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

This paper 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 subjective 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.