Connectivity of the Feasible and Sublevel Sets of Dynamic Output Feedback Control with Robustness Constraints

This letter considers the optimization landscape of linear dynamic output feedback control with H∞ robustness constraints. We consider the feasible set of all the stabilizing full-order dynamical controllers that satisfy an additional H∞ robustness constraint. We show that this H∞ -constrained set has at most two path-connected components that are diffeomorphic under a mapping defined by a similarity transformation. Our proof technique utilizes a classical change of variables in H∞ control to establish a surjective mapping from a set with a convex projection to the H∞ -constrained set. This proof idea can also be used to establish the same topological properties of strict sublevel sets of linear quadratic Gaussian (LQG) control and optimal H∞ control. Our results bring positive news for gradient-based policy search on robust control problems.