Kết quả mới về tính ổn định của hệ rời rạc không chắc chắn với độ trễ biến thiên = new result on the stability of nonlinea runcertain discrete-time systems with time-varying time delay

NEWRESULTSONTHESTABILITYOFNONLINEARUNCERTAIN

DISCRETE-TIMESYSTEMSWITHTIME-VARYINGTIMEDELAY

Tran Thanh Tung

Thai Nguyen University of Economics and Business Administration - Thai Nguyen University

ABSTRACT

Theproblemsofstabilityandstabilizabilityforaclassofnonlineardiscrete-timesystemswithvarying

timedelayareconsidered.Theaimofthispaperistopresentanewstabilitysufficientconditionsfor

nonlinearuncertaindiscrete-timesystemswithtime-varyingtimedelay.Severalexamplearepresented

toillustratethevarioustheorems.

Keywords: Discrete systems, uncertainty, Lyapunov function, linear matrix inequal￾ity.

1 INTRODUCTION

It is known that, because of the finite switch￾ing speed, memory effects and so on, time

delay is frequently encountered in technology

and nature. It extensively exists in various

mechanical, biological, physical, chemical en￾gineering, economic systems, and becomes an

important source of instability and oscillation.

This makes the design and hardware imple￾mentation of delayed system become difficult.

Thus, the studies on stability for delayed sys￾tem 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 sys￾tem, and many excellent papers and mono￾graphs have been available (see [1-12]). Gen￾erally speaking, these so-far obtained stability

results for delayed system can be mainly classi￾fied into two types: that is, delay-independent

and delay-dependent. Since sufficiently consid￾ered the information of the size of time delay,

delay-dependent criteria may be less conserva￾tive than delay-independent ones when time de￾lay is small. For delay-dependent type, the size

of allowable upper bound of time delay always

be regarded as an important criterion to dis￾criminate the quality between different results.

Recently, the problem of delay-dependent sta￾bilization analysis for delayed system has re￾ceived considerable attention, and many impor￾tant results are obtained (see [3-7]). In [8], 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 crite￾ria. For reducing conservatism of the criteria

obtained in [8-9].

To the best of the authors knowledge, the

delay-dependent time delay case for the class

of discrete-time nonlinear systems with interval

varying time delay has not been fully investi￾gated and this will be the subject of this paper.

In this paper, by using the Lyapunov method,

a sufficient condition for the asymptotic stabil￾ity of uncertain nonlinear discrete-time systems

with interval time-varying delay is derived in

terms of LMIs, which can be solved by various

efficient convex optimization algorithms [2,3,4].

The paper is organized as follows. After Intro￾duction, in Section 2 we give technical lemmas

needed for the proof of the main result. Suffi￾cient conditions for asymptotic stability of the

system and a numerical example to illustrate

the effectiveness of our conditions are presented

in Section 3. The paper ends with conclusions

and cited references.

Notations.

0Tel: 0943822828, e-mail: [email protected]