Delay－dependent absolute stability analysis of improved Lurie control system with multiple delays

， ， ，

（1.School of Mathematics and System Science，Shenyang Normal University，Shenyang 110034，China；2.Math department，Shenyang Guangquan Middle School，Shenyang 110141，China）

0 Introduction

It is well known that time delay is often an significant source for instability in various engineering systems［1－14］.The stability problem of Lurie control system with time－delay has been studied by many researchers［11－14］.

In this paper，we study the stability problem for Lurie control system with multiple delays，respectively.Based on the division of the delay，The system matrixBi（i＝1，2，…，m）are decomposed.an suitable of Lyapunov functional is proposed to study this class of system.Some improved delay－dependent Lyapunov functionals are derived by employing an integral－inequality.An example is presented to illustrate the effectiveness for some existing results.

1 Problem statement

Consider the Lurie control system with multiple time delays of the form

wherex（t）∈Rnis state vector；A，Bi（i＝1，2，…，m）∈Rn×n，D，C∈Rn×mare consent matrices；hi≥0（i＝1，2，…，m）are time delays.φ（t）is a continuous vector valued initial condition.The nonlinearitiesfj（·）（i＝1，2，…，m）satisfy the finite sector condition

In order to receive the main results，which begin with the following lemma

Lemma［2］Given any constant appropriately dimensioned matricesR，N，X，and scalarh＞0 and vector valued functionsf（x）andηsuch that the following integration is well defined，then

2 Stability analysis

In order to improve the bound of the discrete－delayhi，let us decompose the matrixBiasBi＝Bi1＋Bi2（i＝1，2，…，m），whereBi1，Bi2（i＝1，2，…，m）are constant matrice.Then the original system∑can be represented in the form of a descriptor system with discrete and distributed delays

namel

with the initial conditon∑.

LetD（xt）be a new operator，we have

In order to guarantee that the difference operatorD（xt）：C［－max｛τi｝，0］→Rngiven by（4）is stable（i.e.difference equationD（xt）＝0is asymptotically）.

TheoremGiven positive scalarshi，system ∑ with nonlinearity located in the finite sector［0，K］is absolutely stable if there exist positive definite matricesP＞0，Qi1＞0，Qi2＞0，Zi1＞0，Zi2＞0，Ki＞0（i＝1，2，…，m），diagonal matrixR＝diag｛r1，r2，…，rm｝≥0，a scalarε＞0and appropriately dimensioned matricesMij＝…］T，i＝1，2，…，m；j＝1，2，…，N，satisfying

ProofFor∑ with nonlinearity located in the sector［0，K］，condition（2）is equivalent to

Let us choose the following Lyapunov functional candidate for the system∑，

whereP＞0，Qi1＞0，Qi2＞0，Zi1＞0，Zi2＞0，Ki＞0（i＝1，2，…，m）and diagonal matrixR＝diag｛r1，r2，…，rm｝≥0are to be determined.

Then，taking the derivative ofV（t）with respect to along the solution of system ∑，at the same time，employing the lemma，schur complete and other mathematical technology，This theorem can be proved.The process can be omitted.

WhenD≡0，system ∑can reduce to the following linear system with multiple delays

Take the Lyapunov functional asV（t）＝V1（t）＋V2（t）＋V3（t）＋V5（t），whereV1（t），V2（t），V3（t）andV5（t）are defined in（6）～（10）.

system ∑′is asymptotically stable.We can use theorom to prore this.

3 Illustrative example

ExampleConsider the system （1）with

This example was given in［10－14］，The maximum value ofhmaxfor absolute stability of system（1）form is shown in table 1.Now we use the criterion in this paper to study the problem.Let us decompose matrixB1asB1＝B11＋B12as well as it did in［14］，where

Solving LMI（5），table 1shows that the absolute stability criterion in this paper gives a much less conservative result than these in Refs.［10－14］.

Table 1 Maximum upper bound of h for different Methods

4 Conclusion

The stability of Lurie control systems with multiple time－delays is investigated.By dividing the delay interval into n segments and choosing proper Lyapunov functional，the delay－dependent absolutely stable condition for Lurie control systems with multiple time－delay is received.The numerical example has shown significant improvement over some existing results.

［1］JACK H.Theory of Functional Differential Equation［M］.New York：Spring，1977.

［2］HE Yong，WANG Qingguo，XIE Lihua，et al.Further improvemen of free－weighting matrices technique for systems with timevarying delay［J］.Automat Control，2007，52（2）：293－299.

［3］HE Yong，WU Min，SHE Jinhua.Parameter－dependent Lyapunov functional for stability of time－delay systems with polytopic type uncertainties［J］.Automat Control，2004，49（5）：828－832.

［4］WU Min，HE Yong，SHE Jinhua.New delay－dependent stability criteria and stabilizing method for neutral systems［J］.Automat Control，2004，49（12）：2266－2271.

［5］TIAN Junkang，ZHONG Shouming，XIONG Lianglin.Delay－dependent absolute stability of Lurie control systems with multiple time－delays［J］.Appl Math Comput，2007，188（1）：397－384.

［6］GAO Jie，XU Zhaodi.Guaranteed cost control of uncertain linear systems with both state and input delays［J］.Journal of Shenyang Normal University Natural Science Edition，2009，27（4）：408－412.

［7］CAO Jiuwen，ZHONG Shouming，HU Yuanyuan.Delay－dependent condition for absolute stability of Lurie control systems with multiple time delays and nonlinearities［J］.J Math Anal Appl，2008，338（1）：497－504.

［8］CAO Jiuwen，ZHONG Shouming.New delay－dependent condition for absolute stability of Lurie control systems with multiple timedelays and nonlinearities［J］.Appl Math Comput，2007，194（1）：250－258.

［9］QIU Fang，ZHANG Quanxin.Absolute stability analysis of Lurie control system with multiple delays：An integral－equality approach［J］.Nonlinear Anal，2011，12（3）：1475－1484.

［10］MIAN Xiaohong.Delay dependent conditions for absolute stability of Lurie type control systems［J］.Acta Automat Sinica，1999，25（4）：564－566.

［11］XU Bingji，LIAO Xiaoxin.Absolute stability criteria of delay－dependent for Lurie control systems［J］.Acta Automat Sinica，2002，28（2）：317－320.

［12］CHEN Dongyan，LIU Weihua.Delay－dependent robust stability for Lurie control systems with multiple time delays［J］.Contr Theor Appl，2005，22（3）：499－502.

［13］YANG Bin，CHEN Mianyun.Delay－dependent criterion for absolute stability of Lurie type control systems with time delays［J］.Contr Theor Appl，2001，18（6）：929－931.

［14］TIAN Junkang，ZHONG Shouming，XIONG Lianglin.Delay－dependent absolute stability of Lurie control systems with multiple time－delays［J］.Appl Math Comput，2007，188（1）：379－384.