Hybrid approaches for regional Takagi Sugeno static output feedback fuzzy controller design

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1 Available online at Expert Systems with Applications Expert Systems with Applications (009) Hybrid approaches for regional Takagi Sugeno static output feedback fuzzy controller design Hung-Yuan Chung *, Sheng-Ming Wu Department of Electrical Engineering, National Central University, Chung-li 0, Taiwan, ROC Abstract This paper proposes a Takagi Sugeno (T S) fuzzy region model to relax the original one. Such switching concept has got rid of the complicated analysis of parallel distributed compensation (PDC). By mixing genetic algorithm (GA) and linear matrix inequality (LMI), we present a new hybrid approach about the static output feedback controller design. It is unlike other researches that involve abstruse mathematic transformations and system constraints that are difficult to find. In this paper, we fix the static output feedback gains by GA to solve the non-convex problem. It is proved that the existence of a set of solvable non-linear matrix inequality (NLMI) suffices to guarantee the stabilization of T S fuzzy region system in H 1 sense. Numerical examples are given to illustrate the effectiveness of the algorithm and validate the new method. Ó 00 Elsevier Ltd. All rights reserved. Keywords: Takagi Sugeno fuzzy region systems; Static output feedback control; Non-linear matrix inequality; Genetic algorithms; Linear matrix inequality solver 1. Introduction * Corresponding author. Tel.: x55. address: hychung@ee.ncu.edu.tw (H.-Y. Chung). T S fuzzy model (Tanaka & Wang, 001) is essentially a non-linear model. It can describe the dynamical characteristics of a complex non-linear system. In the past, the majority of T S fuzzy controller designs were developed by using the concept of parallel distributed compensation (PDC) (Chang, 001; Chang & Sun, 00; Chang, Sun, & Chung, 00; Gahinet, Nemirovski, Laub, & Chilali, 1995; Li, Wang, Niemann, & Tanaka, 000; Tanaka, Kosaki, & Wang, 1998; Tanaka & Sugeno, 199; Tanaka & Wang, 001) and the Lyapunov stability criterion. This kind of problems can be converted into linear matrix inequalities (LMIs) which are solved by LMI optimization (Boyd, Ghaoui, Feron, & Balakrishnan, 199; Gahinet et al., 1995; Tanaka & Wang, 001). However, there are some design disadvantages occurring in static output control when we employ the PDC concept and LMI solver for coping with problems of T S fuzzy systems. In spite of the usefulness of T S fuzzy control, its main drawback comes from the lack of a simple control design methodology. Particularly, the static output feedback controller design of PDC concept for a T S fuzzy system is not easy, due to the fact that many rule interference effects are increased. This phenomenon may cause a time-consuming procedure, difficult analysis, as well as no solution. This paper attempts to relax the PDC problems by combining fuzzy region concept and robust compensation. The concept of fuzzy region is employed to partition the original plant rules into several fuzzy regions (Tanaka, Iwasaki, & Wang, 001; Wang & Sun, 00; Wu, Sun, Chung, & Chang, 005) so that the membership function of consequent is crisp. Hence, the troubles of rule interference effects will disappear. Secondly, the static output feedback fuzzy control design becomes much more difficult and complex than state feedback one because it belongs to a non-linear matrix inequalities (NLMIs) problem. Therefore, few literatures 095-1/$ - see front matter Ó 00 Elsevier Ltd. All rights reserved. doi:10.101/j.eswa

2 H.-Y. Chung, S.-M. Wu / Expert Systems with Applications (009) (Garcia, Pradin, & Zeng, 001; Liu & Fang, 00; Lo & Lin, 00) can find a way in which it always has extra constraints to be attached to convert the NLMI problem into LMI one. In light of the aforementioned concerns, this paper deals with the non-convex issue in a non-linear system which was presented by the T S fuzzy region system with H 1 sense. We attempt to mix a GA (Abramson, 00; Goldberg, 1989; Herrera & Lozano, 000; Iwasaki, Miwa, & Matsui, 005) with an LMI solver to attain this end. The GA is employed to seek suitable feedback gains from a prescribed fitness function. A good initial population will speed up the computation for the following GA/LMI algorithms. Since the feedback gains are given, the Lyapunov stability inequalities of static output feedback syntheses can be dealt with the LMI solver. To carry on, GA will support LMI to tune the solutions until all stability conditions are satisfied. The proposed method, hybrid algorithm and regional fuzzy system, tackles the static output feedback fuzzy control problems with a simple idea and clear mathematical derivations.. Preliminaries and problem statement First, a general T S fuzzy model and a static output feedback fuzzy controller are introduced. Then Lyapunov criteria are employed to derive the stability conditions. At the end of this section, we shall point out some difficulties in designing a static output feedback fuzzy controller..1. Takagi Sugeno fuzzy systems and stability conditions Takagi Sugeno fuzzy model is described by fuzzy IF- rules, which represent local linear input output relations of a non-linear system. Consider the following T S fuzzy model (Tanaka & Wang, 001): Plant Rule i ðpr i Þ IF x 1 ðkþ is M i1...and x n ðkþ is M in ¼A i þb i ; yðkþ ¼C i where i ¼ 1; ;...; r; M in is a fuzzy set and r is the total number of IF- rules; R nx ; R nu and yðkþ R n y are denoted as the state, input and output vector, respectively; All the matrices of (1) are assumed to have appropriate and known dimensions. After executing the defuzzification, the final output of (1) is inferred as follows: P r ¼ x ifa i þb i g P r x i yðkþ ¼ ¼ Xr P r x ic i P r x i h i fa i þb i g; ¼ Xr h i C i ; ð1þ ðþ ðþ where x i x i ðþ and h i h i ðþ; x i ¼ Q n M ijðx j ðkþþ, and x i is the weight of the ith rule. M ij ðx j ðkþþ is the grade of membership of x j ðkþ in M ij. In general, the T S fuzzy controller that stabilize the fuzzy model (1) is designed by the parallel distributed compensation (PDC) concept (Chang, 001; Chang & Sun, 00; Chang et al., 00; Li et al., 000; Tanaka et al., 1998; Tanaka & Sugeno, 199; Tanaka & Wang, 001). Based on the PDC concept, the static output feedback fuzzy controller is shown as follows: Controller Rule i IF x 1 ðkþ is M i1...and x n ðkþ is M in ðþ u ðkþ ¼F i yðkþ; i ¼ 1; ;...; r After executing the defuzzification, the overall fuzzy controller is represented by P r P r ¼ x ix j F i C j P r P r x ix j ¼ Xr X r h i h j ðf i C j Þ: Substituting (5) into (), we have ¼ Xr X r X r k¼1 h i h j h k R ijk ; where R ijk ¼ðA i þ B i F j C k Þ. Based on the Lyapunov stability criterion, one can obtain the following stability theorem for the closed-loop fuzzy system (). Theorem 1. The equilibrium of the discrete fuzzy system described by () is globally asymptotically stable if there exists a P > 0 such that the following conditions are satisfied: R T iii PR iii P < 0; i ¼ 1; ;...; r; ðþ R T iij PR iij P 0; i j ¼ 1; ;...; r; ð8þ R T ijk PR ijk P 0; i ¼ 1; ;...; r ; j where ¼ i þ 1;...; r 1 and k ¼ j þ 1;...; r; R iii R iii R iij R iij þ R iji þ R jii ; R ijk R ijk þ R iji þ R jik þ R jki þ R kij þ R kji 8t s.t. h i \ h j \ h k /, where / denotes a null set. ð5þ ðþ ð9þ ð10þ ð11þ Proof. The Proof of Theorem 1 can be found in Appendix A. h.. Problem description It is well recognized that the LMI optimization is a powerful tool in solving the T S fuzzy control problems, but

3 1 H.-Y. Chung, S.-M. Wu / Expert Systems with Applications (009) one can find that two difficult questions were not easy to handle in the T S fuzzy static output feedback design. (i) PDC concept. It was quite obvious that the rule interference effects have already increased to three indexes (i, j, k). Thus, the PDC stability derivation becomes too complicated and confused to design output feedback gains. (ii) NLMI problems. Clearly, the set of stability inequalities () (9) cannot be directly expressed in terms of the LMIs because C i is located between decision parameters F i and P. For the above-mentioned reasons, the T S fuzzy region system is presented to relax the PDC designed process in the following section and then the detail of NLMI problems can be found in Section.. Regional T S fuzzy control systems Definition 1. The membership functions of the fuzzy model (1) are defined in Fig. 1. It shows that the x n ðkþ can be partitioned into several fuzzy regions by cutting the membership functions at the operating points p 1n ; p n ;...; p qn of x n ðkþ. It can be regarded as the crisp membership function defined as follows: Region jn ¼ 1; p jn < x n ðkþ p ðjþ1þn ; 0; else; where j ¼ 1; ;...; ðq 1Þ. ð1þ According to Definition 1 and considering the H 1 control problem, the original T S fuzzy model can be converted into the following formulation of the T S fuzzy region model (TSFRM)..1. Construction of the fuzzy region systems Region Plant Rule j ðrpr j Þ IF x 1 ðkþ is Region j1...and x n ðkþ is Region jn Local Plant Rule i ðpr i Þ IF x 1 ðkþ is M i1... and x n ðkþ is M in A i½jš B w;i½jš B u;i½jš z 1 ðkþ 5 ¼ C 1;i½jŠ D 1w;i½jŠ D 1u;i½jŠ 5 wðkþ 5; yðkþ C i½jš 0 0 ð1þ where j ¼ 1; ;...; r g and r g is the amount of fuzzy regions; z 1 ðkþ R nz1 is the controlled output vectors for H 1 norm. In order to simplify the system (1), we let A i½jš B w;i½jš B u;i½jš H i½jš C 1;i½jŠ D 1w;i½jŠ D 1u;i½jŠ 5; C i½jš 0 0 This matrix denotes the ith subsystem in the jth region plant rule; where all matrices or vectors of H i½jš are assumed to be of appropriate and known dimensions. After performing the defuzzification, the final outputs of TSFRM are shown below z 1 ðkþ yðkþ 5 ¼ Xrg X r fired h R j h ih i½jš wðkþ 5; ð1þ where, X r g h R i ¼ 1; h R i h R i ðþ ¼ x R i ðþ x R j ðq 1Þ x R j ðþ Y Region jn ðx n ðkþþ; n¼1 X r g x R i ðþ; ð15þ ð1þ Region jn ðx n P ðkþþ is the grade of membership of x n ðtþ in rfired Region jn ; h i ¼ 1, h i h i ðþ and r fired stands for the total number of fired plant rules in each region. Example. Assume x ðkþ and x ðkþ are non-linear states. x ðkþ can be separated into Region jn, (j ¼ 1; ; ; ) and Fig.. Example of illustrated fuzzy region concept (-rule). Fig. 1. The relationship between triangular-type membership function and fuzzy regions.

4 H.-Y. Chung, S.-M. Wu / Expert Systems with Applications (009) x ðkþ can be separated into Region jn, (j ¼ 1). Based on Definition 1 and Fig., RPR 1 is composed of PR 1, PR, PR and PR. Similarly, RPR involves PR, PR, PR and PR 8, etc. where PR i means the plant rule of the original T S fuzzy model. Remark 1. Fig. shows that the T S fuzzy model can be divided into four fuzzy regions. Note that this case is not the only way for this reduced problem. Designers can further combine the adjacent fuzzy regions into a new one... Regional fuzzy controller and stabilization conditions Based on concept of fuzzy region, each RPR j has its corresponding regional controller. In this paper, a static output feedback region controller is designed for system (1) to guarantee the H 1 performance where the controller is given below Region Controller Rule j ðrcr j Þ IF x 1 ðkþ is Region j1...and x n ðkþ is Region jn u ðkþ ¼F ½jŠ y ðkþ; j ¼ 1; ;...; r g : ð1þ The final output of T S fuzzy region controller is computed by P rg ¼ xr j F ½jŠyðkÞ P rg ¼ Xr h R xr j F ½jŠyðkÞ: ð18þ j Recalling (1) and (18), one obtains the closed-loop system (i) the closed-loop system (1) is asymptotically stable; (ii) the norm of the transfer function from wðkþ to z 1 ðkþ satisfies: kt z1wk 1 < 1: ðþ Stability is one of the most important problems in the analysis and synthesis of control systems. In the sequel, we introduce how to derive the stability condition for the T S fuzzy region system. Theorem. The closed-loop fuzzy system (1) is asymptotically stable with the H 1 performance (), if there exists symmetric matrix X > 0 such that the following inequalities are satisfied. XA T i½jš 0 I < 0; ðþ 5 0 C 1;i½jŠ X D 1w;i½jŠ I where j ¼ 1; ;...; r g and i ¼ 1; ;...; r fired. Proof. The Proof of Theorem can be found in Appendix B. h The settlement of interference effect is not enough because the non-convex problem still exists. For example: the symmetric matrix X and static output feedback gains F ½iŠ are unknown matrices which are separated by B u;i½jš, (i.e. XB u;i½jš F ½jŠ C i½jš ) that is not jointly convex in the variables Original genetic algorithm is based on binary branch, which is similar to the chromosome structure of the biol- ¼ Xr g z 1 ðkþ X r g h R j hr ¼1 ( X r ) fired A i½jš þ B u;i½jš F ½ Š C i½jš B w;i½jš h i C 1;i½jŠ þ D 1u;i½jŠ F ½ Š C i½jš D 1w;i½jŠ wðkþ ; j r g : ð19þ Remark. Because Region jn is a crisp membership function, it has the following property: 1; while j ¼ ; h R j \ h R ¼ where / is a null set: /; else; ð0þ The above equation tell us that the closed-loop system (19) is able to cancel the interference effect because only one region plant rule will be fired at any time. Following this property and then (19) can be rewritten as z 1 ðkþ ¼ Xrg h R j ( ) X r fired A i½jš B w;i½jš h i C 1;i½jŠ D 1w;i½jŠ ; wðkþ ð1þ where A i½jš ¼ A i½jš þ B u;i½jš F ½jŠ C i½jš ; C 1;i½jŠ ¼ C 1;i½jŠ þ D 1u;i½jŠ F ½jŠ C i½jš. The design goal of this T S fuzzy control problems is to find the static output feedback gains F ½iŠ such that ðx; F ½jŠ Þ. Unfortunately, the popular method, LMI solver, can not solve this kind of problems. For this reason, we purpose a simple and flexible method to solve this NLMI problem in the coming section.. Searching strategy for static output controller The key of above problem is to fix the static output feedback gains F ½jŠ. In this section, the proposed hybrid procedure combines the reliability properties of the GA and the typical search heuristics with the efficiency and accuracy of LMI solving methods. Hence, the idea is utilizing GA to find suitable fuzzy region controllers with requisite condition and then put them into LMIs for seeking the common positive definite matrix X..1. Generating the initial population

5 1 H.-Y. Chung, S.-M. Wu / Expert Systems with Applications (009) ogy, but the length of binary code is too long that they occupy the larger memory of a computer, especially in many unknown parameters. Therefore, our genetic algorithm works with a real-code representation for the population. The initial population is randomly generated but it is necessary to satisfy the below requisite condition kða i½jš Þ¼maxðabsðeigðA i½jš ÞÞÞ < 1; ðþ where eigðþ stands for eigenvalues of matrix (), absðþ means absolute value of () and maxðþ represents the maximum value of (). The structure of initial populations generating is shown in the following. Substitute randomly F ½jŠ into () FOR p ¼ 1 : h FOR j ¼ 1 : r g Compute kða i½jš Þ IF kða i½jš Þ < 1 F ½jŠ ðpþ ¼F ½jŠ ELSE ð5þ Run GA process END END FðpÞ ¼ F ½1Š ðpþ F ½Š ðpþ F ½rgŠ ðpþ END where h was selected by designer, it is the number of population. Remark. Because the initial population F ½jŠ is created randomly, it is not apt to make the system stable and find the symmetric matrix X. The kða i½jš Þ is an index that ensure the poles of each fuzzy region system were located within unit circle. This procedure goes on until the number of individual of the population is reached which are performed by MATLAB GA tool (Abramson, 00). Good initial populations will contribute to the following LMI conditions. The flowchart of initial population is demonstrated in Fig.... Mixed GA/LMI algorithm When F ½jŠ is given, the condition () of Theorem can be dealt with the LMI solver via auxiliary convex problem. In the MATLAB s LMI control toolbox (Abramson, 00), the function feasp is designed for solving this kind of problems that the output arguments are t min and X feas, where t min denotes the minimum scalar value t and X feas denotes the decision variable which can be converted into X by using the function decmat. Rearrange (), itcan be reformulated as Minimize t subject to XA T i½jš B T < t min I: ðþ w;i½jš 0 I 5 0 C 1;i½jŠ X D 1w;i½jŠ I Substituting the initial population FðpÞ, which was generated by GAs in Section.1, into the LMI conditions () in sequence, the system (1) is asymptotically stable if and only if t min < 0. Therefore, the following equation is used as an evaluation index. fitnessðtþ ¼t min < 0: ðþ To carry on, we use MATLAB s GA toolbox (Abramson, 00) to tune region controllers and seek the t min until it Fig.. The structure if initial population.

6 H.-Y. Chung, S.-M. Wu / Expert Systems with Applications (009) becomes negative. The index t min < 0 means there exists a Lyapunov matrix X for the system (1)... Minimize H 1 performance Given a scalar c > 0, according to Theorem, one can obtain the static output feedback gains with H 1 performance kt z1wk 1 < c ð8þ and then Theorem can be modified as the following theory. Theorem. The closed-loop fuzzy system (1) is asymptotically stable with the H 1 performance c, if there exists c > 0, X > 0 and Z ii such that the following inequalities are satisfied. Minimize c subject to XA T i½jš 0 I 5 < Z ii; ð9þ 0 C 1;i½jŠ X D 1w;i½jŠ c I Z Z 0... < 0; ð0þ Z ii where j ¼ 1; ;...; r g and i ¼ 1; ;...; r fired. Proof. The proof of stability conditions with H 1 performance is similar to Theorem. From(8) we know that kt z1 wk 1 < c is equal to kc 1 T z1 wk 1 < 1 so the matrices C 1;i½jŠ, D 1u;i½jŠ, D 1w;i½jŠ in system (1) can be replaced by c 1 C 1;i½jŠ, c 1 D 1u;i½jŠ, c 1 D 1w;i½jŠ, respectively. Design the H 1 controller for this new system, the inequalities () is re-derived as follows Finally, this paper adds the matrices Z ii (Liu & Fang, 00) in order to relax the stability conditions. If we let each stability condition is smaller than Z ii instead of zero, the solving process and solutions are even more relaxed than the past one. The inequalities () can be relaxed as (9). h Since the value of fitnessðtþ is negative, it represents this optimization problem is feasibility. Then the feedback gains can be calculated by using the fittest chromosome and the positive definite matrix X is solved at the same time. Finally, if H 1 performance can be minimized by Theorem, the goal of this paper is achieved. From the above analysis, the algorithm structure is shown below. Step 1: Generate the initial population F ½jŠ. (i) satisfy kða i½jš Þ < 1, (ii) keep the superior population F ½jŠ ðpþ ¼F ½jŠ to fill FðpÞ ¼½F ½1Š ðpþ F ½Š ðpþ F ½rgŠðpÞ Š T by GA. Step : Substitute FðpÞ into Theorem to find X > 0; Z ii and evaluate fitnessðtþ. Step : (i) If fitnessðtþ < 0, rearrange the population with superior individuals FðpÞ. (ii) If not, perform the GA process, selection; crossover; mutation; new generation and then go to Step. (iii) The algorithm stops if any one of the following conditions occurs: (a) fitnessðtþ < 0, (b) generation limit, (c) stall time limit. Step 5: Substitute the fittest FðpÞ; X and Z ii into LMIs (9), (0) to find c. Step : Convert FðpÞ into F ½jŠ ðpþ and show the final solutions F ½jŠ ; X and c. XðA i½jš þ B u;i½jš F ½jŠ C i½jš Þ T B T < 0; ð1þ w;i½jš 0 I 5 0 c 1 ðc 1;i½jŠ þ D 1u;i½jŠ F ½jŠ C i½jš ÞX c 1 D 1w;i½jŠ I The left-hand side of inequalities (1) can be pre- and postmultiplied by diagfi; I; I; cg. The inequalities is re-derived as follows XA T i½jš 0 I < 0: ðþ 5 0 C 1;i½jŠ X D 1w;i½jŠ c I In this paper routines provided with MATLAB s GA toolbox (Abramson, 00) and the hybrid algorithm structure is demonstrated in Fig.. In the next section, we use a numerical example to prove that the proposed method is feasible and effective. 5. A numerical example Consider a discrete non-linear model as follows:

7 1 H.-Y. Chung, S.-M. Wu / Expert Systems with Applications (009) A i B w;i B u;i T i C 1;i D 1w;i D 1u;i 5; C i 0 0 and the matrices of T i for i ¼ 1; ;...; 10 are given below: 0:5 0 0:5 A 1 ¼ A ¼ 0:a 0 0:5 5; 0 1:5 0: 0:5 0 0:5 A ¼ A ¼ 0:a 0 0:5 5; 1:5 sinðaþ 1:5 0: 0:5 0 0:5 A ¼ A 8 ¼ 0 0 0:5 5; 0 1:5 0: 0:5 0 0:5 A ¼ A 9 ¼ 0:a 0 0:5 5; 1:5 sinðaþ 1:5 0: 0:5 0 0:5 A 5 ¼ A 10 ¼ 0:a 0 0:5 5; 0 1:5 0: Fig.. The flowchart of mixed GA/LMI algorithm. B 1 ¼ B ¼¼B 5 ¼ ½0 0 1=ð1 0:5Þ Š T ; 8 x 1 ðk þ 1Þ ¼0:5x 1 ðkþþ0:5x ðkþ >< x ðk þ 1Þ ¼0:x 1 ðkþx ðkþþ0:5x ðkþ x ðk þ 1Þ ¼1:5 sinðx ðkþþx 1 ðkþþ1:5x ðkþ 0:x ðkþþ þ 0:1wðkÞ ð1 0:5cosðx ðkþþþ ; ðþ y 1 ðkþ ¼x 1 ðkþþ0:5cosðx ðkþþx ðkþ >: y ðkþ ¼x ðkþ where the non-linear state are x ðkþ ð a; aþ and x ðkþ ð b; bþ. Based on T S fuzzy modeling techniques, the non-linear model () can be represented as the following fuzzy model: PR 1 : IF x ðkþ is M 1 and x ðkþ is M 1 z 1 ðkþ 5 ¼ T 1 wðkþ5; yðkþ. PR 10 : IF x ðkþ is M 5 and x ðkþ is M z 1 ðkþ yðkþ 5 ¼ T 10 wðkþ5; where the membership functions and the structure of fuzzy model PR 1 PR 10 are shown in Fig. ; the matrix T i is defined as follows: B ¼ B ¼¼B 10 ¼ ½0 0 1=ð1 0:5bÞ Š T ; 1 0 0:5 cosðbþ C 1 ¼ C 5 ¼ C ¼ C 10 ¼ ; C ¼ C ¼ C ¼ C 9 ¼ 1 0 0:5b ; C ¼ C 8 ¼ ; B w;i ¼ 0 5: : :1 0 C 1;i ¼ ; D 1w;i ¼ 0 5; D 1u;i ¼ 1 5; 0 0 0:5 0 0 where i ¼ 1; ;...; 10; b ¼ cosð88 Þ and we assume that the range of non-linear states are a ¼ p and b ¼ p=, respectively.

8 H.-Y. Chung, S.-M. Wu / Expert Systems with Applications (009) Case 1: Region-based design (-region) Based on the definitions of TSFRM, the above general T S fuzzy model can be represented as the -rule TSFRM. ¼ H ½1Š wðkþ 5; RPR 1 :IFx ðtþ is Region 1 and x ðtþ is Region 1 Local Plant Rule 1 (PR 1 ) IF x (t) ism 1 and x (t) ism 1 ¼ H 1½1Š wðkþ 5; ðaþ Local Plant Rule (PR ) IF x (t) ism and x (t) ism 1 ¼ H ½1Š wðkþ 5; Local Plant Rule (PR ) IF x (t) ism 1 and x (t) ism Table 1 Relationship between -rule TSFRM ðh i½jš Þ and general T S fuzzy model ðt i Þ i ¼ 1 i ¼ i ¼ i ¼ H i½1š T 1 T T T H i½š T T T T 8 H i½š T T T 8 T 9 H i½š T 5 T T 9 T 10 Table Specifications of GA tools Property Value Inial population F Population size 10 Fitness limit of kða i½jš Þ 0.9 Fitness limit of fitnessðtþ Generations 100 Stall generations 50 Stall times 0 Mutation probability 0. Crossover probability 0.8 Local Plant Rule (PR ) IF x (t) ism and x (t) ism ¼ H ½1Š wðkþ 5; RPR :IFx. ðtþ is Region and x ðtþ is Region 1 RPR :IFx ðtþ is Region and x ðtþ is Region 1 Local Plant Rule 1 (PR 1 ) IF x (t) ism and x (t) ism 1 ¼ H 1½Š wðkþ 5; ðbþ.. Local Plant Rule (PR ) IF x (t) ism 5 and x (t) ism ¼ H ½Š wðkþ 5; ðcþ where the relationship between H i½jš and T i is as shown in Table 1. Suppose that an individual of the population was randomly generated by the procedure Step 1. The initial population of must satisfy the requisite condition () to ensure the stability of each fuzzy region system. The parameters of Table Relationship between -rule TSFRM ðh i½jš Þ and general T S fuzzy model ðt i Þ i ¼ 1 i ¼ i ¼ i ¼ i ¼ 5 i ¼ H i½1š T 1 T T T T T 8 H i½š T T T 5 T 8 T 9 T 10 Table kða i½jš Þ for each fuzzy region system Initial FðpÞ p ¼ 1 p ¼ p ¼ p ¼ p ¼ 5 p ¼ p ¼ p ¼ 8 p ¼ 9 p ¼ 10 RPR 1 kða i½jš Þ RPR kða i½jš Þ RPR kða i½jš Þ RPR kða i½jš Þ

9 18 H.-Y. Chung, S.-M. Wu / Expert Systems with Applications (009) Table 5 kða i½jš Þ for each fuzzy region system Initial FðpÞ p ¼ 1 p ¼ p ¼ p ¼ p ¼ 5 p ¼ p ¼ p ¼ 8 p ¼ 9 p ¼ 10 RPR 1 kða i½jš Þ RPR kða i½jš Þ GA process and the range of kða i½jš Þ for each fuzzy region system are shown in Tables and, respectively. According to the evaluation index (), random population was tuned by GA process and it became an excellent initial population. Substitute the above into LMIs (9) and (0) in turns and employ the function feasp to find a matrix X > 0 until fitness function () becomes negative. Because of the result t min ¼ : < 0, the evaluation process terminated after that the static output feedback gains F ½iŠ and positive define matrix X were solved simultaneously F ½1Š ¼½0:05 1:18 Š; F ½Š ¼½ 0:15 0:19 Š; F ½Š ¼½ 0:18 1:59 Š; F ½Š ¼½ 0:8 1:01 Š; 0:1 0:0950 0:108 X ¼ 0:0950 0: 0:080 5: 0:108 0:080 0:5 Since F ½jŠ and X were found, the parameter c of H 1 norm can be minimized by Theorem. Case : Region-based design (-region) By combining the adjacent regions, the original fuzzy model can be converted into the -region TSFRM: RPR 1 :IFx ðtþ is Region 1 and x ðtþ is Region 1 Local Plant Rule 1 (PR 1 ) IF x (t) ism 1 and x (t) ism 1 ¼ H 1½1Š wðkþ 5; ð5aþ. Local Plant Rule (PR ) IF x (t) ism and x (t) ism ¼ H ½1Š wðkþ 5; RPR :IFx ðtþ is Region and x ðtþ is Region 1 Local Plant Rule 1 (PR 1 ) IF x (t) ism and x (t) ism 1 ¼ H 1½Š Fig.. Simulated output responses (Case ). wðkþ. Local Plant Rule (PR ) IF x (t) ism 5 and x (t) ism ¼ H ½Š wðkþ 5: 5; ð5bþ Fig. 5. Simulated output responses (Case 1). The relationship between H i½jš and T i is exhibited in Table. Following the design procedures of Case 1, the range of kða i½jš Þ for each fuzzy region system is shown in Table 5. According to the design process by Theorem, the TSFRC with region controller rules and symmetric matrix are shown as

10 H.-Y. Chung, S.-M. Wu / Expert Systems with Applications (009) F ½1Š ¼½0:1 1:1 Š; F ½Š ¼½ 0:119 0:98 Š: 0:119 0: 0:011 X ¼ 0: 0:1 0:00 5 0:011 0:00 0:808 based on Theorem, it is easy to obtain the minimized parameter c ¼ 0:59 of H 1 norm. For the original non-linear model with xð0þ ¼½0: 0:5 0:8 Š T, Figs. 5 and show the responses of Cases 1 and, respectively.. Conclusions Because of the PDC concept, the rule interference effects increase and then stability analysis becomes complicated for static output feedback design. Therefore, we propose the T S fuzzy region system to relax the rule interference effects of PDC. Although rules were reduced, the non-convex problem still exists. A new approach of static output feedback controller design has been presented where mixed GA and LMI algorithm are used to solve NLMI problems. In this algorithm, GA is employed to seek F ½jŠ such that the LMI solver is able to find a X > 0andc satisfying the Lyapunov stability inequalities. The contributions of this paper are that (i) rules of fuzzy system are reduced, (ii) the synthetic procedures only use simple mathematical derivations and fundamental control concept and (iii) none of extra constraints is attached. The superiorities of this method are verified in the numerical example. Acknowledgement The authors would like to express their sincere gratitude for the financial support of the National Science Council of Republic of China under contract NSC 95-1-E We also thank Prof. Wen-Jer Chang and Dr. Chein-Chung Sun for their helpful comments that improved the quality of this work. Appendix A Proof of Theorem 1. Consider a candidate of the Lyapunov function V ðþ ¼ T P, where P > 0. Then DV ðþ ¼ T P T P ( ) ¼ T Xr X r X r h i h j h k ðr T ijk PR ijk PÞ k¼1 ( ) ¼ T Xr h i ðrt iii PR iii PÞ ( ) X r þ T h i h jðr T iij PR iij PÞ i ( ) þ T Xr X r 1 X r h i h j h k ðr T ijk PR ijk PÞ : j¼iþ1 k¼jþ1 Clearly, if () (9) hold, DV ðþ < 0 at 0. h Appendix B Proof of Theorem. From (Gahinet, 199), one can infer that kt z1 wk 1 < 1ifA i½jš is strictly Hurwitz and there exists a symmetric P > 0 with P 1 A T i½jš P 0 I < 0: ðþ 5 0 C 1;i½jŠ D 1w;i½jŠ I According to property of Schur complement (Boyd et al., 199), () is hold if and only if the following inequalities are satisfied. P 1 A i½jš B w;i½jš A T i½jš P 0 0 I P 1 A i½jš B w;i½jš A T i½jš P 0 0 I 5 < 0; ðþ 5 CT 1;i½jŠ D T 1w;i½jŠ < 0: C 1;i½jŠ D 1w;i½jŠ ð8þ Using Schur complement again, the matrix P 1 A i½jš B w;i½jš A T i½jš P 0 5 is hold if and only if P 0 þ 0 I 0 I A T i½jš P A i½jš B w;i½jš < 0 is satisfied. Hence, (8) can be rewritten as follows P 0 þ AT i½jš P A 0 I B T i½jš w;i½jš þ CT i½jš < 0 D T 1w;i½jŠ C i½jš D 1w;i½jŠ B w;i½jš! AT i½jš P 0 P A B T i½jš B w;i½jš 0 I w;i½jš C T i½jš þ < 0: D T 1w;i½jŠ C i½jš D 1w;i½jŠ ð9þ Suppose there exist a Lyapunov function V ðkþ ¼ x T ðkþp, P > 0, kt z1wk 1 < 1 and define J N ¼ P N 1 k¼0 ðzðkþ T zðkþ w T ðkþwðkþþ In the following, without loss a generality we assume zero initial condition ðxð0þ ¼0Þ. Similar to the derivation of Yong-Yan and Frank (000), we can find that any non-zero wðkþ l ½0; NŠ J N ¼ XN 1 ðzðkþ T zðkþ w T ðkþwðkþþvðk þ 1Þ VðkÞÞ k¼0 þ V ðnþ:

11 10 H.-Y. Chung, S.-M. Wu / Expert Systems with Applications (009) The above condition becomes J N ¼ XN 1 ðx T ðkþða T i½jš PA i½jš P þ C T i½jš C i½jšþ k¼0 þ x T ðkþða T i½jš PB w;i½jš þ C T i½jš D 1w;i½jŠÞwðkÞþw T ðkþ ð PA i½jš þ D T 1w;i½jŠ C i½jšþþw T ðkþ ð B w;i½jš þ D T 1w;i½jŠ D 1w;i½jŠ IÞwðkÞÞ þ V ðnþ: ð0þ After simplifying (0), it yields J N XN 1 T A T i½jš P 0 P A k¼0 wðkþ B T i½jš B w;i½jš 0 I w;i½jš C T i½jš! þ C i½jš D 1w;i½jŠ : ð1þ wðkþ D T 1w;i½jŠ From (9), it implies J N < 0 if the LMI s () hold. Finally, the left-hand side of inequalities () can be pre- and postmultiplied by S T and S, respectively, where S ¼ diagfi; P 1 ; I; Ig. It yields P 1 P 1 A T i½jš P 1 0 I < 0; ðþ 5 0 C 1;i½jŠ P 1 D 1w;i½jŠ I where X ¼ P 1 and () is equivalent to (). Therefore, the inequality () holds for all N > 0. In other words, we have that z l ½0; NŠ, for any non-zero w l ½0; NŠ, and kzk < kwk. It can be concluded that the closed-loop fuzzy system (1) is asymptotically stable with a guaranteed performance in the H 1 sense and thus the proof is completed. h References Abramson, M. A. (00). Genetic algorithm and direct search toolbox. Natick, MA: The Math Work Inc. Boyd, S., Ghaoui, L. E., Feron, E., & Balakrishnan, V. (199). Linear matrix inequalities in system and control theory. Philadelphia, PA: SIAM. Chang, W. J. (001). Model-based fuzzy controller-design with common observability Gramian assignment. Journal of Dynamic Systems Measurement and Control ASME, 1(1), Chang, W. J., & Sun, C. C. (00). Constrained fuzzy controller-design of discrete Takagi Sugeno fuzzy models. Fuzzy Sets and Systems, 1(1), 55. Chang, W. J., Sun, C. C., & Chung, H. Y. (00). Fuzzy controller design for discrete controllability canonical Takagi Sugeno fuzzy systems. IEE Proceedings Control Theory and Applications, 151(), Gahinet, P. (199). Explicit controller formulas for LMI-based H-infinity synthesis. Automatica, (), Gahinet, P., Nemirovski, A., Laub, A. J., & Chilali, M. (1995). LMI control toolbox. The Math Work Inc. Garcia, G., Pradin, B., & Zeng, F. (001). Stabilization of discrete time linear systems by static output feedback. IEEE Transactions on Automatic Control, (1), Goldberg, D. E. (1989). Genetic algorithms in search, optimization, and machine learning. Reading, MA: Addison-Wesley Inc. Herrera, F., & Lozano, M. (000). Gradual distributed real-coded genetic algorithms. IEEE Transactions on Evolutionary Computation, (1),. Iwasaki, M., Miwa, M., & Matsui, N. (005). GA-based evolutionary identification algorithm for unknown structured mechatronic systems. IEEE Transactions on Industrial Electronics, 5(1), Liu, Y. S., Fang, C. H. (00). A new LMI-based approach to relaxed quadratic stabilization of T S fuzzy control systems. In IEEE International Conference on Systems Man and Cybernetics (pp. 55 0). Li, J., Wang, H. O., Niemann, D., & Tanaka, K. (000). Dynamic parallel distributed compensation for Takagi Sugeno fuzzy-systems An LMI approach. Information Sciences, 1( ), Lo, J. C., & Lin, M. L. (00). Robust H-infinity nonlinear control via fuzzy static output feedback. IEEE Transactions on Circuits and Systems Part I: Fundamental Theory and Applications, 50(11), Tanaka, K., Iwasaki, M., & Wang, H. O. (001). Switching control of an R/C Hovercraft Stabilization and smooth switching. IEEE Transaction on Systems Man and Cybernetics Part B, 1(), Tanaka, K., Kosaki, T., & Wang, H. O. (1998). Backing control problem of a mobile robot with multiple trailers Fuzzy modeling and LMIbased design. IEEE Transactions on Systems Man and Cybernetics Part C, 8(), 9. Tanaka, K., & Sugeno, M. (199). Stability analysis and design of fuzzy control-systems. Fuzzy Sets and Systems, 5(), Tanaka, K., & Wang, H. O. (001). Fuzzy control systems design and analysis: A linear matrix inequality approach. NY: John Wiley & Son, Inc.. Wang, W. J., & Sun, C. H. (00). A relaxed stability criterion for T S fuzzy discrete systems. IEEE Transactions on Systems Man and Cybernetics Part B, (5), Wu, S. M., Sun, C. C., Chung, H. Y., & Chang, W. J. (005). Discrete H / H inf nonlinear controller design based on fuzzy region concept and Takagi Sugeno fuzzy framework. IEEE Transactions on Circuit Systems Part: I, 5(1), Yong-Yan, C., & Frank, P. M. (000). Robust H-inf disturbance attenuation for a class of uncertain discrete-time fuzzy systems. IEEE Transactions on Fuzzy Systems, 8(), 0 15.