Robust finite-time H control for uncertain discrete jump systems with time delay

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Abstract

This paper addresses the robust finite-time H control problem for a family of uncertain discrete-time Markovian jump systems with time delay. Initially, the concepts of stochastic finite-time boundedness and stochastic H finite-time stabilization are given. Then, a stochastic finite-time H controller is designed such that the underlying closed-loop discrete jump time-delay system is stochastic finite-time boundedness and satisfies a prescribed H performance level in some given finite-time interval. Based on Lyapunov approach, sufficient criteria on the stochastic H finite-time stabilization are established for the solvability of the problem, which can be handled by a feasibility problem in the form of linear matrix inequalities with a fixed parameter. Finally, simulation examples are presented to demonstrate the validity of the developed techniques.

Introduction

Stochastic systems have become an important research topic since it has been well recognized that the class of dynamic systems have variable structures subject to stochastic abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, sudden environment changes, and so on. Meanwhile, Markovian jump systems, which are referred to as one special family of hybrid systems and stochastic systems and modeled by a set of subsystems with transitions among the models determined by a Markov chain taking values in a finite set, have received considerable attention in the control community. In the last few decades, many results and a large variety of control problems have been widely studied, such as stochastic Lyapunov stability [1], [2], [3], optimal control [4], [5], robust control [6], [7], H control [8], [9], [10], [11], [12], H filtering design [13], [14], [15], [16], guaranteed cost control [17], sliding mode control [18], [19] and tracking control [20]. For more results on this issue, the readers to refer [21], [22] and the references therein.

It is well known that the classical Lyapunov theory, just as was mentioned above, deals mainly with the state convergence property of the systems in infinite time interval. Often Lyapunov stability is enough for practical applications, but there are some cases where large values of the state are not acceptable in the presence of saturations [23], [24], [25]. To tackle this transient performance of control dynamics, some early results on finite-time stability (FTS) or short-time stability were found in the references [26], [27], [28], [29]. In addition, it is necessary to point out that finite-time stability and Lyapunov asymptotic stability are independent concepts: a system could be FTS but not Lyapunov asymptotically stable, and a Lyapunov asymptotical stable system may not be FTS since the transient of a system response may exceed the bound [23]. Based on Lyapunov function approach or linear matrix inequality (LMI) techniques, varieties of results on finite-time stability, finite-time boundedness and finite-time stabilization were obtained for continuous- or discrete-time systems including linear systems [30], [31], [32], [33], [34], nonlinear systems [35], neural networks systems [36], [37], switching systems [38], [39] and singular systems [40], [41]. For more details of the literature related to finite-time stability, finite-time boundedness and finite-time H control, the reader is referred to [42], [43], [44], [45], [46] and the references therein. It is worth mentioned that, in many practical applications, discrete-time Markovian jump time-delay systems become more important than their continuous-time counterparts when implementing the control laws in a digital way. However, to date and to the best of our knowledge, the problem of finite-time H control of uncertain discrete-time Markovian time-delay systems has not been investigated and it motivates the main purpose of our study.

As we know, control problem of time-delay systems has been a topic of recurring interest over the past decades since time delays are often the main causes for instability and performance deterioration of systems and encountered in many practical systems such as chemical processes, communication networks, economics and mechanics [47], [48], [49]. Due to the presence of time delay, there are latest research methods to reduce the conservativeness in stabilization and performance index analysis of time-delay systems, such as slack matrix method, delay-divisioning method and input–output method, see the references [50], [51], [52], [53]. In this paper, we concentrate on the robust finite-time H control problem for a family of discrete-time Markovian jump time-delay systems with parametric uncertainties and time-varying norm-bounded disturbance. Our results are totally different from those previous results, although some studies on robust finite-time stabilization for discrete-time systems have been investigated and the results mainly focused on the systems without time delay, see the references [32], [42], [54], [55]. The main contribution of this paper is to design a stochastic finite-time H controller which ensures stochastic finite-time boundedness of the uncertain discrete-time Markovian jump time-delay system and a prescribed H performance level can be achieved in the given finite-time interval. Sufficient criteria are presented for the solvability of the problem, which can be handled by a feasibility problem in the form of LMIs with a fixed parameter. Numerical examples demonstrate the validity of the proposed methods.

The structure of this paper is organized as follows. Section 2 is devoted to problem statement and preliminaries. The results on the results of stochastic finite-time H control are provided for a class of uncertain discrete-time Markovian jump systems with time delay in Section 3. Section 4 presents numerical examples to demonstrate the validity of the proposed methodology. The conclusions are drawn in Section 5.

Notations. In the sequel, Rn, Rn×m and Zk0 denote the sets of n component real vectors, n×m real matrices and the set of nonnegative integers, respectively. E{.} stands for the mathematical expectation with some probability measure P. The symbol is used to denote a matrix which can be inferred by symmetry and diag{} stands for a block-diagonal matrix. Notations sup and inf denote the supremum and infimum, respectively. λmin(P) and λmax(P) denote the smallest and the largest eigenvalue of matrix P, respectively. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

Section snippets

Problem statement

Consider the following discrete-time Markovian jump system (DMJS) with time delay described byx(k+1)=A(rk)x(k)+Ah(rk)x(k-h)+B(rk)u(k)+G(rk)w(k),z(k)=C(rk)x(k)+Ch(rk)x(k-h)+D1(rk)u(k)+D2(rk)w(k),x(j)=φ(j),j{-h,,0},where x(k)Rn is the state variable, u(k)Rm is the control input, z(k)Rp is the control output of the system, h is a positive integer denoting the constant delay time of the state in the system, φ(j) is the initial conditions. The stochastic jump process {rk,k0} is a

Main results

In this section, firstly we provide stochastic H finite-time stabilization analysis of the nominal time-delay DMJS (1a)–(1c). Then, these results will be extended to the uncertain time-delay DMJS. LMI conditions are established to show that the nominal or uncertain time-delay DMJS (6a) and (6b) is finite-time boundedness and the output z(k) and disturbance w(k) satisfies the constraint condition (8).

Theorem 1

The time-delay DMJS (6a) is SFTB with respect to (δx,ϵ,Ri,N,d), if there exist scalars μ1 and ϵ

Numerical examples

Example 1

This example is used to show stochastic H finite-time stabilization for the nominal DMJS. Consider a time-delay DMJS with two modes (1a)–(1c) described as follows:

• Mode 1:A1=1.5001,Ah1=1000.5,B1=01,G1=01,C1=0.50,Ch1=00.5,D11=0.5,D21=1,

• Mode 2:A2=1101,Ah2=010-0.5,B2=10,G2=01,C2=0.50,Ch2=00.5,D12=0.1,D22=1.In addition, the transition rate matrix is given byΠ=0.70.30.40.6.Given the initial values for R1=R2=I2,δx=1,N=5,d=1 and h=1, by Theorem 3, the optimal bound with minimum value of ϵ2+γ2

Conclusions

This paper studied the robust finite-time H control problem for a family of uncertain DMJSs with time delay. Firstly, the concepts of stochastic finite-time boundedness and stochastic H finite-time stabilization are given. Then, a stochastic finite-time H controller is designed to ensure stochastic finite-time boundedness of the resulting closed-loop time-delay DMJS with a prescribed H performance level in the given finite-time interval. Sufficient criteria are presented for the solvability

Acknowledgements

This work was supported by Doctoral Foundation of Henan University of Technology under Grant 2009BS048, supported by Foundation of Henan Educational Committee under Grant 2011A120003 and 2011B110009, and supported by the Natural Science Foundation of Henan Province of China under Grant 122300410221. The authors thank the referees and the editors for their constructive criticism which helped improving the presentation of the paper.

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