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

TL;DR

This paper investigates the topological structure of the set of stabilizing controllers with robustness constraints, showing it has at most two connected components, which aids gradient-based optimization methods.

Contribution

It establishes the connectivity properties of the feasible and sublevel sets in robust control, providing theoretical insights that support gradient-based control design.

Findings

Feasible set has at most two path-connected components.

The set is diffeomorphic under a similarity transformation.

Results extend to LQG and optimal \\mathcal{H}_\infty control.

Abstract

This paper 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 subjective 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.