A Characterization of Lyapunov Inequalities for Stability of Switched Systems

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

doi:10.1109/TAC.2017.2671345

A Characterization of Lyapunov Inequalities for Stability of Switched Systems

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

Research output: Contribution to journal › Article › peer-review

35 Scopus citations

Abstract

We study stability criteria for discrete-time switched systems and provide a meta-theorem that characterizes all Lyapunov theorems of a certain canonical type. For this purpose, we investigate the structure of sets of LMIs that provide a sufficient condition for stability. Various such conditions have been proposed in the literature in the past 15 years. We prove in this note that a family of language-theoretic conditions recently provided by the authors encapsulates all the possible LMI conditions, thus putting a conclusion to this research effort. As a corollary, we show that it is PSPACE-complete to recognize whether a particular set of LMIs implies stability of a switched system. Finally, we provide a geometric interpretation of these conditions, in terms of existence of an invariant set.

Original language	English (US)
Article number	7858590
Pages (from-to)	3062-3067
Number of pages	6
Journal	IEEE Transactions on Automatic Control
Volume	62
Issue number	6
DOIs	https://doi.org/10.1109/TAC.2017.2671345
State	Published - Jun 2017

All Science Journal Classification (ASJC) codes

Control and Systems Engineering
Computer Science Applications
Electrical and Electronic Engineering

Keywords

Linear matrix inequalities
lyapunov methods
set theory
stability
switching systems (control)

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10.1109/TAC.2017.2671345

Cite this

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title = "A Characterization of Lyapunov Inequalities for Stability of Switched Systems",

abstract = "We study stability criteria for discrete-time switched systems and provide a meta-theorem that characterizes all Lyapunov theorems of a certain canonical type. For this purpose, we investigate the structure of sets of LMIs that provide a sufficient condition for stability. Various such conditions have been proposed in the literature in the past 15 years. We prove in this note that a family of language-theoretic conditions recently provided by the authors encapsulates all the possible LMI conditions, thus putting a conclusion to this research effort. As a corollary, we show that it is PSPACE-complete to recognize whether a particular set of LMIs implies stability of a switched system. Finally, we provide a geometric interpretation of these conditions, in terms of existence of an invariant set.",

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language = "English (US)",

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AU - Jungers, Raphaël M.

AU - Ahmadi, Amir Ali

AU - Parrilo, Pablo A.

AU - Roozbehani, Mardavij

PY - 2017/6

Y1 - 2017/6

N2 - We study stability criteria for discrete-time switched systems and provide a meta-theorem that characterizes all Lyapunov theorems of a certain canonical type. For this purpose, we investigate the structure of sets of LMIs that provide a sufficient condition for stability. Various such conditions have been proposed in the literature in the past 15 years. We prove in this note that a family of language-theoretic conditions recently provided by the authors encapsulates all the possible LMI conditions, thus putting a conclusion to this research effort. As a corollary, we show that it is PSPACE-complete to recognize whether a particular set of LMIs implies stability of a switched system. Finally, we provide a geometric interpretation of these conditions, in terms of existence of an invariant set.

AB - We study stability criteria for discrete-time switched systems and provide a meta-theorem that characterizes all Lyapunov theorems of a certain canonical type. For this purpose, we investigate the structure of sets of LMIs that provide a sufficient condition for stability. Various such conditions have been proposed in the literature in the past 15 years. We prove in this note that a family of language-theoretic conditions recently provided by the authors encapsulates all the possible LMI conditions, thus putting a conclusion to this research effort. As a corollary, we show that it is PSPACE-complete to recognize whether a particular set of LMIs implies stability of a switched system. Finally, we provide a geometric interpretation of these conditions, in terms of existence of an invariant set.

KW - Linear matrix inequalities

KW - lyapunov methods

KW - set theory

KW - stability

KW - switching systems (control)

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A Characterization of Lyapunov Inequalities for Stability of Switched Systems

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