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**New delay-dependent stability criteria for systems with interval delay.**
*(English)*
Zbl 1168.93387

Summary: This paper provides a new delay-dependent stability criterion for systems with a delay varying in an interval. With a different Lyapunov functional defined, a tight upper bound of its derivative is given. The resulting criterion has advantages over some previous ones in that it involves fewer matrix variables but has less conservatism, which is established theoretically. Examples are provided to demonstrate the advantage of the stability result.

### MSC:

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

93C05 | Linear systems in control theory |

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

15A39 | Linear inequalities of matrices |

### Keywords:

time-varying delay; delay-dependent; Lyapunov functional; stability; linear matrix inequality (LMI)
Full Text:
DOI

### References:

[1] | Fridman, E.; Shaked, U., A descriptor system approach to \(H^\infty\) control of linear time-delay systems, IEEE transactions automatic control, 47, 253-270, (2002) · Zbl 1364.93209 |

[2] | Gu, K. (2000). An integral inequality in the stability problem of time-delay systems. In Proceedings of 39th IEEE conference on decision and control (pp. 2805-2810) |

[3] | Han, Q.-L.; Gu, K., Stability of linear systems with time-varying delay: A generalized discretized Lyapunov functional approach, Asian journal of control, 3, 170-180, (2001) |

[4] | Hale, J., Functional differential equations, (1977), Springer-Verlag New York · Zbl 0222.34003 |

[5] | He, Y.; Wang, Q.; Lin, C.; Wu, M., Delay-range-dependent stability for systems with time-varying delay, Automatica, 43, 371-376, (2007) · Zbl 1111.93073 |

[6] | Jiang, X.; Han, Q.L., On \(H^\infty\) control for linear systems with interval time-varying delay, Automatica, 41, 2099-2106, (2005) · Zbl 1100.93017 |

[7] | Jiang, X.; Han, Q.L.; Liu, S.; Xue, A., A new \(H^\infty\) stabilization criterion for networked control systems, IEEE transactions automatic control, 53, 1025-1032, (2008) · Zbl 1367.93179 |

[8] | Jing, X.; Tan, D.; Wang, Y., An LMI approach to stability of systems with severe time-delay, IEEE transactions automatic control, 49, 1192-1195, (2004) · Zbl 1365.93226 |

[9] | Kao, C.Y.; Lincoln, B., Simple stability criteria for systems with time-varying delays, Automatica, 40, 1429-1434, (2004) · Zbl 1073.93047 |

[10] | Kim, J.H., Delay and itâ€™ time-derivative dependent robust stability of time-delayed linear systems with uncertainty, IEEE transactions on automatic control, 46, 789-792, (2001) · Zbl 1008.93056 |

[11] | Lee, Y.S.; Moon, Y.S.; Kwon, W.H.; Park, P.G., Delay-dependent robust \(H^\infty\) control for uncertain systems with a state-delay, Automatica, 40, 65-72, (2004) · Zbl 1046.93015 |

[12] | Li, T.; Guo, L.; Sun, C.; Zhang, B., Exponential stability of recurrent neural networks with time-varying discrete and distributed delays, Nonlinear analysis: real world applications, (2008) |

[13] | Moon, Y.S.; Park, P.G.; Kwon, W.H.; Lee, Y.S., Delay-dependent robust stabilization of uncertain state-delayed systems, International journal of control, 74, 1447-1455, (2001) · Zbl 1023.93055 |

[14] | Niculescu, S.I., Neto, A.T., Dion, J.M., & Dugard, L. (1995). Delay-dependent stability of linear systems with delayed state: An LMI approach. In Proc. 34th IEEE conf. decision and control (pp. 1495-1496) |

[15] | Park, P., A delay-dependent stability criterion for systems with uncertain time-invariant delays, IEEE transactions on automatic control, 44, 876-877, (1999) · Zbl 0957.34069 |

[16] | Park, P.; Ko, J.W., Stability and robust stability for systems with a time-varying delay, Automatica, 43, 1855-1858, (2007) · Zbl 1120.93043 |

[17] | Richard, J.P., Time-delay systems: an overview of some recent advances and open problems, Automatica, 39, 1667-1694, (2003) · Zbl 1145.93302 |

[18] | Shao, H., Delay-dependent approaches to globally exponential stability for recurrent neural networks, IEEE transactions on circuits systems II, 55, 591-595, (2008) |

[19] | Shao, H., Delay-range-dependent robust \(H^\infty\) filtering for uncertain stochastic systems with mode-dependent time delays and Markovian jump parameters, Journal of mathematical analysis and applications, 342, 1084-1095, (2008) · Zbl 1141.93025 |

[20] | Shao, H., Delay-dependent stability for recurrent neural networks with time-varying delays, IEEE transactions on neural networks, 19, 1647-1651, (2008) |

[21] | Shao, H., Shi, X., & Wang, H. (2008). Improvements on delay-dependent stability criteria for linear delayed systems. In Proceedings of the 7th word conference on intelligent control and automation |

[22] | Suplin, V., Fridman, E., & Shaked, U. (2004). A projection approach to \(H^\infty\) control of time-delay systems. In Proc. IEEE conf. decision control (pp. 4548-4553) |

[23] | Wu, J.; Chen, T.; Wang, L., Delay-dependent robust stability and \(H^\infty\) control for jump linear systems with delays, Systems and control letters, 55, 937-948, (2006) |

[24] | Wu, M.; He, Y.; She, J.; Liu, G., Delay-dependent criteria for robust stability of time-varying delay systems, Automatica, 40, 1435-1439, (2004) · Zbl 1059.93108 |

[25] | Xie, L.; de Souza, C.E., Criteria for robust stability and stabilization of uncertain linear systems with state-delay, Automatica, 33, 1622-1657, (1997) |

[26] | Xu, S.; Lam, J., Improved delay-dependent stability criteria for time-delay systems, IEEE transactions automatic control, 50, 384-387, (2005) · Zbl 1365.93376 |

[27] | Xu, S.; Lam, J., On equivalence and efficiency of certain stability criteria for time-delay systems, IEEE transactions automatic control, 52, 95-101, (2007) · Zbl 1366.93451 |

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