- Open Access
Delay-dependent non-fragile robust dissipative filtering for uncertain nonlinear stochastic singular time-delay systems with Markovian jump parameters
© Ma and Yan; licensee Springer 2013
Received: 15 March 2013
Accepted: 24 April 2013
Published: 10 May 2013
The problem of delay-dependent non-fragile and robust dissipative filtering is investigated for a class of uncertain nonlinear stochastic singular time-delay Itô-type systems with Markovian jump parameters. With the system uncertainty and the filter gain perturbations, the closed-loop filtering error system is robust asymptotically stable and satisfies the dissipation performance. By constructing a Lyapunov-Krasovskii function and applying the Itô differential formula to compute the differential function along the system, a sufficient condition for the existence of a robust and non-fragile dissipative filter is derived in the form of linear matrix inequality (LMI). An illustrative numerical example is provided to demonstrate the effectiveness of the proposed approach.
As an important kind of hybrid systems, Markovian jump systems have received increasing attention in the past few years due to the fact that they have strong practical relevance in mechanical systems, economics systems with human operators, and other engineering areas [1–4]. Partially mode-dependent filter design problem was tackled for discrete-time Markovian jump systems with partly unknown transition probabilities in .
Note that stochastic systems with Brownian motions, governed by the Itô differential equations, have attracted many researchers’ attention over the past decades due to the extensive application of stochastic modeling in mechanical systems, economics, and other areas. In , the filtering problem was solved for uncertain stochastic time-delay systems with sector-bounded nonlinearities. In , the joint state filtering and parameter estimation problem was solved for linear stochastic time-delay systems. In , the robust filtering problem was solved for discrete nonlinear stochastic systems with time-varying delay. In , the problem of non-fragile filtering was thoroughly studied for uncertain stochastic time-delay systems.
For the stochastic case, the Markovian switching problem has received considerable attention, and a number of traditional approaches have been proposed in the literature. In , the delay-range-dependent robust filtering problem was neatly solved for uncertain stochastic systems with mode-dependent time delays and Markovian jump parameters. In , the non-fragile filtering problem for uncertain stochastic time-delay systems with Markovian jump parameters was thoroughly studied. In , the robust exponential filtering was exploited to uncertain stochastic time-delay systems with Markovian switching and nonlinearities. It is well known that stochastic time-delay systems with Markovian jump parameters play a very important role in digital signal analysis and processing. However, despite its importance, up to now, the delay-dependent non-fragile robust dissipative filtering problem for general uncertain stochastic time-delay systems with Markovian jump parameters has not been fully investigated and the relevant results have been very few.
On the other hand, a great deal of attention has been devoted to the study of singular systems over the past decades. Singular systems are referred to as descriptor systems, implicit systems, generalized state-space systems, or semi-state systems. The singular Markovian jump time-delay systems have also been investigated by many researchers. In , the robust exponential stability was studied for uncertain singular Markovian jump time-delay systems. In , the delay-dependent filtering problem for singular Markovian jump time-delay systems was thoroughly investigated.
The notion of dissipativity plays an important role in systems, circuits, networks, and control theory. The dissipativity contains small gain and passivity as its special cases and is mainly used for stability analysis for nonlinear systems. The main motivation for the study of a general dissipative control problem is that it offers flexibility for gain and phase performance trade off. Moreover, it is effective to deal with robust and nonlinear control. In , the passivity control for a kind of T-S fuzzy descriptor system is presented. In , the dissipative filtering for discrete fuzzy systems was thoroughly studied. In , the dissipative filtering for linear discrete-time systems via LMI was studied. In , the robust dissipative filtering for continuous-time polytopic uncertain neutral systems was studied. Unfortunately, to the best of the authors’ knowledge, up to now, the delay-dependent non-fragile robust dissipative filtering for uncertain nonlinear stochastic singular time-delay systems with Markovian jump parameters has rarely been reported.
In this paper, the delay-dependent non-fragile robust dissipative filter design method for uncertain nonlinear stochastic singular time-delay systems with Markovian jump parameters is considered. The system under study involves parameter uncertainties, stochastic disturbances, time-varying delays and inherent sector-like nonlinearities. Note that, among different descriptions of the nonlinearities, the so-called sector nonlinearity  has gained much attention for deterministic systems, the control analysis and model reduction problems have been studied; see [20, 21]. By establishing a Lyapunov-Krasovskii function and applying the Itô differential formula, a new delay-dependent bounded real lemma for nonlinear stochastic singular time-delay and Markovian jump systems is derived. Neither model transformation nor bounding technique for cross terms is involved. Based on the obtained bounded real lemma, the existence condition of a robust dissipative filter and a filter design method are presented by LMI approach. Also, it is shown that the proposed filter design method is widely applicable to singular systems and non-singular systems by a numerical example.
2 System description and preliminary lemma
where , are known real constant scalars, . denotes a vector-valued initial continuous function defined on the interval .
where , , , , , , , , , are known real constant matrices and is the uncertain time-varying matrix satisfying .
where are known real constant matrices, and , are symmetric positive definite matrices.
Remark 1 When E is a unit matrix, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , system (1) was studied in . The system in this paper is a class of stochastic time-delay systems broader than others.
where , , , , are known real constant matrices and is unknown time-varying matrix function satisfying .
Remark 2 When is a unit matrix, and , , , , , the filter (6) was studied in . The objective in this paper improves the function of the filter.
where Q, S, R are appropriately dimensioned, and Q, R are symmetric matrices.
Definition 1 
the augmented system (9) with is robust asymptotically stable for all uncertainties;
the filtering error system is strict robust dissipative.
Our aim is to determine parameters , , , , , such that system (6) is a robust dissipative filter for uncertain system (1).
Lemma 1 
Lemma 2 (Schur complement)
3 Main results
and , , , , . , , are governed by t, so we have .
where , , , .
Via Lyapunov-Krasovskii, according to (13), if and , system (9) is robust asymptotically stable.
it is easy to get .
When , there must exist small enough such that . Hence, there exists for all and , we have . By the Schur complement formula, (13) is completed. □
Based on the sufficient conditions above, the following criterion can be obtained readily.
Remark 3 The robust dissipative filter design problem is solved in Theorem 2 for the uncertain stochastic singular time-delay systems with Markovian jump parameters. We derive an LMI-based sufficient condition for the existence of full-order filters that ensure asymptotic stability and strict robust dissipativity of the resulting filtering error system and reduce the effect of the disturbance input on the estimated signal to a prescribed level for all admissible uncertainties. The feasibility of the filter design problem can be readily checked by the solvability of an LMI, which is dependent on the upper bound of the time-varying delays. The solvability of such a delay-dependent LMI can be readily checked by resorting to the Matlab LMI Tool box. In the next section, an illustrative example is provided to show the potential of the proposed techniques.
Remark 4 When the finite set , the uncertain linear stochastic system (1) is a generalized uncertain stochastic singular time-delay system without Markovian jump parameters.
4 Numerical value example
Example In this section, a numerical example is presented to demonstrate the usefulness of the developed method on the design of a robust dissipative filter for the uncertain stochastic singular time-delay systems with Markovian jump parameters.