### Abstract

In this paper, we investigate some qualitative properties for time-controlled switched systems consisting of several linear discrete-time subsystems. First, we study exponential stability of the switched system with commutation property, stable combination and average dwell time. When all subsystem matrices are commutative pairwise and there exists a stable combination of unstable subsystem matrices, we propose a class of stabilizing switching laws where Schur stable subsystems (if exist) are activated arbitrarily while unstable ones are activated in sequence with their duration time periods satisfying a specified ratio. For more general switched system whose subsystem matrices are not commutative pairwise, we show that the switched system is exponentially stable if the average dwell time is chosen sufficiently large and the total activation time ratio between Schur stable and unstable subsystems is not smaller than a specified constant. Secondly, we use an average dwell time approach incorporated with a piecewise Lyapunov function to study the L_{2} gain of the switched system. We show that when all subsystems are Schur stable and achieve an L_{2} gain smaller than a positive scalar γ0, (1) if all subsystems have a common Lyapunov function in the sense of L_{2} gain, then the switched system achieves the same L_{2} gain γ0 under arbitrary switching; (2) if there does not exist a common Lyapunov function, then the switched system under an average dwell time scheme achieves a weighted L_{2} gain γ0, and the weighted L_{2} gain approaches normal L_{2} gain if the average dwell time is chosen sufficiently large.

Original language | English |
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Title of host publication | Proceedings of the American Control Conference |

Pages | 1880-1885 |

Number of pages | 6 |

Volume | 3 |

DOIs | |

Publication status | Published - 2002 |

Externally published | Yes |

Event | 2002 American Control Conference - Anchorage, AK Duration: 2002 May 8 → 2002 May 10 |

### Other

Other | 2002 American Control Conference |
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City | Anchorage, AK |

Period | 02/5/8 → 02/5/10 |

### Fingerprint

### ASJC Scopus subject areas

- Control and Systems Engineering

### Cite this

*Proceedings of the American Control Conference*(Vol. 3, pp. 1880-1885) https://doi.org/10.1109/ACC.2002.1023907

**Qualitative analysis of discrete-time switched systems.** / Zhai, Guisheng; Hu, Bo; Yasuda, Kazunori; Michel, Anthony N.

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

*Proceedings of the American Control Conference.*vol. 3, pp. 1880-1885, 2002 American Control Conference, Anchorage, AK, 02/5/8. https://doi.org/10.1109/ACC.2002.1023907

}

TY - GEN

T1 - Qualitative analysis of discrete-time switched systems

AU - Zhai, Guisheng

AU - Hu, Bo

AU - Yasuda, Kazunori

AU - Michel, Anthony N.

PY - 2002

Y1 - 2002

N2 - In this paper, we investigate some qualitative properties for time-controlled switched systems consisting of several linear discrete-time subsystems. First, we study exponential stability of the switched system with commutation property, stable combination and average dwell time. When all subsystem matrices are commutative pairwise and there exists a stable combination of unstable subsystem matrices, we propose a class of stabilizing switching laws where Schur stable subsystems (if exist) are activated arbitrarily while unstable ones are activated in sequence with their duration time periods satisfying a specified ratio. For more general switched system whose subsystem matrices are not commutative pairwise, we show that the switched system is exponentially stable if the average dwell time is chosen sufficiently large and the total activation time ratio between Schur stable and unstable subsystems is not smaller than a specified constant. Secondly, we use an average dwell time approach incorporated with a piecewise Lyapunov function to study the L2 gain of the switched system. We show that when all subsystems are Schur stable and achieve an L2 gain smaller than a positive scalar γ0, (1) if all subsystems have a common Lyapunov function in the sense of L2 gain, then the switched system achieves the same L2 gain γ0 under arbitrary switching; (2) if there does not exist a common Lyapunov function, then the switched system under an average dwell time scheme achieves a weighted L2 gain γ0, and the weighted L2 gain approaches normal L2 gain if the average dwell time is chosen sufficiently large.

AB - In this paper, we investigate some qualitative properties for time-controlled switched systems consisting of several linear discrete-time subsystems. First, we study exponential stability of the switched system with commutation property, stable combination and average dwell time. When all subsystem matrices are commutative pairwise and there exists a stable combination of unstable subsystem matrices, we propose a class of stabilizing switching laws where Schur stable subsystems (if exist) are activated arbitrarily while unstable ones are activated in sequence with their duration time periods satisfying a specified ratio. For more general switched system whose subsystem matrices are not commutative pairwise, we show that the switched system is exponentially stable if the average dwell time is chosen sufficiently large and the total activation time ratio between Schur stable and unstable subsystems is not smaller than a specified constant. Secondly, we use an average dwell time approach incorporated with a piecewise Lyapunov function to study the L2 gain of the switched system. We show that when all subsystems are Schur stable and achieve an L2 gain smaller than a positive scalar γ0, (1) if all subsystems have a common Lyapunov function in the sense of L2 gain, then the switched system achieves the same L2 gain γ0 under arbitrary switching; (2) if there does not exist a common Lyapunov function, then the switched system under an average dwell time scheme achieves a weighted L2 gain γ0, and the weighted L2 gain approaches normal L2 gain if the average dwell time is chosen sufficiently large.

UR - http://www.scopus.com/inward/record.url?scp=0036349370&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036349370&partnerID=8YFLogxK

U2 - 10.1109/ACC.2002.1023907

DO - 10.1109/ACC.2002.1023907

M3 - Conference contribution

AN - SCOPUS:0036349370

SN - 0780372980

VL - 3

SP - 1880

EP - 1885

BT - Proceedings of the American Control Conference

ER -