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Distributionally Robust Model Order Reduction for Linear Systems

Le Liu, Yu Kawano, Yangming Dou, Ming Cao

TL;DR

This work addresses robust model order reduction (MOR) for linear, discrete-time systems when the input distribution is uncertain within a Wasserstein ambiguity set. It develops a Stackelberg-game formulation between the reduced-order model and the uncertain distribution, and leverages Gelbrich distance as a tractable surrogate to convert the problem into nested convex optimization (DROMOR). Theoretical results show stability and error guarantees for known covariance cases and equivalence to Gelbrich-based outer problems under distributional uncertainty, accompanied by a convex relaxation that yields a computable, provably bounded reduced model. Numerical experiments on a mechanical system indicate that the proposed method outperforms traditional balanced truncation under distributional uncertainty, demonstrating practical robustness gains. The framework paves the way for scalable, distributionally robust MOR and suggests extensions to nonlinear systems.

Abstract

In this paper, we investigate distributionally robust model order reduction for linear, discrete-time, time-invariant systems. The external input is assumed to follow an uncertain distribution within a Wasserstein ambiguity set. We begin by considering the case where the distribution is certain and formulate an optimization problem to obtain the reduced model. When the distribution is uncertain, the interaction between the reduced-order model and the distribution is modeled by a Stackelberg game. To ensure solvability, we first introduce the Gelbrich distance and demonstrate that the Stackelberg game within a Wasserstein ambiguity set is equivalent to that within a Gelbrich ambiguity set. Then, we propose a nested optimization problem to solve the Stackelberg game. Furthermore, the nested optimization problem is relaxed into a nested convex optimization problem, ensuring computational feasibility. Finally, a simulation is presented to illustrate the effectiveness of the proposed method.

Distributionally Robust Model Order Reduction for Linear Systems

TL;DR

This work addresses robust model order reduction (MOR) for linear, discrete-time systems when the input distribution is uncertain within a Wasserstein ambiguity set. It develops a Stackelberg-game formulation between the reduced-order model and the uncertain distribution, and leverages Gelbrich distance as a tractable surrogate to convert the problem into nested convex optimization (DROMOR). Theoretical results show stability and error guarantees for known covariance cases and equivalence to Gelbrich-based outer problems under distributional uncertainty, accompanied by a convex relaxation that yields a computable, provably bounded reduced model. Numerical experiments on a mechanical system indicate that the proposed method outperforms traditional balanced truncation under distributional uncertainty, demonstrating practical robustness gains. The framework paves the way for scalable, distributionally robust MOR and suggests extensions to nonlinear systems.

Abstract

In this paper, we investigate distributionally robust model order reduction for linear, discrete-time, time-invariant systems. The external input is assumed to follow an uncertain distribution within a Wasserstein ambiguity set. We begin by considering the case where the distribution is certain and formulate an optimization problem to obtain the reduced model. When the distribution is uncertain, the interaction between the reduced-order model and the distribution is modeled by a Stackelberg game. To ensure solvability, we first introduce the Gelbrich distance and demonstrate that the Stackelberg game within a Wasserstein ambiguity set is equivalent to that within a Gelbrich ambiguity set. Then, we propose a nested optimization problem to solve the Stackelberg game. Furthermore, the nested optimization problem is relaxed into a nested convex optimization problem, ensuring computational feasibility. Finally, a simulation is presented to illustrate the effectiveness of the proposed method.

Paper Structure

This paper contains 11 sections, 6 theorems, 37 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Distributionally robust model order reduction
  3. Main results
  4. Model Order Reduction for Certain Distribution
  5. Distributionally Robust Model Order Reduction for Uncertain Distribution
  6. Tractable Convex Optimization Solution
  7. Numerical Experiment
  8. Conclusion
  9. Proof of Theorem \ref{['thm:opt']}
  10. Proof of Theorem \ref{['thm:opt_equiv']}
  11. Proof of Theorem \ref{['lem:constraint']}

Key Result

Theorem III.1

Consider the system sys:origin, where the inputs follow a known i.i.d. distribution ${\mathbb P}$ with covariance $Q \in \mathbb{S}_{+}^m$. Given a reduced dimension $1 \leq r < n$, if there exist a scalar $\gamma \geq 0$, a positive definite matrix $P_{ {\mathcal{D}}}$, a positive semi-definite mat where $\Psi_1 = AP_{1, {\mathcal{D}}}A^{\top} - P_{1, {\mathcal{D}}} + B Q B^{\top}, \\ \Psi_2 = A

Figures (2)

  • Figure 1: Performance Results by DROMOR and Balanced Truncation.
  • Figure 2: Absolute Approximation Error by DROMOR and Balanced Truncation.

Theorems & Definitions (13)

  • Theorem III.1
  • proof
  • Definition III.2
  • Lemma III.3
  • proof
  • Corollary III.4
  • proof
  • Theorem III.5
  • proof
  • Lemma III.6
  • ...and 3 more