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Reachable and observable sets for switched systems via generalized Lyapunov equations: application to switched descriptor systems

Mattia Manucci, Benjamin Unger

TL;DR

This work addresses reachability and observability for switched DAEs with jumps and impulses, where standard MOR techniques struggle due to singular E and discontinuities. It reformulates the problem as a switched descriptor system using piecewise-smooth distributions and a quasi-Weierstrass decoupling, then connects these dynamics to two generalized Lyapunov equations to obtain gramians. The main contributions are explicit recursive characterizations of reachable and unobservable sets for the reformulated system and the demonstration that the GLE solutions bound these sets (R ⊆ span(P), O ⊆ span(Q)) under mild stability assumptions, enabling balancing-based MOR. The results provide a principled, computable route to reduced-order modelling of switched DAEs in applications where inputs, state jumps, and impulses are intrinsic, with relevance to control of robotics, power systems, and related domains.

Abstract

In a recent work [Manucci, Unger, ArXiv e-print 2404.10511, 2024], the authors propose using two generalized Lyapunov equations (GLEs) to derive a balancing-based model order reduction~(MOR) method for a general class of switched differential-algebraic equations (DAEs). This work explains why these GLEs provide solutions suitable for MOR by showing that the image set of the solutions of the two GLEs always encloses the reachable and observable set of a suitably defined switched system with the same input to output map of the switched DAE system.

Reachable and observable sets for switched systems via generalized Lyapunov equations: application to switched descriptor systems

TL;DR

This work addresses reachability and observability for switched DAEs with jumps and impulses, where standard MOR techniques struggle due to singular E and discontinuities. It reformulates the problem as a switched descriptor system using piecewise-smooth distributions and a quasi-Weierstrass decoupling, then connects these dynamics to two generalized Lyapunov equations to obtain gramians. The main contributions are explicit recursive characterizations of reachable and unobservable sets for the reformulated system and the demonstration that the GLE solutions bound these sets (R ⊆ span(P), O ⊆ span(Q)) under mild stability assumptions, enabling balancing-based MOR. The results provide a principled, computable route to reduced-order modelling of switched DAEs in applications where inputs, state jumps, and impulses are intrinsic, with relevance to control of robotics, power systems, and related domains.

Abstract

In a recent work [Manucci, Unger, ArXiv e-print 2404.10511, 2024], the authors propose using two generalized Lyapunov equations (GLEs) to derive a balancing-based model order reduction~(MOR) method for a general class of switched differential-algebraic equations (DAEs). This work explains why these GLEs provide solutions suitable for MOR by showing that the image set of the solutions of the two GLEs always encloses the reachable and observable set of a suitably defined switched system with the same input to output map of the switched DAE system.
Paper Structure (10 sections, 10 theorems, 58 equations)

This paper contains 10 sections, 10 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Differential-algebraic equations
  4. Solution space for switched DAE
  5. Reformulation of switched descriptor systems
  6. Reachability and observability
  7. Reachability for switched systems with input-state-dependent jumps
  8. Observability set for state-dependent impulses
  9. Reachable and observable sets via GLE
  10. Conclusions

Key Result

Theorem 3.1

The matrix pair $(\bm{E}_j,\bm{A}_j)\in\mathbb{R}^{n\times n}\times\mathbb{R}^{n\times n}$ is regular if and only if there exist nonsingular real-valued matrices $\bm{S}_j,\bm{T}_j$, such that where $\bm{N}_j\in\mathbb{R}^{n_{\bm{N}_j} \times n_{\bm{N}_j}}$ is nilpotent with nilpotency index $\nu_j$ and $\bm{J}_j\in\mathbb{R}^{n_{\bm{J}_j} \times n_{\bm{J}_j} }$, with $n_{\bm{J}_j}=n-n_{\bm{N}_j}

Theorems & Definitions (26)

  • Theorem 3.1: Quasi-Weierstrass Form, BerIT12
  • Definition 3.2: Piecewise-smooth distributions
  • Lemma 3.3: Trenn2012
  • Theorem 3.4: Hos22 and HosT23
  • Definition 3.5
  • Definition 3.6
  • Definition 4.1
  • Remark 4.2
  • Lemma 4.3
  • proof
  • ...and 16 more