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

TL;DR

This paper formulates the $ ext{H}_2$-optimal control problem with passivity constraints as a convex optimization over the Youla parameter, enabling globally optimal solutions for passive systems.

Contribution

It introduces a convex optimization framework for $ ext{H}_2$ control with passivity constraints using the Youla parameter, allowing for globally optimal solutions in passive systems.

Findings

The problem reduces to a convex optimization over the Youla parameter.

Global optimality can be achieved through infinite-dimensional convex optimization.

Finite-dimensional truncation provides near-optimal solutions, demonstrated on a vibration suppression example.

Abstract

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.