Reduced-order $H_{\infty }$ filtering for singular systems

Abstract

This paper solves the problem of reduced-order $H_{\infty }$ filtering for singular systems. The purpose is to design linear filters with a specified order lower than the given system such that the filtering error dynamic system is regular, impulse-free (or causal), stable, and satisfies a prescribed $H_{\infty }$ performance level. One major contribution of the present work is that necessary and sufficient conditions for the solvability of this problem are obtained for both continuous and discrete singular systems. These conditions are characterized in terms of linear matrix inequalities (LMIs) and a coupling non-convex rank constraint. Moreover, an explicit parametrization of all desired reduced-order filters is presented when these inequalities are feasible. In particular, when a static or zeroth-order $H_{\infty }$ filter is desired, it is shown that the $H_{\infty }$ filtering problem reduces to a convex LMI problem. All these results are expressed in terms of the original system matrices without decomposition, which makes the design procedure simple and directly. Last but not least, the results have generalized previous works on $H_{\infty }$ filtering for state-space systems. An illustrative example is given to demonstrate the effectiveness of the proposed approach.

Introduction

One of the most famous filtering technique is the celebrated Kalman filter, which minimizes the error variance in the state estimation. The Kalman filter has been widely used in various fields of control and signal processing [1]. It is noted that the Kalman filtering approach requires that the system under consideration has known dynamics described by a certain well-posed model, and the external noises are white processes with known statistical properties [1], [6], which are often not satisfied in practical applications. Furthermore, it has been shown that the standard Kalman filtering algorithms cannot guarantee satisfactory performance when uncertainties arise in a system model [7]. To deal with these difficulties, an alternative approach called $H_{\infty }$ filtering has been proposed. The purpose is the design of a filter such that the error system is stable and its $H_{\infty }$-norm is lower than a prescribed level. The advantage of the $H_{\infty }$ filtering technique in comparison with the Kalman filtering approach is that no statistical assumptions on the exogenous noises are needed, and the filter is more robust when there exists additional parameter uncertainty in a system [10], [23], which makes this approach useful in many practical applications. A great number of results on the $H_{\infty }$ filtering have been proposed in the literature in both the deterministic and stochastic contexts; see e.g. [11], [13], [19], [20], [22], [28], and the references therein.

On the other hand, the problem of reduced-order filter design, which is concerned with finding a filter whose order is lower than that of the system according to some given criteria, has received much attention in the past decades. The motivation for the study of this problem stems from some practical considerations, such as the limitation of computing power, real-time requirements, fast data processing with a process of limited power, etc. Various approaches have been proposed and many results on this issue have been reported in the literature. For example, by using the direction optimization method, some conditions for the design of reduced-order filters were obtained in [3], while in [29] two convergent algorithms were proposed for the minimization of the integral squared impulse response error between a full-order digital filter and a reduced one, which were further applied to the general MIMO case. Recently, the reduced-order $H_{\infty }$ filtering problem was investigated in [21], where discrete time-varying systems were considered and the design procedure involving the solution to a nonlinear difference equation was presented. For the continuous case, the reduced-order $H_{\infty }$ filtering problem was solved by converting the problem to a general distance one in [4]. Very recently, the reduced-order$H_{\infty }$ filtering problem was studied in [15], in which necessary and sufficient conditions were derived for both discrete and continuous systems, and all the parameters of a desired reduced-order filter can be obtained by solving certain linear matrix inequalities (LMIs) and a coupling non-convex rank constraint. These results were further extended to stochastic systems in [25].

Singular systems have been widely studied in the past years due to their extensive applications in modelling and control of electrical circuits, power systems, economics and other areas. Singular systems are also referred to as descriptor systems, implicit systems, generalized state-space systems, differential-algebraic systems or semi-state systems [8], [17], [26], [27], [31]. The filtering problem for singular systems has also been investigated by many researchers. For example, a simple optimal filter was obtained in [2] by changing an estimable discrete-time linear singular system to an equivalent standard system via orthogonal transformations, while in [30], an optimal recursive filtering approach for singular stochastic discrete-time systems was proposed by resorting to a time-domain innovation analysis method. It is also worth mentioning that the reduced-order filtering problem for singular systems subject to unknown disturbances was addressed in [9], [24], respectively, where several conditions for the design of reduced-order observers were obtained by using different approaches. It is noted that in both [9], [24] no $H_{\infty }$ performance was considered. To the best of the authors’knowledge, the problem of reduced-order $H_{\infty }$ filtering for singular systems has not been fully investigated so far, which still remains open and unsolved.

In this paper, we deal with the problem of reduced-order $H_{\infty }$ filtering for singular systems. Attention is focused on the design of linear filters, which is with a specified order lower than the order of the system under consideration, such that the filtering error dynamical system is regular, impulse-free (for the continuous case) or causal (for the discrete case) and stable, while satisfying a prescribed $H_{\infty }$ performance level. Both continuous and discrete singular systems are considered. In terms of certain LMIs and a coupling non-convex rank constraint set, necessary and sufficient conditions for the solvability of this problem are obtained. These non-convex inequalities can be solved by the so-called alternating projection algorithm proposed in [14]. When these inequalities are feasible, an explicit parametrization of all desired reduced-order filters is presented. In particular, when the static or zeroth-order $H_{\infty }$ filtering is concerned, the solvability conditions reduce to a convex LMI feasibility problem which can be handled efficiently by using standard numerical algorithms [5], and a simple parametrization of all the desired filters is also given. It is worth pointing out that all the results are derived without decomposing the original system matrices. Therefore, the desired reduced-order filter can be constructed directly. It is shown that the results obtained in this paper can be viewed as extensions of existing results on reduced-order $H_{\infty }$ filtering for state-space systems to singular systems. Finally, an illustrative example is given to show the applicability of the proposed approach.

Notation

Throughout this paper, for real symmetric matrices X and $Y$, the notation $X⩾Y$ (respectively, $X>Y$) means that the matrix $X-Y$ is positive semi-definite (respectively, positive definite). I is the identity matrix with appropriate dimension. $M^{\mathrm{T}}$ and $M^{+}$ represent the transpose and the Moore-Penrose inverse of matrix $M$. For a given stable continuous-time transfer function matrix $G(s)$, its $H_{\infty }$ norm is defined by $\parallel G(s){\parallel }_{\infty }={\sup }_{\omega \in [0,\infty )}{\sigma }_{\max }(G(j\omega ))$, while for a given stable discrete-time transfer function matrix $G(z)$, its $H_{\infty }$ norm is defined by $\parallel G(z){\parallel }_{\infty }={\sup }_{\omega \in [0,2\pi )}{\sigma }_{\max }(G({\mathrm{e}}^{j\omega }))$, where ${\sigma }_{\max }(·)$ represents the maximum singular value of a matrix. ${\mathcal{L}}_2[0,\infty )$ stands for the space of square integrable functions on $[0,\infty )$, while $l_2[0,\infty )$ refers to the space of square summable infinite vector sequences over $[0,\infty )$. For a matrix $M\in {\mathbb{R}}^{n\times m}$ with rank $r$, the orthogonal complement $M^{\perp }$ is defined as a (possibly non-unique) $(n-r)\times n$ matrix such that $M^{\perp }M=0$ and $M^{\perp }{M}^{\perp \mathrm{T}}>0$. The notation $D_{\mathrm{int}}(0,1)$ is the interior of the unit disk with center at the origin. $\lambda (E,A)=\{z|\det (\mathit{zE}-A)=0\}$. Matrices, if not explicitly stated, are assumed to have compatible dimensions.