Evolutionary design of static output feedback controller for Takagi Sugeno fuzzy systems

Transcription

1 Evolutionary design of static output feedback controller for Takagi Sugeno fuzzy systems H.-Y. Chung, S.-M. Wu, F.-M. Yu and W.-J. Chang Abstract: The design of a static output feedback fuzzy controller for nonlinear systems based on Takagi Sugeno (T S) fuzzy models is addressed. To avoid complex mathematical derivations and conservative results, a genetic algorithm (GA) is integrated with a linear matrix inequality (LMI) optimisation to seek the static output feedback gains that satisfy the Lyapunov stability inequalities with a decay rate constraint. To do so, the fitness function of the GA must be made up of constraints that are derived from some fundamental control theories, the Lyapunov stability criteria and LMI solver. To improve the computational efficiency of the GA and LMI solver, a fitness function, called the hierarchical fitness function structure, is built on a hierarchical structure such that the GA can in turn deal with stability inequalities. The relaxed syntheses of static output feedback fuzzy control are then easy to implement without using complex mathematical derivations. 1 Introduction Nonlinear control syntheses based on the Takagi Sugeno (T S) fuzzy model have been successfully developed in the past decade [1 11]. Most of these designs [1 4, 9 11] are based on linear matrix inequality (LMI) techniques [1, 13], which concentrate on transferring various performance constraints and Lyapunov inequalities into LMI representations. Afterwards, a LMI solver can be employed to compute the feedback gains of each fuzzy rule and the common positive definite matrix to satisfy all Lyapunov inequalities. Unfortunately, the static output feedback fuzzy control design becomes much more difficult and complex than the state feedback design because it is a nonlinear matrix inequalities (NLMIs) problem. Therefore, only a few investigators [1, 4, 7] have found a way to convert the NLMI problem into an LMI problem. Unfortunately, the mathematical derivations of these approaches are too complex to apply in the real world. Also, these design results are conservative because some extra constraints have to be attached when converting the NLMIs into LMIs. The purpose of this paper is to develop a simple and powerful method to solve the static output feedback gains of each fuzzy rule such that all the Lyapunov stability inequalities can be satisfied with a decay rate constraint. It is well known that the system performance depends on a closed-loop pole placement that can be shifted by feedback gains. Therefore, the novel idea is to satisfy the Lyapunov stability conditions and performance constraints by tuning the feedback gains of each fuzzy rule. We attempt to # The Institution of Engineering and Technology 7 doi:1.149/iet-cta:67 Paper first received 18th May 6 and in revised form 15th January 7 H.-Y. Chung and S.-M. Wu are with the Department of Electrical Engineering, National Central University, Chung-li 3, Taiwan, Republic of China F.-M. Yu is with the Department of Computer Science and Information Engineering, St. John s University, Taipei 499, Taiwan, Republic of China W.-J. Chang is with the Department of Marine Engineering, National Taiwan Ocean University, Keelung, Taiwan, Republic of China hychung@ee.ncu.edu.tw 196 combine a genetic algorithm (GA) [14 17] with an LMI solver to achieve this end. The GA is employed to seek suitable feedback gains from a prescribed range. Since the feedback gains are given, the Lyapunov stability inequalities of the static output feedback syntheses can be dealt with using the LMI solver. Therefore, the turning mechanism of the GA is composed of the stability information, that is, how to define the fitness function by using the computing results of the LMI solver is the key. It should be noted that the computational efficiency of the LMI solver and the GA drop severely if the static output feedback gains cannot stabilise the corresponding plant rule in advance. Therefore, sequential stability conditions are developed to overcome the above problem, which means that the LMI solver is utilised to check the global stability after all plant rules of the T S fuzzy model are stabilised; that is, the maximum eigenvalue of each closed-loop subsystem is placed on the left plane of the complex axis [18]. Based on the concept of hierarchical structure, the sequential stability conditions are converted into an objective fitness function. This kind of fitness function is called the hierarchical fitness function structure (HFFS). By replacing the objective function with the HFFS, many evolutionary algorithms can be employed to implement the proposed idea. Nevertheless, a GA is recommended because it has been embedded in many optimisation software packages. In addition, it is not necessary to redesign the algorithm structure when changing the performance constraints. Based on the combination of the GA and the LMI solver, we can tackle the static output feedback fuzzy control problems without suffering complex mathematical derivations and attaching extra constraints. Consequently, the design result of the proposed method is much more relaxed than those of previous works even if the design result is a suboptimal solution. The efficiency and validity are demonstrated via a numerical example. Technical background First, a general T S fuzzy model and a static output feedback fuzzy controller are introduced. The Lyapunov criteria IET Control Theory Appl., 7, 1, (4), pp

2 are employed to derive the stability conditions with a constraint on the decay rate. At the end of this section, we point out some of the difficulties in designing a static output feedback fuzzy controller..1 Takagi Sugeno fuzzy model and fuzzy controller The Takagi Sugeno fuzzy model is described by fuzzy IF THEN rules, which represent local linear input output relations of a nonlinear system. Consider the following T S fuzzy model [3, 11] Plant Rule i IF z 1 (t) is M i1 and z n (t) is M in THEN _x(t) ¼ A i x(t) þ B i u(t), y(t) ¼ C i x(t) where i ¼ 1,,..., r, M in is a fuzzy set and r is the total number of IF THEN rules; z(t) ¼ [z 1 (t), z (t),..., z n (t)] is the premise input vector that is composed of the system states; x(t) ¼< n x, u(t) ¼< n u and y(t) ¼< 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 _x(t) ¼ y(t) ¼ ¼ Xr v i{a i x(t) þ B i u(t)} v i (1) h i {A i x(t) þ B i u(t)} () v ic i x(t) v i ¼ Xr h i C i x(t) (3) n where v i ; v i (z(t)) and h i ; h i (z(t)); v i ¼ P j¼1 M ij (z j (t)), and v i is the weight of the ith rule. M ij (z j (t)) is the grade of membership of z j (t) inm ij. To stabilise the fuzzy model (1), many fuzzy controller designs are based on the parallel distributed compensation (PDC) concept [11]. This concept is to design a compensator for each rule of the T S fuzzy model, meaning that the fuzzy controller shares the same fuzzy sets as the fuzzy model (1). Based on the PDC concept, the static output feedback fuzzy controller is Controller Rule i IF z 1 (t) is M i1 and z n (t) is M in THEN u(t) ¼ G i y(t), i ¼ 1,,..., r After executing the defuzzification, for overall fuzzy controller is represented by u(t) ¼ ¼ Xr j¼1 v i v j ðg i C j Þx(t) j¼1 X r j¼1 v i v j h i h j (G i C j )x(t) (4) (5) Substituting (5) into (), we have _x(t) ¼ Xr X r where A cl ; Pr X r j¼1 k¼1 j¼1 k¼1 h i h j h k (A i B i G j C k )x(t) ¼ A cl x(t) (6) h i h j h k (A i B i G j C k ):. Stability conditions and constraint on decay rate It is well known that the Lyapunov-based stability criterion is a powerful and popular method for analysing the stability of a nonlinear system. Based on the Lyapunov stability criterion, the following stability theorem can be obtained for the closed-loop fuzzy system (6). Theorem 1: The equilibrium of the continuous fuzzy system described by (6) is globally asymptotically stable if there exists a P. such that the following conditions are satisfied Q T i P þ PQ i, for i ¼ 1,,..., r (7) R T ijp þ PR ij for i, j ¼ 1,,..., r and i, j (8) S ijk P þ PS ijk for i, j, k ¼ 1,,..., r, and j, k (9) where Q i ; A i B i G i C i (1) R ij ; (A i B i G j C j ) þ (A j B j G i C i ) (11) S ijk ; (A i B i G j C k ) þ (A i B i G k C j ) (1) except the pair (i, j, k) such that h i h j h k ¼, 8t. Proof: Consider a candidate of the Lyapunov function V(x(t)) ¼ x(t) T Px(t), where P.. Then _V(x(t)) ¼ _x(t) T Px(t) þ x(t) T Pẋ(t) 8 8 ¼ x(t) T Xr X r X r (A i B i G j C k ) T 99 < < P = = h i h j h : k : ;; x(t) j¼1 k¼1 þp(a i B i G j C k ) < ¼ x(t) T Xr < h j h [(A i B i G j C j )T P = = i j¼1 : : ;; x(t) þp(a i B i G j C j )] ( P þ x(t) T r ) h j h k [S T ijkp þ PS ijk ] x(t) Xr h i j,k IET Control Theory Appl., Vol. 1, No. 4, July 7 197

3 8 < ¼ x(t) T : þ x(t) T 8 >< ¼ x(t) T >: þ x(t) T j¼1 9 h i h j [(A i B i G j G j ) T P = ; x(t) þp(a i B i G j C j )] Pr j,k h i h j h k [S T ijk P þ PS ijk ] h 3 i [(A i B i G i C i ) T P þp(a i B i G i C i )] þ Pr i,j Pr j,k 9 h i h j [R T ijp þ PR ij ] >; h i h j h k [S T ijkp þ PS ijk ] x(t) >= x(t) x(t) Clearly, if (7) (9) hold, V (x(t)), atx(t) =. A To improve the speed of responses, the constraint of the decay rate has been considered in many controller designs [3, 19]. Therefore, it is easy to infer the following theorem to deal with the constraint on the decay rate. Theorem : The equilibrium of the continuous fuzzy system described by (6) is asymptotically stable on the decay rate a for a., if there exists a P. such that Q T i P þ PQ i þ ap, for i ¼ 1,,..., r (13) R T ij P þ PR ij þ ap fori, j ¼ 1,,..., r and i, j (14) S T ijkpþps ijk þap fori,j,k ¼1,,...,r and j,k (15) except the pair (i, j, k) such that h i h j h k ¼, 8t. Proof: It follows directly from Theorem 1. A Clearly, the control objective is to search for the static output feedback gains of each rule, G i, that are able to satisfy all the conditions of Theorem 1 or. It is well known that LMI optimisation is a powerful tool for solving the T S fuzzy control problems. However, the set of stability inequalities (7) (9) or (13) (15) cannot be directly expressed in terms of the LMIs because C i is located between the decision parameters G i and P. Few investigations were given in this non-convex problem. Hence, the purpose of this research is to develop a simple and flexible method to design a static output feedback fuzzy controller. The design results can be more relaxed than previous works because the constraints caused by the mathematical derivations can be removed. 3 Main results In this work, the design concept is to seek G i with a GA and then to obtain a P. with a LMI optimisation. Note that we do not directly seek G i and P. for (7) (9) or (13) (15) with a multi-objective GA (MOGA) [14] because the computing efficiency of an MOGA drops greatly when tackling many objectives or solving lots of decision variables simultaneously; that is, the GA s method requires a lot of computer time when we regard all Lyapunov inequalities as an objective set. Also, the algorithm structure becomes much more complex than a single-objective one when using a MOGA to implement the proposed idea. Fortunately, the NLMIs (7) (9) or (13) (15) can be regarded as LMI problems after the G i is given. That is 198 why we use a GA to seek G i and then apply the LMI solver to solve a P.. It should be noted that many evolutionary or optimisation algorithms can be employed to seek G i. Nevertheless, we prefer the GA over other numerical algorithms because it possesses the following advantages: (i) in terms of the control field, the GA is much more popular than other optimisation algorithms because it does not have many mathematical requirements for optimisation problems; (ii) it does not need a redesign of the algorithm structure when the objective function is changed, that is, the proposed idea can deal with various performance constraints by modifying the fitness function; (iii) the GA optimisation toolbox (GAOT) has already been formulated for many optimisation software packages, for example, MATLAB 7.X [15]. First, we briefly introduce the rudiments of the LMI solver in the MATLAB LMI toolbox [1] and explain why we redefine the T S fuzzy controller. Second, we use the fundamental control concept to define the sequential stability conditions and show how to convert them into a fitness value, which is used to improve the computing efficiency of the GA and the LMI solver. Based on the above statements, a rough flowchart of the proposed idea is shown in Fig Rudiments of the LMI solver and redefinition of the fuzzy controller When G i and a are given, the conditions (13) (15) of Theorem can be dealt with by the LMI solver via the following auxilliary convex problem [9, 13] Minimise b subject to Q T i P þ PQ i þ ap, bi R T ijp þ PR ij þ ap, bi S T ijkp þ PS ijk þ ap, bi In the MATLAB s LMI control toolbox [1], the function feasp is designed for solving the kind of problems in which the output arguments are b min and X feas, where b min denotes the minimum scalar value b and X feas denotes the decision variable that can be converted into Pby using the function decmat. Remark 1: The function feasp will be terminated when b, is achieved. This implies that the given G i and a can find a P. to satisfy (13) (15). Clearly, b min corresponds to a stability indicator of (13) (15), and the objective becomes to seek G i to achieve b min,. The fundamental control concept shows that the decay rate constraint corresponds to l(a cl ) a,, where l ( ) denotes a set of eigenvalues of. This decay rate Fig. 1 Rough design flowchart IET Control Theory Appl., Vol. 1, No. 4, July 7

4 constraint can be redefined as l(q i ) a (16) l(r ij ) a (17) l(s ijk ) a (18) The GA s computational efficiency drops greatly when the total number of elements of G i is large. To improve the above problems, we redefine the fuzzy controller as Controller Rule i IF z 1 (t) is M i1 and z n (t) is M in THEN u(t) ¼ (G n þ DG i )y(t), i ¼ 1,,..., r (19) where G i ; (G n þ DG i ); G n corresponds to stabilising the nominal model of (1), that is, l(ā n ) a,, where A n ; A n B n G n C n ; A n ¼ 1 r A i, B n ¼ 1=r B i, C n ¼ 1=r C i Remark : The purpose of the controller representation (19) is to improve the searching efficiency. That is G n can be regarded as a result of roughly tuning G i. After G n achieves l (Ā n ) a,, G i is obtained by finely tuning G n that is, regulation of DG i. In the rough-tuning stage, the computing efficiency is promoted because of total number of decision variables of the GA is reduced to n u n y rather than n u n y r. In the fine-tuning strategy, these suitable DG i can be found rapidly because the searching space is greatly reduced. 3. Hierarchical fitness function structure (HFFS) According to the above statements, one can find that the following conditions must be satisfied before tackling (13) (15)with the LMI solver. Requisite 1: l d (Ā n ) a,, Requisite : l d (Q i ) a,, Requisite 3: l d (R ij ) a,, Requisite 4: l d (S ijk ) a,, where l d ( ) ; Re max (l( )) denotes the dominant pole of. Once the above requisites hold, the LMI solver can be employed to deal with (13) (15), that is Requisite 5: The LMI solver is applied to solve (13) (15) until b min, holds. Next, we focus on converting the above sequential stability conditions into a one-objective fitness function such that the GA can deal with the sequential requisites in turn. The fitness function is called the HFFS, which is composed of the following steps Step 1: Convert each stability requisite into the following sub-fitness function which is defined as d 1 ; (d 1 a l d ( A n ))=d 1 (1) d ; (d a l d (Q i ))=d () d 3 ; (d 3 a l d (R ij ))=d 3 (3) d 4 ; (d 4 a l d (S ijk ))=d 4 (4) d 5 ; d 5 b min d 5 (5) where d i denotes the worst value of the ith requisite, for example, d 1 stands for the maximum dominant pole of A n when G n is randomly given; b min is the output argument of the LMI solver that applied to solve Theorem. Remark 3: By randomly generating G n from a prescribed range, it is easy to obtain the distribution of l(ā n ), which contains the distribution of l(q i ), l(r ij ), and l(s ijk ). Therefore, the value of d 1 d 4 is defined as d 1. Following in this way, d 5 stands for the maximum valve of b min. According to (1) (5), the requisite examiner is the parameter d i and it can be enlarged by using the scaling factor. Clearly d i. 1 implies that the ith requisite is satisfied, and the evaluation process carries on dealing with the next sub-fitness function. Step : Merge all sub-fitness functions with the following hierarchical structure FOR i ¼ 1:h Compute fitness i IF fitness i, k or i ¼ 5 fitness ¼ (i 1) k þ fitness i break; END END (6) where h ¼ 5, fitness i has been defined in (1) (5) and fitness denotes the fitness value of the HFFS. The relationship between the final fitness value and the stability requisites is demonstrated in Fig.. Obviously, fitness. 5k implies that G i is able to stabilise the fuzzy model with the decay rate a. 3.3 Mixed GA/LMI algorithm According to Remark, the GA has to seek G n and DG i to obtain G i. Therefore, the GA used in this paper is fitness i ; k d i () where k is a scaling factor used to amplify d i ; fitness i denotes the fitness value of the ith requisite; d i is designed for converting the ith stability requisite into a real value, Fig. Relationship between fitness value and stability requisites IET Control Theory Appl., Vol. 1, No. 4, July 7 199

5 partitioned into the rough-tuning stage G n and the finetuning one DG i The algorithm structure is shown below. Beginning of Mixed GA/LMI algorithm Step 1: Call Initialisation for G n and set g idx ¼. DO WHILE (any one condition of the examination procedure holds) Step : Conversion (if necessary) Convert each individual into decision variables. Step 3: Evaluation Evaluate the fitness value of the HFFS. Step 4: Examination Case 1(g idx ¼ ): Call Initialisation for DG i, when fitness. k. Case (g idx ¼ ): The superior individual is found when fitness. 5k Case 3 The algorithm stops if any one of the following conditions occurs: (i) Generation time limit. (ii) Stall time limit. Step 5: Selection Rearrange the population with superior individuals. Step 6: Reproduction, Crossover & mutation Generate next generation. END DO Beginning of Initialisation for G n 1. Generate a population for G n. End of Initialisation for G n. Beginning of Initialisation for DG i 1. Convert the individual possed of fitness. k into G n.. Fix G n and define g idx ¼ Generate a population for DG i. 4. GOTO Step. End of Initialisation for DG i END of algorithm. Remark 4: Case 1 of the examination procedure is used to switch the rough-tuning state to the fine-tuning one, that is, population can be regenerated for seeking DG i when a qualified G n is found. The above algorithm structure is demonstrated in Fig. 3. Clearly, any type of GAs can be employed to implement the proposed idea as long as we modify the examination procedure and replace the one-objective fitness function with the HFFS. It is important to emphasise that the GA only needs to find a suitable G i to satisfy all stability conditions of Theorem. Therefore, the contribution of the proposed idea is not diminished even if the GA returns to the sub-optimal solution. The advantages of the mixed GA/LMI algorithm can be arranged in the following remark. Remark 5: From the algorithm point-of-view, the GA is able to deal with the sequential constraints solver such that G i and P. can be dealt with respectively. The design result of the proposed method is more relaxed than pure LMI-based ones because it does not attach any extra constraints. From the implementation point-of-view, high gain control is no longer a problem because the search space of G i is bounded. 4 Numerical example Consider a nonlinear system given by _x 1 (t) ¼ x (t) þ sin (x 3 (t)) þ (x 1(t) þ 1)u(t) _x (t) ¼ x 1 (t) þ x (t) _x 3 (t) ¼ x 1(t)x (t) þ x 1 (t) _x 4 (t) ¼ sin (x 3 (t)) y 1 (t) ¼ 3x (t) þ sin (x 3 (t)) y (t) ¼ 1:5x 1 (t) þ x 4 (t) (7) where x 1 (t) [ [a a] and x 3 (t) [ [b b]. The fuzzy modelling for the above nonlinear system was done in [3], and the T-S fuzzy model is Plant Rule 1 IF x 1 (t) ism 11 and x 3 (t) ism 13 THEN _x(t) ¼ A 1 x(t) þ B 1 u(t), y(t) ¼ C 1 x(t) (8) Plant Rule IF x 1 (t) ism 11 and x 3 (t) ism 3 THEN _x(t) ¼ A x(t) þ B u(t), y(t) ¼ C x(t) (9) Plant Rule 3 IF x 1 (t) ism 1 and x 3 (t) ism 13 THEN _x(t) ¼ A 3 x(t) þ B 3 u(t), y(t) ¼ C 3 x(t) (3) Plant Rule 4 IF x 1 (t) ism 1 and x 3 (t) ism 3 THEN _x(t) ¼ A 4 x(t) þ B 4 u(t), y(t) ¼ C 4 x(t) (31) Fig. 3 Demonstration of mixed GA/LMI algorithm _x(t) ¼ X4 h i (x(t)) A i x(t) þ B i u(t) (3) 11 IET Control Theory Appl., Vol. 1, No. 4, July 7

6 where Table 1: BCGA specifications x(t) ¼ [x 1 (t) x (t) x 3 (t) x 4 (t)] T Property Value M 11 (x 1 (t)) ¼ x 1=a M 1 (x 1 (t)) ¼ 1 M 11 (x 1 (t)) M 13 (x 3 (t)) ¼ b sin (x 3) sin (b)x 3 =x 3 (b sin (b)) x 3 = 1 x 3 ¼ M 3 (x 3 (t)) ¼ 1 M 13 (x 3 (t)) þ a 3 1 A 1 ¼ a 7 5, B 1 ¼ , C 1 ¼ 3 1 sin (b)=b 1 þ a 3 1 A ¼ a 7 5, B ¼ , sin (b)=b 3 sin (b)=b C ¼ A 3 ¼ , B 3 ¼ , C 3 ¼ sin (b)=b 1 1 A 4 ¼ , B 4 ¼ , sin (b)=b 3 sin (b)=b C 4 ¼ According to definition (19), we can infer that 3 1 sin (b) 3 1 þ b 1 þ a 1 A n ¼ 1 a, B = n ¼ sin (b) 5 þ b 3 ( 1 3 sin (b) þ C n ¼ 4 b ) 5 In this example, we assume that a ¼.8 and b ¼.6; the range of each element of G n and DG i are given by jg n j, 15 j Dg i j, 3, where g n and Dg i denote an arbitrary element of G n and DG i. The binary code GA (BCGA) is applied to seek G n and DG i ; its specifications are listed in Table 1. In this example, parameters of these specifications have to be Population size 1 Bit length of each decision 8 variable Execution times 5 Stall times 5 Mutation probability.5 Crossover probability.9 given according to some experience. Of course, there are several approaches [14, ] that can be employed to define these specifications so as to improve the computational efficiency. However, the focus of this paper is to solve the static output feedback controller by combining the GA and the LMI solver, and so we do not explain further how to find the optimal values for these parameters of the GAs. All HFFS parameters for this example are given an Table, in which the parameters d 1 d 4 are given to Fig. 4. Since the details of the BCGA can be found in many textbooks, the following statements only focus on illustrating the procedure that needs to be interpreted. Suppose that an individual of the population generated by the procedure Initialisation of G n has been converted as G n ¼ [ 13:4 4:573 ] (33) Substituting (33) into () (1), we have fitness 1 ¼ 1.39 and l( A n ) ¼ :5549, : :75li, 4:86 : Because of fitness 1. k, case 1 of Examination is switched to the fine-tuning state to seek DG i. In the finetuning stage, the GA calls the procedure Initialisation of DG i to hold on to G n and to regenerate a new population for DG i. Suppose that an individual of the DG i population has been converted into DG 1 ¼ [ 1:15 :5977 ], DG ¼ [1:8984 :638 ] (34) DG 3 ¼ [ 1:653 :194 ], DG 4 ¼ [ :565 :73 ] (35) where G i ¼ G n þ DG i. Substituting the above G i into (), we have l d (Q i ) ¼.9 and d ¼.993. Because of fitness ¼ 9.93, k the evaluation process of HFFS is terminated in the nd stage and the fitness value of HFFS, fitness ¼ 19.93, is obtained by using the hierarchical structure (6). After going through 13 iterations, we have l d Table : HFFS specifications K h d 1 d 4 d 5 a Value IET Control Theory Appl., Vol. 1, No. 4, July 7 111

7 Fig. 4 Distribution of l(ā n ) for randomly given G n Fig. 6 Response of control force (R ij ) ¼.44, l d (S ijk ) ¼.466, b min ¼.1141 and G 1 ¼ [ 14:3479 3:578], G ¼ [ 11:564 3:5459] (36) G 3 ¼ [ 14:7753 4:7668], G 4 ¼ [ 1:185 3:3637] (37) 3 6:875 46:599 14:8399 5:8 46: : : :6914 P ¼ : : : : :8 14: : :868 (38) Obviously, matrix P is symmetric and positive definite. Furthermore, it can be easily checked that the stability conditions (13) (15) are satisfied. Therefore, the feedback control system (7) with feedback gains (36) (37) is asymptotically stable. Figs. 5 and 6 show the responses of y(t) and u(t), for the original nonlinear model with x() ¼ [.3.3 ] T. 5 Conclusions In this paper, we have described a mixed GA/LMI algorithm to solve the static output feedback fuzzy control problems. In this algorithm, a GA is employed to seek G i such that the LMI solver is able to find a P. satisfying the Lyapunov stability inequalities. Therefore, the fitness function is composed of sequential stability requisites that are derived from the fundamental control concepts and the rudiments of the LMI optimisation. To deal with these stability requisites in turn, a fitness function of the HFFS is built on a hierarchical structure. The designers can use their favourite GAs to implement this algorithm just by replacing the fitness function with HFFS. The contributions of this paper are that: (i) no extra constraints are attached, (ii) the synthetic procedures only use simple mathematical derivations and fundamental control concepts, and (iii) it is much more flexible than related works, because G i and P. can be solved respectively. In other words, all LMI-based T S fuzzy syntheses with static output feedback free of redefining the stability constraints. The advantages of this method are demonstrated in the numerical example. 6 Acknowledgment The authors would like to thank the associate editor and reviewers for their constructive comments upon which the presentation of this paper was greatly improved. The authors would also like to express their sincere gratitude for the financial support of the National Science Council of the Republic of China under contrast NSC E Finally, we would like to thank Dr. Chein-Chung Sun for his helpful comments that have also improved the quality of this work. Fig Simulated output responses 7 References 1 Lo, J.C., and Lin, M.L.: Robust H-infinity nonlinear control via fuzzy static output feedback, IEEE Trans. Circuits Syst. I: Fundam. Theory Appl., 3, 5, (11), pp Tanaka, K., Kosaki, T., and Wang, H.O.: Backing control problem of a mobile robot with multiple trailers Fuzzy modeling and LMI-based design, IEEE Trans. Syst., Man Cybern. C, 1998, 8, (3), pp Tanaka, K., Ikeda, T., and Wang, H.O.: Fuzzy regulators and fuzzy observers relaxed stability conditions and LMI-based designs, IEEE Trans. Fuzzy Syst., 1998, 6, (), pp IET Control Theory Appl., Vol. 1, No. 4, July 7

8 4 Chang, Y.C., Chen, S.S., Su, S.F., and Lee, T.T.: Static output feedback stabilization for nonlinear interval time-delay systems via fuzzy control approach, Fuzzy Sets Syst., 4, 148, (3), pp Chang, W.J., Sun, C.C., and Chung, H.Y.: Fuzzy controller design for discrete controllability canonical Takagi Sugeno fuzzy systems, IEE Proc., Control Theory Appl., 4, 151, (3), pp Chang, W.J.: Model-based fuzzy controller design with common observability Gramian assignment, ASME J. Dyn. Syst. Meas. Control, 1, 13, (1), pp Chang, W.J., and Sun, C.C.: Constrained fuzzy controller-design of discrete Takagi Sugeno fuzzy models, Fuzzy Sets Syst., 3, 133, (1), pp Chang, W.J.: Fuzzy controller-design via the inverse solution of Lyapunov equations, ASME J. Dyn. Syst. Meas. Control, 3, 15, (1), pp Chen, B.S., Tseng, C.S., and Usang, H.J.: Mixed H H 1 fuzzy output-feedback control design for nonlinear dynamic-systems an LMI approach, IEEE Trans. Fuzzy Syst.,, 8, (3), pp Chen, B.S., Chen, Y.Y., and Lin, C.L.: Nonlinear fuzzy H 1 guidance law with saturation of actuators against maneuvering targets, IEEE Trans. Control Syst. Technol.,, 1, (6), pp Wang, H.O., Tanaka, K., and Griffin, M.F.: An approach to fuzzy control of nonlinear systems: stability and design issues, IEEE Trans. Fuzzy Syst., 1996, 4, (1), pp Gahinet, P., Nemirovski, A., Laub, A.J., and Chilali, M.: LMI Control Toolbox (The Math Work Inc, Natick, MA, 1995) 13 Werner, H.: Multi-model robust control by fast output sampling an LMI approach, Automatica, 1998, 34, (1), pp Deb, K.: Multi-objective optimization using evolutionary algorithms (John Wiley & Sons, Inc, NY, USA, 4) 15 Abramson, M.A.: Genetic algorithm and direct search toolbox (The Math Work Inc, Natick, MA, 4) 16 Man, K.F., Tang, K.S., Kwong, S., and Halang, W.A.: Genetic algorithms for control and signal processing (Advances in Industrial Control, Springer, London, UK, 1997) 17 Goldberg, D.E.: Genetic algorithms in search, optimization, and machine learning (Addison-Wesley Inc, Reading, MA, 1989) 18 Kuo, B.C.: Automatic control system (Prentice-Hall, Inc, Englewood Cliffs, NJ, 1995, 7th edn.) 19 Chilali, M., Gahinet, P., and Apkarian, P.: Robust pole-placement in LMI regions, IEEE Trans. Autom. Control, 1999, 44, (1), pp Yang, Q.W., Jiang, J.P., and Chen, G.: How to select optimal control parameters for genetic algorithms. IEEE Int. Symp. on Industrial Electronics,, pp Komartsova, L.G.: Research of a genetic algorithms designer. IEEE Int. Conf. on Artifical Intellignece Systems,, pp Gates, G.H., Merkle, L.D., Lamont, G.B., and Pachter, R.: Simple genetic algorithm parameter selection for protein structure prediction. IEEE Int. Conf. on Evolutionary Computing, 1995, pp IET Control Theory Appl., Vol. 1, No. 4, July 7 113