JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 111, No. 2, pp. 273–303, November 2001 ( 2001) Multi-Input Multi-Output Ellipsoidal State Bounding 1 C. DURIEU, 2 E ´ .WALTER, 3 AND B. POLYAK 4 Abstract. Ellipsoidal state outer bounding has been considered in the literature since the late sixties. As in the Kalman filtering, two basic steps are alternated: a prediction phase, based on the approximation of the sum of ellipsoids, and a correction phase, involving the approxi- mation of the intersection of ellipsoids. The present paper considers the general case where K ellipsoids are involved at each step. Two measures of the size of an ellipsoid are employed to characterize uncertainty, namely, its volume and the sum of the squares of its semiaxes. In the case of multi-input multi-output state bounding, the algorithms pre- sented lead to less pessimistic ellipsoids than the usual approaches incorporating ellipsoids one by one. Key Words. Bounded noise, ellipsoidal bounding, identification, set- membership estimation, state estimation. 1. Introduction In the literature, most parameter or state estimation problems involving an explicit characterization of uncertainty are solved via a stochastic approach, with the perturbations assumed to be random and usually white and Gaussian. However, often the statistics of these perturbations are not known and sometimes it is more natural to assume that they belong to known compact sets, with no other hypothesis on their distributions. Then, 1 This work was supported by the International Association for the Promotion of Cooperation with Scientists from the New Independent States of the Former Soviet Union, Grant INTAS- RFBR-97-10782. 2 Maı ˆtre de Confe´rences, Laboratoire d’E ´ lectricite´, Signaux et Robotique, Centre National de la Recherche Scientifique and E ´ cole Normale Supe´rieure de Cachan, Cachan, France. 3 Directeur de Recherche, Laboratoire des Signaux et Syste`mes, Centre National de la Recherche Scientifique, Ecole Supe´rieure d’E ´ lectricite´, and Universite´ de Paris-Sud, Gif-sur-Yvette, France. 4 Head of Laboratory, Institute of Control Science, Russian Academy of Sciences, Moscow, Russia. 273 0022-3239011100-0273$19.500  2001 Plenum Publishing Corporation