A Natural-Gradient Approach for Nonlinear Stochastic Systems with Parameter Uncertainty

Abstract

Controlling nonlinear stochastic systems with parametric uncertainty is a fundamental challenge in modern control theory. This paper presents a comprehensive theoretical framework for a natural-gradient method applied to polynomial chaos theory. We focus on quadratic regulator problems characterized by both parametric uncertainty and additive stochastic disturbances. We extend existing polynomial chaos approaches from linear systems to general nonlinear dynamics. To achieve this, we develop new mathematical tools to handle the complex interactions between nonlinearity, parameter uncertainty, and noise. The framework provides local convergence guarantees for the proposed natural gradient algorithm. Furthermore, it offers practical computational strategies while carefully characterizing the theoretical limitations in the nonlinear setting.