Elsevier

Automatica

Volume 44, Issue 7, July 2008, Pages 1918-1922
Automatica

Technical communique
H filtering for discrete-time systems with randomly varying sensor delays

https://doi.org/10.1016/j.automatica.2007.10.026Get rights and content

Abstract

This paper investigates an H filtering problem for discrete-time systems with randomly varying sensor delays. The stochastic variable involved is a Bernoulli distributed white sequence appearing in measured outputs. This measurement mode can be used to characterize the effect of communication delays and/or data-loss in information transmissions across limited bandwidth communication channels over a wide area. H filtering of this class of systems is used to design a filter using the measurements with random delays to ensure the mean-square stochastic stability of the filtering error system and to guarantee a prescribed H filtering performance. A sufficient condition for the existence of such a filter is presented in terms of the feasibility of a linear matrix inequality (LMI). Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.

Introduction

Estimation of dynamic systems has found many practical applications and has attracted a lot of attention during the last decades. H filtering is introduced as an alternative to classical Kalman filtering when the statistical property of noise sources is unknown or unavailable (Nagpal & Khargonekar, 1991). H filtering is concerned with the design of estimators which ensure a bound on the L2-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 H 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 H filtering problem for systems with delays has gained growing interest. A delay-independent H 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 H control problem for this class of systems. In the meantime, the H 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 H filtering problem for a class of stochastic systems was investigated in Xu and Chen (2002). The authors of Berman and Shaked (2006) presented an H 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 H 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 H filtering performance is achieved.

This paper is organized as follows. Section 2 formulates the H filtering problem. In Section 3, a novel H 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, Z+ denotes the set of positive integers; Rn denotes the n-dimensional Euclidean space; Rm×n denotes the set of all m×n real matrices. A real symmetric matrix P>0(0) denotes P being a positive definite (or positive semi-definite) matrix, and A>()B means AB>()0. I 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 ‘τ’ represents the transpose. For a matrix N, Nτ stands for the transpose of matrix N1. A star ‘’ is used as an ellipsis for a corresponding transposed block matrix that is induced by symmetry. The notation l2[0,) represents the space of square summable infinite vector sequences with the usual norm 2. Suppose that the vectors vkRn̄, a sequence v={vk}l2[0,) if v2=k=1vkτvk<. Prob{.} stands for the occurrence probability of an event; E{.} denotes the expectation operator with respect to some probability measure.

Section snippets

Problem formulation

Consider a discrete-time system Σ: xk+1=Axk+Aωωkzk=Lxk+Lωωk where xkRn is the state; ωkRp̄ is the deterministic disturbance signal in l2[0,); zkRq is the signal to be estimated; A, Aω, L and Lω are known constant matrices with compatible dimensions. The measurement with random delays is given by yk=Cxkyck=(1θk)yk+θkyk1 where ykRp is the output, yckRp is the measured output, C is a known matrix, and the stochastic variable θk is a Bernoulli distributed white sequence taking the values of

H filter design

Based on Lemma 3 we will give a sufficient condition for the existence of an H filter in the form of (7) and present a method to construct the filter.

Theorem 5

Consider the system of(1), (2). Given a scalarγ>0, there exists a filter in the form of(7)such that the filtering error systemΣ̃is exponentially mean-square stable withHfiltering performanceγ, if there exist positive definite matricesX,Z,Qand matricesĀf,B̄f,C̄f,D̄fsuch thatIn this case, two nonsingular constant matricesMandNcan always be

A design example

In this section, we present a numerical example to illustrate the theory developed in Section 3. Consider the system Σ with parametric matrices as follows: A=[0.20.050.020.3],Aω=[0.10.2]L=[0.50.7],C=[10.6],Lω=0.3. The objective is to design a filter in the form of (7) for this system such that the filtering error system Σ̃ is exponentially mean-square stable and an optimal H filtering performance γ is achieved. By using the Matlab LMI Control Toolbox to solve the convex optimization

Conclusions

This paper considers the H 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 H filtering performance is guaranteed. An LMI-based filter design approach has been developed for this class of systems. It has been shown that the H 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.

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