LI Bo-ren

(School of Computer，Dongguan University of Technology，Dongguan 523808，China）

ROBUST STABILIZATION OF UNCERTAIN STOCHASTIC SYSTEMS WITH TIME-VARYING DELAY AND NONLINEARITY

LI Bo-ren

(School of Computer，Dongguan University of Technology，Dongguan 523808，China）

In this paper，we study with robust stabilization problem of uncertain stochastic time-varying delay systems with nonlinear perturbation.Constructing a suitable Lyapunov-Krasovskii functional and employ the free weighting matrix method，in terms of the linear matrix inequality（LMI）technique，we design a memoryless state feedback controller，and obtain delay dependent robust stabilization criterion for the uncertain stochastic time-varying delay systems.A numerical example and its simulation curve are given to show that the proposed theoretical result is effective.

free-weighting matrices；nonlinear perturbation；time-varying delay；feedback control

2010 MR Subject Classification：93C10；93D09

Document code：AArticle ID：0255-7797（2016）05-0898-11

1 Introduction

The problem of the stabilization of time-delayed systems was often explored in recent years.Time delays are common in engineering processes.They frequently arose in chemical processes，in long transmission lines and in pneumatic，hydraulic and rolling mill systems. The problem of stability analysis in time-delayed systems was one of the main concerns of research into the attributes of such systems.Many works on this subject were published［1-7］.Depending on the information about the size of time-delays of the systems，criteria for time-delay systems can be classified into two categories，namely，delay-independent criteria ［1，2］and delay-dependent criteria［3-7］.Generally speaking，for the cases of small delays ，the latter ones are less conservative than the former ones.To obtain delay-dependent conditions，many efforts were made in the literature，among which the model transformation and bounding technique for cross terms［8］were often used.However，it is well known that these two kinds of methods are the main sources of conservatism.Recently，in order to reduce the conservatism，a free-weighting matrix method was proposed in［9，10］to investigatedelay-dependent stability，in which neither model transformation nor bounding technique is involved.

In recent years，the non-fragile control problem was an attractive topic in theory analysis and practical implement，because of perturbations often appearing in the controller gain，which may result from either the actuator degradations or the requirements for readjustment of controller gains.The non-fragile control concept is how to design a feedback control that will be insensitive to some error in gains of feedback control［11］.Xu et al.［12］concerned the problem s of robust non-fragile stochastic stabilization and H∞control for uncertain timedelay stochastic systems with time-varying norm-bounded parameter uncertainties in both the state and input matrices，when the delay was assumed to be constant.Zhang et al.［13］dealt with the same problem for uncertain nonlinear stochastic systems at the time-varying delay case.However，there was the restriction that time-derivative of time-varying delay must be less than one，which limits the application scope of the existing results.Wang et al.［14］dealt with the problems of non-fragile robust stochastic stabilization and robust H∞control for uncertain stochastic nonlinear single time-varying delay systems.By introducing the homogeneous domination approach to stochastic systems，Liu et al.［15］investigated a class of stochastic feedforward nonlinear systems with time-varying delay.By constructing delaypartitioning dependent Lyapunov–Krasovskii functional with reciprocally convex approach，Xia et al.［16］dealt with the problem of state robust H∞tracking control for uncertain stochastic systems with interval time-varying delay.

In this paper，our objective is to solve the problem of robust stabilization of uncertain stochastic systems with time-varying delay and nonlinearity.Parameter uncertainty in the state and input matrices，It is assumed to be norm bounded.Time delay is unknown，but in the known range changes with time.The goal of this paper is to design a memoryless state feedback controller，for all admissible parametric uncertainties，and make the closedloop system is robustly stochastically stable.The present results are derived by choosing an appropriate Lyapunov functional and by making use of free-weighting matrices method. Numerical example and its simulation curve are given to show the proposed theoretical result is effective.

NotationThrough this paper，the superscript T stands for matrix transposition；Rndenotes the n-dimensional Euclidean space，Rn×mis the set of n×m real matrices，I is the identity matrix of appropriate dimensions；the notation X＞0（respectively，X≥0），for X∈Rn×nmeans that the matrix X is real positive definite（respectively，positive semidefinite）；the symbol?is used to denote the transposed elements in the symmetric positions of a matrix.Matrices，if the dimensions are not explicitly stated，are assumed to have compatible dimensions for algebraic operation.

2 System Descriptions and Preliminaries

Consider the following uncertain linear stochastic differential delay system with nonlin-ear perturbation and parameter uncertainties

where x（t）∈Rnis the state vector，u（t）∈Rnis the control input，φ（t）is a continuoustime real valued function representing the initial condition of the system，and ω（t）is onedimensional Brownian motion defined on a complete probability space（?，F(xiàn)，P）satisfying E｛dω（t）｝=0，E｛dω（t）2｝=dt.In the system descriptive equation（2.1），the time-varying matrices are given by A（t）=A+△A（t），A1（t）=A1+△A1（t），B1（t）=B1+△B1（t），C（t）=C+△C（t），C1（t）=C1+△C1（t），and B2（t）=B2+△B2（t），where A，A1，B1，C，C1and B2are known constant matrices and△A（t），△A1（t），△B1（t），△C（t），△C1（t）and△B2（t）are unknown matrices representing time-varying parametric uncertainties in the system.They are assumed to be norm-bounded of the form

where D1，D2，E1，E2and E3are known real constant matrices with appropriate dimensions and F（t）is unknown time-varying matrix which is Lebesgue measurable satisfying FT（t）F（t）≤I，?t.The time-varying delay h（t）is a differentiable function satisfying the following condition

where h andμare constant scalars.The term σ（t，x（t），x（t-h（t）））∈Rnrepresents the unknown nonlinear perturbation with respect to the state x（t）and the delayed state x（th（t）），which is assumed to be bounded with the following form

where α，β are the known non-negative constants.

Before formulating the problems to be coped with，we first introduce the following concept of robust stability for system（2.1）.

Definition 1The uncertain stochastic system in（2.1）with u（t）=0 is said to be robustly stochastically stable if there exists a positive scalar?＞0 such that

for all admissible uncertainties△A（t），△A1（t），△B1（t），△C（t），△C1（t）and△B2（t）.

The objective of this paper is to develop delay-dependent stochastic stabilization criterion for the existence of a memoryless state feedback controller for system（2.1）satisfying the time-varying delay（2.3）.The state feedback controller is given by

where K being the controller gain to be designed.Following lemma is indispensable for deriving the criterion.

Lemma 1For any symmetric positive-definite matrices G and Z，of appropriate dimensions，the following inequality holds

ProofSince Z＞0，we have（Z-G）Z-1（Z-G）≥0.The proof follows immediately. Lemma 2［17］Given appropriately dimensioned matrices ψ，D，E with ψ=ψT.Then

holds for all F（t）satisfying FT（t）F（t）≤I if and only if for some η＞0，

3 Main Results

Now we provide a novel delay-dependent stabilization criterion for system（2.1）as follows

Theorem 1For given positive scalars h，μand λ，if there exist symmetric positivedefinite matrices X，S1，S2，Z，appropriately dimensioned matrices Y，Uj，Vj（j=1，2，3），and positive scalars ε1，ε2，such that the following LMI hold

where

Then the uncertain linear stochastic differential delay system（2.1）with time-varying parametric uncertainties（2.2）and nonlinear perturbation（2.4）is robust stabilization，in this case，an appropriate memoryless state feedback controller can be chosen by

Proof Substituting the state feedback controller（2.5）into system（2.1），we obtain the resulting closed-loop system as

where

Now，choose a Lyapunov functional candidate as

where P，Q1，Q2and R are symmetric positive-definite matrices to be chosen. By It?o's differential formula，we obtain stochastic differential as follows

where

From the Leibniz-Newton formula，the following equations are true for any matrices M and N with appropriate dimensions O

where

On the other hand， the following equation is also true

For any positive scalar δ，it follows from（2.4）that

where ζ（t）=σ（t，x（t），x（t-h（t）））.

Combining（3.3）-（3.7），we can obtain the following inequality

where

Since R＞0，then the last two parts in inequality（3.8）are all less than 0.So，taking the mathematical expectation on both sides of equation（3.2）and using inequality（3.8），since E｛F（dω（t））｝=0，we can obtain that

It remains to show that Ξ（t）+hMR-1MT+hNR-1NT＜0.Using Schur complement formula，we see that Ξ（t）+hMR-1MT+hNR-1NT＜0 if and only if the following matrix inequality holds

where

Then premultiplying and postmultiplying inequality（3.10）by

where

and Θ13，Θ23and Θ33are defined in inequality（3.1）.

Noting equation（2.2），and let Y=KX，inequality（3.11）can be written as

where

For given scalar λ＞0，the nonlinear term-hXZ-1X in the matrix inequality（3.12）can be rewritten as-h（λX）（λ2Z）-1（λX）.Therefore，by Lemma 1，we have the inequality -hXZ-1X≤hλ2Z-2hλX.Applying Lemma 2 and Schur complement to inequality （3.12），we can obtain the LMI（3.1）stated in Theorem 1，which means that system（2.1）under control law u（t）=Y X-1x（t）is robust stabilization.This completes the proof.

Remark 1When the differential of h（t）is unknown，and the delay h（t）satisfies 0≤h（t）≤h，by setting S1=0，a delay-dependent and rate-independent criterion for robust stabilization of systems（2.1）from Theorem 1 can be obtained.

Table 1：（MAUB）h of the time-varying delay h（t）for differentμ.

Remark 2When α=0，β=0，a uncertain linear stochastic differential delay system criterion without nonlinear perturbation for robust stabilization of systems（2.1）from Theorem 1 can be obtained.

4 Numerical Example

In this section，in order to demonstrate the effectiveness of the proposed method，we provide the following numerical example.

Example 1 Consider the uncertain nonlinear single time-delay system（2.1）with the following parameters

By using matlab solver feasp，for givenμ=0.5，λ=0.2，the feasibility upper bound of h（t）is 0.3108.Choosing h=0.3，according to Theorem 1，solve LMI in inequality（3.1），and get a set of solutions as follows

Therefore the robust problem is solvable，and the memoryless feedback gains in control are computed as

Figure 1：Trajectory of the solution to such system in Example 1

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具有非線性擾動(dòng)的不確定隨機(jī)時(shí)變時(shí)滯系統(tǒng)的魯棒鎮(zhèn)定

李伯忍
（東莞理工學(xué)院計(jì)算機(jī)學(xué)院，廣東東莞523808）

本文研究了具有非線性擾動(dòng)的不確定隨機(jī)時(shí)變時(shí)滯系統(tǒng)的魯棒鎮(zhèn)定的問題.構(gòu)造了適當(dāng)?shù)腖yapunov-Krasovskii泛函并利用自由權(quán)矩陣方法，借助于線性矩陣不等式（LMI）技術(shù)，設(shè)計(jì)了一個(gè)無記憶狀態(tài)反饋控制器，并獲得了不確定隨機(jī)時(shí)變時(shí)滯系統(tǒng)的時(shí)滯依賴魯棒鎮(zhèn)定判據(jù).數(shù)值例子及其仿真曲線表明所提出的理論結(jié)果是有效的.

自由權(quán)矩陣；非線性擾動(dòng)；時(shí)變時(shí)滯；反饋控制

MR（2010）主題分類號：93C10；93D09O23；O29

date：2014-12-07Accepted date：2015-04-07

Supported by the Guangdong Province Natural Science Foundation Project （2016A030313130）and the National Natural Science Foundation Project of China（11371154）.

Biography：Li Boren（1980-），male，born at Yugan，Jiangxi，doctor，major in analysis and synthesis of the time-delay and uncertain system.