TY - JOUR

T1 - Lie-algebraic stability conditions for nonlinear switched systems and differential inclusions

AU - Margaliot, Michael

AU - Liberzon, Daniel

PY - 2006/1

Y1 - 2006/1

N2 - We present a stability criterion for switched nonlinear systems which involves Lie brackets of the individual vector fields but does not require that these vector fields commute. A special case of the main result says that a switched system generated by a pair of globally asymptotically stable nonlinear vector fields whose third-order Lie brackets vanish is globally uniformly asymptotically stable under arbitrary switching. This generalizes a known fact for switched linear systems and provides a partial solution to the open problem posed in [D. Liberzon, Lie algebras and stability of switched nonlinear systems, in: V. Blondel, A. Megretski (Eds.), Unsolved Problems in Mathematical Systems and Control Theory, Princeton University Press, NJ, 2004, pp. 203-207.]. To prove the result, we consider an optimal control problem which consists in finding the "most unstable" trajectory for an associated control system, and show that there exists an optimal solution which is bang-bang with a bound on the total number of switches. This property is obtained as a special case of a reachability result by bang-bang controls which is of independent interest. By construction, our criterion also automatically applies to the corresponding relaxed differential inclusion.

AB - We present a stability criterion for switched nonlinear systems which involves Lie brackets of the individual vector fields but does not require that these vector fields commute. A special case of the main result says that a switched system generated by a pair of globally asymptotically stable nonlinear vector fields whose third-order Lie brackets vanish is globally uniformly asymptotically stable under arbitrary switching. This generalizes a known fact for switched linear systems and provides a partial solution to the open problem posed in [D. Liberzon, Lie algebras and stability of switched nonlinear systems, in: V. Blondel, A. Megretski (Eds.), Unsolved Problems in Mathematical Systems and Control Theory, Princeton University Press, NJ, 2004, pp. 203-207.]. To prove the result, we consider an optimal control problem which consists in finding the "most unstable" trajectory for an associated control system, and show that there exists an optimal solution which is bang-bang with a bound on the total number of switches. This property is obtained as a special case of a reachability result by bang-bang controls which is of independent interest. By construction, our criterion also automatically applies to the corresponding relaxed differential inclusion.

KW - Differential inclusion

KW - Global asymptotic stability

KW - Lie bracket

KW - Maximum principle

KW - Optimal control

KW - Switched nonlinear system

UR - http://www.scopus.com/inward/record.url?scp=28444484618&partnerID=8YFLogxK

U2 - 10.1016/j.sysconle.2005.04.011

DO - 10.1016/j.sysconle.2005.04.011

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AN - SCOPUS:28444484618

SN - 0167-6911

VL - 55

SP - 8

EP - 16

JO - Systems and Control Letters

JF - Systems and Control Letters

IS - 1

ER -