Policy Optimization over Submanifolds for Constrained Feedback Synthesis

Shahriar Talebi, Mehran Mesbahi

Abstract: In this paper, we study linearly constrained policy optimizations over the manifold of Schur stabilizing controllers, equipped with a Riemannian metric that emerges naturally in the context of optimal control problems. We provide extrinsic analysis of a generic constrained smooth cost function, that subsequently facilitates subsuming any such constrained problem into this framework. By studying the second order geometry of this manifold, we provide a Newton-type algorithm with local convergence guarantees that exploits this inherent geometry without relying on the exponential mapping nor a retraction. The algorithm hinges instead upon the developed stability certificate and the linear structure of the constraints. We then apply our methodology to two well-known constrained optimal control problems. Finally, several numerical examples showcase the performance of the proposed algorithm.