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$\mathcal{H}_\infty$ model order reduction for quadratic output systems

Birgit Hillebrecht, Benjamin Unger

TL;DR

This work addresses MOR for linear time-invariant systems with quadratic outputs (LTIQO) by defining an $\mathcal{H}_\infty$-norm that accounts for both linear and quadratic outputs. An optimization-based MOR framework using a leveled least squares surrogate, with adaptive level $\gamma$ and frequency sampling, yields nonintrusive ROMs that minimize the $\mathcal{H}_\infty$-error. A condensed structure-preserving parametrization for port-Hamiltonian systems is developed to guarantee stability and respect the Hamiltonian structure in reduced models. Numerical experiments on a mass-spring-damper example and a PH system show that the proposed method outperforms state-of-the-art MOR approaches, with benefits in both linear and quadratic output channels.

Abstract

Linear time-invariant quadratic output (LTIQO) systems generalize linear time-invariant systems to nonlinear regimes. Problems of this class occur in multiple applications naturally, such as port-Hamiltonian systems, optimal control, and stochastical problems. We introduce an $\mathcal{H}_\infty$-norm for LTIQO systems with one or multiple outputs and propose an algorithm to optimize a reduced order model (ROM) to be close in the $\mathcal{H}_\infty$-norm to a given full order model. We illustrate the applicability and the performance with an established numerical example and compare the resulting ROMs with results from balanced truncation and $\mathcal{H}_2$-focussed algorithms.

$\mathcal{H}_\infty$ model order reduction for quadratic output systems

TL;DR

This work addresses MOR for linear time-invariant systems with quadratic outputs (LTIQO) by defining an -norm that accounts for both linear and quadratic outputs. An optimization-based MOR framework using a leveled least squares surrogate, with adaptive level and frequency sampling, yields nonintrusive ROMs that minimize the -error. A condensed structure-preserving parametrization for port-Hamiltonian systems is developed to guarantee stability and respect the Hamiltonian structure in reduced models. Numerical experiments on a mass-spring-damper example and a PH system show that the proposed method outperforms state-of-the-art MOR approaches, with benefits in both linear and quadratic output channels.

Abstract

Linear time-invariant quadratic output (LTIQO) systems generalize linear time-invariant systems to nonlinear regimes. Problems of this class occur in multiple applications naturally, such as port-Hamiltonian systems, optimal control, and stochastical problems. We introduce an -norm for LTIQO systems with one or multiple outputs and propose an algorithm to optimize a reduced order model (ROM) to be close in the -norm to a given full order model. We illustrate the applicability and the performance with an established numerical example and compare the resulting ROMs with results from balanced truncation and -focussed algorithms.
Paper Structure (19 sections, 8 theorems, 72 equations, 5 figures, 2 tables, 2 algorithms)

This paper contains 19 sections, 8 theorems, 72 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The H infty-norm
  3. Optimization-based MOR for the H infty norm
  4. A leveled least squares approach
  5. Adapting the level $\gamma$
  6. Choice of the frequency points
  7. Gradient-based minimization
  8. Algorithm
  9. Real parameterization and gradients
  10. Structure preserving MOR for pH systems
  11. Numerical Example
  12. Mass-spring-damper
  13. PH system with quadratic output
  14. Summary and open questions
  15. Derivatives of the linear contributions
  16. ...and 4 more sections

Key Result

Theorem 2.2

The $\mathcal{H}_\infty$-norm of the LTIQO-system eq::LTIMQO has the following properties:

Figures (5)

  • Figure 1: Comparison of the $\mathcal{H}_\infty$-error for varying reduced dimensions $r$ of our method, called Name, in comparison with BT and TSIA.
  • Figure 2: Frobenius norm of the pointwise transfer function error $\mathbf{G}_{\mathrm{e},2}(\omega_1, \omega_2)$ for varying frequencies $\omega_1$ and $\omega_2$ for a ROM with $r=12$. Top: $\mathcal{H}_\infty$ opt. Bottom: BT. The white marks are the elements of $\Omega_2$. The red line is the cross-section plotted in \ref{['fig::crossection']}.
  • Figure 3: Sampling of the map $\omega_2\mapsto \mathbf{G}_{\mathrm{e},2}(0,\omega_2)$ for ROMs of dimension $r=12$.
  • Figure 4: Output $\mathcal{L}_2$-error between FOM and ROM for $\mathcal{H}_\infty$ opt. and TSIA over a time interval $\mathbb{T} =[0, 100]$ for the input $u_{1, 2} = [\sin(s_{1,2} t)\cos(s_{1,2} t), \cos(s_{1,2} t)\sin(s_{1,2} t)]$ with $s_1=0.02$ and $s_2 = 4.1$, and $u_3$ being a linear chirp signal in the first component and a quadratic one in the second component and different reduced dimensions.
  • Figure 5: Comparison of the $\mathcal{H}_\infty$-error for ROMs of varying sizes generated by TSIA (without preserving the pH structure) and with $\mathcal{H}_\infty$ opt. with the structure-preserving parametrization \ref{['eqn:parameterization:pH']}. The contributions of the linear and quadratic part of the transfer function are displayed separately.

Theorems & Definitions (17)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 7 more