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
$\mathcal {H}_{\infty }$
robustness constraints. We consider the feasible set of all the stabilizing full-order dynamical controllers that satisfy an additional
$\mathcal {H}_{\infty }$
robustness constraint. We show that this
$\mathcal {H}_{\infty }$
-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
$\mathcal {H}_{\infty }$
control to establish a surjective mapping from a set with a convex projection to the
$\mathcal {H}_{\infty }$
-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
$\mathcal {H}_{\infty }$
control. Our results bring positive news for gradient-based policy search on robust control problems.