On the Gap Between H2 Optimal Control and Disturbance Decoupling

TL;DR

This paper investigates the fundamental differences between disturbance decoupling and H2 optimal control in LTI systems, revealing limitations of SDP-based methods and proposing a new BMI-based approach for direct DD controller synthesis.

Contribution

It introduces a BMI-constrained optimization framework and a DC algorithm that directly enforces DD conditions, bridging geometric DD theory with optimization-based control design.

Findings

The SDP-based H2 minimization often fails to recover DD controllers.

The proposed BMI approach effectively computes DD controllers with improved disturbance rejection.

Numerical experiments show the method's superiority in power network disturbance rejection.

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.