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

*(English)*Zbl 1114.93086Summary: 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 |

PDF
BibTeX
XML
Cite

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

Full Text:
DOI

##### 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. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.