Asynchronous H control for nonhomogeneous higher-level Markov jump systems

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Abstract

This paper investigates about asynchronous H control for nonhomogeneous higher-level Markov jump systems, where a higher-level transition probability matrix is considered. The asynchronous H controller designed is subject to an independent Markov chain. Using a linear matrix inequality approach, sufficient conditions are established such that system with controller is stochastically stable. A numerical example is presented to demonstrate the effectiveness of our method developed.

Introduction

In the mid 1960s, Markov jump systems (MJSs) were first proposed and attracted widelywide attention, due to its widely application in economic systems, solar systems, etc [2], [3], [6], [7], [8], [9], [16]. Since then, it has been a hot research topic because of their great potentials in applications. MJSs can describe the systems whose structure or parameters may change abruptly if external environment is altered or machine breaks down. Thus, a lot of complicated systems such as communication systems, chemical processes, electric systems can reasonably modeled by MJSs, and a great number of results about MJSs have been presented, and these results cover stability and stabilization [13], [26], filtering [17], [24] and fault detection [22]. As special extensions of MJSs, nonhomogeneous Markov jump systems [10], and semi-Markov jump systems [11], [15] also have attracted a lot of attention.

Many articles concerning control for MJSs have been delivered in the past decades, see [4], [5], [12] and later on, work has been done on the asynchronous technique. Asynchronous controller not only can characterize the case that controller mode is different from systems mode, but is also able to make use of the known information of mode to ensure that controller is effective. In recent years, the number of articles about asynchronous filter has been increasing [18], [20], [27]. However, most of asynchronous filters or controllers are designed based on current system model. Wu investigated asynchronous l2l filtering for Markov jump systems in [18]. Meanwhile, some asynchronous filters proposed are subject to a Bernoulli distribution, such as [28], [29]. On the other hand, the results about asynchronous controller are also obtained [1], [14], [19]. Extensions of these results can be found in [14], [19]. In this paper, the transition probability matrix of asynchronous H controller does not depend on current system mode, in other words, if system current mode is unavailable, the asynchronous H controller can still work. Furthermore, we deal with nonhomogeneous MJSs in this paper.

Homogeneous MJSs mean MJSs whose transition probability matrices are fixed constant matrix. However, in reality, most transition probability matrices are not fixed constantly, and so nonhomogeneous MJSs need to be introduced. In the past several years, some results on nonhomogeneous MJSs are obtained [21], [23]. Yin analyzed stochastic stability and designed an H controller for constrained nonhomogeneous MJSs in [21]. However, most articles about nonhomogeneous MJSs do not explain how transition probability matrices vary. In 2009, Zhang took variational form of transition probability matrices into consideration in investigating the problem of H estimation for piecewise homogeneous MJSs [25]. To date, the problem of nonhomogeneous higher-level MJSs whose transition probability matrices follow stochastic variation has never been investigated. In this technical note, transition probability matrices of nonhomogeneous MJSs are subject to a higher-level Markov chain.

The rest of this paper is divided into five parts. Section 2 models the problem and gives some preliminaries. In Section 3, sufficient conditions such that nonhomogeneous higher-level MJSs are stochastically stable are proposed. An asynchronous H controller for nonhomogeneous higher-level MJSs is designed in Section 4. Section 5 provides a numerical example to demonstrate the effectiveness. Some conclusions are given in the last section.

Notation. Rn and E{·} denote the n-dimensional Euclidean space and mathematical statistical set, respectively; AT means the transpose of the matrix A; L2n[0,) stands for the space of n-dimensional square integral functions over [0, ∞); a positive-definite matrix is denoted by P > 0; diag{...} stands for a block-diagonal matrix; I is the unit matrix with appropriate dimension, and * denotes the symmetric term in a symmetric matrix.

Section snippets

Problem statement and preliminaries

We consider an underlying probability space (Ω, F, P) and the following discrete-time nonhomogeneous higher-level MJSs:{x(k+1)=A(rk)x(k)+B(rk)u(k)+C(rk)ω(k),z(k)=D(rk)x(k)+E(rk)ω(k),where x(k) ∈ Rn is the system state vector, u(k) ∈ Rp is the control input vector, and ω(k)L2p[0,) is external disturbance vector, z(k) ∈ Rn is system controlled output vector. We use {rk, k ≥ 0} to represent the concerned discrete-time Markov chain, which is defined on a finite set Λ1={1,2,3,,N} and mode

Stability analysis

In this section, sufficient conditions are given under which MJS (1) is stochastically stable.

Theorem 1

Let u(k)=0 and ω(k)=0. System (1) is stochastically stable if there exist positive definite symmetric matrices Pi,m such that it holdsAiT·I·n=1Mπmnl=1wξmlj=1Nπijl·I·Pj,n·AiPi,m<0,i=1,,N.

Proof

Construct a Lyapunov function as followsVi,m=xT(k)Pi,mx(k),then we haveΔVi,m=E{xT(k+1)Pj,nx(k+1)}xT(k)Pi,mx(k)=xT(k)[AiT·I·n=1Mπmnl=1wξmlj=1Nπijl·I·Pj,n·AiPi,m]x(k)=xT(k)Ωx(k),whereΩ=AiT·I·n=1Mπmnl=1wξmlj=1N

Controller design

We construct an asynchronous controller as follows:u(k)=[σ(φk)Ki+(1σ(φk))Ki1]x(k),where {φk, k ≥ 0} is subject to an independent Markov chain, which is defined on the finite set Λσ={1,2}, and independent mode transition probabilities πpv(k)=P{φk+1=v|φk=p}. For simplicity, let φk=p and σk) be denoted by σp, where σp satisfies σ1=1,σ2=0 and independent transition probability matrix satisfiesΠσ=[1ααβ1β],0<α1,0<β1.

Detailed application of asynchronous controller is as follows: if the system

Numerical example

Consider a system of discrete-time MJSs and an asynchronous controller, both of which have two jumping modes, define byA1=[0.90.450.90.9],A2=[0.90.290.91.26],B1=[0.51.1],B2=[0.61.4],C1=[0.10.2],C2=[0.10.2],D1=[0.50.4],D2=[0.30.1],E1=0.9,E2=0.9,Π1=[0.650.350.40.6],Π2=[0.40.60.550.45],Πq=[0.40.60.450.55],Πσ=[0.80.20.70.3].We assume the external disturbance satisfies ω(k)=exp(0.1*k)*sin(0.034*k) and initial mode is x(0)=[0.550.4].

Solving LMI (10), we haveK1=[0.92420.6393],K2=[0.85270.6778],

Conclusion

In this paper, an asynchronous H controller for nonhomogeneous higher-level MJSs is designed, which is under a stochastic variation. The asynchronous phenomenon is subject to another independent Markov chain. The controller gain is determined by a linear matrix inequality system. Two numerical examples are solved to demonstrate the effectiveness and advantages of the controller designed. To make the results have broader applications, we will try to extend the results to practical Markov jump

Acknowledgment

This work has been partially supported by the National Natural Science Foundation of PR China (No. 61773011, 61773183), the Ministry of Education of China under the 111 Project B12018, Australian ARC grant and Curtin Fellowship.

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