Linear matrix inequality approach to robust stability of uncertain nonlinear discrete-Time systems - Tran Nguyen Binh

LINEAR MATRIX INEQUALITY APPROACH TO ROBUST STABILITY OF UNCERTAIN NONLINEAR DISCRETE-TIME SYSTEMS Tran Nguyen Binh∗ Thai Nguyen University of Economics and Business Administration - Thai Nguyen University, Vietnam abstract This paper deals with the problem of asymptotic stability for a class of nonlinear discrete-time systems with time-varying delay. The time-varying delay is assumed to be belong to a given interval, in which the lower bound of delay is not restricted to zero. A linear matrix inequality (LMI) approach to asymptotic stability of the system is presented. Based on constructing improved Lyapunov functionals, delay-depenent criteria for the asymptotic stability of the system are established via linear matrix inequalities. A numerical example is given to show the effectiveness of the result.. Keywords: Stability, discrete systems, uncertainty, Lyapunov function, linear matrix inequality. 1 Introduction The stability analysis of time-delay uncertain systems is a topic of great practical importance, which has attracted a lot of interest over the decades, e.g. see [1, 5, 6]. Also, system un- certainties arise from many sources such as un- avoidable approximation, data errors and aging of systems and so the stability issue of uncer- tain time-delay systems has been investigated by many researchers [5, 6, 7], where the Lyapunov functional method is certainly used as the main tool. However, the conditions obtained in these papers must be solved upon a grid on the pa- rameter space, which results in testing a finite number of linear matrix inequalities (LMIs). In the case, the result using finite griding points are unreliable and the numerical complexity of the tests grows rapidly. In [5,6], to reduce the conservatism of the stability condition the au- thors proposed a legitimate Lyapunov functional which employs free weighting matrices. To the best of the authors knowledge, the delay-dependent time delay case for the class of discrete-time nonlinear systems with interval varying time delay has not been fully investi- gated and this will be the subject of this paper. In this paper, by using the Lyapunov method, a sufficient condition for the asymptotic stabil- ity of uncertain nonlinear discrete-time systems with interval time-varying delay is derived in terms of LMIs, which can be solved by various efficient convex optimization algorithms [2,3,4]. The paper is organized as follows. After Intro- duction, in Section 2 we give technical lemmas needed for the proof of the main result. Suf- ficient conditions for asymptotic stability of the system and a numerical example to illustrate the effectiveness of our conditions are presented in Section 3. The paper ends with conclusions and cited references. Notations. The following notations will be used throughout this paper. R+ denotes the set of all real non-negative numbers; Rn denotes the n-dimensional space with the scalar prod- uct of two vectors 〈x, y〉 or xT y; Rn×r denotes the space of all matrices of (n × r)− dimen- sion. AT denotes the transpose of A; a ma- trix A is symmetric if A = AT . Matrix A is semi-positive definite (A ≥ 0) if 〈Ax, x〉 ≥ 0, for all x ∈ Rn;A is positive definite (A > 0) if 〈Ax, x〉 > 0 for all x 6= 0; A ≥ B means A − B ≥ 0. λ(A) denotes the set of all eigen- values of A; λmin(A) = min{Reλ : λ ∈ λ(A)}. 2 Preliminaries Consider an uncertain nonlinear discrete-time systems with time-varying delay of the form x(k + 1) = (A+ ∆A)x(k) + (B + ∆B)x(k − h(k)) + f(k, x(k), x(k − h(k))), k ∈ Z+ x(k) = φ(k), k ∈ [−h, ..., 0], (2.1) 0*Tel: 0984411299, e-mail: nguyenbinh.tueba@gmail.com Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên where x(k) ∈ Rn is the state; A,B are given ma- trices; φ(k) is the initial function with the norm ‖ φ ‖= max i∈[−h,0] ‖ φ(i) ‖; The uncertainties satisfy the following condition: ∆A = DaFa(k)Ea,∆B = DbFb(k)Eb, ∆C = DcFc(k)Ec, where Ea, Eb, Ec, Da, Db, Dc are given con- stant matrices with appropriate dimensions, Fa(k), Fb(k) and Fc(k)) are unknown but satisfy FTa (k)Fa(k) ≤ I, FTb (k)Fb(k) ≤ I, FTc (k)Fc(k) ≤ I. The nonlinear purterbation f(k, x(k), x(k − h(k))) satisfies the following condition: fT (.)f(.) ≤ α2[xT (k)GTGx(k) + xT (k − h(k))HTHx(k − h(k))]. (2.2) The time delay function h(k) satisfies the follow- ing condition 0 < h1 ≤ h(k) ≤ h2. Definition 2.1. The system (2.1) is robustly stable if the zero solution of the system is asymp- toticaaly stable for all uncertainties. The following technical lemmas will be used in the proof of the results. Lemma 2.1. Let E,H and F be any constant matrices of appropriate dimensions and FTF ≤ I. For any  > 0, we have EFH +HTFTET ≤ EET + −1HTH. Lemma 2.2. For every P ∈ Rn×n;M ∈ Rn×m and Q is a symmetric positive definite matrix, we have( P MT M −Q ) < 0⇔ P +MTQM < 0. 3 Main results The following result gives what conditions have to be satisfied to guarantee that the system (2.1) is stable. Theorem 3.1. System (2.1) is robustly stable if there exist symmetric positive definite matrices P,Q, and a matrix Ω, a positive number  such that the following LMIs hold: Q ≤ R, (3.1) Y1 Y2 −ΩTDb √ ΩTDb √ ΩTDa −ΩT ∗ Y3 −ΩTDb √ ΩTDb √ ΩTDa −ΩT ∗ ∗ Y4 0 0 0 ∗ ∗ ∗ −I 0 0 ∗ ∗ ∗ ∗ −I 0 ∗ ∗ ∗ ∗ ∗ −I  , < 0 (3.2) where Y1 = G TG+Q+ (h− h)R− P −ATΩ − ΩTA+ −1ETa Ea; Y2 = Ω T −ATΩ; Y3 = P + Ω T + Ω; Y4 =  −1ETb Eb +H TH −Q. Proof. Let A˜ = A+ ∆A, B˜ = B + ∆B. Consider the following Lyapunov-Krasovskii function V (x(k)) = V1(x(k)) + V2(x(k)) + V3(x(k)), where V1(x(k)) = x T (k)Px(k); V2(x(k)) = k−1∑ l= k−h(k) xT (l)Qx(l); V3(x(k)) = −h+1∑ l=−h+2 k−1∑ m=k+l−1 xT (m)Rx(m). We have V (x(k)) ≥ λmin(P ). (3.3) The difference of V1(x(k)) gives ∆V1(x(k)) =V1(x(k + 1))− V1(x(k)) =xT (k + 1)Px(k + 1)− xT (k)Px(k). Let us denote x(k + 1) = y(k), and z(k) = [x(k), y(k), x(k − h(k)), f(.)], S =  P 0 0 0 Ω Ω 0 0 0 0 I 0 0 0 0 I  . Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên Using expression (2.1): 0 = −y(k) + A˜x(k) + B˜x(k − h(k)) + f(.), we have ∆V1(x(k)) = zT (k)  P + ΩT A˜ +A˜TΩ −ΩT +A˜TΩ ΩT B˜ ΩT ∗ −Ω−ΩT + P 0 0 ∗ ∗ 0 0 ∗ ∗ ∗ 0  z(k). (3.4) The difference of ∆V2(x(k)) gives ∆V2(x(k)) = V2(x(k + 1))− V2(x(k)) = k∑ l=k+1−h(k+1) xT (l)Qx(l)− k−1∑ l=k−h(k) xT (l)Qx(l) = xT (k)Qx(k)− xT (k − h(k))Qx(k − h(k)) + k−h∑ l=k+1−h(k+1) xT (l)Qx(l) + k−1∑ l=k+1−h xT (l)Qx(l) − k−1∑ l=k+1−h(k) xT (l)Qx(l). (3.5) The difference of ∆V3(x(k)) gives ∆V3(x(k)) = −h+1∑ l=−h+2 k∑ m=k+l xT (m)Rx(m) − −h+1∑ l=−h+2 k−1∑ m=k+l−1 xT (m)Rx(m) = (h− h)xT (k)Rx(k)− k−h∑ l=k+1−h xT (l)Rx(l). (3.6) Since 0 ≤ h1 ≤ h(k) ≤ h2,∀k ∈ Z+ and Q ≤ R, we have: k−1∑ l=k+1−h1 xT (l)Qx(l) ≤ k−1∑ l=k+1−h(k) xT (l)Qx(l); k−h1∑ l=k+1−h(k+1) xT (l)Qx(l) ≤ k−h1∑ l=k+1−h2 xT (l)Qx(l); k−h1∑ k+1−h2 xT (l)Qx(l) ≤ k−h1∑ k+1−h2 xT (l)Rx(l). (3.7) From (3.4)-(3.7) it follows that ∆V (xk) ≤ zT (k)Mz(k), (3.8) where M = M11 −A˜TΩ + ΩT −ΩT B˜ −ΩT −ΩT A˜+ Ω m2P + ΩT + Ω −ΩT B˜ −ΩT −B˜TΩ −B˜TΩ −Q 0 −Ω −Ω 0 0  , M11 = Q+ (h2 − h1)R− P − A˜TΩ− ΩT A˜. We note that condition (3.2) equivalent to zT (k)  −GTG 0 0 0 0 0 0 0 0 0 −HTH 0 0 0 0 I  z(k) ≤ 0 (3.9) Taking (3.8), (3.9) into acount we obtain ∆V (xk) ≤ zT (k)M˜z(k), (3.10) where, M˜ =  GTG 0 0 0 0 0 0 0 0 0 HTH 0 0 0 0 −I +M. (3.11) On the other hand, we can expresse the matrix M as follows: M = W˜+ Q+ (h2 − h1)R− P ΩT 0 −ΩT Ω P + ΩT + Ω 0 −ΩT 0 0 −Q 0 −Ω −Ω 0 0  , (3.12) where W˜ =  −A˜TΩ− ΩT A˜ −A˜TΩ −ΩT B˜ 0 −ΩT A˜ 0 −ΩT B˜ 0 −B˜TΩ −B˜TΩ 0 0 0 0 0 0  Note that W˜ = W +WT+ −ATΩ− ΩTA −ATΩ −ΩTB 0 −ΩTA 0 −ΩTB 0 −BTΩ −BTΩ 0 0 0 0 0 0  , (3.13) Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên where W = −ΩTDaFa(k)Ea 0 −ΩTDbFb(k)Eb 0 −ΩTDaFa(k)Ea 0 −ΩTDbFb(k)Eb 0 0 0 0 0 0 0 0 0  . (3.14) Applying Lemma 2.1, Lemma 2.3 gives From the relations (3.11)-(3.13), (3.16) it follows that M˜ ≤ M˜11 M˜12 −ΩTB −ΩT ∗ M˜22 −ΩTB −ΩT ∗ ∗ Y4 0 ∗ ∗ ∗ −I , (3.15) where, M˜11 = G TG+Q+ (h2 − h1)R− P −ATΩ − ΩTA+ −1ETa Ea + ΩT (DaD T a +DbD T b )Ω M˜12 = Ω T −ATΩ + ΩT (DaDTa +DbDTb )Ω M˜22 = P + Ω T + Ω + ΩT (DaD T a +DbD T b )Ω Y4 =  −1ETb Eb +H TH −Q. Using Lemma 2.1, Lemma 2.3 the inequality (3.17) is equivalent to Y1 Y2 √ ΩTDa √ ΩTDb −ΩTB −ΩT ∗ Y3 √ ΩTDa √ ΩTDb −ΩTB −ΩT ∗ ∗ −I 0 0 0 ∗ ∗ ∗ −I 0 0 ∗ ∗ ∗ ∗ Y4 0 ∗ ∗ ∗ ∗ ∗ −I  (3.16) < 0. Then, using Lemma 2.1, Lemma 2.3, the in- equality (3.2) is equivalent to (3.18). Therefore, we have ∆V (xk) < 0, k = 1, 2, ...., combined with the condition (3.3) and the Lya- punov stability theorem [1], implies the asymp- totic stability of the system. The proof is com- pleted. To illustrate the effectiveness of the obtained re- sult, we consider the following numerical exam- ple. Example 3.1. Consider the discrete -time sys- tem with time-varying delay (2.1), where h1 = 4, h2 = 23 and A = ( 0.0312 0.0410 0.0100 0.0380 ) , B = ( 0.0210 0.0220 0.0300 0.0200 ) , Da = ( 0.0130 0.0250 0.0500 0.0730 ) , Db = ( 0.0330 0 0 0.0200 ) , Ea = ( 0.0290 0.0200 0.0500 0.0630 ) , Eb = ( 0.0125 0.0270 0.0230 0.0210 ) , G = ( 0.2500 −0.2000 −0.1250 0.1000 ) , H = ( 0 0.4000 0 0.5000 ) , Fa(k) = ( 0 sin(k) 0.2cos(k) 0 ) , Fb(k) = ( cos(k) 0 0 3/10sin(k) ) , f(.) = 0.05sin(k)Gx(k)+0.02cos(k)Hx(k−h(k)). By LMI toolbox of Matlab, we find that the LMIs of Theorem 3.1 are satisfied with  = 1, and P = ( 0.1511 0.0013 0.0013 0.1517 ) , Q = ( 0.0039 0.0001 0.0001 0.0040 ) , R = ( 0.0057 −0.0 −0.0 0.0058 ) ,Ω = (−0.004 0.0 0.0 −0.004 ) . By Theorem 3.1, the system is asymptotic sta- bility. 4 Conclusion This paper has proposed delay-dependent con- ditions for the robust stability of nonlinear un- certain discrete-time system with interval time- varying delay. The conditions are presented in terms of linear matrix inequalities. Acknowledgments. This work was supported by the National Foundation for Science and Technology Development, Vietnam. Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên References [1] R.P. Agarwal, Difference Equations and In- equalities, Second Edition, Marcel Dekker, New York, 2000. [2] S. Boyd, L.El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. [3] A. Czornik, P. Mokry, A. Nawrat, On the exponential exponents of discrete linear sys- tems Linear Algebra and its Applications, 433 (2010), 867-875. [4] M. C. de Oliveira, J. C. Geromel, Liu Hsu, LMI characterization of structural and robust stability: the discrete-time case, Linear Algebra and its Applications, 296 (1999), 27-38. [5] V.N. Phat, Robust stability and stabiliz- ability of uncertain linear hybrid systems with state delays, IEEE Trans. CAS II, 52 (2005), 94-98. [6] J.H. Park, Robust Decentralized Stabiliza- tion of Uncertain Large-Scale Discrete-Tine Systems with Delays, Journal of Optimiza- tion Theory and Applications 113 (2002) 105-119 [7] E.K. Boukas, State feedback stabilization of nonlinear discrete - time systems with time - varying time delay, Nonlinear Analysis 66 (2007) 1341 - 1350. Tãm t¾t TiÕp cËn bÊt ®¼ng thøc ma trËn tuyÕn tÝnh ®Ó nghiªn cøu tÝnh æn ®Þnh m¹nh cña hÖ phi tuyÕn kh«ng ch¾c ch¾n víi thêi gian rêi r¹c Trong bµi b¸o nµy chóng t«i ®Ò cËp tíi bµi to¸n æn ®Þnh tiÖm cËn cho líp hÖ rêi r¹c phi tuyÕn víi ®é trÔ biÕn thiªn. §é trÔ lµ biÕn thiªn vµ lín h¬n kh«ng. Bµi b¸o còng x©y dùng vµ c¶i tiÕn hµm Lyapunov ®Ó nghiªn cøu tÝnh æn ®Þnh tiÖm cËn cña hÖ th«ng qua bÊt ®¼ng thøc ma trËn tuyÕn tÝnh. Mét vµi vÝ dô trong bµi b¸o thÓ hiÖn tÝnh hiÖu qu¶ cña kÕt qu¶. Tõ khãa: HÖ rêi r¹c, æn ®Þnh, kh«ng æn ®Þnh, hµm Lyapunov, bÊt ®¼ng thøc ma trËn tuyÕn tÝnh Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên