research-article

Generating Unstable Trajectories for Switched Systems via Dual Sum-Of-Squares Techniques

Authors:

Raphaël M. Jungers,

Pablo A. ParriloAuthors Info & Claims

HSCC '16: Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control

Pages 51 - 60

https://doi.org/10.1145/2883817.2883821

Published: 11 April 2016 Publication History

Abstract

The joint spectral radius (JSR) of a set of matrices characterizes the maximal asymptotic growth rate of an infinite product of matrices of the set. This quantity appears in a number of applications including the stability of switched and hybrid systems. Many algorithms exist for estimating the JSR but not much is known about how to generate an infinite sequence of matrices with an optimal asymptotic growth rate. To the best of our knowledge, the currently known algorithms select a small sequence with large spectral radius using brute force (or branch-and-bound variants) and repeats this sequence infinitely.

In this paper we introduce a new approach to this question, using the dual solution of a sum of squares optimization program for JSR approximation. Our algorithm produces an infinite sequence of matrices with an asymptotic growth rate arbitrarily close to the JSR. The algorithm naturally extends to the case where the allowable switching sequences are determined by a graph or finite automaton. Unlike the brute force approach, we provide a guarantee on the closeness of the asymptotic growth rate to the JSR. This, in turn, provides new bounds on the quality of the JSR approximation. We provide numerical examples illustrating the good performance of the algorithm.

Supplementary Material

ZIP File (hscc14.zip)

This directory contains code to recreate the examples from Benoît Legat and Raphaël M. Jungers and Pablo A. Parrilo, "Generating unstable trajectories for Switched Systems via Dual Sum-Of-Squares techniques" in Hybrid Systems Computational & Control, ACM (2016). Also a Read Me file with instructions

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References

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Cited By

Klett CAbate MCoogan SFeron E(2021)A Numerical Method to Compute Stability Margins of Switching Linear Systems2021 American Control Conference (ACC)10.23919/ACC50511.2021.9483141(864-869)Online publication date: 25-May-2021
https://doi.org/10.23919/ACC50511.2021.9483141
Xu XAckmese B(2021)Approximation of the Constrained Joint Spectral Radius via Algebraic LiftingIEEE Transactions on Automatic Control10.1109/TAC.2020.302058066:7(3386-3392)Online publication date: Jul-2021
https://doi.org/10.1109/TAC.2020.3020580
Legat BParrilo PJungers R(2019)An Entropy-Based Bound for the Computational Complexity of a Switched SystemIEEE Transactions on Automatic Control10.1109/TAC.2019.290262564:11(4623-4628)Online publication date: Nov-2019
https://doi.org/10.1109/TAC.2019.2902625
Show More Cited By

Index Terms

Generating Unstable Trajectories for Switched Systems via Dual Sum-Of-Squares Techniques
1. Computer systems organization
  1. Embedded and cyber-physical systems
    1. Robotics
      1. Robotic control
2. Mathematics of computing
  1. Mathematical analysis
    1. Mathematical optimization
      1. Continuous optimization
        Convex optimization
        Semidefinite programming

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cover image ACM Conferences

HSCC '16: Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control

April 2016

324 pages

ISBN:9781450339551

DOI:10.1145/2883817

Program Chairs:
Alessandro Abate
University of Oxford, UK
,
Georgios Fainekos
Arizona State University, USA

Copyright © 2016 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 11 April 2016

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HSCC'16: 19th International Conference on Hybrid Systems: Computation and Control

April 12 - 14, 2016

Vienna, Austria

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HSCC '16 Paper Acceptance Rate 28 of 65 submissions, 43%;

Overall Acceptance Rate 153 of 373 submissions, 41%

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Cited By

Klett CAbate MCoogan SFeron E(2021)A Numerical Method to Compute Stability Margins of Switching Linear Systems2021 American Control Conference (ACC)10.23919/ACC50511.2021.9483141(864-869)Online publication date: 25-May-2021
https://doi.org/10.23919/ACC50511.2021.9483141
Xu XAckmese B(2021)Approximation of the Constrained Joint Spectral Radius via Algebraic LiftingIEEE Transactions on Automatic Control10.1109/TAC.2020.302058066:7(3386-3392)Online publication date: Jul-2021
https://doi.org/10.1109/TAC.2020.3020580
Legat BParrilo PJungers R(2019)An Entropy-Based Bound for the Computational Complexity of a Switched SystemIEEE Transactions on Automatic Control10.1109/TAC.2019.290262564:11(4623-4628)Online publication date: Nov-2019
https://doi.org/10.1109/TAC.2019.2902625
Dercole FRossa F(2019)Tree-Based Algorithms for the Stability of Discrete-Time Switched Linear Systems Under Arbitrary and Constrained SwitchingIEEE Transactions on Automatic Control10.1109/TAC.2018.288714264:9(3823-3830)Online publication date: Sep-2019
https://doi.org/10.1109/TAC.2018.2887142
Gomes CLegat BJungers RVangheluwe H(2019)Stable Adaptive Co-simulation: A Switched Systems ApproachIUTAM Symposium on Solver-Coupling and Co-Simulation10.1007/978-3-030-14883-6_5(81-97)Online publication date: 15-May-2019
https://doi.org/10.1007/978-3-030-14883-6_5
Gomes CJungers RLegat BVangheluwe H(2018)Minimally, Constrained Stable Switched Systems and Application to Co-Simulation2018 IEEE Conference on Decision and Control (CDC)10.1109/CDC.2018.8619223(5676-5681)Online publication date: Dec-2018
https://doi.org/10.1109/CDC.2018.8619223
Athanasopoulos NSmpoukis KJungers R(2017)Invariant Sets Analysis for Constrained Switching SystemsIEEE Control Systems Letters10.1109/LCSYS.2017.27148401:2(256-261)Online publication date: Oct-2017
https://doi.org/10.1109/LCSYS.2017.2714840
Dercole FDella Rossa F(2017)A simple tree-based algorithm for deciding the stability of discrete-time switched linear systems2017 IEEE 56th Annual Conference on Decision and Control (CDC)10.1109/CDC.2017.8264443(5298-5303)Online publication date: Dec-2017
https://doi.org/10.1109/CDC.2017.8264443

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