Robust H control of Takagi Sugeno fuzzy systems with state and input time delays

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Fuzzy Sets and Systems 6 (29 43 422 www.elsevier.

University, Qingdao 266, PR China b Faculty of Electrical Engineering, Lakehead University, Thunder Bay, Ont.

1 Fuzzy Sets and Systems 6 ( Robust H control of Takagi Sugeno fuzzy systems with state and input time delays Bing Chen a,, Xiaoping Liu b, Chong Lin a,kefuliu b a Institute of Complexity Science, Qingdao University, Qingdao 266, PR China b Faculty of Electrical Engineering, Lakehead University, Thunder Bay, Ont., Canada P7B 5E Received 3 April 27; received in revised form 25 March 28; accepted 28 March 28 Available online 8 April 28 Abstract This paper addresses the robust H fuzzy control problem for nonlinear uncertain systems with state and input time delays through Takagi Sugeno (T S fuzzy model approach. The delays are assumed to be interval time-varying delays, and no restriction is imposed on the derivative of time delay. Based on Lyapunov Krasoviskii functional method, delay-dependent sufficient conditions for the existence of an H controller are proposed in linear matrix inequality (LMI format. Illustrative examples are given to show the effectiveness and merits of the proposed fuzzy controller design methodology. Crown Copyright 28 Published by Elsevier B.V. All rights reserved. Keywords: H control; T S fuzzy system; Time delay; Delay-dependent stabilization; Uncertainty. Introduction During the last two decades, the problem of stability analysis and stabilization of nonlinear systems in Takagi and Sugeno [2] fuzzy model has been extensively studied, and a lot of significant results on stabilization and H control via linear matrix inequality (LMI approach have been reported, see [ 3,22,23,7,8,5,8,25,35,34,27], and the reference therein. So far, Takagi Sugeno (T S fuzzy model has become a popular and effective approach to control complex and ill-defined systems for which the application of conventional techniques is infeasible. In view of time delays frequently occurring in practical dynamic systems, T S fuzzy model was first used to deal with the stability analysis and control synthesis of nonlinear time delay systems in [4]. Afterwards, many people have devoted a great deal of effort to both theoretical research and implementation techniques for T S fuzzy systems with time delays. In these works, stability analysis and fuzzy controller design methods are provided in the sense of delay-independent stability [4,33], and in the sense of delay-dependent stability [9,5,6,27]. Generally speaking, the delay-independent approach provides the stability conditions irrespective of the size of the delay. As a result, it may be lead to comparatively conservative stability analysis results, particularly when the delay is small. More recently, further delay-dependent stabilization results are presented for T S fuzzy systems with interval time delay in [3,24,2]. The robust stabilization problem has been addressed in [24], and LMI-based stability criteria and stabilization conditions have been proposed based on Lyapunov Krasoviskii functional method, while the H control Corresponding author. Tel.: address: chenbing958@yahoo.com.cn (B. Chen. 65-4/$ - see front matter Crown Copyright 28 Published by Elsevier B.V. All rights reserved. doi:.6/j.fss

2 44 B. Chen et al. / Fuzzy Sets and Systems 6 ( problem has been studied in [3,2], and some sufficient conditions for the existence of an H controller have been presented by means of LMI format. However, all the aforementioned results have not considered the effect of control input delay on the resulting systems. Hence, these approaches mentioned above may be invalid when they are used to control the systems with input delays. As well-known, input delays are often encountered in many industrial processes. The presence of input delay, if not taken into account in the controller deign, may cause instability or serious deterioration in the performance of the resulting control systems. In addition, in modern industrial systems, sensors, controllers and plants are often connected over a network medium. The controller signals are transmitted through a network, therefore, the network-induced time delay is usually inevitable. In view of this, more recently, many controller design schemes have been presented for linear systems with input delay [9,32,3]. Then, there are a few results on stabilization of T S fuzzy systems with state and input delays. To the best of author s knowledge, the literature by Lin et al. [7] is the only one, which deals with the stability and stabilization of T S fuzzy systems with state and input delays. Based on Razumikhin Theorem, the delay-dependent stability and stabilization conditions have been developed in terms of LMIs. However, uncertainties and external perturbations have not been considered in [7]. Motivated by the above concerns, we will consider the problem of robust H control for T S fuzzy systems with both state and input delays. It is assumed that state and input delays are time-varying delays with both upper and lower bounds. there is no restriction imposed on the derivative of time delay. Delay-dependent sufficient conditions for stability and stabilization will be derived in terms of LMI format, and systematic design procedure for an H fuzzy controller will be proposed for uncertain T S fuzzy systems with state and input delays. At last, three examples are given to illustrate the effectiveness and feasibility of our results. Throughout this paper, identity matrix, of appropriate dimensions, will be denoted by I. The notation X > (respectively, X, for X R n n means that the matrix X is real symmetric positive definite (respectively, positive semi-definite. If not explicitly stated, all matrices are assumed to have compatible dimensions for algebraic operations. The symbol * in a matrix A R n n stands for the transposed elements in the symmetric positions. The superscripts T and denote the matrix transpose and inverse, respectively. 2. Problem statement Consider a nonlinear time delay system represented by T S fuzzy model as follows: Plant Rule i: IFθ (tisn i θ p (tisn ip THEN ẋ(t = (A i + ΔA i (tx(t + (A i + ΔA i (tx(t τ(t + B wi w(t +(B i + ΔB i (tu(t + (B i + ΔB i (tu(t σ(t, z(t = C i x + C i x(t τ(t + D i u(t + D i u(t σ(t, x(t = (t, t [ d, ], where N ij is the fuzzy set, x(t R n stands for the state vector, u(t R m denotes the control input vector, z(t R q is the controlled output, and w(t R p, which is assumed to belong to L 2 [,, denotes the external perturbation. A i, A i R n n, B i, B i R n m, B wi R n d, C i and C i R q s, D i and D i R q m are known constant matrices. Constant d is the upper bound of τ(tandσ(t. Scalar k is the number of IF THEN rules. θ (t, θ 2 (t,..., θ p (t are the premise variables. It is assumed that the premise variables do not depend on the input u(t. ΔA i (t, ΔA i (t, ΔB i (t and ΔB i (t denote the time-varying uncertain matrices satisfying [ΔA i, ΔA i, ΔB i, ΔB i ] = G i F i (t[e i, E i, E bi, E bi ], where G i, E i, E i, E bi and E bi are known constant matrices of appropriate dimensions. F(t is an unknown matrix function satisfying the inequality F T (tf(t I. τ(tandσ(t denote state and input delays, respectively. In this paper, delays τ(tandσ(t are assumed to be interval type delay, i.e., there exist known positive constants τ m, τ M, σ m and σ M such that τ m τ(t τ M, σ m σ(t σ M,

3 B. Chen et al. / Fuzzy Sets and Systems 6 ( and let d = max{τ M, σ M }. Given a pair of (x(t, u(t, the final output of the fuzzy system is inferred as follows: ẋ(t = z(t = h i (θ(t[(a i + ΔA i x(t + (A i + ΔA i x(t τ(t i= +(B i + ΔB i u(t + (B i + ΔB i u(t σ(t + B wi w(t], h i (θ(t[c i x + C i x(t τ(t + D i u(t + D i u(t σ(t], i= x(t = (t, t [ d, ], ( where h i (θ(t = μ i (θ(t/ k i= μ i (θ(t, μ i (θ(t = p j= N ij(θ j (t and N ij (θ j (t is the degree of the membership of θ j (t in fuzzy set N ij. In this paper, it is assumed that μ i (θ(t fori =, 2,..., k and k i= μ i (θ(t > forall t. Therefore, h i (θ(t (for i =, 2,..., k and k i= h i (θ(t =. For simplicity, the following notations will be used: Ā ij = A ij + ΔA ij, A ij = A i + B i K j, ΔA ij = ΔA i + ΔB i K j, Ā i = A i + ΔA i, B i = B i + ΔB i, h i = h i (θ(t, h i (σ = h i (θ(t σ, τ = τ(t, σ = σ(t. To design a state feedback fuzzy controller, the ith fuzzy control rule is presented as follows: Control Rule i: IFθ (tisn i θ p (tisn ip THEN u(t = K i x(t, i =, 2,..., k. Hence, the overall fuzzy control law is represented by u(t = h i (θ(tk i x(t, i= (2 where K i (i =, 2,..., k are the local control gains. Remark. The existence of input delay leads to the term k i= h i (θ(t σk i x(t σ. Therefore, a natural and essential assumption is that all h i (θ(t s are well defined for t [ σ M, ], and also satisfy the equality k i= h i (θ(t = with h i (θ(t fort [ σ M, ] for i =, 2,..., k. Associated with the control law (2, the resulting closed-loop system can be expressed as follows: ẋ(t = h i (θ(t A i + ΔA i + h j (θ(t(b i + ΔB i K j x(t i= j= +(A i + ΔA i x(t τ + h s (θ(t σ(b i + ΔB i K s x(t σ + B wi w(t = z(t = s= h i h j h s (σ(ā ij x(t + Ā i x(τ + B i K s x(t σ + B wi w(t, (3 h i h j h s (σ((c i + D i K j x + C i x(t τ + D i K s x(t σ. (4

4 46 B. Chen et al. / Fuzzy Sets and Systems 6 ( Definition. Systems (3 (4 is said to be robustly asymptotically stable with an H norm bound γ > if there exists a feedback control law u(t such that the following holds: ( System (3 with w(t is asymptotically stable. (2 Under the assumption of zero initial condition, the controlled output z(t satisfies z(t 2 γ w(t 2 for any nonzero w(t L 2 [,. The objective of this paper is to determine the feedback control law u(t = k i= K i x(t such that systems (3 (4 to be robust asymptotically stable with an H norm bound γ >. To state our main results, the following lemmas are useful in the proofs of our results. Lemma (Park and Kwon [2]. For any constant matrix M >, any scalars a and b with a < b, and any vector function x(t :[a, b] R n such that the integrals concerned are well defined, then [ b T [ b ] b x(sds] M x(sds (b a x T (smx(sds. a a a Lemma 2 (Wang et al. [26]. Let D, E, and F(t be real matrices of appropriate dimensions, and F(t satisfying F T (tf(t I. Then, the following inequality holds for any constant ε > : DF(tE + E T F T (td T εdd T + ε E T E. 3. H performance analysis In this section, we will derive the delay-dependent LMI conditions for H performance analysis for the systems (3 (4. To this end, define τ = 2 (τ M + τ m, δ = 2 (τ M τ m, σ = 2 (σ M + σ m, β = 2 (σ M σ m. (5 Then, by the above notations, we have that τ(t [τ δ, τ +δ], τ δ, σ(t [σ β, σ +β]andσ β. Apparently, when δ =, i.e., τ m = τ M, τ(t becomes a constant delay. If δ = τ, then the routine case for the time delay, i.e., τ(t τ M, is covered. Obviously, the similar results also hold for σ(t in the cases of β = andβ = σ. In the following, it is assumed that the feedback gain matrices K i (i =, 2,..., k are known. Then, the following conclusion can be obtained. Theorem. For given scalars τ >, δ >, σ >, β > and γ >, as well as the given matrices K i (i =, 2,..., k, systems (3 (4 are robustly asymptotically stable with an H norm bound γ if there exist matrices T l >, H l >, P, P 2, P 3, P 22, P 23, P 33, Q q, N q, L q, M q and W q (l =, 2, 3, 4, q =, 2,..., 6 so that the following LMIs hold, for i, j, s k: where Φ( rl + ΔΦ + Ξ T Ξ Γ QB wi Ω γ 2 I P P 2 P 3 P 22 P 23 >, P 33 Ω = diag ( τ T 2, τ T 3, δ T 4, σ H 2, <, (6 σ H 3, β H 4, (7

5 B. Chen et al. / Fuzzy Sets and Systems 6 ( Γ = [Z N L Z 2 M W ], Ξ = [C i + D i K j C i D i K s ], Z T = [PT 22 + P 23 P22 T PT 2 P 23], Q T = [Q T QT 2 QT 3 QT 4 QT 5 QT 6 ], Z2 T = [PT 23 + PT 33 PT 23 PT 3 PT 33 ], N T = [N T N 2 T N 3 T N 4 T N 5 T N 6 T ], L T = [L T LT 2 LT 3 LT 4 LT 5 LT 6 ], MT = [M T MT 2 MT 3 MT 4 MT 5 MT 6 ], W T = [W T W 2 T W 3 T W 4 T W 5 T W 6 T ], ΔΦ = QG i F i (te ij + (QG i F i (te ij T, E ij = [E i + E bi K j E i E bi K s ], and Φ( rl is a symmetric block-matrix with rl being its element at the position (r, l, and defined as follows for r, l 6: = P 2 + P2 T + P 3 + P3 T + T + H + τ T 2 + σ H 2 + N + N T + M + M T + Q A ij + Aij T QT, 2 = N2 T + MT 2 L + Q A i + Aij T QT 2, 3 = P 2 N + N3 T + MT 3 + L + Aij T QT 3, 4 = P + N4 T + MT 4 Q + Aij T QT 4, 5 = N5 T + MT 5 W + Q B i K s + Aij T QT 5, 6 = P 3 + N6 T M + M6 T + W + Aij T QT 6, 22 = L 2 L T 2 + Q 2 A i + A T i QT 2, 23 = N 2 + L 2 L T 3 + AT i QT 3, 24 = L T 4 Q 2 + A T i QT 4, 25 = L T 5 W 2 + Q 2 B i K s + A T i QT 5, 26 = M 2 L T 6 + W 2 + A T i QT 6, 33 = N 3 N3 T + L 3 + L T 3 T, 34 = N4 T + LT 4 Q 3, 35 = N5 T + LT 5 W 3 + Q 3 B i K s, 36 = N6 T M 3 + L T 6 + W 3, 44 = τ T 3 + σ H 3 + 2δT 4 + 2βH 4 Q 4 Q T 4, 45 = W 4 + Q 4 B i K s Q T 5, 46 = M 4 + W 4 Q T 6, 55 = W 5 W5 T + Q 5 B i K s + Ks T BT i QT 5, 56 = M 5 + W 5 W6 T + K s T BT i QT 6, 66 = H M 6 M6 T + W 6 + W6 T. Proof. Consider the following Lyapunov function candidate: V = V + V 2 + V 3, (8 where V = x T P x + 2x T P 2 x(sds + 2x T P 3 x(sds + x T (sdsp 22 t τ t σ t τ t +2 x T (sdsp 23 x(sds + x T (sdsp 33 x(sds, t τ t σ t σ t σ V 2 = x T (st x(sds + x T (θt 2 x(θdθds t τ τ t+s + x T (sh x(sds + x T (θh 2 x(θdθds, t σ σ t+s t τ x(sds

6 48 B. Chen et al. / Fuzzy Sets and Systems 6 ( τ V 3 = ẋ T +δ (θt 3 ẋ(θdθds + ẋ T (θt 4 ẋ(θdθds τ t+s τ δ t+s σ + ẋ T +β (θh 3 ẋ(θdθds + ẋ T (θh 4 ẋ(θdθds. σ t+s σ β t+s Differentiating V gives V = x T (t(p 2 + P2 T + P 3 + P3 T + T + H + τ T 2 + σ H 2 x(t +2x T (tp ẋ(t 2x T (tp 2 x(t τ 2x T (tp 3 x(t σ x T (t τ T x(t τ x T (t σ H x(t σ +ẋ T (t(τ T 3 + 2δT 4 + σ H 3 + 2βH 4 ẋ(t +2x T (t(p 22 + P23 T x(sds + 2x T (t(p 23 + P 33 x(sds t τ t σ t 2x T (t τ P 22 x(sds 2x T (t τ P 23 x(sds t τ t σ t 2x T (t σ P23 T x(sds 2x T (t σ P 33 x(sds t τ t σ t +2ẋ T (tp 2 ẋ(sds + 2ẋ T (tp 3 ẋ(sds t τ t σ t x T (st 2 x(sds x T (sh 2 x(sds t τ t σ τ ẋ T +δ (st 3 ẋ(tds ẋ T (st 4 ẋ(tds t τ t τ δ σ ẋ T +β (sh 3 ẋ(tds ẋ T (sh 4 ẋ(tds. (9 t σ t σ β Define e T (t = [x T (t x T (t τ x T (t τ ẋ T (t x T (t σ x T (t σ ], Z T = [PT 22 + P 23 P22 T PT 2 P 23], Z2 T = [PT 23 + PT 33 PT 23 PT 3 PT 33 ]. Then, (9 can be rewritten as follows: [ ] V = e T Φ Φ (t e(t Φ 2 +2e(t T Z x(sds + 2e(t T Z 2 x(sds x T (st 2 x(sds t τ t σ t τ τ x T (sh 2 x(sds ẋ T +δ (st 3 ẋ(tds ẋ T (st 4 ẋ(tds t σ t τ t τ δ σ ẋ T +β (sh 3 ẋ(tds ẋ T (sh 4 ẋ(tds ( t σ t σ β with Φ = P 2 + P2 T + P 3 + P3 T + T + H + τ T 2 + σ H 2, Φ = [ P 2 P P 3 ], Φ 2 = diag(, T, Ψ,, H, Ψ = τ T 3 + 2δT 4 + σ H 3 + 2βH 4.

7 B. Chen et al. / Fuzzy Sets and Systems 6 ( To establish the delay-dependent stability conditions, the free-weighting matrix approach [,] will be used in the following. By Newton Leibniz formula, we have the equality below: = x(t x(t τ ẋ(sds. t τ Furthermore, define a matrix N T = [N T N 2 T N 3 T N 4 T N 5 T N 6 T ] with appropriate dimensions. Then, the following equality can be verified easily: ( = 2e T (tn x(t x(t τ ẋ(sds t τ = 2e T (tn[i I ]e(t 2e T (tn ẋ(sds t τ = e T (t(σ N + Σ T N e(t 2eT (tn ẋ(sds t τ ( with Σ N = [N N ]. Similarly, the following equalities (2 (5 can be obtained. For matrix L T = [L T ( LT 2 LT 3 LT 4 LT 5 LT 6 ] with L i having compatible dimensions, τ = 2e T (tl x(t τ x(t τ ẋ(sds = e T (t(σ L + Σ T L e(t 2eT (tl τ t τ t τ ẋ(sds, (2 where Σ L = [ L L ]. For matrix M T = [M T ( MT 2 MT 3 MT 4 MT 5 MT 6 ] with M i being of compatible dimensions, = 2e T (tm x(t x(t σ ẋ(sds t σ t = e T (t(σ M + Σ T M e(t 2eT (tm ẋ(sds, t σ (3 where Σ M = [M M]. For matrix W T = [W T W 2 T W 3 T W 4 T W 5 T W 6 T] with W i having compatible dimensions, ( σ = 2e T (tw x(t σ x(t σ ẋ(sds = e T (t(σ W + Σ T W e(t 2eT (tw σ t σ t σ ẋ(sds, (4 where Σ W = [ W W]. Finally, for matrix Q T = [Q T QT 2 QT 3 QT 4 QT 5 QT 6 ] with Q i having compatible dimensions, = 2e T (tq h i h j h s (σ(ā ij x(t + Ā i x(t τ + B i K s x(t σ + B wi w(t ẋ(t = = + h i h j h s (σ2e T (tq[a ij A i I B i K s ]e(t h i h j h s (σ2e T (tqm i F i (t[e i + E bi K j E i E bi K s ]e(t h i h j h s (σ2qe T (tb wi w(t h i h j h s (σe T (t(σ + ΔΦe(t h i h j h s (σ2qe T (tb wi w(t, (5

8 4 B. Chen et al. / Fuzzy Sets and Systems 6 ( where ΔΦ = QG i F i (te ij + (QG i F i (te ij T with E ij = [E i + E bi K j E i E bi K s ], and Σ = Σ(Σ rl isa symmetric block-matrix and its block-elements Σ rl are defined as follows for r, l 6: Σ = Q A ij + Aij T QT, Σ 2 = Q A i + Aij T QT 2, Σ 3 = Aij T QT 3, Σ 4 = Q + Aij T QT 4, Σ 5 = Q B i K s + Aij T QT 5, Σ 6 = Aij T QT 6, Σ 22 = Q 2 A i + A T i QT 2, Σ 23 = A T i QT 3, Σ 24 = Q 2 + A T i QT 4, Σ 25 = Q 2 B i K s + A T i QT 5, Σ 26 = A T i QT 6, Σ 33 =, Σ 34 = Q 3, Σ 35 = Q 3 B i K s, Σ 36 =, Σ 44 = Q 4 Q T 4, Σ 45 = Q 4 B i K s Q T 5, Σ 46 = Q T 6, Σ 55 = Q 5 B i K s Ks T BT i QT 5, Σ 56 = Ks T BT i QT 6, Σ 66 =. Then, combining ( (5 yields the following equality: V = h i h j h s (σe T (t(φ( rl + ΔΦe(t +2e(t T Z x(sds + 2e(t T Z 2 x(sds 2e T (tn ẋ(sds t τ t σ t τ t τ σ 2e T (tl ẋ(sds 2e T (tm ẋ(sds 2e T (tw ẋ(sds t τ t σ t σ x T (st 2 x(sd x T (sh 2 x(sds ẋ T (st 3 ẋ(sds t τ t σ t τ τ ẋ T +δ σ (sh 3 ẋ(sds ẋ T +β (st 4 ẋ(sds ẋ T (sh 4 ẋ(sds t σ t τ δ t σ β h i h j h s (σ2e T (tqb wi w(t, (6 where matrix Φ( rl andδφ are the same ones defined in (6. Note that T 4 >. Thus when τ τ, we have τ +δ τ ẋ T (st 4 ẋ(sds = ẋ T (st 4 ẋ(sds t τ δ t τ δ τ +δ t τ τ ẋ T (st 4 ẋ(sds τ t τ ẋ T (st 4 ẋ(sds ẋ T (st 4 ẋ(sds. (7 t τ Since τ [τ δ, τ + δ], applying Lemma to τ t τ ẋ T (st 4 ẋ(sds yields τ ẋ T (st 4 ẋ(sds ( τ ( τ ẋ T (sds T 4 ẋ(sds (8 t τ δ t τ t τ = ( τ ( τ ẋ T (sds T 4 ẋ(sds. δ t τ t τ Consequently, (7 and(8 imply that for τ τ, τ +δ ẋ T (st 4 ẋ(sds ( τ ( τ ẋ T (sds T 4 ẋ(sds. (9 t τ δ δ t τ t τ On the other hand, it can be clearly seen that (9 is also true for the case of τ τ >. Similar to (9, the following inequality holds for any σ [σ β, σ + β]: σ +β ẋ T (sh 4 ẋ(sds ( σ ( σ ẋ T (sds H 4 ẋ(sds. (2 β t σ β t σ t σ

9 B. Chen et al. / Fuzzy Sets and Systems 6 ( In addition, by applying Lemma 2 to the corresponding integral terms in (6, the following inequalities are obtained: x T (st 2 x(sds ( ( x T (sds T 2 x(sds, (2 t τ τ t τ t τ t x T (sh 2 x(sds ( ( x T (sds H 2 x(sds, (22 t σ σ t σ t σ t ẋ T (st 3 ẋ(sds ( ( ẋ T (sds T 3 ẋ(sds, (23 t τ τ t τ t τ t ẋ T (sh 3 ẋ(sds ( ( ẋ T (sds H 3 ẋ(sds. (24 t σ σ t σ t σ Then, substituting (9 (24into(6 results in that [ ] T [ ][ ] e(t Φ( rl + ΔΦ Γ e(t V h i h j h s (σ η(t Ω η(t + h i h j h s (σe T (tqb wi Bwi T Qe(tγ 2 + γ 2 w T (tw(t, (25 where the inequality 2e T (tqb wi w(t e T (tqb wi Bwi T Qe(tγ 2 + γ 2 w T (tw(tisused,η(tisdefinedby [ τ σ ] η T (t = x T (sds ẋ T (sds ẋ T (sds x T (sds ẋ T (sds ẋ T (sds, t τ t τ t τ t σ t σ t σ and the matrices Γ and Ω are the same ones defined in (6. Let Ξ = [C i + D i K j C i D i K s ]. Then, according to the definition of z(t, we have z T (tz(t = h i h j h s (σe T (tξ T h l h p h q (σξe(t Furthermore, h i h j h s (σe T (tξ T Ξe(t. l= p= q= z T (tz(t γ 2 w T (tw(t h i h j h s (σe T (tξ T Ξe(t γ 2 w T (tw(t + V V = + + h i h j h s (σe T (tξ T Ξe(t γ 2 w T (tw(t h i h j h s (σ [ e(t η(t ] T [ Φ( rl + ΔΦ Γ Ω ][ ] e(t η(t h i h j h s (σγ 2 e T (tqb wi Bwi T Qe(t + γ2 w T (tw(t V h i h j h s (σ [ e(t η(t ] T [ Θ Γ Ω with Θ = Φ( rl + ΔΦ + Ξ T Ξ + γ 2 QB wi B T wi QT. ][ ] e(t V (26 η(t

10 42 B. Chen et al. / Fuzzy Sets and Systems 6 ( At present stage, by applying Schur complement to (6, we have that [ ] Θ Γ < (27 Ω which means that z T (tz(t γ 2 w T (tw(t V. Consequently, integrating both sides of (28 from to T gives T z T (tz(tdt T which, together with zero initial condition, implies that z T (tz(tdt γ 2 w T (tw(tdt V (T + V ( (29 γ 2 w T (tw(tdt. As a result, z(t 2 γ w(t holds. When w(t, it follows from (25that V <, which ensures the asymptotical stability of the closed-loop system. The proof is thus completed. Because of the existence of the parameter uncertainty ΔΦ in (6, Theorem cannot be directly used to determine the performance of the closed-loop system. However, the following theorem provides a sufficient condition for the system to be robustly asymptotically stable with an H norm bound γ. (28 Theorem 2. For given scalars τ >, δ >, σ >, β > and γ >, as well as the given matrices K i (i =, 2,..., k, systems (3 (4 are robustly asymptotically stable with an H norm bound γ > if there exist matrices T l >, H l >, P, P 2, P 3, P 22, P 23, P 33, Q q, N q, L q, M q and W q (l =, 2, 3, 4, q =, 2,..., 6, and scalars ε ijs > so that the following LMIs hold, for i = i, j, s k: Φ( rl + Ξ T Ξ + ε ijs Eij T E ij Γ QB wi QG i Ω γ 2 I ε ijs I P P 2 P 3 P 22 P 23 >, P 33 where Φ( ij, E ij, Γ and Ω are defined as in Theorem. Proof. In light of the structure of ΔΦ,wehave <, (3 Φ( rl + ΔΦ = Φ( rl + QG i F(tE ij + Eij T FT (tgi T Q. (32 Applying Lemma 2 to (32 produces Φ( rl + ΔΦ Φ( rl + εijs QG i Gi T QT + ε ijs Eij T E ij. Therefore, a sufficient condition for (6 holding is Φ( rl + Ξ T Ξ + εijs QG i G i Q T + ε ijs Eij T E ij Γ QB wi Ω <. (33 γ 2 I Obviously, by Schur complement, (3 is equivalent to (33. This means that under conditions (3 and(3, (6 and (7 hold. Thus, the proof is completed by Theorem. (3

11 4. Robust H fuzzy controller design B. Chen et al. / Fuzzy Sets and Systems 6 ( Based on Theorem 2, we will present a systematic procedure for designing the robust H fuzzy controller. The suggested fuzzy controller (2 can guarantee systems (3 (4 to be robustly asymptotically stable with an H norm bound γ >. Theorem 3. For given scalars τ >, δ >, σ >, β >, γ > and a p (p = 2,..., 6, a 4, systems (3 (4 is robust asymptotically stable with an H norm bound γ >, and the feedback gain matrices are given by K i = F i X, i =, 2,..., k, if there exist matrices T l >, H l >, P, P 2, P 3, P 22, P 23, P 33, N q, L q, M q, W q (l =, 2, 3, 4, q =, 2,..., 6, and X, as well as matrices F i and scalars α ijs > so that the following LMIs hold, for i, j, s k: Φ Γ ϒ ϒ Ω γ 2 I <, Ψ P P 2 P 3 P 22 P 23 >, (35 P 33 where ϒ T = [Bwi T a 2 Bwi T a 3 Bwi T a 4 Bwi T a 5 Bwi T a 6 Bwi T [ ], ] ϒ T = Ei X + E bi F j E i X E bi F s, C i X + D i F j C i X D i F s ( Ω = diag T 2, T 3, τ τ δ T 4, H 2, H 3, σ σ Ψ = diag(α ijs I, I, Γ = [ Z N L Z 2 M W ], Z T = [ P 22 T + P 23 P 22 T P 2 T P 23 ], Z 2 T = [ P 23 T + P 33 T P 23 T P 3 T P 33 T ], N T = [ N T N 2 T N 3 T N 4 T N 5 T N 6 T ], L T = [ L T L T 2 L T 3 L T 4 L T 5 L T 6 ], M T = [ M T M 2 T M 3 T M 4 T M 5 T M 6 T ], W T = [ W T W 2 T W 3 T W 4 T W 6 T ] W T 5 β H 4 and Φ = Φ( rl is a symmetric block-matrix with rl being its element at the position (r, l, and defined as follows for r, l 6: = P 2 + P 2 T + P 3 + P 3 T + T + H + τ T 2 + σ H 2 + N + N T + M + M T + A i X + B i F j + XAi T + Fj T BT i + α ijs G i Gi T, 2 = N 2 T + M 2 T L + A i X + a 2 XAi T + a 2 Fj T BT i + a 2 α ijs G i Gi T, 3 = P 2 N + N 3 T + M 3 T + L + a 3 XAi T + a 3 Fj T BT i + a 3 α ijs G i Gi T, 4 = P + N 4 T + M 4 T X + a 4 XAi T + a 4 Fj T BT i + a 4 α ijs G i Gi T, 5 = N 5 T + M 5 T W + B i F s + a 5 XAi T + a 5 Fj T BT i + a 5 α ijs G i Gi T, 6 = P 3 + N 6 T M + M 6 T + W + a 6 XAi T + a 6 Fj T BT i + a 6 α ijs G i Gi T, 22 = L 2 L T 2 + a 2 A i X + XA T i a 2 + a 2 a 2 α ijs G i Gi T,,

12 44 B. Chen et al. / Fuzzy Sets and Systems 6 ( = N 2 + L 2 L T 3 + XAT i a 3 + a 2 a 3 α ijs G i Gi T, 24 = L T 4 a 2 X + XA T i a 4 + a 2 a 4 α ijs G i Gi T, 25 = L T 5 W 2 + a 2 B i F s + XA T i a 5 + a 2 a 5 α ijs G i Gi T, 26 = M 2 L T 6 + W 2 + XA T i a 6 + a 2 a 6 α ijs G i Gi T, 33 = N 3 N 3 T + L 3 + L T 3 T + a 3 a 3 α ijs G i Gi T, 34 = N 4 T + L T 4 a 3 X + a 3 a 4 α ijs G i Gi T, 35 = N 5 T + L T 5 W 3 + a 3 B i F s + a 3 a 5 α ijs G i Gi T, 36 = N 6 T M 3 + L T 6 + W 3 + a 3 a 6 α ijs G i Gi T, 44 = τ T 3 + σ H 3 + 2δ T 4 + 2β H 4 a 4 X a 4 X + a 4 a 4 α ijs G i Gi T, 45 = W 4 + a 4 B i F s a 5 X + a 4 a 5 α ijs G i Gi T, 46 = M 4 + W 4 a 6 X + a 4 a 6 α ijs G i Gi T, 55 = W 5 W 5 T + a 5 B i F s + Fs T BT i a 5 + a 5 a 5 α ijs G i Gi T, 56 = M 5 + W 5 W 6 T + FT s BT i a 6 + a 5 a 6 α ijs G i Gi T, 66 = H M 6 M 6 T + W 6 + W 6 T + a 6a 6 α ijs G i Gi T. Proof. Notice that (34 implies that X is a nonsingular matrix. So, let Q = X. Thus, to prove Theorem 3, we can show that (34 and(35 imply(3 and(3. To this end, let Q p = a p Q (p = 2,..., 6, ε ijs = αijs, and define the following variables: T = Q T Q T, T 2 = Q T 2 Q T, T 3 = Q T 3 Q T, T 4 = Q T 4 Q T, H = Q H Q T, H 2 = Q H 2 Q, H 3 = Q H 3 Q T, H 4 = Q H 4 Q T, P = Q P Q T, P 2 = Q P 2 Q T, P 3 = Q P 3 Q T, P 22 = Q P 22 Q T, P 23 = Q P 23 Q T, P 33 = Q P 33 Q T, K i = F i Q, i =, 2,..., k, N l = Q N l Q T, L l = Q L l Q T, M l = Q M l Q T, W l = Q W l Q T, l =, 2,..., 6. Then, pre- and post-multiplying both sides of (34 byσ = diag Q, Q, I, I, I and its transpose, respec- }{{} 2 tively, shows that Σ(34Σ T <. Then, it can be verified by Schur complement that Σ(34Σ T < is equivalent to (33. Consequently, from the proof of Theorem 2, (33 ensures that (3 is true. In addition, (3 is immediately obtained by pre- and post-multiplying both sides of (35 with diag(q, Q, Q and its transpose. Therefore, by Theorem 2, the proof is completed. Remark 2. Unlike in Theorem 2, in order to determine the feedback gain matrices, we have to set Q i = a i Q (i = 2, 3, 4, 5, 6 to obtain the LMI conditions. The parameters a i (i = 2, 3, 4, 5, 6 should be given prior to solve LMI (34. How to choose these design parameters to optimize is still an open problem. If the common form for Q i (i =, 2, 3, 4, 5, 6 are retained (i.e., a i s are not introduced, one has to employ nonlinear programming to solve nonrestrict LMI conditions which brings much computational burden and may lead to conservative results. Remark 3. To reduce the conservatism of the stability criteria, some slack variables are introduced. Then, it should be pointedout that the works in [28 3] show that when the slack variables are introduced, some ones among them maybe not useful for improving the results. And the redundant slack variables will lead to the burdensome computation. Therefore, it is important to known which slack variables are redundant to reduce the conservatism of the stability criteria. However, to expressly known the effect of each slack variable to improvement of the stability criteria is very difficult, and is still an opened problem which will be studied further. In what follows, we consider the robust stabilization problem of system (3 with w(t. And the following corollary can be derived from Theorem 3 directly.

13 B. Chen et al. / Fuzzy Sets and Systems 6 ( Corollary. For given scalars τ >, δ >, σ >, β >, and scalars a ρ (ρ = 2, 3, 4, 5, 6, a 4, if there exist matrices T l >, H l >, P, P 2, P 3, P 22, P 23, P 33, N q, L q, M q, W q (l =, 2, 3, 4, q = 2, 3, 4, 5, 6 and X, as well as matrices F i and scalars α ijs > such that for i, j, s k, Φ Γ ˆϒ Ω <, α ijs I P P 2 P 3 P 22 P 23 >, P 33 (36 (37 where ˆϒ T = [E i X + E bi F j matrices K i are given by E i X E bi F s ], and Φ, Γ and Ω are defined as in Theorem 3, then the feedback gain K i = F i X, i =, 2,..., k and the resulting closed-loop system (3 is asymptotically stable. In addition, another especial case of (3isu(t σ. For this case, the following corollary can be obtained from the proof of Theorems 3 by setting P 3 =, P 23 =, P 33 =, H l = (for l =, 2, 3, 4, M =, W =, as well as N i =, L i = andq i = fori = 5, 6. Corollary 2. For given scalars τ >, δ >, γ > and a p (p = 2, 3, 4, systems (3 (4 are robust asymptotically stable with an H norm bound γ > and the feedback gain matrices are given by K i = F i X, i =, 2,..., k, if there exist matrices T l >, P, P 2, P 22, N q, L q (l =, 2, 3, 4, q =, 2, 3, 4, and X, as well as matrices F i and scalars α ij > so that the following LMIs hold, for i < j k: Φ Γ ĪB wi ϒ(i, j Ω γ 2 I <, i k, (38 Ψ Φ(i, j + Φ( j, i Γ Ī B wi + B wj ϒ(i, j + ϒ( j, i Ω γ 2 <, i < j k, (39 I Ψ [ ] P P 2 >, (4 P 22 where [ Ei X + E Γ = [ Z N L], ϒT = bi F j E i X C i X + D i F j C i X Z T = [ P 22 T P 22 T P 2 T ], N T = [ N T N 2 T N 3 T N 4 T ( ], L T = [ L T L T 2 L T 3 L T 4 ], Ω = diag T 2 T 3 τ τ δ T 4, Ī T = [I a 2 I a 3 I a 4 I ], Ψ = diag(α ij I, I ],

14 46 B. Chen et al. / Fuzzy Sets and Systems 6 ( and with Φ(i, j = = P 2 + P 2 T + T + τ T 2 + N + N T + A i X + B i F j + XAi T + Fj T BT i + α ij G i Gi T, 2 = N 2 T L + a 2 A i X T + a 2 XAi T + a 2 Fj T BT i + a 2 α ij G i Gi T, 3 = P 2 N + N 3 T + L + a 3 XAi T + a 3 Fj T BT i + a 3 α ij G i Gi T, 4 = P + N 4 T X + a 4 XAi T + a 4 Fj T BT i + a 4 α ij G i Gi T, 22 = L 2 L T 2 + a 2 A i X T + XA T i a 2 + a 2 a 2 α ij G i Gi T, 23 = N 2 + L 2 L T 3 + XAT i a 3 + a 2 a 3 α ij G i Gi T, 24 = L T 4 a 2 X + XA T i a 4 + a 2 a 4 α ij G i Gi T, 33 = N 3 N 3 T + L 3 + L T 3 T + a 3 a 3 α ij G i Gi T, 34 = N 4 T + L T 4 Q 3 + a 3 a 4 α ij G i Gi T, 44 = τ T 3 + 2δ T 4 a 4 X a 4 X + a 4 a 4 α ij G i Gi T. 5. Computer simulation In this section, some examples are used to illustrate the effectiveness of the proposed results, and to compare with the existing results. Example. Consider the following uncertain T S fuzzy system with state and input delays: Rule : IF θ(t = x 2 (t + a(v t/2lx (t + ( a(v t/2lx (t σ is about, THEN ẋ(t = (A + ΔA x(t + (A + ΔA x(t τ(t + (B + ΔB u(t + (B + ΔB u(t σ(t. Rule 2: IF θ(t = x 2 (t + a(v t/2lx (t + ( a(v t/2lx (t σ is about π or π, THEN ẋ(t = (A 2 + ΔA 2 x(t + (A 2 + ΔA 2 x(t τ(t + (B 2 + ΔB 2 u(t + (B 2 + ΔB 2 u(t σ(t, where a v t ( a v t Lt A = a v t, A Lt = a v2 t 2 v t 2Lt t a v t Lt ( a v t Lt ( a v2 t 2 2Lt, ( a v t Lt A 2 = a v t Lt, A Lt 2 = ( a v t, Lt ad v2 t 2 dv t ( a dv2 t 2 2Lt t 2Lt v t v t B = lt, B 2 = lt, B =.B, B 2 =.B 2,

15 B. Chen et al. / Fuzzy Sets and Systems 6 ( x x 2 x Time (Sec Fig.. Response of state for Example. G i = [ ] T, E i = E i = [. ], E bi =.5, E bi =.5, i =, 2 and l = 2.8, L = 5.5, v =., t = 2., t =.5, a =.7, d = t /π. When all the uncertainties are removed from the above system, this system is just Example 5.2 in [7]. According to [7],take ( h =, h 2 = h. + exp( 3(θ(t +.5π For the case where ΔA i =, ΔA i =, ΔB i =, and ΔB i =, the following fuzzy control law is proposed in [7] for τ = σ =.: u(t = h [ ]x(t + h 2 [ ]x(t, (4 which guarantees the asymptotic stability of the resulting closed-loop system. Because the uncertainties have not been taken into account, the control law (4 cannot, theoretically, be used to stabilize this uncertain system. Then, Corollary can be used to solve the robust stabilization problem for the above system. By setting the parameter as a p =.25 (p = 2, 3, 5, 6 and a 4 =.5, and τ =.235 and σ =, solving LMIs (36 (37 givesδ max =.235, β max = and the feedback gains as follows: K = [ ], K 2 = [ ], which implies that the regarding fuzzy controller guarantees the asymptotic stability of the closed-loop system for τ(t [, 2.427] and σ(t [, 2]. The simulation is run under the initial conditions as (t = [4,, 2] for t [ 2.427, ], and u(t = fort [ 2, ] with σ(t = + sin(t andτ(t =.2 +.2sin(t. The simulation results are shown by Figs., 2. Fig. shows the state response of the system. Fig. 2 displays the control input signal. In addition, for given σ m and τ m by using Corollary in this paper, we can get τ M, σ M and the feedback gains as shownintable. Example 2. Consider the following T S fuzzy system: Plant Rule i: IF x 2 (tish i,then ẋ(t = (A i + ΔA i x + (A i + ΔA i x(t τ(t + (B i + ΔB i u(t +(B i + ΔB i u(t σ(t + B wi w(t, z(t = C i x(t + C i x(t τ(t + D i u(t + D i u(t σ(t, i =, 2, (42

16 48 B. Chen et al. / Fuzzy Sets and Systems 6 ( u Time (Sec Fig. 2. Control input curve for Example. Table σ m σ M τ m τ M Feedback gain matrices [ ], [ ] [ ], [ ] [ ], [ ] where h = ( /( + exp( 6x 2 +.5π(/( + exp( 6x 2.5π and h 2 = h, and the system matrices are [ ] [ ] [ ] [ ]. A =, A. 2 =, B..2 =, B =,.25 [ ] [ ].. A 2 =, A =, B.5 2 = B, B 2 = B, [ ] B w = B w2 =, C = [.5.5], C 2 = [.35.25], D = D 2 =., C = [.5.5], C 2 = [.35.25], D = D 2 =., [ ] [ ] G = G 2 =, E.3 = E 2 =,.4 [ ].5.35 E = E 2 =, E.8.45 b = E b2 = [.5.5], E b = E b2 = [.25.75]. Note that both the state delay and the input delay appear in the system dynamics and the control output equations. Given γ =, then for τ m = andσ m =, we apply Theorem 3 with a p =.25 (p = 2, 3, 5, 6 and a 4 =.5 to solve (34 and(35, and it was found that the maximal upper bounds of τ(t andσ(t areτ M =.996 and σ M =.324, and the feedback gain matrices are K = [ ], K 2 = [ ], for τ m = and σ m =, we have that τ M =.684 and σ M =.3782 and the feedback matrices are K = [ ], K 2 = [ ]. Thus for the case of τ(t.684 and σ(t.3782, take the disturbance input as w(t = 4e.t sin(t, σ(t = +.37 cos(2t and τ(t = +.68 cos(2t, the simulation is carried out under the initial conditions (t = [4, 3] and u(t = fort. Fig. 3 shows the state response of the closed-loop systems (42 and Fig. 4 displays the control input curve. It is seen from Fig. 3 that the closed-loop system is asymptotically stable.

17 B. Chen et al. / Fuzzy Sets and Systems 6 ( x x Time (Sec Fig. 3. Response of state for Example 2. 3 u Time (Sec Fig. 4. Control input curve for Example 2. Example 3. For the special case of u(t σ(t, the robust H fuzzy control problem is studied in [3],inwhichthe H fuzzy controller design procedure is proposed. The following example is taken from [3]. Consider the uncertain nonlinear time-delay system described as follows: ẋ = x (2 + sin 2 x 2 + x 2 +.x (t τ(t +.2x 2 (t τ(t +c(tx cos 2 x 2 + u + ( + sin 2 (x 2 w(t, ẋ 2 = x 2 x 2 ( cos 2 x 2 +.2x (t τ(t sin 2 x 2.5x 2 (t τ(t +.5u 2 +.c(tx 2, where c(t is an uncertain parameter satisfying c(t [.2,.2]. According to [3], by selecting the membership functions as M (x 2 = sin 2 x 2, M 2 (x 2 = cos 2 x 2, then the above nonlinear system can be represented by the following T S fuzzy model.

18 42 B. Chen et al. / Fuzzy Sets and Systems 6 ( x x Time (Sec Fig. 5. Response of state for Example u u 2 Rule. IF x 2 is M, THEN Time (Sec Fig. 6. Control input curve for Example 3. ẋ = (A + ΔA x + (A + ΔA x(t τ(t + (B + ΔB u + B w w(t, z = C x + D u(t, Rule 2. IF x 2 is M 2, THEN ẋ = (A 2 + ΔA 2 x + (A 2 + ΔA 2 x(t τ(t + (B 2 + ΔB 2 u + B w2 w(t, z = C 2 x + D 2 u(t, where [ ] [ ] [ ] [ ] 3. 2 A =, A =, B.2.5 =, B.5 w =, [ ] [ ] [ ] [ ] 2..2 A 2 =, A 2 =, B.5 2 =, B.5 w2 =,

19 B. Chen et al. / Fuzzy Sets and Systems 6 ( Table 2 τ m τ M [ ]. C = C 2 =, D. = D 2 = I, G = G 2 = [ ] [.2.2 E =, E 2 = [ ], ], E i =, E bi =, i =, 2. For this system, setting γ = andτ =, the method in [3] suggests the maximum allowed bound δ = This implies that the allowable interval for τ(tis[.7642,.2358]. Then, applying Corollary 2 with a i =.5 (i = 2, 3 and a 4 =.25, it was found that the maximal allowable value of δ is When setting δ =.2358, the maximum allowed bound of τ is and the feedback gain matrices are K = [ ].76.74, K = [ ] , which means that the regarding fuzzy controller guarantees the asymptotic stability of the closed-loop system for τ(t [ ]. Take the disturbance input as w(t = 2e.t sin(t, and time-varying delay as τ(t = sin(2t which satisfies τ(t.57. Then, the simulation is carried out under the initial conditions (t = [.5, 2.5] for t <. Fig. 5 shows the state response of the resulting closed-loop systems and Fig. 6 displays the control input curve. In addition, for given different τ m, by using Corollary 2 we have the corresponding τ M, as shown in Table Conclusion In this paper, we have studied the H control problem for T S fuzzy systems with state and input delays. Both the state time delay and input time delay are time-varying delays and no restriction is imposed on the derivative of time delays. Based on Lyapunov Krasoviskii functional method, a new design scheme of H fuzzy controller is derived. The main contribution lies in that delay-dependent conditions are presented in terms of LMI format. The effectiveness of the proposed results has been illustrated by some examples. Acknowledgment This work is partially supported by the Natural Sciences and Engineering Research Council of Canada, the National Natural Science Foundation of China (Nos , , the Taishan Scholar programme and the Natural Science Foundation of Shandong Province (No. Y26G4. References [] W. Assawinchaichote, S.K. Nguang, P. Shi, H output feedback control design for uncertain fuzzy singularly perturbed systems: an lmi approach, Automatica 4 (2 ( [2] S.G. Cao, N.W. Rees, G. Feng, Analysis and design for a class of fuzzy control systems using dynamic fuzzy-state-space models, IEEE Trans. Fuzzy Systems 7 (2 ( [3] S.G. Cao, N.W. Rees, G. Feng, H control of uncertain fuzzy continuous-time systems, IEEE Trans. Fuzzy Systems 8 (2 ( [4] Y.Y. Cao, P.M. Frank, Analysis and synthesis of nonlinear time-delay system via fuzzy control approach, IEEE Trans. Fuzzy Systems 8 (2 ( [5] C.L. Chen, G. Feng, D. Sun, Y. Zhu, H output feedback control of discrete-time fuzzy systems with application to chaos control, IEEE Trans. Fuzzy Systems 3 (4 ( [7] G. Feng, H controller design of fuzzy dynamic systems based on piecewise Lyapunov functions, IEEE Trans. Systems Man Cybernet. Part B 34 ( ( [8] G. Feng, H controller synthesis of fuzzy dynamic systems based on piecewise Lyapunov functions and based on bilinear matrix inequalities, IEEE Trans. Fuzzy Systems 3 ( (

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Delay-Dependent Stability Criteria for Linear Systems with Multiple Time Delays

Delay-Dependent Stability Criteria for Linear Systems with Multiple Time Delays Yong He, Min Wu, Jin-Hua She Abstract This paper deals with the problem of the delay-dependent stability of linear systems