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Stability analysis for neutral Markovian jump systems with partially unknown transition probabilities

https://doi.org/10.1016/j.jfranklin.2012.04.003Get rights and content

Abstract

This paper addresses the problem of the delay-dependent stability for neutral Markovian jump systems with partial information on transition probability. The time delays discussed in this paper are time-varying delays. Combined the new constructed Lyapunov functional with the introduced free matrices, and using the analysis technique of matrix inequalities, the delay-dependent stability conditions are obtained. The obtained results are formulated in terms of LMIs, which can be easily checked in practice by Matlab LMI control toolbox. Three numerical examples are given to show the validity and potential of the developed criteria.

Highlights

► We concern the neutral MJS with partially unknown transition probabilities. ► The new method makes full use of the information of transition probabilities. ► The conditions are easily computed and less conservative based on LMIs.

Introduction

As well known, neutral systems are frequently encountered in various engineering systems, including population, ecology, distributed networks containing lossless transmission lines, heat exchangers, and repetitive control. For the last two decades, a great deal of attention has been drawn to the study of neutral systems, such as [1], [2], [3], [4], [5], [12], and the references therein. Since many complex systems can be modeled preferably as switched neutral systems, recently, the stability and stabilization for switched neutral systems have received much more attention [3], [6], [7], [8], [9], [10], [11], [12], [13], and the references therein. In [3], [12], we designed switched rule for the switched neutral systems to overcome the constraints on the radius of neutral matrix, improved the conditions theretofore. For all practical purposes, we have discussed the delay-dependent BIBO stability analysis for the switched neutral systems in [9].

However, just as most of the related authors for studying the switched neutral systems, we have not considered the Markovian switching, an important and special stochastic switching style. Fortunately, there are some reports about the stability and stabilization for Markovian neutral systems in the literature, such as [14], [15], [16], [17], [18], [27], [28], [29], and the references therein. The exponential stability for uncertain neutral systems with Markov jumps was discussed in [17], and the corresponding systems with interval time-varying delays were also studied by Balasubramaniam et al. [15]. The problem of stabilization for Markovian neutral systems was discussed in [14], [16], [18]. It should be noted that all these results are obtained provided that the complete knowledge of the transition probabilities can be given. Moreover, since the transition probabilities in the jumping process determine the system behavior to a large extent, and because the ideal knowledge on the transition probabilities is definitely expected to simplify the system analysis and design, many analyses and synthesis results of the other dynamic systems have been reported [20], [21], [22], [23], [24], [25], [26] and the reader is referred to the references therein. By constructing a novel Lyapunov–Krasovskii functional with the idea of partitioning the time delay, and a sufficient condition was derived from the H performance for Markovian jump linear systems with norm-bounded parameter uncertainties and time-varying delays in [23]. For uncertain Markovian jump systems with time delay, Li et al. [26] presented the parameter-dependent stability conditions in the form of linear matrix inequalities (LMIs). Not only that Zhu et al. [25] studied the stochastic stability of delayed recurrent neural networks with both Markovian jump parameters and nonlinear disturbances.

However, the likelihood of obtaining such available knowledge is actually questionable, and the cost is probably expensive. So, rather than having a large complexity to measure or estimate all the transition probabilities, it is significant and necessary, from control perspectives, to further study more general jump systems with partly unknown transition probabilities. Recently, many results on the Markovian jump systems with partly unknown transition probabilities are obtained [30], [31], [32], [33], [34], [35]. Most of these results just require the knowledge of the known elements, such as the bounds or structures of uncertainties, and some of the unknown elements need not be considered. It is a great progress on the analysis of Markovian jump systems. However, few of these papers have considered the effect of delay to the stability or stabilization conditions. To the best of the authors’ knowledge, the problem of delay-dependent stability results for Markovian jumping neutral systems with time-varying delays has not been investigated and it is very challenging.

In this paper, the delay dependent stability problem of neutral Markovian jump linear systems with partly unknown transition probabilities is investigated. First, by introducing some free matrices and using the novel analysis technique of matrix inequalities, stability condition for the nominal neutral Markovian systems is presented, based on a class of Lyapunov functionals. Second, to decrease the conservativeness, for the special case of complete unknown transition probabilities, the common Lyapunov functional is constructed for the stability analysis. Third, as a supplementary study, the stability result for uncertain neutral Markovian jump systems is presented. All the sufficient conditions are formed in terms of LMIs, which can be easily calculated by the MATLAB LMI control toolbox [38]. Furthermore, the number of matrix inequalities conditions obtained in this paper is much more than some existing results which may increase the complexity of computation, due to the introduced free matrices based on the lemmas about matrix inequalities introduced in this paper. However, it would decrease the conservative for the delay-dependent stability conditions. Finally, three numerical examples are provided to illustrate the validity of our results. In order to illustrate the proposed results, one of these three examples is a real-world example, that is, partial element equivalent circuit (PEEC) model with Markovian jump presented in [15].

Section snippets

Problem statement and preliminaries

Consider the following neutral systems with Markovian jump parameters:x˙(t)C(rt)x˙(tτ(t))=A(rt)x(t)+B(rt)x(tr(t)),x(t0+θ)=φ(θ),θ[ρ,0],where x(t)Rn is the state vector, τ(t) and r(t) are the time-varying delays which satisfy 0τ(t)τ,τ˙(t)τd,0r(t)r,r˙(t)rd, ρ=max(τ,r), and φ(θ) is the initial condition function. {rt},t0 is a right-continuous Markov process on the probability space taking values in a finite state space, ={1,2,,N} with generator I=(λij),i,j given byPr{rt+Δ=j|rt=i}=λ

Main results

This section will state the stability analysis for neutral Markovian jump systems with partly unknown transition rate. With the introduced free matrices and novel matrix inequalities analysis, the stability conditions are presented.

Theorem 3.1

The system (1) with a partly unknown transition rate matrix (3) is stochastically stable if there exist matrices Pi>0,Q1i>0,Q2i>0,Q3>0,Q4>0,R1>0,R2>0,R3>0, any matrices Wi=WiT,Mi=MiT,Si=SiT,Fk,Gk,Nk(k=1,,6) with appropriate dimensions satisfying the following

Extension to uncertain neutral Markov jump systems

In this section, we will consider the uncertain neutral Markov jump systems with partially unknown transition probabilities as follows:x˙(t)C(rt)x˙(tτ(t))=(A(rt)+ΔA(rt))x(t)+(B(rt)+ΔB(rt))x(tr(t)).A(rt)andB(rt) are the known mode dependent constant matrices with appropriate dimensions. While ΔA(rt) and ΔB(rt) are the time-varying but norm-bounded uncertainties satisfying(ΔA(rt)ΔB(rt))=LrtFrt(t)(H1(rt)H2(rt)),where Lrt,H1(rt),andH2(rt) are the known mode-dependent matrices with appropriate

Examples

In order to show the effectiveness of the approaches presented in the above section, three numerical examples are provided.

Example 1

Consider the MJLS (1) with four operation modes whose state matrices are listed as following:A1=1.150.751.501.50,A2=2.150.491.502.10,A3=1.300.151.501.80,A4=1.900.341.501.65,B1=1.200.120.240.25,B2=1.450.960.471.57,B3=0.580.680.130.96,B4=0.671.501.391.23,C1=0.150.060.500.50,C2=0.250.350.050.65,C3=0.330.070.190.36,C4=0.230.160.020.57.The partly

Conclusion

The delay-dependent stability for neutral Markovian jump systems with partly known transition probabilities has been investigated. Based on a new class of stochastic Lyapunov–Krasovskii functionals constructed, and combined with the technique of analysis for matrix inequalities, some new stability criteria are obtained. The main contribution of this paper contains the following twofold: one is the extension of delay-dependent stability conditions for Markovian jump delay systems to Markovian

References (41)

  • L. Xiong et al.

    New stability and stabilization for switched neutral control systems

    Chaos, Solitons and Fractals

    (2009)
  • T.-F. Li et al.

    Stability and L2-gain analysis for switched neutral systems with mixed time-varying delays

    Journal of the Franklin Institute

    (2011)
  • P. Balasubramaniam et al.

    Exponential stability results for uncertain neutral systems with interval time-varying delays and Markovian jumping parameters

    Applied Mathematics and Computation

    (2010)
  • Z. Shu et al.

    Robust stabilization of Markovian delay systems with delay-dependent exponential estimates

    Automatica

    (2006)
  • G.-L. Wang et al.

    Design of reduced-order H filtering for Markovian jump systems with mode-dependent time delays

    Signal Processing

    (2009)
  • H.-Y. Zhao et al.

    H guaranteed cost control for uncertain Markovian jump systems with mode-dependent distributed delays and input delays

    Journal of the Franklin Institute

    (2009)
  • X.-D. Zhao et al.

    New robust delay-dependent stability and H analysis for uncertain Markovian jump systems with time-varying delays

    Journal of the Franklin Institute

    (2010)
  • Y.-Y. Cao et al.

    Delay-dependent stochastic stability and H analysis for time-delay systems with Markovian jumping parameters

    Journal of the Franklin Institute

    (2003)
  • Q.-X. Zhu et al.

    Stochastically asymptotic stability of delayed recurrent neural networks with both Markovian jump parameters and nonlinear disturbances

    Journal of the Franklin Institute

    (2010)
  • H.-Y. Li et al.

    Parameter-dependent robust stability for uncertain Markovian jump systems with time delay

    Journal of the Franklin Institute

    (2011)
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    This work was supported by the Mathematical Tianyuan Foundation of China (Grant No. 11126305), the Natural Science Foundation of Yunnan Province (Grant No. 2011FZ172), the Youth Foundation of Yunnan University of Nationalities (Grant No. 11QN07).

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