Automatica 47 (2011) 466–476 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Linear minimum mean square filter for discrete-time linear systems with Markov jumps and multiplicative noises ✩ Oswaldo L.V. Costa ∗ , Guilherme R.A.M. Benites Departamento de Engenharia de Telecomunicações e Controle, Escola Politécnica da Universidade de São Paulo, 05508-970-São Paulo, SP, Brazil article info Article history: Available online 15 February 2011 Keywords: Filtering theory Kalman filters Multiplicative noise Jump process Riccati equations abstract In this paper we obtain the linear minimum mean square estimator (LMMSE) for discrete-time linear systems subject to state and measurement multiplicative noises and Markov jumps on the parameters. It is assumed that the Markov chain is not available. By using geometric arguments we obtain a Kalman type filter conveniently implementable in a recurrence form. The stationary case is also studied and a proof for the convergence of the error covariance matrix of the LMMSE to a stationary value under the assumption of mean square stability of the system and ergodicity of the associated Markov chain is obtained. It is shown that there exists a unique positive semi-definite solution for the stationary Riccati- like filter equation and, moreover, this solution is the limit of the error covariance matrix of the LMMSE. The advantage of this scheme is that it is very easy to implement and all calculations can be performed offline. © 2011 Elsevier Ltd. All rights reserved. 1. Introduction Linear systems subject to Markov jumps and multiplicative noises have been receiving a great deal of attention lately. This is due mainly to the fact that this kind of formulation has found many applications in engineering and finance. Some examples of such systems can be found in nuclear fission and heat transfer, population models and immunology, portfolio optimization, etc. (see, for instance, Costa & Kubrusly, 1996, Costa & de Paulo, 2007, Dragan & Morozan, 2002, Dragan & Morozan, 2006a,b, Geromel, Gonçalves, & Fioravanti, 2009 and Gershon & Shaked, 2006 and references therein for H 2 and H ∞ control problems, optimal filtering, robust stability and stabilizability conditions, predictive model-based control, etc.). The filtering problem of this class of systems has also attracted a great deal of interest in the last past years under different hypothesis and performance criterions. For systems with only multiplicative noise we can mention (Chow & Birkemeier, 1990), in which it was considered that the influence of multiplicative noises affects only the measurements of the model, and a recursive ✩ The first author received financial support from CNPq (Brazilian National Research Council), grant 301067/2009-0. The material in this paper was partially presented at 49th IEEE Conference on Decision and Control, December 15–17, 2010, Atlanta, Georgia, USA. This paper was recommended for publication in revised form under the direction of Editor Berç Rüstem. ∗ Corresponding author. Tel.: +55 1130915771; fax: +55 1130915718. E-mail addresses: oswaldo@lac.usp.br (O.L.V. Costa), guilherme@riskoffice.com.br (G.R.A.M. Benites). structure was achieved by combining the previous estimate with a recursive innovation, which yields a linear combination of the most recent data samples and the previous estimate. The results in Chow and Birkemeier (1990) were somehow generalized in Zhang and Zhang (2007) to consider correlated additive noises. In Carravetta, Germani, and Raimondi (1997) the multiplicative noise affects only the state model and the theory developed covers linear systems with nonstationary and non-Gaussian noises. The authors were able to define a filter for systems with multiplicative state noises which is optimal in a class of polynomial transformations. In Yang, Wang, and Hung (2002) the authors considered a discrete time-varying system with both additive and multiplicative noises. The problem addressed is to design a linear system that yields an estimation error variance with an optimized guaranteed upper bound for all admissible uncertainties. The sufficient conditions for designing such a filter were derived in terms of two Riccati difference equations. The filtering and control problem for systems subject to multiplicative noises under the H ∞ criterion has been studied in Gershon, Shaked, and Yaesh (2001). For systems with only Markov jumps in the parameters and when only an output of the system is available, so that the values of the jump parameter are not known, the problem of optimal and sub-optimal filtering has been addressed in Ackerson and Fu (1970), Bar-Shalom and Li (1993), Blom and Bar-Shalom (1988), Chang and Athans (1978), Dufour and Elliott (1997) and Tugnait (1982) among other authors, under the hypothesis of Gaussian distribution for the disturbances, and by Zhang (1999, 2000) for the non-Gaussian case. Since the optimal estimator requires exponentially increasing memory and computation with time, sub-optimal algorithms are required. In the papers mentioned before the authors considered non-linear 0005-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2011.01.015