A PSO-Tuning Method for Design of Fuzzy PID ... - Semantic Scholar

Journal of Vibration and Control OnlineFirst, published on December 20, 2007 as doi:10.1177/1077546307080038

A PSO-Tuning Method for Design of Fuzzy PID Controllers CHIA-NAN KO Graduate School of Engineering Science and Technology, National Yunlin University of Science and Technology, Douliou, Yunlin 640, Taiwan ([email protected])

CHIA-JU WU Department of Electrical Engineering, National Yunlin University of Science and Technology, Douliou, Yunlin 640, Taiwan (Received 16 September 20061 accepted 8 March 2007)

Abstract: A novel tuning method is proposed for the design of fuzzy PID controllers for multivariable systems. In the proposed method, a PID controller is expressed in terms of fuzzy rules, in which the input variables are the error signals and their derivatives, while the output variables are the PID gains. In this manner, the PID gains are adaptive and the fuzzy PID controller has more flexibility and capability than conventional versions with fixed gains. A particle swarm optimization (PSO) method is proposed for tuning of the fuzzy PID controller, in which the centers and the widths of the Gaussian membership functions, the number of fuzzy control rules, and the PID gains are all parameters to be determined simultaneously. Meanwhile, based on the concept of multiobjective optimization, ways of defining the fitness function of the PSO to include different performance criteria are also discussed. To show the feasibility and validity of the resulting fuzzy PID controller, illustrative experimental results for a multivariable seesaw system are included.

Keywords: Particle swarm optimization, fuzzy PID controllers, multiobjective optimization, multivariable systems.

NOMENCLATURE c1 c2 c1 max c1 min c2 max c2 min ei e1i fi g hi Jb Jr

cognitive parameter social parameter maximum value of c1 minimum value of c1 maximum value of c2 minimum value of c2 error signal derivative of ei objective function gravity acceleration normalized objective function polar moment inertia of beam polar moment inertia of rollers

Journal of Vibration and Control, 00(0): 000–000, 2007

DOI: 10.1177/1077546307080038

1 2007 SAGE Publications Los Angeles, London, New Delhi, Singapore 2

Copyright 2007 by SAGE Publications.

2 C.-N. KO and C.-J. WU KP1 KI 1 KD M mb mr N ni P Plbest Pgbest preal pbinar y R r u V 23 1 1 3 2 1 3 3 3 1 3 M 4 Xi xk 5 6k 7 kl 7 kr 8 9 9max 9min

proportional gain, integral gain, and derivative gain number of objective functions mass of base and motor mass of rollers combined number of fitness functions number of input membership functions particle position (solution) local best solution global best solution real particle binary particle radius of rollers position of motor control signal particle velocity weight vector input membership function input linguistic variable angle of the beam center (mean value) of input membership function left width (standard deviation) of input membership function right width (standard deviation) of input membership function torque of motor inertia weight maximum value of 9 minimum value of 9

1. INTRODUCTION Control theory research has been emphasized recently, and many methods have been proposed for the design of controllers. Although these approaches are theoretically elegant, it should be pointed out that they are not very extensively used in practice. Most industrial processes nowadays are still controlled by PID controllers. Conventional PID controllers have been widely applied in industrial process control for about half a century because of their simple structure and convenience of implementation (Bennett, 19931 Chen, 1996). To implement such a controller, the proportional gains, integral gains, and derivative gains must be determined. Among the existing gain tuning techniques, the method proposed by Ziegler and Nichols (1942) is probably the best-known and most popular. However, a conventional PID controller may have poor control performance for nonlinear and/or complex systems for which there are no precise mathematical models. To overcome these difficulties, various types of modified conventional PID controller, such as auto-tuning and adaptive PID controllers have been developed. Na (2001), Keel et al. (2003), and Cervantes et al. (2004) proposed methods of tuning the PID controller based on mathematical models, but complex mathematical computation is generally required in tuning procedures. The methods of Whidborne and Istepanian (2001) and Lin et al. (2003), in

A PSO-TUNING METHOD FOR DESIGN OF FUZZY PID CONTROLLERS 3

which the PID gains are fixed, have the disadvantage that they usually lack flexibility and capability. Research on fuzzy logic controllers (FLCs) has increased recently, as FLCs allow a simple and human approach to controller design and do not require precise mathematical modeling knowledge. As the controlled systems become more complex, deriving their mathematical models becomes more difficult. Therefore, fuzzy controllers provide reasonable and effective alternatives to conventional controllers and many researchers have attempted to combine conventional PID controllers with fuzzy logic. Despite the significant improvement these fuzzy PID controllers represent over their classical counterparts, it should be noted that they still have disadvantages. For example, the locations of the peaks of the membership functions are fixed and not adjustable (Misir et al., 19961 Chalhoub and Bazzi, 20041 Li et al., 2005), or the fuzzy control rules are hand-designed (Sun and Er, 20041 Shi and Trabia, 20051 Tao and Taur, 2005). Furthermore, for nonlinear multivariable systems, the problem of reducing the number of fuzzy rules remains unsolved. A multivariable system can be deconstructed into several single-input single-output systems in order to reduce the number of fuzzy rules (Li and Shieh, 20001 Wu et al., 20021 Sun and Er, 2004). However, the drawback is that the interdependency between the variables is neglected, as a result of which the designed controller may not work well for some cases. Several evolutionary algorithms have been proposed recently to search for optimal PID controller parameters. Among them, genetic algorithm (GA) has received great attention. However, recent research has identified some deficiencies in GA performance (Eberhart and Shi, 19981 Fogel, 2000). Moreover, the premature convergence of GA degrades its performance and reduces its search capability (Fogel, 2000). Particle swarm optimization (PSO) is an evolutionary algorithms first introduced by Kennedy and Eberhart (1995). This method was developed through simulation of a simplified social system, and has been found to be robust in solving continuous nonlinear optimization problems (Angeline, 19981 Shi and Eberhart, 19991 Parsopoulos and Vrahatis, 2002). In this article, the PSO technique will be adopted to perform parameter tuning of fuzzy PID controllers. It is adopted for several reasons. The most notable of these are its characteristics of stable convergence (Gaing, 2004), the fact that it can generate a highquality solution in a shorter calculation time than other stochastic methods (Boeringer and Werner, 20041 Chatterjee et al., 2005), and that it has already seen successful applications in many nonlinear and highly complex problems (Yoshida et al., 20001 Gaing, 2004). In our proposed PSO-tuning method, in addition to determining the centers and the widths of the Gaussian membership functions, the number of fuzzy control rules, and the PID gains simultaneously, the best way to define the fitness function is also discussed. The basic idea is to use the concept of multiobjective optimization (Ishibuchi and Murata, 19981 Leung and Wang, 20001 Ho et al., 2005) to enable different performance criteria to be included in the fitness function in a systematic way. In this manner, the proposed method is fully capable of creating a fuzzy PID controller and eliminates the need for human input in the design process. To show the flexibility and capability of the proposed method, a multivariable seesaw system is used as an illustrative example. From the experimental results, one can see that the designed fuzzy PID controller is more versatile than a conventional one. The rest of this article is organized as follows. A fuzzy PID control algorithm is proposed in Section 2. The PSO-tuning method and proposals for defining the fitness function are

4 C.-N. KO and C.-J. WU

Figure 1. Block diagram for the fuzzy PID controller, in which a PSO algorithm is used to search for the optimal parameters in the fuzzy tuning rules.

described in Section 3. The experimental results and conclusion are given in Sections 4 and 5 respectively.

2. FUZZY PID CONTROLLERS In a classical PID control system, the time-domain form of a PID controller is usually expressed as 1 u2t4 4 K P e2t4 5 K I e2t4dt 5 K D e2t4 1 (1) where u(t) is the control signal, e(t) is the error signal, and K P , K I , and K D denote the proportional gain, the integral gain, and the derivative gain, respectively. The parameter tuning problem of a PID controller can be considered as selecting values of K P , K I , and K D that give the desired output response. Though many tuning methods have been proposed previously, the PID gains are always fixed after tuning. Therefore, the designed PID controllers lack flexibility and cannot efficiently control systems that have changing parameters. To alleviate the drawback, a fuzzy PID controller is introduced. The block diagram of the proposed fuzzy PID controller is shown in Figure 1. As shown in the block diagram, the input variables of the fuzzy rules are the error signals and their derivatives, while the output variables are the PID gains. Since a multi-input multi-output (MIMO) system can always be decomposed into a set of multi-input single-output (MISO) system, only the case of MISO systems is considered. Moreover, since the experimental example that will be discussed later has two error signals and one output, the fuzzy PID control rules are expressed as If e1 is X 11 and is X 21 and e2 is X 31 and e12 is X 41 1 1111 1111 1111 1 K I 1 4 Y I1111 then K P1 4 Y P1 1 1 3 3 3 1 K I 2 4 Y I 2 1 K D2 4 Y D2

A PSO-TUNING METHOD FOR DESIGN OF FUZZY PID CONTROLLERS 5 If e1 is X 11 and e11 is X 21 and e2 is X 31 and e12 is X 4n 4 1 111n 4 111n 4 4 4 1 K I 1 4 Y I111n 1 3 3 3 1 K I 2 4 Y I111n 1 K D2 4 Y D2 then K P1 4 Y P1 1 2

If e1 is X 11 and e11 is X 21 and e2 is X 32 and e12 is X 41 1 1121 1121 1121 then K P1 4 Y 1121 P1 1 K I 1 4 Y I 1 1 3 3 3 1 K I 2 4 Y I 2 1 K D2 4 Y D2



If e1 is X 11 and e11 is X 21 and e2 is X 3n3 and e12 is X 4n4 1 11n 3 n 4 11n 3 n 4 3 n4 3 n4 1 K I 1 4 Y I11n 1 3 3 3 1 K I 2 4 Y I11n 1 K D2 4 Y D2 then K P1 4 Y P1 1 2



n

If e1 is X 1 1 and e11 is X 2n 2 and e2 is X 3n 3 and e12 is X 4n 4 1 1 n2 n3 n4 1 K I 1 4 Y nI 11n2 n3 n 4 1 3 3 3 1 K I 2 4 Y nI 12n2 n3 n 4 1 then K P1 4 Y nP1

n1 n2 n3 n4 K D2 4 Y D2

(2)

where e1 , e11 , e2 , and e12 are the error signals and their derivatives1 X 11 1 X 12 1 3 3 3 1 X 1n1 1 X 21 1 X 22 1 3 3 3 1 X 2n2 1 X 31 1 X 32 1 3 3 3 1 X 3n 3 1 and X 41 1 X 42 1 3 3 3 1 X 4n4 are the membership functions of e1 , e11 , n1 n2 n3 n4 1111 1112 e2 , and e12 1 K P1 , K I 1 , K D1 , K P2 , K I 2 , and K D2 are the PID gains1 Y P1 1 Y P1 1 3 3 3 1 Y P1 1 n n n n n n n n n n n n 1 2 3 4 1 2 3 4 1 2 3 4 1112 1111 1112 1111 1112 1111 Y I1111 1 Y 1 3 3 3 1 Y 1 Y 1 Y 1 3 3 3 1 Y 1 Y 1 Y 1 3 3 3 1 Y 1 Y I1 D1 P2 1 I1 D1 D1 P2 P2 I2 1 n1 n2 n3 n4 n1 n2 n3 n4 1112 1111 1112 YI 2 1 3 3 3 1 YI 2 1 and Y D2 1 Y D2 1 3 3 3 1 Y D2 are real numbers to be determined1 and n 1 1 n 2 1 n 3 1 and n 4 denote the numbers of input membership functions. The membership functions of an FLC are usually parametric functions such as triangular functions, trapezoidal functions, Gaussian functions, and singletons. Though the proposed method is equally applicable to all these kinds of membership functions, Gaussian functions are used as the antecedent fuzzy sets in this article. This means that input membership functions are represented as 6 7 2 82 9 3 xk 6 6 mk k 3 3 exp 6 if xk 7 6 mk k 3 mk 3 7 kl 4 X km k 2xk 4 4 6 9 7 3 m k 82 3 6 6 x 3 k k 3 exp 6 3 if xk  6 mk k 5 7 mkrk for k 4 11 21 3 3 3 1 41 1 7 m3 7 n31

1 7 m1 7 n11

1 7 m4 7 n4

1 7 m2 7 n21 (3)

where x k represents the input linguistic variables, and 6 mk k 1 7 mklk , and 7 mkrk denote the values of the centers, left widths, and right widths, respectively, of the input membership functions, as shown in Figure 2. For the output membership functions, singleton sets are adopted. In

6 C.-N. KO and C.-J. WU

Figure 2. Gaussian membership functions of the input linguistic variables e1 2t4, e11 2t4, e2 2t4, and e12 2t4.

the defuzzification process, the center of gravity method (Wang, 1997) is used to determine the output crisp value from n3

n2

n4 n1



K P1 4

i jkl 9i jkl Y P1 i41 j41 k41 l41 n3

n2

n4 n1



(4)

9i jkl

i41 j41 k41 l41 n3

n2

n4 n1



KI1 4

i jkl 9i jkl Y I 1 i41 j41 k41 l41 n3

n2

n4 n1



(5)

9i jkl

i41 j41 k41 l41 n3

n2

n4 n1



K D1 4

jkl 9i jkl Y iD1

i41 j41 k41 l41 n3

n2

n4 n1



(6) 9i jkl

i41 j41 k41 l41 n3

n2

n4 n1



K P2 4

jkl 9i jkl Y iP2

i41 j41 k41 l41 n3

n2

n4 n1



(7) 9i jkl

i41 j41 k41 l41 n3

n2

n4 n1



KI2 4

i jkl

9i jkl Y I 2

i41 j41 k41 l41 n3

n2

n4 n1

i41 j41 k41 l41

(8) 9i jkl

A PSO-TUNING METHOD FOR DESIGN OF FUZZY PID CONTROLLERS 7

n3

n2

n4 n1



K D2 4

jkl 9i jkl Y iD2 i41 j41 k41 l41 n3

n2

n4 n1



(9)

9i jkl

i41 j41 k41 l41

where j

9i jkl 4 min2X 1i 2e1 4, X 2 2e11 4, X 3k 2e2 4, X 4l 2e12 44.

(10)

From equations (4) to (10), one can determine the fuzzy PID gains. If the PID control law is used, the control signal determined as n3

n2

n4 n1



u2t4 4

n3

n2

n4 n1



jkl 9i jkl Y iP1

i41 j41 k41 l41 n3

n2

n4 n1



9i jkl

e1 2t4 5

9i jkl Y iIjkl 1 i41 j41 k41 l41 n3

n2

n4 n1



5

i41 j41 k41 l41 n1

n2

n3

n4

jkl 9i jkl Y iD1

i41 j41 k41 l41 n3

n2

n4 n1



9i jkl

e11 2t4 5

5

9i jkl

e2 2t4

i41 j41 k41 l41 n3

n2

n4 n1



9i jkl Y iIjkl 2 1

i41 j41 k41 l41 n3

n2

n4 n1



jkl 9i jkl Y iP2

i41 j41 k41 l41 n3

n2

n4 n1



i41 j41 k41 l41 n3

n2

n4 n1



e1 2t4dt

9i jkl

i41 j41 k41 l41 n1

n2

n3

n4

1

9i jkl

e2 2t4dt 5

i41 j41 k41 l41

jkl 9i jkl Y iD2 i41 j41 k41 l41 e12 2t4. n3

n2

n4 n1



(11)

9i jkl

i41 j41 k41 l41

From the above description, one can find that the gains of the fuzzy PID controller are adaptive such that the controller should have more flexibility and capability than conventional ones. However, it is very difficult, if not impossible, to determine the parameters n 1 1 jkl i jkl i jkl i jkl i jkl mk mk mk n 2 1 n 3 1 n 4 , Y iP1 1 Y iIjkl 1 1 Y D1 , Y P2 1 Y I 2 1 Y D2 , 6 k 1 7 kl 1 and 7 ky directly. Therefore, a PSO-based method is proposed, which will search for the optimal values of these parameters in an iterative manner.

3. PARTICLE SWARM OPTIMIZATION PSO is a population-based stochastic searching technique developed by Kennedy and Eberhart (1995). The searching process behind the algorithm was inspired by social behaviors of animals such as bird flocking and fish schooling. It is similar to the continuous GAs in that it begins with a random population matrix and searches for the optima by updating generations. However, the PSO has no evolution operations such as crossover and mutation. The rows of the matrix are called particles, and are much the as same as chromosomes in GAs.

8 C.-N. KO and C.-J. WU

They contain the continuous variable values. Each particle moves in the search space with a velocity. The velocity value directs the movement of the particle through the problem space by following the current best particle. The particles update their velocities and positions based on the local best and global best solutions as follows (Ratnaweera et al., 2004): V2k 5 14 4 92k 5 14 3 V2k4 5 c1 2k 5 14 3 r1 3 2Plbest 2k4 6 P2k44 5 c2 2k 5 14 3 r2 3 2Pgbest 6 P2k44 P2k 5 14 4 P2k4 5 V2k 5 14

(12) (13)

Where V2k4 and V2k514 denote the particle velocities at iterations k and 2k5141 respectively, P2k4 and P2k514 denote the particle positions at iteration k and 2k5141 respectively, 92k514 denotes the inertia weight at iteration 2k 5 141 r1 and r2 are random numbers between 0 and 1, c1 2k 5 14 is the cognitive parameter, c2 2k 5 14 is the social parameter at iteration 2k 5 141 Plbest 2k4 is the local best solution at iteration k1 and Pgbest is the global best solution of the group. The inertia weight 9 in equation (12) represents the degree of momentum of the particles. This parameter is important for balancing between local and global explorations. For early iterations, it can be set higher, so that the particles are allowed to have much exploration capability and pursue an aggressive search of the solution space. Once the algorithm is found to converge more and more towards the optimum, this coarse tuning is gradually converted to a finer and finer tuning by making 9 smaller and smaller in later iterations. In this article, a linearly adaptable inertia weight 9 is employed (Shi and Eberhart, 1999), which starts with a high value 9max and linearly decreases to 9min at the maximum number of iterations. This means that 92k 5 14 is calculated from 92k 5 14 4 9max 6

9max 6 9min 3 iter itermax

(14)

where itermax is the maximum number of iterations (generations) and iter is the number of iterations so far. The constants c1 and c2 in equation (12) represent the weights of the stochastic acceleration terms that pull each particle toward the local best and global best positions. With a large cognitive component and small social component at the beginning, particles are allowed to move around the search space, instead of moving toward the best solution. In the latter part of the optimization, on the other hand, a small cognitive component and large social component are used, to allow the particles to converge on the global optima. In this article, we use linearly time-varying acceleration coefficients over the evolutionary procedure. The cognitive parameter c1 starts with a high value c1 max and linearly decreases to c1 min , while the social parameter c2 starts with a low value c2 min and linearly increases to c2 max . Therefore, the acceleration coefficients c1 2k 5 14 and c2 2k 5 14 can be expressed as follows (Ratnaweera et al., 2004): c1 2k 5 14 4 c1 max 6

c1 max 6 c1 min 3 iter itermax

(15)

A PSO-TUNING METHOD FOR DESIGN OF FUZZY PID CONTROLLERS 9

c2 2k 5 14 4 c2 min 6

c2 min 6 c2 max 3 iter

itermax

(16)

Where itermax is the maximum number of iterations (generations) and iter is the number of iterations so far. 3.1. Particle Representations jkl Before applying the PSO-tuning method, the encoding of the parameters n 1 1 n 2 1 n 3 1 n 4 , Y iP1 1 i jkl i jkl i jkl i jkl mk mk mk Y iIjkl 1 Y , Y 1 Y 1 Y , 6 1 7 1 and 7 must be considered. In the proposed method, k kl kr 1 D1 P2 I2 D2 a mixed coding method is used, in which n 1 , n 2 1 n 3 1 and n 4 are encoded as binary numbers i jkl i jkl i jkl i jkl i jkl i jkl and Y P1 1 Y I 1 1 Y D1 , Y P2 1 Y I 2 1 Y D2 , 6 mk k 1 7 mklk 1 7 mkrk are encoded as real numbers. This means that the positions of particles are represented as

P 4 [ p1 1 p2 1 3 3 3 1 p4n1 54n 2 54n3 54n4 56n 1 n2 n3 n 4 ] 4 [pbinar y preal ]

(17)

where pbinar y preal

4 [ p1 1 p2 1 3 3 3 1 pn1 5n 2 5n 3 5n4 ]

(18)

4 [ pn 1 5n2 5n3 5n4 51 1 3 3 3 1 p4n1 54n2 54n3 54n 4 56n1 n 2 n3 n4 ]

(19)

The particle pbinar y contains binary variables taking the value of one or zero. The values of p1 to pn 1 5n2 5n3 5n4 are used to indicate which of the membership functions X 11 1 X 12 1 3 3 3 1 X 1n 1 , X 21 1 X 22 1 3 3 3 1 X 2n2 , X 31 1 X 32 1 3 3 3 1 X 3n3 , and X 41 1 X 42 1 3 3 3 1 X 4n4 are activate. When applying the PSO-tuning method, the round-off technique is used to obtain  pi 4

pi 8 0 5

1

if

0

otherwise

for 1 7 i 7 2n 1 5 n 2 5 n 3 5 n 4 4

(20)

One can take the case of n 1 4 5, n 2 4 5, n 3 4 5, and n 4 4 5 as an illustrative example. If pi 4 1 for 1 7 i 7 201 then it implies that all of the input membership functions are activated and there are 625 (4 5 9 5 9 5 9 5) fuzzy rules in the fuzzy PID controller produced. On the other hand, if p1 p2 p3 p4 p5 4 01001, p6 p7 p8 p9 p10 4 01101, p11 p12 p13 p14 p15 4 10110, and p16 p17 p18 p19 p20 4 10010, then it indicates that only the membership functions X 12 1 X 15 , X 22 1 X 23 1 X 25 , X 31 1 X 33 1 X 34 , and X 41 1 X 44 are taken into account and there are 36 (4 2 9 3 9 3 9 2) fuzzy rules in the resulting fuzzy PID controller. In the real particles preal , the values of pn1 5n 2 5n 3 5n4 51 to p2n 1 52n2 52n3 52n 4 , p2n1 52n 2 52n3 52n4 51 to p3n 1 53n2 53n3 53n4 , and p3n1 53n 2 53n3 53n4 51 to p4n 1 54n2 54n3 54n4 are used to represent the values of 6 mk k 1 7 mklk 1 and 7 mkrk , respectively. In addition, the values i jkl p4n1 54n 2 54n3 54n4 51 to p4n 1 54n2 54n3 54n4 56n 1 n2 n3 n 4 are used to represent the values of Y P1 1 Y Ii 1jkl 1 i jkl i jkl i jkl i jkl Y D1 , Y P2 1 Y I 2 1 and Y D2 , for 1 7 i 7 n 1 1 1 7 j 7 n 2 1 1 7 k 7 n 3 , 1 7 l 7 n 4 .

10 C.-N. KO and C.-J. WU

3.2. Fitness

In the time-domain, the fitness function of a PID controller can include performance criteria such as the overshoot M P , the rise time tr , the settling time ts , and the steady-state error E ss (Ogata, 2002). However, it should be pointed out that the definition of fitness function may be different from person to person, and that there thus is no systematic way to define a proper fitness function in the PSO. To cope with this difficulty, a systematic way to define the fitness function based on the concept of multiobjective optimization (Ishibuchi and Murata, 19981 Leung and Wang, 20001 Ho et al., 2005) is proposed here. Assuming that one is interested in M different performance criteria, which are denoted by f 1 2x41 f 2 2x41 3 3 3 , f M 2x41 then the fitness function can be defined as f itness 4 Maximize f 1 2x41 f 2 2x41 3 3 3 , f M 2x4 x

(21)

where x 4 2x1 1 x2 1 3 3 3 1 x n 4 is a variable vector in a real and n-dimensional space, is the feasible solution space, and f 1 2x41 f 2 2x41 3 3 3 , f M 2x4 are the M objective functions to be maximized simultaneously. For example, if one wants to minimize the overshoot and the steady-state error simultaneously, then the fitness can be defined as f itness 4 Maximize x

1 1 and

MP E ss

(22)

From the above illustration, it can be seen that the fitness function can be defined systematically using the method in equation (21). Finding an optimal solution to this multiobjective problem is not an easy task, however, although several algorithms have been proposed (Ishibuchi and Murata, 19981 Leung and Wang, 20001 Ho et al., 2005). Ishibuchi and Murata (1998) proposed one feasible way to search from various directions to find the optimal solution, in which fitness function (21) is rewritten as f itness 4 3 1 h 1 2x4 5 3 2 h 2 2x4 5 3 3 3 5 3 M h M 2x4

(23)

where

h i 2x4 4

f i 2x4 max f i 2y4

for i 4 11 21 3 3 3 1 M

(24)

y

23 1 1 3 2 1 3 3 3 1 3 M 4 is the weight vector and 3i 4

ri r1 5 r2 5 3 3 3 5 r M

for i 4 11 21 3 3 3 1 M

(25)

in which r1 1 r2 1 3 3 3 1 r M are nonnegative random real numbers. Then N sets of weight vectors, which are represented as 23 11 1 3 21 1 3 3 3 1 3 M1 41 23 12 1 3 22 1 3 3 3 1 3 M2 41 3 3 3 1 23 1N 1 3 2N 1 3 3 3 1 3 M N 41 respectively, are generated randomly as de-

A PSO-TUNING METHOD FOR DESIGN OF FUZZY PID CONTROLLERS 11

scribed in equation (25). Corresponding to these weight vectors, M different fitness functions are defined as shown in equation (23) and N optimal solutions, which are represented as xopt11 1 xopt12 1 3 3 3 1 xopt1N 1 respectively, are obtained by applying the PSO method. Ishibuchi and Murata (1998) did not discuss fusing these solutions to obtain the optimal solution of the multiobjective problem in equation (21), however, and, a novel method is therefore developed here. The basic idea is to determine the average weight vector of the N sets of weight vectors using N

3 i 4

3 i j 3 h i 2xopt1 j 4

j41 N

for i 4 11 21 3 3 3 1 M

(26)

h i 2xopt1 j 4

j41

With this new set of weight vectors, the corresponding fitness function is defined as f itness 4 3 1 h 1 2x4 5 3 2 h 2 2x4 5 3 3 3 5 3 M h M 2x4

(27)

with the f itness acting as the performance criterion in the PSO procedure to find the optimal solution. This solution is then considered to be the optimal solution of the multiobjective problem in equation (21). 3.3. Searching Procedure

The searching procedure of the proposed PSO-tuning approach is itemized as follows: Algorithm A: Step 1: Choose the population size, the maximum iteration number, and the number of N to generate different weight vectors in equation (25). Step 2: j 4 0

Step 3: j 4 j 5 1

Step 4: Generate a weight vector 23 1 j 1 3 2 j 1 3 3 3 1 3 M j 4 as described in equation (25) and define the corresponding fitness function as shown in equation (23). Step 5: Produce an initial population of position and velocity particles randomly within the feasible range. Step 6: Evaluate the fitness value for each particle of the population using the fitness function. Step 7: Select the local best for each particle by ranking the fitness values. If the best value of all current local best solutions is better than the previous global best solution, then update the value of the global best solution.

12 C.-N. KO and C.-J. WU

Figure 3. The seesaw system.

Step 8: Modify the velocity and position of each particle by using the updated inertia weight, the local best particle, and the global best particle according to equations (12) and (13). Step 9: Repeat Steps 6 to 8 until the number of iterations reaches the maximum iteration number. Then determine the corresponding optimal solution xopt1 j

Step 10: If j N , then go to Step 3. Otherwise, determine the average weight vector 23 1 1 3 2 1 3 3 3 1 3 M 4 as described in equation (26). Step 11: Define the fitness function as shown in equation (27) and apply the procedure described in Steps 5 to 9 to determine the optimal solution. Step 12: The solution obtained in Step 10 is considered to be the optimal solution of the multiobjective problem in equation (21).

4. EXPERIMENTAL EXAMPLE AND RESULTS 4.1. Seesaw System Setup

To demonstrate the feasibility of the proposed approach to multivariable systems, the seesaw system shown in Figure 3 (and in more detail in the schematic diagram in Figure 4) is used for illustration (Wu, 1999). The system contains a personal computer, a DC motor and three small rollers mounted on a base, two optical encoders, a D/A converter, a power amplifier, and a beam supported by an inverted triangle. Using the two optical encoders, the personal computer can read the angle of the beam and the position of the DC motor. The control

A PSO-TUNING METHOD FOR DESIGN OF FUZZY PID CONTROLLERS 13

Figure 4. A schematic diagram of the seesaw system, in which r and 5 denote the position of the motor and the angle of the beam, respectively.

voltage, determined according to the PID control law, is sent via the D/A converter and the power amplifier to the DC motor to make it rotate clockwise or counterclockwise. In this manner, the motor is free to move along the timing belt on the beam. Bumpers are installed at each end of the beam to prevent the motor from falling off. In the seesaw system, the input and output state variables are 8 (t), r(t), and 5(t), which denote the torque of the motor, the position of the motor, and the angle of the beam, respectively. By applying Newton’s second and third laws, the dynamic equations of the seesaw system can be derived as 8 7 Jr  5 2m b 5 m r 4g sin 52t4 m b 5 m r 6 2 r2t4 R 6 2m b 5 m r 4r2t451 2t4 5 2

8 2t4 4 0 R

(28)

1 r2t4  5 22m b 5 m r 4r 2t452t41 2m b 5 m r 4r 2 2t4 5 Jb 52t4 5 2m b 5 m r 4gr2t4 cos 52t4 4 0

(29)

where m b is the combined mass of the base and motor, m r is the mass of the three rollers combined, Jr is the polar moment inertia of the three rollers, R is the radius of the roller, g is the acceleration due to gravity, and Jb is the polar moment inertia of the beam. The values of m b 1 m r 1 Jr 1 R, g1 and Jb can be determined in the laboratory by measurement (Wu, 1999). Substituting these measured values into equations (28) and (29), one obtains r2t4  4

2 64 753 sin 5 2t4 5 0 485r2t451 2t4 6 1 053 9 102 8 2t4 0 451

(30)

 52t4 4

1 r 2t4 6 4 753r2t4 cos 52t4 60 97r2t452t41

0 485r 2 2t4 5 0 161

(31)

14 C.-N. KO and C.-J. WU 1 In equations (30) and (31), the ranges 1 5(t),  used for r(t),r(t),  and 5(t) are[60.38 m,  0.38 m], [60.76 m/sec, 0.76 m/sec], 6 rad1 rad , and 6 radsec1 radsec , 12 12 6 6 respectively. Since the DC motor will slip to either end of the beam under the influence of gravity the initial state and the desired final state are [r1 r1 1 51   unless the system is balanced, 1 4 [01 01 01 0]1 respectively. Meanwhile, the 15] 4 60 38 m1 01  rad1 0 and [r1 r1 1 51 5] 12 input torque 8 2t4 of the motor is also assumed to be in the range [60.1 Nm, 0.1 Nm] due to the physical limitations of power amplifiers. 4.2. PSO-Tuning the Fuzzy PID Controllers

In the seesaw system, the desired values of r(t) and 5(t) are denoted by rd and 5 d . If the PID control law is employed, then the input-output relation of the seesaw system is expressed as 1 8 2t4 4 K P1 e1 2t4 5 K I 1

1 e1 2t4dt 5 K D1 e11 2t4 5 K P2 e2 2t4 5 K I 2

e2 2t4dt 5 K D2 e12 2t4 (32)

where e1 2t4 4 rd 6 r2t4 and e2 2t4 4 5 d 6 5 2t4 are the error signals, e11 2t4 4 r1d 6 r2t4 1 1 and e12 2t4 4 51 d 6 52t4 are the derivatives of the error signals, and K P1 , K I 1 , K D1 , K P2 , K I 2 , and K D2 denote the proportional gains, the integral gains, and the derivative gains that correspond to each error signal. Unlike the classical PID controller in equation (1), the PID gains in equation (32) are adaptive and expressed as shown in equations (4) through (9). In the experiment, the goal is to use the proposed approach to tune the PID gains in equation (32) such that the DC motor can be driven from one end of the beam to the balance state. The population size and maximum iteration number for Algorithm A are set to 20 and 1000, respectively, and the values of 9max , 9min , c1 max , c1 min , c2 max , and c2 min in equations (14), (15), and (16) are set to 0.9, 0.2, 2.5, 0.5, 2.5, and 0.5, respectively. These values are determined based on the work of Ratnaweera et al. (2004), in which an iterative procedure and several benchmarks are used to determine the best ranges of 9, c1 , and c2 . Moreover, it is assumed that the values of n 1 , n 2 , n 3 , and n 4 are all set to five, and the singletons of the output linguistic variables are all real numbers in the range [610, 10]. In this manner, each particle has 3830 elements, consisting of 20 binary-coded elements and 3810 realcoded elements. Among the real elements, 60 elements are used to encode the Gaussian membership functions for the error signals and their derivatives, and 3750 real elements are used to encode the PID gains. In designing the fuzzy PID controller, the primary goal is to drive the seesaw system from the given initial state to the desired final state. However, if the number of fuzzy rules is large, then a heavy computation burden and huge memory requirement are inevitable. Therefore, the primary goal and the need to minimize the number of fuzzy rules should be taken into account simultaneously in defining the fitness function. This means that two performance criteria are chosen: 1

f1 4  2 t e1 2t4 5 e22 2t4 dt

(33)

A PSO-TUNING METHOD FOR DESIGN OF FUZZY PID CONTROLLERS 15

Figure 5. The membership functions of the input linguistic variables, determined by applying Algorithm A with the fitness function defined in equation (35).

1   7 8 8 7 892

n3 n1 n2 n4





15 pi 3 1 5 pj 3 1 5 pk 3 1 5 pl

f 2 4 67

i41

j41

k41

(34)

l41

Where pi , p j , pk , and pl are the binary elements used to indicate which of the membership functions are active. Based on the multiobjective optimization concept described in Section 3.2, the fitness function can then be defined as f itness 4 3 1 h 1 5 3 2 h 2

(35)

16 C.-N. KO and C.-J. WU

Figure 6. Plots of r2t4 of the seesaw system generated by the fuzzy tuning rules in Table 1 (dotted line) and fixed PID gains (solid line).

where 3 1 1 3 2 are nonnegative random weights satisfying 3 1 53 2 4 11 h 1 and h 2 are normalized functions corresponding to f 1 and f 2 . From definition (35), a fitness value evaluating the performance of the fuzzy PID controller can be calculated, with a higher fitness value denoting a better performance. 4.3. Experimental Results

According to the procedure in Algorithm A, using N 4 10 and the fitness function defined in equation (35), the optimal membership functions for e1 2t4, e11 2t4, e2 2t4, and e12 2t4 are determined to be those shown in Figure 5. Meanwhile, the fuzzy tuning rules generated are those shown in Table 1. The plots of r(t) and 5(t) corresponding to these fuzzy PID tuning rules are shown in Figures 6 and 7, respectively. The experimental results using fixed PID gains (i.e., with fixed values of K P1 , K I 1 , K D1 , K P2 , K I 2 , and K D2 , determined by the PSO searching procedure using the performance criterion defined in equation (33)) are given in the same figures, for comparison. By comparing the plots in Figures 6 and 7, one can easily see that the fuzzy PID controller designed using the method described here has a better performance than the conventional one. This is the expected result, since the PID gains are adaptive.

A PSO-TUNING METHOD FOR DESIGN OF FUZZY PID CONTROLLERS 17

Table 1. Fuzzy tuning rules for the PID controller in equation (32). The membership functions X 11 1 X 13 1 X 14 1 X 22 1 X 24 1 X 31 1 X 34 1 X 42 1 X 45 are shown in Figures 5a to 5d.

If e1 is X 11 , e11 is X 22 , e2 is X 31 , and e12 is X 42 , then K P1 4 69.2008, K I 1 4 8 77731 K D1 4 0 44281 K P2 4 1 9529, K I 2 4 9 07991 K D2 4 3 8213 If e1 is X 11 , e11 is X 22 , e2 is X 31 , and e12 is X 45 , then K P1 4 62.18081 K I 1 4 67.24921 K D1 4 0.5826, K P2 4 67.7883, K I 2 4 8 4978, K D2 4 5.0772 If e1 is X 11 , e11 is X 22 , e2 is X 34 , and e12 is X 42 , then K P1 4 68.0126, K I 1 4 68.7805, K D1 4 3.3457, K P2 4 65.1350, K I 2 4 9 5705, K D2 4 64.9733 If e1 is X 11 , e11 is X 22 , e2 is X 34 , and e12 is X 45 , then K P1 4 67.1450, K I 1 4 66.7460, K D1 4 68.7336, K P2 4 5 8542, K I 2 4 2 6483, K D2 4 61.1857 If e1 is X 11 , e11 is X 24 , e2 is X 31 , and e12 is X 42 , then K P1 4 8.0805, K I 1 4 9.5924, K D1 4 64.3047, K P2 4 68.7172, K I 2 4 69.4726, K D2 4 8.1841 If e1 is X 11 , e11 is X 24 , e2 is X 31 , and e12 is X 45 , then K P1 4 9.1396, K I 1 4 4.8199, K D1 4 8.7782, K P2 4 8.1462, K I 2 4 63.4771, K D2 4 64.1988 If e1 is X 11 , e11 is X 24 , e2 is X 34 , and e12 is X 42 , then K P1 4 5.3515, K I 1 4 4.1158, K D1 4 68.0839, K P2 4 62.2318, K I 2 4 67.8415, K D2 4 5 7188 If e1 is X 11 , e11 is X 24 , e2 is X 34 , and e12 is X 45 , then K P1 4 66.8315, K I 1 4 69.25347, K D1 4 64.6317, K P2 4 7.2134, K I 2 4 6 9904, K D2 4 4.2324 If e1 is X 13 , e11 is X 22 , e2 is X 31 , and e12 is X 42 , then K P1 4 68.6416, K I 1 4 69.1266, K D1 4 4.9847, K P2 4 6.7575, K I 2 4 63.1934, K D2 4 7.9194 If e1 is X 13 , is X 22 , e2 is X 31 , and e12 is X 45 , then K P1 4 9.2949, K I 1 4 68.4661, K D1 4 64.0513, K P2 4 66.3973, K I 2 4 4 6093, K D2 4 6.6527 If e1 is X 13 , e11 is X 22 , e2 is X 34 , and e12 is X 42 , then K P1 4 6.6395, K I 1 4 7.1689, K D1 4 6.1842, K P2 4 7.8576, K I 2 4 68.4072, K D2 4 66.6333 If e1 is X 13 , e11 is X 22 , e2 is X 34 , and e12 is X 45 , then K P1 4 6.9439, K I 1 4 6.3816, K D1 4 7.8787, K P2 4 8.0229, K I 2 4 68.1975, K D2 4 63.6756

18 C.-N. KO and C.-J. WU

Table 1. Fuzzy tuning rules for the PID controller in equation (32). The membership functions X 11 1 X 13 1 X 14 1 X 22 1 X 24 1 X 31 1 X 34 1 X 42 1 X 45 are shown in Figures 5a to 5d. (continued)

If e1 is X 13 , e11 is X 24 , e2 is X 31 , and e12 is X 42 , then K P1 4 3.4858, K I 1 4 9.1949, K D1 4 7.5127, K P2 4 7.5627, K I 2 4 8 3658, K D2 4 68.7758 If e1 is X 13 , e11 is X 24 , e2 is X 31 , and e12 is X 45 , then K P1 4 67.3796, K I 1 4 63.8700, K D1 4 66.8397, K P2 4 65.9459, K I 2 4 69.0189, K D2 4 68.9590 If e1 is X 13 , e11 is X 24 , e2 is X 34 , and e12 is X 42 , then K P1 4 8.2232, K I 1 4 4.4823, K D1 4 5.6195, K P2 4 65.3191, K I 2 4 69.0835, K D2 4 5.0251 If e1 is X 13 , e11 is X 24 , e2 is X 34 , and e12 is X 45 , then K P1 4 68.3183, K I 1 4 8.6695, K D1 4 3.7772, K P2 4 7.7778, K I 2 4 4 4323, K D2 4 64.6852 If e1 is X 14 , e11 is X 22 , e2 is X 31 , and e12 is X 42 , then K P1 4 65.0423, K I 1 4 67.6919, K D1 4 67.3488, K P2 4 2.7069, K I 2 4 68.8068, K D2 4 67.1813 If e1 is X 14 , e11 is X 22 , e2 is X 31 , and e12 is X 45 , then K P1 4 67.0280, K I 1 4 68.8860, K D1 4 63.5346, K P2 4 66.7768, K I 2 4 68.8234, K D2 4 66.9528 If e1 is X 14 , e11 is X 22 , e2 is X 34 , and e12 is X 42 , then K P1 4 2 1418, K I 1 4 9.2890, K D1 4 66.8786, K P2 4 62.6200, K I 2 4 9 1483, K D2 4 62.5443 If e1 is X 14 , e11 is X 22 , e2 is X 34 , and e12 is X 45 , then K P1 4 5.5704, K I 1 4 8.2148, K D1 4 60 08181 K P2 4 6.0953, K I 2 4 7 5193, K D2 4 9.1351 If e1 is X 14 , e11 is X 24 , e2 is X 31 , and e12 is X 42 , then K P1 4 9.0192, K I 1 4 8.3970, K D1 4 61 1143, K P2 4 60 2028, K I 2 4 9 0610, K D2 4 2.2465 If e1 is X 14 , e11 is X 24 , e2 is X 31 , and e12 is X 45 , then K P1 4 9.0292, K I 1 4 69.1721, K D1 4 60.7889, K P2 4 66.4975, K I 2 4 6 8545, K D2 4 7.7849 If e1 is X 14 , e11 is X 24 , e2 is X 34 , and e12 is X 42 , then K P1 4 67.6690, K I 1 4 8.4994, K D1 4 61.2316, K P2 4 7.9659, K I 2 4 4 2544, K D2 4 5.0272 If e1 is X 14 , e11 is X 24 , e2 is X 34 , and e12 is X 45 , then K P1 4 62 43021 K I 1 4 1 0806, K D1 4 5.3510, K P2 4 7.57671 K I 2 4 65.6015, K D2 4 4 8651

A PSO-TUNING METHOD FOR DESIGN OF FUZZY PID CONTROLLERS 19

Figure 7. Plots of 52t4 of the seesaw system generated by the fuzzy tuning rules in Table 1 (dotted line) and fixed PID gains (solid line).

5. CONCLUSIONS In this article, a PSO-based approach toward auto-tuning of a fuzzy PID controller is proposed. The PSO-based technique is adopted because it can generate a high-quality solution with more stable convergence characteristics and lower calculation time than other stochastic methods. The feasibility of the PSO-tuning method is verified using experimental tests on a multivariable seesaw example, in which the adaptive PID gains obtained allow the designed fuzzy PID controller better performance than a fixed-gain PID controller achieves. In addition to tuning the fuzzy PID controller, another advantage of the proposed method is the ability to define the fitness function systematically. The basic idea is to use the concept of multiobjective optimization to include different performance criteria in defining the fitness function. This is illustrated in the experimental example, where balancing the seesaw system and minimizing the number of fuzzy rules are achieved simultaneously by utilizing the proposed method. This produces a design for a fuzzy PID controller with as few as rules as possible. As shown in Table 1, good performance can be obtained using only 24 fuzzy rules, allowing the fuzzy PID controller to be implemented very easily in practice because of its simplicity. Acknowledgment. This work was supported in part by the National Science Council, Taiwan, R.O.C., under grants NSC94-2218-E-224-002 and NSC94-2213-E-224-033.

20 C.-N. KO and C.-J. WU

REFERENCES Angeline, P. J., 1998, “Using selection to improve particle swarm optimization,” in Proceedings of the IEEE International Conference on Evolutionary Computation, Anchorage, AK, pp. 84–89. Bennett, S., 1993, “Development of the PID controller,” IEEE Control Systems Magazine 13, 58–65. Boeringer, D. W. and Werner, D. H., 2004, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Transactions on Antennas and Propagation 52(3), 771–779. Cervantes, I., Garrido, R., Jose, A. R., and Martinez, A., 2004, “Vision-based PID control of planar robots,” IEEE Transactions on Mechatronics 9(1), 132–136. Chalhoub, N. G. and Bazzi, B. A., 2004, “Fuzzy logic control for an integrated system of a micro-manipulator with a single flexible beam,” Journal of Vibration and Control 10(5), 755–776. Chatterjee, A., Pulasinghe, K., Watanabe, K., and Izumi, K., 2005, “A particle-swarm-optimized fuzzy-neural network for voice-controlled robot systems,” IEEE Transactions on Industrial Electronics 52(6), 1478–1489. Chen, G., 1996, “Conventional and fuzzy PID controller: An overview,” International Journal of Intelligent Control Systems 1, 235–246. Eberhart, R. C. and Shi, Y., 1998, “Comparison between genetic algorithms and particle swarm optimization,” in Proceedings of the IEEE International Conference on Evolutionary Computation, Anchorage, AK, pp. 611–616. Fogel, D. B., 2000, Evolutionary Computation: Toward a New Philosophy of Machine Intelligence, 2nd Edition, IEEE Press, New York. Gaing, Z. L., 2004, “A particle swarm optimization approach for optimum design of PID controller in AVR system,” IEEE Transactions on Energy Conversion 19(2), 384–391. Ho, S. L., Yang, S. Y., Ni, G. Z., Lo, E. W. C., and Wong, H. C., 2005, “A particle swarm optimization-based method for multiobjective design optimizations,” IEEE Transactions on Magnetics 41(5), 1756–1759. Ishibuchi, H. and Murata, T., 1998, “A multi-objective genetic local search algorithm and its application to flowshop scheduling,” IEEE Transactions on Systems, Man, and Cybernetics-Part C: Applications and Review 28(3), 392–403. Keel, L. H., Rego, J. I., and Bhattacharyya, S. P., 2003, “A new approach to digital PID controller design,” IEEE Transactions on Automatic Control 48(4), 687–692. Kennedy, J. and Eberhart, R. C., 1995, “Particle swarm optimization,” in Proceedings of the IEEE International Conf erence on Neural Networks, Vol. IV, Perth, Australia, pp. 1942–1948. Leung, Y.W. and Wang, Y., 2000, “Multiobjective programming using uniform design and genetic algorithm,” IEEE Transactions on Systems, Man, and Cybernetics-Part C: Applications and Review 30(3), 293–304. Li, H. X., Zhang, L., Cai, K. Y., and Chen, G., 2005, “An improved robust fuzzy-PID controller with optimal fuzzy reasoning,” IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics 35(6), 1283–1294. Li, T. H. S. and Shieh, M. Y., 2000, “Design of a GA-based fuzzy PID controller for non-minimum phase systems,” Fuzzy Sets and Systems 111(2), 183–197. Lin, L., Jan, H. Y., and Shieh, N. C., 2003, “GA-based multiobjective PID control for a linear brushless dc motor,” IEEE Transactions on Mechatronics 8(1), 56–65. Misir, D., Malki, H. A., and Chen, G., 1996, “Design and analysis of a fuzzy proportional-integral-derivative controller,” International Journal of Fuzzy Sets and Systems 79, 297–314. Na, M. G., 2001, “Auto-tuned PID controller using a model predictive control method for the stream generator water level,” IEEE Transactions on Nuclear Science 48(5), 1664–1671. Ogata, K., 2002, Modern Control Engineering, 4th Edition, Prentice-Hall, Englewood Cliffs, NJ. Ratnaweera, A., Halgamuge, S. K., and Watson, C., 2004, “Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients,” IEEE Transactions on evolutionary computation 8(3), 240–255. Parsopoulos, K. E. and Vrahatis M. N., 2002, “Recent approaches to global optimization problems through Particle Swarm Optimization,” Natural Computing 1(2–3), 235–306. Shi, L. Z. and Trabia, M. B., 2005, “Comparison of distributed PD-like and importancebased fuzzy logic controllers for a two-link rigid-flexible manipulator,” Journal of Vibration and Control 11(6), 723–747. Shi, Y. and Eberhart, R. C., 1999, “Empirical study of particle swarm optimization,” in Proceedings of the Congress on Evolutionary Computation, Washington, DC, pp. 1945–1950. Sun, Y. L. and Er, M. J., 2004, “Hybrid fuzzy control of robotics systems,” IEEE Transactions on Fuzzy Systems 12(6), 755–765.

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Tao, C. W. and Taur, J. S., 2005, “Robust fuzzy control for a plant with fuzzy linear model,” IEEE Transactions on Fuzzy Systems 13(1), 30–41. Wang, L. X., 1997, A Course in Fuzzy Systems and Control, Prentice-Hall, Englewood Cliffs, NJ. Whidborne, J. F. and Istepanian, R. S. H., 2001, “Genetic algorithm approach to designing finite-precision controller structures,” IEE Proceedings of Control Theory and Applications 148(5), 377–382. Wu, C. J., 1999, “Quasi time-optimal PID control of multivariable systems: A seesaw example,” Journal of the Chinese Institute of Engineers 22(5), 617–625. Wu, C. J., Liu, G. Y., Cheng, M.Y., and Lee, T. L., 2002, “A neural-network-based method for fuzzy parameter tuning of PID controllers,” Journal of the Chinese Institute of Engineers 25(3), 265–276. Yoshida, H., Kawata, K., and Fukuyama, Y., 2000, “A particle swarm optimization for reactive power and voltage control considering voltage security assessment,” IEEE Transactions on Power Systems 15(4), 1232–1239. Ziegler, J. G. and Nichols, N. B., 1942, “Optimum setting for automatic controller,” ASME Transactions 64, 759– 768.