research-article

A Scalable Algebraic Method to Infer Quadratic Invariants of Switched Systems

Authors:

Xavier Allamigeon,

Stéphane Gaubert,

Éric Goubault,

Sylvie PutotAuthors Info & Claims

ACM Transactions on Embedded Computing Systems (TECS), Volume 15, Issue 4

Article No.: 69, Pages 1 - 20

https://doi.org/10.1145/2932187

Published: 01 September 2016 Publication History

Abstract

We present a new numerical abstract domain based on ellipsoids designed for the formal verification of switched linear systems. Unlike the existing approaches, this domain does not rely on a user-given template. We overcome the difficulty that ellipsoids do not have a lattice structure by exhibiting a canonical operator overapproximating the union. This operator is the only one that permits the performance of analyses that are invariant with respect to a linear transformation of state variables. It provides the minimum volume ellipsoid enclosing two given ellipsoids. We show that it can be computed in O(n³) elementary algebraic operations. We finally develop a fast nonlinear power-type algorithm, which allows one to determine sound quadratic invariants on switched systems in a tractable way, by solving fixed-point problems over the space of ellipsoids. We test our approach on several benchmarks, and compare it with the standard techniques based on linear matrix inequalities, showing an important speedup on typical instances.

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Cited By

Allamigeon XGaubert SGoubault EPutot SStott N(2017)A Fast Method to Compute Disjunctive Quadratic Invariants of Numerical ProgramsACM Transactions on Embedded Computing Systems10.1145/312650216:5s(1-19)Online publication date: 27-Sep-2017
https://dl.acm.org/doi/10.1145/3126502
Gaubert SStott N(2017)Tropical Kraus maps for optimal control of switched systems2017 IEEE 56th Annual Conference on Decision and Control (CDC)10.1109/CDC.2017.8263839(1330-1337)Online publication date: Dec-2017
https://doi.org/10.1109/CDC.2017.8263839

Index Terms

A Scalable Algebraic Method to Infer Quadratic Invariants of Switched Systems
1. Software and its engineering
  1. Software organization and properties
    1. Software functional properties
      1. Formal methods
2. Theory of computation
  1. Semantics and reasoning
    1. Program reasoning
      1. Invariants

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Published In

cover image ACM Transactions on Embedded Computing Systems

ACM Transactions on Embedded Computing Systems Volume 15, Issue 4

Special Issue on ESWEEK2015 and Regular Papers

August 2016

411 pages

ISSN:1539-9087

EISSN:1558-3465

DOI:10.1145/2982215

Editor:
Sandeep K. Shukla
Indian Institute of Technology, India

Issue’s Table of Contents

Copyright © 2016 ACM.

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Association for Computing Machinery

New York, NY, United States

Journal Family

ACM Journals for the Design of Smart and Connected Systems

Publication History

Published: 01 September 2016

Accepted: 01 April 2016

Revised: 01 April 2016

Received: 01 December 2015

Published in TECS Volume 15, Issue 4

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PGMO programme of EDF and FMJH
Agence Nationale de la Recherche
ICODE
Complex Systems Engineering Research Chair of Ecole polytechnique, THALES, FX, DGA, DASSAULT AVIATION, DCNS Research, ENSTA ParisTech, Télécom ParisTech, Fondation ParisTech, FDO ENSTA

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Cited By

Allamigeon XGaubert SGoubault EPutot SStott N(2017)A Fast Method to Compute Disjunctive Quadratic Invariants of Numerical ProgramsACM Transactions on Embedded Computing Systems10.1145/312650216:5s(1-19)Online publication date: 27-Sep-2017
https://dl.acm.org/doi/10.1145/3126502
Gaubert SStott N(2017)Tropical Kraus maps for optimal control of switched systems2017 IEEE 56th Annual Conference on Decision and Control (CDC)10.1109/CDC.2017.8263839(1330-1337)Online publication date: Dec-2017
https://doi.org/10.1109/CDC.2017.8263839

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