Estimation and analysis of nonlinear stochastic systems

The algebraic and geometric structure of certain classes of nonlinear stochastic systems is exploited in order to obtain useful stability and estimation results. First, the class of bilinear stochastic systems (or linear systems with multiplicative noise) is discussed. The stochastic stability of bilinear systems driven by colored noise is considered; in the case that the system evolves on a solvable Lie group, necessary and sufficient conditions for stochastic stability are derived. Approximate methods for obtaining sufficient conditions for the stochastic stability of bilinear systems evolving on general Lie groups are also discussed. The study of estimation problems involving bilinear systems is motivated by several practical applications involving rotational processes in three dimensions. Two classes of estimation problems are considered. First it is proved that, for systems described by certain types of Volterra series expansions or by certain bilinear equations evolving on nilpotent or solvable Lie groups, the optimal conditional mean estimator consists of a finite dimensional nonlinear set of equations. Finally, the theory of harmonic analysis is used to derive suboptimal estimators for bilinear systems driven by white noise which evolve on compact Lie groups or homogeneous spaces. THESIS SUPERVISOR: Alan S. Willsky TITLE: Assistant Professor of Electrical Engineering