### Abstract

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 [1]. 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. By construction, our criterion also automatically applies to the corresponding relaxed differential inclusion.

Original language | English (US) |
---|---|

Pages (from-to) | 4619-4624 |

Number of pages | 6 |

Journal | Proceedings of the IEEE Conference on Decision and Control |

Volume | 5 |

State | Published - Dec 1 2004 |

Event | 2004 43rd IEEE Conference on Decision and Control (CDC) - Nassau, Bahamas Duration: Dec 14 2004 → Dec 17 2004 |

### Fingerprint

### Keywords

- Differential inclusion
- Global asymptotic stability
- Lie bracket
- Maximum principle
- Optimal control
- Switched nonlinear system

### ASJC Scopus subject areas

- Control and Systems Engineering
- Modeling and Simulation
- Control and Optimization

### Cite this

*Proceedings of the IEEE Conference on Decision and Control*,

*5*, 4619-4624.

**A Lie-algebraic condition for stability of switched nonlinear systems.** / Margaliot, Michael; Liberzon, Daniel M.

Research output: Contribution to journal › Conference article

*Proceedings of the IEEE Conference on Decision and Control*, vol. 5, pp. 4619-4624.

}

TY - JOUR

T1 - A Lie-algebraic condition for stability of switched nonlinear systems

AU - Margaliot, Michael

AU - Liberzon, Daniel M

PY - 2004/12/1

Y1 - 2004/12/1

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 [1]. 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. 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 [1]. 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. 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=14544305191&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=14544305191&partnerID=8YFLogxK

M3 - Conference article

AN - SCOPUS:14544305191

VL - 5

SP - 4619

EP - 4624

JO - Proceedings of the IEEE Conference on Decision and Control

JF - Proceedings of the IEEE Conference on Decision and Control

SN - 0191-2216

ER -