Available online at www.sciencedirect.com Automatica 40 (2004) 1171–1179 www.elsevier.com/locate/automatica Ellipsoidal parameter or state estimation under model uncertainty  Boris T. Polyak a ; * ,SergeyA.Nazin a ,C ecile Durieu b ,EricWalter c a Institute of Control Sciences, Russian Academy of Sciences, 65 Profsojuznaja st., Moscow 117997, Russia b SATIE, CNRS-ENS de Cachan, 61 av. du Pr esident Wilson, 94235 Cachan, France c Laboratoire des Signaux et System es, CNRS-Sup elec-UPS, 91192 Gif-sur-Yvette, France Received 4 November 2002; accepted 18 February 2004 Abstract Ellipsoidal outer-bounding of the set of all feasible state vectors under model uncertainty is a natural extension of state estimation for deterministic models with unknown-but-bounded state perturbations and measurement noise. The technique described in this paper applies to linear discrete-time dynamic systems; it can also be applied to weakly non-linear systems if non-linearity is replaced by uncertainty. Many diculties arise because of the non-convexity of feasible sets. Combined quadratic constraints on model uncertainty and additive disturbances are considered in order to simplify the analysis. Analytical optimal or suboptimal solutions of the basic problems involved in parameter or state estimation are presented, which are counterparts in this context of uncertain models to classical approximations of the sum and intersection of ellipsoids. The results obtained for combined quadratic constraints are extended to other types of model uncertainty. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Bounded noise; Ellipsoidal bounding; Parameter estimation; Set-membership uncertainty; State estimation; Uncertain dynamic systems 1. Introduction In the literature, most parameter or state estimation problems are either treated as deterministic or solved via a stochastic approach, with the state perturbations and measurement noise assumed to be random with no other uncertainty in the model. Kalman ltering is then the most widely applied technique. Often, however, the underlying probabilistic assumptions are not realistic (the main pertur- bation may, for instance, be deterministic). It then seems more natural to assume that the state perturbations and measurement noise are unknown but bounded and to char- acterize the set of all values of the parameter or state vector that are consistent with this hypothesis. This corresponds to guaranteed estimation, rst considered at the end of  Parts of this paper have been presented at the 15th IFAC World Congress, Barcelona, Spain, 21–26 July 2002, and at the 13th IFAC Sym- posium on System Identication, Rotterdam, Netherlands, 27–29 August 2003. This paper was recommended for publication in revised form by Associate Editor Kok Lay Teo under the direction of Editor T. Ba sar. * Corresponding author. Tel.: +7-095-3348829; fax: +7-095-4202016. E-mail addresses: boris@ipu.rssi.ru (B.T. Polyak), snazin@ipu.rssi.ru (S.A. Nazin), durieu@satie.ens-cachan.fr (C. Durieu), walter@lss.supelec.fr (E. Walter). 1960sandtheearly1970s(Schweppe, 1968; Witsenhausen, 1968; Bertsekas & Rhodes, 1971; Schweppe, 1973). One of the main approaches, and the only one to be considered here, aims to compute ellipsoids guaranteed to contain the vector to be estimated given bounds on the perturbations and noise. The Russian school was particularly active in this domain (Kurzhanskii, 1977; Chernousko, 1981, 1994; Kurzhanskii & Valyi, 1997). Important contributions have been presented in Fogel and Huang (1982),inthecontextof parameter estimation. At present, the theory of guaranteed estimation is a well developed and mature area of control theory,(see,e.g.thebooks Milanese, Norton, Piet-Lahanier, Walter, 1996; Walter & Pronzato, 1997), special issues of journals (Walter, 1990; Norton, 1994, 1995) and the refer- ences therein. Recent results regarding ellipsoidal state esti- mation in a MIMO context can be found in Durieu, Walter, and Polyak (2001). However, most of the works mentioned above deal with problems where the plant model (its structure, in the case of parameterestimation)isassumedtobepreciselyknownand where the uncertainty only relates to state perturbations and measurement noise. This assumption seems unrealistic for many real-life problems. The lack of precise information is the fundamental paradigm of modern control theory, where the concept of robustness plays a key role. The goal of this 0005-1098/$-see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2004.02.014