ROBUST STABILITY TEST FOR UNCERTAIN DISCRETE-TIME SYSTEMS: A DESCRIPTOR SYSTEM APPROACH

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1 Latin American Applied Research 41: (211) ROBUS SABILIY ES FOR UNCERAIN DISCREE-IME SYSEMS: A DESCRIPOR SYSEM APPROACH W. ZHANG,, H. SU, Y. LIANG, and Z. HAN Engineering raining Center, Shanghai University of Engineering Science, 2162, Shanghai, China wizzhang@gmail.com; yanlianggm@gmail.com School of Electronic, Information and Electrical Engineering, Shanghai Jiao ong University, 224 Shanghai, China, zzhan@sjtu.edu.cn Department of Control Science and Engineering, Huazhong University of Science and echnology, 4374 Wuhan, China, Corresponding author, houshengsu@gmail.com Abstract his paper presents a new and less conservative condition for the robust stability test of discrete-time polytopic systems by using a descriptor system approach. he stability condition is formulated in terms of a set of linear matrix inequalities and can be easily adapted for robust controller synthesis. he developed results can be viewed as a discretetime counterpart of the continuous-time one proposed by Cao and Lin (24). his also enables us to offer a unified framework, namely the so-called descriptor system approach, for the analysis and synthesis of both discrete-time and continuous-time uncertain linear systems. Simulation examples are given to illustrate the theoretical results we established. Keywords Discrete-time systems, robust stability, parameter-dependent Lyapunov functions, descriptor system approach I. INRODUCION he problem of robust stability analysis and controller synthesis for uncertain systems has been extensively investigated in recent years. he Lyapunov-based approach is one of the most popular methods for solving this problem in the existing literature (see, e.g., Boyd et al., 1994; Feron et al., 1996; Gahinet et al., 1996; Oliveira et al., 1998; Geromel and Korogui, 26; Su and Zhang, 29; Zhang et al., 21a; Zhang et al., 21b; and the references therein). However, it is known that the traditional quadratic stability analysis usually leads to conservative results, especially in the case where the uncertainty is time invariant or slowly time-varying. o overcome this drawback, parameter dependent Lyapunov functions (PDLFs) were proposed in Feron et al. (1996) and Gahinet et al. (1996), where the stability conditions were formulated in terms of linear matrix inequalities (LMIs). Generally, the stability conditions based on PDLFs are less conservative than those resulted from a parameter independent Lyapunov functions (see, e.g., Daafouz and Bernussou, 21; Lin et al., 26; Gao et al., 27). he PDLFs-based approach has become a powerful tool in the analysis and design of linear uncertain systems since the pioneer work of Oliveira et al. (1999a, 1999b). By introducing a slack variable, Oliveira et al. (1999a, 1999b) proposed a new LMI condition for robust stability test of discrete polytopic systems. More importantly, the condition can be easily adapted for controller synthesis since it exhibits a kind of decoupling between the Lyapunov and the system matrices. he results were extended to the continuous-time case by Apkarian et al. (21). Recently, another extension was proposed in Cao and Lin (24) by applying a descriptor system approach, which was originally proposed by Fridman and Shaked (22) to study the stability and H control of time-delay systems. In general, there are two advantages of this approach. First, it can significantly reduce the conservatism by introducing some slack variables. Second, it can be easily applied to solve the problem of controller synthesis. In this paper we revisit the problem of robust stability analysis and synthesis for uncertain discrete-time systems. We obtain a new and less conservative robust stability condition, which encompasses the known result proposed by Oliveira et al. (1999a) as a special case. he condition can also be viewed as a discretetime counterpart of the continuous-time results given by Cao and Lin (24). Also, the stability conditions can be easily adapted for controller synthesis of polytopic systems. herefore, it is interesting to note that our results together with the work Cao and Lin (24) present a new framework for the analysis and synthesis of uncertain linear systems. More precisely, we show that the descriptor system transformation is an efficient approach for the stability analysis and controller synthesis of both discrete-time and continuous-time polytopic systems. We finally use two numerical examples to illustrate the theoretical results. 359

2 Latin American Applied Research 41: (211) Notations: R n denotes the n-dimensional Euclidean space; R m n is the set of all m n real matrices. he superscript stands for matrix transposition. For real symmetric matrices X and Y, the notation X > Y means that the matrix X Y is positive definite. I is an identity matrix with appropriate dimension. In symmetric block matrices, we use an asterisk to represent a term that is induced by symmetry. II. SABILIY ANALYSIS his section introduces a descriptor system transformation to analyze the asymptotical stability of linear discrete-time systems. A new necessary and sufficient stability condition is obtained for such systems. One known result by Oliveira et al. (1999a) is recovered. Consider the following linear discrete-time system x(k + 1) Ax(k), (1) where x(k) R n is the state. From Lyapunov stability theory we know that a necessary and sufficient condition for asymptotical stability of system (1) is that there exists a matrix P P > satisfying A P A P <. (2) In order to obtain another stability criterion, we first make a descriptor system transformation for system (1). As it was done for continuous-time systems in Cao and Lin (24), we rewrite (1) as a descriptor system I x(k + 1) y(k + 1) I A I x(k), (3) where x(k + 1). For simplicity, we denote Let E I I, Ā, x(k) A I P P1, P 2 P 3 x(k). where P 1 P1 > and P i R n n, i 1, 2, 3. Let us now define a Lyapunov function candidate for system (1) as follows V (k) x (k)ep x(k) x (k)p 1 x(k). (4) hen we have the following necessary and sufficient condition for the asymptotic stability of system (1), which can be viewed as a discrete-time counterpart of the continuous-time case by Cao and Lin (24). Lemma 1. System (1) is asymptotically stable if and only if there exist matrices P 1 >, P 2 and G G such that A Ξ GA P 1 A P 2 A G P 1 P 2 P2 <. (5) + G Proof. (Sufficiency) Let the Lyapunov function candidate be given in (4). Denote V (k) V (k + 1) V (k). Let G P 3 + P3. hen we have V (k) x (k + 1)P 1 x(k + 1) x (k)ep x(k) x(k + 1) x(k + 1) P x (k)ep x(k) P Ax(k) x (k)ep x(k) Ax(k) 1 2 x (k) Ā ( P + P ) Ā EP P E x(k) 1 2 x (k)ξ x(k). Note that if (5) holds, then V (k) < for all x(k), and hence (1) is asymptotically stable according to the Lyapunov stability theory. (Necessity) If system (1) is asymptotically stable, then there exists a matrix P 1 P1 > such that A P 1 A P 1 <. (6) By using the Schur complement, (6) is equivalent to P1 A P 1 <. (7) P 1 A P 1 Let P 2 2P 1 and P 3. hen we have P1 A Ξ 2 P 1 <. P 1 A P 1 hus the matrices P 1, P 2 and G G satisfy the LMI (5). Remark 1. In the proof of Lemma 1, two slack variables P 2 and G are introduced, the purpose of which is to decouple the product between the Lyapunov matrix P 1 and the system matrix A. Moreover, by letting G we will recover one of the main results in Oliveira et al. (1999a). Corollary 1. System (1) is asymptotically stable if and only if there exist P P > and H such that P A H HA H + H >. (8) P Proof. (Sufficiency) By using the Schur complement, (8) is equivalent to P A H HA P H H <. (9) Let 2P 1 P, P 2 H, and G. hen from (9) we can obtain (5). Hence, it follows from Lemma 1 that system (1) is asymptotically stable. 36

3 W. ZHANG, H. SU, Y. LIANG, Z. HAN (Necessity) If system (1) is asymptotically stable, then there exist matrices P 1 >, P 2, G G such that LMI (5) holds. By multiplying (5) with M I A on the left and M on the right, we get A P 1 A P 1 <. hus, by choosing H H P 1 > and letting P P 1, one can obtain (9) by using the Schur complement. III. ROBUS SABILIZAION FOR POLYOPIC SYSEMS In the above section, all the involving matrices of the systems are assumed to be known. However, in many physical systems it is very difficult to obtain the exact model of systems. So in this section we consider a class of uncertain discrete systems described as follows x(k + 1) A(θ)x(k) + B(θ)u(k), (1) where x(k) R n, u(k) R m, A(θ) R n n and B(θ) R n m. θ θ 1, θ 2,..., θ N R N is a vector of parameters. Assume the dynamic matrix A(θ) and the input matrix B(θ) belong to a convex polytopic set defined as { A B A(θ) B(θ) : A(θ) B(θ) } θ i A i B i, θ N, (11) where N is a unit simplex given by { } N θ : θ i 1, θ i, i 1,..., N. (12) Based on Lemma 1, we have the following sufficient condition for the robust stability of system (1). Lemma 2. System (1) is robustly stable if there exist matrices P 1 (θ) >, P 2 (θ) and G(θ) G (θ) such that A Γ θ (θ)g(θ)a(θ) P 1 (θ) A (θ)p 2 (θ) A (θ)g(θ) P 1 (θ) P 2 (θ) P2 <. (13) (θ) + G(θ) However, Lemma 2 can not be directly applied to test the robust stability of system (1). In order to obtain new and more appliable stability condition, we resort to the following PDLF V (k, θ) x (k)p 1 (θ)x(k), (14) where P 1 (θ) P 1 (θ) >. For system (1), a simple selection of P 1 (θ) is P 1 (θ) θ i P 1i, P 1i >. (15) he following theorem proposes a criterion to test the robust stability of system (1). heorem 1. System (1) is robustly stable if there exist matrices P 1i >, P 2i, i 1, 2,..., N, and G G such that A i GA j + A j GA i P 1i P 1j A i P 2j + A j P 2i A i G A j G P 1i + P 1j P 2i P2i P 2j P2j + 2G < (16) hold for all 1 i j N. Remark 2. he proof of heorem 1 can be obtained directly from that of the following heorem 2, so we omit it here. Moveover, it is worth mentioning that heorem 1 encompasses heorem 2 in Oliveira et al. (1999a) as a special case. hat is, by letting G and P 2i P 2, i 1, 2,..., N, heorem 1 will recover heorem 2 in Oliveira et al. (1999a). heorem 1 can be further improved by using a relaxed LMI method (Gao et al., 27). We sum it up as the following theorem. heorem 2. System (1) is robustly stable in uncertainty domain (11) if there exist matrices P 1i >, P 2i, W ij, J ij and G G such that the following LMIs hold: W ij + W ji J ij J ij <, 1 i j N, (17) where J 11 J 12 J 1N J 22 J 2N J... <, (18). J NN A W ij i GA j P 1i A i P 2j A i G P2j A i GA i P 1i P 2i P2i + G. (19) Proof. Let G(θ) G. Let P 1 (θ) and P 2 (θ) be given by P 1 (θ) θ i P 1i, P 2 (θ) θ i P 2i. Substituting these matrices into (13) we have Γ θ,j1 A θ i θ i GA j P 1i A i P 2j A i G j P2j A i GA i P 1i P 2i P2i + G N 1 θi 2 W ii + ji+1 θ i θ j (W ij + W ji ), (2) 361

4 Latin American Applied Research 41: (211) where W ij is given in (19). On the other hand, (17) is equivalent to W ii < J ii, i 1,..., N, (21) W ij + W ji < J ij + J ij, 1 i < j N. (22) hen from (2)-(22) we have Γ θ N 1 θi 2 J ii + ji+1 θ i θ j (J ij + J ij) ξ Jξ, (23) where ξ θ 1 I θ 2 I... θ N I and J is given in (18). Inequalities (18) and (23) guarantee Γ θ <, and therefore the proof is completed. Corollary 2. System (1) is robustly stable in uncertainty domain (11) if there exist matrices P 1i >, P 2i, J ij with Xij Y J ij ij (24) Z ij S ij satisfying (18) and such that the following LMIs hold for all 1 i j N, P1,ij X ij Xij A i P 2j + A j P 2i Y ij Zij P 1,ij P 2,ij P2,ij S ij Sij <. (25) where P 1,ij P 1i + P 1j, P 2,ij P 2i + P 2j. (26) Proof. By letting G, one can easily derive Corollary 2 from heorem 2. herefore, the detail proof is omitted here. As usual, the state feedback for system (1) can be obtained through solving an LMI problem. heorem 3. System (1) is robustly stabilizable in uncertainty domain (11) if there exist matrices P 1i >, P 2, L, J ij satisfying (18) and P1,ij X ij Xij A i P 2 + A j P 2 + B i L + B j L Y ij Zij P 1,ij 2P 2 2P2 S ij Sij < (27) for all 1 i j N, where J ij and P 1,ij are given by (24) and (26), respectively. If (27) is feasible, then the stabilizing state feedback gain can be computed as K LP 1 2. imag real Figure 1: Eigenvalues of A(θ) for ρ and various θ. IV. SIMULAIVE EXAMPLES his section gives two examples to show the effectiveness of the results obtained in Section III.. he first example is borrowed from Ramos and Peres (21). Example 1. Consider system (1) with N 2. he system matrix A(θ) parameterized by ρ is assumed to be given by (ρa 1, ρa 2 ) with 1 A 1 1, and 1 A We test the stability of the system by using quadratic stability, heorem 2 in Oliveira et al. (1999a), Lemma 1 in Ramos and Peres (21), and our results, respectively. We compute the maximum value of ρ such that the system (ρa 1, ρa 2 ) is robustly stable. he comparison results are given in able 1. he value of r in able 1 denotes the maximum radius of the circle that contains the eigenvalues of the uncertain system. Note that the radius obtained from our results is very close to the unit circle. Figure 1 depicts the eigenvalues of A(θ) for ρ and various θ (where θ 1.2s, θ 2 1 θ 1, and s, 1, 2,..., 5). As it was proposed in Ramos and Peres (21), one can get the value of r by computing r max λ i θ 1 (ρa 1 ) + θ 2 (ρa 2 ), i Remark 3. If (27) is feasible, then P 2 + P2 S 11 >. herefore, P 2 is nonsingular. > P 11 where θ 1 + θ 2 1, θ 1, θ

5 W. ZHANG, H. SU, Y. LIANG, Z. HAN able 1: Calculation results of stability bound ρ and radius bound r Methods ρ r Quadratic stability Oliveira et al. (1999a) Ramos and Peres (21) heorem heorem Corollary x(k) x 1 (k) x 2 (k) x 3 (k) Note that r < r sup 1 guarantees the robust stability of A(θ). herefore, we can calculate the supremum of ρ with ρ sup From able 1, we can see that heorems 1 and 2, and Corollary 2 in this paper all provide the best estimation for the robust stability domain of the system in this example. Example 2. Consider the polytopic system (1) with N 2. We assume the system matrix A(θ) parameterized by ρ is given by the pair (ρa 1, ρa 2 ) with A , A and the input matrices are B(θ) with B , B Note that the result in Ramos and Peres (21) can not be applied to the controller synthesis of this system. In fact, Ramos and Peres (21, 22) provided only the robust stability results for polytopic systems. By heorem 3, we can obtain the allowable maximum bound of ρ is Moreover, the stabilizing state feedback gain matrix is K Figure 2 displays the state response of the closed-loop system given in Example 2. he simulation result shows the effectiveness of the feedback design given in this paper. V. CONCLUSIONS We have studied the robust stability problem for polytopic discrete-time systems by introducing a descriptor system approach. A new LMI robust stability condition is established. he condition is less conservative than the known results proposed by Oliveira et al. (1999a) and Ramos and Peres (21). Also it can be ime in second Figure 2: State response of the closed-loop system in Example 2. applied to study the controller synthesis of polytopic systems. he comparison results with the known ones are illustrated by numerical examples. Moreover, our results together with the recent work by Cao and Lin (24) provide a unified framework to analysis and synthesis for both discrete-time and continuous-time polytopic systems. In other words, we can deal with such problem by applying the descriptor system approach. As an extension, the proposed approach could be applied to the analysis and design of linear timedelay systems with polytopic uncertainties. VI. ACKNOWLEDGMEN he authors would like to thank the anonymous reviewers and the subject editor Prof. Jorge A. Solsona for their valuable comments and constructive suggestions. his work was supported by the National Natural Science Foundation of China under Grant No and in part by the Fundamental Research Funds for the Central Universities, HUS: Grant No. 21QN4. REFERENCES Apkarian, P., H.D. uan and J. Bernussou, Continuous time analysis, eigenstructure assignment, and H 2 synthesis with enhanced linear matrix inequalities (LMI) characterizations, IEEE rans. Automat. Control, 46, (21). Boyd, S., L.E. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control heory, Philadelphia, PA: SIAM (1994). Cao, Y.Y. and Z.L. Lin, A descriptor system approach to robust stability analysis and controller synthesis, IEEE rans. Automat. Control, 49, (24). Daafouz, J. and J. Bernussou, Parameter dependent Lyapunov functions for discrete-time systems 363

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