**A New Methodology for Tuning PID‐Type Fuzzy Logic Controllers Scaling Factors Using Genetic Algorithm of a Discrete‐Time System**

Wafa Gritli, Hajer Gharsallaoui and Mohamed Benrejeb

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.69604

#### Abstract

[3] Dintsis D, Simankov V. The control automation system method based on combined implementation of fuzzy logic and discrete automation systems. In: European Modelling Symposium EMS2013 - 03-C Methodologies, Tools and Operations Research; November 2013; Manchester. Print ISBN:978-1-4799-2577-3 INSPEC Accession Number:14199480. DOI:

[4] Ferdinand D.S. Flexible Learning Environments-Theories-Trends-Issues. Monography. University of West-Indies. West Indies. St.-Augustine. DOI: 10.13140/RG.2.1.3958.2488.

[5] Dintsis D. Implementing Fuzzy Sets For "Big Data" Analysis Based on Large Training Centre Feedbacks. In: IEEE AEIT Annual 2015 Conference. Naples, Italy. Publisher: IEEE.

[6] Dintsis D, Bredikhin A. Virtual Learning for People with Hearing Impairs. In: IEEE AEIT

[7] LERN International Award Winners 2016. https://www.flipsnack.com/LearningResourcesNetwork/2016-lern-international-award-winners.html. 2016. LERN. Chicago. [Accessed:

Annual 2015 Conference. Naples, Italy. Publisher: IEEE. October, 2015.

10.1109/EMS.2013.15. IEEE. p. 87–90

88 Modern Fuzzy Control Systems and Its Applications

2016. 32p

October, 2015.

2017-01-11]

In this chapter, a proportional-integral derivative (PID)-type fuzzy logic controller (FLC) is proposed for a discrete-time system in order to track a desired trajectory generated using the flatness property. In order to improve the performance of the proposed controller, genetic algorithm (GA) based on minimizing the integral of the squared error (ISE) is used for tuning the input and output PID-type FLC scaling factors online. The considered controller is applied to an electronic throttle valve (ETV). GA tuning shows a better and robust performance compared to Simulink design optimization (SDO) algorithm in terms of tracking a desired trajectory with disturbances rejection.

Keywords: PID-type FLC, scaling factors, genetic algorithm, integral of the squared error, Simulink design optimization technique, flatness, electronic throttle valve

## 1. Introduction

Fuzzy logic control (FLC) has been widely used in many successful industrial applications. The first FLC algorithm was implemented by Mamdani in 1974. Unlike conventional control, which is based on mathematical model of a plant, an FLC usually embeds the intuition and experience of a human operator and may provide a nonlinear relationship induced by membership functions, rules and defuzzification. In that respect, FLC has been reported to be successfully used for a number of complex and nonlinear systems and are proved to be more robust and their performances are less sensitive to parametric variations than conventional controllers.

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the literature, various types such as proportional integral (PI), proportional derivative (PD) and proportional-integral derivative (PID) of FLCs have been proposed. For example, PI-type FLCs have been successfully implemented in many physical applications such the control of the temperature and pressure of a steam engine and control the steering and speed of an automobile. However, performance of PI-type FLCs for higher order systems and nonlinear systems may be poor due to the large overshoot and the excessive oscillation. PD-type FLCs are suitable for a limited class of systems and they are not recommendable in the presence of measurement noise and sudden load disturbances. Theoretically, PID-type FLCs provide a good performance. However, there are difficulties associated with the generation of an efficient rule base and the tuning of parameters.

In the proposed PID-type FLC, the design of parameters within two groups: structural parameters and tuning parameters. Basically, structural parameters include input/output (I/O) variables to fuzzy inference, fuzzy linguistic sets, membership functions, fuzzy rules, inference mechanism and defuzzification mechanism, which are usually determined during offline design. Tuning parameters include I/O scaling factors (SF) and parameters of membership functions (MF), which can be calculated during online adjustments of the controller in order to enhance the process performance [1].

The appropriate selection of input and output scaling factors is very important because they have significant effects on the dynamic of fuzzy controller. This leads researchers to explore the best method in searching optimum PID-type FLC parameters. Various strategies or methods have been used up to now. In Ref. [2], Qiao and Mizumoto proposed a peak observer mechanism-based method to adjust the PID-type FLC parameters. This self-tuning mechanism decreases the equivalent integral control component of the fuzzy controller gradually with the system response process time. Furthermore, Woo et al. [3] developed a method based on two empirical functions evolved with the system's error information. In Ref. [1], Guzelkaya et al. proposed a technique that adjusts the scaling factors, corresponding to the derivative and integral components, using a fuzzy inference mechanism. However, the major disadvantages of all these PID-type FLC tuning method are the difficult choice of their relative parameters and mechanisms. To overcome these difficulties, differential search algorithm (DSA) meta-heuristic technique is proposed for systematically tuning the scaling factors of the PID-type FLC in Ref. [4]. The fuzzy control design is formulated as a constrained optimization problem, which is efficiently solved based on an improved DSA. In this proposed technique, different optimization criteria such as integral square error (ISE) and maximum overshoot are considered in order to guarantee more robustness and performance control objectives.

In this chapter, a genetic algorithm (GA)-based heuristic optimization technique has been implemented to obtain better performance compared to the Simulink design optimization (SDO) technique. GA is based upon minimizing the error between the output system and the desired trajectory starting from a flat output variable generated using the flatness property. Various performance indices can be used. In this study, the integral of squared error (ISE) index has been used in order to minimize the error between the output and the desired flat trajectory. The methods are applied in a discrete-time framework to an electronic throttle valve as a case of study.

## 2. PID-type fuzzy logic controller description

In the literature, various types such as proportional integral (PI), proportional derivative (PD) and proportional-integral derivative (PID) of FLCs have been proposed. For example, PI-type FLCs have been successfully implemented in many physical applications such the control of the temperature and pressure of a steam engine and control the steering and speed of an automobile. However, performance of PI-type FLCs for higher order systems and nonlinear systems may be poor due to the large overshoot and the excessive oscillation. PD-type FLCs are suitable for a limited class of systems and they are not recommendable in the presence of measurement noise and sudden load disturbances. Theoretically, PID-type FLCs provide a good performance. However, there are difficulties associated with the generation of an effi-

In the proposed PID-type FLC, the design of parameters within two groups: structural parameters and tuning parameters. Basically, structural parameters include input/output (I/O) variables to fuzzy inference, fuzzy linguistic sets, membership functions, fuzzy rules, inference mechanism and defuzzification mechanism, which are usually determined during offline design. Tuning parameters include I/O scaling factors (SF) and parameters of membership functions (MF), which can be calculated during online adjustments of the controller in order

The appropriate selection of input and output scaling factors is very important because they have significant effects on the dynamic of fuzzy controller. This leads researchers to explore the best method in searching optimum PID-type FLC parameters. Various strategies or methods have been used up to now. In Ref. [2], Qiao and Mizumoto proposed a peak observer mechanism-based method to adjust the PID-type FLC parameters. This self-tuning mechanism decreases the equivalent integral control component of the fuzzy controller gradually with the system response process time. Furthermore, Woo et al. [3] developed a method based on two empirical functions evolved with the system's error information. In Ref. [1], Guzelkaya et al. proposed a technique that adjusts the scaling factors, corresponding to the derivative and integral components, using a fuzzy inference mechanism. However, the major disadvantages of all these PID-type FLC tuning method are the difficult choice of their relative parameters and mechanisms. To overcome these difficulties, differential search algorithm (DSA) meta-heuristic technique is proposed for systematically tuning the scaling factors of the PID-type FLC in Ref. [4]. The fuzzy control design is formulated as a constrained optimization problem, which is efficiently solved based on an improved DSA. In this proposed technique, different optimization criteria such as integral square error (ISE) and maximum overshoot are considered in order to guarantee more robustness and performance control

In this chapter, a genetic algorithm (GA)-based heuristic optimization technique has been implemented to obtain better performance compared to the Simulink design optimization (SDO) technique. GA is based upon minimizing the error between the output system and the desired trajectory starting from a flat output variable generated using the flatness property. Various performance indices can be used. In this study, the integral of squared error (ISE) index has been used in order to minimize the error between the output and the desired flat trajectory. The methods are applied in a discrete-time framework to an electronic throttle valve

cient rule base and the tuning of parameters.

90 Modern Fuzzy Control Systems and Its Applications

to enhance the process performance [1].

objectives.

as a case of study.

In this study, we will deal with fuzzy PID-type controllers formed using one PD-type FLC with an integrator at the output.

The PID-type fuzzy logic controller structure is shown in Figure 1 [5], where and are the input scaling factors are α and β the output scaling factors.

The inputs variables, well known as the error ek between the desired trajectory and the measure , as well as the error variation Δek given by Eqs. (1) and (2) where Te is the sampling period.

$$e\_k = \mathcal{y}\_k^d - \mathcal{y}\_k \tag{1}$$

$$
\Delta e\_k = \frac{e\_k - e\_{k-1}}{T\_s} \tag{2}
$$

The output variable of such a controller is the variation of the control signal which can be defined as Eq. (3).

$$
\Delta \boldsymbol{\mu}\_k = \frac{\boldsymbol{\mu}\_k - \boldsymbol{\mu}\_{k-1}}{T\_\varepsilon} \tag{3}
$$

The output of the PID-type fuzzy is given by Eq. (4) [5]

$$\begin{aligned} \mu\_k &= \alpha \Delta \mu\_k + \beta \left\| \, \Delta \mu\_k \, dt \\ &= \alpha (A + PK\_\epsilon e\_\lambda + DK\_\rho \Delta e\_\lambda) \\ &+ \beta \left\| \left( A + PK\_\epsilon e\_\lambda + DK\_\iota \Delta e\_\lambda \right) dt \right. \\ &= \alpha A + \beta A t + (\alpha K\_\epsilon P + \beta K\_\rho D) e\_\lambda \\ &+ \beta K\_\epsilon P \left[ e\_\epsilon dt + \alpha K\_\epsilon D \Delta e\_\lambda \right. \end{aligned} \tag{4}$$

Thus, the equivalent control components of the PID-type FLC are such that

Proportional gain:

Integral gain:

Figure 1. PID-type FLC.


Table 1. Fuzzy rules-base.

Derivative gain:

where the terms and are given by Eqs. (5) and (6) [2].

$$P = \frac{\Delta \mu\_{(l+l)f} - \Delta \mu\_y}{e\_{i+l} - e\_i} \tag{5}$$

$$D = \frac{\Delta \mu\_{\epsilon(f \star 1)} - \Delta \mu\_y}{\Delta e\_{j+1} - \Delta e\_j} \tag{6}$$

The fuzzy controllers with a product-sum inference method, centroid defuzzification method and triangular uniformly distributed membership functions for the inputs and a crisp output proposed in Refs. [2, 6] are used in our case of study.

Table 1 gives the linguistic levels, assigned to the variables ek, Δek and Δuk, as follows: NL: negative large; N: negative; ZR: zero; P: positive; PL: positive large.

### 3. Scaling factors tuning using genetic algorithm

In this work, a new method is proposed for tuning the coefficients of PID-type FLCs. This method adjusts the input scaling factor corresponding to the derivative coefficient and the output scaling factor corresponding to the integral coefficient of the PID-type FLC using genetic algorithm, as shown in Figure 2. The integral of squared error (ISE) index has been used in order to minimize the error between the output and the desired flat trajectory.

#### 3.1. Genetic algorithm

Genetic algorithm was first proposed by Holland [7]. It is a heuristic optimization technique inspired by the mechanism of natural selection. It is used in order to solve highly complex problems. GA starts with an initial population containing a number of parameters, where each one is regarded as the genes of a chromosome and can be structured by a string of concatenated values. Each chromosome represents a solution of the problem and its performance is evaluated based on fitness function.

In the beginning, an initial chromosome population is randomly generated. The chromosomes are candidate solutions to the problem. Then, the fitness values of all chromosomes are evaluated by calculating the objective function. So, a group of the best chromosomes is selected based on the fitness of each individual. In this 'surviving' population, the genetic operators of A New Methodology for Tuning PID‐Type Fuzzy Logic Controllers Scaling Factors Using Genetic Algorithm… http://dx.doi.org/10.5772/intechopen.69604 93

Figure 2. PID-type FLC scaling factors tuning.

ð5Þ

ð6Þ

Derivative gain:

Table 1. Fuzzy rules-base.

92 Modern Fuzzy Control Systems and Its Applications

3.1. Genetic algorithm

mance is evaluated based on fitness function.

where the terms and are given by Eqs. (5) and (6) [2].

proposed in Refs. [2, 6] are used in our case of study.

negative large; N: negative; ZR: zero; P: positive; PL: positive large.

3. Scaling factors tuning using genetic algorithm

The fuzzy controllers with a product-sum inference method, centroid defuzzification method and triangular uniformly distributed membership functions for the inputs and a crisp output

ek\Δek N ZR P N NL N ZR ZR N ZR P P ZR P PL

Table 1 gives the linguistic levels, assigned to the variables ek, Δek and Δuk, as follows: NL:

In this work, a new method is proposed for tuning the coefficients of PID-type FLCs. This method adjusts the input scaling factor corresponding to the derivative coefficient and the output scaling factor corresponding to the integral coefficient of the PID-type FLC using genetic algorithm, as shown in Figure 2. The integral of squared error (ISE) index has been

Genetic algorithm was first proposed by Holland [7]. It is a heuristic optimization technique inspired by the mechanism of natural selection. It is used in order to solve highly complex problems. GA starts with an initial population containing a number of parameters, where each one is regarded as the genes of a chromosome and can be structured by a string of concatenated values. Each chromosome represents a solution of the problem and its perfor-

In the beginning, an initial chromosome population is randomly generated. The chromosomes are candidate solutions to the problem. Then, the fitness values of all chromosomes are evaluated by calculating the objective function. So, a group of the best chromosomes is selected based on the fitness of each individual. In this 'surviving' population, the genetic operators of

used in order to minimize the error between the output and the desired flat trajectory.

crossover and mutation are applied in order to create the next population solution. The above steps are repeated until a specific termination criterion is found.


To compute the fitness of each chromosome, the objective functions are used. Many authors use integral of time multiplied by absolute error (ITAE), mean of the squared error (MSE), integral of absolute error (IAE) and integral of the squared error (ISE) as performance index [9, 10].

In this chapter, the method of tuning PID-type FLC parameters using GA consists in finding the optimal I/O scaling factors, which minimize the defined objective function, chosen as the ISE in order to specify more performance in terms of tracking a desired trajectory.

If is the desired trajectory and is the output trajectory, then error e(t) is

$$e(t) = \boldsymbol{\upnu}^{\boldsymbol{\uprho}}(t) - \boldsymbol{\upnu}(t) \tag{7}$$

and the ISE can be defined by

$$ISE = \bigcup\_{0}^{\circ} e(t)^2 \, dt \tag{8}$$

Fitness function is taken as inverse of error, i.e., performance index.

$$Fitness\ value = \frac{1}{Performance\ index} \tag{9}$$

#### 3.2. Tuning procedure

The overall flowchart for optimization using GA is shown in Figure 3. Initially, a number of populations N have been generated for the scaling factors α and β. Each individual of these N sets in the current population is evaluated using the objective function ISE. Based on the values of the objective function, out of these N possible solutions, the good solutions are retained and the others are eliminated. A new population is formed by applying the genetic operators (reproduction, crossover and mutation) to these selected individuals. This process of

Figure 3. Flowchart of the GA optimization algorithm.

production of a new generation and its evaluation is performed repetitively. The algorithm continues until the population converges to the stop criterion.

## 4. Flatness and trajectory planning

ð9Þ

3.2. Tuning procedure

94 Modern Fuzzy Control Systems and Its Applications

Figure 3. Flowchart of the GA optimization algorithm.

The overall flowchart for optimization using GA is shown in Figure 3. Initially, a number of populations N have been generated for the scaling factors α and β. Each individual of these N sets in the current population is evaluated using the objective function ISE. Based on the values of the objective function, out of these N possible solutions, the good solutions are retained and the others are eliminated. A new population is formed by applying the genetic operators (reproduction, crossover and mutation) to these selected individuals. This process of

The flat property has been introduced in Ref. [11] for continuous-time nonlinear systems. It can be stated in a discrete-time version which leads for the design of a control which ensures a tracking of a desired trajectory. One major property of differential flatness is that the state and the input variables can be directly expressed, without integrating any differential equation, in terms of the flat output and a finite number of its derivatives. The flatness approach will be used in this chapter in a discrete-time framework.

The studied dynamic linear discrete system is described by Eq. (10).

$$A(q)\mathcal{y}\_k = B(q)\mathcal{u}\_k\tag{10}$$

where q is the forward operator, and are the input and the output, respectively, and and are polynomials defined by

$$A(q) = q'' + a\_{n-1}q''^{-1} + \dots + a\_1q + a\_0 \tag{11}$$

$$B(q) = b\_{n-1}q^{n-1} + \dots + b\_1q + b\_0 \tag{12}$$

where the parameters and are constants, i=0,1,…,n�1. The partial state of such a dynamic system can be considered as a discrete flat output which can be expressed as a function of input and output signals as following

$$A(q)z\_k = u\_k \tag{13}$$

$$B(q)z\_k = \mathcal{y}\_k\tag{14}$$

Often, the real output signal to be controlled is not a flat output. Then, it is necessary to plan a desired trajectory for the flat output [11] and to consider thereafter the relation (14).

The open loop control law can be determined by the following relations [12].

$$u^{d}(t) = f(z^{d}(t), \dots, z^{d^{(r+1)}}(t))\tag{15}$$

$$\mathbf{y}^d(t) = \mathbf{g}(\boldsymbol{z}^d(t), \dots, \boldsymbol{z}^{d^{(\prime)}}(t)) \tag{16}$$

where and g are the vectorial functions. Then, it is sufficient to find a desired continuous flat trajectory that must to be differentiable at the order.

The polynomial interpolation technique is used in order to plan the desired flat trajectory . Let consider the state vector containing the desired continuous flat output and its successive derivatives. The expression of can be given in Eq. (17); and are the two moments known in advance.

$$Z^d(t) = M\_1(t - t\_0)c\_1(t\_0) + M\_2(t - t\_0)c\_2(t\_0, t\_f) \tag{17}$$

where and are such as [12]

$$M\_1 = \begin{pmatrix} 1 & t & \cdots & \frac{t^{n-1}}{(n-1)!} \\ 0 & 1 & \cdots & \frac{t^{n-2}}{(n-2)!} \\ \vdots & \ddots & \ddots & \vdots \\ 0 & \cdots & 0 & 1 \end{pmatrix} \tag{18}$$

$$M\_2 = \begin{pmatrix} t'' & \frac{t''}{n!} & \frac{t''^{n+1}}{(n+1)!} & \cdots & \frac{t^{2^{n-1}}}{(2n-1)!} \\ \frac{t''^{n-1}}{(n-1)!} & \frac{t''}{n!} & \cdots & \frac{t^{(n-2)}}{(n-2)!} \\ \vdots & \ddots & \ddots & \vdots \\ \vdots & \cdots & \frac{t^{n-1}}{(n-1)!} & \frac{t''}{n!} \end{pmatrix} \tag{19}$$

and the vectors and defined by

$$c\_1 = Z^d(t\_0) \tag{20}$$

$$c\_2 = M\_2^{-1}(t\_f - t\_0)(Z^d(t\_f) - M\_1(t\_f - t\_0)Z^d(t\_0))\tag{21}$$

Then, the output desired trajectory is defined. In the discrete-time framework, the real output has asymptotically to track this such as Eq. (22).

$$\mathbf{y}\_k^d = B(q)\mathbf{z}\_k^d \tag{22}$$

In the following section of this chapter, the efficiency of the proposed methodology for tuning PID-type FLC scaling factors has been validated on a discrete-time system: an electronic throttle valve for a defined desired trajectory generated using the flatness property and compared to the Simulink design optimization (SDO) technique.

#### 5. Case of study: electronic throttle valve (ETV)

Throttle valve is one of the most important devices in the engine management system. In conventional engine, the amount of airflow into the combustion system has been adjusted by the throttle valve, which is connected mechanically to an accelerator pedal [13]. The electronic throttle body (ETB) regulates air inflow into the car engine. Compared to the mechanical throttle, a well-controlled ETB can reduce fuel consumption.

#### 5.1. System modelling

ð17Þ

ð18Þ

ð19Þ

ð20Þ

ð21Þ

ð22Þ

where and are such as [12]

96 Modern Fuzzy Control Systems and Its Applications

and the vectors and defined by

output has asymptotically to track this such as Eq. (22).

pared to the Simulink design optimization (SDO) technique.

5. Case of study: electronic throttle valve (ETV)

throttle, a well-controlled ETB can reduce fuel consumption.

Then, the output desired trajectory is defined. In the discrete-time framework, the real

In the following section of this chapter, the efficiency of the proposed methodology for tuning PID-type FLC scaling factors has been validated on a discrete-time system: an electronic throttle valve for a defined desired trajectory generated using the flatness property and com-

Throttle valve is one of the most important devices in the engine management system. In conventional engine, the amount of airflow into the combustion system has been adjusted by the throttle valve, which is connected mechanically to an accelerator pedal [13]. The electronic throttle body (ETB) regulates air inflow into the car engine. Compared to the mechanical The case of the ETV is described in Figure 4.

The electrical part is modelled by Eq. (23)

$$u(t) = L\frac{d}{dt}i(t) + Ri(t) + k\_\nu \phi\_w(t)\tag{23}$$

where L is the inductance R is the resistance u(t) and i(t) are the voltage and the armature current, respectively, is an electromotive force constant and is the motor rotational speed.

The mechanical part of the throttle is modelled by a gear reducer characterized by its reduction ratio such as Eq. (24)

$$\mathcal{Y} = \frac{C\_{\text{g}}}{C\_{\text{L}}} \tag{24}$$

where is the load torque and Cg is the gear torque. The mechanical part is modelled according to Eq. (25), such that [14, 15].

$$J\frac{d}{dt}\,\alpha\_{\mathfrak{m}}(t) = C\_{\mathfrak{v}} - C\_{f} - C\_{\mathfrak{v}} - C\_{\mathfrak{o}} \tag{25}$$

and

$$\frac{d}{dt}\theta(t) = (180 \wedge \pi \,/\, \gamma)\alpha\_m(t) \tag{26}$$

where is the throttle plate angle, J is the overall moment of inertia, is the electrical torque where K<sup>e</sup> is a constant, Cf is the torque caused by mechanical friction, Cr is the spring resistive torque and is the torque generated by the airflow. The electronic throttle valve involves two complex nonlinearities due to the nonlinear spring torque and the friction

Figure 4. Electronic throttle valve system.

torque . They are given by their static characteristics [16]. The static characteristic of the nonlinear spring torque Cr is defined by

$$C\_r = \frac{k\_r}{\gamma} (\theta - \theta\_0) + D \operatorname{sgn}(\theta - \theta\_0) \tag{27}$$

For , is the spring constant, is a constant, is the default position and sgn(.) is the following signum function

$$\text{sgn}(\theta - \theta\_0) = \begin{cases} 1, & \text{if } \theta \ge \theta\_0 \\ -1, & \text{else} \end{cases} \tag{28}$$

The friction torque function of the angular velocity of the throttle plate can be expressed as

$$C\_f = f\_\text{,} \phi + f\_c \text{sgn}(\phi) \tag{29}$$

where fv and fc are two constants. By substituting in Eq. (25) the expressions of Cg, Cf and Cr and by neglecting the torque generated by the airflow Ca, the two nonlinearities and and the two constants and fv the transfer function of the linear model becomes (30) [15].

$$H(q) = \frac{(180 \land \pi \, / \gamma \,)k\_\circ}{J L q^\circ + J R q^2 + (k\_\circ k\_\circ + L k\_\circ) q + R k\_\circ} \tag{30}$$

with ks = (180 / π / γ<sup>2</sup> )kr and q as the Laplace operators.

#### 5.2. Simulation results

The identified parameters of are given in Table 2 at 25�C temperature [15].

The corresponding discrete-time transfer function is given by Eq. (31) for the sample time = 0.002s.

$$H(q^{-1}) = \frac{0.007833q^{-1} + 0.01396q^{-2} + 0.0007724q^{-3}}{1 - 1.948q^{-1} + 0.954q^{-2} - 0.006182q^{-3}}\tag{31}$$


Table 2. Model's parameters.

The desired continuous time flat trajectory can be computed according to the following polynomial form

$$z^{\prime}(k) = \begin{cases} \frac{cst1}{B(\mathbf{l})}, & \text{if } 0 \le k \le k\_0 \\ \text{Poly}\_1(k), & \text{if } k\_0 \le k \le k\_1 \\ \frac{cst2}{B(\mathbf{l})}, & \text{if } k\_1 \le k \le k\_2 \\ \text{Poly}\_2(k), & \text{if } k\_2 \le k \le k\_3 \\ \frac{cst1}{B(\mathbf{l})}, & \text{if } k \ge k\_3 \end{cases} \tag{32}$$

where and are constants, are the instants of transitions, is the static gain between the flat output and the output signal for each operating mode and are polynomials calculated using the technique of polynomial interpolation.

The desired trajectory is then given in Figure 5.

torque . They are given by their static characteristics [16]. The static characteristic of the

For , is the spring constant, is a constant, is the default position and sgn(.) is

The friction torque function of the angular velocity of the throttle plate can be expressed as

where fv and fc are two constants. By substituting in Eq. (25) the expressions of Cg, Cf and Cr and by neglecting the torque generated by the airflow Ca, the two nonlinearities and and the two constants and fv the transfer function of the linear model becomes (30) [15].

)kr and q as the Laplace operators.

The identified parameters of are given in Table 2 at 25�C temperature [15].

The corresponding discrete-time transfer function is given by Eq. (31) for the sample time

Parameters Values R(Ω) 2.8 L(H) 0.0011 ke (N.m/A) 0.0183 kv (v/rad/s) 0.0183

γ 16.95

) <sup>4</sup> � <sup>10</sup>�<sup>6</sup>

ð27Þ

ð28Þ

ð29Þ

ð30Þ

ð31Þ

nonlinear spring torque Cr is defined by

98 Modern Fuzzy Control Systems and Its Applications

the following signum function

with ks = (180 / π / γ<sup>2</sup>

5.2. Simulation results

= 0.002s.

J (kg.m<sup>2</sup>

Table 2. Model's parameters.

The obtained optimal I/O scaling factors for Simulink design optimization technique and GA are summarized in Table 3.

The obtained results are given in Figures 6–9.

Figure 5. Desired trajectory.


Table 3. PID-type fuzzy scaling factors values.

Figure 6. System outputs using Simulink design optimization.

Figure 7. System outputs using genetic algorithm.

A New Methodology for Tuning PID‐Type Fuzzy Logic Controllers Scaling Factors Using Genetic Algorithm… http://dx.doi.org/10.5772/intechopen.69604 101

Figure 8. Tracking errors.

SF\Method SDO GA Ke 0.1144 0.0086 Kd 1.5997 0.4612 α 2.6239 0.1108 β 0.0001 0.1934

Table 3. PID-type fuzzy scaling factors values.

100 Modern Fuzzy Control Systems and Its Applications

Figure 6. System outputs using Simulink design optimization.

Figure 7. System outputs using genetic algorithm.

Figure 9. Control signals.

Figures 8 and 9 show the responses with Simulink design optimization technique and GA tuning using ISE criterion. Based on a comparative analysis, better results were there obtained with the GA tuning method.

All results, for obtained scaling factors values, are acceptable and show the effectiveness of the proposed GA tuning method in terms of the tracking desired trajectory with disturbances rejection in comparison with the SDO technique.

## 6. Conclusion

In this chapter, an optimization technique was introduced to tune the parameters of PID-type fuzzy logic controller (FLC). The idea is to use the genetic algorithm (GA)-based heuristic optimization technique in order to solve highly complex problems. In order to specify more robustness and performance of the proposed GA-tuned PID-type FLC, optimization criteria such as integral square error (ISE) is considered.

The proposed controller is applied to an electronic throttle valve (ETV) in the discrete-time framework in order to track a desired trajectory starting from a flat output generated using flatness property. The performance comparison with the Simulink design optimization (SDO) technique shows the efficiency of the proposed GA-tuned approach in terms of tracking a desired trajectory with disturbances rejection.

## Author details

Wafa Gritli\*, Hajer Gharsallaoui and Mohamed Benrejeb

\*Address all correspondence to: wafa\_gritli@yahoo.fr

National Engineering School of Tunis, Tunis, Tunisia

## References


[7] Holland JJ. Adaptation in Natural and Artificial Systems, University of Michigan Press; I975

6. Conclusion

102 Modern Fuzzy Control Systems and Its Applications

Author details

References

(3):227–236

such as integral square error (ISE) is considered.

desired trajectory with disturbances rejection.

Wafa Gritli\*, Hajer Gharsallaoui and Mohamed Benrejeb

\*Address all correspondence to: wafa\_gritli@yahoo.fr

National Engineering School of Tunis, Tunis, Tunisia

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ing & Information Technology (CEIT'14); 2014. pp. 329–334

In this chapter, an optimization technique was introduced to tune the parameters of PID-type fuzzy logic controller (FLC). The idea is to use the genetic algorithm (GA)-based heuristic optimization technique in order to solve highly complex problems. In order to specify more robustness and performance of the proposed GA-tuned PID-type FLC, optimization criteria

The proposed controller is applied to an electronic throttle valve (ETV) in the discrete-time framework in order to track a desired trajectory starting from a flat output generated using flatness property. The performance comparison with the Simulink design optimization (SDO) technique shows the efficiency of the proposed GA-tuned approach in terms of tracking a

[1] Guzelkaya M, Eksin I, Yesil E. Self-tuning of PID-type fuzzy logic controller coefficients via relative rate observer. Engineering Applications of Artificial Intelligence. 2003;16

[2] Qiao WZ, Mizumoto M. PID type fuzzy controller and parameters adaptive method.

[3] Woo ZW, Chung HY, Lin JJ. A PID type fuzzy controller with self-tuning scaling factors.

[4] Toumi F, Bouallègue S, Haggège J, Siarry P. Differential search algorithm-based approach for PID-type fuzzy controller tuning. In: International Conference on Control, Engineer-

[5] Gritli W, Gharsallaoui H, Benrejeb M. PID-type fuzzy scaling factors tuning using genetic algorithm and Simulink design optimization for electronic throttle valve. In: International Conference on Control, Decision and Information Technologies (CoDIT'16); 6-8 April;

[6] Galichet S, Foulloy L. Fuzzy controllers: Synthesis and equivalences. IEEE Transactions


**Applications of Fuzzy in Energy and Power Systems**

## **Chapter 6**

## **Applications of the Fuzzy Logic to the Energy Conversion Systems on Board of UAVs**

Dinca Liviu and Corcau Jenica Ileana

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67992

#### Abstract

This chapter intends to present some applications of the fuzzy controllers to the automatic control of the DC-to-DC converters type boost, buck and buck-boost. For the mathematical modelling of these converters, one used averaged models; taking into account that controlled parameter is the average output voltage. One considered only the continuous conduction mode in the mathematical model. For each converter, one makes a short description of the principal scheme and the functioning mode and presents the averaged model. One obtains the transfer functions and finally builds fuzzy controllers in order to stabilize the output voltage with respect to the input voltage variations. The control is realized by modifying the duty cycle of the PWM command pulses. Obtained systems, both in closed loop and in open loop, are implemented in MATLAB/Simulink and simulations results are also presented.

Keywords: DC-to-DC converters, fuzzy logic control, averaged models, UAV, power systems

## 1. Introduction

Type UAV (Unmanned Air Vehicle) platforms have realised in the last period a special development due to their large area of applicability. UAVs can perform a variety of missions like military observation, combat, crops observations, disaster areas surveillance, in conditions of very high econommic and energy efficiency against classical aviation. UAVs offers many advantages like reduced manufacturing and operating costs, low energy consumption, possibility to access safely in hazardous areas for manned aircrafts, ability to penetrate spaces more restricted, inaccessible to classic aircraft, and due to technological developments obtained recently, the ability to achieve continuous flight missions with durations of the order of several days to several weeks. In this sense, the development of UAVs is expected to play an important role in High Altitude Pseudo-Satellites (HAPS) with continuous flight duration of the order of 5 years.

© 2017 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.

Achieving such high flight duration was possible due to the use of hybrid power sources that use at least two types of energy sources. In this sense, the most used energy sources are presently high capacity batteries or super capacitors that are recharged from renewable energy sources, especially high-efficiency photovoltaic cells developed in the last period. Fuel cells present a high efficiency, so it represents an alternative for the electrically powered manned airplanes. This solution is extensively studied and planned to be used in the near future. Airbus Company has already developed an electric airplane for two flights that last approximately for 1 hour. Currently, it is working to develop a plane for four persons with hybrid power source (high capacity battery plus two-stroke combustion engine that acts as an electric generator). Flight duration is expected to be extended to two hours. Research department of Airbus is considering replacing the thermal engine with the fuel cell to further expand the duration of flight.

Titan Aerospace was tested in August 2013, a concept vehicle for future high-altitude SOLAR 50 and 60 UAVs. SUN SOLAR 50 was scheduled to fly in 2014. SOLAR 50 was conceived as a UAV which uses high-capacity batteries and 3000 solar cells as power sources. The maximum flight duration of 50 SOLAR is expected to be 5 years at last. It is designed to be equipped with telecommunications and recognition systems, atmospheric sensors, etc. SUN 60 is planned for a payload of 125 kg [1–3].

The High Altitude Long Endurance Boeing (HALE) UAS 2012 aims at developing high altitude UAS for missions to ensure transmissions for the disposal of existing infrastructure and prolonged surveillance missions to the areas of interest.

Phantom Eye Project uses hydrogen internal combustion engine with hydrogen stored cryogenically. Solar Eagle Project, of Boeing as well, expects the use of high efficiency solar cells and SOFC type fuel cells. Wingspan aircraft is expected to be 120 m and propulsion will use six electric motors with permanent magnets [1–3].

Project Zephyr 7 High Altitude Pseudo-satellites from Airbus Defence & Space is to be used for surveillance, communications and monitoring services on surfaces in the order of tens of thousands km<sup>2</sup> . This UAV holds the record for the longest flight—336 hours. Solar energy has been used for the propulsion and storage in lithium-ion battery with 3 kWh capacities to drive at night [1–3].

Since the parameters of power sources such as solar cells or fuel cells vary greatly in relation to the operating conditions, and loads useful on board UAVs (cameras, sensors, communications systems, etc.) usually require stabilized voltage, power buses of UAVs need to be maintained at a constant voltage.

This can be achieved by using power converters to adapt output sources to the requirements of bus power sources. Most sources on board UAVs are DC sources and loads are mostly (except some propulsion motors) of DC types. Following are required DC-to-DC converter types which have a higher efficiency and weight as low as possible aboard UAVs. The most convenient in this sense are the converters that have not transformed, part of this class are buck, boost and buck-boost converters. They are suitable for use on UAVs because they adapt the power source voltage to the power bus voltage by simply varying the duty cycle of the command pulses.

For the converters control, there are many techniques in literature, more or less sophisticated, among these are fuzzy techniques. These techniques offer a robust control and have an advantage which is related to human logic that these techniques do not require to know a detailed mathematical model of the system and can be tuned relatively easily granted by numerical simulations.

In this chapter, development of models used in the study of electrical systems of UAVs follows the modelling of three types of DC-DC converters, used on board UAVs, and development of fuzzy controllers is used to stabilize the output voltage.

Mathematical models developed for these converters follow the behaviour in terms of the averaged parameters (voltage and current) and neglect their ripples.

Although all models are being averaged, technique for obtaining these models is the introduction of the circuit voltage or current sources to make up in terms of DC parameters switch, coil or diodes present in converters. We consider this method more viable in terms of logic than writing equations split by two periods of operation of the converter and then their average as shown in Ref. [4].

### 2. Boost converter

Achieving such high flight duration was possible due to the use of hybrid power sources that use at least two types of energy sources. In this sense, the most used energy sources are presently high capacity batteries or super capacitors that are recharged from renewable energy sources, especially high-efficiency photovoltaic cells developed in the last period. Fuel cells present a high efficiency, so it represents an alternative for the electrically powered manned airplanes. This solution is extensively studied and planned to be used in the near future. Airbus Company has already developed an electric airplane for two flights that last approximately for 1 hour. Currently, it is working to develop a plane for four persons with hybrid power source (high capacity battery plus two-stroke combustion engine that acts as an electric generator). Flight duration is expected to be extended to two hours. Research department of Airbus is considering replacing the thermal engine with the fuel cell to further expand the

Titan Aerospace was tested in August 2013, a concept vehicle for future high-altitude SOLAR 50 and 60 UAVs. SUN SOLAR 50 was scheduled to fly in 2014. SOLAR 50 was conceived as a UAV which uses high-capacity batteries and 3000 solar cells as power sources. The maximum flight duration of 50 SOLAR is expected to be 5 years at last. It is designed to be equipped with telecommunications and recognition systems, atmospheric sensors, etc. SUN 60 is planned for

The High Altitude Long Endurance Boeing (HALE) UAS 2012 aims at developing high altitude UAS for missions to ensure transmissions for the disposal of existing infrastructure and

Phantom Eye Project uses hydrogen internal combustion engine with hydrogen stored cryogenically. Solar Eagle Project, of Boeing as well, expects the use of high efficiency solar cells and SOFC type fuel cells. Wingspan aircraft is expected to be 120 m and propulsion will use six

Project Zephyr 7 High Altitude Pseudo-satellites from Airbus Defence & Space is to be used for surveillance, communications and monitoring services on surfaces in the order of tens of

been used for the propulsion and storage in lithium-ion battery with 3 kWh capacities to drive

Since the parameters of power sources such as solar cells or fuel cells vary greatly in relation to the operating conditions, and loads useful on board UAVs (cameras, sensors, communications systems, etc.) usually require stabilized voltage, power buses of UAVs need to be maintained

This can be achieved by using power converters to adapt output sources to the requirements of bus power sources. Most sources on board UAVs are DC sources and loads are mostly (except some propulsion motors) of DC types. Following are required DC-to-DC converter types which have a higher efficiency and weight as low as possible aboard UAVs. The most convenient in this sense are the converters that have not transformed, part of this class are buck, boost and buck-boost converters. They are suitable for use on UAVs because they adapt the power source voltage to the power bus voltage by simply varying the duty cycle of the

. This UAV holds the record for the longest flight—336 hours. Solar energy has

duration of flight.

108 Modern Fuzzy Control Systems and Its Applications

a payload of 125 kg [1–3].

thousands km<sup>2</sup>

at night [1–3].

at a constant voltage.

command pulses.

prolonged surveillance missions to the areas of interest.

electric motors with permanent magnets [1–3].

The principal scheme of the boost converter is shown in Figure 1a and b equivalent scheme for continuous mode is presented. Functioning of this converter contains two steps. In the first step when the switch is in position 1, the inductor current rises and the inductor accumulate energy in its electromagnetic field. In the second step when the switch is in position 2, the inductor is now in series with the input source and pushes its energy on the capacitor. Inductor current drops, so the inductor behaves as voltage source in series with the input source and the capacitor will be charged to a voltage higher than the input voltage. For this reason, the boost converter raises the input voltage [5, 6].

Diagram in Figure 1a is equivalent in terms of DC components with the diagram in Figure 1b. Given that, coil behaviour was modelled in terms of averaged voltage with a voltage source dependent on the duty cycle of the control pulses UL(d), and the current through the switch was modelled by a current source dependent on the duty cycle of control pulses IT(d).

Coil voltage was found to be UL <sup>¼</sup> <sup>d</sup>�Uin <sup>1</sup>�<sup>d</sup> so as to respect the converter transfer characteristic in steady state. Regarding the currents, it was considered that IT = ID in terms of the average

Figure 1. Boost converter. (a) Principal scheme, (b) Equivalent scheme for continuous mode.

values. In stationary regime, coil L behaves like a voltage source UL(d). At a duty cycle variation, in transitory regime. In these conditions, one can write relations:

$$\begin{cases} \frac{dI\_{\text{in}}}{dt} = -\frac{1}{L} \mathcal{U}\_{\text{\textdegree C}} + \frac{\mathcal{U}\_{\text{in}}}{L(1-d)}\\ \qquad \frac{d\mathcal{U}\_{\text{\textdegree C}}}{dt} = \frac{1}{2\mathcal{C}} I\_{\text{in}} - \frac{\mathcal{U}\_{\text{\textdegree C}}}{\mathcal{C}\mathcal{R}s} \end{cases} \tag{1}$$

For linearization, notations, Iin ¼ Iin0 þ ΔIin, UC ¼ UC<sup>0</sup> þ ΔUC, Uin ¼ Uin0 þ ΔUin, d ¼ d<sup>0</sup> þ Δd have been used. Relation obtained after linearization is (2)

$$\begin{cases} \begin{bmatrix} \Delta I\_{\text{in}} \\ \Delta \boldsymbol{U}\_{\text{C}} \end{bmatrix} = \begin{bmatrix} 0 & -\frac{1}{L} \\ \frac{1}{2\mathcal{C}} & -\frac{1}{CR\_{\text{S}}} \end{bmatrix} \begin{bmatrix} \Delta I\_{\text{in}} \\ \Delta \boldsymbol{U}\_{\text{C}} \end{bmatrix} + \begin{bmatrix} 1 & \boldsymbol{U}\_{\text{in}} \\ \frac{L(1-d)}{\mathcal{U}} & \frac{L(1-d)^{2}}{\mathcal{L}} \end{bmatrix} \begin{bmatrix} \Delta \boldsymbol{I}\_{\text{in}} \\ \Delta \boldsymbol{d} \end{bmatrix} \\\ \begin{bmatrix} \Delta \boldsymbol{U}\_{\text{out}} \end{bmatrix} = \begin{bmatrix} 0 & 1 \end{bmatrix} \begin{bmatrix} \Delta \boldsymbol{I}\_{\text{in}} \\ \Delta \boldsymbol{U}\_{\text{C}} \end{bmatrix} + \begin{bmatrix} 0 & 0 \end{bmatrix} \begin{bmatrix} \Delta \boldsymbol{I}\_{\text{in}} \\ \Delta \boldsymbol{d} \end{bmatrix} \end{cases} (2)$$

Using this system, one can deduce the transfer functions given in Eqs. (3) and (4)

$$H\_1(\mathbf{s}) = \frac{\Delta U\_{\text{out}}(\mathbf{s})}{\Delta U\_{\text{in}}(\mathbf{s})} = \frac{\frac{1}{2LC(1-d)}}{\mathbf{s}^2 + \mathbf{s}\frac{1}{CRs} + \frac{1}{2LC}} = \frac{k\_1}{(\mathbf{s} - \mathbf{s}\_1)(\mathbf{s} - \mathbf{s}\_2)}\tag{3}$$

$$H\_2(\mathbf{s}) = \frac{\Delta U\_{\text{out}}(\mathbf{s})}{\Delta d(\mathbf{s})} = \frac{\frac{U\_{\text{in}}}{2\mathcal{L}\mathcal{C}(1-d)^2}}{\mathbf{s}^2 + \mathbf{s}\frac{1}{\mathcal{C}\mathcal{R}\_{\text{S}}} + \frac{1}{2\mathcal{L}\mathcal{C}}} = \frac{k\_2}{(\mathbf{s} - \mathbf{s}\_1)(\mathbf{s} - \mathbf{s}\_2)}\tag{4}$$

In Figure 2, simulation scheme of boost converter in open loop is shown. They were used ΔUin and Δd as input, the output voltage was considered as ΔUout. Simulated boost converter parameters were as follows <sup>L</sup> <sup>¼</sup> <sup>47</sup> <sup>μ</sup>F, C <sup>¼</sup> <sup>2</sup>:2 mF, RS <sup>¼</sup> <sup>1</sup> <sup>Ω</sup>, d <sup>¼</sup> <sup>0</sup>:5, Uin <sup>¼</sup> 24 V: In the simulations using transfer functions, variations of output voltage about output voltage reference,

Figure 2. Open loop simulation scheme of boost converter.

when there are some variations in input voltage and duty cycle compared to reference values, are obtained. To capture the absolute values, some sum blocks are introduced in Figure 2 whose variations overlap reference values achieved through the transfer or the absolute values of the input parameters. Simulation results are presented in Figure 3a–c.

values. In stationary regime, coil L behaves like a voltage source UL(d). At a duty cycle

<sup>L</sup> UC <sup>þ</sup>

For linearization, notations, Iin ¼ Iin0 þ ΔIin, UC ¼ UC<sup>0</sup> þ ΔUC, Uin ¼ Uin0 þ ΔUin, d ¼ d<sup>0</sup> þ Δd

þ

ΔUC � �

1 2LCð1�dÞ <sup>s</sup><sup>2</sup> <sup>þ</sup> <sup>s</sup> <sup>1</sup>

> Uin 2LCð1�dÞ 2

<sup>s</sup><sup>2</sup> <sup>þ</sup> <sup>s</sup> <sup>1</sup>

In Figure 2, simulation scheme of boost converter in open loop is shown. They were used ΔUin and Δd as input, the output voltage was considered as ΔUout. Simulated boost converter parameters were as follows <sup>L</sup> <sup>¼</sup> <sup>47</sup> <sup>μ</sup>F, C <sup>¼</sup> <sup>2</sup>:2 mF, RS <sup>¼</sup> <sup>1</sup> <sup>Ω</sup>, d <sup>¼</sup> <sup>0</sup>:5, Uin <sup>¼</sup> 24 V: In the simulations using transfer functions, variations of output voltage about output voltage reference,

CRS <sup>þ</sup> <sup>1</sup> 2LC

CRS <sup>þ</sup> <sup>1</sup> 2LC

2 4

<sup>2</sup><sup>C</sup> <sup>I</sup>in � UC CRS

Uin Lð1 � dÞ

1 Lð1 � dÞ

Uin Lð1 � dÞ 2 3 5

ΔUin Δd � �

<sup>ð</sup><sup>s</sup> � <sup>s</sup>1Þð<sup>s</sup> � <sup>s</sup>2<sup>Þ</sup> <sup>ð</sup>3<sup>Þ</sup>

<sup>ð</sup><sup>s</sup> � <sup>s</sup>1Þð<sup>s</sup> � <sup>s</sup>2<sup>Þ</sup> <sup>ð</sup>4<sup>Þ</sup>

0 0

<sup>¼</sup> <sup>k</sup><sup>1</sup>

<sup>¼</sup> <sup>k</sup><sup>2</sup>

Δd � �

þ ½ 0 0 � <sup>Δ</sup>Iin

ð1Þ

ð2Þ

variation, in transitory regime. In these conditions, one can write relations:

dIin dt ¼ � <sup>1</sup>

8 >><

>>:

have been used. Relation obtained after linearization is (2)

1 <sup>2</sup><sup>C</sup> � <sup>1</sup> CRS

<sup>H</sup>1ðsÞ ¼ <sup>Δ</sup>Uoutðs<sup>Þ</sup>

<sup>H</sup>2ðsÞ ¼ <sup>Δ</sup>Uoutðs<sup>Þ</sup>

Figure 2. Open loop simulation scheme of boost converter.

<sup>Δ</sup>Uinðs<sup>Þ</sup> <sup>¼</sup>

<sup>Δ</sup>dðs<sup>Þ</sup> <sup>¼</sup>

<sup>0</sup> � <sup>1</sup> L

ΔIin ΔUC � �

110 Modern Fuzzy Control Systems and Its Applications

8 >>>>><

>>>>>:

¼

2 6 4 dUC dt <sup>¼</sup> <sup>1</sup>

3 7 5

<sup>Δ</sup>Uout ½ �¼ ½ � 0 1 <sup>Δ</sup>Iin

Using this system, one can deduce the transfer functions given in Eqs. (3) and (4)

ΔIin ΔUC � � Fuzzy control technique for DC-DC converters is applied in some studies in literature [7–10]. Other control techniques for these converters are studied in Refs. [11–17].

Closed loop scheme of the boost converter, implemented in MATLAB/Simulink is shown in Figure 4. One used a fuzzy controller with two inputs, output voltage error with respect to the

Figure 3. Boost converter response: (a) input voltage variation; (b) duty cycle variation; (c) output voltage variation.

Figure 4. Closed loop simulation scheme for boost converter.

nominal output voltage and its integral, so one can say that this is a PI fuzzy controller. The advantage of the fuzzy controllers is that one can modify conveniently the control characteristics slopes in order to obtain the desired system behaviour. Usually one wishes a higher slope for points far from the origin and smaller slope for points near the origin. In this way, when the system is far from the nominal functioning point, one can accelerate its return to the nominal point. When it comes near the nominal point, parameters variations are sluggish in order to obtain a good stability in the nominal point. Fuzzy system in Figure 4 has a particularity for an error and its integral are defined by two separate fuzzy controllers with four membership functions on input and output. In this way, one can simplify the controller tuning. There are necessarily eight inference rules (four rules for each controller) in contrast with the classical fuzzy controller with two inputs and one output having 16 inference rules. This difference increases once the membership function for each input increases. In Figure 5, the membership functions for fuzzy proportional controller P and its control surface are presented.

In Figure 6, the membership functions for fuzzy integrator controller I and its control surface are presented. Converter operation with these fuzzy controllers has been tested to a step type signal and ramp type signal. In hybrid power systems of UAVs containing fuel cell or solar cells, parameters variations are not very sudden, so it is exciting to study the converters behaviour at sluggish signals, such as the ramp type which is taken into consideration here. In Figure 7, the boost converter behaviour at step signal is shown and in Figure 8, its behaviour to a ramp signal with a slope of 3 V/s is shown.

Figure 5. P-fuzzy controller. (a) Input membership functions; (b) Output membership functions; (c) Control surface.

Figure 6. I-fuzzy controller. (a) Input membership functions; (b) Output membership functions; (c). Control surface.

Linguistic terms for both the controllers and for both inputs and outputs are NBIG (large negative), NMC (small negative), PMC (small positive) and PBIG (big positive). For both controllers, inference rules are: If input is NBIG, then output is NBIG; If input is NMC, then output is NMC; If input is PMC, then output is PMC; If input is PBIG, then output is PBIG.

nominal output voltage and its integral, so one can say that this is a PI fuzzy controller. The advantage of the fuzzy controllers is that one can modify conveniently the control characteristics slopes in order to obtain the desired system behaviour. Usually one wishes a higher slope for points far from the origin and smaller slope for points near the origin. In this way, when the system is far from the nominal functioning point, one can accelerate its return to the nominal point. When it comes near the nominal point, parameters variations are sluggish in order to obtain a good stability in the nominal point. Fuzzy system in Figure 4 has a particularity for an error and its integral are defined by two separate fuzzy controllers with four membership functions on input and output. In this way, one can simplify the controller tuning. There are necessarily eight inference rules (four rules for each controller) in contrast with the classical fuzzy controller with two inputs and one output having 16 inference rules. This difference increases once the membership function for each input increases. In Figure 5, the membership

functions for fuzzy proportional controller P and its control surface are presented.

iour to a ramp signal with a slope of 3 V/s is shown.

112 Modern Fuzzy Control Systems and Its Applications

In Figure 6, the membership functions for fuzzy integrator controller I and its control surface are presented. Converter operation with these fuzzy controllers has been tested to a step type signal and ramp type signal. In hybrid power systems of UAVs containing fuel cell or solar cells, parameters variations are not very sudden, so it is exciting to study the converters behaviour at sluggish signals, such as the ramp type which is taken into consideration here. In Figure 7, the boost converter behaviour at step signal is shown and in Figure 8, its behav-

Figure 5. P-fuzzy controller. (a) Input membership functions; (b) Output membership functions; (c) Control surface.

Figure 6. I-fuzzy controller. (a) Input membership functions; (b) Output membership functions; (c). Control surface.

For defining the controllers in this case, it is intended to simplify them as much so as to reduce the computation time required. In this purpose, one eliminated the membership function corresponding to linquistic term ZERO, which is used in the usual manner in fuzzy controllers. By conveniently change of the membership function one obtained control surfaces in concordance with the followed strategy - lower slope around the origin and greater slope farther origin.

The behaviour of the converter closed loop is one adequate operation of power systems on UAVs. Peak voltages at step input are attenuated and voltage returns quickly enough to the prescribed value. For the ramp input, deviation from set point is only 0.025 V, which is a very good performance.

Figure 7. Closed loop behaviour of boost converter: (a) input variation; (b) duty cycle variation; (c) output voltage variation.

Figure 8. Boost converter behaviour at ramp signal: (a) input voltage variation; (b) output voltage variation.

#### 3. Buck converter

Principal scheme of the buck converter is shown in Figure 9. This converter has also two steps in its functioning. In the second step when the switch is in position 1, the capacitor loads through the inductor and inductor current increases, so the inductor behaves as a voltage source with inverse polarity with respect to the input voltage. In this manner on the capacitor, one can obtain a voltage smaller than the input voltage. In the second step when the switch is in position 2, energy accumulated in the inductor is pushed on the capacitor and thus Inductor current decreases, so it behaves as a voltage source with the same polarity as the input voltage but with a smaller value. So, the buck converter decreases the input voltage. In Figure 10, equivalent scheme for continuous mode is presented.

Following Figure 10, one can write the following relations: Uin � UL ¼ Uout ¼ d�Uin, iT þ iD ¼ iL ¼ iS, Uin�iT ¼ Uout�iS: After several transformations, following system of equations give:

$$\begin{cases} \mathcal{U}\_{\rm out} = \mathcal{U}\_{\rm in} d - L \frac{d}{dt} \left( \frac{I\_T}{d} \right) \\\\ \frac{\mathcal{U}\_{\rm out}}{R\_S} = I\_T \frac{1}{d} - \mathcal{C} \frac{d\mathcal{U}\_{\rm out}}{dt} \end{cases} \tag{5}$$

By linearization, like the case of the boost converter, resulted linearized model is:

$$\begin{cases} \Delta I\_T = -\frac{d\_0}{L} \Delta l I\_{\text{out}} + \frac{d\_0^2}{L} \Delta l I\_{\text{in}} + \frac{V\_{\text{in}0} \cdot d\_0}{L} \Delta d + \frac{I\_{T0}}{d\_0} \Delta \dot{d} \\\\ \Delta I\_{\text{out}} = \frac{1}{\mathbb{C} \cdot d\_0} \Delta I\_T - \frac{1}{R\_S \cdot \mathbb{C}} \Delta l I\_{\text{out}} - \frac{I\_{T0}}{\mathbb{C} \cdot d\_0^2} \Delta d \end{cases} \tag{6}$$

Applying Laplace in system zero initial conditions, Eq. (6) results in transfer functions of the form

$$\frac{\Delta U\_{\text{out}}}{\Delta U\_{\text{in}}} = \frac{R\_S \cdot d\_0}{s^2 (R\_S \cdot L \cdot \text{C}) + L \cdot s + R\_S} \tag{7}$$

$$\frac{\Delta \mathcal{U}\_{\text{out}}}{\Delta d} = \frac{R\_S \cdot \mathcal{U}\_{\text{in}0}}{s^2 (R\_S \cdot L \cdot \mathcal{C}) + L \cdot s + R\_S} \tag{8}$$

Buck converter parameters considered in this chapter are <sup>L</sup> <sup>¼</sup> <sup>56</sup>:<sup>5</sup> <sup>μ</sup>F, C <sup>¼</sup> <sup>166</sup>:<sup>7</sup> <sup>μ</sup>F, RS <sup>¼</sup> <sup>2</sup> <sup>Ω</sup>, d ¼ 0:5, Uin0 ¼ 24 V, Uout0 ¼ 12 V: Simulation scheme for the buck converter with transfer functions, open loop is shown in Figure 11. Simulation results are presented in Figure 12. Utilizing a closed loop scheme and the same fuzzy controllers, like in the case of boost converter shown in Figure 4, closed-loop simulation results are obtained shown in Figure 13. For the buck converter control, a classical PI fuzzy controller has been designed with two inputs, the error between the output voltage and the voltage prescribed and integral of this error. Simulation scheme in this case is the one in Figure 14. The membership functions and control surface of this Applications of the Fuzzy Logic to the Energy Conversion Systems on Board of UAVs http://dx.doi.org/10.5772/67992 115

Figure 9. Principal scheme of buck converter.

3. Buck converter

114 Modern Fuzzy Control Systems and Its Applications

equivalent scheme for continuous mode is presented.

ΔIT �

8 >>>><

>>>>:

form

¼ � <sup>d</sup><sup>0</sup>

ΔUout �

Principal scheme of the buck converter is shown in Figure 9. This converter has also two steps in its functioning. In the second step when the switch is in position 1, the capacitor loads through the inductor and inductor current increases, so the inductor behaves as a voltage source with inverse polarity with respect to the input voltage. In this manner on the capacitor, one can obtain a voltage smaller than the input voltage. In the second step when the switch is in position 2, energy accumulated in the inductor is pushed on the capacitor and thus Inductor current decreases, so it behaves as a voltage source with the same polarity as the input voltage but with a smaller value. So, the buck converter decreases the input voltage. In Figure 10,

Following Figure 10, one can write the following relations: Uin � UL ¼ Uout ¼ d�Uin, iT þ iD ¼ iL ¼ iS, Uin�iT ¼ Uout�iS: After several transformations, following system of equations give:

dt

<sup>d</sup> � <sup>C</sup> dUout dt

IT d � �

Vin0 � d<sup>0</sup> <sup>L</sup> <sup>Δ</sup><sup>d</sup> <sup>þ</sup>

RS � <sup>C</sup>ΔUout � IT<sup>0</sup>

IT<sup>0</sup> d0 Δ \_ d

<sup>C</sup> � <sup>d</sup><sup>2</sup> 0 Δd ð5Þ

ð6Þ

ð7Þ

ð8Þ

<sup>U</sup>out <sup>¼</sup> <sup>U</sup>in<sup>d</sup> � <sup>L</sup> <sup>d</sup>

¼ IT 1

> d2 0 <sup>L</sup> <sup>Δ</sup>Uin <sup>þ</sup>

<sup>Δ</sup>IT � <sup>1</sup>

Applying Laplace in system zero initial conditions, Eq. (6) results in transfer functions of the

<sup>¼</sup> RS � <sup>d</sup><sup>0</sup>

<sup>Δ</sup><sup>d</sup> <sup>¼</sup> RS � <sup>U</sup>in0

Buck converter parameters considered in this chapter are <sup>L</sup> <sup>¼</sup> <sup>56</sup>:<sup>5</sup> <sup>μ</sup>F, C <sup>¼</sup> <sup>166</sup>:<sup>7</sup> <sup>μ</sup>F, RS <sup>¼</sup> <sup>2</sup> <sup>Ω</sup>, d ¼ 0:5, Uin0 ¼ 24 V, Uout0 ¼ 12 V: Simulation scheme for the buck converter with transfer functions, open loop is shown in Figure 11. Simulation results are presented in Figure 12. Utilizing a closed loop scheme and the same fuzzy controllers, like in the case of boost converter shown in Figure 4, closed-loop simulation results are obtained shown in Figure 13. For the buck converter control, a classical PI fuzzy controller has been designed with two inputs, the error between the output voltage and the voltage prescribed and integral of this error. Simulation scheme in this case is the one in Figure 14. The membership functions and control surface of this

s<sup>2</sup>ðRS � L � CÞ þ L � s þ RS

s<sup>2</sup>ðRS � L � CÞ þ L � s þ RS

Uout RS

By linearization, like the case of the boost converter, resulted linearized model is:

<sup>L</sup> <sup>Δ</sup>Uout <sup>þ</sup>

<sup>¼</sup> <sup>1</sup> C � d<sup>0</sup>

ΔUout ΔUin

ΔUout

8 >>><

>>>:

Figure 10. Equivalent scheme for continuous mode of buck converter.

Figure 11. Open loop simulation scheme for buck converter.

controller are presented in Figure 15. The inference rules of fuzzy PI controller here are defined in table of this Figure 16. Buck converter behaviour with this control is shown in Figure 17. Note that the peak voltage obtained from step input is reduced to one third of the peak voltage obtained with the controller in the previous case (fuzzy controllers P and I are put in parallel), and the time to restore the prescribed value is also diminished.

Figure 12. Open loop behaviour of buck converter: (a) input voltage variation; (b) duty cycle variation; (c) output voltage variation.

Figure 13. Closed loop behaviour of buck converter with two fuzzy controllers in parallel P+I: (a) input voltage variation; (b). output voltage variation.

Figure 14. Closed loop simulation scheme for buck converter with one PI fuzzy controller with two inputs.

Applications of the Fuzzy Logic to the Energy Conversion Systems on Board of UAVs http://dx.doi.org/10.5772/67992 117

Figure 15. PI fuzzy controller with two inputs: (a) membership functions for error input; (b) membership functions for integral error input; (c) membership functions for output; (d) control surface.


Figure 16. Inference rules for PI fuzzy controller with two inputs.

Figure 12. Open loop behaviour of buck converter: (a) input voltage variation; (b) duty cycle variation; (c) output voltage

Figure 13. Closed loop behaviour of buck converter with two fuzzy controllers in parallel P+I: (a) input voltage variation;

Figure 14. Closed loop simulation scheme for buck converter with one PI fuzzy controller with two inputs.

variation.

116 Modern Fuzzy Control Systems and Its Applications

(b). output voltage variation.

So, we can say that this second fuzzy PI controller with two inputs provides better behaviour, but at the expense of greater computation time, so microcontroller that is implemented should have better performance.

Figure 17. Closed loop behaviour of buck converter with PI fuzzy controller with two inputs: (a) input voltage variation; (b) output voltage variation.

#### 4. Buck-boost converter

Principle scheme of the buck-boost converter is in Figure 18. This converter outputs a higher or a lower voltage than the input voltage but with inverse polarity, so it is known as controllerinverter. It also functions in two steps. In step 1, when the switch is in position 1, the diode is inversely polarized and the current flows from the source through the inductor and charges its electromagnetic field. In the second step, when the switch is in position 2, the energy stored in the inductor is released on the capacitor and the output load with inverse polarity of the voltage.

From the point of view of the average values, it was considered a voltage drop occurs on the switch. Also one considered the coil average voltage is equal with continuous output voltage. Following Figure 19, one can write the following relations: iS þ iC ¼ iD, iD þ iT ¼ iL, UL ¼ <sup>U</sup>in� <sup>d</sup> <sup>d</sup>�<sup>1</sup> , iTð<sup>d</sup> � <sup>1</sup>Þ ¼ iD�d. It has considered among others stationary input-output characteristics of buck-boost converter. In addition, one can write the following equations:

$$
\mathcal{U}\mathcal{U}\_{\rm in} + \mathcal{U}\_T - \mathcal{U}\_L = -L\frac{d\dot{\imath}\_L}{dt} \tag{9}
$$

$$dI\_{\rm in} = -L\frac{d\dot{\imath}\_L}{dt} + R\_{\rm S}\dot{\imath}\_{\rm S} \tag{10}$$

Figure 18. Principle scheme of buck-boost converter.

Applications of the Fuzzy Logic to the Energy Conversion Systems on Board of UAVs http://dx.doi.org/10.5772/67992 119

Figure 19. Equivalent scheme for continuous mode of buck-boost converter.

$$\mathbf{C}\frac{d\mathbf{U}\_{\rm out}}{dt} + \frac{\mathbf{U}\_{\rm out}}{R\_S} + \dot{\mathbf{r}}\_T = \dot{\mathbf{r}}\_L \tag{11}$$

After several transformations, we obtain:

4. Buck-boost converter

118 Modern Fuzzy Control Systems and Its Applications

Figure 18. Principle scheme of buck-boost converter.

(b) output voltage variation.

<sup>U</sup>in� <sup>d</sup>

Principle scheme of the buck-boost converter is in Figure 18. This converter outputs a higher or a lower voltage than the input voltage but with inverse polarity, so it is known as controllerinverter. It also functions in two steps. In step 1, when the switch is in position 1, the diode is inversely polarized and the current flows from the source through the inductor and charges its electromagnetic field. In the second step, when the switch is in position 2, the energy stored in the inductor is released on the capacitor and the output load with inverse polarity of the voltage.

Figure 17. Closed loop behaviour of buck converter with PI fuzzy controller with two inputs: (a) input voltage variation;

From the point of view of the average values, it was considered a voltage drop occurs on the switch. Also one considered the coil average voltage is equal with continuous output voltage. Following Figure 19, one can write the following relations: iS þ iC ¼ iD, iD þ iT ¼ iL, UL ¼

<sup>U</sup>in <sup>þ</sup> UT � UL ¼ �<sup>L</sup> diL

<sup>U</sup>in ¼ �<sup>L</sup> diL

tics of buck-boost converter. In addition, one can write the following equations:

<sup>d</sup>�<sup>1</sup> , iTð<sup>d</sup> � <sup>1</sup>Þ ¼ iD�d. It has considered among others stationary input-output characteris-

dt <sup>ð</sup>9<sup>Þ</sup>

dt <sup>þ</sup> RSiS <sup>ð</sup>10<sup>Þ</sup>

$$\mathcal{U}\_{\rm out} - \mathcal{U}\_{\rm in} \frac{d}{d-1} = -L \cdot \mathbb{C} \frac{d^2 \mathcal{U}\_{\rm out}}{dt^2} \cdot \frac{1}{d-1} + \frac{L}{R\_S} \frac{d \mathcal{U}\_{\rm out}}{dt} \cdot \frac{1}{d-1} \tag{12}$$

By linearization and applying Laplace transform in null initial conditions, transfer functions result as follows:

$$\frac{\Delta \mathcal{U}\_{\rm out}(\mathbf{s})}{\Delta \mathcal{U}\_{\rm in}(\mathbf{s})} = \frac{\frac{d\_0}{d\_0 - 1}}{-L \cdot \mathbb{C} \frac{d\_0}{d\_0 - 1} \mathbf{s}^2 - \frac{L}{R\_S} \cdot \frac{d\_0}{d\_0 - 1} \mathbf{s} + \mathbf{1}} \tag{13}$$

$$\frac{\Delta \mathcal{U}\_{\rm out}(\mathbf{s})}{\Delta d(\mathbf{s})} = \frac{-\mathcal{U}\_{\rm in0} \frac{1}{(d\_0 - 1)^2}}{-L \cdot \mathbb{C} \frac{d\_0}{d\_0 - 1} \mathbf{s}^2 - \frac{L}{\mathbb{R} \cdot \mathbf{s}} \cdot \frac{d\_0}{d\_0 - 1} \mathbf{s} + 1} \tag{14}$$

Parameters used in the simulation buck-boost converter are: L ¼ 56:5 μF, C ¼ 166:7 μF, RS <sup>¼</sup> <sup>2</sup> <sup>Ω</sup>, d <sup>¼</sup> <sup>0</sup>:5, Uin0 <sup>¼</sup> 15 V, Uout0 ¼ �15 V: Scheme simulation for buck-boost converter in open circuit is similar to that for buck converter in Figure 11. Responses to step signals applied to both voltage input, and on the duty cycle are shown in Figure 20. Although for this converter two versions of fuzzy controllers were made, one version with two fuzzy controllers disposed in parallel, one for the proportional component and one for the integrative component, and a second version with a single fuzzy controller with two inputs (proportional and integrative) and one output (duty cycle variation). For the first variant, fuzzy controllers are shown in Figures 21 and 22. Buck-boost converter case has chosen five membership functions, one of which corresponds to the term linguistic ZERO (ZE in Figures 21 and 22), the other membership functions corresponding to the same linguistic terms as with previous fuzzy regulators. The inference rules in this case were the same for both controllers: If the input is NBIG, then output is NBIG; If the input is NMIC, then output is NMIC; If the input is ZE, then output is ZE; If the input is PMIC, then output is PMIC; If the input is PBIG, then output is PBIG.

Figure 20. Buck-boost converter behaviour in open loop: (a) input voltage step; (b) output voltage at input voltage step; (c) duty cycle step; (d) output voltage at duty cycle step.

Figure 21. P fuzzy controller for buck-boost converter: (a) membership functions for input; (b) membership functions for output; (c) control surface.

Closed-loop simulation scheme for this case is similar to that for boost converter in Figure 4 but with transfer functions and controllers shown in buck-boost converter. Buck-boost converter's response in a closed loop and step input are shown in Figure 23.

The behaviour of the converter for sinusoidal input voltage variation has been studied in this case which is shown in Figure 24.

Applications of the Fuzzy Logic to the Energy Conversion Systems on Board of UAVs http://dx.doi.org/10.5772/67992 121

Figure 22. I fuzzy controller for buck-boost converter: (a) membership functions for input; (b) membership functions for output; (c) control surface.

Figure 23. Closed loop behaviour of buck-boost converter with two fuzzy controllers in parallel (P+I): (a) input voltage variation; (b) output voltage variation.

Figure 24. Closed loop behaviour of buck-boost converter with two fuzzy controllers in at sinusoidal input: (a) input voltage variation; (b) output voltage variation.

Closed-loop simulation scheme for this case is similar to that for boost converter in Figure 4 but with transfer functions and controllers shown in buck-boost converter. Buck-boost con-

Figure 21. P fuzzy controller for buck-boost converter: (a) membership functions for input; (b) membership functions for

Figure 20. Buck-boost converter behaviour in open loop: (a) input voltage step; (b) output voltage at input voltage step;

The behaviour of the converter for sinusoidal input voltage variation has been studied in this

verter's response in a closed loop and step input are shown in Figure 23.

case which is shown in Figure 24.

output; (c) control surface.

(c) duty cycle step; (d) output voltage at duty cycle step.

120 Modern Fuzzy Control Systems and Its Applications

We notice here a better behaviour of fuzzy controller at step signal, the restore time to prescribed value is lower than the other controller presented converters but notice a worsened behaviour for sinusoidal signal response. At zero-crossing of the input signal, one obtained jumps in output voltage. The amplitude is not large, about 0.3 V, but these jumps means higher

Figure 25. PI fuzzy controller with two inputs for buck-boost converter: (a) membership functions for proportional input; (b) membership functions for integral input; (c) membership functions for output; (d) control surface.


Figure 26. Inference rules for PI fuzzy controller.

harmonics induced on power bus and thus worsening power quality on board of the UAVs. A second version of fuzzy controller designed for buck-boost converter is that of a controller with two inputs and one output, with five membership functions on each input and the output. This fuzzy controller is shown in Figure 25. Inference rules for this fuzzy controller are presented in table in Figure 26.

One observes a good response at step input in this case; but at sinusoidal input, voltage peaks appear again at input zero crossing. That means again worse energy quality on board.

## 5. Conclusions

Figure 25. PI fuzzy controller with two inputs for buck-boost converter: (a) membership functions for proportional input;

(b) membership functions for integral input; (c) membership functions for output; (d) control surface.

Figure 26. Inference rules for PI fuzzy controller.

122 Modern Fuzzy Control Systems and Its Applications

This paper presents some applications of the fuzzy control technique to DC-DC converters in usual configurations (boost, buck, buck-boost). For each converter, first averaged mathematical models are presented, obtained upon some equivalent schemes for continuous regime. Upon these averaged models, transfer functions for each converter are deduced. These transfer functions are applicable for the transitory regimes in the small perturbation hypothesis. One has to take into account that the averaged models and transfer functions are available for the continuous conduction regime for each converter. In order to obtain the continuous conduction regime, each of the studied converters needs a big enough switching frequency. This frequency does not appear explicitly in the transfer functions, but is determined by the circuit components.

Fuzzy controllers developed here for the studied converters are less usual. First, for the boost and buck converters, one used fuzzy controllers with four membership functions two on each input and output. In order to diminish the calculus volume for each functioning step as much as possible, one used triangular membership functions at the middle of the interval and trapezoidal membership functions at extremities. Although, one followed to reduce the number of membership functions.

For a more stable behaviour of the converters, one tried to reduce the control surface slope around the origin and to keep a bigger slope far from the origin, in order to accelerate converter return to the desired functioning point. By a convenient choose of the membership functions, one could reach this purpose in conditions of a small number of membership functions, as one can see in Figures 5c, 6c and less in Figure 15d.

To reduce further the calculus volume at each functioning step and to simplify inference rules definition, one tested the possibility to decompose a PI fuzzy controller with two inputs in two simpler fuzzy controllers, each of them with one input. These new fuzzy controllers are disposed in parallel and one has as input the error between the real output voltage and the desired output voltage, and the other has as input the integral of this error. In this way, one can reduce the inference rules from n · n to 2 · n where n is the number of the membership functions. Fuzzy controllers defined for the boost converter were used successfully to control the buck converter.

For buck converter, one designed a classical PI fuzzy controller with two inputs and one output, with four membership functions, two on each input an on the output (see Figure 15). Performances obtained with this controller are improved. The peaks on the output voltage are reduced, and time to return to the prescribed voltage is also reduced (see Figure 17). In Figure 17, one observes response slope changes when the controller passes from a control interval to another and the return acceleration when the integrative component reaches the limit of membership function PMC. One observes further response of slope diminishing when the error decreases and smooth approach of the real output voltage to the prescribed one.

For the buck-boost converter, one tested classical fuzzy converters with odd number of membership functions on each input (one of them corresponding to the linguistic term ZERO). One aimed to obtain for this controller a small slope of the control surface near the origin and a bigger slope far from it. Designed controllers can be seen in Figures 21, 22 and 25. Buck-boost converter behaviour is good when step input is used, as one can see in Figures 23 and 27. But when sinusoidal input is used, problems appear. In Figures 24 and 27 one can see the output voltage peaks when the input passes through zero. That means worse quality of energy on board.

As an important conclusion, one can say that control with fuzzy technique offers a special flexibility to the controller design. One can reach very good performances even for simplified configurations of the controller. Possibility to obtain a convenient control surface for each

Figure 27. Buck-boost converter behaviour with PI fuzzy controller: (a) voltage step input; (b) response to step input; (c) sinusoidal voltage input; (d) response to sinusoidal input.

application is an advantage that cannot be reached with other control techniques. Although, in some cases, design of a good fuzzy controller needs some experience and testing of many variants, in order to select the better one. Simplistic definition of the membership functions and inference rules can lead either to inconvenient control surfaces or to linear ones. In the second case, the fuzzy controller could be replaced very well with a classical P or PI controller, with same performances.

## Author details

reduced, and time to return to the prescribed voltage is also reduced (see Figure 17). In Figure 17, one observes response slope changes when the controller passes from a control interval to another and the return acceleration when the integrative component reaches the limit of membership function PMC. One observes further response of slope diminishing when the error decreases and smooth approach of the real output voltage to the prescribed one.

124 Modern Fuzzy Control Systems and Its Applications

For the buck-boost converter, one tested classical fuzzy converters with odd number of membership functions on each input (one of them corresponding to the linguistic term ZERO). One aimed to obtain for this controller a small slope of the control surface near the origin and a bigger slope far from it. Designed controllers can be seen in Figures 21, 22 and 25. Buck-boost converter behaviour is good when step input is used, as one can see in Figures 23 and 27. But when sinusoidal input is used, problems appear. In Figures 24 and 27 one can see the output voltage peaks when the input passes through zero. That means worse quality of energy on board.

As an important conclusion, one can say that control with fuzzy technique offers a special flexibility to the controller design. One can reach very good performances even for simplified configurations of the controller. Possibility to obtain a convenient control surface for each

Figure 27. Buck-boost converter behaviour with PI fuzzy controller: (a) voltage step input; (b) response to step input; (c)

sinusoidal voltage input; (d) response to sinusoidal input.

Dinca Liviu\* and Corcau Jenica Ileana

\*Address all correspondence to: ldinca@elth.ucv.ro

Department of Electrical, Energetic and Aerospace Engineering, Faculty of Electrical Engineering, University of Craiova, Craiova, Romania

## References


## **Fuzzy Logic Energy Management for a Residential Power System Using Renewable Energy Sources**

Stefan Breban, Ioana Gros, Calin Marginean and Petre Teodosescu

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.69486

#### **Abstract**

[9] A. G. Perry, G. Feng, Y.-F. Liu, P.C. Sen. A New Design Method for PI-like Fuzzy Logic Controllers for DC-to-DC Converters. In: 35th Annual IEEE Power Electronics Specialists

[10] A. Gad, M. Farooq. Application of Fuzzy Logic in Engineering Problems. Proceedings of the 27th annual conference of the IEEE Industrial Electronics Society (IECON 01), Denver,

[11] B. Johansson. DC-DC converters-dynamic model design and experimental verification [thesis]. Lund University: Doctoral Dissertation in Industrial Automation Department of

[12] M. Salimi, A. Zakipour. Direct voltage regulation of DC-DC buck converter in a wide range of operation using adaptive input-output linearization. In: IEEE Transactions on Electrical and Electronic Engineering, January 2015; pp. 85–91. DOI: ISSN 1931–4973. [13] J. Chen, D. Maksimovic, R. Erickson. Buck-boost PWM converters having two independently controlled switches. In: Proc. IEEE Power Electronics Specialists Conference; Van-

[14] L. Guo. Design and implementation of digital controllers for buck and boost converters using linear and nonlinear control methods [dissertation]. Auburn University, Alabama;

[15] H. Fadali. Fuel cell distributed generation, power conditioning, control and energy management [dissertation]. Master of Applied Science in Electrical and Computer Engineer-

[16] P. Hemachander, A. D. VimalRaj, M. Sudhakaran. Analysis design and implementation f soft single switched boost converter. In: International Journal of Computer Applications;

[17] P. S. Priambodo, D. Sukoco, W. Purnomo, H. Sudibyo, D. Hartanto., Electric Energy Management and Engineering in Solar Cell System. Chapter 12 in book "Solar cells – Research and Application Perspectives" editor Arturo Morales-Acevedo, published by INTECH on

Conference; 2004; pp. 3751–3757.

126 Modern Fuzzy Control Systems and Its Applications

CO, 29 November-2 December 2001; p. 2044–2049.

couver Canada, June 17–21, 2001; pp. 736–741.

2006.

ing, Ontario, Canada; 2008.

26(11), July 2011; pp. 38–46.

6th March 2013; pp. 327–351.

Industrial Electrical Engineering and Automation; 2004.

A fuzzy logic energy management algorithm is proposed for a hybrid wind/photovoltaic (PV) power generation unit, an electric vehicle battery, and a heat pump for household applications. The proposed concept refers to two independent power systems—a light electric vehicle and a household that interact through light, interchangeable batteries; moreover, they are powered from a renewable energy system comprising PV panels, wind generator, and appropriate MPPT-based converters. The main features of the concept are the heat pump load that produces thermal energy, as the main electric load of the system, and the storage element that is alternately used by the vehicle, which can be recharged from renewable sources. The presented algorithm allows the implementation, by means of fuzzy tools, of an appropriate energy management control system in order to obtain maximum utilization of the renewable energy. The results show that most of the energy required to charge the battery and to feed the heat pump can be covered from renewable sources.

**Keywords:** fuzzy logic energy management, PVs, wind turbine, heat pump, electric vehicle, battery

## **1. Introduction**

Over the last decades, renewable energy sources have witnessed major (annual) growth rates, mainly the solar energy ones, which offer competitive, environmental friendly, low-cost solutions, accessible at a mass production level. The renewable energy producing units located close to energy loads are advantageous as the transportation energy losses are practically eliminated. The energy needed for a typical residential home is relatively small and can be covered mainly from renewable sources.

© 2017 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.

Considering the two major renewable energy sources, such as the wind and sun, the efficient energy harvesting from these sources is one major target of the energy industry. Although these energies seem to be free, with low negative impact, only the way these energies are extracted and used can state if the system is a sustainable system or not. In view of this, one can state that a system is efficiently harvesting the energy from these sources in a sustainable way if the overall efficiency of the system is high and if the energy produced is managed in such a way that these systems have minimum negative effect on the power grid. In other words, it is more efficient to use or store locally the energy produced from renewable sources instead of injecting the energy into the grid. In this way, the power grid is not perturbed by a small production facility with high dynamic behavior. Moreover, if the system is designed in an overall cost-efficient way, the number of PV panels, the wind turbine power, and battery storage capacity can all be minimized in order to obtain an optimized solution in which the power grid can be used only as an energy buffer supply.

The renewable systems of the future will have to integrate all the energy-dependent applications into a system that can be centrally controlled to obtain the best cost-performance balance. In view of this, one interesting way is to combine the electric car and the home electric systems. Concepts like vehicle-to-grid are recently being introduced [1], imposing that the energy storage of the electric vehicle can be integrated into the home electric system to maximize the overall system performance. An overview on solar heat pump systems is presented in Ref. [2]. Garcia et al. [3] present an optimal energy management system for standalone wind turbine/photovoltaic/hydrogen/battery hybrid based on fuzzy logic. Also, Ben Salah et al. [4] and Athari and Ardehali [5] are using fuzzy logic to facilitate the integration of renewable energy into residential or decentralized small power grid applications. Several papers, written by the same research group, present optimal control of different combinations of power sources that feed a heat pump and other loads. A wind/PV + power grid combination is used in Ref. [6], whereas a PV/diesel/battery + power grid combination and a fuel cell/wind/ PV + power grid combination are used in Refs. [7] and [8], respectively. These three papers present cost savings derived from optimal control strategies.

The following sections present the power system under study, the presentation of the fuzzy logic energy management, the simulation results, and a conclusion.

## **2. Power system under study**

This chapter proposes a concept in which two power systems interact by means of two interchangeable small batteries: a lightweight electric vehicle and a household partially powered from renewable energy sources. The overall schematic of the studied system is represented in **Figure 1**. The power generation system is composed of renewable energy conversion equipment: PV panels and a wind generator, both components being connected at the outputs on a DC-Link by means of MPPT-based converters. This DC-Link can be considered the main power line of the system on which also all the DC house loads are connected. This power system is also connected to the grid with a bidirectional DC-AC converter. The Fuzzy Logic Energy Management for a Residential Power System Using Renewable Energy Sources http://dx.doi.org/10.5772/intechopen.69486 129

**Figure 1.** Schematic of the power system.

Considering the two major renewable energy sources, such as the wind and sun, the efficient energy harvesting from these sources is one major target of the energy industry. Although these energies seem to be free, with low negative impact, only the way these energies are extracted and used can state if the system is a sustainable system or not. In view of this, one can state that a system is efficiently harvesting the energy from these sources in a sustainable way if the overall efficiency of the system is high and if the energy produced is managed in such a way that these systems have minimum negative effect on the power grid. In other words, it is more efficient to use or store locally the energy produced from renewable sources instead of injecting the energy into the grid. In this way, the power grid is not perturbed by a small production facility with high dynamic behavior. Moreover, if the system is designed in an overall cost-efficient way, the number of PV panels, the wind turbine power, and battery storage capacity can all be minimized in order to obtain an optimized solution in which the

The renewable systems of the future will have to integrate all the energy-dependent applications into a system that can be centrally controlled to obtain the best cost-performance balance. In view of this, one interesting way is to combine the electric car and the home electric systems. Concepts like vehicle-to-grid are recently being introduced [1], imposing that the energy storage of the electric vehicle can be integrated into the home electric system to maximize the overall system performance. An overview on solar heat pump systems is presented in Ref. [2]. Garcia et al. [3] present an optimal energy management system for standalone wind turbine/photovoltaic/hydrogen/battery hybrid based on fuzzy logic. Also, Ben Salah et al. [4] and Athari and Ardehali [5] are using fuzzy logic to facilitate the integration of renewable energy into residential or decentralized small power grid applications. Several papers, written by the same research group, present optimal control of different combinations of power sources that feed a heat pump and other loads. A wind/PV + power grid combination is used in Ref. [6], whereas a PV/diesel/battery + power grid combination and a fuel cell/wind/ PV + power grid combination are used in Refs. [7] and [8], respectively. These three papers

The following sections present the power system under study, the presentation of the fuzzy

This chapter proposes a concept in which two power systems interact by means of two interchangeable small batteries: a lightweight electric vehicle and a household partially powered from renewable energy sources. The overall schematic of the studied system is represented in **Figure 1**. The power generation system is composed of renewable energy conversion equipment: PV panels and a wind generator, both components being connected at the outputs on a DC-Link by means of MPPT-based converters. This DC-Link can be considered the main power line of the system on which also all the DC house loads are connected. This power system is also connected to the grid with a bidirectional DC-AC converter. The

power grid can be used only as an energy buffer supply.

128 Modern Fuzzy Control Systems and Its Applications

present cost savings derived from optimal control strategies.

**2. Power system under study**

logic energy management, the simulation results, and a conclusion.

storage element is a battery from a lightweight electric vehicle. This small capacity battery is exchangeable with a second battery that is installed in the vehicle. The heat pump load is a key element in the proposed system, mainly because it will be the main electric load in balancing the energy from the renewable sources. Also, a key element of the proposed system is that the battery storage capacity is kept to a minimum; the two batteries are alternately used by the vehicle and are recharged mainly from the renewable sources. The battery can be used by the home electric system only when the power grid is unavailable. It must be stated that, to obtain a feasible approach, the battery has to be as light as possible in order to be easily exchanged.

The main objective of this study is to develop an energy management control system, based on fuzzy logic, which performs in such way that all the energy from the renewable sources is used, with minimum or no energy injection into the power grid and with a small energy storage capacity. The electrical energy can also be obtained from the power grid. In this way, the best cost-performance tradeoff with minimum storage capacity and maximum utilization of the renewable energy can be obtained.

#### **3. Fuzzy logic energy management**

The fuzzy logic energy management is developed with the aim of splitting the renewable energy power to be either stored into the battery or transformed into thermal energy, with a multiplication factor by a heat pump. The fuzzy algorithm is using as inputs the battery state of charge (SoC) and the required heat energy (hot water and heating during cold periods). The output is the battery power (*Pbatt*), the heat pump power (*Php*) being calculated with the simple Eq. (1):

$$P\_{hy} = 1 \text{--} P\_{\text{Aut}} \tag{1}$$

**Figure 2** presents the two inputs and one output of the fuzzy logic supervisor.

**Figures 3**–**5** present the membership functions (MFs) of the two inputs and output. The implementation of the fuzzy logic supervisor is done using Fuzzy logic toolbox and Matlab/ Simulink® software from Mathworks®. It should be noted that those membership functions are built considering that the sum of them is on the entire interval 1 and the variation on both axes is expressed in per unit.

The rule base of the fuzzy algorithm is presented in **Table 1**. It contains nine rules according to the fact that each input has three MFs, all of them being considered in the rule-editing phase.

**Figure 2.** Fuzzy logic energy management inputs and output.

**Figure 3.** MFs for the first input of the fuzzy logic management system.

Fuzzy Logic Energy Management for a Residential Power System Using Renewable Energy Sources http://dx.doi.org/10.5772/intechopen.69486 131

**Figure 4.** MFs for the second input of the fuzzy logic management system.

**3. Fuzzy logic energy management**

130 Modern Fuzzy Control Systems and Its Applications

simple Eq. (1):

axes is expressed in per unit.

level

Thermal energy

Battery SoC

**Figure 2.** Fuzzy logic energy management inputs and output.

**Figure 3.** MFs for the first input of the fuzzy logic management system.

The fuzzy logic energy management is developed with the aim of splitting the renewable energy power to be either stored into the battery or transformed into thermal energy, with a multiplication factor by a heat pump. The fuzzy algorithm is using as inputs the battery state of charge (SoC) and the required heat energy (hot water and heating during cold periods). The output is the battery power (*Pbatt*), the heat pump power (*Php*) being calculated with the

*P hp* = 1−*P batt* (1)

**Figures 3**–**5** present the membership functions (MFs) of the two inputs and output. The implementation of the fuzzy logic supervisor is done using Fuzzy logic toolbox and Matlab/ Simulink® software from Mathworks®. It should be noted that those membership functions are built considering that the sum of them is on the entire interval 1 and the variation on both

The rule base of the fuzzy algorithm is presented in **Table 1**. It contains nine rules according to the fact that each input has three MFs, all of them being considered in the rule-editing phase.

Inference Battery power

**Figure 2** presents the two inputs and one output of the fuzzy logic supervisor.

**Figure 5.** The output of the fuzzy logic management system.


**Table 1.** Rule base of the fuzzy logic power management.

These rules were established considering the next principle: the thermal energy level and the battery SoC should increase similarly if they have a similar level (in per unit), but if one is lower than the other, it will be favored in order to reach a similar level (in per unit).

The defuzzification method used is centroid, which returns the center of area under the response surface. **Figure 6** presents the response surface generated for this fuzzy logic supervisor. As it can be seen from this figure, the output variation does not reach the full range between 0 and 1 (its range is between 0.163 and 0.837). Thus, in order to extend the output range, the output result is subtracted with 0.163 and then multiplied with 1.485.

**Figure 6.** Response surface for the fuzzy logic supervisor.

## **4. Simulation results**

The following figures present the simulation results for a power system with the following assumptions:


The defuzzification method used is centroid, which returns the center of area under the response surface. **Figure 6** presents the response surface generated for this fuzzy logic supervisor. As it can be seen from this figure, the output variation does not reach the full range between 0 and 1 (its range is between 0.163 and 0.837). Thus, in order to extend the output

The following figures present the simulation results for a power system with the following

• The electric vehicle battery has a capacity of 1 kWh with an SoC of 40% when the charge

• It is considered that the vehicle is an ultralight version having two interchangeable batter-

• The heat pump has a coefficient of performance (COP) of 3, thus it consumes one-third

• Wind turbine and PV panels have an installed power of 1 kW each.

ies: one that equips the vehicle and the other left at home for charging.

electrical energy and generates three times more heat energy.

range, the output result is subtracted with 0.163 and then multiplied with 1.485.

**4. Simulation results**

**Figure 6.** Response surface for the fuzzy logic supervisor.

132 Modern Fuzzy Control Systems and Its Applications

assumptions:

process starts.

**Figures 7** and **8** present the power and energy produced by the wind/PV hybrid system. The PV power curve has similar variations with some classic production curves for a sunny day, and the wind power has approximate variations according to a wind measurement from Brasov area (Romania).

The necessary electrical and thermal powers required are presented in **Figure 9**. The electrical power consumption is taken from a figure presented in Ref. [9] and the thermal power is estimated considering 30 kWh of needed energy (for a cold period) and no energy consumption during workhours.

**Figure 10** presents the power grid failures. Even if it is unlikely to have two grid voltage drops during 1 day, it was considered in the simulation for demonstration purposes. Each voltage drop has a period of 15 min, first, from 14h30 to 14h45 and second from 21h30 to 21h45.

**Figures 11** and **12** present the thermal energy level and the battery SoC, respectively, of the heat pump and battery powers, for the considered application. Both the thermal energy level and the SoC have an initial level. Due to the fact that at 18h00, the thermal power consumption restarts, and even if the heat pump is still working, the thermal energy level decreases (**Figure 11**). At the same time (18h00), the batteries are switched, the charged one is placed on the electric vehicle and the discharged one is put to charge. In order to use mainly the renewable energy to charge the battery and considering that the thermal energy demand is high, during the remaining hours of the day (18h00–24h00), the charging of the battery is stopped.

**Figure 7.** Renewable sources power curves.

**Figure 8.** Renewable energy production.

**Figure 9.** Estimated consumption of thermal and electrical powers.

**Figure 10.** Power grid failure.

Fuzzy Logic Energy Management for a Residential Power System Using Renewable Energy Sources http://dx.doi.org/10.5772/intechopen.69486 135

**Figure 11.** Thermal energy level and the battery SoC.

By the end of the day, the thermal energy level and the battery SoC are lower compared with the situation at the beginning of the day. It is clear that not all required energy can be obtained from the renewable sources, thus, some has to come from the power grid to arrive to the necessary level. From **Figure 12** one can observe that the first power grid drop is easily covered from renewable energy sources, just a sudden drop in the heat pump power can be observed. But the second one takes a lot of energy from the battery, which arrives to an SoC of about 5%. If the power grid loss would be longer, the feeding of all the loads would not be possible, and if considered appropriate, only the mandatory loads (pumps, lighting, etc.) should be fed. Or, if charging the discharged one is started at 18h00, the SoC would not decrease that much (**Figure 13**). Of course, in this case, the thermal energy level will decrease. For this situation, the heat pump and battery working powers are changed (**Figure 14**).

**Figure 12.** Heat pump power and battery power.

**Figure 10.** Power grid failure.

**Figure 8.** Renewable energy production.

134 Modern Fuzzy Control Systems and Its Applications

**Figure 9.** Estimated consumption of thermal and electrical powers.

**Figure 13.** Thermal energy level and the battery SoC for the immediate charging of the discharged battery.

**Figure 14.** Heat pump and battery powers for the immediate charging of the discharged battery.

#### **5. Conclusion**

A fuzzy logic energy management algorithm has been proposed and validated by simulations, for a household application. This algorithm allows the distribution of the renewable energy to charge a battery and also to feed a heat pump that produces thermal energy. The results show that the battery charges to around 97%, and the thermal energy level from renewable sources is around 88% for the first case (discharged battery is not charged during the evening) and around 83% for the second (discharged battery is charged during the evening). The rest of the needed energy should be covered from the power grid.

## **Author details**

Stefan Breban\*, Ioana Gros, Calin Marginean and Petre Teodosescu

\*Address all correspondence to: stefan.breban@emd.utcluj.ro

Technical University of Cluj-Napoca, Cluj-Napoca, Romania

## **References**

**5. Conclusion**

136 Modern Fuzzy Control Systems and Its Applications

A fuzzy logic energy management algorithm has been proposed and validated by simulations, for a household application. This algorithm allows the distribution of the renewable energy to charge a battery and also to feed a heat pump that produces thermal energy. The results show that the battery charges to around 97%, and the thermal energy level from renewable sources is around 88% for the first case (discharged battery is not charged during the evening) and around 83% for the second (discharged battery is charged during the evening). The rest of the

**Figure 14.** Heat pump and battery powers for the immediate charging of the discharged battery.

**Figure 13.** Thermal energy level and the battery SoC for the immediate charging of the discharged battery.

needed energy should be covered from the power grid.


## **Robust Adaptive Fuzzy Control for a Class of Switching Power Converters**

Cheng-Lun Chen

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67895

#### Abstract

This chapter provides the reader with a control-centric modeling and analysis approach along with a nonlinear control design for a class of switching power converters. A comprehensive model combining the respective state variable models of the interval subsystems is established. Comparison with PSpice simulation justifies the credibility of the model. Based on this model, internal/BIBO stability can be studied for each interval subsystem. Moreover, controllability and observability can also be analyzed to help determine subsequent control configuration. The established model is further investigated for advanced control design, i.e., robust adaptive fuzzy control.

Keywords: adaptive fuzzy control, dc-dc power converter, modeling, switching power converter

## 1. Introduction

Switching power converters are increasingly taking over conventional linear power converters due to their being compact, lightweight, high efficiency, and larger input voltage range. With the rapid advancement and popularity of personal computers, mobile communication devices, and automotive electronics, the need for stability and efficiency of converters is rising. Among the switching power converters, the phase-shifted pulse-width modulated (PSPWM) fullbridge soft switched power converter [1, 2] and corresponding alteration [3–8] have become a widely used circuit topology due to various beneficial characteristics, e.g., reduction of switching losses and stresses, and elimination of primary snubbers. The circuit is capable of high switching frequency operation with improved power density and conversion efficiency.

© 2017 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.

Feedback control has been incorporated into switching power converters to not only stabilize, but also improve the performance robustness of the output voltage. In spite of its advantageous features, feedback control for soft switched PSPWM full-bridge converters is still confined to simple linear time-invariant design, e.g., proportional-integral (PI) or lead-lag compensators based on a linearized model [9–12]. As pointed out by [13, 14], due to the increased number of topological stages and the PWM duty cycle being affected by input voltage, output voltage, and load current, the dynamics of a PSPWM full-bridge converter is much more sophisticated than that of a simple buck converter. A trade-off needs to be made regarding whether a simple model (e.g., linearized model) or a complex one (e.g., switched model) is to be established for the purpose of control design.

For model-based control design, a mathematical model of appropriate sophistication and capture of desired dynamics is normally the initial step. Such models can usually be obtained by simplifying a complex model, i.e., model reduction, or linearizing a nonlinear around specified operation point. For design of linear controllers, transfer function is a matured modeling tool. For more advanced control design, state variable model is usually a prerequisite. Various modeling approaches for switching power converters of complex topology have been proposed in the literature [15–20]. Most of them have been successful in terms of modeling the "local" behavior (i.e., small signal model) or analysis of the fundamental characteristics. However, few of the results can carry over to the next stage of control design. To be specific, a variety of crucial information for control design cannot be retrieved from those "not control centric" modeling approaches. That essential information includes stability, controllability, and observability of the open-loop system.

This chapter will provide the reader with a control-centric modeling and analysis approach along with robust adaptive fuzzy control design for switching power converters of complex topology and resort to PSPWM full-bridge power converters as a design example. The outline of the chapter is as follows:

Sections 2 and 3 demonstrate how to establish a control-centric mathematical model for a PSPWM full-bridge soft switched power converter system. The set of differential equations and the corresponding state variable model are established for each operation interval. The subsystem models for all intervals are integrated to form a comprehensive model. Numerical simulation of the model is performed and compared to that of the corresponding PSpice model to verify its validity. Section 4 will perform stability analysis for the system. Specifically, stability analysis is performed for each interval subsystem (of the established model) to determine whether the subsystem is internally/BIBO stable. Section 5 will conduct controllability/observability analysis for the system. Controllability and observability of the subsystems are analyzed to determine which signals/variables can actually be manipulated by control effort and which can be estimated using output feedback control structure. The established comprehensive model is further exploited for advanced control design. For example, by getting rid of uncontrollable and unobservable variables and dynamics, an LPV gain-scheduling control design may be conducted as in Ref. [21]. Model reduction and robust adaptive fuzzy control design are presented in Sections 6 and 7. Conclusion and future work are given in Section 8.

## 2. Control-centric mathematical model

Feedback control has been incorporated into switching power converters to not only stabilize, but also improve the performance robustness of the output voltage. In spite of its advantageous features, feedback control for soft switched PSPWM full-bridge converters is still confined to simple linear time-invariant design, e.g., proportional-integral (PI) or lead-lag compensators based on a linearized model [9–12]. As pointed out by [13, 14], due to the increased number of topological stages and the PWM duty cycle being affected by input voltage, output voltage, and load current, the dynamics of a PSPWM full-bridge converter is much more sophisticated than that of a simple buck converter. A trade-off needs to be made regarding whether a simple model (e.g., linearized model) or a complex one (e.g., switched

For model-based control design, a mathematical model of appropriate sophistication and capture of desired dynamics is normally the initial step. Such models can usually be obtained by simplifying a complex model, i.e., model reduction, or linearizing a nonlinear around specified operation point. For design of linear controllers, transfer function is a matured modeling tool. For more advanced control design, state variable model is usually a prerequisite. Various modeling approaches for switching power converters of complex topology have been proposed in the literature [15–20]. Most of them have been successful in terms of modeling the "local" behavior (i.e., small signal model) or analysis of the fundamental characteristics. However, few of the results can carry over to the next stage of control design. To be specific, a variety of crucial information for control design cannot be retrieved from those "not control centric" modeling approaches. That essential information includes stability, controllability, and observability of the

This chapter will provide the reader with a control-centric modeling and analysis approach along with robust adaptive fuzzy control design for switching power converters of complex topology and resort to PSPWM full-bridge power converters as a design example. The outline

Sections 2 and 3 demonstrate how to establish a control-centric mathematical model for a PSPWM full-bridge soft switched power converter system. The set of differential equations and the corresponding state variable model are established for each operation interval. The subsystem models for all intervals are integrated to form a comprehensive model. Numerical simulation of the model is performed and compared to that of the corresponding PSpice model to verify its validity. Section 4 will perform stability analysis for the system. Specifically, stability analysis is performed for each interval subsystem (of the established model) to determine whether the subsystem is internally/BIBO stable. Section 5 will conduct controllability/observability analysis for the system. Controllability and observability of the subsystems are analyzed to determine which signals/variables can actually be manipulated by control effort and which can be estimated using output feedback control structure. The established comprehensive model is further exploited for advanced control design. For example, by getting rid of uncontrollable and unobservable variables and dynamics, an LPV gain-scheduling control design may be conducted as in Ref. [21]. Model reduction and robust adaptive fuzzy control design are presented in Sections 6 and 7. Conclusion and

model) is to be established for the purpose of control design.

140 Modern Fuzzy Control Systems and Its Applications

open-loop system.

of the chapter is as follows:

future work are given in Section 8.

In this section, operation of a PSPWM full-bridge dc-dc power converter will be briefly described. Note that there are eight operation intervals. Due to switching, operation of adjacent intervals is discontinuous. This implies that the parameters and initial conditions change when the converter switches. It will be demonstrated how a comprehensive control-oriented state variable model for each operation interval can be established for subsequent analysis and numerical simulation. The circuit diagram of the converter is shown in Figure 1. Figure 2 is the waveform timing for various signals in the converter, where iLlK is the primary current, vab is the voltage between a and b, iL is the secondary current, vs is the secondary voltage, QA,QB,QC, and QD are the four switches, ΔD is the duty cycle loss, and ZVS delay is the dead time.

## 2.1. Positive half cycle: trailing-leg (passive-to-active) transition <sup>ð</sup>t<sup>0</sup>et1<sup>Þ</sup>

During this operation interval, only QD is conducting. Figure 3 shows the equivalent circuit of state 1. Initial conditions are vCA ðt0Þ ¼ vi, vCB ðt0Þ ¼ 0, vCC ðt0Þ ¼ vi, and vCD ðt0Þ ¼ 0. During this interval, C<sup>A</sup> is discharging and CB is charging until vC<sup>A</sup> equals zero. QAis turned on at zero voltage. Utilizing Kirchhoff's voltage law (KVL) and Kirchhoff's current law (KCL), we arrive at the following set of differential equations:

$$\frac{d\upsilon\_{\mathbb{C}\_{\mathbb{A}}}(t)}{dt} = \frac{\dot{\imath}\_{L\_{k}}}{\mathbb{C}\_{A} + \mathbb{C}\_{\mathbb{B}}},\\\frac{d\upsilon\_{\mathbb{C}\_{\mathbb{B}}}(t)}{dt} = -\frac{\dot{\imath}\_{L\_{k}}}{\mathbb{C}\_{A} + \mathbb{C}\_{\mathbb{B}}},\\\frac{d\upsilon\_{\mathbb{C}\_{\mathbb{C}}}(t)}{dt} = 0,\\\frac{d\upsilon\_{\mathbb{C}\_{\mathbb{D}}}(t)}{dt} = 0 \tag{1}$$

$$-\frac{d\dot{\mathbf{u}}\_{L\mathbf{k}}}{dt} = \frac{d\dot{\mathbf{u}}\_{L}}{dt} = \frac{-1}{n^{2}L\_{\mathrm{lk}} + L}\mathbf{v}\_{\mathrm{C}} + \frac{n}{n^{2}L\_{\mathrm{lk}} + L}\mathbf{v}\_{\mathrm{C}\_{A}} + \frac{-n}{n^{2}L\_{\mathrm{lk}} + L}\mathbf{v}\_{\mathrm{i}\prime}\frac{d\mathbf{v}\_{\mathrm{C}}(t)}{dt} = \frac{1}{\mathbf{C}}\dot{\mathbf{u}}\_{\mathrm{L}} - \frac{1}{RC}\mathbf{v}\_{\mathrm{C}}\tag{2}$$

Figure 1. Circuit topology of a PSPWM full-bridge converter.

Figure 2. Waveform timing for a PSPWM full-bridge converter.

Figure 3. The equivalent circuit of operation interval 1 <sup>ð</sup>t<sup>0</sup>et1Þ.

Define x ⇀ðtÞ¼½ iLlK <sup>ð</sup>t<sup>Þ</sup> iLðt<sup>Þ</sup> vCðt<sup>Þ</sup> vC<sup>A</sup> <sup>ð</sup>t<sup>Þ</sup> vC<sup>B</sup> <sup>ð</sup>t<sup>Þ</sup> vC<sup>C</sup> <sup>ð</sup>t<sup>Þ</sup> vC<sup>D</sup> <sup>ð</sup>tÞ� <sup>T</sup>, where iLlK is leakage inductance current, iL is inductance current, vC is output voltage, vC<sup>A</sup> is the voltage across CA, vCB is the voltage across CB, vCC is the voltage across CC, and vCD is the voltage across CD. Therefore, a state variable model can be obtained as follows:

$$
\stackrel{\circ}{\dot{\mathfrak{X}}}(t) = \begin{bmatrix} 0 & 0 & \frac{n}{n^2 L\_{\mathbb{R}} + L} & \frac{-n^2}{n^2 L\_{\mathbb{R}} + L} & 0 & 0 & 0 \\ 0 & 0 & \frac{-1}{n^2 L\_{\mathbb{R}} + L} & \frac{n}{n^2 L\_{\mathbb{R}} + L} & 0 & 0 & 0 \\ 0 & \frac{1}{C} & -\frac{1}{RC} & 0 & 0 & 0 & 0 \\ \frac{1}{C\_{\mathbb{A}} + C\_{\mathbb{B}}} & 0 & 0 & 0 & 0 & 0 \\ -\frac{1}{C\_{\mathbb{A}} + C\_{\mathbb{B}}} & 0 & 0 & 0 & 0 & 0 \\ -\frac{1}{C\_{\mathbb{A}} + C\_{\mathbb{B}}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \stackrel{\circ}{\dot{\mathfrak{X}}}(t) + \begin{bmatrix} n^2 \\ n^2 L\_{\mathbb{A}} + L \\ -n \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} v\_i(t) \quad (3)$$

where vi is input voltage, N2=N<sup>1</sup> ¼ n is the transformer turns ratio, L is inductance, C is capacitance, and R is resistance.

## 2.2. Positive half cycle: active region <sup>ð</sup>t<sup>1</sup>et2<sup>Þ</sup>

Figure 2. Waveform timing for a PSPWM full-bridge converter.

Modern Fuzzy Control Systems and Its Applications

Figure 3. The equivalent circuit of operation interval 1 <sup>ð</sup>t<sup>0</sup>et1Þ.

During this operation interval, QA and QD are conducting. Initially, due to leakage inductance, the secondary side will experience a short period of "no energy" passing through, called duty cycle loss. Figure 4 shows the equivalent circuit for this period. Initial conditions are vCA ðt1Þ ¼ 0, vCB ðt1Þ ¼ vi, vCC ðt1Þ ¼ vi, and vCD ðt1Þ ¼ 0. Similarly, we may derive a set of differential equations and the corresponding state variable model for this short period is

$$
\dot{\vec{\dot{x}}}(t) = \begin{bmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -\frac{1}{L} & 0 & 0 & 0 & 0 \\
0 & \frac{1}{C} & -\frac{1}{RC} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix} \dot{\vec{\mathbf{x}}}(t) + \begin{bmatrix}
1 \\ L\_{lk} \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0
\end{bmatrix} v\_i(t) \tag{4}
$$

After the short period of duty cycle loss, energy passes through the transformer again. Figure 5 shows the equivalent circuit. Initial conditions are vCA ðt12Þ ¼ 0, vCB ðt12Þ ¼ vi, vCC ðt12Þ ¼ vi, and vCD ðt12Þ ¼ 0. We may derive a set of differential equations and the corresponding set of differential equations and state variable model are

Figure 4. The equivalent circuit of the period of duty cycle loss <sup>ð</sup>t<sup>1</sup>et2Þ.

Figure 5. The equivalent circuit of operation interval 2 <sup>ð</sup>t<sup>1</sup>et2Þ.

$$
\dot{\vec{\chi}}(t) = \begin{bmatrix}
0 & 0 & -\frac{n}{n^2 L\_{\text{kk}} + L} & 0 & 0 & 0 & 0 \\
0 & 0 & -\frac{1}{n^2 L\_{\text{kk}} + L} & 0 & 0 & 0 & 0 \\
0 & \frac{1}{C} & -\frac{1}{RC} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix} \vec{\chi}(t) + \begin{bmatrix} n^2 \\ n \\ n \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} v\_i(t) \tag{5}
$$

The duration of duty cycle loss may be derived based on falling range of iL is equal to rising range of iL, i.e.,

$$\frac{t\_{\text{passive}}}{n^2 L\_{\text{ik}} + L} \upsilon\_{\text{C}} + \frac{\Delta D}{L} \upsilon\_{\text{C}} = \left(\frac{-1}{n^2 L\_{\text{ik}} + L} \upsilon\_{\text{C}} + \frac{n}{n^2 L\_{\text{ik}} + L} \upsilon\_{\text{i}}\right) \times (t\_{\text{active}} - \Delta D)\phi = \frac{t\_{\text{passive}}}{t\_{\text{passive}} + t\_{\text{active}}} \times 180^\circ \quad (6)$$

where ΔD is duty cycle loss time, tpassive is the time of passive region, tactive is the time of active region, and φ is the difference of phase between QA and QC.

## 2.3. Positive half cycle: leading-leg (active-to-passive) transition <sup>ð</sup>t<sup>2</sup>et3<sup>Þ</sup>

During this operation interval, only QA is conducting. Initial conditions are vCA ðt2Þ ¼ 0, vCB ðt2Þ vi, vCC ðt2Þ ¼ vi, and vCD ðt2Þ ¼ 0. In this interval, CC is discharging and CD is charging until vCC equals zero. QC is turned on at zero voltage. Applying KVL/KCL in a similar way, we obtain

$$
\begin{bmatrix}
\dot{\tilde{\mathbf{x}}}(t) \\
\dot{\tilde{\mathbf{x}}}(t) \\
\dot{\tilde{\mathbf{x}}}(t) \\
\dot{\tilde{\mathbf{x}}}(t) \\
\begin{bmatrix}
\dot{\tilde{\mathbf{x}}}(t) \\
0 & 0 \\
0 & \frac{1}{C} & -\frac{1}{RC} \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
\frac{n^{2}}{n^{2}L\_{\rm R} + L} \\
\dot{\boldsymbol{n}} \\
\dot{\boldsymbol{n}}(t) + \begin{bmatrix}
\dot{n} \\
\dot{n} \\
\dot{n}^{2}L\_{\rm R} + L \\
0 \\
0 \\
0 \\
0
\end{bmatrix}
v\_{\rm }(t) \quad(7)
$$

## 2.4. Positive half cycle: Passive region <sup>ð</sup>t<sup>3</sup>et4<sup>Þ</sup>

During this interval, QA and QC are conducting. Initial conditions are vCA ðt3Þ ¼ 0, vCB ðt3Þ ¼ vi, vCC ðt3Þ ¼ 0, and vCD ðt3Þ ¼ vi. Applying KVL/KCL in a similar way, we obtain

$$
\stackrel{\circ}{\underline{\dot{\pi}}}(t) = \begin{bmatrix}
0 & 0 & -\frac{n}{n^2 L\_{\text{fl}} + L} & 0 & 0 & 0 & 0 \\
0 & 0 & -\frac{1}{n^2 L\_{\text{fl}} + L} & 0 & 0 & 0 & 0 \\
0 & \frac{1}{C} & -\frac{1}{RC} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix} \stackrel{\circ}{\underline{\dot{\pi}}}(t) \tag{8}
$$

#### 2.5. Negative half cycle

\_ x ⇀ðtÞ ¼

range of iL, i.e.,

0 0 � <sup>n</sup>

Figure 4. The equivalent circuit of the period of duty cycle loss <sup>ð</sup>t<sup>1</sup>et2Þ.

Modern Fuzzy Control Systems and Its Applications

0 0 � <sup>1</sup>

<sup>C</sup> � <sup>1</sup>

<sup>1</sup>

Figure 5. The equivalent circuit of operation interval 2 <sup>ð</sup>t<sup>1</sup>et2Þ.

<sup>n</sup><sup>2</sup>Llk <sup>þ</sup> <sup>L</sup> <sup>0000</sup>

n2 n<sup>2</sup>Llk þ L n n<sup>2</sup>Llk þ L 

viðtÞ ð5Þ

x ⇀ðtÞ þ

<sup>n</sup><sup>2</sup>Llk <sup>þ</sup> <sup>L</sup> <sup>0000</sup>

00 0 0000 00 0 0000 00 0 0000 00 0 0000

RC <sup>0000</sup>

The duration of duty cycle loss may be derived based on falling range of iL is equal to rising

The subsequent four operation intervals basically "mirror" those in positive cycle. Therefore, the derivations are omitted for brevity.

### 3. Solution and numerical simulation

Numerical simulation based on a typical PSPWM full-bridge power converter circuit (with parameters: transformer turns ratio n = 0.5, Vi ¼ 160 volt, Vo ¼ 50 volt, R = 6 Ω, C = 940 μF, L = 300 μH, Llk ¼ 20μH, CA ¼ CB ¼ CC ¼ CD ¼ 5nF, f <sup>s</sup> ¼ 50 kHz) in our laboratory is performed. A "realistic" model of the circuit is built using PSpice, and the developed mathematical model is realized using MATLAB/Simulink. Comparison of the simulation results validates the correctness and effectiveness of the established model.

### 4. Stability analysis

#### 4.1. Zero-state response

A SISO system ðA, B, CÞ with proper rational transfer function GðsÞ ¼ CðsI � AÞ �1 B is BIBO stable if and only if every pole of GðsÞ has a negative real part or, equivalently, lies inside the left-half s-plane. For both positive and negative half cycles, we can obtain the following transfer function for each operation interval:

• Trailing-leg (passive-to-active) transition:

$$\mathbf{G}\_{1} = \frac{-n\mathbf{R}(\mathbf{C}\_{\mathbf{A}} + \mathbf{C}\_{\mathbf{B}})\mathbf{s}}{\mathbf{R}\mathbf{C}(\mathbf{C}\_{\mathbf{A}} + \mathbf{C}\_{\mathbf{B}})(n^{2}L\_{\mathbf{k}} + L)\mathbf{s}^{3} + (\mathbf{C}\_{\mathbf{A}} + \mathbf{C}\_{\mathbf{B}})(n^{2}L\_{\mathbf{k}} + L)\mathbf{s}^{2} + (R(\mathbf{C}\_{\mathbf{A}} + \mathbf{C}\_{\mathbf{B}}) + n^{2}\mathbf{R}\mathbf{C})\mathbf{s} + n^{2}} \quad (9)$$
  $pole = p11, p12, p13 \text{ (in complicated form)}$ 


$$\begin{aligned} G\_2 &= \frac{nR}{RC(n^2L\_{\text{lk}} + L)s^2 + (n^2L\_{\text{lk}} + L)s + R} \\ pole &= \frac{-(n^2L\_{\text{lk}} + L) \pm \sqrt{(n^2L\_{\text{lk}} + L)^2 - 4R^2C(n^2L\_{\text{lk}} + L)}}{2RC(n^2L\_{\text{lk}} + L)} \end{aligned} \tag{10}$$

• Leading-leg (active-to-passive) transition:

$$\mathbf{G}\_3 = \frac{n\mathbf{R}(\mathbf{C}\_\mathbf{C} + \mathbf{C}\_D)\mathbf{s}}{\mathbf{R}\mathbf{C}(\mathbf{C}\_\mathbf{C} + \mathbf{C}\_D)(n^2L\_\mathbb{K} + L)\mathbf{s}^3 + (\mathbf{C}\_\mathbf{C} + \mathbf{C}\_D)(n^2L\_\mathbb{K} + L)\mathbf{s}^2 + (R(\mathbf{C}\_\mathbf{C} + \mathbf{C}\_D) + n^2\mathbf{R}\mathbf{C})\mathbf{s} + n^2} \tag{11}$$
  $pole = p31, p32, p33 (in complicated form)$ 

• Passive region: no input:

Due to "mirroring" operation, corresponding intervals in positive and negative half cycles will have the same transfer function. The pole location for the operation interval of trailing-leg and leading-leg transition depends further on the values of the circuit elements. The poles for the operation interval of active region have negative real parts due to

$$\left(\left(n^{2}L\_{\text{lk}} + L\right) > \sqrt{\left(n^{2}L\_{\text{lk}} + L\right)^{2} - 4\mathcal{R}^{2}\mathcal{C}\left(n^{2}L\_{\text{lk}} + L\right)}\tag{12}$$

Hence, the system is BIBO stable within this interval. Note that there is pole/zero cancelation for all intervals, which implies that each interval subsystem is either uncontrollable or unobservable.

#### 4.2. Zero-input response

3. Solution and numerical simulation

146 Modern Fuzzy Control Systems and Its Applications

transfer function for each operation interval: • Trailing-leg (passive-to-active) transition:

pole ¼ p11, p12, p13 ðin complicated formÞ

• Active region (duty cycle loss): G<sup>2</sup> loss ¼ 0

• Leading-leg (active-to-passive) transition:

pole ¼ p31, p32, p33ðin complicated formÞ

• Passive region: no input:

<sup>G</sup><sup>1</sup> <sup>¼</sup> �nRðC<sup>A</sup> <sup>þ</sup> <sup>C</sup>BÞ<sup>s</sup>

<sup>G</sup><sup>2</sup> <sup>¼</sup> nR

pole <sup>¼</sup> �ðn<sup>2</sup>Llk <sup>þ</sup> <sup>L</sup>Þ �

<sup>G</sup><sup>3</sup> <sup>¼</sup> nRðCC <sup>þ</sup> CDÞ<sup>s</sup>

4. Stability analysis

4.1. Zero-state response

• Active region:

validates the correctness and effectiveness of the established model.

A SISO system ðA, B, CÞ with proper rational transfer function GðsÞ ¼ CðsI � AÞ

stable if and only if every pole of GðsÞ has a negative real part or, equivalently, lies inside the left-half s-plane. For both positive and negative half cycles, we can obtain the following

RCðC<sup>A</sup> þ CBÞðn<sup>2</sup>Llk þ LÞs<sup>3</sup> þ ðC<sup>A</sup> þ CBÞðn<sup>2</sup>Llk þ LÞs<sup>2</sup> þ ðRðC<sup>A</sup> þ CBÞ þ n<sup>2</sup>RCÞs þ n<sup>2</sup>

ðn<sup>2</sup>Llk þ LÞ

RCðCC þ CDÞðn<sup>2</sup>Llk þ LÞs<sup>3</sup> þ ðCC þ CDÞðn<sup>2</sup>Llk þ LÞs<sup>2</sup> þ ðRðCC þ CDÞ þ n<sup>2</sup>RCÞs þ n<sup>2</sup>

Due to "mirroring" operation, corresponding intervals in positive and negative half cycles will have the same transfer function. The pole location for the operation interval of trailing-leg and

2RCðn<sup>2</sup>Llk þ LÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Cðn<sup>2</sup>Llk þ LÞ

<sup>2</sup> � <sup>4</sup>R<sup>2</sup>

RCðn<sup>2</sup>Llk þ LÞs<sup>2</sup> þ ðn<sup>2</sup>Llk þ LÞs þ R

q

�1

B is BIBO

ð9Þ

ð10Þ

ð11Þ

Numerical simulation based on a typical PSPWM full-bridge power converter circuit (with parameters: transformer turns ratio n = 0.5, Vi ¼ 160 volt, Vo ¼ 50 volt, R = 6 Ω, C = 940 μF, L = 300 μH, Llk ¼ 20μH, CA ¼ CB ¼ CC ¼ CD ¼ 5nF, f <sup>s</sup> ¼ 50 kHz) in our laboratory is performed. A "realistic" model of the circuit is built using PSpice, and the developed mathematical model is realized using MATLAB/Simulink. Comparison of the simulation results

> The equation x\_ ¼ Ax is marginally stable if and only if all eigenvalues of A have zero or negative real parts and those with zero real parts are simple root of the minimal polynomial of A. The equation x\_ ¼ Ax is asymptotically stable if and only if all eigenvalues of A have negative real parts. For both positive and negative half cycles, we can obtain the following set of eigenvalues for each operation interval:

• Trailing-leg (passive-to-active) transitions:

$$A\_{1} = \begin{bmatrix} 0 & 0 & \frac{n}{n^2L\_{\mathbb{R}}+L} & -n^2 & 0 & 0 & 0\\ 0 & 0 & \frac{-1}{n^2L\_{\mathbb{R}}+L} & \frac{n}{n^2L\_{\mathbb{R}}+L} & 0 & 0 & 0\\ 0 & 0 & \frac{-1}{n^2L\_{\mathbb{R}}+L} & \frac{n}{n^2L\_{\mathbb{R}}+L} & 0 & 0 & 0\\ 0 & \frac{1}{\overline{C}} & \overline{RC} & 0 & 0 & 0 & 0\\ \frac{1}{\overline{C}\_{\mathbb{A}}+C\_{\mathbb{B}}} & 0 & 0 & 0 & 0 & 0 & 0\\ \frac{-1}{\overline{C}\_{\mathbb{A}}+C\_{\mathbb{B}}} & 0 & 0 & 0 & 0 & 0 & 0\\ \frac{-1}{\overline{C}\_{\mathbb{A}}+C\_{\mathbb{B}}} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}, \lambda = 0,0,0,0,\lambda\,11,\lambda\,12,\lambda\,13 \qquad (13)$$

• Active region (duty cycle loss):

$$\lambda = 0,0,0,0,0, \frac{-1 \pm \sqrt{(L - 4R^2 \mathcal{C})/L}}{2RC} \tag{14}$$

• Active region:

$$\lambda = 0, 0, 0, 0, 0, \frac{-(n^2 L\_{\text{lk}} + L) \pm \sqrt{(n^2 L\_{\text{lk}} + L)^2 - 4\mathcal{R}^2 \mathcal{C} (n^2 L\_{\text{lk}} + L)}}{2\mathcal{R}\mathcal{C} (n^2 L\_{\text{lk}} + L)}\tag{15}$$

• Leading-leg (active-to-passive) transitions:

$$
\lambda = 0, 0, 0, 0, \lambda \mathbf{31}, \lambda \mathbf{32}, \lambda \mathbf{33} \text{(in complicated form)} \tag{16}
$$

• Passive region:

$$\lambda = 0,0,0,0,0, \frac{-(n^2 L\_{\text{lk}} + L) \pm \sqrt{(n^2 L\_{\text{lk}} + L)^2 - 4R^2 \mathcal{C} (n^2 L\_{\text{lk}} + L)}}{2 \text{RC} (n^2 L\_{\text{lk}} + L)} \tag{17}$$

Since all intervals have zero eigenvalue, we need to determine whether zero is a simple root of the minimal polynomial of A. The minimal polynomial (in x) for each operation interval (positive or negative half cycle) is summarized as follows:

• Trailing-leg (passive-to-active) transitions:

$$\mathbf{x}^4 + \frac{\mathbf{1}}{R\mathbf{C}}\mathbf{x}^3 + \frac{n^2\mathbf{C} + \mathbf{C\_A} + \mathbf{C\_B}}{\mathbf{C(C\_A + C\_B)}(n^2\mathbf{L\_{ik}} + \mathbf{L})}\mathbf{x}^2 + \frac{n^2}{R\mathbf{C(C\_A + C\_B)}(n^2\mathbf{L\_{ik}} + \mathbf{L})}\mathbf{x} \tag{18}$$

• Active region (duty cycle loss):

$$\mathbf{x}^3 + \frac{1}{RC}\mathbf{x}^2 + \frac{1}{LC}\mathbf{x} \tag{19}$$

• Active region:

$$\mathbf{x}^3 + \frac{1}{RC}\mathbf{x}^2 + \frac{1}{C(n^2L\_{\text{lk}} + L)}\mathbf{x} \tag{20}$$

• Leading-leg (active-to-passive) transitions:

$$\mathbf{x}^4 + \frac{1}{RC}\mathbf{x}^3 + \frac{n^2\mathbf{C} + \mathbf{C}\_{\mathbf{C}} + \mathbf{C}\_{\mathbf{D}}}{\mathbf{C}(\mathbf{C}\_{\mathbf{C}} + \mathbf{C}\_{\mathbf{D}})(n^2\mathbf{L}\_{\mathbf{R}} + \mathbf{L})}\mathbf{x}^2 + \frac{n^2}{RC(\mathbf{C}\_{\mathbf{C}} + \mathbf{C}\_{\mathbf{D}})(n^2\mathbf{L}\_{\mathbf{R}} + \mathbf{L})}\mathbf{x} \tag{21}$$

• Passive region:

$$\left(\mathbf{x}^3 + \frac{1}{RC}\mathbf{x}^2 + \frac{1}{C(n^2L\_{\text{lk}} + L)}\mathbf{x}\right) \tag{22}$$

Although all operation intervals have different state matrix (A), corresponding intervals in positive and negative half cycles actually possess the same set of eigenvalues. The eigenvalues for the operation interval of trailing-leg and leading-leg transition depend further on the values of the circuit elements. Both intervals of active (including duty cycle loss) and passive region are marginally stable due to

$$n^2L\_{\rm lk} + L > \sqrt{\left(n^2L\_{\rm lk} + L\right)^2 - 4R^2C(n^2L\_{\rm lk} + L)}, \quad 1 > \sqrt{\frac{L - 4R^2C}{L}}\tag{23}$$

and zero is the simple root of the minimal polynomial.

#### 5. Controllability/observability analysis

λ ¼ 0, 0, 0, 0, λ31, λ32, λ33ðin complicated formÞ ð16Þ

ðn<sup>2</sup>Llk þ LÞ

<sup>x</sup><sup>2</sup> <sup>þ</sup>

1

1 Cðn<sup>2</sup>Llk þ LÞ

<sup>x</sup><sup>2</sup> <sup>þ</sup>

1 Cðn<sup>2</sup>Llk þ LÞ

Although all operation intervals have different state matrix (A), corresponding intervals in positive and negative half cycles actually possess the same set of eigenvalues. The eigenvalues for the operation interval of trailing-leg and leading-leg transition depend further on the values of the circuit elements. Both intervals of active (including duty cycle loss) and passive

q

Since all intervals have zero eigenvalue, we need to determine whether zero is a simple root of the minimal polynomial of A. The minimal polynomial (in x) for each operation interval (positive

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n2 RCðC<sup>A</sup> þ CBÞðn<sup>2</sup>Llk þ LÞ

n2 RCðCC þ CDÞðn<sup>2</sup>Llk þ LÞ

Cðn<sup>2</sup>Llk þ LÞ

LC <sup>x</sup> <sup>ð</sup>19<sup>Þ</sup>

x ð20Þ

x ð22Þ

x ð18Þ

x ð21Þ

<sup>2</sup>RCðn<sup>2</sup>Llk <sup>þ</sup> <sup>L</sup><sup>Þ</sup> <sup>ð</sup>17<sup>Þ</sup>

<sup>2</sup> � <sup>4</sup>R<sup>2</sup>

• Passive region:

λ ¼ 0, 0, 0, 0, 0,

148 Modern Fuzzy Control Systems and Its Applications

or negative half cycle) is summarized as follows: • Trailing-leg (passive-to-active) transitions:

• Leading-leg (active-to-passive) transitions:

<sup>x</sup><sup>4</sup> <sup>þ</sup> 1 RCx<sup>3</sup> <sup>þ</sup>

region are marginally stable due to

<sup>x</sup><sup>4</sup> <sup>þ</sup> 1 RC <sup>x</sup><sup>3</sup> <sup>þ</sup>

• Active region (duty cycle loss):

• Active region:

• Passive region:

�ðn<sup>2</sup>Llk <sup>þ</sup> <sup>L</sup>Þ �

<sup>n</sup><sup>2</sup><sup>C</sup> <sup>þ</sup> <sup>C</sup><sup>A</sup> <sup>þ</sup> <sup>C</sup><sup>B</sup> CðC<sup>A</sup> þ CBÞðn<sup>2</sup>Llk þ LÞ

> <sup>x</sup><sup>3</sup> <sup>þ</sup> 1 RC <sup>x</sup><sup>2</sup> <sup>þ</sup>

<sup>x</sup><sup>3</sup> <sup>þ</sup> 1 RCx<sup>2</sup> <sup>þ</sup>

<sup>n</sup><sup>2</sup><sup>C</sup> <sup>þ</sup> CC <sup>þ</sup> CD CðCC þ CDÞðn<sup>2</sup>Llk þ LÞ

> <sup>x</sup><sup>3</sup> <sup>þ</sup> 1 RCx<sup>2</sup> <sup>þ</sup>

The stability analysis indicates that all interval subsystems have uncontrollable or unobservable modes. We may decompose the state variable model of each subsystem into controllable and uncontrollable parts, and follow by decomposing each part into observable and unobservable parts to obtain

$$
\begin{bmatrix}
\dot{\overline{X}}\_{\alpha} \\
\dot{\overline{X}}\_{\sigma} \\
\dot{\overline{X}}\_{\sigma} \\
\dot{\overline{X}}\_{\sigma}
\end{bmatrix} = \begin{bmatrix}
\overline{\overline{A}}\_{\alpha} \\
\overline{\overline{A}}\_{21} + \frac{1}{\overline{A}\_{\sigma}} - \overline{\overline{A}}\_{23} - \frac{\overline{A}\_{13}}{\overline{A}\_{24}} - \frac{0}{\overline{A}\_{24}} \\
0 \\
0 \\
0 \\
\end{bmatrix} \begin{bmatrix}
\overline{\overline{X}}\_{\alpha} \\
\overline{\overline{X}}\_{\sigma} \\
\overline{\overline{X}}\_{\sigma} \\
\overline{\overline{X}}\_{\sigma}
\end{bmatrix} + \begin{bmatrix}
\overline{B}\_{\alpha} \\
\overline{B}\_{\sigma} \\
0 \\
0
\end{bmatrix} u,\\
y = \left[\begin{array}{cc} \overline{\overline{X}}\_{\alpha} \\
0 & \overline{C}\_{\sigma} & 0 \end{array} \right] \begin{bmatrix}
\overline{\overline{X}}\_{\alpha} \\
\overline{\overline{X}}\_{\sigma} \\
\overline{\overline{X}}\_{\sigma} \\
\overline{\overline{X}}\_{\sigma}
\end{bmatrix} \tag{24}
$$

The observability matrices of the controllable part for each operation interval (positive or negative half cycle) are summarized as follows:

• Trailing-leg (passive-to-active) transition:

$$O\_{c1} = \begin{bmatrix} 0 & \frac{-n}{\mathcal{C}(n^2 L\_{\mathbb{K}} + L)} & \frac{n}{R\mathcal{C}^2(n^2 L\_{\mathbb{K}} + L)}\\ \frac{-n}{\mathcal{C}(n^2 L\_{\mathbb{K}} + L)} & \frac{n}{R\mathcal{C}^2(n^2 L\_{\mathbb{K}} + L)} & \frac{n(n^2 \mathcal{C} + \mathcal{C}\_A + \mathcal{C}\_B)}{\mathcal{C}^2(\mathcal{C}\_A + \mathcal{C}\_B)(n^2 L\_{\mathbb{K}} + L)^2} - \frac{n}{R^2 \mathcal{C}^3(n^2 L\_{\mathbb{K}} + L)}\\ \frac{n}{R\mathcal{C}^2(n^2 L\_{\mathbb{K}} + L)} & \text{or} 1.1 & \text{oc1.2} \end{bmatrix} \tag{25}$$

• Active region (duty cycle loss):

$$O\_{\rm c2loss} = [0] \tag{26}$$

• Active region:

$$\mathbf{O}\_{\ell 2} = \begin{bmatrix} 0 & \frac{n}{\mathbb{C}(n^2 L\_{\mathbb{K}} + L)} \\ n & \frac{\mathbb{C}(n^2 L\_{\mathbb{K}} + L)}{-n} \\ \overline{\mathbb{C}(n^2 L\_{\mathbb{K}} + L)} & \overline{\mathbb{R}\mathbb{C}^2(n^2 L\_{\mathbb{K}} + L)} \end{bmatrix} \tag{27}$$

• Leading-leg (active-to-passive) transitions:

$$\mathbf{O}\_{\mathcal{C}} = \begin{bmatrix} 0 & \frac{n}{\mathcal{C}(n^{2}L\_{\mathbb{K}} + L)} & \frac{-n}{R\mathcal{C}^{2}(n^{2}L\_{\mathbb{K}} + L)} \\\\ \frac{n}{\mathcal{C}(n^{2}L\_{\mathbb{K}} + L)} & \frac{-n}{R\mathcal{C}^{2}(n^{2}L\_{\mathbb{K}} + L)} & \frac{n}{R^{2}\mathcal{C}^{3}(n^{2}L\_{\mathbb{K}} + L)} - \frac{n(n^{2}\mathcal{C} + \mathcal{C}\_{\mathcal{C}} + \mathcal{C}\_{\mathcal{D}})}{\mathcal{C}^{2}(\mathcal{C}\_{\mathbb{C}} + \mathcal{C}\_{\mathcal{D}})(n^{2}L\_{\mathbb{K}} + L)^{2}} \\\\ \frac{-n}{R\mathcal{C}^{2}(n^{2}L\_{\mathbb{K}} + L)} & \text{oc3.1} & \text{oc3.2} \end{bmatrix} \tag{28}$$

Table 1 summarizes the rank of the observability matrix. The state variables of both controllable and observable are listed in Table 2. Since equivalent transformation does not affect the eigenvalues, Eq. (24) has the same set of eigenvalues as in stability analysis. For the operation intervals of trailing/leading leg and active region, uncontrollable or unobservable states (vC<sup>A</sup> ,vC<sup>B</sup> ,vC<sup>C</sup> , and vC<sup>D</sup> ) are marginally stable corresponding to zero eigenvalue. Therefore, those states will stay constant within those intervals, which matches what is observed in numerical simulation. For the intervals of duty cycle loss and passive region, uncontrollable (iL, vC) states are asymptotically stable, which also matches what is observed during simulation.


Table 1. Rank of the observability matrix for the controllable part.


Table 2. State variables of both controllable and observable.

### 6. Model reduction

Oc<sup>3</sup> ¼

<sup>0</sup> <sup>n</sup>

Table 1. Rank of the observability matrix for the controllable part.

Table 2. State variables of both controllable and observable.

RC<sup>2</sup>

n Cðn<sup>2</sup>Llk þ LÞ

150 Modern Fuzzy Control Systems and Its Applications

�n

ðn<sup>2</sup>Llk þ LÞ

RC<sup>2</sup>

Cðn<sup>2</sup>Llk þ LÞ

�n

ðn<sup>2</sup>Llk þ LÞ

�n

C2

Interval Rank

Active region (duty cycle loss) 0 Active region 2 Leading-leg (active-to-passive) transitions 3

Active region (duty cycle loss) 0 Active region 2 Leading-leg (active-to-passive) transitions 3

Interval Controllable and observable

Active region iL, vC Leading-leg (active-to-passive) transitions iL , vC, vCD

Active region iL, vC Leading-leg (active-to-passive) transitions iL , vC, vCD

ðn<sup>2</sup>Llk þ LÞ

� <sup>n</sup>ðn<sup>2</sup><sup>C</sup> <sup>þ</sup> CC <sup>þ</sup> CD<sup>Þ</sup>

ðCC þ CDÞðn<sup>2</sup>Llk þ LÞ

2

ð28Þ

RC<sup>2</sup>

n

oc3:1 oc3:2

Table 1 summarizes the rank of the observability matrix. The state variables of both controllable and observable are listed in Table 2. Since equivalent transformation does not affect the eigenvalues, Eq. (24) has the same set of eigenvalues as in stability analysis. For the operation intervals of trailing/leading leg and active region, uncontrollable or unobservable states (vC<sup>A</sup> ,vC<sup>B</sup> ,vC<sup>C</sup> , and vC<sup>D</sup> ) are marginally stable corresponding to zero eigenvalue. Therefore, those states will stay constant within those intervals, which matches what is observed in numerical simulation. For the intervals of duty cycle loss and passive region, uncontrollable (iL, vC) states

are asymptotically stable, which also matches what is observed during simulation.

Positive half cycle Trailing-leg (passive-to-active) transitions 3

Negative half cycle Trailing-leg (passive-to-active) transitions 3

Positive half cycle Trailing-leg (passive-to-active) transitions iL , vC, vCB

Negative half cycle Trailing-leg (passive-to-active) transitions iL , vC, vCB

ðn<sup>2</sup>Llk þ LÞ

R2 C3 The goal is to obtain a low dimensional model that encompasses the imperative response characteristics of the comprehensive model. The reduced model is then utilized for subsequent control design. For control of the "steady-state" response, we neglect the transition intervals and take only the active region and passive region into consideration. Define d is duty cycle (ON) of the converter and d<sup>0</sup> ¼ 1 � d (OFF). Assuming that Llk ≪ L, we may derive the following differential inclusion model

$$
\begin{bmatrix}
\frac{d\dot{u}\_L(t)}{dt} \\
\frac{d\upsilon\_o(t)}{dt}
\end{bmatrix} = \begin{bmatrix}
0 & -\frac{1}{L} \\
1 & -\frac{1}{R\mathbb{C}}
\end{bmatrix} \begin{bmatrix}
\dot{i}\_L(t) \\
\upsilon\_o(t)
\end{bmatrix} + \begin{bmatrix}
\frac{nd}{L} \\
0
\end{bmatrix} \upsilon\_i(t) \tag{29}
$$

Let x<sup>1</sup> ¼ iL, x<sup>2</sup> ¼ vo, u ¼ d, y ¼ vo

$$
\begin{bmatrix} \dot{\mathbf{x}}\_1 \\ \dot{\mathbf{x}}\_2 \end{bmatrix} = \begin{bmatrix} 0 & \frac{-1}{L} \\ \frac{1}{L} & -1 \\ \frac{1}{C} & \frac{-1}{RC} \end{bmatrix} \begin{bmatrix} \mathbf{x}\_1 \\ \mathbf{x}\_2 \end{bmatrix} + \begin{bmatrix} \frac{n\upsilon\_i}{L} \\ 0 \end{bmatrix} u,\\ \ddot{\mathbf{y}} = \frac{-1}{RC^2} \mathbf{x}\_1 + \left( \frac{1}{R^2 \mathcal{C}^2} - \frac{1}{LC} \right) \mathbf{x}\_2 + \frac{n\upsilon\_i}{L\mathcal{C}} u \tag{30}
$$

## 7. Indirect adaptive fuzzy control for uncertain switching power converters subject to external disturbances

In the following, we propose a robust adaptive fuzzy tracking controller for the PSPWM fullbridge soft switched power converter. Although the controller is designed based on the reduced model, its effectiveness and performance are subsequently verified with the comprehensive model.

Indirect adaptive fuzzy control with sliding model

Based on the input-output linearization concept, Eq. (30) can be represented by

$$f(\mathbf{y}^{(2)}) = f(\mathbf{x}) + g(\mathbf{x})u,\\ f(\mathbf{x}) = \frac{-1}{R\mathbf{C}^2}\mathbf{x}\_1 + \left(\frac{1}{R^2\mathbf{C}^2} - \frac{1}{L\mathbf{C}}\right)\mathbf{x}\_2,\ g(\mathbf{x}) = \frac{n\mathbf{v}\_i}{L\mathbf{C}}\tag{31}$$

The control objective is to force y to follow a given bounded reference signal ym, under the constraint that all signals involved must be bounded. Let e ¼ ym � y, e ¼ ðe,e\_Þ <sup>T</sup> and <sup>k</sup> ¼ ðk2, <sup>k</sup>1<sup>Þ</sup> T be such that all roots of the polynomial <sup>s</sup><sup>2</sup> <sup>þ</sup> <sup>k</sup>1<sup>s</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> are in the open left half-plane. If the functions f and g are known, then the control law

$$\mu = \frac{1}{g(\mathbf{x})} [-f(\mathbf{x}) + y\_m^{\ \ \ \ \ \ \mathbf{z}} + \mathbf{k}^T \mathbf{e}] \tag{32}$$

applied to Eq. (31) results in

$$
\varepsilon^{(2)} + k\_1 \dot{\varepsilon} + k\_2 \varepsilon = 0 \tag{33}
$$

which implies that lim<sup>t</sup>!<sup>∞</sup> <sup>e</sup>ðtÞ ¼ 0 a main objective of control.

However, f and g are unknown. We replace f and g in Eq. (32) by the fuzzy logic systems ^<sup>f</sup> <sup>ð</sup>xjθf<sup>Þ</sup> and <sup>g</sup>^ðxjθgÞ. The resulting control law

$$\mu\_{\varepsilon} = \frac{1}{\hat{g}(\mathbf{x}|\boldsymbol{\theta}\_{\mathcal{S}})} \left[ -\hat{f}(\mathbf{x}|\boldsymbol{\theta}\_{f}) + \boldsymbol{y}\_{m}^{\ \ \ \ \mathbf{z}} + \boldsymbol{k}^{T}\boldsymbol{e} \right] \tag{34}$$

is the so-called certainty equivalent controller. We use

$$
\mu = \mu\_c + \mu\_s \tag{35}
$$

where the additional control term us is called a supervisory control for stability. Substituting Eq. (35) into Eq. (31), we obtain the error equation

$$\dot{\mathbf{e}} = \Lambda\_{\epsilon} \mathbf{e} + \mathbf{b}\_{\epsilon} \left[ \hat{f}(\mathbf{x}|\boldsymbol{\theta}\_{f}) - f(\mathbf{x}) + (\hat{\mathbf{g}}(\mathbf{x}|\boldsymbol{\theta}\_{\mathcal{S}}) - \mathbf{g}(\mathbf{x})) \boldsymbol{\mu}\_{\epsilon} - \mathbf{g}(\mathbf{x}) \boldsymbol{\mu}\_{\mathcal{s}} \right] \tag{36}$$

where

$$
\Lambda\_{\mathfrak{c}} = \begin{bmatrix} 0 & 1 \\ -k\_2 & -k\_1 \end{bmatrix}, \qquad b\_{\mathfrak{c}} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \tag{37}
$$

Since <sup>Λ</sup><sup>c</sup> is a stable matrix (jsI � <sup>Λ</sup>cj ¼ <sup>s</sup><sup>n</sup> <sup>þ</sup> <sup>k</sup>1sðn�1<sup>Þ</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> kn which is stable), we know that there exists a unique positive definite symmetric n · n matrix P which satisfies the Lyapunov equation:

$$
\Lambda\_{\mathfrak{c}}{}^{T}P + P\Lambda\_{\mathfrak{c}} = -Q \tag{38}
$$

where Q is an arbitrary 2 · 2 positive definite matrix. Let Ve <sup>¼</sup> <sup>1</sup> <sup>2</sup> <sup>e</sup>TPe, in order for xi <sup>¼</sup> <sup>y</sup>ði�1<sup>Þ</sup> <sup>m</sup> �eði�1<sup>Þ</sup> to be bounded, we require that Ve must be bounded, which means we require that <sup>V</sup>\_ <sup>e</sup> <sup>≤</sup> <sup>0</sup> when Ve is greater than a large constant V. Using Eq. (35) and Eq. (38), we have

$$\begin{split} \dot{V}\_{\varepsilon} &= -\frac{1}{2} \mathbf{e}^{T} \mathbf{Q} \mathbf{e} + \mathbf{e}^{T} P \mathbf{b}\_{\varepsilon} [\hat{f}(\mathbf{x}|\boldsymbol{\theta}\_{f}) - f(\mathbf{x})) + (\hat{\mathbf{g}}(\mathbf{x}|\boldsymbol{\theta}\_{\mathcal{S}}) - \mathbf{g}(\mathbf{x})) u\_{\varepsilon} - \mathbf{g}(\mathbf{x}) u\_{\mathrm{s}}] \\\\ &\leq -\frac{1}{2} \mathbf{e}^{T} \mathbf{Q} \mathbf{e} + |\mathbf{e}^{T} P \mathbf{b}\_{\varepsilon}| [|\hat{f}(\mathbf{x}|\boldsymbol{\theta}\_{f})| + |f(\mathbf{x})| + |\hat{g}(\mathbf{x}|\boldsymbol{\theta}\_{\mathcal{S}}) u\_{\varepsilon}| + |\mathbf{g}(\mathbf{x}) u\_{\varepsilon}|] - \mathbf{e}^{T} P \mathbf{b}\_{\varepsilon} \mathbf{g}(\mathbf{x}) u\_{\mathrm{s}} \end{split} \tag{39}$$

In order to design the us such that the right-hand side of Eq. (39) is not positive, we need to know the bounds of f and g. That is, we have to make the following assumption.

Assumption: We can determine functions f <sup>U</sup>ðxÞ, gUðx<sup>Þ</sup> and gLðxÞsuch that <sup>j</sup>fðxÞj <sup>≤</sup> <sup>f</sup> <sup>U</sup>ðx<sup>Þ</sup> and gLðx<sup>Þ</sup> <sup>≤</sup> <sup>g</sup>ðx<sup>Þ</sup> <sup>≤</sup> gUðx<sup>Þ</sup> for <sup>x</sup><sup>∈</sup> <sup>R</sup><sup>2</sup> , where f <sup>U</sup>ðx<sup>Þ</sup> <sup>&</sup>lt; <sup>∞</sup>, gUðx<sup>Þ</sup> <sup>&</sup>lt; <sup>∞</sup>, and gLðx<sup>Þ</sup> <sup>&</sup>gt; 0 for <sup>x</sup>∈R<sup>2</sup> .

Based on f <sup>U</sup>ðxÞ, gUðx<sup>Þ</sup> and gLðxÞ, and by observing Eq. (39), we choose the supervisory control us as

$$u\_{\varepsilon} = \begin{cases} \text{sgn}(\mathbf{e}^T \mathbf{P} \mathbf{b}\_{\varepsilon}) \frac{1}{g\_L(\mathbf{x})} [|\hat{f}(\mathbf{x}|\boldsymbol{\theta}\_f)| + |f^{\mathcal{U}}(\mathbf{x})| + |\hat{\mathbf{g}}(\mathbf{x}|\boldsymbol{\theta}\_{\mathcal{S}}) u\_{\varepsilon}| + |\mathbf{g}^{\mathcal{U}}(\mathbf{x}) u\_{\varepsilon}|], & V\_{\varepsilon} \ge \overline{V} \\ & 0, & V\_{\varepsilon} \le \overline{V} \end{cases} \tag{40}$$

Substituting Eq. (40) into Eq. (39) and considering the case Ve > V, we have

$$\dot{W}\_e \le -\frac{1}{2} \mathbf{e}^T Q \mathbf{e} + |\mathbf{e}^T P \mathbf{b}\_c| [|\hat{f}| + |f| + |\hat{g}u\_c| + |gu\_c| - \frac{\mathcal{g}}{\mathcal{g}\_L} (|\hat{f}| + f^{II} + |\hat{g}u\_c| + |\mathcal{g}^{II}u\_c|)]$$

$$\le -\frac{1}{2} \mathbf{e}^T Q \mathbf{e} < 0 \tag{41}$$

In summary, using the control Eq. (35), we can guarantee that Ve ≤V < ∞. Since P is positive definite, the boundedness of Ve implies the boundedness of e, which in turn implies the boundedness of x.

We employ the following fuzzy logic system:

e

However, f and g are unknown. We replace f and g in Eq. (32) by the fuzzy logic systems

<sup>g</sup>^ðxjθg<sup>Þ</sup> �^<sup>f</sup> <sup>ð</sup>xjθfÞ þ ym

where the additional control term us is called a supervisory control for stability. Substituting

which implies that lim<sup>t</sup>!<sup>∞</sup> <sup>e</sup>ðtÞ ¼ 0 a main objective of control.

uc <sup>¼</sup> <sup>1</sup>

<sup>e</sup>\_ <sup>¼</sup> <sup>Λ</sup>c<sup>e</sup> <sup>þ</sup> <sup>b</sup><sup>c</sup> ^<sup>f</sup> <sup>ð</sup>xjθfÞ � <sup>f</sup>ðxÞ þ <sup>g</sup>^ðxjθ<sup>g</sup>

<sup>Λ</sup><sup>c</sup> <sup>¼</sup> 0 1

Λc

when Ve is greater than a large constant V. Using Eq. (35) and Eq. (38), we have

<sup>e</sup>TQ<sup>e</sup> <sup>þ</sup> <sup>e</sup>TPbc½ð^<sup>f</sup> <sup>ð</sup>xjθfÞ � <sup>f</sup>ðxÞÞ þ ðg^ðxjθgÞ � <sup>g</sup>ðxÞÞuc � <sup>g</sup>ðxÞus�

know the bounds of f and g. That is, we have to make the following assumption.

where Q is an arbitrary 2 · 2 positive definite matrix. Let Ve <sup>¼</sup> <sup>1</sup>

�k<sup>2</sup> �k<sup>1</sup> � �

Since <sup>Λ</sup><sup>c</sup> is a stable matrix (jsI � <sup>Λ</sup>cj ¼ <sup>s</sup><sup>n</sup> <sup>þ</sup> <sup>k</sup>1sðn�1<sup>Þ</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> kn which is stable), we know that there exists a unique positive definite symmetric n · n matrix P which satisfies the Lyapunov

�eði�1<sup>Þ</sup> to be bounded, we require that Ve must be bounded, which means we require that <sup>V</sup>\_ <sup>e</sup> <sup>≤</sup> <sup>0</sup>

<sup>e</sup>TQ<sup>e</sup> þ jeTPbcj½jð^<sup>f</sup> <sup>ð</sup>xjθfÞj þ jfðxÞj þ jg^ðxjθgÞucjþjgðxÞucj� � <sup>e</sup>TPbcgðxÞus

In order to design the us such that the right-hand side of Eq. (39) is not positive, we need to

^<sup>f</sup> <sup>ð</sup>xjθf<sup>Þ</sup> and <sup>g</sup>^ðxjθgÞ. The resulting control law

152 Modern Fuzzy Control Systems and Its Applications

is the so-called certainty equivalent controller. We use

Eq. (35) into Eq. (31), we obtain the error equation

where

equation:

<sup>V</sup>\_ <sup>e</sup> ¼ � <sup>1</sup> 2

> <sup>≤</sup> � <sup>1</sup> 2

<sup>ð</sup>2<sup>Þ</sup> <sup>þ</sup> <sup>k</sup>1e\_ <sup>þ</sup> <sup>k</sup>2<sup>e</sup> <sup>¼</sup> <sup>0</sup> <sup>ð</sup>33<sup>Þ</sup>

u ¼ uc þ us ð35Þ

ð34Þ

ð36Þ

ð37Þ

ð39Þ

<sup>ð</sup>2<sup>Þ</sup> <sup>þ</sup> <sup>k</sup><sup>T</sup><sup>e</sup>

� � � <sup>g</sup>ðxÞÞuc � <sup>g</sup>ðxÞus

1 � �

TP <sup>þ</sup> <sup>P</sup>Λ<sup>c</sup> ¼ �<sup>Q</sup> <sup>ð</sup>38<sup>Þ</sup>

<sup>2</sup> <sup>e</sup>TPe, in order for xi <sup>¼</sup> <sup>y</sup>ði�1<sup>Þ</sup> <sup>m</sup>

h i

h i

, bc <sup>¼</sup> <sup>0</sup>

$$\hat{f}(\mathbf{x}|\boldsymbol{\theta}\_{\boldsymbol{f}}) = \sum\_{l=1}^{M} \boldsymbol{\theta}\_{l} \boldsymbol{\xi}\_{l}(\mathbf{x}) \;= \boldsymbol{\theta}^{T} \boldsymbol{\xi}(\mathbf{x}) \tag{42}$$

where θ ¼ ðθ1,…,θMÞ <sup>T</sup>, <sup>ξ</sup>ðxÞ¼ðξ1ðxÞ,…,ξMðxÞÞ<sup>T</sup>, <sup>ξ</sup>lðx<sup>Þ</sup> is the fuzzy basis function defined by

$$\xi\_l(\mathbf{x}) = \prod\_{i=1}^2 \mu\_{F\_i^l}(\mathbf{x}\_i) \bigg/ \sum\_{l=1}^M \prod\_{i=1}^2 \mu\_{F\_i^l}(\mathbf{x}\_i) \tag{43}$$

θ<sup>l</sup> are adjustable parameters, and μFl i are given membership functions.

We present the detailed design steps of the adaptive fuzzy controller.


$$\text{if } \mathbf{x}\_1 \text{ is } A\_1^{l\_1} \text{ and } \mathbf{x}\_2 \text{ is } A\_2^{l\_2}, \quad \text{then } \hat{f} \text{ is } E^{l\_1 \ldots l\_2}.$$

There are <sup>Y</sup><sup>2</sup> <sup>i</sup>¼<sup>1</sup> qi rules to construct fuzzy systems <sup>g</sup>^ðxjθgÞ:

$$\text{if } \propto\_1 \text{ is } B\_1^{l\_1} \text{ and } \propto\_2 \text{ is } B\_2^{l\_2}, \quad \text{then } \hat{\otimes} \text{ is } H^{l\_1 \ldots l\_2}$$

Using product-inference rule, singleton fuzzifier, and center average defuzzifier, we obtain

$$\hat{f}(\mathbf{x}|\boldsymbol{\theta}\_{\boldsymbol{f}}) = \boldsymbol{\theta}\_{\boldsymbol{f}}^{T}\boldsymbol{\xi}\_{\boldsymbol{f}}(\mathbf{x}), \hat{g}(\mathbf{x}|\boldsymbol{\theta}\_{\boldsymbol{\mathcal{S}}}) \ = \boldsymbol{\theta}\_{\boldsymbol{\mathcal{S}}}^{T}\boldsymbol{\xi}\_{\boldsymbol{\mathcal{S}}}(\mathbf{x})\tag{44}$$

where

$$\xi\_f^l(\mathbf{x}) = \frac{\prod\_{i=1}^2 \mu\_{A\_i^l}(\mathbf{x}\_i)}{\sum\_{l=1}^{p\_1 \times p\_2} \prod\_{i=1}^2 \mu\_{A\_i^l}(\mathbf{x}\_i)}, \xi\_g^l(\mathbf{x}) = \frac{\prod\_{i=1}^2 \mu\_{B\_i^l}(\mathbf{x}\_i)}{\sum\_{l=1}^{p\_1 \times p\_2} \prod\_{i=1}^2 \mu\_{B\_i^l}(\mathbf{x}\_i)} \tag{45}$$

Our next task is to develop an adaptive law to adjust the parameters in the fuzzy logic systems for the purpose of forcing the tracking error to converge to zero.

Define

$$\begin{array}{l} \boldsymbol{\theta}\_{f}^{\*} = \operatorname\*{arg\,min}\_{\boldsymbol{\theta}/} [\sup\_{\mathbf{x} \in \boldsymbol{R}^{2}} | \hat{f}(\mathbf{x}|\boldsymbol{\theta}\_{f}) - f(\mathbf{x}) |] \\ \boldsymbol{\theta}\_{\mathcal{S}}^{\*} = \operatorname\*{arg\,min}\_{\boldsymbol{\theta}\_{\mathcal{S}}} [\sup\_{\mathbf{x} \in \boldsymbol{R}^{2}} | \hat{g}(\mathbf{x}|\boldsymbol{\theta}\_{\mathcal{S}}) - g(\mathbf{x}) |] \end{array} \tag{46}$$

Define the minimum approximation error

$$
\omega = (\hat{f}(\mathbf{x}|\boldsymbol{\theta}\_f^\*) - f(\mathbf{x})) + (\hat{g}(\mathbf{x}|\boldsymbol{\theta}\_g^\*) - g(\mathbf{x}))u\_\varepsilon \tag{47}
$$

Then the error equation can be rewritten as

$$\dot{\mathbf{e}} = \Lambda\_{\epsilon} \mathbf{e} + \mathbf{b}\_{\epsilon} [ (\hat{f}(\mathbf{x}|\boldsymbol{\theta}\_{f})) - \hat{f}(\mathbf{x}|\boldsymbol{\theta}\_{f}^{\*}) ) + (\hat{g}(\mathbf{x}|\boldsymbol{\theta}\_{\mathcal{S}}) - \hat{g}(\mathbf{x}|\boldsymbol{\theta}\_{\mathcal{S}}^{\*})) \mu\_{\epsilon} + \omega] \tag{48}$$

Substituting Eq. (43) into Eq. (47), we have

$$\dot{\mathbf{e}} = \Lambda\_{\mathbf{c}} \mathbf{e} + \mathbf{b}\_{\mathbf{c}} \boldsymbol{\omega} + \mathbf{b}\_{\mathbf{c}} [(\boldsymbol{\Theta}\_{\circ} - \boldsymbol{\Theta}\_{\circ}^{\*})^{\mathrm{T}} \boldsymbol{\xi}\_{\circ}(\mathbf{x}) + (\boldsymbol{\Theta}\_{\circ} - \boldsymbol{\Theta}\_{\circ}^{\*})^{\mathrm{T}} \boldsymbol{\xi}\_{\circ}(\mathbf{x}) \boldsymbol{\mu}\_{\circ}] \tag{49}$$

Consider the Lyapunov function candidate

$$V = \frac{1}{2} \mathbf{e}^T \mathbf{P} \mathbf{e} + \frac{1}{2\gamma\_1} (\boldsymbol{\Theta}\_f - \boldsymbol{\Theta}\_f^\*)^T (\boldsymbol{\Theta}\_f - \boldsymbol{\Theta}\_f^\*) + \frac{1}{2\gamma\_2} (\boldsymbol{\Theta}\_\mathcal{g} - \boldsymbol{\Theta}\_\mathcal{g}^\*)^T (\boldsymbol{\Theta}\_\mathcal{g} - \boldsymbol{\Theta}\_\mathcal{g}^\*) \tag{50}$$

where γ<sup>1</sup> and γ2are positive constants. The time derivative of V along the trajectory of Eq. (48) is

$$\dot{V} = -\frac{1}{2}\mathbf{e}^T \mathbf{Q} \mathbf{e} + \mathbf{e}^T P \mathbf{b}\_\ell \mathbf{w} + \frac{1}{\mathcal{V}\_1} (\boldsymbol{\Theta}\_f - \boldsymbol{\Theta}\_f^\*)^T [\dot{\boldsymbol{\Theta}}\_f + \boldsymbol{\gamma}\_1 \mathbf{e}^T P \mathbf{b}\_\ell \boldsymbol{\xi}\_\ell(\mathbf{x})]$$

$$+ \frac{1}{\mathcal{V}\_2} (\boldsymbol{\Theta}\_\mathcal{S} - \boldsymbol{\Theta}\_\mathcal{S}^\*)^T [\dot{\boldsymbol{\Theta}}\_\mathcal{S} + \boldsymbol{\gamma}\_2 \mathbf{e}^T P \mathbf{b}\_\ell \boldsymbol{\xi}\_\mathcal{S}(\mathbf{x}) \boldsymbol{u}\_\ell] \tag{51}$$

If we choose the adaptive law

$$\dot{\boldsymbol{\Theta}}\_{f} = -\boldsymbol{\gamma}\_{1}\boldsymbol{\mathbf{e}}^{T}\boldsymbol{P}\mathbf{b}\_{c}\boldsymbol{\xi}\_{f}(\mathbf{x}),\\\dot{\boldsymbol{\Theta}}\_{\mathcal{S}} = -\boldsymbol{\gamma}\_{2}\boldsymbol{\mathbf{e}}^{T}\boldsymbol{P}\mathbf{b}\_{c}\boldsymbol{\xi}\_{\mathcal{S}}(\mathbf{x})\boldsymbol{u}\_{c} \tag{52}$$

then from Eq. (50) we have

$$\dot{V} = -\frac{1}{2}\mathbf{e}^T \mathbf{Q} \mathbf{e} + \mathbf{e}^T P \mathbf{b}\_c \omega \tag{53}$$

This is the best we can hope to get because the term eTPbcω is of the order of the minimum approximation error. If <sup>ω</sup> <sup>¼</sup> 0, that is, the searching spaces for ^<sup>f</sup> and <sup>g</sup>^ are so big that <sup>f</sup> and <sup>g</sup> are included in them, then we have V\_ ≤ 0. Eq. (51) cannot guarantee θ<sup>f</sup> and θ<sup>g</sup> are bounded, so we use projection algorithm. If the parameter vectors θ<sup>f</sup> and θ<sup>g</sup> are within the constraint sets or on the boundaries of the constraint sets but moving toward the inside of the constraint sets, then use the simple adaptive law Eq. (51). Otherwise, if the parameter vectors are on the boundaries of the constraint sets but moving toward the outside of the constraint sets, then use the projection algorithm to modify the adaptive law Eq. (51). such that the parameter vectors will remain inside the constraint sets.

$$\Omega\_f = \left\{ \theta\_f \in R \prod\_{i=1}^2 p\_i ||\theta\_f|| \le M\_f \right\} \tag{54}$$

$$\Omega\_{\mathcal{S}} = \left\{ \theta\_{\mathcal{S}} \in \mathbb{R} \prod\_{i=1}^{2} q\_i \vert\_{\mid} 0 < \varepsilon \le \Vert \theta\_{\mathcal{S}} \Vert \le M\_{\mathcal{S}} \right\} \tag{55}$$

where Ω<sup>f</sup> and Ω<sup>g</sup> are constraint sets for θ<sup>f</sup> and θg, Mf , Mg, ε are constants

$$\dot{\boldsymbol{\Theta}}\_{f} = \begin{cases} -\boldsymbol{\gamma}\_{1} \mathbf{e}^{T} \mathbf{P} \mathbf{b}\_{\varepsilon} \boldsymbol{\xi}\_{f}(\mathbf{x}) & \text{if } (||\boldsymbol{\Theta}\_{f}|| < M\_{f}) \text{ or } (||\boldsymbol{\Theta}\_{f}|| = M\_{f} \text{ and } \mathbf{e}^{T} \mathbf{P} \mathbf{b}\_{\varepsilon} \boldsymbol{\xi}\_{f}(\mathbf{x}) \ge 0) \\\ P\{ -\boldsymbol{\gamma}\_{1} \mathbf{e}^{T} \mathbf{P} \mathbf{b}\_{\varepsilon} \boldsymbol{\xi}\_{f}(\mathbf{x}) \} & \text{if } (||\boldsymbol{\Theta}\_{f}|| = M\_{f} \text{ and } \mathbf{e}^{T} \mathbf{P} \mathbf{b}\_{\varepsilon \varepsilon} \boldsymbol{\xi}\_{f}(\mathbf{x}) < 0) \end{cases} \tag{56}$$

where the projection operator Pf�g is defined as

$$P\{-\gamma\_1 \mathbf{e}^T \mathbf{P} \mathbf{b}\_\varepsilon \xi\_f(\mathbf{x})\} = -\gamma\_1 \mathbf{e}^T \mathbf{P} \mathbf{b}\_\varepsilon \xi\_f(\mathbf{x}) + \gamma\_1 \mathbf{e}^T \mathbf{P} \mathbf{b}\_\varepsilon \frac{\boldsymbol{\theta}\_f \mathbf{e}\_f^T \xi\_f(\mathbf{x})}{\left\|\mathbf{e}\_f\right\|^2} \tag{57}$$

Whenever an element θgi of θ<sup>g</sup> ¼ ε, use

$$\dot{\boldsymbol{\theta}}\_{\mathcal{g}^i} = \begin{cases} -\boldsymbol{\gamma}\_2 \mathbf{e}^T \mathbf{P} \mathbf{b}\_c \boldsymbol{\xi}\_{\mathcal{g}^i}(\mathbf{x}) \boldsymbol{u}\_c & \text{if } \mathbf{e}^T \mathbf{P} \mathbf{b}\_c \boldsymbol{\xi}\_{\mathcal{g}^i}(\mathbf{x}) \boldsymbol{u}\_c < 0 \\\ 0 & \text{if } \mathbf{e}^T \mathbf{P} \mathbf{b}\_c \boldsymbol{\xi}\_{\mathcal{g}^i}(\mathbf{x}) \boldsymbol{u}\_c \ge 0 \end{cases} \tag{58}$$

where ξgiðxÞ is the ith component of ξgðxÞ.

Otherwise, use

^<sup>f</sup> <sup>ð</sup>xjθfÞ ¼ <sup>θ</sup><sup>T</sup>

Y<sup>2</sup> <sup>i</sup>¼<sup>1</sup> <sup>μ</sup>Al i ðxiÞ , ξl <sup>g</sup>ðxÞ ¼

<sup>f</sup> ¼ arg min<sup>θ</sup><sup>f</sup> ½sup<sup>x</sup> <sup>∈</sup>R<sup>2</sup> j

Y<sup>2</sup> <sup>i</sup>¼<sup>1</sup> <sup>μ</sup>Al i ðxiÞ

for the purpose of forcing the tracking error to converge to zero.

<sup>ω</sup> ¼ ð^<sup>f</sup> <sup>ð</sup>xjθ�

<sup>e</sup>\_ <sup>¼</sup> <sup>Λ</sup>c<sup>e</sup> <sup>þ</sup> <sup>b</sup>c½ð^<sup>f</sup> <sup>ð</sup>xjθfÞÞ � ^<sup>f</sup> <sup>ð</sup>xjθ�

e\_ ¼ Λce þ bcω þ bc½ðθ<sup>f</sup> � θ�

ðθ<sup>f</sup> � θ� f Þ

<sup>e</sup>TQ<sup>e</sup> <sup>þ</sup> <sup>e</sup>TPbc<sup>ω</sup> <sup>þ</sup>

þ 1 γ2

1 2γ<sup>1</sup>

X<sup>p</sup><sup>1</sup> · <sup>p</sup><sup>2</sup> l¼1

θ�

θ�

where

Define

ξl <sup>f</sup>ðxÞ ¼

154 Modern Fuzzy Control Systems and Its Applications

Define the minimum approximation error

Then the error equation can be rewritten as

Substituting Eq. (43) into Eq. (47), we have

Consider the Lyapunov function candidate

<sup>V</sup>\_ ¼ � <sup>1</sup> 2

<sup>V</sup> <sup>¼</sup> <sup>1</sup> 2 <sup>e</sup>TP<sup>e</sup> <sup>þ</sup>

If we choose the adaptive law

then from Eq. (50) we have

<sup>f</sup> <sup>ξ</sup>fðxÞ, <sup>g</sup>^ðxjθgÞ ¼ <sup>θ</sup><sup>T</sup>

Our next task is to develop an adaptive law to adjust the parameters in the fuzzy logic systems

<sup>f</sup> Þ � fðxÞÞ þ ðg^ðxjθ�

f Þ

<sup>T</sup>ðθ<sup>f</sup> � <sup>θ</sup>�

where γ<sup>1</sup> and γ2are positive constants. The time derivative of V along the trajectory of Eq. (48) is

ðθ<sup>f</sup> � θ� f Þ

<sup>T</sup>½θ\_ <sup>g</sup> <sup>þ</sup> <sup>γ</sup>2<sup>e</sup>

<sup>e</sup>TQ<sup>e</sup> <sup>þ</sup> <sup>e</sup>

1 γ1

ðθ<sup>g</sup> � θ� gÞ

<sup>V</sup>\_ ¼ � <sup>1</sup> 2 Y<sup>2</sup> <sup>i</sup>¼<sup>1</sup> <sup>μ</sup>B<sup>l</sup> i ðxiÞ

Y<sup>2</sup> <sup>i</sup>¼<sup>1</sup> <sup>μ</sup>B<sup>l</sup> i ðxiÞ

X<sup>p</sup><sup>1</sup> · <sup>p</sup><sup>2</sup> l¼1

^<sup>f</sup> <sup>ð</sup>xjθfÞ � <sup>f</sup>ðxÞj�

<sup>f</sup> ÞÞ þ ðg^ðxjθgÞ � g^ðxjθ�

<sup>T</sup>ξfðxÞþðθ<sup>g</sup> � <sup>θ</sup>�

<sup>f</sup> Þ þ <sup>1</sup> 2γ<sup>2</sup> gÞ

ðθ<sup>g</sup> � θ� gÞ

<sup>θ</sup>\_ <sup>f</sup> ¼ �γ1eTPbcξfðxÞ, <sup>θ</sup>\_ <sup>g</sup> ¼ �γ2eTPbcξgðxÞuc <sup>ð</sup>52<sup>Þ</sup>

<sup>T</sup>½θ\_ <sup>f</sup> <sup>þ</sup> <sup>γ</sup>1eTPbcξfðxÞ�

<sup>g</sup> <sup>¼</sup> arg min<sup>θ</sup><sup>g</sup> <sup>½</sup>sup<sup>x</sup>∈R<sup>2</sup> <sup>j</sup>g^ðxjθgÞ � <sup>g</sup>ðxÞj� <sup>ð</sup>46<sup>Þ</sup>

<sup>g</sup> ξgðxÞ ð44Þ

<sup>g</sup>Þ � gðxÞÞuc ð47Þ

<sup>T</sup>ðθ<sup>g</sup> � <sup>θ</sup>�

TPbcξgðxÞuc� ð51<sup>Þ</sup>

TPbc<sup>ω</sup> <sup>ð</sup>53<sup>Þ</sup>

<sup>g</sup>ÞÞuc þ ω� ð48Þ

<sup>T</sup>ξgðxÞuc� ð49<sup>Þ</sup>

<sup>g</sup>Þ ð50Þ

ð45Þ

$$\dot{\boldsymbol{\theta}}\_{\mathcal{S}} = \begin{cases} -\mathcal{V}\_{2}\boldsymbol{e}^{T}\boldsymbol{P}\boldsymbol{b}\_{\mathcal{E}}\boldsymbol{\xi}\_{\mathcal{S}}(\mathbf{x})\boldsymbol{u}\_{\mathcal{E}} & \text{if } (\lVert\boldsymbol{\theta}\_{\mathcal{S}}\rVert < M\_{\mathcal{S}}) \text{ or } (\lVert\boldsymbol{\theta}\_{\mathcal{S}}\rVert = M\_{\mathcal{S}} \text{ and } \mathbf{e}^{T}\boldsymbol{P}\boldsymbol{b}\_{\mathcal{E}}\boldsymbol{\xi}\_{\mathcal{S}}(\mathbf{x})\boldsymbol{u}\_{\mathcal{E}} \ge 0) \\\ P\{ -\gamma\_{2}\boldsymbol{e}^{T}\boldsymbol{P}\boldsymbol{b}\_{\mathcal{E}}\boldsymbol{\xi}\_{\mathcal{S}}(\mathbf{x}) \} & \text{if } (\lVert\boldsymbol{\theta}\_{\mathcal{S}}\rVert = M\_{\mathcal{S}} \text{ and } \mathbf{e}^{T}\boldsymbol{P}\boldsymbol{b}\_{\mathcal{E}}\boldsymbol{\xi}\_{\mathcal{S}}(\mathbf{x})\boldsymbol{u}\_{\mathcal{E}} < 0) \end{cases} \tag{59}$$

where the projection operator Pf�g is defined as

$$P\{-\gamma\_2 \mathbf{e}^T P \mathbf{b}\_\varepsilon \xi\_\mathbf{g}(\mathbf{x}) u\_\varepsilon\} = -\gamma\_2 \mathbf{e}^T P \mathbf{b}\_\varepsilon \xi\_\mathbf{g}(\mathbf{x}) u\_\varepsilon + \gamma\_2 \mathbf{e}^T P \mathbf{b}\_\varepsilon \frac{\boldsymbol{\theta}\_\mathbf{g} \boldsymbol{\theta}\_\mathbf{g}^T \xi\_\mathbf{g}(\mathbf{x}) u\_\varepsilon}{\left\| \boldsymbol{\theta}\_\mathbf{g} \right\|^2} \tag{60}$$

Theorem:

1. kθfðtÞk ≤ Mf ,kθgðtÞk ≤ Mg, all elements of θ<sup>g</sup> ≥ ε,

$$\|\|\mathbf{x}(t)\|\| \le \|\mathbf{y}\_m\|\| + \left(\frac{2\overline{V}}{\lambda\_{P\text{min}}}\right)^{\frac{1}{2}}\tag{61}$$

$$|u(t)| \le \frac{1}{\varepsilon} \left( M\_f + |y\_m^2| + ||\mathbb{k}|| \left( \frac{2\overline{V}}{\lambda\_{P\text{min}}} \right)^{\frac{1}{2}} \right) + \frac{1}{g\_L(\mathbf{x})} $$

$$\left[ M\_f + |f^{\mathcal{U}}(\mathbf{x})| + \frac{1}{\varepsilon} \left( M\_{\mathcal{S}} + g^{\mathcal{U}} \right) \left( M\_f + |y\_m^2| + ||\mathbb{k}|| \left( \frac{2\overline{V}}{\lambda\_{P\text{min}}} \right)^{\frac{1}{2}} \right) \right] \tag{62}$$

for all t ≥ 0, where λPmin is the minimum eigenvalue of P, and ym ¼ ðym, y\_ <sup>m</sup>,…, <sup>y</sup>ðn�1<sup>Þ</sup> <sup>m</sup> <sup>Þ</sup> T.

$$\mathbf{2}$$

$$\int\_{0}^{t} \left\| \mathbf{e}(\tau) \right\|^{2} d\tau \le a + b \int\_{0}^{t} \left| \omega(\tau) \right|^{2} d\tau \tag{63}$$

for all t ≥ 0, where a and b are constants, and ω is the minimum approximation error defined by Eq. (46)

3. If <sup>ω</sup> is squared integrable, that is, <sup>ð</sup><sup>∞</sup> 0 jωðtÞj<sup>2</sup> dt <sup>&</sup>lt; <sup>∞</sup>, then lim<sup>t</sup>!<sup>∞</sup> keðtÞk ¼ 0

Design example:

The parameters of the converter are listed in Figure 6. Consider the following system:

Figure 6. Photo and parameters of a PSPWM full-bridge power converter circuit. Transformer turns ratio n = 0.5. V<sup>i</sup> = 160 volt, V<sup>0</sup> = 50 volt, R = 6Ω, C = 940μF L = 300μH, Llk = 20μH CA = CB = CC =CD = 5nF, fs = 50 kHz.

#### Robust Adaptive Fuzzy Control for a Class of Switching Power Converters http://dx.doi.org/10.5772/67895 157

$$y^{(2)} = f(\mathbf{x}) + \mathbf{g}(\mathbf{x})u\tag{64}$$

$$f(\mathbf{x}) = \frac{-1}{R\mathbb{C}^2}\mathbf{x}\_1 + \left(\frac{1}{R^2\mathbb{C}^2} - \frac{1}{LC}\right)\mathbf{x}\_2 \cdot \mathbf{g}(\mathbf{x}) = \frac{nv\_i}{LC} \tag{65}$$

The design steps of the adaptive fuzzy controller are provided in the following:

$$\begin{aligned} \boldsymbol{u} &= \boldsymbol{u}\_{\boldsymbol{c}} + \boldsymbol{u}\_{\boldsymbol{s}} \quad \boldsymbol{u}\_{\boldsymbol{c}} = \frac{1}{\hat{\boldsymbol{g}}(\mathbf{x}|\boldsymbol{\theta}\_{\mathcal{G}})} \Big[ -\hat{f}(\mathbf{x}|\boldsymbol{\theta}\_{f}) + \boldsymbol{y}\_{\boldsymbol{m}}{}^{(2)} + \mathbf{k}^{\mathrm{T}}\boldsymbol{e} \Big] \\ \boldsymbol{u}\_{\boldsymbol{s}} &= \begin{cases} \text{sgn}(\mathbf{e}^{\mathrm{T}}\mathbf{P}\mathbf{b}\_{\boldsymbol{c}}) \frac{1}{\mathcal{G}\_{\boldsymbol{L}}(\mathbf{x})} \left[ |\hat{f}(\mathbf{x}|\boldsymbol{\theta}\_{f})| + |f^{\mathrm{II}}(\mathbf{x})| + |\hat{g}(\mathbf{x}|\boldsymbol{\theta}\_{\mathcal{G}})\boldsymbol{u}\_{\boldsymbol{c}}| + |g^{\mathrm{II}}(\mathbf{x})\boldsymbol{u}\_{\boldsymbol{c}}| \right], & \boldsymbol{V}\_{\boldsymbol{c}} \ge \overline{\mathbf{V}} \\ & \mathbf{0}, & \boldsymbol{V}\_{\boldsymbol{c}} \le \overline{\mathbf{V}} \end{cases} \end{aligned} \tag{66}$$

Step 1:

Let

Theorem:

2.

1. kθfðtÞk ≤ Mf ,kθgðtÞk ≤ Mg, all elements of θ<sup>g</sup> ≥ ε,

juðtÞj ≤

Mf þ jf

"

156 Modern Fuzzy Control Systems and Its Applications

defined by Eq. (46)

Design example:

3. If ω is squared integrable, that is,

1 ε 

<sup>U</sup>ðxÞj þ <sup>1</sup> ε

> ðt 0

kxðtÞk ≤ kymk þ

<sup>m</sup>jþkkk

Mg <sup>þ</sup> gU � � Mf þ jy<sup>2</sup>

dτ ≤ a þ b

for all t ≥ 0, where a and b are constants, and ω is the minimum approximation error

ðt 0 jωðτÞj<sup>2</sup>

dt <sup>&</sup>lt; <sup>∞</sup>, then lim<sup>t</sup>!<sup>∞</sup>

Mf þ jy<sup>2</sup>

for all t ≥ 0, where λPmin is the minimum eigenvalue of P, and ym ¼ ðym, y\_

keðτÞk<sup>2</sup>

ð∞ 0

jωðtÞj<sup>2</sup>

The parameters of the converter are listed in Figure 6. Consider the following system:

Figure 6. Photo and parameters of a PSPWM full-bridge power converter circuit. Transformer turns ratio n = 0.5. V<sup>i</sup> = 160

volt, V<sup>0</sup> = 50 volt, R = 6Ω, C = 940μF L = 300μH, Llk = 20μH CA = CB = CC =CD = 5nF, fs = 50 kHz.

� 2V λPmin

 2V λPmin

�1 2

> !1 2 ! þ 1 gLðxÞ

<sup>m</sup>jþkkk

0 !#

 2V λPmin !1 2

<sup>m</sup>,…, <sup>y</sup>ðn�1<sup>Þ</sup> <sup>m</sup> <sup>Þ</sup>

dτ ð63Þ

T.

@ ð62Þ

keðtÞk ¼ 0

ð61Þ

$$
\Lambda\_{\rm t} = \begin{bmatrix} 0 & 1 \\ -100000 & -1000 \end{bmatrix}, \mathcal{Q} = \begin{bmatrix} 200000 & 0 \\ 0 & 1 \end{bmatrix} \tag{67}
$$

Use Λ<sup>c</sup> TP <sup>þ</sup> <sup>P</sup>Λ<sup>c</sup> ¼ �<sup>Q</sup> to find <sup>P</sup>

$$P = \begin{bmatrix} 1150 & 1\\ 1 & 0.0015 \end{bmatrix} \\ \lambda\_{P\text{min}} = 0.00063 \\ \tag{68}$$

Determine the range of x

$$0 < \mathbf{x}\_1 < 20, 0 < \mathbf{x}\_2 < 60, 0 < \|\mathbf{x}\| < 63.24 \tag{69}$$

Determine the range of input and output

$$10 < u \le 90, y\_m = 50\tag{70}$$

Obtain V by

$$\left(\frac{2\overline{V}}{\lambda\_{P\text{min}}}\right)^{\frac{1}{2}} \le 13.24, \overline{V} = \frac{\lambda\_{P\text{min}}}{2} (13.24)^2 = 0.055\tag{71}$$

Find f <sup>U</sup>ðxÞ, gUðxÞ, gLðx<sup>Þ</sup> according to

$$\begin{aligned} |f(\mathbf{x})| &= \left| \frac{-1}{RC^2} \mathbf{x}\_1 + \left( \frac{1}{R^2 C^2} - \frac{1}{LC} \right) \mathbf{x}\_2 \right| \\ &= \left| \frac{-1}{6 \times (940 \times 10^{-6})^2} \mathbf{x}\_1 + \left( \frac{1}{6^2 (940 \times 10^{-6})^2} - \frac{1}{(300 \times 10^{-6})(940 \times 10^{-6})} \right) \mathbf{x}\_2 \right| \\ &= 188,622.3027 \mathbf{x}\_1 + 3,514,662.2403 \mathbf{x}\_2 = f^{\text{ul}}(\mathbf{x}) \\ |g(\mathbf{x})| &= \left| \frac{m\_i}{LC} \right| = \frac{0.5 \times 160}{(300 \times 10^{-6})(940 \times 10^{-6})} = 283,687,943.2624 = g^{\text{ul}}(\mathbf{x}) = \mathbf{g}\_L(\mathbf{x}) \end{aligned} \tag{72}$$

Set the other parameters as

$$M\_{\!\!\!\!=1,000,000,000,\ }M\_{\!\!\!=1} = 1,000,000,000,\ \text{ }\varepsilon = 2,\ \text{ }\!\!\! \_1 = 10,000,000,000,\ \text{ }\!\!\!/ \_2 = 500,000,000 \text{ }\ (73)$$

$$\begin{aligned} \mu\_{F\_1^\circ}(\mathbf{x}\_1) &= \exp\left(-\left(\frac{\mathbf{x}\_1 - \mathbf{0}}{2}\right)^2\right), \mu\_{F\_1^\circ}(\mathbf{x}\_1) = \exp\left(-\left(\frac{\mathbf{x}\_1 - \mathbf{0}}{2}\right)^2\right), \mu\_{F\_1^\circ}(\mathbf{x}\_1) = \exp\left(-\left(\frac{\mathbf{x}\_1 - \mathbf{0}}{2}\right)^2\right), \\\ \mu\_{F\_1^\circ}(\mathbf{x}\_1) &= \exp\left(-\left(\frac{\mathbf{x}\_1 - \mathbf{1}}{2}\right)^2\right), \mu\_{F\_1^\circ}(\mathbf{x}\_1) = \exp\left(-\left(\frac{\mathbf{x}\_1 - \mathbf{1}}{2}\right)^2\right), \mu\_{F\_1^\circ}(\mathbf{x}\_1) = \exp\left(-\left(\frac{\mathbf{x}\_1 - 2\mathbf{0}}{2}\right)^2\right) \end{aligned} \tag{74}$$

$$\begin{split} \mu\_{\mathbb{F}\_2^{\mathbb{F}}}(\mathbf{x}\_2) &= \exp\left(-\left(\frac{\mathbf{x}\_1 - \mathbf{0}}{6}\right)^2\right), \mu\_{\mathbb{F}\_2^{\mathbb{F}}}(\mathbf{x}\_2) = \exp\left(-\left(\frac{\mathbf{x}\_1 - \mathbf{1}}{6}\right)^2\right), \mu\_{\mathbb{F}\_2^{\mathbb{F}}}(\mathbf{x}\_2) = \exp\left(-\left(\frac{\mathbf{x}\_1 - \mathbf{2}}{6}\right)^2\right), \\ \mu\_{\mathbb{F}\_2^{\mathbb{F}}}(\mathbf{x}\_2) &= \exp\left(-\left(\frac{\mathbf{x}\_1 - 3\mathbf{6}}{6}\right)^2\right), \mu\_{\mathbb{F}\_2^{\mathbb{F}}}(\mathbf{x}\_2) = \exp\left(-\left(\frac{\mathbf{x}\_1 - 4\mathbf{8}}{6}\right)^2\right), \mu\_{\mathbb{F}\_2^{\mathbb{F}}}(\mathbf{x}\_2) = \exp\left(-\left(\frac{\mathbf{x}\_1 - 6\mathbf{0}}{6}\right)^2\right) \end{split} \tag{75}$$

$$
\hat{f}(\mathbf{x}|\boldsymbol{\theta}\_{\hat{f}}) = \boldsymbol{\theta}\_{\hat{f}}^T \boldsymbol{\xi}\_{\hat{f}}(\mathbf{x}), \hat{g}(\mathbf{x}|\boldsymbol{\theta}\_{\hat{g}}) \ = \boldsymbol{\theta}\_{\hat{g}}^T \boldsymbol{\xi}\_{\hat{g}}(\mathbf{x}) \tag{76}
$$

$$\xi\_l(\mathbf{x}) = \prod\_{i=1}^2 \mu\_{\mathbb{F}\_i^l}(\mathbf{x}\_i) \bigg/ \sum\_{l=1}^{36} \prod\_{i=1}^2 \mu\_{\mathbb{F}\_i^l}(\mathbf{x}\_i), l = 1, 2...36 \tag{77}$$

Figure 8. t = 0.299~0.3.

Mf ¼ 1, 000, 000, 000, Mg ¼ 1, 000, 000, 000, ε ¼ 2, γ<sup>1</sup> ¼ 10, 000, 000, 000, γ<sup>2</sup> ¼ 500, 000, 000 ð73Þ

2 � �<sup>2</sup> � �

2 � �<sup>2</sup> � �

6 � �<sup>2</sup> � �

6 � �<sup>2</sup> � �

<sup>f</sup> <sup>ξ</sup>fðxÞ, <sup>g</sup>^ðxjθgÞ ¼ <sup>θ</sup><sup>T</sup>

, μF<sup>3</sup> 1

> , μF<sup>6</sup> 1

, μF<sup>3</sup> 2

> , μF<sup>6</sup> 2

ð Þ¼ <sup>x</sup><sup>1</sup> exp � <sup>x</sup>1�<sup>8</sup>

<sup>ð</sup>x1Þ ¼ exp � <sup>x</sup>1�<sup>20</sup>

<sup>ð</sup>x2Þ ¼ exp � <sup>x</sup>1�<sup>24</sup>

<sup>ð</sup>x2Þ ¼ exp � <sup>x</sup>1�<sup>60</sup>

ðxiÞ, l ¼ 1, 2…36

2 � �<sup>2</sup> � �

2

6 � �<sup>2</sup> � �

6

<sup>g</sup> ξgðxÞ ð76Þ

,

� �<sup>2</sup> � � <sup>ð</sup>74<sup>Þ</sup>

,

� �<sup>2</sup> � � <sup>ð</sup>75<sup>Þ</sup>

ð77Þ

<sup>ð</sup>x1Þ ¼ exp � <sup>x</sup>1�<sup>4</sup>

<sup>ð</sup>x1Þ ¼ exp � <sup>x</sup>1�<sup>16</sup>

<sup>ð</sup>x2Þ ¼ exp � <sup>x</sup>1�<sup>12</sup>

<sup>ð</sup>x2Þ ¼ exp � <sup>x</sup>1�<sup>48</sup>

<sup>ð</sup>xi<sup>Þ</sup> <sup>X</sup><sup>36</sup>

.

l¼1 Y<sup>2</sup> <sup>i</sup>¼<sup>1</sup> <sup>μ</sup>Fl i

Numerical simulation is performed by augmenting the controller and parametric adaptive law with the comprehensive open-loop model. Figure 7 is the input, output current, output

Step 2:

μF1 1

μF4 1

μF1 2

μF4 2

where

Step 3:

Figure 7. t = 0~0.3s.

Establish the following fuzzy rules

158 Modern Fuzzy Control Systems and Its Applications

ð Þ¼ <sup>x</sup><sup>1</sup> exp � <sup>x</sup>1�<sup>0</sup>

<sup>ð</sup>x1Þ ¼ exp � <sup>x</sup>1�<sup>12</sup>

<sup>ð</sup>x2Þ ¼ exp � <sup>x</sup>1�<sup>0</sup>

<sup>ð</sup>x2Þ ¼ exp � <sup>x</sup>1�<sup>36</sup>

such that we have 36 rules

2 � �<sup>2</sup> � �

2 � �<sup>2</sup> � �

6 � �<sup>2</sup> � �

6 � �<sup>2</sup> � �

<sup>ξ</sup>lðxÞ ¼ <sup>Y</sup><sup>2</sup>

Use the adaptive law as described in Eq. 51 to Eq. 59

, μF<sup>2</sup> 1

> , μF<sup>5</sup> 1

, μF<sup>2</sup> 2

> , μF<sup>5</sup> 2

^<sup>f</sup> <sup>ð</sup>xjθfÞ ¼ <sup>θ</sup><sup>T</sup>

<sup>i</sup>¼<sup>1</sup> <sup>μ</sup>F<sup>l</sup> i voltage, and tracking error within 0–0.3s. The input is not supplied until 0.05s to allow some transient response. Note that the tracking error converges around 0.12s. Figure 8 is the zoomed input, output current, output voltage, and tracking error within 0.299–0.3s. We see that the output voltage converges to 50V and the tracking error converges to 0.

### 8. Conclusion

This chapter presents a control-oriented modeling and analysis approach for a class of PWM fullbridge power converters. The results can be extended to other categories of switching power converters with complex topology. The proposed modeling and analysis approach provides an assortment of essential information for subsequent control design, including selection of the values of circuit elements, stability characteristics of the open-loop system, controllable and observable signals/variables, and so on. Current research on feedback control of dc-dc power converters mostly focuses on systems with simple circuit topology (buck, boost, or buck/boost). In particular, control for soft switched PSPWM full-bridge converters is still limited to linearized design with PI or lead-lag compensators. The conventional linearized design approaches may overlook critical dynamics due to bilinear terms being neglected. For systems possessing nonlinearities and uncertainties of which accurate mathematical description is difficult to obtain, fuzzy control is definitely a sensible option. Moreover, in this study, we see that desirable properties are achieved (e.g., tracking, robustness) by integrating fuzzy control with parametric adaptation and sliding mode control. For future work, the experimental verification of the proposed control system is currently under progress. It is also a future plan to build a power factor correction (PFC) circuit to shape the input current of off-line power supplies for maximizing the actual power available from the mains. Another motivation to employ PFC is to comply with regulatory requirements.

## Acknowledgements

The author gratefully acknowledges the support from the Ministry of Science and Technology, R.O.C., under grant MOST105-2221-E-005-047.

## Author details

Cheng-Lun Chen

Address all correspondence to: chenc@dragon.nchu.edu.tw

National Chung Hsing University, Taiwan, Republic of China

## References


[9] Cho J-H, Seong H-W, Jung S-M, Park J-S, Moon G-W, Youn M-J, editors. Implementation of digitally controlled phase shift full bridge converter for server power supply. Energy Conversion Congress and Exposition (ECCE), 2010 IEEE; 2010: IEEE Atlanta, Georgia (USA).

Acknowledgements

160 Modern Fuzzy Control Systems and Its Applications

Author details

Cheng-Lun Chen

References

505.

1996;11(4):622–628.

(4):530–534.

2003: IEEE.

R.O.C., under grant MOST105-2221-E-005-047.

Address all correspondence to: chenc@dragon.nchu.edu.tw

National Chung Hsing University, Taiwan, Republic of China

tions on Power Electronics. 1991;6(3):408–418.

actions on Power Electronics. 2003;18(5):1122–1129.

ceedings 1994, Ninth Annual; 1994: IEEE Orlando, Florida (USA).

The author gratefully acknowledges the support from the Ministry of Science and Technology,

[1] Mweene LH, Wright CA, Schlecht MF. A 1 kW 500 kHz front-end converter for a distributed power supply system. IEEE Transactions on Power Electronics. 1991;6(3):398–407. [2] Redl R, Sokal NO, Balogh L. A novel soft-switching full-bridge DC/DC converter: analysis, design considerations, and experimental results at 1.5 kW, 100 kHz. IEEE Transac-

[3] Brunoro M, Vieira JLF. A high-performance ZVS full-bridge DC-DC 0-50-V/0-10-A power supply with phase-shift control. IEEE Transactions on Power Electronics. 1999;14(3):495–

[4] Cho J-G, Sabate JA, Hua G, Lee FC. Zero-voltage and zero-current-switching full bridge PWM converter for high-power applications. IEEE Transactions on Power Electronics.

[5] Hua G, Lee FC, Jovanovic MM. An improved full-bridge zero-voltage-switched PWM converter using a saturable inductor. IEEE Transactions on Power Electronics. 1993;8

[6] Jang Y, Jovanovic MM, editors. A new family of full-bridge ZVS converters. Applied Power Electronics Conference and Exposition, 2003 APEC'03 Eighteenth Annual IEEE;

[7] Jang Y, Jovanovic MM, Chang Y-M. A new ZVS-PWM full-bridge converter. IEEE Trans-

[8] Redl R, Balogh L, Edwards DW, editors. Optimum ZVS full-bridge dc/dc converter with PWM phase-shift control: analysis, design considerations, and experimental results. Applied Power Electronics Conference and Exposition, 1994 APEC'94 Conference Pro-


## **Fuzzy Optimization Control: From Crisp Optimization**

Makoto Katoh

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67969

#### Abstract

This section shows interesting contents from the development results of author's past crisp optimization combustion control concerning real boilers of fossil power plants to the upper and lower separation new fuzzy optimization control system plan. The fuzzy decision-type optimization is for elevators and the fuzzy table-like control with zero is for a single-element level control of one tank model. In addition, other researchers' recent researches concerning other applications are introduced to maintain fairness and balance.

Keywords: crisp optimization control, fuzzy optimization control

## 1. Introduction

In section 1, the section hierarchy and abstract of this chapter is introduced. Moreover, sometimes, abstract of many studies on many kinds of fuzzy optimization control systems are also introduced and discussed. Current interests [1] are mathematically steady and will become active in the future, whereas Mandani type fuzzy control is comprehensible and has been installed already to a marketing personal computer control system for a coupled tank level [2].

In section 2, the composition of the optimizing control system, which the author and the colleagues of an enterprise had developed [3] and the recent evolution of optimization part are introduced. The first real process application version consisted of the optimal search part with the restriction by an upper computer, a usual cascade control by decentralization digital control system and various input parts and an interface part for them. The optimization search

method was developed from some general nonlinear programming to an integer programming that combined a local search and a boundary search using a simple pattern. Afterward, the optimization part has been enhanced to methods using the double patterns with more evaluation cells [4] and to methods in many books for optimization technique [5–7] except for boundary pattern search. In addition, the concept of cellular automata [8] and the technique of Q learning [9] were taken and it has been enhanced in the three universities and an institute of technology.

Zhang, Maeda, and Kawachi [10] presented an optimization model in order to allocate irrigation of water, which is withdrawn from a river, to paddy field blocks in irrigation system. A fuzzy linear programming is employed in order to solve the fuzzy decision-type optimization in the model formulation for dealing with uncertainties due to randomness of hydrologic and hydraulic parameters and fuzziness in management goals.

Multi various patterns and the other concept for optimization are introduced in a literature by the author [11].

Then, recent literature by Fujita, Tani, and Kawamura et al. [12] on fuzzy optimum control theory based on fuzzy maximization decision method of built structure are introduced.

In Section 3, a single input single output (SISO) feed-forward and feedback control system with a Table Base Controller with Zero (TBCZ) as one of Table Base System (TBS) is proposed using crisp number, which is expected to provide some advantages.

There, a feed-forward controller refers to an inverse transfer function of the controlled object.

It is proposed that a control table for three inputs of PID and three types of membership functions are not in fuzzy sets, but rather in crisp sets in expectation of the some advantages:

Some simulation results and evaluations are also shown there.

In Section 4, a fuzzy decision method based on crisp numbers with not only 1D-2D fuzzy evaluation membership functions and 1D-2D fuzzy restriction membership functions but also 3D description of their membership functions with overview plan, and a search method on the overview plan is proposed as a kind of fuzzy optimization part.

## 2. Examples of crisp and fuzzy optimization control systems

#### 2.1. A crisp example in real plants

The optimizing control is divided into three parts, the optimization part in the upper system, the digital control part in the lower system, and the interface part where they are connected to former two parts. The method of the search for the combination pattern of the local search part and the boundary search part [4] was used as an upper system.

Figure 1 shows a system block diagram of the optimum combustion control system (MHI Operation Support System), which is a real example of crisp optimization controls [3]. The output chart of optimization part is shown in Figure 2 [3] and control logic part is shown in Figure 3 [11]. The patent [11] has more information on the optimization control.

Figure 1. System block diagram of an optimization control.

method was developed from some general nonlinear programming to an integer programming that combined a local search and a boundary search using a simple pattern. Afterward, the optimization part has been enhanced to methods using the double patterns with more evaluation cells [4] and to methods in many books for optimization technique [5–7] except for boundary pattern search. In addition, the concept of cellular automata [8] and the technique of Q learning [9] were taken and it has been enhanced in the three universities and an institute of

Zhang, Maeda, and Kawachi [10] presented an optimization model in order to allocate irrigation of water, which is withdrawn from a river, to paddy field blocks in irrigation system. A fuzzy linear programming is employed in order to solve the fuzzy decision-type optimization in the model formulation for dealing with uncertainties due to randomness of hydrologic and

Multi various patterns and the other concept for optimization are introduced in a literature by

Then, recent literature by Fujita, Tani, and Kawamura et al. [12] on fuzzy optimum control theory based on fuzzy maximization decision method of built structure are introduced.

In Section 3, a single input single output (SISO) feed-forward and feedback control system with a Table Base Controller with Zero (TBCZ) as one of Table Base System (TBS) is proposed

There, a feed-forward controller refers to an inverse transfer function of the controlled object. It is proposed that a control table for three inputs of PID and three types of membership functions are not in fuzzy sets, but rather in crisp sets in expectation of the some advantages:

In Section 4, a fuzzy decision method based on crisp numbers with not only 1D-2D fuzzy evaluation membership functions and 1D-2D fuzzy restriction membership functions but also 3D description of their membership functions with overview plan, and a search method on the

The optimizing control is divided into three parts, the optimization part in the upper system, the digital control part in the lower system, and the interface part where they are connected to former two parts. The method of the search for the combination pattern of the local search part

Figure 1 shows a system block diagram of the optimum combustion control system (MHI Operation Support System), which is a real example of crisp optimization controls [3]. The output chart of optimization part is shown in Figure 2 [3] and control logic part is shown in

hydraulic parameters and fuzziness in management goals.

using crisp number, which is expected to provide some advantages.

Some simulation results and evaluations are also shown there.

overview plan is proposed as a kind of fuzzy optimization part.

and the boundary search part [4] was used as an upper system.

2.1. A crisp example in real plants

2. Examples of crisp and fuzzy optimization control systems

Figure 3 [11]. The patent [11] has more information on the optimization control.

technology.

164 Modern Fuzzy Control Systems and Its Applications

the author [11].

This example is very important, for horizontally developing the optimizing control with the three-layer structures in the future, and as an example of applying a real machine of the optimizing control in a thermal power generation process that is a multivariable system with large changes though it has not arrived at "fuzzy" optimizing control yet.

However, the initially developed artificial intelligent language and the detail search algorithm of the local search agent and the boundary search agent could not open for secret of knowhow. Then, the author has developed a new evolutionary cellular automata algorithm using

Figure 2. An example of optimization search for maximizing a boiler efficiency in OFA rate and O2 concentration plane.

Figure 3. Eco outlet O2 feedback control logic with optimization signal.

different patterns to low cost squares in the upper optimization control system though they are cost up. They are introduced in the next paragraph.

#### 2.2. A crisp evolutionary algorithm for hybrid optimization control

In this paragraph, a crisp evolutionary algorithm for hybrid optimization control by multiagents is shown, because reliability and adaptability of the above hybrid optimization must be needed to increase and open.

Figures 4 and 5 show the cellular (remaining to use fuzzy number) structure of local search agents to search for the peak in the boundary line and boundary search agents to search for the peak out of the boundary line. Moreover, increasing reliability and adaptability of these search methods are shown [8].

Figure 4. Cellular structure of local search agents for reliability and adaptability.

Figure 5. Cellular structure of boundary search agents for reliability and adaptability.

different patterns to low cost squares in the upper optimization control system though they are

In this paragraph, a crisp evolutionary algorithm for hybrid optimization control by multiagents is shown, because reliability and adaptability of the above hybrid optimization

Figures 4 and 5 show the cellular (remaining to use fuzzy number) structure of local search agents to search for the peak in the boundary line and boundary search agents to search for the peak out of the boundary line. Moreover, increasing reliability and adaptability of these search

cost up. They are introduced in the next paragraph.

Figure 3. Eco outlet O2 feedback control logic with optimization signal.

must be needed to increase and open.

166 Modern Fuzzy Control Systems and Its Applications

methods are shown [8].

2.2. A crisp evolutionary algorithm for hybrid optimization control

The following features are obtained by using eight cells group in an agent for evaluation of double circles as rotated machines as shown in Figure 4.


The following features are obtained by using six cells group in an agent for evaluation of double squares as moving cars with six wheels as shown in Figure 5.


Oppositely, there are the following features in the quadrangle (square) agents adopted with an initial real system.


Figure 6 shows local and boundary search algorithm using cellular automata.

The cellular automata algorithm has the following features.

Figure 6. Cellular automata algorithm for local and boundary search algorithm. (a) Local search. (b) Boundary search.

1. Multistart points can set to any positions.

The following features are obtained by using eight cells group in an agent for evaluation of

The following features are obtained by using six cells group in an agent for evaluation of

3. Application to bifurcated boundary by using agents with different priority evaluation order Oppositely, there are the following features in the quadrangle (square) agents adopted with an

Figure 6. Cellular automata algorithm for local and boundary search algorithm. (a) Local search. (b) Boundary search.

double circles as rotated machines as shown in Figure 4.

double squares as moving cars with six wheels as shown in Figure 5.

Figure 6 shows local and boundary search algorithm using cellular automata.

1. Adaptability to movement of mountains

2. Evaluation to continuous values

168 Modern Fuzzy Control Systems and Its Applications

1. Decrease of derailment probability

1. It is easy to develop because algorithm is easy.

The cellular automata algorithm has the following features.

2. There is a room to enlarge fuzzy circle.

2. Evaluation to discrete values

initial real system.

3. Application to multipeak

2. Four steps of center transition, self-organization of all cells to the moved agent, in or out state judgment, are repeated.

In the case of four-cell agents like four wheel cars in Figure 2, center line must be replaced to front line in automata (b) of Figure 6.

#### 2.3. A fuzzy example of built structure

On the other hand, Fujita, Tani, and Kawamura et al. at Kobe University [12] execute fuzzy optimum control theory based on fuzzy maximization decision method (target conditions and restriction condition are expressed by fuzzy membership functions) to built structure, and it is called intelligent active control. Not only target response displacement and target control power but also structure identification values obtained from the responses of the structure and earthquake vibration forecast obtained from earthquake input measurement were used for the structural response forecast.

They decided the parameters using the fuzzy maximization decision method. Then, the effect of controls were examined by experiment and simulation about two methods for addition of control power added the feedback only of the ground vibration input acceleration power and the relative acceleration power's from base of structure.

They reported that the feedback control with optimal coefficients can imorove the rate of improvement by 30% higher or more.

This report feels the difficulty for execution of fuzzy optimization control though it can encourage the execution.

In the following section, a table base controller with zero (TBCZ) is introduced for easiness of tuning of membership functions more than conventional fuzzy control.

## 3. A fuzzy-like table base PID controller with zero

The purpose of this section is to design and evaluate a Table Base Controller with Zero (TBCZ) as one of Table Base System (TBS) [13].

#### 3.1. Table base controller with zero

#### 3.1.1. TBCZ configuration

In this paragraph, we propose the following SISO feed-forward and feedback control system with a TBCZ in Figure 7, which is expected to provide the following advantages:


Figure 7. Configuration of a TBCZ.

Moreover, M/A means a manual/auto switch station, s means a differential operator, and 1/s means an integral operator in Figure 7.

#### 3.1.2. Table of TBC

Figure 8 shows a table of TBCZ and rectangular membership functions instead of usual triangle membership functions. Here, SUMi contains not only the proportional scaling factor SFp and the differentiation scaling factor SFd but also the integrator scaling factor SFi. Finally, the scaling factor of the input SFu must not be overlooked.

The rectangular membership functions are easy on computation because they are crisp.

Figure 8. A table of TBCZ and membership functions.

#### 3.2. Modeling, parameter tuning, and evaluation

Here, assumed characteristic of up-down symmetry without hysteresis, and neglected pump with long nose tube reached in water of tank and PWM control dynamics, which is sufficiently fast.

Then, we modeled the control using the following simple linear transfer functions:

$$\text{G(s)} = \frac{\text{Ke}^{-1s}}{\text{Ts} + 1};\\\text{K} = \text{30, T} = \text{20, L} = \text{2} \tag{1}$$

$$H(\mathbf{s}) = \frac{\mathbf{K}\_s}{T\_s \mathbf{s} + 1}; \mathbf{K}\_s = 1, T\_s = 0.1\tag{2}$$

#### 3.2.1. Parameter tuning

Moreover, M/A means a manual/auto switch station, s means a differential operator, and 1/s

Figure 8 shows a table of TBCZ and rectangular membership functions instead of usual triangle membership functions. Here, SUMi contains not only the proportional scaling factor SFp and the differentiation scaling factor SFd but also the integrator scaling factor SFi. Finally,

The rectangular membership functions are easy on computation because they are crisp.

means an integral operator in Figure 7.

170 Modern Fuzzy Control Systems and Its Applications

Figure 8. A table of TBCZ and membership functions.

Figure 7. Configuration of a TBCZ.

the scaling factor of the input SFu must not be overlooked.

3.1.2. Table of TBC

3.2.1.1. Scaling factor in the TBCZ

The followings are examples of scaling factors in the TBCZ.

$$\begin{aligned} \text{P} &= \frac{1}{\text{SF}\_p}, \quad \text{D} = \frac{1}{\text{SF}\_d}, \quad \text{I} = \frac{\text{s}}{\text{SF}\_i}, \quad \text{e}\_0 = 0\\ \frac{1}{\text{SF}\_p} &= 0.15, \quad \frac{1}{\text{SF}\_i} = 0.0091, \quad \frac{1}{\text{SF}\_d} = 0.1 \end{aligned} \tag{3}$$

### 3.2.2. Performances of 3 � 3 and 2 � 2 tables of TBCZ

An evaluation of the mean integral square error and input (MISEI) was compared for various values of the terminator ε in Figure 9.

Figure 9. MISEI vs. ε (in a table of TBCZ).

The performance of ε = 0.01 is considered to be equivalent to that of the 2 2 table of TBCZ, it is one point except for line from 0.1 to 0.4 in Figure 9. Thus, the 3 3 table of TBCZ is superior than the 2 2 table of TBCZ since this table allows superior performance tuning. If performance is valued, then a larger table is better, although this results in high-cost and increased complexity. Performance of MISEI is superior in the case of smaller Center ZERO. There is a minimum point of MISEI on the edge of out of scope.

#### 3.3. Other considerations to TBCZ

The robust modification of the experimental PID tuning method of Ziegler Nichols could be used here. Figure 10 shows performance MISEI of a double loop.

The FFC mix rate in the double loop case must be less than half that in the single loop case.

The decoupling [14] study is omitted in this paragraph, then refer to in reference [13].

#### 3.4. Subconclusion

A new concept of a TBCZ with rectangular membership functions based on crisp sets, which was featured by ZERO's in the rule table like the fuzzy-like control table as one of TBC and a feed-forward control line were proposed, and simulation results are presented for a tank level control as an example. Then, superior evaluation based on MISEI and performance was obtained.

The membership functions of the proposed TBCZ were able to easily tune only terminators, which mean the size of ZERO's through the evaluation of MISEI.

Figure 10. MISEI vs. ε (in a table of TBCZ).

Making and tuning of the controller are easier than in conventional fuzzy control because the membership functions are rectangular without common parts in crisp sets, and the only control rules are ZERO and SUMi in a 3 � 3 control rule table.

The development of a robust compensator of integrator for no-overshoot property is a future theme. The readers can find the literature [15] on conventional system.

## 4. Fuzzy decision-type optimization

The performance of ε = 0.01 is considered to be equivalent to that of the 2 2 table of TBCZ, it is one point except for line from 0.1 to 0.4 in Figure 9. Thus, the 3 3 table of TBCZ is superior than the 2 2 table of TBCZ since this table allows superior performance tuning. If performance is valued, then a larger table is better, although this results in high-cost and increased complexity. Performance of MISEI is superior in the case of smaller Center ZERO. There is a

The robust modification of the experimental PID tuning method of Ziegler Nichols could be

The FFC mix rate in the double loop case must be less than half that in the single loop case.

A new concept of a TBCZ with rectangular membership functions based on crisp sets, which was featured by ZERO's in the rule table like the fuzzy-like control table as one of TBC and a feed-forward control line were proposed, and simulation results are presented for a tank level control as an example. Then, superior evaluation based on MISEI and performance was

The membership functions of the proposed TBCZ were able to easily tune only terminators,

The decoupling [14] study is omitted in this paragraph, then refer to in reference [13].

minimum point of MISEI on the edge of out of scope.

used here. Figure 10 shows performance MISEI of a double loop.

which mean the size of ZERO's through the evaluation of MISEI.

3.3. Other considerations to TBCZ

172 Modern Fuzzy Control Systems and Its Applications

Figure 10. MISEI vs. ε (in a table of TBCZ).

3.4. Subconclusion

obtained.

It proposes a fuzzy decision-making type optimization technique in this paragraph as an example of the problem on elevators.

Wada and Kato propose an optimization technique of fuzzy rules according to the situation by using behavior acquisition based on emotional memory that uses fuzzy sets on pleasantness and unpleasantness of a robot. And the robot is made to acknowledge pleasantness and unpleasantness by using a source of light and an experiment toward a comfortable goal evading the obstacle is done [16].

The fuzzy decision-type optimization for an elevator is proposed firstly in ordinary 2D membership description [17]. In this paragraph, it is enhanced to 3D description.

The other fuzzy optimization studies [18–22] are interesting for this study.

#### 4.1. Maximizing decision probability methods

In this paragraph, firstly, the problem of fuzzy decision-type optimization (maximization) with subjects is defined generally.

Assuming that x1 and x2 are fuzzy numbers, membership functions μ<sup>1</sup> and μ<sup>2</sup> are introduced according to x1 and x2 and λ is a scalar for λ-cut.

$$\begin{array}{ll}\underset{\begin{subarray}{c}\mathbf{x}\in X\\\mathbf{x}\in X\end{subarray}}{\text{maximize}} & \min\{\mu\_{1}(\mathbf{x}\_{1}), \ \mu\_{2}(-\mathbf{x}\_{1})\} \Leftrightarrow\\\underset{\begin{subarray}{c}\mathbf{x}\in X\\\mathbf{x}\in X\end{subarray}}{\text{maximize}} & \lambda\\\text{subject} & \alpha\_{1}\mathbf{x}\_{1} + \beta\_{1} \geq \lambda, \ -\alpha\_{2}\mathbf{x}\_{2} + \beta\_{2} \geq \lambda, \ \mathbf{x}\in X\end{array} \tag{4}$$

This technique can be used for fuzzy deciding. For example, it is decided using this technique whether an "almost crowded" elevator should pass over a certain floor with "long queuing length". These two fuzzy sets "almost crowded" and "long queuing length" are described by using fuzzy numbers of passengers in the elevators and queues in the floor.

x1 is defined to the number of passengers ride on an elevator, and x2 is defined to the number of queuing in an elevator lobby of a floor. An example of detail equations and measurement method of person numbers can be referred to the literature [23].

The fuzzy decision method is called to maximizing (min-max) decision because it is optimize when the product (minimum) of two membership functions is max as in Figure 11.

where

$$\begin{aligned} \lambda\_{\text{max}} &= \max\_{\left\{ z\_{1'}^0 z\_2^0 \right\}} \min\_{\left\{ z\_{1'}^0 z\_2^0 \right\}} \left\{ \mu(\mathbf{x}\_1), \mu(-\mathbf{x}\_2) \right\} \\ \lambda\_c &= \frac{1}{3} \lambda\_{\text{max}} \qquad \lambda\_0 = 0 \end{aligned} \tag{5}$$

The notation of the trapezoid membership function is described by the three terms set [[left\_terminator, center, right\_terminator]] as same as the triangle membership function (center means the position of grade 1, and terminator means the position of grade 0) and is inserted the end point of right and left terminator by infinity mark ∞, �∞ as the following G and C. Then, the product (minimum calculation) D of G and C is described, if you devise it as multiplying the scalar corresponding the max value μD(x\*); x\* means center of D, then you can write as follows.

$$G = [\![\mathbf{g}\_{1'}, \mathbf{g}\_{'}\!]\_{'}\!] \qquad\qquad \mathsf{C} = [\![\![\![\![\mathbf{c}, \mathbf{c}]\!]\_{'}\!]\_{'}\!]\_{'}\!] \tag{6}$$

$$D = \mu\_D(\mathbf{x}^\*) \left[ \mathbb{S}\_{1'} \mathbf{x}^\*, \mathbf{c}\_2 \right] \tag{7}$$

where x\*, μD(x\*) can be obtained easily because they are coordinate values of a cross point of two lines in the case of triangle or trapezoidal membership functions.

In multiobjective decision-making, generally it is common to narrow down to the only optimum solution by using preference function from among plural noninferior Pareto solutions, and depending on the shape of this preference function, decision maker's preference is divided into whether it is risk avoidance type, risk-oriented type, or risk neutral type.

Figure 11. Membership functions of fuzzy sets "Almost crowded" and "Long Queuing Length" with a variable λ-level on fuzzy passage decision of "Maybe Pass a Waiting Floor".

When the preference function is convex, the case is the risk avoidance type. When it is concave, the case is the risk-oriented type.

When the case has multipurpose Gi (i = 1,…,n) and numerous restrictions Cj (j = 1,…,m), their common set D is defined equally as follows.

$$D = G\_1 \cap G\_2 \cap \dots \cap G\_n \cap C\_1 \cap C\_2 \cap \dots \cap C\_m \tag{8}$$

When getting a minimum of the respective membership function μGi, μCj, the membership function of D is found as μD.

$$
\mu\_D = \mu\_{G\_1} \wedge \mu\_{G\_2} \wedge \dots \wedge \mu\_{G\_n} \wedge \mu\_{C\_1} \wedge \mu\_{C\_2} \wedge \dots \wedge \mu\_{C\_n} \tag{9}
$$

When adopting notation like Eq. (7), it can be written as follows.

$$D = \mu\_D(\mathbf{x}^\*) \left[ \{ \min \mathbf{x}, \mathbf{s}.t., \mu\_D(\mathbf{x}) = 0 \}, \mathbf{x}^\*, \{ \max \mathbf{x}, \mathbf{s}.t., \mu\_D(\mathbf{x}) = 0 \} \right] \tag{10}$$

Here, about (x\*, μD(x\*)), because they are the coordinates of an intersection point of two straight lines, they can be found easily when finding the set of which D is composed. About both end points, the finding method is like the same.

#### 4.2. Expendability of the proposed method

#### 4.2.1. 3D description

where

174 Modern Fuzzy Control Systems and Its Applications

can write as follows.

λmax ¼ max z0 <sup>1</sup>, <sup>z</sup><sup>0</sup> f g<sup>2</sup>

two lines in the case of triangle or trapezoidal membership functions.

into whether it is risk avoidance type, risk-oriented type, or risk neutral type.

<sup>λ</sup><sup>c</sup> <sup>¼</sup> <sup>1</sup> 3

min z0 <sup>1</sup>, <sup>z</sup><sup>0</sup> f g<sup>2</sup>

The notation of the trapezoid membership function is described by the three terms set [[left\_terminator, center, right\_terminator]] as same as the triangle membership function (center means the position of grade 1, and terminator means the position of grade 0) and is inserted the end point of right and left terminator by infinity mark ∞, �∞ as the following G and C. Then, the product (minimum calculation) D of G and C is described, if you devise it as multiplying the scalar corresponding the max value μD(x\*); x\* means center of D, then you

D ¼ μ<sup>D</sup> x� ð Þ g1, x�

where x\*, μD(x\*) can be obtained easily because they are coordinate values of a cross point of

In multiobjective decision-making, generally it is common to narrow down to the only optimum solution by using preference function from among plural noninferior Pareto solutions, and depending on the shape of this preference function, decision maker's preference is divided

Figure 11. Membership functions of fuzzy sets "Almost crowded" and "Long Queuing Length" with a variable λ-level on

fuzzy passage decision of "Maybe Pass a Waiting Floor".

λmax, λ<sup>0</sup> ¼ 0

μð Þ x<sup>1</sup> , μð Þ �x<sup>2</sup> 

<sup>G</sup> <sup>¼</sup> <sup>g</sup>1, g, <sup>∞</sup> , C ¼ �½ � ∞, c, c<sup>2</sup> <sup>ð</sup>6<sup>Þ</sup>

<sup>ð</sup>7<sup>Þ</sup>

, c<sup>2</sup>

ð5Þ

Figure 11 with only a front view becomes Figure 12 with a front view, a side view and an over view when elevator passenger number x<sup>1</sup> and queuing line number x<sup>2</sup> is selected as independent logically double axis if three view description is adopted in this fuzzy optimization. This change may be increase possibility of fuzzy optimization approach.

The following Figure 12 comes next Figure 13 not making the independent double axis of regular axis and reverse axis like Figure 11 but making psychological one axis by describing variable X<sup>3</sup> (the underside is unpleasant and the upper side is pleasant) in addition to physical two axis by describing variables X<sup>1</sup> and X2.

Here, "Almost Crowded" and "Long Queuing Length" are unequality conditions and "Passenger pleasure" is an objective function. It is the fuzzy decision-making method that "pass of an elevator to a floor" is done by maximizing decision method using min-max of three member ship functions (a blue point by minimax method). Whereas it is the crisp decisionmaking that the decision-making is done at the grey point of which pleasant degree is higher slightly in grey area which grade of all membership functions are one. It's a problem that the comfort level of the waiting line isn't considered against the comfort level of the passenger in this fuzzy decision-making area. Though the comfort level of the passenger is sacrificed, it can be said that the comfort level of the waiting line is considered in the crisp decision-making area.

Figure 12. Over view plan on passengers' un-pleasure caused by almost crowded and queuing's unpleasure caused by long queuing length for a fuzzy optimization problem.

Figure 13. Three side plan in physics and psychological crossing at right angle axis of two dimensions on a fuzzy optimization problem.

The three side of plans (over view, front view and side view) are described like Figure 13 instead of the above plan like Figure 12 when X<sup>2</sup> and X<sup>3</sup> out of the three axes are changed like Figure 13. Usually, only front view plan is indicated as Figure 11 and others are omitted for economy and maximum decision policy. Here, overview plan is also indicated because minimum decision policy in overview plan is better than the maximum decision policy as understand if you see.

These plans permit two kinds of optimization methods, minimizing in overview plan and maximizing in front plan.

#### 4.2.2. Search on overview plan

Figure 12. Over view plan on passengers' un-pleasure caused by almost crowded and queuing's unpleasure caused by

Figure 13. Three side plan in physics and psychological crossing at right angle axis of two dimensions on a fuzzy

long queuing length for a fuzzy optimization problem.

176 Modern Fuzzy Control Systems and Its Applications

optimization problem.

For searching minimum point on overview plan less than grade 1 such as Figure 14, mobile method of freely writing lines and circles like dance on the plane in order to evaluate grade values using the following equations with the rotation matrix is proposed here. This is a discrete mobile model which generates left and right double velocities vL and vR of interval d.

$$
\upsilon\_G(k) = \upsilon\_L(k) + \upsilon\_R(k); r = 0.5d \tag{11}
$$

$$
\omega\_G(k) = \frac{1}{r} (|\upsilon\_R(k)| - |\upsilon\_L(k)|) \tag{12}
$$

$$
\Delta\theta\_G(k) = h\omega\_G(k); \theta\_G(0) = \theta\_{G0} \tag{13}
$$

Figure 14. An example pattern of searched course for reduced repeat by a discrete mobile agent.

$$
\theta\_G(k+1) = \theta\_G(k) + \Delta\theta\_G(k)\tag{14}
$$

$$
v\_{\mathbb{G}}(k+1) = \begin{bmatrix}
\cos\Delta\theta\_{\mathbb{G}}(k) & -\sin\Delta\theta\_{\mathbb{G}}(k) \\
\sin\Delta\theta\_{\mathbb{G}}(k) & \cos\Delta\theta\_{\mathbb{G}}(k)
\end{bmatrix} v\_{\mathbb{G}}(k)\tag{15}$$

$$p\_G(k+1) = p\_G(k) + h\upsilon\_G(k+1); p\_G(0) = p\_{G0} \tag{16}$$

where suffix G means center of the mobile object, suffix L means left, suffix R means right, p means position, ω means angle velocity, θ means angle from x-axis, and h means sample time.

Figure 14 shows an example pattern by a mobile agent and the reduced inner copy for design of searched course in the plane.

Addition of inverse kinematics will make easier the process of drawing figures filled in the canvas.

#### 4.3. Action mode, effective, influence, yield analysis (AMEIYA)

A table on action mode effect, influence, yield analysis (AMEIYA) may be obtained by imitating reasoning methods from left-column action mode to right-column effect, influence and yield things.

A Q&A table used to obtain the above table on AMEIYA may also be made by readers referred to Ref. [23].

In the future, new strategic systems for fuzzy optimization control on elevators will expect to born from these tables.

### 5. Conclusions

In this chapter, a fuzzy optimization control system by combining fuzzy decision-type optimization parts and a fuzzy-like table base PID controller parts was proposed separately instead of the conventional united crisp optimization control system.

Double future themes on fuzzy optimization, which may be used in fuzzy optimization control, were also presented. One is three side plans with an overview plan, which omitted conventional studies. The other is a new search method on a plane using a new simple discrete mobile model, which generates left and right double vector velocities.

These two separate parts are expected to be united and to be evaluated. Moreover, optimization based on fuzzy numbers and calculations may be used for absorbing uncertainty of sensors output and digitalizing of input to computer.

Moreover, some idea of strategic systems on fuzzy optimization control was provided for readers in the future.

## Acknowledgements

θGð Þ¼ k þ 1 θGð Þþ k ΔθGð Þk ð14Þ

pGð Þ¼ k þ 1 pGð Þþ k hvGð Þ k þ 1 ; pGð Þ¼ 0 pG<sup>0</sup> ð16Þ

vGð Þk ð15Þ

cos ΔθGðÞ � k sin ΔθGð Þk sin ΔθGð Þk cos ΔθGð Þk

where suffix G means center of the mobile object, suffix L means left, suffix R means right, p means position, ω means angle velocity, θ means angle from x-axis, and h means sample time. Figure 14 shows an example pattern by a mobile agent and the reduced inner copy for design

Addition of inverse kinematics will make easier the process of drawing figures filled in the

A table on action mode effect, influence, yield analysis (AMEIYA) may be obtained by imitating reasoning methods from left-column action mode to right-column effect, influence and

A Q&A table used to obtain the above table on AMEIYA may also be made by readers referred

In the future, new strategic systems for fuzzy optimization control on elevators will expect to

In this chapter, a fuzzy optimization control system by combining fuzzy decision-type optimization parts and a fuzzy-like table base PID controller parts was proposed separately instead of the

Double future themes on fuzzy optimization, which may be used in fuzzy optimization control, were also presented. One is three side plans with an overview plan, which omitted conventional studies. The other is a new search method on a plane using a new simple discrete mobile model,

These two separate parts are expected to be united and to be evaluated. Moreover, optimization based on fuzzy numbers and calculations may be used for absorbing uncertainty of

Moreover, some idea of strategic systems on fuzzy optimization control was provided for

" #

vGð Þ¼ k þ 1

4.3. Action mode, effective, influence, yield analysis (AMEIYA)

conventional united crisp optimization control system.

which generates left and right double vector velocities.

sensors output and digitalizing of input to computer.

of searched course in the plane.

178 Modern Fuzzy Control Systems and Its Applications

canvas.

yield things.

to Ref. [23].

born from these tables.

5. Conclusions

readers in the future.

The author presents cordial acknowledgments to the cooperation of his students Ms. Natsuki Imura, Ms. Xue Li, Mr. Keibun Wang, Mr. Toru Ueno, Mr. Junnichi Sawaki, Mr. Takayuki Ozeki, Mr. Takuma Nishikawa, and Mr. Koichi Wada for their cooperation and presentations in the references on this study. Moreover, he thanks and apologies for Dr. Shuichi Isomura and many colleagues of MHI Co. Ltd, Mr. Mizuno and Mr. Manabe of Hokkaido Power Co. Ltd., many teachers and students of Osaka University, Toin Yokohama University, Osaka Institute of Technology, Hosei University, and family, who cooperated some works, lectures, and a life.

## Author details

Makoto Katoh

Address all correspondence to: makoto.kato@oit.ac.jp

Osaka Institute of Technology, Hosei University, Osaka, Japan

## References


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[9] M. Katoh, R. Simotani and K. Tokushige "Integrated Multiagent Course Search to Goal by ε-Greedy Learning Strategy—Dual Probability Approximation Searching-", Proceedings of

[10] Qin Zhang, Shigeya Maeda and Toshihiko Kawachi, "Fuzzy Optimization Model for Allocating Irrigation Water to Paddy Fields", Transactions of JSIDRE, 2007, No.249,

[12] H. Fujita, A. Tani, H. Kawamura and A. Takizawa, "Study on Activation Methods of Control Forces", Experimental Research on Intelligent Fuzzy Optimal Active Control

[13] M. Katoh and T. Ueno, "Design of an Un-fuzzy Controller for a Tank Level Control", Proceedings of 2013 International Conference on Fuzzy Theory and Its Application, pp.

[14] M. Matsuzawa and M. Saeki, "Entrainment in interpersonal motor coordination of oscillating lower legs", Annual Bulletin of Showa Woman's University, Vol. 11, 9–15,

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[16] K. Wada and M. Katoh, "Behavior Acquisition of Mobile Robots with Emotional Memory

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**Applications of Fuzzy in Navigation Systems**

## **An Approach of Fuzzy Logic H∞ Filter in Mobile Robot Navigation Considering Non-Gaussian Noise**

Hamzah Ahmad, Nur Aqilah Othman and Saifudin Razali

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68866

#### **Abstract**

This chapter has presented an analysis of H∞ filter‐based mobile robot navigation with fuzzy logic to tolerate in non‐Gaussian noise conditions. The technique exploits the infor‐ mation obtained through H∞ filter measurement innovation to reduce the noises or the uncertainties during mobile robot observations. The simulation results depicted that the proposed technique has improved the mobile robot estimation as well as any landmark being observed. Different aspects such as *γ* values, noise parameters, intermittent mea‐ surement data lost and finite escape time issues are also analysed to investigate their effects in estimation. Different fuzzy logic design configurations were also studied to achieve better estimation results. As demonstrated in this work, fuzzy logic offers reliable estimation results compared to the conventional technique.

**Keywords:** navigation, mobile robot, H∞ filter, estimation, fuzzy logic

## **1. Introduction**

Working in a hazardous area has always been an issue to human safety and health. This is a situation where robotics offers an alternate solution to perform any given task. In realiza‐ tion of this problem, an autonomous robot is providing a suitable approach with less human monitoring system. Since the term of 'robot' defined by Karel Capek in 1920s, robotics has experienced a lot of interesting advancements and evolutions.

In general, robotics is classified into several categories stated by the Robotics Institute of America such as the variable‐sequence robot, playback robot, numerical control robot and

intelligent robot. There are also a number of available robotic configurations such as manipu‐ lator robot, SCARA robot, spherical type robot, cartesian robot and cylindrical robot.

The robotic research, development and technology have been immense recently considering various fields and applications. In fact, the technology has spread and is widely used in home‐ based appliances such as the lawn mower robot, vacuum cleaner robot and floor‐washing robot. These domestic robots are sometimes designed to work automatically and indepen‐ dently with their own pre‐described algorithm or system. Considering these conditions, two main features must be made available for the robot to perform their task, which are naviga‐ tion and autonomous capabilities.

Mobile robot navigation is the main concern being focussed in this chapter. In modelling the mobile robot navigation, three approaches are available [1, 2]. First, the mathematical model‐ ling which develops the mathematical model of the robot and the environment through the robot dimensions, configurations and the environment conditions. Therefore, designer must understand clearly the mobile robot configurations as well as the environment conditions.

Sensor information is one of the influential elements during information processing in mobile robot navigation. The information enables better mobile robot confidence in the naviga‐ tion processes. This is the second type of navigation modelling which uses the information extracted from sensors. Several types of sensors can be equipped in a mobile robot to assist the navigation. Compared to the mathematical model, this technique is less complex as sensors provide relative measurements between mobile robot and any observed landmarks.

The third type is focussing on how to tolerate the noises when the mobile robot navigates. Based on probabilistic rules, particularly the Bayes rule, the mobile robot attempts to identify its location and the landmarks observed with some uncertainties. This method consumed lower mathematical computation and does not rely heavily on the sensors information. Owing to these advantages, the third type has been famously applied nowadays. This chapter focusses on the probabilistic technique for mobile robot navigation.

On the aspect of making an autonomous mobile robot, a number of approaches have been proposed especially by using an artificial intelligence technique, for example [3–5]. Jaradat and Abdel‐Hafez [4], for example, have successfully applied the neural network to process information obtained through the sensors, IMU and GPS, to navigate the vehicle. Vehicle‐ measured force and angular velocity are measured via IMU and for the vehicle location, GPS is employed to do the work. Two hidden layer with at least six neurons are designed to cal‐ culate and analyse the vehicle movement. Their results seem to provide a good accuracy of vehicle estimation.

Inspired by the human nature of thinking and based on preceding literatures, the mobile robot can be guaranteed to move with less human monitoring and relies on the artificial intel‐ ligence to observe any environment. In this context, artificial intelligence is recognized as one of the possible techniques to provide efficient solutions. This research applies the fuzzy logic control to improve a probabilistic technique known as H∞ filter by analysing several issues such as finite escape time [6], operating in non‐Gaussian Noise, intermittently on lost mea‐ surement data and parameter effects to the estimation.

## **2. Mobile robot navigation**

intelligent robot. There are also a number of available robotic configurations such as manipu‐

The robotic research, development and technology have been immense recently considering various fields and applications. In fact, the technology has spread and is widely used in home‐ based appliances such as the lawn mower robot, vacuum cleaner robot and floor‐washing robot. These domestic robots are sometimes designed to work automatically and indepen‐ dently with their own pre‐described algorithm or system. Considering these conditions, two main features must be made available for the robot to perform their task, which are naviga‐

Mobile robot navigation is the main concern being focussed in this chapter. In modelling the mobile robot navigation, three approaches are available [1, 2]. First, the mathematical model‐ ling which develops the mathematical model of the robot and the environment through the robot dimensions, configurations and the environment conditions. Therefore, designer must understand clearly the mobile robot configurations as well as the environment conditions.

Sensor information is one of the influential elements during information processing in mobile robot navigation. The information enables better mobile robot confidence in the naviga‐ tion processes. This is the second type of navigation modelling which uses the information extracted from sensors. Several types of sensors can be equipped in a mobile robot to assist the navigation. Compared to the mathematical model, this technique is less complex as sensors

The third type is focussing on how to tolerate the noises when the mobile robot navigates. Based on probabilistic rules, particularly the Bayes rule, the mobile robot attempts to identify its location and the landmarks observed with some uncertainties. This method consumed lower mathematical computation and does not rely heavily on the sensors information. Owing to these advantages, the third type has been famously applied nowadays. This chapter

On the aspect of making an autonomous mobile robot, a number of approaches have been proposed especially by using an artificial intelligence technique, for example [3–5]. Jaradat and Abdel‐Hafez [4], for example, have successfully applied the neural network to process information obtained through the sensors, IMU and GPS, to navigate the vehicle. Vehicle‐ measured force and angular velocity are measured via IMU and for the vehicle location, GPS is employed to do the work. Two hidden layer with at least six neurons are designed to cal‐ culate and analyse the vehicle movement. Their results seem to provide a good accuracy of

Inspired by the human nature of thinking and based on preceding literatures, the mobile robot can be guaranteed to move with less human monitoring and relies on the artificial intel‐ ligence to observe any environment. In this context, artificial intelligence is recognized as one of the possible techniques to provide efficient solutions. This research applies the fuzzy logic control to improve a probabilistic technique known as H∞ filter by analysing several issues such as finite escape time [6], operating in non‐Gaussian Noise, intermittently on lost mea‐

provide relative measurements between mobile robot and any observed landmarks.

focusses on the probabilistic technique for mobile robot navigation.

surement data and parameter effects to the estimation.

lator robot, SCARA robot, spherical type robot, cartesian robot and cylindrical robot.

tion and autonomous capabilities.

186 Modern Fuzzy Control Systems and Its Applications

vehicle estimation.

The celebrated Kalman filter is the famously used approach for mobile robot navigation. The technique which is based on the minimum mean square error utilizes the prior information obtained by the sensor to update its current location. Even though Kalman filter offers reliable estimations, it is still incompetent in an environment that holds non‐Gaussian noise character‐ istics. The noise can be considered as noise when it is not zero mean and holds a characteristic of not uniformly distributed noise. Therefore, a number of new methods are proposed such as the unscented Kalman filter, particle filter and ensemble Kalman filter, which generally use a lot of particles to infer the mobile robot position [2]. These techniques exhibit higher computational cost if the number of particles is increased. In fact, they require a fast processor to calculate and store the information during mobile robot observation. Thus, a simple and robust filter than the above‐mentioned techniques is welcome to solve the navigation issues. Some of the Kalman filter limitations can be listed as follows:


H∞ filter is one of Kalman filter families. The filter is proven to work in non‐Gaussian noise environment and it is assumed that the noises are bounded. By defining the tuning param‐ eters known as *γ*, the filter can provide a better solution than Kalman filter as well as other techniques. In this chapter, the non‐Gaussian noise is assumed to be bounded and acts as a random noise when mobile robot does its observations. It is also important to note that H∞ fil‐ ter is facing one issue, that is the finite escape time where the estimation can be unbounded if the system is not satisfying the designed criteria [7]. Despite these shortcomings, the H∞ filter performance analysis is the main objective in this chapter to propose an alternative solution for mobile robot navigation. To ensure the finite escape time is not perceived during observa‐ tion, fuzzy logic technique is applied for the proposed system. The system design consisting of these approaches is presented in the next section.

## **3. H∞ filter and fuzzy logic technique modelling**

One of the main parts in navigation is the simultaneous localization and mapping (SLAM) problem. It states a situation where the mobile robot builds up an environment based on its sensor readings while at the same time localizes itself on the map [8, 9]. Typically, two mod‐ els that simulate the system are referred to describe how the SLAM problem is being solved. The first model is known as the kinematic model and the latter is defined as measurement model. Both models are essential to define the behaviour of the system with reference to the environmental conditions.

The kinematic model simply illustrates how the mobile robot movements are recorded. The following equation demonstrates the model.

$$X\_{k+1} = F\_k X\_k + B\_k u\_k + w\_k \tag{1}$$

where X is the augmented states of the mobile robot (*x*, *y*) and landmark i (*x*<sup>i</sup> , *y*i ). *F* and *B* are the state transition matrix and the control input matrix, respectively. *u* and *w* are the control input and the associated noise during mobile robot movements.

Mobile robot measures relative distance and angle to any observed landmarks location when moving throughout the environment. The measurement model holds the following equation.


$$y\_{k+1} = H\_k X\_k + v\_k \tag{2}$$

**Table 1.** Kalman filter and H**∞** filter comparison [10].

Here, *y* contains the information of relative distance and angle measurements and *H* is showing the measurement matrix. *v* is the associated noise occurred during measurements.

H∞ filter algorithm is almost similar to the Kalman filter. The only differences are notated on the state covariance and state update. For convenience, the comparison between H∞ filter and Kalman filter is presented in **Table 1**.

### **3.1. Fuzzy Logic‐based navigation**

model. Both models are essential to define the behaviour of the system with reference to the

The kinematic model simply illustrates how the mobile robot movements are recorded. The

*Xk*+<sup>1</sup> = *Fk Xk* + *Bk uk* + *wk* (1)

the state transition matrix and the control input matrix, respectively. *u* and *w* are the control

Mobile robot measures relative distance and angle to any observed landmarks location when moving throughout the environment. The measurement model holds the following equation.

*yk*+<sup>1</sup> = *Hk Xk* + *vk* (2)

(*Xk* , *u k* , *wk* , *k*)

*<sup>k</sup>* <sup>=</sup> *<sup>h</sup>*(*Xk* , *v k* , *k*)

*y <sup>k</sup>* <sup>=</sup> *Dk Xk*

*X* ^

‐ <sup>+</sup> *Kk*+<sup>1</sup> *μk*+<sup>1</sup> *Pk*+<sup>1</sup> <sup>=</sup> <sup>∇</sup>*Fk Pk* (*I*‐ *<sup>γ</sup>*‐<sup>2</sup> *Pk* <sup>+</sup> <sup>∇</sup> *Hk*

, 0, *<sup>k</sup>*) *Kk* <sup>=</sup> *Pk* (*I*‐ *<sup>γ</sup>*‐<sup>2</sup> *Pk* <sup>+</sup> <sup>∇</sup>  *Hk*

*<sup>k</sup>*+<sup>1</sup> = ∇ *Fk X* ^

*<sup>k</sup>* + ∇ *Fk Kk*(*zk* ‐∇ *Hk X*

*<sup>T</sup> Rk*

‐<sup>1</sup> ∇  *Hk Pk*) ‐1 ∇ *Hk <sup>T</sup> Rk* ‐1

^ *<sup>k</sup>*) + *uk*

*<sup>T</sup> Rk* ‐<sup>1</sup> ∇ *Hk Pk*) ‐1 *Ñ F k <sup>T</sup>* <sup>+</sup> *Qk*

, *y*i

). *F* and *B* are

where X is the augmented states of the mobile robot (*x*, *y*) and landmark i (*x*<sup>i</sup>

**The extended Kalman filter The** *H***∞ filter**

, *k*) *z*

, *<sup>k</sup>*) *Xk*+<sup>1</sup> <sup>=</sup> *<sup>f</sup>*

input and the associated noise during mobile robot movements.

environmental conditions.

188 Modern Fuzzy Control Systems and Its Applications

*Xk*+<sup>1</sup> <sup>=</sup> *<sup>f</sup>* (*Xk* , *u k* , *wk*

*z <sup>k</sup>* <sup>=</sup> *<sup>h</sup>*(*Xk* , *v k*

*X* ^ *k*+1 ‐ = *f*(*X* ^ *k* , *uk* , 0, *k*)

*Pk*+1 ‐ = ∇*F X Pk* ∇  *F X <sup>T</sup>* + ∇ *F <sup>w</sup> Qk* ∇ *F w T*

*X* ^ *k*+1 <sup>+</sup> = *X* ^ *k*+1

*Pk*+<sup>1</sup> <sup>+</sup> = (*I*

*μk*+<sup>1</sup> = *zk*+<sup>1</sup> ‐*h*(*X*

*S*(*μμ*)*k*+<sup>1</sup> = ∇*Hi Pk*+<sup>1</sup>

*<sup>n</sup>* ‐ *Kk*+<sup>1</sup> ∇ *Hi* )*Pk*+<sup>1</sup>

‐ ∇*Hi <sup>T</sup>* + *Rk*+<sup>1</sup>

^ *k*+1 ‐

**Table 1.** Kalman filter and H**∞** filter comparison [10].

‐

*Kk*+<sup>1</sup> = *Pk*+<sup>1</sup>

‐ ∇*Hi T* (*S*(*μμ*)*k*+1) ‐1

The system

The prediction step

The update step

following equation demonstrates the model.

In this chapter, the measurement innovation is referred as the main reference in designing the fuzzy logic. Different to Ref. [3] which utilizes the heading angle and relative distance range as their inputs to the system, fuzzy logic is designed in this research to process the angle and distance errors as its inputs. Our objective is to decrease those errors by configuring the fuzzy sets in producing smaller errors. By choosing the output appropriately, the effects or mea‐ surement error due to sensor inaccuracies can be minimized further.

There were also some researchers attempts to applied fuzzy Logic in navigation such as in Refs. [11–17]. Each of them demonstrated that fuzzy logic or artificial intelligence technique is capable to fuse the information obtained from the sensors for mobile robot estimation. Some of the researches are referred to evaluate the performance of this work.

Literature has stated that mobile robot has some confidence on its estimation especially when Kalman filter is applied for estimation [3]. For non‐Gaussian noise characteristic, the sensors reading might be interfered and exhibits bigger error and hence results in bigger measure‐ ment noise covariance, *R*. If the gain *K* is small at all times during observations, then it is pos‐ sible to have smaller measurement errors. Inspired by this fact, fuzzy logic is proposed to find the best value of measurement innovation to pursue lower error. Kobayashi et al. [16] have selected the *P*, *Q* and *R* from fuzzy logic to gain smaller uncertainties. Work by Wang et al. [18] has recognized this as one of the ways to realize smaller measurement noise even when it was being applied to the other H∞ filter family, the Kalman filter. Mamdani technique is used for analysis purposes in determining the output of the system. The technique is proposed as it calculates the output by considering and utilizing the maximum information gained from measurement.

The general design is illustrated in **Figures 1**–**3**, that consists of the input and output and their respective fuzzy sets. The fuzzy sets are changed whenever the mobile robot moves in dif‐ ferent motion and therefore, the values are not included. The following describes the rules of fuzzy logic that are used to define the output of the measurement innovation.


**Figure 1.** Fuzzy logic with inputs and outputs (a) and its associated block diagram to the proposed technique (b).

**Figure 2.** (a) Angle measurement and (b) distance measurement.

**Figure 3.** (a) Fuzzified angle and (b) fuzzified distance measurement.


Generally, Gaussian and triangular membership functions are considered in this chapter for evaluation purposes. Only three fuzzy sets are defined which are divided into three different categories: the negative, normal and positive regions. The scale of each of the fuzzy sets is selected based on the normal condition that has high errors. The value differs with each of the fuzzy sets, and it has been tuned several times to obtain the best estimation results. The tun‐ ing is taking into account the uncertainties behaviour throughout the simulation. Other than that, both angle and distance measurement characteristics are observed prior to the tuning process. Wang et al. [18] designed the membership function of the angle error to be positive at all times. However, random mobile robot's movements may also show a negative angle espe‐ cially when a global coordinate system is being considered. This aspect is one of the major differences between our approach and what they have investigated.

## **4. Results and discussions**

This section describes the performance of the fuzzy logic‐based H∞ filter by considering sev‐ eral factors such as the finite escape time problem, localization and mapping problem and the effects of noise parameters. These three issues are important to be solved concurrently during the navigation. If not, then the estimation results will not be as expected. Analysis is based on the design parameters included in **Table 2**.

**Figures 4** and **5** demonstrate the estimation results when *γ* = 0.7, which determines that fuzzy logic‐based H∞ filter outperforms the normal H∞ filter about the mobile robot and landmarks estimations. Erroneous estimations are perceived for the mobile robot location estimations as well as the landmarks positions. The measurement details are presented in **Figure 2** for each of the estimation error. **Figure 6** illustrates the state covariance update performance. It can also be seen from this figure that the proposed method attempts to avoid the finite escape time from happening.

### **4.1. Effect of changing** *γ*

• IF angle error is positive and distance error is normal, THEN angle is negative.

**Figure 3.** (a) Fuzzified angle and (b) fuzzified distance measurement.

**Figure 2.** (a) Angle measurement and (b) distance measurement.

190 Modern Fuzzy Control Systems and Its Applications

differences between our approach and what they have investigated.

tance is normal.

• IF angle error is positive and distance error is negative, THEN distance is normal.

**Figure 1.** Fuzzy logic with inputs and outputs (a) and its associated block diagram to the proposed technique (b).

• IF angle error is positive and distance error is positive, THEN angle is negative and dis‐

Generally, Gaussian and triangular membership functions are considered in this chapter for evaluation purposes. Only three fuzzy sets are defined which are divided into three different categories: the negative, normal and positive regions. The scale of each of the fuzzy sets is selected based on the normal condition that has high errors. The value differs with each of the fuzzy sets, and it has been tuned several times to obtain the best estimation results. The tun‐ ing is taking into account the uncertainties behaviour throughout the simulation. Other than that, both angle and distance measurement characteristics are observed prior to the tuning process. Wang et al. [18] designed the membership function of the angle error to be positive at all times. However, random mobile robot's movements may also show a negative angle espe‐ cially when a global coordinate system is being considered. This aspect is one of the major The *γ* effect to the proposed technique has also been analysed to ensure that our proposed technique has consistent and reliable results. **Figures 7** and **8** describe the performance of the state estimations for both system of normal H∞ filter and fuzzy logic‐based H∞ filter, respec‐ tively, when *γ* = 0.23, which is smaller than the previous case. It is plotted clearly that normal H∞ filter exhibits erroneous results compared to our proposed technique.

#### **4.2. Effect of initial state covariance**

Generally, in SLAM, mobile robot does not have any prior information about its location or the environment. Therefore, the initial state covariance is designed to pose high uncertain‐ ties. Owing to these conditions, the mobile robot can probably have lost its way and is unable to navigate effectively on the environment. The analysis covers this issue by simulating the results using both normal H∞ filter and fuzzy logic‐based H∞ filter in the next few figures.


**Table 2.** Simulation parameters.

**Figure 4.** The mobile robot movements through the environment. Lighter colour of round shape defines the actual positions while the triangle shape and darker colour round shape presents the H∞ Filter with Fuzzy Logic(FHF) and normal H∞ Filter (HF) estimation performance, respectively.

**Figure 5.** A performance comparison between FHF and normal HF estimations for both mobile robot (a) and landmarks (b) estimations about the errors.

**Figure 6.** The state covariance conditions between H∞ Filter (HF) and H∞ Filter with Fuzzy Logic(FHF). Normal HF exhibits frequent Finite Escape Time(FET) compared to the H∞ Filter with Fuzzy Logic(FHF).

An Approach of Fuzzy Logic H∞ Filter in Mobile Robot Navigation Considering Non-Gaussian Noise http://dx.doi.org/10.5772/intechopen.68866 193

**Figure 7.** The state update conditions of normal HF generate a lot of estimation errors.

**Figure 6.** The state covariance conditions between H∞ Filter (HF) and H∞ Filter with Fuzzy Logic(FHF). Normal HF

**Figure 5.** A performance comparison between FHF and normal HF estimations for both mobile robot (a) and landmarks

**Figure 4.** The mobile robot movements through the environment. Lighter colour of round shape defines the actual positions while the triangle shape and darker colour round shape presents the H∞ Filter with Fuzzy Logic(FHF) and

normal H∞ Filter (HF) estimation performance, respectively.

192 Modern Fuzzy Control Systems and Its Applications

(b) estimations about the errors.

exhibits frequent Finite Escape Time(FET) compared to the H∞ Filter with Fuzzy Logic(FHF).

**Figure 9** demonstrates the results of estimation when mobile robot is in the above‐mentioned condition. As expected, the normal H∞ filter did not deliver good estimation results. On the other hand, the fuzzy logic‐based H∞ filter guarantees a good estimation that can still be pre‐ served with a considerable estimation. In this analysis, the Gaussian membership is used for decision making.

Further inspection is done by using different fuzzy membership with the similar range of fuzzy sets in the case when the mobile robot attempts to localize itself in a given environment. Other associated and related simulation parameters remain unchanged to observe any signifi‐ cant improvement that fuzzy logic can offer. The initial state covariance for the landmarks is now known and mobile robot does not have any information of its location. Even other types

**Figure 8.** The state update conditions of FHF do not encounter high uncertainties or errors during mobile robot navigation.

of membership functions are applied for decisions; the fuzzy logic‐based H∞ filter surpassed the normal filter performance as depicted in **Figure 10**.

Assessment on different mobile robot movements during its observations with the same sim‐ ulation parameters is also conducted to evaluate the consistency of estimation. The results are not disappointing and show a reliable estimation when comparing it to the normal H∞ filter as illustrated in **Figure 11**.

**Figure 9.** The normal H∞ Filter(dotted line) shows erroneous results compared to the true mobile robot path(lighter line) and Fuzzy Logic based H∞ Filter(darker line). Landmarks estimation of normal H∞ Filter(round shape) is also results in erroneous estimation.

#### **4.3. Effect of non‐Gaussian measurement noise**

of membership functions are applied for decisions; the fuzzy logic‐based H∞ filter surpassed

**Figure 8.** The state update conditions of FHF do not encounter high uncertainties or errors during mobile robot navigation.

Assessment on different mobile robot movements during its observations with the same sim‐ ulation parameters is also conducted to evaluate the consistency of estimation. The results are not disappointing and show a reliable estimation when comparing it to the normal H∞ filter

the normal filter performance as depicted in **Figure 10**.

as illustrated in **Figure 11**.

194 Modern Fuzzy Control Systems and Its Applications

H∞ filter is known to be more robust to the extended Kalman filter (EKF) especially when‐ ever non‐Gaussian noise is available. The measurement noise is increased to observe whether the H∞ filter with fuzzy logic still able to preserve a good estimation. **Figure 12** presents the results that still define that the H∞ filter with fuzzy logic (triangular memberships) maintains better performance than the normal H∞ filter. Similar performance is also observed with dif‐ ferent fuzzy memberships even though the results are not included in this chapter.

**Figure 10.** The normal H∞ Filter (dotted line) cannot localize itself as well as the available landmarks even though the landmarks information is given compared to the Fuzzy Logic based H∞ Filter (darker line) with reference to the truth locations (lighter line). Triangle shows the Fuzzy Logic based H∞ Filter and round shape showing the normal H∞ Filter for landmarks estimation.

**Figure 11.** Normal H<sup>∞</sup> Filter (dotted line) produces erroneous results compared to the proposed technique (FHF in darker line) for different mobile robot movements. Triangle shows the Fuzzy Logic based H<sup>∞</sup> Filter and round shape showing the normal H<sup>∞</sup> Filter for landmarks estimation.

The performance analysis between fuzzy logic‐based H∞ filter and extended Kalman filter is also considered. A condition where mobile robot loses its measurement data at random time is referred in this case. Interesting results are obtained in **Figure 13** showing that the H∞ filter with fuzzy logic inference is still better compared to the normal EKF estimation in non‐Gaussian noise. The landmarks estimation for H∞ filter with fuzzy logic outperforms the EKF. Therefore, based on the figure, H∞ filter with fuzzy logic technique offers better solutions when non‐Gaussian noise as well as when measurement data is lost unexpectedly during mobile robot observations.

There are few remarks to be considered in designing the fuzzy logic control for mobile robot navigation. The measurement innovation characteristics must be first examined prior to the estimation to ensure the results achieve the desired conditions. Besides measurement

**Figure 12.** Normal H∞ Filter (dotted line) performance is still low compared to the Fuzzy Logic based H∞ Filter (darker line) with reference to the truth positions(lighter line) for bigger measurement noise. Triangle shows the Fuzzy Logic based H∞ Filter and round shape showing the normal H∞ Filter for landmarks estimation.

An Approach of Fuzzy Logic H∞ Filter in Mobile Robot Navigation Considering Non-Gaussian Noise http://dx.doi.org/10.5772/intechopen.68866 197

**Figure 13.** An analysis when measurement data is randomly unavailable. Fuzzy Logic based H∞ Filter(darker line) performance is surpassing the EKF(triangle) especially on the landmarks estimations with reference to the truth positions(lighter line).

innovation, the noise characteristics can also influence the estimation performance. Therefore, the designer should carefully understand and model the noise according to the environment to be observed. H∞ filter is also sensitive to some parameters as stated by Bolzern and Maroni [7] and those parameters must be studied before conducting further analysis on the proposed technique.

### **5. Concluding remarks**

The performance analysis between fuzzy logic‐based H∞ filter and extended Kalman filter is also considered. A condition where mobile robot loses its measurement data at random time is referred in this case. Interesting results are obtained in **Figure 13** showing that the H∞ filter with fuzzy logic inference is still better compared to the normal EKF estimation in non‐Gaussian noise. The landmarks estimation for H∞ filter with fuzzy logic outperforms the EKF. Therefore, based on the figure, H∞ filter with fuzzy logic technique offers better solutions when non‐Gaussian noise as well as when measurement data is lost unexpectedly during

**Figure 11.** Normal H<sup>∞</sup> Filter (dotted line) produces erroneous results compared to the proposed technique (FHF in darker line) for different mobile robot movements. Triangle shows the Fuzzy Logic based H<sup>∞</sup> Filter and round shape

There are few remarks to be considered in designing the fuzzy logic control for mobile robot navigation. The measurement innovation characteristics must be first examined prior to the estimation to ensure the results achieve the desired conditions. Besides measurement

**Figure 12.** Normal H∞ Filter (dotted line) performance is still low compared to the Fuzzy Logic based H∞ Filter (darker line) with reference to the truth positions(lighter line) for bigger measurement noise. Triangle shows the Fuzzy Logic

based H∞ Filter and round shape showing the normal H∞ Filter for landmarks estimation.

mobile robot observations.

showing the normal H<sup>∞</sup> Filter for landmarks estimation.

196 Modern Fuzzy Control Systems and Its Applications

This research has presented the analysis and study of H∞ filter for mobile robot navigation using fuzzy logic control. The investigation was mainly focussing on the development of the fuzzy logic control to analyse the relative angle and distance measurement as its input to produce smaller error of navigation. Besides, fuzzy logic control was also found to be a possible technique to avoid finite escape time problem in H∞ filter. A number of tests have been conducted for the proposed technique which includes the effects of having different *γ* value, different noise parameters and intermittently data lost. Preliminary results describe that the fuzzy logic‐based H∞ filter is able to tolerate the problem by using only few num‐ ber of rules and fuzzy sets. Thus, the technique can be one of the alternative solutions for navigation.

#### **Author details**

Hamzah Ahmad\*, Nur Aqilah Othman and Saifudin Razali

\*Address all correspondence to: hamzah.ahmd@gmail.com

Faculty of Electrical & Electronics, Universiti Malaysia Pahang, Pekan, Pahang, Malaysia

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## **Indoor Mobile Positioning Using Neural Networks and Fuzzy Logic Control**

Anatoly D. Khomonenko, Sergey E. Adadurov, Alexandr V. Krasnovidow and Pavel A. Novikov

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/68009

#### Abstract

Indoor mobile navigation systems are becoming more prevalent in many areas (transport, public institutions, logistics, etc.). The interior navigation based on the access points, arranged according to the radio fingerprints, is becoming increasingly popular. The model of artificial neural networks (ANN) is often used as a mechanism for storing and processing radio fingerprints. The task of selection of the access point in WLAN network in the case of high user density is quite topical. Such selection must take into account not only the level of the signal received by the mobile device, but also a width in the dedicated channel bandwidth. The main issues related to the creation of program complex for the mobile indoors navigation using neural networks is discussed in the chapter as well as the method of access point selection based on analysis not only the signal level but also the other parameters. To solve this task, fuzzy logic is used.

Keywords: Wi-Fi radio network, neural network, mobile navigation indoor, navigation systems, learning algorithms, mobile devices 802.11k standard, the mobile subscribers, fuzzy logic, MATLAB, frame transmission time

#### 1. Introduction

The modern world cannot exist without precise navigation systems. Satellite navigation systems such as Global Positioning System (GPS) and Global Navigation Satellite System (GLONASS) [1] are widely used in a variety of areas of activity, such as navigation transport, engineering, surveying, and other cellular communication. GPS receivers' consumer level is set in almost all modern phones. In open areas, such receivers allow for positional accuracy in the region of 1–5 m.

© 2017 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.

At the same time, these receivers do not meet the existing demand to navigate indoors. In such circumstances, GPS does not work or provide location data with a very high error of about 100–150 m.

Navigation systems indoors can be applied in many fields. Including navigation inside large shopping malls, warehouses, or different systems, "smart home," in which different home systems (heating, lighting, air conditioning, and so on) can be centrally managed in automatic mode.

In particular, obvious argument for the need for indoor navigation can serve as tasks of navigation in transportation systems, such as airports and railway stations [2]:

(A) For visitors: positioning; search check-in desks/offices/storage rooms and a cafe, parking, taxi, etc.; installation of the route, taking into account number of storeys of buildings; search colleagues inside the airport/train station, social activity; service based on knowledge of the position of the visitor (location-based service).

(B) For airports, railway stations, and tenants: an additional service to visitors; analyst extensive movements of visitors and staff; advertising opportunities geo contextual advertising (location-based advertising) airlines/shops/cafes, etc., as well as products/services; promotions based on the location of the visitor.

There are now possibilities to improve the methods and the practical use of mobile navigation indoors. Let us briefly consider the modern approaches to the solution of this task and proposals for justifying the choice of learning algorithms and neural networks of the individual parameters of the same interests.

Mobile navigation systems indoors are becoming more widespread in many areas (transport, public institutions, logistics, and others.). It is becoming popular navigation based on fingerprint radio access points Wi-Fi. As a mechanism for storing and processing the radio fingerprint is often considered a model of artificial neural networks (ANN). The chapter examines the main issues related to the creation of complex programs for mobile navigation indoors using neural networks, which are one of the parts of fuzzy logic. We justify the choice of learning algorithms ANN navigation mobile devices indoors.

The data for the neural network training in MATLAB taken from the file "train.txt" is used for functional testing of software navigation. To verify the performance of ANN training, the original file should be divided into two sets (training and testing).

Wireless local area networks 802.11k as precise navigation systems have become very popular for several reasons. They operate in the unlicensed frequency bounds and they need not in large time and cost for deployment. The appearance of vast number of mobile devices supporting Wi-Fi technology gives possibilities of free choice and cost saving for the various kinds of users. WLAN networks have a number of advantages over traditional wired networks.

• Much easier and cheaper to deploy a local wireless network in the new location than a traditional network.

• The subscriber does not have to be next to his desk or local network socket-outlet. He can move freely inside the area coverage.

However, the number of users working in the unlicensed frequency bounds increases day by day. In this context, a very important problem is the way by which the wireless device uses to select an access point to connect to the network. Now, the device selects the access point according to signal power. This way allows determining the nearest access point. In other words, currently used 802.11k standard is aimed at the implementation of load balancing of radio Wi-Fi networks. Nevertheless, the high level of the signal does not always mean high network bandwidth. Suppose, for example, that most of the notebooks operating in the certain conference halls connect to WLAN by using the access point which is above the entrance door. In such a case, the number of subscribers connected to it would be up to tens if not hundreds, while other access points would not be fully loaded. As a result, the network bandwidth per subscriber reduces to a significant degree that leads to network reduction of productivity of the network in whole. Hence, the task of selection of the access point in WLAN network in the case of high user density is quite topical. Such selection must take into account not only the level of the signal received by the mobile device, but also a width in the dedicated channel bandwidth that depends on the number of connected subscribers to the access point. In this chapter, we examined the method based on analysis of not only signal level but also other parameters. To solve this task, fuzzy logic is used in the chapter. Constructed membership functions and linguistic rules are examined. Structure of the developed model and simulation results are presented.

## 2. Indoor mobile positioning using neural networks

#### 2.1. Characteristics of modern approach to navigation mobile

First, we note that the modern mobile devices contain a variety of different sensors and receivers. The main of them include:

1. GPS.

At the same time, these receivers do not meet the existing demand to navigate indoors. In such circumstances, GPS does not work or provide location data with a very high error of about

Navigation systems indoors can be applied in many fields. Including navigation inside large shopping malls, warehouses, or different systems, "smart home," in which different home systems (heating, lighting, air conditioning, and so on) can be centrally managed in automatic

In particular, obvious argument for the need for indoor navigation can serve as tasks of

(A) For visitors: positioning; search check-in desks/offices/storage rooms and a cafe, parking, taxi, etc.; installation of the route, taking into account number of storeys of buildings; search colleagues inside the airport/train station, social activity; service based on knowledge of the

(B) For airports, railway stations, and tenants: an additional service to visitors; analyst extensive movements of visitors and staff; advertising opportunities geo contextual advertising (location-based advertising) airlines/shops/cafes, etc., as well as products/services; promotions

There are now possibilities to improve the methods and the practical use of mobile navigation indoors. Let us briefly consider the modern approaches to the solution of this task and proposals for justifying the choice of learning algorithms and neural networks of the individual

Mobile navigation systems indoors are becoming more widespread in many areas (transport, public institutions, logistics, and others.). It is becoming popular navigation based on fingerprint radio access points Wi-Fi. As a mechanism for storing and processing the radio fingerprint is often considered a model of artificial neural networks (ANN). The chapter examines the main issues related to the creation of complex programs for mobile navigation indoors using neural networks, which are one of the parts of fuzzy logic. We justify the choice of

The data for the neural network training in MATLAB taken from the file "train.txt" is used for functional testing of software navigation. To verify the performance of ANN training, the

Wireless local area networks 802.11k as precise navigation systems have become very popular for several reasons. They operate in the unlicensed frequency bounds and they need not in large time and cost for deployment. The appearance of vast number of mobile devices supporting Wi-Fi technology gives possibilities of free choice and cost saving for the various kinds of users. WLAN networks have a number of advantages over traditional wired net-

• Much easier and cheaper to deploy a local wireless network in the new location than a

navigation in transportation systems, such as airports and railway stations [2]:

position of the visitor (location-based service).

learning algorithms ANN navigation mobile devices indoors.

original file should be divided into two sets (training and testing).

based on the location of the visitor.

202 Modern Fuzzy Control Systems and Its Applications

parameters of the same interests.

100–150 m.

mode.

works.

traditional network.


Sensors 1–5 are well represented in many of today's mobile devices that are running operating systems iOS and Android. These systems provide programmatic access to the sensors through its own application program interfaces (API), and any application can receive data from the sensors (with certain insignificant limitations).

Currently, there are several ways to use the mobile navigation device.


Comparative characteristics of the main approaches of navigation using mobile devices [2] are given in Table 1.

In addition to these approaches, following approaches are also included. In Ref. [6], the integration of Wi-Fi and an inertial navigation system is considered. In Refs. [7, 8], mobile navigation services and the use of technology OpenCellID to determine the location of mobile devices are studied.


Table 1. Comparative characteristics of approaches.

The article [9] considered a relatively new approach to the positioning of mobile devices in the premises on the basis of two-dimensional barcodes. In Ref. [10], the model of context-aware computing (context-dependent browser) based on network proximity is considered. This mobile phone is considered as a proximity sensor and geo replaced positional information network proximity. An algorithm for calculating the trajectories of mobile networks on the basis of information about network proximity is also considered.

#### 2.2. Rationale for navigation of mobile devices using neural networks

Employment of navigation system via radio fingerprint from the access points of Wi-Fi consists of two parts:


As a mechanism for storing and processing the radio fingerprint has considered a model of artificial neural networks (ANNs).

This approach is interesting for the following reasons:


The possibilities of modern ANN are the subject of active research. The approach to the use of ANN to solve the above problem is considered in several publications, for example, Refs. [10–12].

#### 2.3. Characteristics of software navigation

Here examined a complex program [13, 14] to navigate through the Wi-Fi signals using a mobile phone. It consists of two components:

1. Mobile application that

Currently, there are several ways to use the mobile navigation device.

that the use of data on the mobile device serving cell towers.

strength to the nearest of them, calculates their relative location.

to implement proven accurate navigation have not yet developed.

which is constructed on the basis of the model orientation in space.

System Dignity Disadvantages

(5 m) Ease of use Good compatibility

Good compatibility

Good compatibility Low power consumption

Accuracy (5 m) Work indoors

iBeacon High accuracy (1–2 m) Ease of use

Table 1. Comparative characteristics of approaches.

satellites might take few minutes.

204 Modern Fuzzy Control Systems and Its Applications

given in Table 1.

devices are studied.

GPS Average precision

GSM Ease of use

Wi-Fi Average

1. GPS (Global Positioning System) and GLONASS (Global Navigation Satellite System) [1]: these are navigation using satellites. Well suited for positioning in the open spaces. Because of the need to be in the field of view of at least three satellites, these are poorly suited for closed premises, as they greatly impair the satellite signal. The initial search for

2. AGPS (Assisted GPS). Navigation using radio signals from cell towers. Typically it used in conjunction with GPS. It accelerates the initial determination of the coordinates by the fact

3. Navigate using radio beacons operating on technology iBeacons (Bluetooth Low Energy) [3]. This is a relatively young technology, the impetus for the development of which will serve standard Bluetooth 4.0 (+) with low power consumption. Tracker is a chip with a radio module, which is a predetermined frequency radio, sends packets with information about themselves. The receiver, knowing the map of the location beacons and signal

4. Navigation on the basis of fingerprints of radio access points Wi-Fi [4]. Such navigation systems are more exploratory in nature. Due to the relative novelty of this approach, ways

5. Inertial navigation system [5]. Navigation is based on data from the inertial sensor device,

Comparative characteristics of the main approaches of navigation using mobile devices [2] are

In addition to these approaches, following approaches are also included. In Ref. [6], the integration of Wi-Fi and an inertial navigation system is considered. In Refs. [7, 8], mobile navigation services and the use of technology OpenCellID to determine the location of mobile

> Inability To work Indoor

Ease of use Good compatibility

The need for network deployment

Limited compatibility High power consumption

The need to deploy BLE-network

	- a. prepares data radio prints for further training of the neural network;
	- b. training and testing neural network.

As an implementation of a neural network, we used multilayer ANN provided free library FANN (Fast Artificial Neural Network Library). To train the ANN learning algorithm, Resilient Propagation (RProp) has been used [15].

Unlike standard algorithm Backprop, RProp uses only partial signs for adjusting the weighting coefficients. The algorithm uses the so-called "training periods" when the correction occurs after the presentation of the balance of the network of examples from the training set.

To determine the amount of correction using the following rule [15]:

$$\Delta\_{\vec{\eta}}^{(t)} = \left\{ \begin{aligned} \eta^{+} \Delta\_{\vec{\eta}}^{(t)} & \frac{\partial E^{(t)}}{\partial \omega\_{\vec{\eta}}} \frac{\partial E^{(t-1)}}{\partial \omega\_{\vec{\eta}}} > 0 \\ \eta^{-} \Delta\_{\vec{\eta}}^{(t)} & \frac{\partial E^{(t)}}{\partial \omega\_{\vec{\eta}}} \frac{\partial E^{(t-1)}}{\partial \omega\_{\vec{\eta}}} < 0 \end{aligned} \right\}, \tag{1}$$

$$0 < \eta^- < 1 < \eta^+ \tag{2}$$

If the partial derivative of the corresponding weight ∂ωi j had changed its sign at the current step, it means that the latest update was great, and the algorithm passed a local minimum and therefore the value of change must be reduced by η and return the previous weight value: in other words, you must make 'roll back'.

$$
\partial \omega\_{i\bar{j}}(t) = \partial \omega\_{i\bar{j}}(t) - \Delta\_{i\bar{j}}^{(t-1)}.\tag{3}
$$

If the sign of the partial derivative is not changed, it is necessary to increase the amount of compensation η<sup>+</sup> to achieve a more rapid convergence. Fixing factors η� and η<sup>+</sup> can be dispensed with global settings of the neural network, which can also be seen as an advantage of the algorithm to the standard algorithm Backprop.

Recommended values are η� = 0.5, η<sup>+</sup> = 1.2, but there are no restrictions on the use of other values for these parameters.

To prevent too large or small weight values, the correction value limit from above the maximum Δmax and below the minimum Δmin value of the correction value, which default, respectively, shall equal 50 and 1.0E–6.

The initial values for all Δij are set to 0.1. Again, this should be seen only as a recommendation, and in the practical implementation it can specify a different value for the initialization.

The current implementation of the navigation software package is a simple one-dimensional classifier, which, by new radio-prints, is able to determine the room in which the mobile device is located.

#### 2.4. Working with software navigation

Working with a program complex navigation is implemented using desktop and mobile software, and includes the following steps:

1. Collection of data on radio-prints in the studied areas. The output of this stage is a set of files, each of which contains a set of vectors (matrix) measuring Wi-Fi signals points (for example, Table 2).

In the first row, there are names of all the access points that were visible in the data collection process. For clarity, we have been selected to work with network names, rather than their mac address. Each line represents the measured signal strength to a point. A value 0 means that the point was not available at this time.

2. Combine all the source files into one.

The result is a common image with the measurements in all the studied areas which is shown in Table 3.

The first column is stored name space in which the measurement was performed. Null values (0) are replaced by 100 (very weak signal).

In addition, a file is created "names.txt," which lists the names of all the networks in the same order in which they are placed in the file of training (step 4).

3. Represented by map matching names with their premises of formal numerical representation:

Cabinet [0] Room [0.5] Kitchen [1]

4. On the basis of the contents of the files of the previous steps 2 and 3, create a file with the data for network training:

60 15 1

As an implementation of a neural network, we used multilayer ANN provided free library FANN (Fast Artificial Neural Network Library). To train the ANN learning algorithm, Resil-

Unlike standard algorithm Backprop, RProp uses only partial signs for adjusting the weighting coefficients. The algorithm uses the so-called "training periods" when the correction occurs

If the partial derivative of the corresponding weight ∂ωi j had changed its sign at the current step, it means that the latest update was great, and the algorithm passed a local minimum and therefore the value of change must be reduced by η and return the previous weight value: in

<sup>∂</sup>ωijðtÞ ¼ <sup>∂</sup>ωijðtÞ � <sup>Δ</sup><sup>ð</sup>t�1<sup>Þ</sup>

If the sign of the partial derivative is not changed, it is necessary to increase the amount of compensation η<sup>+</sup> to achieve a more rapid convergence. Fixing factors η� and η<sup>+</sup> can be dispensed with global settings of the neural network, which can also be seen as an advantage of

Recommended values are η� = 0.5, η<sup>+</sup> = 1.2, but there are no restrictions on the use of other

To prevent too large or small weight values, the correction value limit from above the maximum Δmax and below the minimum Δmin value of the correction value, which default, respec-

The initial values for all Δij are set to 0.1. Again, this should be seen only as a recommendation, and in the practical implementation it can specify a different value for the initialization.

The current implementation of the navigation software package is a simple one-dimensional classifier, which, by new radio-prints, is able to determine the room in which the mobile device

Working with a program complex navigation is implemented using desktop and mobile

∂Εðt�1<sup>Þ</sup> ∂ωij

∂Εðt�1<sup>Þ</sup> ∂ωij

> 0

9 >>>>=

>>>>;

0 < η� < 1 < η<sup>þ</sup> ð2Þ

, ð1Þ

ij : ð3Þ

< 0

after the presentation of the balance of the network of examples from the training set.

ηþΔ<sup>ð</sup>t<sup>Þ</sup> ij , ∂Εðt<sup>Þ</sup> ∂ωij

η�Δ<sup>ð</sup>t<sup>Þ</sup> ij , ∂Εðt<sup>Þ</sup> ∂ωij

To determine the amount of correction using the following rule [15]:

8 >>>><

>>>>:

Δ<sup>ð</sup>t<sup>Þ</sup> ij ¼

ient Propagation (RProp) has been used [15].

206 Modern Fuzzy Control Systems and Its Applications

other words, you must make 'roll back'.

values for these parameters.

tively, shall equal 50 and 1.0E–6.

2.4. Working with software navigation

software, and includes the following steps:

is located.

the algorithm to the standard algorithm Backprop.

86.0 100.0 54.0 69.0 100.0 100.0 88.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0


Table 2. Set of vectors (matrix) measuring signals Wi-Fi points.


Table 3. Common image with the measurements in all the studied areas.

```
0.0
89.0 83.0 56.0 66.0 100.0 100.0 91.0 100.0 100.0 100.0 100.0 100.0
100.0 100.0 100.0
0.0
90.0 71.0 51.0 67.0 100.0 100.0 88.0 100.0 100.0 100.0 100.0 100.0
100.0 100.0 100.0
0.0
```
The first line shall include: the number of hidden layer neurons, the number of neurons in the input layer and output layer.

The following are a couple of lines:


The result of this step is to file "network," which is a trained neural network in the library "FANN." The network name is chosen at random.

The following steps describe the workflow of software application that is running on a mobile device.

	- A. Title.
	- B. The strength of the signal to a predetermined point.

With using the array of file names "names.txt" obtained in the second stage and the current array of visible access points built radio fingerprint, which, in essence, is a simple array of numbers. Create a radio footprint that includes the following steps:

	- a. if the array access point is a point with the current name, the add-in radio signals strength to mark this point of the current iteration of the index;
	- b. otherwise, add the current iteration index of the default value for the missing point that was made in step 2 (100).

#### 2.5. Training a neural network in MATLAB

0.0

0.0

0.0

100.0 100.0 100.0

208 Modern Fuzzy Control Systems and Its Applications

100.0 100.0 100.0

A. The input vector.

a mobile device.

the array of names.

network.

A. Title.

the input layer and output layer.

The following are a couple of lines:

89.0 83.0 56.0 66.0 100.0 100.0 91.0 100.0 100.0 100.0 100.0 100.0

90.0 71.0 51.0 67.0 100.0 100.0 88.0 100.0 100.0 100.0 100.0 100.0

The first line shall include: the number of hidden layer neurons, the number of neurons in

The result of this step is to file "network," which is a trained neural network in the

The following steps describe the workflow of software application that is running on

5. A mobile application launches library "FANN" and passes the file to the trained neural

6. Mobile application starts scanning the surrounding Wi-Fi hotspots. After the completion of the next iteration of the scanning application, an array of the currently available access points is received to which the device may try to connect. Each object of the array, among

With using the array of file names "names.txt" obtained in the second stage and the current array of visible access points built radio fingerprint, which, in essence, is a simple

A. Create an array of numbers (float []) size which determine the number of elements in

a. if the array access point is a point with the current name, the add-in radio signals

b. otherwise, add the current iteration index of the default value for the missing

array of numbers. Create a radio footprint that includes the following steps:

strength to mark this point of the current iteration of the index;

B. The value that the National Assembly should be trained for this vector.

library "FANN." The network name is chosen at random.

other things, contains in its structure the following two fields:

B. The strength of the signal to a predetermined point.

B. Go through the array of names in the cycle:

point that was made in step 2 (100).

The data for the neural network training in MATLAB are taken from the file "train.txt" and used for functional testing of software navigation. To verify the performance of ANN training, the original file should be divided into two sets (training and testing).

By analogy with [16], we make a comparative analysis of the main learning algorithms of the neural network used for the navigation of mobile applications for indoors, with position accuracy and complexity ratios depending on the number of neurons in the hidden layer. The corresponding results are shown in Table 4.

Table 4 uses the following symbols: trainlm—learning algorithm Levenberg-Markarth, trainscg —related gradient method scaled association, and trainbr—Bayesian regularization method. MSE—mean square error, epochs (s)—the number of training cycles, and in brackets the number of seconds.

Based on the analysis of Table 1, we can conclude that the optimal number of hidden layer neurons for a given set of source data is in the range 15–25. It showed the highest accuracy Bayesian regularization algorithm, but it is considerably more time-consuming compared with


Table 4. Characteristics of the complexity and accuracy of learning algorithms NA.

other algorithms. The optimal ratio of accuracy and convergence provides the Levenberg-Marquardt algorithm.

Note that low accuracy is common to all algorithms, which is associated with the original data. Probably, it reduces the dimension of the input vector (data used by Wi-Fi 15 points, most of which are available), and increases the total number of measurements. At the same time in the original data set of vector measurements corresponded to only three different rooms, so get enough accuracy.

## 3. Using fuzzy logic control for selection of the access point

#### 3.1. Introduction

Wireless local area networks of 802.11k standards become very popular due to some reasons. They operate in the unlicensed frequency bounds and they need not require large time and cost for deployment. The appearance of vast number of mobile devices supporting Wi-Fi technology gives possibilities for free choice and cost saving for the various kinds of users. WLAN networks have a number of advantages over traditional wired networks.


However, number of users working in the unlicensed frequency bounds increases day by day. In this context, a very important problem is the way by which the wireless devices are used to select an access point to connect to the network. Now device selects the access point according with signal power [17]. This way allows determining the nearest access point. In other words, currently used 802.11k standard is aimed at the implementation of load balancing of radio Wi-Fi networks. Nevertheless, the high level of the signal does not always mean high network bandwidth. Suppose, for example, that most notebooks operating in the certain conference hall connects to WLAN using access point which is above the entrance door. In such a case, the amount of subscribers connected to it would be to tens if not hundreds, while other access points would not be fully loaded. As a result the network bandwidth per subscriber reduces to a significant degree that leads to the network reduction of productivity of the network in whole. Hence, the task of selection of the access point in WLAN network in the case of high user density is quite topical. Such selection must take into account not only the level of the signal received by the mobile device, but also a width in the dedicated channel bandwidth that depends on the number of connected subscribers to the access point.

Further, examined the method based on analysis not only signal level but also other parameters. To solve this task, fuzzy logic is used (See Ref. [18, 19]). Constructing membership functions and linguistic rules are examined. Structure of the developed model and simulation results are presented.

#### 3.2. Choice of access points for connecting a mobile device

other algorithms. The optimal ratio of accuracy and convergence provides the Levenberg-

Note that low accuracy is common to all algorithms, which is associated with the original data. Probably, it reduces the dimension of the input vector (data used by Wi-Fi 15 points, most of which are available), and increases the total number of measurements. At the same time in the original data set of vector measurements corresponded to only three different rooms, so get

Wireless local area networks of 802.11k standards become very popular due to some reasons. They operate in the unlicensed frequency bounds and they need not require large time and cost for deployment. The appearance of vast number of mobile devices supporting Wi-Fi technology gives possibilities for free choice and cost saving for the various kinds of users.

• Much easier and cheaper to deploy a local wireless network in the new location than with

• The subscriber does not need to be next to his desk or local network socket. He can move

However, number of users working in the unlicensed frequency bounds increases day by day. In this context, a very important problem is the way by which the wireless devices are used to select an access point to connect to the network. Now device selects the access point according with signal power [17]. This way allows determining the nearest access point. In other words, currently used 802.11k standard is aimed at the implementation of load balancing of radio Wi-Fi networks. Nevertheless, the high level of the signal does not always mean high network bandwidth. Suppose, for example, that most notebooks operating in the certain conference hall connects to WLAN using access point which is above the entrance door. In such a case, the amount of subscribers connected to it would be to tens if not hundreds, while other access points would not be fully loaded. As a result the network bandwidth per subscriber reduces to a significant degree that leads to the network reduction of productivity of the network in whole. Hence, the task of selection of the access point in WLAN network in the case of high user density is quite topical. Such selection must take into account not only the level of the signal received by the mobile device, but also a width in the dedicated channel bandwidth that depends on the number of connected subscribers to the

Further, examined the method based on analysis not only signal level but also other parameters. To solve this task, fuzzy logic is used (See Ref. [18, 19]). Constructing membership functions and linguistic rules are examined. Structure of the developed model and simulation

3. Using fuzzy logic control for selection of the access point

WLAN networks have a number of advantages over traditional wired networks.

Marquardt algorithm.

210 Modern Fuzzy Control Systems and Its Applications

enough accuracy.

3.1. Introduction

access point.

results are presented.

a traditional network.

freely inside the area coverage.

Signal level and the bandwidth of a system are linked by the well-known Shannon formula, which allows determining the capacity of the data transmission system:

$$
\mathbf{C} = \Delta \mathbf{F} \times \log\_2(\mathbf{1} + \mathbf{S}/\mathbf{N}).\tag{4}
$$

The total bandwidth allocated in the range, is divided equally among all active subscribers [17]. If the length of the transmitted packet is L bits, one can determine the time required for transmission of a packet:

$$T(M, \mathcal{S}/\mathcal{N}) = \frac{L \times M}{F \times \log\_2(1 + \mathcal{S}/\mathcal{N})} \tag{5}$$

where ΔF = F/M is the network bandwidth, and S/N is signal/noise ratio at the receiver input and M is the number of subscribers already connected. Thus, the functional dependence between bandwidth and the number of connected subscribers is linear. Then the task of selecting the best access point to connect can be formulated as follows:

Find the value of function (5), the Tmax of which does not exceed a predetermined time with the following restrictions: S/N > P<sup>0</sup> and ΔF ≥ ΔFmin, where P<sup>0</sup> is some ratio threshold signal/noise at which the operation of mobile subscriber receiver is possible and ΔFmin is a minimum possible bandwidth width. In other words, the problem is reduced to finding such a pair of values (ΔF, S/N) for which the transmission time has the minimum possible value.

This problem can be solved by various methods:


In the case of choice of any method, the finding of a point of extremum of the considered function is usually required. The type of function (5) is shown in Figure 1.

Figure 1 shows that the function (5) has minimum values for various combinations of parameters and the number of connected subscribers M. Function (5) has no maximum that determines the above-mentioned formulation of the problem of choosing the access point. Table 5 shows the values of the function (5) calculated for various combinations of parameters and M.

Analysis of Table 5 confirms that the high signal level does not always provide an acceptable transmission time. It's known for solve the problem of the selection the access point to connect mobile subscribers on the basis of function (5) using the above methods it is necessary and sufficient that the following conditions are met (as it's shown for instance in Ref. [20, 21]):


Figure 1. Dependence of the packet transmission time from the signal level and the number of subscribers.


Table 5. Values of the time of packet transmission for different ratios of parameters.

Clearly, the condition 1 for the function (5) is performed. To simplify the analysis of the condition (2), it is convenient to consider function 1, taking into account that low values of transmission time are achieved with large bandwidth. Then, for the function (4), condition 2 changes to opposite. Condition 2 for the function (5) is not satisfied due to the fact that <sup>∂</sup><sup>2</sup>C=∂ðΔF<sup>Þ</sup> <sup>2</sup> <sup>¼</sup> 0, which implies that:

$$\begin{array}{c|c|c} \frac{\partial^2 \mathbb{C}}{\left(\Delta F\right)^2} & \frac{\partial^2 \mathbb{C}}{\partial \left(\Delta F\right) \partial \left(\frac{S}{N}\right)} & 0 & \frac{1}{\log\_2 e \left(1 + \frac{S}{N}\right)} \\\\ \frac{\partial^2 \mathbb{C}}{\partial \left(\frac{S}{F}\right) \partial \left(\Delta F\right)} & \frac{\partial^2 \mathbb{C}}{\partial \left(\frac{S}{N}\right)^2} & \frac{1}{\left|\log\_2 e \left(1 + \frac{S}{N}\right)\right|} & \left(\frac{1}{\log\_2 e \left(1 + \frac{S}{N}\right)}\right)^2 \end{array}$$

It follows that the function in question is not concave. However, the requirement of convexity or concavity of the function is a serious restriction that is far from always performed in practical problems. This is why the concept of such functions is generalized by the introduction of pseudo-convex unimodal functions [21]. The f: X ! R function is called the pseudounimodal in the interim ½a, b�⊂X, if ∃ an arbitrary interval I � ⊂½a, b� such that the f function:


The points c and d determined by the following way:

$$c = \inf\_{x \subset I^\*} ; \quad d = \sup\_{x \subset I^\*} . \tag{7}$$

In this case, the interval I\* is the solution of the problem max {fðxÞ : x⊂½a, b�}. In the particular case if c=d, then I � <sup>¼</sup> {x0} and <sup>x</sup> <sup>=</sup> <sup>c</sup> <sup>=</sup> <sup>d</sup>, the <sup>f</sup> function is called unimodal. The example of unimodal function is shown in Figure 2.

Then, if we set

Clearly, the condition 1 for the function (5) is performed. To simplify the analysis of the condition (2), it is convenient to consider function 1, taking into account that low values of transmission time are achieved with large bandwidth. Then, for the function (4), condition 2 changes to opposite. Condition 2 for the function (5) is not satisfied due to the fact that <sup>∂</sup><sup>2</sup>C=∂ðΔF<sup>Þ</sup>

1

S N � �

log2e 1 þ

It follows that the function in question is not concave. However, the requirement of convexity or concavity of the function is a serious restriction that is far from always performed in practical problems. This is why the concept of such functions is generalized by the introduction of pseudo-convex unimodal functions [21]. The f: X ! R function is called the pseudo-

<sup>0</sup> <sup>1</sup>

0

BB@

�

log2e 1 þ

log2e 1 þ

1

S N � �

1

2

� � � � � � � � � � � � � � �

CCA

S N � �

⊂½a, b� such that the f function:

which implies that:

� � � � � � � � � � � � �

∂<sup>2</sup>C ðΔFÞ 2

212 Modern Fuzzy Control Systems and Its Applications

∂<sup>2</sup>C

∂ðΔFÞ

• Strictly increasing in the interval [a, b];

• Strictly decreasing in the interval [d, b].

<sup>∂</sup> <sup>S</sup> F � �

∂<sup>2</sup>C <sup>∂</sup>ðΔFÞ<sup>∂</sup> <sup>S</sup> N � �

Table 5. Values of the time of packet transmission for different ratios of parameters.

� � � � � � � � � � � � �

¼

� � � � � � � � � � � � � � �

Figure 1. Dependence of the packet transmission time from the signal level and the number of subscribers.

30 4E + 6 2.514E � 4 30 8E + 5 1.257E � 3 30 4.444E + 5 2.263E � 3

Signal power, db The bandwidth width, hz The transmission time, s

∂<sup>2</sup>C ∂ðS NÞ 2

unimodal in the interim ½a, b�⊂X, if ∃ an arbitrary interval I

• Equals to some constant, ≤ min{fðcÞ, fðdÞ} on the some interval I\*;

<sup>2</sup> <sup>¼</sup> 0,

≤ 0: ð6Þ

$$a = [
\Delta \, F\_{\rm min} \, P\_0]; \quad b = [
\Delta \, F\_{\rm min} \, P\_{\rm max}].\tag{8}$$

Here Pmax is some maximum possible value of S/N ratio, then unimodality of the function (4) and, consequently, function (5) derived from their definitions. Consequently, the optimization problem has a solution in the following formulation: find the minimum of function (4) in the interval (8). This solution can be found by any of the methods listed above. However, their use is associated with a large number of calculations (solution of the corresponding equations), or with a large amount of stored data, requiring constant modification (various search methods).

In Ref. [18], to solve above-mentioned problem is proposed to use the apparatus of fuzzy logic, free from above disadvantages. Fuzzy logic operators are very similar to conventional Boolean operators and allow simplified algorithms for solving this problem. Complicated mathematical modeling can be replaced by evaluation of membership functions and rules of fuzzy logic [22]. Various approaches to solving this type of problems are considered in Refs. [23, 24]. In Ref. [25] there is an example of controlling the operation of the charger using intelligent controller which applies various algorithms of fuzzy inference. One of the most powerful tools for solving such problems is the MATLAB system, which provides users a various species of software, including the visual ones. With the help of visual programming, the necessary model can be built, and then can run the simulation in program mode.

Figure 2. The example of unimodal function.

#### 3.3. Model of algorithm of determination of the best access point

In accordance with the 802.11 standard, the mobile station scans the individual channels for the detection of the best signal from the access point. APs periodically send a Beacon signal in a broadcast mode. Mobile network station accepts these beacon signals and takes note of the relevant signal level. Thus, the received signal level actually characterizes the relative location of the subscriber and the access point. During this process, a mobile subscriber searches such access point. To determine whether channel is free, the well-known algorithm, which is named clear channel assessment (CCA), is used. Its essence lies in the measurement of the signal power at the antenna and determining the received signal strength (RSSI). If the received signal strength is below a certain threshold, then the channel is declared free, and the MAC layer receives the CTS status. If the power is above the threshold, the data transfer is delayed in accordance with the protocol rules. The standard provides another opportunity to determine the idle channel, which can be used either separately or together with the RSSI measurement method of checking of the carrier. The most appropriate method to use depends on the level of interference in the work area. Various 802.11 standards use one of the five possible CCA modes:


802.11 MAC layer is responsible for the way in which the subscriber is connected to the access point. When a subscriber enters into the coverage area of one or more access points, it selects the access point based on signal power values and observed number of errors, selects one of them and connects thereto. Once the subscriber receives confirmation that it is accepted by the access point, it tunes to a radio channel in which it operates. From time to time, it checks all the channels to see if there is another point which provides access services of higher quality. If such an access point is found, then the subscriber connects to it and readjusts to its frequency.

In accordance with this principle, the diagrams of signal levels and levels of channel diagrams of signal levels and levels of channel congestion can be constructed for each access point. These diagrams look like shown in Figure 3.

Analysis of these diagrams and subject area makes it possible to apply fuzzy logic to make a decision on selecting the best access point to connect. As shown above, a signal power can be characterized through signal/noise ratio, and a congestion level of AP can be described with the help of number of connected subscribers. Then it becomes obvious that proposed model Indoor Mobile Positioning Using Neural Networks and Fuzzy Logic Control http://dx.doi.org/10.5772/68009 215

Figure 3. Diagrams of signal levels and levels of channel congestion.

3.3. Model of algorithm of determination of the best access point

modes:

a certain threshold value;

214 Modern Fuzzy Control Systems and Its Applications

to the 802.11 standard;

during 3.5 ms timeslot;

diagrams look like shown in Figure 3.

In accordance with the 802.11 standard, the mobile station scans the individual channels for the detection of the best signal from the access point. APs periodically send a Beacon signal in a broadcast mode. Mobile network station accepts these beacon signals and takes note of the relevant signal level. Thus, the received signal level actually characterizes the relative location of the subscriber and the access point. During this process, a mobile subscriber searches such access point. To determine whether channel is free, the well-known algorithm, which is named clear channel assessment (CCA), is used. Its essence lies in the measurement of the signal power at the antenna and determining the received signal strength (RSSI). If the received signal strength is below a certain threshold, then the channel is declared free, and the MAC layer receives the CTS status. If the power is above the threshold, the data transfer is delayed in accordance with the protocol rules. The standard provides another opportunity to determine the idle channel, which can be used either separately or together with the RSSI measurement method of checking of the carrier. The most appropriate method to use depends on the level of interference in the work area. Various 802.11 standards use one of the five possible CCA

• solution that channel is free and is based on the detection of channel power which exceeds

• the decision that the channel is free based on the detection of a carrier signal corresponding

• carrier signal detection with message that the ether is free, if neither signal is detected

• detection of the power corresponding to the increased transmission rate in a physical layer

802.11 MAC layer is responsible for the way in which the subscriber is connected to the access point. When a subscriber enters into the coverage area of one or more access points, it selects the access point based on signal power values and observed number of errors, selects one of them and connects thereto. Once the subscriber receives confirmation that it is accepted by the access point, it tunes to a radio channel in which it operates. From time to time, it checks all the channels to see if there is another point which provides access services of higher quality. If such an access point is found, then the subscriber connects to it and readjusts to its frequency.

In accordance with this principle, the diagrams of signal levels and levels of channel diagrams of signal levels and levels of channel congestion can be constructed for each access point. These

Analysis of these diagrams and subject area makes it possible to apply fuzzy logic to make a decision on selecting the best access point to connect. As shown above, a signal power can be characterized through signal/noise ratio, and a congestion level of AP can be described with the help of number of connected subscribers. Then it becomes obvious that proposed model

• carrier signal detection and discovering of power (combination of modes 1 and 2);

and carrier detection at mode 3, but with reference to the ERP.

must have two input linguistic variables (S/N ratio and the degree of loading of the access point). As the set of terms of the first linguistic variable "signal/noise" the following set is used:

$$\mathcal{S}/\mathcal{N} = \{\text{"weak', "medium', "high'}\}.\tag{9}$$

As the set of terms of the second linguistic variable "degree of loading" the following set is used:

$$\text{NUUMUSERS} = \{\text{'few'}, \text{'medium'}, \text{'many'}\}. \tag{10}$$

As the set of terms of the output linguistic variable the following set is used:

RESULT ¼ f'is a candidate for connection'; 'is not a candidate for connection'g ð11Þ

Since the algorithm Mamdami is used for fuzzy inference, the following methods of execution had been chosen for the stages of composition:


In the process of fuzzy inference, it is necessary to select such an access point for which the convolution of the functions of the accessory of the signal/noise ratio and the degree of congestion of the access point give the best result. To implement the fuzzy inference, the creation of the rule base of fuzzy inference system is required. To implement the fuzzy inference, the creation of the rule base of fuzzy inference system is required. Such base of rules for the solution of discussed task looks like:


The following notations are in use here to simplify the formalization of fuzzy productions:


A1 means the S/N ratio is weak, A2 means the S/N ratio is medium, A3 means the S/N ratio is high, B1 means the number of users is few, B2 means the number of users is medium, B3 means the number of users is many, E1 means that the access point is not a candidate for connection, E2 means that the access point is a candidate for connection. Thus, the connection to the access point occurs only when the access point is not overloaded and has an acceptable signal level. The numbers in round brackets mean the weight of rule (0 or 1). The next step in the construction of the model is to determine the membership functions of input and output variables. Kind of membership functions for the signal/noise ratio, the number of subscribers, and the output variable are shown in Figure 4.

For analyzing the adequacy of the developed fuzzy model, the surface fuzzy inference can be quite useful. It allows assessing the impact of changes in the values of the input fuzzy variables on the values of the output fuzzy variables. This surface is shown in Figure 5. A comparison of this figure with Figure 1 allows concluding that the nature of the surface of the fuzzy inference coincides in general with the frame transmission time dependent on the signal level and the number of subscribers. This confirms the adequacy of the proposed model.

Figure 4. Membership functions for input and output variables.

Figure 5. The surface fuzzy inference.

• If the S/N ratio is weak and access point is lightly loaded (the number of users M is few),

• If the S/N ratio is weak and access point is moderately loaded (the number of users M is medium), then the access point is not considered as a candidate for connection.

• If the S/N ratio is weak and access point is heavily loaded (the number of users M is

• If the S/N ratio is medium and access point is lightly loaded (the number of users M is

• If the S/N ratio is medium and access point is moderately loaded (the number of users M is medium), then the access point is not considered as a candidate for connection.

• If the S/N ratio is medium and access point is heavily loaded (the number of users M is

• If the S/N ratio is high and access point is lightly loaded (the number of users M is few),

• If the S/N ratio is high and access point is moderately loaded (the number of users M is

• If the S/N ratio is high and access point is heavily loaded (the number of users M is many),

A1 means the S/N ratio is weak, A2 means the S/N ratio is medium, A3 means the S/N ratio is high, B1 means the number of users is few, B2 means the number of users is medium, B3 means the number of users is many, E1 means that the access point is not a candidate for connection, E2 means that the access point is a candidate for connection. Thus, the connection to the access point occurs only when the access point is not overloaded and has an acceptable signal level. The numbers in round brackets mean the weight of rule (0 or 1). The next step in the construction of

The following notations are in use here to simplify the formalization of fuzzy productions:

many), then the access point is not considered as a candidate for connection.

many), then the access point is not considered as a candidate for connection.

medium), then the access point is considered as a candidate for connection.

then the access point is not considered as a candidate for connection.

1. if (S/N is A1) and (Numusers is B1) then (Result is E1) (1); 2. if (S/N is A1) and (Numusers is B2) then (Result is E1) (1); 3. if (S/N is A1) and (Numusers is B3) then (Result is E1) (1); 4. if (S/N is A2) and (Numusers is B1) then (Result is E2) (1); 5. if (S/N is A2) and (Numusers is B2) then (Result is E1) (1); 6. if (S/N is A2) and (Numusers is B3) then (Result is E1) (1); 7. if (S/N is A3) and (Numusers is B1) then (Result is E2) (1); 8. if (S/N is A3) and (Numusers is B2) then (Result is E2) (1); 9. if (S/N is A3) and (Numusers is B3) then (Result is E1) (1).

then the access point is considered as a candidate for connection.

few), then the access point is considered as a candidate for connection.

then the access point is not considered as a candidate for connection.

216 Modern Fuzzy Control Systems and Its Applications

#### 3.4. Simulation of the algorithm for determining the best access point

For checking the correctness and efficiency of the proposed algorithm, its styling is first carried out. MATLAB provides the ability to download a model developed with the help of visual aids into the MATLAB—program and perform simulation in program mode. The data structure of a software model is shown in Figure 6.

Each access point is represented by the data structure containing the following data fields:


At initialization of this model, these structures combine into a vector. Thus, the sequential number of structure in the vector imitates the Beacon signal. All other fields of structures are set to zero. Also, the vector of signal levels is forming. This vector simulates a signal power at which every access point radiates, and thereby, the location of subscriber is relative to access points in the moment of its connection. The signal levels are forming according with normal distribution and exponential distribution. This adequately reflects the real situation when connecting subscribers grouped in a small area, for example, when entering a room or in some corner of the room, as it is shown in Figure 7.

Simulation process starts with the loading of model into the program. Further, the current values of signal level out of the vector of signal levels are moving into the fields of current values of the signal level in every access point. After this, the values of membership functions are calculated for every access point. Further, the values of membership functions are calculated for every access point and their maximum value is looked for. The access point for which this value is maximal, the value of the field current number of connected subscribers to this point is increased by 1. This simulates the RSSI level correction for the next simulation step. Furthermore, the current time of the frame transmission for this point is calculated according with function (5). This process is repeated for all the elements of the vector of

Figure 6. The data structure of model.

Figure 7. The example of the subscriber's location indoors.

3.4. Simulation of the algorithm for determining the best access point

• Index of the access point (its number in a vector of access points).

• The current number of connected subscribers at this point. • The current time of the frame transmission for this point.

• The current value of the signal level at that point.

a software model is shown in Figure 6.

218 Modern Fuzzy Control Systems and Its Applications

Figure 6. The data structure of model.

For checking the correctness and efficiency of the proposed algorithm, its styling is first carried out. MATLAB provides the ability to download a model developed with the help of visual aids into the MATLAB—program and perform simulation in program mode. The data structure of

Each access point is represented by the data structure containing the following data fields:

At initialization of this model, these structures combine into a vector. Thus, the sequential number of structure in the vector imitates the Beacon signal. All other fields of structures are set to zero. Also, the vector of signal levels is forming. This vector simulates a signal power at which every access point radiates, and thereby, the location of subscriber is relative to access points in the moment of its connection. The signal levels are forming according with normal distribution and exponential distribution. This adequately reflects the real situation when connecting subscribers grouped in a small area, for example, when entering a room or in some corner of the room, as it is shown in Figure 7. Simulation process starts with the loading of model into the program. Further, the current values of signal level out of the vector of signal levels are moving into the fields of current values of the signal level in every access point. After this, the values of membership functions are calculated for every access point. Further, the values of membership functions are calculated for every access point and their maximum value is looked for. The access point for which this value is maximal, the value of the field current number of connected subscribers to this point is increased by 1. This simulates the RSSI level correction for the next simulation step. Furthermore, the current time of the frame transmission for this point is calculated according with function (5). This process is repeated for all the elements of the vector of

signal levels. Thus, after the completion of simulation process, total loading and the frame transmission time in every point would be determined.

The modeling process is based on a method that was considered in Ref. [26]. In this case the sufficient characteristics are current values of congestion of access point and frame transmission time on every simulation step; the final characteristics are the distributions of load of access points, transmission time, mean value, and dispersion of the transmission time. For the confirmation of efficiency of the proposed algorithm, its result was compared with the result of simulation of classic, which is based on the measuring of maximal signal power. The results of the comparison of the analyzed algorithms are shown in Figure 8. In this picture, "Sample Model" means the model is based on the measuring of maximal signal power. This figure shows that the algorithm that uses fuzzy logic (the right-hand side of Figure 8), taking into account the current load on each access point, ensures a more even loading of the entire LAN as a whole. The frame transmission time for the second algorithm is approximately same for all access points (the dispersion is equal to 0.0799). For the first algorithm, the transmission time

Figure 8. The results of the comparison of the analyzed algorithms.

fluctuates strongly (the dispersion is equal to 0.3129). Hence, it follows that accounting the degree of congestion of access points increases the efficiency of the wireless LAN.

Developed in the framework of this chapter, MATLAB—the program cannot be used directly in some hardware for the following reasons:


However, there are various possibilities to convert MatLab-program into equivalent program on one of the high-level programming languages. In particular, there is the possibility of constructing an equivalent program on high level language such as C# or C++. Such program can be installed on the controller or other device that will automatically determine the best access point to connect to the local network.

## 4. Conclusions

We have reviewed the advanced solutions for indoor navigation and briefly reviewed the program complex to navigate through radio fingerprints. Navigation software package uses the model ANN. In addition, by using the MATLAB performed rationale for the selection of ANN learning algorithm and determine the recommended number of neurons of the hidden layer for the considered number of Wi-Fi networks.

As a recommendation, to the use of this software system, one should take into account the importance of using the mac-address of an access point instead of its name because it is not unique. Emerged collision may disrupt the entire complex.

The general advantages of this approach include the fact that the software package can run on the existing infrastructure of Wi-Fi networks that are deployed in a variety of areas, such as residential buildings and shopping malls.

The disadvantages are a feature of this approach lie in the fact that before using it you must manually map the radio fingerprint.

Further studies, in our opinion, are advisable to continue in the following areas.


The task of selection of the access point in WLAN network in the case of high user density is quite topical. Such selection must take into account not only the level of the signal received by the mobile device, but also a width in the dedicated channel bandwidth that depends on the number of connected subscribers to the access point.


## Author details

fluctuates strongly (the dispersion is equal to 0.3129). Hence, it follows that accounting the

Developed in the framework of this chapter, MATLAB—the program cannot be used directly

• MATLAB—program runs in interpreted mode and cannot be loaded directly into the

However, there are various possibilities to convert MatLab-program into equivalent program on one of the high-level programming languages. In particular, there is the possibility of constructing an equivalent program on high level language such as C# or C++. Such program can be installed on the controller or other device that will automatically determine the best

We have reviewed the advanced solutions for indoor navigation and briefly reviewed the program complex to navigate through radio fingerprints. Navigation software package uses the model ANN. In addition, by using the MATLAB performed rationale for the selection of ANN learning algorithm and determine the recommended number of neurons of the hidden

As a recommendation, to the use of this software system, one should take into account the importance of using the mac-address of an access point instead of its name because it is not

The general advantages of this approach include the fact that the software package can run on the existing infrastructure of Wi-Fi networks that are deployed in a variety of areas, such as

The disadvantages are a feature of this approach lie in the fact that before using it you must

• The implementation multivariate classifier for converting radio prints multidimensional

• Conducting research on the influence of other learning algorithms ANN (Quickprop,

• Development of a method for reducing the dimensionality of the input vector radio finger-

• The addition of processing signals from Bluetooth LE beacons to signals from Wi-Fi access points.

• Improving the quality of navigation when combined with inertial navigation system.

Further studies, in our opinion, are advisable to continue in the following areas.

Batch, and Incremental) the speed and accuracy of navigation training.

degree of congestion of access points increases the efficiency of the wireless LAN.

in some hardware for the following reasons:

access point to connect to the local network.

layer for the considered number of Wi-Fi networks.

residential buildings and shopping malls.

manually map the radio fingerprint.

coordinates (x, y, height).

print.

unique. Emerged collision may disrupt the entire complex.

• Developed program is a model.

220 Modern Fuzzy Control Systems and Its Applications

controller.

4. Conclusions

Anatoly D. Khomonenko, Sergey E. Adadurov, Alexandr V. Krasnovidow\*, Pavel A. Novikov

\*Address all correspondence to: alexkrasnovidow@mail.ru

Petersburg State Transport University, St. Petersburg, Russia

## References


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[4] Butakov NA. Applicability inertial navigation in mobile devices. International Journal of

[5] Evennou F, Marx F. Advanced integration of wi-fi and inertial navigation systems for indoor mobile positioning. EURASIP Journal on Applied Signal Processing. 2006;2006:1-

[6] Wang S, Min J, Yi BK. Location based services for mobiles: Technologies and standards.

[7] Dworkina NB, Namiot DE, Dvorkin BA. Mobile navigation services and the use of technology to determine the location OpenCellID. Geomatics. 2010:2;80-87. http://

[8] Abdrakhmanova AM, Namiot DE. Using two-dimensional bar code to create a system of

[9] Namiot DE, Shneps-Shneppe MA. Analysis of the trajectories in mobile networks on the basis of information about network proximity. Automation and Computer Science.

[10] Mok E, Cheung Bernard KS. An improved neural network training algorithm for wi-fi fingerprinting positioning. ISPRS International Journal of Geo-Information. 2013;2:854-

[11] Mehmood H, Tripathi NK, Tipdecho T. Indoor positioning system using artificial neural

[12] Fang SH, Lin TN. Indoor location system based on discriminant-adaptive neural network in IEEE 802.11 environments. IEEE Transactions on Neural Networks. 2008;19(11):1973-

[13] Novikov PA, Khomonenko AD, Yakovlev EL. Justification of the choice of neural networks learning algorithms for indoor mobile positioning. Proceeding CEE-SECR '15 Proceedings of the 11th Central & Eastern European Software Engineering Conference in Russia. Moscow, Russian Federation; 22-24 October 2015; ACM New York, NY, USA©;

[14] Novikov PA, Khomonenko AD, Yakovlev EL. Software for mobile indoor navigation using neural networks. Informatsionno-Upravliaiushchie Sistemy, [Information and Con-

[15] Akobir S. The learning algorithm RProp – mathematical apparatus. http://basegroup.ru/

[16] Khomonenko AD, Yakovlev EL. Neural network approximation of characteristics of multi-channel non-Markovian queuing systems. SPIIRAS Proceedings. 2015;4(41):81-93.

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(In Russian). DOI: http://dx.doi.org/10.15622/sp.41.4.

community/articles/rprop.

Open Information Technologies. 2014;2(5):24-32. ISSN: 2307-8162.

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222 Modern Fuzzy Control Systems and Its Applications

2013;3:S48-S60

868

1978


## **A Fuzzy Logic Approach for Separation Assurance and Collision Avoidance for Unmanned Aerial Systems**

Brandon Cook, Tim Arnett and Kelly Cohen

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/68126

#### Abstract

In the coming years, operations in low altitude airspace will vastly increase as the capabilities and applications of small unmanned aerial systems (sUAS) continue to multiply. Therefore, finding solutions to managing sUAS in highly congested airspace will facilitate sUAS operations. In this study, a fuzzy logic-based approach was used to help mitigate the risk of collisions between aircraft using separation assurance and collision avoidance techniques. The system was evaluated for its effectiveness at mitigating the risk of mid-air collisions between aircraft. This system utilizes only current state information and can resolve potential conflicts without knowledge of intruder intent. The avoidance logic was verified using formal methods and shown to select the correct action in all instances. Additionally, the fuzzy logic controllers were shown to always turn the vehicles in the correct direction. Numerical testing demonstrated that the avoidance system was able to prevent a mid-air collision between two sUAS in all tested cases. Simulations were also performed in a three-dimensional environment with a heterogeneous fleet of sUAS performing a variety of realistic missions. Simulations showed that the system was 99.98% effective at preventing mid-air collisions when separation assurance was disabled (unmitigated case) and 100% effective when enabled (mitigated case).

Keywords: fuzzylogic, UAS, collision avoidance, separation assurance, formal methods, satisfiability modulo theories

## 1. Introduction

In recent times, there have been substantial advances in the capability of mobile robots in several aerospace applications. These advances include autonomous intelligence, surveillance, and reconnaissance (ISR) efforts [1], aerial firefighting [2], and aerial delivery services [3]. However, despite the potential benefits, these advancements are currently being under-utilized due to

© 2017 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.

several unresolved safety issues with integrating these platforms into the National Airspace System (NAS). As a result of these shortcomings, there is a need to develop algorithms that allow a heterogeneous team of small unmanned aerial systems (sUAS) to interact autonomously and perform time-critical tasks in complex environments. As the applications and capabilities of sUAS continue to proliferate, it is imperative to address the safe integration of these vehicles into the NAS.

Most of the work in this area deals with separation assurance, as it typically takes priority in NAS conflict resolution scenarios [4, 5]. However, most methods necessitate the communication of state information between the vehicles in order to properly select resolution actions [6]. For collision avoidance, several intelligent systems have been developed with promising results [7–12], but few have also shown behavioral verification using formal methods [13, 14].

To facilitate real-time control of a large number of sUAS, a fuzzy logic approach was implemented. This approach was utilized to mitigate the risk of losses of separation and ultimately collisions, between the sUAS. In order to generate scenarios to test the sUAS's ability to avoid collisions, a realistic simulation environment was created. This simulation environment was developed in a modular fashion, such that various algorithms could be implemented to coordinate sUAS maneuvers. This enables various vehicle platforms, sensor models, software packages, and traffic management methodologies to be tested and evaluated.

The main goal of this research is to develop a high-level concept of operations for a UAS traffic management (UTM) system. This system must address the challenges of collision avoidance and separation assurance. Each of these platforms will utilize fuzzy logic controllers (FLCs) to enable real-time decision-making and dynamic control. Additionally, the confidence in correct decision-making and avoidance control outputs needs to be extremely high. Therefore, formal methods were employed for behavioral verification.

The remainder of the paper is as follows. In Section 2, background material on some of the methods and tools used in this work is described. Section 3 details the proposed solution, which includes the separation assurance and collision avoidance methods. This includes detailed development procedures for the decision-making and fuzzy avoidance controllers. Section 4 presents the methodology for implementing and evaluating the decision-making and fuzzy avoidance controllers using formal methods. Section 5 then explains the test cases, their implementations, and an overview of the simulation environment and constraints. Section 6 presents results from the formal methods evaluations and simulation runs, and finally, Section 7 discusses conclusions and opportunities for future work.

## 2. Background

#### 2.1. Hybrid fuzzy systems

Most fuzzy inference systems (FISs) involve multiple operations that associate inputs to outputs based on multiple if-then rules that are resolved to a singleton. The output space is typically nonlinear and difficult to describe as a function of the input variables. However, by constraining the FIS to have particular properties and association methods, an explicit expression can be more easily found.

Hybrid systems are systems that have regions of continuous behavior separated by discrete transitions [15]. This is analogous to a subset of FISs that contain membership functions that are constrained to a finite domain. Such FISs can be represented as hybrid systems after an explicit expression is found. This expression maps an input set to an output set using a set of mathematical functions. This is useful due to the number of low-level tools that have been developed for analyzing hybrid systems. Among these are formal methods tools [16], which are described in the following section.

### 2.2. Formal methods

several unresolved safety issues with integrating these platforms into the National Airspace System (NAS). As a result of these shortcomings, there is a need to develop algorithms that allow a heterogeneous team of small unmanned aerial systems (sUAS) to interact autonomously and perform time-critical tasks in complex environments. As the applications and capabilities of sUAS continue to proliferate, it is imperative to address the safe integration of these vehicles into

Most of the work in this area deals with separation assurance, as it typically takes priority in NAS conflict resolution scenarios [4, 5]. However, most methods necessitate the communication of state information between the vehicles in order to properly select resolution actions [6]. For collision avoidance, several intelligent systems have been developed with promising results [7–12], but few have also shown behavioral verification using formal methods [13, 14]. To facilitate real-time control of a large number of sUAS, a fuzzy logic approach was implemented. This approach was utilized to mitigate the risk of losses of separation and ultimately collisions, between the sUAS. In order to generate scenarios to test the sUAS's ability to avoid collisions, a realistic simulation environment was created. This simulation environment was developed in a modular fashion, such that various algorithms could be implemented to coordinate sUAS maneuvers. This enables various vehicle platforms, sensor models, software packages, and traffic manage-

The main goal of this research is to develop a high-level concept of operations for a UAS traffic management (UTM) system. This system must address the challenges of collision avoidance and separation assurance. Each of these platforms will utilize fuzzy logic controllers (FLCs) to enable real-time decision-making and dynamic control. Additionally, the confidence in correct decision-making and avoidance control outputs needs to be extremely high. Therefore, formal

The remainder of the paper is as follows. In Section 2, background material on some of the methods and tools used in this work is described. Section 3 details the proposed solution, which includes the separation assurance and collision avoidance methods. This includes detailed development procedures for the decision-making and fuzzy avoidance controllers. Section 4 presents the methodology for implementing and evaluating the decision-making and fuzzy avoidance controllers using formal methods. Section 5 then explains the test cases, their implementations, and an overview of the simulation environment and constraints. Section 6 presents results from the formal methods evaluations and simulation runs, and finally, Sec-

Most fuzzy inference systems (FISs) involve multiple operations that associate inputs to outputs based on multiple if-then rules that are resolved to a singleton. The output space is typically nonlinear and difficult to describe as a function of the input variables. However, by

the NAS.

226 Modern Fuzzy Control Systems and Its Applications

ment methodologies to be tested and evaluated.

methods were employed for behavioral verification.

tion 7 discusses conclusions and opportunities for future work.

2. Background

2.1. Hybrid fuzzy systems

In systems such as UTM collision avoidance algorithms, the level of confidence that they will always behave as intended needs to be extremely high. Typical methods for evaluating these algorithms usually involve simulation, but simulation and other numerical methods can miss critical cases that result in undesired behavior. To increase the confidence that the avoidance algorithms presented work as intended, formal methods were employed. Formal methods are defined by NASA as "mathematically rigorous techniques and tools for the specification, design, and verification of software and hardware systems." [17] There are numerous types of tools that fall under this definition, but in this work, satisfiability modulo theories (SMT) solvers and model checkers were used.

SMT solvers are tools that extend the Boolean Satisfiability (SAT) problem to first order logic (FOL) sentences and incorporate other theories for evaluating the truth assignments of variables (real values, bitvectors, etc.). If a behavior can be described in FOL, it can be encoded and evaluated by an SMT solver to find truth assignments that violate this behavior. If a behavior is found, it is returned as satisfying the behavioral specification. If there are no possible assignments to the variables that render the specification true, it is then said to be unsatisfiable. Therefore, "safety" properties, properties which should always hold true, can be evaluated by negating a specification that encapsulates the respective behavior. If a satisfying case is found for the negated specification, this means that there are conditions that violate the original specification. If no satisfying cases are found, then the original specification will hold for all possible conditions.

Model checkers are tools that exhaustively check the states of a system to search for combinations of variable assignments that violate behavioral specifications. In finite state systems, they use deductive proofs, and in infinite state systems, they can use inductive methods. These tools can also use SAT or SMT solvers in conjunction with their own search methods for finding counterexample cases. Encoding safety properties in model checkers is slightly different, however, as model checkers typically use some type of temporal operator in conjunction with logical sentences. However, there are methods for relating quantified FOL sentences for safety properties to temporal representations for use in model checkers [18].

In this work, an infinite-state model checker named JKind [19, 20] was used. JKind is a Java implementation of the Kind model checker which uses k-induction. To evaluate the truth assignments for variables within each state, JKind employs SMT solvers. The SMT solver used for this work is Z3 [21], a state-of-the-art SMT solver developed by Microsoft.

## 3. Proposed solution

To ensure that two or more sUAS do not collide with one another, an intruder avoidance system was developed. This avoidance problem is broken down into two sub-systems: strategic (separation assurance) and tactical (collision avoidance). The strategic separation assurance platform uses a centralized approach to coordinate trajectory modifications for sUAS to ensure that vehicles do not get too close to one another. This separation assurance technique is employed when two or more vehicles come within 0.4 nmi laterally of one another (separation alert threshold), 100 ft. vertically, and are predicted to have a loss of separation (LOS), defined by when vehicles come within 0.1 nmi laterally and 100 ft. vertically of one another. This LOS threshold was determined based on the characteristics of the vehicle platforms and feasible sensing abilities of sUAS. Based on the system constraints, the avoidance platform would have roughly 2 sec to resolve maximum closure rate encounters. If the separation assurance system fails to prevent an LOS, the vehicles will employ their onboard sense and avoid systems to prevent a mid-air collision. A mid-air collision occurs when two vehicles come within 60 m of one another. This collision threshold is intentionally conservative to introduce a notion of spatial uncertainty. Since the sensor models provide perfect state information, as described in Section 5, all vehicle locations are precisely known. For this study, the collision avoidance platform uses a de-centralized approach, that is, all vehicles attempt to avoid intruding vehicles independently. Thus, no communication between aircraft is available (i.e., uncoordinated maneuvers). In Table 1, the various distance thresholds used to describe the separation boundaries are shown.

Prior to presenting the details of each avoidance sub-system, Section 3.1 provides an overview of the avoidance system architecture. This overarching logic is used to determine when a vehicle should perform an avoidance maneuver. When deemed necessary, the system will activate the appropriate avoidance platform. In each avoidance platform, a set of heuristics are used to determine the appropriate action to resolve a conflict. These details are presented in Sections 3.2 and 3.3. Once the appropriate action has been decided, an FLC is used to control the vehicle's turn rate in the desired direction. The details of each FLC are shown in Section 3.4.

Finally, it is important to note that these two sub-systems use different approaches when trying to resolve conflicts between aircraft. This is primarily due to the overall purpose each


Table 1. Separation threshold values.

sub-system serves, and the information that is available to each. If vehicles are reporting their state information to a ground-based station, a centralized separation assurance platform could be used to coordinate a trajectory modification to one or more of the vehicles. Thus, coordinated maneuvers are possible. However, when two vehicles are within seconds away from a collision, minimizing the time between sensing the vehicle and performing an action is critical. Therefore, when the collision avoidance system is activated, the vehicles must independently choose the appropriate action using onboard processors. In these collision avoidance scenarios, there is no communication between aircraft. Thus, each sub-system requires a different set of rules to determine the appropriate action.

#### 3.1. Overarching control logic

assignments for variables within each state, JKind employs SMT solvers. The SMT solver used

To ensure that two or more sUAS do not collide with one another, an intruder avoidance system was developed. This avoidance problem is broken down into two sub-systems: strategic (separation assurance) and tactical (collision avoidance). The strategic separation assurance platform uses a centralized approach to coordinate trajectory modifications for sUAS to ensure that vehicles do not get too close to one another. This separation assurance technique is employed when two or more vehicles come within 0.4 nmi laterally of one another (separation alert threshold), 100 ft. vertically, and are predicted to have a loss of separation (LOS), defined by when vehicles come within 0.1 nmi laterally and 100 ft. vertically of one another. This LOS threshold was determined based on the characteristics of the vehicle platforms and feasible sensing abilities of sUAS. Based on the system constraints, the avoidance platform would have roughly 2 sec to resolve maximum closure rate encounters. If the separation assurance system fails to prevent an LOS, the vehicles will employ their onboard sense and avoid systems to prevent a mid-air collision. A mid-air collision occurs when two vehicles come within 60 m of one another. This collision threshold is intentionally conservative to introduce a notion of spatial uncertainty. Since the sensor models provide perfect state information, as described in Section 5, all vehicle locations are precisely known. For this study, the collision avoidance platform uses a de-centralized approach, that is, all vehicles attempt to avoid intruding vehicles independently. Thus, no communication between aircraft is available (i.e., uncoordinated maneuvers). In Table 1, the various distance thresholds used to describe the separation bound-

Prior to presenting the details of each avoidance sub-system, Section 3.1 provides an overview of the avoidance system architecture. This overarching logic is used to determine when a vehicle should perform an avoidance maneuver. When deemed necessary, the system will activate the appropriate avoidance platform. In each avoidance platform, a set of heuristics are used to determine the appropriate action to resolve a conflict. These details are presented in Sections 3.2 and 3.3. Once the appropriate action has been decided, an FLC is used to control the vehicle's turn rate in the desired direction. The details of each FLC are shown in Section 3.4. Finally, it is important to note that these two sub-systems use different approaches when trying to resolve conflicts between aircraft. This is primarily due to the overall purpose each

Threshold label Lateral distance Vertical distance

Separation alert 0.4 nmi 100 ft. LOS 0.1 nmi 100 ft. Collision 60 m 50 ft.

for this work is Z3 [21], a state-of-the-art SMT solver developed by Microsoft.

3. Proposed solution

228 Modern Fuzzy Control Systems and Its Applications

aries are shown.

Table 1. Separation threshold values.

The overarching control logic determines whether to perform a separation assurance maneuver, activate the collision avoidance system, or allow vehicles to continue along their desired trajectories. This logic is shown in flow chart form in Figure 1. First, the system will find the distance separating all aircraft pairs. With this information, a calculation is made to see how much time can pass prior to two vehicles coming within 0.4 nmi. To calculate this value, the current separation, minus the 0.4 nmi threshold, is divided by the maximum closure rate of the aircraft pair. Therefore, if both vehicles moved directly toward one another at their maximum allowable speeds, this is the time it would take them to reach the 0.4 nmi separation threshold. This future time is known as the "time threshold", as shown in Figure 1. Using this time threshold, if two vehicles cannot possibly be within 0.4 nmi of one another, the system will not unnecessarily check if the two aircraft are in conflict. Rather, it remains idle between checks to improve the performance of the system.

Once enough time has passed and an aircraft pair reaches their assigned time threshold, the system will again check their separation. If the two aircraft are still more than 0.4 nmi apart, a new time threshold is calculated and set. However, if the aircraft pair has reached the 0.4 nmi threshold, it will next check to see if an LOS has occurred. If the vehicles have violated the 0.1 nmi LOS threshold, the collision avoidance system is enabled. Otherwise, the separation assurance system may be needed to ensure that two vehicles do not have an LOS. This decision is based on two criteria: if the predicted closest point of approach (pCPA) creates an LOS and if the time to LOS (tLOS) is within 2 min.

If both criteria are met, a final check is used to see if the aircraft pair has already been assigned a separation assurance maneuver. If neither vehicle has been assigned a maneuver to avoid an impending LOS, a resolution advisory is sent from the centralized system to one of the sUAS. However, if the sUAS was already assigned a maneuver and is currently in the middle of its resolution, a check is used to see if turning back toward its preferred heading will cause another predicted LOS. If resuming its originally intended mission will not cause an LOS, it will do so, otherwise, the sUAS will continue on its current bearing.

If neither criterion (pCPA and tLOS) is met, then no separation assurance command is given and the aircraft will continue toward its respective target using its navigation controller. After determining the appropriate separation assurance action, or deciding that no action is needed, the algorithm calculates another time threshold for each aircraft pair.

Figure 1. Control logic flow chart.

#### 3.2. Separation assurance logic

The centralized separation assurance system observes only the current position, heading, and velocity of each vehicle. With this limited information, vehicle intent is unknown. Therefore, the system must be robust to dynamic scenarios and resolve conflicts without the knowledge of other vehicles' goals.

The separation assurance platform will be enabled if three criteria are met: separation less than 0.4 nmi, the pCPA is less than 0.1 nmi, and the tLOS is within 2 min. If two aircraft meet all three of these criteria, the separation assurance platform will activate and assign one of the vehicles a new trajectory in an effort to avoid the predicted LOS.

To determine what action the separation assurance platform should use to avoid a potential LOS, a series of conflict classification techniques are used. For this study, three pieces of information are used to classify all conflict scenarios: relative heading, relative angle, and crossing time. These parameters can be described using the more common parameters: location, speed, and heading. In Figure 2, a sample conflict scenario is shown. Here, the triangular objects each represent an sUAS, the arrows represent the velocities of each aircraft (both magnitude and direction), and the "x" represents the heading intersection point location. The heading intersection point is not to be confused with the pCPA. It is simply the point where the projected headings intersect with one another. For this example, the vehicle with the small circle represents aircraft 1, and the other represents aircraft 2.

The relationship used to describe the heading of vehicle 2 relative to vehicle 1's perspective is shown in Eq. (1).

$$R\_{H\_1} = H\_2 - H\_1 \tag{1}$$

where H<sup>1</sup> is the heading of vehicle 1, H<sup>2</sup> is the heading of vehicle 2, and RH<sup>1</sup> is the relative heading from vehicle 1's perspective. In this study, 0� ≤ Rhi < 360� for all vehicles, where i represents the index for each vehicle. Therefore, if H<sup>1</sup> > H2, a 360� phase shift must be added to RH<sup>1</sup> to remain within the constrained range. Computing the relative heading for the example shown in Figure 2, RH<sup>1</sup> would be 140� (i.e., moving to the left with respect to vehicle 1), whereas, from vehicle 2's perspective, RH<sup>2</sup> would be roughly 220� after applying the 360� phase shift (i.e., moving to the right with respect to vehicle 2).

Similarly, the relative angle between two vehicles describes the relative position of one vehicle with respect to the other. This relationship has been shown in Eq. (2).

Figure 2. Conflict scenario classification information.

Figure 1. Control logic flow chart.

230 Modern Fuzzy Control Systems and Its Applications

$$R\_{A\_1} = \tan^{-1} \left( \frac{y\_2 - y\_1}{x\_2 - x\_1} \right) - H\_1 \tag{2}$$

where x<sup>2</sup> and y<sup>2</sup> are the two-dimensional coordinates of vehicle 2 in the global frame, x<sup>1</sup> and y<sup>1</sup> are the two-dimensional coordinates of vehicle 1 in the global frame, H<sup>1</sup> is the heading of vehicle 1, and RA<sup>1</sup> is the relative angle of vehicle 2 from vehicle 1's perspective. Like the relative heading, the relative angle is constrained. Here, we constrain the relative angle by the following relationship: �180� ≤RA<sup>1</sup> ≤ 180�. Therefore, if the relative angle is less than �180�, a 360� phase shift is added to meet this constraint, or if the angle is greater than 180�, a 360� phase shift is subtracted. Again, using Figure 2 as an example, RA<sup>1</sup> would be roughly �15� after subtracting a 360� phase shift (i.e., vehicle 2 is to the right of vehicle 1 from vehicle 1's perspective), and RA<sup>2</sup> would be roughly 25� (i.e., vehicle 1 is on the left of vehicle 2 from vehicle 2's perspective).

Lastly, the crossing time, t1, can be defined by the relationship shown in Eq. (3). This relationship defines how long it would take for vehicle 1 to reach the heading intersection point, as shown in Figure 2, if they remained on their current trajectory.

$$t\_1 = \text{sign}\left(\vec{V\_1} \cdot \vec{\mathcal{C}\_1}\right) \frac{\text{c}\_1}{\text{v}\_1} \tag{3}$$

where sign is a function used to determine the sign (positive or negative) of an expression, V<sup>1</sup> ! is the velocity of vehicle 1, C<sup>1</sup> ! is the vector used to describe the location of the heading intersection point relative to the vehicle's current position, v<sup>1</sup> is the speed of vehicle 1, and c<sup>1</sup> is the magnitude of the vector C<sup>1</sup> ! .

With these relationships, all possible encounter scenarios can be described. To aid in understanding what each of these parameters represent, Table 2 describes, in linguistic terms, what each range of values represents in the physical conflict scenarios. Here, the term crossing point


Table 2. Linguistic descriptions of encounter scenarios.

is synonymous to the heading intersection point. Each of these descriptions has been listed to describe the motion, location, or crossing time of vehicle 2 from vehicle 1's perspective.

RA<sup>1</sup> <sup>¼</sup> tan �<sup>1</sup> <sup>y</sup><sup>2</sup> � <sup>y</sup><sup>1</sup>

roughly 25� (i.e., vehicle 1 is on the left of vehicle 2 from vehicle 2's perspective).

shown in Figure 2, if they remained on their current trajectory.

!

! .

Table 2. Linguistic descriptions of encounter scenarios.

the velocity of vehicle 1, C<sup>1</sup>

232 Modern Fuzzy Control Systems and Its Applications

magnitude of the vector C<sup>1</sup>

x<sup>2</sup> � x<sup>1</sup> 

where x<sup>2</sup> and y<sup>2</sup> are the two-dimensional coordinates of vehicle 2 in the global frame, x<sup>1</sup> and y<sup>1</sup> are the two-dimensional coordinates of vehicle 1 in the global frame, H<sup>1</sup> is the heading of vehicle 1, and RA<sup>1</sup> is the relative angle of vehicle 2 from vehicle 1's perspective. Like the relative heading, the relative angle is constrained. Here, we constrain the relative angle by the following relationship: �180� ≤RA<sup>1</sup> ≤ 180�. Therefore, if the relative angle is less than �180�, a 360� phase shift is added to meet this constraint, or if the angle is greater than 180�, a 360� phase shift is subtracted. Again, using Figure 2 as an example, RA<sup>1</sup> would be roughly �15� after subtracting a 360� phase shift (i.e., vehicle 2 is to the right of vehicle 1 from vehicle 1's perspective), and RA<sup>2</sup> would be

Lastly, the crossing time, t1, can be defined by the relationship shown in Eq. (3). This relationship defines how long it would take for vehicle 1 to reach the heading intersection point, as

> ! � C<sup>1</sup> ! c<sup>1</sup>

where sign is a function used to determine the sign (positive or negative) of an expression, V<sup>1</sup>

tion point relative to the vehicle's current position, v<sup>1</sup> is the speed of vehicle 1, and c<sup>1</sup> is the

With these relationships, all possible encounter scenarios can be described. To aid in understanding what each of these parameters represent, Table 2 describes, in linguistic terms, what each range of values represents in the physical conflict scenarios. Here, the term crossing point

Relative heading 0� < RH<sup>1</sup> < 180� Moving from right to left

Time to crossing point t<sup>1</sup> > 0 Crossing point in front

Parameter Range Meaning

Relative angle RA<sup>1</sup> > 0� On left

v1

is the vector used to describe the location of the heading intersec-

180� < RH<sup>1</sup> < 360� Moving from left to right

t<sup>1</sup> < 0 Crossing point behind t<sup>1</sup> > t<sup>2</sup> Farther from crossing point t<sup>1</sup> < t<sup>2</sup> Closer to crossing point

RH<sup>1</sup> ¼ 0� Same direction RH<sup>1</sup> ¼ 180� Head-on

RA<sup>1</sup> < 0� On right RA<sup>1</sup> ¼ 0� Straight ahead

t<sup>1</sup> ¼ sign V<sup>1</sup>

� H<sup>1</sup> ð2Þ

ð3Þ

! is Although the primary goal of this system is to ensure safe separation of vehicles, it is also important to try and limit the number of unnecessary flight adjustments. This is particularly important when operating sUAS due to their typical limitations in power and endurance. Because vehicle intent is unknown in this study, a predicted LOS does not guarantee an LOS is imminent. Therefore, there is a tradeoff between using strict and relaxed criteria when determining if a trajectory modification is necessary. The criteria should be relaxed to ensure sUAS do not repetitively perform unnecessary adjustments but strict enough to ensure safe operation.

Aside from optimizing this time to predicted LOS threshold, a second way to limit the number of vehicles that divert from their desired flight paths is to assign vehicles priority. This priority assignment ranks all vehicles in conflict from highest priority to lowest priority. Therefore, the vehicle with the highest priority will continue on its preferred trajectory without modification. However, all vehicles with a lower priority must avoid all other vehicles with a higher priority.

To determine which aircraft has a lower priority, a series of evaluations are made. First, the system will be checked to see if the two aircraft are moving in a similar direction. If two vehicles have a heading within 5 of one another, that is, 355 ≤RH<sup>1</sup> < 360 or 0 ≤ RH<sup>1</sup> ≤ 5, the trailing aircraft will have lower priority. This encounter scenario can be seen in Figure 3. If several aircraft have similar headings, the aircraft furthest behind will be assigned the lowest priority so must avoid all other aircraft. However, the vehicle in the front of the group will have the highest priority and will disregard all other aircraft.

If the vehicles in conflict do not have similar headings (i.e., more than a 5 difference), the vehicle closest to its next waypoint is given priority. Since the separation assurance system logic does not use the location of a vehicle's next waypoint (i.e., intent is unknown), this priority assignment was simply a means to an end. In practice, the priority of each vehicle in these scenarios would be randomly assigned.

To predict whether an LOS will occur, the separation assurance algorithm uses the current location and velocity of each aircraft to calculate a projected flight path for each. Using these projected trajectories, the pCPA between the aircraft is found. If the pCPA will result in an LOS within the next 2 min, a resolution is calculated and employed to prevent the predicted LOS.

Recalling the sample encounter scenario shown in Figure 2, the definitions described by Eqs. (1) through (3), and the constraints shown in Table 2, all encounter scenarios can be described. Figure 4 depicts all the possible conflict scenarios when vehicle 2 is located to the right of vehicle

Figure 3. Encounter example with relative heading within 5.

1. In each diagram, the aircraft with the small circle represents vehicle 1 and the other is vehicle 2. Thus, all parameters used to describe a particular conflict scenario are from the perspective of vehicle 1. Within the scope of the separation assurance system, vehicle 2 is classified as the vehicle that has been assigned the higher priority. Thus, vehicle 1 (lower priority) must perform a maneuver to prevent an LOS.

In Figure 4(a), vehicle 2 is moving to the left, vehicle 1 is approaching from behind, and vehicle 1 is more than 45 sec closer to the heading intersection point. In this case, vehicle 1 would decide to go in front of vehicle 2. However, if vehicle 1 is not at least 45 sec closer, it will go behind.

In Figure 4(b), vehicle 2 is still going to the left, but in this case, it is coming toward vehicle 1. In these scenarios, vehicle 1 must be more than 30 sec closer to the intersection point to go in front. In Figure 4(c), vehicle 2 is to the right of vehicle 1 but is also going to the right. In these instances, the logic will determine vehicle 1 should turn left to avoid a potential LOS. This also holds for when vehicle 2 is located directly in front of vehicle 1. If, however, vehicle 2 is located on the left of vehicle 1 and going left, it will be instructed to turn right. (NOTE: The 30 and 45 sec buffers were selected after testing a handful of design iterations. Optimizing these buffer thresholds is left to future work.)

In Figure 4(d), vehicle 2's heading is parallel and coming toward vehicle 1. If vehicle 2 is directly in front of, or to the left of, vehicle 1, the logic will instruct vehicle 1 to turn right. However, if vehicle 2 is located to the right of vehicle 1, it will be instructed to turn left.

To prevent an aircraft from prematurely exiting an avoidance maneuver, the system checks if reverting to the navigation controller generates another predicted LOS. Since the avoidance controller has only local sensor knowledge (i.e., 90 ≤ RA<sup>1</sup> ≤ 90), switching back to the navigation controller can result in a turning action that generates another predicted LOS. If turning back to its desired target would create another predicted LOS, the avoiding aircraft will continue with its trajectory until authorized to resume its desired mission.

Figure 4. Separation assurance conflict scenario classifications: (a) from behind, (b) coming toward, (c) diverging, and (d) head-on.


Table 3. Summary of separation assurance logic.

1. In each diagram, the aircraft with the small circle represents vehicle 1 and the other is vehicle 2. Thus, all parameters used to describe a particular conflict scenario are from the perspective of vehicle 1. Within the scope of the separation assurance system, vehicle 2 is classified as the vehicle that has been assigned the higher priority. Thus, vehicle 1 (lower priority) must perform

In Figure 4(a), vehicle 2 is moving to the left, vehicle 1 is approaching from behind, and vehicle 1 is more than 45 sec closer to the heading intersection point. In this case, vehicle 1 would decide to go in front of vehicle 2. However, if vehicle 1 is not at least 45 sec closer, it will go

In Figure 4(b), vehicle 2 is still going to the left, but in this case, it is coming toward vehicle 1. In these scenarios, vehicle 1 must be more than 30 sec closer to the intersection point to go in front. In Figure 4(c), vehicle 2 is to the right of vehicle 1 but is also going to the right. In these instances, the logic will determine vehicle 1 should turn left to avoid a potential LOS. This also holds for when vehicle 2 is located directly in front of vehicle 1. If, however, vehicle 2 is located on the left of vehicle 1 and going left, it will be instructed to turn right. (NOTE: The 30 and 45 sec buffers were selected after testing a handful of design iterations. Optimizing these buffer

In Figure 4(d), vehicle 2's heading is parallel and coming toward vehicle 1. If vehicle 2 is directly in front of, or to the left of, vehicle 1, the logic will instruct vehicle 1 to turn right. However, if vehicle 2 is located to the right of vehicle 1, it will be instructed to turn left.

To prevent an aircraft from prematurely exiting an avoidance maneuver, the system checks if reverting to the navigation controller generates another predicted LOS. Since the avoidance controller has only local sensor knowledge (i.e., 90 ≤ RA<sup>1</sup> ≤ 90), switching back to the navigation controller can result in a turning action that generates another predicted LOS. If turning back to its desired target would create another predicted LOS, the avoiding aircraft will

Figure 4. Separation assurance conflict scenario classifications: (a) from behind, (b) coming toward, (c) diverging, and

continue with its trajectory until authorized to resume its desired mission.

a maneuver to prevent an LOS.

234 Modern Fuzzy Control Systems and Its Applications

thresholds is left to future work.)

behind.

(d) head-on.

A summary of the separation assurance logic can be found in Table 3. For all cases, the crossing time is strictly positive. That is, the absolute value of the true crossing time found using Eq. (3) is used for all separation assurance logic.

#### 3.3. Collision avoidance logic

When two vehicles have an LOS, are converging, and are within one another's sensor ranges, the collision avoidance system will be activated. In this study, each sUAS will attempt to avoid all intruders within its sensor range; therefore, no vehicle priority will be assigned. Like the approach used for the separation assurance platform to classify conflict scenarios, the collision avoidance system will use the same inputs to decide the appropriate action (i.e., relative angle, relative heading, and time to crossing point). Here, it is important to note that no two vehicles can communicate with one another. Therefore, the same logic is used onboard each system independently. This means that from each vehicle's perspective, we need to ensure that both vehicles will choose complementary actions, that is, the action will not force the vehicles to turn toward one another.

In Figure 5, all possible encounter scenarios are shown. Here, the black triangle and arrow represent the "ownship" (vehicle 1) location and heading respectively, the filled circle represents the "intruder" (vehicle 2) location, the dashed line connecting the two vehicles represents the relative position, and the other two dashed lines represent intruder headings that are either parallel or perpendicular to the ownship's heading. In this figure, the intruder can have any heading between 0� and 360�.

Using these dashed lines to divide the possible intruder heading into cases, the geometry of each encounter scenario can be broken down into twelve cases, provided that two of the three dashed lines do not coincide with one another. In Figure 5(a), the twelve cases are depicted: two cases where the intruder has a parallel relative heading (vertical line), two cases where the headings are perpendicular (horizontal line), two cases where the intruder heading is directly toward or away from the ownship position (line connecting the two vehicles), and all headings that lie in between these angles each count as one case (i.e., six angle ranges in between the lines). If two of the three dashed lines coincide, this possible geometric space reduces to eight possible cases, as shown in Figure 5(b) and c).

Now that the geometric configurations have been defined, let us now introduce the third characteristic, time to heading intersection point. Unlike the separation assurance platform, the time to the heading intersection point, or "crossing time," can be either positive or negative. Therefore, if the crossing point lies behind the vehicle, the crossing time becomes negative. Using

Figure 5. Classification of all possible encounter scenarios: (a) intruder not straight ahead or on side, (b) intruder on side, (c) intruder straight ahead.

the crossing time, there are up to three possible new situations for each of the twelve cases (or eight cases): the times are equal, the ownship crossing time is greater than that of the intruder, or the intruder crossing time is greater than that of the ownship. All possible crossing time scenarios based on the relative heading and angle have been shown. All instances where the ownship can have a crossing time less than the intruder crossing time have been marked by an "x." The black cross represents scenarios where the intruder crossing time can be less than the ownship crossing time. Finally, the square represents the situations where the two vehicles can have equal crossing times.

Using these dashed lines to divide the possible intruder heading into cases, the geometry of each encounter scenario can be broken down into twelve cases, provided that two of the three dashed lines do not coincide with one another. In Figure 5(a), the twelve cases are depicted: two cases where the intruder has a parallel relative heading (vertical line), two cases where the headings are perpendicular (horizontal line), two cases where the intruder heading is directly toward or away from the ownship position (line connecting the two vehicles), and all headings that lie in between these angles each count as one case (i.e., six angle ranges in between the lines). If two of the three dashed lines coincide, this possible geometric space reduces to eight

Now that the geometric configurations have been defined, let us now introduce the third characteristic, time to heading intersection point. Unlike the separation assurance platform, the time to the heading intersection point, or "crossing time," can be either positive or negative. Therefore, if the crossing point lies behind the vehicle, the crossing time becomes negative. Using

Figure 5. Classification of all possible encounter scenarios: (a) intruder not straight ahead or on side, (b) intruder on side,

possible cases, as shown in Figure 5(b) and c).

236 Modern Fuzzy Control Systems and Its Applications

(c) intruder straight ahead.

Using these three designations, all pairwise encounters where an intruder is not directly in front of or beside the ownship can be described by 20 possible cases, as shown in Figure 5(a). In Figure 5(b), cases where the intruder is directly beside the ownship are shown, resulting in 12 possible cases. Lastly, if the intruder is directly in front of the ownship, 8 additional cases can be attained, as shown in Figure 5(c). Thus, a total of 40 cases can be attained using these three parameters to describe the pairwise encounter space.

Knowing that 40 possible cases have been shown, a total of four conflict classifications can be used to solve all possible encounter scenarios. In Figure 5, the number above each icon represents which of the four conflict classifications the scenario belongs. In Figures 6 and 7, the four conflict scenarios have been shown. In each figure, the ownship (vehicle 1) location is marked by a black triangle and its heading is designated by the black arrow. The small circle represents the location of the intruder (vehicle 2). This intruder can have any heading, but the

Figure 6. Conflict classification #1 where (t<sup>1</sup> > 0) ∨ (t<sup>2</sup> > 0) and: (a) t<sup>1</sup> < t2, (b) t<sup>1</sup> > t2.

Figure 7. Conflict classifications #2, #3, and #4: (a) (t<sup>1</sup> < 0) ∧ (t<sup>2</sup> < 0), (b) (t<sup>1</sup> = t2) ∧ (t<sup>1</sup> 6¼ ∞) ∧ (t<sup>1</sup> > 0), (c) (t<sup>1</sup> = t2) ∧ (t<sup>1</sup> = ∞).

different headings have been separated into different sections, as designated by the shaded regions. In each figure, the shaded regions represent the intruder headings that are excluded by that particular encounter scenario, whereas, the unshaded regions are the possible intruder headings allowed by that encounter scenario. In addition, like Figure 5, there is a dashed line connecting the two vehicles to represent the relative position. Lastly, if a dashed black line is on the boundary of the included and excluded regions, this represents an inclusive boundary (i.e., that heading is included in the possible headings allowed), whereas, the solid black line between the regions represents an exclusive boundary.

Figure 6 shows the first conflict classification scenario. Here, at least one of the vehicles must have a positive crossing time (ti), and they are not equal to one another. As shown in Figure 6(a), the ownship crossing time must be strictly less than that of the intruder. Thus, the intruder crossing time can be negative, or equal to zero (i.e., pointed directly at the ownship). If the intruder heading passes into the excluded region, either both values are negative, or the ownship value must be greater than the intruder value. For all the cases that satisfy this relationship, the ownship would determine to go in front of the intruder. Including the instances where the intruder is to the left of the ownship.

In Figure 6(b), the ownship crossing time must be strictly greater than that of the intruder crossing time. As seen in the figure, the boundary along the relative position line is excluded when the intruder is pointing toward the ownship (i.e., the ownship crossing time is zero, but the intruder crossing time is positive), violating the relationship. However, when the intruder is pointed directly away from the ownship, the boundary is inclusive. In all possible instances, the ownship will determine to go behind the intruder. This includes the cases when the intruder is located to the left of the ownship or is directly in front of the ownship (i.e., t<sup>2</sup> = 0). For all cases represented by conflict classification #1, if one vehicle decides to go in front, the other will decide to go behind, given that they are both within one another's sensor field of view.

In the second conflict classification, as shown in Figure 7(a), both the intruder and the ownship have crossing times less than zero. These negative crossing times, and the fact that the vehicles cannot have the same heading, result in the intruder never sensing the ownship. In these scenarios, if the intruder is on the left, the ownship will turn right. But, if the intruder is on the right, the ownship will turn left.

In Figure 7(b), the scenarios where both the ownship and the intruder have the same time to the crossing point are shown. In this classification case, the intruder cannot have the same heading as the ownship, thus it must be crossing the ownship's path. In addition, this case is restricted to instances where both vehicles have strictly positive crossing times. For these instances, regardless of whether the vehicle is on the left, or on the right, both vehicles will decide to turn right. Since the crossing times are equivalent, and there is no coordination of intent with the other vehicle, both vehicles must choose the same action.

There is one remaining aircraft orientation in the encounter space. This occurs when an intruder is parallel to the ownship, either traveling in the same or opposite direction. This scenario has been shown in Figure 7(c). As seen from this figure, only intruder headings that lie on the dashed line are included. If the intruder is to the right of the ownship, each vehicle will turn to the left (given that the intruder can see the ownship as well). However, if the intruder is directly in front of, or to the left of the ownship, both vehicles will be instructed to turn right.

Although the above classifications describe all possible encounters between moving aircraft, the quad-rotor vehicles can stop and hover. Therefore, a final classification must be described. If an intruder is stationary, the intruder crossing time is set to negative infinity. When this is the case, if the intruder is to the left or directly in front of the ownship, the controller will instruct the vehicle to turn right. However, if the intruder is to the right of the ownship, the logic will instruct the vehicle to turn left.

Table 4 describes the different encounter scenarios using both linguistic and mathematical descriptions, as well as the decided actions. Each of the above conflict classification numbers numerically matches the respective cases in this table. However, the encounter scenario where the intruder vehicle is stationary is referred to as case 0. Like Table 3, the linguistic descriptions in Table 4 represent how the ownship perceives the intruder. Furthermore, all values with the subscript 1 represent the ownship, whereas, all values with the subscript 2 represent the intruder. (NOTE: the \* designation indicates that the crossing time is negative infinity for that vehicle.)

#### 3.4. Fuzzy inference systems

different headings have been separated into different sections, as designated by the shaded regions. In each figure, the shaded regions represent the intruder headings that are excluded by that particular encounter scenario, whereas, the unshaded regions are the possible intruder headings allowed by that encounter scenario. In addition, like Figure 5, there is a dashed line connecting the two vehicles to represent the relative position. Lastly, if a dashed black line is on the boundary of the included and excluded regions, this represents an inclusive boundary (i.e., that heading is included in the possible headings allowed), whereas, the solid black line

Figure 6 shows the first conflict classification scenario. Here, at least one of the vehicles must have a positive crossing time (ti), and they are not equal to one another. As shown in Figure 6(a), the ownship crossing time must be strictly less than that of the intruder. Thus, the intruder crossing time can be negative, or equal to zero (i.e., pointed directly at the ownship). If the intruder heading passes into the excluded region, either both values are negative, or the ownship value must be greater than the intruder value. For all the cases that satisfy this relationship, the ownship would determine to go in front of the intruder. Including the instances where the

In Figure 6(b), the ownship crossing time must be strictly greater than that of the intruder crossing time. As seen in the figure, the boundary along the relative position line is excluded when the intruder is pointing toward the ownship (i.e., the ownship crossing time is zero, but the intruder crossing time is positive), violating the relationship. However, when the intruder is pointed directly away from the ownship, the boundary is inclusive. In all possible instances, the ownship will determine to go behind the intruder. This includes the cases when the intruder is located to the left of the ownship or is directly in front of the ownship (i.e., t<sup>2</sup> = 0). For all cases represented by conflict classification #1, if one vehicle decides to go in front, the other will decide

In the second conflict classification, as shown in Figure 7(a), both the intruder and the ownship have crossing times less than zero. These negative crossing times, and the fact that the vehicles cannot have the same heading, result in the intruder never sensing the ownship. In these scenarios, if the intruder is on the left, the ownship will turn right. But, if the intruder is on

In Figure 7(b), the scenarios where both the ownship and the intruder have the same time to the crossing point are shown. In this classification case, the intruder cannot have the same heading as the ownship, thus it must be crossing the ownship's path. In addition, this case is restricted to instances where both vehicles have strictly positive crossing times. For these instances, regardless of whether the vehicle is on the left, or on the right, both vehicles will decide to turn right. Since the crossing times are equivalent, and there is no coordination of

There is one remaining aircraft orientation in the encounter space. This occurs when an intruder is parallel to the ownship, either traveling in the same or opposite direction. This scenario has been shown in Figure 7(c). As seen from this figure, only intruder headings that lie on the dashed line are included. If the intruder is to the right of the ownship, each vehicle will turn to

to go behind, given that they are both within one another's sensor field of view.

intent with the other vehicle, both vehicles must choose the same action.

between the regions represents an exclusive boundary.

intruder is to the left of the ownship.

238 Modern Fuzzy Control Systems and Its Applications

the right, the ownship will turn left.

Using the methodology described in Sections 3.2 and 3.3, four possible actions can be used to avoid an intruding sUAS: go behind, go in front, turn right, and turn left. When the command


Table 4. Summary of collision avoidance logic.

action is either go in front or go behind, the corresponding FIS is activated to execute the maneuver. Therefore, two FISs were developed for this study: go in front and go behind. If a turn left or turn right command is selected, the turn rate of the vehicle will always be a constant value, either positive or negative, depending on which way it should turn. This constant value is one half of the maximum turn rate for collision avoidance maneuvers and one eighth of the maximum turn rate for separation assurance maneuvers.

Both the go in front and go behind fuzzy systems are of Mamdani-type and were constructed in such a way that the input-output relationship can be described using a simple mathematical representation. By using a hybrid representation, as described in Section 2.1, the fuzzy system can easily be expressed mathematically. In this study, the fuzzy systems have a common architecture: triangular membership functions, normalized inputs and outputs, membership function partitioning, product "and" method, minimum implication method, sum aggregation, and mean of maximum defuzzification. If more than one membership function exists for a particular input or output, membership functions are partitioned such that the endpoints of one membership function coincides with the center points of the neighboring membership functions.

Each FIS was developed using a three-input one-output structure. Each FIS uses the distance separating the two aircraft, their relative heading, and their closure rate as inputs to determine the appropriate turn rate output. Since a heterogeneous system is used in this study, the FIS must provide a sufficient turn rate output to avoid a collision for all vehicle type combinations. By considering the separation and closure rate, the conflict can be solved without expelling more energy than required.

In order to use the FISs for both the separation assurance and the collision avoidance platforms, all inputs and outputs must be normalized. Regardless of the avoidance platform being used, the relative heading and closure rate inputs are always normalized by the same values. The relative heading which falls between 0 and 360 is divided by 360. Thus, a normalized relative heading of 0.5 would represent a head-on encounter. The closure rate is normalized by dividing the true closure rate by the maximum possible closure rate between two vehicles (i.e., 61.762 m/sec in this study). This maximum closure rate would be a result of two fixed wing vehicles approaching one another in a head-on scenario.

The third input, distance, is normalized by 0.4 nmi for separation assurance cases and 0.1 nmi for collision avoidance cases. Lastly, the turn rate output is always between 1 and 1. This output is then scaled by the respective vehicle platform's maximum turn rate. In the case of the collision avoidance system, the output is also multiplied by 1.58. This is to compensate for the fuzzy output not providing a sufficient turn rate command to avoid a collision, especially in head-on scenarios.

Figure 8 shows the structure of each avoidance FIS. Here, the number of membership functions and the corresponding classification can be seen for each input and output. Both the relative heading and the turn rate are partitioned such that the endpoints of the center membership functions coincide with the center points of the adjacent membership functions. As a result of this, and using a product "and" connector, for all possible inputs, at most two of the

A Fuzzy Logic Approach for Separation Assurance and Collision Avoidance for Unmanned Aerial Systems http://dx.doi.org/10.5772/68126 241

Figure 8. Avoidance FIS structure: (a) Input 1, (b) Input 2, (c) Input 3, (d) Output.

three rules will be active at one time. This drastically reduces the possible solution space, making it much easier to represent the system mathematically.

The input and output membership function sets for both the go behind and go in front FISs are identical. However, the rule bases associating the inputs and outputs are opposite, therefore, when one FIS outputs "turn right" (i.e., negative turn rate), the other FIS would output "turn left" (i.e., positive turn rate), and vice versa. In Table 5, the respective rule bases can be seen. Because the other two inputs only have one membership function, they have been excluded from the table.


Table 5. FIS logic.

action is either go in front or go behind, the corresponding FIS is activated to execute the maneuver. Therefore, two FISs were developed for this study: go in front and go behind. If a turn left or turn right command is selected, the turn rate of the vehicle will always be a constant value, either positive or negative, depending on which way it should turn. This constant value is one half of the maximum turn rate for collision avoidance maneuvers and

Both the go in front and go behind fuzzy systems are of Mamdani-type and were constructed in such a way that the input-output relationship can be described using a simple mathematical representation. By using a hybrid representation, as described in Section 2.1, the fuzzy system can easily be expressed mathematically. In this study, the fuzzy systems have a common architecture: triangular membership functions, normalized inputs and outputs, membership function partitioning, product "and" method, minimum implication method, sum aggregation, and mean of maximum defuzzification. If more than one membership function exists for a particular input or output, membership functions are partitioned such that the endpoints of one membership function coincides with the center points of the neighboring membership

Each FIS was developed using a three-input one-output structure. Each FIS uses the distance separating the two aircraft, their relative heading, and their closure rate as inputs to determine the appropriate turn rate output. Since a heterogeneous system is used in this study, the FIS must provide a sufficient turn rate output to avoid a collision for all vehicle type combinations. By considering the separation and closure rate, the conflict can be solved without expelling

In order to use the FISs for both the separation assurance and the collision avoidance platforms, all inputs and outputs must be normalized. Regardless of the avoidance platform being used, the relative heading and closure rate inputs are always normalized by the same values. The relative heading which falls between 0 and 360 is divided by 360. Thus, a normalized relative heading of 0.5 would represent a head-on encounter. The closure rate is normalized by dividing the true closure rate by the maximum possible closure rate between two vehicles (i.e., 61.762 m/sec in this study). This maximum closure rate would be a result of two fixed wing

The third input, distance, is normalized by 0.4 nmi for separation assurance cases and 0.1 nmi for collision avoidance cases. Lastly, the turn rate output is always between 1 and 1. This output is then scaled by the respective vehicle platform's maximum turn rate. In the case of the collision avoidance system, the output is also multiplied by 1.58. This is to compensate for the fuzzy output not providing a sufficient turn rate command to avoid a collision, especially in

Figure 8 shows the structure of each avoidance FIS. Here, the number of membership functions and the corresponding classification can be seen for each input and output. Both the relative heading and the turn rate are partitioned such that the endpoints of the center membership functions coincide with the center points of the adjacent membership functions. As a result of this, and using a product "and" connector, for all possible inputs, at most two of the

one eighth of the maximum turn rate for separation assurance maneuvers.

functions.

more energy than required.

240 Modern Fuzzy Control Systems and Its Applications

head-on scenarios.

vehicles approaching one another in a head-on scenario.


Table 6. Input domains.

To map the fuzzy system to a set of nonlinear expressions, the following process was used. First, the system was discretized into three possible modes based on the domain of the i th input, Di. These modes are described in Table 6. Here, let a square bracket represent an inclusive bound and a round bracket represent an exclusive bound. Based on the structure of the FIS, if input 1 is exactly 0.5 (Mode 2 in Table 6), the output will yield a turn rate of 0. It remains to find the explicit input-to-output mappings for modes 1 and 3.

When the FIS is in mode 1, only the "Left" and "Center" membership functions are active. Thus, the output will be independent of rule 3. Therefore, given that a product "and" method and a mean of maximum aggregation technique is used, the following relationship describes how the output is calculated.

$$\dot{\Psi} = \left[ \left( \prod\_{i=1}^{3} h\_{i\_1} \right) + \left( 1 - \prod\_{i=1}^{3} h\_{i\_2} \right) \right] / 2 \tag{4}$$

where, Ψ\_ , is the turn rate, hi<sup>1</sup> is the degree of membership for the i th input membership function when using rule 1 (i.e., left, close, fast membership functions), and hi<sup>2</sup> is the degree of membership for each input when using rule 2. When substituting the equation of each membership function into Eq. (4), the expression shown in Eq. (5) is found, which can be reduced to the polynomial shown in Eq. (6).

$$\dot{\Psi} = \{ \left[ \left( -2\mu\_1 + 1 \right) \left( -\mu\_2 + 1 \right) \left( \mu\_3 \right) \right] + \left[ 1 - \left( 2\mu\_1 \right) \left( -\mu\_2 + 1 \right) \left( \mu\_3 \right) \right] \}/2 \tag{5}$$

$$\dot{\Psi} = \left( 4\mu\_1\mu\_2\mu\_3 - 4\mu\_1\mu\_3 - \mu\_2\mu\_3 + \mu\_3 + 1 \right) / 2 \tag{6}$$

where μ1, μ2, and μ<sup>3</sup> are inputs 1, 2, and 3, respectively. The polynomial shown in Eq. (6) can be used to map any combination of inputs that belong to D<sup>1</sup> to an output for the go in front FIS.

Using the same methodology for mode 3, Eqs. (7) through (9) can be found.

$$\dot{\Psi} = \left[ \left( -\prod\_{i=1}^{3} h\_{i\_3} \right) + \left( \prod\_{i=1}^{3} h\_{i\_2} - 1 \right) \right] / 2 \tag{7}$$

$$\dot{\Psi} = \{ \left[ -\left( 2\mu\_1 + 1 \right) \left( -\mu\_2 + 1 \right) \left( \mu\_3 \right) \right] + \left[ \left( -2\mu\_1 + 2 \right) \left( -\mu\_2 + 1 \right) \left( \mu\_3 \right) - 1 \right] \}/2 \tag{8}$$

$$\dot{\Psi} = \left(4\mu\_1\mu\_2\mu\_3 - 4\mu\_1\mu\_3 - 3\mu\_2\mu\_3 + 3\mu\_3 - 1\right)/2\tag{9}$$

The polynomial shown in Eq. (9) will map any combination of inputs belonging to D<sup>3</sup> to an output for the go in front FIS.

This same approach was used to map any combination of inputs for the go behind FIS to an output using a polynomial function. Since the rule bases are exactly opposite to one another, the output is the negation of Eqs. (6) and (9). Eqs. (10) and (11) describe the input-output relationships for modes 1 and 3, respectively, for the go behind FIS.

$$\dot{\Psi} = \left( -4\mu\_1\mu\_2\mu\_3 + 4\mu\_1\mu\_3 + \mu\_2\mu\_3 - \mu\_3 - 1 \right) / 2 \tag{10}$$

$$\dot{\Psi} = \left(-4\mu\_1\mu\_2\mu\_3 + 4\mu\_1\mu\_3 + 3\mu\_2\mu\_3 - 3\mu\_3 + 1\right)/2\tag{11}$$

It is important to note that if three or more vehicles are in conflict, each pair of vehicles will be evaluated separately. Thus, for a single ownship, several turn rate outputs will be obtained. Once all respective turn rates are calculated for each intruder, the values are averaged into a single value. This mean turn rate serves as the final vehicle turn rate command.

## 4. Avoidance algorithm verification

To map the fuzzy system to a set of nonlinear expressions, the following process was used. First, the system was discretized into three possible modes based on the domain of the i

Mode # D<sup>1</sup> D<sup>2</sup> D<sup>3</sup> [0,0.5) [0,1] [0,1] [0.5] [0,1] [0,1] (0.5,1] [0,1] [0,1]

input, Di. These modes are described in Table 6. Here, let a square bracket represent an inclusive bound and a round bracket represent an exclusive bound. Based on the structure of the FIS, if input 1 is exactly 0.5 (Mode 2 in Table 6), the output will yield a turn rate of 0. It

When the FIS is in mode 1, only the "Left" and "Center" membership functions are active. Thus, the output will be independent of rule 3. Therefore, given that a product "and" method and a mean of maximum aggregation technique is used, the following relationship describes

> <sup>þ</sup> <sup>1</sup> � <sup>Y</sup> 3

" # !

function when using rule 1 (i.e., left, close, fast membership functions), and hi<sup>2</sup> is the degree of membership for each input when using rule 2. When substituting the equation of each membership function into Eq. (4), the expression shown in Eq. (5) is found, which can be reduced to

where μ1, μ2, and μ<sup>3</sup> are inputs 1, 2, and 3, respectively. The polynomial shown in Eq. (6) can be used to map any combination of inputs that belong to D<sup>1</sup> to an output for the go in front FIS.

> <sup>þ</sup> <sup>Y</sup> 3

" # !

� � � � þ �2μ<sup>1</sup> <sup>þ</sup> <sup>2</sup> � � �μ<sup>2</sup> <sup>þ</sup> <sup>1</sup> � � <sup>μ</sup><sup>3</sup>

i¼1

hi<sup>2</sup> � 1

<sup>Ψ</sup>\_ <sup>¼</sup> <sup>4</sup>μ1μ2μ<sup>3</sup> � <sup>4</sup>μ1μ<sup>3</sup> � <sup>3</sup>μ2μ<sup>3</sup> <sup>þ</sup> <sup>3</sup>μ<sup>3</sup> � <sup>1</sup> � �=<sup>2</sup> <sup>ð</sup>9<sup>Þ</sup>

� � � <sup>1</sup> � � � � <sup>=</sup><sup>2</sup> <sup>ð</sup>8<sup>Þ</sup>

i¼1 hi2

� � � � � � <sup>=</sup><sup>2</sup> <sup>ð</sup>5<sup>Þ</sup>

<sup>Ψ</sup>\_ <sup>¼</sup> <sup>4</sup>μ1μ2μ<sup>3</sup> � <sup>4</sup>μ1μ<sup>3</sup> � <sup>μ</sup>2μ<sup>3</sup> <sup>þ</sup> <sup>μ</sup><sup>3</sup> <sup>þ</sup> <sup>1</sup> � �=<sup>2</sup> <sup>ð</sup>6<sup>Þ</sup>

� � �μ<sup>2</sup> <sup>þ</sup> <sup>1</sup> � � <sup>μ</sup><sup>3</sup>

remains to find the explicit input-to-output mappings for modes 1 and 3.

<sup>Ψ</sup>\_ <sup>¼</sup> <sup>Y</sup> 3

where, Ψ\_ , is the turn rate, hi<sup>1</sup> is the degree of membership for the i

<sup>Ψ</sup>\_ ¼ �2μ<sup>1</sup> <sup>þ</sup> <sup>1</sup> � � �μ<sup>2</sup> <sup>þ</sup> <sup>1</sup> � � <sup>μ</sup><sup>3</sup>

i¼1 hi1 !

� � � � <sup>þ</sup> <sup>1</sup> � <sup>2</sup>μ<sup>1</sup>

Using the same methodology for mode 3, Eqs. (7) through (9) can be found.

3

i¼1 hi3 !

<sup>Ψ</sup>\_ ¼ �<sup>Y</sup>

<sup>Ψ</sup>\_ ¼ � <sup>2</sup>μ<sup>1</sup> <sup>þ</sup> <sup>1</sup> � � �μ<sup>2</sup> <sup>þ</sup> <sup>1</sup> � � <sup>μ</sup><sup>3</sup>

how the output is calculated.

Table 6. Input domains.

242 Modern Fuzzy Control Systems and Its Applications

the polynomial shown in Eq. (6).

th

=2 ð4Þ

=2 ð7Þ

th input membership

To verify that the avoidance algorithm performed as intended, two levels of avoidance control were evaluated. The first was the high-level, decision-making logic that determined which action a vehicle would take during potential collision scenarios as shown in Table 4. The second that was evaluated was the low-level logic in the avoidance FLCs. These FLCs determine the actual vehicle turn rate output after being selected by the decision-making heuristics. For each of these levels, specifications about their behavior were developed and then translated into FOL sentences that could be evaluated by an SMT solver. For the first cases that deal with the avoidance decision-making logic, the specifications and system model were implemented in JKind, with Z3 being used as the SMT solver. The evaluation of the avoidance FISs was performed directly in Z3. The difference was purely a practical one, as the language JKind uses, Lustre [22], is more conducive to easily create a more detailed environment model.

#### 4.1. Collision avoidance decision-making logic

As shown in Table 4, there are several conditions for which there are different output actions for avoidance. The specifications were first translated into FOL sentences. The sentences are shown in the following equations. Note that RAL is a predicate that acts on the relative angle limit to represent the vehicle's sensors detecting an intruder. Since they can only sense intruders between �90�, the relative angle values were limited in the specifications such that being in that range implied that the output action would be the correct one. If the intruder is outside of this range, the specifications do not say what the output action should be. Note that the variables out1 and out2 are the output actions for the two vehicles. Since both vehicles pick actions based on the states and have no knowledge of the other vehicle's selected action, the specifications need to go both ways. Also, the actions are represented by integer values 1–4 such that {1,2,3,4} = {go behind, go in front, turn left, turn right}, respectively.

Although several avoidance actions could be taken by each vehicle, only cases 1, 3, and 4 required formal verification. In these cases, both vehicles can sense one another and will perform some type of avoidance maneuver. Thus, their actions must be such that they do not move closer to one another. In cases 0 and 2, however, there was no need to write requirements because only one vehicle performs an avoidance action. In case 0, vehicle 2 is stationary, so it does not try to avoid vehicle 1. In case 2, vehicle 2 cannot sense vehicle 1, thus, there is no need to check that the two vehicles' actions produce a converging result.

Eqs. (12) through (15) are the specifications for case 1 in Table 4 and are intended to ensure that the vehicles do not both choose to go in front, or go behind.

$$\mathcal{S}\_{1\_1} = \forall \mathcal{R}\_{A\_1} \forall \mathcal{R}\_{A\_2} \forall \mathcal{R}\_{H\_1} \forall \mathcal{R}\_{H\_2} (RAL \to (out1 = 1 \to out2 = 2))\tag{12}$$

$$\mathcal{S}\_{1\_2} = \forall \mathcal{R}\_{A\_1} \forall \mathcal{R}\_{A\_2} \forall \mathcal{R}\_{H\_1} \forall \mathcal{R}\_{H\_2} (RAL \to (out1 = 2 \to out2 = 1))\tag{13}$$

$$\mathcal{S}\_{1\_3} = \forall \mathcal{R}\_{A\_1} \forall \mathcal{R}\_{A\_2} \forall \mathcal{R}\_{H\_1} \forall \mathcal{R}\_{H\_2} (\mathcal{R}AL \to (out2 = 1 \to out1 = 2))\tag{14}$$

$$\mathcal{S}\_{14} = \forall \mathcal{R}\_{A\_1} \forall \mathcal{R}\_{A\_2} \forall \mathcal{R}\_{H\_1} \forall \mathcal{R}\_{H\_2} \left( RAL \to (out2 = 2 \to out1 = 1) \right) \tag{15}$$

Eqs. (16) and (17) are the specifications for case 3 and ensure that the vehicles turn the same way when resolving these particular conflicts. Recall that in this case, both vehicles are at the same time from the crossing point. This specification ensures that they will then be forced into new positions such that the crossing times are not equal and the FISs are then selected.

$$\begin{aligned} S\_{3\_1} &= \forall R\_{A\_1} \forall R\_{A\_2} \forall R\_{H\_1} \forall R\_{H\_2} (RAL \to ((t\_1 = t\_2 \land t\_1 > 0 \land t\_2 > 0 \land ((R\_{H\_1} \ge 0 \land R\_{H\_1} \\ < 180) \lor (R\_{H\_1} > 180 \land R\_{H\_1} < 360))) \to (out1 = 4 \land out2 = 4))) \end{aligned} \tag{16}$$

$$\begin{aligned} S\_{\\$\_2} &= \forall R\_{A\_1} \forall R\_{A\_2} \forall R\_{H\_1} \forall R\_{H\_2} (RAL \to ((t\_1 = t\_2 \land t\_1 > 0 \land t\_2 > 0 \land ((R\_{H\_2} \ge 0 \land R\_{H\_2} \\ < 180) \lor (R\_{H\_2} > 180 \land R\_{H\_2} < 360))) \to (out2 = 4 \land out1 = 4))) \end{aligned} \tag{17}$$

Finally, Eqs. (18) and (19) are for case 4. These two specifications are for cases where the vehicles are head-on, or traveling next to each other in the same direction, respectively. The first specification ensures that while in a head-on encounter, both vehicles turn the same direction (i.e., both left, or both right, forcing them to diverge). The second specification ensures that while traveling in the same direction, the vehicles turn in opposite directions.

$$\begin{aligned} \mathsf{S}\_{4\_1} &= \mathsf{V} \mathsf{R}\_{A\_1} \mathsf{V} \mathsf{R}\_{A\_2} \mathsf{V} \mathsf{R}\_{H\_1} \mathsf{V} \mathsf{R}\_{H\_2} (\mathsf{R}AL \to ((t\_1 = t\_2 \land (\mathsf{R}\_{H\_1} = 180 \land \mathsf{R}\_{H\_2} \\ \qquad &= 180)) \to ((out1 = 3 \land out2 = 3) \lor (out1 = 4 \land out2 = 4))) \end{aligned} \tag{18}$$

$$\begin{aligned} \mathsf{S}\_{4\_2} &= \mathsf{V} \mathsf{R}\_{A\_1} \mathsf{V} \mathsf{R}\_{A\_2} \mathsf{V} \mathsf{R}\_{H\_1} \mathsf{V} \mathsf{R}\_{H\_2} (\mathsf{R}AL \to \left( (t\_1 = t\_2 \land (\mathsf{R}\_{H\_1} = 0 \land \mathsf{R}\_{H\_2} \\ = 0) \right) &\to \left( (out1 = \mathsf{3} \land out2 = 4) \lor (out1 = 4 \land out2 = \mathsf{3}) \right) \end{aligned} \tag{19}$$

where t<sup>1</sup> is the time until the crossing point, RH<sup>1</sup> is the relative heading from vehicle 1 to vehicle 2, and RA<sup>1</sup> is the relative angle from vehicle 1 to vehicle 2. Similarly, t2, RH<sup>2</sup> , and RA<sup>2</sup> are for vehicle 2. In addition to these specifications, another specification was created to ensure that the vehicles always chose one of the valid actions. This specification is not shown but is similar to Eqs. (12) through (19) such that the output actions are one of the possible outcomes (1–4). These specifications were then translated to a temporal representation for evaluation in JKind using previously developed methods [18].

#### 4.2. Avoidance fuzzy inference systems

specifications need to go both ways. Also, the actions are represented by integer values 1–4

Although several avoidance actions could be taken by each vehicle, only cases 1, 3, and 4 required formal verification. In these cases, both vehicles can sense one another and will perform some type of avoidance maneuver. Thus, their actions must be such that they do not move closer to one another. In cases 0 and 2, however, there was no need to write requirements because only one vehicle performs an avoidance action. In case 0, vehicle 2 is stationary, so it does not try to avoid vehicle 1. In case 2, vehicle 2 cannot sense vehicle 1, thus, there is no need

Eqs. (12) through (15) are the specifications for case 1 in Table 4 and are intended to ensure that

Eqs. (16) and (17) are the specifications for case 3 and ensure that the vehicles turn the same way when resolving these particular conflicts. Recall that in this case, both vehicles are at the same time from the crossing point. This specification ensures that they will then be forced into new positions such that the crossing times are not equal and the FISs are then

S<sup>31</sup> ¼ ∀RA1∀RA2∀RH1∀RH<sup>2</sup> RAL ! t<sup>1</sup> ¼ t<sup>2</sup> ∧ t<sup>1</sup> > 0 ∧ t<sup>2</sup> > 0 ∧ RH<sup>1</sup> ≥ 0 ∧ RH<sup>1</sup> ð ðð ðð

S<sup>32</sup> ¼ ∀RA1∀RA2∀RH1∀RH<sup>2</sup> RAL ! t<sup>1</sup> ¼ t<sup>2</sup> ∧ t<sup>1</sup> > 0 ∧ t<sup>2</sup> > 0 ∧ RH<sup>2</sup> ≥ 0 ∧ RH<sup>2</sup> ð ðð ðð

Finally, Eqs. (18) and (19) are for case 4. These two specifications are for cases where the vehicles are head-on, or traveling next to each other in the same direction, respectively. The first specification ensures that while in a head-on encounter, both vehicles turn the same direction (i.e., both left, or both right, forcing them to diverge). The second specification ensures that while traveling in the same direction, the vehicles turn in opposite directions.

S<sup>41</sup> ¼ ∀RA1∀RA2∀RH1∀RH<sup>2</sup> RAL ! t<sup>1</sup> ¼ t<sup>2</sup> ∧ RH<sup>1</sup> ¼ 180 ∧RH<sup>2</sup> ð ðð ð

S<sup>42</sup> ¼ ∀RA1∀RA2∀RH1∀RH<sup>2</sup> RAL ! t<sup>1</sup> ¼ t<sup>2</sup> ∧ RH<sup>1</sup> ¼ 0 ∧RH<sup>2</sup> ð ðð ð

S<sup>11</sup> ¼ ∀RA1∀RA2∀RH1∀RH<sup>2</sup> ðRAL ! ð Þ out1 ¼ 1 ! out2 ¼ 2 Þ ð12Þ

S<sup>12</sup> ¼ ∀RA1∀RA2∀RH1∀RH<sup>2</sup> ðRAL ! ð Þ out1 ¼ 2 ! out2 ¼ 1 Þ ð13Þ

S<sup>13</sup> ¼ ∀RA1∀RA2∀RH1∀RH<sup>2</sup> ðRAL ! ð Þ out2 ¼ 1 ! out1 ¼ 2 Þ ð14Þ

S<sup>14</sup> ¼ ∀RA1∀RA2∀RH1∀RH<sup>2</sup> ðRAL ! ð Þ out2 ¼ 2 ! out1 ¼ 1 Þ ð15Þ

< 180Þ ∨ ðRH<sup>1</sup> > 180 ∧ RH<sup>1</sup> < 360ÞÞÞ ! ðout1 ¼ 4 ∧ out2 ¼ 4ÞÞÞ ð16Þ

< 180Þ ∨ ðRH<sup>2</sup> > 180 ∧ RH<sup>2</sup> < 360ÞÞÞ ! ðout2 ¼ 4 ∧ out1 ¼ 4ÞÞÞ ð17Þ

¼ 180ÞÞ ! ðð Þ out1 ¼ 3 ∧ out2 ¼ 3 ∨ ð Þ out1 ¼ 4 ∧ out2 ¼ 4 ÞÞÞ ð18Þ

¼ 0ÞÞ ! ðð Þ out1 ¼ 3 ∧ out2 ¼ 4 ∨ ð Þ out1 ¼ 4 ∧ out2 ¼ 3 ÞÞÞ ð19Þ

such that {1,2,3,4} = {go behind, go in front, turn left, turn right}, respectively.

to check that the two vehicles' actions produce a converging result.

the vehicles do not both choose to go in front, or go behind.

244 Modern Fuzzy Control Systems and Its Applications

selected.

The method for verifying the avoidance FISs is similar, but the specifications were left in FOL and then implemented directly into Z3. The main reason for this was that less of the system model was needed to check these outputs. The behavior that the specifications needed to encapsulate was that the turn rate output for each FIS needs to always be in the correct direction. The correct direction for each of these cases is shown in Table 7.

These can then be encoded in FOL sentences using the polynomial representation of the FISs shown in Section 3.4. These sentences are then negated to show that there are no possible variable values that make the negated sentences true. The negated FOL sentences are shown in Eqs. (20) through (23). Note that μ1, μ2, μ3, Ψ\_ , and modes 1 and 3 are the same as detailed in Section 3.4.

$$\mathcal{S}\_{\text{beh}nd\_1} = \exists \mu\_1 \exists \mu\_2 \exists \mu\_3 \left( \dot{\Psi} \left( \mu\_1, \mu\_2, \mu\_3 \right) \ge 0 \right) \tag{20}$$

$$\mathcal{S}\_{\text{behind}\downarrow} = \exists \mu\_1 \exists \mu\_2 \exists \mu\_3 \left(\dot{\Psi}\left(\mu\_1, \mu\_2, \mu\_3\right) \le 0\right) \tag{21}$$

$$\mathcal{S}\_{\text{front}\_1} = \exists \mu\_1 \exists \mu\_2 \exists \mu\_3 \left( \dot{\Psi} \left( \mu\_1, \mu\_2, \mu\_3 \right) \le 0 \right) \tag{22}$$

$$\mathcal{S}\_{\text{front}\_3} = \exists \mu\_1 \exists \mu\_2 \exists \mu\_3 \left( \dot{\Psi} \left( \mu\_1, \mu\_2, \mu\_3 \right) \ge 0 \right) \tag{23}$$

These negated sentences were then implemented in Z3. If Z3 finds that these sentences were all unsatisfiable, there are no possible real-valued assignments to μ1, μ2, or μ<sup>3</sup> that allow the FISs to turn the vehicles in an undesired direction.


Table 7. Avoidance FIS outputs.

## 5. Simulation environment descriptions

#### 5.1. Testing environment

Prior to integrating the controllers presented in Section 3 into the full simulation environment, each avoidance platform was tested using pairwise encounter scenarios. This component testing was used to ensure that each was operating as desired. In this section, the methods used to test each controller are described.

#### 5.1.1. Separation assurance

A testing environment was created to evaluate a considerable amount of pairwise encounters between aircraft. This testing environment was used to identify potential controller failures. To accomplish this, various initial relative headings and relative angles were tested. In all cases, the initial location and heading of one aircraft was held constant. Then, by placing the intruding vehicle at a relative angle of 90 and just outside the vehicle's sensing radius, as shown in Figure 9, we evaluated the interactions for 720 initial intruder heading values. This was then repeated four more times by changing the initial relative angle between 90 (to the right) and 0 (straight ahead). A visualization of these scenarios is shown in Figure 9. Note that the radius of the sensing semi-circle for the separation assurance tests was set at 0.4 nmi.

Although intruders can approach a vehicle from the left, that is, a relative angle between 0 and 90, symmetry allows us to limit the initial relative angles to lie between 90 and 0. To verify, a sample scenario was tested for the full field of view. In all cases, the trajectories of the vehicles were symmetric to one another (i.e., when reflected across the vehicle's initial heading).

Figure 9. Separation and avoidance testing scenarios between ownship (triangle) and intruder UAS for multiple different initial positions (circles).

#### 5.1.2. Collision avoidance

5. Simulation environment descriptions

used to test each controller are described.

Prior to integrating the controllers presented in Section 3 into the full simulation environment, each avoidance platform was tested using pairwise encounter scenarios. This component testing was used to ensure that each was operating as desired. In this section, the methods

A testing environment was created to evaluate a considerable amount of pairwise encounters between aircraft. This testing environment was used to identify potential controller failures. To accomplish this, various initial relative headings and relative angles were tested. In all cases, the initial location and heading of one aircraft was held constant. Then, by placing the intruding vehicle at a relative angle of 90 and just outside the vehicle's sensing radius, as shown in Figure 9, we evaluated the interactions for 720 initial intruder heading values. This was then repeated four more times by changing the initial relative angle between 90 (to the right) and 0 (straight ahead). A visualization of these scenarios is shown in Figure 9. Note that the radius

Although intruders can approach a vehicle from the left, that is, a relative angle between 0 and 90, symmetry allows us to limit the initial relative angles to lie between 90 and 0. To verify, a sample scenario was tested for the full field of view. In all cases, the trajectories of the vehicles were symmetric to one another (i.e., when reflected across the vehicle's initial

Figure 9. Separation and avoidance testing scenarios between ownship (triangle) and intruder UAS for multiple different

of the sensing semi-circle for the separation assurance tests was set at 0.4 nmi.

5.1. Testing environment

246 Modern Fuzzy Control Systems and Its Applications

5.1.1. Separation assurance

heading).

initial positions (circles).

The collision avoidance platform was tested in the same manner as the separation assurance. The difference being that the sensing radius for the interaction tests as shown in Figure 9 is 0.1 nmi. Thus, the initial position of each intruder scenario was just outside of this sensing radius.

### 5.2. Full simulation environment

#### 5.2.1. Airspace description

The simulation environment created for this study models a portion of the US airspace over central Ohio. A depiction of the selected airspace can be seen in Figure 10. This airspace covers approximately 2500 square miles where sUAS can operate at a maximum altitude of 400 ft.

#### 5.2.2. Mission types and objectives

Many sUAS will be active throughout this airspace, each with individual missions, such as, precision agriculture, forest monitoring, roadway surveillance, disaster management, and package delivery. During simulation runs, the UAS will travel to various waypoints to fulfill their assigned missions. After visiting all waypoints for a given mission, the aircraft will return to their respective starting locations. For more on the mission types, please refer to Refs. [11, 12].

Figure 10. Simulation platform example.

#### 5.2.3. Scalability and operations

The full simulation environment can accommodate any number of sUAS and missions. However, these numbers were constant throughout testing. A maximum of 184 sUAS can be airborne at any given time. This is a result of having centralized control of the aircraft. In a realistic environment, some of the computationally heavy components can be handled in parallel onboard each individual sUAS.

#### 5.2.4. Aircraft models

Two vehicle platforms were used in all simulations: fixed wing and quad-rotor. Kinematic models were developed for each within the constraints of the environment. These constraints include maximum turn rate, maximum climb/decent rate, minimum/maximum speed, and maximum altitude.

For this study, each aircraft is assumed to climb at 4 ft./sec upon takeoff. The vehicles will continue to climb at this rate until an altitude of 400 ft. is reached. When vehicles are in conflict and are required to make trajectory modifications, all adjustments are constrained to lateral deviations in flight path (i.e., speed and altitude modifications are not used). Therefore, UAS are limited to level, two-dimensional flight during conflict resolution scenarios. Using this assumption, the maximum turn rate for each fixed wing vehicle is described in Eq. (24).

$$
\dot{W}\_{\text{max}} = \frac{g\sqrt{n^2 - 1}}{V} \tag{24}
$$

All fixed wing vehicles travel at 60 knots and have a maximum load factor of 3.5. Therefore, they are constrained to a maximum turn rate of 61.06 deg/sec. The multi-rotor systems, however, can travel at a maximum airspeed of 38 knots. Due to the nature of the quad-rotor sUAS, Eq. (24) does not accurately model the maximum turn rate for this vehicle type and it is assumed that the aircraft can yaw at a maximum rate of 45 deg/sec. Also, each aircraft is assumed to have the capability to detect an intruding aircraft at a distance of 0.1 nmi if within a 180� field of view in front of the aircraft (i.e., �90� ≤ RA<sup>1</sup> ≤ 90�).

As previously described in Table 1, the collision threshold is defined by aircraft coming within 60 m laterally and 50 ft. vertically of one other. To track the vehicles, ground-based sensors for detecting aircraft have been dispersed in the airspace. These sensors, along with telemetry data, provide continuous and reliable vehicle state information (i.e., position and velocity). This capability allows the implementation of global separation assurance practices. More details about the simulation environment and its properties can be found in Refs. [11, 12]. (NOTE: The collision buffer value was used to accommodate for position uncertainties while tracking the sUAS. In this study, the avoidance system has perfect knowledge of all vehicle state information acquired from ground based and onboard sensors.)

## 6. Results

5.2.3. Scalability and operations

248 Modern Fuzzy Control Systems and Its Applications

5.2.4. Aircraft models

maximum altitude.

Eq. (24).

sensors.)

parallel onboard each individual sUAS.

The full simulation environment can accommodate any number of sUAS and missions. However, these numbers were constant throughout testing. A maximum of 184 sUAS can be airborne at any given time. This is a result of having centralized control of the aircraft. In a realistic environment, some of the computationally heavy components can be handled in

Two vehicle platforms were used in all simulations: fixed wing and quad-rotor. Kinematic models were developed for each within the constraints of the environment. These constraints include maximum turn rate, maximum climb/decent rate, minimum/maximum speed, and

For this study, each aircraft is assumed to climb at 4 ft./sec upon takeoff. The vehicles will continue to climb at this rate until an altitude of 400 ft. is reached. When vehicles are in conflict and are required to make trajectory modifications, all adjustments are constrained to lateral deviations in flight path (i.e., speed and altitude modifications are not used). Therefore, UAS are limited to level, two-dimensional flight during conflict resolution scenarios. Using this assumption, the maximum turn rate for each fixed wing vehicle is described in

> <sup>Ψ</sup>\_ max <sup>¼</sup> <sup>g</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>n</sup><sup>2</sup> � <sup>1</sup> <sup>p</sup>

All fixed wing vehicles travel at 60 knots and have a maximum load factor of 3.5. Therefore, they are constrained to a maximum turn rate of 61.06 deg/sec. The multi-rotor systems, however, can travel at a maximum airspeed of 38 knots. Due to the nature of the quad-rotor sUAS, Eq. (24) does not accurately model the maximum turn rate for this vehicle type and it is assumed that the aircraft can yaw at a maximum rate of 45 deg/sec. Also, each aircraft is assumed to have the capability to detect an intruding aircraft at a distance of 0.1 nmi if within

As previously described in Table 1, the collision threshold is defined by aircraft coming within 60 m laterally and 50 ft. vertically of one other. To track the vehicles, ground-based sensors for detecting aircraft have been dispersed in the airspace. These sensors, along with telemetry data, provide continuous and reliable vehicle state information (i.e., position and velocity). This capability allows the implementation of global separation assurance practices. More details about the simulation environment and its properties can be found in Refs. [11, 12]. (NOTE: The collision buffer value was used to accommodate for position uncertainties while tracking the sUAS. In this study, the avoidance system has perfect knowledge of all vehicle state information acquired from ground based and onboard

a 180� field of view in front of the aircraft (i.e., �90� ≤ RA<sup>1</sup> ≤ 90�).

<sup>V</sup> <sup>ð</sup>24<sup>Þ</sup>

#### 6.1. Avoidance systems testing

For each pairwise encounter shown in Figure 9, the closest point of approach (CPA) was found. The CPA is defined as the minimum recorded distance between the two vehicles throughout the entire encounter. For both the collision avoidance and separation assurance platforms, a total of four different vehicle platform trials were conducted: fixed vs. fixed, quad vs. quad, quad vs. fixed, and fixed vs. quad. The first vehicle type designation represents the ownship vehicle's type in Figure 9 (i.e., has the same starting position and heading for all tested cases), whereas, the second vehicle platform designation represents the intruder vehicle's type (i.e., initial conditions change for each tested scenario).

To measure the effectiveness of the avoidance logic, the minimum CPA was recorded for all initial relative angles tested. In addition, the total number of collisions for each case were tallied, and for the separation assurance case, the total number of LOSs. As a reminder, a collision is deemed by two vehicles coming within 60 m of one another, and an LOS is when two vehicles come with 0.1 nmi of one another.

The results for the separation assurance testing have been shown in Table 8. Here, the "angle case" refers to the various initial relative angles, from 90 to 0, respectively. Thus, case 1 represents an intruder directly to the right, and case 5 represents an intruder directly in front of the ownship.

Although all vehicle platform combinations had more than one LOS, no LOS resulted in two vehicles colliding (i.e., a CPA less than 60 m). To try and understand where the separation assurance platform was breaking down to allow an LOS to occur, the CPA results were plotted for each initial intruder heading. When evaluating the results of each vehicle case, it was found that although several LOSs occurred, the failures tended to lie in groups near the same intruder angle depending on the initial position of the intruder. For example, in the fixed vs. quad case, it can be seen that no LOSs occurred in the first three trials. However, once the intruder was positioned such that it was nearly in front of or directly in front of the ownship,


Table 8. Separation assurance testing results.

the separation assurance platform began to fail. This was caused by the fixed wing vehicle traveling at higher speeds than the quad-rotor vehicle. Therefore, when the intruder heading was away from the ownship, the ownship tended to approach the intruder from behind and begin to pass the vehicle. When approaching the vehicle from behind, it proved difficult for the avoidance platform to solve the conflict prior to an LOS in all vehicle configurations, as shown by angle case 5.

Table 9 shows the results of the collision avoidance testing. Each case had a minimum CPA greater than 60 m, implying no collisions were found throughout testing. Although not all possible encounter scenarios have been tested, this shows that the avoidance system logic is quite robust. As seen from the results, the homogeneous quad-rotor and the fixed vs. quad cases showed promising results. They consistently had a higher minimum CPA than the other two cases. The closure rates in the fixed vs. fixed cases were higher than cases involving quadrotors. This resulted in consistently lower minimum CPA values. A noteworthy result is in the fixed vs. fixed scenario for angle case 5. The intruder being head-on and directly in front of the ownship resulted in a CPA of 60.8 m. Although this is close to the collision boundary, this shows that even in the highest closure rate scenario, the collision avoidance system was able to resolve the conflict.

#### 6.2. Formal verification

#### 6.2.1. Avoidance logic

After evaluating all the specifications outlined in Eqs. (12) through (19), JKind returned that they always held. This means that for all possible real-valued assignments to the variables (within the sensor domain limitations), the vehicles will always select the desired output action.

Although the final version of the avoidance logic adhered to all of the specifications, during development there were several cases where JKind found values that violated one or more of the specifications. These counterexamples are invaluable as they identify exact cases that result in undesired behavior. This gives way to corrections based on the counterexample conditions.


Table 9. Collision avoidance testing results.


Table 10. Avoidance logic counter-example values.

As an example of this, one of the conditions that violated a specification found during development is shown in Table 10.

These conditions mean that one vehicle is heading in the exact opposite direction of the other and there is a slight position offset between them as shown in Figure 11. One vehicle selects the turn left action (3) while the other selects the turn right action (4). This implies they are not turning away from each other. The reason for this was that the range of angles that would force vehicle 1 into the correct action was not inclusive on one of its boundaries. This then meant that the conditions forced vehicle 1 into a different action and generated this counterexample.

#### 6.2.2. Avoidance FISs

the separation assurance platform began to fail. This was caused by the fixed wing vehicle traveling at higher speeds than the quad-rotor vehicle. Therefore, when the intruder heading was away from the ownship, the ownship tended to approach the intruder from behind and begin to pass the vehicle. When approaching the vehicle from behind, it proved difficult for the avoidance platform to solve the conflict prior to an LOS in all vehicle configurations, as shown

Table 9 shows the results of the collision avoidance testing. Each case had a minimum CPA greater than 60 m, implying no collisions were found throughout testing. Although not all possible encounter scenarios have been tested, this shows that the avoidance system logic is quite robust. As seen from the results, the homogeneous quad-rotor and the fixed vs. quad cases showed promising results. They consistently had a higher minimum CPA than the other two cases. The closure rates in the fixed vs. fixed cases were higher than cases involving quadrotors. This resulted in consistently lower minimum CPA values. A noteworthy result is in the fixed vs. fixed scenario for angle case 5. The intruder being head-on and directly in front of the ownship resulted in a CPA of 60.8 m. Although this is close to the collision boundary, this shows that even in the highest closure rate scenario, the collision avoidance system was able to

After evaluating all the specifications outlined in Eqs. (12) through (19), JKind returned that they always held. This means that for all possible real-valued assignments to the variables (within the sensor domain limitations), the vehicles will always select the desired output

Although the final version of the avoidance logic adhered to all of the specifications, during development there were several cases where JKind found values that violated one or more of the specifications. These counterexamples are invaluable as they identify exact cases that result in undesired behavior. This gives way to corrections based on the counterexample conditions.

Angle Case Min CPA (m) Min CPA (m) Min CPA (m) Min CPA (m)

 132.1 138.9 154.7 125.3 111.9 125.2 133.3 121.0 101.9 113.5 122.9 111.1 82.9 104.4 103.5 97.8 60.8 96.7 98.9 99.0

Fixed vs. Fixed Quad vs. Quad Fixed vs. Quad Quad vs. Fixed

by angle case 5.

250 Modern Fuzzy Control Systems and Its Applications

resolve the conflict.

6.2. Formal verification

Table 9. Collision avoidance testing results.

6.2.1. Avoidance logic

action.

Similarly, after evaluating the specifications in Eqs. (20) through (23), Z3 showed that they were all unsatisfiable. This shows that the FLCs will always make the ownship turn away from an intruder.

#### 6.3. Full simulation results

To test the algorithms in a dynamic environment, simulations were run both with the separation assurance mitigations and then without. The number of LOSs and number of collisions were recorded in order to directly compare the mitigated and unmitigated cases.

#### 6.3.1. Unmitigated study

For this unmitigated study, the separation assurance system was disabled. The results of this study are shown in Table 11.

Figure 11. Counterexample showing head-on vehicles turning into each other.


Table 11. Results for simulation without separation assurance.

Over the span of 25,116 flight hours, there were 26,576 recorded violations of the 0.4 nmi separation threshold. These resulted in 8263 LOSs. The collision avoidance algorithm was employed for all except for 11 LOS occurrences. In those cases, the vehicles were outside of one another's field of view, thus the collision avoidance system was not used. The collision avoidance system was 99.98% successful at resolving conflicts.

The only collisions that occurred throughout simulation can be attributed to the restriction on the detection sensor field of view and having no memory of state time histories. Therefore, if two vehicles were nearly parallel and directly beside one another, they would turn to resolve the conflict (i.e., turn away from one another). However, this turning again puts each intruder outside of the other vehicle's field of view. The lack of state memory combined with no sensor input caused a switch back to their navigation controllers. The navigation controller caused them to go back toward one another. Since the navigation controller had a higher turn rate output than the avoidance output, this cycle would continue (each vehicle turning away then toward) until they converged and were within 60 m of one another.

#### 6.3.2. Mitigated study

For this study, the separation assurance features were enabled to help mitigate the risk of having an LOS. Although fewer LOSs were expected with the mitigations enabled, some LOSs were expected due to sub-optimal performance in head-on and trailing situations. The results of this mitigated study are shown in Tables 12 and 13.

Table 12 presents the results of the separation assurance platform. In this mitigated study, only 9277 flight hours were recorded. Thus, sUAS were able to complete their respective missions in a shorter period of time. Throughout the simulation aircraft came within 0.4 nmi on 33,550 occasions. However, of those instances, the separation assurance system predicted an LOS to occur within 2 min only 14,750 times. Of these resolution advisories, only 75.74% were successful, resulting in 3579 LOSs and 0.39 LOSs per flight hour. Although this number is slightly larger than the number of LOSs per flight hour in the unmitigated study, this additional layer of avoidance kept vehicles from entering any scenarios that resulted in collision. Thus, the overall safety of the UTM system has been improved.


Table 12. Separation assurance results.


Table 13. Collision avoidance results.

If an LOS occurred and the vehicles were within one another's sensor ranges, the sense and avoid software would activate. The results of the collision avoidance software can be seen in Table 13. Of the 3568 encounters, the collision avoidance software was 100.00% successful at resolving conflicts.

## 7. Conclusion

Over the span of 25,116 flight hours, there were 26,576 recorded violations of the 0.4 nmi separation threshold. These resulted in 8263 LOSs. The collision avoidance algorithm was employed for all except for 11 LOS occurrences. In those cases, the vehicles were outside of one another's field of view, thus the collision avoidance system was not used. The collision

The only collisions that occurred throughout simulation can be attributed to the restriction on the detection sensor field of view and having no memory of state time histories. Therefore, if two vehicles were nearly parallel and directly beside one another, they would turn to resolve the conflict (i.e., turn away from one another). However, this turning again puts each intruder outside of the other vehicle's field of view. The lack of state memory combined with no sensor input caused a switch back to their navigation controllers. The navigation controller caused them to go back toward one another. Since the navigation controller had a higher turn rate output than the avoidance output, this cycle would continue (each vehicle turning away then

For this study, the separation assurance features were enabled to help mitigate the risk of having an LOS. Although fewer LOSs were expected with the mitigations enabled, some LOSs were expected due to sub-optimal performance in head-on and trailing situations. The results

Table 12 presents the results of the separation assurance platform. In this mitigated study, only 9277 flight hours were recorded. Thus, sUAS were able to complete their respective missions in a shorter period of time. Throughout the simulation aircraft came within 0.4 nmi on 33,550 occasions. However, of those instances, the separation assurance system predicted an LOS to occur within 2 min only 14,750 times. Of these resolution advisories, only 75.74% were successful, resulting in 3579 LOSs and 0.39 LOSs per flight hour. Although this number is slightly larger than the number of LOSs per flight hour in the unmitigated study, this additional layer of avoidance kept vehicles from entering any scenarios that resulted in collision. Thus, the overall safety of the

> LOSs Separation assurance success rate

33,550 14,750 3579 75.74% 9277 0.39

success rate

Collisions Collision avoidance

3568 0 100.00% 9277 0

Number of flight

LOSs per flight

Collisions per flight hour

hour

hours

Number of flight hours

avoidance system was 99.98% successful at resolving conflicts.

252 Modern Fuzzy Control Systems and Its Applications

toward) until they converged and were within 60 m of one another.

of this mitigated study are shown in Tables 12 and 13.

Separation assurance maneuvers

6.3.2. Mitigated study

UTM system has been improved.

Table 12. Separation assurance results.

Table 13. Collision avoidance results.

Separation less than

Collision avoidance maneuvers

0.4 nmi

In this work, multiple fuzzy logic controllers and decision-making systems were used in conjunction to prevent potential losses of separation in a congested, three-dimensional airspace. This simulation environment allowed for extensive encounter scenarios between heterogeneous vehicles to test the two conflict resolution systems. First, a sense and avoid system was developed to prevent potential collisions using only current state information and without communication between vehicles. Next, a separation assurance platform was developed to further mitigate the risk of a potential collision. This platform uses global aircraft state information to predict if two aircraft will have an LOS within a given look-ahead time. If an LOS was predicted, the system would issue necessary resolution advisories to the proper aircraft to prevent an LOS.

Once the controllers were developed, numerical simulations and formal methods were used to verify the controllers performed as expected. Using a formal methods approach, we could show that the controller output was always in the correct direction (i.e., always performed as expected). In addition, we were able to verify that in all pairwise encounter scenarios between sUAS, the actions of each vehicle were such that they would never turn toward one another when avoiding a collision.

After a formal methods approach verified the control logic behavior and fuzzy logic controller outputs, numerical simulations were conducted. In all simulations, the avoidance system had perfect knowledge of all vehicle state information (i.e., speed, heading, and location). For the collision avoidance scenarios tested, the fuzzy system was successful at resolving all potential conflicts for both the homogeneous and heterogeneous cases. However, the separation assurance platform had trouble resolving certain types of encounter scenarios. Thus, it sometimes would not prevent an LOS between vehicles. However, when an LOS occurred, the collision avoidance system again prevented any mid-air collisions from occurring.

Several full simulation environment missions were also run to evaluate the effectiveness of the avoidance algorithms. These missions included cases where the separation assurance mitigations were both enabled and disabled. Overall, the results of this experiment were as expected. In the mitigated study, no collisions between aircraft occurred. However, when the mitigations were removed, vehicles encountered scenarios where the collision avoidance system could not prevent a collision. These collisions were not due to the collision avoidance logic or the fuzzy logic controllers, but were attributed to the limited vehicle sensor performance and lack of memory.

For future work, we aim to improve upon the separation assurance techniques to prevent an LOS in all encounter scenarios. Also, representing the system with a higher fidelity model of the environment in the formal methods tools would allow for more complete specifications (i.e., vehicles never lose separation) and then identify cases that violate them. In addition, since the avoidance system had perfect vehicle state information, we would like to introduce a level of uncertainty to the sensor models. Finally, we wish to implement the proposed avoidance software into hardware testing environments.

#### Appendix: Nomenclature


## Author details

Brandon Cook1,2\*, Tim Arnett2 and Kelly Cohen<sup>2</sup>

\*Address all correspondence to: cookb9@mail.uc.edu

1 NASA Ames Research Center, Moffett Field, CA, USA

2 Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati, OH, USA

## References


[3] Bamburry D. Drones: Designed for product delivery. Design Management Review. 2015;26(1):40-48. DOI: 10.1111/drev.10313

(i.e., vehicles never lose separation) and then identify cases that violate them. In addition, since the avoidance system had perfect vehicle state information, we would like to introduce a level of uncertainty to the sensor models. Finally, we wish to implement the proposed avoidance

software into hardware testing environments.

pCPA Predicted Closest Point of Approach

SAT Satisfiability (Boolean) SMT Satisfiability Modulo Theories sUAS Small Unmanned Aerial System tLOS Time to Loss of Separation UAS Unmanned Aerial System UTM UAS Traffic Management

Brandon Cook1,2\*, Tim Arnett2 and Kelly Cohen<sup>2</sup>

Cincinnati, Cincinnati, OH, USA

DOI: 10.1109/AERO.2007.352737

Ecology and Management. 2006;234(1):S263

\*Address all correspondence to: cookb9@mail.uc.edu

1 NASA Ames Research Center, Moffett Field, CA, USA

2 Department of Aerospace Engineering and Engineering Mechanics, University of

[1] Cook K. The silent force multiplier: The history and role of UAVs in warfare. Proceedings of the IEEE Aerospace Conference; 3–10 March 2007; Big Sky, MT, USA. IEEE. 2007:1-7.

[2] Ollero A, Merino L. Unmanned aerial vehicles as tools for forest-fire fighting. Forest

Appendix: Nomenclature

CPA Closest Point of Approach FIS Fuzzy Inference System FLC Fuzzy Logic Controller FOL First Order Logic LOS Loss of Separation NAS National Airspace System

254 Modern Fuzzy Control Systems and Its Applications

Author details

References


## **Precision Improvement in Inertial Miniaturized Navigators Based on Fuzzy Logic Denoising of Sensors Signals**

Teodor Lucian Grigorie and Ruxandra Mihaela Botez

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67966

#### **Abstract**

[19] Hagen GE. Verifying safety properties of Lustre programs: An SMT-based approach. Ann

[20] Ghassabani E, Gacek A, Whalen MW. Efficient generation of inductive validity cores for safety properties. In: Proceedings of the 2016 24th ACM SIGSOFT International Sympo-

[21] De Moura L, Bjørner N. Z3: An efficient SMT solver. In: International Conference on Tools and Algorithms for the Construction and Analysis of Systems. 2008 Mar 29. 337-340.

[22] Halbwachs N, Caspi P, Raymond P, Pilaud D. The synchronous data flow programming

sium on Foundations of Software Engineering. 2016 Nov 1:314-325. ACM

language LUSTRE. Proceedings of the IEEE. 1991 Sep;79(9):1305-1320

Arbor, MI: ProQuest; 2008

256 Modern Fuzzy Control Systems and Its Applications

Berlin Heidelberg: Springer

The chapter presents a new strategy to improve the precision of the inertial navigators processing in a fuzzy manner the signals provided by the miniaturized sensors equipping their inertial measurement units (IMU). To apply the developed technique, the hardware component of the inertial measurement units was specifically designed to include some redundant clusters of inertial sensors disposed in linear configurations on the measurement axes. The algorithm acts at the level of each detection cluster designed to measure an acceleration or angular speed along with an IMU axis by fusing the data obtained from the sensors in respective cluster. Based on the standard deviations of the sensors signals estimated for a data frame with a well-known size, the fuzzy logic mechanism provides a set of weights associated with each sensor in cluster, which are further used to fuse the data acquired from sensors at the current time. The algorithm has an adaptive character, the data frame used to estimate the standard deviations of the sensors signals being permanently updated with the new sensors measurements, and, in this way, the weights associated with each sensor are reestimated at each measurement step.

**Keywords:** inertial navigation, miniaturized sensors, data fusion, fuzzy logic, redundant configurations, noise reduction, precision improvement

## **1. Introduction**

The permanent change at the level of the sensor fabrication technologies in the last two decades aiming especially toward miniaturization produced a significant change in the navigation and

positioning systems field from the point of view of strap-down inertial navigation systems (SDINS). Therefore, more and more applications have been developed and are currently developed starting from this concept and based on miniaturized inertial measurement units and miniaturized data processing systems. These new technologies, MEMS (micro-electro-mechanical systems), NEMS (nano-electro-mechanical systems), MOEMS (micro-opto-electro-mechanical systems), or NOEMS (nano-opto-electro-mechanical systems), are currently successfully applied in the inertial sensors, dedicated to a large part of the SDINS applications. Actually, due to the miniaturization, the SDINS systems application field has been seriously extended beyond the classical navigation systems equipping the vehicles in the aerospace industry, at the level of land-vehicle applications, robotics, medicine, and assisted living [1–4]. The distribution of the inertial sensor fabrication technologies on future applications accuracy classes can be organized as in **Figure 1** [1]. It can be easily observed that the accuracy level for NEMS technologies (possibly NOEMS) are still at the commercial applications level; IOG—integrated optics gyro, IFOG—interferometric fiber optic gyro, HRG—hemispherical resonant gyro, PCF-FOG—photonic crystal fiber-fiber optic gyro.

Included in the dead-reckoning navigation systems category, the SDINS was succeeded on the market due to its availability to provide the vehicle speed and position with abundant dynamic information, due to the possibility to calculate at an extremely high rate comparatively with the GPS system, and due to its excellent short-term performance. Having


**Figure 1.** Inertial sensor technologies on the application accuracy classes.

characteristics complementary to satellite positioning systems, in time, it became an important component of the modern navigation systems near the GPS [5–6]. If the two systems are used in an integrated configuration, for a short time, derived from the quality of the used inertial sensors, the inertial navigator can overcome the nonavailability of GPS signals and can maintain in this way the navigation solution at a high quality; a well-designed INS/GPS navigator provides improved performance in terms of accuracy, availability, and reliability in front of a simple GPS system.

positioning systems field from the point of view of strap-down inertial navigation systems (SDINS). Therefore, more and more applications have been developed and are currently developed starting from this concept and based on miniaturized inertial measurement units and miniaturized data processing systems. These new technologies, MEMS (micro-electro-mechanical systems), NEMS (nano-electro-mechanical systems), MOEMS (micro-opto-electro-mechanical systems), or NOEMS (nano-opto-electro-mechanical systems), are currently successfully applied in the inertial sensors, dedicated to a large part of the SDINS applications. Actually, due to the miniaturization, the SDINS systems application field has been seriously extended beyond the classical navigation systems equipping the vehicles in the aerospace industry, at the level of land-vehicle applications, robotics, medicine, and assisted living [1–4]. The distribution of the inertial sensor fabrication technologies on future applications accuracy classes can be organized as in **Figure 1** [1]. It can be easily observed that the accuracy level for NEMS technologies (possibly NOEMS) are still at the commercial applications level; IOG—integrated optics gyro, IFOG—interferometric fiber optic gyro, HRG—hemispherical resonant gyro, PCF-FOG—pho-

Included in the dead-reckoning navigation systems category, the SDINS was succeeded on the market due to its availability to provide the vehicle speed and position with abundant dynamic information, due to the possibility to calculate at an extremely high rate comparatively with the GPS system, and due to its excellent short-term performance. Having

tonic crystal fiber-fiber optic gyro.

258 Modern Fuzzy Control Systems and Its Applications

**Figure 1.** Inertial sensor technologies on the application accuracy classes.

From another point of view, the inertial navigators are subjected to various error sources, their main deficiency residing in the great accuracy degradation over time mainly due to the quality of the used inertial sensors [7–9]. New miniaturization technologies plays an important role in the reduction of the cost and the volume of the inertial sensors, which, apparently, offers an important advantage for the miniaturized inertial sensors, but, in fact, the fabrication processes of such sensors make these very sensitive to the changes of the environmental conditions: temperature, pressure, electric and magnetic fields, and vibrations. Therefore, the sensors output can vary quickly, widely, and sometimes randomly and is very hard to be modeled. In many cases, this sensitivity leads to the decreased sensor performance, adding more error types, and possibly, with higher values than those of classical sensors [3, 9].

In this context, the need to develop low-cost, small-size, and high-precision INS/GPS integrated navigators, which are able to be used also in GPS challenging environments, generated few research directions in the field; the main two being [4–6, 10–14]: (1) to develop standalone accurate SDINS structures with miniaturized IMUs; (2) to develop new INS/GPS data-fusion methods based on artificial intelligence algorithms, having objectives to overcome the limitations in terms of model dependency, prior knowledge dependency, and linearization dependency.

The increase of the SDINS accuracy in standalone configuration targets, in fact, the improvement of the quality for the signals provided by the IMUs to the navigation processor through the obtaining of high quality miniaturized sensors or through the development of various IMU hardware architectures accompanied by right numerical algorithms able to produce sensor errors estimation and compensation [9, 13, 14]. Two categories of errors parasitize the inertial sensors: deterministic errors and stochastic errors. For the deterministic errors, the specialists conceived various calibration procedures, which can easily estimate and eliminate it. Unfortunately, the stochastic error estimation necessitates a more complex process and they cannot be fully removed from the sensors data. For both categories of inertial sensors (accelerometers and gyros), the most important stochastic errors are caused by noise and by instability of bias and of scale factor [7].

The main difficulty with the noise is related to the impossibility to apply a direct filtering procedure by using classical methods, because, in terms of frequency spectrum, it is superimposed on the band 0–100 Hz of the navigation useful signal [7]. As a consequence, many researches have been initiated to find alternative solutions to limit its impact on the solution of navigation. On the other way, it must not forget that at the level of the noise there are present two components, long term noise (low frequency noise) and short term noise (high frequency noise), each one with its own spectral characteristics and with its own compensation mechanism: optimal low pass filtering or using algorithms that fusing the data from multiple sensors [13, 15–19].

Our identified way to improve the quality of the inertial sensor signals is based on the use of redundant detection clusters with the sensors disposed in linear configurations along each detection axis in IMU (**Figure 2**) [9, 13]. For each cluster, the measured data are subsequently fused to provide to the navigation processor a better measured signal of acceleration or angular speed for the respective axis in IMU. Simultaneously, this structure provides the advantage to have a redundant inertial navigator in terms of the detection unit components, i.e., when one or more sensors in a detection cluster break down they are removed from calculation process but the system still remains operational. The inertial sensors in the same cluster have the sensitivity axes parallel and oriented in the same sense with the detection axis.

The proposed fuzzy logic procedure was mainly designed to produce effects at the level of the sensor noise, but some benefits were also noted at the level of bias effects reduction.

The work exposed here is related to a research project developed in Romania and financed by the Executive Unit for the Financing of Higher Education and University of Scientific Research, which aimed at the strap-down inertial navigators' precision improvement by using redundant sensor networks for their IMUs and new adaptive numerical algorithms for data fusion.

In this chapter, the information is structured as follows: Section 1 is constituted by a short introduction; Section 2 exposes the algorithm basic elements and structure, including the fuzzy logic mechanism; Section 3 shows the numerical simulation results obtained with the developed algorithm for detection clusters with various sizes and sensors types; Section 4 provides the results obtained with some experimental data acquired with detection clusters with six MEMS sensors each (assisted by an integrated GPS/INS navigator as reference positioning system) and processed for a bidimensional INS navigator; and Section 5 highlights the benefits brought by the algorithm in the form of the conclusions.

**Figure 2.** Inertial sensors disposed in linear configurations along each detection axis in IMU.

### **2. Data fusion algorithm basic elements and structure**

To describe the generalized form of the proposed algorithm, it is denoted with *n* the number of collinear sensors included in a detection cluster, as in **Figure 2**. On the other way, if *xi* (*i* = 1 *÷ n*) are the measurements of the *x* quantity provided by the *n* sensors, characterized by the standard deviations σ*<sup>i</sup>* (*i* = 1 *÷ n*), then the weighted mean obtained by applying the data fusion algorithm conducts to the *xe* estimate of the *x* quantity under the next form [14]:

$$\mathbf{x}\_{\varepsilon} = \mathbf{w}\_1 \cdot \mathbf{x}\_1 + \mathbf{w}\_2 \cdot \mathbf{x}\_2 + \dots \cdot \mathbf{w}\_n \cdot \mathbf{x}\_n = \sum\_{i=1}^{n} \mathbf{w}\_i \cdot \mathbf{x}\_p \left(\sum\_{i=1}^{n} \mathbf{w}\_i = \mathbf{1}\right) \tag{1}$$

*wi* (*i* = 1 *÷ n*) are the weights of the *xi* (*i = 1 ÷ n*) measurements. For each sensor, the standard deviation is evaluated starting from the last *m* samples acquired from it. The algorithm has an adaptive character, the data frame used to estimate the standard deviations of the sensors signals being permanently updated with the new sensor measurements, and, in this way, the weights associated with each sensor are reestimated at each measurement step. The estimation of the new weight set is realized based on a fuzzy logic mechanism. The necessary data frame used in the estimation of sensors standard deviations is built using a FIFO (first in first out) buffer with *n* channels and able to memorize *m* successive samples on each channel. In this way, at each measurement time, the last column in the data frame goes out from the buffer and a new one comes in; two consecutive data frames are superposed with (*m* − 1) samples [13]. The standard deviations of the *n*-independent channels result with the next equation [13, 14]: \_\_1

 *σ<sup>i</sup>* (*<sup>k</sup>* ) <sup>=</sup> <sup>√</sup> \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ *m* ∑*<sup>j</sup>*=1 *<sup>m</sup>* [ *xi* (*j* ) −¯¯*xi* (*k* ) ]2 , (*<sup>i</sup>* <sup>=</sup> <sup>1</sup>¯, *<sup>n</sup>* ) ; (2)

*xi* (*j*) is the *j*th measurement provided by the *i*th sensor in the *k*th data frame; σ*<sup>i</sup>* (*k*) is the standard deviation for the *k*th data frame, which characterize the *i*th sensor; and ¯¯ *x i x*′ *i* (*k*) is the arithmetic mean of the *i*th sensor data for the *k*th data frame [14]. The value of σ*<sup>i</sup>* (*k*) is used to fuse the data at the next sensor reading.

According to the proposed algorithm, a fuzzy logic controller is used in each channel of a detection cluster, and, based on the standard deviation, σ*<sup>i</sup>* (*k*) of the *m* consecutive samples acquired from the sensor in the *k*th data frame provides a weight *wi* fuzzy(*k*)(*<sup>i</sup>* <sup>=</sup> <sup>1</sup>¯, *<sup>n</sup>*1, *<sup>n</sup>*) for the respective sensor. Based on the fact that in a weighted sum, the sum of all parts weight should be equal to 1, the sensors weights are recalculated and result with the equations [13, 14]):

$$\text{zw}\_1(\mathbf{k}) = \frac{w\_i^{\text{fuzzy}}(\mathbf{k})}{\sum\_{i=1}^n w\_i^{\text{fuzzy}}(\mathbf{k})} \dots \text{, } \text{zw}\_n(\mathbf{k}) = \frac{w\_{\text{tiny}}^{\text{fuzzy}}(\mathbf{k})}{\sum\_{i=1}^n w\_{\text{tiny}}^{\text{heavy}}(\mathbf{k})} \text{.} \tag{3}$$

As a consequence, by calculating the weighted sum from the data fusion equation, at the time *t <sup>k</sup>*+1, the *xe* estimate of *x* is [13, 14]:

$$\mathbf{x}\_{\iota}(k+1) = w\_{\iota}(k) \cdot \mathbf{x}\_{\iota}(k+1) + w\_{\iota}(k) \cdot \mathbf{x}\_{\iota}(k+1) + \dots + w\_{\iota}(k) \cdot \mathbf{x}\_{\iota}(k+1),\tag{4}$$

i.e.,

Our identified way to improve the quality of the inertial sensor signals is based on the use of redundant detection clusters with the sensors disposed in linear configurations along each detection axis in IMU (**Figure 2**) [9, 13]. For each cluster, the measured data are subsequently fused to provide to the navigation processor a better measured signal of acceleration or angular speed for the respective axis in IMU. Simultaneously, this structure provides the advantage to have a redundant inertial navigator in terms of the detection unit components, i.e., when one or more sensors in a detection cluster break down they are removed from calculation process but the system still remains operational. The inertial sensors in the same cluster have the sensitivity axes parallel and oriented in the same sense with the detection axis.

260 Modern Fuzzy Control Systems and Its Applications

The proposed fuzzy logic procedure was mainly designed to produce effects at the level of the

The work exposed here is related to a research project developed in Romania and financed by the Executive Unit for the Financing of Higher Education and University of Scientific Research, which aimed at the strap-down inertial navigators' precision improvement by using redundant sensor networks for their IMUs and new adaptive numerical algorithms for data fusion. In this chapter, the information is structured as follows: Section 1 is constituted by a short introduction; Section 2 exposes the algorithm basic elements and structure, including the fuzzy logic mechanism; Section 3 shows the numerical simulation results obtained with the developed algorithm for detection clusters with various sizes and sensors types; Section 4 provides the results obtained with some experimental data acquired with detection clusters with six MEMS sensors each (assisted by an integrated GPS/INS navigator as reference positioning system) and processed for a bidimensional INS navigator; and Section 5 highlights the

sensor noise, but some benefits were also noted at the level of bias effects reduction.

benefits brought by the algorithm in the form of the conclusions.

**Figure 2.** Inertial sensors disposed in linear configurations along each detection axis in IMU.

$$\mathbf{x}\_{\circ}(k+1) = \sum\_{l=1}^{\nu} w\_{l}(k) \cdot \mathbf{x}\_{l}(k+1), \left(\sum\_{l=1}^{\nu} w\_{l}(k) = 1\right). \tag{5}$$

Based on the previous considerations, the architecture of the data fusion algorithm results as in **Figure 3** [14]. The chosen fuzzy logic controllers are based on rules set that created

**Figure 3.** Data fusion algorithm.

a proportional dependence between the quality of the sensor signals and their associated weights, e.g., a "P" fuzzy logic controller. Therefore, the implemented fuzzy logic controllers established an inverse proportionality between the calculated standard deviations and sensor-associated weights.

The aim is to obtain a simpler fuzzy controller in order to reduce at minimum the computing time at each sample in order to implement the controller on a cheaper microcontroller. The way to reduce the computing time is to reduce on one hand the number of the membership functions and on the other hand to simplify these functions. The structure of fuzzy controller realized in Matlab/Simulink is presented in **Figure 4** [20, 21].

**Figure 4.** Fuzzy controller structure.

The universe of discourse for the controller input has considered a [0, 2] interval, where nine membership functions (*in1* to *in9*) with Gaussian shapes were uniformly distributed. In the same time, nine membership functions (*out1* to *out9*) with similar shapes were considered in the [0, 1] interval, taken as universe of discourse for the output. **Figure 5** presents the input/output membership functions.

Precision Improvement in Inertial Miniaturized Navigators Based on Fuzzy Logic Denoising of Sensors Signals http://dx.doi.org/10.5772/67966 263

**Figure 5.** Membership functions (mfs) for input and output.

In the fuzzification process, the next nine rules were defined as: (1) If (*input* is *in1*) then (*output* is *out9*); (2) If (*input* is *in2*) then (*output* is *out8*); (3) If (*input* is *in3*) then (*output* is *out7*); (4) If (*input* is *in4*) then (*output* is *out6*); (5) If (*input* is *in5*) then (*output* is *out5*); (6) If (*input* is *in6*) then (*output* is *out4*); (7) If (*input* is *in7*) then (*output* is *out3*); (8) If (*input* is *in8*) then (*output* is *out2*); and (9) If (*input* is *in9*) then (*output* is *out1*). Widely accepted for capturing expert knowledge, a Mamdani controller type was used due to its simple structure of "min-max" operations [21]; and for defuzzification, the centroid method was applied.

#### **3. Numerical simulation results**

a proportional dependence between the quality of the sensor signals and their associated weights, e.g., a "P" fuzzy logic controller. Therefore, the implemented fuzzy logic controllers established an inverse proportionality between the calculated standard deviations and

The aim is to obtain a simpler fuzzy controller in order to reduce at minimum the computing time at each sample in order to implement the controller on a cheaper microcontroller. The way to reduce the computing time is to reduce on one hand the number of the membership functions and on the other hand to simplify these functions. The structure of fuzzy controller

The universe of discourse for the controller input has considered a [0, 2] interval, where nine membership functions (*in1* to *in9*) with Gaussian shapes were uniformly distributed. In the same time, nine membership functions (*out1* to *out9*) with similar shapes were considered in the [0, 1] interval, taken as universe of discourse for the output. **Figure 5** presents the input/output

realized in Matlab/Simulink is presented in **Figure 4** [20, 21].

sensor-associated weights.

**Figure 3.** Data fusion algorithm.

262 Modern Fuzzy Control Systems and Its Applications

membership functions.

**Figure 4.** Fuzzy controller structure.

To simulate the algorithm, some models in Matlab/Simulink for detection clusters with various sizes and sensors types were realized. An example of such model is shown in **Figure 6**, being realized for a cluster with four sensors by the same type, with errors software modeled by the blocks "Model Acc" placed at the input of the simulation model. These models, realized by the authors, are based on the sensor data sheets and on the IEEE equivalent models for the inertial sensors [22, 23]. Accelerometers were modeled as in **Figure 7**, the obtained model having inputs, such as acceleration *ai* , applied along the sensitive axis, and the cross-axis acceleration *ac* , and as output the perturbed acceleration *a* [22, 23].

**Figure 6.** Simulation model with four sensors in cluster.

**Figure 7.** Accelerometers Matlab/Simulink model and its interface.

Studying the data sheets for various acceleration sensors, it was observed that a part of the included parameters is provided by using an interval inside which they can vary arbitrary. For example, for the bias it is provided a maximum absolute value (*B*) which is considered as percent from span, for the cross-axis sensitivity it is provided a maximal value (*kc* ) considered as a percent from the cross-axis acceleration (*ac* ), for the scale factor calibration error the producers give a maximum absolute value (Δ*K*) under the form of a percent from the scale factor (*K*), and for the noise is provided the maximum value of its density. Therefore, according to **Figure 7**, in the model, the "rand(1)" MATLAB function is used three times to generate a random value in the (−*B, B*) interval for the bias, a random value in the (0, *kc* ) interval for the cross-axis sensitivity, and a random value in the (−Δ*K*, Δ*K*) interval for the scale factor calibration error. The noises are realized using the SIMULINK block "Band-Limited White Noise" and MATLAB function "RandSeed" through the generation of a random value of the density in the (80%⋅ν*<sup>d</sup>* , ν*<sup>d</sup>* ) interval [24].

In a simulation example at null accelerations as inputs, graphical characteristics resulted are shown in **Figures 8**–**10**. **Figure 8** shows the sensor measurements from "Acc-1" to "Acc-4" and also the results of the accelerometer data fusion "Acc-f". In the simulation, there was considered just the sensor noise overlapped on an ideal null input for four sensors with a span of 18 g, a bandwidth of 2500 Hz each, and with various noise densities taken aleatory between 280 and 380 μg/Hz1/2. The noise pattern of the data fusion resulted a signal that proves an important reduction of the noise level, a similar observation resulting from **Figure 9** also depicting the evolution in time of the standard deviations for the sensors signals and for the estimate *xe* . On the other way, **Figure 9** proves that the adaptive character of the algorithm is maintained, the standard deviations values being updated at each simulation time step. The best sensor in the cluster was sensor #3, the standard deviation of the fused signal being two times smaller (approx. 3⋅10–3 m/s2 ) than the raw values of the standard deviations of the sensors signals (approx. 6⋅10–3 m/s2 ). **Figure 10**, depicting the values of the sensor weights estimated at each calculation step, also proves that the best sensor in the cluster was the sensor #3, and it receives the biggest values from the four weights during the entire numerical simulation.

Precision Improvement in Inertial Miniaturized Navigators Based on Fuzzy Logic Denoising of Sensors Signals http://dx.doi.org/10.5772/67966 265

**Figure 8.** The sensor measurements and the data fusion results.

Studying the data sheets for various acceleration sensors, it was observed that a part of the included parameters is provided by using an interval inside which they can vary arbitrary. For example, for the bias it is provided a maximum absolute value (*B*) which is considered

producers give a maximum absolute value (Δ*K*) under the form of a percent from the scale factor (*K*), and for the noise is provided the maximum value of its density. Therefore, according to **Figure 7**, in the model, the "rand(1)" MATLAB function is used three times to generate

cross-axis sensitivity, and a random value in the (−Δ*K*, Δ*K*) interval for the scale factor calibration error. The noises are realized using the SIMULINK block "Band-Limited White Noise" and MATLAB function "RandSeed" through the generation of a random value of the density

In a simulation example at null accelerations as inputs, graphical characteristics resulted are shown in **Figures 8**–**10**. **Figure 8** shows the sensor measurements from "Acc-1" to "Acc-4" and also the results of the accelerometer data fusion "Acc-f". In the simulation, there was considered just the sensor noise overlapped on an ideal null input for four sensors with a span of 18 g, a bandwidth of 2500 Hz each, and with various noise densities taken aleatory between 280 and 380 μg/Hz1/2. The noise pattern of the data fusion resulted a signal that proves an important reduction of the noise level, a similar observation resulting from **Figure 9** also depicting the evolution in time of the standard deviations for the sensors signals and for the estimate *xe*

the other way, **Figure 9** proves that the adaptive character of the algorithm is maintained, the standard deviations values being updated at each simulation time step. The best sensor in the cluster was sensor #3, the standard deviation of the fused signal being two times smaller

calculation step, also proves that the best sensor in the cluster was the sensor #3, and it receives

the biggest values from the four weights during the entire numerical simulation.

) than the raw values of the standard deviations of the sensors signals

). **Figure 10**, depicting the values of the sensor weights estimated at each

) consid-

. On

) interval for the

), for the scale factor calibration error the

as percent from span, for the cross-axis sensitivity it is provided a maximal value (*kc*

a random value in the (−*B, B*) interval for the bias, a random value in the (0, *kc*

ered as a percent from the cross-axis acceleration (*ac*

**Figure 7.** Accelerometers Matlab/Simulink model and its interface.

264 Modern Fuzzy Control Systems and Its Applications

) interval [24].

in the (80%⋅ν*<sup>d</sup>*

(approx. 3⋅10–3 m/s2

(approx. 6⋅10–3 m/s2

, ν*<sup>d</sup>*

The sample time for the numerical simulation was 0.01 s, which means a data processing rate of 100 samples/s. Moreover, the data frame size for one detection channel was established to be *m* = 100, i.e., the buffers allow to be obtained data frames of 100 consecutive samples for each channel. In this way, any two consecutive data frames for one channel are overlapped with 99 samples. From **Figure 9**, the initialization phase of the algorithm with a duration of 1 s, time necessary for the first buffer to be full with 100 samples can be easily observed. All simulation results proved a good functioning of the algorithm which provided an important reduction of the sensor noise level, with a higher potential in the improvement of the solution of navigation accuracy when it is used in an inertial navigator.

**Figure 9.** Standard deviations for the sensor signals and for the data fusion signals.

**Figure 10.** Sensor weights.

#### **4. Results obtained with experimental data**

The proposed algorithm was also tested using some experimental data acquired with an IMU equipped with detection clusters, which included six MEMS sensors each (assisted by an integrated GPS/INS navigator as reference positioning system). For the used IMU structure, with six sensors in each detection cluster, the Matlab/Simulink implementation of the data fusion algorithm resulted as shown in **Figure 11** [14]. Grouping the algorithm model resulted in "Fuzzy-logic data fusion" block from the right side of the figure. The data fusion block inputs are the measured data obtained from the sensors in the cluster ("Si" correspond to the

**Figure 11.** Matlab/Simulink implementation of the data fusion algorithm for experimental data.

measurements *xi* , *i* = 1 ÷ 6), whereas its outputs are the fusion signal "S\_f" (the estimate *xe* ), the sensor weights "w1–w6" (*wi* , *i* = 1 ÷ 6), the standard deviation of the fusion signal "Std\_f", and the standard deviations of the sensor data "Std1–Std6".

After the fusion, the obtained data were processed in a bidimensional INS navigator in horizontal plane, software implemented as in **Figure 12** [14]. Therefore, in this testing mechanism, three detection clusters were implied: two accelerometers clusters along the longitudinal and lateral axes of the vehicle and a gyro cluster along the vertical axis of the vehicle. The model included the "Flat Earth to LLA" Matlab/Simulink block in order to obtain the vehicle coordinates also in terms of latitude and longitude. The block "Horizontal plane navigator" in the right side of the figure has been obtained by grouping the navigator software model in the left side of the figure. The inputs of the navigator are the inertial measurements provided by the accelerometers and by the gyro: longitudinal (*f\_xv*) and lateral (*f\_yv*) accelerations in vehicle reference frame, and the angular speed (*w\_zv*) along the vertical axis of the vehicle reference frame. As outputs, it provides information related to the angular orientation of the vehicle in horizontal plane (yaw angle—*psi*), to the vehicle speed and position components in local horizontal frame, *v\_xl* (North speed), *v\_yl* (East speed), *x\_l* (position in North direction), and *y\_l* (position in East direction), and to the latitude and longitude coordinates of the vehicle (*Lat* and *Lon*).

During tests, the IMU data were acquired simultaneously with the data from an integrated GPS/INS navigator used as reference positioning system; both equipments were boarded on a test car. The sensor acquired data were offline processed through fusion but also used together with the fusion signals in a Matlab/Simulink software model implementing seven bidimensional INS navigators ("Horizontal plane navigator" blocks) as in **Figure 13** [14]. It also includes three "Fuzzy-logic data fusion" blocks used to fuse the acquired data in each of the three detection clusters.

The sensors in IMU were organized in six groups, with three sensors each (two accelerometers and one gyro), one sensor from each of the three detection clusters. To avoid the complications in the data processing, the detection groups included sensors with the same number in

**Figure 12.** Software model of the bidimensional INS navigator in horizontal plane.

**4. Results obtained with experimental data**

**Figure 10.** Sensor weights.

266 Modern Fuzzy Control Systems and Its Applications

The proposed algorithm was also tested using some experimental data acquired with an IMU equipped with detection clusters, which included six MEMS sensors each (assisted by an integrated GPS/INS navigator as reference positioning system). For the used IMU structure, with six sensors in each detection cluster, the Matlab/Simulink implementation of the data fusion algorithm resulted as shown in **Figure 11** [14]. Grouping the algorithm model resulted in "Fuzzy-logic data fusion" block from the right side of the figure. The data fusion block inputs are the measured data obtained from the sensors in the cluster ("Si" correspond to the

**Figure 11.** Matlab/Simulink implementation of the data fusion algorithm for experimental data.

**Figure 13.** The evaluation Matlab/Simulink model.

the detection clusters. Six of the seven navigators in **Figure 13** ("Nav1" to "Nav6") processed data from the six detection groups, whereas the seventh navigator ("Horizontal plane navigator – Fused signals") processed data resulted from the fusion procedure applied on each of the three detection clusters. In this way, the evaluation model allows the obtaining of seven solutions of navigation, six solutions based on inertial sensor raw data (nonredundant INSs) and one solution based on inertial sensor fused data. All these solutions are further compared with the navigation solution provided by the integrated GPS/INS navigators which are used as reference positioning system.

For an evaluation situation, the experimental data acquired in each of the three detection clusters and the data fusion results for each cluster are shown in **Figure 14** for the accelerometers in the *x*-axis cluster, in **Figure 15** for the accelerometers in the *y*-axis cluster, and in **Figure 16** for the gyros in *z*-axis cluster. The data fusion signals are the last ones in each figure.

The processing of the acquired data with the model in **Figure 13** conducted at the navigation solutions components shown in the next figures. Together with the navigation solution, components have shown the associated errors, evaluated vis-à-vis of the reference navigation solution provided by the GPS/INS integrated navigator. **Table 1** centralizes the absolute maximal values of the positioning and speed errors for all six nonredundant INSs ("INS1" to "INS6" columns) and for the redundant INS ("Fusion" column), and also the mean values obtained for all these errors ("Mean value" column). In the same time, by calculating the mean of the absolute maximal values of the errors for the six nonredundant INSs in each line of **Table 1** and dividing it with the corresponding value in the "Fusion" column, the ratios in the "Mean/Fus" column have been obtained. Similarly, the "Max/Fus" and "Min/Fus" columns show the ratios between the maximal value of the tabled errors for the six nonredundant INSs in each line of **Table 1** and the corresponding value in the "Fusion" column, respectively, between the minimum value of the tabled errors for the six nonredundant INSs in each line of **Table 1** and the corresponding value in the "Fusion" column.

**Figure 14.** Experimental data and the data fusion results for accelerometers in the *x*-axis cluster.

the detection clusters. Six of the seven navigators in **Figure 13** ("Nav1" to "Nav6") processed data from the six detection groups, whereas the seventh navigator ("Horizontal plane navigator – Fused signals") processed data resulted from the fusion procedure applied on each of the three detection clusters. In this way, the evaluation model allows the obtaining of seven solutions of navigation, six solutions based on inertial sensor raw data (nonredundant INSs) and one solution based on inertial sensor fused data. All these solutions are further compared with the navigation solution provided by the integrated GPS/INS navigators which are used

For an evaluation situation, the experimental data acquired in each of the three detection clusters and the data fusion results for each cluster are shown in **Figure 14** for the accelerometers in the *x*-axis cluster, in **Figure 15** for the accelerometers in the *y*-axis cluster, and in **Figure 16**

The processing of the acquired data with the model in **Figure 13** conducted at the navigation solutions components shown in the next figures. Together with the navigation solution, components have shown the associated errors, evaluated vis-à-vis of the reference navigation solution provided by the GPS/INS integrated navigator. **Table 1** centralizes the absolute maximal values of the positioning and speed errors for all six nonredundant INSs ("INS1" to "INS6" columns) and for the redundant INS ("Fusion" column), and also the mean values obtained for all these errors ("Mean value" column). In the same time, by calculating the mean

for the gyros in *z*-axis cluster. The data fusion signals are the last ones in each figure.

as reference positioning system.

**Figure 13.** The evaluation Matlab/Simulink model.

268 Modern Fuzzy Control Systems and Its Applications

**Figure 15.** Experimental data and the data fusion results for gyros in the *z* axis cluster.

**Figure 16.** Experimental data and the data fusion results for gyros in the *z* axis cluster.

At the level of the vehicle attitude, the characteristics in **Figure 17** were obtained for the yaw angle. Also, from the point of view of the horizontal positioning, **Figure 18** presents the covered distances in the North direction, and **Figure 19** presents the covered distances in East direction. The North speed evolutions during time are depicted in **Figure 20**, whereas the speed evolutions during time in the East direction are shown in **Figure 21**. For each navigator, the incorporated "Flat Earth to LLA" block allows the calculus of the vehicle Latitude and Longitude as shown in **Figures 22** and **23**. Combining the positioning data in a horizontal plane, it resulted the vehicle trajectories (Latitude versus Longitude) and the deviations from the reference trajectory (Latitude errors versus Longitude errors), established by all seven navigators, as in **Figure 24**.

**Figure 25** depicts the time evolution of the vehicle trajectories estimated with all seven navigators, whereas **Figure 26** presents their deviations from the reference trajectory.

Both graphical results and numerical values presented in **Table 1** prove an important positioning precision improvement by using the proposed data fusion mechanism. The errors of the navigation solution are substantially reduced comparatively with the nonredundant navigation solutions. The numerical data in the "Mean/Fus" column of **Table 1** highlighted a decrease of the mean of the absolute maximal values of the errors for the six nonredundant INSs of approximately 16.6 times for the positioning in the North direction, 5.1 times for the positioning in the East direction, 2.8 times for the speed component in North direction, 1.05 times for the speed component in the East direction, and 3.8 times in the angular positioning in the horizontal plane (yaw angle). It is very important to be mentioned that the inertial sensor outputs in the experimental model of the inertial measurement unit were not previously corrected with the biases values or with other errors values. The values in the "Fusion" column of **Table 1**, related to the positioning and speed errors for the redundant navigator after 100 s have shown us that the absolute maximal values of these errors were 32.67 m for the linear


At the level of the vehicle attitude, the characteristics in **Figure 17** were obtained for the yaw angle. Also, from the point of view of the horizontal positioning, **Figure 18** presents the covered distances in the North direction, and **Figure 19** presents the covered distances in East direction. The North speed evolutions during time are depicted in **Figure 20**, whereas the speed evolutions during time in the East direction are shown in **Figure 21**. For each navigator, the incorporated "Flat Earth to LLA" block allows the calculus of the vehicle Latitude and Longitude as shown in **Figures 22** and **23**. Combining the positioning data in a horizontal plane, it resulted the vehicle trajectories (Latitude versus Longitude) and the deviations from the reference trajectory (Latitude errors versus Longitude errors), established by all seven

**Figure 16.** Experimental data and the data fusion results for gyros in the *z* axis cluster.

**Figure 25** depicts the time evolution of the vehicle trajectories estimated with all seven navi-

Both graphical results and numerical values presented in **Table 1** prove an important positioning precision improvement by using the proposed data fusion mechanism. The errors of the navigation solution are substantially reduced comparatively with the nonredundant navigation solutions. The numerical data in the "Mean/Fus" column of **Table 1** highlighted a decrease of the mean of the absolute maximal values of the errors for the six nonredundant INSs of approximately 16.6 times for the positioning in the North direction, 5.1 times for the positioning in the East direction, 2.8 times for the speed component in North direction, 1.05 times for the speed component in the East direction, and 3.8 times in the angular positioning in the horizontal plane (yaw angle). It is very important to be mentioned that the inertial sensor outputs in the experimental model of the inertial measurement unit were not previously corrected with the biases values or with other errors values. The values in the "Fusion" column of **Table 1**, related to the positioning and speed errors for the redundant navigator after 100 s have shown us that the absolute maximal values of these errors were 32.67 m for the linear

gators, whereas **Figure 26** presents their deviations from the reference trajectory.

navigators, as in **Figure 24**.

270 Modern Fuzzy Control Systems and Its Applications

**Table 1.** Error analysis for redundant INS and for nonredundant INSs. positioning in North direction, 39.27 m for the linear positioning in East direction, 2.03 m/s for the speed component in North direction, 7.14 m/s for the speed component in East direction, 2.940∙10–4 degrees for the Latitude positioning, 4.923∙10–4 degrees for the Longitude positioning, and 5.74 degrees for the angular positioning in the horizontal plane (Yaw angle).

The nonredundant inertial navigation system's best configuration was switched between the detection groups associated with the first sensor in each detection cluster (East channel and Yaw angle channel) and with the sixth sensor in each detection cluster (North channel). The worst configurations for the nonredundant INSs are for the second sensor in each detection cluster, the absolute maximal values of errors in North position channel being 990.52 m, 318.17 m in East position channel, 19.40 m/s in North speed channel, 89.141∙10–4 degrees in the Latitude position channel, and 39.886∙10–4 degrees in the Longitude channel. From the point of view of East speed channel and yaw angle channel, the worst nonredundant INS is INS4, the East speed error being 13.68 m/s and the yaw angle error being 31.10 degrees.

**Figure 17.** Yaw angle values and errors.

**Figure 18.** North positions and errors.

Precision Improvement in Inertial Miniaturized Navigators Based on Fuzzy Logic Denoising of Sensors Signals http://dx.doi.org/10.5772/67966 273

**Figure 19.** East positions and errors.

positioning in North direction, 39.27 m for the linear positioning in East direction, 2.03 m/s for the speed component in North direction, 7.14 m/s for the speed component in East direction, 2.940∙10–4 degrees for the Latitude positioning, 4.923∙10–4 degrees for the Longitude position-

The nonredundant inertial navigation system's best configuration was switched between the detection groups associated with the first sensor in each detection cluster (East channel and Yaw angle channel) and with the sixth sensor in each detection cluster (North channel). The worst configurations for the nonredundant INSs are for the second sensor in each detection cluster, the absolute maximal values of errors in North position channel being 990.52 m, 318.17 m in East position channel, 19.40 m/s in North speed channel, 89.141∙10–4 degrees in the Latitude position channel, and 39.886∙10–4 degrees in the Longitude channel. From the point of view of East speed channel and yaw angle channel, the worst nonredundant INS is INS4, the

ing, and 5.74 degrees for the angular positioning in the horizontal plane (Yaw angle).

East speed error being 13.68 m/s and the yaw angle error being 31.10 degrees.

**Figure 17.** Yaw angle values and errors.

272 Modern Fuzzy Control Systems and Its Applications

**Figure 18.** North positions and errors.

**Figure 20.** North speed evolutions during time.

**Figure 21.** East speed evolutions during time.

**Figure 22.** Latitude and errors.

**Figure 23.** Longitude and errors.

**Figure 24.** Vehicle trajectories and deviations from the reference trajectory in horizontal plane.

Precision Improvement in Inertial Miniaturized Navigators Based on Fuzzy Logic Denoising of Sensors Signals http://dx.doi.org/10.5772/67966 275

**Figure 25.** The time evolution of the vehicle trajectories for all seven INSs.

**Figure 26.** Time evolution of the deviations of the vehicle from the reference trajectory.

#### **5. Conclusions**

**Figure 23.** Longitude and errors.

**Figure 22.** Latitude and errors.

274 Modern Fuzzy Control Systems and Its Applications

**Figure 24.** Vehicle trajectories and deviations from the reference trajectory in horizontal plane.

The chapter exposed a new way to reduce the noise in the inertial sensor data based on a fuzzy logic mechanism. As a special need of the developed mechanism is considered to be a redundant inertial measurement unit with inertial sensors disposed in detection clusters with linear configurations, and measuring the same quantity. The proposed algorithm was firstly numerically simulated based on some models in Matlab/Simulink realized for detection of clusters with various sizes and including various sensor types. All simulation results proved a good functioning of the algorithm which provided an important reduction of the sensor noise level, with a higher potential in the improvement of the solution of navigation accuracy when it is used in an inertial navigator. Second, the algorithm was integrated in an INS redundant structure, with an IMU with three detection clusters and six miniaturized sensors in each cluster. An integrated GPS/INS navigator was used as reference positioning system to estimate the errors of the redundant INS in each channel of the solution of navigation. The data acquired from the redundant IMU were used in a software model both in nonredundant configuration, to compute the solutions of navigations for six detection groups which included sensors with the same number in the detection clusters, but also for the INS redundant configuration when our data fusion methodology is preliminary applied. Both graphical results and tabled numerical values proved an important positioning precision improvement by using the proposed data fusion mechanism. The errors of the navigation solution were substantially reduced comparatively with the nonredundant navigation solutions. The numerical data highlighted a decrease of the mean of the absolute maximal values of the errors for the six nonredundant INSs of approximately 16.6 times for the positioning in the North direction, 5.1 times for the positioning in the East direction, 2.8 times for the speed component in North direction, 1.05 times for the speed component in the East direction, and 3.8 times in the angular positioning in the horizontal plane (yaw angle). It is very important to be mentioned that the inertial sensor outputs in the experimental model of the inertial measurement unit were not previously corrected with the biased values or with other errors values.

### **Acknowledgements**

This work was supported by the CNCSIS-UEFISCSU, project PN II-RU, No. 1/28.07.2010, "High-precision strap-down inertial navigators, based on the connection and adaptive integration of the nano and micro inertial sensors in low cost networks, with a high degree of redundance," code TE-102/2010.

## **Author details**

Teodor Lucian Grigorie<sup>1</sup> \* and Ruxandra Mihaela Botez<sup>2</sup>

\*Address all correspondence to: ltgrigorie@yahoo.com

1 University of Craiova, Craiova, Romania

2 École de Technologie Supérieure, Montréal, Canada

## **References**

with linear configurations, and measuring the same quantity. The proposed algorithm was firstly numerically simulated based on some models in Matlab/Simulink realized for detection of clusters with various sizes and including various sensor types. All simulation results proved a good functioning of the algorithm which provided an important reduction of the sensor noise level, with a higher potential in the improvement of the solution of navigation accuracy when it is used in an inertial navigator. Second, the algorithm was integrated in an INS redundant structure, with an IMU with three detection clusters and six miniaturized sensors in each cluster. An integrated GPS/INS navigator was used as reference positioning system to estimate the errors of the redundant INS in each channel of the solution of navigation. The data acquired from the redundant IMU were used in a software model both in nonredundant configuration, to compute the solutions of navigations for six detection groups which included sensors with the same number in the detection clusters, but also for the INS redundant configuration when our data fusion methodology is preliminary applied. Both graphical results and tabled numerical values proved an important positioning precision improvement by using the proposed data fusion mechanism. The errors of the navigation solution were substantially reduced comparatively with the nonredundant navigation solutions. The numerical data highlighted a decrease of the mean of the absolute maximal values of the errors for the six nonredundant INSs of approximately 16.6 times for the positioning in the North direction, 5.1 times for the positioning in the East direction, 2.8 times for the speed component in North direction, 1.05 times for the speed component in the East direction, and 3.8 times in the angular positioning in the horizontal plane (yaw angle). It is very important to be mentioned that the inertial sensor outputs in the experimental model of the inertial measurement unit were not previously corrected with the biased val-

This work was supported by the CNCSIS-UEFISCSU, project PN II-RU, No. 1/28.07.2010, "High-precision strap-down inertial navigators, based on the connection and adaptive integration of the nano and micro inertial sensors in low cost networks, with a high degree of

\* and Ruxandra Mihaela Botez<sup>2</sup>

\*Address all correspondence to: ltgrigorie@yahoo.com

2 École de Technologie Supérieure, Montréal, Canada

1 University of Craiova, Craiova, Romania

ues or with other errors values.

276 Modern Fuzzy Control Systems and Its Applications

redundance," code TE-102/2010.

**Acknowledgements**

**Author details**

Teodor Lucian Grigorie<sup>1</sup>


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2015

## **A Fuzzy Belief-Desire-Intention Model for Agent-Based Image Analysis**

Peter Hofmann

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67899

#### **Abstract**

Recent methods of image analysis in remote sensing lack a sufficient grade of robustness and transferability. Methods such as object-based image analysis (OBIA) achieve satisfying results on single images. However, the underlying rule sets for OBIA are usually too complex to be directly applied on a variety of image data without any adaptations or human interactions. Thus, recent research projects investigate the potential for integrating the agent-based paradigm with OBIA. Agent-based systems are highly adaptive and therefore robust, even under varying environmental conditions. In the context of image analysis, this means that even if the image data to be analyzed varies slightly (e.g., due to seasonal effects, different locations, atmospheric conditions, or even a slightly different sensor), agent-based methods allow to autonomously adapt existing analysis rules or segmentation results according to changing imaging situations. The basis for individual software agents' behavior is a so-called believe-desire-intention (BDI) model. Basically, the BDI describes for each individual agent its goal(s), its assumed current situation, and some action rules potentially supporting each agent to achieve its goals. The chapter introduces a believe-desire-intention (BDI) model based on fuzzy rules in the context of agent-based image analysis, which extends the classic OBIA paradigm by the agent-based paradigm.

**Keywords:** agent-based image analysis, fuzzy believe-desire-intention model, object-based image analysis, fuzzy control system, remote sensing

## **1. Introduction**

Analyzing remote sensing data is strongly bound to methods of image processing and image analysis. In contrast to other imaging techniques, remote sensing as per definition is a method to acquire information about the earth's surface by detecting and analyzing its reflected or emitted electromagnetic radiation and without being in direct contact with it. Besides radiation

© 2017 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.

of the visual spectrum also infrared (optical data) and microwave radiation (RADAR) is used to produce remote sensing images. The remote sensing instruments can be carried by space crafts (usually satellites) or airborne vehicles (airplanes, drones, etc.). In order to gather geo-information from remote sensing data, the produced images need to be analyzed, that is, preprocessed and classified. In this context, image classification means to assign pixels to meaningful object classes of the earth's surface, whereas the delineated and classified objects are finally stored in a geographic information system (GIS) as polygons, lines, or points (vector model). With the continuous increase of remote sensing images' spatial (and radiometric) resolution, image analysis in remote sensing became more and more complex. Until the late 1990s, the majority of remote sensing data was analyzed by means of classification methods taking into account the radiation stored in each single pixel. Meanwhile, rather sophisticated methods of pattern analysis, artificial intelligence, and computer vision are applied.

With the advent of very high resolution (VHR) satellite images, classic methods of image classification, as described above, failed since most of the objects of interest are represented in VHR data by numerous and spectrally inhomogeneous pixels. Moreover, properties such as shape, texture, and spatial context play a rather important role when identifying and delineating objects of interest in this kind of data [1–3]. Thus, more or less simultaneously with the advent of VHR satellite images, object-based image analysis (OBIA) has meanwhile evolved as a new and accepted paradigm for analyzing remote sensing data. In contrast to pixel-based analysis methods, OBIA deals with image objects as the building blocks for analysis. Image objects are initially generated by an arbitrary image segmentation followed by an initial classification of these image segments. The feature space for classification can be very high dimensional describing color, shape, texture, or the spatial context properties for the desired object classes. Numerous classifiers can be applied ranging from simple thresholding to Support Vector Machines (SVM), Bayesian Network Classifiers (BNC), and Artificial Neural Networks (ANN). Fuzzy set assignments are possible, too. For the latter the definition of fuzzy sets and the underlying fuzzy classification rules should reflect the ontology of the desired object classes [4–6]. Thus, the typical workflow of OBIA starts with an initial segmentation (and classification) as described above; followed by an iterative process of knowledge-based segmentation and classification improvement. The latter reflects the so-called task-ontology describing the necessary expert knowledge on image processing, which can be stored as an OBIA rule set and reapplied. However, the more precisely and reliebly remote sensing data has to be analyzed, the more complex are the methods and rule sets. The latter finally reduces the rule sets' transferability. In order to achieve acceptable results for different image data more manual interaction such as changing single rules or manually correcting object borders and/or class assignments is necessary [7, 8]. Consequently, in order to benefit from OBIA's advantages for numerous images or even whole image archives, intelligent, and flexible solutions are necessary, which are capable to autonomously adapt to image variability.

#### **2. Agent-based and multiagent systems**

Agent-based and multiagent systems (MAS) recently show a variety of applications: they range from simulation of complex systems such as social systems [9] and ecosystems [10, 11] to the automation and optimization of complex production systems such as industrial processes [12–14]. In software development, meanwhile the agent-oriented paradigm has evolved as a new paradigm that extends the classic object-oriented approach. Simply spoken, the general differences are: (1) objects behave rather passive than agents, that is, objects only change once they receive an appropriate signal while agents behave proactive and collaborative and (2) agents can be mobile while objects are static. Thus, agents have (individual) goals they intend to achieve; they have sensors and effectors that enable them to become aware about their current status and to interact with their environment. Agents can decide autonomously about their potential next action. The environment agents are interacting with can be of arbitrary complexity ranging from other (human) agents to factory plants, sports fields to traffic situations, etc. When embedded in (collaborative) MAS individual agents often have different roles but common goals. All these abilities allow software agents and MAS to react flexible but robust on unforeseeable changes of their environment.

Since each agent needs to have certain situation awareness, each agent must be capable to appraise its current situation. That is, evaluating the grade of goal achievement and the acting opportunities supporting the agent's goal achievement. This sort of situation awareness is commonly known as the belief-desire-intention (BDI) model [15–19]. Simply spoken, the BDI model allows an agent to analyze its current situation and to choose from a predefined list of plans the most promising one in order to achieve its goals. It is obvious that for the design of software agents and MAS ontologies are required, which are capable to formally describe an agent's or MAS' environment and which allow individual agents to infer their current situation. Further, ontologies are necessary to describe an agent's goals and to infer the most promising action for goal achievement [20]. Casali et al. [21] extend the classic BDI model to a graded BDI (g-BDI) model, which allows each agent to express its preferences among its acting opportunities, while Shen et al. [22] introduce an agent fuzzy decision making (AFDM) approach, which extends the classic BDI model "by making decisions based on quantified fuzzy judgment." Zarandi and Ahmadpour [23] present a fuzzy agent-based expert system for the steel making process that uses a fuzzy described knowledge base.

#### **2.1. Agent-based image analysis**

of the visual spectrum also infrared (optical data) and microwave radiation (RADAR) is used to produce remote sensing images. The remote sensing instruments can be carried by space crafts (usually satellites) or airborne vehicles (airplanes, drones, etc.). In order to gather geo-information from remote sensing data, the produced images need to be analyzed, that is, preprocessed and classified. In this context, image classification means to assign pixels to meaningful object classes of the earth's surface, whereas the delineated and classified objects are finally stored in a geographic information system (GIS) as polygons, lines, or points (vector model). With the continuous increase of remote sensing images' spatial (and radiometric) resolution, image analysis in remote sensing became more and more complex. Until the late 1990s, the majority of remote sensing data was analyzed by means of classification methods taking into account the radiation stored in each single pixel. Meanwhile, rather sophisticated methods of pattern

With the advent of very high resolution (VHR) satellite images, classic methods of image classification, as described above, failed since most of the objects of interest are represented in VHR data by numerous and spectrally inhomogeneous pixels. Moreover, properties such as shape, texture, and spatial context play a rather important role when identifying and delineating objects of interest in this kind of data [1–3]. Thus, more or less simultaneously with the advent of VHR satellite images, object-based image analysis (OBIA) has meanwhile evolved as a new and accepted paradigm for analyzing remote sensing data. In contrast to pixel-based analysis methods, OBIA deals with image objects as the building blocks for analysis. Image objects are initially generated by an arbitrary image segmentation followed by an initial classification of these image segments. The feature space for classification can be very high dimensional describing color, shape, texture, or the spatial context properties for the desired object classes. Numerous classifiers can be applied ranging from simple thresholding to Support Vector Machines (SVM), Bayesian Network Classifiers (BNC), and Artificial Neural Networks (ANN). Fuzzy set assignments are possible, too. For the latter the definition of fuzzy sets and the underlying fuzzy classification rules should reflect the ontology of the desired object classes [4–6]. Thus, the typical workflow of OBIA starts with an initial segmentation (and classification) as described above; followed by an iterative process of knowledge-based segmentation and classification improvement. The latter reflects the so-called task-ontology describing the necessary expert knowledge on image processing, which can be stored as an OBIA rule set and reapplied. However, the more precisely and reliebly remote sensing data has to be analyzed, the more complex are the methods and rule sets. The latter finally reduces the rule sets' transferability. In order to achieve acceptable results for different image data more manual interaction such as changing single rules or manually correcting object borders and/or class assignments is necessary [7, 8]. Consequently, in order to benefit from OBIA's advantages for numerous images or even whole image archives, intelligent, and flexible solu-

tions are necessary, which are capable to autonomously adapt to image variability.

Agent-based and multiagent systems (MAS) recently show a variety of applications: they range from simulation of complex systems such as social systems [9] and ecosystems [10, 11]

**2. Agent-based and multiagent systems**

analysis, artificial intelligence, and computer vision are applied.

282 Modern Fuzzy Control Systems and Its Applications

Although meanwhile matured, the agent-based paradigm and potential applications based on it are not very widespread in the image analysis community, yet. The most applications can be found in the field of image coregistration and image fusion [24–26]. In the field of object detection and delineation reported applications are still rare [27–31]—even more in the field of remote sensing image analysis [32–35].

#### **2.2. Software agents and multiagent systems in OBIA**

As already mentioned, in order to fully exploit the advantages of OBIA there is a strong need for more robustness and transferability of methods. The limiting factors are the rule sets' complexity and the unpredictable variations of the image objects' appearance in remote sensing data. At this background recently the integration of the agent-based paradigm with OBIA are investigated in order to improve its degree of automation, its robustness, and its transferability. MAS seem to have the potential to overcome OBIA's obstacles [33, 36–38]. Especially their ability to react flexibly on environmental perturbations, which is given in the remote sensing domain by varying illuminations, seasons, locations, sensors, and atmospheric conditions, are promising aspects to be investigated. Consequently, Hofmann et al. [38] developed a conceptual framework for agent-based image analysis (ABIA), which suggests two principle ways of integration: (1) adapting already existing OBIA rule sets(e.g., thresholds of single rules) by means of a MAS built by respective rule set adaptation agents (RSAAs) and (2) evolving OBIA image objects to image object agents (IOAs). In the first approach different RSAAs adapt a rule set's rules in order to improve its classification results. As constraint, adaptations must not violate the underlying ontology of the original rule set. The latter is controlled by one or more control agents (CAs), which also give feedback whether a to-bedefined minimum classification quality has been achieved after rule set adaptation (**Figure 1**).

In the second approach after initial segmentation and (fuzzy) classification IOAs build a hierarchical network of IOAs. Each IOA intends to become a best possible member of its assigned class (goal, desire). To achieve this goal, every IOA can change its shape by a number of predefined methods (effectors, intention). Further, every IOA is aware about its topological situation and can communicate within other IOAs in the hierarchical net. The underlying ontology for this approach is given by the (fuzzy) class descriptions (**Figure 2**).

The ABIA framework has been implemented in a typical environment for agent-based modeling (REPAST [39]) as well as in a typical OBIA environment (eCognition) [40], realizing

**Figure 1.** Principle workflow for OBIA rule set adaptation by means of a MAS built by RSAAs (Source: [38]).

the IOA approach. In both implementations the real world objects to be delineated and identified were described as fuzzy sets based on an appropriate ontology. As test scene a very high resolution digital orthophoto (0.08m) has been used together with an appropriate digital surface model (DSM) together with the calculated slope and curvature (slope of slope) per pixel (**Figure 3**).

transferability. MAS seem to have the potential to overcome OBIA's obstacles [33, 36–38]. Especially their ability to react flexibly on environmental perturbations, which is given in the remote sensing domain by varying illuminations, seasons, locations, sensors, and atmospheric conditions, are promising aspects to be investigated. Consequently, Hofmann et al. [38] developed a conceptual framework for agent-based image analysis (ABIA), which suggests two principle ways of integration: (1) adapting already existing OBIA rule sets(e.g., thresholds of single rules) by means of a MAS built by respective rule set adaptation agents (RSAAs) and (2) evolving OBIA image objects to image object agents (IOAs). In the first approach different RSAAs adapt a rule set's rules in order to improve its classification results. As constraint, adaptations must not violate the underlying ontology of the original rule set. The latter is controlled by one or more control agents (CAs), which also give feedback whether a to-bedefined minimum classification quality has been achieved after rule set adaptation (**Figure 1**). In the second approach after initial segmentation and (fuzzy) classification IOAs build a hierarchical network of IOAs. Each IOA intends to become a best possible member of its assigned class (goal, desire). To achieve this goal, every IOA can change its shape by a number of predefined methods (effectors, intention). Further, every IOA is aware about its topological situation and can communicate within other IOAs in the hierarchical net. The underlying

284 Modern Fuzzy Control Systems and Its Applications

ontology for this approach is given by the (fuzzy) class descriptions (**Figure 2**).

**Figure 1.** Principle workflow for OBIA rule set adaptation by means of a MAS built by RSAAs (Source: [38]).

The ABIA framework has been implemented in a typical environment for agent-based modeling (REPAST [39]) as well as in a typical OBIA environment (eCognition) [40], realizing The rule set was intentionally designed to delineate buildings in that particular scene following the ontology as outlined in **Figure 4**.

However, if the rule set is applied without any further adaptations it creates a rather over-segmented image—which would be a typical OBIA use case after reapplying a given rule set on similar images. The BDI model to solve this problem has been implemented as a hybrid model. That is, the class definitions were implemented as fuzzy sets, whereas the next action's decision rules were designed crisp. For the latter simply all three provided actions were virtually executed for each IOA. Every IOA then opted for that action that improved its class membership to "building" at best. In the example demonstrated the final result has been achieved after 17 iteration steps already (**Figure 5**).

**Figure 2.** Scheme of a MAS built up by hierarchically organized IOAs and CAs (Source: [38]).

**Figure 3.** Used image and DSM data for first implementation of the ABIA framework (Source: [40]).

**Figure 4.** Ontology describing buildings and their appearance in the given data (Source: [38]).

**Figure 5.** Segmentation and classification result before (left) and after (right) applying the agent-based optimization approach according to the ABIA framework. Numbers indicate the membership degree to "building" (Source: [38]).

## **3. A fuzzy believe-desire-intention model for agent-based image analysis**

Based on the already achieved results with the relative simple crisp BDI model it has been investigated here whether the IOAs' intentions could also be expressed in fuzzy manner and whether this is of advantage compared to the above described BDI model. For this purpose the existing rule set, which has been developed in a software environment dedicated for OBIA (here: the commercial software eCognition), has been extended by the necessary components. In a standard fuzzy control-system control is given implicitly through the membership functions, e.g., "the colder room temperature the more open heater's valve." In this particular case the used software only allows fuzzy sets (alias classes) to be described in fuzzy manner. Consequently, the agents' acting opportunities had to be expressed as fuzzy sets whereas the membership degree to an "action-class" can be interpreted as the "intention-degree" or the willingness of an agent to perform that particular action. Another difference of the fuzzy BDI model developed here is that it only evaluates the current situation. That is, there is no virtual test for each potential action if and how it would improve an agent's situation (here: its class membership).

#### **3.1. Components of the fuzzy BDI model**

In order to fuzzify each IOA's believe its current status quo after segmentation and classification is analyzed. That is, each IOA had to be enabled not only to know its current class membership degree (degree of goal/desire achievement), but also for each classification rule the degree of its contribution to that particular classification result. The latter would allow each IOA to select an action that improves one of those conditions.

#### *3.1.1. Fuzzy beliefs*

**Figure 3.** Used image and DSM data for first implementation of the ABIA framework (Source: [40]).

286 Modern Fuzzy Control Systems and Its Applications

**Figure 4.** Ontology describing buildings and their appearance in the given data (Source: [38]).

For this purpose, the conditions that build a "building" as described in the ontology were separated into the four categories: color conditions, DSM conditions, and shape conditions. The class "roof"—which represents buildings—consequently was now described through these aggregated conditions whereas the property "Classification Value of …" expresses the membership degree μ for this particular class or the degree of fulfillment (DOF) of that particular condition. Similarly, the condition classes were further deconstructed to classes describing the DOF's property conditions (feature-2-class) or operator conditions ( operator-2-class). **Table 1** shows the cascaded classification scheme. With this decomposition every IOA now can determine the grade each of the fuzzy classification rules or rule groups (color conditions, shape conditions, or context conditions) it contributes to its final class assignment result (= grade of goal achievement). In the example displayed in **Figure 6** the **red** outlined IOA fulfills the criteria for "roof" only by 0.432, whereas the color conditions are fully fulfilled, the shape conditions are fulfilled by 0.555 and the DSM conditions are fulfilled by 0.432.The shape conditions are not fulfilled, because the IOA's rectangular fit value of 0.76 leads to a grade of fulfillment for that condition of 0.555 only.



the membership degree μ for this particular class or the degree of fulfillment (DOF) of that particular condition. Similarly, the condition classes were further deconstructed to classes describing the DOF's property conditions (feature-2-class) or operator conditions ( operator-2-class). **Table 1** shows the cascaded classification scheme. With this decomposition every IOA now can determine the grade each of the fuzzy classification rules or rule groups (color conditions, shape conditions, or context conditions) it contributes to its final class assignment result (= grade of goal achievement). In the example displayed in **Figure 6** the **red** outlined IOA fulfills the criteria for "roof" only by 0.432, whereas the color conditions are fully fulfilled, the shape conditions are fulfilled by 0.555 and the DSM conditions are fulfilled by 0.432.The shape conditions are not fulfilled, because the IOA's rectangular fit value of 0.76

**function type**

AND 0.00 1.00

AND 0.00 1.00

AND 0.00 1.00

AND 0.00 1.00

**Function values**

0.00 1.00

0.00 1.00

0.00 1.00

0.00 1.00

0.00 1.00

0.00 1.00

0.00 1.00

**Lower bound Upper bound**

leads to a grade of fulfillment for that condition of 0.555 only.

**Class Properties Operator Membership** 

Classification value of 'DSM

Classification value of 'Shape

'feature-2-class "Ratio green"'

Classification value of 'operator-2-class "Red-OR-Grey"'

'feature-2-class "Mean difference to neighbors DSM"'

Classification value of 'feature-2-class "Mean Slope-of-Slope"'

Classification value of 'feature-2-class "Upper Bound

Classification value of 'operator-2-class "Border Index-OR-Shape Index"'

'feature-2-class "Lower Bound

Classification value of 'feature-2-class "Rectangular fit"'

Roof Classification value of 'Color conditions'

288 Modern Fuzzy Control Systems and Its Applications

conditions'

conditions'

Color conditions Classification value of

DSM conditions Classification value of

Shape conditions Classification value of

of Area"'

of Area"'

**Table 1.** Cascaded fuzzy classification scheme for the class "roof" as described in **Figure 4.**

**Figure 6.** Evaluation of classification conditions of an IOA.

Similarly, the IOA's DSM conditions show a DOF of 0.432 because the mean difference to neighbors in the DSM is at 1.631 translating to a DOF for that feature of 0.432.

#### *3.1.2. Fuzzy intentions*

Based on these fuzzy beliefs an intended next action can be determined in fuzzy manner based on appropriate fuzzy decision rules—again expressed as fuzzy sets. In the example present the following possible actions were implemented: "grow," "shrink," "smooth," "merge," and "do nothing." While the latter action is obvious and applies only if an IOA has achieved its goal already, the former actions point to procedures eCognition offers and can be easily exchanged or adapted if necessary. In this particular example the actions translate to:


The appropriate intentions are defined as the fuzzy sets: "want grow," "want shrink," "want smooth," "want merge," or "do nothing." The degree or the intensity an agent wants to execute one of these actions can depend on the prior determined classification conditions or on the spatial situation in which IOA is embedded in, or a combination of both. Each of the action intensities is expressed gradually, that is, through an action-membership (**Table 2**). In the example given, an IOA wants to shrink if the situation is similar to that of "want grow;" it wants the more shrink the closer its size is at the upper bound of the area-rule for "building" ("Upper bound of Area"). An IOA wants to do nothing the more it fulfills already


**Table 2.** Definition of intentions as fuzzy sets.

Similarly, the IOA's DSM conditions show a DOF of 0.432 because the mean difference to

Based on these fuzzy beliefs an intended next action can be determined in fuzzy manner based on appropriate fuzzy decision rules—again expressed as fuzzy sets. In the example present the following possible actions were implemented: "grow," "shrink," "smooth," "merge," and "do nothing." While the latter action is obvious and applies only if an IOA has achieved its goal already, the former actions point to procedures eCognition offers and can be easily exchanged or adapted if necessary. In this particular example the actions translate to: • "grow": grow the IOA of concern by one pixel into neighbor IOAs, which are unclassified.

• "smooth": perform a grow-and-shrink sequence by one pixel each starting with shrink.

The appropriate intentions are defined as the fuzzy sets: "want grow," "want shrink," "want smooth," "want merge," or "do nothing." The degree or the intensity an agent wants to execute one of these actions can depend on the prior determined classification conditions or on the spatial situation in which IOA is embedded in, or a combination of both. Each of the action intensities is expressed gradually, that is, through an action-membership (**Table 2**). In the example given, an IOA wants to shrink if the situation is similar to that of "want grow;" it wants the more shrink the closer its size is at the upper bound of the area-rule for "building" ("Upper bound of Area"). An IOA wants to do nothing the more it fulfills already

neighbors in the DSM is at 1.631 translating to a DOF for that feature of 0.432.

• "shrink": shrink the IOA of concern by one pixel.

**Figure 6.** Evaluation of classification conditions of an IOA.

290 Modern Fuzzy Control Systems and Its Applications

• "merge": merge the IOA of concern with its unclassified neighbors.

*3.1.2. Fuzzy intentions*

the "building" criteria. It wants the more grow, the lower its contrast and elevation difference is at its border. Its intention to merge increases, the lower its area is and simultaneously the more its DSM and color criteria are similar with those of its neighbors. Its intention to smooth its border increases the less the shape criteria for building are fulfilled.

As displayed in **Figure 7**, the **red** outlined IOA prefers to grow. However, similar to the ambiguity of class assignments of objects, each IOA can have ambiguous intentions in terms of a favorite, a second favorite, etc. action. Further, as with fuzzy class assignments [41], a minimum intention value should be defined (sensibly not less than 0.5) below which an intended action must be seen as not clearly enough wanted and thus not further considered.

In the example given in **Figure 7** although the IOA's most wanted action is to grow, this action is not the only one it intends to perform. Since the intention value for "want\_grow" is not clearly around 1.0 the IOA seems to be not fully convinced about this action to achieve its goal. Merging (intention = 0.68) seems to be an option although it is second-best. In other words, the IOA could also be satisfied if it merges. The only thing that is clear, is, that the IOA does not want to shrink (intention = 0.0). And its willingness to smooth its border (intention = 0.0298) is even lower than that for doing nothing (intention = 0.356).

**Figure 7.** Grades of intentions for actions an IOA wants to perform for its improvement.

## **4. Conclusion**

The amount of remote sensing data stored in archives is increasing continuously. At the background of an increasing demand for reliable, precise and timely geoinformation, searching these archives, and analyzing the image data stored in them urges for methods of reliable and automated image analysis. While OBIA is an accepted and highly accurate method for analyzing especially VHR remote sensing data, its robustness, and transferability, as well as degree of automation is still low. Major obstacles for automating OBIA and its methods are its sensibility against perturbations, that is, the images' and objects variability. ABIA as an extension of the OBIA paradigm has the potential to overcome these obstacles, since it has the ability to react more flexible and robust on unforeseeable perturbations. Nevertheless, research in this field is still in its beginning.

The example demonstrated in this chapter is just one aspect of this wide research field. It has been demonstrated how a fuzzy BDI model can be implemented in standard OBIA software, which is capable to allow individual IOAs to control their improvement actions within a MAS. Further research needs to be done in learning mechanisms for individual agents as well as in improved individual decision rules. Last but not the least, the performance issues are still another aspect that needs to be investigated in that field.

## **5. Outlook**

the "building" criteria. It wants the more grow, the lower its contrast and elevation difference is at its border. Its intention to merge increases, the lower its area is and simultaneously the more its DSM and color criteria are similar with those of its neighbors. Its intention to smooth

As displayed in **Figure 7**, the **red** outlined IOA prefers to grow. However, similar to the ambiguity of class assignments of objects, each IOA can have ambiguous intentions in terms of a favorite, a second favorite, etc. action. Further, as with fuzzy class assignments [41], a minimum intention value should be defined (sensibly not less than 0.5) below which an intended

In the example given in **Figure 7** although the IOA's most wanted action is to grow, this action is not the only one it intends to perform. Since the intention value for "want\_grow" is not clearly around 1.0 the IOA seems to be not fully convinced about this action to achieve its goal. Merging (intention = 0.68) seems to be an option although it is second-best. In other words, the IOA could also be satisfied if it merges. The only thing that is clear, is, that the IOA does not want to shrink (intention = 0.0). And its willingness to smooth its border

The amount of remote sensing data stored in archives is increasing continuously. At the background of an increasing demand for reliable, precise and timely geoinformation, searching these archives, and analyzing the image data stored in them urges for methods of reliable and automated image analysis. While OBIA is an accepted and highly accurate method for analyzing especially VHR remote sensing data, its robustness, and transferability, as well as degree of automation is still low. Major obstacles for automating OBIA and its methods are its sensibility against perturbations, that is, the images' and objects variability. ABIA

**Figure 7.** Grades of intentions for actions an IOA wants to perform for its improvement.

action must be seen as not clearly enough wanted and thus not further considered.

(intention = 0.0298) is even lower than that for doing nothing (intention = 0.356).

**4. Conclusion**

its border increases the less the shape criteria for building are fulfilled.

292 Modern Fuzzy Control Systems and Its Applications

The implemented fuzzy BDI model acts as the basis for a negotiation model that can be applied on ABIA agents: based on their individual action priorities, IOAs can negotiate their common next action(s) and therefore optimize the overall classification result of multiple but different images of the same kind as it is typical in remote sensing. While the fuzzy BDI model has been implemented for IOAs, it is also applicable in principle for rule set adaptation agents (RSAAs).

## **Author details**

Peter Hofmann

Address all correspondence to: peter.hofmann@sbg.ac.at

Department of Geoinformatics – Z\_GIS, University of Salzburg, Salzburg, Austria

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