Technical communiqueH∞ filtering for discrete-time systems with randomly varying sensor delays☆
Introduction
Estimation of dynamic systems has found many practical applications and has attracted a lot of attention during the last decades. filtering is introduced as an alternative to classical Kalman filtering when the statistical property of noise sources is unknown or unavailable (Nagpal & Khargonekar, 1991). filtering is concerned with the design of estimators which ensure a bound on the -induced gain from disturbance signals to estimation errors. Over the past decades, various approaches, such as the interpolation approach, the Riccati equation-based approach, and the LMI-based approach have been developed to deal with the filtering problem in various settings such as deterministic systems with uncertainties and/or delays as well as various stochastic systems (Xie, Liu, Zhang, & Zhang, 2004). Recently, the study of the filtering problem for systems with delays has gained growing interest. A delay-independent filtering result for discrete-time systems with multiple time delays has been given in Palhares, de Souza, and Peres (2001) while delay-dependent results for this problem have been reported more recently in Gao, Lam, Xie, and Wang (2004) and Gao and Wang (2005).
It is noted that the time delays are assumed to be deterministic in the literature mentioned above. However, they may occur in a randomly varying way in many practical applications as pointed out in Wang, Ho, and Liu (2004). Recently, there has been some attention to the research of systems with randomly varying delays. A randomly varying delayed sensor mode was first introduced in Ray (1994). Since then, some results for randomly delayed systems have been reported in Wang et al. (2004), Yaz and Ray (1996) and Yang, Wang, Hung, and Gani (2006). A variance-constrained filtering approach was proposed for systems with random sensor delays in Wang et al. (2004). And more recently, the authors in Yang et al. (2006) investigated the control problem for this class of systems. In the meantime, the filtering and control problems of stochastic systems have also attracted a lot of attention over the past few years (Hinrichsen & Pritchard, 1998). In Xu and Chen (2003), an LMI-based filter design approach was proposed for impulsive stochastic systems, and based on the projection lemma, the reduced-order filtering problem for a class of stochastic systems was investigated in Xu and Chen (2002). The authors of Berman and Shaked (2006) presented an control scheme for a class of discrete-time nonlinear stochastic systems more recently.
Motivated by the works in Wang et al. (2004) and Yang et al. (2006), this paper focuses on the filtering problem for systems with randomly varying sensor delays. We are interested in designing filters such that for all randomly varying sensor delays, the filtering error system is exponentially mean-square stable and a prescribed filtering performance is achieved.
This paper is organized as follows. Section 2 formulates the filtering problem. In Section 3, a novel filtering approach is proposed. A numerical example is given to demonstrate the effectiveness of the proposed method in Section 4, which is followed by conclusions in Section 5.
Notations: Throughout this paper, denotes the set of positive integers; denotes the -dimensional Euclidean space; denotes the set of all real matrices. A real symmetric matrix denotes being a positive definite (or positive semi-definite) matrix, and means . denotes an identity matrix of appropriate dimension. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations. The superscript ‘’ represents the transpose. For a matrix , stands for the transpose of matrix . A star ‘’ is used as an ellipsis for a corresponding transposed block matrix that is induced by symmetry. The notation represents the space of square summable infinite vector sequences with the usual norm . Suppose that the vectors , a sequence if . stands for the occurrence probability of an event; denotes the expectation operator with respect to some probability measure.
Section snippets
Problem formulation
Consider a discrete-time system : where is the state; is the deterministic disturbance signal in ; is the signal to be estimated; , , and are known constant matrices with compatible dimensions. The measurement with random delays is given by where is the output, is the measured output, is a known matrix, and the stochastic variable is a Bernoulli distributed white sequence taking the values of
filter design
Based on Lemma 3 we will give a sufficient condition for the existence of an filter in the form of (7) and present a method to construct the filter.
Theorem 5 Consider the system of(1), (2). Given a scalar, there exists a filter in the form of(7)such that the filtering error systemis exponentially mean-square stable withfiltering performance, if there exist positive definite matricesand matricessuch thatIn this case, two nonsingular constant matricesandcan always be
A design example
In this section, we present a numerical example to illustrate the theory developed in Section 3. Consider the system with parametric matrices as follows: The objective is to design a filter in the form of (7) for this system such that the filtering error system is exponentially mean-square stable and an optimal filtering performance is achieved. By using the Matlab LMI Control Toolbox to solve the convex optimization
Conclusions
This paper considers the filter design problem for a class of discrete-time systems with randomly varying sensor delays. Attention is focused on the design of a filter such that the filtering error system is mean-square stable and a prescribed level of filtering performance is guaranteed. An LMI-based filter design approach has been developed for this class of systems. It has been shown that the filtering problem can be solved in terms of the feasibility of an LMI. A numerical example
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Guoxiang Gu under the direction of Editor André Tits. This work was supported in part by the National Natural Science Foundation of PR China under Grant 60574080 and 60434020.