Elsevier

Signal Processing

Volume 90, Issue 9, September 2010, Pages 2667-2675
Signal Processing

Improved robust energy-to-peak filtering for uncertain linear systems

https://doi.org/10.1016/j.sigpro.2010.03.011Get rights and content

Abstract

This paper investigates the problem of energy-to-peak filtering for both discrete-time and continuous-time systems with polyhedral uncertainties in the state-space equations. By increasing the flexible dimensions in the solution space for the energy-to-peak optimization, less conservative results on the robust energy-to-peak filtering are obtained. The filter parameters can be readily designed by solving a set of parameter-dependent linear matrix inequalities (LMIs). In comparison with the existing methods, the improvement of the proposed method over the existing results is shown via two numerical examples.

Introduction

The filtering problem is of paramount importance from the application perspective, and it has received considerable attention during the past few decades [1], [2], [3]. Among all the existing work, Kalman filtering [4] has been proven to be very useful and has received continued interest since it was proposed in the 1960s. In Kalman filtering, the system model is assumed to be precisely known and sufficient statistical information, such as the spectral density of the measurement noise is given. When the priori information is not precisely known, the performance of the Kalman filter may degrade [5]. Due to this reason, by following the Kalman filtering and assuming that the noise is bounded, energy-to-energy filtering, energy-to-peak filtering and peak-to-peak filtering were developed. In the energy-to-energy (also called H) filtering, the noise is arbitrary but with bounded energy, an optimal filter can be derived by minimizing the upper bound of the L2 gain from the noise to the estimation error [1], [2], [6], [7], [8], [9]. In peak-to-peak filtering, the worst-case peak value of the estimation error for the bounded peak value of the noise is minimized, see [10], [11] and the references therein.

In energy-to-peak filtering (L2L for continuous-time system and l2l for discrete-time system), a stable filter is designed so that the estimation error is minimized for any bounded energy disturbance, see [12], [13], [14]. In some applications, it is specifically required that the peak value of the estimation error should be less than a certain amount, and thus, the energy-to-peak filter will be a better choice in such applications. Energy-to-peak performance conditions for both continuous-time and discrete-time systems have been studied in [12]. With these conditions, the robust energy-to-peak performance was further analyzed in [13] and an improved approach was introduced in [15]. From then on, the energy-to-peak filtering problem attracted more attention. The strategy has been studied to solve filtering problems under different situations or different kinds of setups, see, e.g. [5], [16], [17], [18], [19], [20], [21], [22], [23]. Moreover, it is well known that almost all existing physical and engineering systems unavoidably include uncertainties and other disturbances due to inaccurate modeling, component aging, measurement errors, exterior conditions or parameter variations. Therefore, in the past few years, robust control and filtering has become a hot topic in engineering literature and constitutes an integral part of control systems and signal processing research [24].

On the other hand, a lot of effort has been paid to improve the estimation performance which results in less conservative designs. Normally, there are two main approaches to decrease the filtering conservatism. The first one is to construct a new Lyapunov functional (polynomially parameter-dependent one [25] or one with new structure [26]) such that more information of the system can be incorporated into the design. The other one is to introduce auxiliary slack variables so that the Lyapunov weighting matrices are decoupled from system matrices. This technique was first presented in [27] and was applied to many filtering applications, see [18], [28]. Recently, these two methods were integrated in the estimator design, see [18], [28], [29], [30] and the reference therein, to name a few. However, most of these work was focused on the H and H2 filtering problems.

In this paper, we study systems that are subject to uncertainties which are assumed to lie within a convex polytope. By using Finsler's lemma, we derive new results for energy-to-peak filtering for both continuous-time and discrete-time uncertain systems. By applying integrated methods, it shows less conservative results can be obtained. This work is an energy-to-peak counterpart of the results in [28], which are related to H2 and H filtering. However, here, we introduce one more auxiliary slack variable, which increases the flexibility of determining the Lyapunov functional candidate for each vertex of the uncertain polytope. Moreover, compared with the work in [15], [18], we consider a more general form of the estimation equation for a class of discrete-time systems, which has new applications in signal estimation. Furthermore, the improvements of the derived results and reduction of conservatism in the performance evaluation are illustrated by two examples.

Notation: The notations used in this paper are fairly standard. Superscript ‘T’ indicates matrix transposition; Rn stands for the n-dimensional Euclidean space, and sym(A) denotes (A+AT) for simplicity. In addition, in symmetric block matrices or long matrix expressions, we use * as an ellipsis for the terms that are introduced by symmetry. Matrices, even if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. For a real symmetric matrix P, the notations P>0, and P<0, mean that P is positive definite, and negative-definite, respectively.

Section snippets

Robust l2l filtering

In this section, the filtering problem for a discrete-time system is formulated firstly and an augmented filtering error system is obtained. Then the stability and l2l performance of the filtering error system is analyzed. Based on the analysis, an l2l filter is designed.

Consider a stable discrete-time system as follows:xk+1=A(α)xk+B(α)ωk,yk=C(α)xk+D(α)ωk,zk=E(α)xk+F(α)ωk,where xkRn is the state vector, ykRq is the measured output, zkRp is the signal to be estimated, and ωkRm is the

Robust L2L filtering

In Section 2, the plant is in discrete-time form. The continuous-time case will be studied in this section. Consider a stable continuous-time system as follows:x˙(t)=A(α)x(t)+B(α)ω(t),y(t)=C(α)x(t)+D(α)ω(t),z(t)=E(α)x(t),where x(t), ω(t), y(t) and z(t) have the identical definitions with system in (1), but for continuous-time case, x˙(t) represents the derivative of x(t). The system matrices are also assumed to locate in the polytope M.

The main objective of this section is to design a stable

Numerical examples

In this section, simulation examples and comparisons are given to demonstrate the effectiveness and improvement of the proposed methods for both discrete-time and continuous-time systems.

Example 1

Consider the following discrete-time system which was used in [18]: xk+1=00.511+β1xk+6010ωk,yk=[10010]xk+[01]ωk,zk=1001xk.Here, β1 represents the system uncertainty and satisfies |β1|0.4. It is noted that the uncertainty in the system can be described by a two-vertex polytope. We aim to tackle with the

Concluding remarks

In this paper, we studied a parameter-dependent approach to robust energy-to-peak filter design. Both discrete-time and continuous-time cases were considered. By using Finsler's lemma, three slack parameters were introduced in the stability analysis for the filtering error systems. Further, these parameters were assumed to have special forms. Under this assumption, it is effective and convenient to design the energy-to-peak filters by solving the LMI-based optimal problem. Numerical examples

Acknowledgements

The authors wish to thank the associate editor and the anonymous reviewers for providing constructive suggestions which have improved the presentation of the paper.

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