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Balancing-based model reduction for switched descriptor systems

Mattia Manucci, Benjamin Unger

TL;DR

This work tackles model order reduction for switched descriptor systems where differential-algebraic dynamics arise in multiple modes. It introduces a reformulation to a switched ODE with jumps and extends balanced truncation to this setting, enabling a large-scale MOR via projection and GLE-based Gramian computation. A core contribution is a certified a-priori error bound that accounts for numerical approximation errors in solving generalized Lyapunov equations and in Gramian computations, supported by LMIs. The numerical framework combines efficient Wong-space computation with stationary Lyapunov solvers, providing reliable error certificates for reduced models. Demonstrations on a constrained mass-spring-damper system and an instationary Stokes problem validate both accuracy and computational efficiency, underscoring the method’s practical impact for engineering applications with switching dynamics.

Abstract

We present a novel certified model order reduction (MOR) algorithm for switched descriptor systems applicable to large-scale systems. Our algorithm combines the idea of [Hossain \& Trenn, Technical report, 2023] to reformulate the switched descriptor system as a switched ordinary differential equation with jumps and an extension of the balanced truncation for switched ODE from [Pontes Duff et al., IEEE Trans.~Automat.~Control, 2020]. Besides being the first MOR method for switched descriptor systems applicable to the large-scale setting, we give a detailed numerical analysis by incorporating the error in the computation of the system Gramians in the a-priori error bound for the output of the reduced system. In more detail, we demonstrate, theoretically and numerically, that the standard error bound is not applicable, and a certificate must account for the numerical approximation errors.

Balancing-based model reduction for switched descriptor systems

TL;DR

This work tackles model order reduction for switched descriptor systems where differential-algebraic dynamics arise in multiple modes. It introduces a reformulation to a switched ODE with jumps and extends balanced truncation to this setting, enabling a large-scale MOR via projection and GLE-based Gramian computation. A core contribution is a certified a-priori error bound that accounts for numerical approximation errors in solving generalized Lyapunov equations and in Gramian computations, supported by LMIs. The numerical framework combines efficient Wong-space computation with stationary Lyapunov solvers, providing reliable error certificates for reduced models. Demonstrations on a constrained mass-spring-damper system and an instationary Stokes problem validate both accuracy and computational efficiency, underscoring the method’s practical impact for engineering applications with switching dynamics.

Abstract

We present a novel certified model order reduction (MOR) algorithm for switched descriptor systems applicable to large-scale systems. Our algorithm combines the idea of [Hossain \& Trenn, Technical report, 2023] to reformulate the switched descriptor system as a switched ordinary differential equation with jumps and an extension of the balanced truncation for switched ODE from [Pontes Duff et al., IEEE Trans.~Automat.~Control, 2020]. Besides being the first MOR method for switched descriptor systems applicable to the large-scale setting, we give a detailed numerical analysis by incorporating the error in the computation of the system Gramians in the a-priori error bound for the output of the reduced system. In more detail, we demonstrate, theoretically and numerically, that the standard error bound is not applicable, and a certificate must account for the numerical approximation errors.
Paper Structure (25 sections, 13 theorems, 82 equations, 4 figures, 1 algorithm)

This paper contains 25 sections, 13 theorems, 82 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Literature review and state-of-the-art
  4. Organization of the manuscript
  5. Notation
  6. Preliminaries
  7. Differential-algebraic equations
  8. Switched descriptor systems
  9. Reachability and Observability
  10. Model Reduction
  11. Model reduction algorithm
  12. Reformulation of the switched system
  13. Handling the general class of switched DAEs
  14. Efficient and certified numerics for large scale systems
  15. Efficient computation of the Wong sequences space
  16. ...and 10 more sections

Key Result

Theorem 2.1

A matrix pair $(\bm{E},\bm{A})\in\mathbb{R}^{n\times n}\times\mathbb{R}^{n\times n}$ is regular if and only if there exists matrices $\bm{S},\bm{T}\in\mathrm{GL}_{n}$ such that where $\bm{N}\in\mathbb{R}^{n_{\bm{N}} \times n_{\bm{N}}}$ is nilpotent with nilpotency index $\nu$ and $\bm{J}\in\mathbb{R}^{n_{\bm{J}} \times n_{\bm{J}} }$, with $n_{\bm{J}}=n-n_{\bm{N}}$.

Figures (4)

  • Figure 1: Constrained mass-spring-damper system with $M=5$ modes. Qualitative comparison of full and reduced outputs (specifically second output) for two different switching paths and input signals.
  • Figure 2: Constrained mass-spring-damper system. The input signal used to produce \ref{['fig3:left']} is $\bm{u}(t)\coloneqq\sin(t)$.
  • Figure 3: Constrained mass-spring-damper system with $M=2$ allowing input dependent state-jumps. The input used is $\bm{u}(t)\coloneqq\sin(t^2+t)$.
  • Figure 4: Instationary Stokes problem with $M=5$. The input used is $\bm{u}(t)\coloneqq\sin(t^2+t)$

Theorems & Definitions (30)

  • Theorem 2.1: Quasi-Weierstrass Form, BerIT12
  • Remark 2.2
  • Theorem 2.3: QWF via Wong sequences, BerIT12
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6: Hos22
  • Definition 2.8
  • Lemma 2.9: See KueT16Hos22
  • Theorem 3.1
  • proof
  • ...and 20 more