then** the input-affine continuous-time fuzzy system equation is:

can be functions of state variables, external disturbance and time.

*<sup>i</sup>*=<sup>1</sup> *gi*(*ϕ*(*t*))[*Aix*(*t*) + *Biu*(*t*) + *Fiw*(*t*)]

*<sup>i</sup>*=<sup>1</sup> *gi*(*ϕ*(*t*))[*Cix*(*t*) + *Diu*(*t*) + *Ziw*(*t*)]

*<sup>i</sup>*=<sup>1</sup> *gi*(*ϕ*(*t*)) <sup>=</sup>

*<sup>i</sup>*=<sup>1</sup> *gi*(*ϕ*(*t*)) <sup>=</sup>

∑*r*

∑*r*

**If** *ϕ*1(*t*) is *Mi*1,*ϕ*2(*t*) is *Mi*2,... and *ϕp*(*t*) is *Mip*,

*<sup>k</sup>*=0�*xk*�<sup>2</sup> <sup>&</sup>lt; <sup>∞</sup> in discrete-time, respectively.

**2. Takagi-Sugeno system model**

Σ<sup>∞</sup>

The *i*

**Model Rule** *i***:**

*<sup>x</sup>*˙(*t*) = <sup>∑</sup>*<sup>r</sup>*

*<sup>y</sup>*(*t*) = <sup>∑</sup>*<sup>r</sup>*

$$\varphi(t) = [\varphi\_1(t), \varphi\_2(t), \dots, \varphi\_p(t)] \tag{3}$$

$$\log\_i(\varphi(t)) = \prod\_{j=1}^p M\_{ij}(\varphi\_j(t)) \tag{4}$$

$$h\_i(\boldsymbol{\varrho}(t)) = \frac{\boldsymbol{g}\_i(\boldsymbol{\varrho}(t))}{\sum\_{i=1}^r \boldsymbol{g}\_i(\boldsymbol{\varrho}(t))}\tag{5}$$

for all time *t*. The term *Mij*(*ϕj*(*t*)) is the grade membership function of *φj*(*t*) in *Mij*. Since, the following properties hold

$$\sum\_{i=1}^{r} \mathcal{g}\_i(\varphi(t)) > 0$$

$$\mathcal{g}\_i(\varphi(t)) \ge 0, i = 1, 2, 3, \dots, r \tag{6}$$

We have

$$\sum\_{i=1}^{r} h\_i(\varphi(t)) = 1$$

$$h\_i(\varphi(t)) \ge 0, i = 1, 2, 3, \dots, r \tag{7}$$

for all time *t*.

It is assumed that the state feedback is available and the non-linear state feedback control input is given by

$$\ln(t) = -\sum\_{i=1}^{r} h\_i(\varphi(t)) K\_i \mathbf{x}(t) \tag{8}$$

Substituting this into the system and performance output equation, we have

$$\dot{\mathbf{x}}(t) = \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_{i}(\boldsymbol{\varrho}(t)) h\_{\rangle}(\boldsymbol{\varrho}(t)) (A\_{i} - B\_{i}K\_{j}) \mathbf{x}(t) + \sum\_{i=1}^{r} h\_{i}(\boldsymbol{\varrho}(t)) F\_{i} \mathbf{w}(t)$$

$$\mathbf{y}(t) = \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_{i}(\boldsymbol{\varrho}(t)) h\_{j}(\boldsymbol{\varrho}(t)) (\mathbf{C}\_{i} - D\_{i}K\_{j}) \mathbf{x}(k) + \sum\_{i=1}^{r} h\_{i}(\boldsymbol{\varrho}(t)) Z\_{i} \mathbf{w}(t) \tag{9}$$

Using the notation

$$\begin{aligned} \mathbf{G}\_{i\dot{j}} &= A\_i - B\_i \mathbf{K}\_{\dot{j}} \\ H\_{i\dot{j}} &= \mathbf{C}\_i - D\_i \mathbf{K}\_{\dot{j}} \end{aligned} \tag{10}$$

then the system equation becomes

$$\dot{\mathbf{x}}(t) = \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_i(\boldsymbol{\varrho}(t)) h\_j(\boldsymbol{\varrho}(t)) \mathbf{G}\_{lj} \mathbf{x}(t) + \sum\_{i=1}^{r} h\_i(\boldsymbol{\varrho}(t)) F\_i \mathbf{w}(t)$$

$$\mathbf{y}(t) = \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_i(\boldsymbol{\varrho}(t)) h\_j(\boldsymbol{\varrho}(t)) H\_{lj} \mathbf{x}(t) + \sum\_{i=1}^{r} h\_i(\boldsymbol{\varrho}(t)) \mathbf{Z}\_i \mathbf{w}(t) \tag{11}$$

#### **2.2. Discrete-time Takagi-Sugeno system model**

At time step *k*, the *i th* rule of the Takagi-Sugeno fuzzy model can be expressed by the following forms:

#### **Model Rule** *i***:**

**If** *ϕ*1(*k*) is *Mi*1,*ϕ*2(*k*) is *Mi*2,... and *ϕp*(*k*) is *Mip*, **Then** the input-affine discrete-time fuzzy system equation is:

$$\begin{aligned} \mathbf{x}(k+1) &= A\_i \mathbf{x}(k) + B\_i \boldsymbol{u}(k) + F\_i \mathbf{w}(k) \\ \mathbf{y}(k) &= \mathbf{C}\_i \mathbf{x}(k) + D\_i \boldsymbol{u}(k) + Z\_i \mathbf{w}(k) \\ &\quad \mathbf{i} = 1, 2, 3, \dots, r \end{aligned} \tag{12}$$

for all *k*.

input is given by

Using the notation

*x*(*k* + 1) =

*y*(*k*) =

then the system equation becomes

*x*(*k* + 1) =

*y*(*k*) =

**3. General performance criteria**

general performance criteria are given below:

Consider the quadratic Lyapunov function

for the following difference inequality

with *Q* > 0, *R* > 0 functions of *x*(*t*).

**3.1. Continuous-time general performance criteria**

*r* ∑ *i*=1

*r* ∑ *i*=1

*r* ∑ *j*=1

*r* ∑ *j*=1

> *r* ∑ *i*=1

*r* ∑ *i*=1

*r* ∑ *j*=1

*r* ∑ *j*=1

It is assumed that the state feedback is available and the non-linear state feedback control

*hi*(*ϕ*(*k*))*Kix*(*k*) (19)

Fuzzy Control of Nonlinear Systems with General Performance Criteria 123

*hi*(*ϕ*(*k*))*Fiw*(*k*)

*hi*(*ϕ*(*k*))*Ziw*(*k*) (20)

*hi*(*ϕ*(*k*))*Ziw*(*k*) (22)

*r* ∑ *i*=1

*r* ∑ *i*=1

*Hij* = *Ci* − *DiKj* (21)

*hi*(*ϕ*(*k*))*Fiw*(*k*)

*r* ∑ *i*=1

*r* ∑ *i*=1

*V*(*t*) = *xT*(*t*)*Px*(*t*) > 0 (23)

*r* ∑ *i*=1

*hi*(*ϕ*(*k*))*hj*(*ϕ*(*k*))(*Ai* − *BiKj*)*x*(*k*) +

*hi*(*ϕ*(*k*))*hj*(*ϕ*(*k*))(*Ci* − *DiKj*)*x*(*k*) +

*Gij* = *Ai* − *BiKj*

*hi*(*ϕ*(*k*))*hj*(*ϕ*(*k*))*Gijx*(*k*) +

*hi*(*ϕ*(*k*))*hj*(*ϕ*(*k*))*Hijx*(*k*) +

In this section, we propose the general performance criteria for non-linear control design, which yields a mixed Non-Linear Quadratic Regular (NLQR) in combination with H<sup>∞</sup> or dissipative performance index. The commonly used system performance criteria, including bounded-realness, positive-realness, sector boundedness and quadratic cost criterion, become special cases of the general performance criteria. Both the continuous-time and discrete-time

*<sup>V</sup>*˙ (*t*) + *<sup>x</sup>T*(*t*)*Qx*(*t*) + *<sup>u</sup>T*(*t*)*Ru*(*t*) + *<sup>α</sup>yT*(*t*)*y*(*t*) <sup>−</sup> *<sup>β</sup>yT*(*t*)*w*(*t*) + *<sup>γ</sup>wT*(*t*)*w*(*t*) <sup>≤</sup> 0 (24)

*u*(*k*) = −

Substituting this into the system and performance output equation, we have

where *<sup>x</sup>*(*k*) ∈ R*<sup>n</sup>* is the state vector, *<sup>u</sup>*(*k*) ∈ R*<sup>m</sup>* is the control input vector, *<sup>y</sup>*(*k*) ∈ R*<sup>q</sup>* is the performance output vector, *<sup>w</sup>*(*k*) ∈ R*<sup>s</sup>* is *<sup>l</sup>*<sup>2</sup> type of disturbance, *<sup>r</sup>* is the total number of model rules, *Mij* is the fuzzy set. The coefficient matrices are *Ai* ∈ R*n*×*n*, *Bi* ∈ R*n*×*m*, *Fi* <sup>∈</sup> <sup>R</sup>*n*×*s*, *Ci* ∈ R*q*×*n*, *Di* ∈ R*q*×*m*, *Zi* ∈ R*q*×*s*. And *<sup>ϕ</sup>*1, ..., *<sup>ϕ</sup><sup>p</sup>* are known premise variables which can be functions of state variables, external disturbance and time.

It is assumed that the premises are not the function of the input vector *u*(*k*), which is needed to avoid the defuzzification process of fuzzy controller. If we use *ϕ*(*k*) to denote the vector containing all the individual elements *ϕ*1(*k*), *ϕ*2(*k*), ..., *ϕp*(*k*), then the overall fuzzy system is

$$\mathbf{x}(k+1) = \frac{\sum\_{i=1}^{r} g\_i(\boldsymbol{\varrho}(k)) A\_i \mathbf{x}(k) + B\_i \boldsymbol{u}(k) + F\_i \boldsymbol{w}(k)}{\sum\_{i=1}^{r} g\_i(\boldsymbol{\varrho}(k))} = \sum\_{i=1}^{r} h\_i(\boldsymbol{\varrho}(k)) A\_i \mathbf{x}(k) + B\_i \boldsymbol{u}(k) + F\_i \boldsymbol{w}(k)$$

$$\mathbf{y}(k) = \frac{\sum\_{i=1}^{r} g\_i(\boldsymbol{\varrho}(k)) \mathbf{C}\_i \mathbf{x}(k) + D\_i \boldsymbol{u}(k) + Z\_i \boldsymbol{w}(k)}{\sum\_{i=1}^{r} g\_i(\boldsymbol{\varrho}(k))} = \sum\_{i=1}^{r} h\_i(\boldsymbol{\varrho}(k)) \mathbf{C}\_i \mathbf{x}(k) + D\_i \boldsymbol{u}(k) + Z\_i \boldsymbol{w}(k) \tag{13}$$

where

$$\varphi(k) = [\varphi\_1(k), \varphi\_2(k), \dots, \varphi\_p(k)] \tag{14}$$

$$g\_i(\boldsymbol{\varrho}(k)) = \prod\_{j=1}^p M\_{ij}(\boldsymbol{\varrho}\_j(k)) \tag{15}$$

$$h\_i(\boldsymbol{\varrho}(k)) = \frac{\boldsymbol{g}\_i(\boldsymbol{\varrho}(k))}{\sum\_{i=1}^r \boldsymbol{g}\_i(\boldsymbol{\varrho}(k))}\tag{16}$$

for all *k*. The term *Mij*(*ϕj*(*k*)) is the grade membership function of *φj*(*k*) in *Mij*. Since, the following properties hold

$$\sum\_{i=1}^{r} g\_i(\varphi(k)) > 0$$

$$g\_i(\varphi(k)) \ge 0, i = 1, 2, 3, \dots, r \tag{17}$$

We have

$$\sum\_{i=1}^{r} h\_i(\varphi(k)) = 1$$

$$h\_i(\varphi(k)) \ge 0, i = 1, 2, 3, \dots, r \tag{18}$$

for all *k*.

4 Fuzzy Controllers

*x*(*k* + 1) = *Aix*(*k*) + *Biu*(*k*) + *Fiw*(*k*) *y*(*k*) = *Cix*(*k*) + *Diu*(*k*) + *Ziw*(*k*)

where *<sup>x</sup>*(*k*) ∈ R*<sup>n</sup>* is the state vector, *<sup>u</sup>*(*k*) ∈ R*<sup>m</sup>* is the control input vector, *<sup>y</sup>*(*k*) ∈ R*<sup>q</sup>* is the performance output vector, *<sup>w</sup>*(*k*) ∈ R*<sup>s</sup>* is *<sup>l</sup>*<sup>2</sup> type of disturbance, *<sup>r</sup>* is the total number of model rules, *Mij* is the fuzzy set. The coefficient matrices are *Ai* ∈ R*n*×*n*, *Bi* ∈ R*n*×*m*, *Fi* <sup>∈</sup> <sup>R</sup>*n*×*s*, *Ci* ∈ R*q*×*n*, *Di* ∈ R*q*×*m*, *Zi* ∈ R*q*×*s*. And *<sup>ϕ</sup>*1, ..., *<sup>ϕ</sup><sup>p</sup>* are known premise variables which

It is assumed that the premises are not the function of the input vector *u*(*k*), which is needed to avoid the defuzzification process of fuzzy controller. If we use *ϕ*(*k*) to denote the vector containing all the individual elements *ϕ*1(*k*), *ϕ*2(*k*), ..., *ϕp*(*k*), then the overall fuzzy system is

> *r* ∑ *i*=1

*r* ∑ *i*=1

*ϕ*(*k*)=[*ϕ*1(*k*), *ϕ*2(*k*), ..., *ϕp*(*k*)] (14)

*gi*(*ϕ*(*k*)) ≥ 0, *i* = 1, 2, 3, ...,*r* (17)

*hi*(*ϕ*(*k*)) ≥ 0, *i* = 1, 2, 3, ...,*r* (18)

*th* rule of the Takagi-Sugeno fuzzy model can be expressed by the following

*i* = 1, 2, 3, ...,*r* (12)

*hi*(*ϕ*(*k*))*Aix*(*k*)+*Biu*(*k*)+*Fiw*(*k*)

*Mij*(*ϕj*(*k*)) (15)

*<sup>i</sup>*=<sup>1</sup> *gi*(*ϕ*(*k*)) (16)

*hi*(*ϕ*(*k*))*Cix*(*k*)+*Diu*(*k*)+*Ziw*(*k*) (13)

**2.2. Discrete-time Takagi-Sugeno system model**

**Then** the input-affine discrete-time fuzzy system equation is:

can be functions of state variables, external disturbance and time.

*<sup>i</sup>*=<sup>1</sup> *gi*(*ϕ*(*k*))*Aix*(*k*)+*Biu*(*k*)+*Fiw*(*k*)

*<sup>i</sup>*=<sup>1</sup> *gi*(*ϕ*(*k*))*Cix*(*k*)+*Diu*(*k*)+*Ziw*(*k*)

*<sup>i</sup>*=<sup>1</sup> *gi*(*ϕ*(*k*)) <sup>=</sup>

*<sup>i</sup>*=<sup>1</sup> *gi*(*ϕ*(*k*)) <sup>=</sup>

*gi*(*ϕ*(*k*)) =

for all *k*. The term *Mij*(*ϕj*(*k*)) is the grade membership function of *φj*(*k*) in *Mij*.

*p* ∏ *j*=1

*hi*(*ϕ*(*k*)) = *gi*(*ϕ*(*k*)) ∑*r*

> *r* ∑ *i*=1

> *r* ∑ *i*=1

*gi*(*ϕ*(*k*)) > 0

*hi*(*ϕ*(*k*)) = 1

∑*r*

∑*r*

**If** *ϕ*1(*k*) is *Mi*1,*ϕ*2(*k*) is *Mi*2,... and *ϕp*(*k*) is *Mip*,

At time step *k*, the *i*

*<sup>x</sup>*(*k*+1) = <sup>∑</sup>*<sup>r</sup>*

*<sup>y</sup>*(*k*) = <sup>∑</sup>*<sup>r</sup>*

Since, the following properties hold

where

We have

**Model Rule** *i***:**

forms:

It is assumed that the state feedback is available and the non-linear state feedback control input is given by

$$\mu(k) = -\sum\_{i=1}^{r} h\_i(\varphi(k)) K\_i x(k) \tag{19}$$

Substituting this into the system and performance output equation, we have

$$\mathbf{x}(k+1) = \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_i(\boldsymbol{\varrho}(k)) h\_j(\boldsymbol{\varrho}(k)) (A\_i - B\_i K\_j) \mathbf{x}(k) + \sum\_{i=1}^{r} h\_i(\boldsymbol{\varrho}(k)) F\_i \mathbf{w}(k)$$

$$\mathbf{y}(k) = \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_i(\boldsymbol{\varrho}(k)) h\_j(\boldsymbol{\varrho}(k)) (C\_i - D\_i K\_j) \mathbf{x}(k) + \sum\_{i=1}^{r} h\_i(\boldsymbol{\varrho}(k)) Z\_i \mathbf{w}(k) \tag{20}$$

Using the notation

$$\begin{aligned} \mathbf{G}\_{\dot{i}\dot{j}} &= A\_{\dot{i}} - B\_{\dot{i}} \mathbf{K}\_{\dot{j}} \\ H\_{\dot{i}\dot{j}} &= \mathbf{C}\_{\dot{i}} - D\_{\dot{i}} \mathbf{K}\_{\dot{j}} \end{aligned} \tag{21}$$

then the system equation becomes

$$\mathbf{x}(k+1) = \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_i(\boldsymbol{\varrho}(k)) h\_j(\boldsymbol{\varrho}(k)) \mathbf{G}\_{ij} \mathbf{x}(k) + \sum\_{i=1}^{r} h\_i(\boldsymbol{\varrho}(k)) F\_i \mathbf{w}(k)$$

$$\mathbf{y}(k) = \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_i(\boldsymbol{\varrho}(k)) h\_j(\boldsymbol{\varrho}(k)) H\_{ij} \mathbf{x}(k) + \sum\_{i=1}^{r} h\_i(\boldsymbol{\varrho}(k)) Z\_i \mathbf{w}(k) \tag{22}$$

### **3. General performance criteria**

In this section, we propose the general performance criteria for non-linear control design, which yields a mixed Non-Linear Quadratic Regular (NLQR) in combination with H<sup>∞</sup> or dissipative performance index. The commonly used system performance criteria, including bounded-realness, positive-realness, sector boundedness and quadratic cost criterion, become special cases of the general performance criteria. Both the continuous-time and discrete-time general performance criteria are given below:

#### **3.1. Continuous-time general performance criteria**

Consider the quadratic Lyapunov function

$$V(t) = \mathbf{x}^T(t)P\mathbf{x}(t) > 0\tag{23}$$

for the following difference inequality

$$\dot{V}(t) + \mathbf{x}^T(t)\mathbf{Q}\mathbf{x}(t) + \mathbf{u}^T(t)\mathbf{R}\mathbf{u}(t) + \mathbf{a}\mathbf{y}^T(t)\mathbf{y}(t) - \beta \mathbf{y}^T(t)\mathbf{w}(t) + \gamma \mathbf{w}^T(t)\mathbf{w}(t) \le 0 \tag{24}$$

with *Q* > 0, *R* > 0 functions of *x*(*t*).

#### 6 Fuzzy Controllers 124 Fuzzy Controllers – Recent Advances in Theory and Applications Fuzzy Control of Nonlinear Systems with General Performance Criteria <sup>7</sup>

Note that upon integration over time from 0 to *Tf* , (24) yields

$$V(T\_f) + \int\_0^{T\_f} [(\mathbf{x}^T(t)\mathbf{Q}\mathbf{x}(t) + \mathbf{u}^T(t)\mathbf{R}\mathbf{u}(t)]dt + \\\\\int\_0^{T\_f} [\mathbf{a}\mathbf{y}^T(t)\mathbf{y}(t) - \beta \mathbf{y}^T(t)\mathbf{w}(t) + \gamma \mathbf{w}^T(t)\mathbf{w}(t)]dt \le V(0) \\\tag{25}$$

which is a mixed *NLQR* − *H*<sup>∞</sup> Design [16–18]. In (19), *γ* can be minimized to achieve a smaller

Fuzzy Control of Nonlinear Systems with General Performance Criteria 125

Other possible performance criteria which can be used in this framework with various design parameters *α*, *β*, *γ* are given in Table.1. Design coefficients *α* and *γ* can be maximized or minimized to optimize the controller behavior. It should also be noted that the satisfaction of any of the criteria in Table 1 will also guarantee asymptotic stability of the controlled system.

**4. Fuzzy LMI control of continuous time non-linear systems with general**

The main results of this chapter are summarized in section 4 and section 5. The following theorem provides the fuzzy LMI control to the continuous time non-linear systems with

**Theorem 1** Given the system model and performance output (2) and control input (8), if there

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

≥ 0 (31)

*<sup>j</sup>* <sup>−</sup> *<sup>M</sup><sup>T</sup>*

*<sup>j</sup>* <sup>−</sup> *<sup>M</sup><sup>T</sup>*

*<sup>i</sup> <sup>D</sup><sup>T</sup> j* ]

*<sup>i</sup> <sup>D</sup><sup>T</sup> j* ]

*<sup>i</sup>* <sup>+</sup> *<sup>M</sup><sup>T</sup> j* )

Λ<sup>15</sup> = *SQT*/2

*β*(*Zi* + *Zj*)*<sup>T</sup>*

*<sup>T</sup>* (32)

*α*1/2[*Zi* + *Zj*]

*<sup>j</sup>* + *AiS* − *BiMj* + *AjS* − *BjMi*]

*<sup>i</sup>* <sup>+</sup> *SC<sup>T</sup>*

*<sup>i</sup>* <sup>+</sup> *SC<sup>T</sup>*

1 2

<sup>Λ</sup><sup>14</sup> <sup>=</sup> <sup>1</sup> 2 (*M<sup>T</sup>*

Λ<sup>11</sup> Λ<sup>12</sup> Λ<sup>13</sup> Λ<sup>14</sup> Λ<sup>15</sup> ∗ Λ<sup>22</sup> Λ<sup>23</sup> 0 0 ∗ ∗ *I* 0 0 ∗∗∗ *<sup>R</sup>*−<sup>1</sup> <sup>0</sup> ∗∗∗∗ *I*

> *<sup>j</sup>* <sup>−</sup> *<sup>M</sup><sup>T</sup> <sup>i</sup> <sup>B</sup><sup>T</sup>*

> > <sup>4</sup> [*SC<sup>T</sup>*

*α*1/2[*SC<sup>T</sup>*

*<sup>i</sup>* <sup>−</sup> *MjD<sup>T</sup>*

*<sup>i</sup>* <sup>−</sup> *MjD<sup>T</sup>*

Λ<sup>22</sup> = −*γI* +

<sup>Λ</sup><sup>23</sup> <sup>=</sup> <sup>1</sup> 2

(*Fi* <sup>+</sup> *Fj*) + *<sup>β</sup>*

<sup>Λ</sup><sup>13</sup> <sup>=</sup> <sup>1</sup> 2

exist matrices *<sup>S</sup>* <sup>=</sup> *<sup>P</sup>*−<sup>1</sup> <sup>&</sup>gt; 0 for all *<sup>t</sup>* <sup>≥</sup> 0, such that the following LMI holds:

*<sup>i</sup>* <sup>+</sup> *SA<sup>T</sup>*

2

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

*<sup>i</sup>* <sup>−</sup> *MjB<sup>T</sup>*

<sup>Λ</sup><sup>12</sup> <sup>=</sup> <sup>−</sup><sup>1</sup>

*α β γ* Performance Criteria 1 0 <0 *NLQR* − H<sup>∞</sup> *Design* 0 1 0 *NLQR* − *Passivity Design* 0 1 >0 *NLQR* − *Input Strict Passivity Design* >0 1 0 *NLQR* − *Output Strict Passivity Design* >0 1 >0 *NLQR* − *Very Strict Passivity*

*l*<sup>2</sup> − *l*<sup>2</sup> or *H*<sup>∞</sup> gain for the closed loop system.

**Table 1.** Various performance criteria in a general framework

**performance criteria**

general performance criteria.

<sup>Λ</sup><sup>11</sup> <sup>=</sup> <sup>−</sup><sup>1</sup>

2 [*SA<sup>T</sup>*

where

By properly specifying the value of weighing matrices *Q*, *R*, *Ci*, *Di*, *Zi* and *α*, *β*, *γ*, mixed performance criteria can be used in non-linear control design, which yields a mixed Non-linear Quadratic Regulator (NLQR) in combination with dissipative type performance index with disturbance reduction capability. For example, if we take *α* = 1, *β* = 0, *γ* < 0, (25) yields

$$V(T\_f) + \int\_0^{T\_f} [(\mathbf{x}^T(t)\mathbf{Q}\mathbf{x}(t) + \boldsymbol{u}^T(t)\mathbf{R}\mathbf{u}(t) + \boldsymbol{y}^T(t)\mathbf{y}(t)]dt + \\
\quad \le V(0) - \gamma \int\_0^{T\_f} [\boldsymbol{w}^T(t)\boldsymbol{w}(t)]dt\tag{26}$$

which is a mixed *NLQR* − *H*<sup>∞</sup> Design [16–18].

Other possible performance criteria which can be used in this framework with various design parameters *α*, *β*, *γ* are given in Table.1. Design coefficients *α* and *γ* can be maximized or minimized to optimize the controller behavior. It should also be noted that the satisfaction of any of the criteria in Table 1 will also guarantee asymptotic stability of the controlled system.

#### **3.2. Discrete-time general performance criteria**

Consider the quadratic Lyapunov function

$$V(k) = \mathbf{x}^T(k)\mathbf{P}\mathbf{x}(k)\tag{27}$$

for the following difference inequality

$$V(k+1) - V(k) + \mathbf{x}^T(k)\mathbf{Q}\mathbf{x}(k) + \mathbf{u}^T(k)\mathbf{R}\mathbf{u}(k) + \mathbf{a}\mathbf{y}^T(k)\mathbf{y}(k) - \boldsymbol{\beta}\mathbf{y}^T(k)\mathbf{w}(k) + \boldsymbol{\gamma}\mathbf{w}^T(k)\mathbf{w}(k) \le 0\tag{28}$$

with *Q* > 0, *R* > 0 functions of *x*(*k*).

Note that upon summation over *k*, (28) yields

$$V(N) + \sum\_{k=0}^{N-1} \left( \mathbf{x}^T(k)\mathbf{Q}\mathbf{x}(k) + \mathbf{u}^T(k)\mathbf{R}\mathbf{u}(k) + \mathbf{a}\mathbf{y}^T(k)\mathbf{y}(k) - \beta \mathbf{y}^T(k)\mathbf{w}(k) + \gamma \mathbf{w}^T(k)\mathbf{w}(k) \right) \le V(0) \tag{29}$$

By properly specifying the value of weighing matrices *Q*, *R*, *Ci*, *Di*, *Zi* and *α*, *β*, *γ*, mixed performance criteria can be used in non-linear control design, which yields a mixed Non-linear Quadratic Regulator (NLQR) in combination with dissipative type performance index with disturbance reduction capability. For example, if we take *α* = 1, *β* = 0, *γ* < 0, (29) yields

$$V(N) + \sum\_{k=0}^{N-1} \left( \mathbf{x}^T(k) \mathbf{Q} \mathbf{x}(k) + \mathbf{u}^T(k) \mathbf{R} \mathbf{u}(k) + \mathbf{a} \mathbf{y}^T(k) \mathbf{y}(k) \right) \le V(0) - \gamma \sum\_{k=0}^{N-1} w^T(k) w(k) \tag{30}$$

which is a mixed *NLQR* − *H*<sup>∞</sup> Design [16–18]. In (19), *γ* can be minimized to achieve a smaller *l*<sup>2</sup> − *l*<sup>2</sup> or *H*<sup>∞</sup> gain for the closed loop system.

Other possible performance criteria which can be used in this framework with various design parameters *α*, *β*, *γ* are given in Table.1. Design coefficients *α* and *γ* can be maximized or minimized to optimize the controller behavior. It should also be noted that the satisfaction of any of the criteria in Table 1 will also guarantee asymptotic stability of the controlled system.


**Table 1.** Various performance criteria in a general framework

## **4. Fuzzy LMI control of continuous time non-linear systems with general performance criteria**

The main results of this chapter are summarized in section 4 and section 5. The following theorem provides the fuzzy LMI control to the continuous time non-linear systems with general performance criteria.

**Theorem 1** Given the system model and performance output (2) and control input (8), if there exist matrices *<sup>S</sup>* <sup>=</sup> *<sup>P</sup>*−<sup>1</sup> <sup>&</sup>gt; 0 for all *<sup>t</sup>* <sup>≥</sup> 0, such that the following LMI holds:

$$
\begin{bmatrix}
\Lambda\_{11} \ \Lambda\_{12} \ \Lambda\_{13} \ \Lambda\_{14} \ \Lambda\_{15} \\
\* \ \Lambda\_{22} \ \Lambda\_{23} \ 0 & 0 \\
\* & \* \ I & 0 & 0 \\
\* & \* & \* \ R^{-1} & 0 \\
\* & \* & \* & \* & I
\end{bmatrix} \geq 0 \tag{31}
$$

where

6 Fuzzy Controllers

By properly specifying the value of weighing matrices *Q*, *R*, *Ci*, *Di*, *Zi* and *α*, *β*, *γ*, mixed performance criteria can be used in non-linear control design, which yields a mixed Non-linear Quadratic Regulator (NLQR) in combination with dissipative type performance index with disturbance reduction capability. For example, if we take *α* = 1, *β* = 0, *γ* < 0, (25)

[(*xT*(*t*)*Qx*(*t*) + *uT*(*t*)*Ru*(*t*) + *yT*(*t*)*y*(*t*)]*dt* +

≤ *V*(0) − *γ*

Other possible performance criteria which can be used in this framework with various design parameters *α*, *β*, *γ* are given in Table.1. Design coefficients *α* and *γ* can be maximized or minimized to optimize the controller behavior. It should also be noted that the satisfaction of any of the criteria in Table 1 will also guarantee asymptotic stability of the controlled system.

*<sup>V</sup>*(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> *<sup>V</sup>*(*k*) + *<sup>x</sup>T*(*k*)*Qx*(*k*) + *<sup>u</sup>T*(*k*)*Ru*(*k*) + *<sup>α</sup>yT*(*k*)*y*(*k*) <sup>−</sup> *<sup>β</sup>yT*(*k*)*w*(*k*) + *<sup>γ</sup>wT*(*k*)*w*(*k*) <sup>≤</sup> <sup>0</sup>

By properly specifying the value of weighing matrices *Q*, *R*, *Ci*, *Di*, *Zi* and *α*, *β*, *γ*, mixed performance criteria can be used in non-linear control design, which yields a mixed Non-linear Quadratic Regulator (NLQR) in combination with dissipative type performance index with disturbance reduction capability. For example, if we take *α* = 1, *β* = 0, *γ* < 0, (29)

(*xT*(*k*)*Qx*(*k*) + *<sup>u</sup>T*(*k*)*Ru*(*k*) + *<sup>α</sup>yT*(*k*)*y*(*k*)) <sup>≤</sup> *<sup>V</sup>*(0) <sup>−</sup> *<sup>γ</sup>*

(*xT*(*k*)*Qx*(*k*) + *<sup>u</sup>T*(*k*)*Ru*(*k*) + *<sup>α</sup>yT*(*k*)*y*(*k*) <sup>−</sup> *<sup>β</sup>yT*(*k*)*w*(*k*) + *<sup>γ</sup>wT*(*k*)*w*(*k*)) <sup>≤</sup> *<sup>V</sup>*(0)

[(*xT*(*t*)*Qx*(*t*) + *uT*(*t*)*Ru*(*t*)]*dt* +

[*αyT*(*t*)*y*(*t*) <sup>−</sup> *<sup>β</sup>yT*(*t*)*w*(*t*) + *<sup>γ</sup>wT*(*t*)*w*(*t*)]*dt* <sup>≤</sup> *<sup>V</sup>*(0) (25)

 *Tf* 0

*V*(*k*) = *xT*(*k*)*Px*(*k*) (27)

*N*−1 ∑ *k*=0

[*wT*(*t*)*w*(*t*)]*dt* (26)

(28)

(29)

*wT*(*k*)*w*(*k*) (30)

Note that upon integration over time from 0 to *Tf* , (24) yields

 *Tf* 0

*V*(*Tf*) +

which is a mixed *NLQR* − *H*<sup>∞</sup> Design [16–18].

**3.2. Discrete-time general performance criteria**

Consider the quadratic Lyapunov function

for the following difference inequality

with *Q* > 0, *R* > 0 functions of *x*(*k*).

*V*(*N*) +

yields

*V*(*N*) +

*N*−1 ∑ *k*=0

*N*−1 ∑ *k*=0

Note that upon summation over *k*, (28) yields

 *Tf* 0

yields

*V*(*Tf*) +

 *Tf* 0

$$\Lambda\_{11} = -\frac{1}{2} [SA\_i^T - M\_jB\_i^T + SA\_j^T - M\_i^T B\_j^T + A\_iS - B\_iM\_j + A\_jS - B\_jM\_i]$$

$$\Lambda\_{12} = -\frac{1}{2}(F\_i + F\_j) + \frac{\beta}{4} [SC\_i^T - M\_jD\_i^T + SC\_j^T - M\_i^T D\_j^T]$$

$$\Lambda\_{13} = \frac{1}{2}a^{1/2} [SC\_i^T - M\_jD\_i^T + SC\_j^T - M\_i^T D\_j^T]$$

$$\Lambda\_{14} = \frac{1}{2}(M\_i^T + M\_j^T)$$

$$\Lambda\_{15} = S\mathcal{Q}^{T/2}$$

$$\Lambda\_{22} = -\gamma I + \frac{1}{2}\beta (Z\_i + Z\_j)^T$$

$$\Lambda\_{23} = \frac{1}{2}a^{1/2} [Z\_i + Z\_j]^T\tag{32}$$

#### 8 Fuzzy Controllers 126 Fuzzy Controllers – Recent Advances in Theory and Applications Fuzzy Control of Nonlinear Systems with General Performance Criteria <sup>9</sup>

using the notation

$$M\_{\dot{i}} = K\_{\dot{i}} P^{-1} = K\_{\dot{i}} \mathcal{S} \tag{33}$$

where

where

<sup>Θ</sup><sup>11</sup> = −(∑

*i* ∑ *j*

*hihjGij*)*TP* <sup>−</sup> *<sup>P</sup>*(∑

By applying Schur complement to inequality (37), we have

⎡ ⎣

Similarly, inequality (39) can also be written as

∗ ∗ *I*

*Φ*<sup>11</sup> *Φ*<sup>12</sup> *α*1/2[∑*<sup>i</sup>* ∑*<sup>j</sup> hihjHij*]

*<sup>Φ</sup>*<sup>11</sup> = −(∑

By applying Schur complement again to (40), we have

∑ *i* ∑ *j*

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *i* ∑ *j*

<sup>∗</sup> *<sup>Φ</sup>*<sup>22</sup> *<sup>α</sup>*1/2[∑*<sup>i</sup> hiZi*]

⎡ ⎣

Equivalently, we have

*i* ∑ *j*

Θ<sup>11</sup> Θ<sup>12</sup> *α*1/2[∑*<sup>i</sup>* ∑*<sup>j</sup> hihjHij*]

<sup>∗</sup> <sup>Θ</sup><sup>22</sup> *<sup>α</sup>*1/2[∑*<sup>i</sup> hiZi*]

∗ ∗ *I*

*T*

⎤ ⎦ − ⎡ ⎣

*hihjGij*)*TP* <sup>−</sup> *<sup>P</sup>*(∑

*i*

[∑*<sup>i</sup> hiKi*]

0 0

> *i* ∑ *j*

<sup>2</sup> [∑ *i* ∑ *j*

*<sup>Φ</sup>*<sup>22</sup> = −*γ<sup>I</sup>* + *<sup>β</sup>*[∑

*<sup>T</sup>* [∑*<sup>i</sup> hiKi*]

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

*<sup>T</sup>* 0

Γ<sup>11</sup> Γ<sup>12</sup> Γ<sup>13</sup> Γ<sup>14</sup> ∗ Γ<sup>22</sup> Γ<sup>23</sup> 0 ∗ ∗ *I* 0 ∗∗∗ *<sup>R</sup>*−<sup>1</sup>

*hiFi*) + *<sup>β</sup>*

*T*

*<sup>Φ</sup>*<sup>12</sup> = −*P*(∑

*Φ*<sup>11</sup> *Φ*<sup>12</sup> *α*1/2[∑*<sup>i</sup>* ∑*<sup>j</sup> hihjHij*]

<sup>∗</sup> *<sup>Φ</sup>*<sup>22</sup> *<sup>α</sup>*1/2[∑*<sup>i</sup> hiZi*]

*hihj* ×

∗ ∗ *I* 0 ∗∗ ∗ *<sup>R</sup>*−<sup>1</sup>

> ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

*hihjGij*) − *<sup>Q</sup>* − [∑

*i*

*T*

*T*

⎤ <sup>⎦</sup> *<sup>R</sup>* �

⎤

*T*

<sup>Θ</sup><sup>12</sup> = −*P*(∑

*i hiKi*]

> <sup>2</sup> [∑ *i* ∑ *j*

Fuzzy Control of Nonlinear Systems with General Performance Criteria 127

<sup>Θ</sup><sup>22</sup> = −*γ<sup>I</sup>* + *<sup>β</sup>*[∑

[∑*<sup>i</sup> hiKi*] 0 0�

*hihjGij*) − *Q*

*hihjHij*] *T*

> *i hiZi*]

*T*

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

*hiFi*) + *<sup>β</sup>*

*TR*[∑ *i hiKi*]

> *i hiZi*]

⎦ ≥ 0 (39)

*hihjHij*] *T*

*<sup>T</sup>* (38)

≥ 0 (40)

*<sup>T</sup>* (41)

≥ 0 (42)

≥ 0 (43)

then inequality (24) is satisfied.

#### **Proof**

By applying system model and performance output (2)(11), and state feedback input (8), the performance index inequality (24) becomes

[ *r* ∑ *i*=1 *r* ∑ *j*=1 *hi*(*ϕ*(*t*))*hj*(*ϕ*(*t*))*Gijx*(*t*) + *r* ∑ *i*=1 *hi*(*ϕ*(*t*))*Fiw*(*t*)]*TPx*(*t*) + *xT*(*t*)*P*[ *r* ∑ *i*=1 *r* ∑ *j*=1 *hi*(*ϕ*(*t*))*hj*(*ϕ*(*t*))*Gijx*(*t*) + *r* ∑ *i*=1 *hi*(*ϕ*(*t*))*Fiw*(*t*)] + *<sup>x</sup>T*(*t*)*Qx*(*t*)+[<sup>−</sup> *r* ∑ *i*=1 *hiϕ*(*t*)*Kix*(*t*)]*TR*[<sup>−</sup> *r* ∑ *i*=1 *hiϕ*(*t*)*Kix*(*t*)] *α*[ *r* ∑ *i*=1 *r* ∑ *j*=1 *hi*(*ϕ*(*t*))*hj*(*ϕ*(*t*))*Hijx*(*t*) + *r* ∑ *i*=1 *hi*(*ϕ*(*t*))*Ziw*(*t*)]*<sup>T</sup>* ×[ *r* ∑ *i*=1 *r* ∑ *j*=1 *hi*(*ϕ*(*t*))*hj*(*ϕ*(*t*))*Hijx*(*t*) + *r* ∑ *i*=1 *hi*(*ϕ*(*t*))*Ziw*(*t*)] −*β*[ *r* ∑ *i*=1 *r* ∑ *j*=1 *hi*(*ϕ*(*t*))*hj*(*ϕ*(*t*))*Hijx*(*t*) + *r* ∑ *i*=1 *hi*(*ϕ*(*t*))*Ziw*(*t*)]*<sup>T</sup>* <sup>×</sup> *<sup>w</sup>*(*t*) <sup>+</sup>*γwT*(*t*)*w*(*t*) <sup>≤</sup> <sup>0</sup> (34)

Inequality (34) is equivalent to

$$\mathbb{E}\left[\mathbf{x}^T(t)\ w^T(t)\right] \times \begin{bmatrix} \Delta\_{11} \ \Delta\_{12} \\ \ast \ \Delta\_{22} \end{bmatrix} \times \begin{bmatrix} \mathbf{x}(t) \\ w(t) \end{bmatrix} \le 0 \tag{35}$$

where

Δ<sup>11</sup> = (∑ *i* ∑ *j hihjGij*)*TP* <sup>+</sup> *<sup>P</sup>*(∑ *i* ∑ *j hihjGij*) + *Q* + [∑ *i hiKi*] *TR*[∑ *i hiKi*] + *α*[∑ *i* ∑ *j hihjHij*] *<sup>T</sup>*[∑ *i* ∑ *j hihjHij*] Δ<sup>12</sup> = *P*(∑ *i hiFi*) + *α*[∑ *i* ∑ *j hihjHij*] *<sup>T</sup>*[∑ *i hiZi*] <sup>−</sup> *<sup>β</sup>* <sup>2</sup> [∑ *i* ∑ *j hihjHij*] *T* Δ<sup>22</sup> = *γI* + *α*[∑ *i hiZi*] *<sup>T</sup>*[∑ *i hiZi*] − *<sup>β</sup>*[∑ *i hiZi*] *<sup>T</sup>* (36)

Inequality (35) can be rewritten as

$$
\begin{bmatrix}
\begin{bmatrix}
\Theta\_{11} \ \Theta\_{12} \\
\ast \ \Theta\_{22}
\end{bmatrix} - a \begin{bmatrix}
\begin{bmatrix}
\sum\_{i} h\_{i} h\_{j} H\_{\overline{ij}} \end{bmatrix}^{T} \\
\begin{bmatrix}
\sum\_{i} h\_{i} \underline{Z\_{i}}
\end{bmatrix}^{T}
\end{bmatrix} \times \begin{bmatrix}
\begin{bmatrix}
\sum\_{i} h\_{i} h\_{\overline{i}} H\_{\overline{ij}}
\end{bmatrix} \begin{bmatrix}
\sum\_{i} h\_{i} \underline{Z\_{i}}
\end{bmatrix}
\end{bmatrix} \ge 0\tag{37}
$$

where

8 Fuzzy Controllers

By applying system model and performance output (2)(11), and state feedback input (8), the

*hi*(*ϕ*(*t*))*hj*(*ϕ*(*t*))*Gijx*(*t*) +

*hi*(*ϕ*(*t*))*hj*(*ϕ*(*t*))*Hijx*(*t*) +

*hi*(*ϕ*(*t*))*hj*(*ϕ*(*t*))*Hijx*(*t*) +

*r* ∑ *i*=1

*r* ∑ *i*=1

*hiϕ*(*t*)*Kix*(*t*)]*TR*[<sup>−</sup>

*r* ∑ *i*=1

> × *x*(*t*) *w*(*t*)

*hihjGij*) + *Q* + [∑

*α*[∑ *i* ∑ *j*

> *<sup>T</sup>*[∑ *i*

*i hiZi*] *<sup>T</sup>*[∑ *i*

*hihjHij*]

Δ<sup>22</sup> = *γI* + *α*[∑

*T*

 ×

*T*

*i hiKi*]

*hiZi*] <sup>−</sup> *<sup>β</sup>*

[∑*<sup>i</sup>* ∑*<sup>j</sup> hihjHij*] [∑*<sup>i</sup> hiZi*]

*hihjHij*]

*hi*(*ϕ*(*t*))*hj*(*ϕ*(*t*))*Gijx*(*t*) +

*Mi* = *KiP*<sup>−</sup><sup>1</sup> = *KiS* (33)

*hi*(*ϕ*(*t*))*Fiw*(*t*)]*TPx*(*t*) +

*r* ∑ *i*=1

*hi*(*ϕ*(*t*))*Fiw*(*t*)] +

*hiϕ*(*t*)*Kix*(*t*)]

*hi*(*ϕ*(*t*))*Ziw*(*t*)]*<sup>T</sup>*

*hi*(*ϕ*(*t*))*Ziw*(*t*)]

*TR*[∑ *i*

*<sup>T</sup>*[∑ *i* ∑ *j*

<sup>2</sup> [∑ *i* ∑ *j*

*hiZi*] − *<sup>β</sup>*[∑

<sup>+</sup>*γwT*(*t*)*w*(*t*) <sup>≤</sup> <sup>0</sup> (34)

≤ 0 (35)

*hiKi*] +

*hihjHij*]

*hihjHij*] *T*

*<sup>T</sup>* (36)

≥ 0 (37)

*i hiZi*]

*hi*(*ϕ*(*t*))*Ziw*(*t*)]*<sup>T</sup>* <sup>×</sup> *<sup>w</sup>*(*t*)

*r* ∑ *i*=1

> *r* ∑ *i*=1

> > *r* ∑ *i*=1

using the notation

**Proof**

then inequality (24) is satisfied.

[ *r* ∑ *i*=1

−*β*[ *r* ∑ *i*=1

Inequality (34) is equivalent to

Δ<sup>11</sup> = (∑ *i* ∑ *j*

Inequality (35) can be rewritten as

 Θ<sup>11</sup> Θ<sup>12</sup> ∗ Θ<sup>22</sup>

where

performance index inequality (24) becomes

*r* ∑ *j*=1

> *r* ∑ *i*=1

*α*[ *r* ∑ *i*=1

> ×[ *r* ∑ *i*=1

Δ<sup>12</sup> = *P*(∑

 − *α* 

*i*

*xT*(*t*) *wT*(*t*)

*hihjGij*)*TP* <sup>+</sup> *<sup>P</sup>*(∑

*hiFi*) + *α*[∑

[∑*<sup>i</sup>* ∑*<sup>j</sup> hihjHij*]

[∑*<sup>i</sup> hiZi*]

*r* ∑ *j*=1

*r* ∑ *j*=1

*<sup>x</sup>T*(*t*)*Qx*(*t*)+[<sup>−</sup>

*r* ∑ *j*=1

> *r* ∑ *j*=1

*hi*(*ϕ*(*t*))*hj*(*ϕ*(*t*))*Hijx*(*t*) +

 × Δ<sup>11</sup> Δ<sup>12</sup> ∗ Δ<sup>22</sup>

> *i* ∑ *j*

*i* ∑ *j*

*xT*(*t*)*P*[

$$\Theta\_{11} = -(\sum\_{i} \sum\_{j} h\_{i} h\_{j} \mathbf{G}\_{ij})^{T} P - P(\sum\_{i} \sum\_{j} h\_{i} h\_{j} \mathbf{G}\_{ij}) - Q - [\sum\_{i} h\_{i} \mathbf{K}\_{i}]^{T} R [\sum\_{i} h\_{i} \mathbf{K}\_{i}]$$

$$\Theta\_{12} = -P(\sum\_{i} h\_{i} \mathbf{F}\_{i}) + \frac{\beta}{2} [\sum\_{i} \sum\_{j} h\_{i} h\_{j} \mathbf{H}\_{ij}]^{T}$$

$$\Theta\_{22} = -\gamma I + \beta [\sum\_{i} h\_{i} \mathbf{Z}\_{i}]^{T} \tag{38}$$

By applying Schur complement to inequality (37), we have

$$
\begin{bmatrix}
\Theta\_{11} \ \Theta\_{12} \ \mathfrak{a}^{1/2} [\Sigma\_i \sum\_{j} h\_i h\_j H\_{ij}]^T \\
\ast \ \Theta\_{22} \ \mathfrak{a}^{1/2} [\Sigma\_i \ h\_i Z\_i]^T \\
\ast \ \mathfrak{a} \end{bmatrix} \ge 0 \tag{39}
$$

Similarly, inequality (39) can also be written as

$$
\begin{bmatrix}
\Phi\_{11} \ \Phi\_{12} \ a^{1/2} [\underline{\sum}\_{i} \sum\_{j} h\_{i} h\_{j} H\_{ij}]^{T} \\
\ast \quad \Phi\_{22} \quad a^{1/2} [\underline{\sum}\_{i} h\_{i} Z\_{i}]^{T} \\
\ast \quad \qquad I
\end{bmatrix} - \begin{bmatrix}
[\underline{\sum}\_{i} h\_{i} K\_{i}]^{T} \\
0 \\
0
\end{bmatrix} R \left[ [\underline{\sum}\_{i} h\_{i} K\_{i}] \ 0 \ 0 \right] \geq 0 \tag{40}
$$

where

$$\Phi\_{11} = -(\sum\_{i} \sum\_{j} h\_{i} h\_{j} \mathbf{G}\_{ij})^{T} P - P(\sum\_{i} \sum\_{j} h\_{i} h\_{j} \mathbf{G}\_{ij}) - \mathcal{Q}$$

$$\Phi\_{12} = -P(\sum\_{i} h\_{i} \mathbf{F}\_{i}) + \frac{\beta}{2} [\sum\_{i} \sum\_{j} h\_{i} h\_{j} H\_{ij}]^{T}$$

$$\Phi\_{22} = -\gamma I + \beta [\sum\_{i} h\_{i} \mathbf{Z}\_{i}]^{T} \tag{41}$$

By applying Schur complement again to (40), we have

$$
\begin{bmatrix}
\Phi\_{11} \ \Phi\_{12} \ a^{1/2} [\Sigma\_i \sum\_j h\_i h\_j H\_{ij}]^T \ [\Sigma\_i \ h\_i K\_i]^T \\
\ast \quad \Phi\_{22} \ \quad a^{1/2} [\Sigma\_i \ h\_i Z\_i]^T & 0 \\
\ast \quad \ast \qquad I & 0 \\
\ast \quad \ast \qquad \ast \qquad R^{-1}
\end{bmatrix} \geq 0 \tag{42}
$$

Equivalently, we have

$$\sum\_{i} \sum\_{j} h\_{i} h\_{j} \times \begin{bmatrix} \Gamma\_{11} \ \Gamma\_{12} \ \Gamma\_{13} \ \Gamma\_{14} \\ \ast & \Gamma\_{22} \ \Gamma\_{23} & 0 \\ \ast & \ast & I & 0 \\ \ast & \ast & \ast & R^{-1} \end{bmatrix} \geq 0 \tag{43}$$

where

$$\Gamma\_{11} = -\frac{1}{2} [(A\_i - B\_i K\_j) + (A\_j - B\_j K\_i)]^T P - \frac{1}{2} P [(A\_i - B\_i K\_j) + (A\_j - B\_j K\_i)] - Q$$

$$\Gamma\_{12} = -\frac{1}{2} P (F\_i + F\_j) + \frac{\beta}{4} [(\mathbf{C}\_i - D\_i K\_j) + (\mathbf{C}\_j - D\_j K\_i)]^T$$

$$\Gamma\_{13} = -\frac{1}{2} \alpha^{1/2} [(\mathbf{C}\_i - D\_i K\_j) + (\mathbf{C}\_j - D\_j K\_i)]^T$$

$$\Gamma\_{14} = -\frac{1}{2} (K\_i + K\_j)^T$$

$$\Gamma\_{22} = -\gamma I + \frac{1}{2} \beta (\mathbf{Z}\_i + \mathbf{Z}\_j)^T$$

$$\Gamma\_{23} = \frac{1}{2} \alpha^{1/2} (\mathbf{Z}\_i + \mathbf{Z}\_j)^T \tag{44}$$

Therefore, we have the following LMI

$$
\begin{bmatrix}
\Gamma\_{11} \ \Gamma\_{12} \ \Gamma\_{13} \ \Gamma\_{14} \\
\* & \Gamma\_{22} \ \Gamma\_{23} \ 0 \\
\* & \* & I \ 0 \\
\* & \* & \* & R^{-1}
\end{bmatrix} \geq 0 \tag{45}
$$

By applying Schur complement again, the final LMI is derived ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

*<sup>i</sup>* <sup>−</sup> *MjB<sup>T</sup>*

<sup>Λ</sup><sup>12</sup> <sup>=</sup> <sup>−</sup><sup>1</sup>

the feedback control can be found to satisfy the chosen criterion.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *<sup>i</sup>* <sup>+</sup> *SA<sup>T</sup>*

2

where

theorem.

**performance criteria**

systems with general performance criteria:

<sup>Λ</sup><sup>11</sup> <sup>=</sup> <sup>−</sup><sup>1</sup>

2 [*SA<sup>T</sup>* Λ<sup>11</sup> Λ<sup>12</sup> Λ<sup>13</sup> Λ<sup>14</sup> Λ<sup>15</sup> ∗ Λ<sup>22</sup> Λ<sup>23</sup> 0 0 ∗ ∗ *I* 0 0 ∗∗∗ *<sup>R</sup>*−<sup>1</sup> <sup>0</sup> ∗∗∗∗ *I*

> *<sup>j</sup>* <sup>−</sup> *<sup>M</sup><sup>T</sup> <sup>i</sup> <sup>B</sup><sup>T</sup>*

> > <sup>4</sup> [*SC<sup>T</sup>*

*α*1/2[*SC<sup>T</sup>*

Hence, if the LMI (49) holds, inequality (24) is satisfied. This concludes the proof of the

Remark 1: For the chosen performance criterion, the LMI (49) need to be solved at each time to find matrices *S*, *M*, by using relation (33), we can find the feedback control gain, therefore,

This section summarizes the main results for fuzzy LMI control of discrete time non-linear

**Theorem 2:** Given the closed loop system and performance output (13), and control input (19), if there exist matrices *<sup>S</sup>* <sup>=</sup> *<sup>P</sup>*−<sup>1</sup> <sup>&</sup>gt; 0 for all *<sup>k</sup>* <sup>≥</sup> 0, such that the following LMI holds:

> ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Ξ<sup>11</sup> Ξ<sup>12</sup> Ξ<sup>13</sup> Ξ<sup>14</sup> Ξ<sup>15</sup> Ξ<sup>16</sup> ∗ Ξ<sup>22</sup> Ξ<sup>23</sup> Ξ<sup>24</sup> 0 0 ∗ ∗ *S* 000 ∗∗∗ *I* 0 0 ∗∗∗∗ *<sup>R</sup>*−<sup>1</sup> <sup>0</sup> ∗∗∗∗ ∗ *I*

**5. Fuzzy LMI control of discrete time non-linear systems with general**

*<sup>i</sup>* <sup>−</sup> *MjD<sup>T</sup>*

*<sup>i</sup>* <sup>−</sup> *MjD<sup>T</sup>*

Λ<sup>22</sup> = −*γI* +

<sup>Λ</sup><sup>23</sup> <sup>=</sup> <sup>1</sup> 2

(*Fi* <sup>+</sup> *Fj*) + *<sup>β</sup>*

<sup>Λ</sup><sup>13</sup> <sup>=</sup> <sup>1</sup> 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

≥ 0 (49)

*<sup>j</sup>* <sup>−</sup> *<sup>M</sup><sup>T</sup>*

*<sup>j</sup>* <sup>−</sup> *<sup>M</sup><sup>T</sup>*

*<sup>i</sup> <sup>D</sup><sup>T</sup> j* ]

*<sup>i</sup> <sup>D</sup><sup>T</sup> j* ]

*<sup>i</sup>* <sup>+</sup> *<sup>M</sup><sup>T</sup> j* )

Λ<sup>15</sup> = *SQT*/2

*β*(*Zi* + *Zj*)*<sup>T</sup>*

≥ 0 (51)

*<sup>T</sup>* (50)

*α*1/2[*Zi* + *Zj*]

*<sup>j</sup>* + *AiS* − *BiMj* + *AjS* − *BjMi*]

Fuzzy Control of Nonlinear Systems with General Performance Criteria 129

*<sup>i</sup>* <sup>+</sup> *SC<sup>T</sup>*

*<sup>i</sup>* <sup>+</sup> *SC<sup>T</sup>*

1 2

<sup>Λ</sup><sup>14</sup> <sup>=</sup> <sup>1</sup> 2 (*M<sup>T</sup>*

By multiplying both sides of the LMI above by the block diagonal matrix *diag*{*S*, *I*, *I*, *I*}, where *S* = *P*−1, and using the notation

$$M\_{\dot{l}} = K\_{\dot{l}} P^{-1} = K\_{\dot{l}} \mathcal{S} \tag{46}$$

we obtain

$$
\begin{bmatrix}
\mathbf{X}\_{11} & \mathbf{X}\_{12} & \mathbf{X}\_{13} & \mathbf{X}\_{14} \\
\ast & \mathbf{X}\_{22} & \mathbf{X}\_{23} & \mathbf{0} \\
\ast & \ast & I & \mathbf{0} \\
\ast & \ast & \ast & \mathbf{R}^{-1}
\end{bmatrix} \geq \mathbf{0} \tag{47}
$$

where

$$X\_{11} = -\frac{1}{2} [SA\_i^T - M\_jB\_j^T + SA\_j^T - M\_i^TB\_j^T + A\_iS - B\_iM\_j + A\_jS - B\_jM\_i] - SQS$$

$$X\_{12} = -\frac{1}{2}(F\_i + F\_j) + \frac{\beta}{4} [SC\_i^T - M\_j^TD\_i^T + SC\_j^T - M\_i^TD\_j^T]$$

$$X\_{13} = \frac{1}{2}a^{1/2} [SC\_i^T - M\_j^TD\_i^T + SC\_j^T - M\_i^TD\_j^T]$$

$$X\_{14} = \frac{1}{2}(M\_i^T + M\_j^T)$$

$$X\_{22} = -\gamma I + \frac{1}{2}\beta (Z\_i + Z\_j)^T$$

$$X\_{23} = \frac{1}{2}a^{1/2} (Z\_i + Z\_j)^T \tag{48}$$

By applying Schur complement again, the final LMI is derived

$$
\begin{bmatrix}
\Lambda\_{11} \ \Lambda\_{12} \ \Lambda\_{13} \ \Lambda\_{14} \ \Lambda\_{15} \\
\* \ \Lambda\_{22} \ \Lambda\_{23} \ 0 & 0 \\
\* & \* \ I & 0 & 0 \\
\* & \* & \* \ R^{-1} & 0 \\
\* & \* & \* & \* & I
\end{bmatrix} \geq 0 \tag{49}
$$

where

10 Fuzzy Controllers

2

*<sup>P</sup>*(*Fi* <sup>+</sup> *Fj*) + *<sup>β</sup>*

<sup>Γ</sup><sup>13</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> 2

*P*[(*Ai* − *BiKj*)+(*Aj* − *BjKi*)] − *Q*

*<sup>α</sup>*1/2[(*Ci* <sup>−</sup> *DiKj*)+(*Cj* <sup>−</sup> *DjKi*)]*<sup>T</sup>*

<sup>Γ</sup><sup>14</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> 2

> 1 2

≥ 0 (45)

≥ 0 (47)

Γ<sup>22</sup> = −*γI* +

<sup>Γ</sup><sup>23</sup> <sup>=</sup> <sup>1</sup> 2

*Mi* = *KiP*<sup>−</sup><sup>1</sup> = *KiS* (46)

*<sup>j</sup>* + *AiS* − *BiMj* + *AjS* − *BjMi*] − *SQS*

*<sup>j</sup> <sup>D</sup><sup>T</sup>*

*<sup>j</sup> <sup>D</sup><sup>T</sup>*

*X*<sup>22</sup> = −*γI* +

*<sup>X</sup>*<sup>23</sup> <sup>=</sup> <sup>1</sup> 2

*<sup>i</sup>* <sup>+</sup> *SC<sup>T</sup>*

*<sup>i</sup>* <sup>+</sup> *SC<sup>T</sup>*

1 2

*<sup>X</sup>*<sup>14</sup> <sup>=</sup> <sup>1</sup> 2 (*M<sup>T</sup>*

*<sup>j</sup>* <sup>−</sup> *<sup>M</sup><sup>T</sup>*

*<sup>j</sup>* <sup>−</sup> *<sup>M</sup><sup>T</sup>*

*<sup>i</sup> <sup>D</sup><sup>T</sup> j* ]

*<sup>i</sup> <sup>D</sup><sup>T</sup> j* ]

*α*1/2(*Zi* + *Zj*)*<sup>T</sup>* (48)

*<sup>i</sup>* <sup>+</sup> *<sup>M</sup><sup>T</sup> j* )

*β*(*Zi* + *Zj*)*<sup>T</sup>*

(*Ki* + *Kj*)*<sup>T</sup>*

*β*(*Zi* + *Zj*)*<sup>T</sup>*

*α*1/2(*Zi* + *Zj*)*<sup>T</sup>* (44)

<sup>4</sup> [(*Ci* <sup>−</sup> *DiKj*)+(*Cj* <sup>−</sup> *DjKi*)]*<sup>T</sup>*

[(*Ai* <sup>−</sup> *BiKj*)+(*Aj* <sup>−</sup> *BjKi*)]*TP* <sup>−</sup> <sup>1</sup>

<sup>Γ</sup><sup>12</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> 2

> ⎡ ⎢ ⎢ ⎣

⎡ ⎢ ⎢ ⎣

*<sup>i</sup>* <sup>+</sup> *SA<sup>T</sup>*

*<sup>X</sup>*<sup>12</sup> <sup>=</sup> <sup>−</sup><sup>1</sup>

Γ<sup>11</sup> Γ<sup>12</sup> Γ<sup>13</sup> Γ<sup>14</sup> ∗ Γ<sup>22</sup> Γ<sup>23</sup> 0 ∗ ∗ *I* 0 ∗∗∗ *<sup>R</sup>*−<sup>1</sup>

*X*<sup>11</sup> *X*<sup>12</sup> *X*<sup>13</sup> *X*<sup>14</sup> ∗ *X*<sup>22</sup> *X*<sup>23</sup> 0 ∗ ∗ *I* 0 ∗∗∗ *<sup>R</sup>*−<sup>1</sup>

> *<sup>j</sup>* <sup>−</sup> *<sup>M</sup><sup>T</sup> <sup>i</sup> <sup>B</sup><sup>T</sup>*

> > (*Fi* <sup>+</sup> *Fj*) + *<sup>β</sup>*

*<sup>X</sup>*<sup>13</sup> <sup>=</sup> <sup>1</sup> 2

2

By multiplying both sides of the LMI above by the block diagonal matrix *diag*{*S*, *I*, *I*, *I*}, where

⎤ ⎥ ⎥ ⎦

> ⎤ ⎥ ⎥ ⎦

<sup>4</sup> [*SC<sup>T</sup>*

*α*1/2[*SC<sup>T</sup>*

*<sup>i</sup>* <sup>−</sup> *<sup>M</sup><sup>T</sup>*

*<sup>i</sup>* <sup>−</sup> *<sup>M</sup><sup>T</sup>*

where

<sup>Γ</sup><sup>11</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> 2

Therefore, we have the following LMI

*S* = *P*−1, and using the notation

*<sup>X</sup>*<sup>11</sup> <sup>=</sup> <sup>−</sup><sup>1</sup>

2 [*SA<sup>T</sup>*

*<sup>i</sup>* <sup>−</sup> *MjB<sup>T</sup>*

we obtain

where

$$\mathbf{A}\_{11} = -\frac{1}{2} [SA\_i^T - M\_jB\_i^T + SA\_j^T - M\_i^TB\_j^T + A\_iS - B\_iM\_j + A\_jS - B\_jM\_i]$$

$$\Lambda\_{12} = -\frac{1}{2}(F\_i + F\_j) + \frac{\beta}{4} [SC\_i^T - M\_jD\_i^T + SC\_j^T - M\_i^TD\_j^T]$$

$$\Lambda\_{13} = \frac{1}{2} \mathbf{a}^{1/2} [SC\_i^T - M\_jD\_i^T + SC\_j^T - M\_i^TD\_j^T]$$

$$\Lambda\_{14} = \frac{1}{2} (M\_i^T + M\_j^T)$$

$$\Lambda\_{15} = \mathbf{S}\mathbf{Q}^{T/2}$$

$$\Lambda\_{22} = -\gamma I + \frac{1}{2}\beta (Z\_i + Z\_j)^T$$

$$\Lambda\_{23} = \frac{1}{2} \mathbf{a}^{1/2} [Z\_i + Z\_j]^T \tag{50}$$

Hence, if the LMI (49) holds, inequality (24) is satisfied. This concludes the proof of the theorem.

Remark 1: For the chosen performance criterion, the LMI (49) need to be solved at each time to find matrices *S*, *M*, by using relation (33), we can find the feedback control gain, therefore, the feedback control can be found to satisfy the chosen criterion.

## **5. Fuzzy LMI control of discrete time non-linear systems with general performance criteria**

This section summarizes the main results for fuzzy LMI control of discrete time non-linear systems with general performance criteria:

**Theorem 2:** Given the closed loop system and performance output (13), and control input (19), if there exist matrices *<sup>S</sup>* <sup>=</sup> *<sup>P</sup>*−<sup>1</sup> <sup>&</sup>gt; 0 for all *<sup>k</sup>* <sup>≥</sup> 0, such that the following LMI holds:

$$
\begin{bmatrix}
\Xi\_{11} \ \Xi\_{12} \ \Sigma\_{13} \ \Sigma\_{14} \ \Xi\_{15} \ \Xi\_{16} \\
\* & \Xi\_{22} \ \Xi\_{23} \ \Xi\_{24} \ 0 & 0 \\
\* & \* & S & 0 & 0 \\
\* & \* & \* & I & 0 & 0 \\
\* & \* & \* & \* & R^{-1} & 0 \\
\* & \* & \* & \* & \* & I
\end{bmatrix} \geq 0 \tag{51}
$$

#### 12 Fuzzy Controllers 130 Fuzzy Controllers – Recent Advances in Theory and Applications Fuzzy Control of Nonlinear Systems with General Performance Criteria <sup>13</sup>

where

$$\Xi\_{11} = S$$

$$\Xi\_{12} = \frac{\beta}{4} (\mathbf{C}\_i \mathbf{S} - D\_i \mathbf{Y}\_j + \mathbf{C}\_j \mathbf{S} - D\_j \mathbf{Y}\_i)^T$$

$$\Xi\_{13} = \frac{1}{2} (A\_i \mathbf{S} - B\_i \mathbf{Y}\_j + A\_j \mathbf{S} - B\_j \mathbf{Y}\_i)^T$$

$$\Xi\_{14} = \frac{1}{2} \mathbf{a}^{1/2} (\mathbf{C}\_i \mathbf{S} - D\_i \mathbf{Y}\_j + \mathbf{C}\_j \mathbf{S} - D\_j \mathbf{Y}\_i)^T$$

$$\Xi\_{15} = \frac{1}{2} (\mathbf{Y}\_i + \mathbf{Y}\_j)^T$$

$$\Xi\_{16} = S \mathbf{Q}^T / 2$$

$$\Xi\_{22} = -\gamma I + \frac{\beta}{2} (\mathbf{Z}\_i + \mathbf{Z}\_j)^T$$

$$\Xi\_{23} = \frac{1}{2} \mathbf{a}^{1/2} (\mathbf{F}\_i + \mathbf{F}\_j)^T$$

$$\Xi\_{24} = \frac{1}{2} \mathbf{a}^{1/2} (\mathbf{Z}\_i + \mathbf{Z}\_j)^T \tag{52}$$

and

$$S(k+1) > S(k) \tag{53}$$

Equivalently,

*xT*(*k*) *wT*(*k*)

*xT*(*k*) *wT*(*k*)

� �(∑*<sup>i</sup>* ∑*<sup>j</sup> hihjGij*) (∑*<sup>i</sup> hiFi*)

� �(∑*<sup>i</sup>* ∑*<sup>j</sup> hihjGij*) (∑*<sup>i</sup> hiFi*)

� �(∑*<sup>i</sup>* ∑*<sup>j</sup> hihjHij*) (∑*<sup>i</sup> hiZi*)

*<sup>Υ</sup>*<sup>11</sup> = *<sup>P</sup>* − *<sup>Q</sup>* − [∑

(∑*<sup>i</sup>* ∑*<sup>j</sup> hihjGij*) (∑*<sup>i</sup> hiFi*)

(∑*<sup>i</sup>* ∑*<sup>j</sup> hihjHij*) (∑*<sup>i</sup> hiZi*)

which can be written, after collecting terms, as

� �(∑*<sup>i</sup>* ∑*<sup>j</sup> hihjHij*) (∑*<sup>i</sup> hiZi*)

−*β* �

�

*α* �

�

*α* �

where

*xT*(*k*) *wT*(*k*)

*xT*(*k*) *wT*(*k*)

Equivalently, we have

*Υ*<sup>11</sup> *Υ*<sup>12</sup> (∑*<sup>i</sup>* ∑*<sup>j</sup> hihjGij*))*<sup>T</sup>* <sup>∗</sup> *<sup>Υ</sup>*<sup>22</sup> (∑*<sup>i</sup> hiFi*)*<sup>T</sup>* ∗ ∗ *<sup>P</sup>*−<sup>1</sup>

− �

*α* �

By applying Schur complement, we obtain

⎤ ⎦ − *α* �

� *Υ*11 *Υ*12 <sup>∗</sup> *<sup>Υ</sup>*22�

⎡ ⎣ �

<sup>×</sup> *<sup>P</sup>* <sup>×</sup> �

�*T* × �

�

*TR*[∑ *i hiKi*]

> *i hiZi*]

*hihjHij*] *T*

> �*T* × �

�*T*

*xT*(*k*) *wT*(*k*)

�*T*

�*T* × �

*i hiKi*]

> <sup>2</sup> [∑ *i* ∑ *j*

*<sup>Υ</sup>*<sup>22</sup> = −*γ<sup>I</sup>* + *<sup>β</sup>*[∑

�*T*

(∑*<sup>i</sup>* ∑*<sup>j</sup> hihjHij*) (∑*<sup>i</sup> hiZi*)

�*T* × �

<sup>×</sup> *<sup>P</sup>* <sup>×</sup> �

*<sup>Υ</sup>*<sup>12</sup> <sup>=</sup> *<sup>β</sup>*

<sup>×</sup> *<sup>P</sup>* <sup>×</sup> �

*xT*(*k*) *wT*(*k*)

<sup>+</sup>*xT*(*k*)[∑

� �

Fuzzy Control of Nonlinear Systems with General Performance Criteria 131

(∑*<sup>i</sup>* ∑*<sup>j</sup> hihjGij*) (∑*<sup>i</sup> hiFi*)

*i hiKi*]

� �(∑*<sup>i</sup>* ∑*<sup>j</sup> hihjHij*) (∑*<sup>i</sup> hiZi*)

*xT*(*k*) *wT*(*k*)

(∑*<sup>i</sup>* ∑*<sup>j</sup> hihjHij*) (∑*<sup>i</sup> hiZi*)

� �

(∑*<sup>i</sup>* ∑*<sup>j</sup> hihjGij*) (∑*<sup>i</sup> hiFi*)

(∑*<sup>i</sup>* ∑*<sup>j</sup> hihjHij*) (∑*<sup>i</sup> hiZi*)

(∑*<sup>i</sup>* ∑*<sup>j</sup> hihjGij*) (∑*<sup>i</sup> hiFi*)

(∑*<sup>i</sup>* ∑*<sup>j</sup> hihjHij*) (∑*<sup>i</sup> hiZi*)

*Υ*11 *Υ*12 <sup>∗</sup> *<sup>Υ</sup>*22� � *<sup>x</sup>*(*k*)

−*P* + *Q* 0 0 *γI*

> *TR*[∑ *i*

� � *x*(*k*) *w*(*k*) � +

*hiKi*]*x*(*k*) +

*w*(*k*) ≤ 0

*w*(*k*) � +

� � *x*(*k*) *w*(*k*) � +

� � *x*(*k*) *w*(*k*) � ≥ 0

*<sup>T</sup>* (58)

� −

� ≥ 0

(∑*<sup>i</sup>* ∑*<sup>j</sup> hihjHij*) (∑*<sup>i</sup> hiZi*)

(56)

(57)

(59)

(60)

� ≥ 0

� � *x*(*k*) *w*(*k*) � +

� � *x*(*k*) *w*(*k*) � +

�*T*

where *S*(*k*) = *P*−1(*k*), then (28) is satisfied with the feedback control gain being found by

$$K(k) = Y(k)P(k)\tag{54}$$

#### **Proof**

The performance index inequality (28) can be explicitly written as

[ *r* ∑ *i*=1 *r* ∑ *j*=1 *hi*(*ϕ*(*k*))*hj*(*ϕ*(*k*))*Gijx*(*k*) + *r* ∑ *i*=1 *hi*(*ϕ*(*k*))*Fiw*(*k*)]*<sup>T</sup>* ×*P* × [ *r* ∑ *i*=1 *r* ∑ *j*=1 *hi*(*ϕ*(*k*))*hj*(*ϕ*(*k*))*Gijx*(*k*) + *r* ∑ *i*=1 *hi*(*ϕ*(*k*))*Fiw*(*k*)] <sup>−</sup>*xT*(*k*)*Px*(*k*) + *<sup>x</sup>T*(*k*)*Qx*(*k*)+[<sup>−</sup> *r* ∑ *i*=1 *hi*(*ϕ*(*k*))*Kix*(*k*)]*TR*[<sup>−</sup> *r* ∑ *i*=1 *hi*(*ϕ*(*k*))*Kix*(*k*)] + *α*[ *r* ∑ *i*=1 *r* ∑ *j*=1 *hi*(*ϕ*(*k*))*hj*(*ϕ*(*k*))*Hijx*(*k*) + *r* ∑ *i*=1 *hi*(*ϕ*(*k*))*Ziw*(*k*)]*<sup>T</sup>* ×[ *r* ∑ *i*=1 *r* ∑ *j*=1 *hi*(*ϕ*(*k*))*hj*(*ϕ*(*k*))*Hijx*(*k*) + *r* ∑ *i*=1 *hi*(*ϕ*(*k*))*Ziw*(*k*)] −*β*[ *r* ∑ *i*=1 *r* ∑ *j*=1 *hi*(*ϕ*(*k*))*hj*(*ϕ*(*k*))*Hijx*(*k*) + *r* ∑ *i*=1 *hi*(*ϕ*(*k*))*Ziw*(*k*)]*<sup>T</sup>* <sup>×</sup> *<sup>w</sup>*(*k*) <sup>+</sup>*γwT*(*k*)*w*(*k*) <sup>≤</sup> 0 (55)

Equivalently,

12 Fuzzy Controllers

<sup>4</sup> (*CiS* <sup>−</sup> *DiYj* <sup>+</sup> *CjS* <sup>−</sup> *DjYi*)*<sup>T</sup>*

(*AiS* <sup>−</sup> *BiYj* <sup>+</sup> *AjS* <sup>−</sup> *BjYi*)*<sup>T</sup>*

*<sup>α</sup>*1/2(*CiS* <sup>−</sup> *DiYj* <sup>+</sup> *CjS* <sup>−</sup> *DjYi*)*<sup>T</sup>*

<sup>Ξ</sup><sup>22</sup> <sup>=</sup> <sup>−</sup>*γ<sup>I</sup>* <sup>+</sup> *<sup>β</sup>*

<sup>Ξ</sup><sup>23</sup> <sup>=</sup> <sup>1</sup> 2

<sup>Ξ</sup><sup>24</sup> <sup>=</sup> <sup>1</sup> 2

where *S*(*k*) = *P*−1(*k*), then (28) is satisfied with the feedback control gain being found by

<sup>Ξ</sup><sup>15</sup> <sup>=</sup> <sup>1</sup> 2

*hi*(*ϕ*(*k*))*hj*(*ϕ*(*k*))*Gijx*(*k*) +

*hi*(*ϕ*(*k*))*hj*(*ϕ*(*k*))*Gijx*(*k*) +

*hi*(*ϕ*(*k*))*Kix*(*k*)]*TR*[<sup>−</sup>

*hi*(*ϕ*(*k*))*hj*(*ϕ*(*k*))*Hijx*(*k*) +

*hi*(*ϕ*(*k*))*hj*(*ϕ*(*k*))*Hijx*(*k*) +

*r* ∑ *i*=1

<sup>Ξ</sup><sup>12</sup> <sup>=</sup> *<sup>β</sup>*

<sup>Ξ</sup><sup>13</sup> <sup>=</sup> <sup>1</sup> 2

The performance index inequality (28) can be explicitly written as

*r* ∑ *j*=1

> *r* ∑ *j*=1

> > *r* ∑ *i*=1

*hi*(*ϕ*(*k*))*hj*(*ϕ*(*k*))*Hijx*(*k*) +

*r* ∑ *i*=1

> *r* ∑ *j*=1

> > *r* ∑ *j*=1

[ *r* ∑ *i*=1

×*P* × [

*α*[ *r* ∑ *i*=1

> ×[ *r* ∑ *i*=1

*r* ∑ *j*=1

<sup>−</sup>*xT*(*k*)*Px*(*k*) + *<sup>x</sup>T*(*k*)*Qx*(*k*)+[<sup>−</sup>

−*β*[ *r* ∑ *i*=1

<sup>Ξ</sup><sup>14</sup> <sup>=</sup> <sup>1</sup> 2 Ξ<sup>11</sup> = *S*

(*Yi* + *Yj*)*<sup>T</sup>*

Ξ<sup>16</sup> = *SQT*/2

<sup>2</sup> (*Zi* <sup>+</sup> *Zj*)*<sup>T</sup>*

*α*1/2(*Fi* + *Fj*)*<sup>T</sup>*

*α*1/2(*Zi* + *Zj*)*<sup>T</sup>* (52)

*S*(*k* + 1) > *S*(*k*) (53)

*K*(*k*) = *Y*(*k*)*P*(*k*) (54)

*r* ∑ *i*=1

> *r* ∑ *i*=1

*r* ∑ *i*=1

> *r* ∑ *i*=1

> > *r* ∑ *i*=1

*hi*(*ϕ*(*k*))*Fiw*(*k*)]*<sup>T</sup>*

*hi*(*ϕ*(*k*))*Fiw*(*k*)]

*hi*(*ϕ*(*k*))*Kix*(*k*)] +

*hi*(*ϕ*(*k*))*Ziw*(*k*)]*<sup>T</sup>*

*hi*(*ϕ*(*k*))*Ziw*(*k*)]

<sup>+</sup>*γwT*(*k*)*w*(*k*) <sup>≤</sup> 0 (55)

*hi*(*ϕ*(*k*))*Ziw*(*k*)]*<sup>T</sup>* <sup>×</sup> *<sup>w</sup>*(*k*)

where

and

**Proof**

$$\begin{aligned} \begin{bmatrix} \mathbf{x}^T(k) \ w^T(k) \end{bmatrix} \begin{bmatrix} -P + Q & 0 \\ 0 & \gamma I \end{bmatrix} \begin{bmatrix} \mathbf{x}(k) \\ w(k) \end{bmatrix} + \\ \begin{bmatrix} \mathbf{x}^T(k) \ w^T(k) \end{bmatrix} \begin{bmatrix} \left(\sum\_i h\_i h\_j G\_{ij}\right) \left(\sum\_i h\_i F\_i\right) \end{bmatrix}^T \times P \times \begin{bmatrix} \left(\sum\_i h\_i h\_j G\_{ij}\right) \left(\sum\_i h\_i F\_i\right) \end{bmatrix} + \\ \begin{bmatrix} \mathbf{x}^T(k) \end{bmatrix} \begin{bmatrix} \sum\_i h\_i K\_i \end{bmatrix}^T \mathbb{R} \begin{bmatrix} \mathbf{x}(k) \\ w(k) \end{bmatrix} + \\ \begin{bmatrix} +\mathbf{x}^T(k) \end{bmatrix} \begin{bmatrix} \sum\_i h\_i K\_i \end{bmatrix}^T \mathbb{R} \begin{bmatrix} \sum\_i h\_i K\_i \end{bmatrix} \mathbf{x}(k) + \\ \begin{bmatrix} \mathbf{x}^T(k) \end{bmatrix} \begin{bmatrix} \left(\sum\_i h\_i h\_i H\_{ij}\right) \left(\sum\_i h\_i K\_i\right) \end{bmatrix}^T \times \begin{bmatrix} \left(\sum\_i h\_i h\_j H\_{ij}\right) \left(\sum\_i h\_i K\_i\right) \end{bmatrix} \begin{bmatrix} \mathbf{x}(k) \\ w(k) \end{bmatrix} + \\ - \boldsymbol{\theta} \begin{bmatrix} \mathbf{x}^T(k) \ \mathbf{w}^T(k) \end{bmatrix} \begin{bmatrix} \left(\sum\_i h\_i h\_j H\_{ij}\right) \left(\sum\_i h\_i Z\_i\right) \end{bmatrix}^T \begin{bmatrix} \mathbf{x}(k) \\ w(k) \end{bmatrix} + \end{aligned} \tag{56}$$

which can be written, after collecting terms, as

$$\begin{aligned} \begin{bmatrix} \mathbf{x}^T(k) \ w^T(k) \end{bmatrix} \begin{bmatrix} \mathbf{Y}\_{11} \ \mathbf{Y}\_{12} \\ \mathbf{w}(k) \end{bmatrix} + \\ \begin{bmatrix} \mathbf{x}^T(k) \ w^T(k) \end{bmatrix} \begin{bmatrix} \left(\sum\_i \sum\_j h\_i h\_j \mathbf{G}\_{ij}\right) \left(\sum\_i h\_i \mathbf{F}\_i\right)^T \times \mathbf{P} \times \left[\left(\sum\_i \sum\_j h\_i h\_j \mathbf{G}\_{ij}\right) \left(\sum\_i h\_i \mathbf{F}\_i\right)\right] \begin{bmatrix} \mathbf{x}(k) \\ w(k) \end{bmatrix} + \\ \begin{bmatrix} \mathbf{a} \left[\mathbf{x}^T(k) \ w^T(k)\right] \left[\left(\sum\_i \sum\_j h\_i h\_j \mathbf{H}\_{ij}\right) \left(\sum\_i h\_i \mathbf{Z}\_i\right)\right]^T \times \left[\left(\sum\_i \sum\_j h\_i h\_j \mathbf{H}\_{ij}\right) \left(\sum\_i h\_i \mathbf{Z}\_i\right)\right] \begin{bmatrix} \mathbf{x}(k) \\ w(k) \end{bmatrix} \ge \mathbf{0} \end{aligned} \end{aligned} \tag{57}$$

where

$$Y\_{11} = P - Q - \left[\sum\_{i} h\_{i} K\_{i}\right]^{T} R \left[\sum\_{i} h\_{i} K\_{i}\right]$$

$$Y\_{12} = \frac{\beta}{2} \left[\sum\_{i} \sum\_{j} h\_{i} h\_{j} H\_{ij}\right]^{T}$$

$$Y\_{22} = -\gamma I + \beta \left[\sum\_{i} h\_{i} Z\_{i}\right]^{T}\tag{58}$$

Equivalently, we have

$$
\begin{bmatrix}
\mathbf{Y}\_{11} & \mathbf{Y}\_{12} \\
\mathbf{\*} & \mathbf{Y}\_{22}
\end{bmatrix} - \begin{bmatrix}
\left(\sum\_{i} \sum\_{j} h\_{i} h\_{j} \mathbf{G}\_{ij}\right) \left(\sum\_{i} h\_{i} \mathbf{F}\_{i}\right)
\end{bmatrix}^{T} \times P \times \left[\left(\sum\_{i} \sum\_{j} h\_{i} h\_{j} \mathbf{G}\_{ij}\right) \left(\sum\_{i} h\_{i} \mathbf{F}\_{i}\right)\right] - \mathbf{0}
$$

$$
\text{as } \left[\left(\sum\_{i} \sum\_{j} h\_{i} h\_{j} \mathbf{H}\_{ij}\right) \left(\sum\_{i} h\_{i} \mathbf{Z}\_{i}\right)\right]^{T} \times \left[\left(\sum\_{i} \sum\_{j} h\_{i} h\_{j} \mathbf{H}\_{ij}\right) \left(\sum\_{i} h\_{i} \mathbf{Z}\_{i}\right)\right] \geq 0
\tag{59}
$$

By applying Schur complement, we obtain

$$
\begin{bmatrix}
\mathbf{Y}\_{11} \ \mathbf{Y}\_{12} \ \left(\sum\_{i} \sum\_{j} h\_{i} h\_{j} \mathbf{G}\_{ij}\right) \\
\ast \ \mathbf{Y}\_{22} \ \left(\sum\_{i} h\_{i} \mathbf{F}\_{i}\right)^{T} \\
\ast \ \ast \ \mathbf{P}^{-1}
\end{bmatrix} - \mathbf{a} \begin{bmatrix}
\left(\sum\_{i} \sum\_{j} h\_{i} h\_{j} \mathbf{H}\_{ij}\right) \left(\sum\_{i} h\_{i} \mathbf{Z}\_{i}\right)
\end{bmatrix}^{T} \times \begin{bmatrix}
\left(\sum\_{i} \sum\_{j} h\_{i} h\_{j} \mathbf{H}\_{ij}\right) \left(\sum\_{i} h\_{i} \mathbf{Z}\_{i}\right)
\end{bmatrix} \ge 0\tag{60}
$$

#### 14 Fuzzy Controllers 132 Fuzzy Controllers – Recent Advances in Theory and Applications Fuzzy Control of Nonlinear Systems with General Performance Criteria <sup>15</sup>

By applying Schur complement again, we obtain

$$
\begin{bmatrix}
\mathbf{Y}\_{11} \ \mathbf{Y}\_{12} \ (\sum\_{i} \sum\_{j} h\_{i} h\_{i} \mathbf{G}\_{ij})^{T} \ a^{1/2} (\sum\_{i} \sum\_{j} h\_{i} h\_{j} \mathbf{H}\_{ij})^{T} \\
\ast \ \mathbf{Y}\_{22} \ (\sum\_{i} h\_{i} \mathbf{F}\_{i})^{T} & a^{1/2} (\sum\_{i} h\_{i} \mathbf{Z}\_{i})^{T} \\
\ast \ \ast \ \mathbf{P}^{-1} & \mathbf{0} \\
\ast \ \ast \ \ast \ \ast \ \mathbf{I} & \mathbf{I}
\end{bmatrix} \geq \mathbf{0} \tag{61}
$$

Equivalently, the following inequality holds

$$
\begin{bmatrix}
\Psi\_{11} & \Psi\_{12} \left(\sum\_{i} \sum\_{j} h\_{i} h\_{j} \mathbf{G}\_{ij}\right)^{\top} & \mathbf{a}^{1/2} \left(\sum\_{i} \sum\_{j} h\_{i} h\_{j} \mathbf{H}\_{ij}\right)^{\top} \\
\ast & \Psi\_{22} \qquad \left(\sum\_{i} h\_{i} F\_{i}\right)^{\top} & \mathbf{a}^{1/2} \left(\sum\_{i} h\_{i} Z\_{i}\right)^{\top} \\
\ast & \mathbf{s} & P^{-1} & \mathbf{0} \\
\ast & \ast & \mathbf{s} & I
\end{bmatrix}
$$

$$
\begin{bmatrix}
(\sum\_{i} h\_{i} K\_{i})^{T} \\
\mathbf{0} \\
\mathbf{0} \\
\mathbf{0} \\
\mathbf{0}
\end{bmatrix} \times \mathbf{R} \times \left[ (\sum\_{i} h\_{i} K\_{i}) \begin{bmatrix} \sum\_{i} h\_{i} K\_{i} \end{bmatrix} \mathbf{0} \ \mathbf{0} \ \mathbf{0} \ \mathbf{0} \right] \geq \mathbf{0} \tag{62}
$$

where

$$\Psi\_{11} = P - Q$$

$$\Psi\_{12} = \frac{\beta}{2} [\sum\_{i} \sum\_{j} h\_i h\_j H\_{ij}]^T$$

$$\Psi\_{22} = -\gamma I + \beta [\sum\_{i} h\_i Z\_i]^T \tag{63}$$

By applying Schur complement one more time, we have

$$
\begin{bmatrix}
\mathbb{1}\_{11} & \mathbb{1}\_{12} \left(\sum\_{i} \bar{h}\_{i} \boldsymbol{h}\_{j} \mathbb{G}\_{ij}\right)^{T} & \mathbf{a}^{1/2} \left(\sum\_{i} \bar{\sum}\_{j} \bar{h}\_{i} \boldsymbol{h}\_{j} \bar{H}\_{ij}\right)^{T} \left(\sum\_{i} \bar{h}\_{i} \boldsymbol{K}\_{i}\right)^{T} \\
\ast & \mathbb{1}\_{22} & \left(\sum\_{i} \bar{h}\_{i} \bar{r}\_{i}\right)^{T} & \mathbf{a}^{1/2} \left(\sum\_{i} \bar{h}\_{i} \bar{\boldsymbol{z}}\_{i}\right)^{T} & \mathbf{0} \\
\ast & \ast & P^{-1} & \mathbf{0} & \mathbf{0} \\
\ast & \ast & \ast & I & \mathbf{0} \\
\ast & \ast & \ast & \ast & \mathbf{R}^{-1}
\end{bmatrix} \ge \mathbf{0}
$$

By factoring out the ∑*<sup>i</sup>* ∑*<sup>j</sup> hi*(*ϕk*)*hj*(*ϕk*) term, we have

$$
\begin{bmatrix}
\Omega\_{11} \ \Omega\_{12} \ \Omega\_{13} \ \Omega\_{14} \ \Omega\_{15} \\
\* & \Omega\_{22} \ \Omega\_{23} \ \Omega\_{24} & 0 \\
\* & \* & P^{-1} & 0 & 0 \\
\* & \* & \* & I & 0 \\
\* & \* & \* & \* & R^{-1}
\end{bmatrix} \geq 0
$$

(64)

where

result is obtained

where

Ω<sup>11</sup> = *P* − *Q*

*T*

Fuzzy Control of Nonlinear Systems with General Performance Criteria 133

<sup>4</sup> [*Hji* <sup>+</sup> *Hij*]

(*Gji* + *Gij*))*<sup>T</sup>*

(*Ki* + *Kj*)*<sup>T</sup>*

<sup>2</sup> (*Zi* <sup>+</sup> *Zj*)*<sup>T</sup>*

*α*1/2(*Zi* + *Zj*)*<sup>T</sup>*

<sup>4</sup> (*CiS* <sup>−</sup> *DiYj* <sup>+</sup> *CjS* <sup>−</sup> *DjYi*)*<sup>T</sup>*

(*AiS* <sup>−</sup> *BiYj* <sup>+</sup> *AjS* <sup>−</sup> *BjYi*)*<sup>T</sup>*

*<sup>α</sup>*1/2(*CiS* <sup>−</sup> *DiYj* <sup>+</sup> *CjS* <sup>−</sup> *DjYi*)*<sup>T</sup>*

<sup>Ξ</sup><sup>22</sup> <sup>=</sup> <sup>−</sup>*γ<sup>I</sup>* <sup>+</sup> *<sup>β</sup>*

<sup>Ξ</sup><sup>23</sup> <sup>=</sup> <sup>1</sup> 2

<sup>Ξ</sup><sup>24</sup> <sup>=</sup> <sup>1</sup> 2

<sup>Ξ</sup><sup>15</sup> <sup>=</sup> <sup>1</sup> 2

(*Fi* + *Fj*)*<sup>T</sup>*

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Ξ<sup>11</sup> = *S*

(*Yi* + *Yj*)*<sup>T</sup>*

Ξ<sup>16</sup> = *SQT*/2

<sup>2</sup> (*Zi* <sup>+</sup> *Zj*)*<sup>T</sup>*

*α*1/2(*Fi* + *Fj*)*<sup>T</sup>*

*α*1/2(*Zi* + *Zj*)*<sup>T</sup>* (68)

(66)

≥ 0 (67)

*α*1/2(*Hij* + *Hji*)*<sup>T</sup>*

<sup>Ω</sup><sup>12</sup> <sup>=</sup> *<sup>β</sup>*

<sup>Ω</sup><sup>13</sup> <sup>=</sup> <sup>1</sup> 2

> <sup>Ω</sup><sup>15</sup> <sup>=</sup> <sup>1</sup> 2

<sup>Ω</sup><sup>23</sup> <sup>=</sup> <sup>1</sup> 2

By pre-multiplying and post-multiplying the matrix with the block diagonal matrix *diag*(*S*, *I*, *I*, *I*, *I*) , where *S* = *P*−1, and applying Schur complement again, the following LMI

> Ξ<sup>11</sup> Ξ<sup>12</sup> Ξ<sup>13</sup> Ξ<sup>14</sup> Ξ<sup>15</sup> Ξ<sup>16</sup> ∗ Ξ<sup>22</sup> Ξ<sup>23</sup> Ξ<sup>24</sup> 0 0 ∗ ∗ *S* 000 ∗∗∗ *I* 0 0 ∗∗∗∗ *<sup>R</sup>*−<sup>1</sup> <sup>0</sup> ∗∗∗∗ ∗ *I*

<sup>Ω</sup><sup>14</sup> <sup>=</sup> <sup>1</sup> 2

<sup>Ω</sup><sup>22</sup> <sup>=</sup> <sup>−</sup>*γ<sup>I</sup>* <sup>+</sup> *<sup>β</sup>*

<sup>Ω</sup><sup>24</sup> <sup>=</sup> <sup>1</sup> 2

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

<sup>Ξ</sup><sup>12</sup> <sup>=</sup> *<sup>β</sup>*

<sup>Ξ</sup><sup>13</sup> <sup>=</sup> <sup>1</sup> 2

<sup>Ξ</sup><sup>14</sup> <sup>=</sup> <sup>1</sup> 2 where

14 Fuzzy Controllers

⎤ ⎥ ⎥ ⎦ ≥ 0

> ⎤ ⎥ ⎥ <sup>⎦</sup> <sup>−</sup>

> > ≥ 0

*<sup>T</sup>* (63)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

≥ 0

(61)

(62)

(64)

(65)

*Υ*<sup>11</sup> *Υ*<sup>12</sup> (∑*<sup>i</sup>* ∑*<sup>j</sup> hihjGij*))*<sup>T</sup> α*1/2(∑*<sup>i</sup>* ∑*<sup>j</sup> hihjHij*)*<sup>T</sup>* <sup>∗</sup> *<sup>Υ</sup>*<sup>22</sup> (∑*<sup>i</sup> hiFi*)*<sup>T</sup> <sup>α</sup>*1/2(∑*<sup>i</sup> hiZi*)*<sup>T</sup>* ∗ ∗ *<sup>P</sup>*−<sup>1</sup> <sup>0</sup> ∗∗ ∗ *I*

Ψ<sup>11</sup> Ψ<sup>12</sup> (∑*<sup>i</sup>* ∑*<sup>j</sup> hihjGij*))*<sup>T</sup> α*1/2(∑*<sup>i</sup>* ∑*<sup>j</sup> hihjHij*)*<sup>T</sup>* <sup>∗</sup> <sup>Ψ</sup><sup>22</sup> (∑*<sup>i</sup> hiFi*)*<sup>T</sup> <sup>α</sup>*1/2(∑*<sup>i</sup> hiZi*)*<sup>T</sup>* ∗ ∗ *<sup>P</sup>*−<sup>1</sup> <sup>0</sup> ∗∗ ∗ *I*

> ⎤ ⎥ ⎥ ⎦

<sup>×</sup> *<sup>R</sup>* <sup>×</sup> �

Ψ<sup>11</sup> = *P* − *Q*

*i hiZi*]

*hihjHij*] *T*

(∑*<sup>i</sup> hiKi*) <sup>000</sup>�

By applying Schur complement again, we obtain

⎡ ⎢ ⎢ ⎣

Equivalently, the following inequality holds

⎡ ⎢ ⎢ ⎣

By applying Schur complement one more time, we have

By factoring out the ∑*<sup>i</sup>* ∑*<sup>j</sup> hi*(*ϕk*)*hj*(*ϕk*) term, we have

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ (∑*<sup>i</sup> hiKi*)*<sup>T</sup>* 0 0 0

<sup>Ψ</sup><sup>12</sup> <sup>=</sup> *<sup>β</sup>*

<sup>2</sup> [∑ *i* ∑ *j*

<sup>Ψ</sup><sup>22</sup> = −*γ<sup>I</sup>* + *<sup>β</sup>*[∑

Ψ<sup>11</sup> Ψ<sup>12</sup> (∑*<sup>i</sup>* ∑*<sup>j</sup> hihjGij*))*<sup>T</sup> α*1/2(∑*<sup>i</sup>* ∑*<sup>j</sup> hihjHij*)*<sup>T</sup>* (∑*<sup>i</sup> hiKi*)*<sup>T</sup>* <sup>∗</sup> <sup>Ψ</sup><sup>22</sup> (∑*<sup>i</sup> hiFi*)*<sup>T</sup> <sup>α</sup>*1/2(∑*<sup>i</sup> hiZi*)*<sup>T</sup>* <sup>0</sup> ∗ ∗ *<sup>P</sup>*−<sup>1</sup> 0 0 ∗∗ ∗ *I* 0 ∗∗ ∗ <sup>∗</sup> *<sup>R</sup>*−<sup>1</sup>

> Ω<sup>11</sup> Ω<sup>12</sup> Ω<sup>13</sup> Ω<sup>14</sup> Ω<sup>15</sup> ∗ Ω<sup>22</sup> Ω<sup>23</sup> Ω<sup>24</sup> 0 ∗ ∗ *<sup>P</sup>*−<sup>1</sup> 0 0 ∗∗∗ *I* 0 ∗∗∗∗ *<sup>R</sup>*−<sup>1</sup>

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ≥ 0

⎡ ⎢ ⎢ ⎣

where

$$\begin{aligned} \Omega\_{11} &= P - Q \\ \Omega\_{12} &= \frac{\beta}{4} [H\_{ji} + H\_{ij}]^T \\ \Omega\_{13} &= \frac{1}{2} (G\_{ji} + G\_{ij})^T \\ \Omega\_{14} &= \frac{1}{2} a^{1/2} (H\_{ij} + H\_{ji})^T \\ \Omega\_{15} &= \frac{1}{2} (K\_i + K\_j)^T \\ \Omega\_{22} &= -\gamma I + \frac{\beta}{2} (Z\_i + Z\_j)^T \\ \Omega\_{23} &= \frac{1}{2} (F\_i + F\_j)^T \\ \Omega\_{24} &= \frac{1}{2} a^{1/2} (Z\_i + Z\_j)^T \end{aligned} \tag{66}$$

By pre-multiplying and post-multiplying the matrix with the block diagonal matrix *diag*(*S*, *I*, *I*, *I*, *I*) , where *S* = *P*−1, and applying Schur complement again, the following LMI result is obtained

$$
\begin{bmatrix}
\Xi\_{11} \ \Xi\_{12} \ \Sigma\_{13} \ \Sigma\_{14} \ \Xi\_{15} \ \Xi\_{16} \\
\* & \Xi\_{22} \ \Xi\_{23} \ \Sigma\_{24} \ 0 & 0 \\
\* & \* & S & 0 & 0 \\
\* & \* & \* & I & 0 & 0 \\
\* & \* & \* & \* & R^{-1} & 0 \\
\* & \* & \* & \* & \* & I
\end{bmatrix} \geq 0 \tag{67}
$$

where

$$\Xi\_{11} = S$$

$$\Xi\_{12} = \frac{\beta}{4}(\mathbf{C}\_i S - D\_i Y\_j + \mathbf{C}\_j S - D\_j Y\_i)^T$$

$$\Xi\_{13} = \frac{1}{2}(A\_i S - B\_i Y\_j + A\_j S - B\_j Y\_i)^T$$

$$\Xi\_{14} = \frac{1}{2}\mathbf{a}^{1/2}(\mathbf{C}\_i S - D\_i Y\_j + C\_j S - D\_j Y\_i)^T$$

$$\Xi\_{15} = \frac{1}{2}(Y\_i + Y\_j)^T$$

$$\Xi\_{16} = S \mathbf{Q}^T / 2$$

$$\Xi\_{22} = -\gamma I + \frac{\beta}{2}(Z\_i + Z\_j)^T$$

$$\Xi\_{23} = \frac{1}{2}\mathbf{a}^{1/2}(F\_i + F\_j)^T$$

$$\Xi\_{24} = \frac{1}{2}\mathbf{a}^{1/2}(Z\_i + Z\_j)^T\tag{68}$$

where *S*(*k*) = *P*−1(*k*), then (28) is satisfied with the feedback control gain being found by

$$K(k) = Y(k)P(k)\tag{69}$$

where

A<sup>1</sup> =

The following values are used in our simulation:

**Figure 1.** Membership functions of Rule 1 and Rule 2.

**Figure 2.** Angle trajectory of the inverted pendulum.

following design parameters are chosen to satisfy:

Mixed *NLQR* − H<sup>∞</sup> criteria:

Mixed *NLQR* − *passivity* criteria:

A<sup>2</sup> =

and Rule 2 is shown below in Fig.1.

 1 *T gT* <sup>4</sup>*L*/3−*amL* <sup>1</sup>

 1 *T* 2*gT <sup>π</sup>*(4*L*/3−*amLδ*<sup>2</sup>) <sup>1</sup>  B<sup>1</sup> =

 B<sup>2</sup> =

*M* = 8*kg*, *m* = 2*kg*, *L* = 0.5*m*, *g* = 9.8*m*/*s*

and the initial condition of *x*1(0) = *π*/6, *x*2(0) = −*π*/6. The membership function of Rule 1

The feedback control gain can be found from (31)(51) by solving the LMI at each time. The

*C* = [1 1], *D* = [1], *Q* = *diag*[1001], *R* = 1, *α* = 1, *β* = 0, *γ* = −5

*C* = [1 1], *D* = [1], *Q* = *diag*[1001], *R* = 1, *α* = 1, *β* = 5, *γ* = 0

 0 <sup>−</sup> *aT* 4*L*/3−*amL*

 <sup>0</sup> <sup>−</sup> *<sup>a</sup>δ<sup>T</sup>* 4*L*/3−*amLδ*<sup>2</sup>  F<sup>1</sup> = *�*1*T �*2*T* 

*with δ* = *cos*(80*o*), *Sampling time T* = 0.001 (72)

F<sup>2</sup> = *�*1*T �*2*T* 

Fuzzy Control of Nonlinear Systems with General Performance Criteria 135

2, *�*<sup>1</sup> = 1, *�*<sup>2</sup> = 0

### **6. Application to the inverted pendulum system**

The inverted pendulum on a cart problem is a benchmark control problem used widely to test control algorithms. A pendulum beam attached at one end can rotate freely in the vertical 2-dimensional plane. The angle of the beam with respect to the vertical direction is denoted at angle *<sup>θ</sup>*. The external force *<sup>u</sup>* is desired to set angle of the beam *<sup>θ</sup>* (*x*1) and angular velocity ˙ *θ* (*x*2) to zero while satisfying the mixed performance criteria. A model of the inverted pendulum on a cart problem is given by [1, 9]:

$$\dot{\mathbf{x}}\_1 = \mathbf{x}\_2 + \varepsilon\_1 w$$

$$\dot{\mathbf{x}}\_2 = \frac{g \sin(\mathbf{x}\_1) - amL x\_2^2 \sin(2\mathbf{x}\_1)/2 - ac \cos(\mathbf{x}\_1)u}{4L/3 - amL \cos^2(\mathbf{x}\_1)} + \varepsilon\_2 w \tag{70}$$

where *x*<sup>1</sup> is the angle of the pendulum from vertical direction, *x*<sup>2</sup> is the angular velocity of the pendulum, *g* is the gravity constant, *m* is the mass of the pendulum, *M* is the mass of the cart, *L* is the length of the center of mass (the entire length of the pendulum beam equals 2*L*), *u* is the external force, control input to the system, *<sup>w</sup>* is the <sup>L</sup><sup>2</sup> type of disturbance, *<sup>a</sup>* <sup>=</sup> <sup>1</sup> *<sup>m</sup>*+*<sup>M</sup>* is a constant, and *�*1.*�*<sup>2</sup> is the weighing coefficients of disturbance.

Due to the system non-linearity, we approximate the system using the following two-rule fuzzy model:

#### **continuous-time fuzzy model**

*Rule 1:* If |*x*1(*t*)| is close to zero, Then *x*˙(*t*) = *A*1*x*(*t*) + *B*1*u*(*t*) + *F*1*w*(*t*)

*Rule 2:* If |*x*1(*t*)| is close to *π*/2, Then *x*˙(*t*) = *A*2*x*(*t*) + *B*2*u*(*t*) + *F*2*w*(*t*)

where

$$A\_{1} = \begin{bmatrix} 0 & 1\\ \frac{\delta}{4L/3 - amL} & 0 \end{bmatrix} B\_{1} = \begin{bmatrix} 0\\ -\frac{a}{4L/3 - amL} \end{bmatrix} F\_{1} = \begin{bmatrix} \varepsilon\_{1} \\ \varepsilon\_{2} \end{bmatrix}$$

$$A\_{2} = \begin{bmatrix} 0 & 1\\ \frac{2g}{\pi(4L/3 - amL\delta^{2})} & 0 \end{bmatrix} B\_{1} = \begin{bmatrix} 0 \\ -\frac{a\delta}{4L/3 - amL\delta^{2}} \end{bmatrix} F\_{1} = \begin{bmatrix} \varepsilon\_{1} \\ \varepsilon\_{2} \end{bmatrix} with \,\delta = \cos(80^{\circ})\tag{71}$$

#### **discrete-time fuzzy model**

*Rule 1:* If |*x*1(*k*)| is close to zero, Then *x*(*k* + 1) = A1*x*(*k*) + B1*u*(*k*) + F1*w*(*k*) *Rule 2:* If |*x*1(*k*)| is close to *π*/2,

Then *x*(*k* + 1) = A2*x*(*k*) + B2*u*(*k*) + F2*w*(*k*)

where

16 Fuzzy Controllers

The inverted pendulum on a cart problem is a benchmark control problem used widely to test control algorithms. A pendulum beam attached at one end can rotate freely in the vertical 2-dimensional plane. The angle of the beam with respect to the vertical direction is denoted at angle *<sup>θ</sup>*. The external force *<sup>u</sup>* is desired to set angle of the beam *<sup>θ</sup>* (*x*1) and angular velocity ˙ *θ* (*x*2) to zero while satisfying the mixed performance criteria. A model of the inverted

where *x*<sup>1</sup> is the angle of the pendulum from vertical direction, *x*<sup>2</sup> is the angular velocity of the pendulum, *g* is the gravity constant, *m* is the mass of the pendulum, *M* is the mass of the cart, *L* is the length of the center of mass (the entire length of the pendulum beam equals 2*L*), *u* is the external force, control input to the system, *<sup>w</sup>* is the <sup>L</sup><sup>2</sup> type of disturbance, *<sup>a</sup>* <sup>=</sup> <sup>1</sup> *<sup>m</sup>*+*<sup>M</sup>* is a

Due to the system non-linearity, we approximate the system using the following two-rule

**continuous-time fuzzy model** *Rule 1:* If |*x*1(*t*)| is close to zero, Then *x*˙(*t*) = *A*1*x*(*t*) + *B*1*u*(*t*) + *F*1*w*(*t*) *Rule 2:* If |*x*1(*t*)| is close to *π*/2, Then *x*˙(*t*) = *A*2*x*(*t*) + *B*2*u*(*t*) + *F*2*w*(*t*)

> 0 1 *g* <sup>4</sup>*L*/3−*amL* <sup>0</sup>

**discrete-time fuzzy model** *Rule 1:* If |*x*1(*k*)| is close to zero, Then *x*(*k* + 1) = A1*x*(*k*) + B1*u*(*k*) + F1*w*(*k*) *Rule 2:* If |*x*1(*k*)| is close to *π*/2, Then *x*(*k* + 1) = A2*x*(*k*) + B2*u*(*k*) + F2*w*(*k*)

 <sup>0</sup> <sup>−</sup> *<sup>a</sup><sup>δ</sup>* 4*L*/3−*amLδ*<sup>2</sup>

 *B*<sup>1</sup> =

> *F*<sup>1</sup> = *�*1 *�*2

 0 <sup>−</sup> *<sup>a</sup>* <sup>4</sup>*L*/3−*amL*  *F*<sup>1</sup> = *�*1 *�*2 

*with δ* = *cos*(80*o*) (71)

<sup>2</sup>*sin*(2*x*1)/2 − *acos*(*x*1)*u*

*K*(*k*) = *Y*(*k*)*P*(*k*) (69)

*x*˙1 = *x*<sup>2</sup> + *�*1*w*

<sup>4</sup>*L*/3 <sup>−</sup> *amLcos*2(*x*1) <sup>+</sup> *�*2*<sup>w</sup>* (70)

where *S*(*k*) = *P*−1(*k*), then (28) is satisfied with the feedback control gain being found by

**6. Application to the inverted pendulum system**

*<sup>x</sup>*˙2 <sup>=</sup> *gsin*(*x*1) <sup>−</sup> *amLx*<sup>2</sup>

constant, and *�*1.*�*<sup>2</sup> is the weighing coefficients of disturbance.

*A*<sup>1</sup> =

 *B*<sup>1</sup> =

pendulum on a cart problem is given by [1, 9]:

fuzzy model:

where

*A*<sup>2</sup> =

 0 1 2*g <sup>π</sup>*(4*L*/3−*amLδ*<sup>2</sup>) <sup>0</sup>

$$\begin{aligned} \mathcal{A}\_{1} &= \begin{bmatrix} 1 & T \\ \frac{\delta T}{4L/3 - amL} & 1 \end{bmatrix} \mathcal{B}\_{1} = \begin{bmatrix} 0 \\ -\frac{aT}{4L/3 - amL} \end{bmatrix} \mathcal{F}\_{1} = \begin{bmatrix} \varepsilon\_{1}T \\ \varepsilon\_{2}T \end{bmatrix} \\ \mathcal{A}\_{2} &= \begin{bmatrix} 1 & T \\ \frac{2gT}{\pi(4L/3 - amL\delta^{2})} & 1 \end{bmatrix} \mathcal{B}\_{2} = \begin{bmatrix} 0 \\ -\frac{a\delta T}{4L/3 - amL\delta^{2}} \end{bmatrix} \mathcal{F}\_{2} = \begin{bmatrix} \varepsilon\_{1}T \\ \varepsilon\_{2}T \end{bmatrix} \end{aligned} \tag{72}$$
 
$$with \ \delta = \cos(80^{\circ}) \text{, sampling time } T = 0.001 \tag{72}$$

The following values are used in our simulation:

$$M = 8 \\ \text{kg}, m = 2 \\ \text{kg}, \text{L} = 0.5 \\ \text{m}, \text{g} = 9.8 \\ \text{m}/\text{s}^2, \text{e}\_1 = 1, \text{e}\_2 = 0$$

and the initial condition of *x*1(0) = *π*/6, *x*2(0) = −*π*/6. The membership function of Rule 1 and Rule 2 is shown below in Fig.1.

**Figure 1.** Membership functions of Rule 1 and Rule 2.

**Figure 2.** Angle trajectory of the inverted pendulum.

The feedback control gain can be found from (31)(51) by solving the LMI at each time. The following design parameters are chosen to satisfy:

Mixed *NLQR* − H<sup>∞</sup> criteria:

$$\mathcal{C} = \begin{bmatrix} 1 & 1 \end{bmatrix}, \mathcal{D} = [1], \mathcal{Q} = \text{diag}\{1001\}, \mathcal{R} = 1, \mathfrak{a} = 1, \mathfrak{b} = 0, \gamma = -\mathfrak{b}$$

Mixed *NLQR* − *passivity* criteria:

$$\mathcal{C} = \begin{bmatrix} 1 & 1 \end{bmatrix}, \mathcal{D} = \begin{bmatrix} 1 \end{bmatrix}, \mathcal{Q} = \operatorname{diag} \left[ 1001 \right], \mathcal{R} = 1, \mathfrak{a} = 1, \mathcal{\mathfrak{b}} = \mathfrak{5}, \gamma = 0$$

simulation studies show that the proposed method provides a satisfactory alternative to the

Fuzzy Control of Nonlinear Systems with General Performance Criteria 137

*Oregon Institute of Technology, Department of Electrical and Renewable Energy Engineering, Klamath*

*Marquette University, Department of Electrical and Computer Engineering, Haggerty Hall of*

*Oregon Institute of Technology, Department of Computer Systems Engineering Technology, Klamath*

[1] Baumann W.T, Rugh W.J (1986) Feedback Control of Non-linear Systems by Extended

[2] Basar T and Bernhard P (1995) H-infinity Optimal Control and Related Minimax Design

[3] Boyd S, Ghaoui L E, Feron E, Balakrishnan V (1994) Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, SIAM, Philadelphia. [4] Huang Y, Lu W-M (1996) Non-linear Optimal Control: Alternatives to Hamilton-Jacobi Equation, Proc. of 35th Conf. on Decision and Control, Kobe, Japan, pp. 3942-3947.

[6] Mohseni J, Yaz E, Olejniczak K (1998) State Dependent LMI Control of Discrete-Time Non-linear Systems, Proc. of the 37th IEEE Conference on Decision and Control, Tampa,

[7] Takagi T, Sugeno M (1985) Fuzzy Identification of Systems and Its Applications to Model

[8] Tanaka K, Sugeno M (1990) Stability Analysis of Fuzzy Systems Using Lyapunov's

[9] Tanaka K, Ikeda T, Wang H.O (1996) Design of Fuzzy Control Systems Based on Relaxed LMI Stability Conditions, the 35th IEEE Conference on Decision and Control, Kobe,

[10] Tanaka K, Wang H.O (2001) Fuzzy Control Systems Design and Analysis, A Linear

[11] Van der Shaft A.J (1993) Non-linear State Space H1 control Theory, in Perspectives in

[13] Wang L.X (1994) Adaptive Fuzzy Systems and Control: Design and Stability Analysis,

Linearization. IEEE Trans. Automatic Control. Vol. AC-31, No.1, pp.40-46.

Problems, A Dynamic Game Approach, 2nd Ed.,Birkhauser, 1995.

[5] Khalil H.K (2002) Non-linear Systems, 3rd Ed., Prentice Hall, N.J.

and Control, IEEE Trans. Syst. Man. Cyber., Vol. 15, pp.116-132.

control, H. J. Trentelman and J. C. Willems, Eds. Birkhauser. [12] Vidyasagar M (2002) Non-linear System Analysis, 2nd Ed., SIAM.

Direct Method, Proc. NAFIPS90, pp. 133-136.

Matrix Inequality Approach, Wiley.

Prentice Hall, Englewood Cliffs, NJ.

existing non-linear control approaches.

*Engineering, Milwaukee, Wisconsin, USA*

*Green Lite Motors Corporation, Portland, OR, USA*

**Author details**

*Falls, Oregon, USA*

*Falls, Oregon, USA*

**8. References**

FL, pp. 4626-4627.

Vol.1, pp. 598-603.

Edwin E. Yaz

James Long

Tim Miller

Xin Wang

**Figure 3.** Angular velocity trajectory of the inverted pendulum.

**Figure 4.** Control input applied to the inverted pendulum.

The mixed criteria control performance results are shown in the Figs.2-4. From these figures, we find that the novel fuzzy LMI control has satisfactory performance. The mixed *NLQR* − *H*∞ criteria control has a smaller overshoot and a faster response than the one with passivity property. The new technique controls the inverted pendulum very well under the effect of finite energy disturbance. It should also be noted that the LMI fuzzy control with mixed performance criteria satisfies global asymptotic stability.

#### **7. Summary**

This chapter presents a novel fuzzy control approach for both of continuous time and discrete time non-linear systems based on the LMI solutions. The Takagi-Sugeno fuzzy model is applied to decompose the non-linear system. Multiple performance criteria are used to design the controller and the relative weighting matrices of these criteria can be achieved by choosing different coefficient matrices. The optimal control can be obtained by solving LMI at each time. The inverted pendulum is used as an example to demonstrate its effectiveness. The simulation studies show that the proposed method provides a satisfactory alternative to the existing non-linear control approaches.

## **Author details**

#### Xin Wang

18 Fuzzy Controllers

The mixed criteria control performance results are shown in the Figs.2-4. From these figures, we find that the novel fuzzy LMI control has satisfactory performance. The mixed *NLQR* − *H*∞ criteria control has a smaller overshoot and a faster response than the one with passivity property. The new technique controls the inverted pendulum very well under the effect of finite energy disturbance. It should also be noted that the LMI fuzzy control with mixed

This chapter presents a novel fuzzy control approach for both of continuous time and discrete time non-linear systems based on the LMI solutions. The Takagi-Sugeno fuzzy model is applied to decompose the non-linear system. Multiple performance criteria are used to design the controller and the relative weighting matrices of these criteria can be achieved by choosing different coefficient matrices. The optimal control can be obtained by solving LMI at each time. The inverted pendulum is used as an example to demonstrate its effectiveness. The

**Figure 3.** Angular velocity trajectory of the inverted pendulum.

**Figure 4.** Control input applied to the inverted pendulum.

performance criteria satisfies global asymptotic stability.

**7. Summary**

*Oregon Institute of Technology, Department of Electrical and Renewable Energy Engineering, Klamath Falls, Oregon, USA*

Edwin E. Yaz

*Marquette University, Department of Electrical and Computer Engineering, Haggerty Hall of Engineering, Milwaukee, Wisconsin, USA*

James Long

*Oregon Institute of Technology, Department of Computer Systems Engineering Technology, Klamath Falls, Oregon, USA*

Tim Miller

*Green Lite Motors Corporation, Portland, OR, USA*

## **8. References**

	- [14] Wang H.O, Tanaka K, Griffin M (1996) An Approach to Fuzzy Control of Non-linear Systems: Stability and Design Issues, IEEE Trans. Fuzzy Syst., Vol. 4, No. 1, pp.14-23.
	- [15] Wang X, Yaz E.E (2009) The State Dependent Control of Continuous-Time Non-linear Systems with Mixed Performance Criteria, Proc. of IASTED Int. Conf. on Identi cation Control and Applications, Honolulu, HI, pp. 98-102.

**A New Method for Tuning PID-Type Fuzzy**

S. Bouallègue, J. Haggège and M. Benrejeb

http://dx.doi.org/10.5772/47139

robustness and effectiveness properties.

to tune the PID-type fuzzy controller parameters.

cited.

**1. Introduction**

Additional information is available at the end of the chapter

**Controllers Using Particle Swarm Optimization**

**Chapter 6**

The complexity of dynamic system, especially when only qualitative knowledge about the process is available, makes it generally difficult to elaborate an analytic model which is sufficiently precise enough for the control. Thus, it is interesting to use, for this kind of systems, non conventional control techniques, such as fuzzy logic, in order to achieve high performances and robustness [8, 15, 20–22, 24, 33, 34]. Fuzzy logic control approach has been widely used in many successful industrial applications which have demonstrated high

In the literature, various Fuzzy Controller (FC) structures are proposed and extensively studied. The particular structure given by Qiao and Mizumoto in [26], namely PID-type FC, is especially established and improved within the practical framework in [11, 16, 31]. Such a FC structure, which retains the characteristics similar to the conventional PID controller, can be decomposed into the equivalent proportional, integral and derivative control components as shown in [26]. In order to improve further the performance of the transient and steady state responses of this kind of fuzzy controller, various strategies and methods are proposed

Indeed, Qiao and Mizumoto [26] designed a parameter adaptive PID-type FC based on a peak observer mechanism. This self-tuning mechanism decreases the equivalent integral control component of the fuzzy controller gradually with the system response process time. On the other hand, Woo et al. [31] developed a method to tune the scaling factors related to integral and derivative components of the PID-type FC structure via two empirical functions and based on the system's error information. In [12, 16], the authors proposed a technique that adjusts the scaling factors, corresponding to the derivative and integral components of the PID-type FC, using a fuzzy inference mechanism. However, the major drawback of all these PID-type FC structures is the difficult choice of their relative scaling factors. Indeed, the fuzzy controller dynamic behaviour depends on this adequate choice. The tuning procedure depends on the control experience and knowledge of the human operator, and it is generally

and reproduction in any medium, provided the original work is properly cited.

©2012 Bouallègue et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,


## **A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization**

S. Bouallègue, J. Haggège and M. Benrejeb

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/47139

## **1. Introduction**

20 Fuzzy Controllers

[14] Wang H.O, Tanaka K, Griffin M (1996) An Approach to Fuzzy Control of Non-linear Systems: Stability and Design Issues, IEEE Trans. Fuzzy Syst., Vol. 4, No. 1, pp.14-23. [15] Wang X, Yaz E.E (2009) The State Dependent Control of Continuous-Time Non-linear Systems with Mixed Performance Criteria, Proc. of IASTED Int. Conf. on Identi

[16] Wang X, Yaz E.E (2010) Robust multi-criteria optimal fuzzy control of continuous-time non-linear systems, Proc. of the 2010 American Control Conference, Baltimore, MD,

[17] Wang X, Yaz E.E, Jeong C.S (2010) Robust non-linear feedback control of discrete-time non-linear systems with mixed performance criteria, Proc. of the 2010 American Control

[18] Wang X, Yaz E.E (2010) Robust multi-criteria optimal fuzzy control of discrete-time non-linear systems, Proc. of the 49th IEEE Conference on Decision and Control, Atlanta,

[19] Wang X, Yaz E.E, Yaz Y.I (2010) Robust and resilient state dependent control of continuous-time non-linear systems with general performance criteria, Proc. of the 49th

IEEE Conference on Decision and Control, Atlanta, Georgia, USA, pp. 603-608. [20] Wang X, Yaz E.E, Yaz Y.I, Robust and resilient state dependent control of discrete time non-linear systems with general performance criteria, Proc. of the 18th IFAC Congress,

cation Control and Applications, Honolulu, HI, pp. 98-102.

Conference, Baltimore, MD, USA,pp. 6357-6362.

USA, pp. 6460-6465.

Georgia, USA, pp. 4269-4274.

Milano, Italy, pp. 10904-10909.

The complexity of dynamic system, especially when only qualitative knowledge about the process is available, makes it generally difficult to elaborate an analytic model which is sufficiently precise enough for the control. Thus, it is interesting to use, for this kind of systems, non conventional control techniques, such as fuzzy logic, in order to achieve high performances and robustness [8, 15, 20–22, 24, 33, 34]. Fuzzy logic control approach has been widely used in many successful industrial applications which have demonstrated high robustness and effectiveness properties.

In the literature, various Fuzzy Controller (FC) structures are proposed and extensively studied. The particular structure given by Qiao and Mizumoto in [26], namely PID-type FC, is especially established and improved within the practical framework in [11, 16, 31]. Such a FC structure, which retains the characteristics similar to the conventional PID controller, can be decomposed into the equivalent proportional, integral and derivative control components as shown in [26]. In order to improve further the performance of the transient and steady state responses of this kind of fuzzy controller, various strategies and methods are proposed to tune the PID-type fuzzy controller parameters.

Indeed, Qiao and Mizumoto [26] designed a parameter adaptive PID-type FC based on a peak observer mechanism. This self-tuning mechanism decreases the equivalent integral control component of the fuzzy controller gradually with the system response process time. On the other hand, Woo et al. [31] developed a method to tune the scaling factors related to integral and derivative components of the PID-type FC structure via two empirical functions and based on the system's error information. In [12, 16], the authors proposed a technique that adjusts the scaling factors, corresponding to the derivative and integral components of the PID-type FC, using a fuzzy inference mechanism. However, the major drawback of all these PID-type FC structures is the difficult choice of their relative scaling factors. Indeed, the fuzzy controller dynamic behaviour depends on this adequate choice. The tuning procedure depends on the control experience and knowledge of the human operator, and it is generally

©2012 Bouallègue et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2 Will-be-set-by-IN-TECH 140 Fuzzy Controllers – Recent Advances in Theory and Applications A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization <sup>3</sup>

achieved based on a classical trials-errors procedure. Up to now, there is neither clear mor systematic method to guide such a choice. So, this tuning problem becomes more delicate and harder as the complexity of the controlled plant increases. Hence, the proposition of a systematic approach to tune the scaling factors of these particular PID-type FC structures is interesting.

*α* and *β* denote the scaling factors associated to the inputs and output of the FC, as shown in

When approximating the integral and derivative terms within the discrete-time framework, we can consider the closed-loop control structure for a discrete-time PID-type FC, as shown

α

A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization 141

β

*Kdk* <sup>=</sup> *Kd*0<sup>Ψ</sup> (*ek*) (1)

Fuzzy Controller

As shown in [11, 16, 26, 31], the dynamic behaviour of this PID-type FC structure is strongly

In order to improve the performances of the considered PID-type FC structure, various self-tuning mechanisms for scaling factors have been proposed in the literature. Two methods

In this self-tuning method [31], the PID-type FC integral and derivative components updating are achieved based on scaling factors *β* and *Kd*, using the information on system's error as

*β<sup>k</sup>* = *β*0Φ (*ek*)

where *β*<sup>0</sup> and *Kd*<sup>0</sup> are the initial values of *β* and *Kd*, respectively, Φ (.) and Ψ (.) are the

In these equations, the parameters to be tuned *φ*1, *φ*2, *ψ*<sup>1</sup> and *ψ*<sup>2</sup> are all positive. The empirical function related to integral component decreases as the error decreases while the function related to derivative factor increases. Indeed, the objective of the function is to decrease the parameter with the change of error. However, the function has an inverse objective to make constant the proportional effect. Hence, the system may not always keep quick reaction

Φ (*ek*) = *φ*<sup>1</sup> |*ek*| + *φ*<sup>2</sup> Ψ (*ek*) = *ψ*<sup>1</sup> (1 − |*ek*|) + *ψ*<sup>2</sup>

dependent on the scaling factors *Ke*, *Kd*, *α* and *β*, difficult and delicate to tune.

Delay Operator z 1

+ +

*uk*

(2)

+ +

Figure 1. The proof of this computation is shown with more details in [26].

*K*

*e*

*K*

<sup>Δ</sup>*ek*

are especially adopted in this chapter.

+ \_

*d*

**Figure 1.** The proposed discrete-time PID-type FC structure.

**2.2. PID-type FC with self-tuning mechanisms**

*2.2.1. Self-tuning via Empirical Functions Tuner Method EFTM*

empirical tuner functions defined, respectively, by:

against the error as demonstrated by Woo et al. in [31].

in Figure 1.

*ek*

Delay operator z 1

follows:

In this study, a new approach based on the Particle Swarm Optimization (PSO) meta-heuristic technique is proposed for systematically tuning the scaling factors of the PID-type FC, both with and without self-tuning mechanisms. This work can be considered as an extension of the results given in [11, 12, 16, 26, 31]. The fuzzy control design is formulated as a constrained optimization problem which is efficiently solved based on a developed PSO algorithm. In order to specify more robustness and performance control objectives of the proposed PSO-tuned PID-type FC, different optimization criteria are considered and compared subject to several various control constraints defined in the time-domain framework.

The remainder of this chapter is organized as follows. In Section 2, the proposed fuzzy PID-type FC structures, both with and without self-tuning scaling factors mechanisms, are presented and discussed within the discrete-time framework. Two adaptive mechanisms for scaling factors tuning are especially adopted. The optimization-based problems of the PID-type FC scaling factors tuning are formulated in Section 3. The developed constrained PSO algorithm, used in solving the formulated problems, is also described. An external static penalty technique is used to deal with optimization constraints. Theoretical conditions for convergence algorithm and parameters choice are established, based on the stability theory of dynamic systems. Section 4 is dedicated to apply the proposed fuzzy control approaches on an electrical DC drive benchmark and a thermal process within an experimental real-time framework based on an Advantech PCI-1710 multi-functions board associated with a PC computer and MATLAB/Simulink environment. Performances on convergence properties of the proposed PSO and the used GAO algorithm, are compared for the known Integral Absolute Error (IAE) and the Integral Square Error (ISE) criterion cases. The real-time fuzzy controllers are developed through the compilation and linking stage, in a form of a Dynamic Link Library (DLL) which is, then, loaded in memory and started-up.

## **2. PID-type fuzzy control design**

In this section, the considered PID-type FC structures are briefly described within the discrete-time framework based on [11, 12, 16, 26, 31].

## **2.1. Discrete-time PID-type FLC**

Proposed by Qiao and Mizumoto in [26] within continuous-time formalism, this particular fuzzy controller structure, called PID-type FC, retains the characteristics similar to the conventional PID controller. This result remains valid while using a type of FC with triangular and uniformly distributed membership functions for the fuzzy inputs and a crisp output, a product-sum inference and a center of gravity defuzzification methods.

Under these conditions, the equivalent proportional, integral and derivative control components of such a PID-type FC are given by *αKe*P + *βKd*D, *βKe*P, and *αKd*D, respectively, as shown in [16, 26, 31]. In these expressions, P and D represent relative coefficients, *Ke*, *Kd*,

*α* and *β* denote the scaling factors associated to the inputs and output of the FC, as shown in Figure 1. The proof of this computation is shown with more details in [26].

When approximating the integral and derivative terms within the discrete-time framework, we can consider the closed-loop control structure for a discrete-time PID-type FC, as shown in Figure 1.

**Figure 1.** The proposed discrete-time PID-type FC structure.

2 Will-be-set-by-IN-TECH

achieved based on a classical trials-errors procedure. Up to now, there is neither clear mor systematic method to guide such a choice. So, this tuning problem becomes more delicate and harder as the complexity of the controlled plant increases. Hence, the proposition of a systematic approach to tune the scaling factors of these particular PID-type FC structures is

In this study, a new approach based on the Particle Swarm Optimization (PSO) meta-heuristic technique is proposed for systematically tuning the scaling factors of the PID-type FC, both with and without self-tuning mechanisms. This work can be considered as an extension of the results given in [11, 12, 16, 26, 31]. The fuzzy control design is formulated as a constrained optimization problem which is efficiently solved based on a developed PSO algorithm. In order to specify more robustness and performance control objectives of the proposed PSO-tuned PID-type FC, different optimization criteria are considered and compared subject

The remainder of this chapter is organized as follows. In Section 2, the proposed fuzzy PID-type FC structures, both with and without self-tuning scaling factors mechanisms, are presented and discussed within the discrete-time framework. Two adaptive mechanisms for scaling factors tuning are especially adopted. The optimization-based problems of the PID-type FC scaling factors tuning are formulated in Section 3. The developed constrained PSO algorithm, used in solving the formulated problems, is also described. An external static penalty technique is used to deal with optimization constraints. Theoretical conditions for convergence algorithm and parameters choice are established, based on the stability theory of dynamic systems. Section 4 is dedicated to apply the proposed fuzzy control approaches on an electrical DC drive benchmark and a thermal process within an experimental real-time framework based on an Advantech PCI-1710 multi-functions board associated with a PC computer and MATLAB/Simulink environment. Performances on convergence properties of the proposed PSO and the used GAO algorithm, are compared for the known Integral Absolute Error (IAE) and the Integral Square Error (ISE) criterion cases. The real-time fuzzy controllers are developed through the compilation and linking stage, in a form of a Dynamic

In this section, the considered PID-type FC structures are briefly described within the

Proposed by Qiao and Mizumoto in [26] within continuous-time formalism, this particular fuzzy controller structure, called PID-type FC, retains the characteristics similar to the conventional PID controller. This result remains valid while using a type of FC with triangular and uniformly distributed membership functions for the fuzzy inputs and a crisp output, a

Under these conditions, the equivalent proportional, integral and derivative control components of such a PID-type FC are given by *αKe*P + *βKd*D, *βKe*P, and *αKd*D, respectively, as shown in [16, 26, 31]. In these expressions, P and D represent relative coefficients, *Ke*, *Kd*,

to several various control constraints defined in the time-domain framework.

Link Library (DLL) which is, then, loaded in memory and started-up.

product-sum inference and a center of gravity defuzzification methods.

**2. PID-type fuzzy control design**

**2.1. Discrete-time PID-type FLC**

discrete-time framework based on [11, 12, 16, 26, 31].

interesting.

As shown in [11, 16, 26, 31], the dynamic behaviour of this PID-type FC structure is strongly dependent on the scaling factors *Ke*, *Kd*, *α* and *β*, difficult and delicate to tune.

#### **2.2. PID-type FC with self-tuning mechanisms**

In order to improve the performances of the considered PID-type FC structure, various self-tuning mechanisms for scaling factors have been proposed in the literature. Two methods are especially adopted in this chapter.

#### *2.2.1. Self-tuning via Empirical Functions Tuner Method EFTM*

In this self-tuning method [31], the PID-type FC integral and derivative components updating are achieved based on scaling factors *β* and *Kd*, using the information on system's error as follows:

$$\begin{array}{l} \beta\_k = \beta\_0 \Phi \left( e\_k \right) \\ K\_{dk} = K\_{d0} \Psi \left( e\_k \right) \end{array} \tag{1}$$

where *β*<sup>0</sup> and *Kd*<sup>0</sup> are the initial values of *β* and *Kd*, respectively, Φ (.) and Ψ (.) are the empirical tuner functions defined, respectively, by:

$$\begin{array}{l}\Phi\left(e\_{k}\right) = \phi\_{1}\left|e\_{k}\right| + \phi\_{2} \\ \Psi\left(e\_{k}\right) = \psi\_{1}\left(1 - \left|e\_{k}\right|\right) + \psi\_{2} \end{array} \tag{2}$$

In these equations, the parameters to be tuned *φ*1, *φ*2, *ψ*<sup>1</sup> and *ψ*<sup>2</sup> are all positive. The empirical function related to integral component decreases as the error decreases while the function related to derivative factor increases. Indeed, the objective of the function is to decrease the parameter with the change of error. However, the function has an inverse objective to make constant the proportional effect. Hence, the system may not always keep quick reaction against the error as demonstrated by Woo et al. in [31].

#### *2.2.2. Self-tuning via Relative Rate Observer Method RROM*

In this self-tuning method [12, 16], the PID-type FC integral and derivative components updating are achieved as follows:

$$\begin{array}{l} \beta\_k = \frac{\beta\_0}{K\_f \delta\_k} \\ K\_{dk} = K\_{d0} K\_{fd} K\_f \delta\_k \end{array} \tag{3}$$

For this RROM self-tuning approach, the uniformly distributed triangular and the symmetrical membership functions, as shown in Figures 2, 3, 4, are assigned for the fuzzy inputs *rk* and |*ek*|, and fuzzy output variable *δk*. The view of the above fuzzy rule-base is

A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization 143

S M F


S SM M L

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

S SM M L

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fuzzy output

*δk*

Fuzzy input | |

*e k*

Fuzzy input

*r k*

illustrated in Figure 5.

0

1

0

0

0.2

0.4

Degree of membership

0.6

0.8

1

0.2 0.4 0.6 0.8

Degree of membership

**Figure 2.** Membership functions for *rk*.

**Figure 3.** Membership functions for |*ek*|.

**Figure 4.** Membership functions for *δk*.

0.2

0.4

Degree of membership

0.6

0.8

1

where *δ<sup>k</sup>* is the output of the fuzzy Relative Rate Observer (RRO) *Kf* is the output scaling factor for *δ<sup>k</sup>* and *Kf d* is the additional parameter that affects only the derivative factor of the FC.

The rule-base for *δk*, as used by Eksin et al. [12] and Güzelkaya et al. [16], is considered for the fuzzy RRO. This fuzzy RRO block has as inputs the absolute values of error |*ek*| and the variable *rk*, defined subsequently, as shown in Table 1.


**Table 1.** Fuzzy rule-base for the variable *δk*.

The linguistic levels assigned to the input |*ek*| and the output variable *δ<sup>k</sup>* are as follows: L (Large), M (Medium), SM (Small Medium) and S (Small). For the input variable *rk*, the following linguistic levels are assigned: F (Fast), M (Moderate) and S (Slow).

The variable *rk*, defined in [12, 16] and called normalized acceleration, gives "relative rate" information about the fastness or slowness of the system response as shown in Table 2. It is defined as follows [16]:

$$r\_k = \frac{\Delta e\_k - \Delta e\_{k-1}}{\Delta e^\*} = \frac{\Delta \left(\Delta e\_k\right)}{\Delta e^\*} \tag{4}$$

where Δ*ek* and Δ (Δ*ek*) are the incremental change in error and the so-called acceleration in error given respectively by:

$$
\Delta e\_k = e\_k - e\_{k-1} \tag{5}
$$

$$
\Delta \left( \Delta e\_k \right) = \Delta e\_k - \Delta e\_{k-1} \tag{6}
$$

In equation (4), the variable Δ*e*∗ is chosen as follows:

$$
\Delta \mathcal{e}^\* = \begin{cases}
\Delta \mathcal{e}\_k & \text{if } \begin{array}{l} |\Delta \mathcal{e}\_k| \ge |\Delta \mathcal{e}\_{k-1}| \\
\Delta \mathcal{e}\_{k-1} & \text{if } \begin{array}{l} |\Delta \mathcal{e}\_k| < |\Delta \mathcal{e}\_{k-1}| \end{array} \end{cases} \tag{7}
$$


**Table 2.** Nature of the system response depending on the variable *rk*.

For this RROM self-tuning approach, the uniformly distributed triangular and the symmetrical membership functions, as shown in Figures 2, 3, 4, are assigned for the fuzzy inputs *rk* and |*ek*|, and fuzzy output variable *δk*. The view of the above fuzzy rule-base is illustrated in Figure 5.

**Figure 2.** Membership functions for *rk*.

4 Will-be-set-by-IN-TECH

In this self-tuning method [12, 16], the PID-type FC integral and derivative components

where *δ<sup>k</sup>* is the output of the fuzzy Relative Rate Observer (RRO) *Kf* is the output scaling factor for *δ<sup>k</sup>* and *Kf d* is the additional parameter that affects only the derivative factor of the

The rule-base for *δk*, as used by Eksin et al. [12] and Güzelkaya et al. [16], is considered for the fuzzy RRO. This fuzzy RRO block has as inputs the absolute values of error |*ek*| and the

> |*ek*|/*rk* S M F S M M L SM SM M L M S SM M L S S SM

The linguistic levels assigned to the input |*ek*| and the output variable *δ<sup>k</sup>* are as follows: L (Large), M (Medium), SM (Small Medium) and S (Small). For the input variable *rk*, the

The variable *rk*, defined in [12, 16] and called normalized acceleration, gives "relative rate" information about the fastness or slowness of the system response as shown in Table 2. It is

where Δ*ek* and Δ (Δ*ek*) are the incremental change in error and the so-called acceleration in

<sup>Δ</sup>*e*<sup>∗</sup> <sup>=</sup> <sup>Δ</sup> (Δ*ek*)

<sup>Δ</sup>*ek i f* <sup>|</sup>Δ*ek*<sup>|</sup> <sup>≥</sup> <sup>|</sup>Δ*ek*−1<sup>|</sup>

Δ*e*<sup>∗</sup> Δ (Δ*ek*) System response Positive Positive Fast Positive Negative Slow Negative Positive Slow Negative Negative Fast

<sup>Δ</sup>*e*<sup>∗</sup> (4)

<sup>Δ</sup>*ek* = *ek* − *ek*−<sup>1</sup> (5) <sup>Δ</sup> (Δ*ek*) = <sup>Δ</sup>*ek* − <sup>Δ</sup>*ek*−<sup>1</sup> (6)

<sup>Δ</sup>*ek*−<sup>1</sup> *i f* <sup>|</sup>Δ*ek*<sup>|</sup> <sup>&</sup>lt; <sup>|</sup>Δ*ek*−1<sup>|</sup> (7)

following linguistic levels are assigned: F (Fast), M (Moderate) and S (Slow).

*rk* <sup>=</sup> <sup>Δ</sup>*ek* <sup>−</sup> <sup>Δ</sup>*ek*−<sup>1</sup>

(3)

*<sup>β</sup><sup>k</sup>* <sup>=</sup> *<sup>β</sup>*<sup>0</sup> *Kf δ<sup>k</sup> Kdk* = *Kd*0*Kf dKf δ<sup>k</sup>*

*2.2.2. Self-tuning via Relative Rate Observer Method RROM*

variable *rk*, defined subsequently, as shown in Table 1.

In equation (4), the variable Δ*e*∗ is chosen as follows:

Δ*e* ∗ =

**Table 2.** Nature of the system response depending on the variable *rk*.

**Table 1.** Fuzzy rule-base for the variable *δk*.

defined as follows [16]:

error given respectively by:

updating are achieved as follows:

FC.

**Figure 3.** Membership functions for |*ek*|.

**Figure 4.** Membership functions for *δk*.

**Figure 5.** View of the fuzzy rule-base for *δk*.

## **3. The proposed PSO-based approach**

In this section, the problem of scaling factors tuning, for all defined PID-type FC structures, is formulated as a constrained optimization problem which is solved using the proposed PSO-based approach.

### **3.1. PID-type FC tuning problem formulation**

The choice of the adequate values for the scaling factors of each PID-type FC structure is often done by a trials-errors hard procedure. This tuning problem becomes difficult and delicate without a systematic design method. To deal with these difficulties, the optimization of these scaling factors is proposed like a promising solution. This tuning problem can be formulated as the following constrained optimization problem:

$$\begin{cases} \underset{\mathbf{x}\in\mathbf{D}}{\text{minimize}} f\left(\mathbf{x}\right) \\ \underset{g\_l\left(\mathbf{x}\right)}{\text{subject to}} \\ g\_l\left(\mathbf{x}\right)\leq 0; \quad \forall l=1,\ldots,n\_{\text{con}} \end{cases} \tag{8}$$

where *D*max, *E*max

as follows:

*3.2.1. Overview*

25, 27, 28, 30].

the local minima problem.

control design, as shown in [2–6].

*ss* , *t* max *<sup>r</sup>* and *t*

max

*minimize x*=(*ϕ*1,*ϕ*2,*ψ*1,*ψ*2)

*<sup>D</sup>* <sup>≤</sup> *<sup>D</sup>*max; *ts* <sup>≤</sup> *<sup>t</sup>*

*minimiser x*=(*Kf* ,*Kf d*)

*subjectto*

*T* ∈**R**<sup>2</sup> +

*<sup>D</sup>* <sup>≤</sup> *<sup>D</sup>*max; *ts* <sup>≤</sup> *<sup>t</sup>*

*subjectto*

*<sup>T</sup>*∈**R**<sup>4</sup> + *f* (*x*)

> max *<sup>s</sup>* ; *tr* ≤ *t*

For the PID-type FC structure with the RROM self-tuning mechanism, the scaling factors to be optimized are *Kf* and *Kf d*. The formulated optimization problem is defined as follows:

*f* (*x*)

max *<sup>s</sup>* ; *tr* ≤ *t*

In this study, the proposed PSO approach is presented and a constrained PSO algorithm is also developed. The convergence conditions of such an algorithm are analyzed and established.

The PSO technique is an evolutionary computation method developed by Kennedy and Eberhart [9]. This recent meta-heuristic technique is inspired by the swarming or collaborative behaviour of biological populations. The cooperation and the exchange of information between population individuals allow solving various complex optimization problems [10,

Without any regularity on the cost function to be optimized, the recourse to this stochastic and global optimization technique is justified by the empirical evidence of its superiority in solving a variety of non-linear, non-convex and non-smooth problems. In comparison with other meta-heuristics, this optimization technique is a simple concept, easy to implement, and a computationally efficient algorithm [10, 27, 30]. The convergence and parameters selection of the PSO algorithm are proved using several advanced theoretical analysis proposed by Ruben and Kamran in [27] and Van den Bergh in [30]. Its stochastic behaviour allows overcoming

Particle swarm optimisation has been enormously successful in several and various industrial domains [18, 19]. It has been used across a wide range of engineering applications. These applications can be summarized around domains of robotics, image and signal processing, electronic circuits design, communication networks, but more especially the domain of plant

max

max

system, and can define some time-domain templates.

⎧ ⎪⎪⎪⎨

⎪⎪⎪⎩

⎧ ⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

**3.2. Particle Swarm Optimization technique**

times respectively, that constraint the step response of the PSO-tuned PID-type FC controlled

In the case of the PID-type FC structure with the EFTM self-tuning mechanism, the scaling factors to be optimized are *ϕ*1, *ϕ*2, *ψ*<sup>1</sup> and *ψ*2. The formulated optimization problem is defined

*<sup>s</sup>* are the specified overshoot, steady state, rise and settling

A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization 145

*<sup>r</sup>* ; *Ess* <sup>≤</sup> *<sup>E</sup>*max

*<sup>r</sup>* ; *Ess* <sup>≤</sup> *<sup>E</sup>*max

*ss*

*ss*

(10)

(11)

where *<sup>f</sup>* : **<sup>R</sup>***<sup>m</sup>* <sup>→</sup> **<sup>R</sup>** the cost function, **<sup>D</sup>** <sup>=</sup> {*<sup>x</sup>* <sup>∈</sup> **<sup>D</sup>***m*; *<sup>x</sup>*min <sup>≤</sup> *<sup>x</sup>* <sup>≤</sup> *<sup>x</sup>*max} the initial search space, which is supposed containing the desired design parameters, and *gl* : **<sup>R</sup>***<sup>m</sup>* <sup>→</sup> **<sup>R</sup>** the problem's constraints.

The optimization-based tuning problem consists in finding the optimal decision variables *x*<sup>∗</sup> = � *x*∗ <sup>1</sup> , *x*<sup>∗</sup> <sup>2</sup> ,..., *x*<sup>∗</sup> *m* �*T* , representing the scaling factors of a given PID-type FC structure, which minimize the defined cost function, chosen as the ISE and IAE performance criteria. These cost functions are minimized, using the proposed constrained PSO algorithm, under various time-domain control constraints such as overshoot *D*, steady state error *Ess*, rise time *tr* and settling time *ts* of the system's step response, as shown in the equations (9), (10) and (11).

Hence, in the case of the PID-type FC structure without self-tuning mechanisms, the scaling factors to be optimized are *Ke*, *Kd*, *α* and *β*. The formulated optimization problem is defined as follows:

$$\begin{cases} \underset{\mathbf{x} = \left(\mathbf{K}\_{\mathbf{c}}, \mathbf{K}\_{\mathbf{d}}, \mathbf{a}, \boldsymbol{\beta}\right)^{\mathsf{T}} \in \mathbb{R}\_{+}^{4}}{\operatorname{\$\bf{x}\$}} \text{ }\_{+}^{4} \\ \underset{\mathbf{b} \in D^{\max}}{\operatorname{\$\bf{x}\$}} \text{ }\_{\operatorname{\bf t}}{\operatorname{\$\bf s}} \leq t\_{\operatorname{s}}^{\max}; \mathbf{t}\_{r} \leq t\_{r}^{\max}; \mathbf{E}\_{\operatorname{ss}} \leq E\_{\operatorname{ss}}^{\max} \end{cases} \tag{9}$$

where *D*max, *E*max *ss* , *t* max *<sup>r</sup>* and *t* max *<sup>s</sup>* are the specified overshoot, steady state, rise and settling times respectively, that constraint the step response of the PSO-tuned PID-type FC controlled system, and can define some time-domain templates.

In the case of the PID-type FC structure with the EFTM self-tuning mechanism, the scaling factors to be optimized are *ϕ*1, *ϕ*2, *ψ*<sup>1</sup> and *ψ*2. The formulated optimization problem is defined as follows:

$$\begin{cases} \underset{\mathbf{x} = \left(\varphi\_1, \varphi\_2, \varphi\_1, \varphi\_2\right)^\mathsf{T} \in \mathbb{R}\_+^4}{} \; \mathsf{f}\left(\mathbf{x}\right) \\ \underset{\mathbf{x} \; \mathbf{b} \; \mathsf{f} \; \mathsf{c} \; \mathsf{f}}{\operatorname{subject to}} \; \mathsf{f}\left(\mathbf{x}\right) \end{cases} \tag{10}$$
 
$$D \le D^{\max}; \mathbf{t}\_s \le t\_s^{\max}; \mathbf{t}\_r \le t\_r^{\max}; \mathbf{E}\_{\mathsf{ss}} \le E\_{\mathsf{ss}}^{\max}$$

For the PID-type FC structure with the RROM self-tuning mechanism, the scaling factors to be optimized are *Kf* and *Kf d*. The formulated optimization problem is defined as follows:

$$\begin{cases} \begin{aligned} &\text{minimiser} \quad f \left( \mathbf{x} \right) \\ &\mathbf{x} = \left( \mathbf{k}\_f, \mathbf{k}\_{fd} \right)^{\top} \in \mathbb{R}\_+^2 \\ &subtext{subject to} \\ &D \le D^{\max}; \mathbf{t}\_s \le t\_s^{\max}; \mathbf{t}\_r \le t\_r^{\max}; E\_{ss} \le E\_{ss}^{\max} \end{aligned} \end{cases} \tag{11}$$

## **3.2. Particle Swarm Optimization technique**

In this study, the proposed PSO approach is presented and a constrained PSO algorithm is also developed. The convergence conditions of such an algorithm are analyzed and established.

#### *3.2.1. Overview*

6 Will-be-set-by-IN-TECH


*rk ek*

In this section, the problem of scaling factors tuning, for all defined PID-type FC structures, is formulated as a constrained optimization problem which is solved using the proposed

The choice of the adequate values for the scaling factors of each PID-type FC structure is often done by a trials-errors hard procedure. This tuning problem becomes difficult and delicate without a systematic design method. To deal with these difficulties, the optimization of these scaling factors is proposed like a promising solution. This tuning problem can be formulated

*gl* (*x*) ≤ 0; ∀*l* = 1, . . . , *ncon*

where *<sup>f</sup>* : **<sup>R</sup>***<sup>m</sup>* <sup>→</sup> **<sup>R</sup>** the cost function, **<sup>D</sup>** <sup>=</sup> {*<sup>x</sup>* <sup>∈</sup> **<sup>D</sup>***m*; *<sup>x</sup>*min <sup>≤</sup> *<sup>x</sup>* <sup>≤</sup> *<sup>x</sup>*max} the initial search space, which is supposed containing the desired design parameters, and *gl* : **<sup>R</sup>***<sup>m</sup>* <sup>→</sup> **<sup>R</sup>** the problem's

The optimization-based tuning problem consists in finding the optimal decision variables

which minimize the defined cost function, chosen as the ISE and IAE performance criteria. These cost functions are minimized, using the proposed constrained PSO algorithm, under various time-domain control constraints such as overshoot *D*, steady state error *Ess*, rise time *tr* and settling time *ts* of the system's step response, as shown in the equations (9), (10) and (11). Hence, in the case of the PID-type FC structure without self-tuning mechanisms, the scaling factors to be optimized are *Ke*, *Kd*, *α* and *β*. The formulated optimization problem is defined

, representing the scaling factors of a given PID-type FC structure,

max

*<sup>r</sup>* ; *Ess* <sup>≤</sup> *<sup>E</sup>*max

*ss*

0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8

δ

**Figure 5.** View of the fuzzy rule-base for *δk*.

PSO-based approach.

constraints.

*x*∗ <sup>1</sup> , *x*<sup>∗</sup>

<sup>2</sup> ,..., *x*<sup>∗</sup> *m* �*T*

*x*<sup>∗</sup> = �

as follows:

**3. The proposed PSO-based approach**

**3.1. PID-type FC tuning problem formulation**

as the following constrained optimization problem:

⎧ ⎪⎪⎪⎨

*minimize x*=(*Ke*,*Kd*,*α*,*β*)

*<sup>D</sup>* <sup>≤</sup> *<sup>D</sup>*max; *ts* <sup>≤</sup> *<sup>t</sup>*

*subjectto*

*<sup>T</sup>*∈**R**<sup>4</sup> + *f* (*x*)

> max *<sup>s</sup>* ; *tr* ≤ *t*

⎪⎪⎪⎩

⎧ ⎪⎨

*minimize <sup>x</sup>*∈**<sup>D</sup>** *<sup>f</sup>* (*x*) *subjectto*

⎪⎩

*k*


0

0.5

1

(8)

(9)

The PSO technique is an evolutionary computation method developed by Kennedy and Eberhart [9]. This recent meta-heuristic technique is inspired by the swarming or collaborative behaviour of biological populations. The cooperation and the exchange of information between population individuals allow solving various complex optimization problems [10, 25, 27, 28, 30].

Without any regularity on the cost function to be optimized, the recourse to this stochastic and global optimization technique is justified by the empirical evidence of its superiority in solving a variety of non-linear, non-convex and non-smooth problems. In comparison with other meta-heuristics, this optimization technique is a simple concept, easy to implement, and a computationally efficient algorithm [10, 27, 30]. The convergence and parameters selection of the PSO algorithm are proved using several advanced theoretical analysis proposed by Ruben and Kamran in [27] and Van den Bergh in [30]. Its stochastic behaviour allows overcoming the local minima problem.

Particle swarm optimisation has been enormously successful in several and various industrial domains [18, 19]. It has been used across a wide range of engineering applications. These applications can be summarized around domains of robotics, image and signal processing, electronic circuits design, communication networks, but more especially the domain of plant control design, as shown in [2–6].

#### 8 Will-be-set-by-IN-TECH 146 Fuzzy Controllers – Recent Advances in Theory and Applications A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization <sup>9</sup>

#### *3.2.2. Basic PSO algorithm*

The basic PSO algorithm uses a swarm consisting of *np* particles (i.e. *x*1, *x*2,..., *xnp* ), randomly distributed in the considered initial search space, to find an optimal solution *<sup>x</sup>*<sup>∗</sup> <sup>=</sup> arg min *<sup>f</sup>* (*x*) <sup>∈</sup> **<sup>R</sup>***<sup>m</sup>* of a generic optimization problem (8). Each particle, that represents a potential solution, is characterised by a position and a velocity given by *x<sup>i</sup> <sup>k</sup>* := *xi*,1 *<sup>k</sup>* , *<sup>x</sup>i*,2 *<sup>k</sup>* ,..., *<sup>x</sup>i*,*<sup>m</sup> k T* and *v<sup>i</sup> <sup>k</sup>* := *vi*,1 *<sup>k</sup>* , *<sup>v</sup>i*,2 *<sup>k</sup>* ,..., *<sup>v</sup>i*,*<sup>m</sup> k T* where (*i*, *<sup>k</sup>*) <sup>∈</sup> 1, *np* <sup>×</sup> [[1, *<sup>k</sup>*max]].

At each algorithm iteration, the *i th* particle position, *<sup>x</sup><sup>i</sup>* <sup>∈</sup> **<sup>R</sup>***m*, evolves based on the following update rules:

$$\mathbf{x}\_{k+1}^{i} = \mathbf{x}\_{k}^{i} + \mathbf{v}\_{k+1}^{i} \tag{12}$$

Similarly to other meta-heuristic methods, the PSO algorithm is originally formulated as an unconstrained optimizer. Several techniques have been proposed to deal with constraints. One useful approach is by augmenting the cost function of problem (8) with penalties proportional to the degree of constraint infeasibility. In this paper, the following external

> *ncon* ∑ *l*=1

where *χ<sup>l</sup>* is a prescribed scaling penalty parameters and *ncon* is the number of problem

1. Define all PSO algorithm parameters such as swarm size *np*, maximum and minimum

3. Increment the iteration number *k*. For each particle apply the update equations (12)

*<sup>k</sup>* <sup>=</sup> *<sup>x</sup><sup>i</sup> k*;

*<sup>k</sup>* <sup>=</sup> *<sup>x</sup><sup>i</sup> k*;

4. If the termination criterion is satisfied, the algorithm terminates with the solution *x*∗ =

In this part, the proposed PSO algorithm is analysed based on results in [27, 30]. Theoretical

Let us replace the velocity update equation (13) into the position update equation (12) to get

*<sup>k</sup>* <sup>+</sup> *<sup>c</sup>*1*r<sup>i</sup>*

*<sup>k</sup>* <sup>+</sup> *<sup>c</sup>*1*r<sup>i</sup>*

� + � *c*1*r<sup>i</sup>* 1,*<sup>k</sup> <sup>c</sup>*2*r<sup>i</sup>* 2,*k*

1,*kp<sup>i</sup>*

1,*kp<sup>i</sup>*

*c*1*r<sup>i</sup>* 1,*<sup>k</sup> <sup>c</sup>*2*r<sup>i</sup>* 2,*k*

*<sup>k</sup>* <sup>+</sup> *<sup>c</sup>*2*r<sup>i</sup>*

*<sup>k</sup>* <sup>+</sup> *<sup>c</sup>*2*r<sup>i</sup>*

2,*kp<sup>g</sup>*

2,*kp<sup>g</sup>*

� � *pi k pg k*

�

*<sup>k</sup>* and *gbestk* represent the best previously fitness of the *i*

Finally, the basic proposed PSO algorithm can be summarized by the following steps:

*<sup>χ</sup><sup>l</sup>* max �

0, *gl* (*x*) 2 �

A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization 147

<sup>0</sup> and velocities *<sup>v</sup><sup>i</sup>*

<sup>0</sup> and *<sup>p</sup><sup>g</sup>* 0.

*<sup>k</sup>* = *ϕ* � *xi k* � : (15)

<sup>0</sup> in the search

*th* particle and the

*<sup>k</sup>* (16)

*<sup>k</sup>* (17)

(18)

*ϕ* (*x*) = *f* (*x*) +

inertia factor values, cognitive *c*<sup>1</sup> and social *c*<sup>2</sup> scaling factors, etc.

2. Initialize the *np* particles with randomly chosen positions *x<sup>i</sup>*

space **D**. Evaluate the initial population and determine *p<sup>i</sup>*

and (13), and evaluate the corresponding fitness values *ϕ<sup>i</sup>*

*<sup>k</sup>* <sup>=</sup> *<sup>ϕ</sup><sup>i</sup>*

*<sup>k</sup>* and *<sup>p</sup><sup>i</sup>*

*<sup>k</sup>* and *<sup>p</sup><sup>g</sup>*

. Otherwise, go to step 3.

conditions for convergence algorithm and parameters choice are established.

1,*<sup>k</sup>* <sup>−</sup> *<sup>c</sup>*2*r<sup>i</sup>*

1,*<sup>k</sup>* <sup>+</sup> *<sup>c</sup>*2*r<sup>i</sup>*

1,*<sup>k</sup>* <sup>+</sup> *<sup>c</sup>*2*r<sup>i</sup>*

1,*<sup>k</sup>* <sup>+</sup> *<sup>c</sup>*2*r<sup>i</sup>*

A similar re-arrangement of the velocity term (13) leads to:

� *c*1*r<sup>i</sup>*

− � *c*1*r<sup>i</sup>* 2,*k* � *xi <sup>k</sup>* <sup>+</sup> *wv<sup>i</sup>*

2,*k* � *xi <sup>k</sup>* <sup>+</sup> *wv<sup>i</sup>*

The obtained equations (16) and (17) can be combined and written in matrix form as:

2,*k* � *w* ⎤ ⎦ � *xi k vi k*

2,*k* � *w*

*<sup>k</sup>* then *pbest<sup>i</sup>*

*<sup>k</sup>* <sup>≤</sup> *gbestk* then *gbestk* <sup>=</sup> *<sup>ϕ</sup><sup>i</sup>*

*3.2.3. The convergence of PSO algorithm analysis*

static penalty technique is used:

constraints *gl* (*x*).

• if *ϕ<sup>i</sup>*

• if *ϕ<sup>i</sup>*

arg min*xi k*

where *pbest<sup>i</sup>*

� *f* � *xi k* � , ∀*i*, *k* �

the following expression:

� *xi k*+1 *vi k*+1

*xi <sup>k</sup>*+<sup>1</sup> = � <sup>1</sup> <sup>−</sup> *<sup>c</sup>*1*r<sup>i</sup>*

*vi <sup>k</sup>*+<sup>1</sup> = −

> � = ⎡ ⎣ 1 − � *c*1*r<sup>i</sup>*

*<sup>k</sup>* <sup>≤</sup> *pbest<sup>i</sup>*

entire swarm, respectively.

$$\mathfrak{w}\_{k+1}^{i} = w\_{k+1}\mathfrak{v}\_{k}^{i} + c\_{1}r\_{1,k}^{i}\left(\mathfrak{p}\_{k}^{i} - \mathfrak{x}\_{k}^{i}\right) + c\_{2}r\_{2,k}^{i}\left(\mathfrak{p}\_{k}^{\mathcal{S}} - \mathfrak{x}\_{k}^{i}\right) \tag{13}$$

where

*wk*+1: the inertia factor,

*c*1, *c*2: the cognitive and the social scaling factors respectively,

*ri* 1,*k*, *<sup>r</sup><sup>i</sup>* 2,*k*: random numbers uniformly distributed in the interval [[0, 1]],

*pi <sup>k</sup>*: the best previously obtained position of the *i th* particle,

*pg <sup>k</sup>* : the best obtained position in the entire swarm at the current iteration *k*.

Hence, the principle of a particle displacement in the swarm is graphically shown in the Figure 6, for a two dimensional design space.

**Figure 6.** Particle position and velocity updates.

In order to improve the exploration and exploitation capacities of the proposed PSO algorithm, we choose for the inertia factor a linear evolution with respect to the algorithm iteration as given by Shi and Eberhart in [28]:

$$w\_{k+1} = w\_{\text{max}} - \left(\frac{w\_{\text{max}} - w\_{\text{min}}}{k\_{\text{max}}}\right)k\tag{14}$$

where *w*max = 0.9 and *w*min = 0.4 represent the maximum and minimum inertia factor values, respectively, *k*max is the maximum iteration number.

Similarly to other meta-heuristic methods, the PSO algorithm is originally formulated as an unconstrained optimizer. Several techniques have been proposed to deal with constraints. One useful approach is by augmenting the cost function of problem (8) with penalties proportional to the degree of constraint infeasibility. In this paper, the following external static penalty technique is used:

$$\log\left(\mathbf{x}\right) = f\left(\mathbf{x}\right) + \sum\_{l=1}^{n\_{\text{con}}} \chi\_l \max\left[0, g\_l\left(\mathbf{x}\right)^2\right] \tag{15}$$

where *χ<sup>l</sup>* is a prescribed scaling penalty parameters and *ncon* is the number of problem constraints *gl* (*x*).

Finally, the basic proposed PSO algorithm can be summarized by the following steps:

	- if *ϕ<sup>i</sup> <sup>k</sup>* <sup>≤</sup> *pbest<sup>i</sup> <sup>k</sup>* then *pbest<sup>i</sup> <sup>k</sup>* <sup>=</sup> *<sup>ϕ</sup><sup>i</sup> <sup>k</sup>* and *<sup>p</sup><sup>i</sup> <sup>k</sup>* <sup>=</sup> *<sup>x</sup><sup>i</sup> k*;

8 Will-be-set-by-IN-TECH

The basic PSO algorithm uses a swarm consisting of *np* particles (i.e. *x*1, *x*2,..., *xnp* ), randomly distributed in the considered initial search space, to find an optimal solution *<sup>x</sup>*<sup>∗</sup> <sup>=</sup> arg min *<sup>f</sup>* (*x*) <sup>∈</sup> **<sup>R</sup>***<sup>m</sup>* of a generic optimization problem (8). Each particle, that represents a potential solution, is characterised by a position and a velocity given by *x<sup>i</sup>*

*<sup>k</sup>* <sup>+</sup> *<sup>v</sup><sup>i</sup>*

*th* particle,

*p*

*k i*

*v p x*

1,*k*

*<sup>k</sup>* <sup>1</sup> ) *r*

*<sup>i</sup> w c*

*k i*

*k* (14)

\_ (

*k*

where (*i*, *<sup>k</sup>*) <sup>∈</sup>

*th* particle position, *<sup>x</sup><sup>i</sup>* <sup>∈</sup> **<sup>R</sup>***m*, evolves based on the following

1, *np*

*<sup>k</sup>*+<sup>1</sup> (12)

<sup>×</sup> [[1, *<sup>k</sup>*max]].

*<sup>k</sup>* ,..., *<sup>v</sup>i*,*<sup>m</sup> k T*

*xi <sup>k</sup>*+<sup>1</sup> <sup>=</sup> *<sup>x</sup><sup>i</sup>*

*<sup>k</sup>* <sup>+</sup> *<sup>c</sup>*1*r<sup>i</sup>* 1,*k pi <sup>k</sup>* <sup>−</sup> *<sup>x</sup><sup>i</sup> k* + *c*2*r<sup>i</sup>* 2,*k pg <sup>k</sup>* <sup>−</sup> *<sup>x</sup><sup>i</sup> k* 

2,*k*: random numbers uniformly distributed in the interval [[0, 1]],

*<sup>k</sup>* : the best obtained position in the entire swarm at the current iteration *k*.

*x*

*k i*

+

1

*v*

*k i*

+

1

Hence, the principle of a particle displacement in the swarm is graphically shown in the

2,*k i r*

........

In order to improve the exploration and exploitation capacities of the proposed PSO algorithm, we choose for the inertia factor a linear evolution with respect to the algorithm

where *w*max = 0.9 and *w*min = 0.4 represent the maximum and minimum inertia factor values,

*x*

*wk*<sup>+</sup><sup>1</sup> = *w*max −

*k i*

.............

*i i*

*v*

*k i*

*w*max <sup>−</sup> *<sup>w</sup>*min *k*max

...............................

*c*<sup>2</sup> *p x <sup>k</sup> g <sup>k</sup>* \_ *<sup>i</sup>* ( ) *<sup>k</sup>* :=

(13)

*3.2.2. Basic PSO algorithm*

*<sup>k</sup>* ,..., *<sup>x</sup>i*,*<sup>m</sup> k T*

*wk*+1: the inertia factor,

update rules:

where

*ri* 1,*k*, *<sup>r</sup><sup>i</sup>*

*pi*

*pg*

At each algorithm iteration, the *i*

and *v<sup>i</sup> <sup>k</sup>* := *vi*,1 *<sup>k</sup>* , *<sup>v</sup>i*,2

*vi*

*<sup>k</sup>*+<sup>1</sup> <sup>=</sup> *wk*<sup>+</sup>1*v<sup>i</sup>*

*<sup>k</sup>*: the best previously obtained position of the *i*

*p*

*k g*

Figure 6, for a two dimensional design space.

**Figure 6.** Particle position and velocity updates.

iteration as given by Shi and Eberhart in [28]:

respectively, *k*max is the maximum iteration number.

*c*1, *c*2: the cognitive and the social scaling factors respectively,

 *xi*,1 *<sup>k</sup>* , *<sup>x</sup>i*,2

> • if *ϕ<sup>i</sup> <sup>k</sup>* <sup>≤</sup> *gbestk* then *gbestk* <sup>=</sup> *<sup>ϕ</sup><sup>i</sup> <sup>k</sup>* and *<sup>p</sup><sup>g</sup> <sup>k</sup>* <sup>=</sup> *<sup>x</sup><sup>i</sup> k*;

where *pbest<sup>i</sup> <sup>k</sup>* and *gbestk* represent the best previously fitness of the *i th* particle and the entire swarm, respectively.

4. If the termination criterion is satisfied, the algorithm terminates with the solution *x*∗ = arg min*xi k* � *f* � *xi k* � , ∀*i*, *k* � . Otherwise, go to step 3.

#### *3.2.3. The convergence of PSO algorithm analysis*

In this part, the proposed PSO algorithm is analysed based on results in [27, 30]. Theoretical conditions for convergence algorithm and parameters choice are established.

Let us replace the velocity update equation (13) into the position update equation (12) to get the following expression:

$$\mathbf{x}\_{k+1}^{i} = \left(1 - c\_1 r\_{1,k}^{i} - c\_2 r\_{2,k}^{i}\right) \mathbf{x}\_k^{i} + w \mathbf{v}\_k^{i} + c\_1 r\_{1,k}^{i} \mathbf{p}\_k^{i} + c\_2 r\_{2,k}^{i} \mathbf{p}\_k^{\mathcal{S}} \tag{16}$$

A similar re-arrangement of the velocity term (13) leads to:

$$\boldsymbol{\sigma}\_{k+1}^{i} = -\left(c\_{1}r\_{1,k}^{i} + c\_{2}r\_{2,k}^{i}\right)\mathbf{x}\_{k}^{i} + w\mathbf{v}\_{k}^{i} + c\_{1}r\_{1,k}^{i}\mathbf{p}\_{k}^{i} + c\_{2}r\_{2,k}^{i}\mathbf{p}\_{k}^{g} \tag{17}$$

The obtained equations (16) and (17) can be combined and written in matrix form as:

$$
\begin{bmatrix} \mathbf{z}\_{k+1}^{i} \\ \mathbf{z}\_{k+1}^{i} \end{bmatrix} = \begin{bmatrix} 1 - \left(c\_{1}r\_{1,k}^{i} + c\_{2}r\_{2,k}^{i}\right)w \\ -\left(c\_{1}r\_{1,k}^{i} + c\_{2}r\_{2,k}^{i}\right)w \end{bmatrix} \begin{bmatrix} \mathbf{z}\_{k}^{i} \\ \mathbf{z}\_{k}^{i} \end{bmatrix} + \begin{bmatrix} c\_{1}r\_{1,k}^{i} \ c\_{2}r\_{2,k}^{i} \\ c\_{1}r\_{1,k}^{i} \ c\_{2}r\_{2,k}^{i} \end{bmatrix} \begin{bmatrix} \mathbf{p}\_{k}^{i} \\ \mathbf{p}\_{k}^{s} \end{bmatrix} \tag{18}
$$

This above expression can be considered as a state-space representation of a discrete-time dynamic linear system, given by:

$$
\mathfrak{H}\_{k+1} = \mathcal{M}\mathfrak{H}\_k + \mathcal{N}\mathfrak{A}\_k \tag{19}
$$

In order to illustrate the efficiency of the proposed PSO algorithm in the resolution of problems (9), (10) and (11), several comparisons with the Genetic Algorithms Optimization GAO-based method [14, 29] are considered. The next section is dedicated to the application of the proposed PSO-tuned PID-FC approaches to an electrical DC drive and a thermal process

A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization 149

In this section, all designed PSO-tuned PID-type FC structures are applied to two different systems such as an electrical DC drive and a thermal PT-326 Process Trainer benchmarks. Real-time implementations and experimental results of these control laws are presented and

The considered benchmark is a 250 watts electrical DC drive, as shown in Figure 15. The machine's speed rotation is 3000 rpm at 180 volts DC armature voltage. The motor is supplied by an AC-DC power converter. The developed real-time application acquires input data (speed of the DC drive) and generates control signal for thyristors of AC-DC power converter (PWM signal). This is achieved using a data acquisition and control system based on a PC computer and a multi-functions data acquisition PCI-1710 board which is compatible with

The considered electrical DC drive can be described by the following model that is used in the

The model's parameters are obtained by an experimental identification procedure and they are summarized in Table 3 with their associated uncertainty bounds. Also, this model is sampled

> Parameters Nominal values Uncertainty bounds *km* 0.05 75 % *τm* 300 ms 75 % *τe* 14 ms 75 %

For all proposed PSO-tuned PID-type FC structures, product-sum inference and center of gravity defuzzification methods are adopted for the FC block. Uniformly distributed and symmetrical membership functions, are assigned for the fuzzy input and output variables.

(<sup>1</sup> <sup>+</sup> *<sup>τ</sup>ms*) (<sup>1</sup> <sup>+</sup> *<sup>τ</sup>es*) (27)

*<sup>G</sup>* (*s*) <sup>=</sup> *km*

with 10 ms sampling time for simulation and experimental setups.

**Table 3.** Identified DC drive model parameters.

The associated fuzzy rule-base is given in Table 4.

*4.1.2. Simulation results*

within a developed real-time framework.

discussed.

*4.1.1. Plant model description*

MATLAB/Simulink [1, 17].

design setup:

**4. Real-time control approach implementation**

**4.1. Control of an electrical DC drive benchmark**

where *y*ˆ*<sup>k</sup>* is the state vector, *u*ˆ*<sup>k</sup>* the external input system, M and N the dynamic and input matrices respectively, defined as:

$$\mathcal{Y}\_{k} = \begin{bmatrix} \mathbf{x}\_{k}^{i} \\ \mathbf{z}\_{k}^{i} \end{bmatrix}; \boldsymbol{\hat{u}}\_{k} = \begin{bmatrix} \mathbf{p}\_{k}^{i} \\ \mathbf{p}\_{k}^{s} \end{bmatrix}; \mathcal{M} = \begin{bmatrix} 1 - \left(c\_{1}r\_{1,k}^{i} + c\_{2}r\_{2,k}^{i}\right)w \\ -\left(c\_{1}r\_{1,k}^{i} + c\_{2}r\_{2,k}^{i}\right)w \end{bmatrix}; \mathcal{N} = \begin{bmatrix} c\_{1}r\_{1,k}^{i} \ c\_{2}r\_{2,k}^{i} \\ c\_{1}r\_{1,k}^{i} \ c\_{2}r\_{2,k}^{i} \end{bmatrix} \tag{20}$$

For a given particle, the convergent behaviour can be maintained while assuming that the external input is constant, as there is no external excitation in the dynamic system. In such a case, as the iterations go to infinity the updated positions and velocities become constants from the *kth* to the (*k* + 1) *th* iteration, given the following equilibrium state:

$$\boldsymbol{\hat{y}}\_{k+1} - \boldsymbol{\hat{y}}\_{k} = \begin{bmatrix} -\left(c\_{1}r\_{1,k}^{i} + c\_{2}r\_{2,k}^{i}\right) & \boldsymbol{w} \\ -\left(c\_{1}r\_{1,k}^{i} + c\_{2}r\_{2,k}^{i}\right) & \boldsymbol{w} - 1 \end{bmatrix} \begin{bmatrix} \mathbf{x}\_{k}^{i} \\ \mathbf{v}\_{k}^{i} \end{bmatrix} + \begin{bmatrix} c\_{1}r\_{1,k}^{i} \ c\_{2}r\_{2,k}^{i} \\ c\_{1}r\_{1,k}^{i} \ c\_{2}r\_{2,k}^{i} \end{bmatrix} \begin{bmatrix} \mathbf{p}\_{k}^{i} \\ \mathbf{p}\_{k}^{0} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \tag{21}$$

which is true only when:

$$\begin{aligned} \mathfrak{x}\_k^i &= \mathfrak{p}\_k^i = \mathfrak{p}\_{k'}^\mathcal{S} \\ \mathfrak{w}\_k^i &= 0 \end{aligned} \tag{22}$$

Therefore, we obtain an equilibrium point, for which all particles tend to converge as algorithm iteration progresses, given by:

$$\mathcal{Y}\_{eq} = \begin{bmatrix} \mathfrak{p}\_{k'}^{\mathcal{S}} \, 0 \end{bmatrix}^{T} \tag{23}$$

So, the dynamic behaviour of the *i th* particle can be analysed using the eigenvalues derived from the dynamic matrix formulation (19) and (20), solutions of the following characteristic polynomial:

$$
\lambda^2 - \left(1 + w - c\_1 r\_{1,k}^i - c\_2 r\_{2,k}^i\right)\lambda + w = 0\tag{24}
$$

The following necessary and sufficient conditions for stability of the considered discrete-time dynamic system (20) are obtained while applying the classical Jury criterion:

$$\begin{array}{l}|w| < 1\\c\_{1}r\_{1,k}^{i} + c\_{2}r\_{2,k}^{i} > 0\\w + 1 - \frac{c\_{1}r\_{1,k}^{i} + c\_{2}r\_{2,k}^{i}}{2} > 0\end{array} \tag{25}$$

Knowing that *r<sup>i</sup>* 1,*k*,*r<sup>i</sup>* 2,*<sup>k</sup>* ∈ [[0, 1]], the above stability conditions are equivalents to the following set of parameter selection heuristics which guarantee convergence for the PSO algorithm:

$$\begin{cases} 0 < c\_1 + c\_2 < 4\\ \frac{c\_1 + c\_2}{2} - 1 < w < 1 \end{cases} \tag{26}$$

While these heuristics provide useful selection parameter bounds, an analysis of the effect of the different parameter settings is achieved and verified by some numerical simulations to determine the effect of such parameters in the PSO algorithm convergence performances.

In order to illustrate the efficiency of the proposed PSO algorithm in the resolution of problems (9), (10) and (11), several comparisons with the Genetic Algorithms Optimization GAO-based method [14, 29] are considered. The next section is dedicated to the application of the proposed PSO-tuned PID-FC approaches to an electrical DC drive and a thermal process within a developed real-time framework.

## **4. Real-time control approach implementation**

In this section, all designed PSO-tuned PID-type FC structures are applied to two different systems such as an electrical DC drive and a thermal PT-326 Process Trainer benchmarks. Real-time implementations and experimental results of these control laws are presented and discussed.

## **4.1. Control of an electrical DC drive benchmark**

#### *4.1.1. Plant model description*

10 Will-be-set-by-IN-TECH

This above expression can be considered as a state-space representation of a discrete-time

where *y*ˆ*<sup>k</sup>* is the state vector, *u*ˆ*<sup>k</sup>* the external input system, M and N the dynamic and input

For a given particle, the convergent behaviour can be maintained while assuming that the external input is constant, as there is no external excitation in the dynamic system. In such a case, as the iterations go to infinity the updated positions and velocities become constants

1,*<sup>k</sup>* <sup>+</sup> *<sup>c</sup>*2*r<sup>i</sup>*

1,*<sup>k</sup>* <sup>+</sup> *<sup>c</sup>*2*r<sup>i</sup>*

*th* iteration, given the following equilibrium state:

⎤ ⎦ � *xi k vi k*

*<sup>k</sup>* <sup>=</sup> *<sup>p</sup><sup>g</sup> k* ,

Therefore, we obtain an equilibrium point, for which all particles tend to converge as

from the dynamic matrix formulation (19) and (20), solutions of the following characteristic

The following necessary and sufficient conditions for stability of the considered discrete-time

1,*<sup>k</sup>* <sup>−</sup> *<sup>c</sup>*2*r<sup>i</sup>*

2,*<sup>k</sup>* > 0

1,*k*+*c*2*r<sup>i</sup>* 2,*k* <sup>2</sup> > 0

2,*k* �

2,*<sup>k</sup>* ∈ [[0, 1]], the above stability conditions are equivalents to the following

2,*k* � *w*

2,*k* � *w*

� + � *c*1*r<sup>i</sup>* 1,*<sup>k</sup> <sup>c</sup>*2*r<sup>i</sup>* 2,*k*

*y*ˆ*k*+<sup>1</sup> = M*y*ˆ*<sup>k</sup>* + N *u*ˆ*<sup>k</sup>* (19)

⎦ ; N =

� *c*1*r<sup>i</sup>* 1,*<sup>k</sup> <sup>c</sup>*2*r<sup>i</sup>* 2,*k*

*c*1*r<sup>i</sup>* 1,*<sup>k</sup> <sup>c</sup>*2*r<sup>i</sup>* 2,*k*

� � *pi k pg k*

*<sup>k</sup>* <sup>=</sup> <sup>0</sup> (22)

*th* particle can be analysed using the eigenvalues derived

<sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>&</sup>lt; *<sup>w</sup>* <sup>&</sup>lt; <sup>1</sup> (26)

� = � 0 0 �

*λ* + *w* = 0 (24)

�

(20)

(21)

(23)

(25)

⎤

*c*1*r<sup>i</sup>* 1,*<sup>k</sup> <sup>c</sup>*2*r<sup>i</sup>* 2,*k*

dynamic linear system, given by:

matrices respectively, defined as:

� ; *u*ˆ*<sup>k</sup>* = � *pi k pg k*

from the *kth* to the (*k* + 1)

*y*ˆ*k*+<sup>1</sup> − *y*ˆ*<sup>k</sup>* =

which is true only when:

polynomial:

Knowing that *r<sup>i</sup>*

⎡ ⎣ − � *c*1*r<sup>i</sup>*

algorithm iteration progresses, given by:

So, the dynamic behaviour of the *i*

1,*k*,*r<sup>i</sup>*

− � *c*1*r<sup>i</sup>*

�

;M =

1,*<sup>k</sup>* <sup>+</sup> *<sup>c</sup>*2*r<sup>i</sup>*

1,*<sup>k</sup>* <sup>+</sup> *<sup>c</sup>*2*r<sup>i</sup>*

*<sup>λ</sup>*<sup>2</sup> <sup>−</sup> � 2,*k* � *w*

2,*k* � *w* − 1

> *xi <sup>k</sup>* <sup>=</sup> *<sup>p</sup><sup>i</sup>*

*vi*

*y*ˆ*eq* = � *pg <sup>k</sup>* , 0 �*T*

<sup>1</sup> <sup>+</sup> *<sup>w</sup>* <sup>−</sup> *<sup>c</sup>*1*r<sup>i</sup>*

dynamic system (20) are obtained while applying the classical Jury criterion:


1,*<sup>k</sup>* <sup>+</sup> *<sup>c</sup>*2*r<sup>i</sup>*

set of parameter selection heuristics which guarantee convergence for the PSO algorithm:

<sup>0</sup> <sup>&</sup>lt; *<sup>c</sup>*<sup>1</sup> <sup>+</sup> *<sup>c</sup>*<sup>2</sup> <sup>&</sup>lt; <sup>4</sup> *<sup>c</sup>*1+*c*<sup>2</sup>

While these heuristics provide useful selection parameter bounds, an analysis of the effect of the different parameter settings is achieved and verified by some numerical simulations to determine the effect of such parameters in the PSO algorithm convergence performances.

*<sup>w</sup>* <sup>+</sup> <sup>1</sup> <sup>−</sup> *<sup>c</sup>*1*r<sup>i</sup>*

⎡ ⎣ 1 − � *c*1*r<sup>i</sup>*

− � *c*1*r<sup>i</sup>*

*y*ˆ*<sup>k</sup>* = � *xi k vi k*

> The considered benchmark is a 250 watts electrical DC drive, as shown in Figure 15. The machine's speed rotation is 3000 rpm at 180 volts DC armature voltage. The motor is supplied by an AC-DC power converter. The developed real-time application acquires input data (speed of the DC drive) and generates control signal for thyristors of AC-DC power converter (PWM signal). This is achieved using a data acquisition and control system based on a PC computer and a multi-functions data acquisition PCI-1710 board which is compatible with MATLAB/Simulink [1, 17].

> The considered electrical DC drive can be described by the following model that is used in the design setup:

$$G\left(\mathbf{s}\right) = \frac{k\_m}{\left(1 + \tau\_m \mathbf{s}\right)\left(1 + \tau\_\ell \mathbf{s}\right)}\tag{27}$$

The model's parameters are obtained by an experimental identification procedure and they are summarized in Table 3 with their associated uncertainty bounds. Also, this model is sampled with 10 ms sampling time for simulation and experimental setups.


**Table 3.** Identified DC drive model parameters.

#### *4.1.2. Simulation results*

For all proposed PSO-tuned PID-type FC structures, product-sum inference and center of gravity defuzzification methods are adopted for the FC block. Uniformly distributed and symmetrical membership functions, are assigned for the fuzzy input and output variables. The associated fuzzy rule-base is given in Table 4.


Cost function Algorithm Best Mean Worst St. dev. ISE PSO 0.0660 0.0765 0.1030 0.015 ISE GAO 0.0820 0.0912 0.1330 0.012 IAE PSO 0.0659 0.0838 0.0946 0.014 IAE GAO 0.0718 0.0814 0.0973 0.013

A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization 151

Cost function Algorithm Best Mean Worst St. dev. ISE PSO 0.0559 0.0792 0.0840 0.013 ISE GAO 0.0822 0.0936 0.1120 0.015 IAE PSO 0.0673 0.0861 0.0993 0.016 IAE GAO 0.0855 0.0905 0.1009 0.008

Indeed, the population size, used in the GAO algorithm, is set as 30 individuals and the maximum generation number is 50. However, the GA parameters, used for MATLAB simulations, are chosen as the Stochastic Uniform selection and the Gaussian mutation methods. The Elite Count is set as 2 and the Crossover Fraction as 0.8. The algorithm stops when the number of generations reaches the specified value for the maximum generation. According to the statistical analysis of Tables 5,6 and 7, we can conclude that the proposed PSO-based approach produces better results in comparison with the standard GAO-based one. Also, while using a Pentium IV, 1.73 GHz and MATLAB 7.7.0, the CPU computation times are about 358 and 364 seconds for ISE and IAE criteria, respectively, for the considered

On the other hand, performances on convergence properties of the proposed PSO and the used GAO algorithm, in term of iterations number's required to find the best solution, are compared for the IAE criterion case, as shown in Figures 8 and 9. While using the proposed PSO-based method, we succeed to obtain the optimal solution within only about 28 iterations. However, the GAO-based method finds the same result after 40 iterations. All these observations can show the superiority of the proposed PSO-based method in comparison with the GAO-based one. Indeed, the quality of the obtained optimal solution, the fastness convergence as well as the simple software implementation is better than those of

In a typical optimization procedure, the scaling parameters *χl*, given in equation (15), will be linearly increased at each iteration step so constraints are gradually enforced. Generally, the quality of the solution will directly depend on the value of the specified scaling parameters. In this paper and in order to make the proposed approach simple, great and constant scaling penalty parameters, equal to 103, are used for simulations. Indeed, simulation results show that with a great values of *χl*, the control system performances are weakly degraded and the effects on the tuning parameters are less meaningful. The PSO algorithm convergence is faster

The robustness of the proposed PSO algorithm convergence, under variation of the cognitive, social and inertia factor parameters, is analysed based on numerical simulations as shown

**Table 6.** Optimization results from 20 trials of problem (10).

**Table 7.** Optimization results from 20 trials of problem (11).

PID-type FC without self-tuning mechanisms structure.

than the case with linearly variable scaling parameters.

the GAO-based approach.

**Table 4.** Fuzzy rule-base for the output *uf z*.

The linguistic levels assigned to the input variables *ek* and Δ*ek*, and the output variable *u f z* are given as follows: N (Negative), Z (Zero), P (Positive), N (Negative), NB (Negative Big) and PB (Positive Big). The view of this rule-base is illustrated in Figure 7.

**Figure 7.** View of fuzzy rule-base for the fuzzy output *uf z*.

The swarm size algorithm's choice is generally a problem-dependent in PSO framework. However, Eberhart and Shi [10] as well as Poli et al. [25] show that this parameter is often set empirically in relation to the dimensionality and perceived difficulty of a considered optimization problem. They suggest that swarm size values in the range 20-50 are quite common. For this purpose, we have tested the proposed PSO algorithm with different values in this range for the case of PID-type FC structure without self-tuning mechanisms. Globally, all the found results are close to each other. But, best values of the fitness are obtained while using a swarm size equal to 30. Henceforth, this value will be adopted for our following works.

In the PSO framework, it is necessary to run the algorithm several times in order to get some statistical data on the quality of results and so to validate the proposed approach. We run the proposed algorithm 20 times and feasible solutions are found in 98 % of trials and within an acceptable CPU computation time for the IAE and ISE criterion cases. The obtained optimization results are summarized in Tables 5, 6 and 7. Besides, the fact that the algorithm's convergence always takes place in the same region of the design space, whatever is the initial population, indicates that the algorithm succeeds in finding a region of the interesting research space to explore. The performances comparison of PSO- and GAO-based approaches is achieved in the same conditions.


**Table 5.** Optimization results from 20 trials of problem (9).


150 Fuzzy Controllers – Recent Advances in Theory and Applications A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization <sup>13</sup> A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization 151

**Table 6.** Optimization results from 20 trials of problem (10).

12 Will-be-set-by-IN-TECH

*ek* / Δ*ek* N Z P N NB N Z Z N Z P P Z P PB

The linguistic levels assigned to the input variables *ek* and Δ*ek*, and the output variable *u f z* are given as follows: N (Negative), Z (Zero), P (Positive), N (Negative), NB (Negative Big) and PB


*ek* <sup>Δ</sup>*ek*

The swarm size algorithm's choice is generally a problem-dependent in PSO framework. However, Eberhart and Shi [10] as well as Poli et al. [25] show that this parameter is often set empirically in relation to the dimensionality and perceived difficulty of a considered optimization problem. They suggest that swarm size values in the range 20-50 are quite common. For this purpose, we have tested the proposed PSO algorithm with different values in this range for the case of PID-type FC structure without self-tuning mechanisms. Globally, all the found results are close to each other. But, best values of the fitness are obtained while using a swarm size equal to 30. Henceforth, this value will be adopted for our following

In the PSO framework, it is necessary to run the algorithm several times in order to get some statistical data on the quality of results and so to validate the proposed approach. We run the proposed algorithm 20 times and feasible solutions are found in 98 % of trials and within an acceptable CPU computation time for the IAE and ISE criterion cases. The obtained optimization results are summarized in Tables 5, 6 and 7. Besides, the fact that the algorithm's convergence always takes place in the same region of the design space, whatever is the initial population, indicates that the algorithm succeeds in finding a region of the interesting research space to explore. The performances comparison of PSO- and GAO-based approaches

> Cost function Algorithm Best Mean Worst St. dev. ISE PSO 0.0193 0.0304 0.0511 0.018 ISE GAO 0.1200 0.1780 0.2410 0.050 IAE PSO 0.0162 0.0261 0.0497 0.016 IAE GAO 0.1892 0.2835 0.3227 0.066



1

(Positive Big). The view of this rule-base is illustrated in Figure 7.

*ufz*

**Figure 7.** View of fuzzy rule-base for the fuzzy output *uf z*.

works.

is achieved in the same conditions.

**Table 5.** Optimization results from 20 trials of problem (9).

**Table 4.** Fuzzy rule-base for the output *uf z*.


**Table 7.** Optimization results from 20 trials of problem (11).

Indeed, the population size, used in the GAO algorithm, is set as 30 individuals and the maximum generation number is 50. However, the GA parameters, used for MATLAB simulations, are chosen as the Stochastic Uniform selection and the Gaussian mutation methods. The Elite Count is set as 2 and the Crossover Fraction as 0.8. The algorithm stops when the number of generations reaches the specified value for the maximum generation.

According to the statistical analysis of Tables 5,6 and 7, we can conclude that the proposed PSO-based approach produces better results in comparison with the standard GAO-based one. Also, while using a Pentium IV, 1.73 GHz and MATLAB 7.7.0, the CPU computation times are about 358 and 364 seconds for ISE and IAE criteria, respectively, for the considered PID-type FC without self-tuning mechanisms structure.

On the other hand, performances on convergence properties of the proposed PSO and the used GAO algorithm, in term of iterations number's required to find the best solution, are compared for the IAE criterion case, as shown in Figures 8 and 9. While using the proposed PSO-based method, we succeed to obtain the optimal solution within only about 28 iterations. However, the GAO-based method finds the same result after 40 iterations. All these observations can show the superiority of the proposed PSO-based method in comparison with the GAO-based one. Indeed, the quality of the obtained optimal solution, the fastness convergence as well as the simple software implementation is better than those of the GAO-based approach.

In a typical optimization procedure, the scaling parameters *χl*, given in equation (15), will be linearly increased at each iteration step so constraints are gradually enforced. Generally, the quality of the solution will directly depend on the value of the specified scaling parameters. In this paper and in order to make the proposed approach simple, great and constant scaling penalty parameters, equal to 103, are used for simulations. Indeed, simulation results show that with a great values of *χl*, the control system performances are weakly degraded and the effects on the tuning parameters are less meaningful. The PSO algorithm convergence is faster than the case with linearly variable scaling parameters.

The robustness of the proposed PSO algorithm convergence, under variation of the cognitive, social and inertia factor parameters, is analysed based on numerical simulations as shown

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> -0.02

The robust stability of the proposed PSO-tuned PID-type FC approach is analysed while considering external disturbances and model uncertainties. According to uncertain bounds on nominal plant parameters, given in Table 1, we are going to consider the following family

Figure 12 shows the step responses of a family of 5 random generated closed-loop uncertain models. The stability robustness of the uncertain plants, under the above considered

0.5

DC drive controlled speed (1000 rpm)

1

1.5

**Figure 11.** Robustness of the proposed PSO algorithm under variations of the inertia factor: IAE

of continuous-time transfer functions supposed including the real studied plant:

; *km* ∈ *k*min *<sup>m</sup>* , *<sup>k</sup>*max *m* , *τ<sup>e</sup>* ∈ *τ*min *<sup>e</sup>* , *<sup>τ</sup>*max *e* , *τ<sup>m</sup>* ∈ *τ*min *<sup>m</sup>* , *<sup>τ</sup>*max *m*

uncertainty types, is guaranteed for all designed PID-type FC structures.

nominal plant uncertain plant

**(IAE criterion case)**

for the PSO- and GAO-based design cases as shown in Figure 13.

**Figure 12.** Stability robustness of the PSO-tuned PID-type fuzzy controlled system under model

Finally, the time-domain performances of all proposed PID-type FC structures, are compared

Besides, Table 8 shows the superiorities of the self-tuning EFTM and RROM PID-type FC structures in relation to the one without self-tuning mechanisms as verified in [16]. Remenber that the considered time-domain constraints for the PID-type FC tuning problems (9), 10)

Iteration

*<sup>w</sup>*max=0.75, *<sup>w</sup>*min=0.4 *<sup>w</sup>*max=0.9, *w*min=0.1 *<sup>w</sup>*max=0.8, *<sup>w</sup>*min=0.2 *<sup>w</sup>*max=0.6, *w*min=0.3 *w*max=0.95, *w*min=0.2 *w*max=0.75, *w*min=0.25

A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization 153

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> <sup>0</sup>

Time (sec)

(28)

nominal plant uncertain plant

**(ISE criterion case)**

0

0.02

0.04

Cost function value

criterion case.

G = 

0.5

DC drive controlled speed (1000 rpm)

1

1.5

*<sup>G</sup>*<sup>ˆ</sup> (*s*) <sup>=</sup> *km*

(1 + *τms*) (1 + *τes*)

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> <sup>0</sup>

Time (sec)

parameters uncertainties and external disturbances.

0.06

0.08

0.1

**Figure 8.** Convergence properties of the proposed PSO algorithm: IAE criterion case.

**Figure 9.** Convergence properties of the standard GAO algorithm: IAE criterion case.

in Figure 10 and Figure 11. The PSO algorithm's convergence is guaranteed within the established domain given by the equation (26).

**Figure 10.** Robustness of the proposed PSO algorithm under variations of the cognitive and social parameters: IAE criterion case.

14 Will-be-set-by-IN-TECH

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.05

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.15

in Figure 10 and Figure 11. The PSO algorithm's convergence is guaranteed within the

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> -0. 1

**Figure 10.** Robustness of the proposed PSO algorithm under variations of the cognitive and social

Iteration

Generation

Iteration

**IAE\*=0.0162**

**IAE\*=0.1892**

*c*1 =0; *c* 2 =3

*c*1 =3; *c* 2 =0

*c*1 =1.75; *c* 2 =0.8

*c*1 =0.5; *c* 2 =2.5

*c*1 =2; *c* 2 =2

*c*1 =1.19; *c* 2 =1.19

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6

Cost function value

parameters: IAE criterion case.

0.25

0.3

Cost function value

established domain given by the equation (26).

0.35

0.4

0.45

**Figure 8.** Convergence properties of the proposed PSO algorithm: IAE criterion case.

**Figure 9.** Convergence properties of the standard GAO algorithm: IAE criterion case.

(a)

Cost function value

**Figure 11.** Robustness of the proposed PSO algorithm under variations of the inertia factor: IAE criterion case.

The robust stability of the proposed PSO-tuned PID-type FC approach is analysed while considering external disturbances and model uncertainties. According to uncertain bounds on nominal plant parameters, given in Table 1, we are going to consider the following family of continuous-time transfer functions supposed including the real studied plant:

$$\mathcal{G} = \left\{ \hat{\mathcal{G}}\left(\mathbf{s}\right) = \frac{k\_{\rm m}}{\left(1 + \tau\_{\rm m}s\right)\left(1 + \tau\_{\rm c}s\right)}; k\_{\rm m} \in \left[k\_{\rm m}^{\rm min}, k\_{\rm m}^{\rm max}\right], \tau\_{\rm c} \in \left[\tau\_{\rm c}^{\rm min}, \tau\_{\rm c}^{\rm max}\right], \tau\_{\rm m} \in \left[\tau\_{\rm m}^{\rm min}, \tau\_{\rm m}^{\rm max}\right] \right\} \tag{28}$$

Figure 12 shows the step responses of a family of 5 random generated closed-loop uncertain models. The stability robustness of the uncertain plants, under the above considered uncertainty types, is guaranteed for all designed PID-type FC structures.

**Figure 12.** Stability robustness of the PSO-tuned PID-type fuzzy controlled system under model parameters uncertainties and external disturbances.

Finally, the time-domain performances of all proposed PID-type FC structures, are compared for the PSO- and GAO-based design cases as shown in Figure 13.

Besides, Table 8 shows the superiorities of the self-tuning EFTM and RROM PID-type FC structures in relation to the one without self-tuning mechanisms as verified in [16]. Remenber that the considered time-domain constraints for the PID-type FC tuning problems (9), 10)

benchmark is given by Figure 15. The designed real-time application acquires input data (speed of the DC drive) and generates control signals for the AC-DC power converter through a thyristors gate drive circuit. This is achieved using a data acquisition and control system based on PC and a multi-function data acquisition PCI-1710 board with 12-bit resolution of A/D converter and up to 100 KHz sampling rate. A thyristors gate drive circuit, based on a multivibrator, is used to generate a triggering burst of high-frequency impulses. A pulse transformer is used to assure the galvanic insulation between the control and power circuits. The acquired speed measure, obtained from tachometer sensor, must be adapted to be applied to the used multi-function PCI-1710 board. The complete electronic circuit diagram of the

D1 D2

 Interfacing of acquirement and adaptation of speed DC motor

The multi-function data acquisition PCI-1710 board allows achieving measurement and controlling functions. This target is used to create a real-time application to let the implemented controller system run while synchronized to a real-time clock. The model of the plant was removed from the simulation model, and instead of it, the input device driver (Analog Input) and the output device driver (Analog Output) were introduced as shown in Figure 17. These device drivers close the feedback loop when moving from simulations to experiments. Device driver's blocks include procedures to access the inputs-outputs board. The real-time controller is developed through the compilation and linking stage, in a form of

The practical implementation of the PSO-tuned PID-type FC approach leads to the experimental results of Figure 17 and Figure 18. The obtained results are satisfactory for a simple, systematic and non-conventional control approach and point out the controller's viability and validate the proposed control approach. The measured speed tracking error of the controlled DC drive is very small (less than 10 % of set point) showing the high performances of the proposed control especially in terms of tracking. On the other hand,

a Dynamic Link Library (DLL) which is then loaded in memory and started-up

Th1 Th2

. . *vs* <sup>~</sup> *is*

 Interfacing of AC-DC power converter control and galvanic insulation

Gate impulses

 Thyristors gate drive circuit

Load

ω

DC

D3 motor .

*v*

A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization 155

Tachometer

DT

designed control system is given in [1, 17].

 **Data Acquisition and Control System**

**Figure 15.** The developed experimental DC drive benchmark.

**Figure 14.** The proposed experimental setup schematic.

**Figure 13.** Time-domain performances comparison of all designed PSO- and GAO-tuned PID-type FC structures: IAE criterion case.

and (11) problems, defined in terms of overshoot, steady state, rise and settling times, have been specified as *D*max = 20%, *E*max *ss* = 0, *t* max *<sup>r</sup>* = 0.25 sec and *t* max *<sup>s</sup>* = 0.75 sec.


**Table 8.** Performances of the PSO-tuned PID-type FC structures: IAE criterion case.

#### *4.1.3. Experimental setup and results*

In order to illustrate the efficiency of the proposed PSO-tuned fuzzy control structures within a real-time framework, the example of the PID-type FC without self-tuning mechanism is considered. The same principle of implementation remains valid for the other PID-type FC structures.

The controlled process is constituted by the single-phase AC-DC power converter and the independent excitation DC motor. A schematic diagram of the experimental setup prepared for testing of the designed controller is shown in Figure 14. The developed experimental benchmark is given by Figure 15. The designed real-time application acquires input data (speed of the DC drive) and generates control signals for the AC-DC power converter through a thyristors gate drive circuit. This is achieved using a data acquisition and control system based on PC and a multi-function data acquisition PCI-1710 board with 12-bit resolution of A/D converter and up to 100 KHz sampling rate. A thyristors gate drive circuit, based on a multivibrator, is used to generate a triggering burst of high-frequency impulses. A pulse transformer is used to assure the galvanic insulation between the control and power circuits. The acquired speed measure, obtained from tachometer sensor, must be adapted to be applied to the used multi-function PCI-1710 board. The complete electronic circuit diagram of the designed control system is given in [1, 17].

**Figure 14.** The proposed experimental setup schematic.

16 Will-be-set-by-IN-TECH

0.2 0.4 0.6 0.8 1 1.2 1.4

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> <sup>0</sup>

=3.6938; ψ1

**Figure 13.** Time-domain performances comparison of all designed PSO- and GAO-tuned PID-type FC

and (11) problems, defined in terms of overshoot, steady state, rise and settling times, have

PSO-tuned PID-type FC structure *D*(%) *tr*(sec) *ts* (sec) *Ess* CPU computation time (sec)

In order to illustrate the efficiency of the proposed PSO-tuned fuzzy control structures within a real-time framework, the example of the PID-type FC without self-tuning mechanism is considered. The same principle of implementation remains valid for the other PID-type FC

The controlled process is constituted by the single-phase AC-DC power converter and the independent excitation DC motor. A schematic diagram of the experimental setup prepared for testing of the designed controller is shown in Figure 14. The developed experimental

max

=3.9495; ψ1

Time (sec)

=1.8558; ψ2

*<sup>r</sup>* = 0.25 sec and *t*

=2.4156; ψ2

 **PSO-tuned PID-type FC with RROM self-tuning mechanism**

DC drive controlled speed (1000 rpm)

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> <sup>0</sup>

GAO: <sup>Κ</sup> <sup>f</sup>

Proposed PSO-based approach GAO-based approach

=0.3326; ΙΑΕ=0.0718

max

*<sup>s</sup>* = 0.75 sec.

=0.4503; ΙΑΕ=0.0659

PSO: Κ<sup>f</sup>

Time (sec)

Proposed PSO-based approach GAO-based approach

=0.6980; Κfd=5.2936; ΙΑΕ=0.0989

=0.5284; Κfd=8.3131; ΙΑΕ=0.0673

 **PSO-tuned PID-type FC with EFTM self-tuning mechanism**

PSO-based proposed approach GAO-based approach

=1.9879; α=1.9844; β=5.2642; ΙΑΕ=0.2933

 **PSO-tuned PID-type FC without self-tuning mechanisms**

=10.7780; α=0.4747; β=32.8991; ΙΑΕ=0.0957

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> <sup>0</sup>

Time (sec)

0.2 0.4 0.6 0.8 1 1.2 1.4

GAO:φ1

PSO: φ1

=1.1327; φ2

*ss* = 0, *t*

**Table 8.** Performances of the PSO-tuned PID-type FC structures: IAE criterion case.

without self-tuning mechanisms 17.5 0.23 0.49 0 364 with EFTM self-tuning mechanism 15 0.21 0.64 0 370 with RROM self-tuning mechanism 7 0.20 0.68 0 392

=1.0329; φ2

DC drive controlled speed (1000 rpm)

0.2 0.4 0.6 0.8 1 1.2 1.4

GAO: Κ<sup>e</sup>

structures: IAE criterion case.

been specified as *D*max = 20%, *E*max

*4.1.3. Experimental setup and results*

structures.

PSO: Κ<sup>e</sup>

=1.4720; Κ <sup>d</sup>

=1.8240; Κ<sup>d</sup>

DC drive controlled speed (1000 rpm)

**Figure 15.** The developed experimental DC drive benchmark.

The multi-function data acquisition PCI-1710 board allows achieving measurement and controlling functions. This target is used to create a real-time application to let the implemented controller system run while synchronized to a real-time clock. The model of the plant was removed from the simulation model, and instead of it, the input device driver (Analog Input) and the output device driver (Analog Output) were introduced as shown in Figure 17. These device drivers close the feedback loop when moving from simulations to experiments. Device driver's blocks include procedures to access the inputs-outputs board. The real-time controller is developed through the compilation and linking stage, in a form of a Dynamic Link Library (DLL) which is then loaded in memory and started-up

The practical implementation of the PSO-tuned PID-type FC approach leads to the experimental results of Figure 17 and Figure 18. The obtained results are satisfactory for a simple, systematic and non-conventional control approach and point out the controller's viability and validate the proposed control approach. The measured speed tracking error of the controlled DC drive is very small (less than 10 % of set point) showing the high performances of the proposed control especially in terms of tracking. On the other hand,

18 Will-be-set-by-IN-TECH 156 Fuzzy Controllers – Recent Advances in Theory and Applications A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization <sup>19</sup>

0 10 20 30 40 50 60 70

the efficiency of the proposed control system.

Acquisition time (sec)

0 10 20 30 40 50 60 70

External load disturbance rejection

Acquisition time (sec)

*e*−*τ<sup>d</sup> <sup>s</sup>* (29)


**Figure 18.** Experimental results of PSO-tuned PID-type FC implementation: fuzzy controller robustness

flowing in the pipe can be adjusted by the mean of an inlet throttle attached to the blower. The process consists of heating the air flowing in the pipe to the desired temperature level. The digital control system generates a 40W signal which determines the amount of electrical power supplied to heating resistor made of 10*K*Ω/7*W* power resistors. According to these settings, the experimental trials show that the controlled air temperature can be varied up to 20◦C from the ambient temperature. The assigned control objective is to regulate the temperature of the air at a desired level, with high tracking performance and under internal disturbances, like model parameters variation, and output disturbances. The temperature sensor can be placed at three different locations on the path of the air flow. A variation in the temperature makes a voltage variation at the sensor's output. The amount of air trough the pipe, adjusted by setting the opening of the throttle, can also be used to generate an output disturbance, in order to test

The controlled system input is the voltage applied to the AC-AC power electronic circuit feeding the heating resistor, and the output is the air flow temperature in the pipe, expressed by a 50 mV/◦C voltage, obtained after amplification of the LM35 temperature sensor's output signal. As shown in [23, 32], this process can be characterized as a non-linear system with a pure time delay. The pure time delay depends on the position of the temperature sensor element inserted into the air stream at any one of the three positions along the tube. When the temperature in the air volume inside the tube is assumed uniform a linear model can be obtained. To identify a numerical model of the considered plant, some experimental trials lead to consider the following transfer function, between the heater input voltage and the sensor's

*<sup>G</sup>* (*s*) <sup>=</sup> *km*

1 + *τs*

where *km* is the DC gain system, *τ* is the time constant system, and *τ<sup>d</sup>* is the time delay system. This obtained plant model is assumed to be the nominal one and will be adopted in PSO-tuned PID-type FC synthesis step. These model's parameters are obtained by an experimental identification procedure and they are summarized in Table 9 with their associated uncertainty bounds. Also, this model is sampled with 2 sec sampling time for simulation and experimental


0

Speed tracking error (1000 rpm)

0.25

0.5

A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization 157

External load disturbance rejection

0

output voltage:

setups.

under external load disturbance.

1

Controlled speed variation (1000 rpm)

2

3

**Figure 16.** PCI-1710 board based real-time implementation of the proposed PSO-tuned PID-type FC structure.

Figure 18 shows the robustness of the proposed PSO-tuned PID-type FC in rejection of an external load disturbance applied on the controlled system. The dynamic of the disturbance rejection is fast and guaranteed.

**Figure 17.** Experimental results of PSO-tuned PID-type FC implementation: fuzzy controller tracking performances.

## **4.2. Control of a thermal process benchmark**

#### *4.2.1. Plant model description*

The thermal process to be controlled, shown in the photography of Figure 20, is based on a known PT-326 Process trainer [13], initially developed with an analog control system and modified in order to be digitally controlled. To power the heating resistor, a single-phase AC-AC converter, is developed [7].

In this prototype, the air drawn from atmosphere by a centrifugal blower is injected, through a heating element, in a polypropylene pipe, and rejected in the atmosphere. The amount of air

18 Will-be-set-by-IN-TECH

<sup>1</sup> Delay operator

Control signal Tracking error

*β*

*α*

Delay operator z

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> -0. 5

Acquisition time (sec)

Analog Output Advantech PCI -1710 [auto ]

Analog Input Advantech PCI -1710 [auto ]

Analog Input

Analog Output

ek

Ke

Kd

**Figure 16.** PCI-1710 board based real-time implementation of the proposed PSO-tuned PID-type FC

Figure 18 shows the robustness of the proposed PSO-tuned PID-type FC in rejection of an external load disturbance applied on the controlled system. The dynamic of the disturbance

> -0. 4 -0. 3 -0. 2 -0. 1 0 0.1 0.2 0.3 0.4 0.5

**Figure 17.** Experimental results of PSO-tuned PID-type FC implementation: fuzzy controller tracking

The thermal process to be controlled, shown in the photography of Figure 20, is based on a known PT-326 Process trainer [13], initially developed with an analog control system and modified in order to be digitally controlled. To power the heating resistor, a single-phase

In this prototype, the air drawn from atmosphere by a centrifugal blower is injected, through a heating element, in a polypropylene pipe, and rejected in the atmosphere. The amount of air

Speed tracking error (1000 rpm)

Trajectory planning block

t traj 1\_c yc

rejection is fast and guaranteed.

structure.

0 0.5 1 1.5 2 2.5 3

performances.

*4.2.1. Plant model description*

AC-AC converter, is developed [7].

Controlled speed variation (1000 rpm)

Controlled system output

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> 0. 5

**4.2. Control of a thermal process benchmark**

Acquisition time (sec)

<sup>1</sup> S-Function Builder

Δek

z

Clock Fuzzy Controller

**Figure 18.** Experimental results of PSO-tuned PID-type FC implementation: fuzzy controller robustness under external load disturbance.

flowing in the pipe can be adjusted by the mean of an inlet throttle attached to the blower. The process consists of heating the air flowing in the pipe to the desired temperature level. The digital control system generates a 40W signal which determines the amount of electrical power supplied to heating resistor made of 10*K*Ω/7*W* power resistors. According to these settings, the experimental trials show that the controlled air temperature can be varied up to 20◦C from the ambient temperature. The assigned control objective is to regulate the temperature of the air at a desired level, with high tracking performance and under internal disturbances, like model parameters variation, and output disturbances. The temperature sensor can be placed at three different locations on the path of the air flow. A variation in the temperature makes a voltage variation at the sensor's output. The amount of air trough the pipe, adjusted by setting the opening of the throttle, can also be used to generate an output disturbance, in order to test the efficiency of the proposed control system.

The controlled system input is the voltage applied to the AC-AC power electronic circuit feeding the heating resistor, and the output is the air flow temperature in the pipe, expressed by a 50 mV/◦C voltage, obtained after amplification of the LM35 temperature sensor's output signal. As shown in [23, 32], this process can be characterized as a non-linear system with a pure time delay. The pure time delay depends on the position of the temperature sensor element inserted into the air stream at any one of the three positions along the tube. When the temperature in the air volume inside the tube is assumed uniform a linear model can be obtained. To identify a numerical model of the considered plant, some experimental trials lead to consider the following transfer function, between the heater input voltage and the sensor's output voltage:

$$G\left(s\right) = \frac{k\_m}{1 + \tau s} e^{-\tau\_d s} \tag{29}$$

where *km* is the DC gain system, *τ* is the time constant system, and *τ<sup>d</sup>* is the time delay system.

This obtained plant model is assumed to be the nominal one and will be adopted in PSO-tuned PID-type FC synthesis step. These model's parameters are obtained by an experimental identification procedure and they are summarized in Table 9 with their associated uncertainty bounds. Also, this model is sampled with 2 sec sampling time for simulation and experimental setups.

#### 20 Will-be-set-by-IN-TECH 158 Fuzzy Controllers – Recent Advances in Theory and Applications A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization <sup>21</sup>


As shown in Figure 21 and Figure 22, the controlled air temperature of the considered thermal process tracks the desired trajectory with high performance in terms of response speed and precision in the two considered cases. The robustness of the proposed control strategy in term of output static disturbance rejection, which caused by the throttle opening, is improved.

0

0

5

10

Tracking error temperature [°C]

**Figure 22.** Experimental result of PSO-tuned fuzzy controlled PT-326 Process Trainer: ISE criterion case.

In this study, a new method for tuning PID-type FC structures, using a constrained PSO-based technique, is proposed and successfully applied to an electrical DC drive and thermal process within a real-time framework. This efficient tool leads to a robust and systematic fuzzy control design approach. The performances comparison, with the standard GAO-based method, shows the efficiency and superiority of the proposed PSO-based approach in terms of the obtained solution qualities, the convergence speed and the simple software implementation

The practical implementation of the PSO-tuned PID-type FC approach, for the considered electrical DC drive and the thermal PT-326 Process Trainer benchmarks, leads to several

15

20

5

10

Tracking error temperature [°C]

**Figure 21.** Experimental result of PSO-tuned fuzzy controlled PT-326 Process Trainer: IAE criterion case.

15

20

A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization 159

<sup>0</sup> <sup>500</sup> <sup>1000</sup> <sup>1500</sup> -5

External output disturbance rejection

External output disturbance rejection

Acquisition time (sec)

<sup>0</sup> <sup>500</sup> <sup>1000</sup> <sup>1500</sup> <sup>5</sup>

Acquisition time (sec)

<sup>0</sup> <sup>500</sup> <sup>1000</sup> <sup>1500</sup> <sup>20</sup>

<sup>0</sup> <sup>500</sup> <sup>1000</sup> <sup>1500</sup> <sup>20</sup>

Acquisition time (sec)

Acquisition time (sec)

External output disturbance rejection

External ouput disturbance rejection

controlled temperature desired temperature

controlled temperature desired temperature

25

25

**5. Conclusion**

of its algorithm.

30

35

Thermal process temperature [°C]

40

45

30

35

Thermal process temperature [°C]

40

45

**Table 9.** Identified Thermal Process model parameters.

For this PSO-tuned PID-type fuzzy control example, we represent only the obtained experimental results. For the numerical simulations step, both IAE and ISE criterion, used for the electrical DC drive control, are investigated for this thermal process example. Same problems (9), (10) and (11) are considered and resolved by the developed constrained PSO algorithm.

## *4.2.2. Experimental setup and results*

The developed real-time application acquires air temperature measure and generates control signals for the triac of AC-AC power converter through a gate drive circuit, as shown in Figure 19. This is achieved using a control system based on PC and the used multi-function data acquisition PCI-1710 board. A triac gate drive circuit is used to generate a Pulse Width Modulation (PWM) control signal synchronized with the zero-crossing of the AC voltage. The acquired air temperature measure is scaled before being applied to the used multi-function PCI-1710 board used to create a real-time application to let the implemented controller system run while synchronized to a real-time clock. This leads to experimental results shown in Figure 21 and Figure 22.

**Figure 19.** The proposed thermal process experimental setup schematic.

**Figure 20.** Developed experimental benchmark of the PT-326 Process Trainer.

As shown in Figure 21 and Figure 22, the controlled air temperature of the considered thermal process tracks the desired trajectory with high performance in terms of response speed and precision in the two considered cases. The robustness of the proposed control strategy in term of output static disturbance rejection, which caused by the throttle opening, is improved.

**Figure 21.** Experimental result of PSO-tuned fuzzy controlled PT-326 Process Trainer: IAE criterion case.

**Figure 22.** Experimental result of PSO-tuned fuzzy controlled PT-326 Process Trainer: ISE criterion case.

## **5. Conclusion**

20 Will-be-set-by-IN-TECH

Parameters Nominal values Uncertainty bounds *km* 20 50 % *τ* 65 sec 50 % *τ<sup>d</sup>* 1 sec 50 %

For this PSO-tuned PID-type fuzzy control example, we represent only the obtained experimental results. For the numerical simulations step, both IAE and ISE criterion, used for the electrical DC drive control, are investigated for this thermal process example. Same problems (9), (10) and (11) are considered and resolved by the developed constrained PSO

The developed real-time application acquires air temperature measure and generates control signals for the triac of AC-AC power converter through a gate drive circuit, as shown in Figure 19. This is achieved using a control system based on PC and the used multi-function data acquisition PCI-1710 board. A triac gate drive circuit is used to generate a Pulse Width Modulation (PWM) control signal synchronized with the zero-crossing of the AC voltage. The acquired air temperature measure is scaled before being applied to the used multi-function PCI-1710 board used to create a real-time application to let the implemented controller system run while synchronized to a real-time clock. This leads to experimental results shown in

> **160° 90°**

 Interfacing of AC-AC power converter control and galvanic insulation

Gate impulses

 Triac gate drive circuit

**Figure 19.** The proposed thermal process experimental setup schematic.

**Figure 20.** Developed experimental benchmark of the PT-326 Process Trainer.

 Interfacing of acquirement and adaptation of Air temperature

Air Output

Disturbance **PT-326 Process Trainer**

Sensor

**Heater**

**Table 9.** Identified Thermal Process model parameters.

 **Data Acquisition and Control System**

*4.2.2. Experimental setup and results*

Figure 21 and Figure 22.

algorithm.

In this study, a new method for tuning PID-type FC structures, using a constrained PSO-based technique, is proposed and successfully applied to an electrical DC drive and thermal process within a real-time framework. This efficient tool leads to a robust and systematic fuzzy control design approach. The performances comparison, with the standard GAO-based method, shows the efficiency and superiority of the proposed PSO-based approach in terms of the obtained solution qualities, the convergence speed and the simple software implementation of its algorithm.

The practical implementation of the PSO-tuned PID-type FC approach, for the considered electrical DC drive and the thermal PT-326 Process Trainer benchmarks, leads to several satisfactory experimental results showing the high performances of the proposed control especially in terms of tracking and robustness.

[9] Eberhart, R.C., Kennedy, J., (1995). A New Optimizer Using Particle Swarm Theory. Proceedings of the 6th International Symposium on Micro Machine and Human Science,

A New Method for Tuning PID-Type Fuzzy Controllers Using Particle Swarm Optimization 161

[10] Eberhart, R.C., Shi, Y., (2001). Particle Swarm Optimization: Developments, Applications and Resources. Proceedings of the IEEE Congress on Evolutionary

[11] Eker, I., Torun, Y., (2006). Fuzzy logic control to be conventional method. Energy

[12] Eksin, I., Güzelkaya, M., Gürleyen, F., (2001). A new methodology for deriving the rule-base of a fuzzy logic controller with a new internal structure. Engineering

[13] Feedback, (1980). Process Trainer PT326, Feedback Instruments Limited, Crowborough,

[14] Goldberg, D.E., (1989). Genetic Algorithms in Search, Optimization, and Machine

[15] Grigorie, L. (editor), (2011). Fuzzy Controllers, Theory and Applications, ISBN

[16] Güzelkaya, M., Eksin, I., Yesil, E., (2003). Self-tuning of PID-type fuzzy logic controller coefficients via relative rate observer. Engineering Applications of Artificial Intelligence

[17] Haggège, J., Ayadi, M., Bouallègue, S., Benrejeb, M., (2010). Design of Fuzzy Flatness-based Controller for a DC Drive. Control and Intelligent Systems 38 (3), pp.

[18] Korosec, P., (editor), (2010). New Achievements in Evolutionary Computation, ISBN

[19] Lazinica, A. (editor), (2009). Particle Swarm Optimization, ISBN 978-953-7619-48-0,

[20] Lee, C.C., (1990). Fuzzy Logic in Control Systems: Fuzzy Logic Controller. Part I. IEEE

[21] Lee, C.C., (1990). Fuzzy Logic in Control Systems: Fuzzy Logic Controller. Part II. IEEE

[22] Mamdani, E.H., (1974). Application of fuzzy logic algorithms for control of simple dynamic plant, Proceedings of the Institute of Electrical Engineering (IEE), 121 (12),

[23] Mohd, R., Yeoh Keat, H., Sahnius, U., Norhaliza, A.W., (2007). Modelling of PT326 Hot Air Blower Trainer Kit Using PRBS Signal and Cross-Correlation Technique, Jurnal

[24] Passino, K.M., Yurkovich, S., (1998). Fuzzy Control. Addison Wesley Longman, Menlo

[25] Poli, R., Kennedy, J., Blackwell, T., (2007). Particle Swarm Optimization: An Overview.

[26] Qiao, W.Z., Mizumoto, M., (1996). PID type fuzzy controller and parameters adaptive

[27] Ruben, E.P., Kamran, B., (2007). Particle Swarm Optimization in Structural Design, in: Chan, F.T.S., Tiwari, M.K. (Eds.), Swarm Intelligence: Focus on Ant and Particle Swarm

Optimization. In-Tech Education and Publishing, Vienna, pp. 373-394.

Transactions on Systems, Man, and Cybernetics 20 (2), pp. 404-418.

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Nagoya, pp. 39-43.

United Kingdom.

16, pp. 227-236.

InTech, Croatia.

pp.1585-1588

Park, California.

Teknologi, vol. 42, pp. 9-22.

Swarm Intelligence, Springer 1, pp. 33-57.

method. Fuzzy Sets and Systems 78, pp. 23-35.

164-172.

Learning. Addison-Wesley.

978-953-307-543-3, InTech, Croatia.

978-953-307-053-7, 2010, InTech, Croatia.

Computation, Seoul, Korea, pp. 81-86.

Conversion and Management 47, pp. 377-394.

Applications of Artificial Intelligence 14, pp. 617-628.

The PSO-tuned PID-type FC structures robustness, under external influences such as the output static disturbances and parametric uncertainties, are proven. The control design methodology is systematic, practical and simple without need to exact analytic plant model description. The obtained simulation and experimental results show the efficiency in terms of performance and robustness of the proposed fuzzy control approach which can be applied in industrial motor control field.

## **Author details**

S. Bouallègue

*Higher Insitute of Industrial Systems of Gabes (ISSIG), Salaheddine Elayoubi Street, 6032 Gabes, Tunisia*

J. Haggège and M. Benrejeb

*National Engineering School of Tunis (ENIT), BP 37, le Belvédère, 1002 Tunis, Tunisia*

All authors are with the Research Laboratory in Automatic Control (LA.R.A) of ENIT.

## **6. References**


[9] Eberhart, R.C., Kennedy, J., (1995). A New Optimizer Using Particle Swarm Theory. Proceedings of the 6th International Symposium on Micro Machine and Human Science, Nagoya, pp. 39-43.

22 Will-be-set-by-IN-TECH

satisfactory experimental results showing the high performances of the proposed control

The PSO-tuned PID-type FC structures robustness, under external influences such as the output static disturbances and parametric uncertainties, are proven. The control design methodology is systematic, practical and simple without need to exact analytic plant model description. The obtained simulation and experimental results show the efficiency in terms of performance and robustness of the proposed fuzzy control approach which can be applied in

*Higher Insitute of Industrial Systems of Gabes (ISSIG), Salaheddine Elayoubi Street, 6032 Gabes,*

[1] Bouallègue, S., Haggège, J., Benrejeb, M., (2012). On a robust real-time H<sup>∞</sup> controller design for an electrical drive, International Journal of Modelling, Identification and

[2] Bouallègue, S., Haggège, J., Ayadi, M., Benrejeb, M., (2012). PID-type fuzzy logic controller tuning based on particle swarm optimization, Engineering Applications of

[3] Bouallègue, S., (2011). Optimisation par essaim particulaire de lois de commande robuste : théorie et mise en œuvre pratique, Editions Universitaires Européennes, ISBN

[4] Bouallègue, S., Haggège, J., Benrejeb, M., (2010). Structured Loop-Shaping H<sup>∞</sup> Controller Design using Particle Swarm Optimization. Proceedings of the 2010 IEEE

[6] Bouallègue, S., Haggège, J., Benrejeb, M., (2011). Particle Swarm Optimization-Based Fixed-Structure H<sup>∞</sup> Control Design. International Journal of Control, Automation, and

[7] Bouallègue, S., Haggège,J., and Benrejeb, M., (2009). Real-Time H<sup>∞</sup> Control Design for a Thermal Process, Proceedings of the 10th International conference on Sciences and Techniques of Automatic control & computer engineering STA'2009-ACS, pp. 949-958,

[8] Bühler, H., (1994). Réglage par logique floue, Presses polytechniques et universitaires

International Conference on Systems, Man, and Cybernetics SMC'10, Istanbul. [5] Bouallègue, S., Haggège, J., Benrejeb, M., (2010). Structured Mixed-Sensitivity H<sup>∞</sup> Design using Particle Swarm Optimization. Proceedings of the 7th IEEE International

Multi-Conference on Systems, Signals and Devices SSD'10, Amman.

*National Engineering School of Tunis (ENIT), BP 37, le Belvédère, 1002 Tunis, Tunisia*

All authors are with the Research Laboratory in Automatic Control (LA.R.A) of ENIT.

especially in terms of tracking and robustness.

industrial motor control field.

J. Haggège and M. Benrejeb

Control, 15 (2), pp. 89-96.

Systems 9 (2), pp. 258-266.

Hammamet, Tunisia.

romandes, Lausanne.

Artificial Intelligence, 25 (3), pp. 484-493.

: 978-613-1-59335-2, Saarbrücken, Germany.

**Author details**

S. Bouallègue

**6. References**

*Tunisia*

	- [28] Shi, Y., Eberhart, R., (1999). Empirical study of particle swarm optimization. Proceedings of the 1999 Congress on Evolutionary Computation, Washington, pp. 1945-1950.
	- [29] The MathWorks Inc., (2009). Genetic Algorithm and Direct Search ToolboxTM: User's Guide. Natick.
	- [30] Van den Bergh, F., (2006). An Analysis of Particle Swarm Optimizers. PhD Thesis, University of Pretoria, Pretoria, South Africa.
	- [31] Woo, Z-W., Chung, H-Y., Lin, J-J., (2000). A PID type fuzzy controller with self-tuning scaling factors. Fuzzy Sets and Systems 115, pp. 321-326.
	- [32] Yesil, E., Güzelkaya, M., Eksin, I., Tekin, O.A., (2008). Online Tuning of Set-point Regulator with a Blending Mechanism Using PI Controller, Turk J Elec Engin, 16 (2), pp. 143-157.

**1. Introduction**

http://dx.doi.org/10.5772/47138

Over the past decades, many advances have been made in the field of control theory which rely on state-space theory. The control design methodology that has been most investigated for the state-feedback control, see for example [1, 2] and the references therein. The state-feedback control design supposes that all the system states are available, which is not always possible in realistic applications. Instead, one has to deal with the absence of full-state information by using observers. From the control point of view, observers can be used as part of dynamical controllers. This observer-based design has been extensively studied in the literature [3, 4]. However, it leads to high-order controllers. As a matter of fact, one has to solve a large problem, which increases numerical computations for large scale systems. Other difficulties may arise, if we consider additional performances, such as disturbance rejection, time delays, uncertainties, etc. Hence, it is more suitable to develop methodologies which involve a design with a low dimensionality. In this context, intensive efforts have been devoted to design low-order controllers [3, 5–7]. In particular, it has been shown that designing reduced order stabilizing controllers can be cast as a static output-feedback stabilization problem. Also, it is recognized that, in general, the static output-feedback control design may not exist for certain systems. Note that an important advantage of these

**Output Tracking Control for Fuzzy Systems** 

**Chapter 7**

**via Static-Output Feedback Design** 

Meriem Nachidi and Ahmed El Hajjaji

Additional information is available at the end of the chapter

controllers is that they are easy to implement without significant numerical burden.

properly cited.

In general, the synthesis of static output-feedback stabilizing controllers is known to be a hard task [5–7]. The main difficulty rises from its nonconvexity. In the literature, some convexification techniques and iterative algorithms have been proposed to handle this problem [3, 5, 7]. A comprehensive survey on static output-feedback stabilization can be found in [6]. The authors show that despite the considerable efforts devoted to solve this problem, there is yet no methodology that can solve it exactly, so it is still an important open topic. However, it has been shown that for SISO (Single-Input Single-Output) systems, this problem can be solved exactly based on an algebraic characterization [8, 9]. Unfortunately, these approaches are valid only for SISO case and cannot be used to take into account additional constraints on the system. In any case, the investigation of this topic within the field of fuzzy control is continuously increasing and leading to many approaches. A most

and reproduction in any medium, provided the original work is properly cited.

©2012 Nachidi and El Hajjaji, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,


## **Output Tracking Control for Fuzzy Systems via Static-Output Feedback Design**

Meriem Nachidi and Ahmed El Hajjaji

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/47138

## **1. Introduction**

24 Will-be-set-by-IN-TECH

[28] Shi, Y., Eberhart, R., (1999). Empirical study of particle swarm optimization. Proceedings of the 1999 Congress on Evolutionary Computation, Washington, pp. 1945-1950. [29] The MathWorks Inc., (2009). Genetic Algorithm and Direct Search ToolboxTM: User's

[30] Van den Bergh, F., (2006). An Analysis of Particle Swarm Optimizers. PhD Thesis,

[31] Woo, Z-W., Chung, H-Y., Lin, J-J., (2000). A PID type fuzzy controller with self-tuning

[32] Yesil, E., Güzelkaya, M., Eksin, I., Tekin, O.A., (2008). Online Tuning of Set-point Regulator with a Blending Mechanism Using PI Controller, Turk J Elec Engin, 16 (2),

[33] Zadeh, L.A., (1994). Soft computing and fuzzy logic, IEEE Software, 11 (6), pp. 48-56.

[34] Zadeh, L.A., (1965). Fuzzy Sets, Information and Control, 8, pp. 338-353.

Guide. Natick.

pp. 143-157.

University of Pretoria, Pretoria, South Africa.

162 Fuzzy Controllers – Recent Advances in Theory and Applications

scaling factors. Fuzzy Sets and Systems 115, pp. 321-326.

Over the past decades, many advances have been made in the field of control theory which rely on state-space theory. The control design methodology that has been most investigated for the state-feedback control, see for example [1, 2] and the references therein. The state-feedback control design supposes that all the system states are available, which is not always possible in realistic applications. Instead, one has to deal with the absence of full-state information by using observers. From the control point of view, observers can be used as part of dynamical controllers. This observer-based design has been extensively studied in the literature [3, 4]. However, it leads to high-order controllers. As a matter of fact, one has to solve a large problem, which increases numerical computations for large scale systems. Other difficulties may arise, if we consider additional performances, such as disturbance rejection, time delays, uncertainties, etc. Hence, it is more suitable to develop methodologies which involve a design with a low dimensionality. In this context, intensive efforts have been devoted to design low-order controllers [3, 5–7]. In particular, it has been shown that designing reduced order stabilizing controllers can be cast as a static output-feedback stabilization problem. Also, it is recognized that, in general, the static output-feedback control design may not exist for certain systems. Note that an important advantage of these controllers is that they are easy to implement without significant numerical burden.

In general, the synthesis of static output-feedback stabilizing controllers is known to be a hard task [5–7]. The main difficulty rises from its nonconvexity. In the literature, some convexification techniques and iterative algorithms have been proposed to handle this problem [3, 5, 7]. A comprehensive survey on static output-feedback stabilization can be found in [6]. The authors show that despite the considerable efforts devoted to solve this problem, there is yet no methodology that can solve it exactly, so it is still an important open topic. However, it has been shown that for SISO (Single-Input Single-Output) systems, this problem can be solved exactly based on an algebraic characterization [8, 9]. Unfortunately, these approaches are valid only for SISO case and cannot be used to take into account additional constraints on the system. In any case, the investigation of this topic within the field of fuzzy control is continuously increasing and leading to many approaches. A most

©2012 Nachidi and El Hajjaji, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2 Will-be-set-by-IN-TECH 164 Fuzzy Controllers – Recent Advances in Theory and Applications Output Tracking Control for Fuzzy Systems Via Static-Output Feedback Design <sup>3</sup>

efficient approach is based on the Linear Matrix Inequality (LMI) technique: see for example [10–12]. Indeed, since the developed interior-point methods [13], LMIs can be solved in polynomial-time, using numerical algorithms [14]. Recently, other approach, based on a projective algorithm has been proposed [15]. Notice that the existing LMI tools have opened an important research area in system and control theory and tackled numerous unsolved problems [14]. Therefore, our main focus in this chapter is the design of static output-feedback controllers using LMI theory for a class of nonlinear systems described by Takagi-Sugeno (T-S) fuzzy models.

to static output-feedback control synthesis, a cone complementarity formulation [7] for T-S fuzzy systems is used combined with an iterative algorithm. This algorithm has to optimize a linear objective function subject to a set of LMIs in each iteration. Thus, controllers are derived that not only ensure stability of the closed-loop system, but also provide a prescribed level of

The main contribution of this chapter is the purpose of a simple procedure reflected by an efficient LMI-based iterative algorithm to solve the fuzzy tracking control problem for nonlinear systems described by discrete-time T-S models. Therefore, since the proposed fuzzy tracking controllers have a low-order character, they are suitable for industrial application. Furthermore, this chapter shows an application to a relevant practical problem, in power engineering and drives field, of the proposed design procedure: guaranteeing a good tracking

Consider a nonlinear system which is approximated by a T-S fuzzy model of the following

where *<sup>x</sup>*(*k*) ∈ �*<sup>n</sup>* is the state vector, *<sup>u</sup>*(*k*) ∈ �*nu* is the input vector, *<sup>w</sup>*(*k*) ∈ �*nw* comprises the bounded external disturbances and *<sup>y</sup>*(*k*) ∈ �*ny* is the system output. *<sup>N</sup>* is the number of IF-THEN rules. *z*1(*k*),... *zp*(*k*) are the premise variables (that comprises states and/or inputs)

where *Mi*, *N*1*<sup>i</sup>* and *N*2*<sup>i</sup>* are known real constant matrices. *F*(*k*) is the uncertainty function that

Thus, the global T-S model is an interpolation of all uncertain subsystems through nonlinear

*N* ∑ *i*=1

*<sup>j</sup>* (*i* = 1, . . . , *N*, *j* = 1, . . . , *p*) are the fuzzy sets. *Ai*, *Bi*, *Ci* and *Ei* are known constant matrices of appropriate size, Δ*Ai*(*k*), Δ*Bi*(*k*) are unknown matrices representing time-varying

[Δ*Ai*(*k*) Δ*Bi*(*k*)] = [*M*1*F*(*k*)*N*1*<sup>i</sup> M*2*F*(*k*)*N*2*i*], *i* = 1, 2, . . . , *N*, (2)

*θi*(*z*)[(*Ai* + Δ*Ai*(*k*))*x*(*k*)+(*Bi* + Δ*Bi*(*k*))*u*(*k*) + *Eiw*(*k*)]

*θi*(*z*)

*αi*(*z*)[(*Ai* + Δ*Ai*(*k*))*x*(*k*)+(*Bi* + Δ*Bi*(*k*))*u*(*k*) + *Eiw*(*k*)] ,

*<sup>F</sup>*(*k*)*TF*(*k*) <sup>≤</sup> *<sup>I</sup>*, <sup>∀</sup>*k*. (3)

,

(4)

*p*,

Output Tracking Control for Fuzzy Systems via Static-Output Feedback Design 165

(1)

*x*(*k* + 1)=(*Ai* + Δ*Ai*(*k*))*x*(*k*)+(*Bi* + Δ*Bi*(*k*))*u*(*k*) + *Eiw*(*k*),

<sup>1</sup> and . . . and *zp*(*k*) is *<sup>μ</sup><sup>i</sup>*

*y*(*k*) = *Cix*(*k*), *i* = 1, . . . , *N*,

parameter uncertainties, and are assumed to be as follows:

output tracking error attenuation.

form:

and *μ<sup>i</sup>*

functions [16]:

*i*

THEN

of the output voltage of DC-DC buck converter [24–26].

**2. Problem formulation and preliminaries**

*th*R*ule:* IF *<sup>z</sup>*1(*k*) is *<sup>μ</sup><sup>i</sup>*

⎧ ⎨ ⎩

satisfies the classical bounded condition:

*N* ∑ *i*=1

*x*(*k* + 1) =

= *N* ∑ *i*=1

*N* ∑ *i*=1

*αi*(*z*)*Cix*(*k*),

*y*(*k*) =

Recently, the study of T-S fuzzy models has attracted the attention of many of researchers: see [16] and references therein. Fuzzy models have local dynamics (i.e., dynamics in different state space regions), that are represented by local linear systems. The overall model of a fuzzy system is then obtained by interpolating these linear models through nonlinear fuzzy membership functions. Unlike conventional modeling techniques, which use a single model to describe the global behavior of a nonlinear system, fuzzy modeling is essentially a multi-model approach, in which simple local linear submodels are designed in the form of a convex combination of local models in order to describe the global behavior of the nonlinear system. This kind of models has proved to be a good representation for a certain class of nonlinear dynamic systems.

Since the work by [17] on stability analysis and state feedback stabilization for fuzzy systems, the Parallel Distributed Compensation (PDC) procedure has extensively been used for the control of such systems: for more details see [16]. The basic idea of this procedure is to design a feedback gain for each local model, and then to construct a global controller from these local gains, so that the global stability of the overall fuzzy system can be guaranteed. The most interesting of this concept is that the obtained stability conditions do not depend on the nonlinearities (membership functions), so that this makes possible to use linear system techniques for nonlinear control design.

Up to now, the stabilization control design for T-S systems is successfully investigated based on state-feedback or static/dynamic output-feedback [18, 19]. However, the design of a controller which guarantees an adequate tracking performance for finite-dimensional systems is more general problem than the stabilization one, and is still attract considerable attentions due to demand from practical dynamical processes in electric, mechanics, agriculture, . . . . One of our main interest in this chapter is solving the static output-feedback tracking problem. Due to the fact that the T-S fuzzy models aggregate a set of local linear subsystems, blended together through nonlinear scalar functions, the static output-feedback control problem can be very complicated to solve. With regard to the literature of fuzzy control, a few recent approaches have dealt with the tracking control design problem for nonlinear systems described by T-S fuzzy model. Generally speaking, the incorporation of linearization techniques and adaptive schemes usually needs system's perfect knowledge and leads to complicated adaptation control laws. In [20], the author has been shown that the use of the feedback linearization strategy [21] may lead to unbounded controllers, since their stability is not guaranteed. To overcome these drawbacks, LMI-based methodologies have been developed for tracking control problem, using observer-based fuzzy controller to deal with the absence of full-state information [22].

In this context, this chapter will tackle the static output-feedback fuzzy tracking control problem, focusing on an *H*∞ tracking performance, related to an output tracking error for all bounded references inputs. The presented results are an extension of already published works for the stabilization case [12, 23]. In fact, to solve the nonconvexity problem, inherent to static output-feedback control synthesis, a cone complementarity formulation [7] for T-S fuzzy systems is used combined with an iterative algorithm. This algorithm has to optimize a linear objective function subject to a set of LMIs in each iteration. Thus, controllers are derived that not only ensure stability of the closed-loop system, but also provide a prescribed level of output tracking error attenuation.

The main contribution of this chapter is the purpose of a simple procedure reflected by an efficient LMI-based iterative algorithm to solve the fuzzy tracking control problem for nonlinear systems described by discrete-time T-S models. Therefore, since the proposed fuzzy tracking controllers have a low-order character, they are suitable for industrial application. Furthermore, this chapter shows an application to a relevant practical problem, in power engineering and drives field, of the proposed design procedure: guaranteeing a good tracking of the output voltage of DC-DC buck converter [24–26].

#### **2. Problem formulation and preliminaries**

2 Will-be-set-by-IN-TECH

efficient approach is based on the Linear Matrix Inequality (LMI) technique: see for example [10–12]. Indeed, since the developed interior-point methods [13], LMIs can be solved in polynomial-time, using numerical algorithms [14]. Recently, other approach, based on a projective algorithm has been proposed [15]. Notice that the existing LMI tools have opened an important research area in system and control theory and tackled numerous unsolved problems [14]. Therefore, our main focus in this chapter is the design of static output-feedback controllers using LMI theory for a class of nonlinear systems described by Takagi-Sugeno (T-S)

Recently, the study of T-S fuzzy models has attracted the attention of many of researchers: see [16] and references therein. Fuzzy models have local dynamics (i.e., dynamics in different state space regions), that are represented by local linear systems. The overall model of a fuzzy system is then obtained by interpolating these linear models through nonlinear fuzzy membership functions. Unlike conventional modeling techniques, which use a single model to describe the global behavior of a nonlinear system, fuzzy modeling is essentially a multi-model approach, in which simple local linear submodels are designed in the form of a convex combination of local models in order to describe the global behavior of the nonlinear system. This kind of models has proved to be a good representation for a certain class of

Since the work by [17] on stability analysis and state feedback stabilization for fuzzy systems, the Parallel Distributed Compensation (PDC) procedure has extensively been used for the control of such systems: for more details see [16]. The basic idea of this procedure is to design a feedback gain for each local model, and then to construct a global controller from these local gains, so that the global stability of the overall fuzzy system can be guaranteed. The most interesting of this concept is that the obtained stability conditions do not depend on the nonlinearities (membership functions), so that this makes possible to use linear system

Up to now, the stabilization control design for T-S systems is successfully investigated based on state-feedback or static/dynamic output-feedback [18, 19]. However, the design of a controller which guarantees an adequate tracking performance for finite-dimensional systems is more general problem than the stabilization one, and is still attract considerable attentions due to demand from practical dynamical processes in electric, mechanics, agriculture, . . . . One of our main interest in this chapter is solving the static output-feedback tracking problem. Due to the fact that the T-S fuzzy models aggregate a set of local linear subsystems, blended together through nonlinear scalar functions, the static output-feedback control problem can be very complicated to solve. With regard to the literature of fuzzy control, a few recent approaches have dealt with the tracking control design problem for nonlinear systems described by T-S fuzzy model. Generally speaking, the incorporation of linearization techniques and adaptive schemes usually needs system's perfect knowledge and leads to complicated adaptation control laws. In [20], the author has been shown that the use of the feedback linearization strategy [21] may lead to unbounded controllers, since their stability is not guaranteed. To overcome these drawbacks, LMI-based methodologies have been developed for tracking control problem, using observer-based fuzzy controller to deal with

In this context, this chapter will tackle the static output-feedback fuzzy tracking control problem, focusing on an *H*∞ tracking performance, related to an output tracking error for all bounded references inputs. The presented results are an extension of already published works for the stabilization case [12, 23]. In fact, to solve the nonconvexity problem, inherent

fuzzy models.

nonlinear dynamic systems.

techniques for nonlinear control design.

the absence of full-state information [22].

Consider a nonlinear system which is approximated by a T-S fuzzy model of the following form:

$$i^{th} \mathcal{R} \underline{u} \underline{e} \text{: IF} \quad z\_1(k) \text{ is } \mu\_1^i \text{ and } \dots \text{ and } z\_p(k) \text{ is } \mu\_{p'}^i$$

$$\text{THEN}\left\{ \begin{aligned} \mathbf{x}(k+1) &= (A\_l + \Delta A\_l(k))\mathbf{x}(k) + (\mathcal{B}\_l + \Delta \mathcal{B}\_l(k))\boldsymbol{\mu}(k) + E\_l \boldsymbol{w}(k), \\\\ \mathbf{y}(k) &= \mathbf{C}\_l \mathbf{x}(k), i = 1, \dots, N\_\prime \end{aligned} \right. \tag{1}$$

where *<sup>x</sup>*(*k*) ∈ �*<sup>n</sup>* is the state vector, *<sup>u</sup>*(*k*) ∈ �*nu* is the input vector, *<sup>w</sup>*(*k*) ∈ �*nw* comprises the bounded external disturbances and *<sup>y</sup>*(*k*) ∈ �*ny* is the system output. *<sup>N</sup>* is the number of IF-THEN rules. *z*1(*k*),... *zp*(*k*) are the premise variables (that comprises states and/or inputs) and *μ<sup>i</sup> <sup>j</sup>* (*i* = 1, . . . , *N*, *j* = 1, . . . , *p*) are the fuzzy sets. *Ai*, *Bi*, *Ci* and *Ei* are known constant matrices of appropriate size, Δ*Ai*(*k*), Δ*Bi*(*k*) are unknown matrices representing time-varying parameter uncertainties, and are assumed to be as follows:

$$\left[\Delta A\_{\mathrm{i}}(\mathrm{k})\,\Delta B\_{\mathrm{i}}(\mathrm{k})\right] = \left[M\_{\mathrm{1}}F(\mathrm{k})N\_{\mathrm{1i}}\,M\_{\mathrm{2}}F(\mathrm{k})N\_{\mathrm{2i}}\right], \; \mathrm{i} = 1,2,\ldots,N,\tag{2}$$

where *Mi*, *N*1*<sup>i</sup>* and *N*2*<sup>i</sup>* are known real constant matrices. *F*(*k*) is the uncertainty function that satisfies the classical bounded condition:

$$F(k)^T F(k) \le I, \quad \forall k. \tag{3}$$

Thus, the global T-S model is an interpolation of all uncertain subsystems through nonlinear functions [16]:

$$\begin{aligned} \mathbf{x}(k+1) &= \frac{\sum\_{i=1}^{N} \theta\_i(z) \left[ (A\_i + \Delta A\_i(k)) \mathbf{x}(k) + (\mathcal{B}\_i + \Delta \mathcal{B}\_i(k)) \boldsymbol{u}(k) + \mathcal{E}\_i \mathbf{w}(k) \right]}{\sum\_{i=1}^{N} \theta\_i(z)}, \\ &= \sum\_{i=1}^{N} \mathbf{a}\_i(z) \left[ (A\_i + \Delta A\_i(k)) \mathbf{x}(k) + (\mathcal{B}\_i + \Delta \mathcal{B}\_i(k)) \boldsymbol{u}(k) + \mathcal{E}\_i \mathbf{w}(k) \right], \\ \mathbf{y}(k) &= \sum\_{i=1}^{N} \mathbf{a}\_i(z) \mathsf{C}\_i \mathbf{x}(k), \end{aligned} \tag{4}$$

$$\text{where } \theta\_{i\cdot} \, i = 1, \dots, N\_{\prime} \text{ is the membership function corresponding to system rule } i \text{, and } a\_i(z) = \theta\_i(z) / \sum\_{i=1}^{N} \theta\_i(z), \text{ fulfills the convex property: } 0 \le a\_i(z) \le 1 \text{ and } \sum\_{i=1}^{N} a\_i(z) = 1.$$

Note that using the so-called sector of nonlinearity approach, a wide number of nonlinear systems can be represented exactly by T-S models in a compact set of the state space. However, with the growing complexity of nonlinear systems, it is useful to take into account the approximations in the dynamical process. Thus, the main objective of the next paragraph is to provide stability conditions that ensure the tracking performance for the uncertain T-S model (4).

## **3.** *H*<sup>∞</sup> **output tracking performance analysis**

This section gives sufficient stability conditions which ensure an *H*∞ output tracking performance of the uncertain system (4) using a fuzzy Lyapunov function. We recall the following lemma which will be used in this section.

**Lemma 3.1.** *[27] Let* A, D, S, W *and* F *be real matrices of appropriate dimension such that* <sup>W</sup> <sup>&</sup>gt; <sup>0</sup> *and* F F*<sup>T</sup>* <sup>≤</sup> *I. Then, for any scalar �* <sup>&</sup>gt; <sup>0</sup> *such that* W − *�*DD*<sup>T</sup>* <sup>&</sup>gt; <sup>0</sup>*, we have* (<sup>A</sup> <sup>+</sup> DFS)*T*W−1(<sup>A</sup> <sup>+</sup> DFS) ≤ A*T*(W − *�*DD*T*)−1<sup>A</sup> <sup>+</sup> *�*−1<sup>S</sup> *<sup>T</sup>*S.

Suppose that the desired trajectory can be generated by the following reference model as follows:

$$\begin{cases} \mathbf{x}\_d(k+1) = \mathbb{A}\mathbf{x}\_d(k) + \mathbb{B}r(k), \\\\ \mathbf{y}\_d(k) = \mathbb{C}\mathbf{x}\_d(k), \end{cases} \tag{5}$$

Combining (4), (5) and (7), the following augmented closed-loop system is obtained

*Ai* + *BiKjCs* −*BiKj***C**

0 **A**

⎡ ⎣ *x*(*k*)

⎤ ⎦ , *w*˜ =

*xd*(*k*)

Hence, to meet the required tracking performance, the effect of *w*˜(*k*) on the tracking error

The following theorem shows that *H*∞ output tracking performances can be guaranteed if

**Theorem 3.1.** *The augmented closed-loop system in (8) achieves the H*∞ *output tracking performance γ, if there exists matrices P*<sup>1</sup> > 0, . . . , *PN* > 0 *and controller gains K*1, ..., *KN such that the*

> ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

�

⎤ ⎥ ⎦ ,

Δ*Ai*(*k*) + Δ*Bi*(*k*)*KjCs* −Δ*Bi*(*k*)*Kj***C**

0 0

⎡ ⎣ *w*(*k*)

⎤ ⎦ ,

*r*(*k*)

*k f* ∑ *k*=0

(*G*1*ijs* + *G*2*ijs*(*k*))*x*˜(*k*) + *Wiw*˜(*k*)

Output Tracking Control for Fuzzy Systems via Static-Output Feedback Design 167

⎤ ⎥ ⎦ , �

*w*˜(*k*)*Tw*˜(*k*), (10)

< 0, 1 ≤ *i*, *j*,*s*,*r* ≤ *N*, (11)

, (8)

(9)

*αi*(*z*)*αj*(*z*)*αs*(*z*)

*x*˜(*k* + 1) =

where

*N* ∑ *i*,*j*,*s*=1

> ⎡ ⎢ ⎣

> > ⎡ ⎢ ⎣

*G*1*ijs* =

*G*2*ijs*(*k*) =

⎡ ⎣ *Ei* 0

⎤ ⎦ , *x*˜ =

*y*(*k*) − *yd*(*k*) should be attenuated below a desired level in the sense of [29]:

(*y*(*k*) <sup>−</sup> *yd*(*k*))*T*(*y*(*k*) <sup>−</sup> *yd*(*k*)) <sup>≤</sup> *<sup>γ</sup>*<sup>2</sup>

0 **B**

*Wi* =

*k f* ∑ *k*=0

*following conditions hold:*

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

*G<sup>T</sup>* <sup>1</sup>*ijs <sup>N</sup>*˜ *<sup>T</sup>*

*W<sup>T</sup>*

∀*k <sup>f</sup>* �= 0, and ∀*w*˜(*k*) ∈ *l*2, *k <sup>f</sup>* is the control final time.

there exist some matrices satisfying certain conditions.

<sup>−</sup>*P*−<sup>1</sup> *<sup>r</sup>* 0 0 *<sup>G</sup>*1*ijs Wi <sup>M</sup>*˜

<sup>0</sup> <sup>−</sup>*�<sup>I</sup>* <sup>0</sup> *<sup>N</sup>*˜*ijs* 0 0

0 0 −*I Hi* 0 0

*<sup>i</sup>* 00 0 <sup>−</sup>*γ*<sup>2</sup> *<sup>I</sup>* <sup>0</sup>

*<sup>M</sup>*˜ *<sup>T</sup>* 00 0 0 <sup>−</sup>*�*−1*<sup>I</sup>*

*<sup>i</sup>* −*Pi* 0 0

*ijs <sup>H</sup><sup>T</sup>*

where, *yd*(*k*) has the same dimension as *<sup>y</sup>*(*k*), *xd*(*k*) and *<sup>r</sup>*(*k*) ∈ �*nr* are respectively the reference state and the bounded reference input, **A**, **B** and **C** are appropriately dimensional constant matrices with **A** Hurwitz.

Since we deal with the static output-feedback control design problem, the fuzzy controller can incorporates information from *y*(*k*) and *yd*(*k*). Thus, the control law which is based on the classical structure of the Parallel Distributed Compensation (PDC) concept [17, 28] shares the same fuzzy sets as the T-S system and can be given as follows:

$$\begin{aligned} \text{i}^{\text{th}} \mathcal{R} & \text{iF} \quad z\_1(k) \text{ is } \mu\_1^i \text{ and } \dots \text{ and } z\_p(k) \text{ is } \mu\_{p\prime}^i\\ \text{THEN} \qquad u(k) &= \mathcal{K}\_l(y(k) - y\_d(k)), \end{aligned} \tag{6}$$

where the the controller gain *Ki* is to be chosen. The overall static output-feedback control law is thus inferred as:

$$\mu(k) = \sum\_{i=1}^{N} \mathfrak{a}\_i(z)\mathcal{K}\_i(y(k) - y\_d(k)). \tag{7}$$

The advantages of the static output-feedback controller (7), is well discussed in the literature [3], [6]. This fact motivates us to use such type of control law avoiding the complex control schemes with an additional observer.

Combining (4), (5) and (7), the following augmented closed-loop system is obtained

$$\mathfrak{x}(k+1) = \sum\_{i,j,s=1}^{N} a\_i(z)a\_j(z)a\_s(z) \left[ (\mathcal{G}\_{1ijs} + \mathcal{G}\_{2ijs}(k))\mathfrak{x}(k) + W\_i\mathfrak{w}(k) \right],\tag{8}$$

where

4 Will-be-set-by-IN-TECH

where *θi*, *i* = 1, . . . , *N*, is the membership function corresponding to system rule *i*, and *αi*(*z*) =

Note that using the so-called sector of nonlinearity approach, a wide number of nonlinear systems can be represented exactly by T-S models in a compact set of the state space. However, with the growing complexity of nonlinear systems, it is useful to take into account the approximations in the dynamical process. Thus, the main objective of the next paragraph is to provide stability conditions that ensure the tracking performance for the uncertain T-S

This section gives sufficient stability conditions which ensure an *H*∞ output tracking performance of the uncertain system (4) using a fuzzy Lyapunov function. We recall the

**Lemma 3.1.** *[27] Let* A, D, S, W *and* F *be real matrices of appropriate dimension such that* <sup>W</sup> <sup>&</sup>gt; <sup>0</sup> *and* F F*<sup>T</sup>* <sup>≤</sup> *I. Then, for any scalar �* <sup>&</sup>gt; <sup>0</sup> *such that* W − *�*DD*<sup>T</sup>* <sup>&</sup>gt; <sup>0</sup>*, we have*

Suppose that the desired trajectory can be generated by the following reference model as

*xd*(*k* + 1) = **A***xd*(*k*) + **B***r*(*k*),

where, *yd*(*k*) has the same dimension as *<sup>y</sup>*(*k*), *xd*(*k*) and *<sup>r</sup>*(*k*) ∈ �*nr* are respectively the reference state and the bounded reference input, **A**, **B** and **C** are appropriately dimensional

Since we deal with the static output-feedback control design problem, the fuzzy controller can incorporates information from *y*(*k*) and *yd*(*k*). Thus, the control law which is based on the classical structure of the Parallel Distributed Compensation (PDC) concept [17, 28] shares

THEN *u*(*k*) = *Ki*(*y*(*k*) − *yd*(*k*)),

where the the controller gain *Ki* is to be chosen. The overall static output-feedback control law

The advantages of the static output-feedback controller (7), is well discussed in the literature [3], [6]. This fact motivates us to use such type of control law avoiding the complex control

<sup>1</sup> and . . . and *zp*(*k*) is *<sup>μ</sup><sup>i</sup>*

*p*,

*αi*(*z*)*Ki*(*y*(*k*) − *yd*(*k*)). (7)

*yd*(*k*) = **C***xd*(*k*),

*N* ∑ *i*=1

*αi*(*z*) = 1.

(5)

(6)

*θi*(*z*), fulfills the convex property: 0 ≤ *αi*(*z*) ≤ 1 and

**3.** *H*<sup>∞</sup> **output tracking performance analysis**

following lemma which will be used in this section.

constant matrices with **A** Hurwitz.

schemes with an additional observer.

is thus inferred as:

*i*

(<sup>A</sup> <sup>+</sup> DFS)*T*W−1(<sup>A</sup> <sup>+</sup> DFS) ≤ A*T*(W − *�*DD*T*)−1<sup>A</sup> <sup>+</sup> *�*−1<sup>S</sup> *<sup>T</sup>*S.

⎧ ⎨ ⎩

the same fuzzy sets as the T-S system and can be given as follows:

*u*(*k*) =

*N* ∑ *i*=1

*th*R*ule:* IF *<sup>z</sup>*1(*k*) is *<sup>μ</sup><sup>i</sup>*

*θi*(*z*)/

model (4).

follows:

*N* ∑ *i*=1

$$\begin{aligned} \mathbf{G}\_{1ijs} &= \begin{bmatrix} A\_i + B\_i K\_j \mathbf{C}\_s - B\_i K\_j \mathbf{C} \\\\ 0 & \mathbf{A} \end{bmatrix}, \\\\ \mathbf{G}\_{2ijs}(k) &= \begin{bmatrix} \Delta A\_i(k) + \Delta B\_i(k) K\_j \mathbf{C}\_s - \Delta B\_i(k) K\_j \mathbf{C} \\\\ 0 & 0 \end{bmatrix}, \\\\ \mathbf{W}\_i &= \begin{bmatrix} E\_i \ 0 \\\\ 0 \ \mathbf{B} \end{bmatrix}, \tilde{\mathbf{x}} = \begin{bmatrix} \mathbf{x}(k) \\\\ \mathbf{x}\_d(k) \end{bmatrix}, \tilde{w} = \begin{bmatrix} w(k) \\\\ r(k) \end{bmatrix}, \end{aligned} \tag{9}$$

Hence, to meet the required tracking performance, the effect of *w*˜(*k*) on the tracking error *y*(*k*) − *yd*(*k*) should be attenuated below a desired level in the sense of [29]:

$$\sum\_{k=0}^{k\_f} \left( y(k) - y\_d(k) \right)^T (y(k) - y\_d(k)) \le \gamma^2 \sum\_{k=0}^{k\_f} \tilde{w}(k)^T \tilde{w}(k),\tag{10}$$

∀*k <sup>f</sup>* �= 0, and ∀*w*˜(*k*) ∈ *l*2, *k <sup>f</sup>* is the control final time.

The following theorem shows that *H*∞ output tracking performances can be guaranteed if there exist some matrices satisfying certain conditions.

**Theorem 3.1.** *The augmented closed-loop system in (8) achieves the H*∞ *output tracking performance γ, if there exists matrices P*<sup>1</sup> > 0, . . . , *PN* > 0 *and controller gains K*1, ..., *KN such that the following conditions hold:*

$$\begin{cases} \begin{bmatrix} -P\_{r}^{-1} & 0 & 0 & G\_{1ijs} & W\_{i} & \bar{M} \\\\ 0 & -\epsilon I & 0 & \bar{N}\_{ijs} & 0 & 0 \\\\ 0 & 0 & -I & H\_{i} & 0 & 0 \\\\ G\_{1ijs}^{T} & \bar{N}\_{ijs}^{T} \ H\_{i}^{T} & -P\_{i} & 0 & 0 \\\\ \end{bmatrix} < 0, \quad 1 \le i, j, s, r \le N,\tag{11} $$

*where*

*<sup>G</sup>*1*ijs and Wi are defined in (9), Hi* = [*Ci* <sup>−</sup> **<sup>C</sup>**] *, <sup>M</sup>*˜ <sup>=</sup> ⎡ ⎣ *M*<sup>1</sup> *M*<sup>2</sup> 0 0 ⎤ <sup>⎦</sup> *and <sup>N</sup>*˜*ijs* <sup>=</sup> ⎡ ⎣ *N*1*<sup>i</sup>* 0 *N*2*iKjCs* −*N*2*iKj***C** ⎤ ⎦ .

*Proof.* Consider the following fuzzy Lyapunov function *V*(*x*˜, *k*) given by

$$V(\mathfrak{x},k) = \mathfrak{x}(k)^T \sum\_{i=1}^N \alpha\_i(z) P\_i \mathfrak{x}(k).$$

The stability of (8) is ensured, under zero initial condition, with guaranteed *H*∞ performance (10) if [29]:

$$
\Delta V(\vec{x},k) + (y(k) - y\_d(k))^T (y(k) - y\_d(k)) - \gamma^2 \vec{w}(k)^T \vec{w}(k) < 0 \tag{12}
$$

where Δ*V*(*x*˜, *k*) is the rate of *V* along the trajectory:

$$
\Delta V(\mathfrak{x}, k) = V(\mathfrak{x}(k+1)) - V(\mathfrak{x}(k)).\tag{13}
$$

where

where

Γ*r ijs* ≡ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

*G<sup>T</sup>* <sup>1</sup>*ijsPr <sup>N</sup>*˜ *<sup>T</sup>*

*W<sup>T</sup>*

M<sup>1</sup> =

M<sup>2</sup> =

*Hz* =

Thus, to proof (12), it is sufficient to show that

The first part of (19) can also be rewritten as

*G*˜ *<sup>z</sup>* = �

*and* N*<sup>z</sup>* =

*G*1*<sup>z</sup> Wz* � , *G*1*<sup>z</sup>* =

> � *N* ∑ *i*,*j*,*s*=1

On the other hand, pre- and post-multiplying (11) by *diag*{*Pr*, *I*, *I*, *I*, *I*, *I*} gives

<sup>−</sup>*Pr* 0 0 *PrG*1*ijs PrWi PrM*˜

<sup>0</sup> <sup>−</sup>*�<sup>I</sup>* <sup>0</sup> *<sup>N</sup>*˜*ijs* 0 0

0 0 −*I Hi* 0 0

*<sup>i</sup> Pr* 00 0 <sup>−</sup>*γ*<sup>2</sup> *<sup>I</sup>* <sup>0</sup>

*<sup>M</sup>*˜ *TPr* 00 0 0 <sup>−</sup>*�*−<sup>1</sup> *<sup>I</sup>*

*<sup>i</sup>* −*Pi* 0 0

*ijs <sup>H</sup><sup>T</sup>*

⎡ ⎢ ⎣

⎡ ⎢ ⎣

*N* ∑ *i*=1 *W<sup>T</sup>*

*Pz* <sup>−</sup> *<sup>H</sup><sup>T</sup>*

*αi*(*z*)*Hi*.

*Gz*(*k*)*TP*<sup>+</sup>*Gz*(*k*) *Gz*(*k*)*TP*<sup>+</sup>*Wz*

⎤ ⎥ ⎦ , *<sup>z</sup> P*+*Wz*

<sup>M</sup><sup>1</sup> − M<sup>2</sup> = (*G*˜ *<sup>z</sup>* <sup>+</sup> *<sup>M</sup>*˜ *<sup>F</sup>*(*k*)N*z*)*TP*+(*G*˜ *<sup>z</sup>* <sup>+</sup> *<sup>M</sup>*˜ *<sup>F</sup>*(*k*)N*z*), (20)

*αi*(*z*)*αj*(*z*)*αs*(*z*)*G*1*ijs*,

� .

< 0, 1 ≤ *i*, *j*,*s*,*r* ≤ *N*. (22)

*N* ∑ *i*,*j*,*s*=1

*αi*(*z*)*αj*(*z*)*αs*(*z*)*N*˜ <sup>1</sup>*ijs* 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎥ ⎦ ,

Output Tracking Control for Fuzzy Systems via Static-Output Feedback Design 169

M<sup>1</sup> − M<sup>2</sup> < 0. (19)

(18)

(21)

*<sup>z</sup> P*+*Gz*(*k*) *W<sup>T</sup>*

*<sup>z</sup> Hz* 0

0 *γ*<sup>2</sup>

By substituting (13) in(12), we have:

*<sup>x</sup>*˜(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>)*TP*<sup>+</sup>*x*˜(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> *<sup>x</sup>*˜(*k*)*TPzx*˜(*k*)+(*y*(*k*) <sup>−</sup> *yd*(*k*))*T*(*y*(*k*) <sup>−</sup> *yd*(*k*)) <sup>−</sup> *<sup>γ</sup>*2*w*˜(*k*)*Tw*˜(*k*) <sup>&</sup>lt; 0,(14)

where

$$P\_z = \sum\_{i=1}^{N} \alpha\_i(z) P\_i \text{ and } P^+ = \sum\_{i=1}^{N} \alpha\_i(z(k+1)) P\_i.$$

Now, let

$$\begin{aligned} \mathbf{G}\_z(k) &= \sum\_{i,j,s=1}^N a\_i(z)a\_j(z)a\_s(z)\mathbf{G}\_{1ijs} + \sum\_{i,j,s=1}^N a\_i(z)a\_j(z)a\_s(z)\mathbf{G}\_{2ijs}(k), \\\\ \mathbf{W}\_z &= \sum\_{i=1}^N a\_i(z)\mathbf{W}\_i. \end{aligned} \tag{15}$$

Then, the inequality (14) can be rewritten as follows

$$\begin{cases} \left[\mathbf{G}\_{\overline{z}}(k)\tilde{\mathbf{x}}(k) + \mathbf{W}\_{\overline{z}}\tilde{w}(k)\right]^{\mathrm{T}}P^{+} \left[\mathbf{G}\_{\overline{z}}(k)\tilde{\mathbf{x}}(k) + \mathbf{W}\_{\overline{z}}\tilde{w}(k)\right] - \tilde{\mathbf{x}}(k)^{\mathrm{T}}P\_{2}\tilde{\mathbf{x}}(k) - \gamma^{2}\tilde{w}(k)^{\mathrm{T}}\tilde{w}(k) + \\\\ \left(y(k) - y\_{d}(k)\right)^{\mathrm{T}}(y(k) - y\_{d}(k)) < 0. \end{cases} \tag{16}$$

By consequence, (16) leads to:

$$
\begin{bmatrix} \tilde{\mathfrak{x}}(k) \\\\ \tilde{w}(k) \end{bmatrix}^T (\mathcal{M}\_1 - \mathcal{M}\_2) \begin{bmatrix} \tilde{\mathfrak{x}}(k) \\\\ \tilde{w}(k) \end{bmatrix} < 0,\tag{17}
$$

where

6 Will-be-set-by-IN-TECH

*N* ∑ *i*=1

The stability of (8) is ensured, under zero initial condition, with guaranteed *H*∞ performance

*<sup>x</sup>*˜(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>)*TP*<sup>+</sup>*x*˜(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> *<sup>x</sup>*˜(*k*)*TPzx*˜(*k*)+(*y*(*k*) <sup>−</sup> *yd*(*k*))*T*(*y*(*k*) <sup>−</sup> *yd*(*k*)) <sup>−</sup> *<sup>γ</sup>*2*w*˜(*k*)*Tw*˜(*k*) <sup>&</sup>lt; 0,(14)

*N* ∑ *i*=1

*N* ∑ *i*,*j*,*s*=1

*<sup>T</sup> <sup>P</sup>*<sup>+</sup> [*Gz*(*k*)*x*˜(*k*) + *Wzw*˜(*k*)] <sup>−</sup> *<sup>x</sup>*˜(*k*)*TPzx*˜(*k*) <sup>−</sup> *<sup>γ</sup>*2*w*˜(*k*)*Tw*˜(*k*)+

*<sup>α</sup>i*(*z*)*Pi* and *<sup>P</sup>*<sup>+</sup> =

*αi*(*z*)*αj*(*z*)*αs*(*z*)*G*1*ijs* +

*αi*(*z*)*Pix*˜(*k*).

<sup>Δ</sup>*V*(*x*˜, *<sup>k</sup>*)+(*y*(*k*) <sup>−</sup> *yd*(*k*))*T*(*y*(*k*) <sup>−</sup> *yd*(*k*)) <sup>−</sup> *<sup>γ</sup>*2*w*˜(*k*)*Tw*˜(*k*) <sup>&</sup>lt; <sup>0</sup> (12)

Δ*V*(*x*˜, *k*) = *V*(*x*˜(*k* + 1)) − *V*(*x*˜(*k*)). (13)

*αi*(*z*(*k* + 1))*Pi*.

*αi*(*z*)*αj*(*z*)*αs*(*z*)*G*2*ijs*(*k*),

<sup>⎦</sup> < 0, (17)

⎡ ⎣

*M*<sup>1</sup> *M*<sup>2</sup>

⎤

<sup>⎦</sup> *and <sup>N</sup>*˜*ijs* <sup>=</sup>

(15)

(16)

0 0

*<sup>G</sup>*1*ijs and Wi are defined in (9), Hi* = [*Ci* <sup>−</sup> **<sup>C</sup>**] *, <sup>M</sup>*˜ <sup>=</sup>

*Proof.* Consider the following fuzzy Lyapunov function *V*(*x*˜, *k*) given by

*V*(*x*˜, *k*) = *x*˜(*k*)*<sup>T</sup>*

*where*

⎡ ⎣

*N*1*<sup>i</sup>* 0

(10) if [29]:

where

Now, let

*N*2*iKjCs* −*N*2*iKj***C**

⎤ ⎦ .

where Δ*V*(*x*˜, *k*) is the rate of *V* along the trajectory:

*Pz* =

*αi*(*z*)*Wi*.

Then, the inequality (14) can be rewritten as follows

⎡ ⎣ *x*˜(*k*)

⎤ ⎦ *T*

(M<sup>1</sup> − M2)

⎡ ⎣ *x*˜(*k*)

⎤

*w*˜(*k*)

*w*˜(*k*)

*N* ∑ *i*,*j*,*s*=1

*N* ∑ *i*=1

By substituting (13) in(12), we have:

*Gz*(*k*) =

*N* ∑ *i*=1

(*y*(*k*) <sup>−</sup> *yd*(*k*))*T*(*y*(*k*) <sup>−</sup> *yd*(*k*)) <sup>&</sup>lt; 0.

*Wz* =

[*Gz*(*k*)*x*˜(*k*) + *Wzw*˜(*k*)]

By consequence, (16) leads to:

$$\begin{aligned} \mathcal{M}\_1 &= \begin{bmatrix} \mathcal{G}\_z(k)^T P^+ \mathcal{G}\_z(k) \ \mathcal{G}\_z(k)^T P^+ W\_z \\\\ W\_z^T P^+ \mathcal{G}\_z(k) & W\_z^T P^+ W\_z \end{bmatrix}' \\\\ \mathcal{M}\_2 &= \begin{bmatrix} P\_z - H\_z^T H\_z \ \mathbf{0} \\\\ \mathbf{0} & \gamma^2 \end{bmatrix}' \\\\ H\_z &= \sum\_{i=1}^N \boldsymbol{\alpha}\_i(z) H\_i \end{aligned} \tag{18}$$

Thus, to proof (12), it is sufficient to show that

*i*=1

$$
\mathcal{M}\_1 - \mathcal{M}\_2 < 0. \tag{19}
$$

The first part of (19) can also be rewritten as

$$\mathcal{M}\_1 - \mathcal{M}\_2 = (\tilde{\mathcal{G}}\_z + \tilde{M}F(k)\mathcal{N}\_z)^T P^+ (\tilde{\mathcal{G}}\_z + \tilde{M}F(k)\mathcal{N}\_z)\_\prime \tag{20}$$

where

$$\begin{aligned} \tilde{\mathbf{G}}\_z &= \left[ \mathbf{G}\_{1z} \ \mathbf{W}\_z \right] \mathbf{G}\_{1z} = \sum\_{i,j,s=1}^N \mathbf{a}\_i(z) \mathbf{a}\_j(z) \mathbf{a}\_s(z) \mathbf{G}\_{1 \text{ij}s}, \\\\mathbf{\mathcal{N}}\_z &= \left[ \sum\_{i,j,s=1}^N \mathbf{a}\_i(z) \mathbf{a}\_j(z) \mathbf{a}\_s(z) \mathbf{N}\_{1 \text{ij}s}, \mathbf{0} \right]. \end{aligned} \tag{21}$$

On the other hand, pre- and post-multiplying (11) by *diag*{*Pr*, *I*, *I*, *I*, *I*, *I*} gives

$$
\Gamma\_{ijs}^{r} \equiv \begin{bmatrix} -P\_r & 0 & 0 & P\_r G\_{1ijs} & P\_r W\_i & P\_r \tilde{M} \\\\ 0 & -\epsilon I & 0 & \tilde{N}\_{ljs} & 0 & 0 \\\\ 0 & 0 & -I & H\_i & 0 & 0 \\\\ G\_{1ijs}^T P\_r \ \tilde{N}\_{ljs}^T \ H\_i^T & -P\_l & 0 & 0 \\\\ \end{bmatrix} \\ \\ \\
\begin{aligned} \left. \begin{aligned} \begin{aligned} 0 \\ \end{aligned} \right|\_{l} \\ \left. \begin{aligned} \begin{aligned} \left. \begin{Bmatrix} 0 \\ \end{Bmatrix} \end{Bmatrix} \right|\_{l} \end{aligned} \\ \begin{aligned} \left. \begin{aligned} \begin{Bmatrix} 0 \\ \end{Bmatrix} \end{aligned} \\ \begin{aligned} \begin{Bmatrix} 0 \\ \end{Bmatrix} \end{Bmatrix} \end{aligned} \\ \begin{aligned} \begin{Bmatrix} \begin{Bmatrix} 0 \\ \end{Bmatrix} \end{Bmatrix} \end{aligned} \\ \begin{aligned} \begin{Bmatrix} \begin{Bmatrix} 0 \\ \end{Bmatrix} \end{Bmatrix} \end{aligned} \\ \begin{aligned} \begin{Bmatrix} \begin{Bmatrix} 0 \\ \end{Bmatrix} \end{Bmatrix} \end{aligned} \end{cases} \end{cases} \end{cases} \begin{aligned} \begin{aligned} \begin{Bmatrix} \begin{Bmatrix} 0 \\ \end{Bmatrix} \end{aligned} \\ \begin{aligned} \begin{Bmatrix} \begin{Bmatrix} 0 \\ \end{Bmatrix} \end{Bmatrix} \end{aligned} \\ \begin{aligned} \begin{Bmatrix} \begin{Bmatrix} 0 \\ \end{Bmatrix} \end{Bmatrix} \end{aligned} \\ \begin{aligned} \begin{Bmatrix} \begin{Bmatrix} 0 \\ \end{Bmatrix} \end{Bmatrix} \end{aligned} \\ \begin{aligned} \begin{Bmatrix} \begin{Bmatrix} 0 \\ \end{Bmatrix} \end{Bmatrix} \end{$$

8 Will-be-set-by-IN-TECH 170 Fuzzy Controllers – Recent Advances in Theory and Applications Output Tracking Control for Fuzzy Systems Via Static-Output Feedback Design <sup>9</sup>

$$\begin{aligned} \text{Since } \sum\_{i=1}^{N} a\_i(z) = \sum\_{r=1}^{N} a\_r(k+1) = 1, \text{(22) can be written as} \\\\ \sum\_{r=1}^{N} a\_r(k+1) \sum\_{i,j,s=1}^{N} a\_i(z)a\_j(z)a\_s(z) \Gamma\_{ijs}^r &\equiv \begin{bmatrix} -P^+ & 0 & 0 & P^+ G\_{1z} & P^+ W\_z & P^+ \tilde{M} \\\\ 0 & -\varepsilon I & 0 & \bar{N}\_z & 0 & 0 \\\\ 0 & 0 & -I & Hz & 0 & 0 \\\\ \bar{G}\_{1z}^T P^+ & \bar{N}\_z^T & H\_z^T & -P\_z & 0 & 0 \\\\ \bar{M}\_z^T P^+ & 0 & 0 & 0 & -\gamma^2 I & 0 \\\\ \bar{M}^T P^+ & 0 & 0 & 0 & 0 & -\varepsilon^{-1} I \end{bmatrix} < 0,\text{ (23)} \end{aligned}$$

*(a):PiQi* = *I*, *i* = 1, . . . , *N.*

⎤

Ω*r ijs* ≡

**Tr**(*PiQi*) = *N* × *n*,

⎦ ≥ 0, 1 ≤ *i* ≤ *N*.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

*G<sup>T</sup>* <sup>1</sup>*ijs <sup>N</sup>*˜ *<sup>T</sup>*

*W<sup>T</sup>*

**Lemma 4.2.** *[12] Consider the following matrix* **A**¯ =

*N* ∑ *s*>*j*≥*i*

> *N* ∑ *s*>*j*≥*i*

Hence, using Lemma 4.2, (26) can be rewritten as follows:

Υ*r*

Φ*r*

Ψ*r*

1*. Then,* **A**¯ *can be expressed as follows*

**A**¯ = *N* ∑ *i*=1 *α*3 *<sup>i</sup>* **A***iii* +

*Moreover, N* ∑ *ijs*=1

*αijs* =

*N* ∑ *i*=1 *α*3 *<sup>i</sup>* + 3

Using *Pr* = *Q*−<sup>1</sup> *<sup>r</sup>* , the stability condition (11) can be rewritten as follows:

<sup>−</sup>*Qr* 0 0 *<sup>G</sup>*1*ijs Wi <sup>M</sup>*˜

<sup>0</sup> <sup>−</sup>*�<sup>I</sup>* <sup>0</sup> *<sup>N</sup>*˜*ijs* 0 0

0 0 −*I Hi* 0 0

*<sup>i</sup>* 00 0 <sup>−</sup>*γ*<sup>2</sup> *<sup>I</sup>* <sup>0</sup>

*<sup>M</sup>*˜ *<sup>T</sup>* 00 0 0 <sup>−</sup>*�*−<sup>1</sup> *<sup>I</sup>*

*<sup>i</sup>* −*Pi* 0 0

Before giving the final formulation of the problem in hand, we suggest to relax the LMIs (26) from the point of view number of LMIs to be satisfied, for this, we suggest to use the following

*αijs*(**A***ijs* + **A***jsi* + **A***sij*) +

*N* ∑ *s*≥*j*>*i*

*ijs* ≤ 0, 1 ≤ *i* ≤ *j* < *s* ≤ *N*, 1 ≤ *r* ≤ *N*,

*ijs* ≤ 0, 1 ≤ *i* < *j* ≤ *s* ≤ *N*, 1 ≤ *r* ≤ *N*,

*αijs* + 3

*iii* < 0, 1 ≤ *i*,*r* ≤ *N*,

*N* ∑ *i*,*j*,*s*=1

*αijs* = 1.

*N* ∑ *s*≥*j*>*i*

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

< 0, 1 ≤ *i*, *j*,*s*,*r* ≤ *N*, (26)

*PrQr* = *I*, 1 ≤ *r* ≤ *N*. (27)

Output Tracking Control for Fuzzy Systems via Static-Output Feedback Design 171

*αijs***A***ijs, where αijs* = *αiαjα<sup>s</sup> and*

*αijs*(**A***sji* + **A***isj* + **A***jis*),

*N* ∑ *i*=1 *α<sup>i</sup>* =

(28)

*ijs <sup>H</sup><sup>T</sup>*

*(b):*

⎧ ⎪⎪⎪⎪⎪⎪⎪⎨

*N* ∑ *i*=1

⎡ ⎣

*Pi I*

*I Qi*

⎪⎪⎪⎪⎪⎪⎪⎩

lemma.

Applying Schur complement on (23), it is straightforward to verify that the condition (23) is equivalent to the following inequalities:

$$\begin{aligned} \left(P^{+}\tilde{\mathbf{G}}\_{z}\right)^{T}\left(P^{+}-\epsilon P^{+}\tilde{\mathcal{M}}\tilde{\mathcal{M}}^{T}P^{+}\right)^{-1}P^{+}\tilde{\mathcal{G}}\_{z}+\epsilon^{-1}\mathcal{N}\_{z}^{T}\mathcal{N}\_{z}-\mathcal{M}\_{2}<0 \quad \text{and} \\ P^{+}-\epsilon P^{+}\tilde{\mathcal{M}}\tilde{\mathcal{M}}^{T}P^{+}>0. \end{aligned} \tag{24}$$

Using (20), (24) and Lemma 3.1, we have

$$\begin{split} \mathcal{M}\_1 - \mathcal{M}\_2 &= (\tilde{\mathbf{G}}\_z + \tilde{\mathbf{M}}F(k)\mathcal{N}\_z)^T P^+ \left( \tilde{\mathbf{G}}\_z + \tilde{\mathbf{M}}F(k)\mathcal{N}\_z \right) \\ &\le \left( P^+ \tilde{\mathbf{G}}\_z \right)^T \left( P^+ - \epsilon P^+ \tilde{\mathbf{M}} \tilde{\mathbf{M}}^T P^+ \right)^{-1} P^+ \tilde{\mathbf{G}}\_z + \epsilon^{-1} \mathcal{N}\_z^T \mathcal{N}\_z - \mathcal{M}\_2 \\ &\quad < 0. \end{split} \tag{25}$$

By consequence

$$\sum\_{k=0}^{k\_f} (y(k) - y\_d(k))^T (y(k) - y\_d(k)) < \gamma^2 \sum\_{k=0}^{k\_f} \tilde{w}(k)^T \tilde{w}(k).$$

Hence, *H*∞ output tracking performance is achieved with the prescribed attenuation level *γ*. On the other hand, it follows from (11) and (25) that Δ*V*(*x*˜) < 0 for *w*˜(*k*) = 0, which leads that the uncertain system (8) with *w*˜(*k*) = 0 is robustly asymptotically stable.

## **4.** *H*<sup>∞</sup> **fuzzy tracking controller synthesis**

In this section, a cone complementarity formulation [7] is used to solve the bilinearity involved in (11). The idea is based on converting the conditions (11) to convex and nonconvex parts and then casting them into an optimization problem subject to some LMIs. For this, first recall the following lemma, which generalizes the result of [7].

**Lemma 4.1.** *[12] Let Pi* ∈ �*n*×*n, Qi* ∈ �*n*×*n, i* <sup>=</sup> 1, . . . , *N be any symmetric positive definite matrices, then the following statements are equivalent:*

$$(a) \colon P\_i Q\_i = I\_\prime \mid i = 1, \ldots, N.$$

$$(b) \colon \begin{cases} \sum\_{i=1}^N \text{Tr}(P\_i Q\_i) = N \times n\_\prime \\\\ \begin{bmatrix} P\_i & I \\\\ I & Q\_i \end{bmatrix} \ge 0, \quad 1 \le i \le N. \end{cases}$$

8 Will-be-set-by-IN-TECH

<sup>−</sup>*P*<sup>+</sup> 0 0 *<sup>P</sup>*+*G*1*<sup>z</sup> <sup>P</sup>*+*Wz <sup>P</sup>*+*M*˜

<sup>0</sup> <sup>−</sup>*�<sup>I</sup>* <sup>0</sup> *<sup>N</sup>*˜ *<sup>z</sup>* 0 0

0 0 −*I Hz* 0 0

*<sup>z</sup> <sup>P</sup>*<sup>+</sup> 00 0 <sup>−</sup>*γ*<sup>2</sup> *<sup>I</sup>* <sup>0</sup>

*<sup>M</sup>*˜ *TP*<sup>+</sup> 00 0 0 <sup>−</sup>*�*−<sup>1</sup> *<sup>I</sup>*

*k f* ∑ *k*=0

*<sup>z</sup>* −*Pz* 0 0

*<sup>z</sup>* N*<sup>z</sup>* − M<sup>2</sup> < 0 and

*z* N*z* − M2

*w*˜(*k*)*Tw*˜(*k*).

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

< 0, (23)

(24)

(25)

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

*G<sup>T</sup>*

*W<sup>T</sup>*

Applying Schur complement on (23), it is straightforward to verify that the condition (23) is

*<sup>P</sup>*<sup>+</sup> <sup>−</sup> *�P*+*M*˜ *<sup>M</sup>*˜ *TP*<sup>+</sup>�−<sup>1</sup> *<sup>P</sup>*+*G*˜ *<sup>z</sup>* <sup>+</sup> *�*−1<sup>N</sup> *<sup>T</sup>*

Hence, *H*∞ output tracking performance is achieved with the prescribed attenuation level *γ*. On the other hand, it follows from (11) and (25) that Δ*V*(*x*˜) < 0 for *w*˜(*k*) = 0, which leads that

In this section, a cone complementarity formulation [7] is used to solve the bilinearity involved in (11). The idea is based on converting the conditions (11) to convex and nonconvex parts and then casting them into an optimization problem subject to some LMIs. For this, first recall the

**Lemma 4.1.** *[12] Let Pi* ∈ �*n*×*n, Qi* ∈ �*n*×*n, i* <sup>=</sup> 1, . . . , *N be any symmetric positive definite*

<sup>1</sup>*zP*<sup>+</sup> *<sup>N</sup>*˜ *<sup>T</sup>*

*<sup>z</sup> H<sup>T</sup>*

*αr*(*k* + 1) = 1, (22) can be written as

*ijs* ≡

*<sup>P</sup>*<sup>+</sup> <sup>−</sup> *�P*+*M*˜ *<sup>M</sup>*˜ *TP*<sup>+</sup>�−<sup>1</sup> *<sup>P</sup>*+*G*˜ *<sup>z</sup>* <sup>+</sup> *�*−1<sup>N</sup> *<sup>T</sup>*

<sup>M</sup><sup>1</sup> − M<sup>2</sup> = (*G*˜ *<sup>z</sup>* <sup>+</sup> *MF*˜ (*k*)N*z*)*TP*+(*G*˜ *<sup>z</sup>* <sup>+</sup> *MF*˜ (*k*)N*z*)

(*y*(*k*) <sup>−</sup> *yd*(*k*))*T*(*y*(*k*) <sup>−</sup> *yd*(*k*)) <sup>&</sup>lt; *<sup>γ</sup>*<sup>2</sup>

the uncertain system (8) with *w*˜(*k*) = 0 is robustly asymptotically stable.

Since

*N* ∑ *r*=1

*N* ∑ *i*=1

*αi*(*z*) =

*αr*(*k* + 1)

*N* ∑ *r*=1

*N* ∑ *i*,*j*,*s*=1

equivalent to the following inequalities:

Using (20), (24) and Lemma 3.1, we have

<sup>≤</sup> (*P*+*G*˜ *<sup>z</sup>*)*<sup>T</sup>* �

*k f* ∑ *k*=0

**4.** *H*<sup>∞</sup> **fuzzy tracking controller synthesis**

following lemma, which generalizes the result of [7].

*matrices, then the following statements are equivalent:*

< 0.

By consequence

*<sup>P</sup>*<sup>+</sup> <sup>−</sup> *�P*+*M*˜ *<sup>M</sup>*˜ *TP*<sup>+</sup> <sup>&</sup>gt; 0.

(*P*+*G*˜ *<sup>z</sup>*)*<sup>T</sup>* �

*αi*(*z*)*αj*(*z*)*αs*(*z*)Γ*<sup>r</sup>*

Using *Pr* = *Q*−<sup>1</sup> *<sup>r</sup>* , the stability condition (11) can be rewritten as follows:

$$
\Omega\_{ijs}^{T} \equiv \begin{bmatrix}
0 & -\epsilon I & 0 & \tilde{N}\_{ijs} & 0 & 0 \\
\\ 0 & 0 & -I & H\_i & 0 & 0 \\
G\_{1ijs}^{T} \ \tilde{N}\_{ijs}^{T} \ H\_i^{T} & -P\_i & 0 & 0 \\
\\ W\_i^{T} & 0 & 0 & 0 & -\gamma^2 I & 0 \\
\\ \tilde{M}^{T} & 0 & 0 & 0 & 0 & -\epsilon^{-1} I
\end{bmatrix} < 0, \qquad 1 \le i, j, s, r \le N,\tag{26}
$$

$$P\_{\mathcal{I}}Q\_{\mathcal{I}} = I, \ 1 \le r \le N. \tag{27}$$

Before giving the final formulation of the problem in hand, we suggest to relax the LMIs (26) from the point of view number of LMIs to be satisfied, for this, we suggest to use the following lemma.

**Lemma 4.2.** *[12] Consider the following matrix* **A**¯ = *N* ∑ *i*,*j*,*s*=1 *αijs***A***ijs, where αijs* = *αiαjα<sup>s</sup> and N* ∑ *i*=1 *α<sup>i</sup>* =

1*. Then,* **A**¯ *can be expressed as follows*

$$\begin{split} \bar{\mathbf{A}} &= \sum\_{i=1}^{N} \mathbf{a}\_{i}^{3} \mathbf{A}\_{iii} + \sum\_{s>j\geq i}^{N} \mathbf{a}\_{ijs} (\mathbf{A}\_{ijs} + \mathbf{A}\_{jsi} + \mathbf{A}\_{sij}) + \sum\_{s\geq j>i}^{N} \mathbf{a}\_{ijs} (\mathbf{A}\_{sji} + \mathbf{A}\_{isj} + \mathbf{A}\_{jis}),\\ &\quad \text{Moreover,} \\ &\sum\_{\substack{j\mid s=1\\j\mid s=1}}^{N} \mathbf{a}\_{ijs} = \sum\_{i=1}^{N} \mathbf{a}\_{i}^{3} + 3 \sum\_{s>j\geq i}^{N} \mathbf{a}\_{ijs} + 3 \sum\_{s\geq j>i}^{N} \mathbf{a}\_{ijs} = 1. \end{split}$$

Hence, using Lemma 4.2, (26) can be rewritten as follows:

$$\begin{array}{ll} \mathbf{Y}\_{\text{iii}}^{r} < 0, & 1 \le i, r \le N\_{\text{\textdegree}} \\\\ \boldsymbol{\Phi}\_{\text{ijs}}^{r} \le 0, & 1 \le i \le j < s \le N\_{\text{\textdegree}} \ 1 \le r \le N\_{\text{\textdegree}} \\\\ \mathbf{Y}\_{\text{ijs}}^{r} \le 0, & 1 \le i < j \le s \le N\_{\text{\textdegree}} \ 1 \le r \le N\_{\text{\textdegree}} \end{array} \tag{28}$$

where,

Υ*r iii* ≡ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ <sup>−</sup>*Qr* 0 0 *<sup>G</sup>*1*iii Wi <sup>M</sup>*˜ <sup>0</sup> <sup>−</sup>*�<sup>I</sup>* <sup>0</sup> *<sup>N</sup>*˜*iii* 0 0 0 0 −*I Hi* 0 0 *G<sup>T</sup>* <sup>1</sup>*iii <sup>N</sup>*˜ *<sup>T</sup> iii <sup>H</sup><sup>T</sup> <sup>i</sup>* −*Pi* 0 0 *W<sup>T</sup> <sup>i</sup>* <sup>000</sup> <sup>−</sup>*γ*<sup>2</sup> *<sup>I</sup>* <sup>0</sup> *<sup>M</sup>*˜ *<sup>T</sup>* 000 0 <sup>−</sup>*�*−1*<sup>I</sup>* ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , Φ*r ijs* ≡ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ <sup>−</sup>3*Qr* 0 0 *Gijs* <sup>+</sup> *Gjsi* <sup>+</sup> *Gsij* <sup>W</sup> <sup>3</sup>*M*˜ <sup>0</sup> <sup>−</sup>3*�<sup>I</sup>* <sup>0</sup> *<sup>N</sup>*˜*ijs* <sup>+</sup> *<sup>N</sup>*˜ *jsi* <sup>+</sup> *<sup>N</sup>*˜ *sij* 0 0 0 0 −3*I Hi* + *Hj* + *Hs* 0 0 ∗ ∗ ∗−(*Pi* + *Pj* + *Ps*) 0 0 <sup>∗</sup> 00 0 <sup>−</sup>3*γ*<sup>2</sup> *<sup>I</sup>* <sup>0</sup> <sup>∗</sup> 00 0 0 <sup>−</sup>3*�*−<sup>1</sup> *<sup>I</sup>* ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , Ψ*r ijs* ≡ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ <sup>−</sup>3*Qr* 0 0 *Gsji* <sup>+</sup> *Gisj* <sup>+</sup> *Gjis* <sup>W</sup> <sup>3</sup>*M*˜ <sup>0</sup> <sup>−</sup>3*�<sup>I</sup>* <sup>0</sup> *<sup>N</sup>*˜*sji* <sup>+</sup> *<sup>N</sup>*˜*isj* <sup>+</sup> *<sup>N</sup>*˜ *jis* 0 0 0 0 −3*I Hi* + *Hj* + *Hs* 0 0 ∗ ∗ ∗−(*Pi* + *Pj* + *Ps*) 0 0 <sup>∗</sup> 00 0 <sup>−</sup>3*γ*<sup>2</sup> *<sup>I</sup>* <sup>0</sup> <sup>∗</sup> 00 0 0 <sup>−</sup>3*�*−<sup>1</sup> *<sup>I</sup>* ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ,

Now, back to our main problem. We suggest to use Lemma 4.1 to handle the nonconvexity

**Theorem 4.1.** *Given a weight β* > 0 *and �* > 0*. The augmented closed-loop system in (8) achieves the H*<sup>∞</sup> *output tracking performance γ, if there exists positive definite matrices P*<sup>1</sup> > 0, . . . , *PN* > 0*, Q*<sup>1</sup> > 0, . . . , *QN* > 0 *and controller gains K*1,..., *KN such that the following optimization problem is*

**Tr**(*PiQi*)+(1 − *β*)*γ*

Output Tracking Control for Fuzzy Systems via Static-Output Feedback Design 173

⎦ ≥ 0, 1 ≤ *i* ≤ *N*.

(29)

involved in (27), as it is clearly shown by the following theorem:

*minimize Ki*,*Pi*,*Qi*,*γ*

subject to:

(28) *and*

*β N* ∑ *i*=1

⎡ ⎣

*Pi I*

⎤

*I Qi*

give a weight *β*, fix a tolerance *ε* (for example *ε* = 10−6) and execute the following steps:

**Remark 4.1.** *In the optimization problem (29), the attenuation level γ is also included in the optimization function. Thus, a multi-objective optimization problem is solved by the Algorithm 4.1.*

In this section, the proposed tracking control scheme is applied to regulate the output voltage of DC-DC converter. The model of a buck converter is described in Fig. 1. Using the Kirchoff laws, the converter of Fig. 1 can be represented by the following discrete-time nonlinear model

*<sup>i</sup>* = *I*, for *i* = 1, . . . , *N*.

*<sup>i</sup> Pi*)+(1 − *β*)*γ*

The following iterative algorithm [7, 12] can be used to linearize the objective function of the

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

*<sup>i</sup>* <sup>=</sup> *<sup>I</sup>* and *<sup>Q</sup>*<sup>0</sup>

**Tr**(*P*∗

⎤

• Step 2: Solve the following LMI optimization:

*<sup>i</sup>* � ≤ *ε*.

Select *β* = *β* − 0.01 and repeat from step 1. Else

*<sup>i</sup>* ←− *Qi* and repeat from step 2.

*<sup>i</sup>* � ≤ *ε*,

*<sup>i</sup> Qi* + *Q*<sup>∗</sup>

⎦ ≥ 0, 1 ≤ *i* ≤ *N*.

*solvable and equal to nx*˜ × *N:*

optimization problem (29).

*β N* ∑ *i*=1

*Pi I*

*I Qi*

⎡ ⎣

• Step 3: If �*Pi* <sup>−</sup> *<sup>Q</sup>*−<sup>1</sup>

While �*Pi* <sup>−</sup> *<sup>Q</sup>*−<sup>1</sup>

*<sup>i</sup>* ←− *Pi*, *Q*<sup>∗</sup>

**5. Illustrative example**

**Algorithm 4.1**

• Step 1: Set *P*<sup>0</sup>

*minimize Ki*,*Pi*,*Qi*,*γ*

*subject to* :

(28) and

Set *P*∗

[24]:

where W = *Wi* + *Wj* + *Ws*.

From Lemma 4.2, It is only sufficient to see that [12]

$$\sum\_{i=1}^{N} a\_i(z) \boldsymbol{\Omega}\_{ij\mathbf{s}}^r = \sum\_{i=1}^{N} a\_i^3(k) \mathbf{Y}\_{\text{iii}}^r + \sum\_{i \le j < s}^{N} a\_i(z) a\_j(z) a\_s(z) \boldsymbol{\Phi}\_{ij\mathbf{s}}^r + \sum\_{i < j \le s}^{N} a\_i(z) a\_j(z) a\_s(z) \mathbf{Y}\_{\text{ij\mathbf{s}}}^r.$$

It should be noted that, Lemma 4.2 is very useful in reducing the number of LMIs to be satisfied. Indeed, (26) leads to *N*<sup>4</sup> LMIs to be satisfied. In contrast, by using Lemma 4.2, this number decreases to (*N*2(*N*<sup>2</sup> + 2))/3.

Now, back to our main problem. We suggest to use Lemma 4.1 to handle the nonconvexity involved in (27), as it is clearly shown by the following theorem:

**Theorem 4.1.** *Given a weight β* > 0 *and �* > 0*. The augmented closed-loop system in (8) achieves the H*<sup>∞</sup> *output tracking performance γ, if there exists positive definite matrices P*<sup>1</sup> > 0, . . . , *PN* > 0*, Q*<sup>1</sup> > 0, . . . , *QN* > 0 *and controller gains K*1,..., *KN such that the following optimization problem is solvable and equal to nx*˜ × *N:*

$$\begin{cases} \underset{\mathbf{K}\_i P\_i, \mathbf{Q}\_i \boldsymbol{\gamma}}{\min} & \beta \sum\_{i=1}^N \text{Tr}(P\_i \mathbf{Q}\_i) + (1 - \beta)\gamma \\\\ & \text{subject to:} \\\\ & (28) \; and \; \begin{bmatrix} P\_i & I \\\\ & I \end{bmatrix} \ge 0, \; 1 \le i \le N. \end{cases} \tag{29}$$

The following iterative algorithm [7, 12] can be used to linearize the objective function of the optimization problem (29).

## **Algorithm 4.1**

10 Will-be-set-by-IN-TECH

<sup>−</sup>*Qr* 0 0 *<sup>G</sup>*1*iii Wi <sup>M</sup>*˜

<sup>0</sup> <sup>−</sup>*�<sup>I</sup>* <sup>0</sup> *<sup>N</sup>*˜*iii* 0 0

0 0 −*I Hi* 0 0

*<sup>i</sup>* <sup>000</sup> <sup>−</sup>*γ*<sup>2</sup> *<sup>I</sup>* <sup>0</sup>

*<sup>M</sup>*˜ *<sup>T</sup>* 000 0 <sup>−</sup>*�*−1*<sup>I</sup>*

<sup>−</sup>3*Qr* 0 0 *Gijs* <sup>+</sup> *Gjsi* <sup>+</sup> *Gsij* <sup>W</sup> <sup>3</sup>*M*˜

<sup>0</sup> <sup>−</sup>3*�<sup>I</sup>* <sup>0</sup> *<sup>N</sup>*˜*ijs* <sup>+</sup> *<sup>N</sup>*˜ *jsi* <sup>+</sup> *<sup>N</sup>*˜ *sij* 0 0

0 0 −3*I Hi* + *Hj* + *Hs* 0 0

∗ ∗ ∗−(*Pi* + *Pj* + *Ps*) 0 0

<sup>∗</sup> 00 0 <sup>−</sup>3*γ*<sup>2</sup> *<sup>I</sup>* <sup>0</sup>

<sup>−</sup>3*Qr* 0 0 *Gsji* <sup>+</sup> *Gisj* <sup>+</sup> *Gjis* <sup>W</sup> <sup>3</sup>*M*˜

<sup>0</sup> <sup>−</sup>3*�<sup>I</sup>* <sup>0</sup> *<sup>N</sup>*˜*sji* <sup>+</sup> *<sup>N</sup>*˜*isj* <sup>+</sup> *<sup>N</sup>*˜ *jis* 0 0

0 0 −3*I Hi* + *Hj* + *Hs* 0 0

∗ ∗ ∗−(*Pi* + *Pj* + *Ps*) 0 0

<sup>∗</sup> 00 0 <sup>−</sup>3*γ*<sup>2</sup> *<sup>I</sup>* <sup>0</sup>

<sup>∗</sup> 00 0 0 <sup>−</sup>3*�*−<sup>1</sup> *<sup>I</sup>*

*αi*(*z*)*αj*(*z*)*αs*(*z*)Φ*<sup>r</sup>*

It should be noted that, Lemma 4.2 is very useful in reducing the number of LMIs to be satisfied. Indeed, (26) leads to *N*<sup>4</sup> LMIs to be satisfied. In contrast, by using Lemma 4.2,

*ijs* +

*N* ∑ *i*<*j*≤*s*

<sup>∗</sup> 00 0 0 <sup>−</sup>3*�*−<sup>1</sup> *<sup>I</sup>*

*<sup>i</sup>* −*Pi* 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

,

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

,

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

,

*αi*(*z*)*αj*(*z*)*αs*(*z*)Ψ*<sup>r</sup>*

*ijs*.

*iii <sup>H</sup><sup>T</sup>*

where,

Υ*r iii* ≡

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

From Lemma 4.2, It is only sufficient to see that [12]

*N* ∑ *i*≤*j*<*s*

*N* ∑ *i*=1 *α*3 *<sup>i</sup>* (*k*)Υ*<sup>r</sup> iii* +

this number decreases to (*N*2(*N*<sup>2</sup> + 2))/3.

Φ*r ijs* ≡

Ψ*r ijs* ≡

where W = *Wi* + *Wj* + *Ws*.

*αi*(*z*)Ω*<sup>r</sup>*

*ijs* =

*N* ∑ *i*=1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

*G<sup>T</sup>* <sup>1</sup>*iii <sup>N</sup>*˜ *<sup>T</sup>*

*W<sup>T</sup>*

give a weight *β*, fix a tolerance *ε* (for example *ε* = 10−6) and execute the following steps:


$$\underset{\begin{subarray}{c}\mathbf{K}\_{i},\mathbf{P}\_{i},\mathbf{Q}\_{i},\boldsymbol{\gamma}\\ \text{subject to}\end{subarray}}{\text{arg}\sum\_{i=1}^{N}\text{Tr}(P\_{i}^{\*}\mathbf{Q}\_{i}+\mathbf{Q}\_{i}^{\*}P\_{i})+(1-\beta)\boldsymbol{\gamma}}$$

$$(28) \text{ and } \begin{bmatrix} P\_i & I \\ & \\ I & Q\_i \end{bmatrix} \ge 0, 1 \le i \le N.$$

• Step 3: If �*Pi* <sup>−</sup> *<sup>Q</sup>*−<sup>1</sup> *<sup>i</sup>* � ≤ *ε*. While �*Pi* <sup>−</sup> *<sup>Q</sup>*−<sup>1</sup> *<sup>i</sup>* � ≤ *ε*, Select *β* = *β* − 0.01 and repeat from step 1. Else Set *P*∗ *<sup>i</sup>* ←− *Pi*, *Q*<sup>∗</sup> *<sup>i</sup>* ←− *Qi* and repeat from step 2.

**Remark 4.1.** *In the optimization problem (29), the attenuation level γ is also included in the optimization function. Thus, a multi-objective optimization problem is solved by the Algorithm 4.1.*

#### **5. Illustrative example**

In this section, the proposed tracking control scheme is applied to regulate the output voltage of DC-DC converter. The model of a buck converter is described in Fig. 1. Using the Kirchoff laws, the converter of Fig. 1 can be represented by the following discrete-time nonlinear model [24]:

where

*Ano*<sup>1</sup> <sup>=</sup> *Ano*<sup>2</sup> <sup>=</sup> *<sup>A</sup>*1+*A*<sup>1</sup>

*A*<sup>1</sup> = *A*<sup>2</sup> =

*A*<sup>1</sup> = *A*<sup>2</sup> =

⎡ ⎢ ⎣

⎡ ⎢ ⎣

*C*<sup>1</sup> = *C*<sup>2</sup> =

⎤

<sup>−</sup> *Ts*

<sup>−</sup> *Ts*

*B*<sup>1</sup> =

*B*<sup>1</sup> =

⎡ ⎣ 1 0

after only 41 iterations:

0 0

0.1, *M*<sup>2</sup> =

<sup>2</sup> , *Bno*<sup>1</sup> <sup>=</sup> *<sup>B</sup>*1+*B*<sup>1</sup>

*<sup>L</sup>* (*RL* <sup>+</sup> *RRc*

*<sup>L</sup>* (*RL* <sup>+</sup> *RRc*

*<sup>L</sup>* (*RMiL* − *Vin* − *VD*)

0

*<sup>L</sup>* (*RMiL* − *Vin* − *VD*)

0

*R R*+*Rc* �

� *RRc R*+*Rc*

<sup>⎦</sup> , *<sup>N</sup>*<sup>11</sup> <sup>=</sup> <sup>10</sup> *<sup>A</sup>*1−*A*<sup>1</sup>

The reference system matrices of (5) is selected as follows

⎡ ⎣

0.5 0

0 0.5

**A** =

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

*P*<sup>1</sup> =

*TsR*

*TsR*

<sup>−</sup> *Ts*

⎡ ⎢ ⎢ ⎢ ⎣

⎡ ⎢ ⎢ ⎣ <sup>−</sup> *Ts*

<sup>2</sup> , *Bno*<sup>2</sup> <sup>=</sup> *<sup>B</sup>*2+*B*<sup>2</sup>

*<sup>C</sup>*(*R*+*RC*) <sup>−</sup> *Ts*

*<sup>C</sup>*(*R*+*RC*) <sup>−</sup> *Ts*

⎤ ⎥ <sup>⎦</sup> , *<sup>B</sup>*<sup>2</sup> <sup>=</sup>

⎤ ⎥

<sup>⎦</sup> , *<sup>B</sup>*<sup>2</sup> <sup>=</sup>

, and *E*<sup>1</sup> = *E*<sup>2</sup> =

Δ*A*1(*k*), Δ*A*2(*k*), Δ*B*1(*k*) and Δ*B*2(*k*) can be represented in the form of (2) with *M*<sup>1</sup> =

<sup>2</sup> , *<sup>N</sup>*<sup>12</sup> <sup>=</sup> *<sup>N</sup>*11, *<sup>N</sup>*<sup>21</sup> <sup>=</sup> *<sup>B</sup>*1−*B*<sup>1</sup>

In this example, the objective is to make the output voltage of the buck converter, i.e. *vo* follow

Let *β* = 0.99 and *�* = 1, using the Algorithm 4.1, the following feasible solution is obtained

0.122328 0.72818 0 −0.070451

0.72818 7.378511 0 −0.550637

0 010

−0.070451 −0.550637 0 2.846761

⎡ ⎣ 0

1

⎤

<sup>⎦</sup> , **<sup>C</sup>** <sup>=</sup> �

0 1 �

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

,

a desired signal to meet the *H*∞ tracking performance of the uncertain system (30).

⎤ ⎦ , **B** =

(*R*+*Rc*)) + <sup>1</sup> <sup>−</sup> *TsR*

(*R*+*Rc*)

<sup>2</sup> , with

*L*(*R*+*RC*)

⎤ ⎥ ⎥ ⎥ ⎦ ,

Output Tracking Control for Fuzzy Systems via Static-Output Feedback Design 175

⎤ ⎥ ⎥ ⎦ ,

*<sup>L</sup>* (*RMiL* − *Vin* − *VD*)

⎤ ⎥ ⎦ ,

> ⎤ ⎥ ⎦ ,

0

*<sup>L</sup>* (*RMiL* − *Vin* − *VD*)

0

<sup>2</sup> , *<sup>N</sup>*<sup>22</sup> <sup>=</sup> *<sup>B</sup>*2−*B*<sup>2</sup>

<sup>2</sup> .

. (33)

*<sup>C</sup>*(*R*+*Rc*) <sup>+</sup> <sup>1</sup>

*L*(*R*+*RC*)

*<sup>C</sup>*(*R*+*Rc*) + <sup>1</sup>

<sup>−</sup> *Ts*

⎡ ⎢ ⎣

> ⎡ ⎢ ⎣

> > � 1 0 � .

<sup>−</sup> *Ts*

) + <sup>1</sup> <sup>−</sup> *TsR*

**Figure 1.** Buck converter circuit.

$$\mathbf{x}(k+1) = \begin{bmatrix} \frac{-T\_s}{L}(R\_L + \frac{R(k)R\_c}{R(k) + R\_c}) + 1 & \frac{-T\_sR(k)}{L(R(k) + R\_c)}\\\\ \frac{T\_sR(k)}{\overline{C}(R(k) + R\_c)} & \frac{-T\_s}{\overline{C}(R(k) + R\_c)} + 1 \end{bmatrix} \mathbf{x}(k) +$$
 
$$\begin{bmatrix} \frac{-T\_s}{L}(R\_M i\_L(k) - V\_{in}(k) - V\_D) \\\\ 0 \end{bmatrix} u(k) + \begin{bmatrix} \frac{-T\_sV\_D}{L} \\\\ 0 \end{bmatrix},\tag{30}$$

$$y(k) = \begin{bmatrix} \frac{R(k)R\_\circ}{(R(k) + R\_\circ)} & \frac{R(k)}{(R(k) + R\_\circ)} \end{bmatrix} \propto (k)\_\prime$$

where *x*(*k*)=[*iL*(*k*) *vc*(*k*)]*<sup>T</sup>* is the state vector, *u*(*k*) is the control vector i.e. the duty cycle of the switched **M**, *y*(*k*) is the output vector i.e. the output voltage and *Ts* is the sampling period *Ts* = 0.001 × 1/ *f*0, with *f*<sup>0</sup> is the resonance frequency of the buck converter (30). *R*(*k*) and *Vin*(*k*) are uncertain parameters satisfying *R*(*k*) ∈ [*R*(*k*), *R*(*k*)], *Vin*(*k*) ∈ [*Vin*(*k*), *Vin*(*k*)]. Table (1) gives the parameter values of the buck converter (Fig. 1). Similar to [24], we assume that the inductor current belongs in a compact set: *iL*(*k*) ∈ [*iL*, *iL*], and select the membership functions as follows

$$a\_1(k) = \frac{-i\_L(k) + \overline{i}\_L}{\overline{i}\_L - \underline{i}\_L}, \quad a\_2(k) = 1 - a\_1(k). \tag{31}$$

The nonlinear system (30) can be represented by the following uncertain T-S model:

$$\begin{aligned} \mathcal{R} & \text{ulle}^{\hat{I}} \text{ if } i\_L(k) \text{ is } \mu\_i \\\\ \text{Then } \begin{cases} \mathbf{x}(k+1) &= (A\_{noi} + \Delta A\_i(k))\mathbf{x}(k) + (B\_{noi} + \Delta B\_i(k))u(k) + E\_l w(k), \\\\ \mathbf{y}(k) &= C\_l \mathbf{x}(k), \; i = 1, 2, \end{cases} \end{aligned} \tag{32}$$

where

12 Will-be-set-by-IN-TECH

**Figure 1.** Buck converter circuit.

*x*(*k* + 1) =

⎡ ⎢ ⎢ ⎢ ⎣

⎡ ⎢ ⎣

*y*(*k*) =

functions as follows

Then

<sup>R</sup>ule*<sup>i</sup>* If *iL*(*k*) is *<sup>μ</sup><sup>i</sup>*

⎧ ⎨ ⎩ −*Ts*

� *R*(*k*)*Rc* (*R*(*k*)+*Rc*)

*<sup>α</sup>*1(*k*) = <sup>−</sup>*iL*(*k*) + *iL*

*y*(*k*) = *Cix*(*k*), *i* = 1, 2,

*iL* − *iL*

The nonlinear system (30) can be represented by the following uncertain T-S model:

*x*(*k* + 1)=(*Anoi* + Δ*Ai*(*k*))*x*(*k*)+(*Bnoi* + Δ*Bi*(*k*))*u*(*k*) + *Eiw*(*k*),

−*Ts*

*TsR*(*k*) *C*(*R*(*k*)+*Rc*)

*<sup>L</sup>* (*RL* <sup>+</sup> *<sup>R</sup>*(*k*)*Rc*

*R*(*k*)+*Rc*

*<sup>L</sup>* (*RMiL*(*k*) − *Vin*(*k*) − *VD*)

0

*R*(*k*) (*R*(*k*)+*Rc*) � *x*(*k*),

where *x*(*k*)=[*iL*(*k*) *vc*(*k*)]*<sup>T</sup>* is the state vector, *u*(*k*) is the control vector i.e. the duty cycle of the switched **M**, *y*(*k*) is the output vector i.e. the output voltage and *Ts* is the sampling period *Ts* = 0.001 × 1/ *f*0, with *f*<sup>0</sup> is the resonance frequency of the buck converter (30). *R*(*k*) and *Vin*(*k*) are uncertain parameters satisfying *R*(*k*) ∈ [*R*(*k*), *R*(*k*)], *Vin*(*k*) ∈ [*Vin*(*k*), *Vin*(*k*)]. Table (1) gives the parameter values of the buck converter (Fig. 1). Similar to [24], we assume that the inductor current belongs in a compact set: *iL*(*k*) ∈ [*iL*, *iL*], and select the membership

) + 1 <sup>−</sup>*TsR*(*k*) *L*(*R*(*k*)+*Rc*)

> −*Ts <sup>C</sup>*(*R*(*k*)+*Rc*) + <sup>1</sup>

> > ⎤ ⎥ ⎦ *u*(*k*) +

⎤ ⎥ ⎥ ⎥ ⎦

⎡ ⎢ ⎣ *x*(*k*)+

−*TsVD L*

⎤ ⎥ ⎦ , (30)

(32)

0

, *α*2(*k*) = 1 − *α*1(*k*). (31)

*Ano*<sup>1</sup> <sup>=</sup> *Ano*<sup>2</sup> <sup>=</sup> *<sup>A</sup>*1+*A*<sup>1</sup> <sup>2</sup> , *Bno*<sup>1</sup> <sup>=</sup> *<sup>B</sup>*1+*B*<sup>1</sup> <sup>2</sup> , *Bno*<sup>2</sup> <sup>=</sup> *<sup>B</sup>*2+*B*<sup>2</sup> <sup>2</sup> , with *A*<sup>1</sup> = *A*<sup>2</sup> = ⎡ ⎢ ⎢ ⎢ ⎣ <sup>−</sup> *Ts <sup>L</sup>* (*RL* <sup>+</sup> *RRc* (*R*+*Rc*) ) + <sup>1</sup> <sup>−</sup> *TsR L*(*R*+*RC*) *TsR <sup>C</sup>*(*R*+*RC*) <sup>−</sup> *Ts <sup>C</sup>*(*R*+*Rc*) <sup>+</sup> <sup>1</sup> ⎤ ⎥ ⎥ ⎥ ⎦ , *A*<sup>1</sup> = *A*<sup>2</sup> = ⎡ ⎢ ⎢ ⎣ <sup>−</sup> *Ts <sup>L</sup>* (*RL* <sup>+</sup> *RRc* (*R*+*Rc*)) + <sup>1</sup> <sup>−</sup> *TsR L*(*R*+*RC*) *TsR <sup>C</sup>*(*R*+*RC*) <sup>−</sup> *Ts <sup>C</sup>*(*R*+*Rc*) + <sup>1</sup> ⎤ ⎥ ⎥ ⎦ , *B*<sup>1</sup> = ⎡ ⎢ ⎣ <sup>−</sup> *Ts <sup>L</sup>* (*RMiL* − *Vin* − *VD*) 0 ⎤ ⎥ <sup>⎦</sup> , *<sup>B</sup>*<sup>2</sup> <sup>=</sup> ⎡ ⎢ ⎣ <sup>−</sup> *Ts <sup>L</sup>* (*RMiL* − *Vin* − *VD*) 0 ⎤ ⎥ ⎦ , *B*<sup>1</sup> = ⎡ ⎢ ⎣ <sup>−</sup> *Ts <sup>L</sup>* (*RMiL* − *Vin* − *VD*) 0 ⎤ ⎥ <sup>⎦</sup> , *<sup>B</sup>*<sup>2</sup> <sup>=</sup> ⎡ ⎢ ⎣ <sup>−</sup> *Ts <sup>L</sup>* (*RMiL* − *Vin* − *VD*) 0 ⎤ ⎥ ⎦ ,

$$\mathbf{C}\_1 = \mathbf{C}\_2 = \begin{bmatrix} \frac{\mathbf{R}\mathbf{R}\_c}{\mathbf{R} + \mathbf{R}\_c} & \frac{\mathbf{R}}{\mathbf{R} + \mathbf{R}\_c} \end{bmatrix}, \text{ and } E\_1 = E\_2 = \begin{bmatrix} 1\\0 \end{bmatrix}.$$

Δ*A*1(*k*), Δ*A*2(*k*), Δ*B*1(*k*) and Δ*B*2(*k*) can be represented in the form of (2) with *M*<sup>1</sup> = 0.1, *M*<sup>2</sup> = ⎡ ⎣ 1 0 0 0 ⎤ <sup>⎦</sup> , *<sup>N</sup>*<sup>11</sup> <sup>=</sup> <sup>10</sup> *<sup>A</sup>*1−*A*<sup>1</sup> <sup>2</sup> , *<sup>N</sup>*<sup>12</sup> <sup>=</sup> *<sup>N</sup>*11, *<sup>N</sup>*<sup>21</sup> <sup>=</sup> *<sup>B</sup>*1−*B*<sup>1</sup> <sup>2</sup> , *<sup>N</sup>*<sup>22</sup> <sup>=</sup> *<sup>B</sup>*2−*B*<sup>2</sup> <sup>2</sup> .

In this example, the objective is to make the output voltage of the buck converter, i.e. *vo* follow a desired signal to meet the *H*∞ tracking performance of the uncertain system (30). The reference system matrices of (5) is selected as follows

$$\mathbf{A} = \begin{bmatrix} 0.5 & 0\\ & \\ 0 & 0.5 \end{bmatrix}, \mathbf{B} = \begin{bmatrix} 0\\ \\ 1 \end{bmatrix}, \mathbf{C} = \begin{bmatrix} 0 \ 1 \end{bmatrix}. \tag{33}$$

,

Let *β* = 0.99 and *�* = 1, using the Algorithm 4.1, the following feasible solution is obtained after only 41 iterations:

$$P\_1 = \begin{bmatrix} 0.122328 & 0.72818 & 0 \ -0.070451 \\\\ 0.72818 & 7.378511 & 0 \ -0.550637 \\\\ 0 & 0 & 1 & 0 \\\\ -0.070451 & -0.550637 & 0 & 2.846761 \end{bmatrix}$$

$$P\_2 = \begin{bmatrix} 0.124832 & 0.7441565 & 0 & -0.09171 \\ 0.7441565 & 7.455168 & 0 & -0.661751 \\ 0 & 0 & 1.11602 & 0 \\ -0.09171 & -0.661751 & 0 & 3.00123 \\\\ \end{bmatrix},$$

$$Q\_1 = \begin{bmatrix} 19.852249 & -1.950697 & 0 & 0.113988 \\ -1.950697 & 0.329190 & 0.015398 \\ 0 & 0 & 1 & 0 \\ 0.113988 & 0.015398 & 0.0357075 \end{bmatrix},$$

$$Q\_2 = \begin{bmatrix} 19.869471 & -1.967944 & 0 & 0.173243 \\ -1.967944 & 0.3317250 & 0 & 0.013007 \\ 0 & 0 & 0.896056 & 0 \\ 0.173243 & 0.013007 & 0 & 0.341358 \end{bmatrix},$$

$$K\_1 = -6.0943; \quad K\_2 = -7.1963;$$

and the *H*<sup>∞</sup> output tracking performance index: *γ* = 2.52. Hence, according to (7), the static output-feedback control law that ensures the desired trajectory tracking for (30) is given as follows:

$$
\mu(k) = (\mu\_1(k)K\_1 + \mu\_2(k)K\_2)(y(k) - y\_d(k)). \tag{34}
$$

<sup>0</sup> 0.005 0.01 0.015 <sup>0</sup>

yd (k) y(k)

Output Tracking Control for Fuzzy Systems via Static-Output Feedback Design 177

x 10 −4

yd (k) y(k)

time (s)

Fig. 3 and Fig. 4 depict a zoom of Fig. 2 at 0 s and between 5 ms and 10 ms respectively. It can be seen that the designed fuzzy static output-feedback controller ensures the robust stability of the nonlinear system (30) and guarantees an acceptable *H*∞ trajectory tracking performance

0 1 2

time (s)

5

**Figure 2.** Response of *y*(*k*) and *yd*(*k*).

0

**Figure 3.** Zoom on Fig. 2 at 0 sec.

2

4

6

8

10

12

14

16

18

level.

10

15

20

25

Fig. 2 shows the evolution of the output signal of the nonlinear system (30), using the fuzzy controller, with an external disturbance input *w*(*k*) defined as *w*(*k*) = *ro* <sup>1</sup>+15(*k*+1) − *TsVD*/*L*, where, *ro* is a random number taken from a uniform distribution over [0, 2], the uncertain parameters are as follow

$$\begin{aligned} R(k) &= \frac{\overline{R} + R}{2} + \frac{\overline{R} - R}{2} \cos(k\pi/T\_s), \\ V\_{in}(k) &= \frac{\overline{V\_{in}} + V\_{in}}{2} + \frac{\overline{V\_{in}} - V\_{in}}{2} \cos(k\pi/T\_s), \end{aligned} \tag{35}$$

and the reference signal *r*(*k*), are supposed to be

$$\begin{cases} r(k) = 12V & \text{for} \quad 0 \le k \le 0.005s, \\\\ r(k) = 6V & \text{for} \quad 0.005 < k \le 0.01s, \\\\ r(k) = 24V & \text{for} \quad k > 0.01s, \end{cases} \tag{36}$$

**Figure 2.** Response of *y*(*k*) and *yd*(*k*).

14 Will-be-set-by-IN-TECH

0.124832 0.7441565 0 −0.09171

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

,

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

,

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

,

<sup>1</sup>+15(*k*+1) − *TsVD*/*L*,

(35)

(36)

0.7441565 7.455168 0 −0.661751

0 0 1.116002 0

−0.09171 −0.661751 0 3.00123

19.852249 −1.950697 0 0.113988

−1.950697 0.329190 0 0.015398

0 0 10

0.113988 0.015398 0 0.357075

19.869471 −1.967944 0 0.173243

−1.967944 0.3317250 0 0.013007

0 0 0.896056 0

0.173243 0.013007 0 0.341358

*u*(*k*)=(*α*1(*k*)*K*<sup>1</sup> + *α*2(*k*)*K*2)(*y*(*k*) − *yd*(*k*)). (34)

*K*<sup>1</sup> = −6.0943; *K*<sup>2</sup> = −7.1963, and the *H*<sup>∞</sup> output tracking performance index: *γ* = 2.52. Hence, according to (7), the static output-feedback control law that ensures the desired trajectory tracking for (30) is given as

Fig. 2 shows the evolution of the output signal of the nonlinear system (30), using the fuzzy

where, *ro* is a random number taken from a uniform distribution over [0, 2], the uncertain

<sup>2</sup> <sup>+</sup> *Vin*−*Vin*

*r*(*k*) = 12*V* for 0 ≤ *k* ≤ 0.005*s*,

*r*(*k*) = 6*V* for 0.005 < *k* ≤ 0.01*s*,

*r*(*k*) = 24*V* for *k* > 0.01*s*,

<sup>2</sup> *cos*(*kπ*/*Ts*),

<sup>2</sup> *cos*(*kπ*/*Ts*),

controller, with an external disturbance input *w*(*k*) defined as *w*(*k*) = *ro*

<sup>2</sup> <sup>+</sup> *<sup>R</sup>*−*<sup>R</sup>*

*R*(*k*) = *<sup>R</sup>*+*<sup>R</sup>*

and the reference signal *r*(*k*), are supposed to be

⎧ ⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

*Vin*(*k*) = *Vin*+*Vin*

*P*<sup>2</sup> =

*Q*<sup>1</sup> =

*Q*<sup>2</sup> =

follows:

parameters are as follow

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

> ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

> ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Fig. 3 and Fig. 4 depict a zoom of Fig. 2 at 0 s and between 5 ms and 10 ms respectively. It can be seen that the designed fuzzy static output-feedback controller ensures the robust stability of the nonlinear system (30) and guarantees an acceptable *H*∞ trajectory tracking performance level.

**Figure 3.** Zoom on Fig. 2 at 0 sec.

**Author details** Meriem Nachidi

Ahmed El Hajjaji

**7. References**

1971.

42:11–71, 1997.

27(1):9–19, 1996.

36(1):216–222, 2006.

*Departamento de Ingeneriía de Sistemas y Automática, University of Carlos III, Madrid, Spain*.

*Laboratoire Modélisation, Information et Systèmes, University of Picardie Jules Verne, Amiens, France*.

Output Tracking Control for Fuzzy Systems via Static-Output Feedback Design 179

[1] B.D.O. Anderson and J.B. Moore. *Linear Optimal Control*. Prentice-Hall, Englewood Cliffs,

[3] J. C. Geromel, C. C. de Souza, and R. E. Skelton. Static output feedback controllers:

[4] R. Johansson and A. Robertsson. Observer-based strict positive real (spr)feedback control

[5] T. Iwasaki and R.E. Skelton. The xy-centring algorithm for the dual lmi problem: a new

[6] V.L. Syrmos, C.T. Abdallah, P. Dorato, and K. Grigoriadis. Static output feedbackUa˚

[7] L. El Ghaoui, F. Oustry, and M. Ait Rami. A cone complementarity linearisation algorithm for static output-feedback and related problems. *IEEE Trans. Automat. Contr.*,

[8] D. Mustafa. LQG optimal scalar static output feedback. *Systems and Control Letters*,

[9] D. Henrion, M. Sebek, and V. Kucera. An algorithm for static output feedback simultaneous stabilization of scalar plants. In *Proceedings of the IFAC World Congress on*

[10] S.-S. Chen, Y.-C. Chang, S.-F. Su, S.-L. Chung, and T.-T. Lee. Robust static output-feedback stabilization for nonlinear discrete-time systems with time delay via

[12] M. Nachidi, A. Benzaouia, F. Tadeo, and M. Ait Rami. LMI-based approach for output-feedback stabilization for discrete time Takagi-Sugeno systems. *IEEE Transactions*

[13] Yu. Nesterov and A. Nemirovsky. *Interior-Point Polynomial Methods in Convex*

[14] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. *Linear matrix inequalities in system*

[15] M. Ait Rami, U. Helmeke, and J. B. Moore. A finite steps algorithm for solving convex

[16] K. Tanaka and H. O. Wang. *Fuzzy Control Systems Design and Analysis: A Linear Matrix*

[17] K. Tanaka and M. Sugeno. Stability analysis and design of fuzzy control systems. *Fuzzy*

feasability problems. *Journal of Global Optimization*, 38:143–160, 2007.

*Inequality Approach*. New York: Wiley , New York: Wiley, 2001.

fuzzy control approach. *IEEE Transactions on Fuzzy Systems*, 13(2):263–272, 2005. [11] D. Huang and S. K. Nguang. Robust H∞ static output feedback control of fuzzy systems: an ilmi approach. *IEEE Transactions on Systems, Man, and Cybernetics, Part-B*,

Stability and convexity. *IEEE Transactions on Automatic Control*, 43, 1998.

approach to fixed-order control design. *Int. J. Control*, 62(6):1257–1272, 1995.

[2] H. Khalil. *Nonlinear Systems*. Pearson Higher Education, 2002.

system design. *Automatica*, 38, 2002.

survey. *Automatica*, 33(2):125–137, 1997.

*Automatic Control, Barcelona, Spain*, 2001.

*on Fuzzy Systems*, 16(5):1188–1196, 2008.

*Sets and Syst.*, 45:135–156, 1992.

*Programming*. Philadelphia, PA: SIAM, 1994.

*and control theory*. Philadelphia, PA: SIAM, 1994.

**Figure 4.** Zoom on Fig. 2 between 5 msec and 10 msec.



#### **6. Conclusion**

In this chapter, the problem of model reference tracking control with a guaranteed *H*∞ performance is solved for uncertain discrete-time fuzzy systems. Based on the fuzzy Lyapunov function and cone complementary formulation, a fuzzy static output controller is calculated to make small as possible as the tracking output error and reject disturbances.

## **Author details**

Meriem Nachidi

16 Will-be-set-by-IN-TECH

yd (k)

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

y(k)

time (s)

Parameter Value Unity Input voltage, *Vin*(*k*) *Vin*(*k*) ∈ [10, 30] *V* Current in the inductance , *iL* -8 - 8 *A* Inductance, *L* 98.58 *μH*

Parasitic resistance of *L*, *RL* 48.5 *m*Ω Capacitor, *C* 202.5 *μF*

Parasitic resistance of *C*, *Rc* 162 *m*Ω

Resistance of Switch, *RM* 0.27 Ω Diode voltage, *VD* 0.82 *V*

Load resistance, *R*(*k*) *R*(*k*) ∈ [2, 10] Ω

In this chapter, the problem of model reference tracking control with a guaranteed *H*∞ performance is solved for uncertain discrete-time fuzzy systems. Based on the fuzzy Lyapunov function and cone complementary formulation, a fuzzy static output controller is calculated to make small as possible as the tracking output error and reject disturbances.

5.8

**Figure 4.** Zoom on Fig. 2 between 5 msec and 10 msec.

**Table 1.** Parameter values of the buck converter.

**6. Conclusion**

5.85

5.9

5.95

6

6.05

6.1

6.15

6.2

x 10 −3 *Departamento de Ingeneriía de Sistemas y Automática, University of Carlos III, Madrid, Spain*.

Ahmed El Hajjaji

*Laboratoire Modélisation, Information et Systèmes, University of Picardie Jules Verne, Amiens, France*.

## **7. References**

	- [18] T. M. Guerra and L. Vermeiren. LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the takagi-sugeno's form. *Automatica*, 40(5):823–829, 2004.
	- [19] S. Xu and J. Lam. Robust H∞ control for uncertain discrete-time-delay fuzzy systems via output feedback controllers. *IEEE Trans. Fuzzy Syst.*, 13:82–93, 2005.
	- [20] H. Ying. Analytical analysis and feedback linearization tracking control of the general takagi-sugeno fuzzy dynamic systems. *IEEE Transactions Systems, Man, Cybern.*, 29:290–298, 1999.
	- [21] C. C. Kung and H. Li. Tracking control of nonlinear systems by fuzzy model-based controller. In *6th IEEE International Conference*, page 623–628, 1997.
	- [22] C. S. Tseng, B. S. Chen, and H. J. Uang. Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model. *IEEE Transaction on fuzzy systems*, 8(2):200–211, 2001.
	- [23] M. Nachidi, F. Tadeo, A. Benzaouia and M. Ait Rami. Static output-feedback for Takagi-Sugeno systems with delays. *International Journal of Adaptive Control and Signal Processing*, 25:295–312, 2011.
	- [24] K.-Y. Lian, J.-J. Liou, and C.-Y Huang. LMI-based integral fuzzy control of DC-DC converters. *IEEE Transactions on Fuzzy systems*, 14:71–80, 2006.
	- [25] A. Balestrino, A. Landi, and L. Sani. Cuk converter global control via fuzzy logic and scaling factors. *IEEE Trans. Indusriel Applications*, 38:406–413, 2002.
	- [26] G. Papafotiou, T. Geyer, and M. Morari. Hybrid modelling and optimal control of swich-mode dc-dc converters. In *IEEE Workshop on Computers in Power Electronics, Champaign, IL, USA*, 2004.
	- [27] Y. Wang, L. Xie, and C. E. de Souza. Robust control of a class of uncertain nonlinear systems. *Syst. Control Lett.*, 19, 1999.
	- [28] H. O. Wang, K. Tanaka, and M. F. Griffin. An approach to fuzzy control of nonlinear systems: Stability and design issues. *IEEE Trans. Fuzzy syst.*, 4(1):14–23, 1996.
	- [29] B.-S. Chen, C.-S. Tseng, and H.-J. Uang. Robustness design of nonlinear dynamic systems via fuzzy linear control. *IEEE Trans. Fuzzy Syst.*, 7:571–585, 1999.

© 2012 Kung et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**FPGA-Based Motion Control IC for Linear Motor** 

The development of a compact and high performance motion controller for the X-Y table of a CNC machine has been an important field in literatures (Groove, 1996; Goto et al., 1996; Hanafi et al., 2003). The typical architecture of the conventional motion control system for X-Y table is shown in Fig. 1, which consists of a central controller, two sets of servo drivers and an X-Y table. The central controller, which usually adopts a float-pointed processor, performs the function of motion trajectory and data communication with servo drivers and with external device. Each servo driver usually use a fixed-pointed processor, some specific ICs and an inverter to perform the functions of position/speed/current control at each single axis of X-Y table and to do the data communication with the central controller. Data communication between two devices uses an analog signal, a bus signal or a serial asynchronous signal. However, the motion control system in Fig.1 has some drawbacks, such as large volume, easy effect by the noise, expensive cost, inflexible, etc. In addition, data communication and handshake protocol between the central controller and servo

In recent years, the FPGA has been widely applied in implementing the digital control system (Cho, 2009; Monmasson et al. 2011; Sanchez-Solano et al. 2007). Besides, an embedded processor IP and an application IP can be developed and downloaded into FPGA to construct a SoPC environment (Altera, 2004; Hall and Hamblem, 2004), allowing the users to design a SoPC module by mixing hardware and software in one FPGA chip (Kung et al. 2004; Kung and Tsai, 2007; Kung and Chen, 2008). Therefore, based on the FPGA technology, we improve the aforementioned drawbacks and integrate the central controller and the controller part of two servo drivers in Fig. 1 into a motion control IC in this study, which is shown in Fig. 2. Our proposed motion control IC has two IPs (Intellectual Properties). One IP performs the functions of the motion trajectory by software. The other IP

and reproduction in any medium, provided the original work is properly cited.

**Drive X-Y Table Using Adaptive Fuzzy Control** 

Ying-Shieh Kung, Chung-Chun Huang and Liang-Chiao Huang

Additional information is available at the end of the chapter

drivers slow down the system executing speed.

http://dx.doi.org/10.5772/48457

**1. Introduction** 

## **FPGA-Based Motion Control IC for Linear Motor Drive X-Y Table Using Adaptive Fuzzy Control**

Ying-Shieh Kung, Chung-Chun Huang and Liang-Chiao Huang

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48457

## **1. Introduction**

18 Will-be-set-by-IN-TECH

[18] T. M. Guerra and L. Vermeiren. LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the takagi-sugeno's form. *Automatica*, 40(5):823–829, 2004. [19] S. Xu and J. Lam. Robust H∞ control for uncertain discrete-time-delay fuzzy systems via

[20] H. Ying. Analytical analysis and feedback linearization tracking control of the general takagi-sugeno fuzzy dynamic systems. *IEEE Transactions Systems, Man, Cybern.*,

[21] C. C. Kung and H. Li. Tracking control of nonlinear systems by fuzzy model-based

[22] C. S. Tseng, B. S. Chen, and H. J. Uang. Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model. *IEEE Transaction on fuzzy systems*, 8(2):200–211,

[23] M. Nachidi, F. Tadeo, A. Benzaouia and M. Ait Rami. Static output-feedback for Takagi-Sugeno systems with delays. *International Journal of Adaptive Control and Signal*

[24] K.-Y. Lian, J.-J. Liou, and C.-Y Huang. LMI-based integral fuzzy control of DC-DC

[25] A. Balestrino, A. Landi, and L. Sani. Cuk converter global control via fuzzy logic and

[26] G. Papafotiou, T. Geyer, and M. Morari. Hybrid modelling and optimal control of swich-mode dc-dc converters. In *IEEE Workshop on Computers in Power Electronics,*

[27] Y. Wang, L. Xie, and C. E. de Souza. Robust control of a class of uncertain nonlinear

[28] H. O. Wang, K. Tanaka, and M. F. Griffin. An approach to fuzzy control of nonlinear systems: Stability and design issues. *IEEE Trans. Fuzzy syst.*, 4(1):14–23, 1996. [29] B.-S. Chen, C.-S. Tseng, and H.-J. Uang. Robustness design of nonlinear dynamic systems

output feedback controllers. *IEEE Trans. Fuzzy Syst.*, 13:82–93, 2005.

controller. In *6th IEEE International Conference*, page 623–628, 1997.

converters. *IEEE Transactions on Fuzzy systems*, 14:71–80, 2006.

scaling factors. *IEEE Trans. Indusriel Applications*, 38:406–413, 2002.

via fuzzy linear control. *IEEE Trans. Fuzzy Syst.*, 7:571–585, 1999.

29:290–298, 1999.

*Processing*, 25:295–312, 2011.

*Champaign, IL, USA*, 2004.

systems. *Syst. Control Lett.*, 19, 1999.

2001.

The development of a compact and high performance motion controller for the X-Y table of a CNC machine has been an important field in literatures (Groove, 1996; Goto et al., 1996; Hanafi et al., 2003). The typical architecture of the conventional motion control system for X-Y table is shown in Fig. 1, which consists of a central controller, two sets of servo drivers and an X-Y table. The central controller, which usually adopts a float-pointed processor, performs the function of motion trajectory and data communication with servo drivers and with external device. Each servo driver usually use a fixed-pointed processor, some specific ICs and an inverter to perform the functions of position/speed/current control at each single axis of X-Y table and to do the data communication with the central controller. Data communication between two devices uses an analog signal, a bus signal or a serial asynchronous signal. However, the motion control system in Fig.1 has some drawbacks, such as large volume, easy effect by the noise, expensive cost, inflexible, etc. In addition, data communication and handshake protocol between the central controller and servo drivers slow down the system executing speed.

In recent years, the FPGA has been widely applied in implementing the digital control system (Cho, 2009; Monmasson et al. 2011; Sanchez-Solano et al. 2007). Besides, an embedded processor IP and an application IP can be developed and downloaded into FPGA to construct a SoPC environment (Altera, 2004; Hall and Hamblem, 2004), allowing the users to design a SoPC module by mixing hardware and software in one FPGA chip (Kung et al. 2004; Kung and Tsai, 2007; Kung and Chen, 2008). Therefore, based on the FPGA technology, we improve the aforementioned drawbacks and integrate the central controller and the controller part of two servo drivers in Fig. 1 into a motion control IC in this study, which is shown in Fig. 2. Our proposed motion control IC has two IPs (Intellectual Properties). One IP performs the functions of the motion trajectory by software. The other IP

© 2012 Kung et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

performs the functions of two axes' position/speed/current controllers by hardware. As the results, this two IP will parallel processing in FPGA, and the hardware/software co-design technology in FPGA can make the motion controller of X-Y more compact, flexible, better performance and less cost. Further, the X-Y table usually leads to the existence of unmodelled dynamics and disturbances which often significantly deteriorate the system performance during a machining process. Many studies attempt to improve the tracking performance in a machining process (Lin et al., 2006; Wang and Lee, 1999). Lin et al. (2006) adopts a recurrent-neural-network sliding-mode controller to improve the motion tracking performance of the X-Y table. Wang and Lee (1999) integrate the cross-coupled control and neural network techniques to achieve a high accuracy of the motion tracking in the linear motor X-Y table. However, due to the complicate computation of the neural-network, the algorithms of above two studies are realized in the PC-based control system.

FPGA-Based Motion Control IC for Linear Motor Drive X-Y Table Using Adaptive Fuzzy Control 183

(AFC) is introduced and adopted in position loop of X-Y table to improve the motion tracking performance under unmodelled uncertainty condition. Thirdly, in implementation, an FPGA embedded by a Nios II processor is used to design the overall circuits of the motion control IC which the scheme of position/speed/current control for two PMLSMs (permanent magnetic linear synchronous motors) is realized by hardware in FPGA and the motion trajectory algorithm for X-Y table is implemented by software using Nios II embedded processor. To reduce the FPGA resource usage, an FSM (Finite state machine) joined by a multiplier, an adder, a LUT (Look-up table), some comparators and registers is used to model the overall AFC algorithm. And VHDL (VHSIC hardware description language) is adopted to describe the FSM. Herein, Altera Stratix II EP2S60, which has 48,352 ALUTs (Adaptive Look-Up Tables), total 2,544,192 RAM bits, and a Nios II embedded processor which has a 32-bit configurable CPU core, 16 M byte Flash memory, 1 M byte SRAM and 16 M byte SDRAM, is used. Therefore, a fully digital motion controller can be implemented by an FPGA using hardware/software co-design technology which will make the motion controller of the X-Y table more compact, flexible and better performance. Finally, an experimental system is set up to verify the performance of the proposed motion

**2. System description of X-Y table and motion controller design** 

**Figure 3.** The architecture of a motion controller system for linear motor drive X-Y table

The internal architecture of the proposed FPGA-based controller system for a linear motor drive X-Y table is shown in Fig. 3, in which the motion trajectory is implemented by software using Nios II embedded processor; the position, speed and current vector controller for two PMLSMs are implemented by hardware in FPGA chip. The mathematical modeling of PMLSM, AFC algorithm and motion trajectory planning are introduced as

control IC for linear motor drive X-Y table.

follows:

**Figure 1.** Conventional motion control system for X-Y table

**Figure 2.** Proposed FPGA-based motion control system for X-Y table

In this chapter, a motion control IC for linear motor drive X-Y table based on FPGA (Field programmable gate array) technology is presented and shown in Fig.3. Firstly, the mathematical model of the X-Y table is defined. Secondly, an adaptive fuzzy controller (AFC) is introduced and adopted in position loop of X-Y table to improve the motion tracking performance under unmodelled uncertainty condition. Thirdly, in implementation, an FPGA embedded by a Nios II processor is used to design the overall circuits of the motion control IC which the scheme of position/speed/current control for two PMLSMs (permanent magnetic linear synchronous motors) is realized by hardware in FPGA and the motion trajectory algorithm for X-Y table is implemented by software using Nios II embedded processor. To reduce the FPGA resource usage, an FSM (Finite state machine) joined by a multiplier, an adder, a LUT (Look-up table), some comparators and registers is used to model the overall AFC algorithm. And VHDL (VHSIC hardware description language) is adopted to describe the FSM. Herein, Altera Stratix II EP2S60, which has 48,352 ALUTs (Adaptive Look-Up Tables), total 2,544,192 RAM bits, and a Nios II embedded processor which has a 32-bit configurable CPU core, 16 M byte Flash memory, 1 M byte SRAM and 16 M byte SDRAM, is used. Therefore, a fully digital motion controller can be implemented by an FPGA using hardware/software co-design technology which will make the motion controller of the X-Y table more compact, flexible and better performance. Finally, an experimental system is set up to verify the performance of the proposed motion control IC for linear motor drive X-Y table.

182 Fuzzy Controllers – Recent Advances in Theory and Applications

**Figure 1.** Conventional motion control system for X-Y table

**Figure 2.** Proposed FPGA-based motion control system for X-Y table

In this chapter, a motion control IC for linear motor drive X-Y table based on FPGA (Field programmable gate array) technology is presented and shown in Fig.3. Firstly, the mathematical model of the X-Y table is defined. Secondly, an adaptive fuzzy controller

performs the functions of two axes' position/speed/current controllers by hardware. As the results, this two IP will parallel processing in FPGA, and the hardware/software co-design technology in FPGA can make the motion controller of X-Y more compact, flexible, better performance and less cost. Further, the X-Y table usually leads to the existence of unmodelled dynamics and disturbances which often significantly deteriorate the system performance during a machining process. Many studies attempt to improve the tracking performance in a machining process (Lin et al., 2006; Wang and Lee, 1999). Lin et al. (2006) adopts a recurrent-neural-network sliding-mode controller to improve the motion tracking performance of the X-Y table. Wang and Lee (1999) integrate the cross-coupled control and neural network techniques to achieve a high accuracy of the motion tracking in the linear motor X-Y table. However, due to the complicate computation of the neural-network, the

algorithms of above two studies are realized in the PC-based control system.

## **2. System description of X-Y table and motion controller design**

The internal architecture of the proposed FPGA-based controller system for a linear motor drive X-Y table is shown in Fig. 3, in which the motion trajectory is implemented by software using Nios II embedded processor; the position, speed and current vector controller for two PMLSMs are implemented by hardware in FPGA chip. The mathematical modeling of PMLSM, AFC algorithm and motion trajectory planning are introduced as follows:

**Figure 3.** The architecture of a motion controller system for linear motor drive X-Y table

#### **2.1. Mathematical model of the PMLSM drive**

The dynamic model of a typical PMLSM can be described in the synchronous rotating reference frame, as follows

$$\frac{d\dot{\mathbf{u}}\_d}{dt} = -\frac{R\_s}{L\_d}\dot{\mathbf{i}}\_d + \frac{\pi}{\pi} \frac{L\_q}{L\_d} \dot{\mathbf{x}}\_p \dot{\mathbf{i}}\_q + \frac{1}{L\_d} \mathbf{v}\_d \tag{1}$$

FPGA-Based Motion Control IC for Linear Motor Drive X-Y Table Using Adaptive Fuzzy Control 185

*a b c*

(7)

(8)

(10)

*i*

*v*

(9)

\* () () () *d dd ek ik ik* (11)

\_ \_ () () *pd pd d v k k ek* (12)

\_ \_ \_ ( ) ( 1) ( 1) *id id id d v k v k k ek* (13)

\_ \_ () () () *d pd id vk v k v k* (14)

*v*

 

 

*e e d e e q*

 

 

rotating *d-q* frame. Further, the formulations among three coordination systems are

*<sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup>*

cos sin sin cos *d e e q e e i i i i*

> cos sin sin cos

In Fig. 3, two digital *PI* controllers are presented in the current loop of PMLSM. For the

the *<sup>d</sup> e* is the error between current command and measured current. The *p d* \_ *k* , *i d* \_ *k* are *P* controller gain and *I* controller gain, respectively. The \_ ( ) *p d v k* , \_ ( ) *i d v k* , () *<sup>d</sup> v k* are the output of *P* controller only, *I* controller only and the *PI* controller, respectively. Similarity, the

*v v v v*

1 0 *<sup>a</sup>*

 

*b c*

*v*

*v*

*v*

1. *Clarke* : stationary *a-b-c* frame to stationary - frame.

3. *Park* : stationary - frame to rotating *d-q* frame.

**4.** <sup>1</sup> *Park* : rotating *d-q* frame to stationary - frame.

example in *d* frame, the formulation is shown as follows.

formulation of *PI* controller in *q* frame is the same.

is the electrical angle.

2. Modified 1 *Clarke* : stationary - frame to stationary *a-b-c* frame.

presented as follows.

where *<sup>e</sup>* 

$$\frac{d\dot{\mathbf{u}}\_q}{dt} = -\frac{\pi}{\pi} \frac{\mathbf{L}\_d}{\mathbf{L}\_q} \dot{\mathbf{x}}\_p \dot{\mathbf{i}}\_d - \frac{\mathbf{R}\_\*}{\mathbf{L}\_q} \dot{\mathbf{i}}\_q - \frac{\pi}{\pi} \frac{\dot{\mathbf{A}}\_f}{\mathbf{L}\_q} \dot{\mathbf{x}}\_p + \frac{\mathbf{1}}{\mathbf{L}\_q} \mathbf{v}\_q \tag{2}$$

where v*d*, v*q* are the d and q axis voltages; *i*d, *i*q, are the d and q axis currents, *R*s is the phase winding resistance; *L*d, *L*q are the d and q axis inductance; *<sup>p</sup> x* is the translator speed; *<sup>f</sup>* is the permanent magnet flux linkage; is the pole pitch. The developed electromagnetic thrust force is given by

$$F\_r = \frac{3\pi}{2\pi} ((L\_d - L\_q)\dot{\imath}\_d + \dot{\varkappa}\_f)\dot{\imath}\_q \tag{3}$$

The current control of a PMLSM drive is based on a vector control approach. That is, if we control *i*d to 0 in Fig.3, the PMLSM will be decoupled, so that control a PMLSM will become easy as to control a DC linear motor. After simplification and considering the mechanical load, the model of a PMLSM can be written as the following equations,

$$F\_s = \frac{3}{2} \frac{\pi}{\pi} \mathcal{A}\_{\dot{\gamma}} \dot{i}\_q \underline{\Delta} \ K\_{\dot{\gamma}} \dot{i}\_q \tag{4}$$

with

$$K\_{\iota} = \frac{\mathfrak{Z}}{2} \frac{\pi}{\pi} \mathcal{A}\_{\iota} \tag{5}$$

and the mechanical dynamic equation of PMLSM in x-axis table is

$$F\_s - F\_\perp = M\_m \frac{d^2 \mathbf{x}\_r}{dt^2} + B\_m \frac{d \mathbf{x}\_r}{dt} \tag{6}$$

where *<sup>e</sup> F* , *Kt* , *Mm* , *Bm* and *<sup>L</sup> F* represent the motor thrust force, the force constant, the total mass of the moving element, the viscous friction coefficient and the external force, respectively. In addition, the current loop of the PMLSM drive in Fig.3 includes PI controller, coordinate transformations of Clark, Modified inverse Clark, Park, inverse Park, SVPWM (Space Vector Pulse Width Muldulation), pulse signal detection of the encoder etc. The coordination transformation of the PMLSM in Fig. 3 can be described in synchronous rotating reference frame. Figure 4 is the coordination system in rotating motor which includes stationary *a-b-c* frame, stationary - frame and synchronously rotating *d-q* frame. Further, the formulations among three coordination systems are presented as follows.

1. *Clarke* : stationary *a-b-c* frame to stationary - frame.

184 Fuzzy Controllers – Recent Advances in Theory and Applications

reference frame, as follows

force is given by

with

**2.1. Mathematical model of the PMLSM drive** 

The dynamic model of a typical PMLSM can be described in the synchronous rotating

v *<sup>q</sup> d s d p q d d dd*

v *q f d s*

where v*d*, v*q* are the d and q axis voltages; *i*d, *i*q, are the d and q axis currents, *R*s is the phase

permanent magnet flux linkage; is the pole pitch. The developed electromagnetic thrust

<sup>3</sup> (( ) ) <sup>2</sup> *<sup>e</sup> d qd f q F L Li i*

The current control of a PMLSM drive is based on a vector control approach. That is, if we control *i*d to 0 in Fig.3, the PMLSM will be decoupled, so that control a PMLSM will become easy as to control a DC linear motor. After simplification and considering the mechanical

*pd q p q q q qq*

 

 

*i xi dt L L L* 

*di R L*

*di L R xi i x dt L L L L*

winding resistance; *L*d, *L*q are the d and q axis inductance; *<sup>p</sup> x* is the translator speed; *<sup>f</sup>*

3 <sup>2</sup> *<sup>e</sup> fq tq F i Ki* 

3 2 *Kt f* 

2 2

*eL m m*

where *<sup>e</sup> F* , *Kt* , *Mm* , *Bm* and *<sup>L</sup> F* represent the motor thrust force, the force constant, the total mass of the moving element, the viscous friction coefficient and the external force, respectively. In addition, the current loop of the PMLSM drive in Fig.3 includes PI controller, coordinate transformations of Clark, Modified inverse Clark, Park, inverse Park, SVPWM (Space Vector Pulse Width Muldulation), pulse signal detection of the encoder etc. The coordination transformation of the PMLSM in Fig. 3 can be described in synchronous rotating reference frame. Figure 4 is the coordination system in rotating motor which includes stationary *a-b-c* frame, stationary - frame and synchronously

*FF M B*

*p p*

*d x dx*

*dt dt*

load, the model of a PMLSM can be written as the following equations,

and the mechanical dynamic equation of PMLSM in x-axis table is

1

(1)

(3)

(4)

(5)

(6)

is the

1

(2)

$$
\begin{bmatrix} \dot{i}\_a \\ \dot{i}\_\rho \end{bmatrix} = \begin{bmatrix} \frac{2}{3} & \frac{-1}{3} & \frac{-1}{3} \\ 0 & \frac{1}{\sqrt{3}} & \frac{-1}{\sqrt{3}} \end{bmatrix} \begin{bmatrix} \dot{i}\_s \\ \dot{i}\_s \\ \dot{i}\_c \end{bmatrix} \tag{7}
$$

2. Modified 1 *Clarke* : stationary - frame to stationary *a-b-c* frame.

$$
\begin{bmatrix} \boldsymbol{\upsilon}\_{\boldsymbol{s}} \\ \boldsymbol{\upsilon}\_{\boldsymbol{s}} \\ \boldsymbol{\upsilon}\_{\boldsymbol{c}} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ \frac{-1}{2} & \frac{\mathsf{g}\mathsf{F}}{2} \\ \frac{-1}{2} & \frac{-\mathsf{g}\mathsf{F}}{2} \end{bmatrix} \begin{bmatrix} \boldsymbol{\upsilon}\_{\boldsymbol{\rho}} \\ \boldsymbol{\upsilon}\_{\boldsymbol{a}} \end{bmatrix} \tag{8}
$$

3. *Park* : stationary - frame to rotating *d-q* frame.

$$
\begin{bmatrix} \dot{i}\_d \\ \dot{i}\_q \end{bmatrix} = \begin{bmatrix} \cos\theta\_\epsilon & \sin\theta\_\epsilon \\ -\sin\theta\_\epsilon & \cos\theta\_\epsilon \end{bmatrix} \begin{bmatrix} \dot{i}\_a \\ \dot{i}\_\beta \end{bmatrix} \tag{9}
$$

**4.** <sup>1</sup> *Park* : rotating *d-q* frame to stationary - frame.

$$
\begin{bmatrix} v\_a \\ v\_\rho \end{bmatrix} = \begin{bmatrix} \cos \theta\_\epsilon & -\sin \theta\_\epsilon \\ \sin \theta\_\epsilon & \cos \theta\_\epsilon \end{bmatrix} \begin{bmatrix} v\_d \\ v\_q \end{bmatrix} \tag{10}
$$

where *<sup>e</sup>* is the electrical angle.

In Fig. 3, two digital *PI* controllers are presented in the current loop of PMLSM. For the example in *d* frame, the formulation is shown as follows.

$$e\_d(k) = \dot{i\_d}(k) - \dot{i\_d}(k)\tag{11}$$

$$
\sigma\_{p\_{-}d}(k) = k\_{p\_{-}d} e\_d(k) \tag{12}
$$

$$
\upsilon\_{i\_{\cdot,d}}(k) = \upsilon\_{i\_{\cdot,d}}(k-1) + k\_{i\_{\cdot,d}}e\_d(k-1) \tag{13}
$$

$$
\upsilon\_d(k) = \upsilon\_{\upsilon\_-d}(k) + \upsilon\_{\upsilon\_-d}(k) \tag{14}
$$

the *<sup>d</sup> e* is the error between current command and measured current. The *p d* \_ *k* , *i d* \_ *k* are *P* controller gain and *I* controller gain, respectively. The \_ ( ) *p d v k* , \_ ( ) *i d v k* , () *<sup>d</sup> v k* are the output of *P* controller only, *I* controller only and the *PI* controller, respectively. Similarity, the formulation of *PI* controller in *q* frame is the same.

**Figure 4.** Transformation between stationary axes and rotating axes

### **2.2. Adaptive fuzzy controller (AFC) in position control loop**

The green dash rectangular area in Fig. 3 presents the architecture of an AFC. It consists of a fuzzy controller, a reference model and a parameter adjusting mechanism. Detailed description of these is as follows.

1. Fuzzy controller (FC):

In Fig.3, the tracking error and the change of the error, *e*, *de* are defined as

$$e(k) = \boldsymbol{\nu}\_m(k) - \boldsymbol{\nu}\_p(k)\tag{15}$$

FPGA-Based Motion Control IC for Linear Motor Drive X-Y Table Using Adaptive Fuzzy Control 187

() 1 () *A A i i*

 *e e*

<sup>i</sup> <sup>j</sup> j,i IF is A and e is B THEN is c *<sup>f</sup> e u* (19)

*ni mj*

. And those *i j*, *c* are adjustable parameters. In addition, by using

(20)

(21)

has

is damping ratio. Furthermore, because the

. The design methodology is described as follows: Firstly, the

(18)

where 1 6 2 \* ( 1) *<sup>i</sup> e i* . Similar results can be obtained in computing the membership

where *i and j =* 0~6, *Ai* and *Bj* are fuzzy number, and *cj,i* is a real number. The graph of the

d. Construct the output of the fuzzy system *uf*(*e*,*de*) by using the singleton fuzzifier, product-inference rule, and central average defuzzifier method. Although there are total 49 fuzzy rules in Fig. 5 will be inferred, actually only 4 fuzzy rules can be effectively excited to generate a non-zero output. Therefore, if an error *e* is located between *ei* and *ei+1*, and an error change *de* is located between *dej* and *dej+1*, only four linguistic values *Ai*, *Ai+1*, *Bj*, *Bj+1* and corresponding consequent values *cj,i*, *cj+1,i*, *cj,i+1*, *cj+1,i+1* can be excited, and the output of the fuzzy system can be inferred by the following

> , 1 1 1 1 , ,

*mn A B i j*

[ ( ) \* ( )] (, ) \* ( )\* ( )

 

*f i j mn nm*

*n m*

 

in (20).

Second order system is usually as the RM in the adaptive control system. Therefore, the

\*2 2

characteristics of no overshoot, fast response and zero steady-state error are the important factors in the design of a PMLSM servo system; therefore, it can be considered as the

(21) matches the requirement of a zero steady-state error condition. Secondly, if we choose

1 , it can guarantee no overshoot condition. Especially, the critical damp value 1

*ss s*

() 2 *m n p n n*

2

 

 

*n m*

*c e de u e de c d e de*

*ni mj <sup>d</sup>* 

( )

 *s*

*A B*

,<sup>1</sup> *<sup>i</sup> <sup>j</sup> n m*

<sup>1</sup> ( ) and <sup>2</sup> *<sup>i</sup>*

c. Select the initial fuzzy control rules, such as,

degree ( ) *<sup>j</sup> <sup>B</sup> de .*

expression:

where , ( )\* ( ) *n m nm A <sup>B</sup> d e de* 

2. Reference model (RM):

where

selective criterion of

 

(18), it is straightforward to obtain <sup>1</sup> <sup>1</sup>

transfer function of the RM in Fig.3 can be expressed as

*<sup>n</sup>* is natural frequency and

*n* and 

*A*

fuzzy rule table and the fuzzification are shown in Fig. 5.

1 1

*ni mj*

 

*ni mj*

*i j*

*i*

*e e*

*<sup>e</sup>* <sup>1</sup>

$$de(k) = e(k) - e(k-1)\tag{16}$$

and *e*, *de* and *uf* are input and output variables of FC, respectively. Besides *<sup>m</sup>* represents *<sup>m</sup> x* or *<sup>m</sup> y* , and *<sup>p</sup>* represents *<sup>p</sup> x* or *<sup>p</sup> y* . The design procedure of the FC is as follows:

a. Take the *e* and *de* as the input variables of the FC, and define their linguist variables as *E* and *dE*. The linguist value of *E* and *dE* are {*A0, A1, A2, A3, A4, A5, A6*} and {*B0, B1, B2, B3, B4, B5, B6*}, respectively. Each linguist value of *E* and *dE* is based on the symmetrical triangular membership function which is shown in Fig.5. The symmetrical triangular membership function are determined uniquely by three real numbers <sup>123</sup> , if one fixes 1 3 *f f* () () 0 and <sup>2</sup> *f*()1 . With respect to the universe of discourse of [-6.6], the numbers for these linguistic values are selected as follows:

$$\begin{aligned} A\_0 &= B\_0: \left\{ -6, -6, -4 \right\}, A\_1 = B\_1: \left\{ -6, -4, -2 \right\}, A\_2 = B\_2: \left\{ -4, -2, 0 \right\}, \\ A\_3 &= B\_3: \left\{ -2, 0, 2 \right\}, A\_4 = B\_4: \left\{ 0, 2, 4 \right\}, A\_5 = B\_5: \left\{ 2, 4, 6 \right\}, A\_6 = B\_6: \left\{ 4, 6, 6 \right\} \end{aligned} \tag{17}$$

b. Compute the membership degree of the *e* and *de*. Figure 5 shows that the only two linguistic values are excited (resulting in a non-zero membership) in any input value, and the membership degree is obtained by

$$
\mu\_{A\_i}(e) = \frac{e\_{i+1} - e}{2} \text{and} \quad \mu\_{A\_{i+1}}(e) = 1 - \mu\_{A\_i}(e) \tag{18}
$$

where 1 6 2 \* ( 1) *<sup>i</sup> e i* . Similar results can be obtained in computing the membership degree ( ) *<sup>j</sup> <sup>B</sup> de .*

c. Select the initial fuzzy control rules, such as,

186 Fuzzy Controllers – Recent Advances in Theory and Applications

**Figure 4.** Transformation between stationary axes and rotating axes

*c*

**stator**

*b*

description of these is as follows.

1. Fuzzy controller (FC):

or *<sup>m</sup> y* , and

fixes 1 3 *f f* () () 0 

 

and <sup>2</sup> *f*()1

and the membership degree is obtained by

the numbers for these linguistic values are selected as follows:

0 0 1 1 2 2

*A B AB AB*

**2.2. Adaptive fuzzy controller (AFC) in position control loop** 

In Fig.3, the tracking error and the change of the error, *e*, *de* are defined as

and *e*, *de* and *uf* are input and output variables of FC, respectively. Besides

The green dash rectangular area in Fig. 3 presents the architecture of an AFC. It consists of a fuzzy controller, a reference model and a parameter adjusting mechanism. Detailed

 

β

*f*

*qf*

*q*

*abc*

**rotor**

: 3-axis **stationary frame** : 2-axis **stationary frame** *d q* : 2-axis **rotating frame**

*f* α

*e* 

*S f*

*f d*

*a*

*e*

*d*

() () () *m p ek k k* 

 *<sup>p</sup>* represents *<sup>p</sup> x* or *<sup>p</sup> y* . The design procedure of the FC is as follows: a. Take the *e* and *de* as the input variables of the FC, and define their linguist variables as *E* and *dE*. The linguist value of *E* and *dE* are {*A0, A1, A2, A3, A4, A5, A6*} and {*B0, B1, B2, B3, B4, B5, B6*}, respectively. Each linguist value of *E* and *dE* is based on the symmetrical triangular membership function which is shown in Fig.5. The symmetrical triangular

membership function are determined uniquely by three real numbers <sup>123</sup>

b. Compute the membership degree of the *e* and *de*. Figure 5 shows that the only two linguistic values are excited (resulting in a non-zero membership) in any input value,

: 6, 6, 4 , : 6, 4, 2 , : 4, 2,0 ,

3 3 44 55 66

*AB AB AB AB* 

: 2,0,2 , : 0,2,4 , : 2,4,6 , : 4,6,6

(15)

 , if one

*<sup>m</sup>* represents *<sup>m</sup> x*

(17)

*de k e k e k* ( ) ( ) ( 1) (16)

. With respect to the universe of discourse of [-6.6],

$$\text{IF } e \text{ is } \mathbf{A}\_{i} \text{ and } \boldsymbol{\Delta e} \text{ is } \mathbf{B}\_{i} \text{ THEN } u\_{/} \text{ is } \mathbf{c}\_{\boldsymbol{\mu}} \tag{19}$$

where *i and j =* 0~6, *Ai* and *Bj* are fuzzy number, and *cj,i* is a real number. The graph of the fuzzy rule table and the fuzzification are shown in Fig. 5.

d. Construct the output of the fuzzy system *uf*(*e*,*de*) by using the singleton fuzzifier, product-inference rule, and central average defuzzifier method. Although there are total 49 fuzzy rules in Fig. 5 will be inferred, actually only 4 fuzzy rules can be effectively excited to generate a non-zero output. Therefore, if an error *e* is located between *ei* and *ei+1*, and an error change *de* is located between *dej* and *dej+1*, only four linguistic values *Ai*, *Ai+1*, *Bj*, *Bj+1* and corresponding consequent values *cj,i*, *cj+1,i*, *cj,i+1*, *cj+1,i+1* can be excited, and the output of the fuzzy system can be inferred by the following expression:

$$
\mu\_f(e, de) = \frac{\sum\_{n=i}^{i+1} \sum\_{m=j}^{j+1} c\_{m,n} [\mu\_{A\_n}(e) \,^\* \mu\_{\mathbb{B}\_n}(de)]}{\sum\_{n=i}^{i+1} \sum\_{m' \atop n' \ne j}^{i+1} \mu\_{A\_n}(e) \,^\* \mu\_{\mathbb{B}\_n}(de)} \stackrel{\Delta}{=} \sum\_{n=i}^{i+1} \sum\_{m=j}^{j+1} c\_{m,n} \,^\* d\_{n,n} \tag{20}
$$

where , ( )\* ( ) *n m nm A <sup>B</sup> d e de* . And those *i j*, *c* are adjustable parameters. In addition, by using (18), it is straightforward to obtain <sup>1</sup> <sup>1</sup> ,<sup>1</sup> *<sup>i</sup> <sup>j</sup> n m ni mj <sup>d</sup>* in (20).

2. Reference model (RM):

Second order system is usually as the RM in the adaptive control system. Therefore, the transfer function of the RM in Fig.3 can be expressed as

$$\frac{\left(\boldsymbol{\nu}\_{\text{u}}\right)}{\left(\boldsymbol{\nu}\_{\text{v}}^{\text{'}}\right)} = \frac{\left.\boldsymbol{\alpha}\_{\text{v}}^{\text{2}}\right|}{\left.\boldsymbol{s}^{2} + 2\xi\boldsymbol{\alpha}\_{\text{u}}\boldsymbol{s} + \boldsymbol{\alpha}\_{\text{v}}^{\text{2}}\right|}\tag{21}$$

where *<sup>n</sup>* is natural frequency and is damping ratio. Furthermore, because the characteristics of no overshoot, fast response and zero steady-state error are the important factors in the design of a PMLSM servo system; therefore, it can be considered as the selective criterion of *n* and . The design methodology is described as follows: Firstly, the (21) matches the requirement of a zero steady-state error condition. Secondly, if we choose 1 , it can guarantee no overshoot condition. Especially, the critical damp value 1 has a fastest step response. Hence, the relation between the rising time *tr* and the natural frequency *<sup>n</sup>* for a step input response in (21) can be derived and shown as follows.

$$(1 + \alpha\_{\underline{n}} t\_r)e^{-\alpha\_{\underline{n}}t\_r} = 0.1\tag{22}$$

FPGA-Based Motion Control IC for Linear Motor Drive X-Y Table Using Adaptive Fuzzy Control 189

represents learning rate. However, following the

(28)

(29)

(31)

(33)

, *r* , *<sup>i</sup> x* , *<sup>i</sup> y* are angle increment, radius, X-axis trajectory

1 1 , *i ii i x x y Sy* (30)

1

1

 

1 1 , *i ii i x x y Sy* (34)

 

1 1 , *i ii i x Sx y y* (32)

(27)

, , ( ) ( ) ( ) *m n p i v nm c k K K Kekd* 

The circular, window and star motion trajectories are typical used as the performance

b. The window motion trajectory is shown in Fig.6. The formulation is derived as follows:

<sup>6</sup> ( : 2 , and ) <sup>4</sup> *<sup>i</sup> i i*

1 1 cos( ), sin( ) *i x ii y <sup>i</sup> xO r yO r* 

<sup>6</sup> ( : , and ) <sup>4</sup> *<sup>i</sup> i i*

2 2 cos( ), sin( ) *i x ii y <sup>i</sup> xO r yO r* 

 

   

  similar derivation with (Kung & Tsai, 2007), the *m n*, *c* can be obtained as

with *m* = *j*, *j*+1, *n* = *i*,*i*+1 and where

**2.3. Motion trajectory planning of X-Y table** 

evaluation of the motion controller for X-Y table.

a. In circular motion trajectory, it is computed by

sin( ) *i i x r*

os( ) *i i y rc*

command and Y-axis trajectory command, respectively.

> 

with *m* = *j*, *j+1* and *n* = *i*,*i+1*.

with *i i* <sup>1</sup> 

a-trajectory:

b-trajectory:

c-trajectory:

d-trajectory:

e-trajectory:

 . Where

Once the *tr* is chosen, the natural frequency *<sup>n</sup>* can be obtained. Furthermore, applying the bilinear transformation, (21) can be transformed to a discrete model by

$$\frac{\text{y}\,\text{y}\,\text{(}\text{z}^{-1}\text{)}}{\text{y}\,\text{(}\text{z}^{-1}\text{)}} = \frac{a\_{\text{o}} + a\_{\text{i}}\,\text{z}^{-1} + a\_{\text{z}}\,\text{z}^{-2}}{1 + b\_{\text{i}}\,\text{z}^{-1} + b\_{\text{z}}\,\text{z}^{-2}}\tag{23}$$

and the difference equation is written as.

$$
\boldsymbol{\psi}\_{m}(\boldsymbol{k}) = -\boldsymbol{b}\_{i}\boldsymbol{\psi}\_{m}(\boldsymbol{k}-1) - \boldsymbol{b}\_{i}\boldsymbol{\psi}\_{m}(\boldsymbol{k}-2) + \boldsymbol{a}\_{i}\boldsymbol{\psi}\_{p}^{\prime}(\boldsymbol{k}) \\
+ \boldsymbol{a}\_{i}\boldsymbol{\psi}\_{p}^{\prime}(\boldsymbol{k}-1) + \boldsymbol{a}\_{i}\boldsymbol{\psi}\_{p}^{\prime}(\boldsymbol{k}-2) \tag{24}
$$

**Figure 5.** The symmetrical triangular membership function of *e* and *de*, fuzzy rule table, fuzzy inference and fuzzification

3. Parameter adjusting mechanism:

The gradient descent method is used to derive the AFC control law in Fig. 3. The objective of the parameters adjustment in FC is to minimize the square error between the mover position and the output of the RM. The instantaneous cost function is defined by

$$f(k+1) = \frac{1}{2} e\_{\
u} (k+1)^2 = \frac{1}{2} \left[ \left. \nu \right|\_{\
u} (k+1) - \left. \nu \right|\_{\
u} (k+1) \right]^2 \tag{25}$$

and the four defuzzifier parameters of *cj,i*, *cj+1,i*, *cj,i+1*, *cj+1,i+1* are adjusted according to

$$
\Delta \mathfrak{c}\_{n,n}(k+1) \propto -\frac{\partial \mathfrak{J}(k+1)}{\partial \mathfrak{c}\_{n,n}(k)} = -\alpha \frac{\partial \mathfrak{J}(k+1)}{\partial \mathfrak{c}\_{n,n}(k)}\tag{26}
$$

with *m* = *j*, *j*+1, *n* = *i*,*i*+1 and where represents learning rate. However, following the similar derivation with (Kung & Tsai, 2007), the *m n*, *c* can be obtained as

$$
\Delta \mathcal{L}\_{n,v}(k) \quad \approx \mathcal{a}(K\_p + K\_i) K\_v e(k) d\_{n,v} \tag{27}
$$

with *m* = *j*, *j+1* and *n* = *i*,*i+1*.

188 Fuzzy Controllers – Recent Advances in Theory and Applications

Once the *tr* is chosen, the natural frequency

and the difference equation is written as.

 

<sup>1</sup> *A0 A1 A2 A3 A4 A5 A6*

**Input of** *e (for i=3)*



*A0 A1 A2 A3 A4 A5 A6 c01 c02 c03 c04 c05 c06 c10 c11 c12 c13 c14 c15 c16 c20 c21 c22 c23 c24 c25 c26 c30 c31 c32 c33 c34 c35 c36 c40 c41 c42 c43 c44 c45 c46 c50 c51 c52 c53 c54 c55 c56 c60 c61 c62 c63 c64 c65 c66*

*e*

A3 (e)

A4 (e)=1- A3 (e)

B1(de)

*(e)*

and fuzzification

*(de)*

**Input of** *de (for j=1)*

1

*B0 B1 B2 B3 B4 B5 B6* B2(de)=1- B1(de)

*de*

6

4 2

3. Parameter adjusting mechanism:

**Fuzzy Rule Table**

*c00 dE*

*E*

*B0 B1 B2 B3 B4 B5 B6*

*de*

0



frequency

a fastest step response. Hence, the relation between the rising time *tr* and the natural

(1 ) 0.1 *n rt n r t e* 

bilinear transformation, (21) can be transformed to a discrete model by

*m p*

 

*e*

( ) () 1

*<sup>n</sup>* for a step input response in (21) can be derived and shown as follows.

1 12 01 2 \* 1 1 2

 

*z a az az z bz bz*

1 2 01 2 ( ) ( 1) ( 2) ( ) ( 1) ( 2) *m m m pp p*

1 *(e)*

(*e*) *Ai* 

(*e*) *Ai<sup>1</sup>* 

**Figure 5.** The symmetrical triangular membership function of *e* and *de*, fuzzy rule table, fuzzy inference

The gradient descent method is used to derive the AFC control law in Fig. 3. The objective of the parameters adjustment in FC is to minimize the square error between the mover position

> <sup>2</sup> 1 1 <sup>2</sup> ( 1) ( 1) ( 1) ( 1) 2 2 *m mp Jk e k*

> > ( 1) ( 1) ( 1) () () *m n*

*Jk Jk c k*

and the four defuzzifier parameters of *cj,i*, *cj+1,i*, *cj,i+1*, *cj+1,i+1* are adjusted according to

 

, ,

*ck ck* 

*m n m n*

*k k* (25)

(26)

and the output of the RM. The instantaneous cost function is defined by

,

1 2

*Ai Ai+1*

Rule 1: *e* is *A3* and *de* is *B1* then *uf* is *c13* Rule 2: *e* is *A3* and *de* is *B2* then *uf* is *c23* Rule 3: *e* is *A4* and *de* is *B1* then *uf* is *c14* Rule 4: *e* is *A4* and *de* is *B2* then *uf* is *c24*

**Fuzzy Inference and Output**

*k bk b k akak ak* (24)

*<sup>e</sup> e ei* = -6+2\*i *ei+1* = -4+2\*i

(22)

(23)

\*\* \*

then

*<sup>n</sup>* can be obtained. Furthermore, applying the

 

*2 <sup>e</sup> <sup>e</sup> <sup>e</sup> <sup>i</sup> <sup>1</sup> Ai* \*

If is located between the and

*e <sup>i</sup> e <sup>i</sup> <sup>1</sup> e*

(*e*) *1* (*e*) *Ai <sup>1</sup> Ai*

where

**Defuzzification**

*d31* =A3(e)\*B1(de)

and *d31* +*d32*+*d41*+*d42*=1

( )

 

*2 4 2 i e*

*13 31 23 32 14 41 24 42 31 32 41 42 <sup>13</sup> <sup>31</sup> <sup>23</sup> <sup>32</sup> <sup>14</sup> <sup>41</sup> <sup>24</sup> <sup>42</sup> <sup>f</sup> c d c d c d c d d d d d <sup>c</sup> <sup>d</sup> <sup>c</sup> <sup>d</sup> <sup>c</sup> <sup>d</sup> <sup>c</sup> <sup>d</sup> <sup>u</sup>* \* \* \* \* \* \* \* \* 

*d41* =A4(e)\*B1(de) =(1-A3(e))\*B1(de) *d32* =A3(e)\*B2(de)=A3(e)\*(1-B1(de)) *d42* = A4(e)\*B2(de)=(1-A3(e))\*(1-B1(de))

#### **2.3. Motion trajectory planning of X-Y table**

The circular, window and star motion trajectories are typical used as the performance evaluation of the motion controller for X-Y table.

a. In circular motion trajectory, it is computed by

$$\mathbf{x}\_i = r \cdot \text{sir}(\theta\_i) \tag{28}$$

$$y\_i = r \cos(\theta\_i) \tag{29}$$

with *i i* <sup>1</sup> . Where , *r* , *<sup>i</sup> x* , *<sup>i</sup> y* are angle increment, radius, X-axis trajectory command and Y-axis trajectory command, respectively.

b. The window motion trajectory is shown in Fig.6. The formulation is derived as follows: a-trajectory:

$$\mathbf{x}\_{i} = \mathbf{x}\_{i-1}, \mathbf{y}\_{i} = \mathbf{S} + \mathbf{y}\_{i-1} \tag{30}$$

b-trajectory:

$$
\theta\_i \colon \frac{6}{4}\pi \to 2\pi,\text{ and }\theta\_i = \theta\_{i-1} + \Delta\theta\}
$$

$$
\propto\_i = O\_{x1} + r\cos(\theta\_i), \\
y\_i = O\_{y1} + r\sin(\theta\_i) \tag{31}
$$

c-trajectory:

$$\mathbf{x}\_{i} = \mathbf{S} + \mathbf{x}\_{i-1}, \mathbf{y}\_{i} = \mathbf{y}\_{i-1} \tag{32}$$

d-trajectory:

$$
\begin{aligned}
\text{(}\theta\_i: \pi \to \frac{6}{4}\pi, \text{ and } \theta\_i = \theta\_{i-1} + \Delta\theta\text{)}\\\\
\text{(}\infty\_i = O\_{\cdot 2} + r\cos(\theta\_i)\_{\cdot}y\_i = O\_{\cdot 2} + r\sin(\theta\_i)\text{)}\end{aligned}
\tag{33}
$$

e-trajectory:

$$\mathbf{x}\_{i} = \mathbf{x}\_{i-1}\prime\prime\_{i} = -\mathbf{S} + \mathbf{y}\_{i-1} \tag{34}$$

f-trajectory:

$$
\begin{aligned}
\{\theta\_i \colon \frac{1}{2}\pi \to \pi \text{ , and } \theta\_i = \theta\_{i-1} + \Delta\theta\} \\\\
\alpha\_i = O\_{\times 3} + r\cos(\theta\_i)\_{\prime} \underline{y}\_i = O\_{\cdot 3} + r\sin(\theta\_i) \end{aligned}
\tag{35}
$$

g-trajectory:

$$\mathbf{x}\_{i} = -\mathbf{S} + \mathbf{x}\_{i-1'} \mathbf{y}\_{i} = \mathbf{y}\_{i-1} \tag{36}$$

FPGA-Based Motion Control IC for Linear Motor Drive X-Y Table Using Adaptive Fuzzy Control 191

*<sup>i</sup> i i <sup>i</sup> xS x yS y* (40)

*<sup>i</sup> i i <sup>i</sup> xS x yS y* (41)

*<sup>i</sup> i i <sup>i</sup> xS x y S y* (42)

*<sup>i</sup> i i <sup>i</sup> x S x yS y* (43)

*x*

1 1 \* sin 54 , \* sin 36 *o o*

1 1 \* sin18 , \* sin72 *o o*

1 1 \* sin18 , \* sin72 *o o*

1 1 \* sin 54 , \* sin 36 *o o*

Where *S* , *<sup>i</sup> x* , *<sup>i</sup> y* are position increment, X-axis trajectory command and Y-axis trajectory

*y*

18

**b e**

**d**

36 54

**3. The design of a motion control IC for linear motor drive X-Y table** 

72

Figure 8 illustrates the internal architecture of the proposed FPGA-based motion control IC for linear motor drive X-Y table. The FPGA uses Altera Stratix II EP2S60, which has 48,352 ALUTs (Adaptive Look-Up Tables), 36 DSP blocks, 144 embedded multipliers, 718 maximum user I/O pins, total 2,544,192 RAM bits, and a Nios II embedded processor which has a 32-bit configurable CPU core, 16 M byte flash memory, 1 M byte SRAM and 16 M byte SDRAM. The Nios II processor can be downloaded into FPGA to construct a SoPC environment. The internal circuit in Fig. 8 comprises a Nios II embedded processor IP (Intellectual Properties) and an application IP. The Nios II processor is depicted to both generate the motion trajectory and collect the response data. The application IP includes the

command, respectively. The motion speed of the table is determined by *S* .

**a**

*Start*

**c**

b-trajectory :

c-trajectory :

d-trajectory :

e-trajectory :

**Figure 7.** Star motion trajectory

h-trajectory:

$$
\begin{aligned}
\{\theta\_i \colon 0 \to \frac{1}{2}\pi \text{ , and } \theta\_i = \theta\_{i-1} + \Delta\theta\} \\\\
\chi\_i = O\_{\ge 4} + r\cos(\theta\_i)\_\nu \underline{y}\_i = O\_{\ge 4} + r\sin(\theta\_i)
\end{aligned}
\tag{37}
$$

i-trajectory:

$$\mathbf{x}\_{i} = \mathbf{x}\_{i-1}\prime\prime\_{i} = \mathbf{S} + \mathbf{y}\_{i-1} \tag{38}$$

where *S* , , *<sup>i</sup> x* , *<sup>i</sup> y* are position increment, angle increment, X-axis trajectory command and Y-axis trajectory command, respectively. In addition, the 1 1 (,) *O O x y* , 2 2 (,) *O O x y* , 3 3 (,) *O O x y* , 4 4 (,) *O O x y* are arc center of b-, d-, f-, and h-trajectory in the Fig. 6 and r is the radius. The motion speed of the table is determined by .

**Figure 6.** Window motion trajectory

c. Star motion trajectory is shown in Fig.7. The formulation is derived as follows: a-trajectory :

$$\mathbf{x}\_{i} = \mathbf{S} + \mathbf{x}\_{i-1}, \qquad \mathbf{y}\_{i} = \mathbf{y}\_{i-1} \tag{39}$$

b-trajectory :

190 Fuzzy Controllers – Recent Advances in Theory and Applications

*y*

**c**

**4 4 (,)** *O O x y* **3 3 (,)** *O O x y*

**g**

 

1

1

, *<sup>i</sup> x* , *<sup>i</sup> y* are position increment, angle increment, X-axis trajectory command

*x*

 

1 1 , *i ii i x x y Sy* (38)

*i*1 **2 2 (,)** *O O x y*

*r*

*i i* <sup>1</sup>

**(,)** *i i x y* **1 1 (,)** *i i x y*

1 1 , *i i ii x Sx y y* (39)

 

1 1 , *i ii i x Sx y y* (36)

(35)

(37)

<sup>1</sup> ( : , and ) <sup>2</sup> *<sup>i</sup> i i*

3 3 cos( ), sin( ) *i x ii y <sup>i</sup> xO r yO r* 

<sup>1</sup> ( : 0 , and ) <sup>2</sup> *<sup>i</sup> i i*

4 4 cos( ), sin( ) *i x ii y <sup>i</sup> xO r yO r* 

and Y-axis trajectory command, respectively. In addition, the 1 1 (,) *O O x y* , 2 2 (,) *O O x y* , 3 3 (,) *O O x y* , 4 4 (,) *O O x y* are arc center of b-, d-, f-, and h-trajectory in the Fig. 6 and r is the radius. The

> .

> > **2 2 (,)** *O O x y*

**e**

c. Star motion trajectory is shown in Fig.7. The formulation is derived as follows:

**f**

**d**

 

 

f-trajectory:

g-trajectory:

h-trajectory:

i-trajectory:

where *S* ,

**Figure 6.** Window motion trajectory

*Start*

a-trajectory :

motion speed of the table is determined by

**1 1 (,)** *O O x y*

**a**

**i**

**b**

**h**

$$\mathbf{x}\_{i} = -\mathbf{S} \,^{\ast}\sin 54^{\circ} + \mathbf{x}\_{i-1} \qquad y\_{i} = -\mathbf{S} \,^{\ast}\sin 36^{\circ} + y\_{i-1} \tag{40}$$

c-trajectory :

$$\mathbf{x}\_{i} = \mathbf{S}^{\*} \sin 18^{\circ} + \mathbf{x}\_{i-1^{\prime}} \qquad y\_{i} = \mathbf{S}^{\*} \sin 72^{\circ} + y\_{i-1} \tag{41}$$

d-trajectory :

$$\mathbf{x}\_{i} = \mathbf{S}^\* \sin 18^\circ + \mathbf{x}\_{i-1'} \qquad y\_{i} = -\mathbf{S}^\* \sin 72^\circ + y\_{i-1} \tag{42}$$

e-trajectory :

$$\mathbf{x}\_{i} = -\mathbf{S} \text{ \* } \sin 54^{\circ} + \mathbf{x}\_{i-1} \qquad y\_{i} = \mathbf{S} \text{ \* } \sin 36^{\circ} + y\_{i-1} \tag{43}$$

Where *S* , *<sup>i</sup> x* , *<sup>i</sup> y* are position increment, X-axis trajectory command and Y-axis trajectory command, respectively. The motion speed of the table is determined by *S* .

**Figure 7.** Star motion trajectory

### **3. The design of a motion control IC for linear motor drive X-Y table**

Figure 8 illustrates the internal architecture of the proposed FPGA-based motion control IC for linear motor drive X-Y table. The FPGA uses Altera Stratix II EP2S60, which has 48,352 ALUTs (Adaptive Look-Up Tables), 36 DSP blocks, 144 embedded multipliers, 718 maximum user I/O pins, total 2,544,192 RAM bits, and a Nios II embedded processor which has a 32-bit configurable CPU core, 16 M byte flash memory, 1 M byte SRAM and 16 M byte SDRAM. The Nios II processor can be downloaded into FPGA to construct a SoPC environment. The internal circuit in Fig. 8 comprises a Nios II embedded processor IP (Intellectual Properties) and an application IP. The Nios II processor is depicted to both generate the motion trajectory and collect the response data. The application IP includes the circuits of two position AFC and speed P controllers as well as two current vector controllers for X-axis and Y-axis table. The sampling frequency of position control loop is designed with 2kHz. The operating clock rate of the designed FPGA controller is 50MHz and the frequency divider generates 50 Mhz (*Clk*), 25 MHz (*Clk-step*), 2 kHz (*Clk-sp*) and 16 kHz (*Clk-cur*) clock to supply all module circuits of application IP in Fig. 8.

FPGA-Based Motion Control IC for Linear Motor Drive X-Y Table Using Adaptive Fuzzy Control 193

*<sup>e</sup>* **[11..0]**

**Application IP**

Clk Clk-cur Clk-step

*\** **[11..0]** *qi*

> Clk Clk-cur Clk-step

**[11..0]** *\* qi*

> + **-**

> > **SD** *<sup>k</sup> de*

*de***(***k***)**

*<sup>i</sup>***<sup>1</sup>** *e*

*<sup>e</sup>* **[11..0]** **Frequency divider**

**current vector controller** 

**current vector controller** 

> & **<sup>i</sup> j j&i**

> > **(***de***)** *Bj*

**( ) \*** *i k <sup>q</sup>* x + *<sup>j</sup> <sup>i</sup> c* **,**

*r*

**s10 s11 s12 s13**

**RS,1**

*di***,** *<sup>j</sup>*

+ **-**

*de <sup>j</sup>***<sup>1</sup>**

**Fuzzification**

**RS,1**

**(***e***)** *Ai* 

α

*u* **(***k***)**

+

**-**

*v* **(***k***)**

x *Kv*

**s26 s27**

*r*

x *Kv*

*<sup>j</sup> ,i c* **<sup>1</sup>**

*di***1***,j*

*r*

*<sup>j</sup> <sup>i</sup> c* **,**

<sup>x</sup> <sup>+</sup> *<sup>i</sup>***,** *<sup>j</sup>***<sup>1</sup>** *<sup>d</sup>*

*r*

*<sup>j</sup> ,i c* **<sup>1</sup>**

**s29 s30 s31 s32 s33 s34**

*<sup>i</sup>***1***, <sup>j</sup>***<sup>1</sup>** *d*

x +

*j,i***<sup>1</sup>** *c*

**Look-up fuzzy table**

*r* *<sup>j</sup>***1***,i***<sup>1</sup>** *c*

*<sup>j</sup>***1***,i***<sup>1</sup>** *c*

x +

*<sup>j</sup> ,i***<sup>1</sup>** *c*

**Circuit of xBDIN[0]**

**yBDIN[0] Circuit of** 

Clk Clk-ctr Clk-sp

Clk-step

**CLK**

**STSB RCB RCA STSA CHB CHA**

**yPWM 1 yPWM 2 yPWM 3 yPWM 4 yPWM 5 yPWM 6**

**yADIN[11] yADIN[0] yBDIN[11]**

**xADIN[11] xADIN[0] xBDIN[11]**

**STSB RCB RCA STSA CHB CHA**

**xPWM 1 xPWM 2 xPWM 3 xPWM 4 xPWM 5 xPWM 6**

> *<sup>j</sup> <sup>i</sup> c* **,** *j,i***<sup>1</sup>** *c <sup>j</sup> ,i c* **<sup>1</sup>** *<sup>j</sup>***1***,i***<sup>1</sup>** *c*

**Look-up Fuzzy rule table**

**Figure 8.** The internal architecture of a motion control IC for linear motor drive X-Y table

+ **-**

*x* **(***k***)** *<sup>p</sup>*

*xm***(***k***)**

*x* **(***k 1***)** *<sup>p</sup>*

+ **-**

*e***(***k* **1)**

*e***(***k***)**

*e***(***k* **1)**

**SD** *ke*

**s9**

*ui*

*K <sup>p</sup>*

*K <sup>p</sup> e***(***k***)**

*Ki*

x +

+ x x

**Defuzzification Computation of velocity and current command, and tuning of fuzzy rule parameters** 

*ui*

**s7 s8**

+ + x +

*uf*

**s20 s21 s23 s24 s28 s19 s22 s25**

*Ki*

**position error and error change** 

*<sup>v</sup>* **(***k***)** *<sup>e</sup>***(***k***)**

+ **-**

*x* **(***k 1***)** *<sup>p</sup>*

*xm***(***k 1***)**

**s6**

+

x

*<sup>i</sup> <sup>j</sup> d* **1,**

*j,i***<sup>1</sup>** *c*

**Computation of reference model output Computation of mover velocity,** 

x

*<sup>j</sup> ,i c* **<sup>1</sup>**

*<sup>i</sup>***,** *<sup>j</sup>***<sup>1</sup>** *d*

**[15..0]** *<sup>p</sup> x*

*<sup>e</sup>* **[11..0]**

*xm* **(***k***)** *xm* **(***k 1***)** *xm***(***k 2***) ( ) \*** *<sup>x</sup> <sup>k</sup> <sup>p</sup>* **( ) \*** *<sup>x</sup> <sup>k</sup> <sup>1</sup> <sup>p</sup>* **( ) \*** *<sup>x</sup> <sup>k</sup> <sup>1</sup> <sup>p</sup>* **( ) \*** *<sup>x</sup> <sup>k</sup> <sup>2</sup> <sup>p</sup>*

*x* **(***k***)** *<sup>p</sup>*

Clk Clk-sp *\* <sup>p</sup> x* **[15..0]** Clk-step

**UART PIO Timer SPI**

**[15..0]**

**Circuit of position adaptive fuzzy controller (AFC) and speed P controller** 

**[15..0]**

\* *p x*

\* *p y*

**Circuit of position adaptive fuzzy controller (AFC) and speed P controller** 

*Altera FPGA (Stratix II EP2S60F672C5ES )*

Clk

*<sup>e</sup>* **[11..0]**

**[15..0]** *<sup>p</sup> y*

\* *<sup>p</sup> y* **[15..0]**

> Clk Clk-sp

Clk-step

**Avalon Bus**

**Nios II Embedded Processor IP**

**QEP detection and transformation**

> x + **-** + **-**

*xm* **(***k 2***)**

**Encoder of x-axis** Clk

+ x

*xm* **(***k 1***)**

**<sup>1</sup>** *b* **<sup>2</sup>** *b*

x

**(***e***)** *Ai* 

x

*<sup>i</sup>***1***, <sup>j</sup>***<sup>1</sup>** *d*

x

*<sup>i</sup> <sup>j</sup> d* **,**

*<sup>j</sup> <sup>i</sup> c* **,**

**s18**

*<sup>i</sup> , <sup>j</sup>***<sup>1</sup>** *d*

*X-axis Position controller*

**CPU On-chip ROM On-chip RAM**

**QEP detection and** 

*Y-axis Position controller*

**Avalon Bus**

**transformation yEncoder-A**

**sram\_cs sram\_we sram\_oe sram\_be[3] sram\_be[2] sram\_be[1] sram\_be[0]**

**xEncoder-A xEncoder-B xEncoder-Z From Linear** 

x

*a***0**

**( ) \*** *<sup>x</sup> <sup>k</sup> <sup>p</sup>*

x

**( ) \*** *<sup>x</sup> <sup>k</sup> <sup>1</sup> <sup>p</sup>* **( ) \*** *<sup>x</sup> <sup>p</sup> <sup>k</sup> <sup>2</sup>*

**1** *a*

+

x

**2** *a*

**s0 s1 s2 s3 s4 s5**

**D[31] D[0]**

**A[0]**

**A[22]**

**yEncoder-B yEncoder-Z From Linear Encoder of y-axis**

**Figure 9.** State diagram of an FSM for describing the AFC in position loop and P controller in speed

x

*<sup>j</sup>***1***,i***<sup>1</sup>** *c*

*di***1,** *<sup>j</sup>***<sup>1</sup>**

loop (for X-axis Table)

**s14**

x + **1 - (***e***)** *Ai* 

**(***de***)** *Bj* 

*di* **,** *<sup>j</sup>* + **-**

**(***de***)** *Bj* 

> *( e ) Ai***<sup>1</sup>**

> > x

**(***de***)** *Bj* 

**1** *( de ) Bj***<sup>1</sup>** 

**s15 s16 s17**

*<sup>i</sup> , <sup>j</sup> d* **<sup>1</sup>**

An FSM is also employed to model the AFC of the position loop and P controller of the speed loop in X-axis table and shown in Fig. 9, which uses one adder, one multiplier, a look-up table, comparators, registers, etc. and manipulates 35 steps machine to carry out the overall computation. With exception of the data type in reference model are 24-bits, others data type are designed with 12-bits length, 2's complement and Q11 format. Although the algorithm of AFC is highly complexity, the FSM can give a very adequate modeling and easily be described by VHDL. Furthermore, steps s0~s6 execute the computation of reference model output; steps s6~s9 are for the computation of mover velocity, position error and error change; steps s9~s12 execute the function of the fuzzification; s13 describe the look-up table and s14~s22 defuzzification; and steps s23~s34 execute the computation of velocity and current command output, and the tuning of fuzzy rule parameters. The SD is the section determination of *e* and *de* and the RS,1 represents the right shift function with one bit. The operation of each step in Fig.9 can be completed within 40ns (25 MHz clock) in FPGA; therefore total 35 steps need a 1.4s operation time. It doesn't loss any control performance for the overall system because the operation time with 1.4s is much less than the sampling interval, 500 s (2 kHz), of the position control loop in Fig.3. In Fig. 8, the QEP circuit and circuit for current vector control refer to (Kung & Tsai, 2007). Further, the Nios II embedded processor IP is depicted to perform the function of the motion trajectory and two-axis position/speed loop controller for X-Y table in software. Figure 10 illustrates the flow charts of the main program and the interrupt service routine (ISR), where the interrupt interval is designed with 2ms. All programs are coded in the C programming language. Then, through the complier and linker operation in the Nios II IDE (Integrated Development Environment), the execution code is produced and can be downloaded to the external Flash or SDRAM via JTAG interface.

Under the proposed design method, the overall resource usage of the proposed motion control IC is listed in Table 1 which the two AFC circuits need 16,110 ALUTs, the Nios II embedded processor IP needs 8,275 ALUTs and 46,848 RAM bits and the application IP needs 22,928 ALUTs and 595,968 RAM bits in FPGA. Therefore, the motion control IC uses 64.5% ALUTs resource and 25.2% RAM resource of Stratix II EP2S60.


**Table 1.** The resource usage of a motion control IC in FPGA

circuits of two position AFC and speed P controllers as well as two current vector controllers for X-axis and Y-axis table. The sampling frequency of position control loop is designed with 2kHz. The operating clock rate of the designed FPGA controller is 50MHz and the frequency divider generates 50 Mhz (*Clk*), 25 MHz (*Clk-step*), 2 kHz (*Clk-sp*) and 16

An FSM is also employed to model the AFC of the position loop and P controller of the speed loop in X-axis table and shown in Fig. 9, which uses one adder, one multiplier, a look-up table, comparators, registers, etc. and manipulates 35 steps machine to carry out the overall computation. With exception of the data type in reference model are 24-bits, others data type are designed with 12-bits length, 2's complement and Q11 format. Although the algorithm of AFC is highly complexity, the FSM can give a very adequate modeling and easily be described by VHDL. Furthermore, steps s0~s6 execute the computation of reference model output; steps s6~s9 are for the computation of mover velocity, position error and error change; steps s9~s12 execute the function of the fuzzification; s13 describe the look-up table and s14~s22 defuzzification; and steps s23~s34 execute the computation of velocity and current command output, and the tuning of fuzzy rule parameters. The SD is the section determination of *e* and *de* and the RS,1 represents the right shift function with one bit. The operation of each step in Fig.9 can be completed within 40ns (25 MHz clock) in FPGA; therefore total 35 steps need a 1.4s operation time. It doesn't loss any control performance for the overall system because the operation time with 1.4s is much less than the sampling interval, 500 s (2 kHz), of the position control loop in Fig.3. In Fig. 8, the QEP circuit and circuit for current vector control refer to (Kung & Tsai, 2007). Further, the Nios II embedded processor IP is depicted to perform the function of the motion trajectory and two-axis position/speed loop controller for X-Y table in software. Figure 10 illustrates the flow charts of the main program and the interrupt service routine (ISR), where the interrupt interval is designed with 2ms. All programs are coded in the C programming language. Then, through the complier and linker operation in the Nios II IDE (Integrated Development Environment), the execution code is

produced and can be downloaded to the external Flash or SDRAM via JTAG interface.

64.5% ALUTs resource and 25.2% RAM resource of Stratix II EP2S60.

**Total**

**Table 1.** The resource usage of a motion control IC in FPGA

**Application IP**

Under the proposed design method, the overall resource usage of the proposed motion control IC is listed in Table 1 which the two AFC circuits need 16,110 ALUTs, the Nios II embedded processor IP needs 8,275 ALUTs and 46,848 RAM bits and the application IP needs 22,928 ALUTs and 595,968 RAM bits in FPGA. Therefore, the motion control IC uses

> **2 x Current loop controller (Current vector control, SVPWM,ADC,QEP)**

**(AFC)**

**Nios II Embedded Processor IP 8,275 46,848**

**IP Module circuit ALUTs**

**16,110 <sup>0</sup> 2 x Adaptive fuzzy controller** 

**31,203 642,816**

**6,818 595,968**

**Memory (bits)**

kHz (*Clk-cur*) clock to supply all module circuits of application IP in Fig. 8.

**Figure 8.** The internal architecture of a motion control IC for linear motor drive X-Y table

**Figure 9.** State diagram of an FSM for describing the AFC in position loop and P controller in speed loop (for X-axis Table)

FPGA-Based Motion Control IC for Linear Motor Drive X-Y Table Using Adaptive Fuzzy Control 195

2 2 ( ) ( ( ) ( )) ( ( ) ( )) *Tk x k x x y k y x mp mp* (44)

(45)

**0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> -5**

**(c)**

**Position command Position response**

**Reference command**

**<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> -10**

**(d)**

**Time (s)**

**Using AFC** 

**Time (s)**

(46)

rate=0.1). The position command is a 4/3Hz square wave signal with 10mm amplitude. In Figs. 11(a) and 12(a), when an 11 kg load is added upon the mover of the X-Y table and the fuzzy control by using a fixed rule table, the position dynamic response in X-axis and Y-axis table exhibits a 12.8% and 23.1% overshoot and severe oscillation, respectively. Accordingly, an AFC is adopted in Fig.3. When the proposed AFC is used with learning rate being 0.1, the tracking results are highly improved and presented in Figs. 11(c) and 12(c). Initially, the position response in X-axis or Y-axis table tracks the output of the reference model with oscillation. After one or two square wave commands, the *ci,j* parameters in fuzzy rule table are tuned to adequate values, and the position response in X-axis or Y-axis table can closely follow the output of the reference model. Further, the tracking motion about circular, window and star trajectory by using FC and AFC are experimented. To evaluate the

> 1 ( )/ *<sup>n</sup>*

 *Tk m n* 

tracking error. In the circular tracking motion, the circle command is with center (25, 25) cm and radius 10cm and its experimental results are shown in Figs. 13~14. In the window tracking motion, the trajectory is designed as Fig.6 and its experimental results are shown in Figs. 15~16. In the star tracking motion, the trajectory is designed as Fig.7 and its experimental results are shown in Figs. 17~18. Further, the tracking performance in Figs 13~18 by using FC and AFC control algorithm are evaluated according to the indices of (44)~(46), and its results are listed in Table 2. Compared with FC, the mean of tracking errors

2

respectively represent instantaneous value, mean and variance of

(() )/

*k m Tk n* 

1

**Figure 11.** (a) Position step response and (b) current response by using the FC as well as (c) position step response and (d) current response by using the AFC in X-axis table

**Time (s)**

**0**

**Iq current (A)**

**Position response**

**(mm)** 

**Using FC** 

**Time (s)**

*n k*

Where *T(k)*, *m* and

**0**

**Reference command**

**Iq current (A)**

**Position response**

**(mm)** 

**0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> -5**

**(a)**

**Position command Position response**

**<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> -10**

**(b)**

tracking performance, the indices are firstly defined as follows.

**Figure 10.** Flow chart of the main and ISR program in Nios II embedded processor

## **4. Experimental results**

The overall experimental system depicted in Fig.3 includes an FPGA (Stratix II EP2S60F672C5) experimental board, two voltage source IGBT inverters and an X-Y table which is driven by two PMLSMs. The PMLSM was manufactured by the BALDOR electric company; and it is a single-axis stage with a cog-free linear motor and a stroke length with 600mm. The parameters of the motor are: *R*s = 27 , *L*d = *L*q = 23.3 mH, Kt = 79.9N/A. The input voltage, continuous current, peak current (10% duty) and continuous power of the PMLSM are 220V, 1.6A, 4.8A and 54W, respectively. The maximum speed and acceleration are 4m/s and 4 g but depend on external load. The moving mass is 2.5Kg, the maximum payload is 22.5Kg and the maximum thrust force is 73N under continuous operating conditions. A linear encoder with a resolution of 5m is mounted on the PMLSM as the position sensor, and the pole pitch is 30.5mm (about 6100 pulses). The inverter has three sets of IGBT power transistors. The collector-emitter voltage of the IGBT is rated 600V; the gateemitter voltage is rated 20V, and the DC collector current is rated 25A and in short time (1ms) is 50A. The photo-IC, Toshiba TLP250, is used in the gate driving circuit of IGBT. Input signals of the inverter are PWM signals from the FPGA device.

To confirm the effectiveness of the proposed AFC in linear drive X-Y table, a realization of position controller based on the FPGA in Fig.3 is constructed and some experiments are evaluated. The control sampling frequency of the current, speed and position loops are designed as 16kHz, 2kHz and 2kHz, respectively. In the motion control IC, two position/speed/current controllers are all realized by hardware in FPGA, and the motion trajectory algorithm is implemented by software using the Nios II embedded processor. The speed controller adopts a P controller and the AFC is used in the position loop. The transfer function of the reference model is selected by a second order system with the natural frequency of 30 rad/s and damping ratio of 1. The step response is first tested to evaluate the performance of the proposed controller. Figures 11 and 12 respectively show the position step responses for X-axis and Y-axis table using the FC (learning rate=0) and AFC (learning rate=0.1). The position command is a 4/3Hz square wave signal with 10mm amplitude. In Figs. 11(a) and 12(a), when an 11 kg load is added upon the mover of the X-Y table and the fuzzy control by using a fixed rule table, the position dynamic response in X-axis and Y-axis table exhibits a 12.8% and 23.1% overshoot and severe oscillation, respectively. Accordingly, an AFC is adopted in Fig.3. When the proposed AFC is used with learning rate being 0.1, the tracking results are highly improved and presented in Figs. 11(c) and 12(c). Initially, the position response in X-axis or Y-axis table tracks the output of the reference model with oscillation. After one or two square wave commands, the *ci,j* parameters in fuzzy rule table are tuned to adequate values, and the position response in X-axis or Y-axis table can closely follow the output of the reference model. Further, the tracking motion about circular, window and star trajectory by using FC and AFC are experimented. To evaluate the tracking performance, the indices are firstly defined as follows.

194 Fuzzy Controllers – Recent Advances in Theory and Applications

**4. Experimental results** 

**Figure 10.** Flow chart of the main and ISR program in Nios II embedded processor

**Start of main program**

**Initial interrupt**

**Initial timer**

**loop**

**of each axis** 

Input signals of the inverter are PWM signals from the FPGA device.

The overall experimental system depicted in Fig.3 includes an FPGA (Stratix II EP2S60F672C5) experimental board, two voltage source IGBT inverters and an X-Y table which is driven by two PMLSMs. The PMLSM was manufactured by the BALDOR electric company; and it is a single-axis stage with a cog-free linear motor and a stroke length with 600mm. The parameters of the motor are: *R*s = 27 , *L*d = *L*q = 23.3 mH, Kt = 79.9N/A. The input voltage, continuous current, peak current (10% duty) and continuous power of the PMLSM are 220V, 1.6A, 4.8A and 54W, respectively. The maximum speed and acceleration are 4m/s and 4 g but depend on external load. The moving mass is 2.5Kg, the maximum payload is 22.5Kg and the maximum thrust force is 73N under continuous operating conditions. A linear encoder with a resolution of 5m is mounted on the PMLSM as the position sensor, and the pole pitch is 30.5mm (about 6100 pulses). The inverter has three sets of IGBT power transistors. The collector-emitter voltage of the IGBT is rated 600V; the gateemitter voltage is rated 20V, and the DC collector current is rated 25A and in short time (1ms) is 50A. The photo-IC, Toshiba TLP250, is used in the gate driving circuit of IGBT.

**Y-axis position command Set parameters**

**Start of ISR ( each 500Hz )**

**Computation of position value for each axis from motion trajectory command**

**Return**

**Output the X-axis and** 

To confirm the effectiveness of the proposed AFC in linear drive X-Y table, a realization of position controller based on the FPGA in Fig.3 is constructed and some experiments are evaluated. The control sampling frequency of the current, speed and position loops are designed as 16kHz, 2kHz and 2kHz, respectively. In the motion control IC, two position/speed/current controllers are all realized by hardware in FPGA, and the motion trajectory algorithm is implemented by software using the Nios II embedded processor. The speed controller adopts a P controller and the AFC is used in the position loop. The transfer function of the reference model is selected by a second order system with the natural frequency of 30 rad/s and damping ratio of 1. The step response is first tested to evaluate the performance of the proposed controller. Figures 11 and 12 respectively show the position step responses for X-axis and Y-axis table using the FC (learning rate=0) and AFC (learning

$$T(k) = \sqrt{(\mathbf{x}\_m(k) - \mathbf{x}\_p(\mathbf{x}))^2 + (\mathbf{y}\_m(k) - \mathbf{y}\_p(\mathbf{x}))^2} \tag{44}$$

$$m = \sum\_{k=1}^{n} T(k) / m \tag{45}$$

$$\sigma = \sqrt{\sum\_{k=1}^{n} (T(k) - m)^2 / n} \tag{46}$$

Where *T(k)*, *m* and respectively represent instantaneous value, mean and variance of tracking error. In the circular tracking motion, the circle command is with center (25, 25) cm and radius 10cm and its experimental results are shown in Figs. 13~14. In the window tracking motion, the trajectory is designed as Fig.6 and its experimental results are shown in Figs. 15~16. In the star tracking motion, the trajectory is designed as Fig.7 and its experimental results are shown in Figs. 17~18. Further, the tracking performance in Figs 13~18 by using FC and AFC control algorithm are evaluated according to the indices of (44)~(46), and its results are listed in Table 2. Compared with FC, the mean of tracking errors

**Figure 11.** (a) Position step response and (b) current response by using the FC as well as (c) position step response and (d) current response by using the AFC in X-axis table

FPGA-Based Motion Control IC for Linear Motor Drive X-Y Table Using Adaptive Fuzzy Control 197

**200 300 400**

**200 300 400**

> **-10 0 10**

> **-10 0 10**

**200 300 400**

**200 300 400**

> **-10 0 10**

**-3.0**

**-10 0 10**

**X-axis Position error**

**Y-axis Position error**

**(mm)** 

**(mm)** 

**X-axis Position**

**Y-axis Position**

**(mm)** 

**(mm)** 

**<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>100</sup>**

**<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>100</sup>**

**6.2 11.5**

**(b)**

**0 1 2 3 4 5**

**0 1 2 3 4 5**

**-5.7**

**(d)**

**<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>100</sup>**

**<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>100</sup>**

**(b)**

**3.0**

**0 1 2 3 4 5**

**0 1 2 3 4 5**

**-2.7**

**(d)**

**Time (s)**

**Time (s)**

**Time (s)**

**4.6**

**-6.2**

**Time (s)**

**Time (s)**

**Time (s)**

**Time (s)**

**3.5**

**Time (s)**

**Figure 13.** Circular trajectory response by using the FC (a) Star trajectory tracking (b) Position tracking in X- and Y-axis table (c) Control efforts in X- and Y-axis table (d) Tracking errors in X- and Y-axis table

**X-axis Position**

**Y-axis Position**

**X-axis Position error**

**Y-axis Position error**

**(mm)** 

**(mm)** 

**(mm)** 

**(mm)** 

**Time (s)**

**Time (s)**

**<sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>350</sup> <sup>140</sup>**

**X-axis Position (mm)** 

**(a)**

**<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> -4**

**<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> -4**

**(c)**

**150 200 250 300 350**

**X-axis Position (mm)** 

**(a)**

**<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> -4**

**<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> -5**

**(c)**

> > **150**

**0**

**Y-axis Iq (A)** 

**5**

**X-axis Iq (A)** 

**200**

**250**

**Y-axis Position (mm)** 

**300**

**350**

**Y-axis Iq (A)** 

**X-axis Iq (A)** 

**Y-axis Position (mm)** 

**Figure 14.** Circular trajectory response by using the AFC (a) Star trajectory tracking (b) Position tracking in X- and Y-axis table (c) Control efforts in X- and Y-axis table (d) Tracking errors

**Time (s)**

**Time (s)**

in X- and Y-axis table

**Figure 12.** (a) Position step response and (b) current response by using the FC as well as (c) position step response and (d) current response by using the AFC in Y-axis table

in circular, winddow and star motion trajectory are significantly reduced about 41.6%, 14.6% and 12.8% and the variance of tracking errors reduced about 33.3%, 64.6% and 47.4% after using AFC. Therefore, it shows that the AFC has a better tracking performance than FC in motion control of linear motor drive X-Y table. Finally, from the experimental results of Figs.11~18, it demonstrates that the proposed AFC and the FPGA-based motion control IC used for the linear motor drive X-Y table is effective and correct.

## **5. Conclusion**

This study successfully presents a motion control IC for linear motor drive X-Y table based on FPGA technology. The works herein are summarized as follows.


However, the experimental results by step response as well as the circular, window and star motion trajectory tracking, has been revealed that the software/hardware co-design technology with the parallel processing well in the motion control system of linear motor drive X-Y table.


**Table 2.** Evaluation of tracking performance using FC and AFC

**Position command Position response**

**<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> -5**

**(a)**

**<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> -10**

**(b)**

**Figure 12.** (a) Position step response and (b) current response by using the FC as well as (c) position step response and (d) current response by using the AFC in Y-axis table

**Time (s)**

**Position command**

**Iq current (A)**

**Position response**

**(mm)** 

**Using FC** 

**Time (s)**

used for the linear motor drive X-Y table is effective and correct.

on FPGA technology. The works herein are summarized as follows.

**Table 2.** Evaluation of tracking performance using FC and AFC

**Circular motion**

**Variance Mean**

**Tracking error (mm)**

**Control algorithm**

**5. Conclusion** 

**Reference command**

**Iq current (A)**

**Position response**

**(mm)** 

in circular, winddow and star motion trajectory are significantly reduced about 41.6%, 14.6% and 12.8% and the variance of tracking errors reduced about 33.3%, 64.6% and 47.4% after using AFC. Therefore, it shows that the AFC has a better tracking performance than FC in motion control of linear motor drive X-Y table. Finally, from the experimental results of Figs.11~18, it demonstrates that the proposed AFC and the FPGA-based motion control IC

This study successfully presents a motion control IC for linear motor drive X-Y table based

1. The functionalities required to build a fully digital motion controller of linear motor drive X-Y table, such as the two current vector controllers, two speed P controllers, and two position AFCs and one motion trajectory planning, have been integrated in one FPGA chip. 2. An FSM joined by one multiplier, one adder, one LUT, or some comparators and registers has been employed to model the overall AFC algorithm, such that it not only is easily implemented by VHDL but also the resources usage can be reduced in the FPGA. 3. The software/hardware co-design technology under SoPC environment has been successfully applied to the motion controller of linear motor drive X-Y table.

However, the experimental results by step response as well as the circular, window and star motion trajectory tracking, has been revealed that the software/hardware co-design technology with the parallel processing well in the motion control system of linear motor drive X-Y table.

> **Star motion**

**Window motion**

**Fuzzy controller (FC)**

**0.60 2.12 3.52 0.40 0.75 1.85 3.51 1.43 1.64 2.05 1.22 1.43**

**Circular motion**

**Star motion**

**<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> -5**

**(c)**

**Reference command Position response**

**<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> -10**

**(d)**

**Time (s)**

**Using AFC** 

**Time (s)**

**Window motion**

**Adaptive fuzzy controller (AFC)**

**Figure 13.** Circular trajectory response by using the FC (a) Star trajectory tracking (b) Position tracking in X- and Y-axis table (c) Control efforts in X- and Y-axis table (d) Tracking errors in X- and Y-axis table

**Figure 14.** Circular trajectory response by using the AFC (a) Star trajectory tracking (b) Position tracking in X- and Y-axis table (c) Control efforts in X- and Y-axis table (d) Tracking errors in X- and Y-axis table

FPGA-Based Motion Control IC for Linear Motor Drive X-Y Table Using Adaptive Fuzzy Control 199

**200 300 400**

> **-10 0 10**

> > **9.0**

**-3.2**

**-10 0 10**

**X-axis tracking error**

**Y-axis tracking error**

**(mm)** 

**200 300 400**

> **-10 0 10**

**-2.3**

**-10 0 10**

**X-axis Position error**

**Y-axis Position error**

**(mm)** 

**(mm)** 

**X-axis Position**

**Y-axis Position**

**350**

**(mm)** 

**(mm)** 

**(mm)** 

**X-axis tracking** 

**Y-axis tracking** 

**350**

**(mm)** 

**(mm)** 

**<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>100</sup>**

**<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>0</sup>**

**(b)**

**0 5 10 15**

**3.5**

**0 5 10 15**

**(d)**

**-11.4**

**<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>100</sup>**

**<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>0</sup>**

**(b)**

**2.2**

**0 5 10 15**

**0 5 10 15**

**(d)**

**Time (s)**

**Time (s)**

**Time (s)**

**Time (s)**

**Time (s)**

**Time (s)**

**Time (s)**

**8.0**

**-10**

**Time (s)**

**Figure 17.** Star trajectory response by using the FC (a) Star trajectory tracking (b) Position tracking in X- and Y-axis table (c) Control efforts in X- and Y-axis table (d) Tracking errors in X- and Y-axis table

**Time (s)**

**Time (s)**

**<sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>0</sup>**

**X-axis position (mm)** 

**(a)**

**<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> -5**

**<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> -5**

**(c)**

**<sup>100</sup> <sup>150</sup> <sup>200</sup> <sup>250</sup> <sup>300</sup> <sup>0</sup>**

**X-axis Position (mm)** 

**(a)**

**<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> -5**

**<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> -5**

**(c)**

**0**

**0**

**(A)** 

**0**

**0**

**Y-axis Iq (A)** 

**5**

**X-axis Iq (A)** 

**5**

**Y-axis Position (mm)** 

**5**

**(A)** 

**X-axis Iq current** 

**Y-axis Iq current** 

**5**

**Y-axis position (mm)** 

**Figure 18.** Star trajectory response by using the AFC (a) Star trajectory tracking (b) Position tracking in X- and Y-axis table (c) Control efforts in X- and Y-axis table (d) Tracking errors in X- and Y-axis table

**Time (s)**

**Time (s)**

**Figure 15.** Window trajectory response by using the FC (a) Star trajectory tracking (b) Position tracking in X- and Y-axis table (c) Control efforts in X- and Y-axis table (d) Tracking errors in X- and Y-axis table

**Figure 16.** Window trajectory response by using the AFC (a) Star trajectory tracking (b) Position tracking in X- and Y-axis table (c) Control efforts in X- and Y-axis table (d) Tracking errors in X- and Y-axis table

**100**

**0**

**100 150 200**

**0**

**-5 0 5**

**Y-axis Iq (A)** 

**X-axis Iq (A)** 

**5**

**250 300**

**Y-axis Position (mm)** 

**Y-axis Iq (A)** 

**X-axis Iq (A)** 

**5**

**150**

**200**

**Y-axis Position (mm)** 

**250**

**300**

**50 100 150 200 250**

**X-axis Position (mm)** 

**(a)**

**<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> -5**

**<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> -4**

**(c)**

**50 100 150 200 250**

**X-axis Position (mm)** 

**(a)**

**<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> -5**

**0 2 4 6 8 10 12**

**(c)**

**<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>0</sup>**

**0 2 4 6 8 10 12**

**(b)**

**3.1**

**0 2 4 6 8 10 12**

**7.7**

**-3.1**

**0 2 4 6 8 10 12**

**-8.0**

**(d)**

**<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>0</sup>**

**0 2 4 6 8 10 12**

**(b)**

**1.8**

**0 2 4 6 8 10 12**

**5.0**

**-1.9**

**0 2 4 6 8 10 12**

**-5.0**

**(d)**

**Time (s)**

**Time (s)**

**Time (s)**

**Time (s)**

**Time (s)**

**Time (s)**

**Time (s)**

**Time (s)**

**100 200 300**

**100 200 300**

> **-10 0 10**

> **-10 0 10**

**100 200 300**

**100 200 300**

> **-10 0 10**

> **-10 0 10**

**X-axis Position**

**Y-axis Position**

**X-axis Position error**

**Y-axis Position error**

**(mm)** 

**(mm)** 

**(mm)** 

**(mm)** 

**Figure 15.** Window trajectory response by using the FC (a) Star trajectory tracking (b) Position tracking in X- and Y-axis table (c) Control efforts in X- and Y-axis table (d) Tracking errors in X- and Y-axis table

**X-axis Position**

**Y-axis Position**

**X-axis Position error**

**Y-axis Position error**

**(mm)** 

**(mm)** 

**(mm)** 

**(mm)** 

**Time (s)**

**Time (s)**

**Figure 16.** Window trajectory response by using the AFC (a) Star trajectory tracking (b) Position tracking in X- and Y-axis table (c) Control efforts in X- and Y-axis table (d) Tracking errors in X- and Y-axis table

**Time (s)**

**Time (s)**

**Figure 17.** Star trajectory response by using the FC (a) Star trajectory tracking (b) Position tracking in X- and Y-axis table (c) Control efforts in X- and Y-axis table (d) Tracking errors in X- and Y-axis table

**Figure 18.** Star trajectory response by using the AFC (a) Star trajectory tracking (b) Position tracking in X- and Y-axis table (c) Control efforts in X- and Y-axis table (d) Tracking errors in X- and Y-axis table

## **Author details**

Ying-Shieh Kung *Department of Electrical Engineering, Southern Taiwan University, Taiwan* 

### Chung-Chun Huang and Liang-Chiao Huang

*Green Energy and Environment Research Laboratories, Industrial Technology Research Institute, Taiwan* 

**Chapter 9** 

© 2012 Zhang and Gao, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Novel Yinger Learning Variable** 

**Universe Fuzzy Controller** 

Additional information is available at the end of the chapter

sense reasoning mode is approximate in nature.

Fuzzy control is a practical alternative for a variety of challenging of challenging control applications because it provides a convenient method for constructing nonlinear controllers via the use of heuristic information. The heuristic information may come from an operator who has acted as a human controller for a process. In the fuzzy control design methodology, a set of rules are written down by the operator on how to control the process, then make these into a fuzzy controller that emulates the decision-making process of the human. In some cases, the heuristic information may come from other novel intelligent applications. In other cases, the heuristic information may come from a control engineer who has performed extensive mathematical modeling, analysis, and development of control algorithms for a particular process. Regardless of where the heuristic control knowledge comes from, fuzzy control provides a user-friendly formalism for representing and implementing the ideas.

Over the past few decades, fuzzy logic theory is widely used: process control, management and decision making, operations research, economies. Dealing with simple 'yes' and 'no' answers is no longer satisfactory enough; a degree of membership (Zadeh, 1965) became a new way of solving problems. Fuzzy logic derives from the truth that the human common

In this chapter we provide a control engineering perspective on novel fuzzy controller. We take a pragmatic engineering approach to the design, analysis, performance evaluation, and implement of fuzzy control system. The chapter is basically broken into five parts. In section 2, we provide an overview of conventional control system design. In section 3 the basic theories of variable universe fuzzy control are been introduced. In section 4, we cover the novel fuzzy controller based on Yinger algorithm. In section 5, we use some examples to show how to design, simulate, and implement these controllers. Finally, in section 6, we

and reproduction in any medium, provided the original work is properly cited.

Ping Zhang and Guodong Gao

http://dx.doi.org/10.5772/48266

**1. Introduction** 

## **Acknowledgement**

The financial support provided by Bureau of Energy is gratefully acknowledged.

## **6. References**


SOPC World, (2004) Altera Corporation.

Wang, G.J. and Lee, T.J. (1999) Neural-network cross-coupled control system with application on circular tracking of linear motor X-Y table. *International Joint Conference on Neural Networks*, pp. 2194-2199.

**Chapter 9** 

## **Novel Yinger Learning Variable Universe Fuzzy Controller**

Ping Zhang and Guodong Gao

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48266

## **1. Introduction**

200 Fuzzy Controllers – Recent Advances in Theory and Applications

Chung-Chun Huang and Liang-Chiao Huang

Prentice Hall, Upper Saddle River, NJ.

*Electron. Magaz.,* vol. 5, no.1, pp.14-26.

153, no. 1, pp. 111-123.

pp. II-329~II-332.

pp.291~312, I-Tech, Vienna.

*Electron.,* vol. 54, no. 4, pp. 1937-1945. SOPC World, (2004) Altera Corporation.

*on Neural Networks*, pp. 2194-2199.

*Advanced Intelligent Mechatronics*, pp. 1188-1193.

the classroom. *IEEE Trans. on Education*, vol.47, no.4, pp.502-507.

*Department of Electrical Engineering, Southern Taiwan University, Taiwan* 

chip, *IEEE Trans. Ind. Electron.,* vol. 56, no.3, pp.856-870.

*Green Energy and Environment Research Laboratories, Industrial Technology Research Institute,* 

Cho, J. U., Le, Q. N. and Jeon, J. W. (2009) An FPGA-based multiple-axis motion control

Groove, M.P. (1996) *Fundamentals of modern manufacturing: materials, process, and systems*.

Goto, S., Nakamura, M. and Kyura, N. (1996) Accurate contour control of mechatronic servo systems using gaussian networks. *IEEE Trans. Ind. Electron.*, vol.43, no. 4, pp. 469-476. Hanafi, D., Tordon, M. and Katupitiya, J. (2003) An active axis control system for a conventional CNC machine. *Proceedings of IEEE/ASME International Conference on* 

Hall, T.S., Hamblen, J.O. (2004) System-on-a-programmable -chip development platforms in

Lin, F.J., SHieh, P.H. and Shen, P.H. (2006) Robust recurrent-neural-network sliding-mode control for the X-Y table of a CNC machine. *IEE Proc. Control Theory Application,* vol.

Monmasson, E., Idkhajine L. and Naouar, M.W. (2011) FPGA-based Controllers. *IEEE Trans.* 

Kung, Y.S., Huang P.G. and Chen, C.W. (2004) Development of a SOPC for PMSM drives. *Proceedings of the IEEE International Midwest Symposium on Circuits and Systems*, vol. II,

Kung, Y.S. and Tsai, M.H. (2007) FPGA-based speed control IC for PMSM drive with adaptive fuzzy control. *IEEE Trans. on Power Electronics*, vol. 22, no. 6, pp. 2476-2486. Kung, Y.S. and Chen, C.S. (2008). Realization of a motion control IC for robot manipulator based on novel FPGA technology. *Robot manipulators, programming, design and control.*

Sanchez-Solano, S., Cabrera, A. J., Baturone, I., Moreno-Velo, F.J., Brox, M. (2007) FPGA implementation of embedded fuzzy controllers for robotic applications. *IEEE Trans. Ind.* 

Wang, G.J. and Lee, T.J. (1999) Neural-network cross-coupled control system with application on circular tracking of linear motor X-Y table. *International Joint Conference* 

The financial support provided by Bureau of Energy is gratefully acknowledged.

**Author details** 

Ying-Shieh Kung

**Acknowledgement** 

**6. References** 

*Taiwan* 

Fuzzy control is a practical alternative for a variety of challenging of challenging control applications because it provides a convenient method for constructing nonlinear controllers via the use of heuristic information. The heuristic information may come from an operator who has acted as a human controller for a process. In the fuzzy control design methodology, a set of rules are written down by the operator on how to control the process, then make these into a fuzzy controller that emulates the decision-making process of the human. In some cases, the heuristic information may come from other novel intelligent applications. In other cases, the heuristic information may come from a control engineer who has performed extensive mathematical modeling, analysis, and development of control algorithms for a particular process. Regardless of where the heuristic control knowledge comes from, fuzzy control provides a user-friendly formalism for representing and implementing the ideas.

Over the past few decades, fuzzy logic theory is widely used: process control, management and decision making, operations research, economies. Dealing with simple 'yes' and 'no' answers is no longer satisfactory enough; a degree of membership (Zadeh, 1965) became a new way of solving problems. Fuzzy logic derives from the truth that the human common sense reasoning mode is approximate in nature.

In this chapter we provide a control engineering perspective on novel fuzzy controller. We take a pragmatic engineering approach to the design, analysis, performance evaluation, and implement of fuzzy control system. The chapter is basically broken into five parts. In section 2, we provide an overview of conventional control system design. In section 3 the basic theories of variable universe fuzzy control are been introduced. In section 4, we cover the novel fuzzy controller based on Yinger algorithm. In section 5, we use some examples to show how to design, simulate, and implement these controllers. Finally, in section 6, we

© 2012 Zhang and Gao, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

explain how to write a computer program to simulate the novel fuzzy control system, using either a high-level language or Matlab.

Novel Yinger Learning Variable Universe Fuzzy Controller 203

When a control engineer is given a control problem, often one of the first tasks is the development of a mathematical model of the process to be controlled, in order to gain a clear understanding of the problem. Basically, there are only a few ways to actually generate the model. We can use first principles of physics to write down a model. Another way is to perform "system identification" via the use of real plant data to produce a model of the system. Sometimes a combined approach is used where we use physics to write down a general different equation that we believe represent the plant behavior, and then we perform experiments on the plant to determine certain model parameters or functions.

Often, more than one mathematical model is produced. A "truth model" is one that is developed to be as accurate as possible so that it can be used in simulation-based evaluations of control systems. It must be understood, however, that there is never a perfect mathematical model for the plant. The mathematical model is an abstraction and hence cannot perfectly represent all possible dynamics of any physical process. This is not to say that we cannot produce models that are "accurate enough" to closely represent the behavior of a physical system. Usually, control engineer to be able to design a controller that will work. Then, they often also need a very accurate model to test the controller in simulation before it is tested in an experimental setting. Hence, lower-order "design model" are also often developed that may satisfy certain assumption yet still capture the essential plant behavior. Indeed, it is quite an art to produce good low-order model that satisfy these constraints. We emphasize that the reason we often need simpler models is that the synthesis techniques for controller often require that the model of the plant satisfy certain

Linear models such as the one in Equation (1) have been used extensively in the past and the

 *x Ax Bu*

In this case u is the m-dimensional input; x is the n-dimensional state; y is the p-dimensional output; and A,B,C and D are matrices of appropriate dimension. Such models are appropriate for use with frequency domain design techniques, the root-locus method, statespace methods, and so on. Sometimes it is assumed that the parameters of the linear model

Much of the current focus in control is on the development of controllers using nonlinear

Where the variables are defined as for the linear model and f and g are nonlinear functions of their arguments. One form of the nonlinear model that has received significant attention

 (,) (,) *x f xu*

are constant but unknown, or can be perturbed form their nominal values.

*<sup>y</sup> Cx Du* (1)

*<sup>y</sup> gxu* (2)

assumptions or there methods generally cannot be used.

control theory for linear system is quite mature.

models of the plant of the form

is

## **2. Conventional control system design**

## **2.1. Introduction**

A control system is a device, or set of devices to manage, command, direct or regulate the behavior of other devices or system. There are two common classes of control systems, with many variations and combinations: logic or sequential controls, and feedback or linear controls. There is also fuzzy logic, which attempts to combine some of the design simplicity of logic with the utility of linear control. Some devices or systems are inherently not controllable. A basic control system is shown in figure 1. The plant is object to be controlled. Its inputs are *u t*( ) , its outputs are *y*( )*t* , and reference input is *r t*( ) .

**Figure 1.** Control system

## **2.2. Mathematical modeling**

The mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines (such as computer science, artificial intelligence), but also in the social sciences (such as economics, psychology, sociology and political science),physicists, engineers, statisticians, operations research analysts and economists use mathematical models most extensively. A model may help to explain a system and to study the effects of different components, and to make predictions about behaviour.

Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include logical models, as far as logic is taken as a part of mathematics. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed.

When a control engineer is given a control problem, often one of the first tasks is the development of a mathematical model of the process to be controlled, in order to gain a clear understanding of the problem. Basically, there are only a few ways to actually generate the model. We can use first principles of physics to write down a model. Another way is to perform "system identification" via the use of real plant data to produce a model of the system. Sometimes a combined approach is used where we use physics to write down a general different equation that we believe represent the plant behavior, and then we perform experiments on the plant to determine certain model parameters or functions.

202 Fuzzy Controllers – Recent Advances in Theory and Applications

**2. Conventional control system design** 

Its inputs are *u t*( ) , its outputs are *y*( )*t* , and reference input is *r t*( ) .

either a high-level language or Matlab.

**2.1. Introduction** 

**Figure 1.** Control system

about behaviour.

**2.2. Mathematical modeling** 

explain how to write a computer program to simulate the novel fuzzy control system, using

A control system is a device, or set of devices to manage, command, direct or regulate the behavior of other devices or system. There are two common classes of control systems, with many variations and combinations: logic or sequential controls, and feedback or linear controls. There is also fuzzy logic, which attempts to combine some of the design simplicity of logic with the utility of linear control. Some devices or systems are inherently not controllable. A basic control system is shown in figure 1. The plant is object to be controlled.

*u t*( ) *r t*( ) *y t*( )

The mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines (such as computer science, artificial intelligence), but also in the social sciences (such as economics, psychology, sociology and political science),physicists, engineers, statisticians, operations research analysts and economists use mathematical models most extensively. A model may help to explain a system and to study the effects of different components, and to make predictions

Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include logical models, as far as logic is taken as a part of mathematics. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental

measurements often leads to important advances as better theories are developed.

Often, more than one mathematical model is produced. A "truth model" is one that is developed to be as accurate as possible so that it can be used in simulation-based evaluations of control systems. It must be understood, however, that there is never a perfect mathematical model for the plant. The mathematical model is an abstraction and hence cannot perfectly represent all possible dynamics of any physical process. This is not to say that we cannot produce models that are "accurate enough" to closely represent the behavior of a physical system. Usually, control engineer to be able to design a controller that will work. Then, they often also need a very accurate model to test the controller in simulation before it is tested in an experimental setting. Hence, lower-order "design model" are also often developed that may satisfy certain assumption yet still capture the essential plant behavior. Indeed, it is quite an art to produce good low-order model that satisfy these constraints. We emphasize that the reason we often need simpler models is that the synthesis techniques for controller often require that the model of the plant satisfy certain assumptions or there methods generally cannot be used.

Linear models such as the one in Equation (1) have been used extensively in the past and the control theory for linear system is quite mature.

$$\begin{aligned} \dot{\mathbf{x}} &= A\mathbf{x} + B\mathbf{u} \\ \mathbf{y} &= \mathbf{C}\mathbf{x} + D\mathbf{u} \end{aligned} \tag{1}$$

In this case u is the m-dimensional input; x is the n-dimensional state; y is the p-dimensional output; and A,B,C and D are matrices of appropriate dimension. Such models are appropriate for use with frequency domain design techniques, the root-locus method, statespace methods, and so on. Sometimes it is assumed that the parameters of the linear model are constant but unknown, or can be perturbed form their nominal values.

Much of the current focus in control is on the development of controllers using nonlinear models of the plant of the form

$$\begin{aligned} \dot{\mathbf{x}} &= f(\mathbf{x}, \boldsymbol{\mu}) \\ \mathbf{y} &= \mathbf{g}(\mathbf{x}, \boldsymbol{\mu}) \end{aligned} \tag{2}$$

Where the variables are defined as for the linear model and f and g are nonlinear functions of their arguments. One form of the nonlinear model that has received significant attention is

$$
\dot{\mathbf{x}} = f(\mathbf{x}) + \mathbf{g}(\mathbf{x})\mu \tag{3}
$$

Novel Yinger Learning Variable Universe Fuzzy Controller 205

Basically,there conventional approaches to control system design offer a variety of ways to utilize information from mathematical model on how to do good control. Sometimes they do not take into account certain heuristic information early in the design process, but use heuristics when the controller is implemented to tune it(tuning is invariably needed since the model used for the controller development is not perfectly accurate).Unfortunately, when using some approaches to conventional control, some engineers become somewhat removed from the control problem, and sometimes this leads to the development of unrealistic control laws. Sometimes in conventional control, useful heuristics are ignored because they do not fit into the proper mathematical framework, and this can cause

The next step in the design process is to perform analysis and performance evaluation. Basically, we need performance evaluation to test that we design does in fact meet the closed-loop specifications. This can be particularly important in safety-critical applications such as the control of a washing machine or an electric shaver, it may not be as important in the sense that failures will not imply the loss of life, so some of the rigorous evaluation methods can sometimes be ignored. Basically, there are three general ways to verify that a control system is operating properly: (1) mathematical analysis based on the use of formal models, (2) simulation-based analysis that most often uses formal models, and (3)

The fuzzy controller block diagram is given in figure 2. The plant outputs are denoted by *y*( )*t* , its input is denoted by *u t*( ) , and the reference input to the fuzzy controller is denoted

*u t*( ) *r t*( ) *y t*( )

Basically, the difficult task of modeling and simulating complex real-world systems for controller systems development, especially when implementation issues are considered, is

problem.

by *r t*( ) .

**2.5. Performance evaluation** 

experimental investigations on the real system.

**3. Variable fuzzy control system design** 

**Figure 2.** Fuzzy controller architecture

**3.1. Fuzzy controller** 

Since it is possible to exploit the structure of this model to construct nonlinear controllers. Of particular with both of the above nonlinear models is the case where f and g are not completely known and subsequent research focuses on robust control of nonlinear system.

Discrete time versions of the above models are also used, and stochastic effect are often taken into account via the addition of a input or other stochastic effects. Under certain assumptions you can linearize the nonlinear model in Equation(2) to obtain a linear one. In this case we sometimes think of the nonlinear model as the truth model, and the linear model that are generated form it as control design model.

There are certain properties of the plant that the control engineer often seeks to identify early in the design process. For instance, the stability of the plant may be analyzed. The effects of certain nonlinearities are also studied. The engineer may want to determine if the plant is controllable to see, for example, if the control input will be able to properly affect the plant; and observable to see, for example, if the chosen sensors will allow the controller observe the critical plant behavior so that it can be compensated Overall, this analysis of the plant's behavior gives the control engineer a fundamental understanding of the plant dynamics.

## **2.3. Performance objectives and design constrains**

Controller design entails constructing a controller to meet the specifications. Often the first issue to address is whether to use open or closed-loop control. Often, need to pay for a sensor for the feedback information and there need to justification for this cost. Moreover, feedback can destabilize the system. Do not develop a feedback controller just because you are used to developing feedback controllers; you may want to consider an open-loop controllers since it may provide adequate performance. Assuming you use feedback control, the closed-loop specifications can involve the following factors: Disturbance rejection properties; Insensitivity to plant parameter variations; Stability; Rise-time.

## **2.4. Controller design**

Conventional control has provided numerous methods for controllers for dynamic system. Some of there are listed below:


Basically,there conventional approaches to control system design offer a variety of ways to utilize information from mathematical model on how to do good control. Sometimes they do not take into account certain heuristic information early in the design process, but use heuristics when the controller is implemented to tune it(tuning is invariably needed since the model used for the controller development is not perfectly accurate).Unfortunately, when using some approaches to conventional control, some engineers become somewhat removed from the control problem, and sometimes this leads to the development of unrealistic control laws. Sometimes in conventional control, useful heuristics are ignored because they do not fit into the proper mathematical framework, and this can cause problem.

## **2.5. Performance evaluation**

204 Fuzzy Controllers – Recent Advances in Theory and Applications

model that are generated form it as control design model.

**2.3. Performance objectives and design constrains** 

**2.4. Controller design** 

Some of there are listed below:

dynamic programming,an so on.

backstepping, and so on.

adaptive control,and so on.

*x f x gxu* () () (3)

Since it is possible to exploit the structure of this model to construct nonlinear controllers. Of particular with both of the above nonlinear models is the case where f and g are not completely known and subsequent research focuses on robust control of nonlinear system. Discrete time versions of the above models are also used, and stochastic effect are often taken into account via the addition of a input or other stochastic effects. Under certain assumptions you can linearize the nonlinear model in Equation(2) to obtain a linear one. In this case we sometimes think of the nonlinear model as the truth model, and the linear

There are certain properties of the plant that the control engineer often seeks to identify early in the design process. For instance, the stability of the plant may be analyzed. The effects of certain nonlinearities are also studied. The engineer may want to determine if the plant is controllable to see, for example, if the control input will be able to properly affect the plant; and observable to see, for example, if the chosen sensors will allow the controller observe the critical plant behavior so that it can be compensated Overall, this analysis of the plant's behavior gives the control engineer a fundamental understanding of the plant dynamics.

Controller design entails constructing a controller to meet the specifications. Often the first issue to address is whether to use open or closed-loop control. Often, need to pay for a sensor for the feedback information and there need to justification for this cost. Moreover, feedback can destabilize the system. Do not develop a feedback controller just because you are used to developing feedback controllers; you may want to consider an open-loop controllers since it may provide adequate performance. Assuming you use feedback control, the closed-loop specifications can involve the following factors: Disturbance rejection

Conventional control has provided numerous methods for controllers for dynamic system.

1. Proportional-integral-derivative(PID) control:Over 90% of the controllers in operation today are PID controllers. This approach is often viewed as simple, reliable,and easy to understand. Often, like fuzzy controller, heuristics are used to tune PID controllers.

3. Optimal control: Linear quadratic regulator,use of Pontryagin's minimum principle or

4. Nonlinear methods: Feedback linearization, Lyapunov redesign, sliding mode control,

5. Adaptive control; model reference adaptive control,self-tuning regulators, nonlinear

properties; Insensitivity to plant parameter variations; Stability; Rise-time.

2. State-space methods: State feedback,observers,and so on.

The next step in the design process is to perform analysis and performance evaluation. Basically, we need performance evaluation to test that we design does in fact meet the closed-loop specifications. This can be particularly important in safety-critical applications such as the control of a washing machine or an electric shaver, it may not be as important in the sense that failures will not imply the loss of life, so some of the rigorous evaluation methods can sometimes be ignored. Basically, there are three general ways to verify that a control system is operating properly: (1) mathematical analysis based on the use of formal models, (2) simulation-based analysis that most often uses formal models, and (3) experimental investigations on the real system.

## **3. Variable fuzzy control system design**

The fuzzy controller block diagram is given in figure 2. The plant outputs are denoted by *y*( )*t* , its input is denoted by *u t*( ) , and the reference input to the fuzzy controller is denoted by *r t*( ) .

**Figure 2.** Fuzzy controller architecture

#### **3.1. Fuzzy controller**

Basically, the difficult task of modeling and simulating complex real-world systems for controller systems development, especially when implementation issues are considered, is well documented. Even if a relatively accurate model of a dynamic system can be developed, it is often too complex to use require restrictive assumptions for the plant. It is for this reason that in practice conventional controllers are often developed via simple models of the plant behavior that satisfy the necessary assumptions, and via the ad hoc tuning of relatively simple linear or nonlinear controllers. Regardless, it is well understood.

Novel Yinger Learning Variable Universe Fuzzy Controller 207

(6)

(5)

*Xx xE xE ii ii ii* ( ) [ ( ) ,( ) ] 

*Y y yU yU* () [ () , () ] 

Novel fuzzy controller is composed of three parts. Firstly, new kind of contractionexpansion factor is established, then local space is optimized, finally novel controller

Many real-world environments in which learning systems have to operate are time-varying. Several aspects of the learning problem can vary, including the mapping to be learned, and the sampling distribution that governs the input-space location of exemplars that make up the input information. In this section, K-Vector Nearest Neighbors (K-VNN) is proposed to

Where *h* is radius of local space ( *<sup>k</sup>* ), and data-window is changed by adjusting it. *DAB* ( ,) is the distance function which is defined by (8), 1 ,, *X XK* are messages to input.

Define 2. Lets *A* [ ,, ] *A A* <sup>1</sup> *<sup>n</sup>* and [,,] <sup>1</sup> *<sup>n</sup> BB B* ,in the Euclidean space, gets distance and

2

( , ) arccos *<sup>T</sup>*

*A B A B*

*<sup>k</sup> X X X DX X h* <sup>1</sup> ,, (, ) *K i im* (7)

2 2

(8)

*A B*

**Figure 3.** Universe compress and expand

dynamically adjust output by rules.

**4.1. Optimal local spaces** 

this problem.

intersection angle:

**4. Novel fuzzy controller based on Yinger algorithm** 

Define 1. Lets *k* is input sets which can be defined to local space as:

*dAB A B*

( ,)

 

 

Fuzzy control provides a formal methodology for representing, manipulating, and implementing a human's heuristic knowledge about how to control a system.

The fuzzy controller block diagram is given in Figure 2, where we show a fuzzy controller embedded in a closed-loop control system. The plant outputs are denoted by y(t), its inputs are denoted by u(t), and the reference input to the fuzzy controller is denoted by r(t).

The fuzzy controller has four main components: (1) The" rule-base" holds the knowledge, in the form of a set of rules are relevant at the current time and then decides what the input to the plant should be, (3) The fuzzification interface simply modifies the inputs so that they can be interpreted and compared to the rules in the rule-base. And (4) the defuzzification interface converts the conclusions reached by the inference mechanism into the inputs to the plant.

To design the fuzzy controller, the control engineer must gather information on how the artificial decision maker should act in the closed-loop system. Sometimes this information can come from a human decision maker who performs the control task, while at other times the control engineer can come to understand the plant dynamics and write down a set of rules about how to control the system without outside help. These "rules" basically say, "If should be some value." A whole set of such "If-Then" rules is loaded into the rule-base, and specifications are met.

## **3.2. Structure of variable adaptive fuzzy controller**

Let *X EE i n <sup>i</sup>* , ( 1,2, , ) be the universe of input variable *xi n <sup>i</sup>* ( 1,2, , ) , and *Y UU* [ ,] be the universe of output variable *y* . (1 ) { } *i ij j m A* stands for a fuzzy partition on *Xi* ,and (1 ) { }*j jm B* defines a fuzzy partition on *Y* . A group of fuzzy inference rules is formed as follow:

If <sup>1</sup> *x* is *A*1*<sup>j</sup>* and <sup>2</sup> *x* is *A*<sup>2</sup> *<sup>j</sup>* and…and *<sup>n</sup> x* is *Anj* then *y* is *<sup>j</sup> B* , *j m* 1,2, ,

The fuzzy logic system can be represented as an n-dimension piecewise interpolation function (, , , ) <sup>2</sup> *<sup>n</sup> Fxx x* :

$$F(\mathbf{x}, \mathbf{x}\_2, \dots, \mathbf{x}\_n) = y(\mathbf{x}, \mathbf{x}\_2, \dots, \mathbf{x}\_n) = \sum\_{j=1}^m \prod\_{i=1}^n A\_{ij}(\mathbf{x}\_i) y\_j \tag{4}$$

Generally speaking, a function : [0,1], ( ) *X xx* can be called a contraction-expansion factor on , *X EE <sup>i</sup>* . The so-called variable universe means *Xi* and*Y* can change with changing variable *<sup>i</sup> x* and *y* expressed by:

Novel Yinger Learning Variable Universe Fuzzy Controller 207

$$X\_i(\mathbf{x}\_i) = \left[ -a(\mathbf{x}\_i)E\_{i'}a(\mathbf{x}\_i)E\_i \right] \tag{5}$$

$$Y(y) = [ -\beta(y)\mathbb{U}, \beta(y)\mathbb{U} ] \tag{6}$$

**Figure 3.** Universe compress and expand

206 Fuzzy Controllers – Recent Advances in Theory and Applications

plant.

specifications are met.

formed as follow:

function (, , , ) <sup>2</sup> *<sup>n</sup> Fxx x* :

factor on

Generally speaking, a function

changing variable *<sup>i</sup> x* and *y* expressed by:

**3.2. Structure of variable adaptive fuzzy controller** 

If <sup>1</sup> *x* is *A*1*<sup>j</sup>* and <sup>2</sup> *x* is *A*<sup>2</sup> *<sup>j</sup>* and…and *<sup>n</sup> x* is *Anj* then *y* is *<sup>j</sup> B* , *j m* 1,2, ,

*Y UU* [ ,] be the universe of output variable *y* .

well documented. Even if a relatively accurate model of a dynamic system can be developed, it is often too complex to use require restrictive assumptions for the plant. It is for this reason that in practice conventional controllers are often developed via simple models of the plant behavior that satisfy the necessary assumptions, and via the ad hoc tuning of relatively simple linear or nonlinear controllers. Regardless, it is well understood.

Fuzzy control provides a formal methodology for representing, manipulating, and

The fuzzy controller block diagram is given in Figure 2, where we show a fuzzy controller embedded in a closed-loop control system. The plant outputs are denoted by y(t), its inputs

The fuzzy controller has four main components: (1) The" rule-base" holds the knowledge, in the form of a set of rules are relevant at the current time and then decides what the input to the plant should be, (3) The fuzzification interface simply modifies the inputs so that they can be interpreted and compared to the rules in the rule-base. And (4) the defuzzification interface converts the conclusions reached by the inference mechanism into the inputs to the

To design the fuzzy controller, the control engineer must gather information on how the artificial decision maker should act in the closed-loop system. Sometimes this information can come from a human decision maker who performs the control task, while at other times the control engineer can come to understand the plant dynamics and write down a set of rules about how to control the system without outside help. These "rules" basically say, "If should be some value." A whole set of such "If-Then" rules is loaded into the rule-base, and

Let *X EE i n <sup>i</sup>* , ( 1,2, , ) be the universe of input variable *xi n <sup>i</sup>*

*j i Fxx x yxx x A x y* (4)

*n n ij i j*

, *X EE <sup>i</sup>* . The so-called variable universe means *Xi* and*Y* can change with

*m n*

1 1

on *Xi* ,and (1 ) { }*j jm B* defines a fuzzy partition on *Y* . A group of fuzzy inference rules is

The fuzzy logic system can be represented as an n-dimension piecewise interpolation

2 2

: [0,1], ( ) *X xx*

(, , , ) (, , , ) ( )

( 1,2, , ) , and

(1 ) { } *i ij j m A* stands for a fuzzy partition

can be called a contraction-expansion

are denoted by u(t), and the reference input to the fuzzy controller is denoted by r(t).

implementing a human's heuristic knowledge about how to control a system.

## **4. Novel fuzzy controller based on Yinger algorithm**

Novel fuzzy controller is composed of three parts. Firstly, new kind of contractionexpansion factor is established, then local space is optimized, finally novel controller dynamically adjust output by rules.

#### **4.1. Optimal local spaces**

Many real-world environments in which learning systems have to operate are time-varying. Several aspects of the learning problem can vary, including the mapping to be learned, and the sampling distribution that governs the input-space location of exemplars that make up the input information. In this section, K-Vector Nearest Neighbors (K-VNN) is proposed to this problem.

Define 1. Lets *k* is input sets which can be defined to local space as:

$$\Omega \Omega\_k \equiv \left\{ X\_{1'} \cdots \prime, X\_K \right\} = \left\{ X\_i \, \middle| \, \mathrm{D}(X\_{i'} X\_m) < h \right\} \tag{7}$$

Where *h* is radius of local space ( *<sup>k</sup>* ), and data-window is changed by adjusting it. *DAB* ( ,) is the distance function which is defined by (8), 1 ,, *X XK* are messages to input.

Define 2. Lets *A* [ ,, ] *A A* <sup>1</sup> *<sup>n</sup>* and [,,] <sup>1</sup> *<sup>n</sup> BB B* ,in the Euclidean space, gets distance and intersection angle:

$$\begin{cases} d(A,B) = \sqrt{\left\|A - B\right\|\_2} \\ \theta(A,B) = \arccos A \end{cases} \tag{8}$$
 
$$\begin{cases} A \parallel\_2 \bullet \parallel B \parallel\_2 \end{cases} \tag{8}$$

According to (7), we can get the distance and intersection angle of *Xi* and *Xd* , from inputoutput specimen choice similar message to *<sup>k</sup>* .

If intersection angle of *Xi* and *Xd* .greater than 90 , thinking *Xi* stray from *Xd* ,and define as follows:

$$\begin{aligned} D(X\_i, X\_d) &= a e^{\left\lfloor -d(X\_i, X\_d) \right\rfloor} + b \sin \left\lceil \varphi(X\_i, X\_d) \right\rceil \\ &\quad \text{(} 0 \le a \le 1/2, 0 \le b \le 1/2\text{)} \end{aligned} \tag{9}$$

Novel Yinger Learning Variable Universe Fuzzy Controller 209

,

 1 2 ( ( )) ( ( )) *et et*

2. Analyze and verify characters

**Figure 4.** Function cluster surface (k>0)

10 0 0.2 0.4 0.6 0.8 1

**Figure 5.** Function cluster surface contour(k>0)

2. Near zero

4. Normality

1. Duality *et E et et* ( ) ( ( )) ( ( )) 

, (0) ( ) 0 *DXXi d*

 () ( ) 1 *E E*

3. Monotonicity: 1 2 *et et E* ( ), ( ) [0, ]if 1 2 *et et* () () then

contraction-expansion factor is satisfied with the requests.

5

4 2


0



0

0.5

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

<sup>1</sup> ( ) ( ) 4 2 *i d <sup>E</sup> x kx kx D X X*

is the primary function can easily realize in nonlinear system. So the new kind of

From (9), we can see, if *Xi* is more similar to *Xd* , (,) *i d dX X e* and sin ( , ) *X Xi d* are more similar to 1, use this method and get the new input set

$$\Omega\_k = \left\langle (X\_1, Y\_1), \dots, (X\_{K'}, Y\_K) \middle| D(X\_1, X\_d) > \dots > D(X\_{k'}, X\_d) \right\rangle \tag{10}$$

From this section, some noise can be deleted by this section.

#### **4.2. Contraction-expansion factor**

Now the popular contraction-expansion factor is <sup>2</sup> ( ) () 1 *kx x ce* ( *c k* (0 1) 0 ), but the algorithm module can not be realized easily by C++ which support some methods by using VC++ accomplish control system. So building up a kind of contraction-expansion factor to nonlinear system is very important.

1. Establish differential equation

Firstly, ( ( )) *e t* is strictly monotonously increasing on [0 1] and monotonously decreasing on [-1 0].

Secondly, *e t*() 0 Then ( ( )) 0.0001 *e t* and *e t*() 1 then ( ( )) 1 *e t* .

Thirdly, ( ( )) ( ) *et k et* , and to the same *e t*( ) , *e t*( ) is larger and ( ( )) *e t* is larger too. From those conditions the differential equation can be build as follow:

$$
\Delta \alpha (e(t)) = ke(t) \Delta e(t) (E - e^2(t)) \tag{11}
$$

get hold of:

$$\alpha(\mathbf{x}) = -\frac{1}{4}k\mathbf{x}^4 + \frac{E}{2}k\mathbf{x}^2 + \mathbf{c} \tag{12}$$

and initialized condition:

when *e t*() 0 then( ( )) ( , ) *i d et DX X* ,and *e t*( ) =E ( ( )) 1 *e t* get hold of:

$$\alpha(\mathbf{x}) = -\frac{1}{4}k\mathbf{x}^4 + \frac{E}{2}k\mathbf{x}^2 + D(X\_{i\_\ast}X\_d) \tag{13}$$

2. Analyze and verify characters

208 Fuzzy Controllers – Recent Advances in Theory and Applications

output specimen choice similar message to *<sup>k</sup>* .

as follows:

According to (7), we can get the distance and intersection angle of *Xi* and *Xd* , from input-

If intersection angle of *Xi* and *Xd* .greater than 90 , thinking *Xi* stray from *Xd* ,and define

(,) (, ) sin ( , )

*i d dX X D X X ae b X X i d i d*

From (9), we can see, if *Xi* is more similar to *Xd* , (,) *i d dX X*

From this section, some noise can be deleted by this section.

similar to 1, use this method and get the new input set

**4.2. Contraction-expansion factor** 

nonlinear system is very important.

1. Establish differential equation

Firstly,

get hold of:

[-1 0].

Secondly, *e t*() 0 Then

and initialized condition:

when *e t*() 0 then

get hold of:

Thirdly, 

Now the popular contraction-expansion factor is

( ( )) 0.0001 *e t* 

From those conditions the differential equation can be build as follow:

( ( )) ( , ) *i d et DX X* ,and *e t*( ) =E

4 2

<sup>1</sup> ( ) ( ) 4 2 *i d*

( ( )) ( ) *et k et* , and to the same *e t*( ) , *e t*( ) is larger and

 <sup>1</sup> 4 2 ( ) 4 2

( ( )) 1 *e t*

,

*<sup>E</sup> x kx kx D X X* (13)

*a b*

(0 1 / 2,0 1 / 2)

*<sup>k</sup>* ( , ), ,( , ) ( , ) ( , ) *X Y X Y DX X DX X* 1 1 *KK d* <sup>1</sup> *k d* (10)

*e* and

<sup>2</sup> ( ) () 1 *kx x ce* ( *c k* (0 1) 0 ), but the

( ( )) 1 *e t* .

( ( )) *e t* is larger too.

<sup>2</sup> ( ( )) ( ) ( )( ( )) *e t ke t e t E e t* (11)

*<sup>E</sup> x kx kx c* (12)

sin ( , ) *X Xi d* are more

(9)

( ( )) *e t* is strictly monotonously increasing on [0 1] and monotonously decreasing on

and *e t*() 1 then

algorithm module can not be realized easily by C++ which support some methods by using VC++ accomplish control system. So building up a kind of contraction-expansion factor to


$$\alpha(\mathbf{x}) = -\frac{1}{4}k\mathbf{x}^4 + \frac{E}{2}k\mathbf{x}^2 + D(\mathbf{X}\_{i\_\prime}\mathbf{X}\_d)^2$$

is the primary function can easily realize in nonlinear system. So the new kind of contraction-expansion factor is satisfied with the requests.

**Figure 4.** Function cluster surface (k>0)

**Figure 5.** Function cluster surface contour(k>0)

Novel Yinger Learning Variable Universe Fuzzy Controller 211

2.4 , 2.4 1.5

*e*

*etc*

, 0 0.6

*etc*

*e*

0.6 ( ) , 1.5 0.6 0.9

1 , 3

*e*

*etc*

1.9 ( ) , 1.9 3 1.1

*<sup>e</sup> Nb e <sup>e</sup>*

1.5 ( ) , 0.6 1.5 0.9

*e*

0.9

*<sup>e</sup> Nm e <sup>e</sup>*

 

0 ,

0.6

*<sup>e</sup> Ps e <sup>e</sup>*

*e*

 

 

0 ,

0 ,

*U*[ 1,1]

*if e is Nb then u is NB* , *if e is Nm then u is NM*

*if e is Ns then u is NS* , *if e is Pb then u is PB*

*if e is Pm then u is PM* , *if e is Ps then u is PS*

We can get the rules as follows:

**Figure 8.** Function

 

0 ,

*e*

0.9

*<sup>e</sup> Ns e <sup>e</sup>*

 

0 ,

*e*

0.9

*<sup>e</sup> Pm e <sup>e</sup>*

 

0 ,

*<sup>e</sup> Nb e <sup>e</sup>*

1 , 3

*e*

*etc*

1.5 , 1.5 0.6

*etc*

0.6 , 0.6 1.5

*etc*

*e*

2.4 ( ) , 1.5 2.4 0.9

*e*

( ) , 0.6 0 0.6

1.9 ( ) , 3 1.9 1.1

**Figure 6.** Function cluster surface (c<1)

**Figure 7.** Function cluster surface contour (c<1)

#### **5. Examples**

Choose the typical non-linear system to the new algorithm.

$$\text{Plant:}\begin{aligned} \text{x(t)} &= \frac{1 - e^{-x(t)}}{1 + e^{-x(t)}} + u(t) \\ y(t) &= x(t) \end{aligned} \tag{14}$$

$$\lim\_{t \to \infty} \|e(t)\| = \lim\_{t \to \infty} \|r(t) - y(t)\| = 0 \tag{15}$$

And () () *<sup>c</sup> ut u t*

*U*[ 1,1]

We can get the rules as follows:

210 Fuzzy Controllers – Recent Advances in Theory and Applications

**Figure 6.** Function cluster surface (c<1)

**Figure 7.** Function cluster surface contour (c<1)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Plant:

Choose the typical non-linear system to the new algorithm.

 

( ) <sup>1</sup> ( ) ( ) <sup>1</sup>

(14)

*et rt yt* (15)

*x t x t <sup>e</sup> x t u t e*

. ( )


() () () 0 lim lim *t t*

And () () *<sup>c</sup> ut u t*

() ()

*yt xt*


伸 伸伸伸伸 缩

0

0.1

0.2

0.3

0.4 0 0.2 0.4 0.6 0.8 1


0

0.5

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

**5. Examples** 

**Figure 8.** Function

$$Nb(e) = \begin{cases} 1 & e \le -3 \\ e + 1.9 \\ -1.1 \\ 0 & \text{etc} \end{cases}, \qquad -3 \le e \le -1.9 \text{ } Nm(e) = \begin{cases} \frac{e + 2.4}{0.9}, & -2.4 \le e \le -1.5 \\ \frac{e + 0.6}{-0.9}, & -1.5 \le e \le -0.6 \\ -0.9 & \text{etc} \end{cases}$$

$$\text{Ns}(e) = \begin{cases} \frac{e + 1.5}{0.9} , & -1.5 \le e \le -0.6\\ \frac{e}{-0.6} , & -0.6 \le e \le 0\\ \frac{e}{0.1} , & \text{etc} \end{cases} \qquad \text{Ps(e)} = \begin{cases} \frac{e}{0.6} , & 0 \le e \le 0.6\\ \frac{e - 1.5}{-0.9} , & 0.6 \le e \le 1.5\\ -0.9 , & \text{etc} \end{cases}$$

$$Pm(e) = \begin{cases} \frac{e - 0.6}{0.9}, & 0.6 \le e \le -1.5\\ \frac{e - 2.4}{-0.9}, & 1.5 \le e \le 2.4\\ 0 & \text{etc} \end{cases} \quad 1.5 \le e \le 2.4 \quad Nb(e) = \begin{cases} 1 & e \ge -3\\ \frac{e - 1.9}{1.1}, & 1.9 \le e \le 3.5\\ 1.1 & \text{etc} \end{cases}$$

$$y\_1 = -1, y\_2 = -0.5, y\_3 = -0.2, y\_4 = 0.2, y\_5 = 0.5, y\_6 = 1$$

$$u\_c(t) = \beta(t) \mathcal{U} \sum\_{j=1}^m \prod\_{i=1}^n A\_{ij}(\frac{e(t)}{\alpha(e(t))}) y\_j$$

$$\text{make } \alpha(e(t)) = -\frac{1}{81} e(t)^4 + \frac{2}{9} e(t)^2 + 0.0001$$

$$\text{and } \lim\_{t \to \infty} \|e(t)\| = 0, \ P\_n = (p\_1, p\_2, \dots, p\_n)^\tau$$

$$\text{then } \beta'(t) = K \sum\_{i=1}^n p\_i e\_i(t)$$

$$\beta(0) = 1, \ P\_n \Big|\_{n=1} = p\_1 = 1, \ k = 2, \ U = 3$$

$$u\_c(t) = \beta(t) L \sum\_{j=1}^m \prod\_{i=1}^n A\_{ij}(\frac{e(t)}{\alpha(e(t))}) y\_j$$

$$u\_c(t) = \beta(t) L \sum\_{j=1}^m \prod\_{i=1}^n A\_{ij}(\frac{e(t)}{\alpha(e(t))}) y\_j$$

$$= (k \int\_0^t e^\tau(t) P\_n dt + \beta(0)) L \sum\_{j=1}^m \prod\_{i=1}^n A\_{ij}(\frac{e(t)}{\alpha(e(t))}) y\_j$$

Novel Yinger Learning Variable Universe Fuzzy Controller 213

In order to make out the advantages of the new function, Let *rt t* ( ) sin the result of

**Figure 10.** Rules

controller (seeFig.11) is formed as follows

**Figure 11.** The contrast of control effect

 

0.5

( )

*r t*

the result of control (see Fig.12 and Fig.13) is formed as follow

From Fig.12, we learn that there are some errors between aim curve (blue) and real curve

e(t)=0. From Fig.13, we can clearly learn that the real curve (black) almost coincide with aim curve (blue). So we say that the variable fuzzy controller is one of the efficient tools for control system. From Fig.11, we can see the difference between the new function and exponential

*etc*

1.5 0 3 6 9

*t and t*

( ( )) *e t* =1. System cannot immediately regulate control strategy to make

Let

(black) because of

 <sup>0</sup> 6( ( ) 1)[ ( ( )) 0.5 ( ( )) 0.2 ( ( )) *<sup>t</sup> e t dt Nb e t Nm e t Ns e t* 0.2 ( ( )) 0.5 ( ( )) ( ( ))] *Ps e t Pm e t Pb e t*

**Figure 9.** Controller

**Figure 10.** Rules

make 

and

12 3 4 5 6 *yy y y y y* 1, 0.5, 0.2, 0.2, 0.5, 1

 1 2 4 2 ( ( )) ( ) ( ) 0.0001 81 9 *et et et*

 1 1 ( ) () () ( ) ( ( )) *m n c ij j j i e t u t tU A y e t*

*P pp p n n* (,, ) 1 2

 

*i t K pe t*

(0) 1 , 1 1 <sup>1</sup> *<sup>n</sup> <sup>n</sup> P p* , *<sup>k</sup>* 2, *<sup>U</sup>* <sup>3</sup>

( ) ( ( ) (0)) ( ) ( ( ))

 1 1 ( ) () () ( ) ( ( )) *m n c ij j j i e t u t tU A y e t*

<sup>0</sup>

*<sup>m</sup> <sup>n</sup> <sup>t</sup>*

 1 1 ( ) () () ( ) ( ( )) *m n c ij j j i e t u t tU A y e t*

1 () () *n*

*i i*

*e t dt Nb e t Nm e t Ns e t* 0.2 ( ( )) 0.5 ( ( )) ( ( ))] *Ps e t Pm e t Pb e t*

1 1

*n ij j j i e t k e t P dt U A y e t*

lim ( ) 0 *<sup>t</sup> e t* ,

> then

<sup>0</sup> 6( ( ) 1)[ ( ( )) 0.5 ( ( )) 0.2 ( ( )) *<sup>t</sup>*

**Figure 9.** Controller

In order to make out the advantages of the new function, Let *rt t* ( ) sin the result of controller (seeFig.11) is formed as follows

**Figure 11.** The contrast of control effect

Let

$$r(t) = \begin{cases} 1.5 & 0 \le t \le 3 \\ 0.5 & \text{etc} \end{cases}$$

the result of control (see Fig.12 and Fig.13) is formed as follow

From Fig.12, we learn that there are some errors between aim curve (blue) and real curve (black) because of ( ( )) *e t* =1. System cannot immediately regulate control strategy to make e(t)=0. From Fig.13, we can clearly learn that the real curve (black) almost coincide with aim curve (blue). So we say that the variable fuzzy controller is one of the efficient tools for control system. From Fig.11, we can see the difference between the new function and exponential

function (conventional function), and the algorithm module with new contraction-expansion factor is applied successfully in Matlab, whose results show that algorithm module is reasonable, adaptive and feasible. In the other hand, the new function can be realized easily by C++ to optimize the controller of complicated nonlinear control system.

Novel Yinger Learning Variable Universe Fuzzy Controller 215

compressor

fan

other facilities

resolve this problem. The variable universe fuzzy control theory has become more and more important in process control. The idea of variable universe fuzzy control is first proposed in refs, and several types of variable universe adaptive fuzzy controller are discussed in ref.

The compressor, condenser, evaporator,capillary and other electro-equipments compose the refrigeration system which is a close circulatory system. R-600a as refrigeration material from the low-pressure liquid to gaseity in evaporator to make the icebox inside temperature lowed through absorbing the heat. In other words, the control system of refrigeration makes R-600a changed by electric power. The simplified model of the refrigerator (see Fig.14) show

Periphery

Electroequipment

Sensor of temperature

Switch of doors

**Figure 14.** Simplified model of the refrigerator

**Figure 15.** Contrast of controller effect

Interface

Chip

The popular refrigerator through driving compressor makes the temperature constant, but there are some disadvantages in the control strategy. If there is minuteness temperature warp in system, control system frequent start-up equipments to modulate inside temperature, and a lot of energy will be wasted. In order to solve this problem, we design

the new control strategy based on the idea of variable universe fuzzy control.

as follow:

**Figure 12.** The simulation curves (( ( )) 1 *e t* )

**Figure 13.** The simulation curves (T=10)

## **6. Practical application**

Refrigerator is one kind of popular home appliance, and it became more and more important to economize the energy. The controller of conventional refrigerator keep anticipative temperature through PTC-relays and compress, but a lot of energy is waste. In this paper the new controller based on variable universe adaptive fuzzy control theory can resolve this problem. The variable universe fuzzy control theory has become more and more important in process control. The idea of variable universe fuzzy control is first proposed in refs, and several types of variable universe adaptive fuzzy controller are discussed in ref.

The compressor, condenser, evaporator,capillary and other electro-equipments compose the refrigeration system which is a close circulatory system. R-600a as refrigeration material from the low-pressure liquid to gaseity in evaporator to make the icebox inside temperature lowed through absorbing the heat. In other words, the control system of refrigeration makes R-600a changed by electric power. The simplified model of the refrigerator (see Fig.14) show as follow:

**Figure 14.** Simplified model of the refrigerator

214 Fuzzy Controllers – Recent Advances in Theory and Applications

**Figure 12.** The simulation curves (

**Figure 13.** The simulation curves (T=10)

**6. Practical application** 

function (conventional function), and the algorithm module with new contraction-expansion factor is applied successfully in Matlab, whose results show that algorithm module is reasonable, adaptive and feasible. In the other hand, the new function can be realized easily by

C++ to optimize the controller of complicated nonlinear control system.

( ( )) 1 *e t* )

Refrigerator is one kind of popular home appliance, and it became more and more important to economize the energy. The controller of conventional refrigerator keep anticipative temperature through PTC-relays and compress, but a lot of energy is waste. In this paper the new controller based on variable universe adaptive fuzzy control theory can The popular refrigerator through driving compressor makes the temperature constant, but there are some disadvantages in the control strategy. If there is minuteness temperature warp in system, control system frequent start-up equipments to modulate inside temperature, and a lot of energy will be wasted. In order to solve this problem, we design the new control strategy based on the idea of variable universe fuzzy control.

**Figure 15.** Contrast of controller effect

A refrigerator plant is a complex nonlinear system and may be prone to instability and oscillatory behaviors. The Fig.15 is the contrast of controller effect by Ying learning (red line), exponent function (blue line).In this section, a fuzzy controller is designed and simulated exclusively to control a refrigerator plant with a new-type function of contractionexpansion factor to optimize the controller of temperature is steady.

Novel Yinger Learning Variable Universe Fuzzy Controller 217

YL-VUF VUF

**Figure 17.** Temperature (YL-VUF and VUF) of steam to body

25

30

35

Temperature(Celsius Degress)

40

45

50

*University Hospital of Gansu Traditional Chinese Medicine, China* 

system. J. Application Research of Computers. 2010,pp. 465--471

drive .2090-4479, Volume 2, Issue 2, 2011, Pages 109-118

on.1949-3029, Volume 2, Issue 4, 2011, Pages 404-413

[1] Aihua Dong. Fuzzy Control Algorithm used in Asynchronous Motor Power Saver. J.

0 5 10 15 20 25

YL-VUF

VUF

Time (minute)

[2] Bin Wang. Fuzzy control algorithm on variable output domain in automotive ACC

[3] Ping zhang. Novel lazy learning variable universe fuzzy controller for temperature system Communications in Computer and Information Science, 2011,Vol58,No4, 364-

[4] YajunGuo, HuoLong; Ain Shams Engineering Journal Self organizing fuzzy sliding mode controller for the position control of a permanent magnet synchronous motor

[5] AdisornThomya, YottanaKhunatorn; Energy Procedia (2011) Design of Control System of Hydrogen and Oxygen Flow Rate for Proton Exchange Membrane Fuel Cell Using

[6] Jabr, H. M.; Lu, D. Design and Implementation of Neuro-Fuzzy Vector Control for Wind-Driven Doubly-Fed Induction Generator.Sustainable Energy, IEEE Transactions

[7] Mehrjerdi, H.; Saad, M.; Ghommam, J. (2011) Hierarchical Fuzzy Cooperative Control and Path Following for a Team of Mobile Robots.Mechatronics, IEEE/ASME

Fuzzy Logic Controller .1876-6102, Volume 9, Issue 000, 2011, Pages 186-197

Transactions on.1083-4435, Volume 16, Issue 5, 2011, Pages 907-917

*Lanzhou University of Technology, China* 

Electric Drive. 2009, pp. 49--21

**Author details** 

Ping Zhang

Guodong Gao

**7. References** 

370

Using control method to explain medical phenomenon is currently a hot subject of research. The traditional Chinese drug fumigation steaming treat protrusion of protrusion of protrusion of lumbar intervertebral disc with steam generated by boiling medicinal herbs, and this process is a typical non-linear, multivariable, and strong coupling. Experienced nurse and doctor cure patient by their experience. So establish a model of this process can discover more factor of the disease, better treat to protrusion of protrusion of protrusion of lumbar intervertebral disc and reduce of energy consumption.

The traditional Chinese drug fumigation fume or steaming treat diseases with fume in moxibustion or with steam generated by boiling medicinal herbs, and its process is a typical non-linear, multivariable, strong coupling. In addition, its characters are difficult to quantitative analysis. So the period of treatment is only determined by experience of doctors. Therefore, there is theoretical and practical significance in studying of traditional Chinese drug fumigation medical data mining.

The illustration of the traditional Chinese drug fumigation machine is shown in Fig.16. Here, the type of machine is MJD-2003 and it has been used 6 years.

**Figure 16.** Drug Fumigation Machine

Doctor treats protrusion of Protrusion of protrusion of lumbar intervertebral disc with steam generated by boiling medicinal herbs at this machine.

Fig.17 is the temperature of steam to body by VUF and YL-VUF. YL-VUF is the blue real line, and VUF is the green dash line. In this picture the aim is 40 Celsius Degrees. The temperature decrease when patient's posture is changed. After 10.725 minute, YL-VUF makes the temperature to 40 Celsius Degrees. On the other hand, VUF almost cost 22.568 minute. DFNN and YL-VUF have the similar frame, but YL-VUF using new local space to forecast. So YL-VUF can avoid over heat.

**Figure 17.** Temperature (YL-VUF and VUF) of steam to body

## **Author details**

216 Fuzzy Controllers – Recent Advances in Theory and Applications

expansion factor to optimize the controller of temperature is steady.

lumbar intervertebral disc and reduce of energy consumption.

Here, the type of machine is MJD-2003 and it has been used 6 years.

Chinese drug fumigation medical data mining.

**Figure 16.** Drug Fumigation Machine

forecast. So YL-VUF can avoid over heat.

generated by boiling medicinal herbs at this machine.

A refrigerator plant is a complex nonlinear system and may be prone to instability and oscillatory behaviors. The Fig.15 is the contrast of controller effect by Ying learning (red line), exponent function (blue line).In this section, a fuzzy controller is designed and simulated exclusively to control a refrigerator plant with a new-type function of contraction-

Using control method to explain medical phenomenon is currently a hot subject of research. The traditional Chinese drug fumigation steaming treat protrusion of protrusion of protrusion of lumbar intervertebral disc with steam generated by boiling medicinal herbs, and this process is a typical non-linear, multivariable, and strong coupling. Experienced nurse and doctor cure patient by their experience. So establish a model of this process can discover more factor of the disease, better treat to protrusion of protrusion of protrusion of

The traditional Chinese drug fumigation fume or steaming treat diseases with fume in moxibustion or with steam generated by boiling medicinal herbs, and its process is a typical non-linear, multivariable, strong coupling. In addition, its characters are difficult to quantitative analysis. So the period of treatment is only determined by experience of doctors. Therefore, there is theoretical and practical significance in studying of traditional

The illustration of the traditional Chinese drug fumigation machine is shown in Fig.16.

Doctor treats protrusion of Protrusion of protrusion of lumbar intervertebral disc with steam

Fig.17 is the temperature of steam to body by VUF and YL-VUF. YL-VUF is the blue real line, and VUF is the green dash line. In this picture the aim is 40 Celsius Degrees. The temperature decrease when patient's posture is changed. After 10.725 minute, YL-VUF makes the temperature to 40 Celsius Degrees. On the other hand, VUF almost cost 22.568 minute. DFNN and YL-VUF have the similar frame, but YL-VUF using new local space to Ping Zhang *Lanzhou University of Technology, China* 

Guodong Gao *University Hospital of Gansu Traditional Chinese Medicine, China* 

## **7. References**

	- [8] Jayasiri, A.; Mann, G. K. I.; Gosine, R. G.(2011) Behavior Coordination of Mobile Robotics Using Supervisory Control of Fuzzy Discrete Event Systems.Systems, Man, and Cybernetics, Part B, IEEE Transactions on .1083-4419, Volume 41, Issue 5, 2011, Pages 1224-1238

Novel Yinger Learning Variable Universe Fuzzy Controller 219

[21] Zhang, Wei;Li, Xue Yong;Li, Li;Lv, Jing Qiao;Chen, Yan Feng;Mao, Xin Hua (2011). Design and Application of Fuzzy Controller .1013-9826 ; 2011;Volume 1076 ;Issue 464

[22] Antonio Sala (2009) .On the conservativeness of fuzzy and fuzzy-polynomial control of

[23] Zhu, Y. .(2011) .Fuzzy Optimal Control for Multistage Fuzzy Systems .1083-4419 ;

[24] Sala, Antonio (2009) .On the conservativeness of fuzzy and fuzzy-polynomial control of

[25] KostasKolomvatsos, StathesHadjiefthymiades; .(2012) . Buyer behavior adaptation based on a fuzzy logic controller and prediction techniques0165-0114, Volume 189,

[26] Vahab Nekoukar; Abbas Erfanian .(2011) .An adaptive fuzzy sliding-mode controller design for walking control with functional electrical stimulation: A computer

[27] Xiangjian Chen; Di Li; Yue Bai; Zhijun Xu .(2011) .Modeling and Neuro-Fuzzy adaptive attitude control for Eight-Rotor MAV .International Journal of Control, Automation and

[28] Han HoChoi, Jin-WooJung; .(2011) .Takagi–Sugeno fuzzy speed controller design for a permanent magnet synchronous motor .0957-4158, Volume 21, Issue 8, 2011, Pages

[29] Mehdi Roopaei; Mansoor Zolghadri Jahromi; Bijan Ranjbar-Sahraei; Tsung-Chih Lin Nonlinear Dynamics .(2011) . Synchronization of two different chaotic systems using novel adaptive interval type-2 fuzzy sliding mode control .0924-090X, Volume 66, Issue

[30] Yin LeeGoh, Agileswari K.Ramasamy, Farrukh HafizNagi, Aidil Azwin ZainulAbidin; Microelectronics Reliability (2011) DSP based overcurrent relay using fuzzy bang–bang

[31] Ana Belén Cara; Héctor Pomares; Ignacio Rojas; Zsófia Lendek; Robert Babu?ka Evolving Systems .(2010) Online self-evolving fuzzy controller with global learning

[32] Mohammad PourmahmoodAghababa; Communications in Nonlinear Science and Numerical Simulation .(2010) Comments on "Fuzzy fractional order sliding mode controller for nonlinear systems" [Commun Nonlinear Sci Numer Simulat 15 (2010)

[33] Shih-YuLi, Zheng-MingGe; Expert Systems with Applications (2011) .Corrigendum to "Generalized synchronization of chaotic systems with different orders by fuzzy logic constant controller" [Expert Systems with Applications 38 (3) (2011) 2302–2310] .0957-

[34] Neng-ShengPai, Her-TerngYau, Chao-LinKuo; Expert Systems with Applications (2012) Comments on "Fuzzy logic combining controller design for chaos control of a rod-type

plasma torch system" .0957-4174, Volume 39, Issue 2, 2012, Pages 2236-2236

nonlinear systems .1367-5788 ; 2009;Volume 33 ; Issue 1 ;Pages 48

nonlinear systems .1367-5788 ; 200904; Volume 33 ; Issue 1 ;Pages 48-58

simulation study .1598-6446, Volume 9, Issue 6, 2011, Pages 1124-1135

controller .0026-2714, Volume 51, Issue 12, 2011, Pages 2366-2373

capabilities .1868-6478, Volume 1, Issue 4, 2010, Pages 225-239

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[21] Zhang, Wei;Li, Xue Yong;Li, Li;Lv, Jing Qiao;Chen, Yan Feng;Mao, Xin Hua (2011). Design and Application of Fuzzy Controller .1013-9826 ; 2011;Volume 1076 ;Issue 464 ;Pages 107

218 Fuzzy Controllers – Recent Advances in Theory and Applications

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[9] Pan, Y.; Er, M. J.; Huang, D.; Wang, Q.(2011) Adaptive Fuzzy Control With Guaranteed Convergence of Optimal Approximation Error.Fuzzy Systems, IEEE Transactions on

[10] Khanesar, M. A.; Kaynak, O.; Teshnehlab, M.(2011)Direct Model Reference Takagi– Sugeno Fuzzy Control of SISO Nonlinear Systems.Fuzzy Systems, IEEE Transactions

[11] Navarro, G.; Manic, M.(2011) FuSnap: Fuzzy Control of Logical Volume Snapshot Replication for Disk Arrays.Industrial Electronics, IEEE Transactions on .0278-0046,

[12] A. A. Niftiyev; C. I. Zeynalov; M. Poormanuchehri (2011) Fuzzy optimal control problem with non-linear functional.Fuzzy Information and Engineering.1616-8658,

[13] KarimTamani[a], RedaBoukezzoula[a], GeorgesHabchi[b];(2011) Application of a continuous supervisory fuzzy control on a discrete scheduling of manufacturing systems.Engineering Applications of Artificial Intelligence.0952-1976, Volume 24, Issue

[14] LaurentVermeiren[a][b][c], Thierry MarieGuerra[a][b][c], HakimLamara[b][c];(2011) Application of practical fuzzy arithmetic to fuzzy internal model control.Engineering Applications of Artificial Intelligence.0952-1976, Volume 24, Issue 6, 2011, Pages 1006-

[15] Andon V.Topalov[a], YesimOniz[b], ErdalKayacan[b], OkyayKaynak[b];(2011) Neurofuzzy control of antilock braking system using sliding mode incremental learning algorithm.Neurocomputing.0925-2312, Volume 74, Issue 11, 2011, Pages 1883-1893 [16] RubiyahYusof[a][1], Ribhan ZafiraAbdul Rahman[b], MarzukiKhalid[a][1], Mohd FaisalIbrahim[c][2](2011) Optimization of fuzzy model using genetic algorithm for process control application.Journal of the Franklin Institute.0016-0032, Volume 348,

[17] Wang, Y.; Wang , D.; Chai, T (2011) Extraction and Adaptation of Fuzzy Rules for Friction Modeling and Control Compensation.Fuzzy Systems, IEEE Transactions

[18] Li, Z.; Cao, X.; Ding , N.(2011) Adaptive Fuzzy Control for Synchronization of Nonlinear Teleoperators With Stochastic Time-Varying Communication Delays.Fuzzy Systems, IEEE Transactions on .1063-6706, Volume 19, Issue 4, 2011, Pages 745-757 [19] M.S. Shahidzadeh;H. Tarzi;M. Dorfeshan (2011) Takagi-Sugeno Fuzzy Control of Adjacent Structures using MR Dampers.Journal of Applied Sciences.1812-5654, Volume

[20] Zhu, Y.(2011) Fuzzy Optimal Control for Multistage Fuzzy Systems.Systems, Man, and Cybernetics, Part B, IEEE Transactions on .1083-4419, Volume 41, Issue 4, 2011, Pages

	- [35] BehnamGanji, Abbas Z.Kouzani; Expert Systems with Applications .(2012) Combined quasi-static backward modeling and look-ahead fuzzy control of vehicles .0957-4174, Volume 39, Issue 1, 2012, Pages 223-233

**Chapter 10** 

© 2012 Hassan, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

**Fuzzy Controllers: A Reliable Component** 

**of Smart Sustainable Structural Systems** 

Structural control has been introduced, several decades ago, as one of the basic forms of smart systems [1, 2]. The structural system's performance is enhanced by the presence of a closed loop feedback controller that employs observed data, about the system's responses, in evaluating and applying corrective actions in order to improve its performance. Initially, conventional control theory has been the backbone of such controllers [1, 2]. Yet, the sheer complexity and size of such structural systems, coupled with the time required for solving the control problem and thus evaluating the necessary corrective actions, limited the applications of such concepts. Needless to say, such systems are intended to operate real time during the occurrence of earthquake events, which are usually over in about few minutes at the most. Recently, smart control algorithms have been introduced in an attempt

Fuzzy control is one of the smart control strategies that were employed in structural control recently [5, 6]. Fuzzy controllers employ a set of input control variables, a rule-base and an inference engine to infer proposed actions aiming at the improvement of the system's performance [7]. Several factors are crucial to a successful fuzzy controller design, namely, membership functions of fuzzy variables, rule-base generation and suitable implication functions [7]. Several membership functions were employed in various applications of fuzzy controllers. It is imperative to select the membership functions that best captures the nature of the modeled variables [7]. The generation of a relevant and suitable rule-base is another major concern, several approaches have been employed, such as relying on expertise of human operators as opposed to designing a smart algorithm which would generate the rule-base, such as neural networks. Finally, appropriate implication functions should be carefully selected in order to reflect the proper and

Maguid H. M. Hassan

http://dx.doi.org/10.5772/48263

**1. Introduction** 

to fill that gap [3, 4].

Additional information is available at the end of the chapter

expected performance of the designed controller.


## **Fuzzy Controllers: A Reliable Component of Smart Sustainable Structural Systems**

Maguid H. M. Hassan

220 Fuzzy Controllers – Recent Advances in Theory and Applications

Volume 39, Issue 1, 2012, Pages 223-233

Volume 39, Issue 1, 2012, Pages 1545-1554

Volume 2, Issue 2, 2011, Pages 99-107

[35] BehnamGanji, Abbas Z.Kouzani; Expert Systems with Applications .(2012) Combined quasi-static backward modeling and look-ahead fuzzy control of vehicles .0957-4174,

[36] Ruey-JingLian; Expert Systems with Applications (2012) Design of an enhanced adaptive self-organizing fuzzy sliding-mode controller for robotic systems .0957-4174,

[37] NordinSaad, M.Arrofiq; Robotics and Computer-Integrated Manufacturing (2012)A PLC-based modified-fuzzy controller for PWM-driven induction motor drive with

constant V/Hz ratio control .0736-5845, Volume 28, Issue 2, 2012, Pages 95-112 [38] B.A.A.Omar, A.Y.M.Haikal, F.F.G.Areed, Ain Shams Engineering Journal (2011) Design adaptive neuro-fuzzy speed controller for an electro-mechanical system .2090-4479,

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48263

## **1. Introduction**

Structural control has been introduced, several decades ago, as one of the basic forms of smart systems [1, 2]. The structural system's performance is enhanced by the presence of a closed loop feedback controller that employs observed data, about the system's responses, in evaluating and applying corrective actions in order to improve its performance. Initially, conventional control theory has been the backbone of such controllers [1, 2]. Yet, the sheer complexity and size of such structural systems, coupled with the time required for solving the control problem and thus evaluating the necessary corrective actions, limited the applications of such concepts. Needless to say, such systems are intended to operate real time during the occurrence of earthquake events, which are usually over in about few minutes at the most. Recently, smart control algorithms have been introduced in an attempt to fill that gap [3, 4].

Fuzzy control is one of the smart control strategies that were employed in structural control recently [5, 6]. Fuzzy controllers employ a set of input control variables, a rule-base and an inference engine to infer proposed actions aiming at the improvement of the system's performance [7]. Several factors are crucial to a successful fuzzy controller design, namely, membership functions of fuzzy variables, rule-base generation and suitable implication functions [7]. Several membership functions were employed in various applications of fuzzy controllers. It is imperative to select the membership functions that best captures the nature of the modeled variables [7]. The generation of a relevant and suitable rule-base is another major concern, several approaches have been employed, such as relying on expertise of human operators as opposed to designing a smart algorithm which would generate the rule-base, such as neural networks. Finally, appropriate implication functions should be carefully selected in order to reflect the proper and expected performance of the designed controller.

© 2012 Hassan, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Fuzzy control, as a heuristic-based control strategy and given the uncertain nature of the problem in question, would definitely require a reliability assessment and assurance algorithms to reinforce its implementation in such a critical application. Successful reliability evaluation of any given system is performed in consecutive steps that start with creating a comprehensive reliability assessment framework, developing a system model, complete definition of potential failure modes, transformation of such failure modes into limit state equations and finally the calculation of the reliability measures for the component and/or system in question.

Fuzzy Controllers: A Reliable Component of Smart Sustainable Structural Systems 223

Smart structural systems are defined as ones that demonstrate the ability to modify their characteristics and/or properties in order to respond favorably to unexpected severe loading conditions [8]. Conventional structural systems are usually designed to resist predefined loading conditions. However, due to the uncertain nature of engineering systems, and the lack of complete and accurate information about some types of highly uncertain loads, such as earthquakes, smart structural systems have emerged as a potential solution for such problems. Instead of designing systems to withstand a single extreme earthquake event that may or may not occur in its lifetime, new designs of smart systems could emerge where the system is capable of responding favorably, in a smart manner, to any type of loading that was not specifically considered at the design stage. The significance of such systems is even further enhanced when modeled systems are unconventional such as historic buildings

As in all engineering endeavors, with a long deep look at god's creations, one can surely develop a lot of smart ideas. For example, if a similarity is drawn between a human trying to balance himself on a shaky table, and a building trying to balance itself on a shaking ground. The first, develops no mathematical models, solves no complicated sets of equations and yet is successfully capable of balancing himself. He simply employs three basic properties of his. First, his sensing capabilities, through his nervous system, which sends messages to his brain, signaling that an adverse effect is about to happen. The brain uses this piece of information and, based on its collection of experiences and reasoning capabilities, develop a balancing solution for the problem. The brain, then, sends specific commands to a set of muscles that are capable of restoring the balance of the human body. The body is balanced throughout a smart procedure that started with data collection about the current state of the body, then, data processing, state identification and problem solving.

If a building is required to balance itself on a shaking ground, in a similar manner, it should employ similar smart procedures. Therefore, for any structural system to behave in a smart manner, it should go through three basic steps. First, it has to realize, somehow, what is going on in terms of adverse effects. Second, it should be able to process this information, i.e., translate that into state identification, and accordingly decide the type of necessary countermeasures. Third, it should have the ability to perform whatever corrective action is required. A structural system, designed as such, should employ three basic components, integrated within its

 First, *Sensors*, which are analogous to the human nervous system, shall be employed in order to measure and register important internal and external information and / or

 Second, *Processors*, which are brain-like units, that are responsible for interpreting the collected data into meaningful state identifications and accordingly necessary corrective

 Third, *Actuators*, which are elements that maintain the capability of adjusting either the system structural characteristics or its own characteristics in order to respond favorably

structure, in order to be able to perform the previously mentioned activities.

and/or structures.

changes.

actions to be taken.

to external excitation.

The final step is action implementation.

In this chapter, the design of fuzzy controllers, tailored for functioning as structural controllers, is outlined together with all necessary definitions of relevant variables, their membership functions, fuzzification and de-fuzzification procedures. The definition of the required inference engine and its underlying rule-base, implication functions and inference mechanisms are also presented. Knowing the importance of reliable performance of such heuristic systems and to ensure their general applicability, a reliability assessment procedure is also outlined to evaluate the reliability of the designed controllers. Finally, other potential applications of fuzzy inference systems are also briefly presented, such applications include, but not limited to, smart abstract deformed shape identification of structural systems under earthquake excitation.

## **2. Smart sustainable structural systems**

## **2.1. Introduction**

Sustainable design entails a range of actions, decisions and procedures that would result in an environmentally friendly structural system. Such a concept has long been ignored in structural engineering and when realized was taken as one that relates to a single dimensional approach which always referred to the use of recyclable materials. Surely, recyclable materials are considered one of the main players in such a design problem, however, structurally speaking this process requires a multi faceted approach that employs higher levels of design decisions and considerations. A sustainable structural system would be one that employs the optimum amount of environmentally friendly construction materials with ensured reliable performance along its expected life time. The keywords here are being recyclable, optimum and reliable. Therefore, when designing a structural system that is expected to withstand uncertain loading conditions, such as earthquake loads, it is more sustainable to design a smart system that is capable of adjusting its own physical and/or engineering characteristics in order to improve its response to such loads, as opposed to a system that is designed to resist loads that it may or may not encounter during its life time. Smart systems, by definition, would result in lighter more optimum systems which definitely would be even more sustainable if they are constructed using a recyclable material, such as structural steel. Even if more invasive materials were used, such as, reinforced concrete, the optimum design coupled with the smart features would result in a more sustainable system. Therefore, it is proposed that if it is possible to design reliable smart structural systems, this would result in a more sustainable structural design.

Smart structural systems are defined as ones that demonstrate the ability to modify their characteristics and/or properties in order to respond favorably to unexpected severe loading conditions [8]. Conventional structural systems are usually designed to resist predefined loading conditions. However, due to the uncertain nature of engineering systems, and the lack of complete and accurate information about some types of highly uncertain loads, such as earthquakes, smart structural systems have emerged as a potential solution for such problems. Instead of designing systems to withstand a single extreme earthquake event that may or may not occur in its lifetime, new designs of smart systems could emerge where the system is capable of responding favorably, in a smart manner, to any type of loading that was not specifically considered at the design stage. The significance of such systems is even further enhanced when modeled systems are unconventional such as historic buildings and/or structures.

222 Fuzzy Controllers – Recent Advances in Theory and Applications

structural systems under earthquake excitation.

**2. Smart sustainable structural systems** 

system in question.

**2.1. Introduction** 

Fuzzy control, as a heuristic-based control strategy and given the uncertain nature of the problem in question, would definitely require a reliability assessment and assurance algorithms to reinforce its implementation in such a critical application. Successful reliability evaluation of any given system is performed in consecutive steps that start with creating a comprehensive reliability assessment framework, developing a system model, complete definition of potential failure modes, transformation of such failure modes into limit state equations and finally the calculation of the reliability measures for the component and/or

In this chapter, the design of fuzzy controllers, tailored for functioning as structural controllers, is outlined together with all necessary definitions of relevant variables, their membership functions, fuzzification and de-fuzzification procedures. The definition of the required inference engine and its underlying rule-base, implication functions and inference mechanisms are also presented. Knowing the importance of reliable performance of such heuristic systems and to ensure their general applicability, a reliability assessment procedure is also outlined to evaluate the reliability of the designed controllers. Finally, other potential applications of fuzzy inference systems are also briefly presented, such applications include, but not limited to, smart abstract deformed shape identification of

Sustainable design entails a range of actions, decisions and procedures that would result in an environmentally friendly structural system. Such a concept has long been ignored in structural engineering and when realized was taken as one that relates to a single dimensional approach which always referred to the use of recyclable materials. Surely, recyclable materials are considered one of the main players in such a design problem, however, structurally speaking this process requires a multi faceted approach that employs higher levels of design decisions and considerations. A sustainable structural system would be one that employs the optimum amount of environmentally friendly construction materials with ensured reliable performance along its expected life time. The keywords here are being recyclable, optimum and reliable. Therefore, when designing a structural system that is expected to withstand uncertain loading conditions, such as earthquake loads, it is more sustainable to design a smart system that is capable of adjusting its own physical and/or engineering characteristics in order to improve its response to such loads, as opposed to a system that is designed to resist loads that it may or may not encounter during its life time. Smart systems, by definition, would result in lighter more optimum systems which definitely would be even more sustainable if they are constructed using a recyclable material, such as structural steel. Even if more invasive materials were used, such as, reinforced concrete, the optimum design coupled with the smart features would result in a more sustainable system. Therefore, it is proposed that if it is possible to design reliable

smart structural systems, this would result in a more sustainable structural design.

As in all engineering endeavors, with a long deep look at god's creations, one can surely develop a lot of smart ideas. For example, if a similarity is drawn between a human trying to balance himself on a shaky table, and a building trying to balance itself on a shaking ground. The first, develops no mathematical models, solves no complicated sets of equations and yet is successfully capable of balancing himself. He simply employs three basic properties of his. First, his sensing capabilities, through his nervous system, which sends messages to his brain, signaling that an adverse effect is about to happen. The brain uses this piece of information and, based on its collection of experiences and reasoning capabilities, develop a balancing solution for the problem. The brain, then, sends specific commands to a set of muscles that are capable of restoring the balance of the human body. The body is balanced throughout a smart procedure that started with data collection about the current state of the body, then, data processing, state identification and problem solving. The final step is action implementation.

If a building is required to balance itself on a shaking ground, in a similar manner, it should employ similar smart procedures. Therefore, for any structural system to behave in a smart manner, it should go through three basic steps. First, it has to realize, somehow, what is going on in terms of adverse effects. Second, it should be able to process this information, i.e., translate that into state identification, and accordingly decide the type of necessary countermeasures. Third, it should have the ability to perform whatever corrective action is required. A structural system, designed as such, should employ three basic components, integrated within its structure, in order to be able to perform the previously mentioned activities.


According to the type and nature of the employed components, several levels of smart systems could be developed. It should be realized that both the actuators and processors, in a human being, have additional levels of smartness based on their nature. For example, the muscles, i.e., the actuators, exert a variable amount of force depending on the signals sent by the brain. The brain, itself, employs highly adaptive thinking and learning techniques in its reasoning process. Therefore, multiple integrated levels of smart structural systems could be realized according to the type and properties of the components used in its development. As the level of integration increases, the level of smartness of the resulting system increases. Figure 1 shows a model of a smart single-degree-of-freedom system, while Figure 2, shows a smart three-story building. Both figures outline the inter-relations among the additional components that drive the performance of the smart system. It should be noted that multidegree-of-freedom systems require a more complex processor that incorporates two main components, i.e., a fuzzy state identifier in addition to the fuzzy controller. The state identifier is required to define the deformed shape of the system, thus, guiding the firing sequence of relevant actuators. The following discussion outlines the properties of each of the three basic components in the sake of providing a comprehensive description of the system under consideration.

Fuzzy Controllers: A Reliable Component of Smart Sustainable Structural Systems 225

applications. The first application requires the first mode of monitoring, while the second

There are several potential sensor technologies that are being used or considered for the smart structural applications in civil engineering. As indicated earlier, a higher level of smartness is attained if smart materials are used as sensors in addition to, or instead of, conventional sensor technologies. Conventional sensor systems are ones that do not posses any smart potential. In other words, such materials are incapable of altering or adjusting their own characteristics in response to external excitation. Their role would be to measure specific state variables and send such records to the processor unit for state identification and corrective action evaluation. Such sensors are very well documented and have been implemented in civil / structural applications for a long time. As an example, Displacement Transducers, Velocity and Acceleration Transducers and Strain Transducers are considered as conventional sensors. Smart sensors, on the other hand, by definition are ones that are capable of altering or adjusting their characteristics in response to external excitation. Such excitation might be temperature, electric current or mechanical movements, piezoelectric

application requires the second mode of monitoring.

**Figure 2.** Smart Sustainable Steel Frame

ceramics and optical fibers are examples of such smart sensors.

**Figure 1.** Smart Single-Degree-of-Freedom System

## **2.2. Sensors**

Sensors are the first component of any smart structural system. The system needs to be able to identify any changes occurring to its state in order to perform any corrective action [8]. The monitoring operation could be implemented in two possible modes. The first mode is a continuous monitoring for a select group of parameters, such as displacements, velocities and accelerations. The second mode is a continuous monitoring for a select group of damage indicators such as cracking, fatigue, corrosion or excessive deflections. The smart structural system application, in question, dictates the required mode of monitoring. Currently, such systems are either designed for structural control, or for structural health monitoring applications. The first application requires the first mode of monitoring, while the second application requires the second mode of monitoring.

**Figure 2.** Smart Sustainable Steel Frame

224 Fuzzy Controllers – Recent Advances in Theory and Applications

system under consideration.

**Figure 1.** Smart Single-Degree-of-Freedom System

**2.2. Sensors** 

According to the type and nature of the employed components, several levels of smart systems could be developed. It should be realized that both the actuators and processors, in a human being, have additional levels of smartness based on their nature. For example, the muscles, i.e., the actuators, exert a variable amount of force depending on the signals sent by the brain. The brain, itself, employs highly adaptive thinking and learning techniques in its reasoning process. Therefore, multiple integrated levels of smart structural systems could be realized according to the type and properties of the components used in its development. As the level of integration increases, the level of smartness of the resulting system increases. Figure 1 shows a model of a smart single-degree-of-freedom system, while Figure 2, shows a smart three-story building. Both figures outline the inter-relations among the additional components that drive the performance of the smart system. It should be noted that multidegree-of-freedom systems require a more complex processor that incorporates two main components, i.e., a fuzzy state identifier in addition to the fuzzy controller. The state identifier is required to define the deformed shape of the system, thus, guiding the firing sequence of relevant actuators. The following discussion outlines the properties of each of the three basic components in the sake of providing a comprehensive description of the

Sensors are the first component of any smart structural system. The system needs to be able to identify any changes occurring to its state in order to perform any corrective action [8]. The monitoring operation could be implemented in two possible modes. The first mode is a continuous monitoring for a select group of parameters, such as displacements, velocities and accelerations. The second mode is a continuous monitoring for a select group of damage indicators such as cracking, fatigue, corrosion or excessive deflections. The smart structural system application, in question, dictates the required mode of monitoring. Currently, such systems are either designed for structural control, or for structural health monitoring There are several potential sensor technologies that are being used or considered for the smart structural applications in civil engineering. As indicated earlier, a higher level of smartness is attained if smart materials are used as sensors in addition to, or instead of, conventional sensor technologies. Conventional sensor systems are ones that do not posses any smart potential. In other words, such materials are incapable of altering or adjusting their own characteristics in response to external excitation. Their role would be to measure specific state variables and send such records to the processor unit for state identification and corrective action evaluation. Such sensors are very well documented and have been implemented in civil / structural applications for a long time. As an example, Displacement Transducers, Velocity and Acceleration Transducers and Strain Transducers are considered as conventional sensors. Smart sensors, on the other hand, by definition are ones that are capable of altering or adjusting their characteristics in response to external excitation. Such excitation might be temperature, electric current or mechanical movements, piezoelectric ceramics and optical fibers are examples of such smart sensors.

## **2.3. Processors**

Processors are brain-like units that are capable of evaluating the current state of the system, based on the data communicated by the sensors, and proposing corrective actions accordingly [8]. The processor shall be capable of operation in one of two modes. The first mode is system monitoring, where the processor only identifies the state of the system without taking any remedial actions. The second mode is system control, where the processor is called upon to identify the state of the system, evaluate the proper corrective action to be taken and implement the suggested action automatically. It should be pointed out that system control mode, by definition, employs system monitoring mode as a subsequent major component. One of the major factors that would dictate the mode of operation of the processor is the type of application and the state parameter being monitored. For example, RC elements might be supplied with a monitoring system, in order to identify any potential damage to a given element, such as steel corrosion, concrete cracking, excessive deflection, etc. The processor, in this case, is only called upon to identify the occurrence of a certain type of damage. Furthermore, any structural system could be supplied with a control system that is capable of suppressing the vibration and balancing the system under wind and earthquake excitations. Thus, a processor operating at the control mode would be required for such an application.

Fuzzy Controllers: A Reliable Component of Smart Sustainable Structural Systems 227

smart actuators which are currently being explored for application in civil engineering systems are Shape Memory Alloys (SMA), Piezoelectric Ceramics, Electro-Rheological (ER) fluids, and Magneto-Rheological (MR) fluids. MR fluids, which are employed in manufacturing MR dampers, have already reached full-scale applications and showing very

Fuzzy controllers are known to employ fuzzy logic and fuzzy set theory in developing their control strategies and evaluating control actions [9, 10, 11, 12, 13]. Fuzzy logic has two primary advantages, as opposed to conventional mathematical algorithms, when employed in control applications. First, it reduces the difficulties of modeling and analysis of extremely complex systems. Second, it is capable of incorporating several qualitative aspects of the human knowledge in the control laws [10, 11, 12, 13]. Fuzzy control is based on the fuzzy set theory which allows for the qualitative, imprecise and/or vague information to be quantitatively included in the evaluation of a representative control action [5, 6, 7, 10, 11, 12, 13]. Such inherent uncertainty would probably be ignored in a conventional mathematical algorithm, thus, rendering inaccurate control forces. Fuzzy set theory utilizes a very important tool in its manipulation procedure, which is the membership function [7]. The membership function, usually takes one of the following forms, i.e., triangular, trapezoidal or Gaussian, in order to evaluate a degree of membership for the element in question. This degree of membership is the major difference between this approach and conventional mathematical methods. Fuzzy control comprises

 Fuzzification: the state variables to be monitored, when measured, have crisp values. These values should be fuzzified, using fuzzy linguistic terms defined by the

Rule-Base: is a collection of If-Then rules describing the control laws governing the

 Inference Engine: comprises two main stages, namely, Implication and Aggregation. The implication procedure evaluates a control action from each applicable rule, given a certain input fuzzy value. The Aggregation procedure evaluates a collective control action, i.e., output, by adding all control actions from all applicable rules in a

 Defuzzification: the resulting control action is in a fuzzified form that could not be applied to any actuator device. Thus, this step evaluates an equivalent crisp value for

The processor, as identified in smart structural applications could perform two main tasks, the first is state identification, if necessary, while the second is control action evaluation. Figure 1 shows a fuzzy controller without a state identifier, while Figure 2 shows a fuzzy dual processor which comprises a fuzzy state identifier and a fuzzy controller. The fuzzy controller would employ the input variables in addition to the output of the fuzzy state

promising results in civil engineering applications [3, 4].

**3. Fuzzy controllers as processors** 

four main components [5, 6, 7, 10, 11, 12, 13];

evaluation of necessary control actions.

the fuzzy collective control action.

predefined manner.

membership functions of the individual fuzzy sets.

## **2.4. Actuators**

Actuators act as the muscles of the structural system. Actuator technology is responsible for the development of materials and/or devices that would either apply control forces to the system or add new characteristics to the structure [8]. Actuators do not necessarily apply balancing forces to the structural system. In case smart actuators are utilized, the system adjusts its structural characteristics without any introduction of external forces, which is the current preferred approach.

All applications that employed conventional actuator technologies were in the field of structural control [1, 2]. Structural control is one of the early applications of smart structural systems. There are three potential schemes of structural control, namely, Passive, Active and Semi-Active [1, 2, 4]. Passive control employs energy dissipation components that are designed for predefined limits and possess no adaptive capabilities. Although most of the practical applications of structural control, currently in operation, are of this primitive type, they do not show efficient performance under real conditions. Active Control employs the basic conventional structure of a smart structural system. It comprises sensors, processors and actuators that are, predominantly, of the conventional type [1, 2]. This type of control exerts an external control force that is utilized in balancing the system in response to external loads. Semi-Active control has received increased attention recently as the most practical and state of the art control system [3, 4]. Semi-Active control employs actuators that are, predominantly, of the smart type. Such actuators cannot inject mechanical energy directly to the system, yet, they have the ability to adjust their properties in a way to optimally adjust the response of the system under unforeseen external events. Some of the smart actuators which are currently being explored for application in civil engineering systems are Shape Memory Alloys (SMA), Piezoelectric Ceramics, Electro-Rheological (ER) fluids, and Magneto-Rheological (MR) fluids. MR fluids, which are employed in manufacturing MR dampers, have already reached full-scale applications and showing very promising results in civil engineering applications [3, 4].

## **3. Fuzzy controllers as processors**

226 Fuzzy Controllers – Recent Advances in Theory and Applications

control mode would be required for such an application.

Processors are brain-like units that are capable of evaluating the current state of the system, based on the data communicated by the sensors, and proposing corrective actions accordingly [8]. The processor shall be capable of operation in one of two modes. The first mode is system monitoring, where the processor only identifies the state of the system without taking any remedial actions. The second mode is system control, where the processor is called upon to identify the state of the system, evaluate the proper corrective action to be taken and implement the suggested action automatically. It should be pointed out that system control mode, by definition, employs system monitoring mode as a subsequent major component. One of the major factors that would dictate the mode of operation of the processor is the type of application and the state parameter being monitored. For example, RC elements might be supplied with a monitoring system, in order to identify any potential damage to a given element, such as steel corrosion, concrete cracking, excessive deflection, etc. The processor, in this case, is only called upon to identify the occurrence of a certain type of damage. Furthermore, any structural system could be supplied with a control system that is capable of suppressing the vibration and balancing the system under wind and earthquake excitations. Thus, a processor operating at the

Actuators act as the muscles of the structural system. Actuator technology is responsible for the development of materials and/or devices that would either apply control forces to the system or add new characteristics to the structure [8]. Actuators do not necessarily apply balancing forces to the structural system. In case smart actuators are utilized, the system adjusts its structural characteristics without any introduction of external forces, which is the

All applications that employed conventional actuator technologies were in the field of structural control [1, 2]. Structural control is one of the early applications of smart structural systems. There are three potential schemes of structural control, namely, Passive, Active and Semi-Active [1, 2, 4]. Passive control employs energy dissipation components that are designed for predefined limits and possess no adaptive capabilities. Although most of the practical applications of structural control, currently in operation, are of this primitive type, they do not show efficient performance under real conditions. Active Control employs the basic conventional structure of a smart structural system. It comprises sensors, processors and actuators that are, predominantly, of the conventional type [1, 2]. This type of control exerts an external control force that is utilized in balancing the system in response to external loads. Semi-Active control has received increased attention recently as the most practical and state of the art control system [3, 4]. Semi-Active control employs actuators that are, predominantly, of the smart type. Such actuators cannot inject mechanical energy directly to the system, yet, they have the ability to adjust their properties in a way to optimally adjust the response of the system under unforeseen external events. Some of the

**2.3. Processors** 

**2.4. Actuators** 

current preferred approach.

Fuzzy controllers are known to employ fuzzy logic and fuzzy set theory in developing their control strategies and evaluating control actions [9, 10, 11, 12, 13]. Fuzzy logic has two primary advantages, as opposed to conventional mathematical algorithms, when employed in control applications. First, it reduces the difficulties of modeling and analysis of extremely complex systems. Second, it is capable of incorporating several qualitative aspects of the human knowledge in the control laws [10, 11, 12, 13]. Fuzzy control is based on the fuzzy set theory which allows for the qualitative, imprecise and/or vague information to be quantitatively included in the evaluation of a representative control action [5, 6, 7, 10, 11, 12, 13]. Such inherent uncertainty would probably be ignored in a conventional mathematical algorithm, thus, rendering inaccurate control forces. Fuzzy set theory utilizes a very important tool in its manipulation procedure, which is the membership function [7]. The membership function, usually takes one of the following forms, i.e., triangular, trapezoidal or Gaussian, in order to evaluate a degree of membership for the element in question. This degree of membership is the major difference between this approach and conventional mathematical methods. Fuzzy control comprises four main components [5, 6, 7, 10, 11, 12, 13];


The processor, as identified in smart structural applications could perform two main tasks, the first is state identification, if necessary, while the second is control action evaluation. Figure 1 shows a fuzzy controller without a state identifier, while Figure 2 shows a fuzzy dual processor which comprises a fuzzy state identifier and a fuzzy controller. The fuzzy controller would employ the input variables in addition to the output of the fuzzy state

identifier in evaluating the appropriate control action. The following sections outline the implementation of fuzzy control in the development of the controller component of the processor while the fuzzy state identifier is discussed in section 5. The reliability of fuzzy controllers when operating within the smart system is a major concern if such a setup is considered for designing systems that are sustainable as defined above. The reliability framework and assessment procedures for fuzzy processors are discussed in section 4 of this chapter.

Fuzzy Controllers: A Reliable Component of Smart Sustainable Structural Systems 229

results of the modeled system. In order to evaluate the range of values, a structural model of the system under consideration should be created and a time history analysis shall be conducted [14, 15, 16]. The results of such an analysis would generate all potential values of velocities and accelerations of control points. These values could be used in identifying several bands within the expected range, correlate these bands to fuzzy variables and evaluate the required parameters, for each band, of the velocity and/or acceleration [17].

*Gaussian Membership Functions*, The Gaussian membership function is fully defined by two main parameters, namely, the average value and the standard deviation. Figure 3 outlines a generic Gaussian membership function with the expected form of the function [7, 8]. When modeling input and/or output variables using a Gaussian membership function, the time history of the modeled variable needs to be evaluated using a finite element model of the structural system under consideration. The resulting time history would allow the segmentation of the variable range into several bands with relevant fuzzy labels and

Average Value

3σ 3σ

suitable standard deviations.

**Figure 3.** Generic Gaussian Membership Function

**Figure 4.** A Singleton Gaussian Fuzzy Variable with Zero Label

## **3.1. Input variables**

Fuzzy controllers usually employ two input variables one is an error measure while the second is a rate of change of that error [7]. In that context, it is usually required of the controller to monitor the performance of the modeled system in order to minimize or even eliminate the error if possible. In case of structural control, this corresponds to the dynamic movement of the controlled system which is generally measured by the velocity of a select set of control points referred to as degrees of freedom [8]. Such degrees of freedom correspond to the floor levels of the framed building shown in Figures 1 & 2 and have assigned sensors to measure their movements. The rate of change of the measured variable, i.e., velocity, would be the acceleration of the control points. Therefore, in structural control applications it is expected to include the velocity and the acceleration of degrees of freedom as input variables to the fuzzy controllers. When dealing with complex systems, having so many degrees of freedom, and in order to attain the objective of reliable, optimum and sustainable systems, it is expected that the control actions would not be required of all actuators, however, a select group of actuators which are identified based on the deformed shape of the controlled building would be fired. Therefore, the input variables to the fuzzy controller should include an additional input variable which classify the current deformed shape of the monitored building. Thus, three dimensional rules would be necessary to drive the inference engine of structural fuzzy controllers. Input variables, as well as, output variables need to be fully defined as part of the design of a fuzzy controller. Such definition would not be complete without the identification of a suitable membership function for each variable.

## *3.1.1. Membership functions*

A major step in defining control variables in fuzzy control applications is the definition of membership functions [7]. Such a task has to be performed in two main underlying steps. The first is the selection of the range of values which the function should cover while the second is the type of membership function to be employed and its relevant parameters. Several membership functions were reported successfully in several fuzzy control applications, such as, triangular, trapezoidal and Gaussian functions [7]. For structural control applications it is expected that either triangular and/or Gaussian membership functions would be suitable for modeling control input and output variables. The proper identification of a representative range of values for properly defining such membership functions should be based on actual results of the modeled system. In order to evaluate the range of values, a structural model of the system under consideration should be created and a time history analysis shall be conducted [14, 15, 16]. The results of such an analysis would generate all potential values of velocities and accelerations of control points. These values could be used in identifying several bands within the expected range, correlate these bands to fuzzy variables and evaluate the required parameters, for each band, of the velocity and/or acceleration [17].

*Gaussian Membership Functions*, The Gaussian membership function is fully defined by two main parameters, namely, the average value and the standard deviation. Figure 3 outlines a generic Gaussian membership function with the expected form of the function [7, 8]. When modeling input and/or output variables using a Gaussian membership function, the time history of the modeled variable needs to be evaluated using a finite element model of the structural system under consideration. The resulting time history would allow the segmentation of the variable range into several bands with relevant fuzzy labels and suitable standard deviations.

**Figure 3.** Generic Gaussian Membership Function

228 Fuzzy Controllers – Recent Advances in Theory and Applications

chapter.

variable.

*3.1.1. Membership functions* 

**3.1. Input variables** 

identifier in evaluating the appropriate control action. The following sections outline the implementation of fuzzy control in the development of the controller component of the processor while the fuzzy state identifier is discussed in section 5. The reliability of fuzzy controllers when operating within the smart system is a major concern if such a setup is considered for designing systems that are sustainable as defined above. The reliability framework and assessment procedures for fuzzy processors are discussed in section 4 of this

Fuzzy controllers usually employ two input variables one is an error measure while the second is a rate of change of that error [7]. In that context, it is usually required of the controller to monitor the performance of the modeled system in order to minimize or even eliminate the error if possible. In case of structural control, this corresponds to the dynamic movement of the controlled system which is generally measured by the velocity of a select set of control points referred to as degrees of freedom [8]. Such degrees of freedom correspond to the floor levels of the framed building shown in Figures 1 & 2 and have assigned sensors to measure their movements. The rate of change of the measured variable, i.e., velocity, would be the acceleration of the control points. Therefore, in structural control applications it is expected to include the velocity and the acceleration of degrees of freedom as input variables to the fuzzy controllers. When dealing with complex systems, having so many degrees of freedom, and in order to attain the objective of reliable, optimum and sustainable systems, it is expected that the control actions would not be required of all actuators, however, a select group of actuators which are identified based on the deformed shape of the controlled building would be fired. Therefore, the input variables to the fuzzy controller should include an additional input variable which classify the current deformed shape of the monitored building. Thus, three dimensional rules would be necessary to drive the inference engine of structural fuzzy controllers. Input variables, as well as, output variables need to be fully defined as part of the design of a fuzzy controller. Such definition would not be complete without the identification of a suitable membership function for each

A major step in defining control variables in fuzzy control applications is the definition of membership functions [7]. Such a task has to be performed in two main underlying steps. The first is the selection of the range of values which the function should cover while the second is the type of membership function to be employed and its relevant parameters. Several membership functions were reported successfully in several fuzzy control applications, such as, triangular, trapezoidal and Gaussian functions [7]. For structural control applications it is expected that either triangular and/or Gaussian membership functions would be suitable for modeling control input and output variables. The proper identification of a representative range of values for properly defining such membership functions should be based on actual

**Figure 4.** A Singleton Gaussian Fuzzy Variable with Zero Label

For example a fuzzy variable with a zero label would have a zero average value and a very narrow standard deviation to simulate the singleton value of zero as shown in Figure 4. While a negative fuzzy variable would have a negative average value, that is equivalent to the range of values the variable takes as indicated by the time history analysis results, and a suitable standard deviation to model the dispersion about this average value as shown in Figure 5.

Fuzzy Controllers: A Reliable Component of Smart Sustainable Structural Systems 231

y

(1)

a

μ()

b

**Figure 6.** Generic Triangular Membership Function

**Figure 7.** Triangular Fuzzy Variable with Zero, Positive & Negative Labels

*NEGATIVE*

mathematically as follows:

The triangular membership functions for variables shown in Figure 7 are defined

<sup>0</sup> ( )

*for all other values of x*

*<sup>x</sup> for LB x <sup>x</sup> LB*

0

 

c

**Figure 5.** A Gaussian Fuzzy Variable with Negative Label

The standard deviation in the first case was selected to be 0.001 to reflect the narrow range and refelect the singleton nature of the zero label, while the standard deviation in Figure 5 was selected to be 1 to reflect the uncertainty associated with the negative label. Depending on the range of input variables some or all of the memebrship functins could be used in modeling the variable.

*Triangular Membership Functions,* The triangular membership function is one of the most widely used and successful membership functions in a wide variety of applications [7, 11, 12, 17]. Figure 6 shows a generic triangular membership function where three basic parameters are necessary in order to fully define the function [17]. The parameters are identified as (a, b and c) in Figure 6 where (a) represents the lower bound of the function, (b) defines the average value and (c) defines the upper bound of the membership function. As in the case of Gaussian membership functions, the range of input values would dictate if the whole function is employed or just a portion is only enough to represent the modeled variable. Moreover, the amount of uncertainty incorporated in the function which is measured by the triangular base of the function, i.e., (c-a), is also problem dependant and should be evaluated based on the actual data resulting from the finite element model of the system. If the same variables, modeled with Gaussian membership functions, are modeled using triangular functions, the zero fuzzy label and the negative fuzzy label would be defined as shown in Figure 7

**Figure 6.** Generic Triangular Membership Function

**Figure 5.** A Gaussian Fuzzy Variable with Negative Label

Negative Average Value

modeling the variable.

defined as shown in Figure 7

Figure 5.

For example a fuzzy variable with a zero label would have a zero average value and a very narrow standard deviation to simulate the singleton value of zero as shown in Figure 4. While a negative fuzzy variable would have a negative average value, that is equivalent to the range of values the variable takes as indicated by the time history analysis results, and a suitable standard deviation to model the dispersion about this average value as shown in

The standard deviation in the first case was selected to be 0.001 to reflect the narrow range and refelect the singleton nature of the zero label, while the standard deviation in Figure 5 was selected to be 1 to reflect the uncertainty associated with the negative label. Depending on the range of input variables some or all of the memebrship functins could be used in

*Triangular Membership Functions,* The triangular membership function is one of the most widely used and successful membership functions in a wide variety of applications [7, 11, 12, 17]. Figure 6 shows a generic triangular membership function where three basic parameters are necessary in order to fully define the function [17]. The parameters are identified as (a, b and c) in Figure 6 where (a) represents the lower bound of the function, (b) defines the average value and (c) defines the upper bound of the membership function. As in the case of Gaussian membership functions, the range of input values would dictate if the whole function is employed or just a portion is only enough to represent the modeled variable. Moreover, the amount of uncertainty incorporated in the function which is measured by the triangular base of the function, i.e., (c-a), is also problem dependant and should be evaluated based on the actual data resulting from the finite element model of the system. If the same variables, modeled with Gaussian membership functions, are modeled using triangular functions, the zero fuzzy label and the negative fuzzy label would be

**Figure 7.** Triangular Fuzzy Variable with Zero, Positive & Negative Labels

The triangular membership functions for variables shown in Figure 7 are defined mathematically as follows:

$$\mu\_{\text{NECATWE}}\left(\mathbf{x}\right) = \begin{cases} \begin{array}{c} \text{x} \\ \text{LB} \\ 0 \text{ } for \text{ all other values of } \mathbf{x} \end{array} \tag{1}$$

$$\mu\_{\text{ZEO}}\left(\mathbf{x}\right) = \begin{cases} -\frac{1}{10^{-6}} \left(-10^{-6} - \mathbf{x}\right) & \text{for } -10^{-6} \le \mathbf{x} < \mathbf{0} \\\ \frac{1}{10^{-6}} \left(10^{-6} - \mathbf{x}\right) & \text{for } 0 \le \mathbf{x} < 10^{-6} \\\ 0 & \text{for all other values of } \mathbf{x} \end{cases} \tag{2}$$

Fuzzy Controllers: A Reliable Component of Smart Sustainable Structural Systems 233

the velocity of predefined degrees of freedom as the error measure while the acceleration, which is the rate of change of the velocity, would be employed as the second input variable [12]. Therefore, structural control applications should at least involve two input variables in their rules. This setup would be enough for s single degree-of-freedom system, as shown in Figure 1, where the movement of a single floor would completely define the deformed shape of the system and thus the necessary action to restore the original shape of the system. In case of multi-degree-of-freedom systems, such as the system shown in Figure 2, a third input variable is necessary in order to provide additional information about the abstract deformed shape of the structural system. The need for that additional variable is reflected in the enhanced fuzzy processor where a fuzzy pattern identifier is integrated with the controller in order to identify the abstract deformed shape of the system [19]. This information is crucial in firing relevant actuators with the proper output value and sequence. Therefore, rules that would drive the operation of a smart sustainable multidegree-of-freedom structural system are expected to employ three input variables and a

In reference to a single degree of freedom system, as shown in Figure 1, a sample rule should include two input variables and a single output variable. The input variables are the velocity and acceleration of the floor level, while the output variable is the voltage which is communicated to an MR damper in order to restore the system's un-deformed shape. The

Where, *D t* = is the velocity at the floor level at a given point in time (t), *D t* = is the acceleration at the floor level at a given point in time (t), V(t) = is the command voltage at a given point in time (t), and BIG, NEGATIVE SMALL and SMALL are fuzzy variables. On the other hand, in reference to the smart system defined in Figure 2, the fuzzy controller would accept the velocity and acceleration of a given degree of freedom in addition to an abstract deformed pattern, as input and produce a voltage value as output. The voltage value is communicated to a specific MR damper, selected based on the abstract deformed shape of the system, which would ultimately result in improving the response of the system under the effect of earthquake excitation. A sample rule, as defined above, could be written

1 1 () B ( ) NEGATIVE SMALL ( ) 2 ( ) SMALL *<sup>j</sup> IF D t is IG AND D t is AND P t is THEN V t is* (5)

Where, *D t <sup>i</sup>* = is the velocity at ith degree of freedom at a given point in time (t), *D t <sup>i</sup>* = is the acceleration of the ith degree of freedom at a given point in time (t), P(t) = is the abstract deformed pattern at a given point in time (t), Vj(t) = is the command voltage to the jth damper, at a given point in time (t), BIG, NEGATIVE SMALL and SMALL are fuzzy variables and 2 is a pre-defined abstract pattern, as evaluated by a smart pattern identifier

*IF D t is IG AND D t is* ( ) B ( ) NEGATIVE SMALL ( ) SMALL *THEN V t is* (4)

single output variable [19].

rule could be defined as follows:

as follows:

[17, 19, 20].

$$\mu\_{\text{pos.mive}} \text{ (x)} = \begin{cases} \frac{\text{x}}{\text{UB}} & \text{for } 0 \le \text{x} \le \text{UB} \\ 0 & \text{for all other values of } \text{x} \end{cases} \tag{3}$$

Where, (.), (.) & (.) *NEGATIVE ZERO POSITIVE* = are the membership functions for the three fuzzy states of an input variable, NEGATIVE, ZERO & POSITIVE respectively; LB & UB = are the lower and upper bounds of the interval holding the range of values of the variable, as evaluated by a time history analysis, at any point in time, LB < 0 and UB > 0; and x = is the value of the variable, at any point in time. For the shown membership functions in Figure 7 the bounds of the interval holding the variable range of values is [-0.55, 0.55].

#### **3.2. Inference engine**

Fuzzy controllers are built on top of an inference engine which employs a rule-base that summarizes the necessary knowledge for inferring actions and an inference engine which performs the evaluation process based on fuzzy logic [7, 12]. The Inference engine comprises inference functions, inference mechanisms and aggregation functions that would combine the results of relevant fired rules into a single fuzzy output variable. There are several types of inference mechanisms, however, the most widely used in control applications is Mamdani's inference [7, 12]. The nature of the problem at hand and its impact on the evaluation of the overall output variable dictates the choice of the relevant inference mechanism [7, 12]. The details of the inference mechanism are beyond the scope of this chapter, however, the method of developing a representative rule-base is further discussed in this section with examples from structural control applications.

It is first important to identify the structure of the rule before building a rule-base. Rules could be multi-dimensional depending on the nature of the problem. In other words, rules could construct a one-to-one mapping between a single input and a single output, or a many-to-one mapping where several input variables are mapped to a single output variable. The number of inputs necessary to infer an output is obviously a problem dependant factor. In structural control applications it is necessary to include at least two measurable input variables in order to infer realistic output values [12]. Usually these two variables are some measure of error and rate of change of that error. The interpretation of an error term would be different from one application to the next. In case of structural control problems, any variable that would measure the movement of the system as a result of dynamic load effects, e.g., earthquakes, would qualify as an error measure. Therefore, it is reasonable to employ the velocity of predefined degrees of freedom as the error measure while the acceleration, which is the rate of change of the velocity, would be employed as the second input variable [12]. Therefore, structural control applications should at least involve two input variables in their rules. This setup would be enough for s single degree-of-freedom system, as shown in Figure 1, where the movement of a single floor would completely define the deformed shape of the system and thus the necessary action to restore the original shape of the system. In case of multi-degree-of-freedom systems, such as the system shown in Figure 2, a third input variable is necessary in order to provide additional information about the abstract deformed shape of the structural system. The need for that additional variable is reflected in the enhanced fuzzy processor where a fuzzy pattern identifier is integrated with the controller in order to identify the abstract deformed shape of the system [19]. This information is crucial in firing relevant actuators with the proper output value and sequence. Therefore, rules that would drive the operation of a smart sustainable multidegree-of-freedom structural system are expected to employ three input variables and a single output variable [19].

232 Fuzzy Controllers – Recent Advances in Theory and Applications

*ZERO*

Where, (.), (.) & (.) *NEGATIVE ZERO POSITIVE*

**3.2. Inference engine** 

6

10

 

*POSITIVE*

 

in this section with examples from structural control applications.

6

0

 

the bounds of the interval holding the variable range of values is [-0.55, 0.55].

6 6

<sup>1</sup> <sup>10</sup> 10 0

*for all other values of x*

*for all other values of x*

= are the membership functions for the three fuzzy

6 6

*x for x*

(2)

(3)

*x x for x*

<sup>0</sup> ( ) 0

*<sup>x</sup> for x UB <sup>x</sup> UB*

states of an input variable, NEGATIVE, ZERO & POSITIVE respectively; LB & UB = are the lower and upper bounds of the interval holding the range of values of the variable, as evaluated by a time history analysis, at any point in time, LB < 0 and UB > 0; and x = is the value of the variable, at any point in time. For the shown membership functions in Figure 7

Fuzzy controllers are built on top of an inference engine which employs a rule-base that summarizes the necessary knowledge for inferring actions and an inference engine which performs the evaluation process based on fuzzy logic [7, 12]. The Inference engine comprises inference functions, inference mechanisms and aggregation functions that would combine the results of relevant fired rules into a single fuzzy output variable. There are several types of inference mechanisms, however, the most widely used in control applications is Mamdani's inference [7, 12]. The nature of the problem at hand and its impact on the evaluation of the overall output variable dictates the choice of the relevant inference mechanism [7, 12]. The details of the inference mechanism are beyond the scope of this chapter, however, the method of developing a representative rule-base is further discussed

It is first important to identify the structure of the rule before building a rule-base. Rules could be multi-dimensional depending on the nature of the problem. In other words, rules could construct a one-to-one mapping between a single input and a single output, or a many-to-one mapping where several input variables are mapped to a single output variable. The number of inputs necessary to infer an output is obviously a problem dependant factor. In structural control applications it is necessary to include at least two measurable input variables in order to infer realistic output values [12]. Usually these two variables are some measure of error and rate of change of that error. The interpretation of an error term would be different from one application to the next. In case of structural control problems, any variable that would measure the movement of the system as a result of dynamic load effects, e.g., earthquakes, would qualify as an error measure. Therefore, it is reasonable to employ

<sup>1</sup> ( ) 10 0 10 <sup>10</sup>

In reference to a single degree of freedom system, as shown in Figure 1, a sample rule should include two input variables and a single output variable. The input variables are the velocity and acceleration of the floor level, while the output variable is the voltage which is communicated to an MR damper in order to restore the system's un-deformed shape. The rule could be defined as follows:

## *IF D t is IG AND D t is* ( ) B ( ) NEGATIVE SMALL ( ) SMALL *THEN V t is* (4)

Where, *D t* = is the velocity at the floor level at a given point in time (t), *D t* = is the acceleration at the floor level at a given point in time (t), V(t) = is the command voltage at a given point in time (t), and BIG, NEGATIVE SMALL and SMALL are fuzzy variables. On the other hand, in reference to the smart system defined in Figure 2, the fuzzy controller would accept the velocity and acceleration of a given degree of freedom in addition to an abstract deformed pattern, as input and produce a voltage value as output. The voltage value is communicated to a specific MR damper, selected based on the abstract deformed shape of the system, which would ultimately result in improving the response of the system under the effect of earthquake excitation. A sample rule, as defined above, could be written as follows:

$$\text{IF } \dot{D}\_i(t) \text{ is BIG AND } \ddot{D}\_i(t) \text{ is NEGATIVE SMALI AND } \mathcal{P}(t) \text{ is 2 THEN } V\_j(t) \text{ is SMALI (5)}$$

Where, *D t <sup>i</sup>* = is the velocity at ith degree of freedom at a given point in time (t), *D t <sup>i</sup>* = is the acceleration of the ith degree of freedom at a given point in time (t), P(t) = is the abstract deformed pattern at a given point in time (t), Vj(t) = is the command voltage to the jth damper, at a given point in time (t), BIG, NEGATIVE SMALL and SMALL are fuzzy variables and 2 is a pre-defined abstract pattern, as evaluated by a smart pattern identifier [17, 19, 20].

## **3.3. Rule-base generation**

The heart of a fuzzy controller is its rule-base. The rule-base houses a collection of IF-THEN rules that summarize the knowledge-base that underpins the decisions made by the fuzzy controller [7, 11]. Being a non-parametric heuristic algorithm, fuzzy controllers are built to simulate a human operator's reasoning when facing a similar control situation. In an effort to design smart sustainable structural systems, that are built to be autonomous systems, the developed rule-base should be capable of handling all potential situations that might arise during the system's expected life time. Such controllers should be designed with self learning capabilities such that their initial rule-bases could be amended and expanded as new experiences and/or situations arise. There are currently several applications and toolboxes that allow the automatic extraction of rules of a given problem, knowing the input/output data sets of the problem without the pre-existing knowledge of a model for the system. This approach might be suitable for ill-defined systems. However, in case of structural systems under earthquake excitation, the system behavior is fully defined and could be identified using finite element models under several types of conventional analysis techniques.

Fuzzy Controllers: A Reliable Component of Smart Sustainable Structural Systems 235

failure probability of zero [21 22, 23]. Yet, it is crucial to be able to evaluate the failure probability of any designed system and attempt to design such systems with predefined and acceptable probabilities of failure [21, 22, 23]. Such acceptable values would be comparable to other failure probabilities humans are accepting and facing in other daily activities [21, 22, 23]. Non-parametric systems are often heuristic in nature and should be carefully analyzed in order to ensure their safe and reliable performance under all expected loading conditions. Smart structural systems, as outlined earlier, comprise sets of integrated components which provide added functionalities to the system, as opposed to conventional structural systems. Despite the fact that some of these components might have been proven reliable, in other applications, their reliable performance as an integral component of such a system needs

In order to develop a comprehensive reliability assessment scheme for smart structural systems, a generic reliability assessment framework needs to be defined. The generic framework functions as a blueprint that identifies the reliability assessment procedures and underlying models, functions and measures that are necessary to perform the reliability

Furthermore, it is crucial to develop proper reliability measures and assessment procedures, at two basic levels. First, individual components shall be investigated, given the appropriate failure conditions that are of concern to the application at hand. Second, the overall system, where all underlying components are integrated and aggregated within a predefined limit state format, shall be investigated in order to evaluate an overall reliability of the resulting system. In this chapter, reliability assessment of a fuzzy controller as a component within a smart structural system is explored. The evaluation of an overall reliability of the system, as a whole, is beyond the scope of this chapter and is addressed in other publications [8, 9].

The main objective of this task is to outline a generic reliability assessment framework for evaluating the reliability of a fuzzy controller, as an integral component of a smart structural system. Figure 8 shows the reliability assessment framework for the fuzzy controller, when operating within a smart structural system. The framework identifies two main paths which are necessary to conduct any reliability assessment. The first identifies the main components which are involved in evaluating the commanded output of the controller, while the second identifies another set of components which are responsible for evaluating what would be the expected output of the fuzzy controller. The output of both paths, i.e., commanded output (supply) and expected output (demand), are the basic inputs to any reliability

The reliability assessment framework, when identifying the components involved in each of the referred paths, pinpoints several systems that need to be analytically modeled in order to be able to perform the reliability assessment as necessary. Figure 8 recognizes the following models; a structural model, i.e., finite element model of the system, fuzzy

validation and confirmation [8, 9].

assessment as per the nature of the problem at hand [8, 9].

**4.1. Reliability assessment framework** 

assessment procedure [9, 21, 22, 23].

Therefore, it is important to start the creation of the rule-base with a set of rules that outlines the basic features of the problem at hand, if an analytical model of the system could be developed. Such rules could be generated using time history analysis results of finite element models of the structural systems under consideration [14, 15, 16]. The rule-base should be designed to incorporate a self-learning mechanism that is capable of expanding the current rule-base with newly generated rules that capture any new situations [11, 12]. There are several platforms that are designed to allow the creation of fuzzy controllers. The most widely used of these is the MATLAB environment with its fuzzy logic toolbox. This toolbox allows the user the ability to design fuzzy inference systems for control applications or any other applications, such as pattern classification. The toolbox has a user interface that allows the extraction of rules given input/output data sets of the modeled system. As mentioned earlier this approach is suitable for systems that are ill-defined and are difficult to model analytically. The MATLAB environment allows the creation of a static fuzzy inference system. In other words, the created rule base is static and will not expand to incorporate newly acquired experiences. Therefore, it is advisable to create an m-file that is capable of extracting new experiences and expanding the initial rule-base as the need arises. This is usually encountered when the system is faced with a set of input variables that do not fire any of the generated rules [11, 12]. The designer should define a mechanism whereby an initial rule, that defines the encountered case, is generated and then fine tuned later using a performance monitoring scheme [11, 12].

## **4. Reliability assessment of fuzzy controllers**

Engineering, by nature, is not an exact science. Engineering systems encounter several sources of uncertainties which render such systems subject to potential failures with certain probabilities. It is rather unrealistic to attempt to design a perfect engineering system with a failure probability of zero [21 22, 23]. Yet, it is crucial to be able to evaluate the failure probability of any designed system and attempt to design such systems with predefined and acceptable probabilities of failure [21, 22, 23]. Such acceptable values would be comparable to other failure probabilities humans are accepting and facing in other daily activities [21, 22, 23]. Non-parametric systems are often heuristic in nature and should be carefully analyzed in order to ensure their safe and reliable performance under all expected loading conditions. Smart structural systems, as outlined earlier, comprise sets of integrated components which provide added functionalities to the system, as opposed to conventional structural systems. Despite the fact that some of these components might have been proven reliable, in other applications, their reliable performance as an integral component of such a system needs validation and confirmation [8, 9].

In order to develop a comprehensive reliability assessment scheme for smart structural systems, a generic reliability assessment framework needs to be defined. The generic framework functions as a blueprint that identifies the reliability assessment procedures and underlying models, functions and measures that are necessary to perform the reliability assessment as per the nature of the problem at hand [8, 9].

Furthermore, it is crucial to develop proper reliability measures and assessment procedures, at two basic levels. First, individual components shall be investigated, given the appropriate failure conditions that are of concern to the application at hand. Second, the overall system, where all underlying components are integrated and aggregated within a predefined limit state format, shall be investigated in order to evaluate an overall reliability of the resulting system. In this chapter, reliability assessment of a fuzzy controller as a component within a smart structural system is explored. The evaluation of an overall reliability of the system, as a whole, is beyond the scope of this chapter and is addressed in other publications [8, 9].

## **4.1. Reliability assessment framework**

234 Fuzzy Controllers – Recent Advances in Theory and Applications

later using a performance monitoring scheme [11, 12].

**4. Reliability assessment of fuzzy controllers** 

The heart of a fuzzy controller is its rule-base. The rule-base houses a collection of IF-THEN rules that summarize the knowledge-base that underpins the decisions made by the fuzzy controller [7, 11]. Being a non-parametric heuristic algorithm, fuzzy controllers are built to simulate a human operator's reasoning when facing a similar control situation. In an effort to design smart sustainable structural systems, that are built to be autonomous systems, the developed rule-base should be capable of handling all potential situations that might arise during the system's expected life time. Such controllers should be designed with self learning capabilities such that their initial rule-bases could be amended and expanded as new experiences and/or situations arise. There are currently several applications and toolboxes that allow the automatic extraction of rules of a given problem, knowing the input/output data sets of the problem without the pre-existing knowledge of a model for the system. This approach might be suitable for ill-defined systems. However, in case of structural systems under earthquake excitation, the system behavior is fully defined and could be identified using finite element models under several types of conventional analysis

Therefore, it is important to start the creation of the rule-base with a set of rules that outlines the basic features of the problem at hand, if an analytical model of the system could be developed. Such rules could be generated using time history analysis results of finite element models of the structural systems under consideration [14, 15, 16]. The rule-base should be designed to incorporate a self-learning mechanism that is capable of expanding the current rule-base with newly generated rules that capture any new situations [11, 12]. There are several platforms that are designed to allow the creation of fuzzy controllers. The most widely used of these is the MATLAB environment with its fuzzy logic toolbox. This toolbox allows the user the ability to design fuzzy inference systems for control applications or any other applications, such as pattern classification. The toolbox has a user interface that allows the extraction of rules given input/output data sets of the modeled system. As mentioned earlier this approach is suitable for systems that are ill-defined and are difficult to model analytically. The MATLAB environment allows the creation of a static fuzzy inference system. In other words, the created rule base is static and will not expand to incorporate newly acquired experiences. Therefore, it is advisable to create an m-file that is capable of extracting new experiences and expanding the initial rule-base as the need arises. This is usually encountered when the system is faced with a set of input variables that do not fire any of the generated rules [11, 12]. The designer should define a mechanism whereby an initial rule, that defines the encountered case, is generated and then fine tuned

Engineering, by nature, is not an exact science. Engineering systems encounter several sources of uncertainties which render such systems subject to potential failures with certain probabilities. It is rather unrealistic to attempt to design a perfect engineering system with a

**3.3. Rule-base generation** 

techniques.

The main objective of this task is to outline a generic reliability assessment framework for evaluating the reliability of a fuzzy controller, as an integral component of a smart structural system. Figure 8 shows the reliability assessment framework for the fuzzy controller, when operating within a smart structural system. The framework identifies two main paths which are necessary to conduct any reliability assessment. The first identifies the main components which are involved in evaluating the commanded output of the controller, while the second identifies another set of components which are responsible for evaluating what would be the expected output of the fuzzy controller. The output of both paths, i.e., commanded output (supply) and expected output (demand), are the basic inputs to any reliability assessment procedure [9, 21, 22, 23].

The reliability assessment framework, when identifying the components involved in each of the referred paths, pinpoints several systems that need to be analytically modeled in order to be able to perform the reliability assessment as necessary. Figure 8 recognizes the following models; a structural model, i.e., finite element model of the system, fuzzy

controller's model, inverse dynamics model and an inverse actuator's model. All these components need to be defined and their analytical models developed in order to conduct the reliability assessment procedure. It should be pointed out that any system definition needs to be conducted in a format that lends itself to the reliability assessment calculations [8, 9]. Reliability assessments are related to the failure to supply what is initially demanded from the system. Potential failure modes, the identification of which is one of the first steps in the reliability assessment procedure, define situations where the analyzed system fails to supply and/or provide the required and/or demanded output. These potential failure modes are usually better expressed in a limit state format since this format lends itself to further developments in order to fully conduct the reliability assessment calculations. The following sections outline procedures for creating models for relevant components, identifying potential failure modes, presenting such failure modes in a limit state formats and finally conducting the reliability evaluation of a fuzzy controller.

Fuzzy Controllers: A Reliable Component of Smart Sustainable Structural Systems 237

() () () () *m xt c xt k xt m x t <sup>g</sup>* (6)

capable of creating such models within one of many finite element software packages that are currently available for research and design purposes. The dynamic equation of motion, shown in Eq. 6 outlines the behavior of a single degree of freedom system under the effect of an earthquake forcing function [14, 15, 16]. Given the time history of the applied earthquake, i.e., ( ) *<sup>g</sup> x t* , the solution of the equation shown in Eq. 6 results in the structural performance

Where, m = floor mass; c = damping constant; k = system stiffness; *xt xt xt* ( ), ( ) & ( ) =are acceleration, velocity and displacement of the floor level, respectively and ( ) *<sup>g</sup> x t* = is the

The second component is the fuzzy controller model which was discussed in the previous sections. The development of such a model could be performed within a platform that supports fuzzy logic such as MATLAB among others. The development of such a model requires the full definition of input and output variables, including their relevant membership functions, the selection of a suitable inference engine, including inference function, inference mechanism and aggregation function. Finally, this also includes the generation of the rule-base necessary to perform any inference in order to evaluate the

The remaining models are relevant to the inverse performance of the system which would start with the controlled structural parameters and back calculate the necessary controller output in order to reach that targeted performance [19, 25]. The inverse system comprise two main components an inverse dynamics model and an inverse actuator model. The inverse dynamics model is simply the dynamic equation of motion which models the behavior of a controlled structural system under the effect of an earthquake loading function. Eq. 6 models the dynamic response of the uncontrolled structural system. Eq. 7, however, includes an additional term that reflects the effect of a control force provided by the fuzzy controller [1, 2]. Eq. 7 should start by introducing a set of required structural parameters, these could be the result of a predefined deformed position which would provide the displacement of the floor level, then by using numerical methods, the accompanying velocity and acceleration could be evaluated and substituted in Eq. 7 to

() () () () *required <sup>g</sup> F m xt x t c xt k xt* (7)

Where, m = floor mass; c = damping constant; k = system stiffness; *xt xt xt* ( ), ( ) & ( ) = are acceleration, velocity and displacement of the floor level, respectively and ( ) *<sup>g</sup> x t* = is the ground acceleration and Frequired = required control force. The identification of such predefined deformed shape would be mostly dependent on the nature of the system in

parameters of the modeled system, i.e., *xt xt xt* ( ), ( ) & ( ) [14, 15, 16].

output of the controller, given the values of input variables.

evaluate the required control force.

question, its size and its level of importance.

ground acceleration.

**Figure 8.** Fuzzy Controller Reliability Assessment Framework

## **4.2. Analytical models**

As indicated earlier, the reliability assessment framework outlines the basic components which are involved in the evaluation of the reliability of the modeled system. The framework, shown in Figure 8, identifies four main components that should be analytically modeled in order to perform the reliability assessment procedure. The first component is a finite element model of the structural system which is necessary in order to evaluate the performance parameters of the system under the effect of an earthquake forcing function. Finite element models of structural systems are very well documented and any structural engineer is capable of creating such models within one of many finite element software packages that are currently available for research and design purposes. The dynamic equation of motion, shown in Eq. 6 outlines the behavior of a single degree of freedom system under the effect of an earthquake forcing function [14, 15, 16]. Given the time history of the applied earthquake, i.e., ( ) *<sup>g</sup> x t* , the solution of the equation shown in Eq. 6 results in the structural performance parameters of the modeled system, i.e., *xt xt xt* ( ), ( ) & ( ) [14, 15, 16].

236 Fuzzy Controllers – Recent Advances in Theory and Applications

conducting the reliability evaluation of a fuzzy controller.

**Figure 8.** Fuzzy Controller Reliability Assessment Framework

As indicated earlier, the reliability assessment framework outlines the basic components which are involved in the evaluation of the reliability of the modeled system. The framework, shown in Figure 8, identifies four main components that should be analytically modeled in order to perform the reliability assessment procedure. The first component is a finite element model of the structural system which is necessary in order to evaluate the performance parameters of the system under the effect of an earthquake forcing function. Finite element models of structural systems are very well documented and any structural engineer is

**4.2. Analytical models** 

controller's model, inverse dynamics model and an inverse actuator's model. All these components need to be defined and their analytical models developed in order to conduct the reliability assessment procedure. It should be pointed out that any system definition needs to be conducted in a format that lends itself to the reliability assessment calculations [8, 9]. Reliability assessments are related to the failure to supply what is initially demanded from the system. Potential failure modes, the identification of which is one of the first steps in the reliability assessment procedure, define situations where the analyzed system fails to supply and/or provide the required and/or demanded output. These potential failure modes are usually better expressed in a limit state format since this format lends itself to further developments in order to fully conduct the reliability assessment calculations. The following sections outline procedures for creating models for relevant components, identifying potential failure modes, presenting such failure modes in a limit state formats and finally

$$m\*\ddot{\mathbf{x}}(t) + c\*\dot{\mathbf{x}}(t) + k\*x(t) = -m\*\ddot{\mathbf{x}}\_{\frac{\pi}{2}}(t)\tag{6}$$

Where, m = floor mass; c = damping constant; k = system stiffness; *xt xt xt* ( ), ( ) & ( ) =are acceleration, velocity and displacement of the floor level, respectively and ( ) *<sup>g</sup> x t* = is the ground acceleration.

The second component is the fuzzy controller model which was discussed in the previous sections. The development of such a model could be performed within a platform that supports fuzzy logic such as MATLAB among others. The development of such a model requires the full definition of input and output variables, including their relevant membership functions, the selection of a suitable inference engine, including inference function, inference mechanism and aggregation function. Finally, this also includes the generation of the rule-base necessary to perform any inference in order to evaluate the output of the controller, given the values of input variables.

The remaining models are relevant to the inverse performance of the system which would start with the controlled structural parameters and back calculate the necessary controller output in order to reach that targeted performance [19, 25]. The inverse system comprise two main components an inverse dynamics model and an inverse actuator model. The inverse dynamics model is simply the dynamic equation of motion which models the behavior of a controlled structural system under the effect of an earthquake loading function. Eq. 6 models the dynamic response of the uncontrolled structural system. Eq. 7, however, includes an additional term that reflects the effect of a control force provided by the fuzzy controller [1, 2]. Eq. 7 should start by introducing a set of required structural parameters, these could be the result of a predefined deformed position which would provide the displacement of the floor level, then by using numerical methods, the accompanying velocity and acceleration could be evaluated and substituted in Eq. 7 to evaluate the required control force.

$$F\_{required} = m\*\left[\ddot{\mathbf{x}}(t) + \ddot{\mathbf{x}}\_{\ddagger}(t)\right] + c\*\dot{\mathbf{x}}(t) + k\*x(t) \tag{7}$$

Where, m = floor mass; c = damping constant; k = system stiffness; *xt xt xt* ( ), ( ) & ( ) = are acceleration, velocity and displacement of the floor level, respectively and ( ) *<sup>g</sup> x t* = is the ground acceleration and Frequired = required control force. The identification of such predefined deformed shape would be mostly dependent on the nature of the system in question, its size and its level of importance.

A second inverse model is necessary in order to translate the required control force into a required voltage which is the output of the fuzzy controller. This model should depend on the actuator, which is proposed in the application under consideration. For the proposed system, an MR damper is employed in applying any required corrective actions to enhance the system's response. MR dampers have very well documented models that outline the relationship between the input voltage to the damper and the resulting force, given the damper parameters and response parameters of the controlled system [26, 27]. The modified Buc-Wen model is the most widely used and accepted within the community of smart materials [26, 27]. The outlined model requires input data relating to the response of the system, i.e., displacement (x) and velocity ( *x* ) , in addition to the applied voltage (v), in order to evaluate the MR damper force that would be applied to the system [26, 27, 28].

#### **4.3. Potential failure modes**

The evaluation of the supplied and demanded fuzzy controller output values is the first step in the reliability assessment calculations. Potential failure modes should be formulated, in order to define situations where the supplied output variable might not satisfy the demanded requirements and thus constitutes a failure condition for the fuzzy controller [9, 19]. The failure modes are formulated in a limit state format in order to lend themselves to the reliability calculations that follow [9, 19]. In order to demonstrate the development of such failure conditions, two potential failure modes are explored for the proposed fuzzy controller. The first is a CRASH failure where the controller fails to produce any voltage signal, i.e., output value, to the MR Damper. The second is a MALFUNCTION failure where the controller produces an inaccurate voltage signal to the MR Damper. The reasons for each failure condition should be explored and all potential situations should be considered in evaluating a representative estimate of the reliability of the system [9, 19, 28].

When evaluating failure conditions all potential situations resulting in such a failure shall be considered and included in the probability of failure to reflect the level of uncertainties involved in the problem [21, 22, 23, 28]. The probability of failure of the controller is, then, evaluated using a Monte-Carlo simulation algorithm where the probability of failure for any given simulation cycle is calculated through the definition of a corresponding limit state as follows [19, 21, 22, 23, 28];

$$LS\_1 = \frac{V\_{\text{sup}\,p\,lied}}{V\_{required}} < \lambda\_1 \tag{8}$$

Fuzzy Controllers: A Reliable Component of Smart Sustainable Structural Systems 239

system's behavior. For the purposes of demonstration if a value of 0.3 is assumed, this means that if the fuzzy controller proposed an output which is less than 30% of the expected value, this controller is considered to have crashed and is not functioning as expected. The reasons for such failure could be due to the lack of relevant rules that handles the set of input values presented to the fuzzy controller. If the rule-base is designed with the ability to expand and learn from experiences, this failure should trigger the creation of additional

In case of a MALFUNCTION failure, which is defined as an inaccurate controller output, a single limit can't properly define such a failure condition and as a result two limits need to be defined. The limits would define an acceptable range within which the output is expected to fall. If the controller's output is below or above that range, the controller is considered to have malfunctioned [28]. This type of failure is addressed by fine tuning currently existing rules within the rule-base. Therefore, in such a case two limits are necessary in order to fully define the failure condition, i.e., a lower limit and an upper limit. Two underlying failure conditions result and two

> sup 21 21

*V*

*V V*

*V*

( ) *fi <sup>N</sup> P t*

Where, Pfi(t) = is the probability of failure of the ith limit state, at any given point in time, N<sup>λ</sup> = is the number of simulation cycles where the processor output resulted in a failure condition, depending on the failure condition in reference to Eqs. (8) & (9), and N = is the

The overall probability of failure of the whole processor should be evaluated, taking into consideration all potential failure combinations. This is accomplished by applying a union operator to evaluate the probability of failure of a single limit state with several underlying scenarios, as well as, the overall probability of failure considering all potential limit states.

*N* 

(10)

*plied required plied required*

(9)

sup 22 22

Where, LS21 and LS22 = are the limit state equations for scenarios 1 & 2 of the second failure mode, Vsupplied = is the supplied voltage command, Vrequired = is the voltage demand as evaluated by the inverse models and λ21 and λ22 = are lower and upper cut off limits respectively. Such limits are functions of the type of problem and relevant failure modes and resulting practical implications [28]. Monte-Carlo simulatin could be employed in generating values for all random variables which are involved in the problem at hand. Thus, each simulation cycle will result in a value for Vsupplied and Vrequired and a corresponding evaluation of LSi. The probability of failure for a given limit state, at any given point in time,

limit state equations could be written to express this failure as shown in Eqs. 9 [28];

*LS*

*LS*

is then evaluated using the following equation [9, 19, 21, 22, 23];

total number of simulation cycles.

rules that are capable of handling such a situation [11, 12].

Where, LS1 = is the limit state equation for the first failure mode, Vsupplied = is the supplied voltage command, Vrequired = is the voltage demand as evaluated by the inverse models and λ1 = is a cut off limit which defines when the supplied voltage is considered out of range. In case of a CRASH failure there is a single cutoff limit that defines when the controller is considered to have produced an insignificant output. These ranges are problem dependant and should be evaluated based on practical experience and the knowledge of the modeled system's behavior. For the purposes of demonstration if a value of 0.3 is assumed, this means that if the fuzzy controller proposed an output which is less than 30% of the expected value, this controller is considered to have crashed and is not functioning as expected. The reasons for such failure could be due to the lack of relevant rules that handles the set of input values presented to the fuzzy controller. If the rule-base is designed with the ability to expand and learn from experiences, this failure should trigger the creation of additional rules that are capable of handling such a situation [11, 12].

238 Fuzzy Controllers – Recent Advances in Theory and Applications

**4.3. Potential failure modes** 

follows [19, 21, 22, 23, 28];

sup

A second inverse model is necessary in order to translate the required control force into a required voltage which is the output of the fuzzy controller. This model should depend on the actuator, which is proposed in the application under consideration. For the proposed system, an MR damper is employed in applying any required corrective actions to enhance the system's response. MR dampers have very well documented models that outline the relationship between the input voltage to the damper and the resulting force, given the damper parameters and response parameters of the controlled system [26, 27]. The modified Buc-Wen model is the most widely used and accepted within the community of smart materials [26, 27]. The outlined model requires input data relating to the response of the system, i.e., displacement (x) and velocity ( *x* ) , in addition to the applied voltage (v), in order to evaluate the MR damper force that would be applied to the system [26, 27, 28].

The evaluation of the supplied and demanded fuzzy controller output values is the first step in the reliability assessment calculations. Potential failure modes should be formulated, in order to define situations where the supplied output variable might not satisfy the demanded requirements and thus constitutes a failure condition for the fuzzy controller [9, 19]. The failure modes are formulated in a limit state format in order to lend themselves to the reliability calculations that follow [9, 19]. In order to demonstrate the development of such failure conditions, two potential failure modes are explored for the proposed fuzzy controller. The first is a CRASH failure where the controller fails to produce any voltage signal, i.e., output value, to the MR Damper. The second is a MALFUNCTION failure where the controller produces an inaccurate voltage signal to the MR Damper. The reasons for each failure condition should be explored and all potential situations should be considered in

When evaluating failure conditions all potential situations resulting in such a failure shall be considered and included in the probability of failure to reflect the level of uncertainties involved in the problem [21, 22, 23, 28]. The probability of failure of the controller is, then, evaluated using a Monte-Carlo simulation algorithm where the probability of failure for any given simulation cycle is calculated through the definition of a corresponding limit state as

> 1 1 *plied required*

(8)

*<sup>V</sup>*

Where, LS1 = is the limit state equation for the first failure mode, Vsupplied = is the supplied voltage command, Vrequired = is the voltage demand as evaluated by the inverse models and λ1 = is a cut off limit which defines when the supplied voltage is considered out of range. In case of a CRASH failure there is a single cutoff limit that defines when the controller is considered to have produced an insignificant output. These ranges are problem dependant and should be evaluated based on practical experience and the knowledge of the modeled

*V*

*LS*

evaluating a representative estimate of the reliability of the system [9, 19, 28].

In case of a MALFUNCTION failure, which is defined as an inaccurate controller output, a single limit can't properly define such a failure condition and as a result two limits need to be defined. The limits would define an acceptable range within which the output is expected to fall. If the controller's output is below or above that range, the controller is considered to have malfunctioned [28]. This type of failure is addressed by fine tuning currently existing rules within the rule-base. Therefore, in such a case two limits are necessary in order to fully define the failure condition, i.e., a lower limit and an upper limit. Two underlying failure conditions result and two limit state equations could be written to express this failure as shown in Eqs. 9 [28];

$$\begin{aligned} LS\_{z1} &= \frac{V\_{supplid}}{V\_{required}} < \lambda\_{z1} \\ LS\_{zz} &= \frac{V\_{supplid}}{V\_{required}} > \lambda\_{z2} \end{aligned} \tag{9}$$

Where, LS21 and LS22 = are the limit state equations for scenarios 1 & 2 of the second failure mode, Vsupplied = is the supplied voltage command, Vrequired = is the voltage demand as evaluated by the inverse models and λ21 and λ22 = are lower and upper cut off limits respectively. Such limits are functions of the type of problem and relevant failure modes and resulting practical implications [28]. Monte-Carlo simulatin could be employed in generating values for all random variables which are involved in the problem at hand. Thus, each simulation cycle will result in a value for Vsupplied and Vrequired and a corresponding evaluation of LSi. The probability of failure for a given limit state, at any given point in time, is then evaluated using the following equation [9, 19, 21, 22, 23];

$$P\_{\circ}(t) = \frac{N\_{\times}}{N} \tag{10}$$

Where, Pfi(t) = is the probability of failure of the ith limit state, at any given point in time, N<sup>λ</sup> = is the number of simulation cycles where the processor output resulted in a failure condition, depending on the failure condition in reference to Eqs. (8) & (9), and N = is the total number of simulation cycles.

The overall probability of failure of the whole processor should be evaluated, taking into consideration all potential failure combinations. This is accomplished by applying a union operator to evaluate the probability of failure of a single limit state with several underlying scenarios, as well as, the overall probability of failure considering all potential limit states.

The probability of failure of both limit states described in Eq. 9 could be evaluated as [19, 21, 22, 23, 24, 28];

$$\begin{aligned} P\_{f\_{21}}(t) &= \frac{N\_{>21}}{N} \\ P\_{f\_{22}}(t) &= \frac{N\_{>21}}{N} \\ P\_{f\_{21}}(t) &= P\left\{ LS\_{21} \cup LS\_{22} \right\} = P\_{f\_{21}}(t) + P\_{f\_{22}}(t) - \left(P\_{f\_{21}}(t) \ \ ^\ast P\_{f\_{22}}(t)\right) \end{aligned} \tag{11}$$

Fuzzy Controllers: A Reliable Component of Smart Sustainable Structural Systems 241

The resulting probability of failure is an instantaneous probability due to the time dependant nature of the problem under consideration. Therefore, a reliability time history could be developed to reflect the time variation of the controller reliability during operation under the effect of a real earthquake event. Figure 10, shows the step by step calculation procedure for evaluating a reliability time history for the fuzzy controller. Figure 11, shows a sample reliability time history diagram. The time history diagram is helpful in allowing the user to visualize the performance of the controller during a real time event and thus identifying events where the performance was unacceptable. The indentified time step where the controller failed to satisfy its expected performance could help in pinpointing the

> Supplied Voltage (Vsupplied)

Other Limit States Pfi

MALFUNCTION Failure

> Other Failure Modes

MALFUNCTION Failure

> Other Failure Modes

In reference to Figure 2, a multi-degree-of-freedom system would require an enhanced fuzzy controller in order to properly suppress any undesirable response of structural systems under earthquake loadings. One of the important enhancements that could be integrated with the controller is a fuzzy pattern identifier [17, 20, 24]. Sustainable Structural systems, as defined above, are bound by three basic characteristics, i.e., recyclable, optimum and reliable. Optimum design of structural systems entails both the minimum amount of material to construct the system itself, in addition to the optimum use of energy resources and any integral elements that are designed to suppress any undesirable responses. By that

CRASH Failure

CRASH Failure

Pf1 Pf21 Pf22

> Required Voltage (Vrequired)

Pf2

Potential Failure Modes

Required Voltage Vrequired

Limit State (1) Vsupplied/Vrequired < λ? Limit State (2-1) Vsupplied/Vrequired < λ<sup>21</sup> Limit State (2-2) Vsupplied/Vrequired >λ<sup>22</sup>

Potential Failure Modes

Limit State (1) Vsupplied/Vrequired < λ<sup>1</sup> Limit State (2-1) Vsupplied/Vrequired < λ<sup>21</sup> Limit State (2-2) Vsupplied/Vrequired >λ<sup>22</sup>

Other Limit States

Limit State (1) Vsupplied/Vrequired < λ? Limit State (2-1) Vsupplied/Vrequired < λ<sup>21</sup> Limit State (2-2) Vsupplied/Vrequired >λ<sup>22</sup>

MALFUNCTION Failure

> Other Failure Modes

CRASH Failure

Required Voltage (Vrequired)

Supplied Voltage (Vsupplied)

( ) ( )

Potential Failure Modes

Other Limit States Pfi

Pf1 Pf21 Pf22

Supplied Voltage (Vsupplied)

Other Limit States Pfi

Pf2

Pf1 Pf21 Pf22

Pf2

Pf2

Pfi

Pf

Pf

Supplied Voltage Vsupplied

Pf1 Pf21 Pf22 Pf

Pf

Potential Failure Modes

λ1

Limit State (1) Vsupplied/Vrequired < Limit State (2-1) Vsupplied/Vrequired < λ<sup>21</sup> Limit State (2-2) Vsupplied/Vrequired >λ<sup>22</sup>

reasons for such unreliable behavior.

Tn

Required Voltage (Vrequired)

T1

T2

T3

**Figure 10.** Reliability Time History Evaluation

**5. Enhanced fuzzy controllers** 

**5.1. Introduction** 

MALFUNCTION Failure

> Other Failure Modes

CRASH Failure

Where, Pf2(t) = is the probability of failure of condition (2), LS21 and LS22 = are underlying limit states as defined in Eq. (11), Pf21(t) & Pf22(t) = are the probabilities of failure of underlying limit states (22 and 21), Nλ21 and Nλ22 = are the number of cycles where the processor output resulted in a failure conditions, i.e., < λ21 and > λ22 respectively in reference to Eq. 9, and N = is the total number of simulation cycles. In order to evaluate the overall probability of failure of the processor, taking into consideration all potential limit states, this could be defined as [19, 21, 22, 23, 24, 28];

$$P\_f(t) = P\left(LS\_1 \cup LS\_2 \cup \dots \cup LS\_i\right) = P\_{f1}(t) + P\_{f2}(t) + \dots + P\_{\hat{\rho}}(t) - \left(P\_{f1}(t) \,^\*P\_{f2}(t) \,^\*\dots \,^\*P\_{\hat{\rho}}(t)\right) \tag{12}$$

Where, Pf(t) = is the overall probability of failure of the processor, LS1, LS2 and LSi = are potential limit states as defined in Eqs. (8 & 9), Pf1(t), Pf2(t) and Pfi(t) = are relevant probabilities of failure of limit states, as defined in Eqs. (10 & 11). The above calculations are performed at a given point in time Ti, which results in an instantaneous reliability measure Pf(t) for the fuzzy controller. Figure 9, shows a block diagram for the instantaneous reliability calculation procedure, taking into consideration all potential failure modes.

**Figure 9.** Instantaneous Reliability Evaluation

The resulting probability of failure is an instantaneous probability due to the time dependant nature of the problem under consideration. Therefore, a reliability time history could be developed to reflect the time variation of the controller reliability during operation under the effect of a real earthquake event. Figure 10, shows the step by step calculation procedure for evaluating a reliability time history for the fuzzy controller. Figure 11, shows a sample reliability time history diagram. The time history diagram is helpful in allowing the user to visualize the performance of the controller during a real time event and thus identifying events where the performance was unacceptable. The indentified time step where the controller failed to satisfy its expected performance could help in pinpointing the reasons for such unreliable behavior.

**Figure 10.** Reliability Time History Evaluation

## **5. Enhanced fuzzy controllers**

## **5.1. Introduction**

240 Fuzzy Controllers – Recent Advances in Theory and Applications

21

*f*

( )

*<sup>N</sup> P t*

( )

*<sup>N</sup> P t*

22

*f*

could be defined as [19, 21, 22, 23, 24, 28];

Required Voltage (Vrequired)

**Figure 9.** Instantaneous Reliability Evaluation

MALFUNCTION Failure

Other Failure Modes

CRASH Failure

21

*N*

*N*

22

22, 23, 24, 28];

The probability of failure of both limit states described in Eq. 9 could be evaluated as [19, 21,

2 21 22 21 22 21 22

Where, Pf2(t) = is the probability of failure of condition (2), LS21 and LS22 = are underlying limit states as defined in Eq. (11), Pf21(t) & Pf22(t) = are the probabilities of failure of underlying limit states (22 and 21), Nλ21 and Nλ22 = are the number of cycles where the processor output resulted in a failure conditions, i.e., < λ21 and > λ22 respectively in reference to Eq. 9, and N = is the total number of simulation cycles. In order to evaluate the overall probability of failure of the processor, taking into consideration all potential limit states, this

*P t P LS LS P t P t P t P t*

*f f f ff*

( ) ( ) ( ) ( )\* ( )

*P t P LS LS LS P t P t P t P t P t P t <sup>f</sup>* ( ) 1 2 *i f* 1 2 ( ) ( ) ( ) ( )\* ( )\* \* ( ) *<sup>f</sup> fi <sup>f</sup>* 1 2*<sup>f</sup> fi* (12)

Supplied Voltage (Vsupplied)

Pf1(t)

Pf21(t)

Pf22(t)

Other Limit States Pfi(t)

Where, Pf(t) = is the overall probability of failure of the processor, LS1, LS2 and LSi = are potential limit states as defined in Eqs. (8 & 9), Pf1(t), Pf2(t) and Pfi(t) = are relevant probabilities of failure of limit states, as defined in Eqs. (10 & 11). The above calculations are performed at a given point in time Ti, which results in an instantaneous reliability measure Pf(t) for the fuzzy controller. Figure 9, shows a block diagram for the instantaneous reliability calculation procedure, taking into consideration all potential failure modes.

Potential Failure Modes

Vsupplied/Vrequired < λ<sup>1</sup>

Vsupplied/Vrequired < λ<sup>21</sup>

Vsupplied/Vrequired >λ<sup>22</sup>

Limit State (1)

Limit State (2-1)

Limit State (2-2)

(11)

Pf2(t)

Pf(t)

In reference to Figure 2, a multi-degree-of-freedom system would require an enhanced fuzzy controller in order to properly suppress any undesirable response of structural systems under earthquake loadings. One of the important enhancements that could be integrated with the controller is a fuzzy pattern identifier [17, 20, 24]. Sustainable Structural systems, as defined above, are bound by three basic characteristics, i.e., recyclable, optimum and reliable. Optimum design of structural systems entails both the minimum amount of material to construct the system itself, in addition to the optimum use of energy resources and any integral elements that are designed to suppress any undesirable responses. By that

it is meant the actuators which are integrated within the system. As defined in Figures 1 and 2, these are selected as MR dampers. The designed fuzzy controller should comprise a scheme whereby an optimum firing procedure for such dampers is employed. Such scheme would rely on information relevant to the deformed shape of the system in order to select only those dampers which could significantly affect the response and thus the deformed shape of the system.

Fuzzy Controllers: A Reliable Component of Smart Sustainable Structural Systems 243

appropriate pattern classification, based on the displacements of individual degrees of freedom, i.e., displacements of individual floors, could be created to drive the fuzzy inference

In order to design a fuzzy inference system that is capable of classifying the deformed pattern of any structural system, a set of potential predefined pattern classifications need to be developed. Such pattern classifications are dependent on the modeled system, its size and its behavior under expected loading conditions. Careful analysis of the modeled system could result in creating such pattern classifications. Figure 12, shows a sample of such potential deformed patterns for a three-degree-of-freedom system. The figure is, by no means, comprehensive, i.e., these are some of the potential deformed patterns a three-

Once such pattern classifications are defined, a fuzzy inference system could be designed in order to classify any similar system into one of the predefined patterns. The fuzzy inference system would rely on a relevant rule-base that would accept the input of the individual floor displacements and assign the appropriate pattern classification accordingly. A sample

> 

Where, LEVEL1, LEVEL2 & LEVEL3 = are fuzzy variables that define the sensor data of relevant story levels, Positive = is a fuzzy value of positive displacement, PATTERN = is the output variable of the fuzzy inference system, 1 = is a fuzzy singleton that defines the

AND LEVEL3 is Positive THEN PATTERN is 1 (13)

IF LEVEL1 is Positive AND LEVEL2 is Positive

system. The following section outlines the main design of a fuzzy pattern identifier.

**5.2. Fuzzy abstract deformed shape identifiers** 

degree-of-freedom system could undergo [17, 20, 24].

**Figure 12.** Sample Potential Asbtract Deformed Patterns

rule could be defined as follows [17, 20, 24]:

**Figure 11.** Sample Reliability Time History Diagram

Figure 2 defines an enhanced fuzzy controller where it accepts three input variables instead of only two as discussed earlier. The third variable relates to the abstract deformed shape of the system [17, 20, 24]. This additional piece of information would help in the selection of the firing sequence of the MR dampers not just how much restoring force they are called upon to produce. Thus, the need for an additional smart component, to operate integrally with the fuzzy controller, that is capable of classifying the deformed pattern of the system, given the sensor data relevant to the actual position of control points, i.e., degrees of freedom of the system.

A fuzzy inference system, comprising the same basic components of fuzzy controllers, could be designed in order to perform the required pattern classification task [17, 20, 24]. The fuzzy inference system should employ the gathered sensor data in testing the closeness of the deformed shape of the system to predefined abstract deformed patterns that are relevant to the modeled system. An inference engine built on top of a rule-base that is used to assign the appropriate pattern classification, based on the displacements of individual degrees of freedom, i.e., displacements of individual floors, could be created to drive the fuzzy inference system. The following section outlines the main design of a fuzzy pattern identifier.

## **5.2. Fuzzy abstract deformed shape identifiers**

242 Fuzzy Controllers – Recent Advances in Theory and Applications

**Figure 11.** Sample Reliability Time History Diagram

freedom of the system.

0.0

0.1

0.2

**Probability of Failure**

0.3

0.4

0.5

shape of the system.

it is meant the actuators which are integrated within the system. As defined in Figures 1 and 2, these are selected as MR dampers. The designed fuzzy controller should comprise a scheme whereby an optimum firing procedure for such dampers is employed. Such scheme would rely on information relevant to the deformed shape of the system in order to select only those dampers which could significantly affect the response and thus the deformed

Figure 2 defines an enhanced fuzzy controller where it accepts three input variables instead of only two as discussed earlier. The third variable relates to the abstract deformed shape of the system [17, 20, 24]. This additional piece of information would help in the selection of the firing sequence of the MR dampers not just how much restoring force they are called upon to produce. Thus, the need for an additional smart component, to operate integrally with the fuzzy controller, that is capable of classifying the deformed pattern of the system, given the sensor data relevant to the actual position of control points, i.e., degrees of

0 5 10 15 20 25 30

**Time (Seconds)**

A fuzzy inference system, comprising the same basic components of fuzzy controllers, could be designed in order to perform the required pattern classification task [17, 20, 24]. The fuzzy inference system should employ the gathered sensor data in testing the closeness of the deformed shape of the system to predefined abstract deformed patterns that are relevant to the modeled system. An inference engine built on top of a rule-base that is used to assign the In order to design a fuzzy inference system that is capable of classifying the deformed pattern of any structural system, a set of potential predefined pattern classifications need to be developed. Such pattern classifications are dependent on the modeled system, its size and its behavior under expected loading conditions. Careful analysis of the modeled system could result in creating such pattern classifications. Figure 12, shows a sample of such potential deformed patterns for a three-degree-of-freedom system. The figure is, by no means, comprehensive, i.e., these are some of the potential deformed patterns a threedegree-of-freedom system could undergo [17, 20, 24].

**Figure 12.** Sample Potential Asbtract Deformed Patterns

Once such pattern classifications are defined, a fuzzy inference system could be designed in order to classify any similar system into one of the predefined patterns. The fuzzy inference system would rely on a relevant rule-base that would accept the input of the individual floor displacements and assign the appropriate pattern classification accordingly. A sample rule could be defined as follows [17, 20, 24]:

$$\begin{aligned} & \text{IF (LEVEL1 is Positive) AND (LEVEL2 is Positive)}\\ & \text{AND (LEVEL3 is Positive) THEN (PATTERN is 1)} \end{aligned} \tag{13}$$

Where, LEVEL1, LEVEL2 & LEVEL3 = are fuzzy variables that define the sensor data of relevant story levels, Positive = is a fuzzy value of positive displacement, PATTERN = is the output variable of the fuzzy inference system, 1 = is a fuzzy singleton that defines the assigned pattern, AND = is the logical operator. In order to ensure the generality of the developed inference system, the pattern classifier should employ a normalized value of the sensor data rather than the actual displacement of the control point. The sensor data is normalized with respect to its maximum input value in order to result in input values within the interval [-1, 1]. In order to classify the abstract shape of a system it is only necessary to identify the relative position of control points with respect to each other rather than their absolute actual position. The normalization function could be defined as follows [17]:

$$\mathbf{L}\_{\circ}^{N}\left(t\right) = \frac{\mathbf{L}\_{\circ}\left(t\right)}{\mathbf{M}\mathbf{A}\mathbf{X}\mathbf{X}\left(\mathbf{L}\_{\circ}\left(\mathbf{t}\right), \mathbf{L}\_{\circ}\left(\mathbf{t}\right), \mathbf{L}\_{\circ}\left(\mathbf{t}\right)\right)}\tag{14}$$

Fuzzy Controllers: A Reliable Component of Smart Sustainable Structural Systems 245

Target Pattern Classification (Y)

**Figure 13.** Linear Compliance Graph

In this chapter sustainable smart structural systems were presented as those which are constructed of recyclable materials, are optimally designed and demonstrate a reliable performance. As such, systems are equipped with fuzzy controllers that allow the structural systems to adjust their response under the effect of highly uncertain loading conditions, i.e., earthquakes. If such loads were considered in the design process it would have resulted in very heavy designs. Moreover, such earthquakes might not even occur during the expected life time of the designed systems. However, if any of these systems is equipped with smart features that adjust its response under the effect of any unseen loading conditions, it would be much lighter, safe and it should be reliable. In addition, the smart characteristics of such systems would minimize, if not eliminate, the amount of damage and destruction and thus the amount of waste, if failures did take place due to the occurrence of unseen earthquake

**6. Conclusions** 

events.

Where, <sup>N</sup> Li *t* = is a normalized sensor output at ith degree of freedom at a given point in time; Li *t* = is the actual sensor output at ith degree of freedom at a given point of time; L t ,L t ,L t <sup>123</sup> = are the three actual sensor outputs at floor levels 1, 2 and 3 respectively and MAX() = is the maximum operator performed over all time instants of a given earthquake record. Therefore, the rule outlined in Eq. 13 should be rewritten to reflect the normalization function as follows;

 NN N 12 3 IF L is Positive AND L is Positive AND L is Positive THEN PATTERN is 1 *tt t* (15)

Where, <sup>N</sup> L1 *t* , <sup>N</sup> L2 *t* and <sup>N</sup> L3 *t* = are the three normalized sensor outputs which identify the displacement, at each individual story level, at any given point in time; and all other variables are as defined above. The integral structure of the enhanced fuzzy controller results in a dual fuzzy processor where its performance and thus its reliability are dependent on the performance of both components [19]. Referring to Figure 2, it is clear that the fuzzy controller accepts inputs from the fuzzy pattern identifier; therefore, new failure conditions for the dual processor should be developed taking into consideration the possibility of the fuzzy pattern identifier providing inaccurate information to the fuzzy controller and thus causing it to fail [19]. Other failure conditions may arise due to the failure of the pattern identifier to evaluate a pattern, given the data that was provided. All such potential failure conditions should be carefully considered when evaluating the reliability of the dual fuzzy processor [19].

The performance of the fuzzy pattern identifier, as for similar pattern classification algorithms, is usually measured by plotting a linear compliance graph and using its geometrical and statistical information in evaluating the system's performance [17, 20]. Figure 13 shows a sample linear compliance graph. The graph is developed using a plot of the target classifications against the fuzzy pattern identifier classifications then generating a trendline of the plotted data [17]. If the trendline has a zero intercept and has a slope of unity, this implies that the fuzzy pattern identifier has perfectly assigned the proper pattern at all time instances. Therefore, the actual slope of the trendline and how close is the plotted data to that line, both are considered acceptable measures of the performance of the fuzzy pattern identifier and could be used in evaluating its reliability [17].

**Figure 13.** Linear Compliance Graph

## **6. Conclusions**

244 Fuzzy Controllers – Recent Advances in Theory and Applications

[17]:

assigned pattern, AND = is the logical operator. In order to ensure the generality of the developed inference system, the pattern classifier should employ a normalized value of the sensor data rather than the actual displacement of the control point. The sensor data is normalized with respect to its maximum input value in order to result in input values within the interval [-1, 1]. In order to classify the abstract shape of a system it is only necessary to identify the relative position of control points with respect to each other rather than their absolute actual position. The normalization function could be defined as follows

> <sup>N</sup> <sup>i</sup>

<sup>123</sup> for all t

12 3 IF L is Positive AND L is Positive AND L is Positive THEN PATTERN is 1 *tt t* (15)

Where, <sup>N</sup> L1 *t* , <sup>N</sup> L2 *t* and <sup>N</sup> L3 *t* = are the three normalized sensor outputs which identify the displacement, at each individual story level, at any given point in time; and all other variables are as defined above. The integral structure of the enhanced fuzzy controller results in a dual fuzzy processor where its performance and thus its reliability are dependent on the performance of both components [19]. Referring to Figure 2, it is clear that the fuzzy controller accepts inputs from the fuzzy pattern identifier; therefore, new failure conditions for the dual processor should be developed taking into consideration the possibility of the fuzzy pattern identifier providing inaccurate information to the fuzzy controller and thus causing it to fail [19]. Other failure conditions may arise due to the failure of the pattern identifier to evaluate a pattern, given the data that was provided. All such potential failure conditions should be carefully considered when evaluating the

The performance of the fuzzy pattern identifier, as for similar pattern classification algorithms, is usually measured by plotting a linear compliance graph and using its geometrical and statistical information in evaluating the system's performance [17, 20]. Figure 13 shows a sample linear compliance graph. The graph is developed using a plot of the target classifications against the fuzzy pattern identifier classifications then generating a trendline of the plotted data [17]. If the trendline has a zero intercept and has a slope of unity, this implies that the fuzzy pattern identifier has perfectly assigned the proper pattern at all time instances. Therefore, the actual slope of the trendline and how close is the plotted data to that line, both are considered acceptable measures of the performance of the fuzzy

Where, <sup>N</sup> Li *t* = is a normalized sensor output at ith degree of freedom at a given point in time; Li *t* = is the actual sensor output at ith degree of freedom at a given point of time; L t ,L t ,L t <sup>123</sup> = are the three actual sensor outputs at floor levels 1, 2 and 3 respectively and MAX() = is the maximum operator performed over all time instants of a given earthquake record. Therefore, the rule outlined in Eq. 13 should be rewritten to reflect the

MAX(L t ,L t ,L t ) *t*

*t* (14)

L

i

NN N

pattern identifier and could be used in evaluating its reliability [17].

L

normalization function as follows;

reliability of the dual fuzzy processor [19].

In this chapter sustainable smart structural systems were presented as those which are constructed of recyclable materials, are optimally designed and demonstrate a reliable performance. As such, systems are equipped with fuzzy controllers that allow the structural systems to adjust their response under the effect of highly uncertain loading conditions, i.e., earthquakes. If such loads were considered in the design process it would have resulted in very heavy designs. Moreover, such earthquakes might not even occur during the expected life time of the designed systems. However, if any of these systems is equipped with smart features that adjust its response under the effect of any unseen loading conditions, it would be much lighter, safe and it should be reliable. In addition, the smart characteristics of such systems would minimize, if not eliminate, the amount of damage and destruction and thus the amount of waste, if failures did take place due to the occurrence of unseen earthquake events.

Smart sustainable structural systems were presented as a simple single-degree-of-freedom system, then, a more complex system was considered. In case of single-degree-of-freedom systems, fuzzy controllers with two input variables and single output variables were discussed. However, in case of more complex systems, the notion of a dual fuzzy processor where a fuzzy pattern identifier feeds additional information to the fuzzy controller was presented. Fuzzy inference systems were discussed in relation to the type of membership functions to be employed in similar applications and the method of generating the necessary rule-bases.

Fuzzy Controllers: A Reliable Component of Smart Sustainable Structural Systems 247

[6] Choi, K-M., Cho, S-W., Jung, H-J. and Lee, I-W. (2004) Semi-Active Fuzzy Control for Seismic Response Reduction using Magneto-rheological Dampers. Earthquake Engng.

[7] Klir, G. J. and Folger T. A. (1988) Fuzzy Sets, Uncertainty, and Information, Prentice Hall, New

[8] Hassan, M.H.M. (2006) A System Model for Reliability Assessment of Smart Structural Systems. Structural Engineering & Mechanics, An International Journal (Techno Press)

[9] Hassan, M.H.M. (2005) Reliability Evaluation of Smart Structural Systems. IMECE2005, ASME International Mechanical Engineering Congress & Exposition, November 5-11,

[10] Hassan, M.H.M, Ayyub, B.M. and Bernold, L. (1991) Fuzzy-based real-time control of construction activities. in Analysis and Management of Uncertainty: Theory and

[11] Ayyub, B.M. and Hassan, M.H.M. (1992) Control of Construction Activities: III. Fuzzy-

[12] Hassan, M.H.M. and Ayyub, B.M. (1997) Structural fuzzy control. in Uncertainty Modeling in Vibration, Control and Fuzzy Analysis of Structural Systems, Edited by

[14] Chopra, A.K. (2007) Dynamics of Structures, Theory and Applications of Earthquake

[15] Paz, M. and Leigh, W. (2004) Structural Dynamics, Theory and Computation, Springer. [16] Hart, G.C. and Wong, K., (2000) Structural Dynamics for Structural Engineers, John

[17] Hassan, M.H.M. (2011) FIS Model of an Abstract Shape Identifier for Structural

[18] Ang, A.H-S., and Tang, W.H. (2007) Probability Concepts in Engineering Emphasis on Applications to Civil and Enviroenmental Engineering , John Wiley & Sons, Inc., New

[19] Hassan, M.H.M. (2012) Reliable Smart Structural Control. 5th European Conference on

[20] Hassan, M.H.M. (2012) Real-Time Smart Shape Identifiers. 4th International Conference on Smart Materials, Strutcures & Systems, CIMTEC 2012, Montecantini Terme, Italy,

[21] Ditlevsen, O. and Madsen, H.O., (1996) Structural Reliability Methods, John Wiley and

[22] Haldar, A. and Mahadevan, S., (2000) Probability, Reliability and Statistical Methods in

[23] Nowak, A.S., and Collins, K.R. (2000) Reliability of Structures McGraw-Hill, Inc.,

Ayyub, B.M., Guran, A. and Haldar, A. World Scientific, Chapter 7, pp. 179-231. [13] Hassan M.H.M. and Ayyub B.M., (1993) A fuzzy controller for construction activities.

Applications, Edited by Ayyub, Gupta and Kanal, Elsevier, pp. 331-349

Based Controller. Civil Engineering Systems vol. 9, pp. 275-297.

Fuzzy Sets and Systems, Vol. 5, No. 3, pp. 253-271.

Engineering, Pearson, Prentice Hall, New Jersey.

Structural Control, EACS 2012, Genoa, Italy, June 18-20.

Engineering Design, John Wiley & Sons, INC., New York.

Wiley & Sons. Inc., New York.

Systems. In Review.

York.

June 10-14.

Boston.

Sons, New York.

Struct. Dyn., 33, p. 723-736.

23, no. 5, pp. 455 - 468.

Orlando, Florida USA.

Jersey.

The reliability of such non-parametric systems is of major concern and thus a reliability assessment framework for evaluating the reliability of fuzzy controllers was presented. Potential failure conditions and limit state equations were presented as the basic tool of formulating the reliability problem of a fuzzy controller. The reliability evaluations were performed instantaneously then a reliability time history was created to suit the time dependent nature of the problem at hand.

Finally, the concept of a fuzzy pattern identifier was presented using a fuzzy inference system which would be coupled with the fuzzy controller to form a dual fuzzy processor. Such structure is necessary in case of complex structural systems where the basic information of the dynamics of control points would not be enough to fully define the problem for the controller to formulate proper decisions.

## **Author details**

Maguid H. M. Hassan *British University in Egypt (BUE), Cairo, Egypt* 

## **7. References**


[6] Choi, K-M., Cho, S-W., Jung, H-J. and Lee, I-W. (2004) Semi-Active Fuzzy Control for Seismic Response Reduction using Magneto-rheological Dampers. Earthquake Engng. Struct. Dyn., 33, p. 723-736.

246 Fuzzy Controllers – Recent Advances in Theory and Applications

dependent nature of the problem at hand.

*British University in Egypt (BUE), Cairo, Egypt* 

problem for the controller to formulate proper decisions.

rule-bases.

**Author details** 

**7. References** 

& Sons Inc.

Pearson Education.

17, no. No. 6, pp. 19-35.

and Materials, pp. 206-209.

Maguid H. M. Hassan

Smart sustainable structural systems were presented as a simple single-degree-of-freedom system, then, a more complex system was considered. In case of single-degree-of-freedom systems, fuzzy controllers with two input variables and single output variables were discussed. However, in case of more complex systems, the notion of a dual fuzzy processor where a fuzzy pattern identifier feeds additional information to the fuzzy controller was presented. Fuzzy inference systems were discussed in relation to the type of membership functions to be employed in similar applications and the method of generating the necessary

The reliability of such non-parametric systems is of major concern and thus a reliability assessment framework for evaluating the reliability of fuzzy controllers was presented. Potential failure conditions and limit state equations were presented as the basic tool of formulating the reliability problem of a fuzzy controller. The reliability evaluations were performed instantaneously then a reliability time history was created to suit the time

Finally, the concept of a fuzzy pattern identifier was presented using a fuzzy inference system which would be coupled with the fuzzy controller to form a dual fuzzy processor. Such structure is necessary in case of complex structural systems where the basic information of the dynamics of control points would not be enough to fully define the

[1] Soong, T.T. (1990) Active Structural Control, Theory & Practice. New York: John Wiley

[2] Connor, J.J. (2003) Introduction to Structural Motion Control. New Jersey: Prentice Hall,

[3] Spencer Jr., B.F. and Sain, M.K. (1998) Controlling Buildings: A New Frontier in Feedback. Control Systems Magazine, IEEE, Special Issue on Engineering Technology,

[4] Spencer Jr., B.F., and Soong, T.T. (1999) New Applications and Development of Active, Semi-Active and Hybrid Control Techniques for the Seismic and Non-Seismic Vibration in the USA. Proceedings of the Inter. Post-SmiRT Conf. on Seismic Isolation, Passive

Energy Dissipation and Active Control of Vibration of Structures, Cheju, Korea. [5] Casciati, F., Faravelli, L. and Yao, T. (1994) Application of fuzzy logic to active structural control. Proceedings of the Second European Conference on Smart Structures

	- [24] Hassan, M.H.M. (2010) Toward Reliability-Based Design of Smart Pattern Identifiers for Semi Active Control Applications. Fifth World Conference on Structural Control and Monitoring, 5WCSCM 2010, Tokyo, Japan, July 12-14.

**Chapter 11** 

© 2012 Chadli and El Hajjaji, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

and reproduction in any medium, provided the original work is properly cited.

**Vehicle Fault Tolerant Control Using** 

M. Chadli and A. El Hajjaji

http://dx.doi.org/10.5772/48262

fault-tolerant control methods.

2000, Dahmani, Chadli and al, 2012).

**1. Introduction** 

Additional information is available at the end of the chapter

**a Robust Output Fuzzy Controller Design** 

Modern life depends increasingly on the availability at all times of services and products provided by technological systems. Many areas, such as communication systems, water supply, power grids, urban transport systems are now completely automated. For such systems, the consequences of faults in component systems can be catastrophic. Reliability of such systems can be increased by ensuring that the faults will not occur, however, this objective unrealistic and often unattainable. In this context, it is very useful to design fault tolerant control systems that are able to tolerate possible faults in such systems to improve reliability and availability. Together with the increasing complexity of engineered systems and rising demands regarding reliability and safety, it is important to develop powerful

A number of surveys are discussed various aspects of fault-tolerant control. For example, Stengel (1991) discusses analytical forms of redundancy using artificial intelligence methods. In (Rauch, 1994) a broad overview over basic methodologies based on classical control techniques (pseudo-inverse methods, adaptive approaches ...) is given with several application examples (aircraft, unmanned underwater vehicles). In (Patton, 1997) (Zhang and Jiang, 2003) surveys on fault-tolerant control methods give a broad summary of the field. In the transport domain, to satisfy increasing safety, many new vehicles are equipped with different driver assisted systems such as Traction Control System (TCS) and Electronic Stabilization Program (ESP) to maintain stability and acceptable performances even when some sensors have failed. These systems use a combination of ABS information, yaw rate, wheel speed, lateral acceleration and steer angle to improve the stabilization of the vehicle in dangerous driving situations and then improve the active safety (Kienck and Nielsen,


## **Chapter 11**

## **Vehicle Fault Tolerant Control Using a Robust Output Fuzzy Controller Design**

M. Chadli and A. El Hajjaji

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48262

## **1. Introduction**

248 Fuzzy Controllers – Recent Advances in Theory and Applications

224.

5, p. 565-575.

Monitoring, 5WCSCM 2010, Tokyo, Japan, July 12-14.

Structures & Systems, Acireale, Sicily, Italy, June 8-13.

[24] Hassan, M.H.M. (2010) Toward Reliability-Based Design of Smart Pattern Identifiers for Semi Active Control Applications. Fifth World Conference on Structural Control and

[25] Hassan, M.H.M. (2008) A Reliability Assessment Model for MR Damper Components within a Structural Control Scheme. Advances in Science & Technology, Vol. 56, pp 218-

[26] Dyke, S. J., Spencer Jr. B. F., Sain, M. K. and Carlson, J. D. (1996) Modeling and Control of Magneto-Rheological Dampers for Seismic Response Reduction. Smart Mater. Struct.,

[27] Spencer Jr., B.F., Dyke, S. J., Sain, M. K. and Carlson, J. D. (1997) Phenomenological

[28] Hassan, M.H.M. (2008) A Reliability Assessment Model for MR Damper Components within a Structural Control Scheme. Third International Conference Smart Materials,

Model for Magneto-rheological Dampers. J. Engrg. Mech. 123 (3), p. 230-238.

Modern life depends increasingly on the availability at all times of services and products provided by technological systems. Many areas, such as communication systems, water supply, power grids, urban transport systems are now completely automated. For such systems, the consequences of faults in component systems can be catastrophic. Reliability of such systems can be increased by ensuring that the faults will not occur, however, this objective unrealistic and often unattainable. In this context, it is very useful to design fault tolerant control systems that are able to tolerate possible faults in such systems to improve reliability and availability. Together with the increasing complexity of engineered systems and rising demands regarding reliability and safety, it is important to develop powerful fault-tolerant control methods.

A number of surveys are discussed various aspects of fault-tolerant control. For example, Stengel (1991) discusses analytical forms of redundancy using artificial intelligence methods. In (Rauch, 1994) a broad overview over basic methodologies based on classical control techniques (pseudo-inverse methods, adaptive approaches ...) is given with several application examples (aircraft, unmanned underwater vehicles). In (Patton, 1997) (Zhang and Jiang, 2003) surveys on fault-tolerant control methods give a broad summary of the field. In the transport domain, to satisfy increasing safety, many new vehicles are equipped with different driver assisted systems such as Traction Control System (TCS) and Electronic Stabilization Program (ESP) to maintain stability and acceptable performances even when some sensors have failed. These systems use a combination of ABS information, yaw rate, wheel speed, lateral acceleration and steer angle to improve the stabilization of the vehicle in dangerous driving situations and then improve the active safety (Kienck and Nielsen, 2000, Dahmani, Chadli and al, 2012).

© 2012 Chadli and El Hajjaji, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The most common approach in coping with such a problem is to separate the overall design in two distinct phases. The first phase concerns "Fault Detection and Isolation" (FDI) problem, which consists in designing filters (dynamical systems) able to detect the presence of faults and to isolate them from other faults/disturbances (Isermann, 2001; Ding, Schneider, Ding and Rehm, 2005; Blanke, Kinnaert, Lunze and Staroswiecki, 2003; Gertler, 1998; Oudghiri, Chadli and ElHajjaji, 2007; Oudghiri, Chadli and ElHajjaji, 2008). The second phase usually consists in designing a supervisory unit. This unit reconfigures the control so as to compensate for the effect of the fault and to fulfill performance constraints. In general, the latter phase is carried out by means of a parameterized controller which is suitably updated by the supervisory unit.

Vehicle Fault Tolerant Control Using a Robust Output Fuzzy Controller Design 251

1

1

observer-based fault tolerant control strategy with simulations of sensor faults and result

**Notation**: symmetric definite positive matrix *P* is defined by *P* 0 , the set 1,2,..,*n* is defined by *nI* and symbol \* denotes the transpose elements in the symmetric positions.

This section introduces concepts and ideas from the field of fault-tolerant control (FTC).

*<sup>w</sup> x t Ax t Bu t B w t*

where **x**(*t*) R*<sup>n</sup>* is the state, **y**(*t*) R*<sup>r</sup>* is the output, **u**(*t*) R*<sup>m</sup>* is the inputs which are measurable, **A**  R*<sup>n</sup>*×*<sup>n</sup>*is the state transition matrix, **B**  R*<sup>n</sup>*×*<sup>m</sup>*is the input distribution matrix, **C**  R*r*×n is the output matrix, **B***<sup>w</sup>* R*<sup>n</sup>*×*<sup>n</sup>*is the disturbance matrix, and **w***1***(t)**  R*<sup>n</sup>* and **w***2***(t)**  R*<sup>r</sup>*

Faults are modelled by changes of system matrices. For example, *Actuator faults* are modelled by modifing input matrix **B***f* by scaling columns or setting to zero of columns in case of actuator failure. The *Sensor faults* are modelled by a modified output matrix **C***f* . This matrix may contain scaled rows due to altered sensor characteristics or zero rows due to failed sensors i.e. the faulty sensor should be switched off. *Plant faults* are modelled by a modified system matrix **A***<sup>f</sup>* . In general, when all types of faults present simultaneously, the

2

( ) ( ) ( ) ( )

*x t Ax t Bu t B w t*

Notice that in almost works, only one type of fault is assumed to have occurred at a time. A general linear controller (*K)* could be designed as a static or dynamic output feedback

In the following paragraphs, brief definitions of terms common in the fault-tolerant control

**Faults.** Faults can cause technical systems to malfunction or operate at reduced performance. Reduced service quality is the consequence. Faults may be triggered internally, such as broken power links in a computer or blocked valves in a chemical batch plant, or externally, such as changes in environmental conditions like a temperature drop

Faults can be further classified by their location in a block diagram. *Actuator faults* affect only actuation systems, such as pumps, valves, stirrers, switches, motors, brakes. They concern the efficiency of inputs on the system. *Plant faults* affect internal plant components,

( ) ( ) ( ) *f ff ff w*

*y t Cx t w t*

*f ff*

community are provided (J. Lunze and J. Richter (2006).

2 ( ) ( ) ( ) ( )

( ) ( ) ( )

*y t Cx t w t*

analysis. Conclusions are given in Section 6.

**2. Preliminaries and some definitions** 

are the disturbances which are unknowns.

faulty system model becomes:

stopping a chemical reaction.

controller.

Consider the following state space representation of linear systems:

Our objective is to develop model-based FTC-scheme for vehicle lateral dynamics. This study is motivated by the practical demands for such monitoring systems that i) automatically and reliably detect and isolate faults from sensors ii) deliver reliable and fault tolerant estimates of the vehicle lateral dynamics and iii) are practically realizable. In this chapter, we propose an observer-based fault tolerant control to detect, identify and accommodate sensor failures. The given method is based on the single failure assumption which states that at most one sensor can fail at any time.

To know the vehicle response, the proposed controller needs to know the yaw rate and the lateral velocity in order to generate the suitable output. If the yaw rate can be directly measurable by a yaw rate sensor (gyroscope), the lateral velocity will have to be estimated using an observer because it is not measurable easily. In this paper, a fuzzy controller is designed by considering the lateral velocity estimated using a nonlinear observer. In the analysis and design, the vehicle lateral will be represented by a switching systems (Chadli and Darouach, 2011) or by a Takagi-Sugeno (T-S) fuzzy model (Takagi and Sugeno, 1985), largely used these last years (Xioodong and Qingling, 2003; Chadli, Maquin and Ragot, 2005; Kirakidis, 2001; Tanaka and Wang, 1998; Chadli and El Hajjaji, 2006; Guerra and al, 2011; Chadli and Guerra, 2012). It is usually referred to as the bicycle model. Moreover, we consider the uncertain Takagi-Sugeno (T-S) fuzzy model to describe the vehicle dynamics in large domains and by the same way to improve the stability of vehicle lateral dynamics (Oudghiri, Chadli and A. ElHajjaji, 2007b; Chadli, ElHajjaji and Oudghiri, 2008). The proposed algorithm is formulated in terms of linear matrix inequalities (LMI) (Boyd and al, 1994) which are easily solvable using classical numerical tools (such as LMI Toolbox for Matlab software).

The subject of this chapter concerns the area of active FTCS for lateral vehicle dynamics that is modeled by uncertain TS fuzzy model. A FDI algorithm based on fuzzy observer is developed and a design method of control law tolerant to some sensors faults is proposed. This chapter is structured as follows. Basic concepts and notions of the FTC field with several general approaches to achieve fault tolerance are described in Sections 2 and 3. In Section 4 applications of control reconfiguration are reviewed briefly. Section 4 describes the vehicle lateral and its representation by uncertain T-S fuzzy model. Section 5 presents the observer-based fault tolerant control strategy with simulations of sensor faults and result analysis. Conclusions are given in Section 6.

**Notation**: symmetric definite positive matrix *P* is defined by *P* 0 , the set 1,2,..,*n* is defined by *nI* and symbol \* denotes the transpose elements in the symmetric positions.

## **2. Preliminaries and some definitions**

250 Fuzzy Controllers – Recent Advances in Theory and Applications

which states that at most one sensor can fail at any time.

updated by the supervisory unit.

Matlab software).

The most common approach in coping with such a problem is to separate the overall design in two distinct phases. The first phase concerns "Fault Detection and Isolation" (FDI) problem, which consists in designing filters (dynamical systems) able to detect the presence of faults and to isolate them from other faults/disturbances (Isermann, 2001; Ding, Schneider, Ding and Rehm, 2005; Blanke, Kinnaert, Lunze and Staroswiecki, 2003; Gertler, 1998; Oudghiri, Chadli and ElHajjaji, 2007; Oudghiri, Chadli and ElHajjaji, 2008). The second phase usually consists in designing a supervisory unit. This unit reconfigures the control so as to compensate for the effect of the fault and to fulfill performance constraints. In general, the latter phase is carried out by means of a parameterized controller which is suitably

Our objective is to develop model-based FTC-scheme for vehicle lateral dynamics. This study is motivated by the practical demands for such monitoring systems that i) automatically and reliably detect and isolate faults from sensors ii) deliver reliable and fault tolerant estimates of the vehicle lateral dynamics and iii) are practically realizable. In this chapter, we propose an observer-based fault tolerant control to detect, identify and accommodate sensor failures. The given method is based on the single failure assumption

To know the vehicle response, the proposed controller needs to know the yaw rate and the lateral velocity in order to generate the suitable output. If the yaw rate can be directly measurable by a yaw rate sensor (gyroscope), the lateral velocity will have to be estimated using an observer because it is not measurable easily. In this paper, a fuzzy controller is designed by considering the lateral velocity estimated using a nonlinear observer. In the analysis and design, the vehicle lateral will be represented by a switching systems (Chadli and Darouach, 2011) or by a Takagi-Sugeno (T-S) fuzzy model (Takagi and Sugeno, 1985), largely used these last years (Xioodong and Qingling, 2003; Chadli, Maquin and Ragot, 2005; Kirakidis, 2001; Tanaka and Wang, 1998; Chadli and El Hajjaji, 2006; Guerra and al, 2011; Chadli and Guerra, 2012). It is usually referred to as the bicycle model. Moreover, we consider the uncertain Takagi-Sugeno (T-S) fuzzy model to describe the vehicle dynamics in large domains and by the same way to improve the stability of vehicle lateral dynamics (Oudghiri, Chadli and A. ElHajjaji, 2007b; Chadli, ElHajjaji and Oudghiri, 2008). The proposed algorithm is formulated in terms of linear matrix inequalities (LMI) (Boyd and al, 1994) which are easily solvable using classical numerical tools (such as LMI Toolbox for

The subject of this chapter concerns the area of active FTCS for lateral vehicle dynamics that is modeled by uncertain TS fuzzy model. A FDI algorithm based on fuzzy observer is developed and a design method of control law tolerant to some sensors faults is proposed. This chapter is structured as follows. Basic concepts and notions of the FTC field with several general approaches to achieve fault tolerance are described in Sections 2 and 3. In Section 4 applications of control reconfiguration are reviewed briefly. Section 4 describes the vehicle lateral and its representation by uncertain T-S fuzzy model. Section 5 presents the This section introduces concepts and ideas from the field of fault-tolerant control (FTC). Consider the following state space representation of linear systems:

$$\begin{aligned} \dot{x}(t) &= A x(t) + B u(t) + B\_w w\_1(t) \\ y(t) &= C x(t) + w\_2(t) \end{aligned}$$

where **x**(*t*) R*<sup>n</sup>* is the state, **y**(*t*) R*<sup>r</sup>* is the output, **u**(*t*) R*<sup>m</sup>* is the inputs which are measurable, **A**  R*<sup>n</sup>*×*<sup>n</sup>*is the state transition matrix, **B**  R*<sup>n</sup>*×*<sup>m</sup>*is the input distribution matrix, **C**  R*r*×n is the output matrix, **B***<sup>w</sup>* R*<sup>n</sup>*×*<sup>n</sup>*is the disturbance matrix, and **w***1***(t)**  R*<sup>n</sup>* and **w***2***(t)**  R*<sup>r</sup>* are the disturbances which are unknowns.

Faults are modelled by changes of system matrices. For example, *Actuator faults* are modelled by modifing input matrix **B***f* by scaling columns or setting to zero of columns in case of actuator failure. The *Sensor faults* are modelled by a modified output matrix **C***f* . This matrix may contain scaled rows due to altered sensor characteristics or zero rows due to failed sensors i.e. the faulty sensor should be switched off. *Plant faults* are modelled by a modified system matrix **A***<sup>f</sup>* . In general, when all types of faults present simultaneously, the faulty system model becomes:

$$\begin{aligned} \dot{\boldsymbol{x}}\_f(t) &= \boldsymbol{A}\_f \boldsymbol{x}\_f(t) + \boldsymbol{B}\_f \boldsymbol{u}\_f(t) + \boldsymbol{B}\_w \boldsymbol{w}\_1(t) \\ \boldsymbol{y}\_f(t) &= \boldsymbol{C}\_f \boldsymbol{x}\_f(t) + \boldsymbol{w}\_2(t) \end{aligned}$$

Notice that in almost works, only one type of fault is assumed to have occurred at a time. A general linear controller (*K)* could be designed as a static or dynamic output feedback controller.

In the following paragraphs, brief definitions of terms common in the fault-tolerant control community are provided (J. Lunze and J. Richter (2006).

**Faults.** Faults can cause technical systems to malfunction or operate at reduced performance. Reduced service quality is the consequence. Faults may be triggered internally, such as broken power links in a computer or blocked valves in a chemical batch plant, or externally, such as changes in environmental conditions like a temperature drop stopping a chemical reaction.

Faults can be further classified by their location in a block diagram. *Actuator faults* affect only actuation systems, such as pumps, valves, stirrers, switches, motors, brakes. They concern the efficiency of inputs on the system. *Plant faults* affect internal plant components,

resulting in changed plant I/O properties, for example clogged pipes or leakages. They concern the system dynamics. *Sensor faults* result in erroneous measurements, such as biased, scaled or simply absent, constant zero readings (Blanke *et al.*, 2003). They concern the measured output of a system.

Vehicle Fault Tolerant Control Using a Robust Output Fuzzy Controller Design 253

**Figure 1.** A classification of fault-tolerant control methods

in the literature.

technique.

Campo and Morari, 1994).

fault tolerance introduces conservatism.

**3.2. Fault accommodation - fault reconfiguration** 

(Ahmed-Zaid *et al.*, 1991; Bodson and Groszkiewicz, 1997).

A typical example of a passive approach is robust controller design, a well-established and researched approach to achieve fault tolerance. Typically, faults that can be modelled as plant uncertainties can be well covered by robust design. A large number of publications concerning the achievement of fault tolerance using various robust design techniques exist

In robustness approaches, a fixed controller is designed to accommodate a class of anticipated component faults or failures. Most robustness approaches are feasible only for faults representable as parameter drift (see for example Fujita and Shimermura, 1988,

The class of faults covered by robust control is in general more limited in comparison to active approaches. In addition, the necessary trade-off between nominal performance and

*Fault accommodation* denotes the case where the variables measured and manipulated by the controller remain unchanged (Blanke *et al.*, 2003). Only the controller internals (including its dynamic order) may change, but the same measurement and actuation signals as in the nominal case must be used. Adaptive control is an example of an accommodation technique

The approach also has its specific limitations. The most serious one concerns the severity of faults and the speed of adaptation. Only faults representable as slowly changing plant parameters can be well accommodated by adjusting controller parameters. Structural damage is not covered. In addition, adaptive control works well in case of slow plant parameter variations in linear plants with respect to signal variation speed. This assumption is very questionable for faults that occur abruptly and rapidly lead out of the region of valid plant linearisation. Adaptive controllers are generally too slow to compensate abrupt faults.

Switching among a bank of predesigned controllers may be used as an accommodation

**Failures.** Failures contrast faults in the following sense. A fault reduces the system performance. The system can in general still serve its purpose, albeit with reduced functionality and/or performance. After a failure, the system provides no service any more. It cancels service availability completely. Faults and failures can occur both at the component level and at the aggregated system level. Fault-tolerant control aims at preventing component faults, component failures or subsystem faults from becoming system failures (Blanke *et al.*, 2003).

**Fault-tolerance.** The term *fault-tolerant system* (FTS) will be used to denote a controlled system which can still serve its purpose in spite of the occurrence of faults, at least for some time and to some degree, until the impaired components can be repaired.

*Fault-tolerant control (FTC)* denotes a framework of methods developed to turn control loops into fault-tolerant systems. The focus is on the design of the automatic control laws. That is, the means to achieve fault-tolerance are specific control design approaches with faulttolerance in mind. The goal is to keep the loop in operation for as long as possible to minimise the cost of down-time. Shutting down a plant may be expensive due to loss of production, or due to resulting plant damage. The latter can be the case in some chemical reactions. As an example, absence of cooling can cause irreversible solidification of the reactor content of a batch process, which means loss of the reactor.

Fault diagnosis is an area of active research of its own. In most parts of this work, the diagnosis task is taken as a prerequisite already solved, as this work focuses on controller adjustment. When considering the joint properties of diagnosis and controller adjustment or in implicit approaches, diagnosis is covered as well.

## **3. Classification of fault-tolerant control**

There already exist several approaches to achieve fault tolerance for control loops. The classification taken here is illustrated in Figure 1.

The classification can be done according to different criteria. The distinction between *passive* and *active* approaches is explained first, followed by *fault accommodation* and *reconfiguration*.

## **3.1. Passive and active FTC**

*Passive* fault tolerance is achieved when the loop remains operational in spite of faults *without changing the controller*. If the controller is changed at fault detection time, for instance by controller parameters or even its structure, the approach is called *active*.

**Figure 1.** A classification of fault-tolerant control methods

measured output of a system.

system failures (Blanke *et al.*, 2003).

resulting in changed plant I/O properties, for example clogged pipes or leakages. They concern the system dynamics. *Sensor faults* result in erroneous measurements, such as biased, scaled or simply absent, constant zero readings (Blanke *et al.*, 2003). They concern the

**Failures.** Failures contrast faults in the following sense. A fault reduces the system performance. The system can in general still serve its purpose, albeit with reduced functionality and/or performance. After a failure, the system provides no service any more. It cancels service availability completely. Faults and failures can occur both at the component level and at the aggregated system level. Fault-tolerant control aims at preventing component faults, component failures or subsystem faults from becoming

**Fault-tolerance.** The term *fault-tolerant system* (FTS) will be used to denote a controlled system which can still serve its purpose in spite of the occurrence of faults, at least for some

*Fault-tolerant control (FTC)* denotes a framework of methods developed to turn control loops into fault-tolerant systems. The focus is on the design of the automatic control laws. That is, the means to achieve fault-tolerance are specific control design approaches with faulttolerance in mind. The goal is to keep the loop in operation for as long as possible to minimise the cost of down-time. Shutting down a plant may be expensive due to loss of production, or due to resulting plant damage. The latter can be the case in some chemical reactions. As an example, absence of cooling can cause irreversible solidification of the

Fault diagnosis is an area of active research of its own. In most parts of this work, the diagnosis task is taken as a prerequisite already solved, as this work focuses on controller adjustment. When considering the joint properties of diagnosis and controller adjustment or

There already exist several approaches to achieve fault tolerance for control loops. The

The classification can be done according to different criteria. The distinction between *passive* and *active* approaches is explained first, followed by *fault accommodation* and

*Passive* fault tolerance is achieved when the loop remains operational in spite of faults *without changing the controller*. If the controller is changed at fault detection time, for instance

by controller parameters or even its structure, the approach is called *active*.

time and to some degree, until the impaired components can be repaired.

reactor content of a batch process, which means loss of the reactor.

in implicit approaches, diagnosis is covered as well.

**3. Classification of fault-tolerant control** 

classification taken here is illustrated in Figure 1.

*reconfiguration*.

**3.1. Passive and active FTC** 

A typical example of a passive approach is robust controller design, a well-established and researched approach to achieve fault tolerance. Typically, faults that can be modelled as plant uncertainties can be well covered by robust design. A large number of publications concerning the achievement of fault tolerance using various robust design techniques exist in the literature.

In robustness approaches, a fixed controller is designed to accommodate a class of anticipated component faults or failures. Most robustness approaches are feasible only for faults representable as parameter drift (see for example Fujita and Shimermura, 1988, Campo and Morari, 1994).

The class of faults covered by robust control is in general more limited in comparison to active approaches. In addition, the necessary trade-off between nominal performance and fault tolerance introduces conservatism.

## **3.2. Fault accommodation - fault reconfiguration**

*Fault accommodation* denotes the case where the variables measured and manipulated by the controller remain unchanged (Blanke *et al.*, 2003). Only the controller internals (including its dynamic order) may change, but the same measurement and actuation signals as in the nominal case must be used. Adaptive control is an example of an accommodation technique (Ahmed-Zaid *et al.*, 1991; Bodson and Groszkiewicz, 1997).

The approach also has its specific limitations. The most serious one concerns the severity of faults and the speed of adaptation. Only faults representable as slowly changing plant parameters can be well accommodated by adjusting controller parameters. Structural damage is not covered. In addition, adaptive control works well in case of slow plant parameter variations in linear plants with respect to signal variation speed. This assumption is very questionable for faults that occur abruptly and rapidly lead out of the region of valid plant linearisation. Adaptive controllers are generally too slow to compensate abrupt faults.

Switching among a bank of predesigned controllers may be used as an accommodation technique.

*Control reconfiguration* is an active approach where both the controller and its measured and manipulated variables may change. Reconfiguration allows the structure of the control loop to be changed in response to faults. This goes beyond structural changes inside the controller by including dynamic signal re-routing of inputs and outputs.

Vehicle Fault Tolerant Control Using a Robust Output Fuzzy Controller Design 255

   

 

*<sup>r</sup>* represent tyre slip-angles at

(3)

(4)

(5)

(6)

(2)

 

*<sup>f</sup>* and

1 () () *fff rrr*

2 () () *fff rrr*

 

 

> 

 

> 

> >

 

 

*F C F C*

*F C F C*

1 2

() ()

1 2


*C C*

*C C*

*f f f f*

*F h <sup>h</sup> f ff*

*F h h r rr*

*f r f r*

where j *h* (j 1,2) is the jth bell curve membership function of fuzzy set *Mj*. They satisfy the

() ()

 

where*Cfi* , *Cri* represent front and rear lateral tire stiffness, which depend on road

 



 

( )sin ( )tan ( )(1- ( )) ( )tan ( ( ) )

 

( )sin ( )tan ( )(1 - ( )) ( )tan ( ( ) ) *f f f f f ff f f*

Coefficients *Di, Ci, Bi* and *Ei* (*i = f,r*) depend on the tire characteristics, road adhesion

1 tan cos

*<sup>a</sup> <sup>f</sup> <sup>r</sup> f f <sup>u</sup>*

*v v u u*

*<sup>f</sup>* is M1 then <sup>1</sup>

*<sup>f</sup>* is M2 then <sup>2</sup>

( (

( (


1 tan cos

To obtain the TS fuzzy model, we have represented the front and rear lateral forces (2) by

*a r r*

*v v u u*

*r u*

*r r r r r rr r r*

*FD C B E E B*

 

*FD C B E E B*

 

and *J* are the mass and the yaw moment of inertia respectively, *<sup>f</sup> a* and *<sup>r</sup> a* are respectively distances of the front and rear axle from the center of gravity, while yaw moment *Mz* is the control input, which must be determined from the control law, *<sup>r</sup> F* and *<sup>f</sup> F* are rear and front lateral forces respectively. They are described by magic formula (Lin, popov and Mcwilliam,

2004) as

coefficient

where *<sup>f</sup>* 

adherence

the following rules:

If

If

The overall forces are obtained by:

 

.

following constraints

is the front steer angle.

 

 

the front and rear of the vehicle respectively. Given that

and the vehicle operational conditions,

## **4. FTC for vehicle dynamics**

## **4.1. Vehicle model**

Vehicle lateral dynamics have been studied since the late 1950's. Segel (Segel, 1956) developed a three-degree-of freedom vehicle model to describe the vehicle directional responses, which includes the yaw, lateral and roll motions. Most of the previous research works on vehicle lateral control have relied on the bicycle model (figure 2) that considers only lateral and yaw motions. It is based on the following assumptions:


The following simplified model is obtained:

$$\begin{aligned} m\left(\dot{\upsilon} + \iota r\right) &= 2\left(F\_f + F\_r\right) \\ J\dot{r} &= 2\left(a\_f F\_f - a\_r F\_r\right) + M\_z \end{aligned} \tag{1}$$

where *u* and *v* ( *v u* ) are components of the vehicle velocity along longitudinal and lateral principle axis of the vehicle body, *r* is yaw rate, denotes the side slip angle, *m*

and *J* are the mass and the yaw moment of inertia respectively, *<sup>f</sup> a* and *<sup>r</sup> a* are respectively distances of the front and rear axle from the center of gravity, while yaw moment *Mz* is the control input, which must be determined from the control law, *<sup>r</sup> F* and *<sup>f</sup> F* are rear and front lateral forces respectively. They are described by magic formula (Lin, popov and Mcwilliam, 2004) as

$$\begin{aligned} F\_f &= D\_f(\mu) \sin \left[ C\_f(\mu) \tan^{-1} \left\{ B\_f(\mu) (1 - E\_f(\mu)) \alpha\_f + E\_f(\mu) \tan^{-1} (B\_f(\mu) \alpha\_f) \right\} \right] \\ F\_r &= D\_r(\mu) \sin \left[ C\_r(\mu) \tan^{-1} \left\{ B\_r(\mu) (1 - E\_r(\mu)) \alpha\_r + E\_r(\mu) \tan^{-1} (B\_r(\mu) \alpha\_r) \right\} \right] \end{aligned} \tag{2}$$

Coefficients *Di, Ci, Bi* and *Ei* (*i = f,r*) depend on the tire characteristics, road adhesion coefficient and the vehicle operational conditions, *<sup>f</sup>* and *<sup>r</sup>* represent tyre slip-angles at the front and rear of the vehicle respectively. Given that

$$\begin{cases} \alpha\_f = -\frac{v}{u} - \tan^{-1}\left(\frac{a\_f}{u}r\cos\left(\frac{v}{u}\right)\right) + \delta\_f\\ \alpha\_r = -\frac{v}{u} + \tan^{-1}\left(\frac{a\_r}{u}r\cos\left(\frac{v}{u}\right)\right) \end{cases} \tag{3}$$

where *<sup>f</sup>* is the front steer angle.

254 Fuzzy Controllers – Recent Advances in Theory and Applications

**4. FTC for vehicle dynamics** 

There is no roll, pitch or bounce

The steering angle is small

**Figure 2.** Bicycle model

where *u* and *v* ( *v u*

The following simplified model is obtained:

lateral principle axis of the vehicle body, *r* is yaw rate,

**4.1. Vehicle model** 

*Control reconfiguration* is an active approach where both the controller and its measured and manipulated variables may change. Reconfiguration allows the structure of the control loop to be changed in response to faults. This goes beyond structural changes inside the

Vehicle lateral dynamics have been studied since the late 1950's. Segel (Segel, 1956) developed a three-degree-of freedom vehicle model to describe the vehicle directional responses, which includes the yaw, lateral and roll motions. Most of the previous research works on vehicle lateral control have relied on the bicycle model (figure 2) that considers

 

*m v ur F F*

2

*f r*

) are components of the vehicle velocity along longitudinal and

(1)

denotes the side slip angle, *m*

*f f rr z*

*Jr a F a F M*

2

controller by including dynamic signal re-routing of inputs and outputs.

only lateral and yaw motions. It is based on the following assumptions:

The relative yaw between the vehicle and the road is small

The tire lateral force varies linearly with the slip angle

To obtain the TS fuzzy model, we have represented the front and rear lateral forces (2) by the following rules:

$$\text{If } \left| a\_f \right| \text{ is } \mathbf{M} \text{ then } \begin{cases} F\_f = \mathbb{C}\_{f1}(\mu) a\_f \\ F\_r = \mathbb{C}\_{r1}(\mu) a\_r \end{cases} \tag{4}$$

$$\text{If } \left| a\_f \right| \text{ is Mz then } \begin{cases} \mathcal{F}\_f = \mathcal{C}\_{f2}(\mu) a\_f \\ \mathcal{F}\_r = \mathcal{C}\_{r2}(\mu) a\_r \end{cases} \tag{5}$$

where*Cfi* , *Cri* represent front and rear lateral tire stiffness, which depend on road adherence .

The overall forces are obtained by:

$$\begin{cases} \mathbf{F}\_f = h\_1(|\alpha\_f|) \mathbf{C}\_{f1}(\mu)\alpha\_f + h\_2(|\alpha\_f|) \mathbf{C}\_{f2}(\mu)\alpha\_f\\ \mathbf{F}\_r = h\_1(|\alpha\_f|) \mathbf{C}\_{r1}(\mu)\alpha\_r + h\_2(|\alpha\_f|) \mathbf{C}\_{r2}(\mu)\alpha\_r \end{cases} \tag{6}$$

where j *h* (j 1,2) is the jth bell curve membership function of fuzzy set *Mj*. They satisfy the following constraints

$$\begin{cases} \sum\_{i=1}^{2} h\_i(\|\alpha\_f\|) = 1\\ 0 \le h\_i(\|\alpha\_f\|) \le 1 \; \forall i = 1, 2 \end{cases} \tag{7}$$

Vehicle Fault Tolerant Control Using a Robust Output Fuzzy Controller Design 257

(15)

(16)

(17)

(18)

(20)

2 2 2

, <sup>2</sup>

*D m* 

*i*

*f* (19)

1 *<sup>i</sup>*

*A A BM B B*

*if i i z fi fi f <sup>i</sup>*

*i f i i z fi f <sup>i</sup>*

0

*C*

*fi*

22 1

*fi f ri r fi f ri r*

0

*J* 

*fi ri fi f ri r*

*C C Ca Ca*

*Ca Ca Ca Ca J Ju*

*B B <sup>i</sup>* 1

0 1

The output vector of system *y* consist of measurements of lateral acceleration *<sup>y</sup> a* and the

The defuzzified output of this T–S fuzzy system is a weighted sum of individual linear

(| |)

(| |)

0 0

*C C df C C df* 

*fi fi i i ri ri i i*

*y h Cx D*

*i f i if <sup>i</sup>*

From the expressions of front and rear forces (4), (5), we note that stiffness coefficients *Cfi* and *Cri* are not constant and vary depending on the road adhesion. To take into account

> (1 ) (1 )

where *di* indicates the deviation magnitude of the stiffness coefficient from its nominal

*xh x A BM B*

*fi ri fi f ri r*

*C C Ca Ca*

2 2

,

2 2

*C mu mu* 

*mu mu <sup>A</sup>*

where , *<sup>T</sup>*

*x vr* , 1 2 , , *<sup>T</sup> <sup>T</sup>*

*i*

*fi*

yaw rate about center of gravity *r*

models

value.

*<sup>y</sup> y yy ar* and

2

*C*

*fi*

*f fi*

*J*

 

these variations, we assume that these coefficients vary as follows:

After some manipulations, the TS fuzzy model can be written as:

( ) (| |) ( )

*xt h x t*

*i f i if <sup>i</sup>*

*h Cx D*

(| |)

( )

*y t*

  2

*mu <sup>B</sup> a C*

*i*

The expressions of membership functions j *h* (j 1,2) used are as follows

$$h\_i\left(\left|\alpha\left(t\right)\_f\right|\right) = \frac{\beta\_i\left(\left|\alpha\_f\left(t\right)\right|\right)}{\sum\_{i=1}^2 \beta\_i\left(\left|\alpha\_f\left(t\right)\right|\right)}, i = 1, 2\tag{8}$$

with

$$\beta\_i \left( \left| \alpha\_f \right| \right) = \frac{1}{\left( 1 + \left| \left\lfloor \frac{\left| \alpha\_f \right| - c\_i}{a\_i} \right\rfloor \right)^{2b\_i}} \tag{9}$$

The membership function parameters and consequence of rules are obtained using an identification method based on the Levenberg-Marquadt algorithm (Lee, Lai and Lin, 2003) combined with the least square method, allow to determine parameters of membership functions ( , , *ii i ab c* ) and stiffness coefficient values

$$a\_1 = 0.5077, \ b\_1 = 3.1893, \ c\_1 = -0.4356, \ a\_2 = 0.4748, \ b\_2 = 5.3907, \ c\_2 = 0.5622 \tag{10}$$

$$\mathbf{C}\_{f1} = 60712.7 \text{ .} \ C\_{f2} = 4814 \text{ .} \ C\_{r1} = 60088 \text{ .} \ C\_{r2} = 3425 \text{ .} \tag{11}$$

Using the above approximation idea of nonlinear lateral forces by TS rules and by considering that

$$
\alpha\_f \cong \frac{-v - a\_f r}{u} + \delta\_{f,v} a\_r \cong \frac{-v + a\_r r}{u} \tag{12}
$$

nonlinear model (1) can be represented by the following TS fuzzy model:

$$\text{If } \|\boldsymbol{\alpha}\_f\| \text{ is } \mathbf{M} \text{ then } \begin{cases} \boldsymbol{\dot{\boldsymbol{x}}} = \mathbf{A}\_1 \mathbf{x} + \mathbf{B}\_1 \mathbf{M}\_z + \mathbf{B}\_{f1} \boldsymbol{\delta}\_f\\ \boldsymbol{y} = \mathbf{C}\_1 \mathbf{x} + \mathbf{D}\_1 \boldsymbol{\delta}\_f \end{cases} \tag{13}$$

$$\text{If } |\alpha\_f| \text{ is M: then } \begin{cases} \dot{\mathbf{x}} = \mathbf{A}\_2 \mathbf{x} + \mathbf{B}\_2 \mathbf{M}\_z + \mathbf{B}\_f \mathbf{2} \delta\_f \\ y = \mathbf{C}\_2 \mathbf{x} + \mathbf{D}\_2 \delta\_f \end{cases} \tag{14}$$

#### Vehicle Fault Tolerant Control Using a Robust Output Fuzzy Controller Design 257

$$\begin{aligned} \text{where } \boldsymbol{x} = \left(\boldsymbol{v}, \boldsymbol{r}\right)^{T}, \ y = \left(y\_{1}, y\_{2}\right)^{T} = \left(a\_{y}, r\right)^{T} \text{ and }\\ \boldsymbol{A}\_{i} &= \begin{pmatrix} -2\frac{\mathsf{C}\_{f} + \mathsf{C}\_{ri}}{mu} & -2\frac{\mathsf{C}\_{f}a\_{f} - \mathsf{C}\_{ri}a\_{r}}{mu^{2}} - 1\\ -2\frac{\mathsf{C}\_{f}a\_{f} - \mathsf{C}\_{ri}a\_{r}}{I} & -2\frac{\mathsf{C}\_{f}a\_{f}^{2} + \mathsf{C}\_{ri}a\_{r}^{2}}{Iu} \end{pmatrix} \\\\ \text{(15)} \end{aligned} \tag{15}$$

256 Fuzzy Controllers – Recent Advances in Theory and Applications

with

considering that

If | |

If | |

2

1 (

*h*

 

0 1 1,2

*h i*


 

1

 

<sup>2</sup>

*<sup>i</sup> i f <sup>b</sup>*

The membership function parameters and consequence of rules are obtained using an identification method based on the Levenberg-Marquadt algorithm (Lee, Lai and Lin, 2003) combined with the least square method, allow to determine parameters of membership

Using the above approximation idea of nonlinear lateral forces by TS rules and by

.

.

 

 , *<sup>r</sup> r*

*f f f v ar u*

nonlinear model (1) can be represented by the following TS fuzzy model:

*<sup>f</sup>* is M1 then

*<sup>f</sup>* is M2 then

*i*

1

2

*h t i*


1

1

*a*

1 11 222 *abc abc* 0.5077, 3.1893, -0.4356, 0.4748, 5.3907, 0.5622 (10)

<sup>1</sup> 60712.7 *Cf* , 2 4814 *Cf* , 1 60088 *Cr* , 2 3425 *Cr* (11)

*v ar u*

11 1

*z ff x Ax BM B*

2

*Mz x Ax B Bf f*

2 2 2

1 1

*y Cx <sup>D</sup> <sup>f</sup>*

2

*y Cx D <sup>f</sup>*

(12)

 

*f i i*

*c*

*t*

*t*

*i f*

, 1,2 *i f*

(7)

(8)

(9)

(13)

(14)

(

*i i f*

The expressions of membership functions j *h* (j 1,2) used are as follows

*i f*

 

functions ( , , *ii i ab c* ) and stiffness coefficient values

$$\mathbf{B}\_{fi} = \begin{pmatrix} 2\mathbf{C}\_{fi} \\ \frac{mu}{m\mu} \\ \frac{2a\_f\mathbf{C}\_{fi}}{I} \end{pmatrix}, \ \mathbf{B}\_i = \mathbf{B} = \begin{pmatrix} 0 \\ 1 \\ \overline{I} \end{pmatrix}, \ D\_i = \begin{pmatrix} \mathbf{C}\_{fi} \\ 2\frac{m}{m} \\ 0 \end{pmatrix} \tag{16}$$

$$\mathbf{C}\_{i} = \begin{pmatrix} -2\frac{\mathbf{C}\_{fi} + \mathbf{C}\_{ri}}{mu} & -2\frac{\mathbf{C}\_{fi}\mathbf{a}\_{f} - \mathbf{C}\_{ri}\mathbf{a}\_{r}}{mu} \\ 0 & 1 \end{pmatrix} \tag{17}$$

The output vector of system *y* consist of measurements of lateral acceleration *<sup>y</sup> a* and the yaw rate about center of gravity *r*

The defuzzified output of this T–S fuzzy system is a weighted sum of individual linear models

$$\begin{cases} \dot{\boldsymbol{x}} &= \sum\_{i=1}^{2} h\_i (\mid \boldsymbol{\alpha}\_{\boldsymbol{f}} \mid ) \left( \boldsymbol{A}\_i \boldsymbol{\chi} + \boldsymbol{B}\_i \boldsymbol{M}\_z + \boldsymbol{B}\_{\boldsymbol{f}\boldsymbol{t}} \boldsymbol{\delta}\_{\boldsymbol{f}} \right) \\ \boldsymbol{y} &= \sum\_{i=1}^{2} h\_i (\mid \boldsymbol{\alpha}\_{\boldsymbol{f}} \mid ) \left( \boldsymbol{C}\_i \boldsymbol{\chi} + \boldsymbol{D}\_i \boldsymbol{\delta}\_{\boldsymbol{f}} \right) \end{cases} \tag{18}$$

From the expressions of front and rear forces (4), (5), we note that stiffness coefficients *Cfi* and *Cri* are not constant and vary depending on the road adhesion. To take into account these variations, we assume that these coefficients vary as follows:

$$\begin{cases} \mathbf{C}\_{fi} = \mathbf{C}\_{fi0} \{ \mathbf{1} + d\_i f\_i \} \\ \mathbf{C}\_{ri} = \mathbf{C}\_{ri0} \{ \mathbf{1} + d\_i f\_i \} \end{cases} \left\| f\_i \right\| \le 1 \tag{19}$$

where *di* indicates the deviation magnitude of the stiffness coefficient from its nominal value.

After some manipulations, the TS fuzzy model can be written as:

$$\begin{cases} \dot{\mathbf{x}}(t) &= \sum\_{i=1}^{2} h\_i(\|\alpha\_{f}\|) \left( (A\_i + \Delta A\_i) \mathbf{x}(t) + BM\_z + \left( B\_{f\_i} + \Delta B\_{f\_i} \right) \delta\_{f\_i} \right) \\\\ \mathbf{y}(t) &= \sum\_{i=1}^{2} h\_i(\|\alpha\_{f\_i}\|) \left( C\_i \mathbf{x} + D\_i \delta\_{f\_i} \right) \end{cases} \tag{20}$$

where *Ai* and *fi B* represent parametric uncertainties represented as follows

$$
\Delta \mathcal{A}\_i = \mathcal{H}\_i \Sigma\_i \begin{pmatrix} t \\ \end{pmatrix} \mathcal{E}\_{Ai} \tag{21}
$$

Vehicle Fault Tolerant Control Using a Robust Output Fuzzy Controller Design 259

(26)

*et xt xt* ˆ (27)

( ) ( )

*i ij ij*

(31)

(32)

*A BK BK*

*A BK BK* 

1 () () () ( ) ˆ *q*

where , *KiI i n* are the constant feedback gains to be determined. We define the error of

*x t h z h z A A B B K xt B B Ket*

*e t h z h z A GC BK e t A BK x t*

.

*i ij*

The global asymptotic stability of the TS fuzzy model (25) is summarized in the following

*Theorem 1:* If there exist symmetric and positive definite matrices *Q* and *P* , some matrices *Ki* and *Gi* such that the following LMIs are satisfied <sup>2</sup> (,) , *<sup>q</sup> ij I i j* , then TS fuzzy system (25) is globally asymptotically stable via TS fuzzy controller (21) based on fuzzy observers

*i ii*

<sup>1</sup> <sup>1</sup>

*A GC*

() ( ( ) ) ( )

*xt h zh z A A xt*

*i j ij ij*

, *i ij ij*

*ij*

0 1

1 0

1

*I*

*A*

() ( ) ( ) ( )

1 1

*i j*

*i ij ij*

 

*A BK BK*

*ii*

*T*

*Ai Bi i ii*

*EQ EM I*

*q q*

*i j i i i ij i ij*

*i j i ij ij i ij*

(28)

(30)

(29)

*i ut zt Kxt h* 

estimation as

where

theorem:

(20):

From systems (20), (21) and (22), we have

*q q*

1 1

1 1

*i j*

The augmented system can be expressed as:

*x*

*e* 

, <sup>0</sup>

*A*

*ij*

*x*

*q q*

*i j*

*i i*

with ( 1,2) *<sup>i</sup> t i* are matrices uncertain parameters such that *<sup>T</sup> i i t tI* , *<sup>i</sup> E* is known real matrix of appropriate dimension that characterizes the structures of uncertainties.

#### **4.2. Output feedback design**

#### a. TS Fuzzy observer structure

Consider the general case of uncertain T-S fuzzy model (Takagi and Sugeno, 1985):

$$\begin{aligned} \dot{\mathbf{x}}(t) &= \sum\_{i=1}^{q} h\_i(\mathbf{z}(t)) \left( (A\_i + \Delta A\_i)\mathbf{x}(t) + (B\_i + \Delta B\_i)\mathbf{u}(t) \right) \\ \mathbf{y}(t) &= \sum\_{i=1}^{q} h\_i(\mathbf{z}(t)) \mathbf{C}\_i \mathbf{x}(t) \end{aligned} \tag{22}$$

with properties

$$\sum\_{i=1}^{q} h\_i(z(t)) = 1, \ h\_i(z(t)) \ge 0 \quad \forall \ i \in I\_q \tag{23}$$

where *q* is the number of sub-models, ( ) *<sup>n</sup> x t* is the state vector, ( ) *<sup>m</sup> u t* is the control input vector, ( ) *<sup>l</sup> y t* is the output vector, ... , , *nn nm ln ABC ii i*  are the ith state matrix, the ith input matrix and the ith output matrix respectively. Vector *z t*( ) is the premise variable depending on measurable variables. *Ai* and , *i n Bi I* are time-varying matrices representing parametric uncertainties in the plant model. These uncertainties are admissibly norm-bounded and structured, defined as

$$
\Delta A\_i = \mathbf{H}\_i \Sigma\_i(\mathbf{t}) E\_{Ai} \quad \Delta B\_i = \mathbf{H}\_i \Sigma\_i(\mathbf{t}) E\_{Bi} \tag{24}
$$

The overall fuzzy observer has the same structure as the TS fuzzy model. It is represented as follows:

$$\begin{cases} \dot{\hat{\boldsymbol{x}}}(t) = \sum\_{i=1}^{q} h\_i(\mathbf{z}(t)) \left( A\_i \hat{\boldsymbol{x}}(t) + B\_i \boldsymbol{u}(t) + G\_i \left( \boldsymbol{y}(t) - \hat{\boldsymbol{y}}(t) \right) \right) \\\\ \dot{\boldsymbol{y}}(t) = \sum\_{i=1}^{q} h\_i(\mathbf{z}(t)) \mathbf{C}\_i \hat{\mathbf{x}}(t) \end{cases} \tag{25}$$

where , *GiI i n* are the constant observer gains to be determined.

#### b. TS Fuzzy controller

Like the fuzzy observer, the TS fuzzy controller is represented as follows

Vehicle Fault Tolerant Control Using a Robust Output Fuzzy Controller Design 259

$$u(t) = -\sum\_{i=1}^{q} h\_i(z(t)) K\_i \hat{x}(t) \tag{26}$$

where , *KiI i n* are the constant feedback gains to be determined. We define the error of estimation as

$$e(t) = \mathbf{x}(t) - \hat{\mathbf{x}}(t) \tag{27}$$

From systems (20), (21) and (22), we have

$$\dot{\mathbf{x}}(\mathbf{t}) = \sum\_{i=1}^{q} \sum\_{j=1}^{q} h\_i(\mathbf{z}) h\_j(\mathbf{z}) \left( \left( A\_i + \Delta A\_i - \left( B\_i + \Delta B\_i \right) K\_j \right) \mathbf{x}(\mathbf{t}) + \left( B\_i + \Delta B\_i \right) K\_j \mathbf{e}(\mathbf{t}) \right) \tag{28}$$

$$\dot{e}(t) = \sum\_{i=1}^{q} \sum\_{j=1}^{q} h\_i(z)h\_j(z) \left( A\_i - G\_i C\_j - \Delta B\_i K\_j \right) e(t) + \left( \Delta A\_i + \Delta B\_i K\_j \right) x(t) \tag{29}$$

The augmented system can be expressed as:

$$\dot{\tilde{\mathbf{x}}}(t) = \sum\_{i=1}^{q} \sum\_{j=1}^{q} h\_i(z) h\_j(z) \left( \tilde{A}\_{ij} + \Delta \tilde{A}\_{ij} \right) \tilde{\mathbf{x}}(t) \tag{30}$$

where

258 Fuzzy Controllers – Recent Advances in Theory and Applications

**4.2. Output feedback design** 

a. TS Fuzzy observer structure

with properties

follows:

b. TS Fuzzy controller

where *Ai* and *fi B* represent parametric uncertainties represented as follows

with ( 1,2) *<sup>i</sup> t i* are matrices uncertain parameters such that *<sup>T</sup>*

1

*i q*

*i*

norm-bounded and structured, defined as

 

*q*

1

1

1

*i q*

*i*

*q*

1

ˆ ˆ () (() () )

*yt h zt Cxt*

where , *GiI i n* are the constant observer gains to be determined.

*i i*

Like the fuzzy observer, the TS fuzzy controller is represented as follows

*i*

*q*

( ) ( ( )) ( )

*yt h zt Cxt*

*i i*

real matrix of appropriate dimension that characterizes the structures of uncertainties.

Consider the general case of uncertain T-S fuzzy model (Takagi and Sugeno, 1985):

( ) ( ( )) ( ) ( ) ( ) ( )

*xt h zt A A xt B B ut*

( ( )) 1, ( ( )) 0

*ii q*

*h zt h zt i I*

where *q* is the number of sub-models, ( ) *<sup>n</sup> x t* is the state vector, ( ) *<sup>m</sup> u t* is the control input vector, ( ) *<sup>l</sup> y t* is the output vector, ... , , *nn nm ln ABC ii i*  are the ith state matrix, the ith input matrix and the ith output matrix respectively. Vector *z t*( ) is the premise variable depending on measurable variables. *Ai* and , *i n Bi I* are time-varying matrices representing parametric uncertainties in the plant model. These uncertainties are admissibly

() *Ai i i Ai t E* , ( ) *i i i Bi B tE* (24)

The overall fuzzy observer has the same structure as the TS fuzzy model. It is represented as

.

ˆˆ ˆ ( ) ( ( )) ( ) ( ) ( ) ( )

*i iii*

*x t h z t Ax t But G y t y t*

*i i i ii*

(23)

*Ai i i Ai t E* (21)

*i i t tI* , *<sup>i</sup> E* is known

(22)

(25)

$$
\tilde{\mathbf{x}} = \begin{pmatrix} \mathbf{x} \\ e \end{pmatrix}, \tilde{A}\_{ij} = \begin{pmatrix} A\_i - \mathbf{B}\_i \mathbf{K}\_j & \mathbf{B}\_i \mathbf{K}\_j \\ 0 & A\_i - \mathbf{G}\_i \mathbf{C}\_j \end{pmatrix}, \quad \Delta \tilde{A}\_{ij} = \begin{pmatrix} \Delta A\_i - \Delta \mathbf{B}\_i \mathbf{K}\_j & \Delta \mathbf{B}\_i \mathbf{K}\_j \\ \Delta A\_i + \Delta \mathbf{B}\_i \mathbf{K}\_j & -\Delta \mathbf{B}\_i \mathbf{K}\_j \end{pmatrix} \tag{31}
$$

The global asymptotic stability of the TS fuzzy model (25) is summarized in the following theorem:

*Theorem 1:* If there exist symmetric and positive definite matrices *Q* and *P* , some matrices *Ki* and *Gi* such that the following LMIs are satisfied <sup>2</sup> (,) , *<sup>q</sup> ij I i j* , then TS fuzzy system (25) is globally asymptotically stable via TS fuzzy controller (21) based on fuzzy observers (20):

$$\begin{pmatrix} \Phi\_{ii} & \* & \* \\\\ E\_{Ai}Q - E\_{Bi}M\_i & -\left(\varepsilon\_{ii}^{-1} + 1\right)^{-1}I & \* \\\\ \mathbf{H}\_i^T & \mathbf{0} & -\left(\varepsilon\_{ii} + 1\right)^{-1}I \end{pmatrix} < \mathbf{0} \tag{32}$$

$$
\begin{pmatrix}
\mathbf{H}\_{ij} & \* & \* & \* & \* & \* \\
E\_{A\_i}Q - E\_{Bi}M\_j & -\left(\varepsilon\_{ij}^{-1} + 1\right)^{-1}I & \* & \* & \* \\
E\_{A\_j}Q - E\_{Bi}M\_j & 0 & -\left(\varepsilon\_{ij}^{-1} + 1\right)^{-1}I & \* & \* & \* \\
\mathbf{H}\_j^T & 0 & 0 & -\varepsilon\_{ij}^{-1} & \* \\
\mathbf{H}\_j^T & 0 & 0 & 0 & -\varepsilon\_{ij}^{-1}
\end{pmatrix} < 0 \tag{33}
$$

$$
\begin{pmatrix}
T\_{ii} & \* & \* \\
E\_{Bi}K\_i & -\left(\varepsilon\_{ii}^{-1} + 1\right)^{-1}I & \* \\
\mathbf{H}\_j^T P & 0 & -\left(\varepsilon\_{ii} + 1\right)^{-1}I
\end{pmatrix} < 0 \tag{34}
$$

$$
\begin{pmatrix}
\Theta\_{ij} & \* & \* & \* \\
E\_{Bi}K\_j & -\left(\varepsilon\_{ij}^{-1} + 1\right)^{-1}I & \* & \* \\
E\_{Bi}K\_j & -\left(\varepsilon\_{ij}^{-1} + 1\right)^{-1}I & \* & \* \\
E\_{Bi}K\_j & 0 & -\left(\varepsilon\_{ij}^{-1} + 1\right)^{-1}I & \* & \* \\
\end{pmatrix} < 0 \tag{35}
$$

$$
\begin{pmatrix}
\mathbf{H}\_j^T & 0 & 0 & -\left(\varepsilon\_{ij} + 1\right)^{-1}I & \* \\
\mathbf{H}\_j^T & 0 & 0 & -\left(\varepsilon\_{ij} + 1\right)^{-1}I & \*
\end{pmatrix} < 0 \tag{36}
$$

Vehicle Fault Tolerant Control Using a Robust Output Fuzzy Controller Design 261

(38)

(39)

1 1 0

). This case leads to four constraints to resolve, whereas the

,

*Corollary 1*: If there exist symmetric and positive definite matrices *Q* and *P* , some matrices *Ki* and *Gi* such that the following LMI are satisfied *<sup>q</sup> i I* , then TS fuzzy system (25) is globally asymptotically stable via TS fuzzy controller (21) based on fuzzy observers

<sup>1</sup> <sup>1</sup>

0 *ii <sup>T</sup> HP I <sup>i</sup>* 

*<sup>T</sup> T T ii i i i QA A Q M B BM I <sup>i</sup>*

*<sup>T</sup> T T T T ii i i i i i i i i A P PA C N N C K B BK*

Result of corollary 1 derive directly from the TS fuzzy model (15) (with common input

The derived stability conditions are LMI on synthesis variables 0, 0, , *i i P Q MN* and


It is important to be able to carry out fault detection and isolation before faults have a drastic effect on the system performance. Even in case of system changes, faults should be detected and isolated. Observer based estimator schemes are used to generate residual signals corresponding to the difference between measured and estimated variables (Chen and

 . However the problem to resolve becomes nonlinear in , *Ki I i q* (inequalities (27)-(28)/(30)-(31)). A method allowing the use of numerical tools to solve these constraints is given in the following.Toresolve the obtained BMI (bilinear matrix inequality) conditions using LMI tools (LMI toolbox of Matlab software for example), we propose to solve

result of theorem 1 leads to six constraints, which means less conservatism.


*ii*

The controller and the observer gains are as defined in (29).

**Proof**: The result is obtained directly from theorem 1.

synthesis conditions (27) (or (30)) sequentially:

matrix <sup>2</sup> , *<sup>i</sup> B Bi I* , and 0 *<sup>f</sup>*

scalars 0 *<sup>i</sup>* 

**5. FTC strategy** 

*Ai ii T*

*E Q I*

0

*i ii*

*H I*

(20):

with

with

$$\boldsymbol{\Theta}\_{ii} = \boldsymbol{Q} \boldsymbol{A}\_i^T + \boldsymbol{A}\_i \boldsymbol{\mathsf{Q}} - \boldsymbol{M}\_i^T \mathbf{B}\_i^T - \mathbf{B}\_i \boldsymbol{M}\_i + \boldsymbol{I}$$

$$\boldsymbol{\Psi}\_{ij} = \boldsymbol{Q} \boldsymbol{A}\_i^T + \boldsymbol{A}\_i \boldsymbol{\mathsf{Q}} + \boldsymbol{\mathsf{Q}} \boldsymbol{A}\_j^T + \boldsymbol{A}\_j \boldsymbol{\mathsf{Q}} - \boldsymbol{M}\_j^T \mathbf{B}\_i^T - \mathbf{B}\_i \boldsymbol{\mathsf{M}}\_j - \boldsymbol{M}\_i^T \mathbf{B}\_j^T - \mathbf{B}\_j \boldsymbol{\mathsf{M}}\_i + \boldsymbol{\mathsf{D}}\_i \boldsymbol{\mathsf{D}}\_i^T + \boldsymbol{\mathsf{D}}\_j \boldsymbol{\mathsf{D}}\_j^T + 2\boldsymbol{I}$$

$$\boldsymbol{T}\_{ii} = \boldsymbol{A}\_i^T \boldsymbol{P} + \boldsymbol{P} \boldsymbol{A}\_i - \mathbf{C}\_i^T \mathbf{N}\_i^T - \mathbf{N}\_i \mathbf{C}\_i + \mathbf{K}\_i^T \mathbf{B}\_i^T \mathbf{B}\_i \mathbf{K}\_i$$

$$\boldsymbol{\Theta}\_{ij} = \boldsymbol{A}\_i^T \boldsymbol{P} + \boldsymbol{P} \boldsymbol{A}\_i + \boldsymbol{A}\_j^T \boldsymbol{P} + \mathbf{P} \boldsymbol{A}\_j - \mathbf{C}\_i^T \mathbf{N}\_j^T - \mathbf{N}\_j \mathbf{C}\_j - \mathbf{C}\_j^T \mathbf{N}\_i^T - \mathbf{N}\_j \mathbf{C}\_i + \mathbf{K}\_i^T \mathbf{B}\_j^T \mathbf{B}\_i \mathbf{K}\_i + \mathbf{K}\_j^T \mathbf{B}\_i^T \mathbf{B}\_i \mathbf{K}\_j$$

The controller and the observer are defined as follows

$$K\_i = M\_i Q^{-1} \tag{36}$$

$$\mathbf{G}\_{i} = \mathbf{P}^{-1} \mathbf{N}\_{i} \tag{37}$$

**Proof**: The proof can be inspired directly from (Chadli & El Hajjaji 2006).

#### **Remarks**

In the case of common input matrix *B* ( *i q B B iI* ), the above result is simplified. The new stability conditions are given in the following corollary

*Corollary 1*: If there exist symmetric and positive definite matrices *Q* and *P* , some matrices *Ki* and *Gi* such that the following LMI are satisfied *<sup>q</sup> i I* , then TS fuzzy system (25) is globally asymptotically stable via TS fuzzy controller (21) based on fuzzy observers (20):

$$\begin{pmatrix} \Phi\_{ii} & \* & \* \\ E\_{Ai}Q & -\left(\boldsymbol{\varepsilon}\_{ii}^{-1} + 1\right)^{-1}I & \* \\ H\_i^T & 0 & -\boldsymbol{\varepsilon}\_{ii}^{-1}I \end{pmatrix} < 0 \tag{38}$$
 
$$\begin{pmatrix} \mathbf{T}\_{ii} & \* \\ H\_i^T \mathbf{P} & -I \end{pmatrix} < 0 \tag{39}$$

with

260 Fuzzy Controllers – Recent Advances in Theory and Applications

*ij*

*T*

*T*

*Ai Bi j ij*

*EQ EM I*

*ii*

*T*

*T*

*Bj i ij*

The controller and the observer are defined as follows

*ij*

*Bi j ij*

*E K I*

*T*

*T*

with

**Remarks** 

<sup>1</sup> <sup>1</sup>

1

*E K I*

*Bi i ii*

*EQ EM I*

*Aj Bj i ij*

<sup>1</sup> <sup>1</sup>

1

*E K I*

<sup>1</sup> <sup>1</sup>

*i ii*

<sup>1</sup> <sup>1</sup>

*P I*

*<sup>T</sup> T T ii i i i i i i QA A Q M B B M I*

<sup>2</sup> *<sup>T</sup> <sup>T</sup> T T T T T T ij i i j j j i i j i j j i i i j j QA A Q QA A Q M B B M M B B M D D D D I*

*T T T T T ii i i i i i i i i i i T A P PA C N N C K B B K*

*<sup>T</sup> <sup>T</sup> T T T T TT TT ij i i j j i j i j j i j i i j j i j i i j <sup>A</sup> P PA A P PA C N N C C N N C K B B K K B B K*

<sup>1</sup> *K MQ i i*

<sup>1</sup> *G PN i i*

In the case of common input matrix *B* ( *i q B B iI* ), the above result is simplified. The

**Proof**: The proof can be inspired directly from (Chadli & El Hajjaji 2006).

new stability conditions are given in the following corollary

*j ij*

00 1

*P I*

*i ij*

0 00

0 0

*j ij*

0 1

*P I*

*i ij*

 

<sup>1</sup> <sup>1</sup>

0 1 0

0 00 1

1 0

1

1

0 0 1

1

1

1

(36)

(37)

(33)

(34)

(35)

$$\begin{aligned} \boldsymbol{\Phi}\_{ii} &= \boldsymbol{\mathcal{Q}} \boldsymbol{A}\_i^T + \boldsymbol{A}\_i \boldsymbol{\mathcal{Q}} - \boldsymbol{M}\_i^T \boldsymbol{\mathcal{B}}^T - \boldsymbol{\mathcal{B}} \boldsymbol{M}\_i + \boldsymbol{I} \\\\ \mathbf{T}\_{ii} &= \boldsymbol{A}\_i^T \boldsymbol{P} + \boldsymbol{P} \boldsymbol{A}\_i - \boldsymbol{\mathsf{C}}\_i^T \boldsymbol{N}\_i^T - \boldsymbol{N}\_i \boldsymbol{\mathsf{C}}\_i + \boldsymbol{\mathsf{K}}\_i^T \boldsymbol{\mathsf{B}}^T \boldsymbol{\mathcal{B}} \boldsymbol{K}\_i \end{aligned}$$

The controller and the observer gains are as defined in (29).

**Proof**: The result is obtained directly from theorem 1.

Result of corollary 1 derive directly from the TS fuzzy model (15) (with common input matrix <sup>2</sup> , *<sup>i</sup> B Bi I* , and 0 *<sup>f</sup>* ). This case leads to four constraints to resolve, whereas the result of theorem 1 leads to six constraints, which means less conservatism.

The derived stability conditions are LMI on synthesis variables 0, 0, , *i i P Q MN* and scalars 0 *<sup>i</sup>* . However the problem to resolve becomes nonlinear in , *Ki I i q* (inequalities (27)-(28)/(30)-(31)). A method allowing the use of numerical tools to solve these constraints is given in the following.Toresolve the obtained BMI (bilinear matrix inequality) conditions using LMI tools (LMI toolbox of Matlab software for example), we propose to solve synthesis conditions (27) (or (30)) sequentially:


### **5. FTC strategy**

It is important to be able to carry out fault detection and isolation before faults have a drastic effect on the system performance. Even in case of system changes, faults should be detected and isolated. Observer based estimator schemes are used to generate residual signals corresponding to the difference between measured and estimated variables (Chen and Patton, 1999). The residual signals are processed using either deterministic (e.g. using fixed or variable thresholds) (Ding, Schneider, Ding and Rehm, 2005) or stochastic techniques (based upon decision theory) (Chen and Liu, 2000). Here, the first one is used.

Vehicle Fault Tolerant Control Using a Robust Output Fuzzy Controller Design 263

*<sup>F</sup>* (41)

*<sup>F</sup>* (42)

(40)

*y*

*r*

where

as:

observer *i*.

For failure of sensor 1

For failure of sensor 2

implied by the two possible values of *F* .

the TS fuzzy model of the vehicle as described in section III.

. 2

1 2

1 2

*i*

1

*i*

The TS fuzzy controller is represented as follows

1

*i*

. 2

*i*

1 1


*a Cx D*

2 2


*r Cx D*

ˆ ˆ (

*y i f i if*



ˆ ˆ (

**Observer-based FDI design** 

*a*

*i if*

1 0

0 1

We also assume that at any time one sensor only fails at the most. This assumption has been

If each , , 1,2 *<sup>l</sup> A C il i i* is observable, then it is possible to construct a TS fuzzy observer for

For observer 1, the state is estimated from the output of the first sensor ( *<sup>y</sup> a* ). It is given

11 1

 

For observer 2, the state is estimated from the output of the second sensor (*r* ). It is given as:

2 2 2

*i f i fi f i z i*

where *<sup>l</sup> Ci* and *<sup>l</sup> Di* are the *lth* rows of matrices *Ci* and *Di* (equations 10) respectively and

the state estimation, the lateral acceleration estimation and yaw rate estimation with

 

 

ˆˆ ˆ (

*x Ax B BM G r r*

2 2

*i f i if*

( , 1,2) *<sup>l</sup> G il <sup>i</sup>* are the constant observer gains to be determined. ˆ*<sup>i</sup> <sup>x</sup>* , ˆ*yi <sup>a</sup>* and ˆ

*i f i fi f i z i y y*

 

ˆˆ ˆ () (

*x t Ax B BM G a a*

1 1

1

2

(43)

(44)

*r* are respectively

*i*

*y C x D Ff*

The method that we propose is illustrated in figure 2, where it can be seen that the FDI functional block uses two observers, each one is driven by a single sensor output. The failure is detected first, and then the faulty sensor is identified. After that, the state variables are reconstructed from the output of the healthy sensor. The lateral control system enters the degraded mode that guaranteed stability and an acceptable level of performance.

Figure 2 shows the block diagram of the proposed closed system, *<sup>T</sup> <sup>y</sup> a r* is the output vector of the system, where *<sup>y</sup> a* denotes the lateral acceleration and *r* is the yaw rate about the center of gravity. Two observer based controllers are designed, one based on the observer that uses the measurement of lateral acceleration *<sup>y</sup> a* and the other one based on the observer that uses the measurement of yaw rate *r* .

**Figure 3.** Block diagram of the observer-based FTC

#### *Assumptions*

Let , 1,2 *<sup>l</sup> C il <sup>i</sup>* denote the *lth* row of matrix *Ci* (12c.). We assume that , *<sup>l</sup> Ai i <sup>C</sup>* are observable, which implies that it is possible to estimate the state through either the first output ( *<sup>y</sup> a* ) or the second one (*r* ) for the vehicle model (15).

Sensor failures are modeled as additive signals to sensor outputs

Vehicle Fault Tolerant Control Using a Robust Output Fuzzy Controller Design 263

$$\mathbf{y} = \begin{pmatrix} a\_y \\ r \end{pmatrix} = \mathbf{C}\_i \mathbf{x} + D\_i \boldsymbol{\mathcal{S}}\_f + Ff \tag{40}$$

where

262 Fuzzy Controllers – Recent Advances in Theory and Applications

the observer that uses the measurement of yaw rate *r* .

**Figure 3.** Block diagram of the observer-based FTC

output ( *<sup>y</sup> a* ) or the second one (*r* ) for the vehicle model (15).

Sensor failures are modeled as additive signals to sensor outputs

Let , 1,2 *<sup>l</sup> C il <sup>i</sup>* denote the *<sup>l</sup>*

*Assumptions* 

Patton, 1999). The residual signals are processed using either deterministic (e.g. using fixed or variable thresholds) (Ding, Schneider, Ding and Rehm, 2005) or stochastic techniques

The method that we propose is illustrated in figure 2, where it can be seen that the FDI functional block uses two observers, each one is driven by a single sensor output. The failure is detected first, and then the faulty sensor is identified. After that, the state variables are reconstructed from the output of the healthy sensor. The lateral control system enters the

vector of the system, where *<sup>y</sup> a* denotes the lateral acceleration and *r* is the yaw rate about the center of gravity. Two observer based controllers are designed, one based on the observer that uses the measurement of lateral acceleration *<sup>y</sup> a* and the other one based on

*th* row of matrix *Ci* (12c.). We assume that , *<sup>l</sup> Ai i <sup>C</sup>* are

observable, which implies that it is possible to estimate the state through either the first

*<sup>y</sup> a r* is the output

(based upon decision theory) (Chen and Liu, 2000). Here, the first one is used.

degraded mode that guaranteed stability and an acceptable level of performance.

Figure 2 shows the block diagram of the proposed closed system, *<sup>T</sup>*

For failure of sensor 1

$$F = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \tag{41}$$

For failure of sensor 2

$$F = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \tag{42}$$

We also assume that at any time one sensor only fails at the most. This assumption has been implied by the two possible values of *F* .

#### **Observer-based FDI design**

If each , , 1,2 *<sup>l</sup> A C il i i* is observable, then it is possible to construct a TS fuzzy observer for the TS fuzzy model of the vehicle as described in section III.

For observer 1, the state is estimated from the output of the first sensor ( *<sup>y</sup> a* ). It is given as:

$$\begin{aligned} \dot{\hat{\boldsymbol{x}}}\_{1}(t) &= \sum\_{i=1}^{2} \mu\_{i}(\mid \boldsymbol{\alpha}\_{\boldsymbol{f}} \mid \mathbb{I}) \Big( A\_{i}\hat{\boldsymbol{x}}\_{1} + B\_{\hat{\boldsymbol{f}}}\boldsymbol{\delta}\_{\boldsymbol{f}} + B\_{i}\boldsymbol{M}\_{z} + G\_{i}^{1} \Big(\boldsymbol{a}\_{\boldsymbol{y}} - \hat{\boldsymbol{a}}\_{\boldsymbol{y}1}\Big) \Big) \\ \hat{\boldsymbol{a}}\_{y1} &= \sum\_{i=1}^{2} \mu\_{i}(\mid \boldsymbol{\alpha}\_{\boldsymbol{f}} \mid \mathbb{I}) \Big(\mathbb{C}\_{i}^{1}\hat{\boldsymbol{x}}\_{1} + D\_{i}^{1}\boldsymbol{\delta}\_{\boldsymbol{f}}\Big) \end{aligned} \tag{43}$$

For observer 2, the state is estimated from the output of the second sensor (*r* ). It is given as:

$$\begin{aligned} \dot{\hat{X}}\_2 &= \sum\_{i=1}^2 \mu\_i (\mid \boldsymbol{\alpha}\_f \mid \mathbf{l}) \Big( \boldsymbol{A}\_i \hat{\mathbf{x}}\_2 + \mathbf{B}\_{\hat{f}} \boldsymbol{\delta}\_f + \mathbf{B}\_i \boldsymbol{M}\_z + \mathbf{G}\_i^2 \Big( \boldsymbol{r} - \hat{\mathbf{r}}\_2 \Big) \Big) \\ \hat{r}\_2 &= \sum\_{i=1}^2 \mu\_i (\mid \boldsymbol{\alpha}\_f \mid \mathbf{l}) \Big( \mathbf{C}\_i^2 \hat{\mathbf{x}}\_2 + \mathbf{D}\_i^2 \boldsymbol{\delta}\_f \Big) \end{aligned} \tag{44}$$

where *<sup>l</sup> Ci* and *<sup>l</sup> Di* are the *lth* rows of matrices *Ci* and *Di* (equations 10) respectively and ( , 1,2) *<sup>l</sup> G il <sup>i</sup>* are the constant observer gains to be determined. ˆ*<sup>i</sup> <sup>x</sup>* , ˆ*yi <sup>a</sup>* and ˆ *i r* are respectively the state estimation, the lateral acceleration estimation and yaw rate estimation with observer *i*.

The TS fuzzy controller is represented as follows

$$M\_z(t) = -\sum\_{i=1}^{2} \mu\_i(\mid \alpha\_f \mid) K\_i \hat{x}\_l(t) \tag{45}$$

Vehicle Fault Tolerant Control Using a Robust Output Fuzzy Controller Design 265

Nominal stiffness Coefficients (N/rad) Cf10 Cf20 Cr10 Cr20

(50)

uncertainties, stiffness coefficients *Cfi* and *Cri* are supposed to be varying depending on road

ar m

0.4 0 0 0.4

*D D*

We point out that only the yaw rate is directly measurable by a yaw rate sensor (gyroscope),

By solving the derived stability conditions of theorem 1, the designed controller and

1 1

2 2

Figure 4 shows the additive signals that represent sensor failures. The first one has been added to sensor 1 output between 2s and 8s, and the second one has been added to sensor 2

5 5

1 2 *K K* 10 -1.1914 1.1616 , 10 -1.2623 1.3102 (51)

1 2 -35.9102 6.2245 , -223.2973 43.8026 *T T G G* (52)

1 2 -50.7356 5.7456 , -28.2271 3.0782 *T T G G* (53)

Values 3214 1740 1.04 1.76 20 60712 4812 60088 3455

U m/s

af m

1 2

the lateral velocity is unavailable and is estimated using the proposed observer.

adhesion.

**Table 1.**

Parameters Iz

with the following uncertainties

observer gains are:

output between 10s and 16s.

**Figure 4.** Failure of sensors

Kg.m2

m kg

with

*l* 1 If sensor 2 fails

*l* 2 If sensor 1 fails

We define the residual signals as

$$R\_{1, \
ay} = \hat{a}\_{y1} - a\_y \ R\_{2, \
ay} = \hat{a}\_{y2} - a\_y \tag{46}$$

$$R\_{1, \, r} = \hat{r}\_1 - r \; R\_{2, \, r} = \hat{r}\_2 - r \tag{47}$$

Note that *R*1, *ay* and *R*1, r are related to observer 1 and *R*2, *ay* and *R*2, r are related to observer 2 with

$$
\begin{pmatrix} \hat{a}\_{y1} \\ \hat{r}\_1 \end{pmatrix} = \sum\_{i=1}^{2} h\_i (\mid \alpha\_f \mid \} \left( \mathbb{C}\_i \, \hat{\mathbf{x}}\_1 + D\_i \, \delta\_f \right) \tag{48}
$$

$$
\begin{pmatrix} \hat{\mathbf{a}}\_{y2} \\ \hat{\mathbf{r}}\_{2} \end{pmatrix} = \sum\_{i=1}^{2} h\_i (\mid \alpha\_f \mid \mathbf{l}) \Big(\mathbf{C}\_i \hat{\mathbf{x}}\_2 + D\_i \, \delta\_f\big) \tag{49}
$$

The FDI scheme developed in this study follows a classical strategy such as the wellestablished observer based FDI methods (Isermann, 2001; Huang and Tomizuka, 2005; Oudghiri, Chadli and El Hajjaji, 2007). The residual signals 1, 1, 2, 2, ,, , *R RR R ay r ay r* are used for the estimation of the model uncertainties and then, for the construction of model uncertainty indicators. The decision bloc is based on the analysis of these residual signals. Indeed faults are detected and then switching operates according to the following scheme:

**Detection**: *if* max R , R , R , R 1,ay 1,r 2,ay 2,r *<sup>h</sup> T then* the fault has occurred where *<sup>h</sup> T* the prescribed threshold is and . denotes the Euclidian norm at each time instant.

**Switching:** *if* R , R > R , R 1,ay 1,r 2,ay 2,r *then* switch to observer 2. *If not* switch to observer 1.

Since model uncertainties and sensor noise also contribute to nonzero residual signals under the normal operation, threshold *<sup>h</sup> T* must be large enough to avoid false alarms while small enough to avoid missed alarms. In this paper, we do not further discuss the selection of the thresholds.

#### **Simulation results**

To show the effectiveness of the proposed FTC based on bank of observer algorithm, we have carried out some simulations using the vehicle model (1) and MATLAB software. In the design, the vehicle parameters considered are given in table 1. To take account of uncertainties, stiffness coefficients *Cfi* and *Cri* are supposed to be varying depending on road adhesion.


#### **Table 1.**

264 Fuzzy Controllers – Recent Advances in Theory and Applications

with

*l* 1 If sensor 2 fails *l* 2 If sensor 1 fails

observer 2 with

observer 1.

thresholds.

**Simulation results** 

We define the residual signals as

2

1 ( ) (| |) ˆ ( ) *z i f il*

 *Kx t*

Note that *R*1, *ay* and *R*1, r are related to observer 1 and *R*2, *ay* and *R*2, r are related to

<sup>ˆ</sup> ( <sup>ˆ</sup> <sup>ˆ</sup> | |) *<sup>y</sup>*

<sup>ˆ</sup> ( <sup>ˆ</sup> <sup>ˆ</sup> | |) *<sup>y</sup>*

The FDI scheme developed in this study follows a classical strategy such as the wellestablished observer based FDI methods (Isermann, 2001; Huang and Tomizuka, 2005; Oudghiri, Chadli and El Hajjaji, 2007). The residual signals 1, 1, 2, 2, ,, , *R RR R ay r ay r* are used for the estimation of the model uncertainties and then, for the construction of model uncertainty indicators. The decision bloc is based on the analysis of these residual signals. Indeed faults

**Detection**: *if* max R , R , R , R 1,ay 1,r 2,ay 2,r *<sup>h</sup> T then* the fault has occurred where *<sup>h</sup> T* the

**Switching:** *if* R , R > R , R 1,ay 1,r 2,ay 2,r *then* switch to observer 2. *If not* switch to

Since model uncertainties and sensor noise also contribute to nonzero residual signals under the normal operation, threshold *<sup>h</sup> T* must be large enough to avoid false alarms while small enough to avoid missed alarms. In this paper, we do not further discuss the selection of the

To show the effectiveness of the proposed FTC based on bank of observer algorithm, we have carried out some simulations using the vehicle model (1) and MATLAB software. In the design, the vehicle parameters considered are given in table 1. To take account of

<sup>2</sup>

<sup>2</sup>

*h Cx D*

*i f i if*

*h Cx D*

*i f i if*

1

2

(49)

(48)

(45)

1, 1 ˆ *R aa ay y y* 2, 2 ˆ *R aa ay y y* (46)

1, 1ˆ *R rr <sup>r</sup>* 2, 2ˆ *R rr <sup>r</sup>* (47)

*i*

*M t*

1

2

*a*

*r*

*a*

*r*

1 1

2 1

*i*

are detected and then switching operates according to the following scheme:

prescribed threshold is and . denotes the Euclidian norm at each time instant.

*i*

with the following uncertainties

$$D\_1 = D\_2 = \begin{pmatrix} 0.4 & 0 \\ 0 & 0.4 \end{pmatrix} \tag{50}$$

We point out that only the yaw rate is directly measurable by a yaw rate sensor (gyroscope), the lateral velocity is unavailable and is estimated using the proposed observer.

By solving the derived stability conditions of theorem 1, the designed controller and observer gains are:

$$K\_1 = 10^5 \begin{pmatrix} -1.1914 & 1.1616 \end{pmatrix}, K\_2 = 10^5 \begin{pmatrix} -1.2623 & 1.3102 \end{pmatrix} \tag{51}$$

$$\mathbf{G}\_1^1 = \begin{pmatrix} \cdot \text{35.9102} & \cdot \text{6.2245} \end{pmatrix}^T, \mathbf{G}\_2^1 = \begin{pmatrix} \cdot \text{223.2973} & \mathbf{43.8026} \end{pmatrix}^T \tag{52}$$

$$\mathbf{G}\_1^2 = \begin{pmatrix} \text{-} \mathbf{50}.7356 & \mathbf{5.7456} \end{pmatrix}^T, \mathbf{G}\_2^2 = \begin{pmatrix} \text{-} \mathbf{28.2271} & \mathbf{3.0782} \end{pmatrix}^T \tag{53}$$

Figure 4 shows the additive signals that represent sensor failures. The first one has been added to sensor 1 output between 2s and 8s, and the second one has been added to sensor 2 output between 10s and 16s.

**Figure 4.** Failure of sensors

Vehicle Fault Tolerant Control Using a Robust Output Fuzzy Controller Design 267

. Indeed, we can see that by using the

<sup>2</sup> x 10-3 Front steering angle

<sup>1</sup> Lateral velocity and its estimated

<sup>1</sup> Yaw velocity and its estimated

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> <sup>18</sup> <sup>20</sup> <sup>0</sup>

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> -1

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> -1

All the simulations are realized on the nonlinear model given in (1) with vehicle speed 20 m/s. The simulation results are given in figures 5 and 6 with and without the FTC strategy. In figure 5 the law control is based on one observer (observer 2) without using the switching bloc. We can see between 10s and 16s that the vehicle lost its performance just after the yaw

Figure 6 shows vehicle state variables and their estimated signals, when the law control is based on the bank of two observers with the switch bloc. We can note that the vehicle remains stable despite the presence of faults, which shows the effectiveness of the proposed

The switching from observer 1 to observer 2 is visualized clearly at t ≈ 8s (figure 7). We notice that switching observers is carried out without loss of control of the system state.

The second simulations are realized to show the importance of the proposed FTC method based on an output fuzzy controller, on the stability of the vehicle dynamics. Simulations propose to show the difference between the vehicle dynamics behaviour with TS fuzzy yaw control based on a fuzzy observer (figure 6) and its behaviour with the linear yaw control based on a linear observer (figure 8). Figure 8 clearly shows that the linear control fails to maintain the stability of the vehicle in presence of sensor faults despite a short magnitude of the additive signal ( *f* 0.1

**Figure 8.** a. Additive signals to sensors output. b. Vehicle states without sensor faults using linear

Using an algorithm based on a bank of two observers, a fault tolerant control has been presented. The vehicle nonlinear model is first represented by an uncertain Takagi-Sugeno

detection sensors faults, the results are better than these with linear control.

proposed fuzzy yaw control based on a fuzzy observer and the algorithm proposed for

1

0

0

rate sensor became faulty.

) and also a very low front steering angle 0.001 *<sup>f</sup>*

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> <sup>18</sup> <sup>20</sup> <sup>0</sup>

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> <sup>18</sup> <sup>20</sup> <sup>0</sup>

0.2 Additive signal to sensor 1 output

0.2 Additive signal to sensor 2 output

control with road friction coefficient fixed at 0.5

**6. Conclusion** 

0.05 0.1 0.15

0.05 0.1 0.15

FTC strategy.

**Figure 5.** Vehicle sates without FTC strategy

**Figure 6.** Vehicle sates with FTC strategy

**Figure 7.** Zooms in of figure 5 at t ≈ 8s

All the simulations are realized on the nonlinear model given in (1) with vehicle speed 20 m/s. The simulation results are given in figures 5 and 6 with and without the FTC strategy. In figure 5 the law control is based on one observer (observer 2) without using the switching bloc. We can see between 10s and 16s that the vehicle lost its performance just after the yaw rate sensor became faulty.

Figure 6 shows vehicle state variables and their estimated signals, when the law control is based on the bank of two observers with the switch bloc. We can note that the vehicle remains stable despite the presence of faults, which shows the effectiveness of the proposed FTC strategy.

The switching from observer 1 to observer 2 is visualized clearly at t ≈ 8s (figure 7). We notice that switching observers is carried out without loss of control of the system state.

The second simulations are realized to show the importance of the proposed FTC method based on an output fuzzy controller, on the stability of the vehicle dynamics. Simulations propose to show the difference between the vehicle dynamics behaviour with TS fuzzy yaw control based on a fuzzy observer (figure 6) and its behaviour with the linear yaw control based on a linear observer (figure 8). Figure 8 clearly shows that the linear control fails to maintain the stability of the vehicle in presence of sensor faults despite a short magnitude of the additive signal ( *f* 0.1 ) and also a very low front steering angle 0.001 *<sup>f</sup>* . Indeed, we can see that by using the proposed fuzzy yaw control based on a fuzzy observer and the algorithm proposed for detection sensors faults, the results are better than these with linear control.

**Figure 8.** a. Additive signals to sensors output. b. Vehicle states without sensor faults using linear control with road friction coefficient fixed at 0.5

### **6. Conclusion**

266 Fuzzy Controllers – Recent Advances in Theory and Applications

**Figure 5.** Vehicle sates without FTC strategy

0 0.1

0 0.2

0 0.2

> 0.08 0.1 0.12 0.14 0.16

> 0.12 0.14 0.16 0.18

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> <sup>18</sup> <sup>20</sup> -0.1

**Lateral velocity and its estimated**

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> <sup>18</sup> <sup>20</sup> -0.2

**Yaw rate and its estimated**

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> <sup>18</sup> <sup>20</sup> -0.2

**Lateral velocity and its estimated**

7.9 <sup>8</sup> 8.1 8.2 8.3 8.4 8.5 0.06

**Yaw rate and its estimated**

Lateral velocity Estimation of lateral velocity

> Yaw rate Estimation of yaw rate

7.9 <sup>8</sup> 8.1 8.2 8.3 8.4 8.5 0.1

**Front Steering Angle**

**Figure 6.** Vehicle sates with FTC strategy

**Figure 7.** Zooms in of figure 5 at t ≈ 8s

Using an algorithm based on a bank of two observers, a fault tolerant control has been presented. The vehicle nonlinear model is first represented by an uncertain Takagi-Sugeno fuzzy model. Then, a robust output feedback controller is designed using LMI terms. Based on the designed robust observer-based controller, a fault tolerant control method is utilized. This method uses a technique based on the switching principle, allowing not only to detect sensor failures but also to adapt the control law in order to compensate the effect of the faults by maintaining the stability of the vehicle and the nominal performances. Simulation results show that the proposed FTC strategy based on robust output TS fuzzy controller are better than these with linear control in spite of a short magnitude of the additive signal and very low front steering angle.

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M. Chadli and A. El Hajjaji *Laboratoire de Modélisation, Information et Système (M.I.S), Amiens, France* 

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*Laboratoire de Modélisation, Information et Système (M.I.S), Amiens, France* 


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**Chapter 12** 

© 2012 Ponce et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Wheelchair and Virtual Environment** 

There are many kinds of diseases and injuries that produce mobility problems. The people affected with any disability must deal with a new lifestyle, specifically people with tetraplegia. According to ICF [1], people with tetraplegia have damages associated to the power of muscles of all limbs, tone of muscles of all limbs, resistance of all muscles of the body and endurance of all muscles of the body. The main objective of this project was to help disabled people to move any member of their body, although this wheelchair can be used for persons with mobility problems, doctors should not recommend it to all patients

Currently, there is not an efficient system that covers the different needs that a person with quadriplegia could have. Their mobility is reduced by physical injury and, depending on the extent of the damage nursing and family assistance is required. Even though many platforms have been developed to address this problem, there is not an integrated system that allows the patient to move autonomously from one place to another, thus limiting the patient to remain at rest all the time. In previous research projects completed in Canada and in the United States, such as wheelchairs controlled with the tongue [2] and a wheelchair controlled with head and shoulder movements [3]. Those systems provide mobility for the person with any injury in functions related to muscle strength. This work offers a different alternative for the patient and aims to build an autonomous wheelchair that can afford enough motion capacity to transport a person with quadriplegia. Different kinds of controls are provided, so the trajectories required by the patient must be controlled using ocular

An existing brand of electric wheelchair was used (the commercial Quickie wheelchair

and reproduction in any medium, provided the original work is properly cited.

because it reduces muscle movement, which could lead to muscular dystrophy.

**Trainer by Intelligent Control** 

Pedro Ponce, Arturo Molina and Rafael Mendoza

Additional information is available at the end of the chapter

movements or voice commands, among others.

model P222 [4] with a Qtronix controller).

http://dx.doi.org/10.5772/48410

**1. Introduction** 

