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**Stability and \(L_{2}\)-gain analysis for switched delay systems: a delay-dependent method.**
*(English)*
Zbl 1114.93086

Summary: In this paper, we study stability and \(L_{2}\)-gain for a class of switched systems with time-varying delays. Sufficient conditions for exponential stability and weighted \(L_{2}\)-gain are developed for a class of switching signals with average dwell time. These conditions are delay-dependent and are given in the form of linear matrix inequalities (LMIs). As a special case of such switching signals, we can obtain exponential stability and normal \(L_{2}\)-gain under arbitrary switching signals. The state decay estimate is explicitly given. Two examples illustrate the effectiveness and applicability of the proposed method.

### MSC:

93D20 | Asymptotic stability in control theory |

93C15 | Control/observation systems governed by ordinary differential equations |

93C05 | Linear systems in control theory |

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\textit{X.-M. Sun} et al., Automatica 42, No. 10, 1769--1774 (2006; Zbl 1114.93086)

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### References:

[1] | Fridman, E.; Shaked, U., An improved stabilization method for linear time-delay systems, IEEE transactions on automatic control, 47, 1931-1937, (2002) · Zbl 1364.93564 |

[2] | Gao, H.J.; Wang, C.H., Delay-dependent robust and filtering for a class of uncertain nonlinear time-delay systems, IEEE transactions on automatic control, 48, 1661-1665, (2003) · Zbl 1364.93210 |

[3] | Hale, J.K.; Lunel, S.M.V., Introduction to functional differential equations, (1993), Springer New York |

[4] | Han, Q.L.; Gu, K.Q., On robust stability of time-delay systems with norm-bounded uncertainty, IEEE transactions on automatic control, 46, 1426-1431, (2001) · Zbl 1006.93054 |

[5] | He, Y.; Wu, M.; She, J.H.; Liu, G.P., Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays, Systems and control letters, 51, 57-65, (2004) · Zbl 1157.93467 |

[6] | He, Y.; Wu, M.; She, J.H.; Liu, G.P., Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties, IEEE transactions on automatic control, 49, 828-832, (2004) · Zbl 1365.93368 |

[7] | Hespanha, J. P., & Morse, A. S. (1999). Stability of switched systems with average dwell-time. 38th IEEE conference on decision and control (pp. 2655-2660), Phoenix, AZ, USA. |

[8] | Johansson, M.; Rantzer, A., Computation of piecewise quadratic Lyapunov functions for hybrid systems, Automatica, 43, 555-559, (1998) · Zbl 0905.93039 |

[9] | Kim, D.K.; Park, P.G.; Ko, J.W., Output-feedback \(H_\infty\) control of systems over communication networks using a deterministic switching system approach, Automatica, 40, 1205-1212, (2004) · Zbl 1056.93527 |

[10] | Liberzon, D., Switching in systems and control, (2003), Birkhauser Boston · Zbl 1036.93001 |

[11] | Meyer, C.; Schroder, S.; De Doncker, R.W., Solid-state circuit breakers and current limiters for medium-voltage systems having distributed power systems, IEEE transactions on power electronics, 19, 1333-1340, (2004) |

[12] | Michiels, W.; Assche, V.V.; Niculescu, S.I., Stabilization of time-delay systems with a controlled time-varying delay and applications, IEEE transactions on automatic control, 50, 493-504, (2005) · Zbl 1365.93411 |

[13] | Sun, Z.D.; Ge, S.S., Switched linear systemsâ€”control and design, (2004), Springer New York |

[14] | Wang, Z.D.; Huang, B.; Unbehauen, H., Robust reliable control for a class of uncertain nonlinear state-delayed systems, Automatica, 35, 955-963, (1999) · Zbl 0945.93605 |

[15] | Xie, G. M., & Wang, L. (2004). Stability and stabilization of switched linear systems with state delay: Continuous-time case. The 16th mathematical theory of networks and systems conference, Catholic University of Leuven. |

[16] | Zhai, G.S.; Hu, B.; Yasuda, K.; Michel, A., Disturbance attenuation properties of time-controlled switched systems, Journal of the franklin institute, 338, 765-779, (2001) · Zbl 1022.93017 |

[17] | Zhai, G. S., Sun, Y., Chen, X. K., & Anthony, N. M. (2003). Stability and \(L_2\) gain analysis for switched symmetric systems with time delay. American control conference (pp. 2682-2687), Denver, CO, USA. |

[18] | Zhao, J., & Hill David, J. (2005). On stability and \(L_2\) gain for switched systems. 44th IEEE conference on decision and control (pp. 3279-3284), Seville, Spain. |

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