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

TL;DR

This paper addresses the problem of achieving optimal $\mathbb{H}_2$ performance when the feedback path is constrained to be output-strictly passive and the plant channel is passive with finite $\mathbb{L}_2$ gain. It shows that the optimization can be recast as a convex problem in the closed-loop Youla parameter, first in a discrete-time domain via a bilinear transform and then in a countably infinite Youla parameter space, with practical finite-dimensional truncations that converge to the global optimum. A suboptimal certainty-equivalence solution provides a computationally light baseline, while a sequence of convex finite-dimensional problems (OP5–OP7) yields provable convergence to the optimum as the truncation order grows. The approach is demonstrated on vibration-suppression examples, where the passive controller achieves near-optimal performance and remains robust to model uncertainties due to passivity. The work offers a principled, convex route to design passivity-constrained controllers with broad potential applications in mechanical and structural systems, while outlining extensions to generalized performance measures and non-passive plants as future directions.

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.

Figures (7)

• Figure 1: Block diagram for feedback system under consideration
• Figure 2: Frequency responses for magnitude of mapping $w \mapsto z$ for the open loop system (solid, thin black), as well as closed-loop with controllers $\hat{K}_0$ (dash, black) and $\hat{K}_{100}^\star$ (solid, thick black), and unconstrained optimum (solid, gray) for Example 1
• Figure 3: Bode plots for $\hat{K}_0$ (dash, black) and $\hat{K}_{100}^\star$ (solid thick, black) and $\hat{K}$ for the unconstrained optimum (solid, gray) for Example 1
• Figure 4: Performance $J_n^\star$, normalized by the performance of the unconstrained $\hat{\mathbb{H}}_2$-optimal controller for Example 1
• Figure 5: Frequency responses for magnitude of mapping $w \mapsto z$ for the open loop system (solid, thin black), as well as closed-loop with controllers $\hat{K}_0$ (dash, black) and $\hat{K}_{100}^\star$ (solid, thick black) for Example 2

Theorems & Definitions (20)

• Remark 1
• Remark 2
• Remark 3
• Definition 1
• Definition 2
• Remark 4
• Remark 5
• Theorem 1
• proof
• Theorem 2