**Design and Simulation of Anfis Controller for Virtual-Reality-Built Manipulator**

Yousif I. Al Mashhadany

314 Fuzzy Controllers – Recent Advances in Theory and Applications

Bin Feng, Guofang Gong, and Huayong Yang.,"Self-tuning parameter fuzzy PID temperature control in a large hydraulic system," International Conference on

Advanced Intelligent Mechatronics, IEEE/ASME,2009, pp.1418-142

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48383

## **1. Introduction**

Fuzzy logic (FL) and artificial neural networks (ANNs), despite their successful use in many challenging control situations, still have drawbacks that limit them to only some applications. Their combined advantages have thus become the subject of much research into ways of overcoming their disadvantages. Neuro-fuzziness is one resulting rapidly emerging field. ANFIS network, proposed by Jang, is one popular neuro-fuzzy system [1-4].

For specific-problem training of an ANFIS network, [1] proposes use of hybrid learning rule, which combines gradient descent technique and least-square estimator (LSE). Being a method of supervised learning, it needs a teaching signal, which can be difficult to provide when the ANFIS network is to be a feedback controller, as the desired control actions that the teaching signal represents are unknown. Literatures have proposed several ANFIS learning methods in which ANFIS is applied as a MIMO controller. Djukanović *et al.*, for example, uses a special ANFIS learning technique called temporal back propagation (TBP); control of a nonlinear MIMO system is by considering both the controller and the plant as a single unit each time step. The method, however, is complex and distinctly computationheavy [5-9].

Another training approach for ANFIS-controller of nonlinear MIMO systems is inverse learning; the ANFIS network is trained to learn the inverse dynamics of the plant it controls. Its success, however, is crucial on three elements: accurate modeling of the original system (a problem when the system is complex), availability of the system's inverse dynamics (they do not always exist), and appropriate distribution of the training data (could be impossible, given the constraints of the system's dynamics). [10-13] has another training approach besides the ones already mentioned.

© 2012 Al Mashhadany, 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.

Present robot navigation systems demand controllers that can solve complex problems under uncertain and dynamic environments. ANFIS garners interest because it offers the benefits of both neural network (NN) and FL, and removes their individual disadvantages by combining them on their common features. ANN is a new motivation for studies into FL. It can be used as a universal learning paradigm in any smooth parameterized models, including fuzzy inference systems [14-15].

Design and Simulation of Anfis Controller for Virtual-Reality-Built Manipulator 317

Virtual reality (VR) has become important to applications in engineering, medicine, statistics, and other areas where 3D images can aid understanding of system complexity. The interactability of a virtual system can in many applications be enhanced by touch sensing. Haptic feedback can convey to a human user, virtual environment forces. It has become useful in tele-surgery, where a master manipulator guides robotic surgical tools while providing realistic-force feedback to the surgeon. It is already available in many systems under development but these often still are specifically developed research

prototypes (i.e., providing specific-force feedbacks and for specific problems) [29, 30].

easy to understand but suits only a few types of manipulators [31, 32].

DOF shoulder joint, 1-DOF elbow joint, and 3-DOF wrist joint [33].

the simulation of the case study, the system design, and the conclusions.

joint *i*-1 can thus be described by a 4×4 homogeneous matrix *i-1Ti* [34]:

**2. Kinematics of the 7-DOF manipulator** 

Kinematics analysis is key to motion control of humanoid manipulators. Its main problems are forward kinematics and inverse kinematics. The inverse kinematics of a 7-DOF manipulator has multiple solutions; obtaining anthropomorphic solutions is thus a problem. Detailed research in it has yet to be found, though works on design, control, and obstacle avoidance in humanoid manipulator exist. In serial manipulators, between forward kinematics and inverse kinematics, solution of the latter is a lot more difficult. Solution methods of inverse kinematics generally are numerical, analytical, or geometric. Numerical method is most widely used, but it cannot obtain all possible solutions. Analytical method can derive all possible solutions but is much more difficult. Geometric method is simple and

A human-manipulator skeleton has 7-DOF mechanism; other degrees of freedom of the arm are performed by the tendon. A 7-DOF anthropomorphic arm has been developed; it has the manipulability criterion and uses 3-DOF planar manipulator theory (i.e., the mechanism of the human limb can be modeled by three moving links). 3-DOF planar mechanism is fundamental to an anthropomorphic arm; its evaluation criteria can be used to analyze the arm's operational performance. 3-DOF planar manipulator theory is a supposition that the three link lengths are of the upper arm, the forearm, and the hand, and the three joints are 3-

This chapter presents the design of an ANFIS controller of a VR manipulator model and simulation of the ANFIS-controlled system's command execution. Simulation results of the 7-DOF human-manipulator show an improved control system. Section II presents the manipulator's kinematic model whereas Section III the structure of the ANFIS controller and its learning methods. Section IV presents the manipulator's VR model, whereas Section V,

Forward kinematics was used to calculate the racket's posture according to the joint angles. It is quite useful for analysis of the manipulator's workspace and verification of the inverse kinematics. Denoting the shoulder width as *D*, the upper arm length as *L1*, and the lower arm length as *L2*, the position of the shoulder is thus P1(0, -*D*, 0) and the position of the elbow is P*2* (*xp2* , *yp2* , *zp2* ). Denoting the joint angles as *q1* … *q7*, the posture of joint *i* relative to

Traditional robot control methods rely on strong mathematical modeling, analysis, and synthesis. Existing approaches suit control of mobile robots operating in unknown environments and performing tasks that require movement in dynamic environments. Operational tasks in unstructured environments such as remote planets and hazardous waste sites, however, are more complex, yet the analytical modeling is inadequate. Many researchers and engineers have tried to solve the navigational problems of mobile robot systems [16-18]. Though fuzzy systems can use knowledge expressed in linguistic rules (and thus could implement expert human knowledge and experience), fuzzy controller lacks a systematic design method. Tuning of membership-function parameters takes time. NN learning techniques can automate the process so development can be hastened and performance improved. The combination of NN and FL has produced neuro-fuzzy controllers and created their present popularity. In real-time autonomous navigation, a robot must be able to sense its environment, interpret the sensed information to obtain knowledge of its position and environment, and plan a route that gets it to the target position from an initial position and with obstacle avoidance and control of its direction and velocity. Ng et al. [19-23] propose a neural-integrated fuzzy controller that integrates FL representation of human knowledge with NN learning to solve nonlinear dynamic control problems. Pham et al. focus on developing intelligent multi-agent robot teams capable of both autonomous action and dynamic-environment collaboration in achieving team objectives. They also propose a neuro-fuzzy adaptive action selection architecture that enables a team of robot agents to achieve adaptive cooperative control of cooperative tasks, track dynamic targets, and push boxes. Crestani et al. defines autonomous navigation in mobile robots as a search process within a navigation environment that contains obstacles and targets, and propose a fuzzy-NN controller that considers navigation direction and navigation velocity as controllable. Rutkowski et al. derived a flexible neuro-fuzzy inference system; their approach increases structural and design flexibility in neuro-fuzzy systems. Hui et al. and Rusu et al. discuss neuro-fuzzy controllers for sensor-based mobile robot navigation. Garbi et al. implemented an adaptive neuro-fuzzy inference system in robotic vehicle navigation [24-26].

Robots are one way to improve industrial automation productivity. Robotic manipulators have been used in routine and dangerous-environment manufacturing jobs. They are highly nonlinear dynamic systems subject to uncertainties. Obtaining accurate dynamic equations for their control laws is thus difficult. Uncertainties in their dynamic models include unknown grasped payloads and unknown frictional coefficients. Adaptive control or modelfree intelligent control has been much proposed as able to compensate for those uncertainties [26-28].

Virtual reality (VR) has become important to applications in engineering, medicine, statistics, and other areas where 3D images can aid understanding of system complexity. The interactability of a virtual system can in many applications be enhanced by touch sensing. Haptic feedback can convey to a human user, virtual environment forces. It has become useful in tele-surgery, where a master manipulator guides robotic surgical tools while providing realistic-force feedback to the surgeon. It is already available in many systems under development but these often still are specifically developed research prototypes (i.e., providing specific-force feedbacks and for specific problems) [29, 30].

Kinematics analysis is key to motion control of humanoid manipulators. Its main problems are forward kinematics and inverse kinematics. The inverse kinematics of a 7-DOF manipulator has multiple solutions; obtaining anthropomorphic solutions is thus a problem. Detailed research in it has yet to be found, though works on design, control, and obstacle avoidance in humanoid manipulator exist. In serial manipulators, between forward kinematics and inverse kinematics, solution of the latter is a lot more difficult. Solution methods of inverse kinematics generally are numerical, analytical, or geometric. Numerical method is most widely used, but it cannot obtain all possible solutions. Analytical method can derive all possible solutions but is much more difficult. Geometric method is simple and easy to understand but suits only a few types of manipulators [31, 32].

A human-manipulator skeleton has 7-DOF mechanism; other degrees of freedom of the arm are performed by the tendon. A 7-DOF anthropomorphic arm has been developed; it has the manipulability criterion and uses 3-DOF planar manipulator theory (i.e., the mechanism of the human limb can be modeled by three moving links). 3-DOF planar mechanism is fundamental to an anthropomorphic arm; its evaluation criteria can be used to analyze the arm's operational performance. 3-DOF planar manipulator theory is a supposition that the three link lengths are of the upper arm, the forearm, and the hand, and the three joints are 3- DOF shoulder joint, 1-DOF elbow joint, and 3-DOF wrist joint [33].

This chapter presents the design of an ANFIS controller of a VR manipulator model and simulation of the ANFIS-controlled system's command execution. Simulation results of the 7-DOF human-manipulator show an improved control system. Section II presents the manipulator's kinematic model whereas Section III the structure of the ANFIS controller and its learning methods. Section IV presents the manipulator's VR model, whereas Section V, the simulation of the case study, the system design, and the conclusions.

## **2. Kinematics of the 7-DOF manipulator**

316 Fuzzy Controllers – Recent Advances in Theory and Applications

including fuzzy inference systems [14-15].

vehicle navigation [24-26].

uncertainties [26-28].

Present robot navigation systems demand controllers that can solve complex problems under uncertain and dynamic environments. ANFIS garners interest because it offers the benefits of both neural network (NN) and FL, and removes their individual disadvantages by combining them on their common features. ANN is a new motivation for studies into FL. It can be used as a universal learning paradigm in any smooth parameterized models,

Traditional robot control methods rely on strong mathematical modeling, analysis, and synthesis. Existing approaches suit control of mobile robots operating in unknown environments and performing tasks that require movement in dynamic environments. Operational tasks in unstructured environments such as remote planets and hazardous waste sites, however, are more complex, yet the analytical modeling is inadequate. Many researchers and engineers have tried to solve the navigational problems of mobile robot systems [16-18]. Though fuzzy systems can use knowledge expressed in linguistic rules (and thus could implement expert human knowledge and experience), fuzzy controller lacks a systematic design method. Tuning of membership-function parameters takes time. NN learning techniques can automate the process so development can be hastened and performance improved. The combination of NN and FL has produced neuro-fuzzy controllers and created their present popularity. In real-time autonomous navigation, a robot must be able to sense its environment, interpret the sensed information to obtain knowledge of its position and environment, and plan a route that gets it to the target position from an initial position and with obstacle avoidance and control of its direction and velocity. Ng et al. [19-23] propose a neural-integrated fuzzy controller that integrates FL representation of human knowledge with NN learning to solve nonlinear dynamic control problems. Pham et al. focus on developing intelligent multi-agent robot teams capable of both autonomous action and dynamic-environment collaboration in achieving team objectives. They also propose a neuro-fuzzy adaptive action selection architecture that enables a team of robot agents to achieve adaptive cooperative control of cooperative tasks, track dynamic targets, and push boxes. Crestani et al. defines autonomous navigation in mobile robots as a search process within a navigation environment that contains obstacles and targets, and propose a fuzzy-NN controller that considers navigation direction and navigation velocity as controllable. Rutkowski et al. derived a flexible neuro-fuzzy inference system; their approach increases structural and design flexibility in neuro-fuzzy systems. Hui et al. and Rusu et al. discuss neuro-fuzzy controllers for sensor-based mobile robot navigation. Garbi et al. implemented an adaptive neuro-fuzzy inference system in robotic

Robots are one way to improve industrial automation productivity. Robotic manipulators have been used in routine and dangerous-environment manufacturing jobs. They are highly nonlinear dynamic systems subject to uncertainties. Obtaining accurate dynamic equations for their control laws is thus difficult. Uncertainties in their dynamic models include unknown grasped payloads and unknown frictional coefficients. Adaptive control or modelfree intelligent control has been much proposed as able to compensate for those

Forward kinematics was used to calculate the racket's posture according to the joint angles. It is quite useful for analysis of the manipulator's workspace and verification of the inverse kinematics. Denoting the shoulder width as *D*, the upper arm length as *L1*, and the lower arm length as *L2*, the position of the shoulder is thus P1(0, -*D*, 0) and the position of the elbow is P*2* (*xp2* , *yp2* , *zp2* ). Denoting the joint angles as *q1* … *q7*, the posture of joint *i* relative to joint *i*-1 can thus be described by a 4×4 homogeneous matrix *i-1Ti* [34]:

$$\begin{aligned} \;^0T\_1 &= \begin{bmatrix} cq\_1 & 0 & sq\_1 & 0 \\ 0 & 1 & 0 & -D \\ -sq\_1 & 0 & cq\_1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}^\prime & \;^1T\_2 = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & cq\_2 & -sq\_2 & 0 \\ 0 & sq\_2 & cq\_2 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}^\prime & \;^2T\_3 = \begin{bmatrix} cq\_3 & -sq\_3 & 0 & 0 \\ sq\_3 & cq\_3 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}^\prime \\ & \;^3T\_4 = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & cq\_4 & -sq\_4 & 0 \\ 0 & sq\_5 & cq\_5 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}^\prime, \;^5T\_5 = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & cq\_6 & -sq\_6 & 0 \\ 0 & sq\_6 & cq\_6 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}^\prime, \;^6T\_7 = \begin{bmatrix} cq\_7 & 0 & sq\_7 & 0 \\ 0 & 1 & 0 & -D \\ -sq\_7 & 0 & cq\_7 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}^\prime \end{aligned} \end{aligned} \tag{1}$$

Where: sqi ≡sin(qi); cqi ≡cos(qi)

Assuming the position of Joint-7 in the fixed coordinate is P3 (*xp3* , *yp3* , *zp3* ) and its pose as described by RPY (Roll Pitch Yaw) angles is (φ, θ, ψ), the posture of Joint-7 can thus be described by a homogenous matrix *0T7*:

$$\begin{aligned} \mathbf{^0T\_7} = \begin{bmatrix} c\rho c\theta & c\rho s\theta s\varphi - s\rho c\varphi & c\rho s\theta c\varphi + s\rho s\varphi & x\_{p3} \\ s\rho c\theta & s\rho s\theta s\varphi + c\rho c\varphi & s\rho s\theta c\varphi - c\rho s\varphi & y\_{p3} \\ -s\theta & c\theta s\varphi & c\theta c\varphi & z\_{p3} \\ 0 & 0 & 0 & 1 \end{bmatrix} \end{aligned} \tag{2}$$

Design and Simulation of Anfis Controller for Virtual-Reality-Built Manipulator 319

(6)

*i T*

(7)

( ) ( ) ( )

*p n p o*

*ik ik ik*

*p o <sup>J</sup> n o a*

Where *ni(nix, niy, niz), oi(oix, oiy, oiz), ai(aix, aiy, aiz)* and *pi(pix, piy,piz)* are the items of matrix *<sup>i</sup>* <sup>1</sup>

*i i yyy y*

*noap TTT*

The parameter *k* stands for the rotational axis of joint *i*. For example, if the joint rotates around the *x*-axis, then *k* is *x*. Racket speed *V* can be calculated by (4) whereas joint space

The inverse kinematics is used to calculate the joint angles according to the racket's posture. The manipulator is redundant, so the elbow's position is not uniquely determined when the racket's posture is given. The motion characteristics of a human arm show the mapping relationships between elbow position and racket posture. In a general configuration of the manipulator (see Fig. 2), any three points on the manipulator's neck, shoulder, elbow, and wrist are not collinear. The position of elbow P2, whose axis is P1P3, is in circle ' *O* . The

by positions P1 and P3, and arm length parameters *L1* and *L2*. Ω1 denotes the plane constructed by points P1, P2, and P3, and Ω2 the plane constructed by points O, P1, and P2. *α* denotes the separation angle between Ω1 and Ω2. Once the position of P3 is known, *α* can uniquely determine P2 elbow position. If the points O, P1, P3 are collinear whereas P1, P2, P3 are not, the plane Ω2 does not exist. Angle *α* can be defined as the separation angle between Ω1 and the horizontal plane. Also, if the points P1, P2, P3 are collinear, the plane Ω1 does not exist and P2 position can be calculated according to positions P1 and P3 and arm length parameters *L1* and *L2*. Considering the general configuration of the manipulator (see Fig. 2), the inverse kinematics can be solved if angle *α* and the racket's posture are given. According to the motion characteristics of human arms, the mapping relationship between *α* and the racket posture can be built offline. Denoting with *i* (*i*1, *i*2, *i*3) the unit vector pointing from P3

.

*q* cannot be uniquely determined because *J* is not a square matrix. This problem can

*i*

11 6 7 7 .......

with <sup>1</sup> ( ) *T T J JJ J* being the Moore-Penrose pseudo inverse matrix of *J*.

speed

.

be solved by Moore-Penrose method:

position of the center point ' ''

'

*i*

*i ik i ik i ik*

0001

*q JV* (8)

(,,) *OOO O xyz* and the radius *r* of the circle are determined

*zzzz*

*noap*

*xxx x*

*noap*

Matrix *0T7* also stands for the racket's posture and can be derived also as:

$$\begin{aligned} ^0T\_7 = ^0T\_1 ^1T\_2 ^2T\_3 ^3T\_4 ^4 ^5T\_5 ^5T\_6 ^6T\_7 \begin{bmatrix} n\_x & o\_x & a\_x & p\_x\\ n\_y & o\_y & a\_y & p\_y\\ n\_z & o\_z & a\_z & p\_z\\ 0 & 0 & 0 & 1 \end{bmatrix} \end{aligned} \tag{3}$$

From (1) and (2), the racket's posture can be calculated as:

$$\begin{cases} \begin{aligned} \varphi &= a \tan 2(n\_y, n\_x) \\ \theta &= a \tan 2(-n\_z, c\varphi n\_x + s\varphi n\_y) \\ \nu &= a \tan 2(s\varphi n\_x - c\varphi n\_y, -s\varphi o\_x + c\varphi o\_y) \end{aligned} & \text{for } \begin{aligned} \chi\_{p3} &= p\_x \\ \chi\_{p3} &= p\_y \end{aligned} \end{cases} \tag{4}$$

with atan2( ) being the four-quadrant inverse tangent function and (3) the manipulator's forward kinematics. All the commands (e.g., the racket's hitting position and hitting speed) are given by the visual system, in the operation space. They must be transformed into joint space values. Jacobian matrix is used to calculate the joint space speed according to the operation space speed. The mapping relationship between them is [35]:

$$V = f(q)\stackrel{\cdot}{q} \tag{5}$$

.

with *V* being the racket's speed in the operation space and *q* the joint space speed. *J* is the 6×7 Jacobian matrix and can be derived by differential transformation method. The *i*th item of *J* is:

$$J\_i = \begin{bmatrix} (p\_i \times n\_i)\_k \\ (p\_i \times o\_i)\_k \\ (p\_i \times o\_i)\_k \\ n\_{ik} \\ o\_{ik} \\ o\_{ik} \end{bmatrix} \tag{6}$$

Where *ni(nix, niy, niz), oi(oix, oiy, oiz), ai(aix, aiy, aiz)* and *pi(pix, piy,piz)* are the items of matrix *<sup>i</sup>* <sup>1</sup> *i T*

318 Fuzzy Controllers – Recent Advances in Theory and Applications

10 0 0 0 0 <sup>0</sup> <sup>0</sup> 0 0 , 0 0 0 01

*sq cq sq cq*

3 4 4 4 5 5 4 5 4 4

described by a homogenous matrix *0T7*:

7

From (1) and (2), the racket's posture can be calculated as:

of *J* is:

operation space speed. The mapping relationship between them is [35]:

with *V* being the racket's speed in the operation space and

 

 

Matrix *0T7* also stands for the racket's posture and can be derived also as:

0 0 123456 7 12 3 4 56 7

*cq sq sq cq T T*

00 0 1

Where: sqi ≡sin(qi); cqi ≡cos(qi)

*sq cq*

1 1 3 3

*cq sq cq sq <sup>D</sup> cq sq sq cq T TT*

0 0 10 0 0 0 0 0 10 <sup>0</sup> <sup>0</sup> 0 0 , , 00 0 0 0 0 10 0 0 0 1 0 0 0 1 0 0 01

Assuming the position of Joint-7 in the fixed coordinate is P3 (*xp3* , *yp3* , *zp3* ) and its pose as described by RPY (Roll Pitch Yaw) angles is (φ, θ, ψ), the posture of Joint-7 can thus be

0 3

*noap T TT T T T T T*

tan 2( , ) &

with atan2( ) being the four-quadrant inverse tangent function and (3) the manipulator's forward kinematics. All the commands (e.g., the racket's hitting position and hitting speed) are given by the visual system, in the operation space. They must be transformed into joint space values. Jacobian matrix is used to calculate the joint space speed according to the

6×7 Jacobian matrix and can be derived by differential transformation method. The *i*th item

.

tan 2( , )

tan 2( , )

 

*sc sss cc ssc cs y <sup>T</sup> s cs cc z*

 

 

 

*cc css sc csc ss x*

 

 

0 0 01

5 6 6 6 6 7

*T T*

0 0 01 0 0 0 1 0 0 0 1

 

 

0001

*y x p x zx y p y x y x y pz*

*a nn x p a n cn sn y p a sa ca so co z p*

3 3 3

*V Jqq* ( ) (5)

*q* the joint space speed. *J* is the

.

*noap*

*xxx x yyy y zzz z*

*noap*

  7 7

(3)

(4)

 

(1)

(2)

*cq sq*

6 6 7 7

3

*p p p*

3

*sq cq sq cq*

*cq sq D*

0 0 0 10 , , 0 0 0 00

10 0 0 0 0

5 5

*cq sq*

0 12 2 2 3 3 1 23 1 1 2 2

$$\mathbf{T}^{i-1}\mathbf{T}\_{\mathbf{T}} = ^{i-1}T\_i \dots \mathbf{T}\_{\mathbf{T}} = \begin{bmatrix} n\_x & o\_x & a\_x & p\_x \\ n\_y & o\_y & a\_y & p\_y \\ n\_z & o\_z & a\_z & p\_z \\ 0 & 0 & 0 & 1 \end{bmatrix} \tag{7}$$

The parameter *k* stands for the rotational axis of joint *i*. For example, if the joint rotates around the *x*-axis, then *k* is *x*. Racket speed *V* can be calculated by (4) whereas joint space speed . *q* cannot be uniquely determined because *J* is not a square matrix. This problem can be solved by Moore-Penrose method:

$$\dot{q} = \int ^+V \tag{8}$$

with <sup>1</sup> ( ) *T T J JJ J* being the Moore-Penrose pseudo inverse matrix of *J*.

The inverse kinematics is used to calculate the joint angles according to the racket's posture. The manipulator is redundant, so the elbow's position is not uniquely determined when the racket's posture is given. The motion characteristics of a human arm show the mapping relationships between elbow position and racket posture. In a general configuration of the manipulator (see Fig. 2), any three points on the manipulator's neck, shoulder, elbow, and wrist are not collinear. The position of elbow P2, whose axis is P1P3, is in circle ' *O* . The position of the center point ' '' ' (,,) *OOO O xyz* and the radius *r* of the circle are determined by positions P1 and P3, and arm length parameters *L1* and *L2*. Ω1 denotes the plane constructed by points P1, P2, and P3, and Ω2 the plane constructed by points O, P1, and P2. *α* denotes the separation angle between Ω1 and Ω2. Once the position of P3 is known, *α* can uniquely determine P2 elbow position. If the points O, P1, P3 are collinear whereas P1, P2, P3 are not, the plane Ω2 does not exist. Angle *α* can be defined as the separation angle between Ω1 and the horizontal plane. Also, if the points P1, P2, P3 are collinear, the plane Ω1 does not exist and P2 position can be calculated according to positions P1 and P3 and arm length parameters *L1* and *L2*. Considering the general configuration of the manipulator (see Fig. 2), the inverse kinematics can be solved if angle *α* and the racket's posture are given. According to the motion characteristics of human arms, the mapping relationship between *α* and the racket posture can be built offline. Denoting with *i* (*i*1, *i*2, *i*3) the unit vector pointing from P3

to P1, and assuming the angle *α* to have been calculated by a well-trained ANN model, the circle ' *O* can thus be expressed as [34-36]:

$$\begin{cases} i\_1(\mathbf{x} - \mathbf{x}\_{\dot{o}^\cdot}) + i\_2(\mathbf{y} - \mathbf{y}\_{\dot{o}^\cdot}) + i\_3(\mathbf{z} - \mathbf{z}\_{\dot{o}^\cdot}) = \mathbf{0} \\ (\mathbf{x} - \mathbf{x}\_{\dot{o}^\cdot})^2 + (\mathbf{y} - \mathbf{y}\_{\dot{o}^\cdot})^2 + (\mathbf{z} - \mathbf{z}\_{\dot{o}^\cdot})^2 = r^2 \end{cases} \tag{9}$$

Design and Simulation of Anfis Controller for Virtual-Reality-Built Manipulator 321

(17)

(14)

(16)

(18)

(19)

0 12 <sup>222</sup> 12 3 <sup>1</sup> 1 00 1

tan 2( , ) ( )

0 123 <sup>333</sup> 12 3 4 <sup>2</sup> 1 00 1

(1 ) tan 2(

<sup>1</sup> 4 56 0 1 23 6 \* 567 1234 7 *TTT TTTT T T*

\* \*\*

tan 2(( , )

ANFIS integrates ANN with FIS. The ANFIS analyzed here was a first-order Takagi Sugeno Fuzzy Model. The analysis has four inputs: front obstacle distance (x1), right obstacle distance (x2), left obstacle distance (x3), and target angle (x4). The output is steering angle.

123 4

*Rule IF x is A x is B x is C and x is D*

: ;; *jkm n*

tan 2( , ) ( )

7 5 12 5 22 5 11 5 21

*q a cq T sq T cq T sq T*

*ij* the column item of *T* 

\* \*\* \*

*<sup>y</sup> D sq L sq cq L x sq cq cq L q a sq sq cq cq sq L sq cq L*

 

 

2 2

*q a k kN*

*p p*

*x z*

*cq L cq L*

21 21

2 21

3 1 2

\* \*

tan 2(( , )

*q a TT k kN*

 

6 5 13 5 23 33

12 34

*ii i i i*

*THEN F p x r x s x t x*

*q a sq T sq T T*

5 13 23

*q a y DL*

*p*

sin(( ) / )

*T T*

*T T*

3 21 2 42 3 12 4 1

1 2 42 2 11 3 21 2 4 2 42 2 4

*y D sq L sq cq cq sq L cq sq*

, )

*x y z TT T T L* (15)

*x y z TT T L* (13)

With P2 position, this can be obtained:

With P3 position, this can be obtained:

Joint-3 angle can thus be derived as:

The angles of Joints 5-7 can thus be derived as:

*th* row and *j*

**3. General structures of ANFIS** 

From (2),

with \*

*ij T* being the *i*

The 'if-then' rules are [26, 37]:

*ppp*

The angles of Joints 1 and 2 can thus be derived as:

1

*ppp*

**Figure 1.** General configuration of the manipulator

The plane Ω2 and the circle ' *O* intersect at two points. One point near the neck is Pw (*xw*, *yw*, *zw*) and angle ∠P2O׳ Pw is the separation angle *α* between Ω1 and Ω2. Denoting Ω2 norm vector as *n* and position as Pw should satisfy this equation:

$$
\overrightarrow{OP\_w}.n = 0\tag{10}
$$

Pw position can be calculated from (9) and (10). P2 position (*xp*2, *yp*2, *zp*2) satisfies this equation:

$$\sqrt{(\mathbf{x}\_{p2} - \mathbf{x}\_m)^2 + (y\_{p2} - y\_m)^2 + (z\_{p2} - z\_m)^2} = 2r\sin\frac{a}{2} \tag{11}$$

P2 position can be calculated from (9) and (11). With P1, P2, P3 positions and the cosine theorem, Joint-4 angle can be calculated as:

$$q\_4 = \pi - a \cos(\frac{L\_1^2 + L\_2^2 - \left\| \overline{P\_1 P\_3} \right\|^2}{2L\_1 L\_2}) \tag{12}$$

With P2 position, this can be obtained:

$$
\begin{bmatrix} x\_{p2} & y\_{p2} & z\_{p2} & 1 \end{bmatrix}^T = {}^0T\_1^1T\_2^2T\_3 \begin{bmatrix} 0 & 0 & -L\_1 & 1 \end{bmatrix}^T \tag{13}
$$

The angles of Joints 1 and 2 can thus be derived as:

$$\begin{aligned} q\_1 &= a \tan 2(\frac{-\chi\_{p\_2}}{cq\_2 L\_1}, \frac{-z\_{p\_2}}{cq\_2 L\_1}) + k \pi & \quad (k \in N) \\ q\_2 &= a \sin((y\_{p\_2} + D) / L\_1) \end{aligned} \tag{14}$$

With P3 position, this can be obtained:

$$
\begin{bmatrix} x\_{p3} & y\_{p3} & z\_{p3} & 1 \end{bmatrix}^T = ^0T\_1 ^1T\_2 ^2T\_3 ^3T\_4 \begin{bmatrix} 0 & 0 & -L\_2 & 1 \end{bmatrix}^T \tag{15}
$$

Joint-3 angle can thus be derived as:

$$\begin{cases} q\_3 = a \tan 2 \zeta \frac{y\_3 + D - sq\_2 L\_1 - sq\_2 c q\_4 L\_2}{c q\_1 c q\_2 s q\_4 L\_2} s q\_1 s q\_2 - \frac{x\_3 + sq\_1 c q\_2 (1 + c q\_4) L\_1}{s q\_2 c q\_1 L\_1} \\ \qquad \qquad \qquad \frac{y\_3 + D - sq\_2 L\_1}{c q\_2 s q\_4 L\_2} - \frac{s q\_2 c q\_4}{c q\_2 s q\_4} \end{cases} \tag{16}$$

From (2),

320 Fuzzy Controllers – Recent Advances in Theory and Applications

circle ' *O* can thus be expressed as [34-36]:

**Figure 1.** General configuration of the manipulator

elbow

theorem, Joint-4 angle can be calculated as:

vector as *n* and position as Pw should satisfy this equation:

4

*zw*) and angle ∠P2O׳

to P1, and assuming the angle *α* to have been calculated by a well-trained ANN model, the

' ''

*o oo*

2 2 22 ( ) ( ) ( )0

zo

O

n

Pw

yo

xo

(9)

. 0 *OP n <sup>w</sup>* (10)

(11)

(12)

' ''

The plane Ω2 and the circle ' *O* intersect at two points. One point near the neck is Pw (*xw*, *yw*,

i

P3 wrist

Pw position can be calculated from (9) and (10). P2 position (*xp*2, *yp*2, *zp*2) satisfies this equation:

2 22 <sup>222</sup> ( ) ( ) ( ) 2 sin <sup>2</sup> *pm pm pm xx yy zz r*

P2 position can be calculated from (9) and (11). With P1, P2, P3 positions and the cosine

*q a L L*

<sup>2</sup> 2 2 1 2 13

1 2 cos( ) <sup>2</sup> *L L PP*

Pw is the separation angle *α* between Ω1 and Ω2. Denoting Ω2 norm

*xx yy zz r* 

*o oo*

( )( )( )

neck shoulder

α

L1

P2

P1

L2

D

*ix x iy y iz z*

12 3

$$T\_5 \, ^4T\_6 \, ^5T\_7^6 = \left[T\_1 \, ^0T\_2 \, ^1T\_3 \, ^2T\_4 \right]^{-1} T\_7^6 = T^\* \tag{17}$$

The angles of Joints 5-7 can thus be derived as:

$$\begin{aligned} q\_5 &= a \tan 2(-T\_{13}^\*, T\_{23}^\*) + k \pi & \quad (k \in N) \\ q\_6 &= a \tan 2((sq\_5 T\_{13}^\* - sq\_5 T\_{23}^\*, T\_{33}^\*) \\ q\_7 &= a \tan 2((-cq\_5 T\_{12}^\* - sq\_5 T\_{22}^\*, cq\_5 T\_{11}^\* + sq\_5 T\_{21}^\*) \end{aligned} \tag{18}$$

with \* *ij T* being the *i th* row and *j ij* the column item of *T* 

#### **3. General structures of ANFIS**

ANFIS integrates ANN with FIS. The ANFIS analyzed here was a first-order Takagi Sugeno Fuzzy Model. The analysis has four inputs: front obstacle distance (x1), right obstacle distance (x2), left obstacle distance (x3), and target angle (x4). The output is steering angle. The 'if-then' rules are [26, 37]:

$$\begin{array}{ccccccccc}\text{Rule:} & \text{IF} & \mathbf{x}\_1 & \text{is} & A\_j \text{:} & \mathbf{x}\_2 & \text{is} & B\_k \text{:} & \mathbf{x}\_3 & \text{is} & \mathbf{C}\_m & \text{and} & \mathbf{x}\_4 & \text{is} & D\_n \\\\ \text{THEN} & F\_i = p\_i \mathbf{x}\_1 + r\_i \mathbf{x}\_2 + s\_i \mathbf{x}\_3 + t\_i \mathbf{x}\_4 & & & & \end{array} \tag{19}$$

Where

$$\begin{aligned} F\_i &= p\_i \mathbf{x}\_1 + r\_i \mathbf{x}\_2 + s\_i \mathbf{x}\_3 + t\_i \mathbf{x}\_4 + u\_i \quad &\text{for} \quad \text{steriring} \quad &\text{angle} \\\ f\_1 &= 1 \quad \text{to} \quad q\_1; \quad k = 1 \quad \text{to} \quad q\_2; \quad m = 1 \quad \text{to} \quad q\_3; \quad n = 1 \quad \text{to} \quad q\_4; \quad \text{and} \quad i = 1 \quad \text{to} \quad q\_1. q\_2 \quad q\_3. q\_4 \end{aligned} \tag{20}$$

Design and Simulation of Anfis Controller for Virtual-Reality-Built Manipulator 323

 

 

(24)

(22)

1

1 12

1 2

with ag, bg, and cg being the parameters for fuzzy membership function. The bell-shaped function changes its pattern with changes to the parameters. This change will give various contours of the bell-shaped function, as needed and in accordance with the data set for the

**Layer 3:** Every node in this layer is a fixed node (circular) labeled 'π'. L2i output is the

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

*For i q q q q and g q q q q*

Each of the second layer's node output represents the firing strength (degree of fulfillment) of the associated rule. The T-norm operator algebraic product {Tap(a,b) = ab} was used to

**Layer 4:** Every node in this layer is a fixed node (circular) labeled "N". The output of the ith node is the ratio of the firing strength of the ith rule (Wi) to the sum of the firing strength of

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1

problem considered.

product of all incoming signals.

obtain the firing strength (Wi).

all the rules [40].

3

This output gives a normalized firing strength.

*Ag b*

*Bg b*

*Bg b*

1

1

1

<sup>1</sup> ( ) ;

*x c a*

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A, B, C, and D are the fuzzy membership sets defined for input variables x1, x2, x3, and x4. q1, q2, q3, and q4 are the number of membership functions, respectively for the fuzzy systems of inputs x1, x2, x3 and x4. fi is the linear consequent functions defined in terms of inputs x1, x2, x3, and x4. qi, ri, si , ti, and ui are consequent parameters of an ANFIS fuzzy model. Same-layer nodes of an ANFIS model have similar functions. Output signals from the nodes of a preceding layer are input signals to a present layer. The output obtained through the node function will be input signals to the next layer (see Figure 2) [38, 39].

**Layer 1:** This is the input layer, which defines obstacles as either static or moving and also the tracker robot's target position. It receives signals from x1, x2, x3, and x4.

**Figure 2.** Structure of a six-layer ANFIS

**Layer 2:** Every node in this layer is an adaptive node (square node) with a particular fuzzy membership function (node function) specifying the degrees to which the inputs satisfy the quantifier. For four inputs, the node outputs are:

$$\begin{aligned} \mathbf{L}\_{2g} &= \mathbf{U}\_{ag}(\mathbf{x}) \quad \text{for} & \mathbf{g} &= \mathbf{1}, \dots, q\_1 & \text{(for } input & \mathbf{x}\_1) \\ \mathbf{L}\_{2g} &= \mathbf{U}\_{Bg}(\mathbf{x}) \quad \text{for} & \mathbf{g} &= q\_1 + 1, \dots, q\_1 + q\_2 & \text{(for } input & \mathbf{x}\_1) \\ \mathbf{L}\_{2g} &= \mathbf{U}\_{Cg}(\mathbf{x}) \quad \text{for} & \mathbf{g} &= q\_1 + q\_2 + 1, \dots, q\_1 + q\_2 + q\_3 & \text{(for } input & \mathbf{x}\_1) \\ \mathbf{L}\_{2g} &= \mathbf{U}\_{Dg}(\mathbf{x}) \quad \text{for} & \mathbf{g} &= q\_1 + q\_2 + q\_3 + 1, \dots, q\_1 + q\_2 + q\_3 + q\_4 & \text{(for } input & \mathbf{x}\_1) \end{aligned} \tag{21}$$

The membership functions considered here for A, B, C, and D are bell-shaped functions and defined as:

$$\begin{aligned} \mu\_{A\_{\mathcal{S}}}(\mathbf{x}) &= \frac{1}{\mathbf{1} + \left[\left(\frac{\mathbf{x} - \mathbf{c}\_{\mathcal{S}}}{a\_{\mathcal{S}}}\right)^2\right]^{b\_{\mathcal{S}}}}; \quad \mathbf{g} = \mathbf{1} \quad \text{to} \quad q\_{1} \\ \mu\_{B\_{\mathcal{S}}}(\mathbf{x}) &= \frac{1}{\mathbf{1} + \left[\left(\frac{\mathbf{x} - \mathbf{c}\_{\mathcal{S}}}{a\_{\mathcal{S}}}\right)^2\right]^{b\_{\mathcal{S}}}}; \quad \mathbf{g} = q\_{1} + \mathbf{1} \quad \text{to} \quad q\_{1} + q\_{2} \\ \mu\_{B\_{\mathcal{S}}}(\mathbf{x}) &= \frac{1}{\mathbf{1} + \left[\left(\frac{\mathbf{x} - \mathbf{c}\_{\mathcal{S}}}{a\_{\mathcal{S}}}\right)^2\right]^{b\_{\mathcal{S}}}}; \quad \mathbf{g} = q\_{1} + q\_{2} + \mathbf{1} \quad \text{to} \quad q\_{1} + q\_{2} + q\_{3} \\ \mu\_{B\_{\mathcal{S}}}(\mathbf{x}) &= \frac{1}{1 + \left[\left(\frac{\mathbf{x} - \mathbf{c}\_{\mathcal{S}}}{a\_{\mathcal{S}}}\right)^2\right]^{b\_{\mathcal{S}}}}; \quad \mathbf{g} = q\_{1} + q\_{2} + q\_{3} + \mathbf{1} \quad \text{to} \quad q\_{1} + q\_{2} + q\_{3} + q\_{4} \\ &= 1 + \left[\left(\frac{\mathbf{x} - \mathbf{c}\_{\mathcal{S}}}{a\_{\mathcal{S}}}\right)^2\right]^{b\_{\mathcal{S}}} \end{aligned} \tag{22}$$

with ag, bg, and cg being the parameters for fuzzy membership function. The bell-shaped function changes its pattern with changes to the parameters. This change will give various contours of the bell-shaped function, as needed and in accordance with the data set for the problem considered.

**Layer 3:** Every node in this layer is a fixed node (circular) labeled 'π'. L2i output is the product of all incoming signals.

$$\begin{aligned} L\_{3i} = \mathcal{W}\_i &= \mathcal{U}\_{\text{ag}}(\mathbf{x})\_\prime \mathcal{U}\_{\text{Bg}}(\mathbf{x})\_\prime \mathcal{U}\_{\text{Cg}}(\mathbf{x})\_\prime \mathcal{U}\_{\text{Cg}}(\mathbf{x}); \\ \text{For } i = 1, \dots, q\_1 + q\_2 + q\_3 + q\_4 \quad \text{and} \quad \mathbf{g} = 1, \dots, q\_1 + q\_2 + q\_3 + q\_4 \end{aligned} \tag{23}$$

Each of the second layer's node output represents the firing strength (degree of fulfillment) of the associated rule. The T-norm operator algebraic product {Tap(a,b) = ab} was used to obtain the firing strength (Wi).

**Layer 4:** Every node in this layer is a fixed node (circular) labeled "N". The output of the ith node is the ratio of the firing strength of the ith rule (Wi) to the sum of the firing strength of all the rules [40].

$$L\_{4i} = \overline{\mathcal{W}\_i} f\_i = \frac{\mathcal{W}\_i}{\sum\_{r=q\_1\cdot q\_2\cdot q\_3\cdot q\_4}^{}}\tag{24}$$

This output gives a normalized firing strength.

322 Fuzzy Controllers – Recent Advances in Theory and Applications

*ii i i i i F p x r x s x t x u for steering angle*

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> 12 34 1 ;1 ; 1 ;1 ; 1 . .

A, B, C, and D are the fuzzy membership sets defined for input variables x1, x2, x3, and x4. q1, q2, q3, and q4 are the number of membership functions, respectively for the fuzzy systems of inputs x1, x2, x3 and x4. fi is the linear consequent functions defined in terms of inputs x1, x2, x3, and x4. qi, ri, si , ti, and ui are consequent parameters of an ANFIS fuzzy model. Same-layer nodes of an ANFIS model have similar functions. Output signals from the nodes of a preceding layer are input signals to a present layer. The output obtained through the node function will be input signals to the next layer (see Figure 2)

**Layer 1:** This is the input layer, which defines obstacles as either static or moving and also

**Layer 2:** Every node in this layer is an adaptive node (square node) with a particular fuzzy membership function (node function) specifying the degrees to which the inputs satisfy the

> 2 1 1 2 1 12 1 2 12 123 1 2 123 1234 1

The membership functions considered here for A, B, C, and D are bell-shaped functions and

*L U x for g q for input x L U x for g q q q for input x L U x for g q q q q q for input x L U x for g q q q q q q q for input x*

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

(21)

the tracker robot's target position. It receives signals from x1, x2, x3, and x4.

(20)

*J to q k to q m to q n to q and i to q q q q*

12 34

**Figure 2.** Structure of a six-layer ANFIS

*g ag g Bg g Cg g Dg*

defined as:

quantifier. For four inputs, the node outputs are:

 

Where

[38, 39].

**Layer 5:** Every node in this layer is an adaptive node (square node) with a node function.

$$L\_{\Xi i} = \overline{\mathcal{W}}\_i f\_i = \overline{\mathcal{W}}\_i (p\_i \mathbf{x}\_1 + r\_i \mathbf{x}\_1 + s\_i \mathbf{x}\_3 + t\_i \mathbf{x}\_4 + u\_i) \tag{25}$$

Design and Simulation of Anfis Controller for Virtual-Reality-Built Manipulator 325

**Figure 3.** The VR-modeled 7-DOF human-arm manipulator

obtained by feedback from the virtual model.

motor with a transfer function similar with that in Equation (1).

**5. Design of the ANFIS controller for the 7-DOF manipulator** 

The design work considered many parameters related to the system's real values (see Table 1 for real limits of the joints of a human arm). The joints motor is considered a real DC-

> 2 <sup>1</sup> . () 6 8 *<sup>m</sup> TF s s s*

Figure 4 is a block diagram of the control system. The inputs to the system design have two sets of targets: orientation (θT1, θT2, θT3) and position (Tx, Ty, Tz). In the proposed technique, each joint has its own controller, so in all, seven identical (same structure, same training algorithm) ANFIS controllers were used. Inputs of the training algorithm were the desired joint angle and the actual angle. The desired values were calculated by using analytical solution of the IKP algorithm shown previously. Actual values of the joint angle were

(27)

with *Wi* being the normalized firing strength form (output) from Layer-3 and {pi, ri, si, ti, ui} the steering-angle parameter set. Parameters in this layer are consequent.

**Layer 6:** The single node in this layer is a fixed node (circular) labeled "Σ". It computes the overall output as the summation of all incoming signals.

$$L\_{6i} = \sum\_{r=1}^{r=q\_1\cdot q\_2\cdot q\_3\cdot q\_4} \overline{\mathcal{W}}\_i f\_i = \frac{\sum\_{i=1}^{i=q\_1\cdot q\_2\cdot q\_3\cdot q\_4} \overline{\mathcal{W}}\_i f\_i}{\sum\_{i=1}^{i=q\_1\cdot q\_2\cdot q\_3\cdot q\_4} \mathcal{W}\_i} \tag{26}$$

This work's ANFIS development has six-dimensional space partitions and q1, q2, q3, and q4 regions. Each region is governed by a fuzzy if-then rule. The first layer is the input layer. The second contains premise or antecedent parameters of the ANFIS and is dedicated to fuzzy sub-space. Consequent parameters of the fifth layer were used to optimize the network. During the forward pass of the hybrid learning algorithm, node outputs go forward until Layer-5 and the consequent parameters are identified by least-square method. In the backward pass, error signals propagate backwards and the premise parameters are updated by gradient descent method [40, 41].

### **4. VR Modeling of the 7-DOF manipulator**

Design requirements for Virtual Reality Modeling Language (VRML) are described in finite processing allocations, autonomy, consistent self-registration, and calculability. VRML design procedure will be presented. Design in VRML depends on the information available to the designer and his imaging of the object. There are two choices for VR design: one is standard configuration such as sphere, cone, cylinder, etc., another is free design by selecting indexed face set button to get many configurations with free rearrangement of points; every real-form design is thus considered to be the latter [42-45], which starts with building parts one by one and comparing the shape's similarity against that of the real manipulator part. Manipulator parts cannot be simulated in VR when the VR library's standard shapes (they are not uniform) are used. Designing thus uses indexed face set in VR. The second choice to be made in design work is very important as it is about connecting all parts to get the final object, and limiting the object's original point. This is the starting point of the design work. The first shape (e.g., the shoulder) is first set, and then the next shape (forearm) is connected to the "children" button. The same procedure is repeated for other parts. Figure 3 presents the full design of the 7-DOF human arm manipulator [46-50].

Design and Simulation of Anfis Controller for Virtual-Reality-Built Manipulator 325

**Figure 3.** The VR-modeled 7-DOF human-arm manipulator

324 Fuzzy Controllers – Recent Advances in Theory and Applications

overall output as the summation of all incoming signals.

updated by gradient descent method [40, 41].

manipulator [46-50].

**4. VR Modeling of the 7-DOF manipulator** 

**Layer 5:** Every node in this layer is an adaptive node (square node) with a node function.

with *Wi* being the normalized firing strength form (output) from Layer-3 and {pi, ri, si, ti, ui}

**Layer 6:** The single node in this layer is a fixed node (circular) labeled "Σ". It computes the

*i rqqqq i*

the steering-angle parameter set. Parameters in this layer are consequent.

1234

<sup>6</sup> ... <sup>1</sup>

*<sup>i</sup> <sup>i</sup> i i iqqqq <sup>r</sup>*

This work's ANFIS development has six-dimensional space partitions and q1, q2, q3, and q4 regions. Each region is governed by a fuzzy if-then rule. The first layer is the input layer. The second contains premise or antecedent parameters of the ANFIS and is dedicated to fuzzy sub-space. Consequent parameters of the fifth layer were used to optimize the network. During the forward pass of the hybrid learning algorithm, node outputs go forward until Layer-5 and the consequent parameters are identified by least-square method. In the backward pass, error signals propagate backwards and the premise parameters are

Design requirements for Virtual Reality Modeling Language (VRML) are described in finite processing allocations, autonomy, consistent self-registration, and calculability. VRML design procedure will be presented. Design in VRML depends on the information available to the designer and his imaging of the object. There are two choices for VR design: one is standard configuration such as sphere, cone, cylinder, etc., another is free design by selecting indexed face set button to get many configurations with free rearrangement of points; every real-form design is thus considered to be the latter [42-45], which starts with building parts one by one and comparing the shape's similarity against that of the real manipulator part. Manipulator parts cannot be simulated in VR when the VR library's standard shapes (they are not uniform) are used. Designing thus uses indexed face set in VR. The second choice to be made in design work is very important as it is about connecting all parts to get the final object, and limiting the object's original point. This is the starting point of the design work. The first shape (e.g., the shoulder) is first set, and then the next shape (forearm) is connected to the "children" button. The same procedure is repeated for other parts. Figure 3 presents the full design of the 7-DOF human arm

...

*L W f*

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1234

...

*iqqqq*

1

1234

1

*i*

*i*

(26)

*W*

*W f*

## **5. Design of the ANFIS controller for the 7-DOF manipulator**

The design work considered many parameters related to the system's real values (see Table 1 for real limits of the joints of a human arm). The joints motor is considered a real DCmotor with a transfer function similar with that in Equation (1).

$$T.F\_m(s) = \frac{1}{s^2 + 6s + 8} \tag{27}$$

Figure 4 is a block diagram of the control system. The inputs to the system design have two sets of targets: orientation (θT1, θT2, θT3) and position (Tx, Ty, Tz). In the proposed technique, each joint has its own controller, so in all, seven identical (same structure, same training algorithm) ANFIS controllers were used. Inputs of the training algorithm were the desired joint angle and the actual angle. The desired values were calculated by using analytical solution of the IKP algorithm shown previously. Actual values of the joint angle were obtained by feedback from the virtual model.

Design and Simulation of Anfis Controller for Virtual-Reality-Built Manipulator 327

**Figure 6.** The surface error and the set of rules for training of the ANFIS controller

**Figure 7.** The triangular built-in memberships function with error signal and change in error signal

**Figure 4.** ANFIS control of the 7-DOF human-arm manipulator

The structure of the ANFIS controller was built in Matlab software Ver.2011b, with two inputs, an error signal, and change in the error. The fuzzy inference method used was Mamdani's, because it is intuitive, widely accepted, and well-suited to human input, and, for the proposed control structure, it gives better results than does Sugeno inference method. In designing the controller, types of membership functions were tried before selecting the best: triangular built-in membership function (trimf). The trial-and-error design approach for the ANFIS controller, i.e., selecting the interference type, the membership function type, and the number of this membership function in the hidden layer, gave optimal results: minimum number of rules and simple simulation. The design uses seven parallel-connected ANFIS to compute the optimal deflection of the joints and get the desired angle. Figures (5-7) show the procedure of the Matlab-fuzzy-toolbox-based design.

**Figure 5.** Mamdani's fuzzy inference and the set of rules for training of the ANFIS controller

**Figure 6.** The surface error and the set of rules for training of the ANFIS controller

**Figure 4.** ANFIS control of the 7-DOF human-arm manipulator

design.

The structure of the ANFIS controller was built in Matlab software Ver.2011b, with two inputs, an error signal, and change in the error. The fuzzy inference method used was Mamdani's, because it is intuitive, widely accepted, and well-suited to human input, and, for the proposed control structure, it gives better results than does Sugeno inference method. In designing the controller, types of membership functions were tried before selecting the best: triangular built-in membership function (trimf). The trial-and-error design approach for the ANFIS controller, i.e., selecting the interference type, the membership function type, and the number of this membership function in the hidden layer, gave optimal results: minimum number of rules and simple simulation. The design uses seven parallel-connected ANFIS to compute the optimal deflection of the joints and get the desired angle. Figures (5-7) show the procedure of the Matlab-fuzzy-toolbox-based

**Figure 5.** Mamdani's fuzzy inference and the set of rules for training of the ANFIS controller

**Figure 7.** The triangular built-in memberships function with error signal and change in error signal

Figure 8 shows in Matlab Simulink GUI window the internal structure of the ANFIS controller and the training output for the input signal, with the various steps used in instructing the manipulator's movement. The ANFIS training used hybrid training algorithm, with the input nodes (3, 3) to the membership functions each having nine rules (see Figure 5). Epoch length was used in training eighty iterations for each sample, with 0.01s Simulink sampling time.

Design and Simulation of Anfis Controller for Virtual-Reality-Built Manipulator 329

Figure 9(a) is a block diagram of the simulation done for the ANFIS-controlled manipulator. Seven ANFIS controllers were used, one each for each degree of freedom of the three joints (3 for the shoulder, 1 for the elbow, and 3 for the wrist). The seven were effective in tracking the trajectory desired for the 7-DOF manipulator. The controller's rules base has 9 rules, each determined by fuzzy neural network (FNN). The desired position and orientation were, in simulation entered as input signal, whereas actual positions for the joints were given as feedback from the output signal. Figure 9(b) shows the results of using both ANFIS and PID controllers on the joints and for the ANFIS controller's implementation into the virtual model. Performance of the ANFIS-controlled joint was better than that of the PIDcontrolled one. The ANFIS controller effected fast response in the manipulator and reduced

**Figure 9.** (a, b, c). Simulation of the ANFIS controller for the VR-implemented 7-DOF manipulator

The manipulator's movements used the link between Matlab-Simulink and VR environment. Computation for the order of movement was done in Matlab-Simulink. The order was then sent to the VR model to implement, with considerations for the axes between

errors, for various complex trajectories of the manipulator.

**6. Simulation results** 

**Figure 8.** The ANFIS structure in Matlab-Simulink GUI and its training response

## **6. Simulation results**

328 Fuzzy Controllers – Recent Advances in Theory and Applications

0.01s Simulink sampling time.

Figure 8 shows in Matlab Simulink GUI window the internal structure of the ANFIS controller and the training output for the input signal, with the various steps used in instructing the manipulator's movement. The ANFIS training used hybrid training algorithm, with the input nodes (3, 3) to the membership functions each having nine rules (see Figure 5). Epoch length was used in training eighty iterations for each sample, with

**Figure 8.** The ANFIS structure in Matlab-Simulink GUI and its training response

Figure 9(a) is a block diagram of the simulation done for the ANFIS-controlled manipulator. Seven ANFIS controllers were used, one each for each degree of freedom of the three joints (3 for the shoulder, 1 for the elbow, and 3 for the wrist). The seven were effective in tracking the trajectory desired for the 7-DOF manipulator. The controller's rules base has 9 rules, each determined by fuzzy neural network (FNN). The desired position and orientation were, in simulation entered as input signal, whereas actual positions for the joints were given as feedback from the output signal. Figure 9(b) shows the results of using both ANFIS and PID controllers on the joints and for the ANFIS controller's implementation into the virtual model. Performance of the ANFIS-controlled joint was better than that of the PIDcontrolled one. The ANFIS controller effected fast response in the manipulator and reduced errors, for various complex trajectories of the manipulator.

**Figure 9.** (a, b, c). Simulation of the ANFIS controller for the VR-implemented 7-DOF manipulator

The manipulator's movements used the link between Matlab-Simulink and VR environment. Computation for the order of movement was done in Matlab-Simulink. The order was then sent to the VR model to implement, with considerations for the axes between

Matlab's and VR's to get the real movement (i.e., for general motion: elbow flexion/extension, elbow rotation (supination/pronation), shoulder adduction/abduction, shoulder flexion/extension, shoulder interior/exterior rotation, wrist flexion/extension, wrist ulnar/radial deviation, shoulder horizontal flexion/extension; for special motion: arm reaching towards the head, arm reaching to the right and the head, arm reaching to the left and the head, and arm reaching to the right) (see Figure 10 for examples).

Design and Simulation of Anfis Controller for Virtual-Reality-Built Manipulator 331

 For any speed, the ANFIS controller has the better transient response and the steadier state response than does a PID controller; even the best-tuned PID controller is unable

 With only 9 neurons, 4 layers, and 9 rules, the proposed ANFIS controller is simpler than other adaptive neuro-fuzzy controllers reported by many other researches. The ANFIS controller does not require an accurate model of the plant. Its relative simplicity makes it fairly easy to construct and implement. High-level knowledge of system is not needed to build a set of rules for a fuzzy controller or for the identification

 The ANFIS controller was used in both simulation and experiment in trajectory tracking of multiple manipulators. Kinematic analysis of the 7-DOF manipulator produced the

 VR implementation of the manipulator was able to show the latter's accuracy. The manipulator can be depended on for many high-accuracy applications such as

Special thanks to Ms. Wirani M. Munawir of Verdana Inc. (verdana.inc@live.com) for her

[1] AWARE M. V, KQTHARI A.G, CHOUBE S.O, (2000), "Application of adaptive neurofuzzy controller (ANFIS) for voltage source inverter fed induction motor drive", IEEE Power Electronics and Motion Control Conference, 2000. Proceedings. IPEMC 2000. The

[2] Choon Y. L, Lee J, (2005)," Multiple Neuro-Adaptive Control of Robot Manipulators Using Visual Cues", IEEE transactions on industrial electronics, Vol. 52, No.1, 320-326. [3] Hui C, Gangquan S, Yanbin Z, Xikui M, (2007),"A Hybrid Controller of Self-Optimizing Algorithm and ANFIS for Ball Mill Pulverizing System", Proceedings of the IEEE

[4] Swasti R. K, Sidhartha P, (2010), "ANFIS Approach for TCSC-based Controller Design System Stability Improvement Design for Power" , IEEE ICCCCT-10, 149-154 [5] Prabu D, Surendra K, Rajendra P, (2011), "Advanced Dynamic Path Control of the Three Links SCARA using Adaptive Neuro Fuzzy Inference System", IN BOOK, Robot Manipulators, Trends and Development, InTech, ch 18, ISBN: 978-953-307-073-5. pp. 399-412 [6] Zhiqiang G, Thomas A. T, James G. D, (2000),"A Stable Self-Tuning Fuzzy Logic Control System for Industrial Temperature Regulation", IEEE Industry Applications

International Conference on Mechatronics and Automation, Harbin, China

to perform well in both slow-speed and high-speed ranges.

calculation for the envelope of similar manipulators.

*Electrical Dept., Al Anbar University, Engineering College, Baghdad, Iraq* 

needed in an NN controller.

collecting of minute/fragile items.

**Author details** 

Yousif I. Al Mashhadany

**Acknowledgement** 

Third International 945-939

Conference, Vol 2, 1232-1240

editing of the text.

**8. References** 

**Figure 10.** Postures of the ANFIS-controlled 7-DOF manipulator

## **7. Conclusion**

This work aimed to design an ANFIS-based controller that overcomes the general problems of FL and NN in a dynamic system. To obtain excellent manipulator postures, training of the ANFIS controller for all the elements (fuzzy interference, membership function, number of neurons, number of rules) was by trial and error. The controller was compared with classical PID controller, in tracking the speed and the joint angle's accuracy. The simulation results allowed these conclusions to be drawn:


## **Author details**

330 Fuzzy Controllers – Recent Advances in Theory and Applications

Matlab's and VR's to get the real movement (i.e., for general motion: elbow flexion/extension, elbow rotation (supination/pronation), shoulder adduction/abduction, shoulder flexion/extension, shoulder interior/exterior rotation, wrist flexion/extension, wrist ulnar/radial deviation, shoulder horizontal flexion/extension; for special motion: arm reaching towards the head, arm reaching to the right and the head, arm reaching to the left

and the head, and arm reaching to the right) (see Figure 10 for examples).

**Figure 10.** Postures of the ANFIS-controlled 7-DOF manipulator

allowed these conclusions to be drawn:

This work aimed to design an ANFIS-based controller that overcomes the general problems of FL and NN in a dynamic system. To obtain excellent manipulator postures, training of the ANFIS controller for all the elements (fuzzy interference, membership function, number of neurons, number of rules) was by trial and error. The controller was compared with classical PID controller, in tracking the speed and the joint angle's accuracy. The simulation results

**7. Conclusion** 

Yousif I. Al Mashhadany *Electrical Dept., Al Anbar University, Engineering College, Baghdad, Iraq* 

## **Acknowledgement**

Special thanks to Ms. Wirani M. Munawir of Verdana Inc. (verdana.inc@live.com) for her editing of the text.

## **8. References**

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Design and Simulation of Anfis Controller for Virtual-Reality-Built Manipulator 333

[22] Ginarsa M, Soeprijanto A, Purnomo M.H, (2009), "Controlling Chaos Using ANFIS-Based Composite Controller (ANFIS-CC) in Power Systems", Instrumentation, Communications, Information Technology, and Biomedical Engineering (ICICI-BME). [23] Yajun Z, Tianyou C, Hong W, (2011), "A Nonlinear Control Method Based on ANFIS and Multiple Models for a Class of SISO Nonlinear Systems and Its Application", IEEE

[24] JafarT, AfsharShamsi J, Muhammad A. D, ( 2012), "A new method for position control of a 2-DOF robot arm using neuro– fuzzy controller", Indian Journal of Science and

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[27] Ming-Y. S, Ke H. C, Chen Y. C, Juing S. C, Jeng H. L, (2007), "ANFIS based Controller Design for Biped Robots", IEEE Proceedings of International Conference on

[28] Widodo B, Achmad J, Djoko P, (2010), Indoor Navigation using Adaptive Neuro Fuzzy Controller for Servant Robot", IEEE Second International Conference on Computer

[29] Greg R. L, (2011), "Haptic Interactions Using Virtual Manipulator Coupling with Applications to Underactuated Systems", IEEE transactions on robotics, Vol. 27, No. 4,

[30] Al-Mashhadany Y. I,(2012), "A Posture of 6-DOF Manipulator By Locally Recurrent Neural Networks (LRNNs) Implement in Virtual Reality", IEEE Symposium on

[31] Jacob R, Joel C. P, Nathan M, Stephen B, Blake H, (2005), "The Human Arm Kinematics and Dynamics During Daily Activities – Toward a 7 DOF Upper Limb Powered Exoskeleton", This work is supported by NSF Grant #0208468 entitled "Neural Control of an Upper Limb Exoskeleton System" - Jacob Rosen (PI) 0-7803-9177-2/05/ IEEE, 532-539 [32] Erico G, Travis D, (2012), "Robotics Trends for 2012", IEEE Robotic & Automation

[33] Shahri M. R. A, Khoshravan H, Naebi A, (2011) " Design ping-pong player robot controller with ANFIS", IEEE Third International Conference on Computational

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TRANSACTIONS ON NEURAL NETWORKS, VOL. 22, NO. 11, 1783-1795

Technology Vol. 5 No. 3, ISSN: 0974- 6846, 2253-2258

Communication and Computing, 727-731

Engineering and Applications, 582-586

Msgazine, 1070-9932/12/, 119-123

Intelligence, Modelling & Simulation, 165-189

ISR(International Symposium on Robotics), 19-21

Robotics and Biomimetics , Kunming, China, 1553-1558

Mechatronics TuA2-B-1 Kumamoto Japan, 1-6

Engineering (ISKE).

730-740.


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332 Fuzzy Controllers – Recent Advances in Theory and Applications

Motion and Control, 399-404

Technologies, ICCCCT-10, 314-320.

Advanced Computer Control, 19-24

39th International, 545-549

Review ,106- 111

Malaysia, 74-78

colloquium (IAPEC), 110-115

Technologies, 138-145

1106-1113.

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[8] Hassanzadeh, S. K, liang G. A, (2002), "Implementation of .I Functional Link Net-ANFIS Controller for a Robot Manipulator", Third International Workshop on Robot

[9] Saifizul A. A., Zainon M. Z, Abu Osman N. A, (2006), "An ANFIS Controller for Visionbased Lateral Vehicle Control System", Control, Automation, Robotics and Vision, 2006.

[10] Ravi S, Balakrishnan P .A, (2010), "Modeling and control of an ANFIS temperature controller for plastic extrusion process", IEEE Communication Control and Computing

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[13] Thair S. M, Mohammed H. M, Tang S. H, Sokchoo N, (2008), "ANFIS Controller with Fuzzy Subtractive Clustering Method to Reduce Coupling Effects in Twin Rotor MIMO System (TRMS) with Less Memory and Time Usage", IEEE International Conference on

[14] Mohammad A*,* (2006), "ANFIS Based Soft-Starting and Speed Control of AC Voltage Controller Fed Induction Motor", 0-7803-9525-5/06/6 IEEE Power India Conference. [15] Srinivasan A, Nigam M. J, (2008), "Adaptive Neuro-Fuzzy Inference System based control of six DOF robot manipulator", Journal of Engineering Science and Technology

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**Chapter 15** 

© 2012 Rodrigues 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,

**Hierarchical Fuzzy Control** 

Additional information is available at the end of the chapter

and Marconi Câmara Rodrigues

http://dx.doi.org/10.5772/48627

identified operational condition.

during its operation according to the learning mechanism.

called hierarchical fuzzy controller (HFC).

**1. Introduction** 

Carlos André Guerra Fonseca, Fábio Meneghetti Ugulino de Araújo

Growing demands for comfort, reliability, accuracy, energy conservation, safety and economy have fueled interest in proposals that can contribute to facilitate high performance control systems design. In terms of vibrations active control, it may represent, for example, a good relationship between the maximum reduction in vibrations transmission between two

The use of more than one controller to provide higher performance for complex systems has attracted interest because in each operation condition, their combination can take advantage of each controller's characteristics. To take advantage of controllers' combination, a supervisor can make a hierarchical classification of controllers' signals, according to the

Advances in artificial intelligence, processing power and data storage, allowed the development of intelligent methods for different characteristics controllers' fusion. The use of intelligent methods allows to the controlled system: adaptability to various operational situations and proper performance, even in the presence of significant uncertainties. Intelligent supervisors are ease to maintain, to reconfigure and could have optimality

This chapter describes a methodology for controllers' combination called controllers hierarchical fusion. In this methodology, a supervisor system is used to obtain a single control signal from the control signals generated simultaneously by two or more controllers. A hierarchical controller's example compounded by one robust controller, one fuzzy controller and one fuzzy supervisor is applied for mechanical vibrations isolation and reference tracking using an electromechanical system proposed in [2]. This controller is

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

systems and the minimum energy expended in order to accomplish this reduction [1].


**Chapter 15** 

## **Hierarchical Fuzzy Control**

Carlos André Guerra Fonseca, Fábio Meneghetti Ugulino de Araújo and Marconi Câmara Rodrigues

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48627

## **1. Introduction**

334 Fuzzy Controllers – Recent Advances in Theory and Applications

www.elsevier.com/locate/ins

Computing, 363–378

730-740.

journal, 440-446

Prints, Uni Münster, 285-296

"Humanoid Robots", InTech ,chapter 1, 1-20

Informatics (ICCCI -2012), Coimbatore, INDIA

R01- NS35673-01, and the Falk Trust

[37] Patricia M, Oscar C, (2005), "Intelligent control of a stepping motor drive using an adaptive neuro–fuzzy inference system", Elsevier Information Sciences 170,133–151

[38] Al-maliki K. H, Wali W.A, Hameed L. J, Turky Y. A, (2011 ), "Force / Motion Control for Constrained Robot Manipulator Using Adaptive Neural Fuzzy Inference System (ANFIS)", it is available with (the last visit at 12/4/2012) http://www.nauss.edu.sa

[39] Oscar C, Patricia M, (2003), "Intelligent adaptive model-based control of robotic dynamic systems with a hybrid fuzzy-neural approach", Elsevier Applied Soft

[40] Mukhtiar S, Chandra A, (2012), "Real Time Implementation of ANFIS Control for Renewable Interfacing Inverter in 3P4W Distribution Network", This article has been

[41] Ouamri B, Ahmed Z, (2012), "Adaptive Neuro-fuzzy Inference System Based Control of Puma 600 Robot Manipulator", International Journal of Electrical and Computer

[42] Greg R. L, (2011), "Haptic Interactions Using Virtual Manipulator Coupling with Applications to Underactuated Systems", IEEE transactions on robotics, Vol. 27, No. 4,

[43] Zappi M., Maltina R., Cerveri,P, (2010), "Modular micro robotic instruments for transluminal endoscopic robotic surgery: new perspectives", 978-1-4244-7101-0110/IEEE

[44] Eckhard F, Jürgen R, (2003), "Integrating Robotics and Virtual Reality with Geo-Information Technology: Chances and Perspectives", In L. Bernhard, A. Sliwinski, K. Senkler (Hrsg.): Geodaten- und Geodienste-Infrastrukturen – von der Forschung zur praktischen Anwendung. Beiträge zu den Münsteraner GI-Tagen 2003, Reihe IfGI

[45] Michal J, Michal P, Michal C , Peter Nand O, (2012), "Towards Incremental Development of Human-Agent-Robot Applications using Mixed-Reality Testbeds", This article has been accepted for publication in IEEE Intelligent Systems but has not yet

[46] Ben C, (2009), "Humanoid Robotic Language and Virtual Reality Simulation" in book

[47] Syed I. A, Nageli V. S, Sangeeta S, Rakshit S, (2012), "Digital Sand Model using Virtual Reality Workbench", IEEE International Conference on Computer Communication and

[48] Manuela C, Fabio S, Silvio P, (2011), "Virtual Reality to Simulate Visual Tasks for Robotic Systems", IN BOOK In tech, ISBN: 978-953-307-518-1 " Virtual Reality"ch 4,, 71-92 [49] Patton J. L, Dawe G, Scharver C, Mussa-Ivaldi1 F. A, Kenyon R, ( 2012 ), "Robotics and Virtual Reality: The Development of a Life-Sized 3-D System for the Rehabilitation of Motor Function" Supported by NIDRR RERC. 0330411Z, NIH 1 R24 HD39627-0, NIH 1

[50] Rong-wen H, Chia-hui L, (2007),"Development of Fuzzy-based Automatic Vehicle Control System in a Virtual Reality Environment", IEEE International Conference on

Emerging Technologies, 10.1109/ICET.2007.4516340, pp 184 – 189.

been fully edited. Some content may change prior to final publication.

/En/DigitalLibrary/Researches/Documents/2011/articles\_2011\_3157.pdf

accepted for publication in a future issue of IEEE journal

Engineering (IJECE) Vol.2, No.1, ISSN: 2088-8708, 90~97

Growing demands for comfort, reliability, accuracy, energy conservation, safety and economy have fueled interest in proposals that can contribute to facilitate high performance control systems design. In terms of vibrations active control, it may represent, for example, a good relationship between the maximum reduction in vibrations transmission between two systems and the minimum energy expended in order to accomplish this reduction [1].

The use of more than one controller to provide higher performance for complex systems has attracted interest because in each operation condition, their combination can take advantage of each controller's characteristics. To take advantage of controllers' combination, a supervisor can make a hierarchical classification of controllers' signals, according to the identified operational condition.

Advances in artificial intelligence, processing power and data storage, allowed the development of intelligent methods for different characteristics controllers' fusion. The use of intelligent methods allows to the controlled system: adaptability to various operational situations and proper performance, even in the presence of significant uncertainties. Intelligent supervisors are ease to maintain, to reconfigure and could have optimality during its operation according to the learning mechanism.

This chapter describes a methodology for controllers' combination called controllers hierarchical fusion. In this methodology, a supervisor system is used to obtain a single control signal from the control signals generated simultaneously by two or more controllers. A hierarchical controller's example compounded by one robust controller, one fuzzy controller and one fuzzy supervisor is applied for mechanical vibrations isolation and reference tracking using an electromechanical system proposed in [2]. This controller is called hierarchical fuzzy controller (HFC).

© 2012 Rodrigues 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.

This electromechanical system can be used to eliminate vibrations in the camera of unmanned vehicles and also to position this camera. It can also be used in manned vehicles for drivers' seat positioning and to eliminate vibrations on it, as shown in Figure 1.

Hierarchical Fuzzy Control 337

**Figure 2.** Electromechanical system

the motor position (*XB*).

originated at the base.

Where:

actuator, is called control system.

The vertical position control of bar's center is made by a servo actuator. This actuator consists of a DC servo motor whose axis is directly coupled to a spindle. The propeller's spindle step is given by *LP*. It represents the direct relationship between motor's rotation angle (*θM*) and control's vertical displacement (*Xu*) imposed to bar's center with reference to

The servo actuator varies the vertical position of bar's center depending on the measured displacements on bar's free end. This is done to isolate the payload from vibrations

A sensor that converts movements into voltage is used to measure vibrations on the payload. Those voltages feed servo motor, thus closing the control loop. Controllers are used to improve control efficiency, reaching thus performance specifications previously determined. This subsystem composed by one (or more) sensors, controllers and a servo-

The nonlinear model used was developed in [1]. For the lever system it was given by:

**x fx u**

**x**

, ,

*y g t t q sen q*

*t tt*

1 2

*l*

(1)

, <sup>2</sup>

**Figure 1.** Application example: active suspension system

Digital simulations are employed in two case studies and the results are compared. On the first case study, the fuzzy controller and the fuzzy supervisor are tuned manually. Genetic algorithms (GA) are used on those systems tuning, in the second case study. Genetic algorithms usage facilitates designer's task and allows tuning parameters' optimization.

Next session describes the electromechanical system used and presents its models developed in [1]. The nonlinear model is used to validate the hierarchical fuzzy controller and in its fuzzy components' tuning, while the linearized model is used for robust control design. Performance criteria's are established at the end of this section.

## **2. Electromechanical system**

Figure 2 details the electromechanical system used for vibration suppression and reference tracking. It consists on an *l* centimeters long bar with *J* inertia angular moment. It is considered that its mass *mB*, is concentrated in its geometric center. This bar works as a lever which is supported in two points by systems with stiffness and damping, given by: *kA*, *kB*, *cA*, *cB*. In one extremity of the bar, a mass, *mA*, called absorbing mass, is used to make a counterbalance with the payload. The payload is represented by a mass, *mC*, on bar's free end. This system part is purely mechanical, being called lever system.

**Figure 2.** Electromechanical system

**Figure 1.** Application example: active suspension system

**2. Electromechanical system** 

This electromechanical system can be used to eliminate vibrations in the camera of unmanned vehicles and also to position this camera. It can also be used in manned vehicles

Digital simulations are employed in two case studies and the results are compared. On the first case study, the fuzzy controller and the fuzzy supervisor are tuned manually. Genetic algorithms (GA) are used on those systems tuning, in the second case study. Genetic algorithms usage facilitates designer's task and allows tuning parameters' optimization.

Next session describes the electromechanical system used and presents its models developed in [1]. The nonlinear model is used to validate the hierarchical fuzzy controller and in its fuzzy components' tuning, while the linearized model is used for robust control

Figure 2 details the electromechanical system used for vibration suppression and reference tracking. It consists on an *l* centimeters long bar with *J* inertia angular moment. It is considered that its mass *mB*, is concentrated in its geometric center. This bar works as a lever which is supported in two points by systems with stiffness and damping, given by: *kA*, *kB*, *cA*, *cB*. In one extremity of the bar, a mass, *mA*, called absorbing mass, is used to make a counterbalance with the payload. The payload is represented by a mass, *mC*, on bar's free

design. Performance criteria's are established at the end of this section.

end. This system part is purely mechanical, being called lever system.

for drivers' seat positioning and to eliminate vibrations on it, as shown in Figure 1.

The vertical position control of bar's center is made by a servo actuator. This actuator consists of a DC servo motor whose axis is directly coupled to a spindle. The propeller's spindle step is given by *LP*. It represents the direct relationship between motor's rotation angle (*θM*) and control's vertical displacement (*Xu*) imposed to bar's center with reference to the motor position (*XB*).

The servo actuator varies the vertical position of bar's center depending on the measured displacements on bar's free end. This is done to isolate the payload from vibrations originated at the base.

A sensor that converts movements into voltage is used to measure vibrations on the payload. Those voltages feed servo motor, thus closing the control loop. Controllers are used to improve control efficiency, reaching thus performance specifications previously determined. This subsystem composed by one (or more) sensors, controllers and a servoactuator, is called control system.

The nonlinear model used was developed in [1]. For the lever system it was given by:

$$\begin{aligned} \dot{\mathbf{x}} &= \mathbf{f}\left(\mathbf{x}(t), \mathbf{u}\left(t\right), t\right) \\ \dot{\mathbf{y}} &= \mathbf{g}\left(\mathbf{x}(t), t\right) = q\_1 - \frac{l}{2} \text{sen}\left(q\_2\right) \end{aligned} \tag{1}$$

Where:

$$\mathbf{x}(t) = \begin{bmatrix} q\_1 \\ \dot{q}\_1 \\ q\_2 \\ \dot{q}\_2 \end{bmatrix}; \quad \mathbf{u}(t) = \begin{bmatrix} \mathbf{x}\_u \\ d \end{bmatrix}; \tag{2}$$

Hierarchical Fuzzy Control 339

<sup>2</sup>

1 *P m uu a m m*

*xx e*

2 1 1 12 11 12 11 12 11 12 12 2 2 112233 1 2

010000 0

3 3 2

000100 0

*x x*

 

> *x x x*

 

11 11 21 21 11 21 2 2 32 42 2 2 5 6 , , , , , *m m x q xx q d x dx q x q d dx x*

 

(13)

 

(14)

*P*

 

12 12 22 22 32 32 11 11 21 21 12 12 11 11 11 21 1 1 1 1 1 1 2 2 2 2 2 2 12 22 22 22 21 11 21 21 2 2 12 22

11 12 11 12 11 17

*L CL a CL a CL a CL a CL a L CL*

*a L CL a CL a CL a CL a CL*

1 11 1 12 2 13 2 14 3 18 12 21 22 21 22 3 19 1 21 1 22 2 23 2 24 21 27 22 31 3 28 3 29 3

*P P P*

*L CL a L CL a L CL a*

*<sup>i</sup> b* are given by:

*b b b b b b ab ab* , , ,,, , *b ab ab*

, ,,, ,

,, , ,,

<sup>1</sup> , ,

*m m*

*T T*

22 21 22 21 22 21 4 4 112233 5 5 <sup>31</sup> 6 6 <sup>3</sup>

000001

*l x*

00000

10 000 2

 

*L K*

The equation that describes servo actuator dynamics is given by:

For robust control project it was used the linearized model founded in [1].

*x x x x aaaaaa x x x x aaaaaa x x x x a*

 

*c*

*<sup>i</sup> <sup>a</sup>* and *jk*

*P*

*y x*

The system states are:

The coefficients *jk*

Where:

And:

<sup>212</sup> cos *DT km k q* (10)

*T T* (11)

32 1

11

 

 

22 22 1 2

0 0

 

0

21

*a*

 

*P*

*m*

*T*

(12)

 

(15)

*e d*

And:

$$\mathbf{f}\left(\mathbf{x}(t),\mathbf{u}(t),t\right) = \begin{bmatrix} \dot{q}\_1\\ \frac{-k\_2\left(T\_{11}+T\_{12}\right) + k\_1\cos\left(q\_2\right)\left(T\_{21}+T\_{22}\right)}{T\_D} \\\ \dot{q}\_2\\ \frac{-m\left(T\_{21}+T\_{22}\right) + k\_1\cos\left(q\_2\right)\left(T\_{11}+T\_{12}\right)}{T\_D} \end{bmatrix} \tag{3}$$

With:

$$k\_1 = \frac{1}{2}(m\_A - m\_C) \tag{4}$$

$$k\_2 = \left(\frac{l}{2}\right)^2 \left(m\_A + m\_C\right) + \frac{1}{12} m\_B \left(a^2 + l^2\right) \tag{5}$$

$$\begin{aligned} T\_{11} &= \left(k\_A \mathcal{S}\_A + k\_B \mathcal{S}\_B - mg\right) - \frac{1}{8} k\_A \left(8q\_1 + 4lsen\left(q\_2\right) - 8d\right) \\ &- lk\_B \left(q\_1 - \mathbf{x}\_{\cup I} - d\right) - c\_B \left(\dot{q}\_1 - \dot{\mathbf{x}}\_{\cup I} - \dot{d}\right) + k\_1 \dot{q}\_2^2 \text{sen}\left(q\_2\right) + m\_m \overline{\mathbf{x}}\_u \end{aligned} \tag{6}$$

$$\begin{aligned} T\_{12} &= -\frac{\frac{1}{16}c\_A \left(8q\_1 + 4lsen\left(q\_2\right) - 8d\right)}{4\left(q\_1 + \frac{l}{2}sen\left(q\_2\right) - d\right)^2 + l^2\left(1 - \cos\left(q\_2\right)\right)^2} \\ &\times \left[8\left(\dot{q}\_1 + \frac{l}{2}\dot{q}\_2\cos\left(q\_2\right) - \dot{d}\right)\left(q\_1 + \frac{l}{2}sen\left(q\_2\right) - d\right) + 2l^2\dot{q}\_2\sec\left(q\_2\right)\left(1 - \cos\left(q\_2\right)\right)\right] \end{aligned} \tag{7}$$

$$T\_{21} = \frac{1}{2} (m\_A - m\_C)g - \frac{1}{8}k\_A \left[ 4 \left( q\_1 + \frac{l}{2} \text{sen}(q\_2) - d \right) l \cos(q\_2) + 2l^2 \left( 1 - \cos(q\_2) \right) \text{sen}(q\_2) \right] \tag{8}$$

$$T\_{22} = -\frac{\frac{1}{16}c\_A \left[4\left(q\_1 + \frac{l}{2}\operatorname{sen}(q\_2) - d\right)l\cos(q\_2) + 2l^2\left(1 - \cos(q\_2)\right)\operatorname{sen}(q\_2)\right]}{4\left(q\_1 + \frac{l}{2}\operatorname{sen}(q\_2) - d\right)^2 + l^2\left(1 - \cos(q\_2)\right)^2} \tag{9}$$

$$= \left[8\left(\dot{q}\_1 + \frac{l}{2}\dot{q}\_2\cos(q\_2) - \dot{d}\right)\left(q\_1 + \frac{l}{2}\operatorname{sen}(q\_2) - d\right) + 2l^2\dot{q}\_2\operatorname{sen}(q\_2)\left(1 - \cos(q\_2)\right)\right]$$

Hierarchical Fuzzy Control 339

$$T\_D = -k\_2 m + \left(k\_1 \cos\left(q\_2\right)\right)^2\tag{10}$$

The equation that describes servo actuator dynamics is given by:

$$
\ddot{\mathbf{x}}\_u + \frac{1}{T\_m} \dot{\mathbf{x}}\_u = \frac{L\_p K\_m}{T\_m} \mathbf{e}\_a \tag{11}
$$

For robust control project it was used the linearized model founded in [1].

$$
\begin{bmatrix}
\dot{x}\_1\\ \dot{x}\_2\\ \dot{x}\_3\\ \dot{x}\_4\\ \dot{x}\_5\\ \dot{x}\_6
\end{bmatrix} = \begin{bmatrix}
0 & 1 & 0 & 0 & 0 & 0\\ -a\_1^{12} & -a\_1^{11} & -a\_2^{12} & -a\_2^{11} & -a\_3^{12} & -a\_3^{11}\\ 0 & 0 & 0 & 1 & 0 & 0\\ -a\_1^{22} & -a\_1^{21} & -a\_2^{22} & -a\_2^{21} & -a\_3^{22} & -a\_3^{21}\\ 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & -a\_3^{31}
\end{bmatrix} \cdot \begin{bmatrix}
x\_1\\ x\_2\\ \dot{x}\_3\\ x\_4\\ x\_5\\ x\_6\\ \dot{x}\_4
\end{bmatrix} + \begin{bmatrix}
0 & \beta\_1^{11}\\ \beta\_1^{12} & \beta\_2^{12}\\ 0 & \beta\_2^{12} & \beta\_3^{12}\\ \beta\_1^{21} & \beta\_2^{22} & 0\\ 0 & 0 & 0\\ \beta\_1^{12} & 0
\end{bmatrix}.
\cdot \begin{bmatrix}
x\_1\\ \dot{x}\_2\\ \dot{x}\_3\\ \dot{x}\_4\\ \dot{x}\_5\\ \dot{x}\_6\\ x\_6\\ x\_7
\end{bmatrix} \cdot \begin{bmatrix}
x\_1\\ \dot{x}\_2\\ \dot{x}\_3\\ \dot{x}\_4\\ \dot{x}\_5\\ \dot{x}\_6\\ x\_6
\end{bmatrix}.
\tag{12}
$$

The system states are:

338 Fuzzy Controllers – Recent Advances in Theory and Applications

**fx u**

, ,

2

12 2

 

2

2 82 *AC A*

16 2

*A*

16

*T*

*T*

*t tt*

And:

With:

 1 1 2 2

*t t*

*q*

**x u**

*q*

; ; *<sup>u</sup>*

cos

1 2 11 12 1 2 21 22

 

*q kT T k q T T T*

cos

2 21 22 1 2 11 12

*q mT T k q T T T*

*D*

<sup>1</sup> <sup>2</sup> *<sup>A</sup> <sup>C</sup>*

2 12 *AC B <sup>l</sup> k m m ma l* 

8 cos 2 1 cos

<sup>1</sup> <sup>4</sup> cos 2 1 cos

 

*T k k mg k q lsen q d*

1 1 12 2

*lk q x d c q x d k q sen q m x*

11 1 2

*BU BU m u*

2 2

 <sup>2</sup> 21 1 2 2 2 2 <sup>1</sup> <sup>4</sup> cos 2 1 cos

1 2 2

*<sup>l</sup> q sen q d l q*

*<sup>l</sup> c q sen q d l q l q sen q*

12 2 1 2 22 2

*l l q q q d q sen q d l q sen q q*

 

4 1 cos

8 cos 2 1 cos

12 2 1 2 22 2

*l l q q q d q sen q d l q sen q q*

*l l T m m g k q sen q d l q l q sen q* 

 

2

8 *AA BB <sup>A</sup>*

1 2

*Ac q lsen q d*

<sup>1</sup> 84 8

1 2 2

2

*<sup>l</sup> q sen q d l q*

4 1 cos

2 2

22 2

2 2

  1

*D*

(2)

(3)

(6)

(7)

(8)

(9)

*<sup>l</sup> k mm* (4)

2 2

2

2

2

2

2 2

12 2 2 2

<sup>1</sup> 84 8

(5)

*q x*

*q d*

11 11 21 21 11 21 2 2 32 42 2 2 5 6 , , , , , *m m x q xx q d x dx q x q d dx x* (13) Where:

$$\begin{aligned} \boldsymbol{\beta}\_{1}^{12} = \boldsymbol{b}\_{1}^{12}, \; \boldsymbol{\beta}\_{1}^{22} = \boldsymbol{b}\_{1}^{22}, \; \boldsymbol{\beta}\_{1}^{32} = \boldsymbol{b}\_{1}^{32}, \; \boldsymbol{\beta}\_{2}^{11} = \boldsymbol{b}\_{2}^{11}, \; \boldsymbol{\beta}\_{2}^{21} = \boldsymbol{b}\_{2}^{21}, \; \boldsymbol{\beta}\_{2}^{12} = \boldsymbol{b}\_{2}^{12} - \boldsymbol{a}\_{1}^{11} \boldsymbol{b}\_{2}^{11} - \boldsymbol{a}\_{2}^{11} \boldsymbol{b}\_{2}^{21},\\ \boldsymbol{\beta}\_{2}^{22} = \boldsymbol{b}\_{2}^{22} - \boldsymbol{a}\_{1}^{21} \boldsymbol{b}\_{2}^{11} - \boldsymbol{a}\_{2}^{21} \boldsymbol{b}\_{2}^{21} \end{aligned} \tag{14}$$

The coefficients *jk <sup>i</sup> <sup>a</sup>* and *jk <sup>i</sup> b* are given by:

$$a\_1^{11} = -\mathbb{C}\mathcal{L}\_{11},\ a\_1^{12} = -\mathbb{C}\mathcal{L}\_{12},\ a\_2^{11} = -\mathbb{C}\mathcal{L}\_{13},\ a\_2^{12} = -\mathbb{C}\mathcal{L}\_{14},\ a\_3^{11} = -\mathbb{L}\_p\mathcal{C}\mathcal{L}\_{18} + \frac{\mathbb{L}\_p\mathcal{C}\mathcal{L}\_{17}}{T\_m},$$

$$a\_3^{12} = -\mathbb{L}\_p\mathcal{C}\mathcal{L}\_{19},\ a\_1^{21} = -\mathbb{C}\mathcal{L}\_{21},\ a\_1^{22} = -\mathbb{C}\mathcal{L}\_{22},\ a\_2^{21} = -\mathbb{C}\mathcal{L}\_{23},\ a\_2^{22} = -\mathbb{C}\mathcal{L}\_{24},\tag{15}$$

$$a\_3^{21} = -\mathbb{L}\_p\mathcal{C}\mathcal{L}\_{28} + \frac{\mathbb{L}\_p\mathcal{C}\mathcal{L}\_{27}}{T\_m},\ a\_3^{22} = -\mathbb{L}\_p\mathcal{C}\mathcal{L}\_{29},\ a\_3^{31} = \frac{1}{T\_m}$$

And:

$$\begin{aligned} b\_1^{12} = \frac{L\_p K\_m \text{CL}\_{17}}{T\_m}, \ b\_2^{11} = \text{CL}\_{15}, \ b\_2^{12} = \text{CL}\_{16}, \ b\_1^{22} = \frac{L\_p K\_m \text{CL}\_{27}}{T\_m}, \ b\_2^{21} = \text{CL}\_{25}, \ b\_2^{22} = \text{CL}\_{26} \end{aligned} \tag{16}$$

$$b\_1^{32} = \frac{K\_m}{T\_m}$$

Where:

$$CL\_{11} = \frac{-k\_2\left(-c\_a - c\_b\right) - \frac{l}{2}k\_1c\_a}{-k\_2m + k\_1^2} \tag{17}$$

Hierarchical Fuzzy Control 341

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

<sup>1</sup>

1

1

<sup>1</sup>

<sup>1</sup>

2 1 2 *a ab*

2 1

2 1

2 1 2 *a ab*

2 1 2 *a ab*

*km k*

1 27 2 2 1

1 28 2 2 1

1 29 2 2 1

*mk k k lk*

 

*km k*

*mc k c c*

*km k*

2 2 *a a l l mk k k*

*km k*

2 2 *a a l l mc k c*

*mk k k lk*

*km k*

22 2

2

23 2

2

24 2

25 2

26 2

*<sup>m</sup> k m CL km k* 

*<sup>b</sup> k c CL km k* 

*<sup>b</sup> k lk CL km k* 

Nonlinear system response to a step reference and for a step disturb was used to determine

Figure 3 shows the nonlinear system in closed loop, without controllers, step response. This response is characterized by the influence of two vibrations modes: one slower and overdamped and the other faster and oscillating. It practically has no overshoot. The settling time, considering an accommodation range of ± 5% of the reference signal amplitude, is more than 12.5s. The rise time from 0 to100% of the reference signal amplitude is greater than 19s. This large difference between the rise time and the settling time highlights the

With the reference fixed at zero, when a 0.01m amplitude step disturbance is injected into the system without the controller, its output goes upper than one and a half the amplitude

*l*

*l*

*l*

*CL*

*CL*

*CL*

*CL*

*CL*

Figure 4 shows the non-controlled system response to a disturbance.

the performance criteria.

influence of the overdamped mode [3].

$$\text{CL}\_{12} = \frac{-k\_2\left(-k\_a - lk\_b\right) - \frac{l}{2}k\_1k\_a}{-k\_2m + k\_1^2} \tag{18}$$

$$CL\_{13} = \frac{\frac{1}{2}k\_2c\_a - \left(\frac{l}{2}\right)^2 k\_1c\_a}{-k\_2m + k\_1^2} \tag{19}$$

$$\text{CL}\_{14} = \frac{\frac{1}{2}k\_2k\_a - \left(\frac{l}{2}\right)^2 k\_1k\_a}{-k\_2m + k\_1^2} \tag{20}$$

$$CL\_{15} = \frac{-k\_2 \left(c\_a + c\_b\right) + \frac{l}{2}k\_1 c\_a}{-k\_2 m + k\_1^2} \tag{21}$$

$$CL\_{16} = \frac{-k\_2\left(k\_a + lk\_b\right) + \frac{l}{2}k\_1k\_a}{-k\_2m + k\_1^2} \tag{22}$$

$$\text{CL}\_{17} = \frac{-k\_2 m\_m}{-k\_2 m + k\_1^2} \tag{23}$$

$$\text{CL}\_{18} = \frac{-k\_2 c\_b}{-k\_2 m + k\_1^2} \tag{24}$$

$$CL\_{19} = \frac{-k\_2lk\_b}{-k\_2m + k\_1^2} \tag{25}$$

$$\text{CL}\_{21} = \frac{\frac{1}{2}m\mathbf{c}\_a + k\_1 \left(-\mathbf{c}\_a - \mathbf{c}\_b\right)}{-k\_2m + k\_1^2} \tag{26}$$

Hierarchical Fuzzy Control 341

$$\text{CL}\_{22} = \frac{\frac{1}{2}mk\_a + k\_1\left(-k\_a - lk\_b\right)}{-k\_2m + k\_1^2} \tag{27}$$

$$CL\_{23} = \frac{\left(\frac{l}{2}\right)^2 mc\_a - \frac{l}{2}k\_1 c\_a}{-k\_2 m + k\_1^2} \tag{28}$$

$$CL\_{24} = \frac{\left(\frac{l}{2}\right)^2 mk\_a - \frac{l}{2}k\_1 k\_a}{-k\_2 m + k\_1^2} \tag{29}$$

$$\text{CL}\_{25} = \frac{-\frac{l}{2}mc\_a + k\_1\left(c\_a + c\_b\right)}{-k\_2m + k\_1^2} \tag{30}$$

$$\text{CLL}\_{26} = \frac{-\frac{l}{2}mk\_a + k\_1\left(k\_a + lk\_b\right)}{-k\_2m + k\_1^2} \tag{31}$$

$$\text{CL}\_{27} = \frac{k\_1 m\_m}{-k\_2 m + k\_1^2} \tag{32}$$

$$CL\_{28} = \frac{k\_1 c\_b}{-k\_2 m + k\_1^2} \tag{33}$$

$$CL\_{29} = \frac{k\_1lk\_b}{-k\_2m + k\_1^2} \tag{34}$$

Nonlinear system response to a step reference and for a step disturb was used to determine the performance criteria.

Figure 3 shows the nonlinear system in closed loop, without controllers, step response. This response is characterized by the influence of two vibrations modes: one slower and overdamped and the other faster and oscillating. It practically has no overshoot. The settling time, considering an accommodation range of ± 5% of the reference signal amplitude, is more than 12.5s. The rise time from 0 to100% of the reference signal amplitude is greater than 19s. This large difference between the rise time and the settling time highlights the influence of the overdamped mode [3].

Figure 4 shows the non-controlled system response to a disturbance.

340 Fuzzy Controllers – Recent Advances in Theory and Applications

Where:

12 17 11 12 22 27 21 22 1 2 15 2 16 1 2 25 2 26

*m m*

*T T*

*CL*

*CL*

*CL*

*CL*

*CL*

*CL*

,, , ,, *P m P m*

*L K CL L K CL <sup>b</sup> b CL b CL b b CL b CL*

*<sup>K</sup> <sup>b</sup> T*

11 2

12 2

13 2

14 2

15 2

16 2

*<sup>m</sup> k m CL*

*<sup>b</sup> k c CL*

*<sup>b</sup> k lk CL*

21 2

*l*

*CL*

 

 

 

 

 

 

 

32 1

*m m*

2 1

*km k*

2 1

*km k*

2 1

2 1

2 2 *a a l l kc kc*

*km k*

2 1

2 1

2 2 *a a l l kk kk*

*km k*

2 1

*km k*

2 1

*km k*

2 17 2 2 1

*km k*

2 18 2 2 1

*km k*

2 19 2 2 1

*km k*

*mc k c c*

*km k*

2 1 2 *a ab*

<sup>1</sup>

2 1 2 *ab a <sup>l</sup> k k lk k k*

2 1 2 *ab a <sup>l</sup> k c c kc*

2 1

2 *ab a <sup>l</sup> k k lk k k*

2

2

2 1 2 *ab a <sup>l</sup> k c c kc*

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

With the reference fixed at zero, when a 0.01m amplitude step disturbance is injected into the system without the controller, its output goes upper than one and a half the amplitude of the injected disturbance. The non-controlled system needs about 12.8s to reject this disturbance on the mentioned condition, considering that the disturbance is sufficiently rejected when the response amplitude is reduced to a range of ± 5% of the injected disturbance amplitude, around zero. Figure 4 shows this response.

Hierarchical Fuzzy Control 343

The settling time should be reduced to at most 20% of the time obtained with the noncontrolled system; The overshoot should be less than 10%; The time required for the controlled system to reject a step disturbance, must be reduced by at least 95%; Furthermore, the response signal may not exceed 40% of disturbance's amplitude; Finally, the control signal generated must respect the servo-actuator saturation limits, that, in this case, is ± 15V. Those specifications were achieved through the use of the hierarchical fuzzy controller. Each controller design aimed to meet some performance specifications. In that way, conflicting specifications were separately addressed, instead of trying, in each project, to get a fit to satisfy conflicting specifications, relaxing those specifications. So the hierarchical fuzzy controller should take the best features of each controller, to meet all the specifications

In vibration control, as well as in several other applications, it is desired that the control system presents robustness to the effects of factors such as: modeling errors, variations in the parameters of the system being controlled, noise and disturbances. There are at least two reasons why the robustness is a desirable feature in the control systems: the need of control systems that operates satisfactorily, even in operating conditions different from the ones considered in the model design; and the possibility to adopt an intentionally simplified project model, to reduce: the time spent in the modeling stage and the resulting controller

Among the main techniques for robust controllers synthesis can be cited: The Linear Quadratic Gaussian / Loop Transfer Recovery (LQG/LTR), H2 and H∞ optimizations, methods based on Lyapunov functions, minmax optimization and Quantitative Feedback

The LQG/LTR controller designed in [1] was used to allow a better comparison between the optimized hierarchical fuzzy controller implemented and the non-optimized developed in [1]. Furthermore the LQG/LTR technique has a simple and systematic design procedure, the controller robustness is ensured by this procedure, even in a broad class modeling errors

This procedure has two steps: initially the target filter loop (TFL) must be projected. It must meet the performance specifications previously established. Once obtained an appropriate TFL, its characteristics are recovered for the transfer function of the loop formed by the

The LTR procedure, initially proposed in [6], suggests that the TFL is achieved through the design of a Linear Quadratic Regulator (LQR) and then recovered by adjusting a Kalman filter. Another way to do it is to set a Kalman filter, to obtain a satisfactory target filter loop,

presence and also the number of design parameters is relatively small [5].

and then project an optimal state feedback, type LQR, to recover the TFL [1].

controller and the nominal model *G sG s K N* .

Given the linearized model in form:

described in this section.

**3. Robust control** 

complexity [4].

Theory (QFT).

**Figure 3.** Electromechanical system step response without controllers and disturbance

**Figure 4.** Non-controlled system response to a step disturbance

Thus, the performance specifications that characterize a satisfactory response to the nonlinear system are: A step reference signal must be tracked without regime error; the rise time should be reduced to at most 10% of the time obtained by the non-controlled system; The settling time should be reduced to at most 20% of the time obtained with the noncontrolled system; The overshoot should be less than 10%; The time required for the controlled system to reject a step disturbance, must be reduced by at least 95%; Furthermore, the response signal may not exceed 40% of disturbance's amplitude; Finally, the control signal generated must respect the servo-actuator saturation limits, that, in this case, is ± 15V.

Those specifications were achieved through the use of the hierarchical fuzzy controller. Each controller design aimed to meet some performance specifications. In that way, conflicting specifications were separately addressed, instead of trying, in each project, to get a fit to satisfy conflicting specifications, relaxing those specifications. So the hierarchical fuzzy controller should take the best features of each controller, to meet all the specifications described in this section.

## **3. Robust control**

342 Fuzzy Controllers – Recent Advances in Theory and Applications

0

Amplitude(m)

0.02

0.04

0.06

Amplitude(m)

0.08

0.1

0.12

0.14

disturbance amplitude, around zero. Figure 4 shows this response.

**Figure 3.** Electromechanical system step response without controllers and disturbance

0 5 10 15 20 25 30

Disturbance Reference Plant output

Disturbance Reference Plant output

Time(s)

**Figure 4.** Non-controlled system response to a step disturbance

Thus, the performance specifications that characterize a satisfactory response to the nonlinear system are: A step reference signal must be tracked without regime error; the rise time should be reduced to at most 10% of the time obtained by the non-controlled system;

0 5 10 15 20 25 30

Time(s)

of the injected disturbance. The non-controlled system needs about 12.8s to reject this disturbance on the mentioned condition, considering that the disturbance is sufficiently rejected when the response amplitude is reduced to a range of ± 5% of the injected

> In vibration control, as well as in several other applications, it is desired that the control system presents robustness to the effects of factors such as: modeling errors, variations in the parameters of the system being controlled, noise and disturbances. There are at least two reasons why the robustness is a desirable feature in the control systems: the need of control systems that operates satisfactorily, even in operating conditions different from the ones considered in the model design; and the possibility to adopt an intentionally simplified project model, to reduce: the time spent in the modeling stage and the resulting controller complexity [4].

> Among the main techniques for robust controllers synthesis can be cited: The Linear Quadratic Gaussian / Loop Transfer Recovery (LQG/LTR), H2 and H∞ optimizations, methods based on Lyapunov functions, minmax optimization and Quantitative Feedback Theory (QFT).

> The LQG/LTR controller designed in [1] was used to allow a better comparison between the optimized hierarchical fuzzy controller implemented and the non-optimized developed in [1]. Furthermore the LQG/LTR technique has a simple and systematic design procedure, the controller robustness is ensured by this procedure, even in a broad class modeling errors presence and also the number of design parameters is relatively small [5].

> This procedure has two steps: initially the target filter loop (TFL) must be projected. It must meet the performance specifications previously established. Once obtained an appropriate TFL, its characteristics are recovered for the transfer function of the loop formed by the controller and the nominal model *G sG s K N* .

> The LTR procedure, initially proposed in [6], suggests that the TFL is achieved through the design of a Linear Quadratic Regulator (LQR) and then recovered by adjusting a Kalman filter. Another way to do it is to set a Kalman filter, to obtain a satisfactory target filter loop, and then project an optimal state feedback, type LQR, to recover the TFL [1].

Given the linearized model in form:

$$\begin{aligned} \dot{\mathbf{x}}(t) &= \mathbf{A}\mathbf{x}(t) + \mathbf{B}\mathbf{u}(t) \\ \mathbf{Y}(t) &= \mathbf{C}\mathbf{x}(t) \end{aligned} \tag{35}$$

Hierarchical Fuzzy Control 345

Disturbance Reference Plant output

With the system controlled only by this robust controller, a step reference with 0.1m amplitude, was tracked without regime error; the rise time from 0 to 100% of the reference, in disturbance absence, was about 0.03s which corresponds to 0.16% of the rise time obtained by the non-controlled system; the settling time for (± 5%) was 0.17s, so, it was reduced to 1.36% of the time obtained with the non-controlled system; the overshoot was 22.4% and the control signal generated to track this reference signal, surpassed the actuator saturation levels. Therefore, with respect to the reference tracking, the controller could not satisfy two performance criteria established, because the overshoot was higher than 10% of the reference signal and some control signals produced, extrapolates the servo actuator saturation levels. Figures 6 and 7 show the system response when controlled only by this

**Figure 6.** System response on step reference tracking, only with the robust controller, and in

With a null reference, a 0,01m step disturbance was injected in the system. The time required for the system to reject this disturbance using only the robust controller, was approximately 0.17s; what represents a 98.67% time reduction when compared to noncontrolled system exposed to the same situation; The response signal maximum amplitude was 17.89% of the disturbance amplitude; the control signal varied within the levels of the servo actuator saturation. So in disturbance rejection, with null reference, the robust controller met all performance requirements described, as could be seen on figures 8 and 9.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time(s)

It was also evaluated the system response, only with the robust controller, to a square wave reference with 0.1m peak to peak, 0.015Hz frequency and 100s duration. The system tracked this reference without regime error, the rise time and the settling time satisfied the performance specifications, but, again, as was expected, the control signal exceeded the actuator saturation limits and the overshoot exceeded the maximum stated in performance

LQG/LTR robust controller.

disturbances absence

0

0.02

0.04

0.06

Amplitude (m)

0.08

0.1

0.12

0.14

criteria, as could be seen on figures 10 and 11.

The Kalman's filter design begins with the solution of the following algebraic Riccati equation:

$$\mathbf{A}\boldsymbol{\Sigma} + \boldsymbol{\Sigma}\mathbf{A}^T + \mathbf{W}\boldsymbol{\Xi}\mathbf{W}^T - \boldsymbol{\Sigma}\mathbf{C}^T\boldsymbol{\Theta}^{-1}\mathbf{C}\boldsymbol{\Sigma} = \mathbf{0} \tag{36}$$

In [1] it was used:

$$\mathbf{W} = \mathbf{B}\begin{pmatrix} \vdots \ 1 \end{pmatrix}; \mathbf{\Xi} = \mathbf{I}; \mathbf{\Theta} = \mu \mathbf{I} \tag{37}$$

Where **B** :,1 corresponds to the first column of the B matrix and is the project's free parameter. This choice was made because the first attempt to select the W matrix must be the matrix related with the control input [5]. As could be seen in [1], this choice proved satisfactory.

In [1] were also used: <sup>6</sup> 10 to obtain the TFL and <sup>12</sup> 10 to recover the TFL, resulting in a LQG/LTR robust controller with the following desired characteristics: good speed in test model controlled response accommodation, when tracking a reference, and principally a good rejection of disturbances. Figure 5 illustrates the TFL obtained and recovered for these values of and .

**Figure 5.** TFL obtained and recovered

As mentioned earlier, this LQG/LTR controller was used to allow a better comparison between the optimized hierarchical fuzzy controller implemented and the non-optimized developed in [1].

With the system controlled only by this robust controller, a step reference with 0.1m amplitude, was tracked without regime error; the rise time from 0 to 100% of the reference, in disturbance absence, was about 0.03s which corresponds to 0.16% of the rise time obtained by the non-controlled system; the settling time for (± 5%) was 0.17s, so, it was reduced to 1.36% of the time obtained with the non-controlled system; the overshoot was 22.4% and the control signal generated to track this reference signal, surpassed the actuator saturation levels. Therefore, with respect to the reference tracking, the controller could not satisfy two performance criteria established, because the overshoot was higher than 10% of the reference signal and some control signals produced, extrapolates the servo actuator saturation levels. Figures 6 and 7 show the system response when controlled only by this LQG/LTR robust controller.

344 Fuzzy Controllers – Recent Advances in Theory and Applications

equation:

In [1] it was used:

satisfactory.

values of

 and .



Phase (degrees)

0

50

Amplitude (dB)

100

In [1] were also used: <sup>6</sup>

**Figure 5.** TFL obtained and recovered

developed in [1].

(35)

**I** (37)

10 to recover the TFL, resulting

Obtained TFL Recovered TFL

is the project's free

<sup>1</sup> 0 *T TT* **AΣ Σ A W ΞW ΣC Θ CΣ** (36)

*ttt*

The Kalman's filter design begins with the solution of the following algebraic Riccati

**W B** :,1 ; ; **Ξ I Θ**

parameter. This choice was made because the first attempt to select the W matrix must be the matrix related with the control input [5]. As could be seen in [1], this choice proved

in a LQG/LTR robust controller with the following desired characteristics: good speed in test model controlled response accommodation, when tracking a reference, and principally a good rejection of disturbances. Figure 5 illustrates the TFL obtained and recovered for these

Bode Diagram

As mentioned earlier, this LQG/LTR controller was used to allow a better comparison between the optimized hierarchical fuzzy controller implemented and the non-optimized

10-1 100 101 102 103

Frequence(rad/s)

10 to obtain the TFL and <sup>12</sup>

Where **B** :,1 corresponds to the first column of the B matrix and

**x Ax Bu**

*t t* 

**Y Cx**

**Figure 6.** System response on step reference tracking, only with the robust controller, and in disturbances absence

With a null reference, a 0,01m step disturbance was injected in the system. The time required for the system to reject this disturbance using only the robust controller, was approximately 0.17s; what represents a 98.67% time reduction when compared to noncontrolled system exposed to the same situation; The response signal maximum amplitude was 17.89% of the disturbance amplitude; the control signal varied within the levels of the servo actuator saturation. So in disturbance rejection, with null reference, the robust controller met all performance requirements described, as could be seen on figures 8 and 9.

It was also evaluated the system response, only with the robust controller, to a square wave reference with 0.1m peak to peak, 0.015Hz frequency and 100s duration. The system tracked this reference without regime error, the rise time and the settling time satisfied the performance specifications, but, again, as was expected, the control signal exceeded the actuator saturation limits and the overshoot exceeded the maximum stated in performance criteria, as could be seen on figures 10 and 11.

Hierarchical Fuzzy Control 347

Disturbance Reference Plant output

**Figure 9.** Robust controller signal for a step disturbance rejection, with a null reference






Amplitude (m)

0

0.02

0.04

0.06

0.08



0

Amplitude (V)

5

10

15

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (s)

**Figure 10.** System response on square wave reference tracking, only with the robust controller

0 10 20 30 40 50 60 70 80 90 100

Time (s)

**Figure 7.** Robust controller signal for a step reference tracking, in disturbances absence

**Figure 8.** System response on step disturbance rejection, only with the robust controller, and with a null reference

**Figure 9.** Robust controller signal for a step disturbance rejection, with a null reference

**Figure 7.** Robust controller signal for a step reference tracking, in disturbances absence

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (s)

Disturbance Reference Plant output

**Figure 8.** System response on step disturbance rejection, only with the robust controller, and with a null

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (s)

reference

Amplitude (m)




Amplitude (V)

0

2000

4000

6000

**Figure 10.** System response on square wave reference tracking, only with the robust controller

Hierarchical Fuzzy Control 349

**Figure 13.** Robust controller signal for a step reference tracking, in uniformly distributed white noise

0 5 10 15 20 25 30

Time (s)

Comparing those results with the first shown, it is concluded that the rise time and the settling time were the same for both situations. In the white noise presence, the system showed a slightly higher overshoot, 23.4%, what is unsatisfactory according to the performance criteria, as well as the control signal applied that extrapolates the actuator saturation limits. So,

Therefore, those requirements should be met by the fuzzy controller and the supervisor

Fuzzy controllers are those that make use of fuzzy logic, which is based on the fuzzy sets' theory. This theory was developed by Zadeh in 1965 [7], to deal with the vague aspect of information through the mathematical representation of expressions commonly used by humans, also called linguistic variables, which give a not exact value to a variable

Fuzzy logic attaches to a statement, not the value 'true' or 'false', but a veracity degree within

Due to its ability to handle uncertainty and imprecision, fuzzy logic has been characterized as one of the current technologies for the successful development of systems to control sophisticated processes, enabling the use of simple controllers to satisfy complex design requirements, even when the model of the system to be controlled has uncertainties [8-14].

The greatest difficulty in creating fuzzy systems is the definition of linguistic terms and rules. One way to solve this problem is to use hybrid approaches as models called neurofuzzy. In a neuro-fuzzy system those parameters are learned with the presentation of

as expected, in both cases the same performance requirements were not satisfied.

must properly combine those two controllers to meet all performance criteria.

presence




Amplitude (V)

0

2000

4000

6000

**4. Fuzzy control** 

a numeric range.

characteristic of the object under observation.

**Figure 11.** Robust controller signal for a square wave reference tracking, in disturbances absence

Finally, the system, only with the robust controller, was tested on tracking a step reference in the presence of uniformly distributed white noise with 0.02m peak to peak. Figures 12 and 13 show the system response and the control signal applied to the plant in this situation.

**Figure 12.** System response on step reference tracking, using only the robust controller, and in uniformly distributed white noise presence

**Figure 13.** Robust controller signal for a step reference tracking, in uniformly distributed white noise presence

Comparing those results with the first shown, it is concluded that the rise time and the settling time were the same for both situations. In the white noise presence, the system showed a slightly higher overshoot, 23.4%, what is unsatisfactory according to the performance criteria, as well as the control signal applied that extrapolates the actuator saturation limits. So, as expected, in both cases the same performance requirements were not satisfied.

Therefore, those requirements should be met by the fuzzy controller and the supervisor must properly combine those two controllers to meet all performance criteria.

## **4. Fuzzy control**

348 Fuzzy Controllers – Recent Advances in Theory and Applications

situation.




Amplitude (V)

0

2000

4000

6000

**Figure 11.** Robust controller signal for a square wave reference tracking, in disturbances absence

**Figure 12.** System response on step reference tracking, using only the robust controller, and in

0 5 10 15 20 25 30

Time (s)

uniformly distributed white noise presence


0

0.02

0.04

0.06

Amplitude (m)

0.08

0.1

0.12

0.14

Finally, the system, only with the robust controller, was tested on tracking a step reference in the presence of uniformly distributed white noise with 0.02m peak to peak. Figures 12 and 13 show the system response and the control signal applied to the plant in this

0 10 20 30 40 50 60 70 80 90 100

Disturbance Reference Plant output

Time (s)

Fuzzy controllers are those that make use of fuzzy logic, which is based on the fuzzy sets' theory. This theory was developed by Zadeh in 1965 [7], to deal with the vague aspect of information through the mathematical representation of expressions commonly used by humans, also called linguistic variables, which give a not exact value to a variable characteristic of the object under observation.

Fuzzy logic attaches to a statement, not the value 'true' or 'false', but a veracity degree within a numeric range.

Due to its ability to handle uncertainty and imprecision, fuzzy logic has been characterized as one of the current technologies for the successful development of systems to control sophisticated processes, enabling the use of simple controllers to satisfy complex design requirements, even when the model of the system to be controlled has uncertainties [8-14].

The greatest difficulty in creating fuzzy systems is the definition of linguistic terms and rules. One way to solve this problem is to use hybrid approaches as models called neurofuzzy. In a neuro-fuzzy system those parameters are learned with the presentation of

training pairs (input, desired output) to a neural network whose nodes basically computes intersection and union operators [15-18]. Another hybrid approach that allows the parameters tuning for fuzzy systems, consists in the use of genetic algorithms [19].

Hierarchical Fuzzy Control 351

(39)

(38)

0 *i i de tes*

With the use of genetic algorithms for tuning of all parameters of fuzzy controller, the designer's task is to limit the search space of GA and find a good setting of its parameters, in

The determination of the limits of the search spaces for the fuzzy controller optimization was based on the results obtained in [1] and in several tests. The population size, the percentage of mutation and the stopping criteria were also determined from several tests.

To obtain the results that will be shown, a square wave was used as reference, allowing a good fit to the fuzzy controller for several references. The genetic algorithm configuration was: population of 30 individuals, all children were generated by recombination with mutation probability of 5% for each gene; the roulette method was used on selection step. The stopping criteria were: maximum number of iterations equal to 100, repeating the best individual for 25% of the generations' maximum number, maintaining the average fitness of the population

For the evaluation of each individual the control of the nonlinear system using only the fuzzy controller, was simulated during 100s. The evaluation function used for this controller

123 1 2 3 1 2

*ev rrr s s s s s*

Where: the "*tri's*" are the rise times, the "*osi's*" are the overshoots, the "*tsi's*" are the settling times, "*em*" is the average error, "*umax*" is the maximum positive amplitude of the control signal above actuator's saturation and "*umin*" is the maximum amplitude of the negative

Higher weights were given to the mean square error and to the rise times because it was observed that they had a lower representation in the evaluation function, than the settling time and the peaks of the control signals above actuator's saturation. Thus allowing to the genetic algorithm, the search for a tune that provides not only short settling times through low control signals, but also small rise times, and that the system does not presents regime errors. Lower weights were given for the settling times and the overshoots, to allow the

Two search spaces were defined for output functions' coefficients determination: one from 0 to 100, for the coefficients of the functions associated with rules that involve in its antecedent the linguistic variables negative big or positive big, and another from 0 to 60 to the

*f ind t t t o o o t t*

. 8 8 8 0,9 0,9 0,9 0,7 0,7

for 10% of the generations' maximum number and mean square error of 10-5.

2 3 max min

*t eu u*

search for fuzzy controllers that give the system a higher speed.

Figures 14 and 15, shows the fuzzy controller optimized membership functions.

0,7 15

*s m*

control signal, below actuator's saturation.

Where: *e* is the error.

tuning, was:

order to obtain the desired results.

*dt* 

A satisfactory definition of the number of membership functions and the degree of overlap between them is fundamental when implementing a fuzzy controller. It directly influences on the next stage, called inference [20].

The inference uses a set of rules that describe the dependence between the linguistic variables of input and output functions. This relationship is usually determined heuristically and consists of two steps: aggregation, when evaluating the 'if' part of each rule, through the operator "and fuzzy," and the composition stage, using the operator "or fuzzy" to considering the different conclusions of the active rules [20, 21].

After the inference from the action to be taken, the classical fuzzy models require a decoding of the linguistic value for the numeric variable output, called defuzzification. This output can represent functions such as adjusting the position of a button, or provide voltage to a particular motor.

The Takagi Sugeno fuzzy controllers do not need a defuzzification step, because they obtain this precise equivalence directly [9, 19]. Therefore they were used to compound the fuzzy hierarchical controller.

For the design and optimization of the fuzzy logic controller it was used the nonlinear model of the physical system, as this model provides a more accurate representation of it.

All available knowledge about the system being controlled is of fundamental importance for the initial stage of designing a fuzzy controller, therefore, knowing the geometrical characteristics, the dynamics and any system particularity, can significantly reduce the project effort [1]. The fuzzy logic controller used has the following structure: Two inputs, which are: the tracking error (the difference between the reference and the system output) and its derivative; an output which is the control signal. For the output variable composition 25 firstorder Sugeno functions are used; five linguistic variables were defined for each input variable: Negative Big, Negative Small, Zero, Positive Small and Positive Big; Triangular membership functions were chosen for the input variables; The probabilistic t-norm and t-conorm operators were chosen; The rule base is composed by 25 rules. For each rule there is a Sugeno output function; For the inference procedure, the Sugeno interpolation model was chosen.

The tuning of this fuzzy controller was made by a genetic algorithm. This algorithm is based on the laws of natural selection and evolution. It searches to an optimal solution in the space of solutions given by the designer, using probabilistic rules for combining solutions in order to improve their quality. It is therefore an efficient search strategy that can be used in optimization or classification problems [22-25].

In the fuzzy controller's optimization, each individual is formed by 70 genes. The first 20 genes represent the input membership functions. The 50 subsequent genes describe the coefficients of the Sugeno output functions, *it* and *is* . Those functions are given by:

Hierarchical Fuzzy Control 351

$$
\left[t\_i \cdot e + s\_i \cdot \frac{de}{dt} + 0\right] \tag{38}
$$

Where: *e* is the error.

350 Fuzzy Controllers – Recent Advances in Theory and Applications

on the next stage, called inference [20].

particular motor.

hierarchical controller.

training pairs (input, desired output) to a neural network whose nodes basically computes intersection and union operators [15-18]. Another hybrid approach that allows the

A satisfactory definition of the number of membership functions and the degree of overlap between them is fundamental when implementing a fuzzy controller. It directly influences

The inference uses a set of rules that describe the dependence between the linguistic variables of input and output functions. This relationship is usually determined heuristically and consists of two steps: aggregation, when evaluating the 'if' part of each rule, through the operator "and fuzzy," and the composition stage, using the operator "or fuzzy" to

After the inference from the action to be taken, the classical fuzzy models require a decoding of the linguistic value for the numeric variable output, called defuzzification. This output can represent functions such as adjusting the position of a button, or provide voltage to a

The Takagi Sugeno fuzzy controllers do not need a defuzzification step, because they obtain this precise equivalence directly [9, 19]. Therefore they were used to compound the fuzzy

For the design and optimization of the fuzzy logic controller it was used the nonlinear model of the physical system, as this model provides a more accurate representation of it.

All available knowledge about the system being controlled is of fundamental importance for the initial stage of designing a fuzzy controller, therefore, knowing the geometrical characteristics, the dynamics and any system particularity, can significantly reduce the project effort [1]. The fuzzy logic controller used has the following structure: Two inputs, which are: the tracking error (the difference between the reference and the system output) and its derivative; an output which is the control signal. For the output variable composition 25 firstorder Sugeno functions are used; five linguistic variables were defined for each input variable: Negative Big, Negative Small, Zero, Positive Small and Positive Big; Triangular membership functions were chosen for the input variables; The probabilistic t-norm and t-conorm operators were chosen; The rule base is composed by 25 rules. For each rule there is a Sugeno output

function; For the inference procedure, the Sugeno interpolation model was chosen.

optimization or classification problems [22-25].

The tuning of this fuzzy controller was made by a genetic algorithm. This algorithm is based on the laws of natural selection and evolution. It searches to an optimal solution in the space of solutions given by the designer, using probabilistic rules for combining solutions in order to improve their quality. It is therefore an efficient search strategy that can be used in

In the fuzzy controller's optimization, each individual is formed by 70 genes. The first 20 genes represent the input membership functions. The 50 subsequent genes describe the

coefficients of the Sugeno output functions, *it* and *is* . Those functions are given by:

parameters tuning for fuzzy systems, consists in the use of genetic algorithms [19].

considering the different conclusions of the active rules [20, 21].

With the use of genetic algorithms for tuning of all parameters of fuzzy controller, the designer's task is to limit the search space of GA and find a good setting of its parameters, in order to obtain the desired results.

The determination of the limits of the search spaces for the fuzzy controller optimization was based on the results obtained in [1] and in several tests. The population size, the percentage of mutation and the stopping criteria were also determined from several tests.

To obtain the results that will be shown, a square wave was used as reference, allowing a good fit to the fuzzy controller for several references. The genetic algorithm configuration was: population of 30 individuals, all children were generated by recombination with mutation probability of 5% for each gene; the roulette method was used on selection step. The stopping criteria were: maximum number of iterations equal to 100, repeating the best individual for 25% of the generations' maximum number, maintaining the average fitness of the population for 10% of the generations' maximum number and mean square error of 10-5.

For the evaluation of each individual the control of the nonlinear system using only the fuzzy controller, was simulated during 100s. The evaluation function used for this controller tuning, was:

$$\begin{aligned} f\_{ev} \left( \text{ind.} \right) &= 8t\_{r1} + 8t\_{r2} + 8t\_{r3} + 0.9o\_{s1} + 0.9 \left| o\_{s2} \right| + 0.9o\_{s3} + 0.7t\_{s1} + 0.7t\_{s2} \\ &+ 0.7t\_{s3} + 15e\_{m}^{2} + u\_{\text{max}} - u\_{\text{min}} \end{aligned} \tag{39}$$

Where: the "*tri's*" are the rise times, the "*osi's*" are the overshoots, the "*tsi's*" are the settling times, "*em*" is the average error, "*umax*" is the maximum positive amplitude of the control signal above actuator's saturation and "*umin*" is the maximum amplitude of the negative control signal, below actuator's saturation.

Higher weights were given to the mean square error and to the rise times because it was observed that they had a lower representation in the evaluation function, than the settling time and the peaks of the control signals above actuator's saturation. Thus allowing to the genetic algorithm, the search for a tune that provides not only short settling times through low control signals, but also small rise times, and that the system does not presents regime errors. Lower weights were given for the settling times and the overshoots, to allow the search for fuzzy controllers that give the system a higher speed.

Figures 14 and 15, shows the fuzzy controller optimized membership functions.

Two search spaces were defined for output functions' coefficients determination: one from 0 to 100, for the coefficients of the functions associated with rules that involve in its antecedent the linguistic variables negative big or positive big, and another from 0 to 60 to the

coefficients of the other functions. The independent terms of output functions were not optimized and were always made equal to zero. The output functions obtained after the tuning can be seen in Table 1.

Hierarchical Fuzzy Control 353

Function name Parameters [t s] S1 [10.93 94.07] S2 [95.34 3.74] S3 [25.50 27.17] S4 [39.12 62.36] S5 [95.86 94.12] S6 [28.08 47.32] S7 [46.76 43.30] S8 [27.71 13.14] S9 [42.55 5.52] S10 [51.22 41.09] S11 [53.64 5.98] S12 [31.55 12.47] S13 [53.31 1.20] S14 [37.24 16.27] S15 [33.20 46.69] S16 [21.36 29.04] S17 [2.97 23.94] S18 [11.98 43.12] S19 [42.94 45.81] S20 [10.77 27.04] S21 [75.33 50.81] S22 [85.35 66.19] S23 [45.64 50.41] S24 [58.09 15.50] S25 [15.18 11.76]

**Table 1.** Output functions' parameters of the optimized fuzzy controller

servo-actuator saturation. Those results are shown in figures 16 and 17.

The control of the electromechanical system made only by the optimized fuzzy controller, presented a poor performance in tracking a 0.1m amplitude step reference, in disturbance absence. The overshoot presented was out of performance specifications (30.60%), and the settling time was almost equal to the uncontrolled system settling time (11.09s). However, the system showed no error at steady state, the rise time was satisfactory, 0.22 s, and the control signal produced was far below the actuator saturation, allowing the use of this controller in the hierarchical control scheme, as a supplier of control signals applicable in situations of great error, where the signals produced by the robust controller extrapolate the

Table 2 shows the fuzzy controller rule base.

**Figure 14.** Fuzzy controller optimized membership functions of error input

**Figure 15.** Fuzzy controller optimized membership functions of error derivative input


**Table 1.** Output functions' parameters of the optimized fuzzy controller

Table 2 shows the fuzzy controller rule base.

352 Fuzzy Controllers – Recent Advances in Theory and Applications

**Figure 14.** Fuzzy controller optimized membership functions of error input

**Figure 15.** Fuzzy controller optimized membership functions of error derivative input


de (m)

tuning can be seen in Table 1.

0

0

0.2

0.4

Membership degree

0.6

0.8

1

0.2

0.4

Membership degree

0.6

0.8

1

coefficients of the other functions. The independent terms of output functions were not optimized and were always made equal to zero. The output functions obtained after the

eNB eNS eZ ePS ePB


e (m)

deNB deNS deZ dePS dePB

The control of the electromechanical system made only by the optimized fuzzy controller, presented a poor performance in tracking a 0.1m amplitude step reference, in disturbance absence. The overshoot presented was out of performance specifications (30.60%), and the settling time was almost equal to the uncontrolled system settling time (11.09s). However, the system showed no error at steady state, the rise time was satisfactory, 0.22 s, and the control signal produced was far below the actuator saturation, allowing the use of this controller in the hierarchical control scheme, as a supplier of control signals applicable in situations of great error, where the signals produced by the robust controller extrapolate the servo-actuator saturation. Those results are shown in figures 16 and 17.


Hierarchical Fuzzy Control 355

Disturbance Reference Plant output

In the rejection of a 0.01m amplitude step disturbance with the null reference, the system response with the fuzzy controller was also unsatisfactory, because its amplitude exceeded in 21% the disturbance amplitude, and it took about 2.01s to reject it, far above the 0.64s, established as a goal. Figures 18 and 19 show the system response and the control signal for

**Figure 18.** System response on step disturbance rejection, only with the fuzzy controller, and with a

0 5 10 15 20 25 30

Time (s)

**Figure 19.** Fuzzy controller signal for a step disturbance rejection, with a null reference

0 5 10 15 20 25 30

Time (s)

this case.

null reference






0

Amplitude (V)

0.5

1

1.5

2

0

5

Amplitude (m)

10

15

x 10-3

**Table 2.** Rule base of fuzzy controller

**Figure 16.** System response on step reference tracking, only with the fuzzy controller, and in disturbances absence

**Figure 17.** Fuzzy controller signal for a step reference tracking, in disturbances absence

In the rejection of a 0.01m amplitude step disturbance with the null reference, the system response with the fuzzy controller was also unsatisfactory, because its amplitude exceeded in 21% the disturbance amplitude, and it took about 2.01s to reject it, far above the 0.64s, established as a goal. Figures 18 and 19 show the system response and the control signal for this case.

354 Fuzzy Controllers – Recent Advances in Theory and Applications

Error

disturbances absence




0

Amplitude (V)

5

10

15

0

0.02

0.04

0.06

Amplitude (m)

0.08

0.1

0.12

0.14

derivative

**Table 2.** Rule base of fuzzy controller

Error

**Figure 16.** System response on step reference tracking, only with the fuzzy controller, and in

0 5 10 15 20 25 30

Time (s)

**Figure 17.** Fuzzy controller signal for a step reference tracking, in disturbances absence

0 5 10 15 20 25 30

Time (s)

eNB eNS eZ ePS ePB

deNB S1 S6 S11 S16 S21 deNS S2 S7 S12 S17 S22 deZ S3 S8 S13 S18 S23 dePS S4 S9 S14 S19 S24 dePB S5 S10 S15 S20 S25

> Disturbance Reference Plant output

**Figure 18.** System response on step disturbance rejection, only with the fuzzy controller, and with a null reference

**Figure 19.** Fuzzy controller signal for a step disturbance rejection, with a null reference

It was also evaluated the system response on a square wave reference tracking in the absence of disturbances and using only the fuzzy controller. As can be seen in figures 20 and 21 the system tracked the reference without regime error, the rise times were acceptable, but the settling times were greater than desirable, moreover, the overshoot and the control signal extrapolated performance specifications. But the fuzzy controller's peak signal was much lower than the robust one.

Hierarchical Fuzzy Control 357

Plant

From these results, it can be concluded that the function of the fuzzy controller is to bring the plant to a situation that favors the use of the robust controller, avoiding the

The multiple controllers' fusion seeks to achieve higher performance than those obtained

The supervisor's task is to find an ideal combination of control signals generated by the controllers designed, in such way that this combination compose the control signal which will effectively act on the plant. To do this, the supervisor evaluates the operating condition in each instant, and then determines an importance hierarchy of each control signal. Therefore, in addition to control signals generated by the controllers, the supervisor must also receive information that enables to evaluate the operating condition at all instants, and then, based on this evaluation, the supervisor will sort, hierarchically, the outputs of the controllers, compounding then the control signal that will act on the plant. This hierarchy is the level of importance associated by the supervisor to each controller in every operating condition. It defines the participation of each controller in the control signal that will be

The fuzzy supervisor used was a Takagi-Sugeno system with: two inputs, which are the same used in the fuzzy controller; 3 linguistic variables (negative, zero and positive), which are represented by trapezoidal membership functions; two output functions, which are zero

Figure 22 illustrates the architecture used for the control signals fusion via hierarchical fuzzy

uR

uF

Hierarchical fuzzy supervisor

us

1-us

y 1-q-1

+ 

**Figure 22.** Control scheme using the fuzzy hierarchical controller

*Fuzzy* controller

1-q-1

Robust controller

extrapolation of control signal limits.

**5. Fuzzy supervisor** 

using only one controller.

applied on the plant.

order functions.

supervisor.

+ Ref

**Figure 20.** System response on square wave reference tracking, only with the fuzzy controller

**Figure 21.** Fuzzy controller signal for a square wave reference tracking, in disturbances absence

From these results, it can be concluded that the function of the fuzzy controller is to bring the plant to a situation that favors the use of the robust controller, avoiding the extrapolation of control signal limits.

## **5. Fuzzy supervisor**

356 Fuzzy Controllers – Recent Advances in Theory and Applications

much lower than the robust one.


Amplitude (V)

Amplitude (m)

It was also evaluated the system response on a square wave reference tracking in the absence of disturbances and using only the fuzzy controller. As can be seen in figures 20 and 21 the system tracked the reference without regime error, the rise times were acceptable, but the settling times were greater than desirable, moreover, the overshoot and the control signal extrapolated performance specifications. But the fuzzy controller's peak signal was

**Figure 20.** System response on square wave reference tracking, only with the fuzzy controller

0 10 20 30 40 50 60 70 80 90 100

Disturbance Reference Plant output

Time (s)

**Figure 21.** Fuzzy controller signal for a square wave reference tracking, in disturbances absence

0 10 20 30 40 50 60 70 80 90 100

Time (s)

The multiple controllers' fusion seeks to achieve higher performance than those obtained using only one controller.

The supervisor's task is to find an ideal combination of control signals generated by the controllers designed, in such way that this combination compose the control signal which will effectively act on the plant. To do this, the supervisor evaluates the operating condition in each instant, and then determines an importance hierarchy of each control signal. Therefore, in addition to control signals generated by the controllers, the supervisor must also receive information that enables to evaluate the operating condition at all instants, and then, based on this evaluation, the supervisor will sort, hierarchically, the outputs of the controllers, compounding then the control signal that will act on the plant. This hierarchy is the level of importance associated by the supervisor to each controller in every operating condition. It defines the participation of each controller in the control signal that will be applied on the plant.

The fuzzy supervisor used was a Takagi-Sugeno system with: two inputs, which are the same used in the fuzzy controller; 3 linguistic variables (negative, zero and positive), which are represented by trapezoidal membership functions; two output functions, which are zero order functions.

Figure 22 illustrates the architecture used for the control signals fusion via hierarchical fuzzy supervisor.

**Figure 22.** Control scheme using the fuzzy hierarchical controller

From the difference between a reference signal, specified by the operator, and the vertical position of the bar's free end, measured by a sensor, it is produced an error signal. With this error signal, the robust controller determines its control action, trying to correct the vertical position of the bar's free end. The fuzzy controller also provides a control signal in an attempt to eliminate the tracking error; for this, it needs this error signal and its derivative. The control signal which actually will act on the plant will be the weighted sum of signals produced by the controllers. The degree of participation of each control action is determined by the supervisor, which uses as well as the fuzzy controller, the error information and its derivative. According to the control signal, the servo-actuator will provide vertical displacements to bar's center, to correct the tracking error.

Hierarchical Fuzzy Control 359

overshoot was 3.9% in [1] it was 7%. The settling time for (± 5%) was 0.22s, less than half that was obtained in [1]. The control signal generated by the optimized hierarchical fuzzy controller to track this reference had lower levels than the ones generated by the nonoptimized hierarchical fuzzy controller. The optimized hierarchical fuzzy controller has used the fuzzy controller for less time, it is because the optimized fuzzy controller provide a faster response than the designed in [1]. Also the transition between controllers was softer with the optimized system. Figures 25, 26 and 27 shows the results obtained with those two

eN eZ eP

structures on the reference tracking in disturbances absence.

**Figure 23.** Fuzzy supervisor optimized membership functions of error input

0

0

0.2

0.4

Membership degree

0.6

0.8

1

0.2

0.4

Membership degree

0.6

0.8

1


e (m)

deN deZ deP

**Figure 24.** Fuzzy Supervisor optimized membership functions of error derivative input


de (m)

The two output functions used are the same presented in [1]. They are described in Table 3.


**Table 3.** Output functions' parameters of the fuzzy supervisor

So, when supervisor output is null, only the robust controller will actuate on the plant, when supervisor output is equal to one, only the fuzzy controller will actuate, for intermediate outputs a combination of those controllers' signals will be applied on the plant.

The supervisor's input membership functions were tuned by a genetic algorithm using the square wave reference and the two controllers. Its evaluation function is given by:

$$\begin{aligned} f\_{ev}\left(\text{ind.}\right) &= t\_{r1} + t\_{r2} + t\_{r3} + o\_{s1} + \left| o\_{s2} \right| + o\_{s3} + t\_{s1} + t\_{s2} + t\_{s3} + e\_m \\ &+ 0,01u\_{\text{max1}} - 0,01u\_{\text{min1}} + 0,01u\_{\text{max2}} - 0,01u\_{\text{min2}} + 0,01u\_{\text{max3}} - 0,01u\_{\text{min3}} \end{aligned} \tag{40}$$

There was no need to give greater weight to the mean square error and to the rise times, as was done for the tuning of the fuzzy controller, because from some tunings, the settling time and the overshoot became very small. The reduction of all performance descriptors along the supervisor tuning was so high that it was necessary to assign lower weights to control signals peaks above the saturation of the servo actuator, to avoid favoring a performance criterion and neglect others.

Figures 23 and 24, shows the fuzzy supervisor optimized membership functions.

The rule base of the supervisor was not optimized by genetic algorithm. It was the same used in [1], as shown in Table 4.

As mentioned the results obtained with the optimized hierarchical fuzzy controller will be compared with the ones obtained by the non-optimized one (presented in [1]). On tracking a 0.1m amplitude step reference, the optimized hierarchical fuzzy controller has satisfied all performance criteria established and presented a more rapid response than the system controlled by the non-optimized hierarchical fuzzy controller. The rise time from 0 to 100% of the reference was approximately 0.22s, which is half the one obtained in [1]. The overshoot was 3.9% in [1] it was 7%. The settling time for (± 5%) was 0.22s, less than half that was obtained in [1]. The control signal generated by the optimized hierarchical fuzzy controller to track this reference had lower levels than the ones generated by the nonoptimized hierarchical fuzzy controller. The optimized hierarchical fuzzy controller has used the fuzzy controller for less time, it is because the optimized fuzzy controller provide a faster response than the designed in [1]. Also the transition between controllers was softer with the optimized system. Figures 25, 26 and 27 shows the results obtained with those two structures on the reference tracking in disturbances absence.

358 Fuzzy Controllers – Recent Advances in Theory and Applications

displacements to bar's center, to correct the tracking error.

**Table 3.** Output functions' parameters of the fuzzy supervisor

.

criterion and neglect others.

used in [1], as shown in Table 4.

From the difference between a reference signal, specified by the operator, and the vertical position of the bar's free end, measured by a sensor, it is produced an error signal. With this error signal, the robust controller determines its control action, trying to correct the vertical position of the bar's free end. The fuzzy controller also provides a control signal in an attempt to eliminate the tracking error; for this, it needs this error signal and its derivative. The control signal which actually will act on the plant will be the weighted sum of signals produced by the controllers. The degree of participation of each control action is determined by the supervisor, which uses as well as the fuzzy controller, the error information and its derivative. According to the control signal, the servo-actuator will provide vertical

The two output functions used are the same presented in [1]. They are described in Table 3.

Function name Parameters [e(t) de(t)/d(t) 1]x[t s 1]T

So, when supervisor output is null, only the robust controller will actuate on the plant, when supervisor output is equal to one, only the fuzzy controller will actuate, for intermediate

The supervisor's input membership functions were tuned by a genetic algorithm using the

max1 min1 max 2 min 2 max 3 min 3

(40)

LTR [0 0 0] FUZ [0 0 1]

outputs a combination of those controllers' signals will be applied on the plant.

square wave reference and the two controllers. Its evaluation function is given by:

 <sup>2</sup> 123 1 2 3123

 

*ev rr r s s ssss m f ind t t t o o o t t t e*

Figures 23 and 24, shows the fuzzy supervisor optimized membership functions.

0,01 0,01 0,01 0,01 0,01 0,01

*uuu uu u*

There was no need to give greater weight to the mean square error and to the rise times, as was done for the tuning of the fuzzy controller, because from some tunings, the settling time and the overshoot became very small. The reduction of all performance descriptors along the supervisor tuning was so high that it was necessary to assign lower weights to control signals peaks above the saturation of the servo actuator, to avoid favoring a performance

The rule base of the supervisor was not optimized by genetic algorithm. It was the same

As mentioned the results obtained with the optimized hierarchical fuzzy controller will be compared with the ones obtained by the non-optimized one (presented in [1]). On tracking a 0.1m amplitude step reference, the optimized hierarchical fuzzy controller has satisfied all performance criteria established and presented a more rapid response than the system controlled by the non-optimized hierarchical fuzzy controller. The rise time from 0 to 100% of the reference was approximately 0.22s, which is half the one obtained in [1]. The

**Figure 23.** Fuzzy supervisor optimized membership functions of error input

**Figure 24.** Fuzzy Supervisor optimized membership functions of error derivative input


Hierarchical Fuzzy Control 361

Optimized supervisor Non-optimized supervisor

**Figure 27.** Comparison of signals generated by the two supervisors in tracking a step reference

performance of the system controlled by the optimized HFC.

0


0

0.02

0.04

Amplitude (m)

0.06

0.08

0.1

0.12

0.2

0.4

0.6

Supervisor output

0.8

1

expected, a better performance was obtained using the optimized HFC.

**Figure 28.** Comparison of the two HFC in tracking a step reference under disturbance

0 5 10 15 20 25 30

Disturbance Reference

Non-optimized HFC response Optimized HFC response

Time (s)

The performance of the optimized HFC was tested on a step reference tracking, in the presence of white noise with 0.02m peak to peak. Figures 28 and 29 show, again, the best

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

To finalize the comparisons, the system was tested on tracking a square wave reference. As

**Table 4.** Rule base of supervisor

**Figure 25.** Comparison of the two hierarchical controllers in tracking a step reference

**Figure 26.** Comparison of control signals generated by the two hierarchical controllers in tracking a step reference

de(t)/dt

**Table 4.** Rule base of supervisor

0

0.02

0.04

0.06

Amplitude (m)

0.08

0.1

0.12

step reference




0

2

Amplitude (V)

4

6

8

10

e (t)

**Figure 25.** Comparison of the two hierarchical controllers in tracking a step reference

0 5 10 15 20 25 30

Disturbance Reference

Non-optimized HFC response Optimized HFC response

> HFC non-optimized HFC optimized

Time (s)

**Figure 26.** Comparison of control signals generated by the two hierarchical controllers in tracking a

0 5 10 15 20 25 30

Time (s)

eN eZ eP

deN FUZ FUZ FUZ deZ FUZ LTR FUZ deP FUZ FUZ FUZ

**Figure 27.** Comparison of signals generated by the two supervisors in tracking a step reference

The performance of the optimized HFC was tested on a step reference tracking, in the presence of white noise with 0.02m peak to peak. Figures 28 and 29 show, again, the best performance of the system controlled by the optimized HFC.

To finalize the comparisons, the system was tested on tracking a square wave reference. As expected, a better performance was obtained using the optimized HFC.

**Figure 28.** Comparison of the two HFC in tracking a step reference under disturbance

Hierarchical Fuzzy Control 363

Optimized HFC HFC

Non-optimized supervisor Optimized supervisor

As could be seen on Figure 31, both hierarchical controllers extrapolated the actuator saturation limits, as it was punctual the use of a saturator may not affect the system

**Figure 31.** Comparison of control signals generated by the two HFC in tracking a square wave

Again the optimized supervisor has used less the fuzzy controller than the non-optimized

0 10 20 30 40 50 60 70 80 90 100

Time (s)

**Figure 32.** Comparison of signals generated by the two supervisors in tracking a square wave reference

32 32.5 33 33.5 34 34.5 35 35.5 36 36.5 37

Time (s)

performance.

reference



0

50

100

Amplitude (V)

150

200

250

supervisor.

0

0.2

0.4

0.6

Supervisor output

0.8

1

**Figure 29.** Comparison of control signals generated by the two HFC in tracking a step reference under disturbance

**Figure 30.** Comparison of the two HFC in tracking a square wave reference

As could be seen on Figure 31, both hierarchical controllers extrapolated the actuator saturation limits, as it was punctual the use of a saturator may not affect the system performance.

362 Fuzzy Controllers – Recent Advances in Theory and Applications

disturbance





Amplitude (m)

0

0.02

0.04

0.06




0

Amplitude (V)

10

20

30

40

**Figure 29.** Comparison of control signals generated by the two HFC in tracking a step reference under

0 5 10 15 20 25 30

HFC optimized HFC non-optimized

> Disturbance Reference HFC Optimized HFC

Time (s)

0 10 20 30 40 50 60 70 80 90 100

Time (s)

**Figure 30.** Comparison of the two HFC in tracking a square wave reference

**Figure 31.** Comparison of control signals generated by the two HFC in tracking a square wave reference

Again the optimized supervisor has used less the fuzzy controller than the non-optimized supervisor.

**Figure 32.** Comparison of signals generated by the two supervisors in tracking a square wave reference

## **6. Conclusion**

One of the main advantages of hierarchical control is to combine different techniques. It allows the supervisor to take the best of each technique.

Hierarchical Fuzzy Control 365

[2] Araújo FMU (1998) Electromechanical System for Vibration Active Control [Master

[3] Araújo FMU and Yoneyama T (2001) Modeling and Control of an Electromechanical Device for Vibrations Active Control. Proceedings of the II National Seminar on Control

[4] Cruz JJ (1988) Contribution to the Study of Robust Stability for Nonlinear Multivariable Regulators [PhD thesis]. Sao Jose dos Campos, Brazil: National Institute of Space

[5] Cruz JJ (1996) Multivariable Robust Control. Sao Paulo: Publisher from Sao Paulo

[6] Doyle JC and Stein G (1981) Multivariable Feedback Design: Concepts for a Classical/Modern Synthesis. IEEE Transactions on Automatic Control, Vol. AC-26, No.

[8] Lee CC (1990) Fuzzy Logic in Control Systems: Fuzzy Logic Controller (Part I). IEEE

[9] Driankov D, Hellendoorn H, Reinfrank M (1993) An Introduction to Fuzzy Control. New

[10] Dutta S (1993) Fuzzy Logic Applications: Technological and Strategic Issues. IEEE

[11] Karr CL, Gentry EJ (1993) Fuzzy Control of Ph Using Genetic Algorithms. IEEE

[12] Chiu S, Chand S (1993) Adaptive Traffic Signal Control Using Fuzzy Logic. Proceedings of the 2nd IEEE International Conference on Fuzzy Systems; March 1993; San Francisco,

[13] Castro JL (1995) Fuzzy Logic Controllers are Universal Approximators. IEEE

[14] Guerra R, Sandri S, Souza MLO (1997) Autonomous Control of Satellites Altitude Using Fuzzy Logic. In: III Brazilian Symposium on Intelligent Automation, pp.337-342,

[15] Kosko B (1992) Neural Networks and Fuzzy System. Englewood Cliffs: Prentice Hall.

[16] Wang L, Mendel J M (1992) Generating Fuzzy Rules by Learning from Examples. IEEE

[17] Jang JSR (1993) ANFIS: Adaptive-Network-Based Fuzzy Inference System. IEEE

[18] Lin CT (1995) A Neural Fuzzy Control System with Structure and Parameter Learning.

[19] Sandri AS, Correa C (1999) Fuzzy Logic. In: V Neural Networks School, São José dos

[20] Shaw IS, Simões MG (1999) Fuzzy Control and Modeling. São Paulo: FAPESP, Editora

[7] Zadeh LA (1965) Fuzzy set. Fuzzy Sets, Information and Control, 8, pp.338-353.

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Campos, Brazil. pp.c073-c090.

Edgard Blücher LTDA.

thesis]. Joao Pessoa, Brazil: Paraiba Federal University.

and Automation. pp.15. Salvador, Brazil.

Research.

University.

1, pp. 4-16.

Vitoria, Brazil.

449 p.

York: Springer-Verlag. 316 p.

The results showed the advantages of using genetic algorithms, such as: making automatic tuning of fuzzy components of the HFC, greatly simplifying the design and allowing the obtaining of optimal controllers and supervisors, which is impossible via manual tuning.

As can be seen, the controllers were designed, relaxing some conflicting performance criteria: on the robust controller design the efforts were concentrated to obtain a rapid response and a rapid accommodation, in tracking references and in disturbance rejection, not worrying about the control signal amplitude, for references tracking. In fuzzy controller design the efforts were concentrated to obtain a rapid response and smaller control signals, but no major requirements for rapid accommodation, which had already been achieved by the robust controller; this way all performance requirements were satisfied through the use of the hierarchical fuzzy controller.

With the use of hierarchical control, the controller design becomes simpler because they are more specific, they do not have to meet conflicting performance criteria.

As a suggestion for future projects can be verified: other control techniques for vibration suppression and tracking reference; new ways to optimize the components of HFC; using more controllers in the composition of HFC; other methods for supervisor project; a better configuration of the proposed genetic algorithms.

## **Author details**

Carlos André Guerra Fonseca

*Informatics Department, Rio Grande do Norte State University, Natal, Brazil Computing Engineering and Automation Department, Rio Grande do Norte Federal University, Natal, Brazil* 

Fábio Meneghetti Ugulino de Araújo

*Computing Engineering and Automation Department, Rio Grande do Norte Federal University, Natal, Brazil* 

Marconi Câmara Rodrigues *Science and Technology School, Rio Grande do Norte Federal University, Natal, Brazil* 

## **7. References**

[1] Araújo FMU (2002) Automatic Intelligent Controllers with Applications in Mechanical Vibration Isolation [PhD thesis]. Sao Jose dos Campos, Brazil: Aeronautical Technology Institute.

[2] Araújo FMU (1998) Electromechanical System for Vibration Active Control [Master thesis]. Joao Pessoa, Brazil: Paraiba Federal University.

364 Fuzzy Controllers – Recent Advances in Theory and Applications

of the hierarchical fuzzy controller.

**Author details** 

*Natal, Brazil* 

*Natal, Brazil* 

**7. References** 

Institute.

Carlos André Guerra Fonseca

Marconi Câmara Rodrigues

Fábio Meneghetti Ugulino de Araújo

configuration of the proposed genetic algorithms.

allows the supervisor to take the best of each technique.

One of the main advantages of hierarchical control is to combine different techniques. It

The results showed the advantages of using genetic algorithms, such as: making automatic tuning of fuzzy components of the HFC, greatly simplifying the design and allowing the obtaining of optimal controllers and supervisors, which is impossible via manual tuning.

As can be seen, the controllers were designed, relaxing some conflicting performance criteria: on the robust controller design the efforts were concentrated to obtain a rapid response and a rapid accommodation, in tracking references and in disturbance rejection, not worrying about the control signal amplitude, for references tracking. In fuzzy controller design the efforts were concentrated to obtain a rapid response and smaller control signals, but no major requirements for rapid accommodation, which had already been achieved by the robust controller; this way all performance requirements were satisfied through the use

With the use of hierarchical control, the controller design becomes simpler because they are

As a suggestion for future projects can be verified: other control techniques for vibration suppression and tracking reference; new ways to optimize the components of HFC; using more controllers in the composition of HFC; other methods for supervisor project; a better

*Computing Engineering and Automation Department, Rio Grande do Norte Federal University,* 

*Computing Engineering and Automation Department, Rio Grande do Norte Federal University,* 

[1] Araújo FMU (2002) Automatic Intelligent Controllers with Applications in Mechanical Vibration Isolation [PhD thesis]. Sao Jose dos Campos, Brazil: Aeronautical Technology

*Science and Technology School, Rio Grande do Norte Federal University, Natal, Brazil* 

more specific, they do not have to meet conflicting performance criteria.

*Informatics Department, Rio Grande do Norte State University, Natal, Brazil* 

**6. Conclusion** 

	- [21] Tsoukalas LH, Uhrig RE (1997) Fuzzy and Neural Approaches in Engineering. New York: Publication, John Wiley & Sons.

**Chapter 0**

**Chapter 16**

**Enhancing Fuzzy Controllers Using**

Nora Boumella, Juan Carlos Figueroa and Sohail Iqbal

Additional information is available at the end of the chapter

the antecedent and the parameters of the consequent.

http://dx.doi.org/10.5772/51608

**1. Introduction**

**Generalized Orthogonality Principle**

In the early days, the parameters of the fuzzy logic systems were fixed arbitrary, thus leading to a large number of possibilities for FLSs. In 1992, it has been shown that linguistic rules can be converted into Fuzzy Basis Functions (FBFs), and numerical rules and its associated FBFs must be extracted from numerical data training. Since that time, a multitude of design methods to construct a FLS are proposed. Some of these methods are intensive on data analysis, some are aimed at computational simplicity, some are recursive and others are offline, but all based on the the same idea: *tune the parameters of a FLS using the numerical training data*. Methods for designing FLSs can be classified into two major categories: A first category where shapes and parameters of the antecedent MFs are fixed ahead of time and training data are used for tuning the consequent parameters, and a second category that consists of fixing the shapes of the antecedent and consequent MFs using training data to tune

Two kinds of FLSs, the Mamdani and the Takagi-Sugeno-Kang (TSK) FLSs are widely used and they are currently adopted by the scientific community. They solely differ in the way the consequent structure is defined. The fact that a TSK FLS does not require a time-consuming

In this chapter, we consider the first category to design a TSK FLS basing on alinear method. Our design approach requires a set of input-output numerical data training pairs. Given linguistic rules of the FLS, we expand this FLS as a series of FBFs that are functions of the FLS inputs. We use the input training data to compute these FBFs. Therefore, the system becomes linear in the FLS consequent parameters, and we consider each set of FBFs as a basis vector which is easy to be optimized. Then follows the consequent parameters optimization via a minimizing process of the error vector - *the output training data minus the FBFs vectors weighted by the consequent parameters* - norm. This minimzation can be obtained by applying the *Generalized Orthogonality Principle* (GOP). Optimization process is carefully analyzed in this chapter and its applications in two major areas of concern are demonstrated including

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

© 2012 Boumella 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,

defuzzification process makes it far more attractive for most of applications.


## **Enhancing Fuzzy Controllers Using Generalized Orthogonality Principle**

Nora Boumella, Juan Carlos Figueroa and Sohail Iqbal

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51608

## **1. Introduction**

366 Fuzzy Controllers – Recent Advances in Theory and Applications

York: Publication, John Wiley & Sons.

Networks School. Natal, Brazil.

University Press.

[21] Tsoukalas LH, Uhrig RE (1997) Fuzzy and Neural Approaches in Engineering. New

[22] Fonseca CAG, Araújo FMU, Maitelli AL, Medeiros AV (2003) Genetic Algorithms for Optimization of a Fuzzy Controller for Vibration Suppression. In: VI Brazilian

[23] Goldbarg MC and Goldbarg EFG (2005) Evolutionary Computation. In: VIII Neural

[24] Holland JH (1970) Robust Algorithms for Adaptation Set in a General Formal Framework. In: IEEE Symposium on Adaptive Processes Decision and Control, 17. Proceedings of the XVII IEEE Symposium on Adaptive Processes Decision and Control. [25] Holland JH (1975) Adaptation in Natural and Artificial Systems. Ann Arbor: Michigan

Symposium on Intelligent Automation, pp. 959-963. Bauru, Brazil.

In the early days, the parameters of the fuzzy logic systems were fixed arbitrary, thus leading to a large number of possibilities for FLSs. In 1992, it has been shown that linguistic rules can be converted into Fuzzy Basis Functions (FBFs), and numerical rules and its associated FBFs must be extracted from numerical data training. Since that time, a multitude of design methods to construct a FLS are proposed. Some of these methods are intensive on data analysis, some are aimed at computational simplicity, some are recursive and others are offline, but all based on the the same idea: *tune the parameters of a FLS using the numerical training data*. Methods for designing FLSs can be classified into two major categories: A first category where shapes and parameters of the antecedent MFs are fixed ahead of time and training data are used for tuning the consequent parameters, and a second category that consists of fixing the shapes of the antecedent and consequent MFs using training data to tune the antecedent and the parameters of the consequent.

Two kinds of FLSs, the Mamdani and the Takagi-Sugeno-Kang (TSK) FLSs are widely used and they are currently adopted by the scientific community. They solely differ in the way the consequent structure is defined. The fact that a TSK FLS does not require a time-consuming defuzzification process makes it far more attractive for most of applications.

In this chapter, we consider the first category to design a TSK FLS basing on alinear method. Our design approach requires a set of input-output numerical data training pairs. Given linguistic rules of the FLS, we expand this FLS as a series of FBFs that are functions of the FLS inputs. We use the input training data to compute these FBFs. Therefore, the system becomes linear in the FLS consequent parameters, and we consider each set of FBFs as a basis vector which is easy to be optimized. Then follows the consequent parameters optimization via a minimizing process of the error vector - *the output training data minus the FBFs vectors weighted by the consequent parameters* - norm. This minimzation can be obtained by applying the *Generalized Orthogonality Principle* (GOP). Optimization process is carefully analyzed in this chapter and its applications in two major areas of concern are demonstrated including

© 2012 Boumella 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.

robotics and dynamic systems. Firstly, we shall show the improved results with analysis upon the application of GOP in the Fuzzy Logic Controller (FLC) for an inverted pendulum. Secondly, we show how a FLS based on this principle enhances the performance of forecaster for the chaotic time series.

## **2. Fuzzy Logic Systems (FLS) basic concepts**

## **2.1. Fuzzy sets**

A *Fuzzy Set* (FS), *F* ∈ *X* is a set of ordered pairs of a generic element *x* and its degree, namely *Membership Function* (MF), *μF*(*x*). Any FS can be represented as follows:

$$F = \{ (\mathbf{x}, \mu\_F(\mathbf{x})) \, | \, \forall \mathbf{x} \in X \} \tag{1}$$

*2.2.2. Fuzzifier*

*2.2.3. Inference*

follows:

a fuzzy set in **X**×*Y*, and can be expressed as:

*R<sup>l</sup>* : *F<sup>l</sup>*

<sup>1</sup> <sup>×</sup> ... <sup>×</sup> *<sup>F</sup><sup>l</sup>*

*μRl*(**x**, *y*) = *μF<sup>l</sup>*

given by the fuzzy set *A***x** whose MF is expressed as [8]

the MF of this output set is expressed as [8]

Finally, the *l*th rule is expressed as follows

*2.2.4. Defuzzifier*

*μBl* (*y*) = sup

returns a crisp measure of the behavior of the FLS.

**x**∈**X**

 *Tp <sup>i</sup>*=1*μXi*

= *Tp <sup>i</sup>*=1*μF<sup>l</sup> i* (*xi*) 

1×*F<sup>l</sup>* <sup>2</sup>×...× *Fl p* 

*μA***<sup>x</sup>** (**x**) = *μX*<sup>1</sup> (*x*1) ...*μXp*

*<sup>μ</sup>Bl* (*y*) <sup>=</sup> *<sup>μ</sup>AX*◦*Rl* (*y*) <sup>=</sup> sup

*μBl*(*y*) = *μGl* (*y*)

[8].

A fuzzifier maps any crisp input **<sup>x</sup>** = (*x*1, ..., *xp*)*<sup>T</sup>* <sup>∈</sup> *<sup>X</sup>*<sup>1</sup> ×···× *Xp* <sup>≡</sup> **<sup>X</sup>** into a fuzzy set *<sup>F</sup>***<sup>x</sup>** in **<sup>X</sup>**

A fuzzy inference engine combines rules from the fuzzy rule base and gives a mapping from input fuzzy sets in **X** to output sets in *Y*. Each rule is interpreted as a fuzzy implication, i.e.,

Usually in Mamdani FLS, the implication is replaced by a *t-norm*, i.e. (product or min). Multiple antecedents are connected by a t-norm, so a rule can be expressed by its MF as

where *<sup>T</sup>* and are *<sup>t</sup>* <sup>−</sup> *norm* operators (*product* or *min*). The p-dimensional input to *<sup>R</sup><sup>l</sup>* is

Each rule detemines a fuzzy set *<sup>B</sup><sup>l</sup>* in *<sup>Y</sup>* which is derived from the sup <sup>−</sup> composition. Then,

**x**∈**X**

*T<sup>p</sup> <sup>i</sup>*=1*μF<sup>l</sup> i* (*xi*) 

(*xi*)

 *Tp <sup>i</sup>*=1*μF<sup>l</sup> i* (*xi*) 

As we pointed out before, the main idea of a Mamdani FLS is to use crisp inputs to make fuzzy inference and finally find a crisp output which represents the behavior of the FLS. The process of finding a crisp output after fuzzification and inference is called *Deffuzification*. This final step consist on find an operation point given the results of the inference process of the FLS, which results on a fuzzy output set, so we need to use a mathematical method which

There are many types of defuzzifiers, but we consider in this paper the *Height Defuzzifier* which replaces each rule output fuzzy set by a singleton at the point having maximum membership

*x*1, *x*2, .., *xp*

 *xp* = *T<sup>p</sup> <sup>i</sup>*=1*μX<sup>l</sup> i*

*<sup>p</sup>* −→ *<sup>G</sup><sup>l</sup>* <sup>=</sup> *<sup>A</sup><sup>l</sup>* −→ *<sup>G</sup><sup>l</sup> <sup>l</sup>* <sup>=</sup> 1, ..., *<sup>M</sup>* (3)

Enhancing Fuzzy Controllers Using Generalized Orthogonality Principle 369

*μGl* (*y*)

*μGl* (*y*) (4)

[*μA***<sup>x</sup>** (**x**) *μRl*(**x**, *y*)] (6)

*μGl* (*y*)

(*xi*) (5)

(7)

*y* ∈ *Y* (8)

where the membership degree of *x*, *μ<sup>F</sup>* (*x*), is constrained to be betwwen 0 and 1 for all *x* ∈ *X*.

## **2.2. Mamdani FLS**

An FLS is an intuitive and numerical system that maps crisp (deterministic) inputs to a crisp output. It is composed of four elements which are depicted in Figure 1. To completly describe this FLS, we need a mathematical formula that maps the crisp input **x** into a crisp output *y* = *f*(**x**), we can obtain this formula by following the signal **x** through the fuzzifier to the inference block and into the defuzzifier. We explain, in this section, the working principle of this formula.

#### *2.2.1. Rules*

The FLS is associated with a set of *IF-THEN* rules with meaningful linguistic interpretations. The *l*th rule of a FLS having *p* inputs *x*1, ..., *xp* and one output *y* ∈ *Y*, *Multiple Input Single Output* (MISO), is expressed as:

$$\text{R}^l: \text{ If } \begin{array}{cccc} \text{x}\_1 & \text{is } \text{F}\_1^l \text{ and} \dots \text{ and } \text{x}\_p \text{ is } \text{F}\_p^l \text{ THEN } \text{y} \text{ is } \text{G}^l \end{array} \tag{2}$$

where *F<sup>l</sup> <sup>i</sup>* (*i* = 1, 2, ..., *p*) are fuzzy antecedent sets wich are represented by their MFs *μF<sup>l</sup> i* , and *G<sup>l</sup>* is a consequent set where *l* = 1, ..., *M* (*M* is the number of rules in the FLS).

**Figure 1.** Block diagram of a fuzzy logic system

#### *2.2.2. Fuzzifier*

2 Will-be-set-by-IN-TECH

robotics and dynamic systems. Firstly, we shall show the improved results with analysis upon the application of GOP in the Fuzzy Logic Controller (FLC) for an inverted pendulum. Secondly, we show how a FLS based on this principle enhances the performance of forecaster

A *Fuzzy Set* (FS), *F* ∈ *X* is a set of ordered pairs of a generic element *x* and its degree, namely

where the membership degree of *x*, *μ<sup>F</sup>* (*x*), is constrained to be betwwen 0 and 1 for all *x* ∈ *X*.

An FLS is an intuitive and numerical system that maps crisp (deterministic) inputs to a crisp output. It is composed of four elements which are depicted in Figure 1. To completly describe this FLS, we need a mathematical formula that maps the crisp input **x** into a crisp output *y* = *f*(**x**), we can obtain this formula by following the signal **x** through the fuzzifier to the inference block and into the defuzzifier. We explain, in this section, the working principle of

The FLS is associated with a set of *IF-THEN* rules with meaningful linguistic interpretations. The *l*th rule of a FLS having *p* inputs *x*1, ..., *xp* and one output *y* ∈ *Y*, *Multiple Input Single*

<sup>1</sup> *and, ... , and xp is F<sup>l</sup>*

*G<sup>l</sup>* is a consequent set where *l* = 1, ..., *M* (*M* is the number of rules in the FLS).

Rules

Inference

*<sup>i</sup>* (*i* = 1, 2, ..., *p*) are fuzzy antecedent sets wich are represented by their MFs *μF<sup>l</sup>*

*F* = {(*x*, *μF*(*x*))|∀*x* ∈ *X*} (1)

Fuzzy output sets

Defuzzifier Output

*<sup>p</sup> THEN y is G<sup>l</sup>* (2)

*y* ∈ *Y*

*i* , and

for the chaotic time series.

**2.1. Fuzzy sets**

**2.2. Mamdani FLS**

this formula.

*2.2.1. Rules*

where *F<sup>l</sup>*

*Output* (MISO), is expressed as:

Crisp inputs

*R<sup>l</sup>* : *If x*<sup>1</sup> *is F<sup>l</sup>*

Fuzzy input sets

**Figure 1.** Block diagram of a fuzzy logic system

Fuzzifier

**2. Fuzzy Logic Systems (FLS) basic concepts**

*Membership Function* (MF), *μF*(*x*). Any FS can be represented as follows:

A fuzzifier maps any crisp input **<sup>x</sup>** = (*x*1, ..., *xp*)*<sup>T</sup>* <sup>∈</sup> *<sup>X</sup>*<sup>1</sup> ×···× *Xp* <sup>≡</sup> **<sup>X</sup>** into a fuzzy set *<sup>F</sup>***<sup>x</sup>** in **<sup>X</sup>** [8].

#### *2.2.3. Inference*

A fuzzy inference engine combines rules from the fuzzy rule base and gives a mapping from input fuzzy sets in **X** to output sets in *Y*. Each rule is interpreted as a fuzzy implication, i.e., a fuzzy set in **X**×*Y*, and can be expressed as:

$$\mathcal{R}^l: \ F\_1^l \times \ldots \times F\_p^l \longrightarrow \mathcal{G}^l = A^l \longrightarrow \mathcal{G}^l \quad l = 1, \ldots, M \tag{3}$$

Usually in Mamdani FLS, the implication is replaced by a *t-norm*, i.e. (product or min). Multiple antecedents are connected by a t-norm, so a rule can be expressed by its MF as follows:

$$
\mu\_{R^l}(\mathbf{x}, y) = \mu\_{F\_1^l \times F\_2^l \times \ldots \times F\_p^l}(\mathbf{x}\_1, \mathbf{x}\_2, \ldots \mathbf{x}\_p) \,\blacktriangleleft \mu\_{G^l}(y)
$$

$$
= \left[ T\_{i=1}^p \mu\_{F\_i^l}(\mathbf{x}\_i) \right] \,\blacktriangleleft \mu\_{G^l}(y) \tag{4}
$$

where *<sup>T</sup>* and are *<sup>t</sup>* <sup>−</sup> *norm* operators (*product* or *min*). The p-dimensional input to *<sup>R</sup><sup>l</sup>* is given by the fuzzy set *A***x** whose MF is expressed as [8]

$$
\mu\_{A\_\mathbf{x}}(\mathbf{x}) = \mu\_{X\_1}(\mathbf{x}\_1) \star \dots \star \mu\_{X\_p}(\mathbf{x}\_p) = T\_{i=1}^p \mu\_{X\_i^l}(\mathbf{x}\_i) \tag{5}
$$

Each rule detemines a fuzzy set *<sup>B</sup><sup>l</sup>* in *<sup>Y</sup>* which is derived from the sup <sup>−</sup> composition. Then, the MF of this output set is expressed as [8]

$$\mu\_{B^l}(y) = \mu\_{A\_X \circ R^l}(y) = \sup\_{\mathbf{x} \in \mathbf{X}} \left[ \mu\_{A\_{\mathbf{x}}}(\mathbf{x}) \, \blacktriangleleft \mu\_{R^l}(\mathbf{x}, y) \right] \tag{6}$$

$$\mu\_{B^l}(y) = \sup\_{\mathbf{x} \in \mathbf{X}} \left[ T\_{i=1}^p \mu\_{X\_i}(\mathbf{x}\_i) \blacktriangleleft \left( \left[ T\_{i=1}^p \mu\_{F\_l^l}(\mathbf{x}\_i) \right] \blacktriangleleft \mu\_{G^l}(y) \right) \right] \tag{7}$$

Finally, the *l*th rule is expressed as follows

$$
\mu\_{B^l}(y) = \mu\_{G^l}(y) \star \left[ T\_{i=1}^p \mu\_{F\_i^l}(\mathfrak{x}\_i) \right] \quad y \in Y \tag{8}
$$

#### *2.2.4. Defuzzifier*

As we pointed out before, the main idea of a Mamdani FLS is to use crisp inputs to make fuzzy inference and finally find a crisp output which represents the behavior of the FLS. The process of finding a crisp output after fuzzification and inference is called *Deffuzification*. This final step consist on find an operation point given the results of the inference process of the FLS, which results on a fuzzy output set, so we need to use a mathematical method which returns a crisp measure of the behavior of the FLS.

There are many types of defuzzifiers, but we consider in this paper the *Height Defuzzifier* which replaces each rule output fuzzy set by a singleton at the point having maximum membership in that output set, *y<sup>l</sup>* , then it calculates the centroid of the resultantF set of these singletons. The crisp output of this defuzzifier is expressed as:

$$y(\mathbf{x}) = f(\mathbf{x}) = \frac{\sum\_{l=1}^{M} \overline{y}^{l} \mu\_{B^{l}}(\overline{y}^{l})}{\sum\_{l=1}^{M} \mu\_{B^{l}}(\overline{y}^{l})} \tag{9}$$

where *φl*(**x**) is called a *Fuzzy Basis Function* (FBF) of the *lth* rule [11], and it is defined as:

This linear combination allows us to view an FLS as series expansions of FBFs [11], [1], [4] and [10] which has the capability of providing a mix of both numerical and linguistic information.

> *M* ∑ *l*=1

> > *M* ∑ *l*=1

This linear combination allows us to view the FLS as series expansions of WFBFs [2]. The WFBFs have also a capability of providing a combination of both numerical and linguistic

We explain in this section how we can obtain, graphically, the optimal scalar that minimizes the norm of an error vector [9].Suppose that we have a set of *N* measurements collected in a

As shown in Figure 2, we can see that the optimal scalar *θ* that minimizes the norm of the

−→*y <sup>T</sup>*−→*φ* −→*φ <sup>T</sup>*−→*φ*

*φl*(**x**)

*p* ∑ *k*=0 *φl k*(**x**)*c<sup>l</sup>*

*<sup>k</sup>*(**x**) is the *kth* Weighted Fuzzy Basis Function (WFBF) of the *lth* rule which is

*<sup>k</sup>*(**x**) = *xkφl*(**x**), *l* = 1, ..., *M*; *k* = 0, ..., *p* (18)

*p* ∑ *k*=0 *cl*

*<sup>l</sup>*=<sup>1</sup> *<sup>f</sup> <sup>l</sup> <sup>l</sup>* <sup>=</sup> 1, ..., *<sup>M</sup>* (15)

Enhancing Fuzzy Controllers Using Generalized Orthogonality Principle 371

*<sup>k</sup>xk* (16)

*<sup>k</sup>* (17)

−→*φ* . The problem is

(21)

(19)

= 0 (20)

*<sup>e</sup>* <sup>⊥</sup> −→*<sup>φ</sup>* . This can be expressed as follows :

*<sup>φ</sup>l*(**x**) = *<sup>f</sup> <sup>l</sup>* ∑*<sup>M</sup>*

*yTSK*(**x**) =

*yTSK*(**x**) =

*N*-vector, −→*y* , gathered for different values collected in another *N*-vector,

min *θ* −→ *y* − *θ* −→*φ* 

, is obtained when −→

*θopt* =

−→*φ* · −→*<sup>y</sup>* <sup>−</sup> *<sup>θ</sup>* −→*φ* 

The crisp output of the TSK FLS in (11) can be expressed as:

*φl*

where *f <sup>l</sup>* is given in (12).

**2.5. Weighted FBF**

It can also be expressed as:

**3. Orthogonality principle**

where *φ<sup>l</sup>*

expressed as [2]:

information.

to find :

error vector,

 −→ *e* = −→ *y* − *θ* −→*φ* 

Solving for *θ* we have:

where *y<sup>l</sup>* is the point having maximum membership in the output set [8].

#### **2.3. Takagi-Sugeno-Kang (TSK) FLS**

A TSK FLS is a special FLS which is also characterized by IF-THEN rules, but its consequent is a polynomial. Its output is a crisp value obtained from computing the polynomial output, so it does not need a defuzzification process. The *lth* rule of a first order type-1 TSK FLS having *p* inputs *x*<sup>1</sup> ∈ *X*1, ..., *xp* ∈ *Xp* and one output *y* ∈ *Y* is expressed as:

$$\mathbf{R}^l: \text{IF } \mathbf{x}\_1 \text{ is } F\_1^l \text{ and } \mathbf{x}\_2 \text{ is } F\_2^l \text{ and...and } \mathbf{x}\_p \text{ is } F\_p^l$$

$$\text{THEN } \ y^l(\mathbf{x}) = c\_0^l + c\_1^l \mathbf{x}\_1 + \dots + c\_p^l \mathbf{x}\_p \tag{10}$$

where *l* = 1, ..., *M*, *c<sup>l</sup> j* (*j* = 0, .., *p*) are the consequent parameters, *y<sup>l</sup>* (**x**) is the output of the *l*th rule, and *F<sup>l</sup> <sup>k</sup>* (*k* = 1, ..., *p*) are type-1 antecedent fuzzy sets.

The output of a TSK FLS is obtained by combining the outputs from the *M* rules in the following form:

$$y\_{TSK}(\mathbf{x}) = \frac{\sum\_{l=1}^{M} f^l(\mathbf{x}) \left( c\_0^l + c\_1^l \mathbf{x}\_1 + \dots + c\_p^l \mathbf{x}\_p \right)}{\sum\_{l=1}^{M} f^l(\mathbf{x})} \tag{11}$$

where *f <sup>l</sup>* (**x**) (*l* = 1, ..., *M*) are the rule firing levels and they are defined as:

$$f^l(\mathbf{x}) = T\_{k=1}^p \mu\_{F\_k^l}(\mathbf{x}\_k) \tag{12}$$

where *T* is a *t* − *norm* operation, i.e. minimum or product operation (Mendel [8]), and **x** is the vector of inputs applied to the TSK FLS.

#### **2.4. Fuzzy basis functions**

For Mamdani FLSs, assuming that all consequent MFs are normalized, i.e., *μGl yl* = 1, and using singleton defuzzification, max-product composition and product implication, then the output of the height defuzzifier (9) becomes:

$$y(\mathbf{x}) = f(\mathbf{x}) = \frac{\sum\_{l=1}^{M} \overline{y}^{l} T\_{i=1}^{p} \mu\_{F\_{l}^{l}}(\mathbf{x}\_{i})}{\sum\_{l=1}^{M} T\_{i=1}^{p} \mu\_{F\_{l}^{l}}(\mathbf{x}\_{i})} \tag{13}$$

The FLS in (13) can be expressed as:

$$y(\mathbf{x}) = f(\mathbf{x}) = \sum\_{l=1}^{M} \overline{y}^{l} \phi\_{l}(\mathbf{x}) \tag{14}$$

where *φl*(**x**) is called a *Fuzzy Basis Function* (FBF) of the *lth* rule [11], and it is defined as:

$$\phi\_l(\mathbf{x}) = \frac{f^l}{\sum\_{l=1}^{M} f^l} \quad l = 1, \dots, M \tag{15}$$

where *f <sup>l</sup>* is given in (12).

4 Will-be-set-by-IN-TECH

*<sup>y</sup>*(**x**) = *<sup>f</sup>*(**x**) = <sup>∑</sup>*<sup>M</sup>*

where *y<sup>l</sup>* is the point having maximum membership in the output set [8].

*p* inputs *x*<sup>1</sup> ∈ *X*1, ..., *xp* ∈ *Xp* and one output *y* ∈ *Y* is expressed as:

*<sup>k</sup>* (*k* = 1, ..., *p*) are type-1 antecedent fuzzy sets.

∑*<sup>M</sup> <sup>l</sup>*=<sup>1</sup> *<sup>f</sup> <sup>l</sup>* (**x**) *cl* <sup>0</sup> <sup>+</sup> *<sup>c</sup><sup>l</sup>*

*f l*

For Mamdani FLSs, assuming that all consequent MFs are normalized, i.e., *μGl*

*y*(**x**) = *f*(**x**) =

*y*(**x**) = *f*(**x**) =

(**x**) (*l* = 1, ..., *M*) are the rule firing levels and they are defined as:

(**x**) = *T<sup>p</sup>*

*R<sup>l</sup>* : IF *x*<sup>1</sup> is *F<sup>l</sup>*

THEN *y<sup>l</sup>*

*yTSK*(**x**) =

, then it calculates the centroid of the resultantF set of these singletons.

*<sup>μ</sup>Bl*(*y<sup>l</sup>* )

<sup>2</sup> and...and *xp* is *<sup>F</sup><sup>l</sup>*

<sup>1</sup>*x*<sup>1</sup> <sup>+</sup> ... <sup>+</sup> *<sup>c</sup><sup>l</sup>*

*pxp* 

*<sup>l</sup>*=<sup>1</sup> *<sup>f</sup> <sup>l</sup>*(**x**) (11)

(*xk*) (12)

<sup>1</sup>*x*<sup>1</sup> <sup>+</sup> ... <sup>+</sup> *<sup>c</sup><sup>l</sup>*

*p*

) (9)

*pxp* (10)

(**x**) is the output of the *l*th

 *yl* 

(*xi*) (13)

*φl*(**x**) (14)

= 1, and

*<sup>l</sup>*=<sup>1</sup> *<sup>y</sup><sup>l</sup>*

∑*<sup>M</sup> <sup>l</sup>*=<sup>1</sup> *<sup>μ</sup>Bl*(*y<sup>l</sup>*

A TSK FLS is a special FLS which is also characterized by IF-THEN rules, but its consequent is a polynomial. Its output is a crisp value obtained from computing the polynomial output, so it does not need a defuzzification process. The *lth* rule of a first order type-1 TSK FLS having

<sup>1</sup> and *<sup>x</sup>*<sup>2</sup> is *<sup>F</sup><sup>l</sup>*

<sup>0</sup> <sup>+</sup> *<sup>c</sup><sup>l</sup>*

The output of a TSK FLS is obtained by combining the outputs from the *M* rules in the

∑*<sup>M</sup>*

*<sup>k</sup>*=1*μF<sup>l</sup> k*

where *T* is a *t* − *norm* operation, i.e. minimum or product operation (Mendel [8]), and **x** is the

using singleton defuzzification, max-product composition and product implication, then the

∑*<sup>M</sup> <sup>l</sup>*=<sup>1</sup> *<sup>y</sup><sup>l</sup> Tp <sup>i</sup>*=1*μF<sup>l</sup> i* (*xi*)

∑*<sup>M</sup> <sup>l</sup>*=<sup>1</sup> *<sup>T</sup><sup>p</sup> <sup>i</sup>*=1*μF<sup>l</sup> i*

> *M* ∑ *l*=1 *yl*

(*j* = 0, .., *p*) are the consequent parameters, *y<sup>l</sup>*

(**x**) = *c<sup>l</sup>*

in that output set, *y<sup>l</sup>*

where *l* = 1, ..., *M*, *c<sup>l</sup>*

rule, and *F<sup>l</sup>*

where *f <sup>l</sup>*

following form:

The crisp output of this defuzzifier is expressed as:

**2.3. Takagi-Sugeno-Kang (TSK) FLS**

*j*

vector of inputs applied to the TSK FLS.

output of the height defuzzifier (9) becomes:

The FLS in (13) can be expressed as:

**2.4. Fuzzy basis functions**

This linear combination allows us to view an FLS as series expansions of FBFs [11], [1], [4] and [10] which has the capability of providing a mix of both numerical and linguistic information.

#### **2.5. Weighted FBF**

The crisp output of the TSK FLS in (11) can be expressed as:

$$y\_{TSK}(\mathbf{x}) = \sum\_{l=1}^{M} \phi\_l(\mathbf{x}) \sum\_{k=0}^{p} c\_k^l \mathbf{x}\_k \tag{16}$$

It can also be expressed as:

$$y\_{TSK}(\mathbf{x}) = \sum\_{l=1}^{M} \sum\_{k=0}^{p} \phi\_k^l(\mathbf{x}) c\_k^l \tag{17}$$

where *φ<sup>l</sup> <sup>k</sup>*(**x**) is the *kth* Weighted Fuzzy Basis Function (WFBF) of the *lth* rule which is expressed as [2]:

$$\phi\_k^l(\mathbf{x}) = \mathbf{x}\_k \phi\_l(\mathbf{x}), \quad l = 1, \dots, M; k = 0, \dots, p \tag{18}$$

This linear combination allows us to view the FLS as series expansions of WFBFs [2]. The WFBFs have also a capability of providing a combination of both numerical and linguistic information.

#### **3. Orthogonality principle**

We explain in this section how we can obtain, graphically, the optimal scalar that minimizes the norm of an error vector [9].Suppose that we have a set of *N* measurements collected in a *N*-vector, −→*y* , gathered for different values collected in another *N*-vector, −→*φ* . The problem is to find :

$$\min\_{\theta} \left\| \begin{array}{c} \exists \text{\raisebox{-0.5pt}{ $\theta \text{--}$ }} \end{array} \right\| \begin{array}{c} \blacksquare \theta \text{--} \blacksquare \theta \text{--} \\\\ \blacksquare \theta \text{--} \end{array} \tag{19}$$

As shown in Figure 2, we can see that the optimal scalar *θ* that minimizes the norm of the error vector, −→ *e* = −→ *y* − *θ* −→*φ* , is obtained when −→ *<sup>e</sup>* <sup>⊥</sup> −→*<sup>φ</sup>* . This can be expressed as follows :

$$
\overrightarrow{\Phi'} \cdot \left(\overrightarrow{y'} - \theta \,\overrightarrow{\phi'}\right) = 0 \tag{20}
$$

Solving for *θ* we have:

$$\theta\_{opt} = \frac{\overrightarrow{\mathcal{Y}}^T \overrightarrow{\Phi}}{\overrightarrow{\Phi}^T \overrightarrow{\Phi}} \tag{21}$$

Now, if each FBF is considered as a basis function, we can compose the following vector:

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

where *M* is the number of rules. We now collect all the *N* training output data in the same

*y*(**x**(1)) *y*(**x**(2)) . . . *y*(**x**(*N*)) ⎞

⎟⎟⎟⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

⎛

*y*1 *y*2 . . . *y<sup>M</sup>* ⎞

⎟⎟⎟⎟⎟⎟⎟⎠

⎜⎜⎜⎜⎜⎜⎜⎝

By considering the *N* equations, a FLS can be expressed in vector-matrix format as follows: −→*y* = **Φ**

To find the optimal vector −→*<sup>θ</sup>* and because of fitting with basis sets, we generalize the presented orthogonality principle to a multi-dimensional basis leading to a GOP. The error

> −→*θ* �

> > **Φ***T*−→*y*

−→*<sup>θ</sup>* =

**<sup>Φ</sup>** = [−→*<sup>φ</sup>* 1,

**<sup>Φ</sup>***<sup>T</sup>* · � −→*<sup>y</sup>* <sup>−</sup> **<sup>Φ</sup>**

⎛

*y*1 *y*2 . . . *y<sup>M</sup>* ⎞

⎟⎟⎟⎟⎟⎟⎟⎠ = � **Φ***T***Φ** �−<sup>1</sup>

where −→*<sup>θ</sup> opt* is a vector which contains the parameters of the consequent, i.e., *<sup>y</sup><sup>l</sup>* in (3).

⎜⎜⎜⎜⎜⎜⎜⎝

−→*<sup>θ</sup> opt* <sup>=</sup>

vector should be perpendicular to all of the basis fuzzy vectors, as shown in Figure 2.

, *j* = 1, 2, ..., *M* (24)

Enhancing Fuzzy Controllers Using Generalized Orthogonality Principle 373

−→*<sup>θ</sup>* (27)

= 0 (29)

−→*<sup>φ</sup>* 2, ..., −→*<sup>φ</sup> <sup>M</sup>*] (28)

(25)

(26)

*φj*(**x**(1)) *φj*(**x**(2)) . . . *φj*(**x**(*N*))

−→*<sup>φ</sup> <sup>j</sup>* <sup>=</sup>

and the parameters of the consequent in a vector −→*<sup>θ</sup>* :

where the fuzzy basis function matrix **Φ** is given by:

In a matrix form, we obtain:

Solving for −→*<sup>θ</sup>* , we have:

vector −→*y* :

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

−→ *y* =

**Figure 2.** Basic Idea of Orthogonality Principle

## **4. FLS design based on GOP**

GOP is an optimization principle which can be applied to both Mamdani and TSK FLSs. Under the premise of fixed shapes and the parameters of the antecedent MFs over the time, then a training dataset is used to tune the consequent parameters. The consequent parameters are *c<sup>l</sup> <sup>k</sup>* (*<sup>l</sup>* <sup>=</sup> 1, ..., *<sup>M</sup>*; *<sup>k</sup>* <sup>=</sup> 0, ..., *<sup>p</sup>*) in (11) for a TSK FLS, and *<sup>y</sup><sup>l</sup>* (*<sup>l</sup>* <sup>=</sup> 1, ..., *<sup>M</sup>*) in (9) for a Mamdani FLS.

#### **4.1. Mamdani FLS design**

Given a collection of *N* input-output numerical data training pairs

$$\left(\mathbf{x}^{(1)}:y^{(1)}\right), \left(\mathbf{x}^{(2)}:y^{(2)}\right), \dots, \left(\mathbf{x}^{(N)}:y^{(N)}\right)$$

where **x**(*i*) and *y*(*i*) are respectively the vector input and scalar output of the FLS given by (13). We have to tune the *y<sup>l</sup>* (*l* = 1, ..., *M*) using these data training. Firstly, we compute the FBFs with training input vectors, then we apply the orthogonality principle on these FBFs and the training output vector.

Equation (14) can be decomposed as follows:

$$\begin{cases} \begin{aligned} \boldsymbol{y}(\mathbf{x}^{(1)}) &= \boldsymbol{f}(\mathbf{x}^{(1)}) = \overline{\mathbf{y}}^{1} \boldsymbol{\phi}\_{1}(\mathbf{x}^{(1)}) + \dots + \overline{\mathbf{y}}^{M} \boldsymbol{\phi}\_{M}(\mathbf{x}^{(1)}) \\ \boldsymbol{y}(\mathbf{x}^{(2)}) &= \boldsymbol{f}(\mathbf{x}^{(2)}) = \overline{\mathbf{y}}^{1} \boldsymbol{\phi}\_{1}(\mathbf{x}^{(2)}) + \dots + \overline{\mathbf{y}}^{M} \boldsymbol{\phi}\_{M}(\mathbf{x}^{(2)}) \\ &\vdots \\ &\vdots \\ \boldsymbol{y}(\mathbf{x}^{(N)}) &= \boldsymbol{f}(\mathbf{x}^{(N)}) = \overline{\mathbf{y}}^{1} \boldsymbol{\phi}\_{1}(\mathbf{x}^{(N)}) + \dots + \overline{\mathbf{y}}^{M} \boldsymbol{\phi}\_{M}(\mathbf{x}^{(N)}) \end{aligned} \end{cases} \tag{22}$$

So we have

$$g(\mathbf{x}^{(i)}) = f(\mathbf{x}^{(i)}) = \sum\_{l=1}^{M} \overline{\mathfrak{Y}}\_l^l \Phi\_l(\mathbf{x}^{(i)}) \quad i = 1, \ldots, N \tag{23}$$

Now, if each FBF is considered as a basis function, we can compose the following vector:

$$\begin{aligned} \stackrel{\textstyle \rightarrow}{\phi}\_{j} = \begin{pmatrix} \phi\_{j}(\mathbf{x}^{(1)})\\ \phi\_{j}(\mathbf{x}^{(2)})\\ \vdots\\ \phi\_{j}(\mathbf{x}^{(N)}) \end{pmatrix}, \quad j = 1, 2, \dots, M \end{aligned} \tag{24}$$

where *M* is the number of rules. We now collect all the *N* training output data in the same vector −→*y* :

$$
\overrightarrow{y}' = \begin{pmatrix} y(\mathbf{x}^{(1)}) \\ y(\mathbf{x}^{(2)}) \\ \vdots \\ \vdots \\ y(\mathbf{x}^{(N)}) \end{pmatrix} \\ \tag{25}
$$

and the parameters of the consequent in a vector −→*<sup>θ</sup>* :

$$
\overline{\theta}^{\flat} = \begin{pmatrix} \overline{y}^{1} \\ \overline{y}^{2} \\ \vdots \\ \overline{y}^{M} \end{pmatrix} \tag{26}
$$

By considering the *N* equations, a FLS can be expressed in vector-matrix format as follows:

$$
\overrightarrow{y'} = \Phi \,\overrightarrow{\theta}\,\tag{27}
$$

where the fuzzy basis function matrix **Φ** is given by:

$$\Phi = [\overrightarrow{\Phi}\_1 \, \overrightarrow{\Phi}\_2 \dots \overrightarrow{\Phi}\_M] \tag{28}$$

To find the optimal vector −→*<sup>θ</sup>* and because of fitting with basis sets, we generalize the presented orthogonality principle to a multi-dimensional basis leading to a GOP. The error vector should be perpendicular to all of the basis fuzzy vectors, as shown in Figure 2.

In a matrix form, we obtain:

$$\Phi^T \cdot \left(\overrightarrow{\mathcal{Y}} - \Phi \,\, \overrightarrow{\theta}^\dagger\right) = 0 \tag{29}$$

Solving for −→*<sup>θ</sup>* , we have:

6 Will-be-set-by-IN-TECH

GOP is an optimization principle which can be applied to both Mamdani and TSK FLSs. Under the premise of fixed shapes and the parameters of the antecedent MFs over the time, then a training dataset is used to tune the consequent parameters. The consequent parameters

**x**(2) : *y*(2)

where **x**(*i*) and *y*(*i*) are respectively the vector input and scalar output of the FLS given by (13). We have to tune the *y<sup>l</sup>* (*l* = 1, ..., *M*) using these data training. Firstly, we compute the FBFs with training input vectors, then we apply the orthogonality principle on these FBFs and the

*y*(**x**(1)) = *f*(**x**(1)) = *y*1*φ*1(**x**(1)) + ... + *yMφM*(**x**(1))

*y*(**x**(2)) = *f*(**x**(2)) = *y*1*φ*1(**x**(2)) + ... + *yMφM*(**x**(2)) . . .

*y*(**x**(*N*)) = *f*(**x**(*N*)) = *y*1*φ*1(**x**(*N*)) + ... + *yMφM*(**x**(*N*))

Given a collection of *N* input-output numerical data training pairs

� , �

**x**(1) : *y*(1)

�

Equation (14) can be decomposed as follows:

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

*y*(**x**(*i*)

) = *f*(**x**(*i*)

) = *M* ∑ *l*=1 *yl <sup>φ</sup>l*(**x**(*i*)

*<sup>k</sup>* (*<sup>l</sup>* <sup>=</sup> 1, ..., *<sup>M</sup>*; *<sup>k</sup>* <sup>=</sup> 0, ..., *<sup>p</sup>*) in (11) for a TSK FLS, and *<sup>y</sup><sup>l</sup>* (*<sup>l</sup>* <sup>=</sup> 1, ..., *<sup>M</sup>*) in (9) for a Mamdani

� , ...., �

**x**(*N*) : *y*(*N*)

�

) *i* = 1, ..., *N* (23)

(22)

**Figure 2.** Basic Idea of Orthogonality Principle

**4. FLS design based on GOP**

**4.1. Mamdani FLS design**

training output vector.

So we have

are *c<sup>l</sup>*

FLS.

$$
\overline{\boldsymbol{\theta}}^{\flat}\_{opt} = \begin{pmatrix} \overline{\boldsymbol{y}}^{1} \\ \overline{\boldsymbol{y}}^{2} \\ \vdots \\ \vdots \\ \overline{\boldsymbol{y}}^{M} \end{pmatrix} = \begin{bmatrix} \boldsymbol{\Phi}^{T}\boldsymbol{\Phi} \end{bmatrix}^{-1} \boldsymbol{\Phi}^{T} \overline{\boldsymbol{y}}^{\flat}
$$

where −→*<sup>θ</sup> opt* is a vector which contains the parameters of the consequent, i.e., *<sup>y</sup><sup>l</sup>* in (3).

where **x**(*i*) =

� 1, *x* (*i*) <sup>1</sup> , ..., *x*

(*i*) *p* �*T*

⎧

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ *φ*1

*φ*1

*φ<sup>M</sup>*

*φ<sup>M</sup>*

*φl <sup>k</sup>*(**x**(1))

*φl <sup>k</sup>*(**x**(2)) . . .

*φl <sup>k</sup>*(**x**(*N*))

and each set of *N* outputs as a vector, the output vector can be expressed as follows :

� −→ *φ*1 <sup>0</sup> ··· −→ *φ*1 *p* � ⎡ ⎢ ⎢ ⎣

� −→ *φ<sup>M</sup>* <sup>0</sup> ··· −→ *φ<sup>M</sup> p* � ⎡ ⎢ ⎢ ⎣

Now we have to tune *p* + 1 parameters for each rule, i.e., *M* vectors of dimension (*p* + 1).

⎞

−→ *φl p* �

*<sup>c</sup>*<sup>1</sup> <sup>+</sup> ··· <sup>+</sup> <sup>Φ</sup>*TSK*,*<sup>M</sup>*

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎛

*cl* 0 . . . *cl p*

⎜⎜⎝

� −→ *φl* <sup>0</sup> , ··· ,

−→

By taking each set of *N* WFBFs as a Weighted Fuzzy Basis Vector, WFBV:

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

−−→*yTSK* =

−→ *c<sup>l</sup>* =

Φ*TSK*,*<sup>l</sup>* =

−−→*yTSK* = Φ*TSK*,1

If we define the *lth* element of Φ*TSK* as Φ*TSK*,*l*, we have:

the output vector (33) becomes :

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−→ *φl <sup>k</sup>* = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

−−→*yTSK* =

. Collecting the *N* equations we obtain:

...

*<sup>p</sup>*(**x**(1))

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

> ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

⎡ ⎢ ⎢ ⎣ *c<sup>M</sup>* 0 . . . *c<sup>M</sup> p*

⎤ ⎥ ⎥ ⎦ (31)

(32)

(33)

⎡ ⎢ ⎢ ⎢ ⎣

*c*1 0 . . . *c*1 *p*

Enhancing Fuzzy Controllers Using Generalized Orthogonality Principle 375

⎤ ⎥ ⎥ ⎥ ⎦

*<sup>p</sup>*(**x**(*N*))

*<sup>p</sup>* (**x**(1))

*<sup>p</sup>* (**x**(*N*))

� *l* = 1, ..., *M k* = 0, ..., *p*

> *cl* 0 . . . *c*1 *p*

> > *c<sup>M</sup>* 0 . . . *c<sup>M</sup> p*

⎤ ⎥ ⎥ ⎦

⎟⎟⎠ , *<sup>l</sup>* <sup>=</sup> 1, ..., *<sup>M</sup>* (34)

−→

, *l* = 1, .., *M* (35)

*c<sup>M</sup>* (36)

+ ··· +

⎤ ⎥ ⎥ ⎦

+ ··· +

<sup>0</sup>(**x**(1)) ··· *<sup>φ</sup>*<sup>1</sup>

<sup>0</sup> (**x**(*N*)) ··· *<sup>φ</sup>*<sup>1</sup>

<sup>0</sup> (**x**(1)) ··· *<sup>φ</sup><sup>M</sup>*

<sup>0</sup> (**x**(*N*)) ··· *<sup>φ</sup><sup>M</sup>*

⎞

⎟⎟⎟⎟⎟⎟⎟⎠ ,

...

**Figure 3.** Basic Idea of Generalized Orthogonality Principle. The error vector should be perpendicular to all of the basis fuzzy vectors.

## **4.2. TSK FLS design**

In the same way, the consequent parameters of a TSK FLS are tuned. The design approach is related to the following problem:

Given a collection of *N* input-output numerical training data pairs:

$$\left(\mathbf{x}^{(1)}:y^{(1)}\right), \left(\mathbf{x}^{(2)}:y^{(2)}\right), \dots, \left(\mathbf{x}^{(N)}:y^{(N)}\right)$$

where **<sup>x</sup>**(*i*) is the (*<sup>p</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> *dimensional* input vector (*<sup>p</sup>* <sup>+</sup> 1 inputs with *<sup>x</sup>*<sup>0</sup> <sup>≡</sup> 1) and *<sup>y</sup>*(*i*) is the scalar output of the FLS given by (11). We have to tune the *c<sup>l</sup> <sup>k</sup>* (*l* = 1, ..., *M*; *k* = 0, ..., *p*) using these data training.

The WFBF vectors are computed using the training input data, then the GOP is applied to the (*p* + 1) combinations of WFBF vectors and the (*p* + 1) of *N*−*dimensional* training output vector.

Using the elements of the input-output training pairs, the TSK output given in (17), can be rewritten as follows:

$$\mathbf{y}\_{TSK}(\mathbf{x}^{(i)}) = \begin{cases} \begin{bmatrix} \Phi\_0^1(\mathbf{x}^{(i)}) \\ \vdots \\ \Phi\_p^1(\mathbf{x}^{(i)}) \end{bmatrix}^T \begin{bmatrix} c\_0^1 \\ \vdots \\ \vdots \\ c\_p^1 \end{bmatrix} \\ + \dots + \\\begin{bmatrix} \Phi\_0^M(\mathbf{x}^{(i)}) \\ \vdots \\ \Phi\_p^M(\mathbf{x}^{(i)}) \end{bmatrix}^T \begin{bmatrix} c\_0^M \\ \vdots \\ \vdots \\ c\_p^M \end{bmatrix} \end{cases} \tag{30}$$

where **x**(*i*) = � 1, *x* (*i*) <sup>1</sup> , ..., *x* (*i*) *p* �*T* . Collecting the *N* equations we obtain:

8 Will-be-set-by-IN-TECH

**Figure 3.** Basic Idea of Generalized Orthogonality Principle. The error vector should be perpendicular to

In the same way, the consequent parameters of a TSK FLS are tuned. The design approach is

where **<sup>x</sup>**(*i*) is the (*<sup>p</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> *dimensional* input vector (*<sup>p</sup>* <sup>+</sup> 1 inputs with *<sup>x</sup>*<sup>0</sup> <sup>≡</sup> 1) and *<sup>y</sup>*(*i*) is the

The WFBF vectors are computed using the training input data, then the GOP is applied to the (*p* + 1) combinations of WFBF vectors and the (*p* + 1) of *N*−*dimensional* training output

Using the elements of the input-output training pairs, the TSK output given in (17), can be

⎤ ⎦

+ ··· +

⎤ ⎦ *T* ⎡ ⎢ ⎢ ⎣

*T* ⎡ ⎢ ⎢ ⎣

*c*1 0 . . . *c*1 *p*

*c<sup>M</sup>* 0 . . . *c<sup>M</sup> p*

⎤ ⎥ ⎥ ⎦

⎤ ⎥ ⎥ ⎦

⎡ ⎣ *φ*1 <sup>0</sup>(**x**(*i*)) ··· *φ*1 *<sup>p</sup>*(**x**(*i*))

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎡ ⎣ *φ<sup>M</sup>* <sup>0</sup> (**x**(*i*)) ··· *φ<sup>M</sup> <sup>p</sup>* (**x**(*i*))

� ,..., �

**x**(*N*) : *y*(*N*)

�

*<sup>k</sup>* (*l* = 1, ..., *M*; *k* = 0, ..., *p*) using

(30)

**x**(2) : *y*(2)

Given a collection of *N* input-output numerical training data pairs:

� , �

**x**(1) : *y*(1)

scalar output of the FLS given by (11). We have to tune the *c<sup>l</sup>*

*yTSK*(**x**(*i*)

) =

�

all of the basis fuzzy vectors.

related to the following problem:

**4.2. TSK FLS design**

these data training.

rewritten as follows:

vector.

$$
\overrightarrow{y\_{TS}} = \begin{cases}
\begin{bmatrix}
\Phi\_0^1(\mathbf{x}^{(1)}) \cdots \Phi\_p^1(\mathbf{x}^{(1)}) \\
\vdots \\
\Phi\_0^1(\mathbf{x}^{(N)}) \cdots \Phi\_p^1(\mathbf{x}^{(N)}) \\
\vdots \\
\vdots \\
\end{bmatrix}
\begin{bmatrix}
c\_0^1 \\
\vdots \\
c\_p^1
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
c\_0^1 \\
\vdots \\
c\_p^1
\end{bmatrix}
$$

$$
\begin{bmatrix}
\Phi\_0^M(\mathbf{x}^{(1)}) \cdots \Phi\_p^M(\mathbf{x}^{(1)}) \\
\vdots \\
\Phi\_0^M(\mathbf{x}^{(N)}) \cdots \Phi\_p^M(\mathbf{x}^{(N)})
\end{bmatrix}
\begin{bmatrix}
c\_0^M \\
\vdots \\
c\_p^M
\end{bmatrix}
$$

By taking each set of *N* WFBFs as a Weighted Fuzzy Basis Vector, WFBV:

$$\begin{aligned} \stackrel{\rightarrow}{\phi\_k^l} = \begin{pmatrix} \phi\_k^l(\mathbf{x}^{(1)})\\ \phi\_k^l(\mathbf{x}^{(2)})\\ \vdots\\ \vdots\\ \phi\_k^l(\mathbf{x}^{(N)}) \end{pmatrix}' \quad \begin{cases} l = 1, \ldots, M\\ k = 0, \ldots, p \end{cases} \end{aligned} \tag{32}$$

and each set of *N* outputs as a vector, the output vector can be expressed as follows :

$$
\overrightarrow{y\_{TSK}} = \begin{cases}
\begin{bmatrix}
\overrightarrow{\phi\_0^1} & \cdots & \overrightarrow{\phi\_p^1}
\end{bmatrix}
\begin{bmatrix}
c\_0^1 \\ \vdots \\ c\_p^1
\end{bmatrix} \\
&+\cdots+\\
\begin{bmatrix}
\overrightarrow{\phi\_0^M} & \cdots & \overrightarrow{\phi\_p^M}
\end{bmatrix}
\begin{bmatrix}
c\_0^M \\ \vdots \\ c\_p^M
\end{bmatrix}
\end{cases}
\tag{33}
$$

Now we have to tune *p* + 1 parameters for each rule, i.e., *M* vectors of dimension (*p* + 1).

$$\begin{aligned} \stackrel{\rightarrow}{c^l} = \begin{pmatrix} c\_0^l \\ \vdots \\ c\_p^l \end{pmatrix}, \quad l = 1, \dots, M \end{aligned} \tag{34}$$

If we define the *lth* element of Φ*TSK* as Φ*TSK*,*l*, we have:

$$\Phi\_{\text{TSK},l} = \left[ \begin{array}{c} \overrightarrow{\phi\_0} \\ \end{array}, \dots, \overrightarrow{\phi\_p} \right], l = 1, \dots, M \tag{35}$$

the output vector (33) becomes :

$$\mathbf{y}\_{TSK} = \Phi\_{TSK,1}\overrightarrow{\mathbf{c}}^{\dagger} + \dots + \Phi\_{TSK,M}\overrightarrow{\mathbf{c}}^{\overrightarrow{M}} \tag{36}$$

#### 10 Will-be-set-by-IN-TECH 376 Fuzzy Controllers – Recent Advances in Theory and Applications Enhancing Fuzzy Controllers Using Generalized Orthogonality Principle <sup>11</sup>

In a matrix form, (36) becomes :

$$\overrightarrow{y\_{TSK}} = \Phi\_{TSK} \left[ \begin{array}{c} \overrightarrow{\text{cJ}} \\ \end{array} \cdots \begin{array}{c} \overrightarrow{\text{cJ}} \end{array} \right]^T \tag{37}$$

**5. FLC design for controlling an inverted pendulum on a cart**

upright, we design a Fuzzy Logic Controller (FLC) using the GOP.

**Figure 5.** A schematic drawing of the inverted pendulum on a cart

�*ml*<sup>2</sup> <sup>4</sup> <sup>+</sup> *<sup>J</sup>*

The Lagrange equation for the position of the cart, *x*, is given by:

(*M*<sup>1</sup> + *m*) *x*¨ +

and *m* = 0.1*Kg*, respectively. The rod has a length *l* = 0.5*m*.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

> *μFl i*

**˙x** =

linearized around *θ* = 0. We chose **x** = �

The Lagrange equation for the position of the pendulum, *θ*, is given by:

� ¨ *θ* + *ml*

> *ml* 2 (¨

*θ* ˙ *θ x x*˙

0 100

6 *<sup>l</sup>*(*m*+4*M*1) <sup>000</sup> 0 001

−3*g*·*m <sup>m</sup>*+4*M*<sup>1</sup> 000

(*xi*) = exp

angle variation and *x*˙ is the cart position variation. The state representation is given by:

⎡ ⎣−<sup>1</sup> 2

*<sup>θ</sup>* cos *<sup>θ</sup>* <sup>−</sup> ˙

where *J* is the moment of inertia of the bar. The masses of the cart and the rod are *M*<sup>1</sup> = 2*Kg*

Since the goal of the control system is to keep the pendulum upright the equations can be

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

**x** +

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

<sup>4</sup> *<sup>m</sup>*+4*M*<sup>1</sup>

*i σFl i*

�<sup>2</sup> ⎤

� *xi* <sup>−</sup> *mFl*

Schematic drawing of an *Inverted pendulum On a Cart* (IPOC) system is depicted in Figure 5. where *x* is the position of the cart, *θ* is the angle of the pendulum with respect to the vertical

*F* is the external acting force in the *x* − *direction*. In order to keep the pendulum

<sup>2</sup> (*x*¨ cos *<sup>θ</sup>* <sup>−</sup> *<sup>g</sup>* sin *<sup>θ</sup>*) = <sup>0</sup> (40)

Enhancing Fuzzy Controllers Using Generalized Orthogonality Principle 377

�*<sup>T</sup>* as the state vector, where ˙

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

*θ*<sup>2</sup> sin *θ*) = *F*(*t*) (41)

*θ* is the pendulum

*F*(*t*) (42)

⎦ (43)

**5.1. Description of the system**

direction and

So the *Weighted Basis Function Matrix* (WBFM) Φ can be defined as:

$$\Phi\_{TSK} = \begin{bmatrix} \Phi\_{TSK,1'} \dots \Lambda\_{TSK,M} \end{bmatrix} \tag{38}$$

The optimal parameters of the consequent conforms a vector, −→ *c<sup>l</sup>* in (34) are obtained when the error vector, −−→*yTSK* <sup>−</sup> <sup>Φ</sup>*TSK* −→ *<sup>c</sup>*<sup>1</sup> ··· −→ *c<sup>M</sup> <sup>T</sup>* , must be perpendicular to all the weighted fuzzy basis vectors, −→ *φl <sup>k</sup>* (*k* = 0, . . . , *p* and *l* = 1, . . . , *M*), which are the columns of the WBFM Φ*TSK*, as shown in Figure 4.

**Figure 4.** Extended Generalized Orthogonality Principle. The error vector −→ *y* − Φ −→ *y*1 *<sup>C</sup>* ··· −→ *y<sup>M</sup> C T* should be perpendicular to all the fuzzy basis vectors, −→ *φl k*

This may be expressed directly in terms of the WBFM Φ as follows:

$$\Phi^T \overrightarrow{y'} - \Phi^T \Phi \left[ \overrightarrow{y'\_{\mathbb{C}}} \dots \overrightarrow{y'\_{\mathbb{C}}} \right]^T = 0 \tag{39}$$

Solving for −→ *y*1 *<sup>C</sup>* ··· −→ *y<sup>M</sup> C T* provides the following −→ *y*1 *<sup>C</sup>* ··· −→ *y<sup>M</sup> C T opt* <sup>=</sup> <sup>Φ</sup> · <sup>Φ</sup>*<sup>T</sup>* −<sup>1</sup> Φ−→ *y*

## **5. FLC design for controlling an inverted pendulum on a cart**

#### **5.1. Description of the system**

10 Will-be-set-by-IN-TECH

 −→ *<sup>c</sup>*<sup>1</sup> ··· −→ *c<sup>M</sup> T*

Φ*TSK*,1, ..., Φ*TSK*,*<sup>M</sup>*

*<sup>k</sup>* (*k* = 0, . . . , *p* and *l* = 1, . . . , *M*), which are the columns of the WBFM Φ*TSK*,

−→

, must be perpendicular to all the weighted fuzzy

*y* − Φ

 −→ *y*1 *<sup>C</sup>* ··· −→ *y<sup>M</sup> C T*

= 0 (39)

should

(37)

(38)

*c<sup>l</sup>* in (34) are obtained when the

−−→*yTSK* = Φ*TSK*

So the *Weighted Basis Function Matrix* (WBFM) Φ can be defined as:

The optimal parameters of the consequent conforms a vector,

 −→ *<sup>c</sup>*<sup>1</sup> ··· −→ *c<sup>M</sup> <sup>T</sup>*

Φ*TSK* =

**Figure 4.** Extended Generalized Orthogonality Principle. The error vector −→

This may be expressed directly in terms of the WBFM Φ as follows:

Φ*T*−→

 −→ *y*1 *<sup>C</sup>* ··· −→ *y<sup>M</sup> C T opt* <sup>=</sup> <sup>Φ</sup> · <sup>Φ</sup>*<sup>T</sup>*

*<sup>y</sup>* <sup>−</sup> <sup>Φ</sup>*T*<sup>Φ</sup>

provides the following

−→ *φl k*

> −→ *y*1 *<sup>C</sup>* ··· −→ *y<sup>M</sup> C T*

> > −<sup>1</sup> Φ−→ *y*

be perpendicular to all the fuzzy basis vectors,

Solving for

 −→ *y*1 *<sup>C</sup>* ··· −→ *y<sup>M</sup> C T*

In a matrix form, (36) becomes :

−→ *φl*

−−→*yTSK* <sup>−</sup> <sup>Φ</sup>*TSK*

error vector,

basis vectors,

as shown in Figure 4.

Schematic drawing of an *Inverted pendulum On a Cart* (IPOC) system is depicted in Figure 5. where *x* is the position of the cart, *θ* is the angle of the pendulum with respect to the vertical direction and *F* is the external acting force in the *x* − *direction*. In order to keep the pendulum upright, we design a Fuzzy Logic Controller (FLC) using the GOP.

**Figure 5.** A schematic drawing of the inverted pendulum on a cart

The Lagrange equation for the position of the pendulum, *θ*, is given by:

$$\left(\frac{ml^2}{4} + f\right)\ddot{\theta} + \frac{ml}{2}(\ddot{x}\cos\theta - g\sin\theta) = 0\tag{40}$$

The Lagrange equation for the position of the cart, *x*, is given by:

$$\left(M\_1 + m\right)\ddot{x} + \frac{ml}{2}\left(\ddot{\theta}\cos\theta - \dot{\theta}^2\sin\theta\right) = F(t) \tag{41}$$

where *J* is the moment of inertia of the bar. The masses of the cart and the rod are *M*<sup>1</sup> = 2*Kg* and *m* = 0.1*Kg*, respectively. The rod has a length *l* = 0.5*m*.

Since the goal of the control system is to keep the pendulum upright the equations can be linearized around *θ* = 0. We chose **x** = � *θ* ˙ *θ x x*˙ �*<sup>T</sup>* as the state vector, where ˙ *θ* is the pendulum angle variation and *x*˙ is the cart position variation. The state representation is given by:

$$\dot{\mathbf{x}} = \begin{bmatrix} 0 & 1 \ 0 \ 0 \\ \frac{6}{l(m + 4M\_1)} \ 0 \ 0 \ 0 \\ 0 & 0 \ 0 \ 1 \\ \begin{bmatrix} -3\underline{\chi}\cdot m \\ \underline{\begin{bmatrix} -3\underline{\chi}\cdot m \\ m + 4M\_1} \ 0 \ 0 \ 0 \end{bmatrix} \end{bmatrix} \mathbf{x} + \begin{bmatrix} 0 \\ \frac{6}{l(m + 4M\_1)} \\ 0 \\ \underline{\begin{bmatrix} 4 \\ m + 4M\_1} \end{bmatrix} \end{bmatrix} F(t) \tag{42}$$
 
$$\mu\_{F\_i^l}(\mathbf{x}\_i) = \exp\left[ -\frac{1}{2} \left( \frac{\mathbf{x}\_i - m\_{F\_i^l}}{\sigma\_{F\_i^l}} \right)^2 \right] \tag{43}$$

### **5.2. FLC structure and design**

We try to keep the pendulum upright regardless the cart's position, i.e., *Pure Angular Position Control System* (PAPCS). Then, the two inputs of the Fuzzy Logic Controller FLC are the angular pendulum position, *θ*, and its derivative, ˙ *θ*, i.e., **x**<sup>1</sup> = � *x*<sup>1</sup> *x*<sup>2</sup> � = � *θ* ˙ *θ* �*<sup>T</sup>* and its output is the applied force to the system *y* =*force*.

**Figure 6.** Fuzzy control system of the PAPCS

In this case, we use a Mamdani FLS with four rules. We use gaussian MF to fuzzify the two controller's inputs (44) and triangular MF to fuzzify the controller output.

$$\mu\_{F\_l^l}(\mathbf{x}\_l) = \exp\left[ -\frac{1}{2} \left( \frac{\mathbf{x}\_l - m\_{F\_l^l}}{\sigma\_{F\_l^l}} \right)^2 \right] \tag{44}$$

**Figure 8.** Membership functions for the second crontroller input ˙

**Figure 9.** Data training and its approximation based on GOP

*of the Time multiplied by Square Error* (*ITSE* = <sup>∞</sup>

(*ISE* = <sup>∞</sup>

<sup>0</sup> [*e*(*t*)]

tuning and no tuning are used.

We evaluate the proposed design by using its error rate. For quantifying the errors, we use three different performance criteria to analyze the rise time, the oscillation behaviour and the behaviour at the end of transition period. These three criteria are: *Integral of Square Error*

Table 1 summarizes the obtained values of *ISE*, *IAE* and *ITSE* of PAPCS using FLC, when

We notice from this table that the errors obtained when tuning is used are all smaller than

Figures 10, 11, 12 and 13 show that the system using tuning is less oscillatory, having a rise time and errors at the end of transition period smaller than those obtained by untuned FLC.

those obtained with untuned FLC. Fig. 11, 12, 13 show the different quantified errors.

<sup>0</sup> *t* [*e*(*t*)]

<sup>2</sup> *dt*)

<sup>0</sup> |*e*(*t*)| *dt*) and *Integral*

<sup>2</sup> *dt*), *Integral of the Absolute value of the Error* (*IAE* = <sup>∞</sup>

*θ*

Enhancing Fuzzy Controllers Using Generalized Orthogonality Principle 379

where *mFl i* and *σF<sup>l</sup> i* are respectively the centers and standard deviations of these MFs*.*

The MFs of the antecedents are depicted in Figures 7 and 8.

**Figure 7.** Membership functions for the first controller input *θ*

Figure 9 shows the 56 data training and the optimal fitting given by the GOP method.

The obtained optimal consequent parameters are

$$\left(\overline{y}^1, \overline{y}^2, \overline{y}^3, \overline{y}^4\right)\_{opt} = (-14.3, -14.23, 9.61, 18.96)\_t$$

Figure 10 shows the response of the pendulum system controlled by the designed FLC to a reference *θref* = 0 with its response at the same reference when it is controlled by untuned FLC. The initial state vector is **x**<sup>0</sup> = � *θ*<sup>0</sup> ˙ *θ*<sup>0</sup> *x*<sup>0</sup> *x*˙0 �*<sup>T</sup>* = � 0.1 0.2 0 0 �*<sup>T</sup>* .

**Figure 8.** Membership functions for the second crontroller input ˙ *θ*

12 Will-be-set-by-IN-TECH

We try to keep the pendulum upright regardless the cart's position, i.e., *Pure Angular Position Control System* (PAPCS). Then, the two inputs of the Fuzzy Logic Controller FLC are the

In this case, we use a Mamdani FLS with four rules. We use gaussian MF to fuzzify the two

� *xi* <sup>−</sup> *mFl*

are respectively the centers and standard deviations of these MFs*.*

*i σFl i*

�<sup>2</sup> ⎤

⎡ ⎣−<sup>1</sup> 2

Figure 9 shows the 56 data training and the optimal fitting given by the GOP method.

Figure 10 shows the response of the pendulum system controlled by the designed FLC to a reference *θref* = 0 with its response at the same reference when it is controlled by untuned

*θ*<sup>0</sup> *x*<sup>0</sup> *x*˙0

*opt* <sup>=</sup> (−14.3, <sup>−</sup>14.23, 9.61, 18.96)

�*<sup>T</sup>* = �

0.1 0.2 0 0 �*<sup>T</sup>*

.

controller's inputs (44) and triangular MF to fuzzify the controller output.

(*xi*) = exp

*μFl i*

The MFs of the antecedents are depicted in Figures 7 and 8.

**Figure 7.** Membership functions for the first controller input *θ*

*y*1, *y*2, *y*3, *y*<sup>4</sup>

�

*θ*<sup>0</sup> ˙

The obtained optimal consequent parameters are �

FLC. The initial state vector is **x**<sup>0</sup> = �

*θ*, i.e., **x**<sup>1</sup> = �

*x*<sup>1</sup> *x*<sup>2</sup>

� = �

⎦ (44)

*θ* ˙ *θ* �*<sup>T</sup>* and its

**5.2. FLC structure and design**

angular pendulum position, *θ*, and its derivative, ˙

output is the applied force to the system *y* =*force*.

**Figure 6.** Fuzzy control system of the PAPCS

where *mFl*

*i*

and *σF<sup>l</sup> i*

**Figure 9.** Data training and its approximation based on GOP

We evaluate the proposed design by using its error rate. For quantifying the errors, we use three different performance criteria to analyze the rise time, the oscillation behaviour and the behaviour at the end of transition period. These three criteria are: *Integral of Square Error* (*ISE* = <sup>∞</sup> <sup>0</sup> [*e*(*t*)] <sup>2</sup> *dt*), *Integral of the Absolute value of the Error* (*IAE* = <sup>∞</sup> <sup>0</sup> |*e*(*t*)| *dt*) and *Integral of the Time multiplied by Square Error* (*ITSE* = <sup>∞</sup> <sup>0</sup> *t* [*e*(*t*)] <sup>2</sup> *dt*)

Table 1 summarizes the obtained values of *ISE*, *IAE* and *ITSE* of PAPCS using FLC, when tuning and no tuning are used.

We notice from this table that the errors obtained when tuning is used are all smaller than those obtained with untuned FLC. Fig. 11, 12, 13 show the different quantified errors.

Figures 10, 11, 12 and 13 show that the system using tuning is less oscillatory, having a rise time and errors at the end of transition period smaller than those obtained by untuned FLC.

No tuning Tuning ISE 0.2338 0.2224 IAE 4.7343 4.0403 ITSE 6.7733 4.4278

Enhancing Fuzzy Controllers Using Generalized Orthogonality Principle 381

**Table 1.** Comparison of performance criteria for of PAPCS using tuned and no tuned FLC.

**Figure 13.** Integral of the time multiplied by square error values of PAPCS of tuned and no tuned consequent parameters. The error at the end of transition period is less important for the tuned FLC

*dt* <sup>=</sup> 0.2*x*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)

We apply the GOP to design an FLS which predicts a time series. The FLS has to predict the future value *x*(*t* + 6) of a Mackey-Glass time series (45) which is volatile. The following four antecedents were used: *x*(*t* − 18), *x*(*t* − 12), *x*(*t* − 6) and *x*(*t*), which are known values of the

The training data are obtained by simulating (45) for *τ* = 17. We use the samples *x*(1001), ··· , *x*(1524) to train the IT2 FLS and the samples *x*(1501), ··· , *x*(2024) for testing. We use two Gaussian MFs per antecedent, so we have then 16 rules. The MFs of the antecedents are Gaussian, where its mean and the standard deviation were obtained from the 524 training samples, *x*(1001), ··· , *x*(1524). Table 2 summarizes the consequent parameters per each rule. Figure 14 displays performance of the FLS in training data, and Figure 15 shows its results on Testing data. Note that the GOP-designed is a better forecaster, since the differences from

Some additional analyses should be performed to verify the goodness of fit of the method (See [5], and [6]), but in this case, the proposed GOP has shown good results, so we can recommend its application to real cases. Time series analysis is an useful topic for many decision makers, so the use of optimal and easy-to-be-implemented techniques, as the proposed one has a wide

<sup>1</sup> <sup>+</sup> *<sup>x</sup>*10(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*) <sup>−</sup> 0.1*x*(*t*) (45)

**6. FLS design for predicting time series**

*dx*(*t*)

original data are small in both training and testing data sets.

time series ([2], [3]).

potential.

**Figure 10.** System responses of PAPCS controlled by a tuned and untuned FLC

**Figure 11.** Integral of square error values of the PAPCS of tuned and no tuned consequent parameters. Rise time of the system is shorter for the tuned FLC

**Figure 12.** Integral of the absolute value of the error values of PAPCS of tuned and no tuned consequent parameters The system is less oscillatory for the tuned FLC before becoming stable


**Table 1.** Comparison of performance criteria for of PAPCS using tuned and no tuned FLC.

**Figure 13.** Integral of the time multiplied by square error values of PAPCS of tuned and no tuned consequent parameters. The error at the end of transition period is less important for the tuned FLC

#### **6. FLS design for predicting time series**

14 Will-be-set-by-IN-TECH

**Figure 11.** Integral of square error values of the PAPCS of tuned and no tuned consequent parameters.

**Figure 12.** Integral of the absolute value of the error values of PAPCS of tuned and no tuned consequent

parameters The system is less oscillatory for the tuned FLC before becoming stable

**Figure 10.** System responses of PAPCS controlled by a tuned and untuned FLC

Rise time of the system is shorter for the tuned FLC

We apply the GOP to design an FLS which predicts a time series. The FLS has to predict the future value *x*(*t* + 6) of a Mackey-Glass time series (45) which is volatile. The following four antecedents were used: *x*(*t* − 18), *x*(*t* − 12), *x*(*t* − 6) and *x*(*t*), which are known values of the time series ([2], [3]).

$$\frac{d\mathbf{x}(t)}{dt} = \frac{0.2\mathbf{x}(t-\tau)}{1+\mathbf{x}^{10}(t-\tau)} - 0.1\mathbf{x}(t) \tag{45}$$

The training data are obtained by simulating (45) for *τ* = 17. We use the samples *x*(1001), ··· , *x*(1524) to train the IT2 FLS and the samples *x*(1501), ··· , *x*(2024) for testing. We use two Gaussian MFs per antecedent, so we have then 16 rules. The MFs of the antecedents are Gaussian, where its mean and the standard deviation were obtained from the 524 training samples, *x*(1001), ··· , *x*(1524). Table 2 summarizes the consequent parameters per each rule.

Figure 14 displays performance of the FLS in training data, and Figure 15 shows its results on Testing data. Note that the GOP-designed is a better forecaster, since the differences from original data are small in both training and testing data sets.

Some additional analyses should be performed to verify the goodness of fit of the method (See [5], and [6]), but in this case, the proposed GOP has shown good results, so we can recommend its application to real cases. Time series analysis is an useful topic for many decision makers, so the use of optimal and easy-to-be-implemented techniques, as the proposed one has a wide potential.

16 Will-be-set-by-IN-TECH 382 Fuzzy Controllers – Recent Advances in Theory and Applications Enhancing Fuzzy Controllers Using Generalized Orthogonality Principle <sup>17</sup>


<sup>1550</sup> <sup>1600</sup> <sup>1650</sup> <sup>1700</sup> <sup>1750</sup> <sup>1800</sup> <sup>1850</sup> <sup>1900</sup> <sup>1950</sup> <sup>2000</sup> <sup>0</sup>

**Figure 15.** Output of the TSK FLS time-series forecaster. The samples *x*(1525), ··· , *x*(2024) are used for

In this chapter we have presented an enhancement method of fuzzy controllers using the generalized orthogonality principle. We applied the method to two different cases: a first one involving control of an inverted pendulum and a second one for fuzzy forecasting. In the first application, numerical rules and their FBFs were extracted from numerical training data. This combination of both linguistic and numerical information simultaneously become FBFs an useful method. Since a specific FLS can be expressed as a linear combination of FBFs, we

In the second study case, we applied the GOP to design a FLS for time series forecasting. The FLS has been applied to a Mackey-Glass time series with better results compared to a

All the FBFs can be seen as a basis vector, which allows to optimize the parameters of the consequents. This means that the error vectors are orthogonal to these FBFs, resulting in the minimization of the magnitudes of these error vectors, and consequently an optimal FLS.

The proposed method has a wide potential in complex forecasting problems ([5], and [6]). Its application to hardware design problems ([7]) can improve the performance of fuzzy

generalized orthogonality principle on FBFs that results in a better FLS.

controllers, so its implementation arises as a new field to be covered.

*Universidad Distrital Francisco Jose de Caldas, Bogota - Colombia*

non-GOP FLS. The results were validated with simulations.

Time (sec)

Mackey Glass Time Series Forecasting

x(t) Test Data GOP−Designed Forcaster

Enhancing Fuzzy Controllers Using Generalized Orthogonality Principle 383

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

testing the GOP design

**Author details** Nora Boumella

Juan Carlos Figueroa

*University of Batna, Batna - Algeria*

**7. Concluding remarks**

x(t) and yTSK(t)

**Table 2.** The optimal TSK FLS consequent parameters obtained by GOP design.

**Figure 14.** Mackey-Glass time series. The samples *x*(1001), ··· , *x*(1524) are used for designing the FLS forecaster

**Figure 15.** Output of the TSK FLS time-series forecaster. The samples *x*(1525), ··· , *x*(2024) are used for testing the GOP design

## **7. Concluding remarks**

16 Will-be-set-by-IN-TECH

*R<sup>l</sup> c*<sup>1</sup> *c*<sup>2</sup> *c*<sup>3</sup> *c*<sup>4</sup> *c*<sup>5</sup> *<sup>R</sup>*<sup>1</sup> <sup>−</sup>3.58 7.94 <sup>−</sup>9.17 0.03 0.78 *<sup>R</sup>*<sup>2</sup> 9.92 <sup>−</sup>10.9 <sup>−</sup>0.02 1.29 <sup>−</sup>7.83 *<sup>R</sup>*<sup>3</sup> 16.05 7.58 12.40 5.25 <sup>−</sup>33.96 *<sup>R</sup>*<sup>4</sup> 8.92 <sup>−</sup>6.78 <sup>−</sup>12.22 3.68 <sup>−</sup>4.13 *<sup>R</sup>*<sup>5</sup> 1.06 4.64 28.42 <sup>−</sup>40.17 0.16 *<sup>R</sup>*<sup>6</sup> 22.57 <sup>−</sup>33.28 <sup>−</sup>12.43 17.13 <sup>−</sup>12.76 *<sup>R</sup>*<sup>7</sup> <sup>−</sup>2.93 <sup>−</sup>7.65 5.73 <sup>−</sup>2.91 <sup>−</sup>1.30 *<sup>R</sup>*<sup>8</sup> 22.88 26.23 <sup>−</sup>15.79 <sup>−</sup>6.19 <sup>−</sup>0.60 *<sup>R</sup>*<sup>9</sup> <sup>−</sup>3.86 4.36 2.04 0.21 4.77 *<sup>R</sup>*<sup>10</sup> 27.72 <sup>−</sup>45.35 24.63 7.92 6.26 *<sup>R</sup>*<sup>11</sup> <sup>−</sup>0.24 <sup>−</sup>4.99 30.94 <sup>−</sup>26.54 6.65 *<sup>R</sup>*<sup>12</sup> 2.36 5.34 <sup>−</sup>26.93 18.21 <sup>−</sup>8.03 *<sup>R</sup>*<sup>13</sup> <sup>−</sup>30.66 13.37 5.27 3.60 1.43 *<sup>R</sup>*<sup>14</sup> 23.62 <sup>−</sup>21.97 <sup>−</sup>3.87 6.04 8.01 *<sup>R</sup>*<sup>15</sup> 3.70 <sup>−</sup>5.07 0.61 <sup>−</sup>0.76 8.38 *<sup>R</sup>*<sup>16</sup> <sup>−</sup>25.05 11.30 <sup>−</sup>0.42 1.27 4.64

**Table 2.** The optimal TSK FLS consequent parameters obtained by GOP design.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

forecaster

x(t)

<sup>1050</sup> <sup>1100</sup> <sup>1150</sup> <sup>1200</sup> <sup>1250</sup> <sup>1300</sup> <sup>1350</sup> <sup>1400</sup> <sup>1450</sup> <sup>1500</sup> <sup>0</sup>

**Figure 14.** Mackey-Glass time series. The samples *x*(1001), ··· , *x*(1524) are used for designing the FLS

Time (sec)

Mackey Glass Time Series

x(t) Training Data GOP Approximation In this chapter we have presented an enhancement method of fuzzy controllers using the generalized orthogonality principle. We applied the method to two different cases: a first one involving control of an inverted pendulum and a second one for fuzzy forecasting. In the first application, numerical rules and their FBFs were extracted from numerical training data. This combination of both linguistic and numerical information simultaneously become FBFs an useful method. Since a specific FLS can be expressed as a linear combination of FBFs, we generalized orthogonality principle on FBFs that results in a better FLS.

In the second study case, we applied the GOP to design a FLS for time series forecasting. The FLS has been applied to a Mackey-Glass time series with better results compared to a non-GOP FLS. The results were validated with simulations.

All the FBFs can be seen as a basis vector, which allows to optimize the parameters of the consequents. This means that the error vectors are orthogonal to these FBFs, resulting in the minimization of the magnitudes of these error vectors, and consequently an optimal FLS.

The proposed method has a wide potential in complex forecasting problems ([5], and [6]). Its application to hardware design problems ([7]) can improve the performance of fuzzy controllers, so its implementation arises as a new field to be covered.

## **Author details**

Nora Boumella *University of Batna, Batna - Algeria*

Juan Carlos Figueroa *Universidad Distrital Francisco Jose de Caldas, Bogota - Colombia* Sohail Iqbal *NUST-SEECS, Islamabad - Pakistan*

## **8. References**


© 2012 Mahmoud 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,

**New Areas in Fuzzy Application** 

"The world is not black and white but only shades of gray." In 1965, Zadeh [1] wrote a seminal paper in which he introduced fuzzy sets, sets with un-sharp boundaries. These sets are considered gray areas rather than black and white in contrast to classical sets which form the basis of binary or Boolean logic. Fuzzy set theory and fuzzy logic are convenient tools for handling uncertain, imprecise, or unmodeled data in intelligent decision-making systems. It has also found many applications in the areas of information sciences and control

In this chapter, we shall discuss two important categories of fuzzy logic nonlinear applications: "Control" and "Trending and Prediction". With respect to Fuzzy Control application, among the huge applications that were published under this category, two new applications are selected in this chapter to focus on the Hierarchal Control application with "multi-input" "multi-output" signals, and another application is selected as application of smart electrical grid. However, for Fuzzy Trending and Prediction Application, c-Mean Fuzzy Cluttering technique is discussed as an introduction for Fuzzy trending algorithm, and then two different applications are introduced. The first application discusses very nonlinear problem to predict the rate of accident for labours work in a construction sector, and the second application is to find a fault in complicated electrical network. All these new applications for fuzzy control and

Control is one of the main the application for the fuzzy controller, especially for the applications that can be easily expressed by linguistically. Many machines now in the market are fuzzy machines. Also the fuzzy logic has take place in the DCS's and PLC's as recognized function to build process controllers. In this chapter we shall select three

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

applications as an example for the new application of fuzzy logic in the control.

Additional information is available at the end of the chapter

fuzzy trending recognition has been found after year 2000.

Muhammad M.A.S. Mahmoud

http://dx.doi.org/10.5772/48781

**1. Introduction** 

systems.

**2. Control** 

## **New Areas in Fuzzy Application**

## Muhammad M.A.S. Mahmoud

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48781

## **1. Introduction**

18 Will-be-set-by-IN-TECH

[1] Berenji, H. & Khedkar, P. [1992]. Learning and tuning fuzzy logic controllers through

[2] Boumella, N., Djouani, K. & Boulemden, M. [2011]. On an Interval Type-2 TSK FLS A1-C1 consequent parameters tuning, *in* IEEE (ed.), *Proc. SSCI 2011 T2FUZZ - 2011 IEEE*

[3] Boumella, N., Djouani, K. & Boulemden, M. [2012]. A robust Interval Type-2 TSK fuzzy logic system design based on chebyshev fitting, *International Journal of Control,*

[4] Boumella, N., Djouani, K. & Iqbal, S. [2009]. A new design of fuzzy logic controller based on generalized orthogonality principle, *in* IEEE (ed.), *Proc. IEEE International Symposium on Computational Intelligence in Robotics and Automation, CIRA 2009*, IEEE, pp. 497–502. [5] Figueroa, J. C. [2009]. An evolutive Interval Type-2 TSK fuzzy logic system for volatile time series identification, *2009 Conference on Systems, Man and Cybernetics*, IEEE, pp. 1–6. [6] Figueroa, J. C., Kalenatic, D. & Lopez, C. A. [2010]. A neuro-evolutive Interval Type-2 TSK fuzzy system for volatile weather forecasting, *Lecture Notes in Computer Science*

[7] Melgarejo, M. & Peña, C. A. [2007]. Implementing Interval Type-2 fuzzy processors I,

[8] Mendel, J. [2001]. *Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New*

[10] Wang, L. [1992]. Fuzzy systems are universal approximators, *in* IEEE (ed.), *Proc. IEEE*

[11] Wang, L. & Mendel, J. M. [1992]. Fuzzy basis functions, universal approximation, and orthogonal least squares learning, *IEEE Trans. Neural Networks* 3(5): 807–814.

[9] Strang, G. [1988]. *Linear Algebra and its Applications, Third edition*, Brooks/Cole.

*Int'l. Conf. on Fuzzy Systems, San Diego, CA*, IEEE, pp. 1163–1170.

reinforcements, *IEEE Trans. Neural Networks* 3(1): 724–740.

*Computational Intelligence Magazine* 2(1): Pág. 63–71.

*Symposium on Advances in Type-2 Fuzzy Logic Systems*, IEEE, pp. 1–6.

Sohail Iqbal

**8. References**

*NUST-SEECS, Islamabad - Pakistan*

*Automation, and Systems* 10(4).

6216: 142–149.

*Directions*, Prentice Hall.

"The world is not black and white but only shades of gray." In 1965, Zadeh [1] wrote a seminal paper in which he introduced fuzzy sets, sets with un-sharp boundaries. These sets are considered gray areas rather than black and white in contrast to classical sets which form the basis of binary or Boolean logic. Fuzzy set theory and fuzzy logic are convenient tools for handling uncertain, imprecise, or unmodeled data in intelligent decision-making systems. It has also found many applications in the areas of information sciences and control systems.

In this chapter, we shall discuss two important categories of fuzzy logic nonlinear applications: "Control" and "Trending and Prediction". With respect to Fuzzy Control application, among the huge applications that were published under this category, two new applications are selected in this chapter to focus on the Hierarchal Control application with "multi-input" "multi-output" signals, and another application is selected as application of smart electrical grid. However, for Fuzzy Trending and Prediction Application, c-Mean Fuzzy Cluttering technique is discussed as an introduction for Fuzzy trending algorithm, and then two different applications are introduced. The first application discusses very nonlinear problem to predict the rate of accident for labours work in a construction sector, and the second application is to find a fault in complicated electrical network. All these new applications for fuzzy control and fuzzy trending recognition has been found after year 2000.

## **2. Control**

Control is one of the main the application for the fuzzy controller, especially for the applications that can be easily expressed by linguistically. Many machines now in the market are fuzzy machines. Also the fuzzy logic has take place in the DCS's and PLC's as recognized function to build process controllers. In this chapter we shall select three applications as an example for the new application of fuzzy logic in the control.

© 2012 Mahmoud 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.

## **2.1. Fuzzy control design for gas absorber system**

In this section, the chapter shall present the research efforts that have been carried out on the control of gas absorbers/gas reactors. It shall also introduce the new approach to a fuzzy control design for a typical gas absorber system. The approach shall incorporate a linear state-estimation to generate the internal knowledge-base that shall store input-output pairs. This collection of pairs shall be then utilized to build a feedback fuzzy controller for the gas absorber.

New Areas in Fuzzy Application 387

*2.1.2. A Gas Absorber System* 

top of the tower) is more pure.

**Figure 1.** Gas absorption column, n stages

Separation processes play an important role in most chemical manufacturing industries. Streams from chemical reactors often contain a number of components; some of these components must be separated from the other components for sale as a final product, or for use in another manufacturing process. A common example of a separation process is gas absorption (also called gas scrubbing, or gas washing) in which a gas mixture is contacted with a liquid (the absorbent or solvent) to selectively dissolve one or more components by mass transfer from the gas to the liquid. Absorption is used to separate gas mixtures; remove impurities, contaminants, pollutants, or catalyst poisons from a gas; or recover valuable chemicals. In general, the species of interest in the gas mixture may be all components, only the component(s) not transferred, or only the component(s) transferred. Absorption is frequently conducted in trayed towers (plate columns), packed columns, spray towers, bubble columns, and centrifugal contactors. A trayed tower is a vertical, cylindrical pressure vessel in which vapor and liquid, which flow counter-currently, are contacted on a series of metal trays or plates; see Fig. 1. Components that enter the bottom of the tower is the gas feed stream are absorbed by the liquid stream, that flows across each tray, over an outlet weir and into a down-comer, so that the gas product stream (leaving the

*A. Brief Account* 

## *2.1.1. Background*

A major direction in systems engineering design has been focused on the use of simplified mathematical models to facilitate the design process. This constitutes the so-called modelbased system design approach, an overview of the underlying techniques can be found in [2]. Most of the available results have thus far overlooked the operational knowledge of the dynamical system under consideration. On the other hand, a knowledge-based system approach [8] has been suggested to deal with the analysis and design problems of different classes of dynamical systems by incorporating both the simplest available model as well as the best available knowledge about the system. For single physical systems, one of the earlier efforts along this direction has been on the development of an expert learning system; see [4-7] and their references. An alternative approach has been on integrating elements of discrete event systems with differential equations [3].

A third approach has been through the use of fuzzy logic control by successfully applying fuzzy sets and systems theory [9]. In the cases where understood there is no acceptable mathematical model for the plant, fuzzy logic controllers [10] are proved very useful and effective. They are generally base on using qualitative rules of thumb, that is, qualitative control rules in terms of vague and fuzzy sentences. It has been pointed out [11] that fuzzy control systems possess the following features:

Hierarchical ordering of fuzzy rules is used to reduce the size of the inference engine. Realtime implementation, or on-line simulation, of fuzzy controllers can help reduce the burden of large-sized rule sets by fusing sensory data before imputing the system's output to the inference engine.

This section is presenting a new approach to fuzzy control design for a gas absorber system. It provides a new and efficient procedure to construct the inference engine by incorporating a linear state-estimator in generating and storing input-output pairs. This collection of pairs is then utilized to build a feedback fuzzy controller. By fine-tuning of the controller parameters, it is shown that the gas absorber system has always a guaranteed stability. Numerical simulation of a six-order gas absorber is carried out and the obtained results show clearly that the proposed estimator-fuzzy controller scheme yields excellent performance.

## *2.1.2. A Gas Absorber System*

#### *A. Brief Account*

386 Fuzzy Controllers – Recent Advances in Theory and Applications

absorber.

*2.1.1. Background* 

inference engine.

performance.

**2.1. Fuzzy control design for gas absorber system** 

elements of discrete event systems with differential equations [3].

control systems possess the following features:

In this section, the chapter shall present the research efforts that have been carried out on the control of gas absorbers/gas reactors. It shall also introduce the new approach to a fuzzy control design for a typical gas absorber system. The approach shall incorporate a linear state-estimation to generate the internal knowledge-base that shall store input-output pairs. This collection of pairs shall be then utilized to build a feedback fuzzy controller for the gas

A major direction in systems engineering design has been focused on the use of simplified mathematical models to facilitate the design process. This constitutes the so-called modelbased system design approach, an overview of the underlying techniques can be found in [2]. Most of the available results have thus far overlooked the operational knowledge of the dynamical system under consideration. On the other hand, a knowledge-based system approach [8] has been suggested to deal with the analysis and design problems of different classes of dynamical systems by incorporating both the simplest available model as well as the best available knowledge about the system. For single physical systems, one of the earlier efforts along this direction has been on the development of an expert learning system; see [4-7] and their references. An alternative approach has been on integrating

A third approach has been through the use of fuzzy logic control by successfully applying fuzzy sets and systems theory [9]. In the cases where understood there is no acceptable mathematical model for the plant, fuzzy logic controllers [10] are proved very useful and effective. They are generally base on using qualitative rules of thumb, that is, qualitative control rules in terms of vague and fuzzy sentences. It has been pointed out [11] that fuzzy

Hierarchical ordering of fuzzy rules is used to reduce the size of the inference engine. Realtime implementation, or on-line simulation, of fuzzy controllers can help reduce the burden of large-sized rule sets by fusing sensory data before imputing the system's output to the

This section is presenting a new approach to fuzzy control design for a gas absorber system. It provides a new and efficient procedure to construct the inference engine by incorporating a linear state-estimator in generating and storing input-output pairs. This collection of pairs is then utilized to build a feedback fuzzy controller. By fine-tuning of the controller parameters, it is shown that the gas absorber system has always a guaranteed stability. Numerical simulation of a six-order gas absorber is carried out and the obtained results show clearly that the proposed estimator-fuzzy controller scheme yields excellent Separation processes play an important role in most chemical manufacturing industries. Streams from chemical reactors often contain a number of components; some of these components must be separated from the other components for sale as a final product, or for use in another manufacturing process. A common example of a separation process is gas absorption (also called gas scrubbing, or gas washing) in which a gas mixture is contacted with a liquid (the absorbent or solvent) to selectively dissolve one or more components by mass transfer from the gas to the liquid. Absorption is used to separate gas mixtures; remove impurities, contaminants, pollutants, or catalyst poisons from a gas; or recover valuable chemicals. In general, the species of interest in the gas mixture may be all components, only the component(s) not transferred, or only the component(s) transferred. Absorption is frequently conducted in trayed towers (plate columns), packed columns, spray towers, bubble columns, and centrifugal contactors. A trayed tower is a vertical, cylindrical pressure vessel in which vapor and liquid, which flow counter-currently, are contacted on a series of metal trays or plates; see Fig. 1. Components that enter the bottom of the tower is the gas feed stream are absorbed by the liquid stream, that flows across each tray, over an outlet weir and into a down-comer, so that the gas product stream (leaving the top of the tower) is more pure.

**Figure 1.** Gas absorption column, n stages

## *B. Assumptions and definitions*

The basic assumptions used are:

A1) The major component of the liquid stream is inert and does not absorb into the gas stream.

New Areas in Fuzzy Application 389

y d x <sup>j</sup> <sup>j</sup> (3)

(4)

(5)

(6)

which expresses a linear relationship between the liquid phase and gas phase compositions at stage j with d being an equilibrium parameter. Using (3) into (2) and

> d x L (L V d) V d x - x - x dt M M M

> d x (L V d) V d L x - x x dt M M M

d x (L V d) L V x x y dt M M M

where A an (nxn) system matrix with a triangular structure, B is an (nxm) input matrix and

0 0 0 0 0

where xf and yn+1 are the known liquid and vapor feed compositions, respectively.

*L M Vd M M L L Vd Vd M MM L L Vd Vd M MM*

For n-stage gas absorber, (4) is valid for j =2,…,n-1. At the extreme stages, we have:

j-1 j j 1

12 f

n n-1 n 1

x (t) A x(t) B u (t) , y(t) C x(t) (7)

0 0 00 0

00 0

(8)

*L L Vd M M*

Under assumption A3), we let

j

1

n

C is an (nxp) output matrix given by:

On combining (3), (4),(5) and (6), we reach the state-space model:

0

*A*

arranging we get:


We now introduce the following variable definitions:


### *C. Dynamic model*

The concept of an equilibrium stage is important for the development of a dynamic model of the absorption tower. An equilibrium stage is represented schematically in Fig. 2. The total amount of solute on stage j is the sum of the solute in the liquid phase and the gas phase (that is, M xj + W yj). Thus the rate of change of the amount of solute is d(M xj + W yj)/dt and the component material balance around stage j can be expressed as:

$$\begin{aligned} \frac{\mathbf{d} \langle \mathbf{M} \,\mathbf{x}\_{\circ} + \mathbf{W} \mathbf{y}\_{\circ} \rangle}{\mathbf{d}t} &=& \mathbf{L} \,\mathbf{x}\_{\circ 1} + \mathbf{V} \,\mathbf{y}\_{\circ 1} \cdot \mathbf{L} \,\mathbf{x}\_{\circ} \cdot \mathbf{V} \,\mathbf{y}\_{\circ} \\ \frac{\mathbf{d} \mathbf{M} \,\mathbf{x}\_{\circ}}{\mathbf{d}t} &\cong& \mathbf{L} \,\mathbf{x}\_{\circ 1} + \mathbf{V} \,\mathbf{y}\_{\circ 1} \cdot \mathbf{L} \,\mathbf{x}\_{\circ} \cdot \mathbf{V} \,\mathbf{y}\_{\circ} \end{aligned} \tag{1}$$

where we assumed that in accumulation, liquid is much more dense than vapor. Under assumption A4), then (1) simplifies into:

$$
\frac{\mathbf{d} \times \mathbf{j}}{\mathbf{d}t} \quad \triangleq \frac{\mathbf{L}}{\mathbf{M}} \mathbf{x}\_{\mathbf{j}+1} + \frac{\mathbf{V}}{\mathbf{M}} \mathbf{y}\_{\mathbf{j}+1} \cdot \frac{\mathbf{L}}{\mathbf{M}} \mathbf{x}\_{\mathbf{j}} \cdot \frac{\mathbf{V}}{\mathbf{M}} \mathbf{y}\_{\mathbf{j}} \tag{2}
$$

$$
\xrightarrow{\mathbf{L} \times \mathbf{i}} \xrightarrow{\mathbf{l} \times \mathbf{y}\_{\mathbf{i}}} \mathbf{V}^{\mathbf{y}\_{\mathbf{i}}}
$$

**Figure 2.** A typical gas absorption stage

Under assumption A3), we let

388 Fuzzy Controllers – Recent Advances in Theory and Applications

A1) The major component of the liquid stream is inert and does not absorb into the gas

A2) The major component of the gas stream is inert and does not absorb into the liquid

A3) Each stage of the process is an equilibrium stage, that is, the vapour leaving a stage is in

The concept of an equilibrium stage is important for the development of a dynamic model of the absorption tower. An equilibrium stage is represented schematically in Fig. 2. The total amount of solute on stage j is the sum of the solute in the liquid phase and the gas phase (that is, M xj + W yj). Thus the rate of change of the amount of solute is d(M xj + W yj)/dt and

d(M x Wy ) L x V y - L x - V y dt

L x V y - L x - V y dt

j-1 j 1 j j

(1)

(2)

j-1 j 1 j j

where we assumed that in accumulation, liquid is much more dense than vapor. Under

d xj L V LV x y - x - y dt M M M M j-1 j 1 j j

thermodynamic equilibrium with the liquid on that stage.

the component material balance around stage j can be expressed as:

j j

j

dM x

assumption A4), then (1) simplifies into:

**Figure 2.** A typical gas absorption stage

*B. Assumptions and definitions* 

The basic assumptions used are:

A4) The liquid molar holdup is constant.

We now introduce the following variable definitions:

 L = moles inert liquid per time: = liquid molar flow rate. V= moles inert vapor per time: = vapor molar flow rate M= moles liquid per stage: = liquid molar holdup per stage W= moles vapor per stage: = vapor molar holdup per stage xj= moles solute (stage j) per mole inert liquid (stage j) yj= moles solute (stage j) per mole inert vapor (stage j)

stream.

stream.

*C. Dynamic model* 

$$\mathbf{y}\_{\mathbf{j}} = \mathbf{d} \cdot \mathbf{x}\_{\mathbf{j}} \tag{3}$$

which expresses a linear relationship between the liquid phase and gas phase compositions at stage j with d being an equilibrium parameter. Using (3) into (2) and arranging we get:

$$\frac{\mathbf{d} \times \mathbf{r}\_{\circ}}{\mathbf{d}\mathbf{t}} = \mathbf{\stackrel{L}{\mathbf{M}}} \times\_{\mathbf{j}+1} \mathbf{\stackrel{(L+V)}{\mathbf{M}}} \mathbf{x}\_{\circ} \mathbf{\stackrel{V}{\mathbf{d}}} \mathbf{x}\_{\circ+1} \tag{4}$$

For n-stage gas absorber, (4) is valid for j =2,…,n-1. At the extreme stages, we have:

$$\frac{\text{d}\,\text{x}\_1}{\text{d}\,\text{t}}\,\text{ } = -\frac{(\text{L} + \text{V}\,\text{d})}{\text{M}}\,\text{x}\_1 \cdot \frac{\text{V}\,\text{d}}{\text{M}}\,\text{x}\_2 \,\text{ } + \,\frac{\text{L}}{\text{M}}\,\text{x}\_f\tag{5}$$

$$\frac{\text{d}\,\text{x}\_{\text{n}}}{\text{d}\,\text{t}}\,\,\,\,- = \frac{\text{(L} + \text{V d)}}{\text{M}}\,\,\text{x}\_{\text{n}} + \frac{\text{L}}{\text{M}}\,\,\text{x}\_{\text{n-1}} + \,\,\frac{\text{V}}{\text{M}}\,\,\text{y}\_{\text{n+1}}\tag{6}$$

where xf and yn+1 are the known liquid and vapor feed compositions, respectively.

On combining (3), (4),(5) and (6), we reach the state-space model:

$$\dot{\mathbf{x}} \text{ (t) } \quad = \mathbf{A} \cdot \mathbf{x}(\mathbf{t}) + \mathbf{B} \,\mathbf{u} \text{ (t) } \qquad \mathbf{y}(\mathbf{t}) \, = \, \mathbf{C} \,\mathbf{x}(\mathbf{t}) \tag{7}$$

where A an (nxn) system matrix with a triangular structure, B is an (nxm) input matrix and C is an (nxp) output matrix given by:

$$A = \begin{bmatrix} -\frac{L+M}{M} & \frac{Vd}{M} & 0 & 0 & 0 & 0 & \cdots & 0\\ \frac{L}{M} & -\frac{L+Vd}{M} & \frac{Vd}{M} & 0 & 0 & \cdots & 0\\ 0 & \frac{L}{M} & -\frac{L+Vd}{M} & \frac{Vd}{M} & 0 & \cdots & 0\\ 0 & & \ddots & & & & 0\\ \vdots & & & \ddots & & & \vdots\\ \vdots & & & & \ddots\\ 0 & & & & \frac{L}{M} & -\frac{L+Vd}{M} \end{bmatrix} \tag{8}$$

0 0 0 *L M B V M* (9) 100000 000001 *<sup>C</sup>* (10)

*Step 3:* 

controller is

Where μA<sup>l</sup>

d=0.5. Thus,

*2.1.4. Simulation studies* 

New Areas in Fuzzy Application 391

Design the fuzzy controller from the 2N+1 fuzzy IF THEN rules (11) using product inference engine, singleton fuzzifier and center average defuzzifier; that is, the designed fuzzy

<sup>y</sup> <sup>μ</sup>A (y) v=-f(y)=

To estimate the range of the input-output pairs {vi, yi}, full order estimator [2] can be used.

Consider a gas absorber system with the following parameters: L=80, M=200, V=100 and

0.65, 0.4, 0.25, 0.5 *L Vd L Vd V M MM M* 

A MATLAB program is written to simulate the gas absorber system. Different positive and negative step input are applied to estimate the outputs. The results of two cases are

illustrated in Fig. 4 and Fig. 5. The tracking behaviour of the outputs is shown.

**Figure 4.** Output response with positive step input signal

(y) are shown in Fig. 3 and y 1 satisfy y (12).

2N+1 1 1 1=1 2N+1 1 1=1

μA (y)

(13)

#### *2.1.3. Fuzzy controller design*

The design of a fuzzy controller can be implemented by the following steps:

*Step1:* 

Supposed that the output y (t) takes values in the interval U = [, ] R . Define 2N+1 fuzzy function Al in U that are consistent and complete with the triangular membership functions shown in Fig. 3. That is, we use the N fuzzy sets A1, ---, AN to cover the negative interval [, 0), the other N fuzzy sets AN+2,---, A2N+1 to cover the positive interval (0, ], and choose the center x N+1 of fuzzy set AN+1 at zero.

**Figure 3.** Membership functions for the fuzzy controller.

*Step 2:* 

Consider the following 2N+1 fuzzy IF-THEN rules:

$$\text{IF y is A}^{\text{l}}\text{, THEN u is B}^{\text{l}}\text{}\tag{11}$$

Where l = 1, 2, ---, 2N+1, and centers <sup>1</sup> y of fuzzy set Bl are chosen such that,

$$\begin{cases} \le 0 & \text{for 1=1,} \cdots, \text{N} \\ \mathbf{y}^1 \Big|\_{} = 0 & \text{for 1=N+1} \\ \ge 0 & \text{for 1=N+2,} \cdots, \text{2N+1} \end{cases} \tag{12}$$

*Step 3:* 

390 Fuzzy Controllers – Recent Advances in Theory and Applications

*2.1.3. Fuzzy controller design* 

center x N+1 of fuzzy set AN+1 at zero.

**Figure 3.** Membership functions for the fuzzy controller.

Consider the following 2N+1 fuzzy IF-THEN rules:

Where l = 1, 2, ---, 2N+1, and centers <sup>1</sup> y of fuzzy set Bl

1

 

0 for1=1, ,N

0 for1= N+2, ,2N+1

y 0 for1= N+1

*Step1:* 

*Step 2:* 

function Al

0

*V M* (9)

(10)

0 0

 

100000 000001 *<sup>C</sup>*

Supposed that the output y (t) takes values in the interval U = [, ] R . Define 2N+1 fuzzy

shown in Fig. 3. That is, we use the N fuzzy sets A1, ---, AN to cover the negative interval [, 0), the other N fuzzy sets AN+2,---, A2N+1 to cover the positive interval (0, ], and choose the

in U that are consistent and complete with the triangular membership functions

l l IF y is A , THEN u is B (11)

are chosen such that,

(12)

*L M*

*B*

The design of a fuzzy controller can be implemented by the following steps:

Design the fuzzy controller from the 2N+1 fuzzy IF THEN rules (11) using product inference engine, singleton fuzzifier and center average defuzzifier; that is, the designed fuzzy controller is

$$\mathbf{v} = \mathbf{f}(\mathbf{y}) = \frac{\sum\_{1=1}^{2\mathbf{N}+1} \overline{\mathbf{y}^1} \mu \mathbf{A}^1(\mathbf{y})}{\sum\_{1=1}^{2\mathbf{N}+1} \mu \mathbf{A}^1(\mathbf{y})} \tag{13}$$

Where μA<sup>l</sup> (y) are shown in Fig. 3 and y 1 satisfy y (12).

To estimate the range of the input-output pairs {vi, yi}, full order estimator [2] can be used.

#### *2.1.4. Simulation studies*

Consider a gas absorber system with the following parameters: L=80, M=200, V=100 and d=0.5.

Thus,

$$\frac{L + Vd}{M} = -0.65 \,\mu \frac{L}{M} = 0.4 \,\mu \frac{V \,d}{M} = 0.25 \,\mu \frac{V}{M} = 0.5 \,V$$

A MATLAB program is written to simulate the gas absorber system. Different positive and negative step input are applied to estimate the outputs. The results of two cases are illustrated in Fig. 4 and Fig. 5. The tracking behaviour of the outputs is shown.

**Figure 4.** Output response with positive step input signal

New Areas in Fuzzy Application 393

**Figure 7.** Controller is tuned to interfere the natural decay of the system

**Figure 8.** The controller is tuned to improve the response of the system.

**Figure 5.** Output response with negative step input signal

From the input-output pair obtained, the behaviour of the system is examined and the ranges of its outputs (controllers' inputs) are predicted. Fig. 6 illustrates a block diagram of the gas absorber and the fuzzy controller array.

**Figure 6.** Block Diagram of gas absorber system and the fuzzy controllers

To control the response of the gas absorber, the range of linguistic values of the output of each feedback fuzzy controller is tuned between (– 3) and (3). Comparison between the output response with fuzzy controller (when the number of linguistic values of the controller input – output pair is three) and without controller is illustrated in Fig.7 and Fig.8.

In Fig.7, the controller is tuned to interfere the natural decay of the system. In Fig. 8, the fuzzy controller is adjusted to improve the response of the gas absorber. It is noted that the response of controlled system has less overshoot, less steady state error and faster compared to the uncontrolled system.

**Figure 7.** Controller is tuned to interfere the natural decay of the system

**Figure 5.** Output response with negative step input signal

the gas absorber and the fuzzy controller array.

to the uncontrolled system.

**Figure 6.** Block Diagram of gas absorber system and the fuzzy controllers

From the input-output pair obtained, the behaviour of the system is examined and the ranges of its outputs (controllers' inputs) are predicted. Fig. 6 illustrates a block diagram of

To control the response of the gas absorber, the range of linguistic values of the output of each feedback fuzzy controller is tuned between (– 3) and (3). Comparison between the output response with fuzzy controller (when the number of linguistic values of the controller input – output pair is three) and without controller is illustrated in Fig.7 and Fig.8. In Fig.7, the controller is tuned to interfere the natural decay of the system. In Fig. 8, the fuzzy controller is adjusted to improve the response of the gas absorber. It is noted that the response of controlled system has less overshoot, less steady state error and faster compared

**Figure 8.** The controller is tuned to improve the response of the system.

## *2.1.5. Discussion*

This section has presented a new and simple fuzzy controller for a gas absorber system to enhance the response of the output. The simulation results have shown that the controller guarantees well-damped behaviour of the controlled gas absorber system.

New Areas in Fuzzy Application 395

and providing qualitative "rules of thumb" (qualitative control rules in terms of vague and

A concerted effort has been made to formally reduce the size of the fuzzy rule base to make fuzzy control attractive to interconnected systems. Two of the difficulties with the design of

The properties that a fuzzy membership function is used to characterize are usually fuzzy. Therefore, we may use different membership functions to characterize the same description. Conceptually, there are two approaches to determine a membership function. The first approach is to use the knowledge of human experts. Usually this approach can only give a rough formula of the membership function; fine-tuning is required. In the second approach, data are collected from various sensors to determine the member ship functions. Specifically, the structures of the membership functions are specified first, then fine-tuning of the membership function parameters should be implemented based on the collected data

In this section, we contribute to the further development of intelligent control techniques of interconnected systems. It provides a new approach to fuzzy control design for interconnected system. The approach consists of two stages: In the first stage, a group of local state estimator is constructed to generate the data base of input-output pairs. In the second stage, an array of feedback fuzzy controllers is designed and implemented to ensure the asymptotic satiability of the interconnected system. Simulation studies on a large-scale system with unstable eigenvalus are carried out to illustrate the features and capability of

In the sequel, the terms large-scale and interconnected are used interchangeable. The term large scale system (LSS) does not have a unique established meaning, but it covers systems that possess several particular feature, such as multiple subsystem, [14,17] multiple control, multiple objectives, decentralized and/or hierarchical information structures. Any LSS includes many variables but their control is faced by a well-know fact [16] that the states are

Many authors have considered the state estimation of large-scale systems in input decentralized fashion. Here we summarize one convenient algorithm [15]. Let the state

<sup>N</sup>

i ii ii ij j x t Ax t Bu t G x (14)

i ii y t C x t , i, j 1, 2, N (15)

fuzzy sentences).

[8].

this new approach.

any fuzzy control system are:

The choice of fuzzy rules.

The shape of the membership functions.

*2.2.2. State estimation of interconnected systems* 

model of the ith subsystem described by

not always available for measurement and state must be estimated.

## **2.2. Large scale fuzzy controller**

In this section, we shall develop a new approach to the control of interconnected system using fuzzy system theory. The approach shall be based on incorporating a group of local estimators on the system level to generate the input-output database. An array of feedback fuzzy controllers shall then be designed to ensure the asymptotic stability of the closed loop system. The developed technique shall be applied to an unstable large-scale system and extensive simulation studies shall be carried out to illustrate the potential of this new approach..

## *2.2.1. Background*

In control engineering research, problems of decentralized control and stabilization of interconnected systems are receiving considerable interest in recent years [14,15] where most of the effort is focused on dealing with the interaction patterns. It is concluded that a systematic approach to deal with the problems of interconnected systems is twofold: first is to base the analysis and design effort on the subsystem level using conventional control methods and second is to deal with interactions effectively. These methods are facilitated, in general, by virtue of several mathematical tools including linearization, delay approximation, decomposition and model reduction. This constitutes the so-called modelbased control system approach for which we have seen numerous techniques [16]. Most of the available results have so far overlooked the operational knowledge of the interconnected system under consideration. In [17], a knowledge-based control system approach has been suggested to deal with the analysis and design problems of interconnected systems by incorporating both the simplest available model as well as the best available knowledge about the system. For single physical systems, one of the earlier efforts along this direction has been on the development of an expert learning system [18-19]. An alternative approach has been on integrating elements of discrete event systems with differential equations [20]. A practically-supported third approach has been through the use of fuzzy logic control by successfully applying fuzzy sets and systems theory [21].

For interconnected systems, the foregoing approach motivates the research into intelligent control by combining techniques of control and systems theory with those from artificial intelligence. The main focus should be on integrating a knowledge base, an approximate (humanlike) reasoning and/or a learning process within a hierarchical structure.

Fuzzy logic controllers [23-25] are generally considered applicable to plants that are mathematically poorly understood (there is no acceptable mathematical model for the plant) and where experienced human operators are available for satisfactorily controlling the plant and providing qualitative "rules of thumb" (qualitative control rules in terms of vague and fuzzy sentences).

A concerted effort has been made to formally reduce the size of the fuzzy rule base to make fuzzy control attractive to interconnected systems. Two of the difficulties with the design of any fuzzy control system are:


394 Fuzzy Controllers – Recent Advances in Theory and Applications

successfully applying fuzzy sets and systems theory [21].

**2.2. Large scale fuzzy controller** 

This section has presented a new and simple fuzzy controller for a gas absorber system to enhance the response of the output. The simulation results have shown that the controller

In this section, we shall develop a new approach to the control of interconnected system using fuzzy system theory. The approach shall be based on incorporating a group of local estimators on the system level to generate the input-output database. An array of feedback fuzzy controllers shall then be designed to ensure the asymptotic stability of the closed loop system. The developed technique shall be applied to an unstable large-scale system and extensive simulation studies shall be carried out to illustrate the potential of this new

In control engineering research, problems of decentralized control and stabilization of interconnected systems are receiving considerable interest in recent years [14,15] where most of the effort is focused on dealing with the interaction patterns. It is concluded that a systematic approach to deal with the problems of interconnected systems is twofold: first is to base the analysis and design effort on the subsystem level using conventional control methods and second is to deal with interactions effectively. These methods are facilitated, in general, by virtue of several mathematical tools including linearization, delay approximation, decomposition and model reduction. This constitutes the so-called modelbased control system approach for which we have seen numerous techniques [16]. Most of the available results have so far overlooked the operational knowledge of the interconnected system under consideration. In [17], a knowledge-based control system approach has been suggested to deal with the analysis and design problems of interconnected systems by incorporating both the simplest available model as well as the best available knowledge about the system. For single physical systems, one of the earlier efforts along this direction has been on the development of an expert learning system [18-19]. An alternative approach has been on integrating elements of discrete event systems with differential equations [20]. A practically-supported third approach has been through the use of fuzzy logic control by

For interconnected systems, the foregoing approach motivates the research into intelligent control by combining techniques of control and systems theory with those from artificial intelligence. The main focus should be on integrating a knowledge base, an approximate

Fuzzy logic controllers [23-25] are generally considered applicable to plants that are mathematically poorly understood (there is no acceptable mathematical model for the plant) and where experienced human operators are available for satisfactorily controlling the plant

(humanlike) reasoning and/or a learning process within a hierarchical structure.

guarantees well-damped behaviour of the controlled gas absorber system.

*2.1.5. Discussion* 

approach..

*2.2.1. Background* 

The properties that a fuzzy membership function is used to characterize are usually fuzzy. Therefore, we may use different membership functions to characterize the same description.

Conceptually, there are two approaches to determine a membership function. The first approach is to use the knowledge of human experts. Usually this approach can only give a rough formula of the membership function; fine-tuning is required. In the second approach, data are collected from various sensors to determine the member ship functions. Specifically, the structures of the membership functions are specified first, then fine-tuning of the membership function parameters should be implemented based on the collected data [8].

In this section, we contribute to the further development of intelligent control techniques of interconnected systems. It provides a new approach to fuzzy control design for interconnected system. The approach consists of two stages: In the first stage, a group of local state estimator is constructed to generate the data base of input-output pairs. In the second stage, an array of feedback fuzzy controllers is designed and implemented to ensure the asymptotic satiability of the interconnected system. Simulation studies on a large-scale system with unstable eigenvalus are carried out to illustrate the features and capability of this new approach.

## *2.2.2. State estimation of interconnected systems*

In the sequel, the terms large-scale and interconnected are used interchangeable. The term large scale system (LSS) does not have a unique established meaning, but it covers systems that possess several particular feature, such as multiple subsystem, [14,17] multiple control, multiple objectives, decentralized and/or hierarchical information structures. Any LSS includes many variables but their control is faced by a well-know fact [16] that the states are not always available for measurement and state must be estimated.

Many authors have considered the state estimation of large-scale systems in input decentralized fashion. Here we summarize one convenient algorithm [15]. Let the state model of the ith subsystem described by

$$\mathbf{x}\_{\mathbf{i}}\left(\mathbf{t}\right) = \mathbf{A}\_{\mathbf{i}}\mathbf{x}\_{\mathbf{i}}\left(\mathbf{t}\right) + \mathbf{B}\_{\mathbf{i}}\mathbf{u}\_{\mathbf{i}}\left(\mathbf{t}\right) + \boldsymbol{\Sigma}^{\mathbf{N}}\mathbf{G}\_{\mathbf{i}\parallel}\mathbf{x}\_{\mathbf{j}} \tag{14}$$

$$\mathbf{y}\_i(\mathbf{t}) = \mathbb{C}\_i \mathbf{x}\_i(\mathbf{t}), \text{i.j.} = \mathbf{1}, \text{2. } \dots \text{N} \tag{15}$$

Where all vectors and matrices are appropriately defined and gi(.) is the interaction function between the ith subsystem and the rest of the system. It is considered that (Ci, Ai) is completely observable for i = 1, 2, …….. N.

The following algorithm finds the optimal states of a large-scale system based on decentralized estimation and control [17]:

*Algorithm 1:* 

*Step 1:* 

Read the matrices Ai, Bi and select Qi 0 and Ri 0 as weighted matrix.

*Step 2:* 

Solve the following 2N algebraic Riccati equations for Hi, Ki

$$\text{Hil}(\text{A}^{\text{T}}\_{\text{i}} + a\text{I}\_{\text{i}}) + (\text{A}\_{\text{i}} + a\text{I}\_{\text{i}})\text{H}\_{\text{i}} - \text{H}\_{\text{i}}\text{D}\_{\text{i}}\text{H}\_{\text{i}} + \text{Q}\_{\text{i}} = \text{0} \tag{16}$$

New Areas in Fuzzy Application 397

(20)

11 12 13 14 15 21 22 23 24 25 31 32 33 34 35 41 42 43 44 45 51 52 53 54 55

> 

25 0.1 0.5

(21)

(22)


0.4 0.2 1 0.5 0.1 r1 r1 r1 r1 r1 A=

r2 r2 r2 r2 r2 0.1 0 0 0.2 0 r2 r2 r2 r2 r2 0 0.2 1 0 0.4 r2 r2 r2 r2 r2 0.6 0.1 0.25 2 0 r2 r2 r2 r2 r2 0.4 0.2 1 0.5 0.1

1100000000 0010100000

0000011000 0000000101

Which is considered to be composed of two-coupled subsystems; each of order 5. The coupling parameters are r1jk and r2jk where j and k take values of 1,2,3,4 and 5. In the sequel,

> A11 G12(r1) B1 x= x+ v G21(r2) A22 B2

For a typical values [4] of r115=-0.1, r124=0.1, r142=0.2, r222=0.1,r242=0.15,r251=0.11 and allothe values of coupling parameters are zeros, we examined the stability of the system by computing the eigenvalues of matrix A. They are {-1.0915, -1.0641, 0.477 + j0.0206, 0.477 – j0.00206, 0.022 + j0.0544, 0.022 – j0.0544, -1.8709 + j0.1713, -1.8709 – j0.1713, -1.9306 + j0.1413, -

(23)

r2 r2 r2 r2 r2 1.5 0.3 0.

B=

11 12 13 14 15

21 22 23 24 25 31 32 33 34 35 41 42 43 44 45 51 52 53 54 55

C=

we refer to the structure of the interconnected system model as:

Where G12(r1) and G21(r2) are the coupling mtrices.

 

$$\text{Ki}(\text{A}^{\text{T}}\_{\text{i}} + a\text{I}\_{\text{i}}) + (\text{A}\_{\text{i}} + a\text{I}\_{\text{i}})\text{K}\_{\text{i}} - \text{K}\_{\text{i}}\text{S}\_{\text{i}}\text{K}\_{\text{i}} + \text{Q}\_{\text{i}} = \text{0} \tag{17}$$

Where Di = CTiCi , Si = Bi R-1i BT

*Step 3:* 

Integrate the following set of N simultaneous equation for ei(t), I = 1, 2 …. N, using the initial condition ei(0) = x i(0)

$$
\begin{pmatrix}
\dot{\mathbf{e}}\_{1} \\
\vdots \\
\dot{\mathbf{e}}\_{N}
\end{pmatrix} = \begin{pmatrix}
\mathbf{A}\_{1}\cdot\mathbf{S}\_{1}\mathbf{K}\_{1} & \cdots & \mathbf{G}\_{1N} \\
\vdots & \ddots & \\
\mathbf{G}\_{N\_{1}} & & \mathbf{A}\_{N}\cdot\mathbf{H}\_{N}\mathbf{D}\_{N}
\end{pmatrix} \begin{pmatrix}
\mathbf{e}\_{1} \\
\vdots \\
\mathbf{e}\_{N}
\end{pmatrix} + \begin{pmatrix}
\mathbf{B}\_{1}\mathbf{v}\_{1} \\
\vdots \\
\mathbf{B}\_{N}\mathbf{v}\_{N}
\end{pmatrix} \tag{18}
$$

*Step 4:* 

Integrate the following set of n simultaneous equations for x 1(t), i = 1, 2, ….. N

$$
\begin{pmatrix}
\dot{\mathbf{x}}\_{1} \\
\vdots \\
\dot{\mathbf{x}}\_{N}
\end{pmatrix} = \begin{pmatrix}
\mathbf{A}\_{1}\cdot\mathbf{S}\_{1}\mathbf{K}\_{1} & \cdots & \mathbf{G}\_{1N} & \mathbf{S}\_{1}\mathbf{K}\_{1}\cdots & 0 \\
\vdots & & \ddots & \\
\mathbf{G}\_{N\_{1}} & & \mathbf{A}\_{N}\cdot\mathbf{S}\_{N}\mathbf{I}\_{N} & 0 & \mathbf{S}\_{N}\mathbf{K}\_{N}
\end{pmatrix} \begin{pmatrix}
\mathbf{x}\_{1} \\
\vdots \\
\mathbf{x}\_{N}
\end{pmatrix} + \begin{pmatrix}
\mathbf{B}\_{1}\mathbf{v}\_{1} \\
\vdots \\
\mathbf{B}\_{N}\mathbf{v}\_{N}
\end{pmatrix} \tag{19}
$$

ẋ N GN1 AN - SNIN 0 SNKN x N BNvN

*Step 5:* 

Generate the input-output pairs {vi, ŷi = ci ˆ i}.

#### *2.2.3. Interconnected system*

Assume the following interconnected system of order 10 [17]:

11 12 13 14 15 21 22 23 24 25 31 32 33 34 35 41 42 43 44 45 51 52 53 54 55 11 12 13 14 15 -1.5 -0.3 -0.25 0.1 0.5 r1 r1 r1 r1 r1 0.1 0 0 0.2 0 r1 r1 r1 r1 r1 0 0.2 1 0 0.4 r1 r1 r1 r1 r1 0.6 0.1 0.25 2 0 r1 r1 r1 r1 r1 0.4 0.2 1 0.5 0.1 r1 r1 r1 r1 r1 A= r2 r2 r2 r2 r2 1.5 0.3 0. 21 22 23 24 25 31 32 33 34 35 41 42 43 44 45 51 52 53 54 55 25 0.1 0.5 r2 r2 r2 r2 r2 0.1 0 0 0.2 0 r2 r2 r2 r2 r2 0 0.2 1 0 0.4 r2 r2 r2 r2 r2 0.6 0.1 0.25 2 0 r2 r2 r2 r2 r2 0.4 0.2 1 0.5 0.1 (20)

$$\mathbf{B} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \end{pmatrix} \tag{21}$$

$$\mathbf{C} = \begin{pmatrix} 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \end{pmatrix} \tag{22}$$

Which is considered to be composed of two-coupled subsystems; each of order 5. The coupling parameters are r1jk and r2jk where j and k take values of 1,2,3,4 and 5. In the sequel, we refer to the structure of the interconnected system model as:

$$
\dot{\mathbf{x}} = \begin{pmatrix}
\text{A11} & \cdots & \text{G12(r1)} \\
\vdots & \ddots & \\
\text{G21(r2)} & & \text{A22}
\end{pmatrix} \mathbf{x} + \begin{pmatrix}
\text{B1} \\
\vdots \\
\text{B2}
\end{pmatrix} \mathbf{v} \\
\tag{23}
$$

Where G12(r1) and G21(r2) are the coupling mtrices.

396 Fuzzy Controllers – Recent Advances in Theory and Applications

completely observable for i = 1, 2, …….. N.

decentralized estimation and control [17]:

Where Di = CTiCi , Si = Bi R-1i BT

condition ei(0) = x i(0)

*Algorithm 1:* 

*Step 1:* 

*Step 2:* 

*Step 3:* 

*Step 4:* 

*Step 5:* 

Where all vectors and matrices are appropriately defined and gi(.) is the interaction function between the ith subsystem and the rest of the system. It is considered that (Ci, Ai) is

The following algorithm finds the optimal states of a large-scale system based on

<sup>T</sup> Hi(A I ) (A I )H H D H Q 0 i i i i i iii i

<sup>T</sup> Ki(A I ) (A I )K K S K Q 0 i i i i i ii i i

Integrate the following set of N simultaneous equation for ei(t), I = 1, 2 …. N, using the initial

1 1 11 1N 1 1 1

e A -S K G e B v

N N N N N N NN

1 1 11 1N 1 1 1 11

x A -S K G S K 0 x B v

N N N NN N N N N N

x G A -S I 0 S K x B v 

e G A -H D e B v 

 

(16)

(17)

(18)

(19)

 

Read the matrices Ai, Bi and select Qi 0 and Ri 0 as weighted matrix.

Solve the following 2N algebraic Riccati equations for Hi, Ki

1

Integrate the following set of n simultaneous equations for x 1(t), i = 1, 2, ….. N

1

Assume the following interconnected system of order 10 [17]:

ẋ N GN1 AN - SNIN 0 SNKN x N BNvN

*2.2.3. Interconnected system* 

Generate the input-output pairs {vi, ŷi = ci ˆ i}.

For a typical values [4] of r115=-0.1, r124=0.1, r142=0.2, r222=0.1,r242=0.15,r251=0.11 and allothe values of coupling parameters are zeros, we examined the stability of the system by computing the eigenvalues of matrix A. They are {-1.0915, -1.0641, 0.477 + j0.0206, 0.477 – j0.00206, 0.022 + j0.0544, 0.022 – j0.0544, -1.8709 + j0.1713, -1.8709 – j0.1713, -1.9306 + j0.1413, -

1.9306 – j0.1413}, and it is quite clear that there are four eigenvalues lie in the open right half of the complex plane and thus the interconnected system is unstable. Further, it is easy to check that the interconnected system is both controllable and observable.

New Areas in Fuzzy Application 399

Ni+1 at zero.

*2.2.5. Design of an array of fuzzy controller* 

*Step 1:* 

*Step 2:* 

Mi

*Step 3:* 

*Step 4:* 

Ni+2, ---, Mi

we use Ni fuzzy set M1i, ---, Mi

and yb-L of the fuzzy set ki

design the fuzzy controller.

The following 2Ni+1 rules are considered

**Figure 11.** Block diagram of the proposed fuzzy feedback controller array.

of the respective subsystem outputs, where i = 1, 2, 3, 4.

In order to build each fuzzy controller, the following steps are implemented:

The range of the inputs to each fuzzy controller [i, i] are driven from the estimated value

2N+1 fuzzy set MLi in [i, i] that are normal, consistent and complete with triangular membership functions [24], are defined for each controller, where L = 1, 2--, 2Ni+1. That is

2Ni+1 to cover the positive internal (0, ], and the center of fuzzy set Mi

ai i bi i i IF y is M or y is M then u is K

Where L = 1, 2,---, 2Ni+1, and ai, bi are the input to the fuzzy controller i, and the center yaiL

Product inference engine, singleton fuzzyfier, and center average defuzzifier are selected to

L are chosen such that

ai bi i

y and y 0 for1= N +1

 


LL L

0 for1=1, ,N

0 for1= N +2, ,2N +1

i

(24)

i i

Ni to cover the negative internal [I, 0), the other Ni fuzzy sets

We are going to treat the interconnected system at hand as being composed of two identical and coupled subsystems. The control system to be designed is such that each subsystem has its own fuzzy negative feedback controller which its input being the output of the respective subsystem (Fig. 11). Each subsystem fuzzy controller is constructed using two fuzzy systems.

## *2.2.4. Estimation of the system state variables and outputs*

A Matlab program is written to implement the computational Algorithm (1) of section 1.2.2 on the interconnected system. Different positive and negative step input are applied to estimate the outputs. The results of two cases are illustrated in Fig. 9 and Fig. 10 It is observed that the outputs tend to track conveniently the input signals.

**Figure 9.** Simulation Results for case 1.

**Figure 10.** Simulation Results for case 2.

## *2.2.5. Design of an array of fuzzy controller*

We are going to treat the interconnected system at hand as being composed of two identical and coupled subsystems. The control system to be designed is such that each subsystem has its own fuzzy negative feedback controller which its input being the output of the respective subsystem (Fig. 11). Each subsystem fuzzy controller is constructed using two fuzzy systems.

**Figure 11.** Block diagram of the proposed fuzzy feedback controller array.

In order to build each fuzzy controller, the following steps are implemented:

*Step 1:* 

398 Fuzzy Controllers – Recent Advances in Theory and Applications

**Figure 9.** Simulation Results for case 1.

**Figure 10.** Simulation Results for case 2.

1.9306 – j0.1413}, and it is quite clear that there are four eigenvalues lie in the open right half of the complex plane and thus the interconnected system is unstable. Further, it is easy to

A Matlab program is written to implement the computational Algorithm (1) of section 1.2.2 on the interconnected system. Different positive and negative step input are applied to estimate the outputs. The results of two cases are illustrated in Fig. 9 and Fig. 10 It is

check that the interconnected system is both controllable and observable.

observed that the outputs tend to track conveniently the input signals.

*2.2.4. Estimation of the system state variables and outputs* 

The range of the inputs to each fuzzy controller [i, i] are driven from the estimated value of the respective subsystem outputs, where i = 1, 2, 3, 4.

#### *Step 2:*

2N+1 fuzzy set MLi in [i, i] that are normal, consistent and complete with triangular membership functions [24], are defined for each controller, where L = 1, 2--, 2Ni+1. That is we use Ni fuzzy set M1i, ---, Mi Ni to cover the negative internal [I, 0), the other Ni fuzzy sets Mi Ni+2, ---, Mi 2Ni+1 to cover the positive internal (0, ], and the center of fuzzy set Mi Ni+1 at zero.

*Step 3:* 

The following 2Ni+1 rules are considered

$$\text{IF } \mathbf{y}\_{\text{ai}} \text{ is } \mathbf{M}\_{\mathbf{i}}^{\text{L}} \text{ or } \mathbf{y}\_{\text{bi}} \text{ is } \mathbf{M}\_{\mathbf{i}}^{\text{L}} \text{ then } \mathbf{u} \text{ is } \mathbf{K}\_{\mathbf{i}}^{\text{L}}.$$

Where L = 1, 2,---, 2Ni+1, and ai, bi are the input to the fuzzy controller i, and the center yaiL and yb-L of the fuzzy set ki L are chosen such that

$$\mathbf{y\_{ai}^{-\mathrm{L}}} \text{ and } \mathbf{y\_{bi}^{-\mathrm{L}}} \begin{cases} \leq -0 & \text{for} \, \mathbf{1} = \mathbf{1}, \cdots, \mathbf{N\_i} \\ = 0 & \text{for} \, \mathbf{1} = \mathbf{N\_i} + \mathbf{1} \\ \geq 0 & \text{for} \, \mathbf{1} = \mathbf{N\_i} + \mathbf{2}, \cdots, \mathbf{2N\_i} + \mathbf{1} \end{cases} \tag{24}$$

*Step 4:* 

Product inference engine, singleton fuzzyfier, and center average defuzzifier are selected to design the fuzzy controller.

## *2.2.6. Simulation results*

The behaviour of the interconnected system outputs after implementing the fuzzy controllers with unity step function input are shown in Fig 12 and Fig: 13. It is clearly evident that the system becomes asymptotically stable by using the negative fuzzy feedback controller array.

New Areas in Fuzzy Application 401

The following figures illustrate the above test cases:

**Figure 14.** Case 2 Outputs y1 against y2

**Figure 15.** Case 2 Outputs y3 against y4

**Figure 16.** Case 3 Outputs y1 against y2

**Figure 17.** Case 3 Outputs y3 against y4

**Figure 18.** Case 4 Outputs y1 against y2

**Figure 12.** Outputs y1 against y2 y2

**Figure 13.** Outputs y3 against y4 y4

## *2.2.7. Performance of the proposed fuzzy feedback controller array*

Now, we examine the effect of coupling matrices on the performance of fuzzy controlled interconnected system. Five additional cases with deferent coupling ranks are implemented. Fine tuning of membership functions was required to adjust their ranges. The following table summarizes the test cases:


**Table 1.** Results summary for 6 test cases.

The following figures illustrate the above test cases:

400 Fuzzy Controllers – Recent Advances in Theory and Applications

The behaviour of the interconnected system outputs after implementing the fuzzy controllers with unity step function input are shown in Fig 12 and Fig: 13. It is clearly evident that the system becomes asymptotically stable by using the negative fuzzy feedback

*2.2.6. Simulation results* 

**Figure 12.** Outputs y1 against y2 y2

**Figure 13.** Outputs y3 against y4 y4

table summarizes the test cases:

**2Norm**

**Table 1.** Results summary for 6 test cases.

**Case No. A11,A2**

*2.2.7. Performance of the proposed fuzzy feedback controller array* 

**G12 Norm** 

**G12 Sparsty**

Now, we examine the effect of coupling matrices on the performance of fuzzy controlled interconnected system. Five additional cases with deferent coupling ranks are implemented. Fine tuning of membership functions was required to adjust their ranges. The following

> **G21 Sparsty**

1 (Fig 12, 13) 2.2529 3/25 0.2 3/25 1.8028 Unstable Stable 2 (Fig. 14, 15) 2.2529 12/25 0.4712 3/25 0.1803 Unstable Stable 3 (Fig. 16, 17) 2.2529 3/25 .2 12/25 0.5341 Unstable Stable 4(Fig. 18, 19) 2.2529 1 3.0361 3/25 0.1803 Unstable Stable 5 (Fig. 20, 21) 2.2529 3/25 .2 1 3.0364 Unstable Stable 6 (Fig.22, 23) 2.2529 1 3.0361 1 3.0417 Unstable Stable

**G21 Norm** 

**System Stability without controller** 

**System Stability with controller** 

controller array.

**Figure 15.** Case 2 Outputs y3 against y4

**Figure 16.** Case 3 Outputs y1 against y2

**Figure 18.** Case 4 Outputs y1 against y2

New Areas in Fuzzy Application 403

This section has developed a new fuzzy control design approach to interconnected system. It has been shown the approach consists of two stages: In stage 1, a group of local state estimator has been constructed to generate the input-output database. Then an array of feedback controllers has been designed and implemented to guarantee the overall asymptotically system stability. Extensive simulation studies have been performed to

This section presents the use of fuzzy logic technique to control the reactive power of a load and hence improve the power factor. A shunt compensator is used, which consists of a reactor in series with a phase controlled Thyristor bridge in parallel with a capacitor. The control composed of two independent fuzzy controllers, the Fuzzy Grouse Controller (FGC) and the Fuzzy Fine Controller (FFC). These fuzzy controllers are used to control the firing angle of the Thyristor Bridge until the source power factor reaches a desired value. Simulations for three different practical study cases are presented and the results show how

Power factor, nowadays, is an important issue. The over increasing utilization of power electronics in all kinds of industry applications and the severe standards requirements are pushing the research toward new solutions to keep the industrial power factor within

A mathematical formulation for the optimal reactive power control is discussed in [27]. Also, the optimized Fuzzy logic and digital PID controllers for a single phase power factor correction converter used in online UPS are demonstrated in [28, 29]. Parameters such as input membership functions, output membership functions, inference rules of fuzzy logic controller and proportional gain, integral gain and derivative PID controller are selected and optimized by genetic algorithms. In additional to that, the applications of a hybrid converter are implemented in [30]. The hybrid converter is basically a converter bridge with two GTOs. A control strategy based on learning is proposed. The learning structure is coded into Fuzzy conditional rules to train a neural network in manipulating the converter variables.

The common method of correction is by means of using static capacitors, whether connected in series or parallel. These are installed as a single unit or as a bank, to regulate the voltage and the reactive power flow at the point of connection. In shunt compensation arrangement, a reactor is connected in parallel with conventional capacitor compensation. The shunt reactor current can be varied via a phase-controlled thyristor bridge connected in series with the shunt reactor [31]. Changing the thyristor firing angle varies the amount. of the current flowing through the reactor. Thus, this thyristor controlled reactor acts as a variable reactor

*2.2.8. Discussion* 

*2.3.1. Background* 

certain ranges [26].

support the developed design approach.

the designed controller is fast and accurate.

**2.3. Power factor correction** 

**Figure 19.** Case 4 Outputs y3 against y4

**Figure 20.** Case 5 Outputs y1 against y2

**Figure 21.** Case 5 Outputs y3 against y4

**Figure 22.** ase 6 Outputs y1 against y2

**Figure 23.** Case 6 Outputs y3 against y4

## *2.2.8. Discussion*

402 Fuzzy Controllers – Recent Advances in Theory and Applications

**Figure 19.** Case 4 Outputs y3 against y4

**Figure 20.** Case 5 Outputs y1 against y2

**Figure 21.** Case 5 Outputs y3 against y4

**Figure 22.** ase 6 Outputs y1 against y2

**Figure 23.** Case 6 Outputs y3 against y4

This section has developed a new fuzzy control design approach to interconnected system. It has been shown the approach consists of two stages: In stage 1, a group of local state estimator has been constructed to generate the input-output database. Then an array of feedback controllers has been designed and implemented to guarantee the overall asymptotically system stability. Extensive simulation studies have been performed to support the developed design approach.

## **2.3. Power factor correction**

This section presents the use of fuzzy logic technique to control the reactive power of a load and hence improve the power factor. A shunt compensator is used, which consists of a reactor in series with a phase controlled Thyristor bridge in parallel with a capacitor. The control composed of two independent fuzzy controllers, the Fuzzy Grouse Controller (FGC) and the Fuzzy Fine Controller (FFC). These fuzzy controllers are used to control the firing angle of the Thyristor Bridge until the source power factor reaches a desired value. Simulations for three different practical study cases are presented and the results show how the designed controller is fast and accurate.

### *2.3.1. Background*

Power factor, nowadays, is an important issue. The over increasing utilization of power electronics in all kinds of industry applications and the severe standards requirements are pushing the research toward new solutions to keep the industrial power factor within certain ranges [26].

A mathematical formulation for the optimal reactive power control is discussed in [27]. Also, the optimized Fuzzy logic and digital PID controllers for a single phase power factor correction converter used in online UPS are demonstrated in [28, 29]. Parameters such as input membership functions, output membership functions, inference rules of fuzzy logic controller and proportional gain, integral gain and derivative PID controller are selected and optimized by genetic algorithms. In additional to that, the applications of a hybrid converter are implemented in [30]. The hybrid converter is basically a converter bridge with two GTOs. A control strategy based on learning is proposed. The learning structure is coded into Fuzzy conditional rules to train a neural network in manipulating the converter variables.

The common method of correction is by means of using static capacitors, whether connected in series or parallel. These are installed as a single unit or as a bank, to regulate the voltage and the reactive power flow at the point of connection. In shunt compensation arrangement, a reactor is connected in parallel with conventional capacitor compensation. The shunt reactor current can be varied via a phase-controlled thyristor bridge connected in series with the shunt reactor [31]. Changing the thyristor firing angle varies the amount. of the current flowing through the reactor. Thus, this thyristor controlled reactor acts as a variable reactor [32]. By varying the firing angle, the total reactive power of the system can be controlled and hence the power factor of the system is improved. The use of fuzzy logic to derive a practical control scheme for a boost rectifier with reactive power factor correction was applied [33]. The control action is primarily derived from a set of linguistic rules used to generate a slow-varying DC signal to determine the PWM ramp function. The proposed technique uses lesser sensing elements than the classical rectifier. A new FACTS controller known as the Bootstrap variable inductance can emulate a variable positive and negative inductance [34]. The bootstrap variable inductance has a variety of FACTS applications such as series compensation of lines, fault current limiting, reactive-power control and load power factor improvement.

New Areas in Fuzzy Application 405

**Figure 24.** A detailed block diagram of the load, the fuzzy controllers, the source and the control

The following procedures describe the steps of designing the power factor controller:

The sizes of the 'Inductance' and the capacitance bank are selected such that their maximum available reactive power (in VAR) is equal to the maximum load reactive power (MLQ). Since the full source voltage is continuously applied on the capacitance bank (assuming the voltage drop across the short cable is negligible), then capacitance value 'C' of bank can be

> 2 source MLQ Farad

However, this is not the case for the inductance since it is connected in series with a full wave controller. The existence of such controller will limit the available maximum reactive consumed by the inductance depending on the firing angle action of the thyristor bridge. Thus, the effect of the single-phase full-wave circuit is considered when determining the size

As listed in [35], the general formulas for the r.m.s value of current (*Iload*) and voltage (*Vload*) across a load, comprises of inductance in series with resistance, controlled by single-phase

r( )(<sup>α</sup> / <sup>ω</sup> t) 1/2 <sup>l</sup> <sup>2</sup> <sup>1</sup> <sup>I</sup> [ {sin( ) sin( ) } ] *source*

2 V <sup>1</sup> <sup>V</sup> [ {sin(<sup>ω</sup> <sup>t</sup> <sup>θ</sup>) sin(α θ)e }d<sup>ω</sup> t] <sup>z</sup> <sup>π</sup>

 

*t e dt*

(26)

(27)

 

<sup>β</sup> r( )(<sup>α</sup> / <sup>ω</sup> t) source 1/2 <sup>l</sup>

*<sup>f</sup>* (25)

2 V

Where Vsource is "source" r.m.s. voltage in Volt and *f* is "source" frequency in Hz.

*C*

full-wave circuit. These formulas are given as follows:

π

*V*

*z*

scheme

*2.3.2.1. Elements sizing* 

determined as follows:

of the inductance.

*Load*

Load

Here in this section, we shall focuses on the use of fuzzy logic sets to control the supply power factor. Unlike the conventional capacitive approach for power factor improvement in ac power system, the proposed control scheme has the advantage to avoid complexities associated with the non-linear mathematical modelling of switching converters. The proposed fuzzy logic controlling scheme consists of two controllers. The first controller (FGC) is designed to give the nearest desired value of the firing angle required to compensate for the source reactive power. However, the output correction of this controller is not efficiently accurate and hence, another correction step is needed. Thus, the second controller (FFC) checks the value of the source power factor and improves it above a pre-set desired value. The discussion includes the following:


## *2.3.2. Fuzzy power factor controller*

Figure 24 illustrates the block diagram for a single-phase variable load, with variable lag power factor, supplied by sinusoidal AC power source. Capacitor bank in parallel with inductance, controlled by single-phase full-wave circuit, are connected in parallel to the load in order to govern the total reactive power of the circuit. Fuzzy controller is designed to tune the firing angle of the single-phase full-wave circuit in order to adjust the voltage applied across the Inductance. By this way, the total source reactive power can be minimized to improve the source power factor. For three-phase circuits, one controller is dedicated for each phase. Here, we considered single-phase circuit for simplicity.

The structure of the controller contains two independent fuzzy controllers: Fuzzy Grouse Controller (FGC) and Fuzzy Fine Controller (FFC). FGC input is the load reactive power, the output of this controller gives the nearest value of the desired firing angle, which required to minimize the source power factor. FFC input is the source power factor. The output of FFC corrects the firing angle of the single-phase full-wave controller until the source power factor reaches or exceeds the pre-setted desired value.

**Figure 24.** A detailed block diagram of the load, the fuzzy controllers, the source and the control scheme

The following procedures describe the steps of designing the power factor controller:

#### *2.3.2.1. Elements sizing*

404 Fuzzy Controllers – Recent Advances in Theory and Applications

desired value. The discussion includes the following:

Description of the design steps of the power factor controller.

each phase. Here, we considered single-phase circuit for simplicity.

factor reaches or exceeds the pre-setted desired value.

Simulation of the proposed technique by testing it for three different study cases.

Figure 24 illustrates the block diagram for a single-phase variable load, with variable lag power factor, supplied by sinusoidal AC power source. Capacitor bank in parallel with inductance, controlled by single-phase full-wave circuit, are connected in parallel to the load in order to govern the total reactive power of the circuit. Fuzzy controller is designed to tune the firing angle of the single-phase full-wave circuit in order to adjust the voltage applied across the Inductance. By this way, the total source reactive power can be minimized to improve the source power factor. For three-phase circuits, one controller is dedicated for

The structure of the controller contains two independent fuzzy controllers: Fuzzy Grouse Controller (FGC) and Fuzzy Fine Controller (FFC). FGC input is the load reactive power, the output of this controller gives the nearest value of the desired firing angle, which required to minimize the source power factor. FFC input is the source power factor. The output of FFC corrects the firing angle of the single-phase full-wave controller until the source power

Illustration of the proposed control scheme.

*2.3.2. Fuzzy power factor controller* 

power factor improvement.

[32]. By varying the firing angle, the total reactive power of the system can be controlled and hence the power factor of the system is improved. The use of fuzzy logic to derive a practical control scheme for a boost rectifier with reactive power factor correction was applied [33]. The control action is primarily derived from a set of linguistic rules used to generate a slow-varying DC signal to determine the PWM ramp function. The proposed technique uses lesser sensing elements than the classical rectifier. A new FACTS controller known as the Bootstrap variable inductance can emulate a variable positive and negative inductance [34]. The bootstrap variable inductance has a variety of FACTS applications such as series compensation of lines, fault current limiting, reactive-power control and load

Here in this section, we shall focuses on the use of fuzzy logic sets to control the supply power factor. Unlike the conventional capacitive approach for power factor improvement in ac power system, the proposed control scheme has the advantage to avoid complexities associated with the non-linear mathematical modelling of switching converters. The proposed fuzzy logic controlling scheme consists of two controllers. The first controller (FGC) is designed to give the nearest desired value of the firing angle required to compensate for the source reactive power. However, the output correction of this controller is not efficiently accurate and hence, another correction step is needed. Thus, the second controller (FFC) checks the value of the source power factor and improves it above a pre-set

The sizes of the 'Inductance' and the capacitance bank are selected such that their maximum available reactive power (in VAR) is equal to the maximum load reactive power (MLQ). Since the full source voltage is continuously applied on the capacitance bank (assuming the voltage drop across the short cable is negligible), then capacitance value 'C' of bank can be determined as follows:

$$C = \frac{\text{MLQ}}{2 \times \pi \times f \times \text{V}\_{\text{source}}^2} \text{ Farad} \tag{25}$$

Where Vsource is "source" r.m.s. voltage in Volt and *f* is "source" frequency in Hz.

However, this is not the case for the inductance since it is connected in series with a full wave controller. The existence of such controller will limit the available maximum reactive consumed by the inductance depending on the firing angle action of the thyristor bridge. Thus, the effect of the single-phase full-wave circuit is considered when determining the size of the inductance.

As listed in [35], the general formulas for the r.m.s value of current (*Iload*) and voltage (*Vload*) across a load, comprises of inductance in series with resistance, controlled by single-phase full-wave circuit. These formulas are given as follows:

$$\mathbf{I}\_{Load} = \frac{\sqrt{2 \times} V\_{source}}{z} \mathbf{\frac{1}{\pi}} \Big| \frac{1}{\pi} \Big| \{ \sin(\alpha \, t - \theta) - \sin(\alpha - \theta)e^{\binom{N}{1}(\alpha \,/\omega - \mathbf{t})} \} \text{dof } t \Big|^{1/2} \tag{26}$$

$$\mathbf{V}\_{\text{Load}} = \frac{\sqrt{2 \times} V\_{\text{source}}}{\mathbf{z}} \mathbf{l} \frac{1}{\pi} \Big| \frac{\beta}{\pi} \Big| \{ \sin(\omega \cdot \mathbf{t} - \Theta) - \sin(\alpha - \Theta) \mathbf{e}^{\int\_{\gamma}^{\zeta} (\alpha \cdot (\omega - \mathbf{t})}) d\omega \, \mathbf{t} \}^{1/2} \tag{27}$$

Where:

 =2f radian/second = Firing angle = Extinction angle (cut-off angle) = tan-1 (l/r) l = Load Inductance r = Load Resistance t = Time z = Load impedance

Since the conducting angle = - cannot exceed, the firing angle may not be less than and the control range of the firing angle is:

$$
\pi \succeq \alpha \ge \theta \tag{28}
$$

New Areas in Fuzzy Application 407

Select an initial value for the inductance as L = 1 *(*2*C)* 

source VAR will determine the desired value of the inductance.

 If the source VAR equals zero*,* then stop the trials. If not, increase slightly the value of the inductance until the source VAR reaches the zero value. Then, this value of the

After using the Simulink model shown in Figure 25, the inductance value is obtained from the iteration. Then, the same model is used again to find the ratio of (QL/MLQ) for a range of () starting from /2 to degree where (QL) is the Inductance reactive power at certain value of (). This ratio (QL/MLQ) verses the firing angle () is then plotted. Typical curve is shown

After that, the resultant nonlinear curve is divided to N sections where each nonlinear section is approximated by the nearest linear section. Thus, N Gross Fuzzy controllers (FGC) are built, one FGC for each section, such that the input of each controller is equal to 1- (QLoad/(MLQ), where (QLoad) is the load reactive power. The output of the FGC is the firing angle (), which results in (QL) approximately equal to MLQ minus QLoad. The accuracy of

In order to design each FGC, let the full range of the input fuzzy membership function FGCMF(in)a for each controller is set to the (QL/MLQ) limits of the respective linearzed section, and the range of the output fuzzy membership function FGCMF(out)b for each controller is set to the () limits of the respective linearzed section. For example, in Case 3 (as will be explained later into details), the first controller is designed to take an action in case of (QL/MLQ) ratio reaches a value between rl=0.95 and 1and, the output firing angle of this controller shall take a value between 90 degree and 1 = 100 degree. However, the second controller is designed to take an action in case of (QL/MLQ) ratio reaches a value between r2 = 0.175 and rl = 0.950 and, the output firing angle of this controller shall take a

The resultant output of each fuzzy controller can be obtained based on the respective

linearzed section using the following functions for fuzzy implication process:

Run the model and record the source VAR.

*2.3.2.2. Fuzzy Grouse Controller (FGC) design* 

**Figure 26.** Typical curve for (QL/MLQ) verses firing angle ()

the resultant (QL) depends on the curve linearization.

value between 1 = 100 degree and 2 = 105degree, and so on.

Triangle type for Membership functions.

Mamdany engine.

 Product for And. Max for Or.

in Figure 26.

For maximum available reactive power consumed by pure inductance MARP where r is ignored in the above equation 24 & 25), the maximum conducting angle is considered. Therefore, the value of the inductance can be calculated from the following equation assuming that =/2 and = (neglecting the impedance of the short Cable and assuming ideal thyristors):

$$\text{MARP} = \left(V\_{load}\right) \times \left(I\_{load}\right) \tag{29}$$

This MARP value must be equal to the maximum load reactive power MLQ for complete compensation. However, the two equations listed above are nonlinear and difficult to solve. Another simple procedure is needed when determining the size of the inductance L.

**Figure 25.** Simulink circuit used to determine and plot the ratio (QL/MLQ) verses firing angle ()in the range from /2 to degree.

Thus, the value of the inductance can be found by a practical and fast method using Simulink tool [26]. Figure 25, illustrates the proposed model that been used. The model consists of an ac power source connected to a reactive power compensator controlled by a thyristor circuit. The procedure to find the value of the inductance L can be summarized by the following steps:


Since the conducting angle = - cannot exceed, the firing angle may not be less than

For maximum available reactive power consumed by pure inductance MARP where r is ignored in the above equation 24 & 25), the maximum conducting angle is considered. Therefore, the value of the inductance can be calculated from the following equation assuming that =/2 and = (neglecting the impedance of the short Cable and assuming

This MARP value must be equal to the maximum load reactive power MLQ for complete compensation. However, the two equations listed above are nonlinear and difficult to solve.

**Figure 25.** Simulink circuit used to determine and plot the ratio (QL/MLQ) verses firing angle ()in the

Thus, the value of the inductance can be found by a practical and fast method using Simulink tool [26]. Figure 25, illustrates the proposed model that been used. The model consists of an ac power source connected to a reactive power compensator controlled by a thyristor circuit. The procedure to find the value of the inductance L can be summarized by

Another simple procedure is needed when determining the size of the inductance L.

παθ (28)

MARP *V I load load* (29)

Where:

= tan-1 (l/r)

t = Time

 =2f radian/second = Firing angle

l = Load Inductance r = Load Resistance

z = Load impedance

ideal thyristors):

range from /2 to degree.

the following steps:

= Extinction angle (cut-off angle)

and the control range of the firing angle is:

 If the source VAR equals zero*,* then stop the trials. If not, increase slightly the value of the inductance until the source VAR reaches the zero value. Then, this value of the source VAR will determine the desired value of the inductance.

#### *2.3.2.2. Fuzzy Grouse Controller (FGC) design*

After using the Simulink model shown in Figure 25, the inductance value is obtained from the iteration. Then, the same model is used again to find the ratio of (QL/MLQ) for a range of () starting from /2 to degree where (QL) is the Inductance reactive power at certain value of (). This ratio (QL/MLQ) verses the firing angle () is then plotted. Typical curve is shown in Figure 26.

**Figure 26.** Typical curve for (QL/MLQ) verses firing angle ()

After that, the resultant nonlinear curve is divided to N sections where each nonlinear section is approximated by the nearest linear section. Thus, N Gross Fuzzy controllers (FGC) are built, one FGC for each section, such that the input of each controller is equal to 1- (QLoad/(MLQ), where (QLoad) is the load reactive power. The output of the FGC is the firing angle (), which results in (QL) approximately equal to MLQ minus QLoad. The accuracy of the resultant (QL) depends on the curve linearization.

In order to design each FGC, let the full range of the input fuzzy membership function FGCMF(in)a for each controller is set to the (QL/MLQ) limits of the respective linearzed section, and the range of the output fuzzy membership function FGCMF(out)b for each controller is set to the () limits of the respective linearzed section. For example, in Case 3 (as will be explained later into details), the first controller is designed to take an action in case of (QL/MLQ) ratio reaches a value between rl=0.95 and 1and, the output firing angle of this controller shall take a value between 90 degree and 1 = 100 degree. However, the second controller is designed to take an action in case of (QL/MLQ) ratio reaches a value between r2 = 0.175 and rl = 0.950 and, the output firing angle of this controller shall take a value between 1 = 100 degree and 2 = 105degree, and so on.

The resultant output of each fuzzy controller can be obtained based on the respective linearzed section using the following functions for fuzzy implication process:


where,

a = 1,2, ... F (fuzzy membership function number) b = F-a+l(fuzzy membership function number) a and b are fuzzy membership function numbers F is the number of the fuzzy membership function.

Simulink automatic switching system SS 1 as shown in Figure 24 is designed to check the active range of (QL/MLQ) in order to select the proper FGC based on that value of (QL/MLQ) ratio.

New Areas in Fuzzy Application 409

*2.3.2.4. Discrete control signal design* 

controlled.

*2.3.3. Case study* 

Simulink Sum Block is used to add the output of GFC to the output of FFC. Sampler system shown in Figure 24 is designed to convert the resultant analog control signal to discrete signal. The sampling time is selected to be greater than the system time constant. The discrete signal is connected to the input for the synchronized pulse generator to control the thyristor firing angle. Accordingly, the network var and the source power factor are

The Simulink circuit shown in Figure 24 is used as a base to study three 'test' cases. These cases are assigned to check the capability of the controller to operate within a considerable

 **Case 1 Case 2 Case 3 Source Voltage (Volt) 120 480 4160** 

Q1(kVAR) 3.197 142.857 142.857

Load Stages Q2 (kVAR) 12.789 571.429 428.571

F (Number of fuzzy membership functions) 7 7 7

**Table 2.** Circuit data and parameters for the test cases

Stage 1 P1 (kW) 2 40 40

Stage 2 P2 (kW) 12 200 240

Stage 3 P3(kW) 32 1200 1240 Q3 (kVAR) 22.38 1000 1000 Stage 4 P4 (kW) 42 2240 2280 Q4 (kVAR) 0 0 0 MLQ (MVAR) 0.02238 1 1

k1 1 1 1 k2 2.8 0.95 0.5 k3 0.44 1 2.3 k4 5.7 4.5 5.3 r1 0.64 0.52 0.95 r2 0.28 0.21 0.2 r3 0.03 0.02 0.04

ALPHA1 (DEGREE) 99 102 100 ALPHA2 (DEGREE) 124 113.5 105 ALPHA3 (DEGREE) 128 125.5 128 N (Number of sections) 4 4 4

variation of power factor values at different loading and voltage level.

## *2.3.2.3. Fuzzy Fine Controller (FFC) design*

Since the FGC output is not accurate due to the linearization process, FFC is designed to tune the firing g angle in order to achieve the desired power factor.

Two fuzzy control1ers are used along with automatic switching system to select the proper controller based on the power factor type (Lead or Lag), which is determined from the source VAR sign (+ or -) as shown in Figure 24.

The input of each FFC controller is the source power factor (PF) and the output is the corrective firing angle (M), which is required to fine tune the FGC output. Since the power factor value varies between zero and one, then the range of each FPC input membership function FFCMF(in)a is [0, 1].

The output membership function of each FFC is FFCMF(out-Lag)a for lagging input and FFCMF(out-Lead)a for leading input and, the magnitude of the firing angle range for the first linearzed section (1) is considered as a base to scale the FFC output as given hereinafter with the fact that any other section can be selected as the base. Also, [0, ∆1] and [-∆1 , 0] are assigned to FFCMF(out-Lag)b and FFCMF(out-Lead)b respectively. In addition to that, N multiplier factors (kN = ∆N*,/* 1*)* are used to scale the FFC output in order to match the magnitude of the firing angle range of the respective linearzed section (∆N). By this method, less number of FFC's are used.

Another Simulink automatic switching system SS2 as shown in Figure 24 synchronized with SS 1 is designed to check the active range of (QL/MLQ) in order to select the proper multiplier factor (KN). The fuzzy implication process functions used for FGC design are used for each FFC design as well, but with the following fuzzy rules:

$$\text{IF (PF)} \text{is } \text{FFCMF (in)}^{\text{a}} \text{THEN}(a\_{\Lambda}) \text{ is } \text{FFCMF (out } \text{Lag} \text{)}^{\text{a}} \tag{30}$$

$$\text{IF (PF)} \text{is } \text{FFCMF (in)}^{\text{a}} \text{THEN}(a\_{\Lambda}) \text{ is } \text{FFCMF (out } \text{Lead} \text{)}^{\text{a}} \tag{31}$$

### *2.3.2.4. Discrete control signal design*

Simulink Sum Block is used to add the output of GFC to the output of FFC. Sampler system shown in Figure 24 is designed to convert the resultant analog control signal to discrete signal. The sampling time is selected to be greater than the system time constant. The discrete signal is connected to the input for the synchronized pulse generator to control the thyristor firing angle. Accordingly, the network var and the source power factor are controlled.

## *2.3.3. Case study*

408 Fuzzy Controllers – Recent Advances in Theory and Applications

 Largest of Maximum for Defuzzification. Fuzzy rule : IF is MF(in)a THEN () is MF(out)b

a = 1,2, ... F (fuzzy membership function number) b = F-a+l(fuzzy membership function number) a and b are fuzzy membership function numbers F is the number of the fuzzy membership function.

*2.3.2.3. Fuzzy Fine Controller (FFC) design* 

source VAR sign (+ or -) as shown in Figure 24.

function FFCMF(in)a is [0, 1].

method, less number of FFC's are used.

for each FFC design as well, but with the following fuzzy rules:

Simulink automatic switching system SS 1 as shown in Figure 24 is designed to check the active range of (QL/MLQ) in order to select the proper FGC based on that value of (QL/MLQ)

Since the FGC output is not accurate due to the linearization process, FFC is designed to

Two fuzzy control1ers are used along with automatic switching system to select the proper controller based on the power factor type (Lead or Lag), which is determined from the

The input of each FFC controller is the source power factor (PF) and the output is the corrective firing angle (M), which is required to fine tune the FGC output. Since the power factor value varies between zero and one, then the range of each FPC input membership

The output membership function of each FFC is FFCMF(out-Lag)a for lagging input and FFCMF(out-Lead)a for leading input and, the magnitude of the firing angle range for the first linearzed section (1) is considered as a base to scale the FFC output as given hereinafter with the fact that any other section can be selected as the base. Also, [0, ∆1] and [-∆1 , 0] are assigned to FFCMF(out-Lag)b and FFCMF(out-Lead)b respectively. In addition to that, N multiplier factors (kN = ∆N*,/* 1*)* are used to scale the FFC output in order to match the magnitude of the firing angle range of the respective linearzed section (∆N). By this

Another Simulink automatic switching system SS2 as shown in Figure 24 synchronized with SS 1 is designed to check the active range of (QL/MLQ) in order to select the proper multiplier factor (KN). The fuzzy implication process functions used for FGC design are used

> a a IF PF is FFCMF in THEN( ) is FFCMF out Lag

 a a IF PF is FFCMF in THEN( ) is FFCMF out Lead 

(30)

(31)

tune the firing g angle in order to achieve the desired power factor.

Proportional for Aggregation.

where,

ratio.

The Simulink circuit shown in Figure 24 is used as a base to study three 'test' cases. These cases are assigned to check the capability of the controller to operate within a considerable variation of power factor values at different loading and voltage level.


**Table 2.** Circuit data and parameters for the test cases

## *2.3.4. Test cases data*

Each test case consists of four load stages. The load stages are selected such that the power factor varies from 0.3 to 1.0 lag. However, for practical cases, the power factor varies between 0.6 to 0.8 and thus the proposed wide range of power factor tested here is to demonstrate the capability of the designed controller. Table 2 summaries the circuit parameters for three voltage levels 120. 480 and 4160 Volts respectively. Figures 27-29 illustrate the linearization process for each case.

New Areas in Fuzzy Application 411

**Figure 29.** Linearization results for case 3

response time of the controller and its accuracy.

The controller is adjusted to correct the power factor of the test cases to a value greater than a desired value of 0.97. This value is the pre-set value and it can be any chosen practical value. Figures 30 -32 illustrate the results for cases 1-3 respectively. These figures show the variation of the load active and reactive power with respect to time for each case. The response of the controller during the test period represented by the firing angle is also shown. In addition to that, source and load power factor values are plotted to check the

*2.3.5. Results* 

**Figure 27.** Linearization results for case1.

**Figure 28.** Linearization results for case 2

**Figure 29.** Linearization results for case 3

#### *2.3.5. Results*

410 Fuzzy Controllers – Recent Advances in Theory and Applications

illustrate the linearization process for each case.

**Figure 27.** Linearization results for case1.

**Figure 28.** Linearization results for case 2

Each test case consists of four load stages. The load stages are selected such that the power factor varies from 0.3 to 1.0 lag. However, for practical cases, the power factor varies between 0.6 to 0.8 and thus the proposed wide range of power factor tested here is to demonstrate the capability of the designed controller. Table 2 summaries the circuit parameters for three voltage levels 120. 480 and 4160 Volts respectively. Figures 27-29

*2.3.4. Test cases data* 

The controller is adjusted to correct the power factor of the test cases to a value greater than a desired value of 0.97. This value is the pre-set value and it can be any chosen practical value. Figures 30 -32 illustrate the results for cases 1-3 respectively. These figures show the variation of the load active and reactive power with respect to time for each case. The response of the controller during the test period represented by the firing angle is also shown. In addition to that, source and load power factor values are plotted to check the response time of the controller and its accuracy.

New Areas in Fuzzy Application 413

Seconds

**Figure 31.** Results of test case 2.

**Figure 30.** Results of test case 1.

**Figure 31.** Results of test case 2.

**Figure 30.** Results of test case 1.

Seconds

New Areas in Fuzzy Application 415

The test results for the three cases show clearly how efficient is the controller. Even when the load reactive power is very small at both high and low power factor, the controller was successful in reaching an accurate level. During the stage where the load power factor is greater than 0.97 and, hence no need for capacitor compensation, the controller will check the source power factor at the beginning of that stage and if it drops below 0. 97, it will take an action in order to eliminate the compensation added in the previous stage. That is why the controller took an action as shown in Figure 30 for the fourth load stage of case no. 1 where the source power factor drops below 0.97. However, if the source power factor stays above the pre-set power factor value of 0.97 during the load stage where the load power factor is greater than 0.97, then no action will be taken as shown in Figure 32 for the fourth load stage of case no. 3. The time required for the controller to improve the power factor in all three cases is relatively short compared with practical applications. In real cases, the power factor does remain unchanged for relatively longer time. The maximum time for power factor correction was 0.35 second recorded in test case no.1. Overall, the graphs show that the controller works satisfactory under different load conditions and when there is no

As mentioned before, power factor correction is really an important issue. The designed controller presented in this section shows an efficient, fast and accurate technique in reactive power control. As seen from the overall structure of the controller, it is applicable for lagging power factor loads. Practically, this is almost true but not always where at rare occasions the power factor of the total load is leading not lagging. This will bring the attention towards generalizing the presented controller such that it will work for both cases. Several issues are also need to be considered in the future such as the dynamics of motors. As known that most connected loads are motors which really necessitate testing this controller under these circumstances. From the test results, it was seen that the speed of the controller depends on the system time constant and hence, a time delay is needed to assure that thc dynamics of the motors reach its equilibrium. Other issues related to Thyristors such as the harmonics are a1so need to be taken care by describing a harmonics filter. In addition to that, protection devices such as relays need to be checked during the controller action. Finally, the work presented was based on a single phase and it can be extended for

Most of the more advanced prediction techniques can be subdivided into two separate tasks. In a rst step, the modelling step, the algorithm uses a set of training data to identify a model of a process, from which the training data could have been obtained. In a second step, the simulation step, the algorithm uses the previously identied model to make predictions outside the training data set. The modelling algorithm can either attempt to identify the true structure of the system, from which the training data were obtained, or it can content itself with identifying any process able to explain the training data set. In the former case, we talk

*2.3.6. Discussion* 

need for capacitor compensation.

three phase system.

**3. Trending and prediction** 

**Figure 32.** Results of test case 3.

### *2.3.6. Discussion*

414 Fuzzy Controllers – Recent Advances in Theory and Applications

**Figure 32.** Results of test case 3.

Seconds

The test results for the three cases show clearly how efficient is the controller. Even when the load reactive power is very small at both high and low power factor, the controller was successful in reaching an accurate level. During the stage where the load power factor is greater than 0.97 and, hence no need for capacitor compensation, the controller will check the source power factor at the beginning of that stage and if it drops below 0. 97, it will take an action in order to eliminate the compensation added in the previous stage. That is why the controller took an action as shown in Figure 30 for the fourth load stage of case no. 1 where the source power factor drops below 0.97. However, if the source power factor stays above the pre-set power factor value of 0.97 during the load stage where the load power factor is greater than 0.97, then no action will be taken as shown in Figure 32 for the fourth load stage of case no. 3. The time required for the controller to improve the power factor in all three cases is relatively short compared with practical applications. In real cases, the power factor does remain unchanged for relatively longer time. The maximum time for power factor correction was 0.35 second recorded in test case no.1. Overall, the graphs show that the controller works satisfactory under different load conditions and when there is no need for capacitor compensation.

As mentioned before, power factor correction is really an important issue. The designed controller presented in this section shows an efficient, fast and accurate technique in reactive power control. As seen from the overall structure of the controller, it is applicable for lagging power factor loads. Practically, this is almost true but not always where at rare occasions the power factor of the total load is leading not lagging. This will bring the attention towards generalizing the presented controller such that it will work for both cases. Several issues are also need to be considered in the future such as the dynamics of motors. As known that most connected loads are motors which really necessitate testing this controller under these circumstances. From the test results, it was seen that the speed of the controller depends on the system time constant and hence, a time delay is needed to assure that thc dynamics of the motors reach its equilibrium. Other issues related to Thyristors such as the harmonics are a1so need to be taken care by describing a harmonics filter. In addition to that, protection devices such as relays need to be checked during the controller action. Finally, the work presented was based on a single phase and it can be extended for three phase system.

## **3. Trending and prediction**

Most of the more advanced prediction techniques can be subdivided into two separate tasks. In a rst step, the modelling step, the algorithm uses a set of training data to identify a model of a process, from which the training data could have been obtained. In a second step, the simulation step, the algorithm uses the previously identied model to make predictions outside the training data set. The modelling algorithm can either attempt to identify the true structure of the system, from which the training data were obtained, or it can content itself with identifying any process able to explain the training data set. In the former case, we talk

about a deep model, whereas models in the latter category are referred to as shallow models.

New Areas in Fuzzy Application 417

(34)

A = {x U| 

Plausibility: The point y\* should represent B` from an intuitive point of view.

Continuity: A small Change in B` should not result in a large change in y\*.

represents the fuzzy set B`.

*3.1.1. Fuzzy C-Means Clustering Algorithm(FCM)* 

to fuzzy membership μij, and cluster centroid Vi.

number of data points and f is a fuzziness index (greater than 1)

a. Initialize memberships μij of Xj belonging to cluster i such that

V =

d. Repeat steps 2 and 3 until the value of Jm is no longer decreasing.

μij , but different choices of initial μij might lead to local minima.

The FCM algorithm is executed by the following steps:

b. Compute the fuzzy centroid Vi from i=1 to i=c using

c. Update the fuzzy memberships μij using

Computational simplicity.

where X = [X1,….,Xn]t

*3.1.2. Cluster validity* 

μ<sup>A</sup> x }

c. Defuzzification is defined as a mapping from fuzzy set B` in V R to crisp point y\* V. Conceptually, the task of the defuzzification is to specify a point in V that best

The following three criteria should be considered in choosing the defuzification method:

The fuzzy c-means (FCM) clustering algorithm is the fuzzy equivalent of the nearest hard clustering algorithm [43,44], which minimizes the following objective function with respect

c n f 2

m ij j j i m ij j

(μ ) X

j i ij ( 2/(m 1)) j i j

The FCM always converges to strict local minimum of Jm starting from an initial guess of

The quality of a clustering is indicated by how closely the data points are associated to the cluster centers, and it is the membership functions, which measure the level of association or

(||(X ,V )||) <sup>μ</sup> (||(X ,V )||)

(μ )

( 2/(m 1))

<sup>m</sup> ij j i I j J ( <sup>μ</sup> ) || X ,V || (35)

ij <sup>j</sup> (<sup>μ</sup> ) 1 (36)

(37)

(38)

is a vector representing the data, c is the number of clusters, n is the

The identied model can be either a quantitative or a qualitative model. A quantitative model operates on the measurement data directly, whereas a qualitative model rst discretizes the measurement data, and then reasons about the discrete classes only. Also the model can be either a parametric model or a nonparametric model. A parametric model maps the knowledge contained in the training data set onto a set of model parameters. During the simulation phase, the training data are no longer needed, since the information contained in them is now stored in the parameter values. A non-parametric model only classies the training data during the modelling phase, and refers back to these classied training data during the simulation phase.

Fuzzy logic are used now a day in many application for diagnostic, prediction forecast and understanding the behaviour of very nonlinear systems such as marketing, electrical load forecast, work load analysis, technical analysis etc…

In this chapter Section 2.1, we shall introduce an important algorithm for classifying "clustering" the data based on fuzzy logic. Then two new fuzzy trending and prediction application shall be discussed. In Section 2.2 Accident rates Estimation Modelling Based on Human Factors shall be introduces, and in the next section 2.3, Fault Location in Distribution Networks shall be introduced.

## **3.1. Clustering algorithm and validity criteria**

Clustering attempts to assess the relationships among patterns of the data set by organizing the patterns into groups or clusters such that patterns within a cluster are more similar to each other than are patterns belonging to different clusters. Many algorithms for hard and fuzzy clustering have been developed to accomplish this[42]. An intimately related important issue is the cluster validity, which deals with the significance of the structure imposed by a clustering method [43].

For fuzzy sets, the following definitions are recalled from [36]:


Such that

$$\sum\_{i} \mu\_{i\mathbf{j}} = 1, \forall\_{\mathbf{j}'} 1 \le \mathbf{j} \le \mathbf{n}.\tag{33}$$

An α-cut of fuzzy set A is a crisp set Aα that contains all the elements in U that have membership value in A greater thanα, that is

$$\mathbf{A}\_{\boldsymbol{\alpha}} = \{ \mathbf{x} \in \mathbf{U} \, | \, \mu\_{\mathcal{A}} \left( \mathbf{x} \right) \ge a \} \tag{34}$$

c. Defuzzification is defined as a mapping from fuzzy set B` in V R to crisp point y\* V. Conceptually, the task of the defuzzification is to specify a point in V that best represents the fuzzy set B`.

The following three criteria should be considered in choosing the defuzification method:


416 Fuzzy Controllers – Recent Advances in Theory and Applications

training data during the simulation phase.

Distribution Networks shall be introduced.

imposed by a clustering method [43].

Such that

that takes values in the interval [0,1].

the set Mfc = {U Vcn ij [0,1]} (32)

membership value in A greater thanα, that is

forecast, work load analysis, technical analysis etc…

**3.1. Clustering algorithm and validity criteria** 

For fuzzy sets, the following definitions are recalled from [36]:

models.

about a deep model, whereas models in the latter category are referred to as shallow

The identied model can be either a quantitative or a qualitative model. A quantitative model operates on the measurement data directly, whereas a qualitative model rst discretizes the measurement data, and then reasons about the discrete classes only. Also the model can be either a parametric model or a nonparametric model. A parametric model maps the knowledge contained in the training data set onto a set of model parameters. During the simulation phase, the training data are no longer needed, since the information contained in them is now stored in the parameter values. A non-parametric model only classies the training data during the modelling phase, and refers back to these classied

Fuzzy logic are used now a day in many application for diagnostic, prediction forecast and understanding the behaviour of very nonlinear systems such as marketing, electrical load

In this chapter Section 2.1, we shall introduce an important algorithm for classifying "clustering" the data based on fuzzy logic. Then two new fuzzy trending and prediction application shall be discussed. In Section 2.2 Accident rates Estimation Modelling Based on Human Factors shall be introduces, and in the next section 2.3, Fault Location in

Clustering attempts to assess the relationships among patterns of the data set by organizing the patterns into groups or clusters such that patterns within a cluster are more similar to each other than are patterns belonging to different clusters. Many algorithms for hard and fuzzy clustering have been developed to accomplish this[42]. An intimately related important issue is the cluster validity, which deals with the significance of the structure

a. A fuzzy set in a universe discourse U is characterized by a membership function A(x)

b. Let X={x1,,xn} be any set, Vcn be the set of real cn matrices U=[ij], c, i, j be integer numbers with 2 c n , 1 i c and 1 j n .Then the fuzzy partition matrix for X is

An α-cut of fuzzy set A is a crisp set Aα that contains all the elements in U that have

<sup>μ</sup>ij j 1, ,1 j n. *<sup>i</sup>* (33)

Continuity: A small Change in B` should not result in a large change in y\*.

#### *3.1.1. Fuzzy C-Means Clustering Algorithm(FCM)*

The fuzzy c-means (FCM) clustering algorithm is the fuzzy equivalent of the nearest hard clustering algorithm [43,44], which minimizes the following objective function with respect to fuzzy membership μij, and cluster centroid Vi.

$$J\_{\mathbf{m}} = \sum\_{\mathbf{l}} \, ^c\_{\mathbf{l}} \sum\_{\mathbf{j}} ^n (\mu\_{\mathbf{ij}})^\mathbf{f} \mid \mathbb{I} \left( \mathbf{X}\_{\mathbf{j}'} \mathbf{V}\_{\mathbf{i}} \right) \mid \mathbb{I}^2 \tag{35}$$

where X = [X1,….,Xn]t is a vector representing the data, c is the number of clusters, n is the number of data points and f is a fuzziness index (greater than 1)

The FCM algorithm is executed by the following steps:

a. Initialize memberships μij of Xj belonging to cluster i such that

$$\sum\_{\mathbf{j}} (\mu\_{\mathbf{i}\mathbf{j}}) = 1 \tag{36}$$

b. Compute the fuzzy centroid Vi from i=1 to i=c using

$$\mathbf{V}\_{\mathbf{i}} = \frac{\sum\_{\mathbf{j}} (\mu\_{\tilde{\mathbf{i}}\tilde{\mathbf{j}}})^{\mathbf{m}} \times \mathbf{X}\_{\mathbf{j}}}{\sum\_{\mathbf{j}} (\mu\_{\tilde{\mathbf{i}}\tilde{\mathbf{j}}})^{\mathbf{m}}} \tag{37}$$

c. Update the fuzzy memberships μij using

$$\mu\_{\rm ij} = \frac{\left( \mid \mid (\mathbf{X}\_{\mathbf{j}}, \mathbf{V}\_{\mathbf{i}}) \mid \mid \right)^{\left( -2/(\mathfrak{m}-1) \right)}}{\sum\_{\mathbf{j}} \left( \mid \mid (\mathbf{X}\_{\mathbf{j}}, \mathbf{V}\_{\mathbf{i}}) \mid \mid \right)^{\left( -2/(\mathfrak{m}-1) \right)}} \tag{38}$$

d. Repeat steps 2 and 3 until the value of Jm is no longer decreasing.

The FCM always converges to strict local minimum of Jm starting from an initial guess of μij , but different choices of initial μij might lead to local minima.

#### *3.1.2. Cluster validity*

The quality of a clustering is indicated by how closely the data points are associated to the cluster centers, and it is the membership functions, which measure the level of association or

classification. If the value of one of the membership is significantly larger than the others for a particular data point, then that point is identified as being a part of the subset of the data represented by the corresponding cluster center. But, each data point has c memberships; so, it is desirable to summarize the information contained in the memberships by a single number, which indicates how well the data point is classified by the clustering. This can be done in a variety of ways; for example, for the data point Xj with memberships {μij, ...,μcj}, one could use any of the following:

$$\text{Index}\mathbf{1} = \sum\_{\mathbf{i}} (\mu\_{\mathbf{i}\mathbf{j}})^2 \tag{39}$$

New Areas in Fuzzy Application 419

Policies. Line management is encouraged to conduct regular inspections of the workplace and employees are trained to behave safely and are given the appropriate protective equipment. The impact of such initiatives could be seen in the overall downward trend in accident statistics from 1990 to 1998/99. After 2000, accident statistics started rising in many UK industrial sectors. In the Quarry Industry, for example, there has been a 60% rise in the

Another study by Stanford University [39] indicated that risk taking is often a normal part of human psychology. We sometimes drive too fast or take chances we should not. Risk which is taken on the job site, however, can be fatal. This study found that many workers believe that taking unnecessary risks is an accepted part of the job process. This risk acceptance attitude leads to carelessness and accidents. The results of the Stanford study show that workers who are likely to have lost-time accidents share similar characteristics. These workers have a negative attitude toward doing their jobs safely, and they accept unnecessary risk and, therefore, do not work safely. Taking unnecessary risks and adopting a poor safety attitude simply makes workers more prone to accident occurrences. The conclusion of the Stanford study clearly supports the contention that employee actions and attitudes can affect the number and type of workplace accidents. Employers can and do address this attitude of risk taking through safety education, safety rules, and training programs. However, no employer can supervise each employee every minute of the work

In [40], the paper focuses on the development and representation of linguistic variables to model risk levels subjectively. These variables are then quantified using fuzzy set theory. In this paper the development of two safety evaluation frameworks, using fuzzy logic approaches for maritime engineering safety based decision support in the concept design stage are presented. An example is used to illustrate and compare the proposed approaches. The paper also suggests that future risk analysis in maritime engineering applications may

The field of fuzzy systems has been making rapid progress over the past decade [36]. There

a. The problem is too complicated for precise description to be obtained, therefore approximation, or fuzziness, must be introduced in order to be a reasonable and net

b. As we move into the information era, knowledge is becoming increasingly important and the need for a theory to formulate human knowledge in a systematic manner

But as a general principle, a good engineering theory should be capable of making use of all the available information effectively. For many practical systems, important information comes from two sources: one source is from human experts who describe their knowledge about the system in natural languages; the other is sensory measurements or mathematical

take full advantages of fuzzy logic approaches to complement existing ones.

are two kinds of justification for fuzzy systems to be used to achieve our objective:

number of fatalities.

day.

traceable model.

becomes the norm not the exception.

models that are derived from to physical laws.

$$\text{Index2} = \sum\_{\text{i}} \mu\_{\text{i}\text{j}} \log(\mu\_{\text{i}\text{j}}) \tag{40}$$

$$\text{Index3=max}\_{i}(\mu\_{ij})\tag{41}$$

$$\mathsf{Index}4 = \min\_{\mathsf{i}} \left( \mu\_{\mathsf{i}\mathsf{j}} \right) / \mathsf{max}\_{\mathsf{i}} (\mu\_{\mathsf{i}\mathsf{j}}) \tag{42}$$

In fact, theses four indices of these are used as measure of the quality of clustering and are the basis for the *validity functional*, *partition coefficient*, *classification entropy, and proportion exponent*, respectively.

To illustrate the use of validity functional, we shall focus on the partition coefficient technique because of its simplicity. It is based on using Sj =Σi (μij)2 as a measure of how well the jth data point has been classified. This is a reasonable indicator because the closer a data point is to a cluster center, the closer Sj is to 1, the maximum value it could have. Conversely, the further away the kth point is from all the cluster centers the closer the value of Sj is to 1/c, the minimum possible value. The partition coefficient is then the average over the data set of the Sj's. In particular, for a data set X={x1,…,xi} and a specific choice of c and m one obtains the output of fuzzy c-means and computes the partition coefficient (PC) by PC=Σj(Σi(μij)2/n. The closer this value is to one the better the data are classified. So, in theory, one computes PC for the outputs of a variety of values of c and m selects the best clustering as the one corresponding to the highest partition coefficient [44,45].

## **3.2. Accident rates estimation modeling based on human factors using fuzzy cmean clustering techniques**

Several individual books [37] and projects shed light on worker accident causation. One study on the Bonneville Dam project, reported that seven times the number of work accidents that had occurred on this project were due to unsafe employee actions rather than to unsafe site conditions. In addition, this study found that the negative attitude of the workers toward safety was a major factor in accident occurrence.

In [38]. Many organizations spend a lot of time and effort trying to improve safety. As well as addressing technical and hardware issues, many conduct safety management system audits to discover deviations from the performance standards set in their Health & Safety Policies. Line management is encouraged to conduct regular inspections of the workplace and employees are trained to behave safely and are given the appropriate protective equipment. The impact of such initiatives could be seen in the overall downward trend in accident statistics from 1990 to 1998/99. After 2000, accident statistics started rising in many UK industrial sectors. In the Quarry Industry, for example, there has been a 60% rise in the number of fatalities.

418 Fuzzy Controllers – Recent Advances in Theory and Applications

corresponding to the highest partition coefficient [44,45].

workers toward safety was a major factor in accident occurrence.

one could use any of the following:

*exponent*, respectively.

**mean clustering techniques** 

classification. If the value of one of the membership is significantly larger than the others for a particular data point, then that point is identified as being a part of the subset of the data represented by the corresponding cluster center. But, each data point has c memberships; so, it is desirable to summarize the information contained in the memberships by a single number, which indicates how well the data point is classified by the clustering. This can be done in a variety of ways; for example, for the data point Xj with memberships {μij, ...,μcj},

In fact, theses four indices of these are used as measure of the quality of clustering and are the basis for the *validity functional*, *partition coefficient*, *classification entropy, and proportion* 

To illustrate the use of validity functional, we shall focus on the partition coefficient technique because of its simplicity. It is based on using Sj =Σi (μij)2 as a measure of how well the jth data point has been classified. This is a reasonable indicator because the closer a data point is to a cluster center, the closer Sj is to 1, the maximum value it could have. Conversely, the further away the kth point is from all the cluster centers the closer the value of Sj is to 1/c, the minimum possible value. The partition coefficient is then the average over the data set of the Sj's. In particular, for a data set X={x1,…,xi} and a specific choice of c and m one obtains the output of fuzzy c-means and computes the partition coefficient (PC) by PC=Σj(Σi(μij)2/n. The closer this value is to one the better the data are classified. So, in theory, one computes PC for the outputs of a variety of values of c and m selects the best clustering as the one

**3.2. Accident rates estimation modeling based on human factors using fuzzy c-**

Several individual books [37] and projects shed light on worker accident causation. One study on the Bonneville Dam project, reported that seven times the number of work accidents that had occurred on this project were due to unsafe employee actions rather than to unsafe site conditions. In addition, this study found that the negative attitude of the

In [38]. Many organizations spend a lot of time and effort trying to improve safety. As well as addressing technical and hardware issues, many conduct safety management system audits to discover deviations from the performance standards set in their Health & Safety

2

ij <sup>i</sup> Index1= ( <sup>μ</sup> ) (39)

ij ij <sup>i</sup> Index2 = <sup>μ</sup> log(<sup>μ</sup> ) (40)

i ij Index3= max (μ ) (41)

i ij i ij Index4= min (μ ) / max (μ ) (42)

Another study by Stanford University [39] indicated that risk taking is often a normal part of human psychology. We sometimes drive too fast or take chances we should not. Risk which is taken on the job site, however, can be fatal. This study found that many workers believe that taking unnecessary risks is an accepted part of the job process. This risk acceptance attitude leads to carelessness and accidents. The results of the Stanford study show that workers who are likely to have lost-time accidents share similar characteristics. These workers have a negative attitude toward doing their jobs safely, and they accept unnecessary risk and, therefore, do not work safely. Taking unnecessary risks and adopting a poor safety attitude simply makes workers more prone to accident occurrences. The conclusion of the Stanford study clearly supports the contention that employee actions and attitudes can affect the number and type of workplace accidents. Employers can and do address this attitude of risk taking through safety education, safety rules, and training programs. However, no employer can supervise each employee every minute of the work day.

In [40], the paper focuses on the development and representation of linguistic variables to model risk levels subjectively. These variables are then quantified using fuzzy set theory. In this paper the development of two safety evaluation frameworks, using fuzzy logic approaches for maritime engineering safety based decision support in the concept design stage are presented. An example is used to illustrate and compare the proposed approaches. The paper also suggests that future risk analysis in maritime engineering applications may take full advantages of fuzzy logic approaches to complement existing ones.

The field of fuzzy systems has been making rapid progress over the past decade [36]. There are two kinds of justification for fuzzy systems to be used to achieve our objective:


But as a general principle, a good engineering theory should be capable of making use of all the available information effectively. For many practical systems, important information comes from two sources: one source is from human experts who describe their knowledge about the system in natural languages; the other is sensory measurements or mathematical models that are derived from to physical laws.

An important task, therefore, is to combine these two types of information into system designs. Therefore, the key question is how to transform human knowledge base into a mathematical formula or model. Essentially, what a fuzzy system does is to perform this transformation in a systematic way.

New Areas in Fuzzy Application 421

*3.2.1.1. Questionnaire design* 

same field made the same judgments.

linguistic evaluations.

work.

*3.2.1.2. The response* 

*3.2.1.3. Limitations* 

only to one sex.

In the design of questionnaire (Appendix) we have selected some certain features of human nature that, we believe, have a great potential on the accident causation in the local market [41]. The first page of the questionnaire concentrated on the personal information of the workers: i.e. 'height', 'weight', 'optical status', 'hearing ability', 'general health', 'education' and 'adherence to safety rules'. In this part we used some linguistic evaluations like 'high', 'low', 'fair', 'good', 'medium' etc., and in some others we have used numerical evaluations like in height, weight, as well as education. Since the objective here is to create a fuzzy model, high accuracy is not important and we considered that the respondents from the

The second page was designed to concentrate on the work information: 'overtime work', 'experience', 'work nature', 'work type', 'hazardous level', 'needs for safety-gears', 'work location' and 'level of boredom'. Again, we have used some linguistic evaluations as well as numerical evaluations. We also concentrated on the managerial factors: 'salary received on time', 'level of training' 'importance' and level of importance placed on safety', with only

The third page was a mix of both external factors: 'noise', 'live with family' and 'communication 'language', and accident history focusing on the number of accidents the laborer has faced during his work in the local construction market, which was the most important data that we needed to develop our model. The severity of accidents has not been

The cases obtained for this study were collected from three different construction companies selected to represent the local market. We have tried to select the cases from different ranks of the workforce, from higher levels to the lower levels to be able to study the different accident level cases that serve the purpose as well as adding versatility and diversity to this

From the original cases that we collected on 95 people, we included only 76 cases and excluded 19 cases, which were incomplete. This has produced a very high response rate that reached 82.1%, which is relatively high, especially as the questionnaire is lengthy and a little bit complicated. More cases could have been obtained. However, since the aim of this study was to develop a model for accidents, the number of cases is not set a priority. Rather, the cases are accumulated and the algorithms are stopped when a cluster validity criterion is

Limitations in this study should be noted. One of the limitations is that we did not include any specific information related to the accident consequences. Another limitation is that this study was made only on males and no female cases were studied, which makes it specific

taken into account since it is not considered a factor influencing the accident rate.

satisfied thereby yielding the optimum number of clusters.

In this part of Section 2, we attempt to use a completely different approach to analyze accidents. A model shall be developed for data collected from an accident rate questionnaire filled-in by laborers working for a reputable construction company. This questionnaire was designed to include information about human factors, as well as other factors such as work type, managerial factors, training, physical factors and the historical accident rate for each labor during his period of employment in this particular construction company, and his experience during his career life time. The collected data shall be split into a training set for model construction and a test for model verification. The training information shall be classified into a number of groups or clusters, the centroids of these clusters were subsequently used to generate a set of rules to develop a fuzzy engine, which can then predict and forecast the rate of accidents. The test cases shall be used to verify and validate the developed model. Discussion on the results obtained from using fuzzy logic techniques shall be carried out.

## *3.2.1. Data organization*

Construction sites are very dynamic and complex by nature, creating the potential for hazards that change constantly. So, what was safe yesterday may no longer be safe today. Thus, safety precautions should be followed and controlled. Unsafe working conditions and accidents are usually warning signs that something is wrong and has to be rectified.

Different government authorities measure safety at construction sites [41], however coordination and sharing of information with each other is still lacking. In addition, the data available on construction site accidents are neither accurate nor complete, due to the absence of a reliable accident reporting and recording system. Incomplete records are due to the poor accident investigation that may be a result of:


For these reasons we endeavored to avoid the normal way of doing the job, and instead, we focused on the laborer, himself, and his accident rates during his years of work experience as an expert source of data. We have tried to design the questionnaire in such a way as to serve our purpose of analyzing the data, and selected the interview method to get the maximum precise data possible.

## *3.2.1.1. Questionnaire design*

420 Fuzzy Controllers – Recent Advances in Theory and Applications

transformation in a systematic way.

shall be carried out.

*3.2.1. Data organization* 

 Narrow interpretation Judgmental behavior.

maximum precise data possible.

poor accident investigation that may be a result of:

 Reluctance of reporters to assert authority. Inexperienced and untrained investigators

 Incomplete or erroneous conclusions. Delays in accident investigations.

An important task, therefore, is to combine these two types of information into system designs. Therefore, the key question is how to transform human knowledge base into a mathematical formula or model. Essentially, what a fuzzy system does is to perform this

In this part of Section 2, we attempt to use a completely different approach to analyze accidents. A model shall be developed for data collected from an accident rate questionnaire filled-in by laborers working for a reputable construction company. This questionnaire was designed to include information about human factors, as well as other factors such as work type, managerial factors, training, physical factors and the historical accident rate for each labor during his period of employment in this particular construction company, and his experience during his career life time. The collected data shall be split into a training set for model construction and a test for model verification. The training information shall be classified into a number of groups or clusters, the centroids of these clusters were subsequently used to generate a set of rules to develop a fuzzy engine, which can then predict and forecast the rate of accidents. The test cases shall be used to verify and validate the developed model. Discussion on the results obtained from using fuzzy logic techniques

Construction sites are very dynamic and complex by nature, creating the potential for hazards that change constantly. So, what was safe yesterday may no longer be safe today. Thus, safety precautions should be followed and controlled. Unsafe working conditions and

Different government authorities measure safety at construction sites [41], however coordination and sharing of information with each other is still lacking. In addition, the data available on construction site accidents are neither accurate nor complete, due to the absence of a reliable accident reporting and recording system. Incomplete records are due to the

For these reasons we endeavored to avoid the normal way of doing the job, and instead, we focused on the laborer, himself, and his accident rates during his years of work experience as an expert source of data. We have tried to design the questionnaire in such a way as to serve our purpose of analyzing the data, and selected the interview method to get the

accidents are usually warning signs that something is wrong and has to be rectified.

In the design of questionnaire (Appendix) we have selected some certain features of human nature that, we believe, have a great potential on the accident causation in the local market [41]. The first page of the questionnaire concentrated on the personal information of the workers: i.e. 'height', 'weight', 'optical status', 'hearing ability', 'general health', 'education' and 'adherence to safety rules'. In this part we used some linguistic evaluations like 'high', 'low', 'fair', 'good', 'medium' etc., and in some others we have used numerical evaluations like in height, weight, as well as education. Since the objective here is to create a fuzzy model, high accuracy is not important and we considered that the respondents from the same field made the same judgments.

The second page was designed to concentrate on the work information: 'overtime work', 'experience', 'work nature', 'work type', 'hazardous level', 'needs for safety-gears', 'work location' and 'level of boredom'. Again, we have used some linguistic evaluations as well as numerical evaluations. We also concentrated on the managerial factors: 'salary received on time', 'level of training' 'importance' and level of importance placed on safety', with only linguistic evaluations.

The third page was a mix of both external factors: 'noise', 'live with family' and 'communication 'language', and accident history focusing on the number of accidents the laborer has faced during his work in the local construction market, which was the most important data that we needed to develop our model. The severity of accidents has not been taken into account since it is not considered a factor influencing the accident rate.

The cases obtained for this study were collected from three different construction companies selected to represent the local market. We have tried to select the cases from different ranks of the workforce, from higher levels to the lower levels to be able to study the different accident level cases that serve the purpose as well as adding versatility and diversity to this work.

#### *3.2.1.2. The response*

From the original cases that we collected on 95 people, we included only 76 cases and excluded 19 cases, which were incomplete. This has produced a very high response rate that reached 82.1%, which is relatively high, especially as the questionnaire is lengthy and a little bit complicated. More cases could have been obtained. However, since the aim of this study was to develop a model for accidents, the number of cases is not set a priority. Rather, the cases are accumulated and the algorithms are stopped when a cluster validity criterion is satisfied thereby yielding the optimum number of clusters.

#### *3.2.1.3. Limitations*

Limitations in this study should be noted. One of the limitations is that we did not include any specific information related to the accident consequences. Another limitation is that this study was made only on males and no female cases were studied, which makes it specific only to one sex.

#### *3.2.1.4. The feature matrix*

The feature matrix (FM) is the most important part of our work in this section , since by using this matrix we have been able to convert the linguistic variables into numerical variables. Thus, we can deal easily with practical cases and reduce the required operations of processing the output.

New Areas in Fuzzy Application 423

In order to obtain the models, the following steps have been implemented:

MATLAB fuzzy toolbox has been used to implement the above three steps [46].

between the number of clusters and the corresponding error where:

**Figure 33.** Relation between number of clusters and the corresponding error

column.

the normalized data.

number of clusters.

*3.2.2.2. Scaling of data* 

the following formula:

Xi = Vector element Xmin = Min. Vector element Xmax = Max. Vector element

Where:

*3.2.2.1. Clustering results and discussions* 

The results can be summarized as follows:

 Each column in the feature matrix is normalized by dividing all the numbers in this column by the maximum number of the absolute values of all the numbers in the said

Cluster validity study is implemented to determine the optimum number of clusters for

FCM technique is implemented to determine the centroids matrix of the selected

a. The optimum number of cluster (twelve) is determined by implementing cluster validity technique. The result is illustrated in Figure 33 which gives the relation

b. Table 9 (Appendix) illustrates the centroide matrix for the optimum number of clusters (twelve), where FW(j) stands for feature weight described in Table 10 (Appendix).

In order to reduce the error in the estimation of the membership functions ranges (e.g. the estimated range of FW5 was from 1 to 5, while from centroide matrix the range for the same feature is found to be from 3.750 to 4.9953) each feature vector element is scaled according to

X X X /X X scaled i min max min (43)

The columns of the FM matrix represent the feature variables which we obtained from the questionnaire. The rows of the matrix represent the different cases of laborers that we selected for interviews. Thus, for each case in the matrix we mapped the linguistic meanings into numbers according to weights we have proposed. For example, in case labelled (S1), the feature weight 1 (FW1) represents the rate of accidents per year of experience, which is the actual representation of the accident rates. FW2 represents the ratio between weight and height (specific weight). FW3 represents optical status and FW4 represents hearing ability. These feature weights are scaled on a scale of five from 1 to 5, to represent the linguistic variables. Therefore, 1 means `very bad`, 2 means `bad`, 3 means `medium`, 4 means `good` and 5 means `very good`. All the other variables are dealt in the same way until the matrix was generated. A sample of the feature matrix is shown in Table3.


**Table 3.** Sample of feature matrix illustrates the weight of some features for the first 10 cases

## *3.2.2. Modelling*

In this stage, FCM techniques will be implemented on the feature matrix after normalizing the data, based on column maximum values for ease and as being more indicative, then deciding the optimum number of clusters, by applying cluster validity techniques. The centroides for these optimum clusters are considered perfect models represent the feature matrix.

In order to obtain the models, the following steps have been implemented:


MATLAB fuzzy toolbox has been used to implement the above three steps [46].

#### *3.2.2.1. Clustering results and discussions*

422 Fuzzy Controllers – Recent Advances in Theory and Applications

The feature matrix (FM) is the most important part of our work in this section , since by using this matrix we have been able to convert the linguistic variables into numerical variables. Thus, we can deal easily with practical cases and reduce the required operations

The columns of the FM matrix represent the feature variables which we obtained from the questionnaire. The rows of the matrix represent the different cases of laborers that we selected for interviews. Thus, for each case in the matrix we mapped the linguistic meanings into numbers according to weights we have proposed. For example, in case labelled (S1), the feature weight 1 (FW1) represents the rate of accidents per year of experience, which is the actual representation of the accident rates. FW2 represents the ratio between weight and height (specific weight). FW3 represents optical status and FW4 represents hearing ability. These feature weights are scaled on a scale of five from 1 to 5, to represent the linguistic variables. Therefore, 1 means `very bad`, 2 means `bad`, 3 means `medium`, 4 means `good` and 5 means `very good`. All the other variables are dealt in the same way until the matrix

was generated. A sample of the feature matrix is shown in Table3.

**Weight/ Height** 

**Table 3.** Sample of feature matrix illustrates the weight of some features for the first 10 cases

In this stage, FCM techniques will be implemented on the feature matrix after normalizing the data, based on column maximum values for ease and as being more indicative, then deciding the optimum number of clusters, by applying cluster validity techniques. The centroides for these optimum clusters are considered perfect models represent the feature

**Optical status** 

S1 1/12 71/160 6/18 5 5 4 5 2 S2 5/12 77/ 170 6/60 5 4 3 5 3 S3 1/21 90/175 6/6 4 5 5 5 5 S4 0/8 54/165 6/60 4 5 2 5 20 S5 6/3.5 68/187 6/36 3 4 3 5 0 S6 10/11 85/177 6/6 4 4 3 5 10 S7 2/19 76/173 6/60 4 5 5 5 14 S8 18/25 72/170 6/18 3 4 4 4 4 S9 45/14 80/176 6/6 4 4 3 4 8 S10 3/20 81/174 6/6 4 4 5 4 1

**Hearing Ability** 

**General Health** 

**Adherence** 

**to Safety** 

**Education** 

**Overtime work** 

*3.2.1.4. The feature matrix* 

of processing the output.

**Feature** 

**Accidents/** 

**experience** 

**Case No**

*3.2.2. Modelling* 

matrix.

The results can be summarized as follows:

a. The optimum number of cluster (twelve) is determined by implementing cluster validity technique. The result is illustrated in Figure 33 which gives the relation between the number of clusters and the corresponding error where:

**Figure 33.** Relation between number of clusters and the corresponding error

b. Table 9 (Appendix) illustrates the centroide matrix for the optimum number of clusters (twelve), where FW(j) stands for feature weight described in Table 10 (Appendix).

#### *3.2.2.2. Scaling of data*

In order to reduce the error in the estimation of the membership functions ranges (e.g. the estimated range of FW5 was from 1 to 5, while from centroide matrix the range for the same feature is found to be from 3.750 to 4.9953) each feature vector element is scaled according to the following formula:

$$\mathbf{X}\_{\text{scaled}} = \mathbf{X}\_{\text{i}} - \mathbf{X}\_{\text{min}} \left( \mathbf{X}\_{\text{max}} - \mathbf{X}\_{\text{min}} \right) \tag{43}$$

Where:

Xi = Vector element Xmin = Min. Vector element Xmax = Max. Vector element

### *3.2.2.3. Model development*

By comparing each row of the centroids matrix with the contents of the questionnaire after scaling, one can infer the structure of the respective model, for example maximum and minimum rate of accident as illustrated in Table 11 (Appendix). From structure, following features can be extracted:

New Areas in Fuzzy Application 425

**Figure 34.** Fuzzy Accident Prediction Flowchart

results are shown as follows in Table 4:

**Table 4.** Standard deviation results for the test cases

results are given in hereinafter.

*3.2.3.1. Relevant results* 

accident rate.

Mamdani inference engine with centroid defuzzification and proportional aggregation is used in the construction of the fuzzy system. MATLAB Fuzzy Toolbox has been used to implement the required fuzzy prediction system. Eight cases have been tested and the

a. Eight additional random cases have been chosen from a local construction company. Features, actual accident rate and output of fuzzy prediction system for each test case are given in Table 12 (Appendix). The standard deviation between the actual accident rate and the predicted accident rate for each test case is calculated then the average standard deviation is obtained in order to determine the validity of the model. The

> **Case Number Standard Deviation** Case#1 0.3536 Case#2 0.2828 Case#3 0.4243 Case#4 0.7071 Case#5 01414 Case#6 0.1414 Case#7 0.2121 Case#8 0.0318 Average 0.2868

b. In (Appendix), Figure 37 shows the output of the prediction system for a case that fires all the linguistic values at the middle. It is observed from this result that the laborers with `average` personal information, `average` work condition, `average` managerial condition and `average` external effects are exposed to `average` annual


## *3.2.3. Accident rate prediction*

Now, fuzzy logic techniques will be implemented using the models obtained in Section 2.2.2, as perfect fuzzy rules, to predict the accident rate for any laborer who works in the construction field. The following flow chart (Figure 34) describes the fuzzy accident prediction system:

The models obtained from the fuzzy c -means the clustering process has been considered as very good and suitable fuzzy rules that govern the relation between the laborers in construction field and the expected annual rate of accident.

The beauty of using this type of clustering is not only to achieve the required models, but also these models are fuzzy, and can be geared in the fuzzy engine.

In order to fuzzify the model variables, a suitable number of Gauss functions (linguistic variable) is selected for each linguistic value so that any rule must fire all the linguistic values and the rules are given as follows [36]:

$$\text{IF } \left( \text{FW1 is } \text{mf}\_{\text{a}}^{-1} \right) \text{ and } \left( \text{FW2 is } \text{mf}\_{\text{b}}^{-2} \right) \text{ and } \dots \left( \text{FW2} \text{ is } \text{mf}\_{\text{v}}^{-2} \right) \text{ THEN } \left( \text{FW2} \text{ is } \text{mf}\_{\text{w}}^{-2} \right) \tag{44}$$

Where,

FW(1to22) : input linguistic variable FW23 : output linguistic variable mf a,b ..w : semantic rule a,b ..w: integer number from 1 to 5

**Figure 34.** Fuzzy Accident Prediction Flowchart

Mamdani inference engine with centroid defuzzification and proportional aggregation is used in the construction of the fuzzy system. MATLAB Fuzzy Toolbox has been used to implement the required fuzzy prediction system. Eight cases have been tested and the results are given in hereinafter.

#### *3.2.3.1. Relevant results*

424 Fuzzy Controllers – Recent Advances in Theory and Applications

By comparing each row of the centroids matrix with the contents of the questionnaire after scaling, one can infer the structure of the respective model, for example maximum and minimum rate of accident as illustrated in Table 11 (Appendix). From structure, following

b. The expected range of accident rate in company varies from 0.1581 to 2.8894 accidents

c. By comparing the above two extreme models, one can easily discover that receiving

d. The highest rate of accidents occurs to non-local workers (Live without their family

e. The twelve models obtained above shall be utilized later as fuzzy rules representing the this local construction company to predict the rate of accident for any person working

Now, fuzzy logic techniques will be implemented using the models obtained in Section 2.2.2, as perfect fuzzy rules, to predict the accident rate for any laborer who works in the construction field. The following flow chart (Figure 34) describes the fuzzy accident

The models obtained from the fuzzy c -means the clustering process has been considered as very good and suitable fuzzy rules that govern the relation between the laborers in

The beauty of using this type of clustering is not only to achieve the required models, but

In order to fuzzify the model variables, a suitable number of Gauss functions (linguistic variable) is selected for each linguistic value so that any rule must fire all the linguistic

 <sup>1</sup> <sup>2</sup> <sup>22</sup> <sup>23</sup> ab v w IF FW1 is mf and FW2 is mf and . . . FW22 is mf THEN FW23 is mf (44)

a. Labours in this construction company can be classified into 12 models.

salary on time has no effective impact on accident rate.

construction field and the expected annual rate of accident.

values and the rules are given as follows [36]:

FW(1to22) : input linguistic variable FW23 : output linguistic variable

a,b ..w: integer number from 1 to 5

mf a,b ..w : semantic rule

also these models are fuzzy, and can be geared in the fuzzy engine.

*3.2.2.3. Model development* 

features can be extracted:

most of the time).

on construction field.

*3.2.3. Accident rate prediction* 

per year.

prediction system:

Where,

a. Eight additional random cases have been chosen from a local construction company. Features, actual accident rate and output of fuzzy prediction system for each test case are given in Table 12 (Appendix). The standard deviation between the actual accident rate and the predicted accident rate for each test case is calculated then the average standard deviation is obtained in order to determine the validity of the model. The results are shown as follows in Table 4:


b. In (Appendix), Figure 37 shows the output of the prediction system for a case that fires all the linguistic values at the middle. It is observed from this result that the laborers with `average` personal information, `average` work condition, `average` managerial condition and `average` external effects are exposed to `average` annual accident rate.

c. Figure 38(Appendix) correlates the laborers general health and specific weight with their annual accident rate considering all other factors are `average`. It is clear that the annual accident rate increases with the significant increase of the specific weight and significant poorness of the general health.

New Areas in Fuzzy Application 427

**3.3. Fault location in distribution networks using fuzzy c-mean clustering** 

In last section of this chapter we shall studies an existing 13.8 kilovolt distribution network which, serves an oil production field spread over an area of approximately sixty kilometers square, in order to locate any fault that may occur anywhere in the network using fuzzy c-

In addition, we shall introduce several methods for normalizing data and selecting the optimum number of clusters in order to classify data. Results and conclusion shall be also

A joint venture oil company possesses two production areas, Area1 and Area2. For each area a power distribution system is provided. The power supply required for the two fields is provided from 130 MVA capacity power generation plant located in Area1. Twenty MW is transmitted to Area2 via 35KM long over head transmission line (OHTL) on wooden

Area2 distribution system consists of three radial overhead transmission lines, 8-10 KM long each, serve submersible oil pumps and other loads scattered in the field (60-kilometer square). These overhead transmission lines have neither differential relays nor sectionalizing fuses. Programmable logic Controller (PLC) is connected to the incomers, outgoings and generators auxiliaries at Area2 power station. This PLC records the running

Due to the aging of the system, remote area problems and harsh desert weather, repeated faults are experienced in the grid. Because of unavailability of differential relays and/or sectionalizing fuses, it is very difficult and long time is consumed to locate any fault in this network. The problem is reflecting passively on the oil productivity of this important area.

In order to measure the features of the faults at 176 nodes of the network; load flow study is implemented to determine the respective power loss for each short circuit case and also short circuit study is carried out to determine the feature vector for each short circuit case.

The results, obtained from the load flow study and the short circuit study, shall be used to form the network *feature matrix,* which will be clustered and analyzed hereinafter. The

Fuzzy clustering technique shall be used to classify the possible fault locations, which can be near to any node in the network or near to a chosen set of nodes based on the operator experience, into groups. The optimum number of groups (clusters) is computed using validity clustering technique. The remaining 13 cases are used as test cases. Euclidean

parameters that selected to build the feature matrix are shown in Table 5.

given to show the feasibility for the using the fuzzy logic to locate the fault location.

poles. Area2 power system also contains two 3.16 MVA stand by generators.

and tripping information for all bus-bar compartments.

**techniques** 

mean classification techniques.

*3.3.1. Network description* 

*3.3.2. Feature Matrix* 


## *3.2.4. Discussion*

The main objective of in this example to generate a human model which reflects the interaction between human factors in addition to other factors such as managerial factors, accident information, and work information, using fuzzy clustering techniques. Secondly, to predict annual rate of accident for any sample of workers by applying fuzzy logic techniques. Some of the important results obtained can be summarized as follows:


## **3.3. Fault location in distribution networks using fuzzy c-mean clustering techniques**

In last section of this chapter we shall studies an existing 13.8 kilovolt distribution network which, serves an oil production field spread over an area of approximately sixty kilometers square, in order to locate any fault that may occur anywhere in the network using fuzzy cmean classification techniques.

In addition, we shall introduce several methods for normalizing data and selecting the optimum number of clusters in order to classify data. Results and conclusion shall be also given to show the feasibility for the using the fuzzy logic to locate the fault location.

## *3.3.1. Network description*

426 Fuzzy Controllers – Recent Advances in Theory and Applications

significant poorness of the general health.

accidents) can be done via the fuzzy engine.

accident of this particular labourer.

suitable workers for any particular task

particular company.

hearing ability.

*3.2.4. Discussion* 

c. Figure 38(Appendix) correlates the laborers general health and specific weight with their annual accident rate considering all other factors are `average`. It is clear that the annual accident rate increases with the significant increase of the specific weight and

d. In Figure 39 (Appendix), the laborers educational level and safety adherence are plotted against their annual accident rate considering all other factors are `average`. It is noticed from this illustration that the laborers with the two educational level extremes are

e. Figure 40 (Appendix) correlates the laborers optical status and hearing ability with their annual accident rate considering all other factors to be `average`. It is clear that the annual accident rate increases with the significant poorness of the optical status and

f. From the last plot (Figure 41 - Appendix), it is clear that if the company is not keen on the level of safety, the annual accident rate will be increased, considering all other factors are `average`. In addition, it is noticeable from the figure that the tasks that are

The main objective of in this example to generate a human model which reflects the interaction between human factors in addition to other factors such as managerial factors, accident information, and work information, using fuzzy clustering techniques. Secondly, to predict annual rate of accident for any sample of workers by applying fuzzy logic

a. Fuzzy clustering techniques can be used to build a model that characterizes the different features of workers in local construction field against their rate of accidents. b. The model obtained from clustering is considered as rules to be used in a fuzzy logic engine to predict the rate of accidents for any worker in the construction field. c. For any specific case, 231 correlations (between any two features and the rate of

d. Optimum training to improve the safety attitude for certain laborer with minimum cost can be estimated by analyzing the correlation between the level of training and rate of

e. By analyzing the correlation between level of safety importance and rate of accidents for the workers in a particular company, the limits of the effective safety improvement

f. A similar technique can be applied to a particular company to predict the rate of accident in order to estimate the insurance rate for the people who work in this

g. Using the accident rate fuzzy prediction techniques companies can select the most

can be predicted in order to evaluate the investment in this direction.

exposed to accidents more than the `average` educational level laborers.

considered to be dangerous and need safety-gear are a source of accidents.

techniques. Some of the important results obtained can be summarized as follows:

A joint venture oil company possesses two production areas, Area1 and Area2. For each area a power distribution system is provided. The power supply required for the two fields is provided from 130 MVA capacity power generation plant located in Area1. Twenty MW is transmitted to Area2 via 35KM long over head transmission line (OHTL) on wooden poles. Area2 power system also contains two 3.16 MVA stand by generators.

Area2 distribution system consists of three radial overhead transmission lines, 8-10 KM long each, serve submersible oil pumps and other loads scattered in the field (60-kilometer square). These overhead transmission lines have neither differential relays nor sectionalizing fuses. Programmable logic Controller (PLC) is connected to the incomers, outgoings and generators auxiliaries at Area2 power station. This PLC records the running and tripping information for all bus-bar compartments.

Due to the aging of the system, remote area problems and harsh desert weather, repeated faults are experienced in the grid. Because of unavailability of differential relays and/or sectionalizing fuses, it is very difficult and long time is consumed to locate any fault in this network. The problem is reflecting passively on the oil productivity of this important area.

## *3.3.2. Feature Matrix*

In order to measure the features of the faults at 176 nodes of the network; load flow study is implemented to determine the respective power loss for each short circuit case and also short circuit study is carried out to determine the feature vector for each short circuit case.

The results, obtained from the load flow study and the short circuit study, shall be used to form the network *feature matrix,* which will be clustered and analyzed hereinafter. The parameters that selected to build the feature matrix are shown in Table 5.

Fuzzy clustering technique shall be used to classify the possible fault locations, which can be near to any node in the network or near to a chosen set of nodes based on the operator experience, into groups. The optimum number of groups (clusters) is computed using validity clustering technique. The remaining 13 cases are used as test cases. Euclidean

distance technique is implemented to find out the group of nodes, which the fault may be found near to, for each test case.

New Areas in Fuzzy Application 429

d. The norms between each test data and the full data in the chosen cluster is examined, accordingly the corresponding cluster for each test data is decided based on the

e. Alpha-cut defuzzification is used with alpha equal to 90% of the average of norms

f. The nearest node to the fault is checked in order to determine whether it is included in

Matlab program is written to implement the above six steps and the results are analyzed

a. The optimum number of cluster (twenty five) is determined by implementing cluster validity technique discussed in 2.1. The result is illustrated in Fig 35 which give the

b. Table 6 shows the effort saved to locate 13 fault cases. From this table, it can be noticed

Effort Saving = 1-Ratio between the number of possible locations to the total

 Percentage of successful trials (Ratio of the cases of including the nearest node in the possible locations to the total number of testing cases\*100%) = 92%. (46) Average effort saving (Summation of the Effort Saving percentages divided by the

It is important to notice that fuzzy cluster technique failed to locate the fault of test data number 4 due to the lack of information near to this location. However, it is expected that

for more available information the performance of this technique will be improved.

relation between the number of clusters and the corresponding error where:

**Figure 35.** The relation between number of clusters and the corresponding error

number of nodes\*100%. (45)

total number of the cases) =75% (47)

Also, we can define the following terms:

minimum norm obtained from the said examination.

between the cluster center and its data.

the possible locations or not.

*3.3.3.2. Results and discussion* 

and summarized as follows:

that:


**Table 5.** Summary of the parameters that are selected to build the feature matrix.

**Assumptions:** The following assumptions are considered:


Assumption (a) is valid since the transmission lines are short [47]. For assumption (b), the same work can be repeated for all other types of failures.

## *3.3.3. Fault location using column maximum normalization*

Now, FCM technique can be implemented after normalizing the data based on column maximum values and deciding the optimum number of clusters. The results shall be analyzed in order to locate any failures may occur in the network.

## *3.3.3.1. Calculation procedures*

To detect the fault using FCM technique with column maximum normalization the following steps have been implemented:


### *3.3.3.2. Results and discussion*

428 Fuzzy Controllers – Recent Advances in Theory and Applications

Set of nods fed from Feeder 1 Set of nods fed from

**Table 5.** Summary of the parameters that are selected to build the feature matrix.

**Assumptions:** The following assumptions are considered:

same work can be repeated for all other types of failures.

*3.3.3. Fault location using column maximum normalization* 

analyzed in order to locate any failures may occur in the network.

found near to, for each test case.

Feeder 1 Short circuit Current red from substation

for any other type of faults.

*3.3.3.1. Calculation procedures* 

the normalized data.

number of clusters.

column.

following steps have been implemented:

Celsius.

distance technique is implemented to find out the group of nodes, which the fault may be

**Feeder 1 Feeder 2 Feeder 3** 

Feeder 2

Feeder 2 Short circuit Current red from substation

Circuit breaker 1 status Circuit breaker 2 status Circuit breaker 3 status

Phase Angel A1 Phase Angel A2 Phase Angel A3 Phase Angel B1 Phase Angel B2 Phase Angel B3 Phase Angel C1 Phase Angel C2 Phase Angel C3 Power dip in Feeder 1 Power dip in Feeder 2 Power dip in Feeder 3 VAR dip in Feeder 1 VAR dip in Feeder 2 VAR dip in Feeder 3

a. The temperature of the network conductors is assumed constant at seventy degree

b. Only symmetrical short circuit is conceded. The same procedures can be implemented

Assumption (a) is valid since the transmission lines are short [47]. For assumption (b), the

Now, FCM technique can be implemented after normalizing the data based on column maximum values and deciding the optimum number of clusters. The results shall be

To detect the fault using FCM technique with column maximum normalization the

a. Each column in the Feature Matrix is normalized by dividing all the numbers in this column by the maximum number of the absolute values of all the numbers in the said

b. Cluster validity study is implemented to determine the optimum number of clusters for

c. FCM technique is implemented to determine the fuzzy partition matrix of the selected

Set of nods fed from Feeder 3

Feeder 2 Short circuit Current red from substation

Matlab program is written to implement the above six steps and the results are analyzed and summarized as follows:

a. The optimum number of cluster (twenty five) is determined by implementing cluster validity technique discussed in 2.1. The result is illustrated in Fig 35 which give the relation between the number of clusters and the corresponding error where:

**Figure 35.** The relation between number of clusters and the corresponding error

	- Effort Saving = 1-Ratio between the number of possible locations to the total number of nodes\*100%. (45)

Also, we can define the following terms:


It is important to notice that fuzzy cluster technique failed to locate the fault of test data number 4 due to the lack of information near to this location. However, it is expected that for more available information the performance of this technique will be improved.


New Areas in Fuzzy Application 431

**OF CLUSTERS** 

a. The optimum number of cluster is determined by implementing cluster validity technique and the results are given in Table 7. It is clear from the Table that the optimum number of clusters varies from case to another, which indicates that for any considerable additional of information, cluster validity study should be implemented

**NUMBER CASE DISCRIPTION OPTIMUM NUMBER** 

b. Figure 36 illustrates the relation between the number of clusters and the corresponding

**Figure 36.** Shows the relation between number of clusters and the corresponding error for case 1

 Effort Saving can be calculated as given in (45) Percentage of successful trails = 100%. See (46)

Average effort saving = 87% See (47)

*c.* Table 8 shows the effort saved to locate 13 fault cases. From the this table , it can be

1 POWER DIP IN FEEDER #1 AND C.B.1 TRIPS 13 2 POWER DIP IN FEEDER #2 AND C.B.2 TRIPS 15 3 POWER DIP IN FEEDER #3 AND C.B.3 TRIPS 7 4 POWER DIP IN FEEDER #1 AND C.B.1 DOES NOT TRIP 10 5 POWER DIP IN FEEDER #2 AND C.B.2 DOES NOT TRIP 11 6 POWER DIP IN FEEDER #2 AND C.B.2 DOES NOT TRIP 5

**Table 7.** Shows the optimum number of clusters for each case identified by the operator

error in case 1, where error is calculated as given in (45).

again to find the new optimum number of clusters.

**CASE** 

noticed that:

#### **Table 6.** Effort Saving

## *3.3.4. Fault location using simple maximum normalization*

Here, FCM technique is implemented after normalizing the data based on the maximum value of the data and deciding the optimum number of clusters. Then, the results are analyzed in order to locate any failures may occur in the network.

#### *3.3.4.1. Calculation procedures*

To detect the fault using FCM technique with simple maximum normalization, the following steps have been implemented:


#### *3.3.4.2. Results*

Matlab program is written to implement the above six steps and the results are analyzed and summarized as follows:

a. The optimum number of cluster is determined by implementing cluster validity technique and the results are given in Table 7. It is clear from the Table that the optimum number of clusters varies from case to another, which indicates that for any considerable additional of information, cluster validity study should be implemented again to find the new optimum number of clusters.


**Table 7.** Shows the optimum number of clusters for each case identified by the operator

430 Fuzzy Controllers – Recent Advances in Theory and Applications

**NUMBER OF POSSIBLE LOCATIONS** 

*3.3.4. Fault location using simple maximum normalization* 

analyzed in order to locate any failures may occur in the network.

minimum norm obtained from the said examination.

**NEAREST NODE EXISTS IN THE POSSIBLE LOCATIONS**

1 38 Yes 77% 2 38 Yes 77% 3 23 Yes 86% 4 2 No 0% 5 35 Yes 79% 6 56 Yes 66% 7 23 Yes 86% 8 49 Yes 70% 9 23 Yes 86% 10 17 Yes 90% 11 20 Yes 88% 12 20 Yes 88% 13 17 Yes 90%

Here, FCM technique is implemented after normalizing the data based on the maximum value of the data and deciding the optimum number of clusters. Then, the results are

To detect the fault using FCM technique with simple maximum normalization, the

a. All the numbers in the feature matrix are normalized by dividing all of them by the

b. The data are preliminary classified into clusters based on the understanding of the

c. Cluster validity study is implemented to determine the optimum number of clusters for

d. FCM technique is implemented to determine the fuzzy partition matrix for the selected

e. The norms between each test data and the full data in the chosen cluster is examined, accordingly the corresponding cluster for each test data is decided based on the

f. Alpha-cut defuzzification is used with alpha equal to the average of norms between the

g. The nearest node to the fault is checked in order to determine whether it is included in

Matlab program is written to implement the above six steps and the results are analyzed

maximum number of the absolute values of all the numbers in the matrix.

**EFFORT SAVING** 

**TESTING CASE NO.** 

**Table 6.** Effort Saving

*3.3.4.1. Calculation procedures* 

network operation.

the selected cluster.

number of clusters.

and summarized as follows:

*3.3.4.2. Results* 

cluster center and its data.

the possible locations or not.

following steps have been implemented:

b. Figure 36 illustrates the relation between the number of clusters and the corresponding error in case 1, where error is calculated as given in (45).

**Figure 36.** Shows the relation between number of clusters and the corresponding error for case 1

	- Effort Saving can be calculated as given in (45)
	- Percentage of successful trails = 100%. See (46)
	- Average effort saving = 87% See (47)


New Areas in Fuzzy Application 433

**FW1 FW2 FW3 FW4 FW5 FW6 FW7 FW8 FW9 FW10 FW11 FW12 FW13 FW14 FW15 FW16 FW17 FW18 FW19 FW20 FW21 FW22 FW23**  0.4899 0.4824 0.5548 4.3318 4.3324 3.9963 4.3311 15.6376 46.6328 53.3672 3.3330 3.6684 3.6677 8.3954 19.9348 71.6699 2.6679 2.6683 3.0055 4.0012 3.3330 5.0219 3.0031 0.3425 0.4881 0.3333 4.9983 4.3328 4.3310 4.6673 0.0000 94.9920 5.0080 3.9983 3.9999 3.3345 23.3133 6.7051 69.9817 2.9999 3.3311 3.3364 4.0020 3.0000 12.0000 3.6656 0.1581 0.5055 0.7761 4.5545 4.6684 3.9956 4.7782 4.2714 96.6805 3.3195 3.5547 3.7724 3.1099 7.8244 45.6152 46.5605 2.8862 2.3357 3.4470 3.9988 3.6650 11.3039 4.1094 2.8894 0.4899 0.5016 4.9953 4.9953 2.5048 3.0046 7.4857 2.5093 97.4907 3.9965 4.4954 2.9979 5.0231 14.9297 80.0472 4.4941 2.5000 2.0000 1.5024 3.0034 6.5188 3.0001 0.3674 0.4435 0.7917 4.7499 4.4997 3.9994 4.2500 8.7491 47.4970 52.5030 3.7497 3.7497 2.7503 21.2531 28.7504 49.9965 3.2497 3.0000 2.7496 3.9992 3.7492 3.3779 3.5003 0.6164 0.4388 0.8158 4.3160 4.3687 2.7888 3.3152 8.4245 5.0013 94.9987 3.8953 3.8950 3.2638 2.3621 0.0000 97.6379 2.9997 2.7371 2.2635 2.8957 3.4735 4.2838 3.2107 1.2917 0.4321 0.5000 4.0000 3.7500 3.0000 3.7500 7.5000 27.5000 72.5000 4.0000 3.2500 3.0000 71.2500 10.0000 18.7500 3.2500 2.5000 2.5000 2.5000 3.7500 3.6250 3.0000 0.3401 0.4600 0.4002 4.3754 4.3744 4.1238 4.8749 3.2411 71.2455 28.7545 3.4994 3.6244 3.2492 6.2440 61.2585 32.4975 2.4996 3.7507 3.0011 3.9998 3.1258 9.3725 4.6244 0.8238 0.4696 0.2803 4.3322 4.3322 2.3357 4.0000 10.6568 6.7131 93.2869 3.6678 3.6678 2.6680 41.5930 11.7266 46.6804 2.6645 2.6676 2.0000 2.6649 3.6645 4.5258 3.6678 0.5278 0.4539 0.5997 4.3992 4.2006 3.0011 4.6002 2.8065 62.0051 37.9949 3.0005 3.0005 3.2006 32.9938 43.0014 24.0048 3.4002 3.5998 2.5999 2.6004 2.4002 10.4017 3.8004 0.2282 0.4517 0.3471 4.7537 4.8743 3.6330 4.8743 3.0413 97.4866 2.5134 3.3767 2.3770 2.3720 4.2728 83.2449 12.4822 2.8797 3.3770 4.1206 3.6233 2.4976 9.4162 4.5074 1.1461 0.4654 0.8339 4.0001 4.2512 3.2512 3.2468 7.4926 27.4917 72.5083 3.8764 3.7486 3.3742 18.1000 13.1229 68.7771 2.8766 2.6225 3.5013 3.7523 3.5017 5.2219 3.2467

**Table 9.** Illustrates the centroide matrix for the optimum number of clusters (twelve),

**Appendix** 

#### **Table 8.** Effort saved

## *3.3.5. Discussion*

The main objective of was to apply fuzzy c-mean clustering technique to locate any 3-phase fault that may occur at any point on actual power distribution network. Two different normalizing methods have been used to process the feature matrix data. The result obtained can be summarized as follows:


## **4. Summary of the chapter and conclusion**

In this chapter we presented five new problems from different application; control, accident analysis, process and electrical network. By using fuzzy logic technique, we succeeded to resolve these problems efficiently. We also introduced different type of normalization of the data. Generating fuzzy rules by either linearization of the curves or by clustering the data were presented as well. Then a method of coloration and prediction of information using the generated fuzzy rules were provided.

## **Appendix**

432 Fuzzy Controllers – Recent Advances in Theory and Applications

**NUMBER OF POSSIBLE LOCATIONS** 

**NEAREST NODE EXISTS IN THE POSSIBLE LOCATIONS** 

1 21 Yes 87% 2 16 Yes 90% 3 23 Yes 86% 4 21 Yes 87% 5 34 Yes 79% 6 14 Yes 91% 7 33 Yes 80% 8 30 Yes 82% 9 33 Yes 80% 10 5 Yes 97% 11 10 Yes 94% 12 10 Yes 94% 13 9 Yes 94%

The main objective of was to apply fuzzy c-mean clustering technique to locate any 3-phase fault that may occur at any point on actual power distribution network. Two different normalizing methods have been used to process the feature matrix data. The result obtained

a. Fuzzy clustering technique can be used to investigate the location of faults in networks. b. Any actual fault can be utilized and be fed back to the database of the clustering system

c. Understanding the network configuration and operation can be utilized to improve the

d. Data matrix comprising the feature vectors should reflect good and adequate

e. For any major network's upgrading or change the clustering should be implemented again by using the new data obtained from complete study of the modified network.

In this chapter we presented five new problems from different application; control, accident analysis, process and electrical network. By using fuzzy logic technique, we succeeded to resolve these problems efficiently. We also introduced different type of normalization of the data. Generating fuzzy rules by either linearization of the curves or by clustering the data were presented as well. Then a method of coloration and prediction of information using the

**EFFORT SAVING** 

**TESTING CASE NO** 

**Table 8.** Effort saved

*3.3.5. Discussion* 

can be summarized as follows:

to improve its performance and efficiency.

**4. Summary of the chapter and conclusion** 

clustering which gives better results.

description of the network.

generated fuzzy rules were provided.



**Table 9.** Illustrates the centroide matrix for the optimum number of clusters (twelve),



New Areas in Fuzzy Application 435

**Feature Case#1 Case#2 Case#3 Case#4 Case#5 Case#6 Case#7 Case#8** 

**Year experience** 2.6 3.2 2.5 2.3 3.1 0 1.1 0.125

**predicted** 2.1 2.8 1.9 1.3 2.9 0.2 1.4 .17 **Weight/Height** 0.465 .454 .42 .466 .429 .49 .51 .415 **Optical status** 6/36 6/6 6/6 6/6 6/12 6/60 6/6 6/6 **Hearing Ability** Medium Good V.Good Good Medium V.Good V.Good V.Good **General Health** Good Good Good Good Good V.Good Good V.Good

**Safety** Fair Fair Fair V. High Low V. High Fair V. High **Education** 16 9 12 8 5 17 10 19 **Overtime work** 0 8 2 1 8 0 3 0 **Mental work** 60 20 50 70 10 100 50 90 **Manual work** 40 80 50 30 90 0 50 10 **Work type** Fait Tough Fair Fair Fair V. Easy Fair V. Easy **Hazard level** V. High High Medium V. High High V. Low Medium Low

**Gear** Sometime Sometime Sometime Always Sometime Rare Rare No need **Indoor Work** 0 20 30 5 0 0 80 0 **Office Work** 80 30 40 45 0 100 20 100 **Outdoor Work** 30 50 30 50 100 0 0 0

**Boredom** Medium High Medium Medium High Low High Medium

**importance** Medium Low Low Low Low V. High High High

**Importance** High Medium Medium High Medium High High V. High **Noise level** High High Medium V. High Medium Low Medium Low **Live with family** 12 12 6 12 0 9 12 12

**language** Good V.Good Medium Good Medium V. Good Good V.Good

Agree Disagree Strongly

Agree Agree Strongly

Agree

Agree Agree Agree Strongly

**Accidents/** 

**Accident rate** 

**Adherence to** 

**Need for Safety** 

**Level of** 

**Level of Training** 

**Level of Safety** 

**Communication** 

**Salary on time** Strongly

**Table 12.** Result of eight test cases

**Table 10.** DESCRIPTION of feature weights,


**Table 11.** Features of the model of Maximum and Minimum accident rate


**Table 12.** Result of eight test cases

434 Fuzzy Controllers – Recent Advances in Theory and Applications

**FW12** Hazard level **Table 10.** DESCRIPTION of feature weights,

Specific weight

high

above average

low

high

good

low

low

average

**Table 11.** Features of the model of Maximum and Minimum accident rate

low

high

tough

high

average low

low

under average

over average

high

high

low

Low

average low

4-6 months per year.

poor

average high

average

above average

above average

high

low

high

low

fair

above average

average high

low

average

average

low

high

Very high

High

high

9-12 months per year

above average

**Minimum accident rate 0.1581** 

**Maximum accident rate 2.8894.** 

Optical status

Hearing ability

General health

Adherence to safety

Education

Overtime work rate

Mental work rate

Manual work rate

Work type

Hazardous level

Needs for safety-gear

Indoor work rate

Office work rate

Outdoor work rate

Level of boredom

Delay on receiving the salary on

time

Level of training

Level of safety importance

Level of noise

**Feature Weight Description Feature Weight Description** 

**FW1** Accidents/ experience FW13 Need for Safety Gear **FW2** Weight/Height FW14 Indoor Work **FW3** Optical status FW15 Office Work **FW4** Hearing Ability FW16 Outdoor Work **FW5** General Health FW17 Level of Boredom **FW6** Adherence to Safety FW18 Salary on time **FW7** Education FW19 Level of Training **FW8** Overtime work FW20 Level of Safety **FW9** Mental work FW21 Noise level **FW10** Manual work FW22 live with family **FW11** Work type FW23 Communication language

**Personal Factors Work Factors Managerial** 

**Factors** 

**External Factors** 

Live with family

Communication language level

New Areas in Fuzzy Application 437

**Figure 38.** Correlation between weight/height, general health and rate of accident

**Figure 39.** Correlation between education level, safety adherence and rate of accident

**Figure 40.** Correlation between optical status, hearing ability and rate of accidents**.** 

**Figure 41.** Correlation between level of safety, need of safety-gear and rate of accident.

**Figure 37.** The output of the prediction system for `average` inputs.

**Figure 38.** Correlation between weight/height, general health and rate of accident

**Figure 37.** The output of the prediction system for `average` inputs.

**Figure 39.** Correlation between education level, safety adherence and rate of accident

**Figure 40.** Correlation between optical status, hearing ability and rate of accidents**.** 

**Figure 41.** Correlation between level of safety, need of safety-gear and rate of accident.

## **Author details**

#### Muhammad M.A.S. Mahmoud

*Received the B.S. degree in Electrical Engineering from Cairo University and the M.Sc. degree from Kuwait University. PH.D Transilvania University of Brasov, Romania He occupies a position of Senior Engineer at Al Hosn Gas Co. UAE* 

New Areas in Fuzzy Application 439

[15] D. D. Siljak, "Decentralized Control of Complex Systems", Academic Press, Boston,

[16] M. S. Mahmoud, "Computer-Operated Systems Control", Marcel Dekker Inc., New

[17] M. Jamshidi, "Large Scale Systems: Modeling, Control and Fuzzy Logic", Prentice-Hall,

[18] M. S. Mahmoud, S.Z. Eid and A .A. Abou-Elseoud, "A Real Time Expert System for Dynamical Processes", IEEE Transactions Systems, Man and Cybernetics, Vol. SMC-19,

[19] M. S. Mahmoud, S. Kotob, and A. A. Abou-Elseoud, "A Learning Rule-Based Control System", Journal of Information and Decision Technologies, Vol. 18, No. 1, January

[20] M. S. Mahmoud, A. A. Abou-Elseoud and S. Kotob, "Development of Expert Control Systems: A Pattern Classification and Recognition Approach", Journal of Intelligent and

[21] L. X. Wang, "A Supervisory Controller for Fuzzy Control Systems that Guarantees", IEEE Trans. Automatic Control, Vol. 39, No. 9, September 1994, pp. 1845-1847. [22] K. M. Passino, and S. Yurkovick, "Fuzzy Control", Addison Wesley, California, 1998. [23] C. C. Lee, "Fuzzy Logic in Control Systems: Fuzzy Logic Controller, Parts I & II", IEEE Systems, Man and Cybernetics, Vol. 20, No. 2, March/April 1990, pp. 404-435. [24] M. Sugeno, and G.T. Kang, Structure Identification of Fuzzy Model, Fuzzy Sets and

[25] Takagi, and M. Sugeno , Fuzzy Identification of Systems and its Applications to Modelling and Control, IEEE Trans. On System, Man,and Cybernetics, Vol. 15, 1985 ,

[26] Suciu, C; Liliana; Dafinca; Kansara M Margineanu ,"Switching capacitor fuzzy controller for power factor correction on inductive circuit", Power Electronics Specialist

[27] K.H . Abdul-Rahman, S.M. Shahidehpour, "Application of Fuzzy sets to optimal reactive power planning with security constrains", TransacTion of Power Systems,

[28] Yu Qin; ShanShan Du, "To design fuzzy and digital controller for a single phase power factor pre-regulator-genetic algorithm approach", Industrial Application Conference,

[29] Yu Qin; ShanShan Du, "Control of single phase power factor pre-regulator for on-line uninterruptible power supply using fuzzy logic control inference", Applied power

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New Jersey, 1997.

1992, pp. 55-66.

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1994, vol. 9, No.2, pp 589-597.

Annual, 1997 IEEE, vol, 2, pp 791-796.

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Conference, 31st Annual, 2000, IEEE, vol. 2.pp 773- 777.

## **5. References**


[15] D. D. Siljak, "Decentralized Control of Complex Systems", Academic Press, Boston, 1991.

438 Fuzzy Controllers – Recent Advances in Theory and Applications

No. 5, September/October 1989, pp. 1101-1105.

Robotic Systems, Vol. 5, No. 2, April 1992, pp.129-146.

Processes, Korean J. Chern. Eng., 15(3), 231(1998)

Theories and Techniques", Marcel Dekker Inc., New York, 1985.

*Received the B.S. degree in Electrical Engineering from Cairo University and the M.Sc. degree from Kuwait University. PH.D Transilvania University of Brasov, Romania He occupies a position of* 

[2] M. S. Mahmoud, "Computer-Operated Systems Control", Marcel Dekker Inc., New

[3] C. C. Lee, Fuzzy Logic in Control Systems: Fuzzy Logic Controller, Parts I& II", IEEE

[4] M. S. Mahmoud, S.Z. Eid and A .A. Abou-Elseoud, "A Real Time Expert System for Dynamical Processes", IEEE Transactions Systems, Man and Cybernetics, Vol. SMC-19,

[5] M. S. Mahmoud, S. Kotob, and A. A. Abou-Elseoud, "A Learning Rule- Based Control System", Journal of Information and Decision Technologies, Vol. 18, No. 1, January

[6] M. S. Mahmoud, A. A. Abou-Elseoud and S. Kotob, "Development of Expert Control Systems: A Pattern Classification and Recognition Approach", Journal of Intelligent and

[7] L. X. Wang, "A Supervisory Controller for Fuzzy Control Systems that Guarantees Stability", IEEE Trans. Automatic Control, Vol. 39, No. 9, September 1994, pp. 1845-

[8] K. M. Passino, and S. Yurkovick, "Fuzzy Control", Addison Wesley, California, 1998. [9] Wang, Li-Xin, "A Course In Fuzzy Systems And Control", Prentice-Hall International,

[10] M. Sugeno, and G.T. Kang, Structure Identification of Fuzzy Model, Fuzzy Sets and

[11] Takagi, and M. Sugeno , "Fuzzy Identification of Systems and its Applications to Modeling and Control", IEEE Trans. On System, Man, and Cybernetics, Vol. 15, 1985 ,

[12] Jeong-Woo Choi, Seung-Mok Ob, Hyun-Goo Choi, SangBaek Lee, Kwang-Soon Lee and Won-Hong Lee Fuzzy Control of Ethanol Concentration for Emulsan Production in a Fed-Batch Cultivation of Acinetobacter calcoaceticus RAG-I Korean 1. Chern. Eng.,

[13] Min Oh and II Moon, Framework of Dynamic Simulation for Complex Chemical

[14] M. S. Mahmoud, M. F. Hassan and M. G. Darwish, "Large Scale Control Systems:

[1] Zadeh, L.A., "Fuzzy Sets", Information and Control, Vol. 12, pp. 338-353, 1965.

Systems, Man and Cybernetics, Vol. 20, No.2, March/April 1990, pp.404-435.

**Author details** 

**5. References** 

York, 1991.

1992, pp. 55-66.

Inc. NJ 07458,USA, 1997.

Systems, 28, 1988, pp. 15-33.

1847.

pp. 116-132.

15(3), 310(1998)

Muhammad M.A.S. Mahmoud

*Senior Engineer at Al Hosn Gas Co. UAE* 

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**Chapter 18** 

© 2012 Seidi 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.

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,

**Fuzzy Control Systems: LMI-Based Design** 

This chapter describes widespread methods of model-based fuzzy control systems. The subject of this chapter is a systematic framework for the stability and design of nonlinear fuzzy control systems. We are trying to build a bridge between conventional fuzzy control and classic control theory. By building this bridge, the strong well developed tools of classic

Model-based fuzzy control, with the possibility of guaranteeing the closed loop stability, is an attractive method for control of nonlinear systems. In recent years, many studies have been devoted to the stability analysis of continuous time or discrete time model based fuzzy control systems (Takagi & Sugeno, 1985; Rhee & Won, 2006; Chen et al., 1993; Wang et al., 1996; Zhao et al., 1996; Tanaka & Wang, 2001; Tanaka et al., 2001). Among such methods, the method of Takagi-Sugeno (Takagi & Sugeno, 1985) has found many applications for modelling complex nonlinear systems (Tanaka & Sano, 1994;Tanaka & Kosaki, 1997;Li et al., 1998). The concept of sector nonlinearity (Kawamoto et al., 1992) provided means for exact approximation of nonlinear systems by fuzzy blending of a few locally linearized subsystems. One important advantage of using such a method for control design is that the closed-loop stability analysis, using the Lyapunov method, becomes easier to apply. Various stability conditions have been proposed for such systems (Tanaka &Wang, 2001), (Ting, 2006), where the existence of a common solution to a set of Lyapunov equations is shown to be sufficient for guaranteeing the closed-loop stability. Some relaxed conditions are also proposed in (Kim & Lee, 2000; Ding et al, 2006; Fang et al., 2006, Tanaka & Ikeda, 1998). Parallel Distributed Compensator (PDC) is a generalization of the state feedback controller to the case of nonlinear systems, using the Takagi-Sugeno fuzzy model (Wang et al., 1996). This method is based on partitioning nonlinear system dynamics into a number of linear subsystems, for which state feedback gains are designed and blended in a fuzzy sense. Takagi-Sugeno model and parallel distributed compensation have been used in many applications successfully (Sugeno & Kang, 1986, Lee et al., 2006, Hong & Langari, 2000, Bonissone et al.,

Morteza Seidi, Marzieh Hajiaghamemar and Bruce Segee

Additional information is available at the end of the chapter

control could be used in model-based fuzzy control systems

http://dx.doi.org/10.5772/48529

**1. Introduction** 


## **Fuzzy Control Systems: LMI-Based Design**

Morteza Seidi, Marzieh Hajiaghamemar and Bruce Segee

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48529

## **1. Introduction**

440 Fuzzy Controllers – Recent Advances in Theory and Applications

FACTS Control Elemenl".1999,l EEE.

Prentice Hall Inc, NJ, 1988.

Technology (A5): 45-58. 2004.

Company, 1989.

Press, 1992.

841-847, August 1991.

Inc., New York, USA, 1994.

Conference, 30th Annual, 1999, IEEE, vol. 1, pp 149- 154.

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Cybernetics, Vol. SMC-11, No. 12, December 1981, pp. 779-785.

[46] The Mathworks, "Fuzzy Logic Toolbox", Boston,1999.

[32] Navd R. Zargari, Yuan Xiao; Bin Wu, "A multilevel thyristor rectifier with improved power faclor", Transaction on Industrial Applications, 1997,vol. 33 pp 1208- 1213. [33] Chung,H.S .H; Tam,E.P.W; Huni,S.Y.R. "Development of a Fuzzy Logic Controller for boost rectifier with acti ve power factor correction", Power Electronics Specialist

[34] Mohammed T. Bi na; David C. Hamill , "The Bootstrap Varible Inductance: A New

[35] Muhammad Harunur Rash id, "Power Electronics Circuit, Devices and Applications",

[38] Dominic Cooper C.Psychol "Human Factors in Accidents", Institute of Quarrying, North Of England- CoalPro Seminar, Ramside Hall, Durham, UK. 12 March 2002

[40] Sii, H.S., Wang, J., Ruxton, T., Yang, J.B., Liu, J. "Fuzzy logic approaches to safety assessment in maritime engineering applications", Journal of Marine Engineering &

[41] KARTAM, N. A. and BOUZ, R. G.. "Fatalities and Injuries in Kuwaiti Construction

[42] A. Kamal, S. M. Eid and M. S. Mahmoud, "Multi-Stage Clustering: An Efficient Technique in Socioeconomic Field Experiments", IEEE Transactions Systems, Man and

[43] BEZDEK, J. C. & PAL K. S. "Fuzzy models for pattern recognition". New York: IEEE

[44] WINDHAM, M. P. "Cluster validity for the fuzzy c-mean clustering algorithm", IEEE Transactions Pattern Analysis and Machine Intelligence, (PAMI-4), 357-363, 1982 . [45] XUANLI LIISA XIE & GERARDO BENI. "A Validity Measure for Fuzzy Clustering". IEEE Transactions Pattern Analysis and Machine Intelligence, vol. (PAMI-13), no. 8, pp.

[47] John J. Granger and William D. Stevenson, Jr., "Power System Analysis", McGraw-Hill,

[36] Wang, Li-Xin, "A Course In Fuzzy Systems And Control", Prentice-Hall, N. J., 1997. [37] ADAMS, J. A. Human factors in engineering. New York, Macmillan Publishing

> This chapter describes widespread methods of model-based fuzzy control systems. The subject of this chapter is a systematic framework for the stability and design of nonlinear fuzzy control systems. We are trying to build a bridge between conventional fuzzy control and classic control theory. By building this bridge, the strong well developed tools of classic control could be used in model-based fuzzy control systems

> Model-based fuzzy control, with the possibility of guaranteeing the closed loop stability, is an attractive method for control of nonlinear systems. In recent years, many studies have been devoted to the stability analysis of continuous time or discrete time model based fuzzy control systems (Takagi & Sugeno, 1985; Rhee & Won, 2006; Chen et al., 1993; Wang et al., 1996; Zhao et al., 1996; Tanaka & Wang, 2001; Tanaka et al., 2001). Among such methods, the method of Takagi-Sugeno (Takagi & Sugeno, 1985) has found many applications for modelling complex nonlinear systems (Tanaka & Sano, 1994;Tanaka & Kosaki, 1997;Li et al., 1998). The concept of sector nonlinearity (Kawamoto et al., 1992) provided means for exact approximation of nonlinear systems by fuzzy blending of a few locally linearized subsystems. One important advantage of using such a method for control design is that the closed-loop stability analysis, using the Lyapunov method, becomes easier to apply. Various stability conditions have been proposed for such systems (Tanaka &Wang, 2001), (Ting, 2006), where the existence of a common solution to a set of Lyapunov equations is shown to be sufficient for guaranteeing the closed-loop stability. Some relaxed conditions are also proposed in (Kim & Lee, 2000; Ding et al, 2006; Fang et al., 2006, Tanaka & Ikeda, 1998). Parallel Distributed Compensator (PDC) is a generalization of the state feedback controller to the case of nonlinear systems, using the Takagi-Sugeno fuzzy model (Wang et al., 1996). This method is based on partitioning nonlinear system dynamics into a number of linear subsystems, for which state feedback gains are designed and blended in a fuzzy sense. Takagi-Sugeno model and parallel distributed compensation have been used in many applications successfully (Sugeno & Kang, 1986, Lee et al., 2006, Hong & Langari, 2000, Bonissone et al.,

© 2012 Seidi 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.

1995). The Linear Matrix Inequality (LMI) technique offers a numerically tractable way to design a PDC controller with objectives such as stability (Wang et al.,1996; Ding et al, 2006; Fang et al., 2006; Tanaka & Sugeno 1992), H∞ control (Lee et al., 2001), H2 control (Lin & Lo, 2003), pole-placement (Jon et al, 1997; Kang & Lee, 1998), and others ( Tanaka & Wang, 2001).

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

The main idea of the Takagi-Sugeno fuzzy modeling method is to partition the nonlinear system dynamics into several locally linearized subsystems, so that the overall nonlinear behavior of the system can be captured by fuzzy blending of such subsystems. The fuzzy rule associated with the i-th linear subsystem for the continuous fuzzy system and the discrete fuzzy system, can then be defined as

Continuous fuzzy system

$$\begin{array}{ll}\text{Rule i :} \text{IF } \mathbf{Z\_1}(t) \text{ is } \mathbf{M\_{ii}}\dots \text{ and } \mathbf{Z\_l}(t) \text{ is } \mathbf{M\_{il}}\\\text{THEN} \qquad \begin{cases} \dot{\mathbf{x}}(t) = A\_i \mathbf{x}(t) + B\_i \boldsymbol{\mu}(t) \\\ y(t) = \mathbf{C\_i} \mathbf{x}(t) \end{cases} \quad \text{i=1,2,...,r} \end{array} \tag{1}$$

Fuzzy Control Systems: LMI-Based Design 443

(4)

(5)

*z t Ax t Bu t*

*i ii*

*z t*

*i ii*

*h z t Ax t Bu t*

*i i*

1

*r i i*

*z t Cx t*

*i*

1 1

> *r i*

0, 1,2,......,

0,

*ij j*

1 1

*w z*

*w z*

1 1

There are generally three approaches to build the fuzzy model: "sector nonlinearity," "local

Figure 1 illustrates the concept of global and local sector nonlinearity. Suppose the original nonlinear system satisfies the sector non-linearity condition (Kawamoto et al., 1992, as cited in Tanaka & Wang, 2001), i.e., the values of nonlinear terms in the state-space equation remain within a sector of hyper-planes passing through the origin. This model guarantees the stability of the original nonlinear system under the control law. A function Φ: R→R is

*j*

*wz M z*

*z t*

*i i*

*h z t Cx t*

1

1

*h z*

*r <sup>i</sup> w z wz i r* 

 

1

said to be sector [a,c] if for all xϵR, y= Φ(x) lies between 1*b x* and <sup>2</sup> *b x* .

1

1

*i*

*r*

*y t*

1

*i*

*r*

1

*i r i i*

*r*

1

*i*

*r*

( 1)

*x t*

Discrete Fuzzy System

Where

It is also true, for all *t*, that

**2.1. Building a fuzzy model** 

*2.1.1. Sector nonlinearity* 

approximation," or a combination of the two.

Discrete Fuzzy System

$$\begin{array}{ll}\text{Rule i :IF } \mathbf{Z\_1(t)} \text{ is } \mathbf{M\_{i1}}\dots \text{ and } \mathbf{Z\_1(t)} \text{is } \mathbf{M\_{il}}\\\text{THEN} \qquad \begin{cases} \mathbf{x(t+1)} = A\_i \mathbf{x(t)} + B\_i \boldsymbol{\mu}(t) \\\ y(t) = C\_i \mathbf{x(t)} \end{cases} \quad \text{i=1,2,...,r} \end{array} \tag{2}$$

where, x t *<sup>n</sup> <sup>R</sup>* is the state vector, u t *<sup>m</sup> <sup>R</sup>* is the input vector, Ai *n n <sup>R</sup>* , i <sup>B</sup> *n m <sup>R</sup>* , Ci *q n <sup>R</sup>* ; *ztzt zt* 1 2 , ,..., *<sup>p</sup>* are nonlinear functions of the state variables obtained from the original nonlinear equation, and *M z ij i* are the degree of membership of *<sup>i</sup> z t* in a fuzzy set *Mij* . Whenever there is no ambiguity, the time argument in *z*(*t*) is dropped. The overall output, using the fuzzy blend of the linear subsystems, will then be as follows:

Continuous fuzzy system

$$\begin{aligned} \dot{X} &= \frac{\sum\_{i=1}^{R} w\_1(z) \{A\_i \ge \mathbf{f}(t) + B\_i u(t)\}}{\sum\_{i=1}^{r} w\_i(z)} = \sum\_{i=1}^{r} h\_1(z) \{A\_i \ge \mathbf{f}(t) + B\_i u(t)\} \\ \mathbf{y}(t) &= \frac{\sum\_{i=1}^{r} w\_1(z) \mathbf{C}\_i \mathbf{x}(t)}{\sum\_{i=1}^{r} w\_1(z)} = \sum\_{i=1}^{r} h\_1(z) \mathbf{C}\_i \mathbf{x}(t) \end{aligned} \tag{3}$$

Discrete Fuzzy System

442 Fuzzy Controllers – Recent Advances in Theory and Applications

**2. Takagi-Sugeno fuzzy model** 

discrete fuzzy system, can then be defined as

Continuous fuzzy system

Discrete Fuzzy System

Continuous fuzzy system

1 1

*r*

*R*

Ci

2001).

1995). The Linear Matrix Inequality (LMI) technique offers a numerically tractable way to design a PDC controller with objectives such as stability (Wang et al.,1996; Ding et al, 2006; Fang et al., 2006; Tanaka & Sugeno 1992), H∞ control (Lee et al., 2001), H2 control (Lin & Lo, 2003), pole-placement (Jon et al, 1997; Kang & Lee, 1998), and others ( Tanaka & Wang,

The main idea of the Takagi-Sugeno fuzzy modeling method is to partition the nonlinear system dynamics into several locally linearized subsystems, so that the overall nonlinear behavior of the system can be captured by fuzzy blending of such subsystems. The fuzzy rule associated with the i-th linear subsystem for the continuous fuzzy system and the

 

*q n <sup>R</sup>* ; *ztzt zt* 1 2 , ,..., *<sup>p</sup>* are nonlinear functions of the state variables obtained from the original nonlinear equation, and *M z ij i* are the degree of membership of *<sup>i</sup> z t* in a fuzzy set *Mij* . Whenever there is no ambiguity, the time argument in *z*(*t*) is dropped. The overall output, using the fuzzy blend of the linear subsystems, will then be as follows:

*i i <sup>r</sup> <sup>i</sup>*

1 1

*X h z Axt But*

1 1

THEN i=1,2,...,r *i i i*

*x t Ax t Bu t y t Cx t*

Rule i : IF Z t is M . . . and Z t is M 1 i1 l il

(1)

(2)

*n n <sup>R</sup>* , i <sup>B</sup> *n m <sup>R</sup>* ,

(3)

 

THEN i=1,2,...,r *i i i*

*x t Ax t Bu t y t Cx t*

 

1

where, x t *<sup>n</sup> <sup>R</sup>* is the state vector, u t *<sup>m</sup> <sup>R</sup>* is the input vector, Ai

*i i <sup>r</sup> <sup>I</sup>*

*<sup>i</sup> <sup>r</sup> <sup>i</sup>*

*y t h zCxt*

*<sup>i</sup> <sup>r</sup> <sup>i</sup>*

*i i*

*w z*

*w z Axt But*

1

*w zCxt*

1 1

*i*

1 1

*w z*

 

Rule i : IF Z t is M . . . and Z t is M <sup>1</sup> i1 l il

$$\begin{aligned} \exp(t+1) &= \frac{\sum\_{i=1}^{r} o\_i \left( z \left( t \right) \right) \left( A\_i \mathbf{x} \left( t \right) + B\_i \mathbf{u} \left( t \right) \right)}{\sum\_{i=1}^{r} o\_i \left( z \left( t \right) \right)} \\ &= \sum\_{i=1}^{r} h\_i \left( z \left( t \right) \right) \left( A\_i \mathbf{x} \left( t \right) + B\_i \mathbf{u} \left( t \right) \right) \\ y \left( t \right) &= \frac{\sum\_{i=1}^{r} o\_i \left( z \left( t \right) \right) \mathbb{C}\_i \mathbf{x} \left( t \right)}{\sum\_{i=1}^{r} o\_i \left( z \left( t \right) \right)} \\ &= \sum\_{i=1}^{r} h\_i \left( z \left( t \right) \right) \mathbb{C}\_i \mathbf{x} \left( t \right) \end{aligned} \tag{4}$$

Where

$$\begin{aligned} w\_1(z) &= \prod\_{j=1}^i M\_{ij} \left( z\_j \right) \\ h\_1(z) &= \frac{w\_1(z)}{\sum\_{i=1}^r w\_1(z)} \end{aligned} \tag{5}$$

It is also true, for all *t*, that

$$\begin{cases} \sum\_{i=1}^r w\_1(z) > 0, \\ w\_1(z) \ge 0, i = 1, 2, ..., r \end{cases}$$

#### **2.1. Building a fuzzy model**

There are generally three approaches to build the fuzzy model: "sector nonlinearity," "local approximation," or a combination of the two.

#### *2.1.1. Sector nonlinearity*

Figure 1 illustrates the concept of global and local sector nonlinearity. Suppose the original nonlinear system satisfies the sector non-linearity condition (Kawamoto et al., 1992, as cited in Tanaka & Wang, 2001), i.e., the values of nonlinear terms in the state-space equation remain within a sector of hyper-planes passing through the origin. This model guarantees the stability of the original nonlinear system under the control law. A function Φ: R→R is said to be sector [a,c] if for all xϵR, y= Φ(x) lies between 1*b x* and <sup>2</sup> *b x* .

Fuzzy Control Systems: LMI-Based Design 445

(7)

region.The sector [b1, b2] consists of two lines blxl and b2xl, where the slopes are bl = 1 and b2=

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

*x xx*

2 4 14 4

*dx x x dx*

<sup>2</sup>

 2 13 1 3 1 sin *i i i z xt Mztbxt* 

From the property of membership functions *Mzt M zt* 11 21 <sup>1</sup> , we can obtain the

1 1 <sup>1</sup> <sup>1</sup> 1 1 <sup>1</sup>

*zt S zt*

2 ( ) sin( ( )) () 0 ( ) 1 ( ( )) <sup>2</sup>

1 1

Similarly we obtain membership functions associated with 2 14 *z t x tx t* () () () . Assume

we have:

*sin z t z t*

<sup>1</sup> <sup>1</sup>

0, .

( ( )) ( ) () 0

1, .

*z t*

*otherwise*

*z t*

*otherwise*

(9)

*M zt sin z t*

1

( ) 2 1 ( ( ))

(8)

.

2 

2 

. It follows that

**Figure 3.** <sup>3</sup> sin ( ) *x t* and its local sector

membership functions

max( ( )) 2 1 *zt d*

We present <sup>3</sup> sin ( ) *x t* is represented as follows:

 

and min( ( )) 2 2 *zt d*

 

2 1 1

*M zt sin z t*

**Figure 1.** a) Global sector nonlinearity, b) Local sector nonlinearity

#### **Example 1**

The well-known nonlinear control benchmark, the ball-and-beam system is commonly used as an illustrative application of various control methods (Wang & Mendel, 1992) depicted in figure 2. Let x1(t) and x2(t) denote the position and the velocity of the ball and let x3(t) and x4(t) denote the angular position and the angular velocity of the beam Then, the system dynamics can be described by the following state-space equation

**Figure 2.** The ball and beam system

$$\begin{aligned} \dot{\mathbf{x}}(t) &= f(\mathbf{x}(t)) + \mathbf{g}(\mathbf{x}(t))u(t) \\ \text{Where} \\ f(\mathbf{x}) &= \begin{bmatrix} \mathbf{x}\_2(t) \\ \mathbf{B}(\mathbf{x}\_1(t)\mathbf{x}\_4^2(t) - \mathbf{G}\sin(\mathbf{x}\_3(t))) \\ \mathbf{x}\_4(t) \\ 0 \end{bmatrix} \text{and } \mathbf{g}(\mathbf{x}) = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \end{aligned} \tag{6}$$

Where 1234 () () () () () *<sup>T</sup> xt x t x t x t x t* and u(t) is torque.

 <sup>3</sup> sin *<sup>x</sup>* and <sup>2</sup> 1 4 *x x* are nonlinear terms in the state-space equation. We define 1 3 *z x* sin and <sup>2</sup> 2 14 *z xx* . Assume 3 2 2 *x* and 1 4 *xx d d* as the region within which the system will operate. Figure 3 shows that 1 3 *zt xt* ( ) sin ( ) and its local sector operating region.The sector [b1, b2] consists of two lines blxl and b2xl, where the slopes are bl = 1 and b2= 2 . It follows that

$$\begin{aligned} \left| \frac{2}{\pi} \mathbf{x} \right| &\leq \left| \sin(\mathbf{x}) \right| \leq \left| \mathbf{x} \right|, \\ \left| -d\mathbf{x}\_4 \leq \mathbf{x}\_1 \mathbf{x}\_4^2 \leq d\mathbf{x}\_4. \end{aligned} \tag{7}$$

**Figure 3.** <sup>3</sup> sin ( ) *x t* and its local sector

444 Fuzzy Controllers – Recent Advances in Theory and Applications

**Figure 1.** a) Global sector nonlinearity, b) Local sector nonlinearity

dynamics can be described by the following state-space equation

The well-known nonlinear control benchmark, the ball-and-beam system is commonly used as an illustrative application of various control methods (Wang & Mendel, 1992) depicted in figure 2. Let x1(t) and x2(t) denote the position and the velocity of the ball and let x3(t) and x4(t) denote the angular position and the angular velocity of the beam Then, the system

> 2 2 14 3 4

*x t*

*Bx tx t G x t f x g x x t*

( ) 0

(6)

0 1

and 1 4 *xx d d* as the region within which the

1 4 *x x* are nonlinear terms in the state-space equation. We define 1 3 *z x* sin

( ( ) ( ) sin( ( ))) 0 ( ) and ( ) ( ) <sup>0</sup>

system will operate. Figure 3 shows that 1 3 *zt xt* ( ) sin ( ) and its local sector operating

( ) ( ( )) ( ( )) ( )

and u(t) is torque.

 

*xt f xt gxt ut*

Where

Where 1234 () () () () () *<sup>T</sup> xt x t x t x t x t*

2 14 *z xx* . Assume 3 2 2 *x*

<sup>3</sup> sin *<sup>x</sup>* and <sup>2</sup>

and <sup>2</sup>

**Example 1** 

**Figure 2.** The ball and beam system

We present <sup>3</sup> sin ( ) *x t* is represented as follows:

$$z\_1 = \sin\left(x\_3\left(t\right)\right) = \left(\sum\_{i=1}^2 M\_i\left(z\_1\left(t\right)\right)b\_i\right)x\_3\left(t\right) \tag{8}$$

From the property of membership functions *Mzt M zt* 11 21 <sup>1</sup> , we can obtain the membership functions

$$M\_1(z\_1(t)) = \begin{cases} z\_1(t) - \left(\frac{\mathcal{Q}}{\mathcal{A}}\right) S \sin(z\_1(t)) & z\_1(t) \neq 0 \\ \hline \left(1 - \frac{\mathcal{Q}}{\mathcal{A}}\right) \sin^{-1}(z\_1(t)) & z\_1(t) \neq 0 \\ & 1 \, & 
otherwise. \end{cases} \tag{9}$$

$$M\_2(z\_1(t)) = \begin{cases} \sin^{-1}(z\_1(t)) - z\_1(t) & z\_1(t) \neq 0 \\ \hline \left(1 - \frac{\mathcal{Q}}{\mathcal{A}}\right) \sin^{-1}(z\_1(t)) & z\_1(t) \neq 0 \\ & 0 \, & 
otherwise. \end{cases}$$

Similarly we obtain membership functions associated with 2 14 *z t x tx t* () () () . Assume max( ( )) 2 1 *zt d* and min( ( )) 2 2 *zt d* we have:

$$\mathbf{x}\_{2}(t) = \mathbf{x}\_{1}(t)\mathbf{x}\_{4}(t) = \left(\sum\_{i=1}^{2} \mathbf{N}\_{i}\left(\mathbf{z}\_{2}\left(t\right)\right)\mathbf{b}\_{i}\right)\mathbf{a}\_{i} \tag{10}$$

$$\begin{aligned} N\_1\left(z\_2\left(t\right)\right) &= \frac{z\_2(t) - a\_2}{a\_1 - a\_2}, \\ N\_2\left(z\_2\left(t\right)\right) &= \frac{a\_1 - z\_2(t)}{a\_1 - a\_2}, \end{aligned} \tag{11}$$

Fuzzy Control Systems: LMI-Based Design 447

(15)

*dt* is stable if and only

(16)

The original system can be partitioned into subsystems by approximation of nonlinear terms about equilibrium points. This approach can have fewer rules and of course less complexity but it cannot guarantee the stability of the original system under the controller. Usually in this approach, construction of a fuzzy membership function requires knowledge of the behavior of the original system and of course different types of membership functions can

Parallel distributed compensation (PDC) is a model-based design procedure introduced in (Wang et al,. 1995). Using the Takagi-Sugeno fuzzy model, a fuzzy combination of the stabilizing state feedback gains, , 1,2,..., , *<sup>i</sup> Fi r* associated with every linear subsystem is used as the overall state feedback controller. The general structure of the controller is then

1 12 2 If is ,and is ,........ ,and is then , 1,2,..., *<sup>i</sup> <sup>i</sup> <sup>p</sup> ip <sup>i</sup> z t M z t M m z t M u Fx t i r* (14)

 <sup>1</sup> 1

*i i <sup>r</sup> <sup>i</sup>*

The Takagi-Sugeno model and the Parallel Distributed Compensation have the same

A variety of problems arising in system and control theory can be reduced to a few standard convex or quasi-convex optimization problems involving linear matrix inequalities (LMIs).

if there exists a positive-definite matrix P such that 0 *<sup>T</sup> A P PA* . The Lypanov inequality,

0

1 ( ) 0, *m*

*i F x F xF* 

*i i*

() .

*z Fx t u h z Fx t*

*i i <sup>r</sup> <sup>i</sup>*

*i i*

1

*r*

number of fuzzy rules and use the same membership functions.

Lyapunov published his theory in 1890 and showed that *<sup>d</sup> x t Ax t*

**4. Stability conditions and control design** 

*P* 0 and 0 *<sup>T</sup> A P PA* is a form of an LMI.

*2.1.2. Local approximation* 

**3. Parallel distributed compensation** 

The output of the controller is represented by

be selected.

as

**4.1. LMI** 

An LMI has the form

The exact TS-fuzzy model-based dynamic system of the ball and beam system can be obtained as following:

$$
\begin{bmatrix}
\dot{\boldsymbol{x}}\_1(t) \\
\dot{\boldsymbol{x}}\_2(t) \\
\dot{\boldsymbol{x}}\_3(t) \\
\dot{\boldsymbol{x}}\_4(t)
\end{bmatrix} = \sum\_{i=1}^2 \sum\_{j=1}^2 M\_i(\boldsymbol{z}\_1(t)) N\_j(\boldsymbol{z}\_2(t)) \times \begin{bmatrix}
\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & -\mathbf{G}\mathbf{b}\_1 & D\boldsymbol{a}\_j \\
0 & 0 & -\mathbf{G}\mathbf{b}\_1 & D\boldsymbol{a}\_j \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{bmatrix} + \begin{bmatrix}
0 \\
0 \\
0 \\
1
\end{bmatrix} u(t) \\
\end{bmatrix} \tag{12}
$$

The fuzzy model has the following 4 rules:

$$\begin{aligned} \text{Rule 1}: \text{ if } z\_1(t) \text{ is } M\_1 \text{ and } z\_2(t) \text{ is } N\_1\\ \text{Then } \dot{x}(t) &= A\_1 x(t) + B\_1 u(t), \\ \text{Rule 2: if } z\_1(t) \text{ is } M\_1 \text{ and } z\_2(t) \text{ is } N\_2\\ \text{Then } \dot{x}(t) &= A\_2 x(t) + B\_2 u(t), \\ \text{Rule 3: if } z\_1(t) \text{ is } M\_2 \text{ and } z\_2(t) \text{ is } N\_1 \\ \text{Then } \dot{x}(t) &= A\_3 x(t) + B\_3 u(t), \\ \text{Rule 4: if } z\_1(t) \text{ is } M\_2 \text{ and } z\_2(t) \text{ is } N\_2 \\ \text{Then } \dot{x}(t) &= A\_4 x(t) + B\_4 u(t) \end{aligned} \tag{13}$$

Where

$$\begin{aligned} &A\_{1}=\begin{bmatrix}0&1&0&0\\0&0&-G\mathbf{b}\_{1}&Da\_{1}\\0&0&0&1\\0&0&0&0\end{bmatrix},A\_{1}=\begin{bmatrix}0&1&0&0\\0&0&-G\mathbf{b}\_{1}&Da\_{2}\\0&0&0&0&1\\0&0&0&0&0\end{bmatrix},\\ &A\_{1}=\begin{bmatrix}0&1&0&0\\0&0&-G\mathbf{b}\_{2}&Da\_{1}\\0&0&0&1\\0&0&0&0\end{bmatrix},A\_{1}=\begin{bmatrix}0&1&0&0\\0&0&-G\mathbf{b}\_{2}&Da\_{2}\\0&0&0&0&1\\0&0&0&0&0\end{bmatrix},\\ &B\_{1}=B\_{2}=B\_{3}=B\_{4}=B=\begin{bmatrix}0\\0\\0\\1\end{bmatrix},z\_{1}=\sin\left(x\_{3}\right)\text{and }z\_{2}=x\_{1}x\_{4}\end{aligned}$$

### *2.1.2. Local approximation*

446 Fuzzy Controllers – Recent Advances in Theory and Applications

2 2

*i j*

The fuzzy model has the following 4 rules:

obtained as following:

 

Where

( ) ,

2 2

( ) ,

1 2 1 2

1 2

*M ztNzt u t*

*j*

1 1

Then ( ) ( ) ( ),

*xt Axt But*

*xt Axt But*

*xt Axt But*

<sup>4</sup> *t But* ) ()

<sup>0</sup> , sin and <sup>0</sup>

Then ( ) ( ) ( ),

Then ( ) ( ) ( ),

2 2

3 3

 

> 

2 14 *x x*

4

l 1 l 2

2 1 2 2

(10)

(11)

(12)

(13)

2

1 () () () *i ii i z t x tx t N z t b*

> 

 

 

2 14 2

*z t Nzt*

*z t Nzt*

The exact TS-fuzzy model-based dynamic system of the ball and beam system can be

( ) 01 0 0 ( ) 0 ( ) 00 b ( ) 0 ( ( )) ( ( )) ( ) ( ) 00 0 1 ( ) <sup>0</sup>

( ) 00 0 0 ( ) 1

1 12 1

01 0 0 01 0 0

00 0 0 00 0 0 01 0 0 01 0 0

00 b 00 b , 00 0 1 00 0 1

00 0 0 00 0 0

0

 

*BBBBB z x z*

*G D G D A A*

1

 

00 b 00 b , , 00 0 1 00 0 1

*zt M zt N*

1 12 2

*zt M zt N*

1 22 1

*zt M zt N*

1 22 2

*zt M zt N*

*G D G D A A*

Then ( ) (

*xt Ax*

1 2

2 2

1 1

*x t x t x t G D x t*

2 i 2

3 1 1 3 4 4

Rule 1 : if is and is

Rule 2: if is and is

Rule 3: if is and is

Rule 4: if is and is

1 1

1 1

1234 1 3

*x t x t x t x t*

1 2

*i j*

The original system can be partitioned into subsystems by approximation of nonlinear terms about equilibrium points. This approach can have fewer rules and of course less complexity but it cannot guarantee the stability of the original system under the controller. Usually in this approach, construction of a fuzzy membership function requires knowledge of the behavior of the original system and of course different types of membership functions can be selected.

## **3. Parallel distributed compensation**

Parallel distributed compensation (PDC) is a model-based design procedure introduced in (Wang et al,. 1995). Using the Takagi-Sugeno fuzzy model, a fuzzy combination of the stabilizing state feedback gains, , 1,2,..., , *<sup>i</sup> Fi r* associated with every linear subsystem is used as the overall state feedback controller. The general structure of the controller is then as

$$\text{If } z\_1(t) \text{is} M\_{i1}, \text{and } z\_2(t) \text{ is } M\_{i2}, \dots, \text{...}\\\text{m/} z\_p(t) \text{ is } M\_{ip} \text{ then } u = -F\_i \mathbf{x}(t), i = 1, 2, \dots, r \quad \text{(14)}$$

The output of the controller is represented by

$$u = -\frac{\sum\_{i=1}^{r} o\_i(\mathbf{z}) F\_i \mathbf{x}\left(t\right)}{\sum\_{i=1}^{r} o\_i} = -\sum\_{i=1}^{r} h\_i(\mathbf{z}) F\_i \mathbf{x}\left(t\right). \tag{15}$$

The Takagi-Sugeno model and the Parallel Distributed Compensation have the same number of fuzzy rules and use the same membership functions.

### **4. Stability conditions and control design**

#### **4.1. LMI**

A variety of problems arising in system and control theory can be reduced to a few standard convex or quasi-convex optimization problems involving linear matrix inequalities (LMIs). Lyapunov published his theory in 1890 and showed that *<sup>d</sup> x t Ax t dt* is stable if and only if there exists a positive-definite matrix P such that 0 *<sup>T</sup> A P PA* . The Lypanov inequality, *P* 0 and 0 *<sup>T</sup> A P PA* is a form of an LMI.

An LMI has the form

$$F(\mathbf{x}) \triangleq F\_0 + \sum\_{i=1}^{m} \mathbf{x}\_i F\_i > \mathbf{0}\_\prime \tag{16}$$

Where , 0,..., *n n <sup>i</sup> FR i m* are the given symmetric matrices and *<sup>m</sup> x R* is the variable and the inequality symbol shows that *F x*( ) is positive definite (Boyd, 1994).

#### **4.2. Stability conditions**

There are a large number of works on stability conditions and control design of fuzzy systems in the literature. A sufficient stability condition for ensuring stability of PDC was derived by Tanaka and Sugeno (Tanaka & Sugeno, 1990; 1992 ).

By substituting the controller output (15) into the TS model for the continuous fuzzy control (4), we have:

$$\dot{\mathbf{x}}\left(t\right) = \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_i \left(\mathbf{z}\left(t\right)\right) h\_j \left(\mathbf{z}\left(t\right)\right) \left(A\_i - B\_i F\_j\right) \mathbf{x}\left(t\right) \tag{17}$$

Fuzzy Control Systems: LMI-Based Design 449

(23)

(25)

(27)

(26)

(24)

that is, a common P has to exist for all subsystems.

The stability of the closed loop system can be derived by using theorem 1 and 2.

*T ii ii*

*T ii ii*

**4.3. Stable controller design** 

Continuous fuzzy system

Find *X* 0 and *Mi* , *i r* 1,2,...,

solvable).

as:

*G PG P*

*G P PG*

Theorem 3: The equilibrium of the continuous fuzzy control system described by (18) is globally asymptotically stable if there exists a common positive definite matrix P such that

0, 2 2

0,

(28)

0,

*GG GG P P*

 

i s.t. h *<sup>j</sup> ij h*

0,

 

i s.t. h *<sup>j</sup> ij h*

By using the following conditions, the solution of the LMI problem for continuous and discrete fuzzy systems gives us the state feedback gains Fi and the matrix P (if the problem is

0.

The conditions (27) and (28) gives us a positive definite matrix *X* and *Mi* (or that there is no solution). From the solution *X* and *Mi* , a common P and the feedback gains can be found

*M B BM M B BM*

*T T T T ji i j i j j i*

i s.t. h *XP ij hj*

Consider a new variable <sup>1</sup> *X P* then the stable fuzzy controller design problem is:

*T T T i i ii i i*

*XA A X M B B M XA A X XA A X*

1

*T T ii jj*

Theorem 4: The equilibrium of the discrete fuzzy control system described by (20) is globally

0, 2 2

*P P*

*T ij ji ij ji*

asymptotically stable if there exists a common positive definite matrix P such that

*T ij ji ij ji*

*GG GG*

or

$$\begin{aligned} \dot{\mathbf{x}}\left(t\right) &= \sum\_{j=1}^{r} h\_i\left(z\left(t\right)\right) h\_i\left(z\left(t\right)\right) \mathbf{G}\_{ii}\mathbf{x}\left(t\right) \\ &+ 2 \sum\_{i=1}^{r} \sum\_{i$$

where *G A BF ij i i j* , Similarly for the discrete fuzzy system we have

$$\mathbf{x}\left(t+1\right) = \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_i\left(z\left(t\right)\right) h\_j\left(z\left(t\right)\right) \left(A\_i - B\_i F\_j\right) \mathbf{x}\left(t\right) \tag{19}$$

or

$$\begin{aligned} \dot{\mathbf{x}}\{t+1\} &= \sum\_{j=1}^{r} h\_i\{\mathbf{z}\{t\}\} h\_i\{\mathbf{z}\{t\}\} \mathbf{G}\_{ii} \mathbf{x}\{t\} \\ &+ 2 \sum\_{i=1}^{r} \sum\_{i$$

Theorem 1: The equilibrium of the continuous fuzzy system (3) with u(t) = 0 is globally asymptotically stable if there exists a common positive definite matrix P such that

$$A\_i^T P + PA\_i < 0, \ i = 1, 2, \dots, r \tag{21}$$

that is, a common P has to exist for all subsystems.

Theorem 2: The equilibrium of the discrete fuzzy system (4) with u(t) = 0 is globally asymptotically stable i f there exists a common positive definite matrix P such that

$$A\_i^T P A\_i - P < 0, \ i = 1, 2, \dots, r \tag{22}$$

that is, a common P has to exist for all subsystems.

448 Fuzzy Controllers – Recent Advances in Theory and Applications

the inequality symbol shows that *F x*( ) is positive definite (Boyd, 1994).

derived by Tanaka and Sugeno (Tanaka & Sugeno, 1990; 1992 ).

1 1

*xt h zt h zt Gxt*

*i i ii*

*i j*

*i j*

1

*j*

*r*

2

1

1

that is, a common P has to exist for all subsystems.

2

1

where *G A BF ij i i j* , Similarly for the discrete fuzzy system we have

1 1

asymptotically stable if there exists a common positive definite matrix P such that

asymptotically stable i f there exists a common positive definite matrix P such that

*xt h zt h zt Gxt*

*i j*

1

*j*

*r*

1

*i ij*

*r r*

*i ij*

*r r*

*<sup>i</sup> FR i m* are the given symmetric matrices and *<sup>m</sup> x R* is the variable and

There are a large number of works on stability conditions and control design of fuzzy systems in the literature. A sufficient stability condition for ensuring stability of PDC was

By substituting the controller output (15) into the TS model for the continuous fuzzy control

*x t h z t h z t A BF x t*

*i j i ij*

*<sup>r</sup> ij ji*

*x t h z t h z t A BF x t*

*i i ii*

*i j*

*<sup>r</sup> ij ji*

Theorem 1: The equilibrium of the continuous fuzzy system (3) with u(t) = 0 is globally

Theorem 2: The equilibrium of the discrete fuzzy system (4) with u(t) = 0 is globally

*h zt h zt xt*

*i j i ij*

*h zt h zt x t*

2

0, 1,2,..., *<sup>T</sup> Ai i P PA i r* (21)

0, 1,2,..., *<sup>T</sup> Ai i PA P i r* (22)

*G G*

(17)

2

(19)

*G G*

(18)

(20)

Where , 0,..., *n n*

**4.2. Stability conditions** 

(4), we have:

or

or

The stability of the closed loop system can be derived by using theorem 1 and 2.

Theorem 3: The equilibrium of the continuous fuzzy control system described by (18) is globally asymptotically stable if there exists a common positive definite matrix P such that

$$\begin{aligned} \mathbf{G}\_{ii}^T \mathbf{P} + \mathbf{P} \mathbf{G}\_{ii} &< \mathbf{0}, \\ \left( \frac{\mathbf{G}\_{ij} + \mathbf{G}\_{ji}}{2} \right)^T \mathbf{P} + \mathbf{P} \left( \frac{\mathbf{G}\_{ij} + \mathbf{G}\_{ji}}{2} \right) &\le \mathbf{0}, \\\\ i &< j \text{ s.t. } \mathbf{h}\_i \cap h\_j \ne \phi \end{aligned} \tag{23}$$

Theorem 4: The equilibrium of the discrete fuzzy control system described by (20) is globally asymptotically stable if there exists a common positive definite matrix P such that

$$\begin{aligned} \mathbf{G}\_{ii}^{T}\mathbf{P}\mathbf{G}\_{ii} - \mathbf{P} &< \mathbf{0},\\ \left(\frac{\mathbf{G}\_{ij} + \mathbf{G}\_{ji}}{2}\right)^{T} \mathbf{P}\left(\frac{\mathbf{G}\_{ij} + \mathbf{G}\_{ji}}{2}\right) - \mathbf{P} &\le \mathbf{0}, \end{aligned} \tag{25}$$

$$i < j \text{ s.t. } \mathbf{h}\_i \cap h\_j \neq \phi \tag{26}$$

#### **4.3. Stable controller design**

By using the following conditions, the solution of the LMI problem for continuous and discrete fuzzy systems gives us the state feedback gains Fi and the matrix P (if the problem is solvable).

Consider a new variable <sup>1</sup> *X P* then the stable fuzzy controller design problem is:

Continuous fuzzy system

Find *X* 0 and *Mi* , *i r* 1,2,...,

$$\begin{aligned} -\mathbf{X}\mathbf{A}\_i^T - \mathbf{A}\_i\mathbf{X} + \mathbf{M}\_i^T\mathbf{B}\_i^T + \mathbf{B}\_i\mathbf{M}\_i &> 0, \\ -\mathbf{X}\mathbf{A}\_i^T - \mathbf{A}\_i\mathbf{X} - \mathbf{X}\mathbf{A}\_j^T - \mathbf{A}\_j\mathbf{X} \\ +\mathbf{M}\_j^T\mathbf{B}\_i^T + \mathbf{B}\_i\mathbf{M}\_j + \mathbf{M}\_i^T\mathbf{B}\_j^T + \mathbf{B}\_j\mathbf{M}\_i &\ge 0. \end{aligned} \tag{27}$$
 
$$\mathbf{X} = \mathbf{P}^{-1} \quad i < j \text{ s.t. } \mathbf{h}\_i \cap \mathbf{h}\_j \neq \boldsymbol{\phi} \tag{28}$$

The conditions (27) and (28) gives us a positive definite matrix  $X$  and  $M\_i$  (or that there is no solution). From the solution  $X$  and  $M\_{i'}$ , a common P and the feedback gains can be found as:

as:

$$P = X^{-1} \ \ \ F\_i = M\_i X^{-1} \tag{29}$$

Fuzzy Control Systems: LMI-Based Design 451

(35)

(36)

(37)

. *u t*

is

is

1 (Ichikawa et al, 1993, as cited in Tanaka &

0

2

The condition that <sup>2</sup> *V xt V xt*

Wang, 2001) for all *x t* can be written as

**4.5. Constraint on control** 

satisfied at all times *t* 0 if the LMIs

Hold, where <sup>1</sup> *X P* and *M FX i i* .

*x* 0 is unknown but the upper bound

Theorem 6: Assume that *x t*

known. Then,

Where <sup>1</sup> *X P*

*T ii ii*

*G PG P*

2

*T ij ji ij ji*

*GG GG*

2 2

The generalized eigenvalue minimization can be found in (Tanaka & Wang, 2001).

Theorem 5: Assume that the initial condition x(0) is known. The constraint <sup>2</sup>

 

*i*

Proofs of theorem 1 and 2 are given in (Tanaka & Wang, 2001)

**4.6. Performance-oriented parallel distributed compensation** 

0

*T*

0

of *x t* is known, i.e., *x t*

, where x(0) is unknown but the upper bound

(38)

1 0 <sup>0</sup>

*x x X X M M I* 

 

2

The above LMI design conditions depend on the initial states. Thus, if the initial states *x* 0 change, this means that the feedback gains Fi must be again determined. To overcome this disadvantage, modified LMI constraints on the control input have been developed, where

1 2 0 0 1 if , *Tx Xx I X*

In the modified PDC proposed in (Seidi & Markazi, 2011), unlike the conventional PDC, state feedback gains associated with every linear subsystem, are not assumed fixed. Instead, based on some pre-specified performance criteria, several feedback gains are designed and

*T i*

0,

<sup>i</sup> s.t. h and <1 *<sup>j</sup> ij h* 

*P P*

Similarly for a discrete fuzzy system the design problem is

Find *X* 0 and *Mi* , *i r* 1,2,...,

$$\begin{aligned} &X - \left(A\_i X - B\_i M\_i\right)^T X^{-1} \left(A\_i X - B\_i M\_i\right) > 0, \\ &X - \frac{1}{4} X \left(A\_i X - B\_i M\_i + A\_j X - B\_j M\_i\right)^T X^{-1} \\ &\times \left(A\_i X - B\_i M\_j + A\_j X - B\_j M\_i\right) X \ge 0. \end{aligned} \tag{30}$$

#### **4.4. Decay rate**

Decay rate is associated with the speed of response. The decay rate fuzzy controller design helps to find feedback gains that provide better setteling time (Tanaka et al,. 1996; 1998a; 1998b).

Continuous fuzzy system: The condition that *V xt V xt* 2 (Ichikawa et al, 1993, as cited in Tanaka & Wang, 2001) for all *x t* can be written as

$$\begin{aligned} \mathbf{G}\_{ii}^T \mathbf{P} + \mathbf{P} \mathbf{G}\_{ii} + 2\alpha \mathbf{P} &< 0\\ \mathbf{P} \left(\frac{\mathbf{G}\_{ij} + \mathbf{G}\_{ji}}{2}\right)^T \mathbf{P} + \mathbf{P} \left(\frac{\mathbf{G}\_{ij} + \mathbf{G}\_{ji}}{2}\right) + 2\alpha \mathbf{P} &\le 0 \end{aligned} \tag{31}$$

Where

$$G\_{ij} = A\_i - B\_i F\_{i\prime} \text{ } \alpha > 0 \text{ and } i < j \text{ s.t. } \mathbf{h}\_i \cap h\_j \neq \phi \tag{32}$$

Therefore, by solving the following generalized eigenvalue minimization problem in X, the largest lower bound on the decay rate that can be found by using a quadratic Lyapunov function:

maximize subject to

$$\begin{aligned} &X > 0, \\ &-\mathbf{X}\mathbf{A}\_i^T + \mathbf{A}\_i\mathbf{X} + \mathbf{M}\_i^T\mathbf{B}\_i^T + \mathbf{B}\_i\mathbf{M}\_i - 2\alpha\mathbf{X} > 0, \\ &-\mathbf{X}\mathbf{A}\_i^T - \mathbf{A}\_i\mathbf{X} - \mathbf{X}\mathbf{A}\_j^T - \mathbf{A}\_j\mathbf{X} + \mathbf{M}\_j^T\mathbf{B}\_j^T + \mathbf{B}\_i\mathbf{M}\_j \\ &+\mathbf{M}\_i^T\mathbf{B}\_j^T + \mathbf{B}\_j\mathbf{M}\_i - 4\alpha\mathbf{X} > 0, \end{aligned} \tag{33}$$

$$\text{Si} < j \text{ s.t. } h\_i \cap h\_j \neq \emptyset, \text{ where } X = P^{-1}, \quad M\_i = F\_i X. \tag{34}$$

Similarly for a discrete fuzzy system:

The condition that <sup>2</sup> *V xt V xt* 1 (Ichikawa et al, 1993, as cited in Tanaka & Wang, 2001) for all *x t* can be written as

$$\begin{aligned} \mathbf{G}\_{ii}^T \mathbf{P} \mathbf{G}\_{ii} - \alpha^2 \mathbf{P} &< \mathbf{0}, \\ \left(\frac{\mathbf{G}\_{ij} + \mathbf{G}\_{ji}}{2}\right)^T \mathbf{P} \left(\frac{\mathbf{G}\_{ij} + \mathbf{G}\_{ji}}{2}\right) - \alpha^2 \mathbf{P} &\le 0 \end{aligned} \tag{35}$$

<sup>i</sup> s.t. h and <1 *<sup>j</sup> ij h* (36)

The generalized eigenvalue minimization can be found in (Tanaka & Wang, 2001).

#### **4.5. Constraint on control**

450 Fuzzy Controllers – Recent Advances in Theory and Applications

Find *X* 0 and *Mi* , *i r* 1,2,...,

**4.4. Decay rate** 

1998b).

Where

function:

maximize

subject to

Similarly for a discrete fuzzy system:

Similarly for a discrete fuzzy system the design problem is

1 4

Continuous fuzzy system: The condition that *V xt V xt* 2

cited in Tanaka & Wang, 2001) for all *x t* can be written as

0,

*X*

*T ii ii* 1 1

*X X AX BM AX BM X*

*T i ii i ii*

*X AX BM X AX BM*

*i i j j ji*

*AX BM AX BM X*

Decay rate is associated with the speed of response. The decay rate fuzzy controller design helps to find feedback gains that provide better setteling time (Tanaka et al,. 1996; 1998a;

2 0

<sup>i</sup> , 0 and s.t. h *G A BF ij i i i <sup>j</sup>*

Therefore, by solving the following generalized eigenvalue minimization problem in X, the largest lower bound on the decay rate that can be found by using a quadratic Lyapunov

*P P P*

*ij h*

2 2

4 0,

*M B BM X*

*XA A X M B B M X*

 

*XA A X XA A X M B B M*

<sup>1</sup> . . , where , . *i j i i i j st h h*

*T T T i i ii i i T T T T i i j j jj i j*

> *T T ij j i*

*GG GG*

*T ij ji ij ji*

*G P PG P*

*i ii j ji*

1

, *i i P X F MX* (29)

1

(Ichikawa et al, 1993, as

(30)

(31)

(32)

(33)

0.

2 0

2 0,

*X P M FX* (34)

*T*

0,

Theorem 5: Assume that the initial condition x(0) is known. The constraint <sup>2</sup> *u t* is satisfied at all times *t* 0 if the LMIs

$$
\begin{bmatrix}
\mathbf{1} & \mathbf{x}\begin{pmatrix} \mathbf{0} \end{pmatrix}^T\\\mathbf{x}\begin{pmatrix} \mathbf{0} \end{pmatrix} & \mathbf{X} \end{bmatrix} \ge \mathbf{0} \\
$$
 
$$
\begin{bmatrix} \mathbf{X} & \mathbf{M}\_i^T\\\mathbf{M}\_i & \mu^2 \mathbf{I} \end{bmatrix} \ge \mathbf{0} \\
$$

Hold, where <sup>1</sup> *X P* and *M FX i i* .

The above LMI design conditions depend on the initial states. Thus, if the initial states *x* 0 change, this means that the feedback gains Fi must be again determined. To overcome this disadvantage, modified LMI constraints on the control input have been developed, where *x* 0 is unknown but the upper bound of *x t* is known, i.e., *x t* .

Theorem 6: Assume that *x t* , where x(0) is unknown but the upper bound is known. Then,

$$\mathbf{x}^T \begin{pmatrix} 0 \end{pmatrix} \mathbf{X}^{-1} \mathbf{x} \begin{pmatrix} 0 \end{pmatrix} \le \mathbf{1} \text{ if } \phi^2 I \le \mathbf{X} \tag{38}$$

Where <sup>1</sup> *X P*

Proofs of theorem 1 and 2 are given in (Tanaka & Wang, 2001)

#### **4.6. Performance-oriented parallel distributed compensation**

In the modified PDC proposed in (Seidi & Markazi, 2011), unlike the conventional PDC, state feedback gains associated with every linear subsystem, are not assumed fixed. Instead, based on some pre-specified performance criteria, several feedback gains are designed and

used for every subsystem. The overall gain associated with each of the subsystems, is then determined by a fuzzy blending of such gains, so that a better closed-loop performance can be achieved. The required membership functions are chosen based on some pre-specified performance indices, for example, a faster response or a smaller control input. In general, the rest of the method for calculating the overall state feedback gain remains similar to the conventional PDC method, as in (14) and (15). Figure 4, depicts the general framework for the proposed method, through which and depending on various performance criteria, different characteristics for the controller can be specified. For example, two different feedback gains could be designed for a typical subsystem; one providing a lower control input with a longer settling time response, and the other a faster response but with a larger control input. The idea is then to select the overall feedback gain for this subsystem as a weighted sum of such gains, where the weights are appropriately adjusted, in a fuzzy sense, during the time evolution of the system response, so that as a whole, a faster response with a lower control input can be achieved. For this purpose, when the magnitude of the control input becomes large, the relative weight of the first feedback gain is increased, so that the magnitude of the control input is kept within the permissible limits. On the other hand, when the control input is well below the permissible limit, the weight of the second feedback gain is increased, for a faster response. The dynamics of the resulting closed-loop control system can be analyzed as follows:

Consider the following Takagi–Sugeno model of the plant

$$\dot{\mathbf{x}} = \sum\_{i=1}^{r} h\_1(\mathbf{z}) \left\{ A\_i \mathbf{x}\left(t\right) + B\_i \boldsymbol{\mu}\left(t\right) \right\} \tag{39}$$

Fuzzy Control Systems: LMI-Based Design 453

(42)

**Figure 4.** General methodology in the proposed PDC method

**Figure 5.** A single link robot with a flexible joint

where , , *i j ij h h*

introduced in (Spong et al., 1987).

**Example 2** 

*T iin iin*

*G A BK ijn i i jn* .

*G P PG*

**Lemma:** The fuzzy control system (39), with the control strategy (41) is globally,

0

 

Consider a single link robot with flexible joint as in Figure 5. This benchmark problem is

2 2

*GG GG P P*

0

asymptotically stable, if there exists a common positive definite matrix P such that

*T ijn jin ijn jin*

The following structure is proposed for the fuzzy controller rules

$$\begin{aligned} \text{if the rule : If } Z\_1(t) \text{ is } M\_{i1} \text{ and } Z\_2(t) \text{ is } M\_{i2}, \dots, \dots, Z\_p(t) \text{ is } M\_{ip}, \newline J(t) \text{ is } H\_{i1}, \dots \text{ and } \newline J(t) \text{ is } H\_{iq} \\ \text{then } u\_i(t) = \left\{ \sum\_{n=1}^q m\_{in} \left( J(t) \right) \mathbf{K}\_{in} \right\} \mathbf{x}(t) \end{aligned} \tag{40}$$

Where *i r* 1,2,..., , *<sup>i</sup> q* is the number of gain coefficients in the *i*th subsystem, *min* is the relevant membership degree for J(t), *Kin* is the *n*th state feedback gain associated with the *i*th subsystem, *Hiq* is the n th membership function for J(t), defined in the ith rule. Here *J t* is a term depicting a selected performance index, for instance, if one wants to limit the magnitude of the control signal *u t*( ) , then *Jt ut* ( ) . Where the control input generated by the PDC controller is in the form of

$$\begin{aligned} \mathbf{u}\left(t\right) &= \sum\_{i=1}^{r} h\_i\left(z\right) \boldsymbol{\mu}\_i\left(t\right) = -\left\{\sum\_{n=1}^{r} h\_i\left(z\right) \mathbf{K}\_i \right\} \mathbf{x}\left(t\right) \\ \mathbf{K}\_i &= \sum\_{n=1}^{q} m\_{in}\left(f\left(t\right)\right) \mathbf{K}\_{in} \end{aligned} \tag{41}$$

**Figure 4.** General methodology in the proposed PDC method

**Lemma:** The fuzzy control system (39), with the control strategy (41) is globally, asymptotically stable, if there exists a common positive definite matrix P such that

$$\begin{aligned} \mathbf{G}\_{i\dot{m}} \,^T \mathbf{P} + \mathbf{P} \mathbf{G}\_{i\dot{m}} &< \mathbf{0} \\ \left( \frac{\mathbf{G}\_{i\dot{m}} + \mathbf{G}\_{j\dot{m}}}{\mathbf{2}} \right)^T \mathbf{P} + \mathbf{P} \left( \frac{\mathbf{G}\_{i\dot{m}} + \mathbf{G}\_{j\dot{m}}}{\mathbf{2}} \right) &\le \mathbf{0} \end{aligned} \tag{42}$$

where , , *i j ij h h G A BK ijn i i jn* .

#### **Example 2**

452 Fuzzy Controllers – Recent Advances in Theory and Applications

Consider the following Takagi–Sugeno model of the plant

1

by the PDC controller is in the form of

*u t m Jt K xt*

 

*q i in in n*

then ( )

1

*i*

The following structure is proposed for the fuzzy controller rules

*r*

used for every subsystem. The overall gain associated with each of the subsystems, is then determined by a fuzzy blending of such gains, so that a better closed-loop performance can be achieved. The required membership functions are chosen based on some pre-specified performance indices, for example, a faster response or a smaller control input. In general, the rest of the method for calculating the overall state feedback gain remains similar to the conventional PDC method, as in (14) and (15). Figure 4, depicts the general framework for the proposed method, through which and depending on various performance criteria, different characteristics for the controller can be specified. For example, two different feedback gains could be designed for a typical subsystem; one providing a lower control input with a longer settling time response, and the other a faster response but with a larger control input. The idea is then to select the overall feedback gain for this subsystem as a weighted sum of such gains, where the weights are appropriately adjusted, in a fuzzy sense, during the time evolution of the system response, so that as a whole, a faster response with a lower control input can be achieved. For this purpose, when the magnitude of the control input becomes large, the relative weight of the first feedback gain is increased, so that the magnitude of the control input is kept within the permissible limits. On the other hand, when the control input is well below the permissible limit, the weight of the second feedback gain is increased, for a faster response. The dynamics of the resulting closed-loop control system can be analyzed as follows:

<sup>1</sup>

Where *i r* 1,2,..., , *<sup>i</sup> q* is the number of gain coefficients in the *i*th subsystem, *min* is the relevant membership degree for J(t), *Kin* is the *n*th state feedback gain associated with the *i*th subsystem, *Hiq* is the n th membership function for J(t), defined in the ith rule. Here *J t* is a term depicting a selected performance index, for instance, if one wants to limit the magnitude of the control signal *u t*( ) , then *Jt ut* ( ) . Where the control input generated

*u t h zu t h z K x t*

*ii i i*

(41)

1 1

*i n*

*r r*

*i in in*

*K m Jt K*

1

*n*

*q*

*Zt M Zt Z t M Jt H Jt H*

*i i p ip i iq*

(40)

*x h z Axt But*

1 12 2 1

i th rule : If is and is M ,......., is , is ,....and is

*i i*

(39)

Consider a single link robot with flexible joint as in Figure 5. This benchmark problem is introduced in (Spong et al., 1987).

**Figure 5.** A single link robot with a flexible joint

The state space equations for the system of Figure 4 are

$$\begin{cases}
\dot{\mathbf{x}}\_1 = \mathbf{x}\_3(t) \\
\dot{\mathbf{x}}\_2 = \mathbf{x}\_4(t) \\
\dot{\mathbf{x}}\_3 = \frac{1}{I} (k(\mathbf{x}\_2(t) - \mathbf{x}\_1(t)) - m\mathbb{g}\mathbf{L}\sin(\mathbf{x}\_1(t))) \\
\dot{\mathbf{x}}\_4 = \frac{1}{I} (u(t) - k(\mathbf{x}\_2(t) - \mathbf{x}\_1(t)))
\end{cases} \tag{43}$$

Fuzzy Control Systems: LMI-Based Design 455

0

0 , 0

*B* 

1

(48)

(50)

) ( 51

Assume *k Nm rad* 100 / , <sup>2</sup> *g* 9.8 / *m s* and other parameters are assumed unity then we

2

1 1

2 2

*A*

If z is ,then ( ) ( )

**1**

*t M z ut Fxt*

If z is ,then ( ) ( )

() () () () *i i*

Using conditions (27) and (28) the stable controller can be obtained by solving below

T T T T T T 1 1 2 2 2 21 1

[-495.76 668.96 14.112 47.388] [-497.23 671.34 14.356 47.552] 42.1464 -50.7108 -1.5337 -3.2007 -50.7108 68.9721 2.4898 4.3456 -1.5337 2.4898 0.2554 0.1719 -3.2007 4.3456 0.1719 0.3527

Using conditions (31) and (32) the stable controller can be obtained by solving the

Figures 6 and 7 show the response of the system and control effort, respectively.

X A A X X A A X M B B M M B B M 0

*u t hFx t h Fx t h F x t*

*t M z ut Fxt*

0 0 10

0 0 01 , 0 100 0 0

100 100 0 0

11 22

(49)

0 0 10

0 0 01 , 109.8 100 0 0

100 100 0 0

Control Rule :

Control Rule 2 :

2

1

*i*

T T 11 1 1 T T 22 2 2

X A A X M B B M 0,

Using the MATLAB LMI Control Toolbox we obtain

1 2  

*F F*

*P*

X A A X M B B M 0,

have

1

*A*

The final output of the controller is

**Case 1: Stable controller design** 

X 0

**Case 2: The decay rate** 

conditions:

conditions

In order to apply the PDC methodology, the fuzzy Takagi-Sugeno Model is developed first (Seidi & Markazi, 2008). The nonlinear expression *Z sin x t* <sup>1</sup> , for <sup>1</sup> *x t pi pi* [ ,] , can be expressed as

$$z = \sin\left(\mathbf{x}\_1(t)\right) = \left(\sum\_{i=1}^2 M\_i\left(z\right)b\_i\right)\mathbf{x}\_1(t) \tag{44}$$

Where, 1 2 *b b* 1, 0 and, hence, the membership functions for *z* are obtained as

$$\begin{aligned} M\_1(z) &= \begin{cases} \frac{z}{\sin^{-1} z}, & z(t) \neq 0 \\ 1, & \text{Otherwise} \end{cases} \\ M\_2(z) &= \begin{cases} \frac{\sin^{-1} z - z}{\sin^{-1} z}, & z(t) \neq 0 \\ 1, & \text{Otherwise} \end{cases} \end{aligned} \tag{45}$$

The resulting fuzzy model would then have the following fuzzy rules:

$$\begin{aligned} \text{Rule 1: If } z(t) \text{ is } M\_1(z) \text{, then } \dot{x}(t) &= A\_1 x(t) + B\_1 u(t) \\ \text{Rule 1: If } z(t) \text{ is } M\_2(z) \text{, then } \dot{x}(t) &= A\_2 x(t) + B\_2 u(t) \end{aligned} \tag{45}$$

Where,

$$A\_1 = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \frac{-k - mgLb\_1}{I} & \frac{k}{I} & 0 & 0 \\ \frac{k}{I} & -\frac{k}{I} & 0 & 0 \end{bmatrix}, \ A\_2 = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \frac{-k - mgLb\_2}{I} & \frac{k}{I} & 0 & 0 \\ \frac{k}{I} & -\frac{k}{I} & 0 & 0 \end{bmatrix} \tag{46}$$

and

$$B\_1 = B\_2 = B = \begin{bmatrix} 0 \ 0 \ 0 \ 0 \ 1 \end{bmatrix}^T. \tag{47}$$

Assume *k Nm rad* 100 / , <sup>2</sup> *g* 9.8 / *m s* and other parameters are assumed unity then we have

$$A\_1 = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -109.8 & 100 & 0 & 0 \\ 100 & -100 & 0 & 0 \end{bmatrix}, \ A\_2 = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 100 & 0 & 0 \\ 100 & -100 & 0 & 0 \end{bmatrix}, B = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix},$$

$$\text{Control Rule } \mathbf{1}:$$

$$\begin{aligned} \text{If } \mathbf{z}(t) \text{ is } M\_1(\mathbf{z}) \text{, then } \boldsymbol{\mu}(t) = -F\_1 \mathbf{x}(t) \\ \text{Control Rule } \mathbf{2}: \\ \text{If } \mathbf{z}(t) \text{ is } M\_2(\mathbf{z}) \text{, then } \boldsymbol{\mu}(t) = -F\_2 \mathbf{x}(t) \end{aligned} \tag{48}$$

The final output of the controller is

*A*

454 Fuzzy Controllers – Recent Advances in Theory and Applications

The state space equations for the system of Figure 4 are

 

 

 

expressed as

Where,

and

 1 1

*k mgLb <sup>k</sup> <sup>A</sup>*

*x xt x xt*

*J*

1

2

The resulting fuzzy model would then have the following fuzzy rules:

0 0 10 0 0 01

*I I k k J J*

 1 : If is ,then 1 : If is ,then

 2 1 1 1 sin *i i i z x t M zb x t* 

, 0

, 0

 

2

*I I k k J J*

*k mgLb <sup>k</sup> <sup>A</sup>*

0 0 10 0 0 01

1, Otherwise

1, Otherwise

 1 1 1 2 2 2

(43)

(44)

(45)

(45)

0 0 ,

(46)

0 0

(47)

In order to apply the PDC methodology, the fuzzy Takagi-Sugeno Model is developed first (Seidi & Markazi, 2008). The nonlinear expression *Z sin x t* <sup>1</sup> , for <sup>1</sup> *x t pi pi* [ ,] , can be

<sup>1</sup> ( )

4 2 1

Where, 1 2 *b b* 1, 0 and, hence, the membership functions for *z* are obtained as

 

> 

*M z Sin z*

1

*<sup>z</sup> z t*

*Rule z t M z x t A x t B u t Rule z t M z x t A x t B u t*

0 0 ,

2

1 2 0,0,0,1 . *<sup>T</sup> BBB*

0 0

*Sin z z z t M z Sin z*

1 1

 

*x ut kx t x t*

<sup>1</sup> ( )

3 21 1

*x k x t x t mgLsin x t <sup>I</sup>*

$$\mathbf{u}(t) = -\sum\_{i=1}^{2} h\_i F\_i \mathbf{x}(t) = h\_1 F\_1 \mathbf{x}(t) + h\_2 F\_2 \mathbf{x}(t) \tag{49}$$

#### **Case 1: Stable controller design**

Using conditions (27) and (28) the stable controller can be obtained by solving below conditions

$$\begin{aligned} &\mathbf{X} > 0\\ &\mathbb{E}\left[-\mathbf{X}\,\mathbf{A}\_{1} - \mathbf{A}\_{1}\mathbf{X} + \mathbf{M}\_{1}^{\mathrm{T}}\mathbf{B}^{\mathrm{T}} + \mathbf{B}\,\mathbf{M}\_{1}\right] > 0, \\ &\mathbb{E}\left[-\mathbf{X}\,\mathbf{A}\_{2} - \mathbf{A}\_{2}\mathbf{X} + \mathbf{M}\_{2}^{\mathrm{T}}\mathbf{B}^{\mathrm{T}} + \mathbf{B}\,\mathbf{M}\_{2}\right] > 0, \\ &\left[-\mathbf{X}\,\mathbf{A}\_{1}^{\mathrm{T}} - \mathbf{A}\_{1}\mathbf{X} - \mathbf{X}\,\mathbf{A}\_{2}^{\mathrm{T}} - \mathbf{A}\_{2}\mathbf{X} + \mathbf{M}\_{2}^{\mathrm{T}}\mathbf{B}^{\mathrm{T}} + \mathbf{B}\,\mathbf{M}\_{2} + \mathbf{M}\_{1}^{\mathrm{T}}\mathbf{B}^{\mathrm{T}} + \mathbf{B}\,\mathbf{M}\_{1}\right] > 0. \end{aligned} \tag{50}$$

Using the MATLAB LMI Control Toolbox we obtain

$$\begin{aligned} F\_1 &= \begin{bmatrix} -495.76 & 668.96 & 14.112 & 47.388 \end{bmatrix} \\ F\_2 &= \begin{bmatrix} -497.23 & 671.34 & 14.356 & 47.552 \end{bmatrix} \\ P &= \begin{bmatrix} 42.1464 & -50.7108 & -1.5337 & -3.2007 \\ -50.7108 & 68.9721 & 2.4898 & 4.3456 \\ -1.5337 & 2.4898 & 0.2554 & 0.1719 \\ -3.2007 & 4.3456 & 0.1719 & 0.3527 \end{bmatrix} \end{aligned}$$

Figures 6 and 7 show the response of the system and control effort, respectively.

#### **Case 2: The decay rate**

Using conditions (31) and (32) the stable controller can be obtained by solving the conditions:

Fuzzy Control Systems: LMI-Based Design 457

(52)

) ( 53

T T T 11 1 1 T T T 22 2 2 T T T T 11 22 2

 

[ X A A X M B B M 2 X] 0 [ X A A X M B B M 2 X] 0 [ X A A X X A A X M B B M M B B M 4 X] 0

10 and by using the MATLAB LMI Control Toolbox we obtain:

[4108.8 6545.2 1271.3 127.77] [4066.9 6502.6 1261.7 127.1] 36.5087 24.0140 6.2135 0.3352 24.0140 30.1341 6.3223 0.5013 6.2135 6.3223 1.4260 0.0995 0.3352 0.5013 0.0995 0.0099

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> -0.2

 

T T 21 1

Figures 8 and 9 show the response of the system and control effort, respectively.

1 2  

*F F*

*P*

Considering

0

**Figure 8.** Response of flexible joint robots x1(t) , case 2.

0.2

0.4

0.6

0.8

1

1.2

**Figure 6.** Response of flexible joint robots x1(t), case 1.

**Figure 7.** Control input for flexible joint robots, case 1.

$$\begin{aligned} & \left[ -\mathbf{X} \,\mathbf{A}\_1^\top - \mathbf{A}\_1 \mathbf{X} + \mathbf{M}\_1^\top \mathbf{B}^\top + \mathbf{B} \,\mathbf{M}\_1 - 2a\mathbf{X} \right] > 0 \\ & \left[ -\mathbf{X} \,\mathbf{A}\_2^\top - \mathbf{A}\_2 \mathbf{X} + \mathbf{M}\_2^\top \mathbf{B}^\top + \mathbf{B} \,\mathbf{M}\_2 - 2a\mathbf{X} \right] > 0 \\ & \left[ -\mathbf{X} \,\mathbf{A}\_1^\top - \mathbf{A}\_1 \mathbf{X} - \mathbf{X} \,\mathbf{A}\_2^\top - \mathbf{A}\_2 \mathbf{X} + \mathbf{M}\_2^\top \mathbf{B}^\top \right] \\ & + \mathbf{B} \,\mathbf{M}\_2 + \mathbf{M}\_1^\top \mathbf{B}^\top + \mathbf{B} \,\mathbf{M}\_1 - 4a\mathbf{X} \right] > 0 \end{aligned} \tag{52}$$

Considering 10 and by using the MATLAB LMI Control Toolbox we obtain:

456 Fuzzy Controllers – Recent Advances in Theory and Applications

**Figure 6.** Response of flexible joint robots x1(t), case 1.

**Figure 7.** Control input for flexible joint robots, case 1.

0

0

50

100

u(t) (N.m)

150

200

250

0.2

0.4

0.6

X1(t) (rad)

0.8

1

1.2

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> 1.2 1.4 1.6 1.8 <sup>2</sup> -0.2

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> 1.2 1.4 1.6 1.8 <sup>2</sup> -50

$$\begin{aligned} F\_1 &= \begin{bmatrix} 4108.8 & 6545.2 & 1271.3 & 127.77 \end{bmatrix} \\ F\_2 &= \begin{bmatrix} 4066.9 & 6502.6 & 1261.7 & 127.1 \end{bmatrix} \\ P &= \begin{bmatrix} 36.5087 & 24.0140 & 6.2135 & 0.3352 \\ 24.0140 & 30.1341 & 6.3223 & 0.5013 \\ 6.2135 & 6.3223 & 1.4260 & 0.0995 \\ 0.3352 & 0.5013 & 0.0995 & 0.0099 \end{bmatrix} \end{aligned}$$

Figures 8 and 9 show the response of the system and control effort, respectively.

**Figure 8.** Response of flexible joint robots x1(t) , case 2.

**Figure 9.** Control input for flexible joint robots, case 2.

#### **Case 3: The decay rate with the constraint on the input**

We design a stable fuzzy controller by considering the decay rate and the constraint on the control input. The design problem of the FJR is defined as follows:

Maximize 

$$\begin{aligned} &X > 0\\ &\left[-XA\_1^T - A\_1X + M\_1^TR^T + BM\_1 - 2\alpha X\right] > 0\\ &\left[-XA\_2^T - A\_2X + M\_2^TR^T + BM\_2 - 2\alpha X\right] > 0\\ &\left[-XA\_1^T - A\_1X - XA\_2^T - A\_2X + M\_2^TR^T\right] > 0\\ &+ BM\_2 + M\_1^TR^T + BM\_1 - 4\alpha X\right] > 0\\ &\left[\begin{array}{c} \text{X} & \text{M}\_1^\text{T}\\ \text{M}\_1 & \mu^2\text{I} \end{array}\right] > 0\\ &\left[\begin{array}{c} \text{X} & \text{M}\_2^\text{T}\\ \text{M}\_2 & \mu^2\text{I} \end{array}\right] > 0\\ &\left[\begin{array}{c} \text{X} - \phi^2\text{I} \end{array}\right] > 0\end{aligned} \tag{5}$$

Fuzzy Control Systems: LMI-Based Design 459

(55)

1 2  

**Figure 10.** System responses of the single-link flexible joint, case 3.

*F F*

0


**Figure 11.** Control input for flexible joint robots, case 3.

0

500

1000

1500

2000

2500

0.2

0.4

0.6

0.8

1

1.2

*P*

0.7301 0.32486 0.096794 0.0034552 0.32486 0.55483 0.10616 0.010209 0.096794 0.10616 0.023049 0.0017139 0.0034552 0.010209 0.0017139 0.00023565

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> -0.2

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> -1000

[327.57 1745 261.86 57.475] [356.05 1739.2 259.77 57.5]

Figures 10 and 11 show the response of the system and control effort, respectively.

Where <sup>1</sup> i i *X P FX* , M , 4600 , 1 .

Using the MATLAB LMI toolbox to solve the LMI conditions (50), we can get the positive definite matrix and a set of gains (51), that make the system stable.

$$\alpha = 0.072401$$

$$P = \begin{bmatrix} 0.7301 & 0.32486 & 0.096794 & 0.0034552\\ 0.32486 & 0.55483 & 0.10616 & 0.010209\\ 0.096794 & 0.10616 & 0.023049 & 0.0017139\\ 0.0034552 & 0.010209 & 0.0017139 & 0.00023565 \end{bmatrix}$$
 
$$\begin{aligned} F\_1 &= [327.57 & 1745 & 261.86 & 57.475] \\ F\_2 &= [356.05 & 1739.2 & 259.77 & 57.5] \end{aligned} \tag{55}$$

Figures 10 and 11 show the response of the system and control effort, respectively.

**Figure 10.** System responses of the single-link flexible joint, case 3.

458 Fuzzy Controllers – Recent Advances in Theory and Applications

**Figure 9.** Control input for flexible joint robots, case 2.

**Case 3: The decay rate with the constraint on the input** 

control input. The design problem of the FJR is defined as follows:

0

*X*

T 1 2

0

0

0

1 .

T 2 2

1

X M

 

M I X M

definite matrix and a set of gains (51), that make the system stable.

M I

2 2

4600 ,

*X I*


Maximize

Where <sup>1</sup>

i i *X P FX* , M ,

0

2000

4000

u(t) (N.m)

6000

8000

10000

12000

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> -4000

We design a stable fuzzy controller by considering the decay rate and the constraint on the

2 0

2 0

(54)

11 1 1

*XA A X M B BM X*

*XA A X M B BM X*

 

*T T T*

*T T T*

22 2 2

11 22 2 21 1

*XA A X XA A X M B*

0 4

Using the MATLAB LMI toolbox to solve the LMI conditions (50), we can get the positive

0.072401

*BM M B BM X*

*T T T T T T*

**Figure 11.** Control input for flexible joint robots, case 3.

#### **Case 4: Performance-oriented parallel distributed compensation**

The following stabilizing feedback gains are chosen using the pole placement method, so that *K*11 and *K*21 produce large magnitude inputs for subsystems 1 and 2, respectively, and *K*22 and *K*21 induce low magnitude inputs for those subsystems. In particular,

$$\begin{aligned} K\_{11} &= \begin{bmatrix} 6667.2 & 4411.9 & 1052.4 & 92.6 \end{bmatrix} \\ K\_{12} &= \begin{bmatrix} -33.321 & 1413.7 & 191.63 & 51.2 \end{bmatrix} \\ K\_{21} &= \begin{bmatrix} 6658.7 & 4332.4 & 1025.4 & 91.1 \end{bmatrix} \\ K\_{22} &= \begin{bmatrix} 72.3 & 1389.8 & 189.6 & 50.6 \end{bmatrix} \end{aligned} \tag{56}$$

Fuzzy Control Systems: LMI-Based Design 461

Applying a unit step reference signal for <sup>1</sup> *x t*( ) , the response history and the corresponding control input are shown in Figures (13) and (14), respectively. Simulation results are

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

investigated for the following three controllers:

**Figure 13.** Response of flexible joint robot x1(t), case 4.

NPDC

LPDC


0

0.2

0.4

x1(t) (rad)

0.6

0.8

1

HPDC

1.2

**Figure 14.** Control input for flexible joint robot, case 4.

LPDC

NPDC



0

2000

4000

u(t) (N.m)

6000

8000

HPDC

10000

12000

The required simple membership functions are selected as in Figure 12, so that, with a decrease in the corresponding plant input, in subsystems 1 and 2 respectively, the overall feedback gains come closer to *K*11 and *K*<sup>21</sup> , and with an increase in the corresponding control input respectively, the overall feedback gains come closer to *K*21 and *K*<sup>22</sup> . Now, the fuzzy rules for the controller are constructed as follows:

Rule 1: If *z t* is *M z* <sup>1</sup> and *u t* is "small" then <sup>11</sup> *ut K xt* Rule 2: If *z t* is *M z* <sup>1</sup> and *u t* is "large" then <sup>12</sup> *ut K xt* Rule 3: If *z t* is *M z* <sup>2</sup> and *u t* is "small" then <sup>21</sup> *ut K xt* Rule 4: If *z t* is *M z* <sup>2</sup> and *u t* is "large" then <sup>22</sup> *ut K xt*

**Figure 12.** Membership functions for the control effort in the flexible joint robots.

A common positive definite matrix, P, satisfying the stability conditions (42) is obtained by solving the LMI problems:

$$P = 10^4 \times \begin{bmatrix} 121710 & 15858 & 2558.5 & 63.525 \\ 15858 & 8624.4 & 1458.4 & 105.36 \\ 2558.5 & 1458.4 & 702.24 & 42.529 \\ 63.525 & 105.36 & 42.529 & 5.0962 \end{bmatrix}$$

Applying a unit step reference signal for <sup>1</sup> *x t*( ) , the response history and the corresponding control input are shown in Figures (13) and (14), respectively. Simulation results are investigated for the following three controllers:

**Figure 13.** Response of flexible joint robot x1(t), case 4.

460 Fuzzy Controllers – Recent Advances in Theory and Applications

**Case 4: Performance-oriented parallel distributed compensation** 

*K K K K*

fuzzy rules for the controller are constructed as follows:

Rule 1: If *z t* is *M z* <sup>1</sup> and *u t* is "small" then <sup>11</sup> *ut K xt*

Rule 2: If *z t* is *M z* <sup>1</sup> and *u t* is "large" then <sup>12</sup> *ut K xt*

Rule 3: If *z t* is *M z* <sup>2</sup> and *u t* is "small" then <sup>21</sup> *ut K xt*

Rule 4: If *z t* is *M z* <sup>2</sup> and *u t* is "large" then <sup>22</sup> *ut K xt*

**Figure 12.** Membership functions for the control effort in the flexible joint robots.

4

10

*P*

solving the LMI problems:

A common positive definite matrix, P, satisfying the stability conditions (42) is obtained by

121710 15858 2558.5 63.525 15858 8624.4 1458.4 105.36

2558.5 1458.4 702.24 42.529 63.525 105.36 42.529 5.0962

The following stabilizing feedback gains are chosen using the pole placement method, so that *K*11 and *K*21 produce large magnitude inputs for subsystems 1 and 2, respectively, and

> 6667.2 4411.9 1052.4 92.6 -33.321 1413.7 191.63 51.2 6658.7 4332.4 1025.4 91.1 72.3 1389.8 189.6 50.6

(56)

 

The required simple membership functions are selected as in Figure 12, so that, with a decrease in the corresponding plant input, in subsystems 1 and 2 respectively, the overall feedback gains come closer to *K*11 and *K*<sup>21</sup> , and with an increase in the corresponding control input respectively, the overall feedback gains come closer to *K*21 and *K*<sup>22</sup> . Now, the

*K*22 and *K*21 induce low magnitude inputs for those subsystems. In particular,

**Figure 14.** Control input for flexible joint robot, case 4.

1. A PDC controller with feedback gains *K*11 and *K*21 providing a high speed response, and with possible high control inputs (HPDC controller).

Fuzzy Control Systems: LMI-Based Design 463

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It is observed that the new controller provides a settling time similar to the HPDC case, with a much lower magnitude for the control input.

## **5. Conclusion**

This chapter deals with approximation of the nonlinear system using Takagi-Sugeno (T-S) models with linear models as rule consequences and a construction procedure of T-S models. Also, the stability conditions and stabilizing control design of parallel distributed compensation (PDC) are discussed. It is seen that PDC a linear control method can be used to control the nonlinear system. Moreover, the stability analysis and control design problems for both continuous and discrete fuzz control systems can be transformed to linear matrix inequality (LMI) problems and they can be solved efficiently by convex programming techniques for LMIs. Design examples demonstrate the effectiveness of the LMI-based designs.

## **Author details**

Morteza Seidi, Marzieh Hajiaghamemar and Bruce Segee *University of Maine, USA* 

## **6. References**


Ichikawa, A. et al. (1993). *Control Hand Book*, Ohmu Publisher, Tokyo, in Japanese.

462 Fuzzy Controllers – Recent Advances in Theory and Applications

level of control input (NPDC controller).

a much lower magnitude for the control input.

Morteza Seidi, Marzieh Hajiaghamemar and Bruce Segee

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*Fuzzy Sets and Systems,* Vol. 57, No. 2, (July 1993) p.p. 125–140.

**5. Conclusion** 

**Author details** 

**6. References** 

*University of Maine, USA* 

and with possible high control inputs (HPDC controller).

1. A PDC controller with feedback gains *K*11 and *K*21 providing a high speed response,

2. A PDC controller with feedback gains *K*22 and *K*21 providing a low speed response, and with a lower control input, as compared with the HPDC case (LPDC controller). 3. Proposed modified PDC controller, providing a fast response, yet with an acceptable

It is observed that the new controller provides a settling time similar to the HPDC case, with

This chapter deals with approximation of the nonlinear system using Takagi-Sugeno (T-S) models with linear models as rule consequences and a construction procedure of T-S models. Also, the stability conditions and stabilizing control design of parallel distributed compensation (PDC) are discussed. It is seen that PDC a linear control method can be used to control the nonlinear system. Moreover, the stability analysis and control design problems for both continuous and discrete fuzz control systems can be transformed to linear matrix inequality (LMI) problems and they can be solved efficiently by convex programming techniques for LMIs. Design examples demonstrate the effectiveness of the LMI-based designs.

Bonissone, P. P., Badami, V., Chaing, K.H., Khedkar, P.S., Marcelle, K. W. & Schutten, M. J. (1995). Industrial applications of fuzzy logic at general electric, *Proceedings of IEEE*, Vol.

Boyd, S., Ghaoui, L. E. & Feron, Eric & Balakrishnan V. (1994). *Linear Matrix Inequalities in System and Control Theory*, SIAM studies in applied mathematics, ISBN 0-89871-334X. Chen , C. L., Chen, P. C. & Chen, C. K. ( 1993). Analysis and design of fuzzy control system,

Ding, B. C., Sun, H. X. & Yang, P. (2006). Further study on LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi–Sugeno's form, *Automatica*,

Fang, C. H., Liu, Y. S., Kau, S. W., Hong, L. & Lee, C. H. (2006). A new LMI-based approach to relaxed quadratic stabilization of T–S fuzzy control systems, *IEEE Transactions on* 

Hong, S.K. & Langari, R. (2000). Robust fuzzy control of a magnetic bearing system subject to harmonic disturbances, *IEEE Transactions on Control System Technology*, Vol. 8, No. 2,

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Tanaka, K. & Sugeno, M. (1990). Stability Analysis of Fuzzy Systems Using Lyapunov's Direct Method, *Proceedings of the North America Fuzzy Information Processing Society NAFIPS'90*, Vol. 1, pp. 133-136, Toronto, Canada, June 1990.

**Chapter 0**

**Chapter 19**

**New Results on Robust** H<sup>∞</sup> **Filter for**

**Uncertain Fuzzy Descriptor Systems**

"parasitic" parameter, typically small time constants, masses, etc.

University of Technology Thonburi, 126 Prachautits Rd., Bangkok 10140, Thailand.

cited.

the "slow" and "fast" modes of the system.

The problem of filter design for descriptor systems system has been intensively studied by a number of researchers for the past three decades; see Ref.[1]-[6]. This is due not only to theoretical interest but also to the relevance of this topic in control engineering applications. Descriptor systems or so called singularly perturbed systems are dynamical systems with multiple time-scales. Descriptor systems often occur naturally due to the presence of small

The main purpose of the singular perturbation approach to analysis and design is the alleviation of high dimensionality and ill-conditioning resulting from the interaction of slow and fast dynamics modes. The separation of states into slow and fast ones is a nontrivial modelling task demanding insight and ingenuity on the part of the analyst. In state space, such systems are commonly modelled using the mathematical framework of singular perturbations, with a small parameter, say *ε*, determining the degree of separation between

In the last few years, many researchers have studied the H<sup>∞</sup> filter design for a general class of linear descriptor systems. In Ref.[3], the authors have investigated the decomposition solution of H<sup>∞</sup> filter gain for singularly perturbed systems. The reduced-order H<sup>∞</sup> optimal filtering for system with slow and fast modes has been considered in Ref.[4]. Although many researchers have studied linear descriptor systems for many years, the H<sup>∞</sup> filtering design for nonlinear descriptor systems remains as an open research area. This is because, in general, nonlinear singularly perturbed systems can not be easily separated into slow and fast subsystems.

Fuzzy system theory enables us to utilize qualitative, linguistic information about a highly complex nonlinear system to construct a mathematical model for it. Recent studies show

\*W. Assawinchaichote is with the Department of Electronic and Telecommnunication Engineering, King Mongkut's

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

©2012 Assawinchaichote, 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,

Additional information is available at the end of the chapter

Wudhichai Assawinchaichote\*

http://dx.doi.org/10.5772/47142

**1. Introduction**

