Policy Optimization in Robust Control: Weak Convexity and Subgradient Methods

Abstract

Robust control seeks stabilizing policies that perform reliably under adversarial disturbances, with ${\mathcal{H}}_{\infty }$ control as a classical formulation. It is known that policy optimization of robust ${\mathcal{H}}_{\infty }$ control naturally lead to nonsmooth and nonconvex problems. This paper builds on recent advances in nonsmooth optimization to analyze discrete-time static output-feedback ${\mathcal{H}}_{\infty }$ control. We show that the ${\mathcal{H}}_{\infty }$ cost is weakly convex over any convex subset of a sublevel set. This structural property allows us to establish the first non-asymptotic deterministic convergence rate for the subgradient method under suitable assumptions. In addition, we prove a weak Polyak-Łojasiewicz (PL) inequality in the state-feedback case, implying that all stationary points are globally optimal. We finally present a few numerical examples to validate the theoretical results.

9 pages, 11 figures