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Energy-Based Approximation of Linear Systems with Polynomial Outputs

Linus Balicki, Serkan Gugercin

TL;DR

The paper addresses MOR for systems with linear dynamics and polynomial outputs (LPO) by proving that controllability and observability energy functions are polynomials, with observability coefficients $\mathbf{w}_k$ found by solving Kronecker-structured linear systems. It develops low-rank tensor methods, including CP decompositions and quadrature-based solvers, to efficiently compute these coefficients in large-scale settings. Building on this, it extends balanced truncation to an energy-based MOR framework that preserves the LPO structure, using optimization on the Stiefel manifold to select a reduced subspace that maximizes observability energy. Validated on a Mass-Spring-Damper and a Convection-Diffusion model, the approach yields accurate input-output behavior with significantly reduced order while maintaining stability.

Abstract

Controllability and observability energy functions play a fundamental role in model order reduction and are inherently connected to optimal control problems. For linear dynamical systems the energy functions are known to be quadratic polynomials and various low-rank approximation techniques allow for computing them in a large-scale setting. For nonlinear problems computing the energy functions is significantly more challenging. In this paper, we investigate a special class of nonlinear systems that have a linear state and a polynomial output equation. We show that the energy functions of these systems are again polynomials and investigate under which conditions they can effectively be approximated using low-rank tensors. Further, we introduce a new perspective on the well-established balanced truncation method for linear systems which then readily generalizes to the nonlinear systems under consideration. This new perspective yields a novel energy-based model order reduction procedure that accurately captures the input-output behavior of linear systems with polynomial outputs via a low-dimensional reduced order model. We demonstrate the effectiveness of our approach via two numerical experiments.

Energy-Based Approximation of Linear Systems with Polynomial Outputs

TL;DR

The paper addresses MOR for systems with linear dynamics and polynomial outputs (LPO) by proving that controllability and observability energy functions are polynomials, with observability coefficients found by solving Kronecker-structured linear systems. It develops low-rank tensor methods, including CP decompositions and quadrature-based solvers, to efficiently compute these coefficients in large-scale settings. Building on this, it extends balanced truncation to an energy-based MOR framework that preserves the LPO structure, using optimization on the Stiefel manifold to select a reduced subspace that maximizes observability energy. Validated on a Mass-Spring-Damper and a Convection-Diffusion model, the approach yields accurate input-output behavior with significantly reduced order while maintaining stability.

Abstract

Controllability and observability energy functions play a fundamental role in model order reduction and are inherently connected to optimal control problems. For linear dynamical systems the energy functions are known to be quadratic polynomials and various low-rank approximation techniques allow for computing them in a large-scale setting. For nonlinear problems computing the energy functions is significantly more challenging. In this paper, we investigate a special class of nonlinear systems that have a linear state and a polynomial output equation. We show that the energy functions of these systems are again polynomials and investigate under which conditions they can effectively be approximated using low-rank tensors. Further, we introduce a new perspective on the well-established balanced truncation method for linear systems which then readily generalizes to the nonlinear systems under consideration. This new perspective yields a novel energy-based model order reduction procedure that accurately captures the input-output behavior of linear systems with polynomial outputs via a low-dimensional reduced order model. We demonstrate the effectiveness of our approach via two numerical experiments.
Paper Structure (18 sections, 8 theorems, 99 equations, 2 figures, 2 algorithms)

This paper contains 18 sections, 8 theorems, 99 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Energy Functions
  3. Tensor Algebra Preliminaries
  4. Canonical Decomposition
  5. Symmetric Tensors
  6. Computing the Observability Energy Function
  7. Low-Rank Solutions to Linear Systems
  8. Exploiting Symmetries
  9. Energy-Based MOR
  10. Balanced Truncation
  11. An Optimization Perspective on Balanced Truncation
  12. Energy-Based Reduction of LPO Systems
  13. Numerical Examples
  14. Mass-Spring-Damper System
  15. Convection Diffusion Model
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\Lambda(\textbf{A}) \subset \mathbb{C}_-$. Then the observability energy function of the system in eq:posys is the unique solution to the Lyapunov PDE in eq:observabilityPDE and satisfies the formula and thus is a degree-$2d$ polynomial. A set of polynomial coefficients $\left\{ \textbf{w}_j \right\}_{j=2}^{2d}$, which defines $\mathcal{E}_o$, is given by the unique solutions to the linear s

Figures (2)

  • Figure 1: Comparison of time-domain responses for mass-spring-damper system.
  • Figure 2: Comparison of time-domain responses for convection diffusion model.

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • Proposition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 3 more