H control for discrete-time singular Markov jump systems subject to actuator saturation

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Abstract

In this paper, the H control problem for discrete-time singular Markov jump systems with actuator saturation is considered. A sufficient condition is obtained which guarantees that the discrete-time singular Markov jump system with actuator saturation is regular, causal, bounded state stable, and satisfies the H performance. With this condition, and based on singular value decomposition approach, the design method of H state feedback controller is developed by using linear matrix inequalities (LMIs) optimization problem. Numerical examples are given to illustrate the effectiveness of the proposed methods.

Introduction

Markov jump systems represent a class of stochastic systems, which are popular in modeling many practical systems with abrupt random changes in their structures and parameters, such as manufacturing systems, networked control systems, etc. In recent years, Markov jump systems have attracted a lot of researchers and many problems have been solved [1], [2], [3], [4], [5], [6], [7]. Singular systems are also referred to as descriptor systems, implicit systems or generalized state-space systems, which can better describe the behavior of some physical systems than standard state-space ones and have extensive applications in electrical networks, circuit systems, chemical processes, economic systems and other areas [8]. Hence a great number of fundamental notions and theories for singular systems have been researched, such as stability, stabilization, H control problem [9], [10], [11], [12], [13] and so on. Recently, stability and control problems for singular Markov jump systems have also been widely researched, and many results have been obtained [14], [15], [16], [17], [18], [19].

On the other hand, actuator saturation can lead to poor performance of the closed-loop systems and sometimes destabilizes the systems. The analysis and design for systems with actuator saturation have received a lot of attentions [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34]. The robust control for uncertain Markov jump systems with actuator saturation has been discussed in [25], [26], respectively, and the definition of domain of attraction in mean square sense is introduced. For singular systems with actuator saturation, readers may refer to [27], [28], [29], [30], [31], [32], [33]. For example, Lin and Lv [29] not only gave the sufficient conditions for the stability of continuous-time singular systems with actuator saturation, but also gave the estimation of the domain of attraction and the design of state feedback gain matrix via LMI technique. L2/L1 control for continuous-time singular systems with actuator saturation was discussed in [30]. Ji et al. [31] discussed the stability for discrete-time singular systems with actuator saturation in terms of LMIs technique. In [29], [30], [31], the conditions obtained depend on the standard form of singular systems, i.e. the matrix E=diag{Ir,0}. Ma and Boukas [32] discussed the stability and H control problems for discrete-time singular systems with actuator saturation based on LMIs method. In [33], the robust stabilization for discrete-time singular systems with actuator saturation was discussed, and the given condition is only valid for the case of E being nonsingular. For discrete-time singular Markov jump systems with actuator saturation, the result is few [34]. Ma et al. [34] discussed the stability for discrete-time singular Markov jump systems with actuator saturation based on LMIs with some chosen scalars. The H control problem for discrete-time singular Markov jump systems with actuator saturation is also an important problem, and it is not simple extension to stability for the systems. To the best of our knowledge, it has not been investigated in the literature.

In this paper, the H control problem for discrete-time singular Markov jump systems with actuator saturation is considered. First, a sufficient condition is given which guarantees that the discrete-time singular Markov jump system with actuator saturation is regular, causal, bounded state stable, and satisfies the H performance. Then, with this condition, and based on singular value decomposition approach, the design method of H state feedback controller is developed, and the controller can be obtained by solving LMIs optimization problem. Last, numerical examples are given to illustrate the effectiveness of the proposed methods.

Notations: Throughout this paper, for real symmetric matrices X and Y, the notation XY (respectively, X>Y) means that the matrix XY is semipositive definite (respectively, positive definite). I is the identity matrix with appropriate dimension, the superscript “T” represents the transpose, diag{} denotes a block-diagonal matrix. x refers to Euclidean norm of the vector x, that is x2=xTx. Z denotes the set of non-negative integer numbers, and E{·} denotes the mathematical expectation. denotes the matrix entries implied by symmetry of a matrix.

Section snippets

Problem description and preliminaries

Consider discrete-time singular Markov jump systems subject to actuator saturation with the following dynamics:Ex(k+1)=A(rk)x(k)+B(rk)sat(u(k))+Bw(rk)w(k),z(k)=C(rk)x(k)+D(rk)sat(u(k))+Dw(rk)w(k),where kZ, x(k)Rn is the system state, u(k)Rp is the control input, and sat: RpRp is the standard saturation function defined as follows: sat(u(k))=[sat(u1(k))sat(u2(k))sat(up(k))]T,where without loss of generality, sat(ui(k))=sign(ui(k))min{1,|ui(k)|} [25], [26]. Here the notation of sat(·) is

H performance analysis

In this section, a sufficient condition that system (4) satisfies the H performance is provided.

First, in order to solve the H control problem for system (1) by using LMI approach, we assume that q=n.

Remark 2

Generally, qn. If q<n, then let w^(k)=w(k)w¯(k)Rn, w^(k)Wμ, and B¯w(i)=[Bw(i)0],D¯w(i)=[Dw(i)0].If q>n, then let x^(k)=x(k)x¯(k)Rq, and E^=E000,A^(i)=A(i)00I,B^(i)=B(i)0,C¯(i)=[C(i)0],B^w(i)=Bw(i)0.Therefore, the characteristic and input–output relation of system (1) do not change.

Theorem 1

Let F(i)=F¯(

Examples

In this section, we give three numerical examples to show the effectiveness of the proposed conditions.

Example 1

Consider the H control for system (1) with the following coefficient matrices: E=462121000,A(1)=11.20.60.90.550.550.250.10.25,B(1)=0.30.20.05,Bw(1)=0.10.20.3,D(1)=1,C(1)=[0.100.2],Dw(1)=0.3,A(2)=0.80.801.80.61.610.40.6,B(2)=1.20.80.6,Bw(2)=0.30.20.5,D(2)=2,C(2)=[245],Dw(2)=1,The mode switching is governed by a Markov chain that has the following transition probability p11=0.4,p12=0.6;p21=

Conclusions

In this paper, the H control problem for discrete-time singular Markov jump systems with actuator saturation is considered. A sufficient condition is obtained which guarantees that the discrete-time singular Markov jump system with actuator saturation is regular, causal, bounded state stable, and satisfies the H performance. Based on singular value decomposition approach and LMIs technique, the design method of H state feedback controller is developed, and the controller can be obtained by

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    This work is supported by National Natural Science Foundation of China (61074037, 61034007).

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