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Singularly Perturbed Hybrid Systems for Analysis of Networks With Frequently Switching Graphs

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Title: Singularly Perturbed Hybrid Systems for Analysis of Networks With Frequently Switching Graphs
Authors: Aneel Tanwani, Hyungbo Shim, Andrew R. Teel
Contributors: Tanwani, Aneel, Équipe Méthodes et Algorithmes en Commande (LAAS-MAC), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT), Department of Electrical and Computer Engineering Seoul National University, Seoul National University Seoul (SNU), Center for Control, Dynamical-Systems, and Computation Santa Barbara (CCDC), University of California Santa Barbara (UC Santa Barbara), University of California (UC)-University of California (UC), National Research Foundation of Korea (Grant Number: 00165417)AFOSR (Grant Number: FA9550-21-1-0452), ANR-20-JSTM-0001,CyphAI,Méthodes formelles pour l'analysis et le développement de systèmes cyber-physiques intégrant l'intelligence artificielle(2020)
Source: IEEE Transactions on Automatic Control. 70:4344-4359
Publisher Information: Institute of Electrical and Electronics Engineers (IEEE), 2025.
Publication Year: 2025
Subject Terms: [INFO.INFO-SY] Computer Science [cs]/Systems and Control [cs.SY], Heterogeneous agents, Hybrid systems, Discrete-time switched systems, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], [INFO.INFO-SY]Computer Science [cs]/Systems and Control [cs.SY], [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS], [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Singular perturbations, Practical stability, Nonlinear oscillators
Description: For a class of hybrid systems, where jumps occur frequently, we analyze the stability of system trajectories in view of singularly perturbed dynamics. The specific model we consider comprises an interconnection of two hybrid subsystems, a timer which triggers the jumps, and some discrete variables to determine the index of the jump maps. The flow equations of these variables are singularly perturbed differential equations and, in particular, a smaller value of the singular perturbation parameter leads to an increase in the frequency of the jump instants. For the limiting value of this parameter, we consider a decomposition which comprises a quasi-steady-state system modeled by a differential equation without any jumps and a boundary-layer system described by purely discrete dynamics. Under appropriate assumptions on the quasi-steady-state system and the boundary-layer system, we derive results showing practical stability of a compact attractor when the jumps occur sufficiently often. As an application of our results, we discuss the control design problem in a network of second-order continuous-time coupled oscillators, where each agent communicates the information about its position to some of its neighbors at discrete times. Using the results developed in this article, we show that if the union of the communication graphs being used for information exchange between agents is connected, then the oscillators achieve practical consensus.
Document Type: Article
File Description: application/pdf
ISSN: 2334-3303
0018-9286
DOI: 10.1109/tac.2024.3523242
Access URL: https://laas.hal.science/hal-04855233v1
https://laas.hal.science/hal-04855233v1/document
https://doi.org/10.1109/tac.2024.3523242
Rights: IEEE Copyright
CC BY
Accession Number: edsair.doi.dedup.....a54e0451d7450f86c5e921db9243bed1
Database: OpenAIRE
Description
Abstract:For a class of hybrid systems, where jumps occur frequently, we analyze the stability of system trajectories in view of singularly perturbed dynamics. The specific model we consider comprises an interconnection of two hybrid subsystems, a timer which triggers the jumps, and some discrete variables to determine the index of the jump maps. The flow equations of these variables are singularly perturbed differential equations and, in particular, a smaller value of the singular perturbation parameter leads to an increase in the frequency of the jump instants. For the limiting value of this parameter, we consider a decomposition which comprises a quasi-steady-state system modeled by a differential equation without any jumps and a boundary-layer system described by purely discrete dynamics. Under appropriate assumptions on the quasi-steady-state system and the boundary-layer system, we derive results showing practical stability of a compact attractor when the jumps occur sufficiently often. As an application of our results, we discuss the control design problem in a network of second-order continuous-time coupled oscillators, where each agent communicates the information about its position to some of its neighbors at discrete times. Using the results developed in this article, we show that if the union of the communication graphs being used for information exchange between agents is connected, then the oscillators achieve practical consensus.
ISSN:23343303
00189286
DOI:10.1109/tac.2024.3523242