# New Trends in Optimal Filtering and Control for Polynomial and Time-Delay Systems

• Authors
• Michael Basin
Book

Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 380)

1. Front Matter
2. Michael Basin
Pages 1-45
3. Michael Basin
Pages 47-74
4. Michael Basin
Pages 75-130
5. Michael Basin
Pages 131-173
6. Michael Basin
Pages 175-197
7. Back Matter

### Introduction

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 respect to obser- tions (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 Ku- ner equation can be reduced to a ?nite-dimensional closed system of ?ltering eq- tions for a certain number of lower conditional moments. The most famous result, the Kalman-Bucy ?lter [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 ?ltering equations. However, the optimal nonlinear ?nite-dimensional ?lter can be - tained in some other cases, if, for example, the state vector can take only a ?nite number of admissible states [91] or if the observation equation is linear and the drift term in the 2 2 state equation satis?es the Riccati equation df /dx + f = x (see [15]). The complete classi?cation of the “general situation” cases (this means that there are no special - sumptions on the structure of state and observation equations and the initial conditions), where the optimal nonlinear ?nite-dimensional ?lter exists, is given in [95].

### Keywords

Nonlinear Polynomial Systems Optimal Filtering Optimal control Sliding Mode Control Stochastic Systems Time-Delay Systems Trend

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-540-70803-2
• Copyright Information Springer-Verlag Berlin Heidelberg 2008
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages Engineering
• Print ISBN 978-3-540-70802-5
• Online ISBN 978-3-540-70803-2
• Series Print ISSN 0170-8643
• Series Online ISSN 1610-7411
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