$H_2/H_{\infty}$ Control for Stochastic Differential Systems with Partial Observation

TL;DR

This paper develops a control strategy for stochastic systems with partial observation, balancing performance and robustness using differential game theory, Kalman filtering, and Riccati equations, demonstrated on a UAV example.

Contribution

It introduces a novel $H_2/H_ extinfty$ control framework for partially observed stochastic systems, including new filtering equations and robustness conditions.

Findings

Derived optimal filtering equations under partial observation.

Established a Stochastic Bounded Real Lemma for robustness.

Connected Nash equilibrium existence to Riccati equation solvability.

Abstract

This paper investigates the $H_{2}/H_{\infty}$ control problem for linear stochastic differential systems under partial observation. Unlike existing studies that assume full state accessibility, we consider the scenario where the controller has access only to an observation process. The objective is to design a controller that balances the $H_2$ performance criterion with the $H_\infty$ robustness requirement under worst-case disturbances, formulated as a nonzero-sum differential game. Using the Kalman filtering method, we derive the corresponding optimal filtering equation. Furthermore, a Stochastic Bounded Real Lemma under the partial observation framework is established, providing necessary and sufficient conditions for the $H_\infty$ robustness constraint. We also show the connection between the existence of a Nash equilibrium and the solvability of the cross-coupled Riccati equations, and illustrate the effectiveness of the proposed approach through a numerical example involving an unmanned aerial vehicle (UAV).