When is a set of LMIs a sufficient condition for stability

Amir Ali Ahmadi; Raphaël M. Jungers; Pablo A. Parrilo; Mardavij Roozbehani

doi:10.3182/20120620-3-DK-2025.00098

When is a set of LMIs a sufficient condition for stability

Amir Ali Ahmadi, Raphaël M. Jungers, Pablo A. Parrilo, Mardavij Roozbehani

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

6 Scopus citations

Abstract

We study stability criteria for discrete time switching systems and provide a meta-theorem that characterizes all the LMI-based Lyapunov theorems. For this purpose, we investigate the structure of sets of LMIs that are a sufficient condition for stability (i.e., such that any switching system which satisfies these LMIs is stable). Different such LMI conditions have been proposed in the last fifteen years, and we prove in this paper that a family of conditions recently provided by us actually encapsulates all the possible valid conditions. As a byproduct, we show that it is PSPACE-complete to recognize whether a particular set of LMIs implies the stability of a switching system.

Original language	English (US)
Title of host publication	ROCOND'12 - 7th IFAC Symposium on Robust Control Design
Pages	313-318
Number of pages	6
Edition	PART 1
DOIs	https://doi.org/10.3182/20120620-3-DK-2025.00098
State	Published - Sep 17 2012
Externally published	Yes
Event	7th IFAC Symposium on Robust Control Design, ROCOND'12 - Aalborg, Denmark Duration: Jun 20 2012 → Jun 22 2012

Publication series

Name	IFAC Proceedings Volumes (IFAC-PapersOnline)
Number	PART 1
Volume	7
ISSN (Print)	1474-6670

Other

Other	7th IFAC Symposium on Robust Control Design, ROCOND'12
Country	Denmark
City	Aalborg
Period	6/20/12 → 6/22/12

All Science Journal Classification (ASJC) codes

Control and Systems Engineering

Keywords

Convex optimization
Finite automata
Hybrid systems
Joint spectral radius
Linear matrix inequalities

Access to Document

10.3182/20120620-3-DK-2025.00098

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title = "When is a set of LMIs a sufficient condition for stability",

abstract = "We study stability criteria for discrete time switching systems and provide a meta-theorem that characterizes all the LMI-based Lyapunov theorems. For this purpose, we investigate the structure of sets of LMIs that are a sufficient condition for stability (i.e., such that any switching system which satisfies these LMIs is stable). Different such LMI conditions have been proposed in the last fifteen years, and we prove in this paper that a family of conditions recently provided by us actually encapsulates all the possible valid conditions. As a byproduct, we show that it is PSPACE-complete to recognize whether a particular set of LMIs implies the stability of a switching system.",

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author = "Ahmadi, {Amir Ali} and Jungers, {Rapha{\"e}l M.} and Parrilo, {Pablo A.} and Mardavij Roozbehani",

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Ahmadi, AA, Jungers, RM, Parrilo, PA & Roozbehani, M 2012, When is a set of LMIs a sufficient condition for stability. in ROCOND'12 - 7th IFAC Symposium on Robust Control Design. PART 1 edn, IFAC Proceedings Volumes (IFAC-PapersOnline), no. PART 1, vol. 7, pp. 313-318, 7th IFAC Symposium on Robust Control Design, ROCOND'12, Aalborg, Denmark, 6/20/12. https://doi.org/10.3182/20120620-3-DK-2025.00098

When is a set of LMIs a sufficient condition for stability. / Ahmadi, Amir Ali; Jungers, Raphaël M.; Parrilo, Pablo A.; Roozbehani, Mardavij.

ROCOND'12 - 7th IFAC Symposium on Robust Control Design. PART 1. ed. 2012. p. 313-318 (IFAC Proceedings Volumes (IFAC-PapersOnline); Vol. 7, No. PART 1).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

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