Optimal $\mathbb{H}_2$ Control with Passivity-Constrained Feedback: Convex Approach

We consider the $\mathbb{H}_2$-optimal feedback control problem, for the case in which the plant is passive with bounded $\mathbb{L}_2$ gain, and the feedback law is constrained to be output-strictly passive. In this circumstance, we show that this problem distills to a convex optimal control problem, in which the optimization domain is the associated Youla parameter for the closed-loop system. This enables the globally-optimal controller to be solved as an infinite-dimensional but convex optimization. Near-optimal solutions may be found through the finite-dimensional convex truncation of this infinite-dimensional domain. The idea is demonstrated on a simple vibration suppression example.