Robust Filtering for Linear Polytopic Discrete-Time Periodic Systems ⋆ Carlos E. de Souza ∗ Daniel F. Coutinho ∗∗ ∗ Department of Systems and Control, Laborat´ orio Nacional de Computac ¸˜ ao Cient´ ıfica – LNCC/MCT, Petr´ opolis, RJ 25651-075, Brazil (e-mail: csouza@lncc.br). ∗∗ Group of Automation and Control Systems, Pontif´ ıcia Universidade Cat´ olica do Rio Grande do Sul (PUCRS), Porto Alegre, RS 90619-900, Brazil (e-mail: dcoutinho@pucrs.br). Abstract: This paper is concerned with the problems of robust H and guaranteed variance filtering for linear discrete-time periodic systems with polytopic-type parameter uncertainty in the matrices of the system state-space model. Filtering methods are derived for designing linear periodic asymptotically stable filters with either a prescribed upper-bound on the ℓ 2 -gain from the noise signals to the estimation error, in the H case, or a guaranteed upper-bound on the average steady-state variance of the estimation error variance, for the guaranteed variance filtering, in spite of the parameter uncertainty. The proposed methods are based on parameter-dependent Lyapunov functions and are given in terms of linear matrix inequalities. The potentials of the proposed filtering methods are demonstrated by an example. Keywords: Robust H filtering, guaranteed variance filtering, linear periodic systems, uncertain systems, discrete-time systems, parameter-dependent Lyapunov function. 1. INTRODUCTION Over the past three decades periodic systems have been attract- ing significant interest within the control community. One of the motivations for this interest is the fact that cyclic processes arise very often in nature and engineering and applications of periodic systems may be found in a large spectrum of different fields, such as economics, biology, aeronautics, control and filtering of linear systems subject to cyclostationary noise, and control of multirate plants; see, Bittanti [1986], Bittanti and Colaneri [2009], and the references therein. A considerable amount of attention has been paid to linear periodic systems and a number of important results on con- trol and filtering problems have been reported in the litera- ture. As for control design, techniques have been developed to solve a variety of problems, as for instance, pole place- ment (Colaneri [1991]), characterization of all stabilizing con- trollers (Bittanti and Colaneri [1999]), robust stabilization (de Souza and Trofino [2000], Farges et al. [2007]), robust track- ing (Grasselli et al. [1996]), H control (Colaneri and de Souza [1992]), H 2 control (Wi´ sniewski and Stoustrup [2001]), and robust H 2 control (Farges et al. [2007]). In the context of filtering, the minimum mean-square state estimation (i.e. the celebrated periodic Kalman filter) and properties of the associ- ated periodic Riccati equation, have been widely studied during the 80’s and early 90’s (see, e.g. Bittanti et al. [1988], de Souza [1991], Bittanti et al.. [1991], De Nicolao [1992], and the references therein). On the other hand, the H filtering problem for continuous-time periodic systems has been solved in Xie and de Souza [1993] in terms of a periodic Riccati equation, whereas the discrete-time case has been treated in Bittanti and ⋆ This work was supported by CNPq, Brazil, under grants 30.3440/2008-2/PQ and 30.1461/2008-2/PQ. Cuzzola [2001] using a linear matrix inequality (LMI) ap- proach. The aforementioned filtering methods are restricted to systems with an uncertainty-free state-space model and as such they do not provide a guaranteed performance in the presence of parameter uncertainty. In the case of uncertain systems, a robust H filtering method for linear continuous-time periodic systems with norm-bounded parameter uncertainty has been developed in Xie et al. [1991] using a Riccati equation approach. In this paper we focus on robust filtering for linear discrete- time periodic systems subject to polytopic-type parameter un- certainty in the all the matrices of the state-space model. Two robust filtering problems are considered. For the first one, the problem of robust H filtering, we address the design of a linear periodic asymptotically stable filter that provides a prescribed upper-bound on the ℓ 2 -gain from the noise signals to the esti- mation error, in spite of the parameter uncertainty. On the other hand, for the second problem, the one of guaranteed variance filtering, the filter is required to ensure a guaranteed upper- bound for the average steady-state variance of the estimation error variance. The proposed filter designs build on uncertainty- dependent Lyapunov functions based on periodic matrices and are tailored via LMIs. An example is presented to illustrate the effectiveness of the developed robust filtering methods. Notation. Z is the set of integers, R n is the n-dimensional Euclidean space, R m × n is the set of m ×n real matrices, I n is the n×n identity matrix, diag{···} stands for a block-diagonal matrix, and a matrix S k is denoted N-periodic, 0 < N ∈ Z, if S k+N = S k , ∀ k ∈ Z. The transpose of a matrix S is denoted by S T , tr ( · ) is the matrix trace, and S k > 0(S k ≥ 0) for a real periodic matrix S k means that S k = S T k and positive definite (positive semidefinite) for all k ∈ Z. ℓ 2 stands for the space of square summable vector sequences over [0, ) with norm ‖·‖ 2 . Periodic Control Systems — PSYCO 2010 Antalya, Turkey, August 26-28, 2010 ISBN 978-3-902661-86-9/11/$20.00 © 2010 IFAC 1