The Delay-Dependent for Discrete-Time Systems

Problem statement: This study is concerned with the problem of delaydependent stability analysis for discrete-time systems with interval-li ke time-varying delays. Conclusion/Recommendations: The problem is solved by applying a novel Lyapunov functional and an improved delay-dependent stability criterion is obtained in terms of a linear matrix inequality.


INTRODUCTION
Recently, the problem of delay-dependent stability analysis for time-delay systems has received considerable attention and lots of significant results have been reported; see, for example, Chen et al. (2003), Bellomo and Bellouquid, 2006;Brezzi et al., 2007;El-Said and EL-Sherbeny, 2005;Lekhmissi, 2006) and the references therein. Among these references, we note that the delay-dependent stability problem for discrete-time systems with interval-like time-varying delays (i.e., the delay d (k) satisfies 0<d m ≤d(k) ≤d M ) has been studied in (Alfaris et al., 2008;Bezzarga and Bucur, 2005;Brezzi et al., 2006), where some LMI-based stability criteria have been presented by constructing appropriate Lyapunov functionals and introducing free-weighting matrices. It should be pointed out that the Lyapunov functionals considered in these references are more restrictive due to the ignorance of the term Gao and Chen (2007) and Gao et al. (2004). The ignorance of these terms may lead to considerable conservativeness.
On the other hand, in the study of stabilization for the discrete-time linear systems, traditional idea of the control schemes is to construct a control signal according to the current system state (Bay and Phat, 2002;Callier and Desoer, 1991). However, as pointed out in (Waziri et al., 2010) in practice there is often a system that itself is not time-delayed but time-delayed may exist in a channel from system to controller. A typical example for the existence of such delays is the measurement and the network transmission of signals. In this case, a time-delayed controller is naturally taken into account. It is worth noting that the closed-loop system resulting from a delayed controller is actually a time-delay system. Therefore, stability results of time-delay systems could be applied to design timedelayed controller.
The present study a new Lyapunov functional, an improved delay-dependent stability criterion for discrete-time systems with time-varying delays is presented in terms of LMIs. It is shown that the obtained result is less conservative than those in. Example is providing, respectively, to demonstrate the reduced conservatism of the proposed stability criterion and the effectiveness of the proposed design method.
Preliminaries: Lemma 2.1 (Agarwal, 2000) the zero solution of difference system is asymptotic stability if there exists a positive definite function n V(x(k)) : along the solution of the system. In the case the above condition holds for all x (k)∈V δ , we say that the zero solution is locally asymptotically stable.

Improved stability criterion:
In this section, we give a novel delay-dependent stability condition for discretetime systems with interval-like time-varying delays. Now, consider the following system: Proof: Consider the Lyapunov function where, P,G being symmetric positive definite solutions of (2). Then difference of V(x(k)) along trajectory of solution of (1) is given by: [Ax(k) Bx(k h(k))] P[Ax(k) Bx(k h(k))] x (k)Px(k) x (k)[A PA P]x(k) x (k)A PBx(k h(k)) x (k h(k))B PAx(k) x (k h(k))B PBx(k h(k)) On the above estimation we use: h (k)≤h and : T T h(k)x (k)Gx(k) x (k)Gx(k) ≤ By the condition (2), ∆V(y(k)) is negative definite, namely there is a number β>0 such that 2 V(y(k)) y(k) , ∆ ≤ −β and hence, the asymptotic stability of the system immediately follows from Lemma 2.1. This completes the proof.

CONCLUSION
In this study, an improved delay-dependent stability condition for discrete-time linear systems with interval-like time-varying delays has been presented in terms of an LMI.