Elsevier

Signal Processing

Volume 89, Issue 4, April 2009, Pages 605-614
Signal Processing

Deconvolution filtering for stochastic systems via homogeneous polynomial Lyapunov functions

https://doi.org/10.1016/j.sigpro.2008.10.008Get rights and content

Abstract

This paper deals with the robust H and L2L deconvolution filtering problems for stochastic systems with polytopic uncertainties. The purpose is to design a full-order deconvolution filter such that (i) the deconvolution error system is robustly exponentially mean-square stable with a prescribed decay rate and (ii) an H or L2L performance of the deconvolution error system is guaranteed. Based on a homogeneous polynomial parameter-dependent matrix (HPPDM) approach, sufficient conditions for the solvability of these problems are given in terms of linear matrix inequalities (LMIs). Such conditions are dependent on the decay rate, which enables one to design robust deconvolution filters by selecting the decay rates according to different practical conditions. In addition, when these LMIs are feasible, a design procedure of the desired filters is developed and an exponential estimate for the deconvolution error system is given. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed design methods.

Introduction

Over the past years, considerable attention has been devoted to the study of the filtering problem for polytopic uncertain systems. The purpose of this problem is to design a stable filter such that, for all parameter uncertainties residing in a polytope, the error system is asymptotically stable and ensures a certain performance constraint. Generally, there are two approaches to solving this problem: the quadratic (parameter-independent) Lyapunov function (QLF) approach and the parameter-dependent Lyapunov function (PDLF) approach. The QLF approach is relatively simple and easily used for the filter design [1], [2], [3]. In addition, such an approach has special advantages to the study of robust and gain-scheduled filtering problems for systems with time-varying uncertainties [4], [5]. However, the QLF approach may produce conservative results because the fixed Lyapunov matrix must be used for the entire uncertainty domain [6], [7]. In order to reduce such conservatism, the PDLF approach has been used and a number of important results have been reported in the literature; see, for example, [6], [8], [9], [10], [11], [12]. We note that the parameter-dependent matrices used in these works have the same structures as the system matrices; that is, they are linearly dependent on the uncertain parameters. Very recently, another approach to constructing the parameter-dependent matrices, known as the homogeneous polynomial parameter-dependent matrix (HPPDM) approach, has been proposed in [13], [14], [15]. By using this approach, the parameter-dependent matrices are represented in the form of homogeneous polynomials with respect to the uncertain parameters, which are more general than the linear parameter-dependent matrices. It has been shown in [13], [14], [15] that, as the degree of the polynomials increases, less conservative robust stability conditions are obtained. It is also worth noting that the HPPDM approach has been generalized to the robust and parameter-dependent filtering problems for polytopic uncertain systems [16], [17], [18].

The deconvolution problem is another important research topic that has extensive applications in data transmission, equalization, reverberation cancellation, seismic deconvolution, image restoration, speech processing, and other areas [19], [20], [21], [22], [23], [24]. The objective of the deconvolution filtering problem is to design a filter to estimate the unknown input signal of a system using available measurements, which is much different from the aforementioned filtering problems where we estimate the system states. Over the past 10 years, the deconvolution filtering problem for linear systems has received much attention. For example, a game-theory approach was proposed in [25], [26] for the H deconvolution filter design. In [23], [27], the full-order and reduced-order H deconvolution filters were designed, respectively. In [28], linear matrix inequality (LMI)-based results on the H2 deconvolution filtering problem were presented. In [29], the problem of finite horizon H deconvolution filtering for linear time-varying systems was studied. In [22], 2-D H deconvolution filters were designed for 2-D digital systems based on the LMI approach.

On the other hand, the stochastic systems have been extensively studied and a lot of results have been reported for different research topics, such as stability and stabilization [30], [31], [32], [33], sliding mode control [34], [35], proportional-integral-derivative (PID) control [36], H model reduction [37], H control and filtering [38], [39], [40], [41], [42], and L2L filtering [43]. It is worth noting that the HPPDM approach has been applied in [42] to study the robust H filter design problem for a class of discrete-time stochastic systems with polytopic uncertainties. Very recently, the Hdeconvolution filtering problem for stochastic systems with interval uncertainties has been investigated in [44], where some valuable conditions for the design of the H deconvolution filters were presented via LMIs. However, to the best of our knowledge, the problems of robust H and L2L deconvolution filtering for stochastic systems with polytopic uncertainties have not been addressed in the literature, which still remain open and unsolved. In addition, in many practical applications, one is concerned not only with the stability of a system but also with the decay rate (also called convergence rate). Therefore, how to design a deconvolution filter for a system such that the deconvolution error system converges with a specified decay rate is also an interesting research topic.

For the first time, this paper investigates the robust H and L2L deconvolution filtering problems for stochastic systems with time-invariant polytopic uncertainties. First, parameter-dependent conditions, which are also dependent on a specified decay rate, for the existence of the robust H and L2L deconvolution filters are derived. Second, the parameter-dependent conditions are transformed into LMIs by using the HPPDM approach and the combinatoric mathematics techniques. Finally, based on the proposed LMIs, a design procedure of the desired filters is developed and an exponential estimate for the deconvolution error system is given. We also provide two examples to demonstrate the effectiveness of the proposed design methods.

The contribution of this paper is threefold: (i) a design procedure of the robust H and L2L deconvolution filters for polytopic uncertain stochastic systems is proposed for the first time; (ii) the exponential estimate is considered in the filter design; (iii) the recently developed HPPDM approach is applied, and thus the presented results are generally less conservative.

Notations: Throughout this paper, Rn denotes the n-dimensional Euclidean space, and Rm×n denotes the set of all m×n real matrices. A real symmetric matrix P>0(0) denotes that P is a positive definite (or positive semi-definite) matrix, and A>()B means A-B>()0. I denotes an identity matrix of appropriate dimension. The superscript T represents the transpose. * is used as an ellipsis for terms that are induced by symmetry. L2[0,) is the space of square-integrable vector functions over [0,). |·| denotes the Euclidean norm for vectors, · denotes the spectral norm for matrices, and ·2 stands for the usual L2[0,) norm. We use diag{} to represent a block-diagonal matrix. Let (Ω,F,{Ft}t0,P) be a complete probability space with a filtration {Ft}t0 satisfying the usual conditions (i.e., the filtration contains all P-null sets and is right continuous). E{·} denotes the expectation operator with respect to the probability measure P. To use the HPPDM approach, we let K(r) denote the set of s-tuples obtained as all possible combinations of nonnegative integers ki, i=1,2,,s, such that i=1ski=r; that is, K(r)={k=k1k2ks|kiZ+,i=1ski=r}, where Z+ is the set of nonnegative integers. In addition, r! denotes the factorial, π(k)=i=1s(ki!), and Ii denotes a combination of 0 and 1 with the i-th element being 1 and the other elements being 0; i.e., Ii=00100. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

Section snippets

Problem formulation

Consider the following uncertain stochastic system:(Σ):dx(t)=[A(α)x(t)+B(α)u(t)]dt+f(t,x(t),u(t))dω(t),x(0)=x0,y(t)=C(α)x(t)+D(α)u(t),where x(t)Rn is the state, u(t)Rm is the unknown input that belongs to L2[0,), y(t)Rp is the measured output, and ω(t)Rl is an l-dimensional Brownian motion defined on a complete probability space (Ω,F,P) relative to an increasing family (Ft)t>0 of σ-algebras FtF, where Ω is the sample space, F is the σ-algebra of subsets of the sample space and P is the

Robust H deconvolution filter design

Here we study the robust H deconvolution filtering problem for system (Σ). We first present some parameter-dependent conditions for the solvability of this problem, and then transform them into LMIs by using the HPPDM approach. First, for system (Σe) we give the following result.

Lemma 1

For prescribed scalars σ>0 and γ>0, system (Σe) is robustly exponentially mean-square stable with a decay rate σ and condition (11) is satisfied under zero initial conditions and all nonzero u(t)L2[0,), if there

Numerical examples

In this section, we provide two examples to demonstrate the effectiveness of the proposed design methods. Example 1 is for the robust H deconvolution filtering problem, while Example 2 is for the robust L2L deconvolution filtering problem.

Example 1

Consider the following stochastic system:dx(t)=-0.13+0.5a-3-4x(t)+-0.5a0.9au(t)dt+0.5000.5x(t)+0.10.1u(t)dω(t),y(t)=[0.80.8(1+a)]x(t)+(0.45-0.5a)u(t),where a is a bounded constant uncertain parameter satisfying |a|1. This system can be rewritten as (Σ)

Conclusions

In this paper, we have investigated the robust H and L2L deconvolution filtering problems for stochastic systems with time-invariant polytopic uncertainties. Based on the HPPDM approach and using the combination mathematics techniques, decay-rate-dependent conditions for the solvability of the two problems have been presented in terms of LMIs. By using the solutions of these LMIs, the desired robust deconvolution filters have been designed and the exponential estimates of the deconvolution

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    This work was supported in part by the Program for the National Science Foundation for Distinguished Young Scholars of PR China under Grant 60625303, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20060288021, the RGC HKU 7029/05P, the Natural Science Foundation of Jiangsu Province under Grant BK2008047, and the Research Innovation Program for Graduate Students in Jiangsu Province under Grant CX07B_114z.

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