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Least Squares Model Reduction: A Two-Stage System-Theoretic Interpretation

Alberto Padoan

TL;DR

For linear time-invariant systems with transfer function $W(s)=C(sI-A)^{-1}B$, the paper reframes least-squares moment matching as a two-stage procedure: first build a surrogate that exactly enforces interpolation constraints, then project to a reduced-order model. This system-theoretic view leverages output regulation theory and Krylov projections to connect LS moment matching with Sylvester-equation and projection-based realizations, and it revisits the Smith–Lucas LS method as a natural instance within this framework. The main contributions are: (i) a unified two-step interpretation that links classical MM and LS MM methods, (ii) a characterization of LS MM through a constrained optimization problem, and (iii) a clear structure showing how zero-interpolation cases (Smith–Lucas) fit into the surrogate-two-step paradigm. The framework clarifies the design space for LS MM and paves the way for extending the approach to nonlinear and time-varying settings while preserving key interpolatory properties.

Abstract

Model reduction simplifies complex dynamical systems while preserving essential properties. This paper revisits a recently proposed system-theoretic framework for least squares moment matching. It interprets least squares model reduction in terms of two steps process: constructing a surrogate model to satisfy interpolation constraints, then projecting it onto a reduced-order space. Using tools from output regulation theory and Krylov projections, this approach provides a new view on classical methods. For illustration, we reexamine the least-squares model reduction method by Lucas and Smith, offering new insights into its structure.

Least Squares Model Reduction: A Two-Stage System-Theoretic Interpretation

TL;DR

For linear time-invariant systems with transfer function , the paper reframes least-squares moment matching as a two-stage procedure: first build a surrogate that exactly enforces interpolation constraints, then project to a reduced-order model. This system-theoretic view leverages output regulation theory and Krylov projections to connect LS moment matching with Sylvester-equation and projection-based realizations, and it revisits the Smith–Lucas LS method as a natural instance within this framework. The main contributions are: (i) a unified two-step interpretation that links classical MM and LS MM methods, (ii) a characterization of LS MM through a constrained optimization problem, and (iii) a clear structure showing how zero-interpolation cases (Smith–Lucas) fit into the surrogate-two-step paradigm. The framework clarifies the design space for LS MM and paves the way for extending the approach to nonlinear and time-varying settings while preserving key interpolatory properties.

Abstract

Model reduction simplifies complex dynamical systems while preserving essential properties. This paper revisits a recently proposed system-theoretic framework for least squares moment matching. It interprets least squares model reduction in terms of two steps process: constructing a surrogate model to satisfy interpolation constraints, then projecting it onto a reduced-order space. Using tools from output regulation theory and Krylov projections, this approach provides a new view on classical methods. For illustration, we reexamine the least-squares model reduction method by Lucas and Smith, offering new insights into its structure.

Paper Structure

This paper contains 13 sections, 6 theorems, 29 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Moment matching
  4. Moment matching via Sylvester equations
  5. Moment matching via Krylov projections
  6. Least squares moment matching
  7. Models achieving least squares moment matching
  8. A system-theoretic interpretation
  9. Connections with the literature
  10. Moment matching via Sylvester equations
  11. Moment matching via Krylov projections
  12. Least squares moment matching method at zero
  13. Conclusion

Key Result

lemma 1

astolfi2010model Consider system eq:system-linear. Suppose Assumptions ass:minimality and ass:signal-generator-linear hold. Then there is a non-singular matrix ${T\in\mathbb{R}^{\nu\times \nu}}$ such that where ${\Pi \in \mathbb{R}^{n\times \nu}}$ is the solution of the Sylvester equation

Figures (2)

  • Figure 1: Least squares model reduction by moment matching as a two-step process using surrogate models.
  • Figure 2: Least squares model reduction by moment matching as a two-step process using surrogate signal generators.

Theorems & Definitions (7)

  • lemma 1
  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4
  • theorem 5
  • proof