On the Gap Between H2 Optimal Control and Disturbance Decoupling

Ruirui Ma; Sarah H. Q. Li

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arXiv · 2603.20438

On the Gap Between H2 Optimal Control and Disturbance Decoupling

Ruirui Ma·Sarah H. Q. Li

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Abstract

We study the relationship between disturbance decoupling (DD) and H2 optimal control for linear time-invariant (LTI) systems, revealing a fundamental gap between DD subspace constraints and semi-definite program (SDP)-based H2 minimization. We show that DD is equivalent to the existence of zero H2 gain without requiring internal stability, whereas SDP-based H2 minimization strictly optimizes over stabilizing controllers and therefore fails to recover DD controllers when the closed-loop dynamics may be marginally stable. Moreover, we show that the trace representation of H2 norms further biases solutions away from complete DD. Motivated by this, we formulate a bilinear matrix inequality (BMI)-constrained optimization program that directly enforces the DD subspace condition to compute DD controllers. We propose a difference-of-convex (DC) iterative algorithm that preserves DD and stability at every iteration, and establish its convergence to Karush-Kuhn-Tucker (KKT) points under standard constraint qualification conditions. Numerical experiments on a four bus power network demonstrate that the proposed algorithm achieves significantly better disturbance rejection while enabling optimization of additional performance metrics. The resulting framework establishes a computationally tractable link between geometric DD theory and optimization-based controller design.

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Ruirui Ma, Sarah H. Q. Li. 2026-03-20. On the Gap Between H2 Optimal Control and Disturbance Decoupling. https://arxiv.org/abs/2603.20438

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