Non-fragile delay feedback control for neutral stochastic Markovian jump systems with time-varying delays

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Abstract

This paper is concerned with the problem of non-fragile delay feedback control for neutral stochastic Markovian jump systems (NSMJSs) with time-varying delays. The main aim is to design non-fragile and mode-dependent delay feedback controller (DFC) both in the drift part and in the diffusion part to realize closed-loop NSMJS is stochastically stable. By constructing mode-dependent and delay-dependent comprehensive Lyapunov-Krasovskii functional, the stabilization conditions are provided in terms of linear matrix inequalities (LMIs). Simulation examples including a lossless transmission line model (LTLM) are utilized to verify the validity and usefulness of the proposed method.

Introduction

Over the past a few decades, Itô stochastic systems have received much attention due to these systems emphasize the effects of random noises interference, which can model various kinds of practical systems subjected to some types of environmental noises, for instance, economic systems, mechanical systems, population systems, ecosystems, and so on [1], [2], [3]. Since time delays are often confronted in lots of engineering and physical systems, which frequently lead to performance degradation even instability [4], [5], [6], [7], [8], [9], [10], [11], the problems of stability and control for Itô stochastic systems with retarded time delays have been extensively investigated during the past decades [1], [2], [12]. The aforesaid retarded delay stochastic systems can be described by the following stochastic differential delay equation (SDDE):dx(t)=f(x(t),x(tτ),t)dt+g(x(t),x(tτ),t)dB(t),where B(t) is l dimensional Brownian motion defined on the given complete probability space (Ω,F,P) with a natural filtration {Ft}t0. Borel-measurable functions f(·), g(·) are the drift part and the diffusion part of Itô stochastic system (1), respectively. f(·), g(·) both satisfy the Lipschitz condition and linear growth condition, and it is well known that the SDDE is a special but important class of stochastic functional differential equations (SFDEs).

Moreover, neutral type delays are often encountered in a variety of real systems such as water or steam pipes, lossless transmission lines, heat exchanges, population ecology, etc., [1], [2]. Therefore, recently, neutral stochastic differential systems (NSDSs) have been intensively addressed by the scholars at home and abroad in [5], [13], [14], [15], [16], [17], etc. The NSDS involves the delays in state as well as in the derivative of the state, which can be depicted as follows:d[x(t)N(x(tτ))]=f(x(t),x(tτ),t)dt+g(x(t),x(tτ),t)dB(t).To mention but a few famous works, [2], [14], [15], [16] introduced the Lipschitz condition and linear growth condition for existence and uniqueness of solutions, and these literature focused on stability in probability, almost sure exponential stability, moment exponential stability. Meanwhile, [5], [13] considered the stochastic stabilization and H control with uncertain parameters.

Noting that Markovian jump systems (MJSs) are an important kind of hybrid systems, which can model many practical systems encountering abrupt changes in structures or parameters caused by component failures or repairs, abrupt environment disturbances, etc. MJSs have been deeply studied and extensively employed in various fields such as BM/C3 (battle management command, control and communication), solar thermal system, power systems, etc., and a great deal of influential results have been published in some famous journals; see, for instance, [11], [14], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]. Itô stochastic systems meet with abrupt changes in structures and/or parameters, which result in the well-known stochastic Markovian jump systems (SMJSs) [30], [31], [32], [33], [34], [35], [36], [37], [38], and recently, neutral stochastic Markovian jump systems (NSMJSs) have received much attention such as [14], [39] and the references therein. NSMJSs can be represented by the following model:d[x(t)N(x(tτ))]=f(x(t),x(tτ),t,rt)dt+g(x(t),x(tτ),t,rt)dB(t),where {rt} denotes the Markovian jump mode and takes values in a finite set S={1,2,,N}.

It is worth pointing out that, in practice, the designed controller is more realistic if it depends on not only the current state but also the past state due to a time delay between the feedback control arrival and the state observation time (see [40]). However, [39], [40] just only considered the regular delay state feedback control (SFC) in the drift part of a Itô stochastic system or a NSMJS. [31] pointed out that the stochastic delay state feedback (SDSF) in the diffusion part of a Itô stochastic system can realize sample-path stabilization because of the preserved original state in average. Therefore, it is of important theoretical and practical significance to design delay state feedback controller and delay output injection feedback controller (see [41]) both in the drift term and in the diffusion term of a Itô stochastic system.

On the other hand, the topic of non-fragile (also known as resilient) control and filtering has been focused not only on theory analysis but also on practical applications. Non-fragile control/filter is to design a feedback controller or filter such that the controller/filter is insensitive to some uncertainty in gains [42]. The inevitable non-fragile phenomenon stems from numerical roundoff errors, the imprecision inherent, or the actuator degradation, etc., and in the past decades, the non-fragile control problem has been widely studied such as [43], [44], [45], [46], [47] and the references therein.

To the best of the authors’ knowledge, so far, there is little literature about non-fragile delay SFC for NSMJSs both in the drift term and in the diffusion term. The mentioned NSMJS plant can be represented as follows:d[x(t)N(x(tτ))]=f(x(t),x(tτ(t,rt)),u(x(t),x(tτ(t,rt)),t,rt))dt+g(x(t),x(tτ(t,rt)),u(x(t),x(tτ(t,rt)),t,rt))dB(t),where u( · ) is the control input, the time delays in drift and diffusion parts are time-variant and mode-dependent. The problem of non-fragile delay feedback control for aforesaid NSMJSs is still open and challenging, which motivates the present research.

In this paper, we will study the non-fragile delay feedback control for NSMJSs with time-varying delays. The main aim is to design non-fragile and mode-dependent delay feedback controller both in the drift part and in the diffusion part such that closed-loop NSMJS is stochastically stable. By constructing mode-dependent and delay-dependent comprehensive Lyapunov-Krasovskii functional, we will provide the stabilization conditions in terms of LMIs. Simulation examples including a lossless transmission line model are utilized to verify the validity and usefulness of the proposed method.

The main contributions of this work are epitomized as follows: (1) non-fragile regular delay feedback control in drift part and non-fragile stochastic delay feedback control in diffusion part are discussed simultaneously; (2) mode-dependent and time-variant features of time delays are considered both in drift and diffusion parts; (3) delay-dependent, mode-dependent and more comprehensive Lyapunov-Krasovskii functional is constructed to realize the desired delay feedback controller designing.

Notation. Sym(B) means B+BT, diag{ · } stands for a block diagonal matrix. (*) implies a symmetry term in symmetric matrices, | · | denotes the Euclidean vector norm and the induced norm, ρ(N) means the spectral radius of N.

Section snippets

Problem formulation and preliminaries

Given a complete probability space (Ω,F,P) with a natural filtration {Ft}t0, we consider the linear NSMJS with time-varying delays described by the following model :{d[x(t)Nx(tτ)]=[A(rt)x(t)+Ad(rt)x(tτ(t,rt))+B(rt)u(x(t),x(tτ(t,rt)))]dt+[C(rt)x(t)+Cd(rt)x(tτ(t,rt))+D(rt)u(x(t),x(tτ(t,rt)))]dB(t),x(t)=φ(t),t[τ,0],where x(t)Rn is the state, φ(t) is the vector-valued initial function, u(·)Rm is the control input and |N| < 1.

B(t) is a scalar Wiener process defined on the given

Main results

Theorem 1

The unforced NSMJS (5) is stochastically stable, if for all i ∈ S, there exist matrices Pi > 0, R > 0, Q > 0 such that the following LMIs are satisfied:Πi=[Π11iΠ12iPiAdiCiTPi*Π22iNTPiAdi0**(1μi)QCdiTPi***Pi]<0,where

Π11i=sym(PiAi)+j=1NπijPj+R+(1+ητ)Q, Π12i=AiTPiNj=1NπijPjN, Π22i=R+NTj=1NπijPjN.

Proof

Defining {xt=x(t+θ),τθ0}, then stochastic process {(xt, rt), t ≥ τ} is a markovian process with initial state (φ( · ), r0). Selecting a stochastic Lyapunov-Krasovskii functional candidate for

Simulation examples

Example 1

Consider the NSMJS (5) with the following parameters:A1=[1112],A2=[0.5111],Ad1=[0.030.20.20.05],Ad2=[0.20.120.030.2],C1=[0.30.50.20.6],C2=[0.30.40.10.7],Cd1=[0.10.20.140.15],Cd2=[0.10.20.140.15],B1=[0.120.020.050.23],B2=[0.130.010.030.22],D1=[0.010.020.010.03],D2=[0.020.030.020.04],N=[0.020.010.020.04],M21=M22=[0.10.1500.2],N31=N32=[0.010.150.10.05],N41=N42=[0.020.050.030.1],Π=[0.020.020.10.1],μ1=0.75,μ2=0.35.

Let τ=1.2, solving LMIs (28) in Theorem 2, we get ε1=11.5873, ε2

Conclusions

In this paper, we have considered the problem of non-fragile delay feedback control for NSMJSs with time-varying delays. We have realized the main aim of designing non-fragile and mode-dependent delay feedback controller both in the drift part and in the diffusion part such that closed-loop NSMJS is stochastically stable. By constructing mode-dependent and delay-dependent comprehensive Lyapunov-Krasovskii functional, the stabilization conditions have been provided in terms of LMIs. Simulation

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China under Grants 61773191, 61573177, 61673215, 61773207; the Natural Science Foundation of Shandong Province for Outstanding Young Talents in Provincial Universities under Grant ZR2016JL025; the 333 Project (BRA2017380); the Program for Changjiang Scholars and Innovative Research Team in University (No. IRT13072); PAPD; the Key Laboratory of Jiangsu Province; Undergraduate Education Reform Project of higher

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