New Trends in Optimal Filtering and Control for Polynomial and Time-Delay Systems
0. 1 Introduction Although the general optimal solution of the ?ltering problem for nonlinear state and observation equations confused with white Gaussian noises is given by the Kushner equation for the conditional density of an unobserved state with resp
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Michael Basin
New Trends in Optimal Filtering and Control for Polynomial and Time-Delay Systems
ABC
Series Advisory Board F. Allgöwer, P. Fleming, P. Kokotovic, A.B. Kurzhanski, H. Kwakernaak, A. Rantzer, J.N. Tsitsiklis
Author Michael Basin Dept. of Physical and Mathematical Sciences Autonomous University of Nuevo Leon Av. Universidad s/n. Ciudad Universitaria San Nicolás de los Garza, Nuevo León, C.P. 66451 Mexico E-Mail: [email protected]
ISBN 978-3-540-70802-5
e-ISBN 978-3-540-70803-2
DOI 10.1007/978-3-540-70803-2 Lecture Notes in Control and Information Sciences
ISSN 0170-8643
Library of Congress Control Number: 2008931011 c 2008
Springer-Verlag Berlin Heidelberg
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Preface
0.1 Introduction Although the general optimal solution of the filtering problem for nonlinear state and observation equations confused with white Gaussian noises is given by the Kushner equation for the conditional density of an unobserved state with respect to observations (see [48] or [41], Theorem 6.5, formula (6.79) or [70], Subsection 5.10.5, formula (5.10.23)), there are a very few known examples of nonlinear systems where the Kushner equation can be reduced to a finite-dimensional closed system of filtering equations for a certain number of lower conditional moments. The most famous result, the Kalman-Bucy filter [42], is related to the case of linear state and observation equations, where only two moments, the estimate itself and its variance, form a closed system of filtering equations. However, the optimal nonlinear finite-dimensional filter can be obtained in some other cases, if, for example, the state vector can take only a finite number of admissible states [91] or if the observation equation is linear and the drift term in the state equation satisfies the Riccati equation d f /dx + f 2 = x2 (see [15]). The complete classification of the “general situation” cases (this means that there are no special assumptions on the structure of state and observation equations and the initial conditions), where the optimal nonlinear finite-dimensional filter exists, is gi
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