### Abstract

We analyze stability for switched systems which are composed of both continuous-time and discrete-time subsystems. By considering a Lie algebra generated by all subsystem matrices, we show that if all subsystems are Hurwitz/Schur stable and this Lie algebra is solvable, then there is a common quadratic Lyapunov function for all subsystems and thus the switched system is exponentially stable under arbitrary switching. When not all subsystems are stable and the same Lie algebra is solvable, we show that there is a common quadratic Lyapunov-like function for all subsystems and the switched system is exponentially stable under a dwell time scheme. Two numerical examples are provided to demonstrate the result.

Original language | English |
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Title of host publication | Proceedings of the 2006 IEEE International Conference on Networking, Sensing and Control, ICNSC'06 |

Pages | 362-367 |

Number of pages | 6 |

Publication status | Published - 2006 |

Externally published | Yes |

Event | 2006 IEEE International Conference on Networking, Sensing and Control, ICNSC'06 - Ft. Lauderdale, FL Duration: 2006 Apr 23 → 2006 Apr 25 |

### Other

Other | 2006 IEEE International Conference on Networking, Sensing and Control, ICNSC'06 |
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City | Ft. Lauderdale, FL |

Period | 06/4/23 → 06/4/25 |

### Fingerprint

### Keywords

- Arbitrary switching
- Common quadratic lyapunov (lyapunov-iike) functions
- Dwell time scheme
- Exponential stability
- Lie algebra
- Switched systems

### ASJC Scopus subject areas

- Computer Networks and Communications
- Control and Systems Engineering

### Cite this

*Proceedings of the 2006 IEEE International Conference on Networking, Sensing and Control, ICNSC'06*(pp. 362-367). [1673173]

**An extension of lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems.** / Zhai, Guisheng; Xu, Xuping; Lin, Hai; Liu, Derong.

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

*Proceedings of the 2006 IEEE International Conference on Networking, Sensing and Control, ICNSC'06.*, 1673173, pp. 362-367, 2006 IEEE International Conference on Networking, Sensing and Control, ICNSC'06, Ft. Lauderdale, FL, 06/4/23.

}

TY - GEN

T1 - An extension of lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems

AU - Zhai, Guisheng

AU - Xu, Xuping

AU - Lin, Hai

AU - Liu, Derong

PY - 2006

Y1 - 2006

N2 - We analyze stability for switched systems which are composed of both continuous-time and discrete-time subsystems. By considering a Lie algebra generated by all subsystem matrices, we show that if all subsystems are Hurwitz/Schur stable and this Lie algebra is solvable, then there is a common quadratic Lyapunov function for all subsystems and thus the switched system is exponentially stable under arbitrary switching. When not all subsystems are stable and the same Lie algebra is solvable, we show that there is a common quadratic Lyapunov-like function for all subsystems and the switched system is exponentially stable under a dwell time scheme. Two numerical examples are provided to demonstrate the result.

AB - We analyze stability for switched systems which are composed of both continuous-time and discrete-time subsystems. By considering a Lie algebra generated by all subsystem matrices, we show that if all subsystems are Hurwitz/Schur stable and this Lie algebra is solvable, then there is a common quadratic Lyapunov function for all subsystems and thus the switched system is exponentially stable under arbitrary switching. When not all subsystems are stable and the same Lie algebra is solvable, we show that there is a common quadratic Lyapunov-like function for all subsystems and the switched system is exponentially stable under a dwell time scheme. Two numerical examples are provided to demonstrate the result.

KW - Arbitrary switching

KW - Common quadratic lyapunov (lyapunov-iike) functions

KW - Dwell time scheme

KW - Exponential stability

KW - Lie algebra

KW - Switched systems

UR - http://www.scopus.com/inward/record.url?scp=34250209921&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34250209921&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:34250209921

SN - 1424400651

SN - 9781424400652

SP - 362

EP - 367

BT - Proceedings of the 2006 IEEE International Conference on Networking, Sensing and Control, ICNSC'06

ER -