Stabilization analysis for discrete-time systems with time delay

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Abstract

The stabilization problem for a class of discrete-time systems with time-varying delay is investigated. By constructing an augmented Lyapunov function, some sufficient conditions guaranteeing exponential stabilization are established in forms of linear matrix inequality (LMI) technique. When norm-bounded parameter uncertainties appear in the delayed discrete-time system, a delay-dependent robust exponential stabilization criterion is also presented. All of the criteria obtained in this paper are strict linear matrix inequality conditions, which make the controller gain matrix can be solved directly. Three numerical examples are provided to demonstrate the effectiveness and improvement of the derived results.

Introduction

It is known that, because of the finite switching speed, memory effects and so on, time delay is frequently encountered in technology and nature. It extensively exists in various mechanical, biological, physical, chemical engineering, economic systems, and becomes an important source of instability and oscillation. This makes the design and hardware implementation of delayed system become difficult. Thus, the studies on stability for delayed system such as neural networks, switched system, bilinear system etc, are of great significance. There has been a growing research interest on the stability analysis problems for delayed system, and many excellent papers and monographs have been available (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]). Generally speaking, these so-far obtained stability results for delayed system can be mainly classified into two types: that is, delay-independent and delay-dependent. Since sufficiently considered the information of the size of time delay, delay-dependent criteria may be less conservative than delay-independent ones when time delay is small. For delay-dependent type, the size of allowable upper bound of time delay always be regarded as an important criterion to discriminate the quality between different results.

Recently, the problem of delay-dependent stabilization analysis for delayed system has received considerable attention, and many important results are obtained (see [13], [14], [15], [16], [17]). In [18], Gao et al. investigated the stabilization problem for a class of discrete-time system with time-varying state delay, and obtained some delay-dependent output-feedback stabilization criteria. For reducing conservatism of the criteria obtained in [18], Refs. [19], [20], [21] established some improved criteria by using different methods, respectively. As pointed out in [22] that the Lyapunov functionals constructed in [18], [19], [20], [21] are restrictive due to the ignorance of some important items. By constructing a new Lyapunov functional, Zhang et al. (see [22]) derived an improved stabilization criterion, which is less conservative than those obtained in [18], [19], [20], [21]. It should be pointed out that the controller design criteria established in [18], [22] are not strict LMI ones, which make the controller gain matrix K can not be solved directly. On the other hand, compared with traditional Lyapunov method, augmented Lyapunov technique can introduce free-weighting or semi-free-weighting matrix more effectively, this may further reduce the conservatism in above problem considered in [18], [19], [20], [21], [22].

Motivated by above discussions, the main aim of this paper is to study the exponential stabilization problem for a class of discrete-time system with time-varying delay. Combined with linear matrix inequality (LMI) technique, by constructing a new augmented Lyapunov functional, some sufficient conditions ensuring exponential stabilization are established. In addition, based on the proposed stabilization results, we also give a delay-dependent robust exponential stabilization criterion for the concerned system with norm-bounded parameter uncertainties. All of the criteria obtained in this paper are strict linear matrix inequality conditions, which make the controller gain matrix K can be solved directly. Finally, three numerical examples are provided to demonstrate the effectiveness and improvement of the derived results.

Notation

The notations are used in our paper except where otherwise specified. · denotes a vector or a matrix norm; R,Rn are real and n-dimension real number sets, respectively; N+ is positive integer set. I is identity matrix; ∗ represents the elements below the main diagonal of a symmetric block matrix; Real matrix P>0(<0) denotes P is a positive-definite (negative definite) matrix; N[a,b]={a,a+1,,b}; λmin(λmax) denotes the minimum (maximum) eigenvalue of a real matrix.

Section snippets

Preliminaries

Consider the following discrete-time system with time-varying delay described byx(k+1)=Ax(k)+Bx(k-τ(k))+Cu(k),kN+,y(k)=Dx(k),x(k)=φ(k),-τMk0,where x(k)=[x1(k),x2(k),,xn(k)]TRn denotes the state vector; u(k)=[u1(k),u2(k),,un(k)]TRm is the control input vector; y(k)=[y1(k),y2(k),,yn(k)]TRl is the control output vector; Positive integer τ(k) represents the transmission delay satisfying 0<τmτ(k)τM, where τm,τM are known positive integers representing the lower and upper bounds of the

Main results

In this section, by using augmented Lyapunov method and free-weighting matrix technique, we give some novel delay-dependent exponential stabilization conditions for system (3).

Theorem 3.1

For given positive integers τmτ(k)τM, system (1) is said to be exponentially stabilized by the local control law (2) with feedback gain matrix K, if there exist positive-definite matrices Q, R, H, W, positive-definite diagonal matrices Z1,Z2, arbitrary matrices P1,P2,G1,G2 of appropriate dimensions, such that the following

Numerical examples

In this section, three numerical examples are presented to show the validity and improvement of the main results derived above.

Example 1

Consider delayed discrete-time system in (3) with parameters given byC=0.8000.7,A=0.71.30.31.6,B=10.10.20.1,τm=1,τM=4.One can check that LMI (4) in Theorem 3.1, LMI (21) in Corollary 3.1 and LMI (28) in Corollary 3.2 are feasible. By Matlab LMI Toolbox, a feasible solution to LMI (4) is obtained as follows:Q11=1.1560-1.2322-1.23221.9847,Q12=-0.0133-0.0079-0.0130-0.0615,Q

Conclusion

By constructing an augmented Lyapunov–Krasovskii function, some new delay-dependent conditions ensuring exponential stabilization or robust exponential stabilization are obtained. Compared with some previous results established in the literature cited therein, the new criteria derived in this paper are less conservative. All of these new criteria are expressed in the forms of strict LMIs, which make the controller design become easy. Numerical examples show that the new results are valid.

Acknowledgements

This work was supported by the program for New Century Excellent Talents in University (NCET-06-0811), National Basic Research Program of China (2010CB732501), and National Natural Science Foundation of China (60702071).

References (25)

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