Optimal Stochastic H∞ Controller Synthesis via LQR Theory

Abstract

Existing published works on H∞ and mixed H∞/H2 control generally solve the problem via game theory and saddle-point solutions. We provide an alternative solution via the stochastic LQR approach. Specifically, we consider the problem of H∞ state regulation problem for continuous-time systems which are subjected to both energy-bounded deterministic disturbances and stochastic white noises, and uncertain initial system state x(0) satisfying E{x(0)} = 0 and E{x(0)x(0)T} ≤ X0. We derive feedback laws which minimize sup||w(t)||2 ≤ 1E{||Π1/2x(t)||T2 + ||R1/2u(t)||T2}, where Π and R are positive-definite weight matrices. We further demonstrate how the approach may be extended to solving the robust stochastic LQR control with uncertainty in the system A matrix.