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On the Gap Between H2 Optimal Control and Disturbance Decoupling

Ruirui Ma, Sarah H. Q. Li

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.

On the Gap Between H2 Optimal Control and Disturbance Decoupling

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.
Paper Structure (13 sections, 6 theorems, 21 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 13 sections, 6 theorems, 21 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Preliminaries
  4. Bridging $H_2$ minimization and DD
  5. Zero $H_2$ norm and DD equivalence
  6. SDP and DD equivalence under internal stability
  7. Optimization-based Synthesis of DDPS
  8. Local optimality via constraint qualification.
  9. Solving for DDPF
  10. Power Network Optimal Control
  11. Case Study 1: DDPF verification
  12. Case Study 2: DDPF vs SDP-$H_2$
  13. Conclusion

Key Result

Proposition 1

A state feedback controller $F\in {\mathbb{R}}^{m\times n}$ is a DD controller for eqn:lti if and only if there exist matrices $X\in {\mathbb{R}}^{k \times k}$, $V\in {\mathbb{R}}^{n \times k}$ such that

Figures (4)

  • Figure 1: The impulse response and cumulative output error $e_{cum}(10)$. Noise magnitude $d(t)\sim\mathcal{N}(0,2^l),7\leq \ell \leq 20$.
  • Figure 2: Four-bus power network example. The (undirected) edges represents transmission lines between buses.
  • Figure 3: The state evolution and output difference plot.
  • Figure 4: Cumulative time domain output error comparison over exponentially increasing noise variance.

Theorems & Definitions (13)

  • Proposition 1: Sarsilmaz2024trentelman2001control
  • Lemma 1
  • proof
  • Example 1: Disjoint DD and SDP controllers
  • Lemma 2
  • proof
  • Example 2: Computational Gap between DD and SDP
  • Theorem 1: Robinson's CQ Satisfaction
  • proof
  • Remark 1
  • ...and 3 more