### Abstract

In this paper, we study ℒ_{2} gain property for a class of switched systems which are composed of both continuous-time LTI subsystems and discrete-time LTI subsystems. Under the assumption that all subsystems are Hurwitz/Schur stable and have the ℒ_{2} gain less than 7, we discuss the ℒ_{2} gain that the switched system could achieve. First, we consider the case where a common Lyapunov function exists for all subsystems in ℒ_{2} sense, and show that the switched system has the ℒ_{2} gain less than the same level 7 under arbitrary switching. As an example in this case, we analyze switched symmetric systems and derive the common Lyapunov function clearly. Next, we use a piecewise Lyapunov function approach to study the case where no common Lyapunov function exists in ℒ_{2} sense, and show that the switched system achieves an ultimate (or weighted) ℒ_{2} gain under an average dwell time scheme.

Original language | English |
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Pages | 2483-2488 |

Number of pages | 6 |

Publication status | Published - 2004 Dec 1 |

Externally published | Yes |

Event | SICE Annual Conference 2004 - Sapporo, Japan Duration: 2004 Aug 4 → 2004 Aug 6 |

### Conference

Conference | SICE Annual Conference 2004 |
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Country | Japan |

City | Sapporo |

Period | 04/8/4 → 04/8/6 |

### Keywords

- ℒ gain
- Average dwell time
- Common Lyapunov function
- Piecewise Lyapunov function
- Switched system

### ASJC Scopus subject areas

- Control and Systems Engineering
- Computer Science Applications
- Electrical and Electronic Engineering

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## Cite this

*ℒ*. 2483-2488. Paper presented at SICE Annual Conference 2004, Sapporo, Japan.

_{2}gain analysis for switched systems with continuous-time and discrete-time subsystems