Reciprocally convex approach to stability of systems with time-varying delays.

*(English)*Zbl 1209.93076Summary: Whereas the upper bound lemma for matrix cross-product, introduced by Park (1999) and modified by Y. S. Moon, P. Park, W. H. Kwon and Y. S. Lee [Int. J. Control 74, No. 14, 1447–1455 (2001; Zbl 1023.93055)], plays a key role in guiding various delay-dependent criteria for delayed systems, Jensen’s inequality has become an alternative as a way of reducing the number of decision variables. It directly relaxes the integral term of quadratic quantities into the quadratic term of the integral quantities, resulting in a linear combination of positive functions weighted by the inverses of convex parameters. This paper suggests the lower bound lemma for such a combination, which achieves performance behavior identical to approaches based on the integral inequality lemma but with much less decision variables, comparable to those based on Jensen’s inequality lemma.

##### MSC:

93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |

93D20 | Asymptotic stability in control theory |

93D99 | Stability of control systems |

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##### References:

[1] | Fridman, E.; Shaked, U., Delay-dependent stability and \(\mathcal{H}_\infty\) control: constant and time-varying delays, International journal of control, 76, 1, 48-60, (2003) · Zbl 1023.93032 |

[2] | Gu, K.; Kharitonov, V.L.; Chen, J., Stability of time-delay systems, (2003), Birkhäuser · Zbl 1039.34067 |

[3] | He, Y.; Wang, Q.-G.; Lin, C.; Wu, M., Delay-range-dependent stability for systems with time-varying delay, Automatica, 43, 2, 371-376, (2007) · Zbl 1111.93073 |

[4] | Jiang, X.; Han, Q.-L., On \(\mathcal{H}_\infty\) control for linear systems with interval time-varying delay, Automatica, 41, 12, 2099-2106, (2005) · Zbl 1100.93017 |

[5] | Ko, J.W.; Park, P., Delay-dependent robust stabilization for systems with time-varying delays, International journal of control, automation, and systems, 7, 5, 711-722, (2009) |

[6] | Moon, Y.S.; Park, P.; Kwon, W.H.; Lee, Y.S., Delay-dependent robust stabilization of uncertain state-delayed systems, International journal of control, 74, 14, 1447-1455, (2001) · Zbl 1023.93055 |

[7] | Park, P., A delay-dependent stability criterion for systems with uncertain time-invariant delays, IEEE transactions on automatic control, 44, 4, 876-877, (1999) · Zbl 0957.34069 |

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

[9] | Shao, H., New delay-dependent stability criteria for systems with interval delay, Automatica, 45, 3, 744-749, (2009) · Zbl 1168.93387 |

[10] | Wu, M.; Feng, Z.-Y.; He, Y., Improved delay-dependent absolute stability of lur’e systems with time-delay, International journal of control, automation, and systems, 7, 6, 1009-1014, (2009) |

[11] | Zhu, X.-L.; Yang, G.-H.; Li, T.; Lin, C.; Guo, L., LMI stability criterion with less variables for time-delay systems, International journal of control, automation, and systems, 7, 4, 530-535, (2009) |

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