Unified filters design for singular Markovian jump systems with time-varying delays

Abstract

This paper considers the problem of unified mode-dependent and mode-independent filters design for continuous-time singular systems with Markovian jumping parameters and time-varying delays. The main objective of this paper is to design the $L_2-L_{\infty }$, $H_{\infty }$, passivity, strict (Q, S, R)-dissipativity, very-strict passivity filters in a unified framework so as to ensure that the filter error system is stochastically admissible and extended dissipative. By tuning the weighting matrices, the extended dissipativity performance can reduce to the $L_2-L_{\infty }$ performance, $H_{\infty }$ performance, passivity, strict (Q, S, R)-dissipativity and very-strict passivity, respectively. A mode-dependent and delay-dependent stochastic Lyapunov–Krasovskii functional is proposed to reflect the information of Markovian jump modes and the time-varying delays, and a set of linear matrix inequalities (LMIs) are utilized to derive sufficient conditions which guarantee that the desired unified filters can be constructed. Three numerical examples including an oil catalytic cracking process (OCCP) are employed to verify the usefulness and effectiveness of the main results obtained.

Introduction

Over the past decades, singular systems (also referred to as generalized state-space systems, differential-algebraic systems, descriptor systems, implicit systems, or semistate systems) have acquired much attention due to their extensive practical applications in various kinds of engineering systems, such as electrical circuits, large-scale electric networks, flexible robots, economics, chemical processes and other areas [1], [2]. Compared with regular systems, the singular systems are more complicated because the stability, regularity and non-impulsiveness (for continuous case) or causality (for discrete case) have to be considered concurrently. Moreover, singular systems provide a more natural description of dynamic systems than the regular systems since the structure of physical systems can be preserved more accurately by considering non-dynamic constraints and impulsive behavior [3], [4]. Up to now, a considerable number of fundamental concepts and results have been extended successfully from regular systems to singular systems; see, e. g., [5], [6], [7], [8], [9], [10], [11] and the references therein.

On the other hand, Markovian jump systems (MJSs) have attracted growing attention in recent years due to the important ability to appropriately portray a great deal of practical systems with abrupt changes in their structures and parameters, which caused by sudden environmental disturbances, repairs, disconnection and failures of some components, etc. As a special kind of hybrid stochastic systems, MJSs have been widely applied to depict networked control systems, manufacturing systems, fault-tolerant control systems, and so on [12], [13], [14]. If singular systems undergo abrupt changes in the parameters and structures, which gives rise to the well-known singular Markovian jump systems (SMJSs). Recently, much attention has been paid to SMJSs, and lots of achievements on the control and filtering have been published; see, for instance, [15], [16], [17], [18], [19], [20] and the references therein.

It should be pointed out that the well-known filtering of dynamic systems is a branch of state estimation theory, which is used to construct a filer such that the filtering error system is stable, furthermore, the output error and the disturbance input satisfy a prescribed performance level. Until now, there are a great deal of significant outcomes on the filtering problems such as H₂ filtering, $H_{\infty }$ filtering, $H_2/H_{\infty }$ filtering, $L_2-L_{\infty }$ filtering, dissipative filtering, etc. [9], [10], [11], [13], [19], [20], [21], [22], [23], [24]. Noting that dissipativity is a generalization of the concept of passivity in electrical networks and other dynamical systems, which denotes that the increase in stored energy in the system is never larger than the supplied energy from the outside environment.

Since the concept of dissipative theory was introduced and further extended in [25], [26], [27], [28], [29], [30], [31], it has been an important approach to the analysis and the synthesis of various kinds of control systems owing to the fact that dissipative theory includes some well-known performance indices such as $H_{\infty }$ performance, passivity and positive realness. To mention a few, [28] established the new results for the delay-independent and delay-dependent problems of dissipative analysis and state-feedback synthesis for a class of nonlinear systems with time-varying delays and convex polytopic uncertainties; [29] developed the complete results on the problem of dissipative analysis for a class of switched systems with time-varying delays; based on discretized Jensen inequality and lower bounds lemma, [30] proposed the problem of dissipativity analysis for discrete-time stochastic neural networks with discrete and finite distributed delays that is described by discrete-time Markov chain. Very recently, dissipative control and filtering for singular and MJSs have been studied; see, for example, [23] investigated the robust reliable dissipative filtering for discrete delay singular systems; [24] studied mixed $H_{\infty }$ and passive filtering for singular systems with time delays; [32], [33] dealt with passivity and passification for MJSs; while [34], [35], [36] considered dissipative control for SMJSs.

Meanwhile, the $L_2-L_{\infty }$ performance (also known as energy-to-peak performance or extended H₂ performance) is a very important index and the $L_2-L_{\infty }$ filtering for various control systems has received considerable attention since [21]. Specially, [8], [22] investigated the $L_2-L_{\infty }$ filtering problem for singular systems by using linear matrix inequality approach. Unfortunately, they did not give the preconditions that ensure the $L_2-L_{\infty }$ performance of singular systems can be effectively exhibited. Another factor we must take into consideration is that the aforementioned $L_2-L_{\infty }$ performance cannot be included in the dissipativity. Then, a natural and meaningful question arises: how to deal with the dissipative filtering and $L_2-L_{\infty }$ filtering problems for singular systems in a unified framework. Its significance is that by tuning the weighting matrices, the extended dissipativity could reduce to the $L_2-L_{\infty }$ performance, $H_{\infty }$ performance, passivity and dissipativity, respectively.

Until quite recently, [14] gave a positive answer to the interesting question of whether is it possible to solve the dissipative filtering and $L_2-L_{\infty }$ filtering problems for MJSs in a unified framework. With the extended dissipativity performance index (which includes dissipativity and $L_2-L_{\infty }$ performance index as special cases) was introduced in [14], the idea has been used in the works such as [31], [33], to deal with extended dissipative analysis and control for neural networks with time-varying delays and sampled-data Markov jump systems in a unified framework, respectively. However, to the best of the authors’ knowledge, so far, there are few reports on the unified $L_2-L_{\infty }$ filtering and dissipative filtering for singular systems, not to mention for SMJSs. This motivates the present research.

It is worth pointing out that the unavoidable time delays often damage the performance and stability of dynamic systems as well as SMJSs, so the research for SMJSs with time delays is of considerable importance [2], [3], [4], [8], [37]. In this paper, we will deal with the problem of unified filters design for singular Markovian jump time-varying delay systems based on extended dissipativity performance index introduced in [14], so that the gap between $L_2-L_{\infty }$ and dissipativity performance can be filled. Our main aim is to design the $L_2-L_{\infty }$, $H_{\infty }$, passivity, strict (Q, S, R)-dissipativity, very-strict passivity filters in a unified framework so as to guarantee that the filter error system is stochastically admissible and extended dissipative. By tuning the weighting matrices, the extended dissipativity performance will reduce to the $L_2-L_{\infty }$ performance, $H_{\infty }$ performance, passivity, strict (Q, S, R)-dissipativity and very-strict passivity, respectively. A mode-dependent and delay-dependent stochastic Lyapunov–Krasovskii functional is employed to reflect the information of Markovian jump modes and the time-varying delays, and a set of linear matrix inequalities are applied to derive sufficient conditions which ensure that the desired unified filters can be constructed. Three numerical examples including an oil catalytic cracking process (OCCP) are used to expound the usefulness and effectiveness of the main results obtained.

The main contributions of this paper are summarized as follows: (1) the unified filters including $L_2-L_{\infty }$, $H_{\infty }$, passivity, strict (Q, S, R)-dissipativity, very-strict passivity filters have been designed for singular Markovian jump time-varying delay systems based on extended dissipativity; (2) a mode-dependent and delay-dependent stochastic Lyapunov–Krasovskii functional is employed to reflect the information of Markovian jump modes and the time-varying delays; (3) the preconditions are introduced so as to ensure that the $L_2-L_{\infty }$ performance of singular systems can be effectively demonstrated; (4) the idea behind this paper could be utilized to design mode-dependent and mode-independent unified filters for parameters uncertain SMJSs with bounded or partly unknown transition rates.

Notation: The notation used throughout this paper is justly standard. I denotes the identity matrix of appropriate dimensions. M^T represents transpose of the matrix M. $X>0$ implies that symmetric matrices X is positive definite. ${\mathbb{R}}^n$ and ${\mathbb{R}}^{n\times m}$ stand for the n-dimensional Euclidean space and set of all $n\times m$ real matrices, respectively. Ker(A) represents the kernel space(null space) of matrix A. $\left(\Omega ,\mathcal{F},\mathcal{P}\right)$ means a complete probability space, $\mathcal{E}\left\{·\right\}$ represents expectation operator with respect to the given probability measure $\mathcal{P}$, ${\mathcal{L}}_2[0,\infty )$ refers to the space of square-integrable vector functions over $[0,\infty )$. $\Vert ·\Vert$ denotes the spectral norm for matrices, $\left|·\right|$ stands for the Euclidean norm in ${\mathbb{R}}^n$, ${\Vert ·\Vert }_2$ represents the usual ${\mathcal{L}}_2[0,\infty )$ norm, ${\Vert ·\Vert }_{E_2}$ refers to the norm in ${\mathcal{L}}_2\left(\right(\Omega ,\mathcal{F},\mathcal{P}),[0,\infty \left)\right)$. ${〈z,\omega 〉}_t$ means ${\int }_0^{t}z{(s)}^T\omega \left(s\right)\mathit{ds}$. Let $\mathit{diag}\{\cdots \}$ represent a block diagonal matrix, $(⁎)$ denote a term that is induced by symmetry in block symmetric matrices or long matrix expressions.