RELAXED STABILIZATION CONDITIONS FOR SWITCHING T-S FUZZY SYSTEMS WITH PRACTICAL CONSTRAINTS. Received January 2011; revised July 2011

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1 International Journal of Innovative Computing, Information and Control ICIC International 2012 ISSN Volume 8, Number 6, June 2012 pp RELAXED STABILIZATION CONDITIONS FOR SWITCHING T-S FUZZY SYSTEMS WITH PRACTICAL CONSTRAINTS Chung-Hsun Sun 1, Shih-Wei Lin 2 and Yin-Tien Wang 1 1 Department of Mehanial and Eletro-mehanial Engineering Tamkang University No. 151, Yingzhuan Road, Tamsui Distrit, New Taipei City 25137, Taiwan { hsun; ytwang }@mail.tku.edu.tw 2 Nulear Instrumentation Division Institute of Nulear Energy Researh Atomi Energy Counil No. 1000, Wenhua Rd., Jiaan Village, Longtan Township, Taoyuan 3256, Taiwan wwlin@iner.gov.tw Reeived January 2011; revised July 2011 Abstrat. Stability issues in Takagi-Sugeno (T-S) fuzzy ontrol system have been widely investigated using the Lyapunov theorem. With respet to stability analysis, the groupfired rules and the maximal width of state steps are applied to deriving a more relaxed stability riterion for T-S fuzzy disrete models. For the stabilization problem, however, the ontroller synthesis and the analysis of the maximal width of state steps annot be performed simultaneously. Evolutionary algorithms or the iterative linear matrix inequality (ILMI) methods an be employed to solve this problem, but the omputational demand is high. In this study, the pratial onstraints of a ontrol system are adopted in plae of the iterative proedures of the ILMI method. The method redues the omputational burden of the previous stabilization design. Finally, two simulation results are arried out to demonstrate the effetiveness of the proposed ontroller design method. Keywords: Relaxed onditions, T-S fuzzy disrete system, Input onstraint 1. Introdution. During the last deade, based on Takagi-Sugeno (T-S) fuzzy system and Lyapunov theorem, the fuzzy ontrol issues have been extensively explored [1-13]. Tanaka and Sugeno employed a single quadrati Lyapunov funtion V (k) = x T (k)px(k) to verify the stability of a T-S fuzzy disrete system [1]. In [1], the stability riterion states that if a ommon positive definite matrix P an be found to satisfy all Lyapunov inequalities, then the stability of the T-S fuzzy disrete system is guaranteed. The ommon positive definite matrix P searhing problem an be onverted into a linear matrix inequality (LMI) problem, whih an be solved effiiently by MATLAB LMI toolbox. Other studies [1-19] have utilized a pieewise quadrati Lyapunov funtion V (k) = j δ jx T (k)p j x(k) to derive stability riteria for T-S fuzzy disrete systems. To derive these riteria using the pieewise quadrati Lyapunov funtion, the T-S fuzzy system should firstly be transformed into an equivalent swithing T-S fuzzy system. Eah swithing region orresponds to a pieewise Lyapunov funtion V (k) = x T (k)p j x(k) and the stability riterion an be derived. In [16], the stability riterion is that if all the positive definite matries P j an be found to satisfy the orresponding Lyapunov inequalities, the stability of the T-S fuzzy system is guaranteed. Although the pieewise quadrati Lyapunov funtion yields more Lyapunov inequalities than a single quadrati Lyapunov funtion does, the pieewise quadrati Lyapunov funtion derives a more relaxed stability riterion. Also, the positive definite matries P j an be solved by LMI method. 133

2 13 C.-H. SUN, S.-W. LIN AND Y.-T. WANG Aording to the universal approximation theorem, the T-S fuzzy model an approximate any nonlinear system. In general, the more rules there are, the smaller modelling error is. Nevertheless, while a fuzzy system has large rules, the signal/pieewise Lyapunov funtions may not be found even using the LMI tool. Therefore, large fuzzy rules indue onservative stability onditions. Hene, the relaxation of the stability onditions in T-S fuzzy systems with large fuzzy rules is proposed. Applying ideas of group-fired rules and the maximal width of state steps, enables the positive definite matries P j to satisfy fewer orresponding Lyapunov inequalities. Then, more relaxed stability riteria for disrete T-S fuzzy systems are proposed [1]. However, with respet to stabilization, the analysis of the maximal state step width and the design of the ontroller annot be performed simultaneously. Presently, sholars solve this problem by omputing the maximal width of state steps and ontrol gains iteratively. Consequently, the problem of finding positive definite matries P j is transformed into an iterative linear matrix inequality (ILMI) problem [1,20,21], but the omputational requirement is heavy. In this study, to redue the omputing demand of the aforementioned stabilization method, the iterative proedures will be replaed by a diret sheme. The onept of input onstraint is onsidered in this diret sheme. In pratial appliations, a ontrol signal always has a definite range, whih is alled the input onstraint [2,3,23-27]. The onstraint of a ontrol input an be applied to estimating the maximal width of state steps. Then, aording to the ideas of group-fired rules, one an find out the positive definite matries P j that satisfy orresponding Lyapunov inequalities. Additionally, all of the input onstraints and Lyapunov inequalities an be represented in LMI form and solved simultaneously. Aordingly, no iterative omputational sheme is required. Then, a novel methodology with a low omputing demand and relaxed stabilization onditions is proposed. Finally, the effetiveness of the approah is demonstrated in two simulation results. 2. Stability Conditions for Swithing T-S Fuzzy Systems. This setion firstly reviews the stability riterion for a T-S fuzzy disrete system based on a pieewise quadrati Lyapunov funtion. Next, ideas of group-fired rules and the maximal width of state steps are applied to introduing a relaxed stability riterion. Consider the following T-S fuzzy disrete system. Rule l: If x 1 (k) is M l1, x 2 (k) is M l2,..., x n (k) is M ln, then x(k + 1) = A l x(k) + B l u(k) (1) where l = 1, 2,..., r. r and n are the numbers of rules and state variables, respetively. x(k) = [x 1 (k), x 2 (k),..., x n (k)] T is a state vetor. u(k) = [u 1 (k), u 2 (k),..., u m (k)] T is an input vetor. M lj is a fuzzy set. A l and B l are the system matrix and the input matrix, respetively. The final output of the above system is inferred as: r x(k + 1) = w l (k)[a l x(k) + B l u(k)] (2) where w l (k) = n h=1 M lh(x h (k))/ r n h=1 M lh(x h (k)). To prove the stability riterion using a pieewise quadrati Lyapunov funtion, the T-S fuzzy system should be transformed into an equivalent swithing T-S fuzzy system in advane. The swithing T-S fuzzy system is onstruted from risp region rules and loal fuzzy rules. In eah region rule (say sub-region), the state vetor x(k) fires the same loal fuzzy rules meaning that the state spae X is divided into some non-overlapping sub-regions. For example, in Figure 1, the universal set of the premise variable x h is divided into several intervals. Figure 2 presents an example of membership funtions and the assoiated rules table for x(k) R 2 [17]. For example, the states that are in the

3 RELAXED STABILIZATION CONDITIONS FOR SWITCHING T-S FUZZY SYSTEMS 135 M lj ( xj) Ml1 Ml2 Ml3 Ml M l 5 x j Figure 1. The partition of membership funtions M 21 M 22 x 2 x 1 Ij2 M M M M 13 M 1 15 I 11 LR11 LR12 S 1 LR LR13 1 I I I1 LR LR31 LR32 S 2 S 3 S LR LR LR21 22 LR23 2 LR33 3 LR1 LR2 LR3 LR Figure 2. The geometry of a swithing T-S fuzzy system sub-region S 1, fire the same loal rules {LR 11, LR 12, LR 13, LR 1 }. The system (1) an be rewritten as an equivalent swithing T-S fuzzy system in the following form [3,17,20]. Region rule j: If x(k) S j, then Loal Plant rule LR jl : If x 1 (k) is M jl1, x 2 (k) is M jl2,..., x n (k) is M jln, then x(k + 1) = A jl x(k) + B jl u(k) (3) where l = 1, 2,..., ; j = 1, 2,..., s. S j represents the j-th sub-region; s is the number of partitioned sub-regions; LR jl denotes the l-th rule in S j, and is the number of loal fuzzy rules in the sub-region S j. The state spae X is divided into s non-overlapping partitions. Aordingly, s j=1 S j = X, S i Sj = φ for i, j = 1, 2,..., s. The final output of the swithing T-S fuzzy system is inferred as x(k + 1) = w jl (k)[a jl x(k) + B jl u(k)], for x(k) S j () where w jl (k) = n h=1 M lh(x h (k))/ n h=1 M lh(x h (k)). Any state x(k) in the subregion S j an be represented as x(k) = w jl (k)x v jl, for x(k) S j (5) where x v jl denotes the vertex of the sub-region S j. Consider, for example, the geometry of a 2-D state spae. Figure 3 displays the membership funtions and verties of the sub-region S j, and the any state x(k) S j an be represented as follows [17]. x(k) = w jl (k)x v jl, for x(k) S j (6) Then, the pieewise quadrati Lyapunov funtion is employed to prove the stability. The

4 136 C.-H. SUN, S.-W. LIN AND Y.-T. WANG v xj x 2 v xj 3 S j v xj 2 x 1 I j2 I j1 v xj 1 Figure 3. The geometry of 2-D sub-region pieewise quadrati Lyapunov funtion andidate for eah sub-region S j is of the form V j (k) = x(k) T P j x, for x(k) S j (7) The andidate Lyapunov funtion for the overall swithing T-S fuzzy system is s s V (k) = g j (k)v j (k) = g j (k)x(k) T P j x (8) j=1 { 1, x(k) Sj where g j (k) =. Therefore, the stability riterion has been obtained 0, otherwise based on pieewise quadrati Lyapunov funtion. Theorem 2.1. [16-18] Consider the swithing T-S fuzzy disrete system (). If positive definite matries P j exist suh that the LMIs j=1 A T jlp j A jl P j < 0, i, j = 1, 2,..., s; l = 1, 2,...,. (9) hold, then the swithing T-S fuzzy disrete system () is asymptotially stable. Notably, that the matrix P j in (9) is assoiated with the pieewise quadrati Lyapunov funtion for x(k) S j. And the P i in (9) is assoiated with the pieewise quadrati Lyapunov funtion for x(k +1) S i. Define a set Ω to denote the possible state transition from S j to S i [18]. Ω { j, i x(k) S j, x(k + 1) S i } (10) In ase j = i, it means that x(k) and x(k + 1) are in the same sub-region. In ase j i, it means that state x(k) will go into S i from S j. In Theorem 2.1, Ω = { j, i i, j = 1, 2,..., s} meaning that any two onseutive states, x(k) and x(k + 1), may loate anywhere of the state spae X. If the relative positions of two onseutive states, x(k) and x(k + 1), ould be estimated, fewer j, i pairs in set Ω are assoiated with Theorem 2.1. It implies that the number of LMIs (9) an be redued. Next, the maximal width of state steps is utilized to estimate relative positions of x(k) and x(k + 1). x(k) = x(k + 1) x(k) [ βj ] = w jl (k)a jl I x(k), for x(k) S j (11) where I is an identity matrix with appropriate dimensions. Aording to (5), and 0 w jl 1, (11) an be rewritten as [ βj ][ βj x(k) w jl (k)a jl I w jl (k)xjl] v max (Ajl I) x v jl, for x(k) S j (12) l

5 RELAXED STABILIZATION CONDITIONS FOR SWITCHING T-S FUZZY SYSTEMS 137 Equation (12) yields the maximal width of state steps. Therefore, the possible state transitions from S j to S i, i.e., Ω, an be obtained aording to the following relation. x(k) δ j min l (I jl ), for x(k) S j (13) where I jl is the width of sub-region S j, as shown in Figure 2 and Figure 3. δ j is the minimum positive integer that satisfies (13). Consider, for example, a swithing fuzzy system with four sub-regions S j. Figure shows the possible transitions of the two onseutive states x(k) and x(k + 1), that is Ω = { j, i x(k) S j, x(k + 1) S i }. If all δ j = 1, then Ω = { 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 3, 3, 3,,, 3,, }. If all δ j = 2, then Ω = { 1, 1, 1, 2, 1, 3, 2, 1, 2, 2, 2, 3, 2,, 3, 1, 3, 2, 3, 3, 3,,, 2,, 3,, }. If all δ j = 3, the set Ω is as Theorem 2.1, Ω = { j, i i, j = 1, 2,..., s}. Now, the set Ω assoiated with the maximal width of state steps is defined as follows. Ω { j, i j = 1, 2,..., s; i = j(±1, 2,..., δ j ); i > 0} (1) Aording to the information on possible state transitions, i.e., (1), then a relaxed stability riterion based on a pieewise quadrati Lyapunov funtion is obtained. S 1 S 2 S 3 S S 1 S 2 S 3 S (a) all# " 1; x( k)! x( k 1) (b) all# " 2; x( k)! x( k 1) j j Figure. Possible transitions of two onseutive states Theorem 2.2. [1,17] Consider the swithing T-S fuzzy disrete system (). If there exist positive definite matries P j for j = 1, 2,..., s, suh that the LMIs A T jlp j A jl P j < 0, i, j Ω; l = 1, 2,...,. (15) hold, then the swithing T-S fuzzy disrete system () is asymptotially stable. Notably, the set Ω = { j, i j = 1, 2,..., s; i = j ± (1, 2,..., δ j ); i > 0} and δ j are obtained from (12) and (13). Remark 2.1. In Theorem 2.1, all possible j, i pairs, for i, j = 1, 2,..., s, should be onsidered in heking the stability of a T-S fuzzy swithing system. However, in Theorem 2.2 only the j, i pairs, for j = 1, 2,..., s; i = j ± (1, 2,..., δ j ); i > 0, have to be heked. Theorem 2.2 yields a more relaxed result than Theorem Controller Synthesis for Swithing T-S Fuzzy Systems. The aim of this setion is to design a ontroller for the swithing T-S fuzzy system based on the above relaxed stability onept. The well-known parallel distributed ompensation (PDC) is employed to stabilize the swithing T-S fuzzy system. To apply above relaxed stability onept to ontroller synthesis, the PDC fuzzy ontroller should also be rewritten as a swithing fuzzy ontroller: Region rule j: If x(k) S j, Loal Plant rule LR jl : If x 1 (k) is M jl1, x 2 (k) is M jl2,..., x n (k) is M jln, then u(k) = F jl x(k), (16)

6 138 C.-H. SUN, S.-W. LIN AND Y.-T. WANG l = 1, 2,..., ; j = 1, 2,..., s. The final output of the swithing fuzzy ontroller is inferred as: u(k) = w jl (k)f jl x(k), x(k) S j (17) By () and (17), the final output of the losed-loop swithing fuzzy system is x(k + 1) = w jp (k)w jq (k)[a jp B jp F jq ]x(k), for x(k) S j (18) p=1 q=1 In the relaxed stabilization problem, the maximal width of state steps is x(k) = x(k + 1) x(k) β j β j w jp (k)w jq (k) [A jp B jp F jq ] I w jp (k)x v jp p=1 q=1 p=1 max (A jp B jp F jq I) x v jl, for x(k) S j (19) p,q,l To obtain the possible state transitions from S j to S i, i.e., Ω, we try to find the minimum positive integer δ j as (13) to satisfy x(k) max p,q,l (A jp B jp F jq I) x v jl δ j min(i jp ), for x(k) S j (20) p where I jp is the width of sub-region S j, as shown in Figure 2 and Figure 3, Consequently, a orollary of Theorem 2.2 applies. Corollary 3.1. Consider the swithing T-S fuzzy disrete system (). positive definite matries P j for j = 1, 2,..., s, suh that the LMIs If there exist (A jp B jp F jq ) T P j (A jp B jp F jq ) P i < 0, i, j Ω; l = 1, 2,..., (21) hold, then the swithing T-S fuzzy disrete system () is asymptotially stable. Notably, Ω = { j, i j = 1, 2,..., s; i = j ± (1, 2,..., δ j ); i > 0}, where δ j is obtained from (20). Remark 3.1. To estimate δ j in (20), the feedbak ontrol gain F jl must be determined in advane. Then, δ j an be obtained and the stability is heked aording to Corollary 3.1. If the stability onditions of the losed-loop system are not met, the above proess must be repeated. This design proedure involves an iterative proess and heavy omputational requirement. Most reently developed stabilization theorems for the T-S fuzzy system, however, an obtain the ontrol gains and ensure the stability of the losed-loop system simultaneously. A more effetive stabilization sheme is presented below. In pratial appliations, the limited power, driver iruit devies, mehanism, approximation error and other limitations ause the outputs of a ontroller to be always with definite ranges that are defined as the input onstraints [2,3,23-27]. Aordingly, the input onstraint (say u(k) 2 < µ) is employed firstly to estimate δ j. Then, ontroller synthesis and stability heking an be ahieved simultaneously.

7 RELAXED STABILIZATION CONDITIONS FOR SWITCHING T-S FUZZY SYSTEMS 139 Aording to (), x(k) for x(k) S j is rewritten as x(k) = x(k + 1) x(k) β j = w jl (k) [ A jl + B jl u(k) ] x(k) β j = w jl (k)a jl I x(k) + w jl (k)b jl u(k) (22) β j w jl (k)a jl I x(k) + w jl (k)b jl u(k) for x(k) S j Aording to (5) and assume u(k) u(k) 2 < µ, then β j x(k) w jl (k)a jl I w jl (k)x v jl + w jl (k)b jl u(k) max (Ajl I) x v Bjl jl + max µ l l δ j min(i jp ), for x(k) S j (23) p where I is an identity matrix with appropriate dimensions; I jp is the width of sub-region S j, as displayed in Figure 2 and Figure 3, and δ j is the minimum positive integer satisfying (23). Consequently, the set Ω = { j, i j = 1, 2,..., s; i = j ±(1, 2,..., δ j ); i > 0} as (1) is obtained. Furthermore, by the following lemma, the input onstraints u(k) 2 < µ an be represented in LMI form and solved simultaneously with stabilization problem. Lemma 3.1. For the system (2), onsider the single quadrati Lyapunov funtion V (k) = x(k) T Px(k). Assume that the unknown initial ondition x(0) is bounded, i.e., x(0) < φ. The onstraint u(k) 2 < µ is enfored at all time k 0 if the LMIs hold, where Q = P 1, K j = F l Q and j = 1, 2,..., r. Q > φ 2 I (2) [ ] Q K T l µ 2 > 0 (25) I K l Therefore, we have the following main result. Theorem 3.1. Consider the swithing T-S fuzzy disrete system (). The swithing T-S fuzzy disrete system is asymptotially stable, if there exist positive definite matries P j for j = 1, 2,..., s, suh that the LMIs Q j > φ 2 I (26) [ ] Qj K jq µ 2 > 0 (27) I [ ] Q j > 0, j, i Ω; 1 p q β A jp Q j B jp K jq + A jq Q j B jq K jp Q j (28) i hold, where Q j = P 1, K jq = F jq Q j, and φ is the upper bound of the unknown initial ondition x(0), i.e., x(0) < φ. The asterisk indiates the transposed element for symmetri positions. The set Ω = { j, i j = 1, 2,..., s; i = j ± (1, 2,..., δ j ); i > 0} and δ j are obtained from (23).

8 10 C.-H. SUN, S.-W. LIN AND Y.-T. WANG Proof: Firstly, for the onveniene of presentation, (18) is rewritten as x(k + 1) = w jp (k)w jq (k)a jpqx(k), for x(k) S j (29) p=1 q=1 where A jpq = A jp B jp F jq. Considering the pieewise Lyapunov funtion () and taking the differene between V (k + 1) and V (k), we have s s V (k) = V (k+1) V (k) = g i (k+1)x(k+1) T P i x(k+1) g j (k)x(k) T P i x(k) (30) i=1 j=1 Aording to g j (k) = { 1, x(k) Sj 0, otherwise and Ω = { j, i x(k) S j, x(k + 1) S i }, then V (k) = x(k + 1) T P i x(k + 1) x(k) T P j x(k) [ (A ) ] = w jp (k)w jq (k)w ju (k)w jv (k)x(k) T T jpq Pi A jpq P j x(k) p=1 q=1 u=1 v=1 1 β j [ (A w jp (k)w jq (k)x(k) T jpq + Ajqp) T ( ) ] Pi A jpq + A jqp Pj x(k) p=1 q=1 = 1 β j [ (A (w jp (k)) 2 x(k) T jpp + Ajpp) T ( ) ] Pi A jpp + A jpp Pj x(k) p=1 + 2 β j [ (A w jp (k)w jq (k)x(k) T jpq +Ajqp) T ( ) ] Pi A jpq +A jqp Pj x(k) (31) p=1 q>p Then V (k) < 0, if the following LMIs hold: ( ) A jpp + A T ( ) jpp Pi A jpp + A jpp Pj < 0; j, i Ω; p = 1, 2,...,. (32) ( ) A jpq + A T ( ) jqp Pi A jpq + A jqp Pj < 0; j, i Ω; 1 p < q. (33) Combining (32) with (33) and using Shur omplements, then we have the following equivalent LMIs. [ ] P j > 0, j, i Ω; 1 p q β A jp B jp F jq + A jq B jq F j. (3) jp P 1 i Multiplying (3) on the right and left by blok-diag[p 1 I] and defining Q j = P 1 j and K jq = F jq Q j, then we an obtain (28). Moreover, aording to the onept of the maximal width of state steps and input onstraints, i.e., u(k) u(k) 2 < µ, then δ j an be obtained from (23). Consequently, Ω = { j, i j = 1, 2,..., s; i = j ±(1, 2,..., δ j ); i > 0} summarizes the relation of any two onseutive states x(k) S j and x(k + 1) S i. Equations (26) and (27) are orollaries of Lemma 3.1. The proof is ompleted.. Examples. This setion presents two examples to prove the effetiveness of the proposed idea. Example.1. Consider a nonlinear T-S fuzzy system with eight rules as follows: Rule l: If x 1 (k) is M l1, then x(k + 1) = A l x(k) + B l tan(u(k)) (35)

9 RELAXED STABILIZATION CONDITIONS FOR SWITCHING T-S FUZZY SYSTEMS 11 where [ ] [ ] [ ] A 1 = = =, [ ] [ ] [ ] A = = =, [ ] [ ] A 7 = = B 1 = B 2 = = B 8 = [0.1 0.] T. Moreover, membership funtions of this system are shown as Figure 5. M11 M12 M13 M1 M M x x 2 Figure 5. Membership funtions of Example.1 Firstly, onvert the nonlinear T-S fuzzy system (35) system to a swithing T-S fuzzy system as (3), where s = 3 and =. Additionally, verties x v jl of eah sub-region S j are as follows: x v 11 = [ 3 1] T, x v 12 = [ 1 1] T, x v 13 = [ 3 1] T, x v 1 = [ 1 1] T, for S 1 x v 21 = [ 1 1] T, x v 22 = [1 1] T, x v 23 = [ 1 1] T, x v 2 = [1 1] T, for S 2 x v 31 = [1 1] T, x v 32 = [3 1] T, x v 33 = [1 1] T, x v 3 = [3 1] T, for S 3 Therefore, we have φ = max i,j x v jl = Also, if the ontrol input is small enough, this nonlinear funtion tan(u(k)) approximates to u(k). In the approximation of tangent funtion (tan(u(k))), u(k) 2 < 0.5 is reasonable. Therefore, we determine the input onstraint µ = 0.5 in this example. Then, the nonlinear T-S fuzzy system (35) system is transformed into a swithing T-S fuzzy system as Region rule j: If x(k) S j, then Loal Plant rule LR jl : If x 1 (k) is M jl1, x 2 (k) is M jl2, then x(k + 1) = A jl x(k) + B jl u(k) (36) where l = 1, 2, 3, and j = 1, 2, 3. A 11 = A 1 12 = A 2 13 = A 5 1 = A 6, A 21 = A 2 22 = A 3 23 = A 6 2 = A 7 31 = A 3 32 = A 33 = A 7, A 3 = A 8 and B jl = [0.1 0.] T for l = 1, 2, 3, and j = 1, 2, 3. Notably, feasible solutions to the stability analysis problem annot be found for Theorem 2.1 using Matlab LMI toolbox. However, the stability an be guaranteed using Theorem 2.2. This example also demonstrates the effetiveness of the ideas of group-fired rules and the maximal width of state steps. Aording to µ = 0.5 and (23), δ 1 = δ 2 = δ 3 = 1 an be obtained by using (23). Additionally, Ω = { 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 3, 3 }. Then, the ontroller F 11 F 3 an be solved based on Theorem 3.1. F 11 = [ ], F 21 = [ ], F 31 = [ ], F 12 = [ ], F 22 = [ ], F 32 = [ ], F 13 = [ ], F 23 = [ ], F 33 = [ ], F 1 = [ ], F 2 = [ ], F 3 = [ ].

10 12 C.-H. SUN, S.-W. LIN AND Y.-T. WANG Figure 6 plots the simulation results in eight different initial onditions, whih proves that Theorem 3.1 an provide an asymptotially stable ontrol. x 2 x 1 Figure 6. Simulation results of eight different initial onditions (Example.1) Remark.1. Based on Theorem 3.1, only seven j, i Ω pairs should be onsidered for Example.1. However, all possible j, i pairs for i, j = 1, 2, 3, i.e., nine pairs, should be onsidered based on Theorem 3 of [16] for Example.1. Therefore, this work proposes a more relaxed stabilization riterion. Remark.2. Based on Theorem 1 of [17], the ontrol gains F jq and j, i Ω annot be solved simultaneously. That is, in [17] some δ j should be assumed in advane to obtain Ω = { j, i j = 1, 2,..., s; i = j ± (1, 2,..., δ j ); i > 0} and then solve Lyapunov inequalities to obtain F jq and P j. Next, a real δ j is derived from the feedbak ontrol gains F jq. If the predetermined δ j and real δ j are ontraditory, another δ j should be assumed and repeat ontrol design. However, in this work, the real δ j is derived from input onstraint (say u(k) 2 < µ) and then F jq and P j are obtained based on Theorem 3.1. Consequently, no iterative omputational sheme is required. Then, a novel methodology with a low omputing demand and relaxed stabilization onditions is proposed. Example.2. Consider an autonomous vehile system [28] as shown in Figure 7. x 2 u x 1 l x 3 Figure 7. Autonomous vehile system The kinemati equations of the autonomous vehile system are presented as follows: x 1 (k + 1) = x 1 (k) + vt tan(u(k)) (37) l x 2 (k + 1) = x 2 (k) + vt sin(x 1 (k)) (38) x 3 (k + 1) = x 3 (k) + vt os(x 1 (k)) (39)

11 RELAXED STABILIZATION CONDITIONS FOR SWITCHING T-S FUZZY SYSTEMS 13 where x 1 (k) is the angle of the vehile. x 2 (k) and x 3 (k) are the vertial and horizontal positions of the enter of the vehile, respetively. l is the length of the vehile, v is the forward speed of the vehile, u(k) is the steering angle, and t is the sampling time. In this study, l = 0.5(m), v = 1(m/s) and t = 0.5(s). Furthermore, the ontrol purpose is lim k x 1 (k) = 0 and lim k x 2 (k) = 0. As Example.1, if u(k) 2 < 0.5, the nonlinear funtion tan(u(k)) approximates to u(k). Therefore, the ontrol objet is desribed as [ ] [ ] [ ] x1 (k + 1) x = 1 (k) vt/l + u(k) (0) x 2 (k + 1) x 2 (k) + vt sin(x 1 (k)) 0 By using loal approximation method [29] to replae the nonlinear term sin(x 1 (k)) in (0), then we have the swithing T-S fuzzy system for the autonomous vehile as: Region Rule j: If x 1 (k) S j, Loal Fuzzy rule LR jl : If x 1 (k) is M jl, then (1) then x(k + 1) = A jl x(k) + Bu(k) [ ] [ ] where l = 1, 2; j = 1, 2,..., 8. A 11 = A 82 = gvt 1 2 = A 51 = = [ ] [ ] A 21 = A 72 = A 81 = = A 31 = A 62 = A 71 = = [ ] [ ] 1 0 vt/l A 1 = A 52 = A 61 =, B = and g = /π for avoid unontrollable situation [28]. Moreover, membership funtions and swithing regions of this system are shown as Figure 8. Sine µ = 0.5, then δ 1 = δ 2 = = δ 8 = 1 an be obtained by using S1 S2 S3 S S 5 S6 S S 7 8 M11 M12 M21 M22 M31 M32 M1 M2 M51 M52 M61 M62 M71 M72 M81 M82 x 1 ( k)! 3!! 2! Figure 8. Membership funtions and swithing regions of Example.2 (23). Additionally, Ω = { 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 3, 3, 3,,, 3,,,, 5, 5,, 5, 5, 5, 6, 6, 5, 6, 6, 6, 7, 7, 6, 7, 7, 7, 8, 8, 7, 8, 8 }. Next, the autonomous vehile system an be stabilized aording to Theorem 3.1. Figure 9 summarizes the simulation results with four different initial states (x 2 ), x(0) = [0, 10, 0] T, x(0) = [0, 5, 0] T, x(0) = [0, 5, 0] T and x(0) = [0, 10, 0] T, respetively. Also, Figure 10 shows the trajetories of the vehile with four different initial states (angles/x 1 ), x(0) = [π/2, 10, 0] T, x(0) = [ π/2, 10, 10] T, x(0) = [0 ; 10, 0] T and x(0) = [ 0.99π, 10, 10] T, respetively. In Figure 9 and Figure 10, the blok denotes the instant attitude of the vehile. Notably, to avoid the omplexity, only partial instant attitudes of vehile show in the following figures. Simulation results indiate that the ontrol design based on Theorem 3.1 an stabilize the autonomous vehile system. 5. Conlusions. This paper addressed the relaxed stabilization problem for a T-S fuzzy disrete system. A swithing PDC fuzzy ontroller is designed under some suffiient onditions in Theorem 3.1, suh that the whole losed-loop systems are asymptotially

12 1 C.-H. SUN, S.-W. LIN AND Y.-T. WANG 10 ) 5 x 2 ( m x ( m ) Figure 9. Simulation of four different initial onditions (Example.2) 10 ) x 2( m x( ) m Figure 10. Simulation of four different initial onditions (Example.2) stable. The ideas of group-fired rules and the maximal width of the state steps are onsidered. The pratial input onstraint is also adopted to redue omputational effort required by the stabilization design. The stable ontrol law an be solved using the Matlab LMI toolbox. Finally, a numerial example and a pratial appliation are presented to demonstrate the effetiveness of the proposed method. Aknowledgment. This work was supported by the National Siene Counil of Taiwan under the Grant NSC E and NSC E The authors also gratefully aknowledge the helpful omments and suggestions of the reviewers, whih have improved the presentation. REFERENCES [1] G. Feng survey on analysis and design of model-based fuzzy ontrol systems, IEEE Trans. on Fuzzy Syst., vol.1, pp , [2] K. Tanaka and H. O. Wang, Fuzzy Control System Design and Analysis: A Linear Matrix Inequality Approah, Wiley, New York, [3] K. Tanaka, M. Iwasaki and H. O. Wang, Swithing ontrol of an R/C hoverraft: Stabilization and smooth swithing, IEEE Trans. on Syst., Man, Cybern. B, vol.31, pp , [] D. J. Choi and P. G. Park, H state-feedbak ontroller design for disrete-time fuzzy systems using fuzzy weighting-dependent Lyapunov funtions, IEEE Trans. on Fuzzy Syst., vol.11, pp , [5] R. J. Wang, W. W. Lin and W. J. Wang, Stabilizability of linear quadrati state feedbak for unertain fuzzy time-delay systems, IEEE Trans. on Syst., Man, Cybern. B, vol.3, pp , 200.

13 RELAXED STABILIZATION CONDITIONS FOR SWITCHING T-S FUZZY SYSTEMS 15 [6] C. S. Tseng, B. K. Lee and Y. F. Li, Robust fuzzy filter design for a lass of nonlinear disrete-time unertain systems, Int. J. Fuzzy Syst., vol.8, pp , [7] C. W. Park, B. S. Kim and J. B. Lee, Digital stabilization of fuzzy systems with time-delay and its appliation to baking up ontrol of a truk-trailer, Int. J. Fuzzy Syst., vol.9, pp.1-21, [8] H. Chung, S. Wu and W. Chang, Regional fuzzy ontrol for nonlinear ship steering systems, International Journal of Innovative Computing, Information and Control, vol., no.7, pp , [9] J. S. Chiou et al., Stability analysis and ontroller design of the nonlinear swithed systems via T-S disrete-time fuzzy model, Int. J. Fuzzy Syst., vol.11, pp , [10] L. Luoh, Control design of T-S fuzzy large-sale systems, International Journal of Innovative Computing, Information and Control, vol.5, no.9, pp , [11] Z. Gao, B. Jiang, P. Shi and Y. Xu, Fault aommodation for near spae vehile attitude dynamis via T-S fuzzy models, International Journal of Innovative Computing, Information and Control, vol.6, no.11, pp , [12] X.-H. Chang and G.-H. Yang, Relaxed results on stabilization and state feedbak H ontrol onditions for T-S fuzzy systems, International Journal of Innovative Computing, Information and Control, vol.7, no., pp , [13] Y. Xu, B. Jiang, G. Tao and Z. Gao, Fault aommodation for near spae hypersoni vehile with atuator fault, International Journal of Innovative Computing, Information and Control, vol.7, no.5(a), pp , [1] W. J. Wang and C. H. Sun, Relaxed stability and stabilization onditions for a T-S fuzzy disrete system, Fuzzy Sets and Syst., vol.156, pp , [15] M. Johansson. Rantzer and K. E. Årzén, Pieewise quadrati stability of fuzzy systems, IEEE Trans. on Fuzzy Syst., vol.7, pp , [16] L. Wang and G. Feng, Pieewise H ontroller design of disrete time fuzzy systems, IEEE Trans. on Syst., Man, Cybern. B, vol.3, pp , 200. [17] W. J. Wang, Y.-J. Chen and C. H. Sun, Relaxed stabilization riteria for disrete-time T-S fuzzy systems based on swithing fuzzy model and pieewise Lyapunov funtion, IEEE Trans. on Syst., Man, Cybern. B, vol.37, pp , [18] D. J. Choi and P. G. Park, Guaranteed ost ontroller design for disrete-time swithing fuzzy system, IEEE Trans. on Syst., Man, Cybern. B, vol.3, pp , 200. [19] I. Zamani and M. H. Zarifn approah for stability analysis of T-S fuzzy systems via pieewise quadrati stability, International Journal of Innovative Computing, Information and Control, vol.6, no.9, pp.01-05, [20] H. Y. Chung and S. M. Wu, Hybrid approahes for regional Takagi-Sugeno stati output feedbak fuzzy ontroller design, Expert Syst. Appl., vol.36, pp , [21] G. Feng, C. L. Chen, D. Sun and Y. Zhu, Controller synthesis of fuzzy dynami systems based on pieewise Lyapunov funtions and bilinear matrix inequalities, IEEE Trans. on Fuzzy Syst., vol.13, pp.9-105, [22] K. Tanaka, T. Hori and H. O. Wang multiple Lyapunov funtion approah to stabilization of fuzzy ontrol systems, IEEE Trans. on Fuzzy Syst., vol.11, pp , [23] T. Zhang, G. Feng and J. H. Lu, Fuzzy onstrained min-max model preditive ontrol based on pieewise Lyapunov funtions, IEEE Trans. on Fuzzy Syst., vol.15, pp , [2] Y. J. Chen, W. J. Wang and C. L. Chang, Guaranteed ost ontrol for an overhead rane with pratial onstraints: Fuzzy desriptor system approah, Eng. Appl. Artif. Intell., vol.22, pp , [25] H. P. Du and N. Zhang, Fuzzy ontrol for nonlinear unertain eletrohydrauli ative suspensions with input onstraint, IEEE Trans. on Fuzzy Syst., vol.17, pp , [26] S. H. Kim, C. H. Lee and P. G. Park, State-feedbak ontrol for fuzzy systems with input saturation via fuzzy weighting-dependent Lyapunov funtions, Comput. Math. Appl., vol.57, pp , [27] M. Khairy. L. Elshafei and H. M. Emara, LMI based design of onstrained fuzzy preditive ontrol, Fuzzy Sets and Syst., vol.161, pp , [28] K. Tanaka and M. Sano, Trajetory stabilization of a model ar via fuzzy ontrol, Fuzzy Sets and Syst., vol.70, pp , [29] M. C. M. Teixeira and S. H. Żak, Stabilizing ontroller design for unertain nonlinear systems using fuzzy models, IEEE Trans. on Fuzzy Syst., vol.7, pp , 1999.