##
**New results on stability analysis for systems with discrete distributed delay.**
*(English)*
Zbl 1331.93166

Summary: The integral inequality technique is widely used to derive delay-dependent conditions, and various integral inequalities have been developed to reduce the conservatism of the conditions derived. In this study, a new integral inequality is devised that is tighter than existing ones. It is used to investigate the stability of linear systems with a discrete distributed delay, and a new stability condition is established. The results can be applied to systems with a delay belonging to an interval, which may be unstable when the delay is small or nonexistent. Three numerical examples demonstrate the effectiveness and the smaller conservatism of the method.

### MSC:

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

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

93D30 | Lyapunov and storage functions |

Full Text:
DOI

### References:

[1] | Chen, W. H.; Zheng, W. X., Delay-dependent robust stabilization for uncertain neutral systems with distributed delays, Automatica, 43, 1, 95-104, (2007) · Zbl 1140.93466 |

[2] | Fridman, E.; Shaked, U., An improved stabilization method for linear time-delay systems, IEEE Transactions on Automatic Control, 47, 11, 1931-1937, (2002) · Zbl 1364.93564 |

[3] | Gu, K.; Kharitonov, V. L.; Chen, J., Stability of time-delay systems, (2003), BirkhĂ¤user Boston · Zbl 1039.34067 |

[4] | Han, Q. L., Absolute stability of time-delay systems with sector-bounded nonlinearity, Automatica, 41, 12, 2171-2176, (2005) · Zbl 1100.93519 |

[5] | Kao, C. Y.; Rantzer, A., Stability analysis of systems with uncertain time-varying delays, Automatica, 43, 6, 959-970, (2007) · Zbl 1282.93203 |

[6] | Kim, J. H., Note on stability of linear systems with time-varying delay, Automatica, 47, 9, 2118-2121, (2011) · Zbl 1227.93089 |

[7] | Park, P. G.; Ko, J. W., Stability and robust stability for systems with a time-varying delay, Automatica, 43, 10, 1855-1858, (2007) · Zbl 1120.93043 |

[8] | Park, M. J.; Kwon, O. M.; Park, J. H.; Lee, S. M.; Cha, E. J., Stability of time-delay systems via Wirtinger-based double integral inequality, Automatica, 55, 5, 204-208, (2015) · Zbl 1377.93123 |

[9] | Seuret, A.; Gouaisbaut, F., Wirtinger-based integral inequality: application to time-delay systems, Automatica, 49, 9, 2860-2866, (2013) · Zbl 1364.93740 |

[10] | Seuret, A., & Gouaisbaut, F. (2014). Complete Quadratic Lyapunov functionals using Bessel-Legendre inequality. In: 2014 European control conference, ECC, Strasbourg, France, June 24-27, 2014, pp. 448-453. |

[11] | Sun, J.; Liu, G. P.; Chen, J.; Rees, D., Improved delay-range-dependent stability criteria for linear systems with time-varying delays, Automatica, 46, 2, 466-470, (2010) · Zbl 1205.93139 |

[12] | Wu, M.; He, Y.; She, J. H.; Liu, G. P., Delay-dependent criteria for robust stability of time-varying delay systems, Automatica, 40, 8, 1435-1439, (2004) · Zbl 1059.93108 |

[13] | Zeng, H. B.; He, Y.; Wu, M.; She, J., Free-matrix-based integral inequality for stability analysis of systems with time-varying delay, IEEE Transactions on Automatic Control, (2015) |

[14] | Zhang, X. M.; Han, Q. L., New stability criterion using a matrix-based quadratic convex approach and some novel integral inequalities, IET Control Theory & Applications, 8, 12, 1054-1061, (2014) |

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.