Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Online Model Order Reduction of Linear Systems via $(γ,δ)$-Similarity

Shivam Bajaj, Carolyn L. Beck, Vijay Gupta

TL;DR

The paper develops a disturbance-robust framework for online model order reduction of continuous-time linear systems using a $\big(\gamma,\delta\big)$-similarity concept. It formulates a BMI-based optimization to obtain a $\big(\gamma,\delta\big)$-ROM that minimizes $\gamma+\delta$, preserves stability, and provides explicit error bounds for open- and closed-loop behavior, including interconnected and modular systems. The approach is compatible with existing methods like Balanced Truncation and Moment Matching, and can be integrated as an add-on to yield robust ROMs with provable interconnection and closed-loop guarantees. Numerical results on a double-pendulum, a spring-mass-damper network, and a building model demonstrate robustness to disturbances and potential benefits for online simulation and digital twin applications.

Abstract

Model order reduction aims to determine a low-order approximation of high-order models with least possible approximation errors. For application to physical systems, it is crucial that the reduced order model (ROM) is robust to any disturbance that acts on the full order model (FOM) -- in the sense that the output of the ROM remains a good approximation of that of the FOM, even in the presence of such disturbances. In this work, we present a framework for online model order reduction for a class of continuous-time linear systems that ensures this property for any $\mathcal{L}_2$ disturbance. Apart from robustness to disturbances in this sense, the proposed framework also displays other desirable properties for model order reduction: (1) a provable bound on the error defined as the $L_2$ norm of the difference between the output of the ROM and FOM, (2) preservation of stability, (3) compositionality properties and a provable error bound for arbitrary interconnected systems, (4) a provable bound on the output of the FOM when the controller designed for the ROM is used with the FOM, and finally, (5) compatibility with existing approaches such as balanced truncation and moment matching. Property (4) does not require computation of any gap metric and property (5) is beneficial as existing approaches can also be equipped with some of the preceding properties. The theoretical results are corroborated on numerical case studies, including on a building model.

Online Model Order Reduction of Linear Systems via $(γ,δ)$-Similarity

TL;DR

The paper develops a disturbance-robust framework for online model order reduction of continuous-time linear systems using a -similarity concept. It formulates a BMI-based optimization to obtain a -ROM that minimizes , preserves stability, and provides explicit error bounds for open- and closed-loop behavior, including interconnected and modular systems. The approach is compatible with existing methods like Balanced Truncation and Moment Matching, and can be integrated as an add-on to yield robust ROMs with provable interconnection and closed-loop guarantees. Numerical results on a double-pendulum, a spring-mass-damper network, and a building model demonstrate robustness to disturbances and potential benefits for online simulation and digital twin applications.

Abstract

Model order reduction aims to determine a low-order approximation of high-order models with least possible approximation errors. For application to physical systems, it is crucial that the reduced order model (ROM) is robust to any disturbance that acts on the full order model (FOM) -- in the sense that the output of the ROM remains a good approximation of that of the FOM, even in the presence of such disturbances. In this work, we present a framework for online model order reduction for a class of continuous-time linear systems that ensures this property for any disturbance. Apart from robustness to disturbances in this sense, the proposed framework also displays other desirable properties for model order reduction: (1) a provable bound on the error defined as the norm of the difference between the output of the ROM and FOM, (2) preservation of stability, (3) compositionality properties and a provable error bound for arbitrary interconnected systems, (4) a provable bound on the output of the FOM when the controller designed for the ROM is used with the FOM, and finally, (5) compatibility with existing approaches such as balanced truncation and moment matching. Property (4) does not require computation of any gap metric and property (5) is beneficial as existing approaches can also be equipped with some of the preceding properties. The theoretical results are corroborated on numerical case studies, including on a building model.

Paper Structure

This paper contains 23 sections, 20 theorems, 75 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Motivation
  4. Contributions
  5. Preliminaries, Motivation, and Problem Statement
  6. Preliminaries
  7. Balanced Truncation
  8. Moment Matching
  9. $(\gamma,\delta)$-Similar Systems
  10. Problem Statement
  11. $(\gamma,\delta)$-Model Order Reduction
  12. Model Order Reduction of Interconnected Systems
  13. Systems with Arbitrary Interconnection Structure
  14. Systems with Series, Parallel, or Feedback Connections
  15. Closed-Loop Stability of Original System
  16. ...and 8 more sections

Key Result

Lemma 1

There exists a $\gamma>0$ such that the system $\Sigma^n$ is $(\gamma,\delta)$-similar to itself for any $\delta>0$.

Figures (10)

  • Figure 1: Difference between the outputs (error) of the FOM and its ROM.
  • Figure 2: Illustration of the modular approach for system $\Sigma$ consisting of $N=2$ subsystems. Subsystem $\Sigma_1^{r_1}$ and $\Sigma_2^{r_2}$ are ROMs of $\Sigma_1^{n_1}$ and $\Sigma_2^{n_2}$, respectively. $w_{12}, w_{21}$ (resp. $w_{12}', w_{21}'$) and $z_{12}, z_{21}$ (resp. $z_{12}', z_{21}'$) are the internal inputs and outputs, respectively, to system $\Sigma$ (resp. $\Sigma_{\text{mod}}$). $u_1,u_2$ (resp. $u_1', u_2'$) and $y_1, y_2$ (resp. $y_1', y_2'$) are the external inputs and outputs, respectively, to system $\Sigma$ (resp. $\Sigma_{\text{mod}}$).
  • Figure 3: Parallel interconnection of $N=2$ systems. System $\Sigma$ consists of two FOMs $\Sigma_1^{n_1}$ and $\Sigma_2^{n_2}$ connected in parallel. For $i\in \{1,2\}$, $\Sigma_i^{r_i}$ denotes a $(\gamma_i,\delta_i)$-ROM of $\Sigma_i^{n_i}$. Systems $\Sigma_i^{r_i}$ are connected in parallel to obtain $\Sigma_{\text{mod}}$.
  • Figure 4: Series interconnection of $N=2$ systems. System $\Sigma$ consists of two FOMs $\Sigma_1^{n_1}$ and $\Sigma_2^{n_2}$ connected in series. For $i\in \{1,2\}$, $\Sigma_i^{r_i}$ denotes a $(\gamma_i,\delta_i)$-ROM of $\Sigma_i^{n_i}$. Systems $\Sigma_i^{r_i}$ are connected in series to obtain $\Sigma_{\text{mod}}$.
  • Figure 5: Controller $\Sigma_K$ designed for ROM $\Sigma^r$ connected in feedback with ROM $\Sigma^r$ and original system $\Sigma^n$.
  • ...and 5 more figures

Theorems & Definitions (47)

  • Definition 1
  • Lemma 1
  • Theorem II.1
  • Lemma 2
  • Definition 2
  • Theorem III.1
  • proof
  • Theorem III.2
  • proof
  • Remark 1
  • ...and 37 more