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Interconnection-based Model Reduction for Linear Hybrid Systems

Zirui Niu, Giordano Scarciotti, Alessandro Astolfi

TL;DR

The paper extends moment matching to linear hybrid systems by formulating hybrid moments for direct and swapped interconnections and deriving corresponding hybrid Sylvester equations. It then constructs reduced-order hybrid models that match these moments, including a two-sided design that achieves simultaneous direct and swapped interconnection matching, with a periodic-jump simplification. A numerical example validates the theory and shows two order-$\nu=1$ reduced models that reproduce the interconnection responses. The work provides a systematic method for obtaining low-order hybrids that preserve interconnection-driven steady-state behavior, enabling scalable simulation and control of complex hybrid dynamics.

Abstract

In this paper, we address the model reduction problem for linear hybrid systems via the interconnection-based technique called moment matching. We consider two classical interconnections, namely the direct and swapped interconnections, in the hybrid setting, and we present families of reduced-order models for each interconnection via a hybrid characterisation of the steady-state responses. By combining the results for each interconnection, the design of a reduced-order model that achieves moment matching simultaneously for both interconnections is studied. In addition, we show that the presented results have simplified counterparts when the jumps of the hybrid system are periodic. A numerical simulation is finally given to illustrate the results.

Interconnection-based Model Reduction for Linear Hybrid Systems

TL;DR

The paper extends moment matching to linear hybrid systems by formulating hybrid moments for direct and swapped interconnections and deriving corresponding hybrid Sylvester equations. It then constructs reduced-order hybrid models that match these moments, including a two-sided design that achieves simultaneous direct and swapped interconnection matching, with a periodic-jump simplification. A numerical example validates the theory and shows two order- reduced models that reproduce the interconnection responses. The work provides a systematic method for obtaining low-order hybrids that preserve interconnection-driven steady-state behavior, enabling scalable simulation and control of complex hybrid dynamics.

Abstract

In this paper, we address the model reduction problem for linear hybrid systems via the interconnection-based technique called moment matching. We consider two classical interconnections, namely the direct and swapped interconnections, in the hybrid setting, and we present families of reduced-order models for each interconnection via a hybrid characterisation of the steady-state responses. By combining the results for each interconnection, the design of a reduced-order model that achieves moment matching simultaneously for both interconnections is studied. In addition, we show that the presented results have simplified counterparts when the jumps of the hybrid system are periodic. A numerical simulation is finally given to illustrate the results.
Paper Structure (11 sections, 10 theorems, 76 equations)

This paper contains 11 sections, 10 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and problem formulation
  3. On the Notion of Steady-state Response
  4. Review of Moment Matching
  5. Problem Formulation
  6. Moment Matching for the Direct Interconnection
  7. Moment Matching for the Swapped Interconnection
  8. Two-Sided Moment Matching
  9. The Special Case of Periodic Jumps
  10. A numerical example
  11. Conclusion

Key Result

Theorem 2.4

Ast:10aref:padoan2017geometricref:mao2024model. Consider system (equ:LTIsystem) and suppose $\sigma(A) \subset \mathbb{C}_{<0}$. Let $S \in \mathbb{R}^{\nu \times \nu}$ and $Q \in \mathbb{R}^{\nu \times \nu}$ be any two matrices with simple eigenvalues and disjoint spectra satisfying $\sigma(S) \sub

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.6
  • Remark 2.7
  • Lemma 3.1
  • Theorem 3.2
  • Remark 3.3
  • Definition 3.4
  • ...and 14 more