Date of Award

Winter 1992

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Electrical/Computer Engineering

Program/Concentration

Electrical Engineering

Committee Director

Joseph L. Hibey

Committee Member

Oscar R. Gonzalez

Committee Member

Griffith J. McRee

Committee Member

Carl A. Schulz

Abstract

We discuss topics in the theory of nonlinear stochastic control, estimation, and decision via a probabilistic approach using measure transformations and martingale theory. First, we investigate the problem of estimating a diffusion process using coordinate transformations and measure transformations, both locally and globally; this is the analog of nonlinear coordinate and state feedback transformations used to obtain exact linearization in nonlinear deterministic control problems. Our results are new in that we use a probabilistic approach rather than a purely geometric one, and also in that we derive representations when the processes are defined locally rather than just globally. A gauge transformation then leads to a Feynman-Kac formula that is related to the unnormalized conditional density and subsequent bounds of filter estimates, where some of these bounds are extensions of pre-existing results while others are presented here for the first time. Second, we present new methods and new results in obtaining a minimum principle for partially observed diffusions using calculus of variations when the control variable is present only in the drift coefficient and correlation exists between state and observation noise, and then when the control variable exists in both drift and diffusion coefficients and no correlation exists. Here the problem is formulated as one of complete information, but instead of considering the unnormalized conditional density as the new state, this density is decomposed into two measure-valued processes and leads to a separation principle reminiscent of the linear-quadratic-Gaussian problem and stochastic flows of Euclidean processes. Third, we study the decision problem using likelihood-ratio tests and evaluate the performance using Chernoff bounds. We present new results by expressing both likelihood-ratios and error-probabilities in terms of a ratio of two unnormalized conditional densities where each satisfies a stochastic differential equation that in some cases can be solved in closed form.

DOI

10.25777/49xw-2678

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