ARTICLES

Robust stability test for uncertain discrete-time systems: a descriptor system approach

W. Zhang^†,†, H. Su^§, Y. Liang^†, and Z. Han^†

^† Engineering Training Center, Shanghai University of Engineering Science, 201620, Shanghai, China
Email: wizzhang@gmail.com; yanlianggm@gmail.com
^† School of Electronic, Information and Electrical Engineering, Shanghai Jiao Tong University, 200240
Shanghai, China, Email: zzhan@sjtu.edu.cn
^§ Department of Control Science and Engineering, Huazhong University of Science and Technology, 430074
Wuhan, China, Corresponding author, Email: houshengsu@gmail.com

Abstract — This paper presents a new and less conservative condition for the robust stability test of discrete-time polytopic systems by using a descriptor system approach. The stability condition is formulated in terms of a set of linear matrix inequalities and can be easily adapted for robust controller synthesis. The developed results can be viewed as a discretetime counterpart of the continuous-time one proposed by Cao and Lin (2004). This 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. INTRODUCTION

The problem of robust stability analysis and controller synthesis for uncertain systems has been extensively investigated in recent years. The 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, 2006; Su and Zhang, 2009; Zhang et al., 2010a; Zhang et al., 2010b; 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. To 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, 2001; Lin et al., 2006; Gao et al., 2007).

The 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. The results were extended to the continuous-time case by Apkarian et al. (2001). Recently, another extension was proposed in Cao and Lin (2004) by applying a descriptor system approach, which was originally proposed by Fridman and Shaked (2002) to study the stability and H₈ 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. The condition can also be viewed as a discretetime counterpart of the continuous-time results given by Cao and Lin (2004). Also, the stability conditions can be easily adapted for controller synthesis of polytopic systems. Therefore, it is interesting to note that our results together with the work Cao and Lin (2004) 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.

Notations: Rⁿ denotes the n-dimensional Euclidean space; R^m×n is the set of all m × n real matrices. The superscript "T " 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. STABILITY ANALYSIS

This 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

(1)

where x(k) ? Rⁿ 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^T > 0 satisfying

(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 (2004), we rewrite (1) as a descriptor system

(3)

where x(k +1) = y(k). For simplicity, we denote

Let

where and P_i ? R^{n × n}, i = 1, 2, 3. Let us now define a Lyapunov function candidate for system (1) as follows

(4)

Then 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 (2004).

Lemma 1. System (1) is asymptotically stable if and only if there exist matrices P₁ > 0, P₂ and G = G^T such that

(5)

Proof. (Sufficiency) Let the Lyapunov function candidate be given in (4). Denote ? V (k)= V (k + 1) - V (k). Let G = P₃ + . Then we have

Note that if (5) holds, then ? V (k) < 0 for all , and hence (1) is asymptotically stable according to the Lyapunov stability theory.

(Necessity) If system (1) is asymptotically stable, then there exists a matrix such that

(6)

By using the Schur complement, (6) is equivalent to

(7)

Let P₂ = 2P₁ and P₃ = 0. Then we have

Thus the matrices P₁, P₂ and G = G^T satisfy the LMI (5).

Remark 1. In the proof of Lemma 1, two slack variables P₂ and G are introduced, the purpose of which is to decouple the product between the Lyapunov matrix P₁ and the system matrix A. Moreover, by letting G = 0 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 and H such that

(8)

Proof. (Sufficiency) By using the Schur complement, (8) is equivalent to

(9)

Let 2P₁ = P₀, P₂ = H^T, and G = 0. Then from (9) we can obtain (5). Hence, it follows from Lemma 1 that system (1) is asymptotically stable.

(Necessity) If system (1) is asymptotically stable, then there exist matrices P₁ > 0, P₂, G = G^T such that LMI (5) holds. By multiplying (5) with M = [I  A^T] on the left and M^T on the right, we get A^TP₁A - P₁ < 0. Thus, by choosing H = H^T = P₁ > 0 and letting P₀ = P₁, one can obtain (9) by using the Schur complement.

III. ROBUST STABILIZATION FOR POLYTOPIC SYSTEMS

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

(10)

where x(k) ? Rⁿ, u(k) ? R^m, A(?) ? R^{n × n} and B(?) ? R^{n × m}. ? = [?₁, ?₂,..., ?_N] ? R^N is a vector of parameters. Assume the dynamic matrix A(?) and the input matrix B(?) belong to a convex polytopic set defined as

	(11)

where ?_N is a unit simplex given by

(12)

Based on Lemma 1, we have the following sufficient condition for the robust stability of system (10).

Lemma 2. System (10) is robustly stable if there exist matrices P₁(?) > 0, P₂(?) and G(?) = G^T(?) such that

	(13)

However, Lemma 2 can not be directly applied to test the robust stability of system (10). In order to obtain new and more appliable stability condition, we resort to the following PDLF

(14)

where . For system (10), a simple selection of P₁(?) is

(15)

The following theorem proposes a criterion to test the robust stability of system (10).

Theorem 1. System (10) is robustly stable if there exist matrices P_1i > 0, P_2i, i = 1, 2,..., N, and G = G^T such that

	(16)

hold for all 1 = i = j = N.

Remark 2. The proof of Theorem 1 can be obtained directly from that of the following Theorem 2, so we omit it here. Moveover, it is worth mentioning that Theorem 1 encompasses Theorem 2 in Oliveira et al. (1999a) as a special case. That is, by letting G = 0 and P_2i = P₂, i = 1, 2,..., N, Theorem 1 will recover Theorem 2 in Oliveira et al. (1999a).

Theorem 1 can be further improved by using a relaxed LMI method (Gao et al., 2007). We sum it up as the following theorem.

Theorem 2. System (10) is robustly stable in uncertainty domain (11) if there exist matrices P_1i > 0, P_2i, W_ij, J_ij and G = G^T such that the following LMIs hold:

	(17)
	(18)

where

(19)

Proof. Let G(?) = G. Let P₁(?) and P₂(?) be given by

Substituting these matrices into (13) we have

	(20)

where W_ij is given in (19). On the other hand, (17) is equivalent to

	(21)
	(22)

Then from (20)-(22) we have

(23)

where and J is given in (18). Inequalities (18) and (23) guarantee G _? < 0, and therefore the proof is completed.

Corollary 2. System (10) is robustly stable in uncertainty domain (11) if there exist matrices P_1i > 0, P_2i, J_ij with

(24)

satisfying (18) and such that the following LMIs hold for all 1 = i = j = N,

	(25)

where

(26)

Proof. By letting G = 0, one can easily derive Corollary 2 from Theorem 2. Therefore, the detail proof is omitted here.

As usual, the state feedback for system (10) can be obtained through solving an LMI problem.

Theorem 3. System (10) is robustly stabilizable in uncertainty domain (11) if there exist matrices P_1i > 0, P₂, L, J_ij satisfying (18) and

	(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 .

Remark 3. If (27) is feasible, then . Therefore, P₂ is nonsingular.

IV. SIMULATIVE EXAMPLES

This section gives two examples to show the effectiveness of the results obtained in Section III. The first example is borrowed from Ramos and Peres (2001).

Example 1. Consider system (10) with N = 2. The system matrix A(?) parameterized by ? is assumed to be given by (?A₁, ?A₂) with

and

We test the stability of the system by using quadratic stability, Theorem 2 in Oliveira et al. (1999a), Lemma 1 in Ramos and Peres (2001), and our results, respectively. We compute the maximum value of ? such that the system (?A₁, ?A₂) is robustly stable. The comparison results are given in Table 1. The value of r in Table 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(?) for ? = 1.97219 and various ? (where ?₁ = 0.02s, ?₂ = 1 - ?₁, and s = 0, 1, 2,..., 50).

Table 1: Calculation results of stability bound ? and radius bound r

Figure 1: Eigenvalues of A(?) for ? = 1.97219 and various ?.

As it was proposed in Ramos and Peres (2001), one can get the value of r by computing

where

Note that r < r_sup = 1 guarantees the robust stability of A(?). Therefore, we can calculate the supremum of ? with ?_sup = 1.9722. From Table 1, we can see that Theorems 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 (10) with N = 2. We assume the system matrix A(?) parameterized by ? is given by the pair (?A₁, ?A₂) with

and the input matrices are B(?) with

Note that the result in Ramos and Peres (2001) can not be applied to the controller synthesis of this system. In fact, Ramos and Peres (2001, 2002) provided only the robust stability results for polytopic systems. By Theorem 3, we can obtain the allowable maximum bound of ? is 3.7411. Moreover, the stabilizing state feedback gain matrix is

Figure 2 displays the state response of the closed-loop system given in Example 2. The simulation result shows the effectiveness of the feedback design given in this paper.

Figure 2: State response of the closed-loop system in Example 2.

V. CONCLUSIONS

We have studied the robust stability problem for poly topic discrete-time systems by introducing a descriptor system approach. A new LMI robust stability condition is established. The condition is less conservative than the known results proposed by Oliveira et al. applied to study the controller synthesis of polytopic systems. The comparison results with the known ones are illustrated by numerical examples. Moreover, our results together with the recent work by Cao and Lin (2004) 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 time delay systems with polytopic uncertainties.

VI. ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewers and the subject editor Prof. Jorge A. Solsona for their valuable comments and constructive suggestions. This work was supported by the National Natural Science Foundation of China under Grant No. 61074003 and in part by 'the Fundamental Research Funds for the Central Universities', HUST: Grant No. 2010QN040.

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Received: June 20, 2010
Accepted: December 17, 2010
Recommended by subject editor: Jorge Solsona