Comparison of Non-deterministic Stable Linear Systems by (γ,δ)-Similarity

Armin Pirastehzad*, Arjan van der Schaft, Bart Besselink

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

We introduce (&#x03B3;,&#x03B4;)-similarity, a notion of system comparison that measures to what extent two stable linear dynamical systems behave similarly in an input-output sense. This behavioral similarity is characterized by measuring the sensitivity of the difference between the two output trajectories in terms of the external inputs to the two potentially non-deterministic systems. As such, (&#x03B3;,&#x03B4;)-similarity is a notion that characterizes <italic>approximation</italic> of input-output behavior, whereas existing notions of simulation target equivalence. Next, as this approximation is specified in terms of the <inline-formula><tex-math notation="LaTeX">$\mathcal {L}_{2}$</tex-math></inline-formula> signal norm, (&#x03B3;,&#x03B4;)-similarity allows for integration with existing methods for analysis and synthesis of control systems, in particular, robust control techniques. We characterize the notion of (&#x03B3;,&#x03B4;)-similarity as a linear matrix inequality feasibility problem and derive its interpretation in terms of transfer matrices. Our study on the compositional properties of (&#x03B3;,&#x03B4;)-similarity shows that the notion is preserved through series and feedback interconnections. This highlights its potential application in compositional reasoning, namely abstraction and modular synthesis of large-scale interconnected dynamical systems. We further illustrate our results in an electrical network example.

Original languageEnglish
Pages (from-to)8617-8632
Number of pages16
JournalIEEE Transactions on Automatic Control
Volume69
Issue number12
Early online date18-Jun-2024
DOIs
Publication statusPublished - Dec-2024

Keywords

  • Abstraction
  • approximation
  • Cognition
  • compositional reasoning
  • Control theory
  • Dynamical systems
  • Linear systems
  • non-deterministic systems
  • Sensitivity
  • simulation relation
  • Trajectory
  • Vectors

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