L2-L Filtering for neutral Markovian switching system with mode-dependent time-varying delays and partially unknown transition probabilities

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Abstract

This paper investigates the problem of L2-L filtering for neutral Markovian switching systems with partially unknown transition probabilities for different system mode and delay mode. The system under consideration involves discrete and mode-dependent time-varying delays. Based on the Lyapunov–Krasovskii functional, an approach to design a filter such that the filtering error system is stochastically stable with a prescribed L2-L performance. By using free weighting matrices, free-connection weighting matrix method and convex combination approach, sufficient conditions for the existence of L2-L filters are expressed in terms of linear matrix inequalities (LMIs), which can be solved by using Matlab LMI control toolbox. A numerical example is given to illustrate the effectiveness and potential of the proposed method.

Introduction

It is well known that Markovian jump systems (MJSs) have been widely studied in the past decades. The application of MJSs has the advantage of modeling many practical dynamic systems subject to abrupt variation in their structures, such as manufacturing systems, networked control systems, fault-tolerant control systems, etc. MJSs which initially introduced by Krasovskii and Lidskii in 1961 [1] is a special class of hybrid systems, which is specified by two components, the first component refers to the mode, which is described by a continuous-time finite-state Markovian process, and second one refers to the state which is represented by a system of differential equations. The stability analysis, controller synthesis and filtering for MJSs with the assumption that the elements of transition probabilities are completely known have been widely studied in the literature, see for example [2], [3], [4], [5], [6], [7]. A recent extension is to consider the systems with uncertain transition probabilities [8], in which the robust methodologies are adopted to cope with the parameter uncertainties in the transition probabilities matrix. Different from the aforementioned references, in most cases the transition probabilities of MJSs are not exactly known. A typical example can be found in networked control systems, where the packet dropouts can be vague and random in different running periods of networks – all or part of the elements in the desired transition probabilities matrix are hard or costly to obtain. The same problem may occur in other practical systems such as the DC motor in position control servomechanisms [9]. Therefore, it is significant and necessary to study more general jump systems with partly unknown transition probabilities. Recently, many results on the MJSs with partly unknown transition probabilities have been obtained, see [10], [11], [12], [13], [14], [15], [16], [17]. Most of these results just requires the knowledge of the known elements, such as the bounds or structures of uncertainties. Further, some of the unknown elements in the transition probability matrix need not be considered.

Time delay is very common in practical dynamical systems such as manufacturing systems, networked control systems, chemical systems, telecommunication, economic systems and biological systems, etc., The presence of time delay can degrade the performance of the system and can even destabilize the system. Many dynamic systems not only depend on the present and the past states but also involve the derivatives with delays. Such systems can be modeled by functional differential equations of the neutral type. Practical examples of neutral systems include the distributed networks containing lossless transmission lines, population ecology, processes including steam or water pipes, heat exchanges, etc. Stability and stabilization for such systems (neutral systems) have been widely investigated by many authors [18], [19], [20], [21], [22]. Results on stability analysis of uncertain neutral systems have been considered in [23], [24], [25], [26]. The authors in [27], [28], [29] have investigated the stability of neutral systems with multiple delays.

The problem of filtering or state estimation has been widely studied in the fields of signal processing and control application. The aims of H and L2-L filters are to design estimators which minimize the L2 induced and energy-to-peak gains from the exterior noises to the filtering errors. Many useful results on estimation and filtering for all kinds of dynamic systems have been reported. H filtering for nonlinear neutral systems have been presented in [30], [31], while the authors in [32] have investigated the problem of L2-L filtering for neutral jump system with time-varying parameters and in [33], authors have dealt with L2-L filtering for MJSs with time-varying delays and partly unknown transition probabilities. To the best of the authors’ knowledge, the problem of L2-L filtering for neutral Markovian switching system and partially unknown transition probabilities with different system mode and delay mode has not been investigated in any of the existing literature and it is very challenging problem for investigation.

Hence, in this paper, we discuss the problem of L2-L performance for neutral Markovian switching system with mode-dependent delays in which the delay mode and system mode depend on different jumping modes. By using the Lynapunov-Krasovskii functional approach, free-connection weighting matrix method and lower bound lemma for a reciprocally convex combination of scalar positive functions, a new mode delay-dependent sufficient condition on stochastic stability with the L2-L performance is derived in terms of LMIs. Based on this, we have presented the existence of the desired L2-L filter, which guarantees the filtering error system to be stochastically stable with a prescribed level of noise attenuation. The system considered in this paper is a more general system, which can be transformed into a neutral form of Partial Element Equivalent Circuit (PEEC) model.

The rest of this paper is organized as follows. Section 2 provides the problem description and preliminaries. The analysis of stability and L2-L filter synthesis for the filtering error system is given in Section 3. In Section 4, numerical example is given to illustrate the effectiveness of the derived results. Finally, Section 5 concludes the paper.

Notation: Throughout this paper, Rn denotes the n dimensional Euclidean space, λmax(Q) and λmin(Q) denote, respectively, the maximal and minimal eigenvalue of matrix Q. (Ω,F,P) is a probability space, Ω is the sample space, F is the σ-algebra of the sample space and P is the probability measure on F. E{·} refers to the expectation operator with respect to some probability measure P. We use diag{·,·,·} as a block-diagonal matrix. A>0(<0) means A is a symmetric positive (negative) definite matrix, A-1 denotes the inverse of matrix A. AT denotes the transpose of matrix A,I is the identity matrix with compatible dimension.

Section snippets

System description and Definitions

Consider the following neutral systems with Markovian jump parameters:ẋ(t)-Ad(rt)ẋ(t-τd(t,st))=A(rt)x(t)+Ah(rt)x(t-τh(t,st))+B(rt)ω(t)y(t)=C(rt)x(t)+Ch(rt)x(t-τh(t,st))+D(rt)ω(t)z(t)=L(rt)x(t),x(t)=ϕ(t),t[-τ,0],where x(t)Rn is the state vector, ω(t)Rl is the noise signal which belongs to L2[0,);y(t)Rp is the measurement output; z(t)Rq is the output signal to be estimated; ϕ(t) is a compatible vector-valued initial function defined on [-τ,0]; A(rt),Ah(rt),Ad(rt),B(rt),C(rt),Ch(rt),D(rt)

Main results

In this section, we are concerned with the problem of L2-L filtering for neutral Markovian jump systems with partly unknown transition probabilities.

Theorem 3.1

Given scalars τh,τd,τhj,τdj,μhj, and μdj, the filtering error system (7) is stochastically stable with an L2-L performance γ for any time delay τhj(t),τdj(t) satisfying (4), if there exist matrices Pij0,Qmij>0,Rm>0 and matrices Fmi=FmiT,(m=1,2,3,4),Sm,(m=1,2),Nˆ with appropriate dimensions such that for each iS1,jS2 the following LMIs are

Numerical example

In this section, a numerical example is provided to illustrate the effectiveness of the derived theoretical results.

Example 4.1

Consider the neutral Markovian jump system (1) with three operation modes whose state matrices are defined asA1=-300.6-1,Ah1=-1.20.3-1-1.3,Ad1=-0.100-0.2,B1=0.7-0.1,C1=0.9-2.1,Ch1=1.50,D1=0.6,L1=0.90.1;A2=-111-0.6-3.3,Ah2=-2.2-1.30.7-2.1,Ad2=-0.100-0.2,B2=0.9-0.9,C2=-0.6-0.5,Ch2=1.2-0.9,D2=-0.2;L2=-1.10.5;A3=-1.50.20.5-1.1,Ah3=-0.20.6-0.2-0.2,Ad3=-0.500-0.5,B3=0.8-0.2,C3=0.8-2.2,Ch

Conclusion

In this paper, the problem of L2-L filtering for neutral Markovian switching systems and partially unknown transition probabilities for different system mode and delay mode has been investigated. By choosing a new class of Lyapunov–Krasovskii functional, a sufficient condition has been derived to propose the stochastic stability criteria with an L2-L performance by using LMI technique, free-weighting matrix method, free connection weighting matrices method and convex optimization approach.

Acknowledgement

The work of J.H. Park was supported by 2013 Yeungnam University Research Grant.

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