IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 3, MARCH 2001 569 Robust Filter Design for Uncertain Linear Systems with Multiple Time-Varying State Delays Carlos E. de Souza, Senior Member, IEEE, Reinaldo Martinez Palhares, Member, IEEE, and Pedro Luis Dias Peres, Member, IEEE Abstract—The problem of robust filtering for contin- uous-time uncertain linear systems with multiple time-varying delays in the state variables is investigated in this paper. The uncertain parameters are supposed to belong to a given convex bounded polyhedral domain. The aim is to design a stable linear filter assuring asymptotic stability and a prescribed per- formance level for the filtering error system, irrespective of the uncertainties and the time delays. Sufficient conditions for the existence of such a filter are established in terms of linear matrix inequalities, which can be efficiently solved by means of powerful convex programming tools with global convergence assured. An example illustrates the proposed methodology. Index Terms— filtering, linear matrix inequalities, robust filtering, time delays. I. INTRODUCTION D URING the last few decades, many authors have inves- tigated the problem of signal estimation based on cor- rupted measurements. In particular, when the noise sources are assumed to be arbitrary signals with bounded energy (or average power), the filtering approach provides both a guaranteed noise attenuation level and robustness against unmodeled dy- namics; see, e.g., [1]–[3] and the references therein. The problem of robust filtering consists of determining an asymptotically stable filter, based on an uncertain signal model, which ensures that the filtering error dynamics is asymptotically stable and that the -induced gain from the noise signals to the filtering error remains bounded by a pre- scribed value for all allowed uncertainties. In the case of linear state-space models with norm-bounded parameter uncertainty, the robust filter design has been addressed mainly through Riccati equation techniques [4]–[6] and in terms of linear matrix inequalities (LMIs) [7]. Very recently, an LMI approach for robust filtering for linear state-space models with polytopic-type uncertainty has been proposed in [8], whereas Manuscript received March 20, 2000; revised November 3, 2000. This work was supported in part by Conselho Nacional de Desenvolvimento Científico e Tecnológico—CNPq, Brazil, under Grants 301653/96-8/PQ and PRONEX 15/98—Control of Dynamical Systems, for C. E. de Souza, 300596/98-7/PQ for R. M. Palhares, and 304604/89-5/PQ for P. L. D. Peres. The associate editor coordinating the review of this paper and approving it for publication was Dr. Paulo J. S. G. Ferreira. C. E. de Souza is with the Department of Systems and Control, Laboratório Nacional de Computação Científica—LNCC, Petrópolis, Brazil. R. M. Palhares is with Pontifical Catholic University of Minas Gerais, Grad- uate Program in Electrical Engineering, Belo Horizonte, Brazil. P. L. D. Peres is with the School of Electrical and Computer Engineering, University of Campinas, Campinas, Brazil (e-mail: peres@dt.fee.unicamp.br). Publisher Item Identifier S 1053-587X(01)01412-X. [9] treated the design of robust filter with pole placement constraints. In the context of state-space models with time-delays, the problem of robust filtering has received less attention in the literature. As is well known [10], the existence of time de- lays is commonly encountered in many dynamic systems, and their presence must be taken into account in a realistic filter de- sign. Moreover, stability and noise attenuation level guaranteed by an filter design without considering non-negligible time delays may collapse in the presence of time delays. Time delays arise in several signal processing related problems, such as, for instance, echo cancellation, local loop equalization, multipath propagation in mobile communications, and array signal pro- cessing, [11]–[15]. Furthermore, note that state delays appear in any filtering application where the sensors are subject to time delays, which can arise due to transport phenomena (e.g., of ma- terial, energy or information) or required numerical processing of the measurements. In such situations, state delays will appear in the measurement equation of the state-space model. In [16], an filtering methodology for precisely known systems with a single time-delayed measurement was pro- posed. On the other hand, the design of filters for precisely known continuous-time systems with time-delayed states was addressed in [17], where a sufficient condition based on an algebraic Riccati equation has been presented. To the authors’ knowledge, the problem of robust filtering for continuous-time state-delayed systems has not yet been fully investigated in the literature. In this paper, the problem of robust filtering for con- tinuous-time linear systems subject to parameter uncertainty in all the matrices of the system state-space model and multiple time-varying state delays is investigated. The uncertain param- eters are assumed to belong to a given convex bounded poly- tope. Sufficient conditions for the existence of an asymptot- ically stable linear filter that ensures asymptotically stability and a prescribed performance for the filtering error system irrespective of the uncertainties and the time delays are pro- vided in terms of LMIs. The suitable filter is obtained through a convex optimization problem, which can be efficiently solved via recently developed algorithms [18]. When compared with the robust filter design for polytopic-type uncertainty and without time delays [8], the additional complexity in the pro- posed filter design is relatively small, considering the perfor- mance benefits that can be achieved. The notation used in the paper is fairly standard: The boldface characters and stand for the identity and the zero matrices of appropriate dimensions, respectively. denotes the space 1053–587X/01$10.00 © 2001 IEEE