Asymptotic Nonlinear Filtering and Large Deviations with Application to Observer Design

Abstract

An important problem in control theory is the design of observers for nonlinear control systems. By observer we mean a deterministic dynamical system which uses observed information to compute an estimate of the state of the control system in such as way that the error decays to zero. Baras and Krishnaprasad have proposed that an observer design might result from a study of an asymptotic nonlinear filtering problem obtained by adding small noise terms to the equations defining the control system. The purpose of this thesis is to study this asymptotic filtering problem and to develop observer designs based on their idea. Asymptotic nonlinear filtering problems have been studied by several authors, and are closely related to large deviations (Wentzell-Freidlin theory). We prove using vanishing viscosity and control theoretic methods a logarithmic limit result for solutions of the Zakai equation. This limit is characterized by a Hamilton-Jacobi equation which, as noted by Hijab, arises in Mortensen's deterministic minimum energy estimation. We make a careful study of this equation in the light of the relatively recent theory of viscosity solutions due to Crandall and Lions. We study the weak limit of the conditional measures and filter. Inspired by Hijab's large deviation result for pathwise conditional measures, we obtain a large deviation principle "in probability" for the conditional measures, and also a large deviation principle for the distributions of these measures. This asymptotic analysis suggests that the limiting filter is a candidate observer. We present an exact infinite dimensional observer for uncontrolled observable systems. In the case of uncontrolled nonlinear dynamics and linear observations, Bensoussan obtained a finite dimensional observer which is an approximation to the limiting filter. A detectability condition was used to prove exponential decay of the error, provided the initial condition lies in a bounded region. We extend his approach to the general case of controlled nonlinear dynamics and nonlinear observation. In particular, we obtain an observer for a class of fully nonlinear systems with no constraints on the initial conditions. The Benes case is considered.