Delay-dependent robust stabilisation of discrete-time systems with time-varying delay

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1 Delay-dependent robust stabilisation of discrete-time systems with time-varying delay X.G. Liu, R.R. Martin, M. Wu and M.L. Tang Abstract: The stability of discrete systems with time-varying delay is considered. New delaydependent stability criteria are devised, which are dependent on the minimum and maximum delay bounds. An initial analysis leads to a criterion depending on an inequality involving certain matrices that can be freely chosen. By carefully choosing them to reflect the appropriate relationship between states at differing times, a stricter criterion is thereby obtained. Furthermore, new results for delay-dependent robust stabilisation of uncertain systems with time-varying delay are provided on the basis of a linear matrix inequality (LMI) framework. As the conditions obtained for the existence of admissible controllers are not expressed using strict LMI conditions, a cone complementary linearisation procedure is used to find suitable controllers. Finally, the results obtained, including the stability analysis, static output-feedback stabilisation and dynamic output feedback stabilisation are further extended to discrete time-delay systems having uncertain but norm-bounded parameters. Numerical examples demonstrate the validity of the approach proposed. 1 Introduction Time delays often appear in control systems and are often a source of instability and oscillations in such systems. Assessing and controlling the stability of such systems with delay are of theoretical and practical importance. Increasing attention has been paid to the problem of feedback stabilisation of systems with state delay. Most of the results obtained have been derived using delay-independent approaches (see, e.g. Li and De Souza [1]). As the time delay is not taken into consideration using these approaches to design controllers, the results are generally more conservative than ones using a delay-dependent approach. However, previous delay-dependent methods for systems with time-varying delays have mainly considered the continuous systems [ 1]. Relatively, few papers have considered the time-varying delay case for discrete-time systems [1]. Recently, Gao et al. [1] studied delay-dependent output-feedback stabilisation of discrete-time systems with time-varying state delay. A new stability condition was proposed, which is dependent on the delay bounds. Their results are based on an inequality on the inner product of two vectors proved by Moon et al. [8], which we repeat as Lemma 1 here. Given the system state x(k), where k is discrete time, with time-dependent delay d(k), this inequality is typically used to evaluate the bounds on # The Institution of Engineering and Technology IEE roceedings online no. doi:1.19/ip-cta: aper first received 9th June and in revised form 9th December X.G. Liu and M.L. Tang are with the School of Mathematical Science and Computing Technology, Central South University, Changsha, Hunan 18, eople s Republic of China R.R. Martin and X.G. Liu are with the School of Computer Science, Cardiff University, Cardiff, UK M. Wu and X.G. Liu are with the School of Information Science and Engineering, Central South University, Changsha, Hunan 18, eople s Republic of China liuxgliuhua@1.com a weighted cross-product between x(k) and the difference x(k) x(k d(k)), needed in the analysis of the delaydependent stability problem. The use of this inequality leads to conservatism in the delay-dependent stability conditions obtained. This paper presents a new approach to establish a stricter delay-dependent stability criterion for time-varying delay systems, using relations between all system states x(k), without requiring any system model transformation. An initial criterion is found on the basis of an inequality involving various matrices that can be freely chosen, and an improved criterion is then found by carefully choosing these matrices to reflect the correlation between system states at differing delays. Moon s inequalities are not needed in our approach. Our new stability condition is very simple. Going further, cone complementary linearisation algorithms as used in El Ghaoui et al. [1] are exploited to enable us to solve the inequalities needed to provide static and dynamic output-feedback stabilisation of such systems. Numerical examples are given to demonstrate that the asymptotic stability results derived in this paper are effective and less conservative than those derived by Gao et al. [1]. roblem description Consider the following discrete-time system with a time-varying delay in the state xðk þ 1Þ ¼ AxðkÞ þ A 1 xðk dðkþþ þ BuðkÞ yðkþ ¼ CxðkÞ þ C 1 xðk dðkþþ xðkþ ¼ fðkþ for k ¼ d max ; d max þ 1;... ; where k is discrete time, x(k) [ R n the state vector, y(k) [ R m the measured output and u(k) [ R l the controlled input. A, A 1, C and C 1 are system matrices with compatible dimensions. d(k), appearing in both the dynamic and measurement equations, is the state delay, as frequently ð1þ IEE roc.-control Theory Appl., Vol. 1, No., November 89

2 encountered in various engineering systems. f(k), k ¼ d max, d max þ 1,...,, is a given initial condition sequence. A natural assumption on d(k) can be made as follows. Assumption 1: The time delay d(k) is assumed to vary with time between some limits, satisfying d min d(k) d max, where d min and d max are positive constants representing minimum and maximum delays, respectively. The time-varying delay d(k) reduces to a constant delay d when d min ¼ d max ¼ d. In this paper, it is assumed that the state variables are not fully measurable, that is, we know only partial information about x(k), for example, several components of x(k), and that we are interested in designing output-feedback controllers such that the resulting closed-loop system is asymptotically stable. In order to analyse the performance of discrete time-delay systems, we introduce the following definitions of stability and asymptotic stability for discrete systems. Definition 1: The discrete time-delay system given in (1), when u(k) ¼, is said to be stable if, for any 1., there is a d(1). such that jx(k) j, 1, k., when sup jfðsþj, dð1þ d max s In addition, if lim k!1 jx(k) j ¼ for any initial conditions, then the system given in (1) with u(k) ¼ is said to be asymptotically stable. Assumption : We assume that the matrices A and A 1 in the system given in (1) have the following forms A ¼ A þ DA; A 1 ¼ A 1 þ DA 1 where A and A 1 are known constant matrices of appropriate dimensions and DA and DA 1 real-valued time-varying matrix functions representing norm-bounded admissible uncertainties. Definition : The uncertain time-delay system given in (1) under Assumption is said to be robustly stable if the trivial solution x(k) ¼ of the functional difference equation associated with the system given in (1) with u(k) ¼ is globally uniformly asymptotically stable for all admissible uncertainties. Moon et al. [8] proved the following lemma, which we use later. Lemma 1: Assume that a [ R a, b [ R b and N [ R ab. Then, for any matrices X [ R aa, Y [ R ab and [ R bb, the following inequality holds a T Nb a T X Y N a b Y T N T b provided that matrices X, Y and satisfy X Y Y T. The goal of the rest of this paper is to establish new asymptotic stability criteria and new robust stability criteria and to develop a procedure to design stabilising output feedback controllers for discrete systems with time-varying delay. For simplicity, in the rest of the paper, in symmetric block matrices or long matrix expressions, we use to represent a term that is induced by symmetry. 9 Asymptotic stability analysis In this section, we aim to establish an asymptotic stability criterion for the system given in (1), when u(k) ¼, using the Lyapunov method combined with the linear matrix inequality (LMI) technique as previously described [1]. Thus, the stability analysis result is based on the following unforced system xðk þ 1Þ ¼ AxðkÞ þ A 1 xðk dðkþþ xðkþ ¼ fðkþ for k ¼ d max ; d max þ 1;... ; This criterion can be stated in the following form. Theorem 1: The unforced system given in () with timevarying delay is asymptotically stable if there exist n n matrices., Q., R, S and T satisfying the following LMI G R þ S T R þ T T ða þ A 1 Þ T Q S S T S T T T T T A T 1 where G ¼ þ (d max d min þ 1)Q þ R þ R T. ðþ, ðþ roof: Let y(k) ¼ x(k þ 1) x(k) and h(k) ¼ k1 m¼k d(k) y(m). Choose as a Lyapunov functional candidate where VðkÞ ¼ V 1 ðkþ þ V ðkþ þ V ðkþ V 1 ðkþ ¼ x T ðkþxðkþ V ðkþ ¼ V ðkþ ¼ i¼k dðkþ dx minþ1 j¼ d max þ i¼kþj 1 and and Q are positive definite matrices to be determined. Define DV ¼ V(k þ 1) V(k) and so on. Then, using (), we have DV 1 ¼ x T ðk þ 1Þxðk þ 1Þ x T ðkþxðkþ DV ¼ ¼ x T ðkþ½ða þ A 1 Þ T ða þ A 1 Þ Š xðkþ þ x T ðkþða þ A 1 Þ T A 1 yðmþ " # T " # þ A 1 yðmþ A 1 yðmþ ¼ x T ðkþ½ða þ A 1 Þ T ða þ A 1 Þ Š xðkþ x T ðkþ ða þ A 1 Þ T A 1 hðkþ þ hðkþ T A T 1 A 1hðkÞ X k i¼kþ1 dðkþ1þ i¼k dðkþ ¼ x T ðkþqxðkþ x T ðk dðkþþqxðk dðkþþ þ i¼k dðkþ1þþ1 i¼k dðkþþ1 ðþ ðþ ðþ IEE roc.-control Theory Appl., Vol. 1, No., November

3 As we have Note that DV ¼ i¼k dðkþ1þþ1 ¼ i¼k d min þ1 þ k d X min i¼k dðkþ1þþ1 i¼k dðkþþ1 þ k d X min i¼k d max þ1 DV x T ðkþqxðkþ x T ðk dðkþþqxðk dðkþþ þ dx minþ1 j¼ d max þ k d X min i¼k d max þ1 ðþ ð8þ ½x T ðkþqxðkþ x T ðk þ j 1ÞQxðk þ j 1ÞŠ ¼ ðd max d min Þx T ðkþqxðkþ In addition k d X min i¼k d max þ1 xðkþ xðk dðkþþ hðkþ ¼ ð9þ ð1þ Therefore for any appropriately dimensioned matrices R, S and T, we have the following equation ½x T ðkþr þ x T ðk dðkþþs þ hðkþ T TŠ ½xðkÞ xðk dðkþþ hðkþš ¼ ð11þ It follows by adding (), inequality (8), and (9) and (11) that DV ¼ DV 1 þ DV þ DV x T ðk dðkþþsxðk dðkþþ þ x T ðk dðkþþ½ S T T ŠhðkÞ hðkþ T ThðkÞ ¼ x T ðkþ½ða þ A 1 Þ T ða þ A 1 Þ þ ðd max d min þ 1ÞQ þ RŠ xðkþ þ x T ðkþ½ R þ S T Š xðk dðkþþ þ x T ðkþ ½ ða þ A 1 Þ T A 1 R þ T T Š hðkþ þ x T ðk dðkþþ½ Q SŠ xðk dðkþþ þ x T ðk dðkþþ½ S T T Š hðkþ þ hðkþ T ½A T 1 A 1 TŠ hðkþ ¼ j ðkþ T VjðkÞ ð1þ where we define j(k) T ¼ [x T (k), x T (k d(k)), h(k) T ] and V 11 R þ S T ða þ A 1 Þ T A 1 R þ T T V ¼ Q S S T S T T A T 1 A 1 T T T ð1þ where V 11 ¼ (A þ A 1 ) T (A þ A 1 ) þ (d max d min þ 1) Q þ R þ R T. From this, it follows that the inequality V, guarantees that DV, for all non-zero j(k). Hence, V, guarantees that the unforced system given in () is asymptotically stable for all time-varying delay d(k) satisfying d min d(k) d max. By Schur complement, V, is equivalent to LMI (). This completes the proof of Theorem 1. A Remark 1: Note that Theorem 1 only depends on the difference between the maximum and minimum delay bounds, that is it only depends on the delay interval and not on the actual delays themselves. Thus, Theorem 1 is not a delaydependent sufficient condition for asymptotic stability of the system given in (). Theorem 1 presents a stability result that depends on the difference between the maximum and minimum delay bounds. Thus, for the constant delay case, as minimum and maximum bounds in Assumption 1 are identical, d min ¼ d max ¼ d: Theorem 1 does not depend on the delay, which gives the following. x T ðkþ½ða þ A 1 Þ T ða þ A 1 Þ Š xðkþ x T ðkþða þ A 1 Þ T A 1 hðkþ þ hðkþ T A T 1 A 1hðkÞ þ x T ðkþqxðkþ x T ðk dðkþþqxðk dðkþþ þ k d X min i¼k d max þ1 k d X min i¼k d max þ1 þ ðd max d min Þx T ðkþqxðkþ þ x T ðkþrxðkþ þ x T ðkþ½ R þ S T Š xðk dðkþþ þ x T ðkþ½ R þ T T ŠhðkÞ Corollary 1: The unforced system given in () with constant delay d(k) ¼ d is asymptotically stable if there exist n n matrices., Q., R, S and T satisfying the following LMI þ Q þ R þ R T R þ S T Q S S T R þ T T ða þ A 1 Þ T S T T T T T A T 1, ð1þ IEE roc.-control Theory Appl., Vol. 1, No., November 91

4 Further asymptotic stability analysis From the proof of Theorem 1, it can be seen that the freely choosable matrices R, S and T only depend on the relationship between two system states x(k) and x(k d(k)), and they do not depend on the relationship between x(k) and x( j), j ¼ k d(k) þ 1, k d(k) þ,..., k 1. In order to further reduce the conservatism of Theorem 1, the relationship between state x(k) and any state x( j), j ¼ k d(k),..., k 1, should be considered. As y(k) ¼ x(k) x(k 1), the relationship between state x(k) and any state x( j), j ¼ k d(k),..., k 1, can be analysed through the relation between x(k) and y( j), j ¼ k d(k),..., k 1. In this section, the freely choosable matrices are selected to reflect these relations, and an improvement of Theorem 1 is obtained. Theorem : The unforced system with time-varying delay given in () is asymptotically stable if there exist n n matrices., Q.,., Y and W satisfying the following LMI ^V 11 ða þ A 1 Þ T A 1 Y þ W T W W T Q A T A 1 Y d max ða IÞ T A T 1 A 1 W d max A T 1, ð1þ where Vˆ 11 ¼ A T (A þ A 1 ) þ (A þ A 1 ) T A þ Y þ Y T T þ (d max d min þ 1)Q. roof: Let y(k) ¼ x(k þ 1) x(k). Then, x(k d(k)) ¼ x(k) k1 m¼kd(k) y(m). Choose as a Lyapunov functional candidate where VðkÞ ¼ V 1 ðkþ þ V ðkþ þ V ðkþ þ V ðkþ V 1 ðkþ ¼ x T ðkþxðkþ V ðkþ ¼ V ðkþ ¼ i¼k dðkþ dx minþ1 j¼ d max þ i¼kþj 1 ð1þ ¼ ½xðkÞ T A T þ xðk dðkþþ T A T 1 Š½ðA þ A 1ÞxðkÞ A 1 yðmþš x T ðkþxðkþ ¼ xðkþ T A T ða þ A 1 ÞxðkÞ xðkþ T A T A 1 þ xðk dðkþþ T A T 1 ða þ A 1ÞxðkÞ xðk dðkþþ T A T 1 A 1 yðmþ þ xðkþ T Y½xðkÞ xðk dðkþþš þ xðk dðkþþ T W ½xðkÞ xðk dðkþþš xðkþ T Y yðmþ xðk dðkþþ T W yðmþ x T ðkþ xðkþ yðmþ ¼ xðkþ T ½A T ða þ A 1 Þ þ YŠ xðkþ þ xðkþ T ½ðA þ A 1 Þ T A 1 Y þ W T Š xðk dðkþþ þ xðkþ T ½ A T A 1 YŠ xðk dðkþþ T Wxðk dðkþþ þ xðk dðkþþ T ½ W A T 1 A 1Š ¼ 1 dðkþ yðmþ x T ðkþxðkþ fxðkþ T ½A T ða þ A 1 Þ yðmþ þ Y Š xðkþ þ xðkþ T ½ðA þ A 1 Þ T A 1 Y þ W T Š xðk dðkþþ þ xðkþ T ½ A T A 1 YŠdðkÞyðmÞ xðk dðkþþ T Wxðk dðkþþ þ xðk dðkþþ T ½ W A T 1 A 1ŠdðkÞyðmÞg ð1þ V ðkþ ¼ X 1 j¼ d max m¼kþj y T ðmþyðmþ and, Q and are positive definite matrices to be determined. roceeding as before, defining DV ¼ V(k þ 1) V(k) ¼ DV 1 þ DV þ DV þ DV, we have DV 1 ¼ x T ðk þ 1Þxðk þ 1Þ x T ðkþxðkþ ¼ ½AxðkÞ þ A 1 xðk dðkþþš T ½AxðkÞ þ A 1 xðk dðkþþš x T ðkþxðkþ Furthermore X 1 DV ¼ ½ y T ðkþyðkþ y T ðk þ iþyðk þ iþš i¼ d max ¼ d max y T ðkþyðkþ y T ðmþyðmþ m¼k d max d max y T ðkþyðkþ y T ðmþyðmþ ð18þ 9 IEE roc.-control Theory Appl., Vol. 1, No., November

5 Thus, adding inequalities (8), (9) and (18) gives DV þ DV þ DV ðd max d min þ 1Þx T ðkþqxðkþ x T ðk dðkþþqxðk dðkþþ þ d max y T ðkþyðkþ y T ðmþyðmþ ¼ ðd max d min þ 1Þx T ðkþqxðkþ x T ðk dðkþþqxðk dðkþþ þ d max ½x T ðk þ 1Þ x T ðkþš ½xðk þ 1Þ xðkþš ¼ ðd max d min þ 1Þx T ðkþqxðkþ x T ðk dðkþþqxðk dðkþþ þ d max ½AxðkÞ þ A 1 xðk dðkþþ xðkþš T ½AxðkÞ þ A 1 xðk dðkþþ xðkþš y T ðmþ yðmþ ¼ ðd max d min þ 1Þx T ðkþqxðkþ x T ðk dðkþþqxðk dðkþþ þ d max ½x T ðkþða IÞ T ða IÞxðkÞ þ x T ðkþða IÞ T A 1 xðk dðkþþ þ xðk dðkþþ T A T 1 A 1xðk dðkþþš y T ðmþyðmþ y T ðmþyðmþ ¼ x T ðkþ½ðd max d min þ 1ÞQ þ d max ða IÞ T ða IÞŠ xðkþ þ x T ðkþd max ða IÞ T A 1 xðk dðkþþ þ x T ðk dðkþþ½ Q þ d max A T 1 A 1Š xðk dðkþþ ¼ 1 dðkþ y T ðmþyðmþ fx T ðkþ½ðd max d min þ 1ÞQ þ d max ða IÞ T ða IÞŠ xðkþ þ x T ðkþd max ða IÞ T A 1 xðk dðkþþ þ x T ðk dðkþþ½ Q þ d max A T 1 A 1Š xðk dðkþþ dðkþy T ðmþyðmþg 1 dðkþ fx T ðkþ½ðd max d min þ 1ÞQ þ d max ða IÞ T ða IÞŠ xðkþ þ x T ðkþd max ða IÞ T A 1 xðk dðkþþ Therefore þ x T ðk dðkþþ½ Q þ d max A T 1 A 1Š xðk dðkþþ dðkþ d max y T ðmþdðkþyðmþg DV 1 dðkþ xðkþ T ½A T ða þ A 1 Þ þ Y Š xðkþ þ xðkþ T ½ðA þ A 1 Þ T A 1 Y þ W T Š xðk dðkþþ þ xðkþ T ½ A T A 1 YŠdðkÞyðmÞ xðk dðkþþ T Wxðk dðkþþ þ xðk dðkþþ T ½ W A T 1 A 1ŠdðkÞyðmÞ þ x T ðkþ ½ðd max d min þ 1ÞQ þ d max ða IÞ T ða IÞŠ xðkþ þ x T ðkþd max ða IÞ T A 1 xðk dðkþþ þ x T ðk dðkþþ½ Q þ d max A T 1 A 1Š xðk dðkþþ dðkþy T ðmþdðkþ yðmþ ¼ 1 dðkþ d max xðkþ T ½A T ða þ A 1 Þ þ Y ð19þ þ ðd max d min þ 1ÞQ þ d max ða IÞ T ða IÞŠ xðkþ þ xðkþ T ½ðA þ A 1 Þ T A 1 Y þ W T þ d max ða IÞ T A 1 Š xðk dðkþþ þ xðkþ T ½ A T A 1 YŠdðkÞyðmÞ þ xðk dðkþþ T ½ W Q þ d max A T 1 A 1Š xðk dðkþþ þ xðk dðkþþ T ½ W A T 1 A 1ŠdðkÞyðmÞ dðkþy T ðmþdðkþ yðmþ d max which we may write in the form DV 1 dðkþ jðk; mþ T Vjðk; mþ ðþ where j(k, m) T ¼ [x T (k), x T (k d(k)), d(k)y T (m)] and 1 V 11 ða þ A 1Þ T A 1 Y þ W T V ¼ þ d max ða IÞ T A 1 W W T Q þ d max A T 1 A 1 1 AT A 1 Y 1 AT 1 A 1 W ð1þ d max and V 11 ¼ A T (A þ A 1 ) þ Y þ Y T þ (d max d min þ 1) Q þ d max (A I) T (A I). Therefore the inequality V, guarantees that DV, for all non-zero j(k, m). V, guarantees that the unforced system given in () is asymptotically stable for all timevarying delays d(k) satisfying d min d(k) d max. IEE roc.-control Theory Appl., Vol. 1, No., November 9

6 Replacing by, the LMI V, is equivalent to V 11 ða þ A 1 Þ T A 1 Y þ W T þ d max ða IÞ T A 1 W W T Q þ d max A T 1 A 1 A T A 1 Y A T 1 A 1 W d max, ðþ where V 11 ¼ A T (A þ A 1 ) þ (A þ A 1 ) T A þ Y þ Y T T þ (d max d min þ 1)Q þ d max (A I) T (A I). By Schur complement, inequality () becomes ^V 11 ða þ A 1 Þ T A 1 Y þ W T W W T Q A T A 1 Y ða IÞ T A T 1 A 1 W A T 1 1, ðþ d max 1 d max Replacing by d max, inequality () is equivalent to ^V 11 ða þ A 1 Þ T A 1 Y þ W T W W T Q A T A 1 Y d max ða IÞ T A T 1 A 1 W d max A T 1, ðþ where Vˆ 11 ¼ A T (A þ A 1 ) þ (A þ A 1 ) T A þ Y þ Y T T þ (d max d min þ 1)Q. This completes the proof of Theorem. A Multiplying both sides of inequality () by diag(i, I, d max I, d max I) leads to the following. Corollary : The unforced system given in () with timevarying delay is asymptotically stable if there exist n n matrices., Q.,., Y and W satisfying the following LMI 9 ^V 11 ða þ A 1 Þ T A 1 Y þ W T W W T Q d max A T A 1 d max Y d max ða IÞ T d max A T 1 A 1 d max W d max A T 1 d max, d max where Vˆ 11 ¼ A T (A þ A 1 ) þ (A þ A 1 ) T A þ Y þ Y T T þ (d max d min þ 1)Q. Remark : As Theorem 1 only depends on the delay interval, it is not a delay-dependent sufficient criterion. However, Theorem depends not only on the delay interval but also directly on the maximum delay bounds, so it is a delay-dependent sufficient criterion. Delay-dependent sufficient criteria generally tend to be less conservative than delay-independent criteria, especially when the delay is small, as they depend on more specific information. Remark : In the proof of Theorem 1, when we calculate the forward difference of the Lyapunov functional, the freely choosable matrices only reflect the relationship between two system states x(k) and x(k d(k)). Theorem 1 does not make use of the relationships between x(k) and x(j), j ¼ k d(k) þ 1,..., k 1. However, Theorem takes these relations between x(k) and x( j), j ¼ k d(k) þ 1,..., k 1, into account by selecting slack matrices to reflect the associated correlations between x(k) and y( j), j ¼ k d(k) þ 1,..., k 1. Remark : The Lyapunov functional in Theorem only contains V (k) in addition to the Lyapunov functional in Theorem 1. As the freely choosable matrices introduced in these two theorems do not change the values of the forward differences of these two Lyapunov functionals, comparison of these two forward differences shows that Theorem is less conservative than Theorem 1. In fact, from inequalities (1) and (), we have jðkþ T VjðkÞ þ d max y T ðkþyðkþ dðkþ ¼ 1 dðkþ jðk; mþ T Vjðk; mþ d max y T ðmþyðmþ Noting that the positive definite matrix in Theorem can be freely chosen, if V,, we can choose the positive definite matrix such that the spectral radius r() is sufficiently small and thus V, implies V,. In contrast, if is a positive definite matrix such that d max y T ðkþyðkþ dðkþ d max y T ðmþyðmþ, then V, cannot guarantee that V,. Therefore Theorem is less conservative than Theorem 1. As LMI (1) in Theorem contains A T (A þ A 1 ) and (A þ A 1 ) T A 1, LMI (1) can never be a LMI when system matrices A and A 1 contain norm-bounded parameter uncertainties. This makes it difficult to directly deal with the robust stability of time-delay systems. In order to simplify the robust stability analysis of uncertain systems with time-varying delay, we thus give Theorem. Theorem : The unforced system with time-varying delay given in () is asymptotically stable if there exist n n matrices., Q.,., Y and W satisfying IEE roc.-control Theory Appl., Vol. 1, No., November

7 the following LMI ~V Y þ W T ða þ A 1 Þ T W W T Q A T 1 A T Y d max ða IÞ T W d max A T 1 A 1, ðþ where Ṽ ¼ Y þ Y T T þ (d max d min þ 1)Q. roof: Given that Vˆ 11 ¼ A T A þ (A þ A 1 ) T (A þ A 1 ) A 1 T A 1 þ Y þ Y T T þ (d max d min þ 1)Q, by Schur complement, inequality () is equivalent to A T A A T 1 A 1 þ V ~ Y þ W T W W T Q A T 1 A 1 ða þ A 1 Þ T A T A 1 Y d max ða IÞ T A T 1 AT 1 A 1 W d max A T 1, Applying Schur complement again, we obtain A T 1 A 1 þ V ~ Y þ W T ða þ A 1 Þ T W W T Q A T 1 A 1 A T 1 A T Y d max ða IÞ T A T 1 A 1 W d max A T 1, ðþ A 1 A T 1 A 1 where Ṽ ¼ Y þ Y T T þ (d max d min þ 1)Q. Inequality () can be rewritten in the following form ~V Y þ W T ða þ A 1 Þ T A T W W T Q A T 1 Y d max ða IÞ T W d max A T 1 A 1 A 1 A 1 A 1 I A 1 A 1 A 1, T I I where Ṽ ¼ Y þ Y T T þ (d max d min þ 1)Q. As diag(,, I, I,, I) is a positive definite matrix, this inequality holds, providing that the following inequality holds ~V Y þ W T ða þ A 1 Þ T W W T Q A T 1 A T Y d max ða IÞ T W d max A T 1, A 1 This completes the proof of Theorem. The proof of Theorem makes it clear that Theorem is more conservative than Theorem, but it is very useful in practice. As an application of Theorem, we will directly prove the new asymptotic stability criterion given in A IEE roc.-control Theory Appl., Vol. 1, No., November 9

8 Theorem. Let X 11 X 1 X 1 X 1 X X ¼ X X X X X and T ¼ Y T ; W T ; ; A T 1 T Theorem : The unforced system with time-varying delay given in () is asymptotically stable if there exist n n matrices., Q.,., X, Y and W satisfying the following LMIs ~V þ X 11 Y þ W T þ X 1 ða þ A 1 Þ T þ X 1 W W T Q þ X A T 1 þ X and þ X A T þ X 1 d max ða IÞ T X d max A T 1 X, ðþ þ X X 11 X 1 X 1 X 1 Y X X X W X X X A 1 ð8þ where Ṽ ¼ Y þ Y T T þ (d max d min þ 1)Q. roof: As is a positive definite matrix, using Schur complement, inequality (8) is equivalent to X 11 X 1 X 1 X 1 X X X X X X Y W 1 Y T W T A T 1 A 1 or X 11 X 1 X 1 X 1 X X X X X 9 X Y W 1 Y T W T A T 1 A 1 that is Furthermore X T 1 T T X T 1 T T so ~V þ X 11 Y þ W T þ X 1 ða þ A 1 Þ T þ X 1 W W T Q þ X A T 1 þ X þ X A T þ X 1 d max ða IÞ T X d max A T 1 X þ X " # X T 1 T T, After a simple computation, using Schur complement, we obtain ~V Y þ W T ða þ A 1 Þ T W W T Q A T 1 A T Y d max ða IÞ T W d max A T 1 A 1. where V ¼ Y þ Y T T þ (d max d min þ 1)Q. Using Theorem completes the proof of Theorem. A Systems with polytopic uncertainties Given a system with uncertainty, in the simple case where the uncertainty is represented by ranges of parameters, this leads to a polytopic region in the parameter space representing the system. Thus, various previous work has focused on time-delay systems with polytopic-type uncertainties []. In this section, we investigate the robust stability of discrete-time system with polytopic-type uncertainties. Consider a linear system with a time-varying delay xðk þ 1Þ ¼ AxðkÞ þ A 1 xðk dðkþþ xðkþ ¼ fðkþ; k ¼ d max ; d max þ 1;... ; ð9þ where x(k) [ R n is the state vector. Suppose the matrices A and A 1 are subject to uncertainties such that they satisfy the IEE roc.-control Theory Appl., Vol. 1, No., November

9 following real convex polytopic model ( X n A A 1 [ AðbÞ A1 ðbþ ¼ X n j¼1 b j ¼ 1; b j ) j¼1 b j ~A j ~A 1j ; where ~A j and ~A 1j, j ¼ 1,..., n, are constant matrices with appropriate dimensions and b j, j ¼ 1,..., n the time-invariant uncertainties and the time delay d(k) is a time-varying discrete function satisfying d min d(k) d max. Theorem 1 can readily be extended to such a system with polytopic uncertainties as follows. Theorem : The time-varying delay linear system with polytopic uncertainties given in (9) is robustly stable if there exist n n matrices., Q., R, S and T satisfying the following LMI G R þ S T R þ T T ð A ~ j þ A ~ 1j Þ T Q S S T S T T T T T ~A T 1j, ðþ where G ¼ þ (d max d min þ 1)Q þ R þ R T and j ¼ 1,..., n. roof: Let y(k) ¼ x(kþ1)x(k) and h(k) ¼ k1 m¼k d(k) y(m). Choose as a Lyapunov functional candidate where VðkÞ ¼ V 1 ðkþ þ V ðkþ þ V ðkþ V 1 ðkþ ¼ x T ðkþxðkþ V ðkþ ¼ V ðkþ ¼ i¼k dðkþ dx minþ1 j¼ d max þ i¼kþj 1 ð1þ and Q are positive definite matrices to be determined. Noting that n j¼1 b j ¼ 1, the proof of Theorem is easily completed along similar lines to the proof of Theorem 1. A Static output-feedback stabilisation The static output-feedback stabilisation problem is to design a controller uðkþ ¼ KyðkÞ ðþ where K is an appropriately dimensioned matrix to be determined, y(k) the measured output, u(k) the controlled input and k an integer representing discrete time. Taking this controller and the time-varying delay system given in (1), the following closed-loop system is obtained xðk þ 1Þ ¼ AxðkÞ þ A 1 xðk dðkþþ where xðkþ ¼ fðkþ; k ¼ d max ; d max þ 1;... ; ðþ A ¼ A þ BKC; A1 ¼ A 1 þ BKC 1 ðþ The purpose of this section is to determine the matrix gain K such that the closed-loop system given in () is asymptotically stable. The following theorem presents a sufficient condition for the existence of such a controller. Theorem : For the system given in (1) with Assumption 1, a stabilising static output-feedback controller u(k) ¼ Ky(k) exists if there exist n n matrices., Q., L., R, S and T satisfying the following matrix inequality G R þ S T R þ T T Q S S T S T T T T T with the side condition ða þ BKC þ A 1 þ BKC 1 Þ T ða 1 þ BKC 1 Þ T L L ¼ I where G ¼ þ (d max d min þ 1)Q þ R þ R T., ðþ ðþ roof: From Theorem 1, the closed-loop system given in () is asymptotically stable if there exist n n matrices., Q., R, S and T satisfying the following matrix inequality G R þ S T R þ T T ð A þ A 1 Þ T Q S S T S T T T T T A T 1, ðþ where G ¼ þ (d max d min þ 1)Q þ R þ R T. A congruence transformation of inequality () by diag(i, I, I, 1 ) together with the substitution of the matrices defined in () leads to G R þ S T R þ T T Q S S T S T T T T T ða þ BKC þ A 1 þ BKC 1 Þ T ða 1 þ BKC 1 Þ T 1, ð8þ Defining L ¼ 1 this completes the proof of Theorem. A If C ¼ I and C 1 ¼ in the system given in (1), the static output-feedback control law u(k) ¼ Ky(k) reduces to the well-known state feedback control law u(k) ¼ Kx(k). In this case, Theorem can be directly used to solve the state feedback control problem by simply setting C ¼ I and C 1 ¼. It should be noted that the condition obtained in Theorem is not a strict LMI condition because of the side condition in (). However, we can solve this non-convex feasibility IEE roc.-control Theory Appl., Vol. 1, No., November 9

10 problem by formulating it as a sequential optimisation problem subject to LMI constraints, using the cone complementarity approach [1]. This gives the following nonlinear minimization problem involving LMI conditions instead of the original non-convex feasibility problem formulated in Theorem. roblem 1 (Static output-feedback stabilisations): Find matrices., Q., L., R, S, T, K such that the following nonlinear minimisation problem can be solved MinimisejtrðLÞ nj Subject to G R þ S T R þ T T Q S S T S T T T T T and ða þ BKC þ A 1 þ BKC 1 Þ T ða 1 þ BKC 1 Þ T I L I L, where G ¼ þ (d max d min þ 1)Q þ R þ R T. ð9þ Trying to minimise jtr(l) nj is easier than trying to directly solve the original non-convex feasibility problem. However, it is not always possible to find a globally optimal solution. Instead, we aim to find a feasible solution satisfying inequalities () and (9) under the condition jtr(l) nj, d for some sufficiently small scalar d, rather than trying to find a solution such that tr(l) is exactly equal to n. Using ideas of El Ghaoui et al. [1], we proceed as follows to find the gain matrix K of the static output-feedback stabilising controller, using the following algorithm. Algorithm 1 (Static output-feedback stabilisation): Step 1. Compute an initial feasible set (, Q, L, R, S, T ) satisfying () and (9). Set k ¼. Step. Solve the following LMI problem: find ( s, Q s, L s, R s, S s, T s, K), which minimise tr(l k þ k L) subject to () and (9). Step. If ~G R s þ S T s R s þ T T s Q s S s S T s S s T T s T s T T s ða þ BKC þ A 1 þ BKC 1 Þ T ða 1 þ BKC 1 Þ T, 1 s T where G ¼ s þ (d max d min þ 1)Q s þ R s þ R s and jtr( s L s ) nj, d for some sufficiently small scalar d, then output the feasible solution (, Q, L, R, S, T) ¼ ( s, Q s, L s, R s, S s, T s ) and stop. 98 Step. If k. N where N is the maximum number of iterations allowed, give up and stop. Step. Set k ¼ k þ 1, ( k, Q k, L k, R k, S k, T k ) ¼ ( s, Q s, L s, R s, S s, T s ) and go to Step. The convergence of this algorithm has been investigated by El Ghaoui et al. [1]. Using a large number of examples, they have shown that this algorithm generally performs well, although it may fail to find a solution in some particular cases. Following similar lines to the proof of Theorem, we can now deal with the static output-feedback stabilisation problem using Theorem. The following result gives a new sufficient condition for the existence of static outputfeedback controller such that the closed-loop system given in () is asymptotically stable. Theorem : For the system given in (1) with Assumption 1, a stabilising static output-feedback controller u(k) ¼ Ky(k) exists if there exist n n matrices., Q.,., L., M., Y and W satisfying the following conditions ~V Y þw T ðaþa 1 þbkðc þc 1 ÞÞ T W W T Q ða 1 þbkc 1 Þ T L ðaþbkcþ T Y d max ðaþbkc IÞ T W d max ða 1 þbkc 1 Þ T L ða 1 þbkc 1 Þ M L ¼ I; M ¼ I, where Ṽ ¼ Y þ Y T T þ (d max d min þ 1)Q. The condition in Theorem is again not a strict LMI condition, as L ¼ I and M ¼ I. By following similar lines to those used in Algorithm 1, or the SOFS algorithm [1], the non-convex feasibility problem in Theorem can be again solved by formulating it as a sequential optimisation problem. Dynamic output-feedback stabilisation In this section, the dynamic output-feedback case is considered. A static output-feedback controller only uses the current output as input. However, the input to a dynamic output-feedback controller takes the output history into account. Our aim is to obtain a dynamic output-feedback control law C d ^xðk þ 1Þ ¼ Ac ^xðkþ þ BcyðkÞ uðkþ ¼ C c ^xðkþ þ D c yðkþ ^xðkþ ¼ ; k ðþ where ^x(k) [ R r represents the internal state of the dynamic output-feedback controller and A c, B c, C c and D c are appropriately dimensioned dynamic output-feedback controller matrices to be determined. Note that here we are interested not only in the full-order compensation problem (when IEE roc.-control Theory Appl., Vol. 1, No., November

11 r ¼ n, the dimension of the state vector x(k)), but also in the reduced-order compensation problem (when 1, r, n). By defining j T (k) ¼ [x T (k), ^x T (k)], the closed-loop system for the system given in (1) can be written as jðk þ 1Þ ¼ AjðkÞ þ A 1 jðk dðkþþ jðkþ ¼ ½fðkÞ T ; Š T ; k ¼ d max ; d max þ 1;... ; ð1þ where A ¼ A þ BD cc BC c ; A 1 ¼ A 1 þ BD c C 1 B c C A c B c C 1 ðþ Our goal is to determine matrices A c, B c, C c and D c for the dynamic output-feedback controller () such that the closed-loop system given in (1) is asymptotically stable. Theorem 8: Consider the system given in (1) with Assumption 1. The closed-loop system given in (1) is asymptotically stable if there exist (n þ r) (n þ r) matrices., Q., R, S and T satisfying G R þ S T R þ T T ð A þ A 1 Þ T Q S S T S T T T T T A T 1, where Ḡ ¼ þ (d max d min þ 1)Q þ R þ R T. ðþ The proof of Theorem 8 follows similar lines to the proof of Theorem 1. The following theorem presents the corresponding solution to the dynamic output-feedback problem. Theorem 9: Consider the system given in (1) with Assumption 1. A stabilising dynamic output-feedback controller () exists if there exist (n þ r) (n þ r) matrices., L., Q., R, S and T satisfying G R þ S T R þ T T ð A þ A 1 Þ T Q S S T S T T T T T A T 1 L, ðþ and L ¼ I, where Ḡ ¼ þ (d max d min þ 1) Q þ R þ R T and A ¼ A þ BD cc BC c ; A1 ¼ A 1 þ BD c C 1 B c C A c B c C 1 The proof of Theorem 9 follows similar lines to the proof of Theorem. As in the static output-feedback case, the conditions in Theorem 9 are again not strict LMI conditions. By following a similar approach to that used in the previous section, this non-convex feasibility problem can be solved by formulating it as a sequential optimisation problem subject to LMI constraints expressed as follows. roblem (Dynamic output-feedback stabilisation): Minimise jtr(l) n rj subject to inequality () and I I L A solution to this minimisation problem that satisfies the conditions in Theorem 9 can easily be found by suitably modifying Algorithm 1. 8 Norm-bounded uncertain systems The analysis and synthesis results developed in the earlier sections can be further extended to cope with uncertain systems in which the system matrices in (1) have normbounded uncertainty meeting the following assumption. Assumption : Assume that the matrices A, A 1 and B in the system given in (1) have the following forms A ¼ A þ D A; A 1 ¼ A 1 þ D A 1 ; B ¼ B þ D B where A, A 1 and B are known constant matrices of appropriate dimensions and DA, DA 1, and DB are real-valued time-varying matrix functions representing norm-bounded parameter uncertainties satisfying D A; D A 1 ; D B ¼ HFðkÞ E1 ; E ; E ðþ where F(k) is a real matrix function representing uncertainty satisfying F T ðkþfðkþ I ðþ and H, E 1, E and E are known real constant matrices of appropriate dimensions. These matrices specify how the uncertain parameters in F(k) enter the nominal matrices A, A 1 and B The following lemma can be found in Xie [1]. Lemma : Given matrices Q ¼ Q T, H, E and R ¼ R T. of appropriate dimensions, Q þ HFE þ E T F T H T, for all F satisfying F T F R, if and only if there exists some l. such that Q þ lhh T þ l 1 E T RE, Using Lemma, we now prove the following theorem. Theorem 1: Given an unforced system as in () with Assumptions 1 and, this system is robustly asymptotically stable if there exist n n matrices., Q., R, S and T satisfying the following LMI N R þ S T R þ T T ðe 1 þ E Þ T E Q S S T S T T T T T þ E T E ða þ A 1 Þ T A T 1, ðþ H where N ¼ þ (d max d min þ 1)Q þ R þ R T þ (E 1 þ E ) T (E 1 þ E ). roof: Robustly asymptotic stability for the system in (1) under Assumptions 1 and follows from LMI (). Replace A and A 1 by A þ DA and A 1 þ DA 1 respectively, I IEE roc.-control Theory Appl., Vol. 1, No., November 99

12 in inequality () of Theorem 1 and substitute (), which gives G R þ S T R þ T T ða þ A 1 Þ T Q S S T S T T T T T A T 1 E T 1 þ ET þ FT ðkþ½ H T Š þ E T FðkÞ ½E 1 þ E E Š, ð8þ H By Lemma, there exists some l. for inequality (8) such that G R þ S T R þ T T ða þ A 1 Þ T Q S S T S T T T T T A T 1 E T 1 þ ET þ l 1 ½E 1 þ E E Š þ l H E T ½ HT Š. ð9þ Multiplying both sides of inequality (9) by l and replacing l, lq, lr, ls, and lt by, Q, R, S and T, respectively gives G R þ S T R þ T T ða þ A 1 Þ T Q S S T S T T T T T A T 1 E T 1 þ ET þ ½E 1 þ E E Š þ E T ½ HT Š. ðþ H Using Schur complement, inequality () is obtained. Applying Theorem 1, static output-feedback stabilisation can be further extended to discrete time-delay systems with norm-bounded uncertain parameters. A Theorem 11: Consider the system given in (1) with Assumptions 1 and. A robustly stabilising static outputfeedback controller in the form of () exists if there exist n n matrices., L., Q., R, S and T satisfying L ¼ I and the following LMI G R þ S T R þ T T Q S S T S T T T T T ða þ A 1 þ B KðC þ C 1 ÞÞ T ða 1 þ B KC 1 Þ T L H I ðe 1 þ E þ E KðC þ C 1 ÞÞ T ðe þ E KC 1 Þ T I where G ¼ þ (d max d min þ 1)Q þ R þ R T., ð1þ roof: By replacing A, A 1, E 1 and E by A þ B KC, A 1 þ B KC 1, E 1 þ E KC and E þ E KC 1 in inequality () of Theorem 1 and applying Schur complement, Theorem 11 can easily be proved using a congruence transformation of diag(i, I, I, 1, I, I). A Similarly, dynamic output-feedback stabilisation can also be further extended to discrete time-delay systems with norm-bounded uncertain parameters. Theorem 1: Consider the system given in (1) with Assumptions 1 and. A robustly stabilising dynamic output-feedback controller in the form of () exists if there exist (n þ r) (n þ r) matrices., L., Q., R, S and T satisfying L ¼ I and G R þ S T R þ T T Q S S T S T T T T T ða þ A 1 Þ T G 1 þ G A T 1 G L H I I, ðþ IEE roc.-control Theory Appl., Vol. 1, No., November

13 where G ¼ þ (d max d min þ 1)Q þ R þ R T A ¼ A þ B D c C B C c ; A1 ¼ A 1 þ B D c C 1 B c C A c B c C 1 " # " # G 1 ¼ ET 1 þ CT D T c ET ; G ¼ ET þ CT 1 DT c ET ðþ Similar theorems to Theorems and and Theorems 8 to 1 can be obtained using Theorem instead of Theorem 1. For brevity, these theorems are omitted. Remark : Gao and Wang [1] proposed an important delay-dependent approach to robust H 1 filtering for uncertain discrete-time state-delayed systems. Both full-order and reduced-order filters guaranteeing a prescribed noise attenuation level for all possible delays and parameter uncertainties are given. Gao et al. [18] investigated the problem of H 1 model reduction for linear discrete-time state-delay systems. For a given stable system, with delay-independent and dependent approaches, they constructed reduced-order models, which guarantee the corresponding error system to be asymptotically stable and have a prescribed H 1 error performance. However, their delay-dependent approaches to robust H 1 filtering and H 1 model reduction are both based on the inequality in Lemma 1, and so the relationships between states at differing times are not taken into account. Our methods can also be used to produce less conservative results for robust H 1 filtering and model reduction. 9 Numerical examples As demonstrations of the effectiveness of our methods, we briefly consider the following three simple examples. Example 1 demonstrates the effectiveness of the asymptotic stability criteria in Theorems 1 and in that they may readily be used to find a solution to the problem posed. It also demonstrates that Theorem is much less conservative than Theorem 1. Example 1: Consider the following discrete-time system with time-varying delay in the state xðk þ 1Þ ¼ : xðkþ þ :1 xðk dðkþþ : : : :1 where d(k) represents the time-varying state delay. Assume that the minimum delay bound of d(k) is d min ¼. Using Theorem 1 to evaluate this system, we find that it is asymptotically stable for a maximum delay bound d max ¼ 1. When d(k) ¼ 1, then 1 1: :9 ¼ Q ¼ 1 1 :9 : 1:1 : : :889 R ¼ S ¼ : 91: :889 9:81 191: :8 T ¼ :8 9: Using the approach in Theorem to consider the same problem, we find that for asymptotic stability of the same system, the maximum allowable delay bound is d max ¼ 1. We can still find feasible solutions for LMI (1) when d max ¼ 1 8: : 189:9 :89 ¼ Q ¼ : : :89 : 1: 1:9 19:8 :8 ¼ Y ¼ 1:9 :19 :8 :1 : :1 W ¼ :1 :19 Example demonstrates that our method is less conservative than existing methods. Example : Consider the following discrete-time system with a time-varying delay in the state, originally given by Gao et al. [1] xðk þ 1Þ ¼ :8 : :9 xðkþ þ :1 : :1 xðk dðkþþ We assume that the minimum delay bound of d(k) to be d min ¼, as in Gao et al. [1]. The maximum delay bound such that this above system is asymptotically stable is found, using the method of Gao et al. [1], to be d max ¼. However, the maximum bound obtained using Theorem is d max ¼ 1; the maximum bound obtained using Theorem is also d max ¼ 1. Both bounds are a significant improvement over those achieved by Gao et al. [1]. Remark : It should be noted that the maximum allowable delay bounds obtained in these two examples are conservative, as the sufficient asymptotic stability conditions in this paper still are conservative. Some conservatism of these sufficient conditions results from replacing d max by d(k) [see, e.g. the last inequality of inequality (18)] or from replacing d(k) with d max [see, e.g. inequality (19)] when we compute the forward difference of the Lyapunov functional. If the system itself is very robust to time-varying delay, this will cause more conservatism when replacing d(k) with a very large value d max in inequality (19). To illustrate the effectiveness of the proposed methods for controller design, we consider the stabilisation of the following system when d max. 1. Example : Consider the following discrete-time system with a time-varying delay in the state xðk þ 1Þ ¼ :8 xðkþ þ :1 : :9 : :1 xðk dðkþþ þ 1 uðkþ : yðkþ ¼ 1 1 xðkþ þ 1 xðk dðkþþ Assume that minimum delay bound is d min ¼. Theorems and cannot ascertain that this system with u(k) ¼ is asymptotically stable for d max ¼ 11: the maximum delay bound of 1 is obtained using either Theorem or Theorem for the system in Example to be asymptotically stable. Here, our purpose is to design a static output-feedback controller u(k) ¼ Ky(k) such that the resulting closed-loop system is asymptotically stable. Using Theorem and SOFS algorithm [1], it is found that this system is stabilisable for all 11 d max 1. When d max ¼ 1, a stabilising controller gain is given by K ¼ [.19,.18]. IEE roc.-control Theory Appl., Vol. 1, No., November 1

14 1 Conclusion The problem of output-feedback stabilisation for discretetime systems with time-varying state delay was investigated in this paper. A new stability condition, dependent on the minimum and maximum delay bounds, was given. New techniques have been developed to eliminate the terms containing the product of the Lyapunov matrices and system matrices in the difference of the Lyapunov functionals. In addition, the problems of robust asymptotic stability and the problems of stabilisation by static and dynamic output-feedback controllers have been solved using LMIs. As the conditions obtained for the existence of admissible controllers are not strict LMI conditions, the cone complementary method has been exploited to solve the nonconvex feasibility problem. Numerical examples have been given to demonstrate the validity and effectiveness of our approach. 11 Acknowledgments The authors would like to thank the reviewers for their valuable comments and constructive suggestions. This work is supported by the National Natural Science Foundation of China under grant no. 1 and grant no. 1, and the Teaching and Research Award rogram for Outstanding Young Teachers in Higher Education Institutions of MOE, eople s Republic of China (TRAOYT). 1 References 1 Li, X., and De Souza, C.E.: Criteria for robust stability and stabilization of uncertain linear systems with state-delay, Automatica, 199,, pp. 1 1 Cao, Y.Y., Sun, Y.X., and Lam, J.: Delay-dependent robust H 1 control for uncertain systems with time-varying delays, IEE roc. Control Theory Appl., 1998, 1, (), pp. 8 Chen, W.H., Guan,.H., and Lu, X.: Delay-dependent guaranteed cost control for uncertain discrete-time systems with delay, IEE roc. Control Theory Appl.,, 1, (), pp. 1 1 Gao, H., and Wang, C.: Robust L L 1 filtering for uncertain systems with multiple time-varying state delays, IEEE Trans. Circuits Syst.,,, (), pp He, Y., Wu, M., She, J.H., and Liu, G..: Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays, Syst. Control Lett.,, 1, (1), pp. He, Y., Wu, M., She, J.H., and Liu, G..: arameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties, IEEE Trans. Autom. Control,, 9, (), pp Kharitonov, V., and Melhor, A.D.: On delay-dependent stability conditions, Syst. Control Lett.,,, (1), pp. 1 8 Moon, Y.S., ark,., Kwon, W.H., and Lee, Y.S.: Delay-dependent robust stabilization of uncertain state delayed systems, Int. J. Control, 1,, (1), pp Niculescu, S.I., Neto, A.T., Dion, J.M., and Dugard, L.: Delaydependent stability of linear system with delayed state: an LMI approach. roc. th IEEE Conf. on Decision Control, 199, pp ark,.: A delay-dependent stability criterion for systems with uncertain time-invariant delays, IEEE Trans. Autom. Control, 1999,, (), pp Wu, M., He, Y., She, J.H., and Liu, G..: Delay-dependent criteria for robust stability of time-varying delay systems, Automatica,,, pp Xie, L., and De Souza, C.E.: Delay-dependent robust stability and stabilization of uncertain linear delay systems: a linear matrix inequality approach, IEEE Trans. Autom. Control, 199,, (8), pp Wang,., and Huang, B.: Robust H 1 observer design of linear state delayed system with parametric uncertainty: the discrete time case, Automatica, 1999,, pp Gao, H., Lam, J., Wang, C., and Wang, Y.: Delay-dependent outputfeedback stabilisation of discrete-time systems with time-varying state delay, IEE roc. Control Theory Appl.,, 11, (), pp El Ghaoui, L.E., Oustery, F., and Ait Rami, M.: A cone complementarity linearization algorithm for static output feedback and related problems, IEEE Trans. Autom. Control, 199,, (8), pp Xie, L.: Output feedback H 1 control of system with parameter uncertainty, Int. J. Control, 199,, (), pp. 1 1 Gao, H., and Wang, C.: A delay-dependent approach to robust H 1 filtering for uncertain discrete-time state-delayed systems, IEEE Trans. Signal rocess.,,, (), pp Gao, H., Lam, J., Wang, C., and Xu, S.: H 1 model reduction for discrete time-delay systems: delay-independent and dependent approaches, Int. J. Control,,, (), pp. 1 IEE roc.-control Theory Appl., Vol. 1, No., November