Robust stability for switched systems with time-varying delay and nonlinear perturbations

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(School of Mathematics and Systems Science, Shenyang Normal University, Shenyang 110034, China)

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Robust stability for switched systems with time-varying delay and nonlinear perturbations

DUJuanjuan,LIUYuzhong

(School of Mathematics and Systems Science, Shenyang Normal University, Shenyang 110034, China)

This paper considers the robust exponential stability for switched systems with time-varying delay and nonlinear perturbations. A Lyapunov-Krasovskii function, which takes the range information of the time-varying delay into account, is proposed to analyze the stability. Furthermore, we also consider the effect of time-varying delay on the stability of switched systems. In the analysis of switched systems, free-weighting matrices is employed to improve the solvability of problems and make the result be less conservative. The switching strategy is projected by using average dwell time method, and nonlinear perturbations are limited to common constraint without loss generality. The sufficient condition of delay-range-dependent exponential stability for switched systems is presented by using Lyapunov stability theory. However, this condition is not easy to verify. This problem can be solved easily by transforming them into equivalent linear matrix inequalities (LMIS) using Schur complement lemma. Finally, switched law of robust exponential stability is obtained for switched systems with time-varying delay and nonlinear perturbations.

switched systems; robust exponential stability; average dwell time; time-varying delay; nonlinear perturbations

0 Introduction

Switched systems are a special class of hybrid systems, which consist of several subsystems and a switching law. The switching law decides the active subsystems at each instant time. In recent decades, switched systems have been paid more attentions by control theorist and engineers, on the other hand, systems with time delay is ubiquitous in engineering[1-4]. With the development of networked control technology, many efforts have been made to investigate the stability of systems with interval time-varying delay[5-9].

Most of the results for switched systems centralize on analysis and design of the stability. A class of switched systems with stably convex combination of structural matrix was introduced in[2]. Switched systems with time-varying delay were studied by average dwell time method in[3]. A sufficient condition of the switched systems with time-delay was given in[4]. A novel Lyapunov-Krasovskii functional was studied in[7]. In[10], the switched systems with mixed delayed and nonlinear perturbations were proved. Delay-range-dependent stability was investigated in [11] by using the free-weighting matrix approach[12-13].

In practice, owing to the presence of some uncertainties due to environmental noise, uncertain or slowly varying parameters, etc. the problem of robust stability of time-varying systems under nonlinear perturbations has received increasing attention in [14-18].

In this paper, we deal with delay-range-dependent stability problem for switched systems with time-varying delay and nonlinear perturbations. The sufficient conditions of delay-range-dependent exponential stability for switched systems with time-varying delay are presented by using average dwell time method and free weighting matrix method. The proposed stability criteria are given in the form of Linear Matrix Inequalities (LMIs).

Notations:Rndenotes then-dimensional Euclidean space. The superscript “T”stands for matrix transposition. I is an identity matrix with appropriate dimension. “*” represents the symmetric elements in symmetric matrix.

1 Systems description

Consider the following switched linear systems, which consist ofmsubsystems, with time-varying delay and nonlinear perturbations.

Wherex(t)∈Rnis the state vector;σ(t):[0,+∞)→M={1,2,…,m} is switching signal;mis a natural number, denotes the number of the subsystems,Ai∈Rn×n,Adi∈Rn×nare constant matrices with appropriate dimensions fori-thsubsystems; the time delayτ(t) is a time-varying continues function that satisfies

Whereh1andh2are constants representing respectively the lower and upper bounds of the delay,τis a positive constant. The initial conditionφ(t) is a continuous vector-valued function. Moreover the functionf(x(t),t) andg(x(t-τ(t),t) are unknown and denote the nonlinear perturbations with respect to the current statex(t) and delayed statex(t-τ(t)) respectively. They satisfy thatf(0,t)=0,g(0,t)=0 and

Whereα≥0 andβ≥0 are known scalars,Fand Gare known constant matrices.

In this paper, we investigate the stability problem of system (1) with the interval time-varying satisfying (2) and the nonlinear perturbationsf(x(t),t) andg(x(t-τ(t),t) satisfying (3) and (4). Our main objective is to derive new delay -range -dependent stability conditions under which system (1) is exponentially stable.

Definition 1[3]The equilibriumx*=0 of systems (1) is said to be exponentially stable underσ(t), if the solutionx(t) of systems (1) satisfies

‖x(t)‖≤κ‖xt0‖ce-λ(t-t0)?t≥t0

Forconstantκ≥1 andλ>0, where

Definition 2[6]For any real numbersT2>T1≥0, letNσ(T1,T2) denotes the number of switching ofσ(t) over time interval (T1,T2). IfNσ(T1,T2)≤N0+(T2-T1)/Taholds forTa>0,N0≥0, and thenTais called average dwell time. As commonly used in the literature, we chooseN0=0.

2. Main results

Furthermore the decay of the state is

‖x(t)‖

Whereμ>0, satisfies

Proof:Choose piecewise Lyapunov function candidate

Moreover,formatricesXandYwithappropriatedimensions,wehave

Ontheotherhand,foranyscalarsε1≥0,ε2≥0,itfollows(3)and(4)

Where

V(xt)=Vσ(t)(xt)≤e-λ(t-tk)Vσ(t)(xtk)

Fromthedefinitionofa,bandV(t),thefollowinginequalitieshold,

then

Withthedefineofγ,theformula(8)issatisfied.Sothetheoremholds.

3 Conclusion

We have studied the problem of exponential stability for switched systems with time-varying delay and nonlinear perturbations, by using the average dwell time method and free weighting matrix method, the sufficient conditions are given. Under suitable switching law, the exponential stability of the switched systems can be guaranteed and the criterion is in the form of LMIS which can be solved easily.

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1673-5862(2015)03-0341-05

一類帶有時(shí)變時(shí)滯和非線性擾動(dòng)的切換系統(tǒng)的魯棒指數(shù)穩(wěn)定性

杜娟娟, 劉玉忠

(沈陽(yáng)師范大學(xué) 數(shù)學(xué)與系統(tǒng)科學(xué)學(xué)院, 沈陽(yáng) 110034)

討論了一類帶有時(shí)變時(shí)滯和非線性擾動(dòng)的切換系統(tǒng)的魯棒指數(shù)穩(wěn)定性問(wèn)題。通過(guò)構(gòu)造新的李雅普諾夫-克拉索夫斯基函數(shù)研究切換系統(tǒng)的穩(wěn)定性,同時(shí)考慮了時(shí)變時(shí)滯對(duì)系統(tǒng)穩(wěn)定性的影響。在系統(tǒng)分析過(guò)程中,采用自由權(quán)矩陣的方法,提高問(wèn)題的可解性并使結(jié)果具有更小的保守性,切換策略采用平均駐留時(shí)間的方法,未知的非線性擾動(dòng)采用通常的限制方法。根據(jù)Lyapunov穩(wěn)定性定理,得到了切換系統(tǒng)時(shí)滯依賴魯棒指數(shù)穩(wěn)定性的充分條件。該判定條件不易檢驗(yàn),利用Schur補(bǔ)引理可以把這個(gè)條件化成等價(jià)的易于求解的線性矩陣不等式形式,從而獲得該類系統(tǒng)魯棒穩(wěn)定性的切換控制策略。

切換系統(tǒng); 魯棒穩(wěn)定性; 平均駐留時(shí)間; 時(shí)變時(shí)滯; 非線性擾動(dòng)

TP273 Document code: A

10.3969/ j.issn.1673-5862.2015.03.006

Received date: 2015-05-14.

Supported: Project supported by the National Natural Science Foundation(11201313).

Biography: DU Juanjuan(1989-), female, was born in Chaoyang city of Liaoning province, postgraduate students of Shenyang Normal University; LIU Yuzhong(1963-), male, was born in Xinbin city of Liaoning province, professor and postgraduates instructor of Shenyang Normal University, doctor.