2110 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 6, JUNE 2006 Robust Filtering for Discrete-Time Linear Systems With Uncertain Time-Varying Parameters Carlos E. de Souza, Senior Member, IEEE, Karina A. Barbosa, and Alexandre Trofino Neto Abstract—This paper deals with the problem of robust fil- tering for linear discrete-time state-space models with uncertain time-varying parameters. The parameters enter affinely into the state-space model matrices, and their admissible values and varia- tions are assumed to belong to given intervals. A method is derived for designing a linear stationary asymptotically stable filter with a prescribed performance, in spite of large parameter uncer- tainty. The proposed method incorporates information on avail- able bounds on both the admissible values and variation of the uncertain parameters and is based on a Lyapunov function with quadratic dependence on the parameters. The filter design is given in terms of linear matrix inequalities. Index Terms—Discrete-time systems, filtering, linear sys- tems, robust filtering, time-varying uncertain parameters. I. INTRODUCTION T HE problem of filtering for linear discrete-time state- state models has been the subject of extensive research over the past decade; see, e.g., [8], [9], [13], [20], and refer- ences therein. In filtering, the noise sources are arbitrary deterministic signals with bounded energy, or average power, and a filter is sought which ensures a prescribed upper bound on the -induced gain from the noise signals to the estimation error. This filtering approach is very appropriate to applications where the statistics of the noise signals are not exactly known. In the past few years, considerable interest has been de- voted to the problem of robust filters for uncertain linear discrete-time systems, namely, the design of filters with a guar- anteed performance for linear discrete-time state-space models with significant parameter uncertainty. In the case of norm-bounded parameter uncertainty, robust filters have been developed using either a Riccati equation approach (see, e.g., [16] and [18]), or linear matrix inequality (LMI) tech- niques [10]. On the other hand, linear discrete-time state-space Manuscript received November 22, 2004; revised July 21, 2005. This work was supported in part by Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq, Brazil, under PRONEX Grant 0331.00/00 and IM-AGIMB. The work of C. E. de Souza and A. Trofino Neto was supported in part by CNPq under Grants 30.2317/02-3/PQ and 305665/03-0/PQ. The work of K. A. Bar- bosa was supported by CAPES, Brazil, under the PRODOC program. The as- sociate editor coordinating the review of this manuscript and approving it for publication was Prof. Mariane R. Petraglia. C. E. de Souza and K. A. Barbosa are with the Department of Systems and Control, Laboratório Nacional de Computação Científica, Petrópolis, RJ, Brazil (e-mail: csouza@lncc.br; karinab@lncc.br). A. Trofino Neto is with the Department of Automation and Systems, Universidade Federal de Santa Catarina, Florianópolis, SC, Brazil (e-mail: trofino@das.ufsc.br). Digital Object Identifier 10.1109/TSP.2006.874349 models with polytopic-type parameter uncertainty have been recently treated in, for instance, [5], [6], [12], and [19] using an LMI methodology. With exception of [5] and [19], the aforementioned robust filtering methods have the feature of being based on the notion of quadratic stability, namely, a Lyapunov function, in- dependent of the uncertain parameters, is used to guarantee sta- bility and performance properties for all admissible parameters. This approach is numerically appealing; however, it can be quite conservative as the Lyapunov function is parameter indepen- dent. Moreover, stability and performance hold even when the parameters change arbitrarily fast, which entails an overdesign for many applications. An alternative to reduce the conservatism of the quadratic stability methods is to use a Lyapunov func- tion which depends on the uncertain parameters, referred to as parameter-dependent Lyapunov function, which has been intro- duced for robust control design. Very recently, robust filters based on parameter-dependent Lyapunov functions have been proposed in [5] and [19] for discrete-time linear systems with polytopic-type parameter uncertainty. These works are, how- ever, restricted to the case of time-invariant parameters, and the Lyapunov functions are affine in the parameters. This paper, which is based on [1], considers the problem of robust filtering for linear discrete-time state-space models subject to uncertain time-varying parameters that enter affinely into the state-space model matrices. The parameters value and variation are unknown; however, they are assumed to be con- strained to given intervals. The problem addressed is the de- sign of a linear stationary asymptotically stable filter with a prescribed upper bound on the -gain from the noise signals to the estimation error for all admissible uncertain parameters. The proposed robust filter design is based on a parameter-de- pendent Lyapunov function with quadratic dependence on the uncertain parameters and is given in terms of a standard LMI problem, which can be numerically solved very efficiently [2]. The motivation for using a quadratic parameter-dependent Lya- punov function is that it allows for achieving better performance than the affine one, in particular, in the case of time-varying pa- rameters, as shown by the example in Section IV. The new fil- tering method incorporates information on available bounds on the variation of the uncertain parameters and is less conservative than those based on quadratic stability. In particular, it includes the quadratic stability approach as a special case. This paper is organized as follows. Section II introduces the signal model used in the paper, the problem statement, and some preliminary results. Section III addresses the robust 1053-587X/$20.00 © 2006 IEEE