Dissipative filtering for singular Markovian jump systems with generally hybrid transition rates

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Highlights

  • A Wirtinger-type free-matrix-based integral inequality (WFMII) is proposed. The slack matrices introduced in WFMII are all coupled by time-varying delay, which makes up the gap of FMBII under utilization on slack matrices. These lead to less conservative results.

  • The GHTRs are considered, which relax the traditional assumption in MJSs that estimate error must be completely symmetric. Such a TR covers the existing TR types as its special cases, which leads to more general and practical models.

  • A double-boundary approach is proposed to deal with GHTRs, which can make full use of the GHTRs so as to loosen the conservatism.

Abstract

This paper studies the dissipative filtering of singular Markovian jump systems (SMJSs) with generally hybrid transition rates (GHTRs). The transition rates of the mode jumps are considered to be generally hybrid, which relax the traditional assumption in Markov jump systems that estimate errors must be completely symmetric. The introduced generally hybrid transition rates (GHTRs) make these systems more general and realistic. In order to deal with the GHTRs, a new approach named double-boundary approach is proposed. Then, a new integral inequality named Wirtinger-type free-matrix-based integral inequality (WFMII) is proposed to estimate Lyapunov-Krasovskii functional (LKF), in which some delay-product-type matrices are produced to fully link the relationship among time-varying delay and system states. Based on these ingredients, an explicit expression of the desired filter can be given to ensure the filtering error system to be stochastically admissible and strictly dissipative. The further examination to demonstrate the feasibility of the presented method is given by designing a filter of a two-loop circuit network.

Introduction

Markovian jump systems (MJSs) have received considerable attention owing to their potential in describing various physical models with abrupt environment changes or random disturbance. These systems can be applied to many practical engineering problems such as networked control systems, manufacturing process, aircraft systems and so on [1], [2], [3], [4], [5]. On the other hand, singular systems are also known as implicit systems, descriptor systems or semi-state systems, which have been excessively studied due to their significance in both practical applications and theoretical research [6], [7], [8], [9], [10], [11]. While in practical applications, it is inevitable that a singular system may bear abrupt changes in system structure or model parameters, which motivates the research on singular Markovian jump systems (SMJSs). During the past few decades, SMJSs have gotten considerable attention from research communities due to their wide range of applications in many practical engineering areas, such as aircraft modeling, economic systems, circuit systems, power systems and robotics [12], [13]. It is noteworthy that the SMJSs are more general and complex compared to the regular MJSs because they can better describe practical engineering systems than the regular ones. Therefore, rather than the regular MJSs, it is more significant and necessary to study and analyze the qualitative properties such as filter design of SMJSs [14], [15]. Additionally, it is known that time delay inherently arises in many practical nonlinear systems, and it has been shown that the existence of time delay may cause instability, chaotic mode, and degraded performance [16], [17], [18], [19], [20], [21]. Thus, the study of SMJSs with time delay is very important [22], [23].

As an important kind of state estimation techniques, filter design of dynamic systems is of importance, and many works have been reported [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36]. To name a few, in [26], the dissipative-based filtering problem has been investigated for fuzzy networked control systems with switching communication channels. The problem of finite-time reliable filtering has been considered for Takagi-Sugeno fuzzy semi-Markovian jump systems [27]. In [28], by combining delay-product matrices with augmented state-related terms in Lyapunov-Krasovskii functional (LKF), a less conservative H filtering is obtained for fuzzy system. To be concrete, for the term d(t)ηT(t)Mη(t), d(t)M is the delay-product matrix and η(t) is the augmented state-related term. Wherein the coupling information between augmented state-related term and time delay can be fully linked, which leads to less conservative results [37]. It should be pointed out that dissipative filtering covers H filtering [12], [13], [38] and passive filtering [33], [39] as its special cases, and many results have been obtained [14], [15], [30], [40]. To name a few, by using Jensen-based integral inequality (JBII) [41], the dissipative filtering has been derived for SMJSs with time delay [14], [40]. It is known that Wirtinger-based integral inequality (WBII) [42] is tighter than JBII. Thus, by using WBII, the dissipative filtering has been investigated for SMJSs with time delay [30]. Since free-matrix-based integral inequality (FMBII) [43] is tighter than WBII, the dissipative filtering has been investigated for SMJSs [15] by using FMBII. Recalling the main idea in [28], introducing delay-product matrix is an effective way for the reduction of conservatism. However, the delay-product matrices in [28] are positive definite. However, positive definite matrices are themselves constrained. Recalling the matrix structure in FMBII [43] such as a three-dimensional matrix [H11H12H13*H22H23**H33]0, it can be seen from [43] that the delay-product matrices mainly focus on the semi-positive definite matrices H11 and H22. However, the slack matrices H12, H13, and H23 have not been fully coupled with time-varying delay in FMBII. Thus, making full use of the slack matrices in FMBII deserves a further study.

Note that transition rates (TRs) determine the transition among different modes for MJSs, which will affect the system stability and performance. Originally, for simplicity, these TRs are assumed to be known exactly [14], [40]. However, the exact information of TRs is not easy to be obtained such that this assumption will restrict the wide engineering applications. Therefore, studying uncertain TRs attracts many researchers’ attention. The first type of the uncertain TRs is bounded uncertain TRs (BUTRs) [44], [45]. In this type, the exact values of their bounds (upper bounds and lower bounds) should be exactly known. However, the exactly bounds are also difficult to be obtained. Then, partly unknown TRs (PUTRs) are proposed and have been widely investigated such as in [15], [46]. In this type, exact elements are also necessary such that PUTRs are difficult to apply in some actual situations. As a general case, generally uncertain transition rate (GUTR) covering completely known TR (CKTR), PTUR and BUTR as its special cases have been widely reported [30], [47], [48]. Concretely, a GUTR matrix can be expressed as[π˜11+Δ11?π˜1N+Δ1N????π˜N2+ΔN2?]where π˜ij and Δij[δij,δij] denote the known estimate value and estimate error of the uncertain transition rate πij, respectively. δij is the given bound of Δij. “?” represents the unknown transition rate. Note that the estimate error is symmetric. Actually, the symmetric type estimate error is limited. Moreover, in order to solve the estimate error Δij, a common technique in the literature mentioned above is a traditional inequality, i.e., ε(Q+QT)ε2T+QT1QT. However, both the combination of this traditional estimation method and the aforementioned symmetric type estimate error in the previous works such as [30], [47], [48] lead to certain degree of conservatism and limitations, which need to be further improved.

With above analysis, this paper is concerned with the dissipative filtering for SMJSs with generally hybrid transition rates (GHTRs). The main contributions are as follows:

(I) A Wirtinger-type free-matrix-based integral inequality (WFMII) is proposed. The slack matrices introduced in WFMII are all coupled by time-varying delay, which makes up the gap of FMBII under utilization on slack matrices. These lead to less conservative results.

(II) The GHTRs are considered, which relax the traditional assumption in MJSs that estimate error must be completely symmetric. Such a TR covers the existing TR types as its special cases, which leads to more general and practical models.

(III) A double-boundary approach is proposed to deal with GHTRs, which can make full use of the GHTRs so as to loosen the conservatism.

Notation: Throughout this paper, Rn represents the n-dimensional Euclidean space. XT denotes the transpose of X. ’*’ in LMIs represents the symmetric term of the matrix. Co represents a convex set. E(·) represents the mathematical expectation. He[X] means X+XT; col[X,Y] denotes [XT,YT]T. We define x,Mxd=0dxT(s)Mx(s)ds.

Section snippets

System description and preliminaries

Consider the following singular Markovian jump systems (SMJSs) with generally hybrid transition rates (GHTRs):{Ec˙(t)=A(rt)c(t)+Ad(rt)c(td(t))+B(rt)w(t)y(t)=C(rt)c(t)+Cd(rt)c(td(t))+D(rt)w(t)z(t)=L(rt)c(t)+Ld(rt)c(td(t))+F(rt)w(t)c(t)=ϕ(t),t[h,0]where c(t)Rn, w(t)Rnw(belonging to L2[0,)), y(t)Rny, z(t)Rnz, and ϕ(t) are the system state, disturbance input, measurement output, controlled output, and initial condition, respectively. The matrix ERn×n is singular, i.e., rank(E)=r<n. If ran

Main results

In this section, we aim to design a filter (5) for SMJSs (1) with GHTRs by using WFMII such that the filtering error system (6) is stochastically admissible and dissipative.

Theorem 1

For given scalars σ, h and μ, filtering error system (6) with given filters (5) is stochastically admissible and dissipative if there exist symmetric positive-definite matrices Pi, M1, M2, Q1, Q2, R, symmetric matrices H11, H22, G11, G22, any matrices Zi, Wi, H12, H13, H23, G12, G13, G23 such that (7), (9) and the following

Numerical examples

In this section, the advantages of the proposed dissipative criterion and designed filters are demonstrated by two examples.

Example 1

Consider SMJSs (34) with the following parametersA1=[0.4972000.9541],Ad1=[1.0101.541500.5449]A2=[0.5121000.7215],Ad2=[0.85211.972100.4321]L11=[0.10.3],L12=[0.20.15]B11=[0.50.4],B12=[0.30.2],Ldi=Fi=0the singular matrix and transition rate matrix are given asE=[1000],Π=[0.45+Δ110.45+Δ12π2π2]where Δ11[δ1,0.05]. The following three cases will be given for different π2

Conclusions

This paper studies the dissipative filtering of singular Markovian jump systems (SMJSs) with generally hybrid transition rates (GHTRs). The estimate error in the GHTRs are considered to be general such that GHTRs cover the existing completely known, partly unknown, bounded uncertain, and generally uncertain TRs as their special cases. A double-boundary approach is proposed to deal with the GHTRs, which avoids some conservative techniques. By inserting two delay-dependent terms, Wirtinger-type

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 61973070, Liaoning Revitalization Talents Program under Grant XLYC1802010, in part by SAPI Fundamental Research Funds under Grant 2018ZCX22, and in part by the Fundamental Research Funds for the Central Universities under Grant N2104003.

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