Solution to global stability of fuzzy regulators via ... - Semantic Scholar

Applied Soft Computing 4 (2004) 25–34

Solution to global stability of fuzzy regulators via evolutionary computation Lanka Udawatta a,1 , Keigo Watanabe b,∗ , Kazuo Kiguchi b , Kiyotaka Izumi b a

b

Department of Production and Control Technology, Saga 840-8502, Japan Department of Advanced Systems Control Engineering, Saga University, Saga 840-8502, Japan Received 8 February 2002; received in revised form 29 January 2003; accepted 30 May 2003

Abstract A novel approach for solving fuzzy model-based stability problems via evolutionary computation (EC) is presented. Gain scheduling problem of a multi-model fuzzy system that satisfies the Lyapunov stability criteria is solved. The generalized eigenvalue problem (GEVP) can be directly introduced to EC in searching positive definite (PD) or positive semi-definite (PSD) matrices, by making a penalty for an individual that violates the inequality condition in order to solve the nonlinear constraints or linear matrix inequalities (LMIs). Four examples for illustrating the proposed methodology are included and the results show the effectiveness. © 2003 Elsevier B.V. All rights reserved. Keywords: Fuzzy controllers; System stability; Evolutionary computation; Linear matrix inequalities

1. Introduction Many theoretical and practical models of artificial intelligence (AI) are gleaned from nature, either by observing human intelligent behavior or by deriving algorithms from natural systems. In AI, fuzzy logic and fuzzy set theory provide a rich and meaningful addition to standard logic, defining a useful rule base, especially for the purpose of engineering control. The basic idea of “fuzzy logic control” (FLC) was suggested by Zadeh [1] and the first implementation of an FLC was reported by Mamdani and Assilian [2,3]. Later, Takagi-Sugeno (TS) fuzzy models became more ∗ Corresponding author. Tel.: +81-952-28-8587; fax: +81-952-28-8587. E-mail addresses: [email protected] (L. Udawatta), [email protected] (K. Watanabe). 1 Present address: Graduate School of Science and Engineering, Saga University, 1-Honjomachi, Saga 840-8502, Japan.

popular due to its simplicity and easiness of implementing on practical controllers [4,5]. The main feature of TS type fuzzy model is that it expresses the local dynamics of each fuzzy rule by linear dynamical model. In fact, research over the past two decades shows a rapid development of fuzzy model-based control (FMC) theory that brings up scientist into a new era in controlling nonlinear systems [6–9]. The one of the core issues of the fuzzy model-based controllers is stability. Recent advances in stability theory on fuzzy controllers [9–12] allowed handling nonlinear control problems. These stability conditions, based on well-known Lyapunov theory, were derived and postulated with matrix inequalities. These matrix inequalities, especially well known and well developed linear matrix inequalities are categorized under one of the promising techniques for analyzing stability criteria with rapidly growing enhanced computing facilities [13–17]. Gain scheduling and stability

1568-4946/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2003.05.009

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L. Udawatta et al. / Applied Soft Computing 4 (2004) 25–34

the EC algorithm is demonstrated in Section 3. The proposed evolutionary computational algorithm for solving stability constraints is presented in Section 4. Finally, results and conclusions are given in Sections 5 and 6, respectively.

2. Stability of fuzzy model-based controllers TS fuzzy models consist of a set of IF-THEN rules for an approximated system [5]. The ith plant rule of each subsystem for a continuous-time system (CS) and discrete-time systems (DS) are given by CS IF z1 (t) is Mi1 and . . . and zp (t) is Mip ˙ = Ai x(t) + Bi u(t) x(t) THEN y(t) = Ci x(t), i = 1, . . . , r

(1)

DS IF z1 (t) is Mi1 and . . . and zp (t) is Mip x(t + 1) = Ai x(t) + Bi u(t) THEN y(t + 1) = Ci x(t), i = 1, . . . , r

(2)

where r is the number of fuzzy rules and Mij (i = 1, . . . , r and j = 1, . . . , p) are the fuzzy sets. The state vector is x(t) ∈ Rn , input vector is u(t) ∈ Rm and the output vector is given by y(t) ∈ Rq . Ai , Bi and Ci are the system parameter matrices of the ith fuzzy model. For a given state, z1 (t), . . . , zp (t) are the premise variables (or antecedent inputs). For the simplicity, TS fuzzy model is represented in Fig. 1. Fuzzy model

Rule 1 Rule 2

Fuzzy controller Linear Model 1 Linear Model 2

Rule r

Model 1 Model 2

…

…

checking of fuzzy regulators via linear matrix inequalities (LMIs) play an important role in designing such fuzzy model-based regulators [9]. These stability conditions were further improved in [10] by relaxed quadratic criteria. For solving these LMIs, recently developed powerful semi-definite programming tools based on convex optimization algorithms are available to date in mathematical literature [9,11]. An exact solution method for a special class of cone-preserving LMIs was developed [14] and optimum solution can be computed by this method faster than general purpose LMI solvers. This research was basically focused on efficient computation of getting the optimal solution and it is limited to a certain class of LMIs. In [17], semi-definite programs whose data depend on some unknown but bounded perturbation parameters were considered and provided sufficient conditions which guarantee that the robust solution is unique and continuous (Holder-stable) with respect to the unperturbed problem’s data. In these studies, solutions were obtained after formulating a set of LMIs. Still there are a considerable number of researches in this field, especially for solving LMI constraints. In this paper, authors focus on developing an algorithm for solving matrix inequalities (including LMIs) using evolutionary computation (EC). Here, GEVP can be directly introduced into EC in searching positive definite (PD) or positive semi-definite (PSD) matrices, by making a penalty for an individual that violates the inequality condition in order to solve the matrix inequalities. Therefore, the designer has the flexibility of selecting either a set of matrix inequalities or LMIs, which lead to obtain a global solution in the optimization process when it comes to stability analysis, as explained in this paper. This method also has the capability of introducing any number of constraints with sub-optimizing goals. In addition to those, one of the advantages of this algorithm is that the LMIs of the fuzzy model-based controller scheme can be directly solved without using any particular software tool for solving LMIs. Furthermore, this proposed algorithm can test the system stability under parameter variations in advance, introducing additional variables as constraints. Rest of the paper is organized as follows. Section 2 provides a theoretical background to stability conditions on FMC. The representation of positive definite (PD) and positive semi-definite (PSD) matrices for

Linear Model r

Model r

Fig. 1. TS fuzzy model and controller.


27

Subjecting to the parallel-distributed compensation, we can design the following fuzzy regulators [18] in CS: Regulator rule i: IF z1 (t) is Mi1 and . . . and zp (t) is Mip

there exists a common positive definite matrix P such that

THEN u(t) = −Ki [x(t) − x r ] + ur

By introducing feedback gains Ki (i = 1, 2, . . . , r) in (4), the stability conditions can be formulated by the following set of equations for both CS and DS [9]. For a CS: Gij + Gji T T Gii P + PGii < 0, P 2 Gij + Gji ≤ 0, i < j (8) +P 2

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

for the fuzzy models (1), where x r is a state reference trajectory, ur is the corresponding input trajectory, and Ki is the local feedback gain matrix. Thus, the fuzzy regulator rules have linear state-feedback laws in the consequent parts and the overall fuzzy regulator can be reduced to r u(t) = − hi (z(t))Ki [x(t) − x r] + ur (4) i=1

ATi P + PAi < 0

for

CS

ATi PAi − P < 0

for

DS

For a DS:

where z(t) = [z1 (t), . . . , zp (t)] wi (z(t)) =

p

GTii PGii − P < 0,

Mij (zj (t))

(5) (6)

j=1

wi (z(t)) hi (z(t)) = r l=1 wl (z(t))

(7)

for all t, in which Mij (zj (t)) denotes the confidence (or grade) of membership of zj (t) in Mij . Following the same manner, we can also design a fuzzy regulator in DS. 2.1. Basic stability conditions In practice, most of the available systems are nonlinear and it is necessary to model them in order to apply linear TS fuzzy models. Neglecting higher order terms, we obtain a linearized model around any arbitrary point. Derivations of Ai and Bi for (1) by using the nonlinear dynamic equation are explained in [28]. The stability of a nonlinear control system is systematically checked by the well-known Lyapunov stability theorems. Here, the basic stability criteria of the open-loop system of (1) is explained. The equilibrium of the fuzzy system described by the TS fuzzy model (1) or (2) is asymptotically stable in the large if

− P ≤ 0,

Gij + Gji 2

T

P

Gij + Gji 2

i 0 so that V˙ (x(t)) ≤ −2αV(x(t)) in CS or V(x(t)) ≤ (α2 − 1)V(x(t)) in DS for all trajectories [9]. Tanaka et al. [9] have introduced that the equilibrium of the fuzzy system described by the TS fuzzy model (1) or (2) is asymptotically stable in the large if there exists a common positive definite matrix P and semi-definite matrix Q such that for a CS: GTii P + PGii + (s − 1)Q + 2αP < 0

Gij + Gji 2

T

P +P

Gij + Gji 2

+ 2αP ≤ 0,

(10)

−Q

i 0, and for a DS: GTii PGii − α2 P + (s − 1)Q < 0

Gij + Gji 2

T P

Gij + Gji 2

− α2 P − Q ≤ 0,

(12)

i 0, Y ≥ 0 X,Y,M1 ,... ,Mr

−XATi −Ai X + MiT BiT + Bi Mi −(s − 1)Y − 2αX > 0

(14)

2Y −XATi − Ai X − XATj − Aj X + MjT BiT + Bi Mj +MiT BjT

+ Bj Mi − 4αX ≥ 0,

i 0, Y ≥ 0

X,Y,M1 ,... ,Mr

βX − (s − 1)Y

XATi − MiT BiT

Ai X − B i Mi

X

βX + Y 1 2 {Ai X + Aj X − Bi Mj

Notation 1. Let Θi (i = 1, . . . , N) be positive definite and positive semi-definite matrices such that Θi (x) > 0,

i = 1, . . . , n1

(18)

Θi (x) ≥ 0,

i = n1 + 1, . . . , N

(19)

where N denotes the number of PD plus PSD matrices required for assuring the closed-loop stability. These Θi (i = 1, . . . , N) are the matrices which are equal to the L.H.S. of the matrix inequalities (8), (9), (or (10)–(13) or (14) and (15)) associated with PD and PSD matrices. Notation 2. Let Θ(x) be N(x) 0 Θ(x) = 0 M(x) where



  N(x) =  

Θ1 (x) ..

  M(x) =  

Θn1 +1 (x)

0 ..

.

   

(16)

1 2 {Ai X + Aj X − Bj Mj

− B j Mi }



ΘN (x)

0 >0

   

. Θn1 (x)

0 



0

X

where X = P −1 , Mi = Ki X and Y = XQX, and β is selected as α2 = β(α < 1) so that 0 ≤ β < 1. 3. Representation of PD and PSD matrices In this section, we present a methodology which leads to solve Eqs. (8)–(17) using EC. As explained in Section 2, direct set of Lyapunov equations or LMIs, which represent the stability conditions of the closed-loop control system, consist of a set of PD and PSD matrices.

− B j M i }T

≥ 0,

i 0,

i = 1, . . . , n1 .

(20)

For Θi (x), i = n1 + 1, . . . , N to be PSD, it needs Λmin (Θi (x)) ≥ 0,

i = n1 + 1, . . . , N

(21)

when Θi (x), i = 1, . . . , N are symmetric matrices.


29

FLC Gains K i , (i = 1,2,..., r ) , Common P and Q Decay Rate α

GENES

Other Parameters

Elements of Θ i ( x), (i = 1,2,..., N )

Fig. 2. Representation of genes.

It is easily found that the matrix Θ(x) > 0, if Θ(x) satisfies the condition: Λmin (Θ(x)) > 0

(22)

whereas the matrix Θ(x) > 0, if Θ(x) satisfies the condition: Λmin (Θ(x)) ≥ 0.

(23)

Note that Λmin (Θ(x)) can be easily found by the following fact: Fig. 3. Evolutionary algorithm.

Λmin (Θ(x)) = min {Λmin (Θ1 (x)), . . . , Λmin (Θn1 (x)), Λmin (Θn1+1 (x)), . . . , Λmin (ΘN (x))} .

(24)

As mentioned before, solutions for the matrix inequalities (8)–(15) are obtained by an EC algorithm. Feedback gains, common P and common Q, decay rate α and other parameters related to the control system are encoded into genes as shown in Fig. 2. Here, we use EC concept introduced in [20]. Moreover, end-user can employ any other EC techniques such as genetic algorithms (GAs) [19]. The optimization process terminates after predefined number of generations, after achieving the desired α (or any other optimization goal). We use the following evolutionary strategy (ES) algorithm explained in the next section.

4. Evolutionary computational algorithm In this section, we develop an EC algorithm for solving matrix inequalities (including LMIs) using ES. Even though this algorithm is constructed using ESs, end-user can employ any of the EC techniques like GA or evolutionary programming, which can be developed for, constrained optimization problems [19].

Solving for generalized eigenvalue problem (GEVP) of (8), (9), (or (10)–(13) or (14)–(15)) is considered as a methodology for searching of optimum common P and common Q (common X and Y).This is systematically developed for determining PD and PSD matrices X and Y, respectively, by using ES algorithm as shown in Fig. 3. Here a1 ,. . . aµ are elements of P, Q and Ki or X, Y and Mi (i = 1, . . . , N) of the LMIs, i.e. the Eqs. (8)–(12) or (14) and (15). Recombined population ak (t) will be obtained by the recombination operation r on population P(t). Mutation operator m will produce ak

(t). P (t) is the new population with x (t) individuals after these two operations. Next population individuals, i.e. x (t + 1), will be selected by evaluating the fitness function Ft considering both x(t) and x (t). The following fitness function is defined in order to fulfill the GEVP of matrix inequalities:       ζ, fitness :

N     pk ,  k=1

if

N

pk ≥ 0

k=1

otherwise.

(25)

30


Here, ζ is a constant and pk is a penalty consisting of Λmin (Θk (x)). In this evolutionary algorithm, all the design parameters of the ES were used as defined in [20] and designer has also the flexibility of using any EC algorithm. Furthermore, various types of constraints can be brought into to the fitness function (25) when it uses ES or any other EC optimization method [19,22].

5. Results In this section, we present three different examples in order to illustrate the design procedure. 5.1. Example 1 The following nonlinear DS is taken into consideration [25]: x1 (t + 1) = −1.4x12 + x2 + 1 + u x2 (t + 1) = 0.3x1 .

(26)

Three linear models around the fixed point (0.6314, 0.1894) are given below: −1.6278 1 1 A1 = , B1 = 0.3 0 0 −1.7678 1 1 A2 = , B2 = 0.3 0 0 −1.9078 1 1 A3 = , B3 = . 0.3 0 0 Note here that the details of system (26) are discussed in [25]. Triangular fuzzy rules given in Fig. 4 are used for examples 1 and 2. Gain scheduling of the above regulator problem can be formulated as an optimization problem with the stability conditions of (9). The optimization algorithm presented in Fig. 3 was applied to the above system (Ai x(t) + Bi u(t) for i = 1, 2, 3) in order to determine the desired feedback gains of the controller. The following common P was obtained after 25 generations:   301.4360 79.9530 . P = 79.9530 687.1344

Fig. 4. Fuzzy rules for the system (26).

Feedback gains of the fuzzy controller are given below: K1 = [5.8877

5.4931] ,

K3 = [3.7798

3.1232] .

K2 = [7.8224

2.4708] ,

Fig. 5 shows the corresponding best fitness values of each generation. 5.2. Example 2 In this example, same linearized models used in example 1 were employed. The fuzzy controller gains were obtained by optimizing the stability and speed of

Fig. 5. Evolutionary history of example 1.


response conditions with (12)–(13). Here, P2 and Q2 matrices could be obtained as follows:   633.277 29.3221 , P = 29.3221 656.3649   55.4408 30.5980  Q= 30.5980 76.9469

together with β = 0.9581, where β=α2 and 0 ≤ β < 1. The gains K1 , K2 and K3 were obtained as below providing the global stability of the fuzzy control system: K1 = [−0.8292

1.0092] ,

K2 = [−2.1818

0.5875] ,

K3 = [−2.0672

1.4699] .

The evolution of the best fitness is given in Fig. 6. 5.3. Example 3 This example is focused on how to employ LMIs in order to determine the gains of the systems. For this purpose, we consider the example of dc motor controlling an inverted pendulum via a gear train as

31

explained in [9]. The nonlinear model was presented by the following two linear models: 

0

1

 A1 =   9.8 0 

0

 A2 =  0 0



0

 1  ,

0 −10

−10

1

0

0 −10





   B1 =   0  10



 1  ,

0



0



   B2 =   0 .

−10

20

The optimization algorithm presented in (14) and (15) was applied to the system and the following common P and Q were obtained: 

3328.76

47.0789

2.2565



 P =  47.0789

98.0633

 0.6499  ,

2.2565

0.6499

0.0599

6978.08

18.8664

0.6184



 Q=  18.8664 0.6184



95.7969

 0.2937  .

0.2937

0.01398

Following gains were obtained at α = 2.2068: K1 = [66.9921

90.0324

6.6415] ,

K2 = [64.7128

77.7657

11.3569] .

5.4. Example 4 Here, we present a more complicated example compared to the previous one. The following nonlinear CS is taken into consideration: x˙ 1 = −δ(x1 − x2 ) + u1 , x˙ 3 = x1 x2 − bx3 .

Fig. 6. Evolutionary history of example 2.

x˙ 2 = −x1 x3 + rx1 − x2 , (27)

Here, the parameters δ, r, and b have the values δ = 10, r = 28, and b = 8/3, respectively. Four linearized models corresponded to the system explained in (27) around the selected fixed-point (0, 0, 0) are given below [24]:

32




−10

10

 A1 =   29.2241

 3.0044  , −2.6667

−1

−1.7353 −3.0044   1    B1 =  0 0 

−10

 A2 =   29.2265 −1.7354   1    B2 =  0   0 

−10  A3 =   29.2253 3.4675   1    B3 =  0   0 

−10

 A4 =   24.3219 0

10



0



0

 −3.0029  ,

−1 3.0029

−2.6667 Fig. 7. Evolutionary history of example 3.



0.6446

 Q=  0.0679 10

0.7350



0

 −1 −8.4839  , 8.4839 −2.6667

10 −1 0

0



 −8.4882  , −2.6667

  1    B4 =  0. 0

0.0679

0.7350



7.6874

 0.8041  .

0.8041

0.7429

Following gains were obtained at α = 0.129: K1 = [82.3694

52.8638

32.2497] ,

K2 = [55.2723

57.1389

−4.8593] ,

K3 = [23.5727

69.6078

7.3486] ,

K4 = [22.0980

48.7338

3.47513].

Moreover, evolutionary history of the optimization process for example 3 is given in Fig. 7 and it is confirmed from this simulation that the algorithm can

The optimization algorithm presented in (14) and (15) was applied to the system (Ai x(t) + Bi u1 (t) for i = 1, 2, 3, 4) in order to determine the desired gains of the feedback controller. The LMIs were solved by using an optimization technique based on EC as explained in Sections 3 and 4.The following common P and Q were obtained:   51642.3 1.1181 2.3941    P =  1.1181 99442.7 4.0974  , 2.3940 4.0974 90146.4 Fig. 8. Eigenvalues of 12 LMI constraints under example 3.


be used for optimizing GEVPs. In fact, the results of [21–27] were obtained under this algorithm. Furthermore, minimum eigenvalues of 12 LMI constraints after the optimization process are given in Fig. 8. It is confirmed that after 500 generations optimal solution for 12 LMIs of (14)–(15) has been achieved via the proposed scheme, because all eigenvalues are greater than or equal to zero.

6. Conclusion This paper has described a novel concept for solving fuzzy controller stability problems on matrix inequalities via evolutionary computation (EC). The generalized eigenvalue problem (GEVP) could be directly introduced to evolutionary computation (EC) in searching positive definite (PD) or positive semi-definite (PSD) matrices, by making a penalty for an individual that violates the inequality condition in order to solve the non-linear constraints or LMIs. Four examples were conducted in simulations so as to illustrate the proposed methodology. The results demonstrated the effectiveness of the proposed algorithm for solving matrix inequalities. An EC based algorithm has a number of advantages. It can quickly scan a vast solution set. Moreover, this EC approach is appropriate for handling any number of constraints of the control system, in addition to the stability criteria guaranteed by the Lyapunov non-linear constraints or LMIs. For an example, if the designer wants to keep the gains at high (or law) it can be easily achieved by setting upper and lower limits of the genes of EC program. In addition to those, one of the advantages of this algorithm is that the LMIs of the fuzzy model-based controller scheme can be directly solved without using any particular software tool for solving LMIs. On the other hand, any of EC algorithms produces a sub-optimal solution and we may not even know whether it has a solution or not. References [1] L.A. Zadeh, Fuzzy sets, J. Inform. Control 8 (1965) 338–353. [2] E.H. Mamdani, Fuzzy sets, 20 years of fuzzy control: experiences gained and lessons learnt, in: Proceedings of IEEE International Conference on fuzzy systems, 1993, pp. 339–344.

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[3] H. Ying, TITO Mamdani fuzzy PI/PD controllers as nonlinear, variable gain PI/PD controllers, Int. J. Fuzzy Syst. 3 (2000) 192–197. [4] A. Jadbabaaie, Robust, non-fragile controller synthesis using model-based fuzzy systems: a linear matrix inequality approach, Master Thesis, University of New Mexico, October 1997. [5] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst., Man Cybernet. 15 (1) (1985) 116–132. [6] R. Palm, D. Driankov, Model Based Fuzzy Control, SpringerVerlag, Berlin, 1997. [7] H. Kiendl, Design of advanced fuzzy systems, in: Proceedings of the TOOLMET’99 Symposium—Tool Environments and Development Methods for Intelligent Systems, Finland, 1999. [8] D. Driankov, R. Palm, Advances in Fuzzy Control, Springer-Verlag, Berlin, 1998. [9] K. Tanaka, T. Ikeda, H.O. Wang, Fuzzy regulators and fuzzy observers: relaxed stability condition and LMI-based designs, IEEE Trans. Fuzzy Syst. 6 (2) (1998) 250–265. [10] E. Kim, H. Lee, New approach to relaxed quadratic stability condition of fuzzy control, IEEE Trans. Fuzzy Syst. 8 (5) (2000) 523–534. [11] S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM, Philadelphia, PA, 1994. [12] X. Ma, Z. Sun, Y. He, Analysis and design of fuzzy controllers and fuzzy observers, IEEE Trans. Fuzzy Syst. 6 (1) (1998) 41–51. [13] S. Limanond, J. Si, Neural-network based control design: an LMI approach, IEEE Trans. Neural Networks 9 (6) (1998) 1422–1429. [14] P.A. Parrilo, S. Khatri, On cone-invariant linear matrix inequalities, IEEE Trans. Automatic Control 45 (8) (2000) 1558–1563. [15] L. Gurvits, M. Mesbahi, A polynomial-time algorithm for solving certain classes of rank minimization problem, in: Proceedings of the American Control Conference, 1999. [16] P.R. Apkarian, H.D. Tuan, Relaxations of parameterized LMIs with control applications, Int. J. Robust Nonlinear Control 9 (1999) 59–84. [17] L. El Ghaoui, F. Oustry, H. Lebret, Robust solutions to uncertain semi-definite programs, SIAM J. Optimization 9 (1) (1998) 33–52. [18] K. Tanaka, H.O. Wang, Fuzzy Control Systems Design and Analysis, Wiley, New York, NY, 2001, pp. 83–96. [19] S. Koziel, Z. Michalewicz, Evolutionary algorithms, homomorphous mapping, and constrained parameter optimization, Evol. Comput. 7 (1) (1999) 20–43. [20] T. Nanayakkara, K. Watanabe, K. Izumi, Evolving in dynamic environment through adaptive chaotic mutation, in: Proceedings of the 4th International Symposium on Artificial Life and Robotics, vol. 2, Oita, Japan, 1997, pp. 520–523. [21] L. Udawatta, K. Watanabe, K. Kiguchi, K. Izumi, Control of robot manipulators via chaotic attractors and fuzzy model-based regulators, in: Proceedings of the 5th International Symposium on Artificial Life and Robotics, vol. 2, Oita, Japan, 2000, pp. 673–676.

34


[22] J. Kim, H. Myung, Evolutionary programming techniques for constrained optimization problems, IEEE Trans. Evol. Comput. 1 (2) (1997) 129–140. [23] L. Udawatta, K. Watanabe, K. Kiguchi, K. Izumi, Control of underactuated manipulators via chaos, in: Proceedings of the Robotics and Mechatronics, Japan, 2000, 1A1-28-027 (CD), pp. 1–2. [24] L. Udawatta, K. Watanabe, K. Kiguchi, K. Izumi, Fuzzy-chaos hybrid controller for 3-dimensional attractors, in: Proceedings of 6th International Conference on Soft Computing (IIZUKA 2000), Iizuka, Japan, 2000, pp. 459–464. [25] K. Watanabe, L. Udawatta, K. Kiguchi, K. Izumi, Control of chaotic systems using fuzzy model-based regulators, in: Proceedings of Sixth International Workshop on Rough Sets, Fuzzy Sets and Data Mining Granular Computing

(RSFDGrC99), Lecture Notes in Artificial Intelligence, Springer-Verlag, Berlin, 1999, pp. 248–256. [26] K. Watanabe, L. Udawatta, K. Izumi, A Solution of fuzzy regulator problems via evolutionary computations, in: Proceedings of SICE’99, Morioka, Japan, 1999, pp. 835–836 (in Japanese). [27] L. Udawatta, K. Watanabe, K. Kiguchi, K. Izumi, Global stability condition of fuzzy model-based controllers via evolutionary computation, in: Proceedings of IEEE World Congress on Computational Intelligence, Honolulu, Hawaii, 2002, pp. 238–244. [28] L. Udawatta, K. Watanabe, K. Kiguchi, K. Izumi, FuzzyChaos hybrid controller for controlling of nonlinear systems, IEEE Trans. Fuzzy Syst. 10 (3) (2002) 401–411.