668 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 3, MARCH 2004 An Improved Delay-Dependent Filtering of Linear Neutral Systems Emilia Fridman and Uri Shaked, Fellow, IEEE Abstract—An improved delay-dependent filtering design is proposed for linear, continuous, time-invariant systems with time delay. The resulting filter is of the Luenberger observer type, and it guarantees that the -norm of the system, relating the exoge- nous signals to the estimation error, is less than a prescribed level. The filter is based on the application of the descriptor model trans- formation and Park’s inequality for the bounding of cross terms. The advantage of the new filtering scheme is clearly demonstrated via simple examples. I. INTRODUCTION T HE filtering problem for linear systems with delay- dependent [1]–[3] and (more conservative) delay-indepen- dent [4], [5] designs have received a lot of attention recently. The prevailing methods are based on bounded real lemmas (BRLs) in terms of Riccati algebraic equations or linear matrix inequal- ities (LMIs), which guarantee a prescribed attenuation level. Recently, a new approach to filtering has been introduced [6]. This approach is based on representing the system by a de- scriptor type model [7] and on deriving a BRL for the corre- sponding adjoint system. The new BRL was found to be very efficient, and it considerably reduced the achievable attenuation level as compared with other results reported in the literature. By assuming a Leunberger-type estimator [8], the new BRL was applied to the resulting estimation error system. In spite of the advantage of the new filter design, it still entails a significant amount of conservatism stemming from the overbounding of mixed terms in the proof of the BRL in [6]. A new overbounding technique has recently been proposed that produces tighter bounds [9]. In the present paper, this tech- nique is applied to reduce the overdesign entailed in the ap- proach of [6]. The treatment is also extended to the more general class of neutral-type systems with multiple delays. It is shown, via simple examples, that the resulting schemes significantly improve the estimation results. Notation: Throughout the paper, the superscript “ ” stands for matrix transposition, denotes the -dimensional Eu- clidean space, is the set of all real matrices, and the notation , for , means that is symmetric and positive definite. The space of functions in that are square integrable over is denoted by , and denotes . Manuscript received December 20, 2001; revised April 20, 2003. This work was supported by the Ministry of Absorption of Israel and by C&M Maus Chair at Tel Aviv University. The associate editor coordinating the review of this paper and approving it for publication was Dr. Zhi-Quan (Tom) Luo. The authors are with the Department of Electrical Engineering–Systems, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: emilia@eng.tau.ac.il). Digital Object Identifier 10.1109/TSP.2003.822287 II. PROBLEM FORMULATION Consider the following system: (1a,b) where is the system state vector, is the exogenous disturbance signal, and , , , 2, and are constant time delays. The matrices , , , , and are constant matrices of appropriate dimensions. For simplicity only, two delays and and one are consid- ered; however, the results can easily be generalized to any finite number of delays: and . Equation (1) describes a system of neutral type since it con- tains derivatives in delayed states. In the case of , (1) is a retarded-type system (see, e.g., [10]). Neutral systems are encountered in the modeling of lossless transmission lines, or in dynamical processes, including steam and water pipes (see, e.g., [10] and the references therein). Unlike retarded systems, linear neutral systems may be destabilized by small changes of the delay [10]. To guarantee robustness of the results with respect to small changes of delay, we assume that the difference equation is asymptotically stable for all values of or, equivalently, the following. A1) is a Schur–Cohn stable matrix, i.e., all the eigenvalues of are inside the unit circle. Given the measurement equation (2) where is the measurement vector and the matrices and are constant matrices of appropriate dimension, a filter of the following Luenberger observer form is sought: (3) This form of the observer is known to produce an estimation error that is independent of the system trajectory, and it only depends on the initial condition of the system state [8]. It is therefore widely used in many practical estimation applications (e.g Kalman filtering). 1053-587X/04$20.00 © 2004 IEEE