Comparison of bounding methods for stability analysis of systems with time-varying delays.

*(English)*Zbl 1364.93738Summary: Integral inequalities for quadratic functions play an important role in the derivation of delay-dependent stability criteria for linear time-delay systems. Based on the Jensen inequality, a reciprocally convex combination approach was introduced by P. Park et al. [Automatica 47, No. 1, 235–238 (2011; Zbl 1209.93076)] for deriving delay-dependent stability criterion, which achieves the same upper bound of the time-varying delay as the one on the use of the Y. S. Moon’s et al. [Int. J. Control 74, No. 14, 1447–1455 (2001; Zbl 1023.93055)] inequality. Recently, a new inequality called Wirtinger-based inequality that encompasses the Jensen inequality was proposed by A. Seuret et al. [“Stability of systems with fast-varying delay using improved Wirtinger’s inequality”, in: Proceedings of the 52nd IEEE conference on decision and control. Los Alamitos, CA: IEEE Computer Society. 946–951 (2013; doi:10.1109/CDC.2013.6760004)] for the stability analysis of time-delay systems. In this paper, it is revealed that the reciprocally convex combination approach is effective only with the use of Jensen inequality. When the Jensen inequality is replaced by Wirtinger-based inequality, the Moon’s et al. inequality [loc. cit.] together with convex analysis can lead to less conservative stability conditions than the reciprocally convex combination inequality. Moreover, we prove that the feasibility of an LMI condition derived by the Moon’s et al. inequality as well as convex analysis implies the feasibility of an LMI condition induced by the reciprocally convex combination inequality. Finally, the efficiency of the methods is demonstrated by some numerical examples, even though the corresponding system with zero-delay as well as the system without the delayed term are not stable.

##### Keywords:

integral inequalities for quadratic functions; delay-dependent stability criteria; linear time-delay systems; Jensen inequality; Wirtinger-based inequality; Moon’s inequality; convex analysis
PDF
BibTeX
XML
Cite

\textit{K. Liu} and \textit{A. Seuret}, J. Franklin Inst. 354, No. 7, 2979--2993 (2017; Zbl 1364.93738)

Full Text:
DOI

##### References:

[1] | Briat, C., Convergence and equivalence results for the jensen’s inequality-application to time-delay and sampled-data systems, IEEE Trans. Automation Control, 56, 7, 1660-1665, (2011) · Zbl 1368.26020 |

[2] | Fridman, E., Introduction to time-delay systems, (2014), Birkhäuser Basel |

[3] | Fridman, E.; Shaked, U., A descriptor system approach to H_∞ control of linear time-delay systems, IEEE Trans. Automatic Control, 47, 2, 253-270, (2002) · Zbl 1364.93209 |

[4] | Fridman, E.; Shaked, U., Delay-dependent stability and H_∞ control: constant and time-varying delays, Int. J. Control, 76, 1, 48-60, (2003) · Zbl 1023.93032 |

[5] | Fridman, E.; Shaked, U.; Liu, K., New conditions for delay-derivative-dependent stability, Automatica, 45, 11, 2723-2727, (2009) · Zbl 1180.93080 |

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

[7] | Gyurkovics, É., A note on Wirtinger-type integral inequalities for time-delay systems, Automatica, 61, 44-46, (2015) · Zbl 1327.93169 |

[8] | 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 |

[9] | He, Y.; Wang, Q. G.; Xie, L.; Lin, C., Further improvement of free-weighting matrices technique for systems with time-varying delay, IEEE Trans. Automatic Control, 52, 2, 293-299, (2007) · Zbl 1366.34097 |

[10] | Hien, L. V.; Trinh, H., New finite-sum inequalities with applications to stability of discrete time-delay systems, Automatica, 71, 197-201, (2016) · Zbl 1343.93079 |

[11] | Liu, K.; Fridman, E., Delay-dependent methods and the first delay interval, Syst. Control Lett., 64, 1, 57-63, (2014) · Zbl 1283.93140 |

[12] | Liu, K.; Seuret, A.; Xia, Y. Q., Stability analysis of systems with time-varying delays via the second-order Bessel-Legendre inequality, Automatica, 76, 138-142, (2017) · Zbl 1352.93079 |

[13] | Liu, K.; Suplin, V.; Fridman, E., Stability of linear systems with general sawtooth delay, IMA J. Math. Control Inf., 27, 4, 419-436, (2010) · Zbl 1206.93080 |

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

[15] | Park, P. G., A delay-dependent stability criterion for systems with uncertain time-invariant delays, IEEE Trans. Automatic Control, 44, 4, 876-877, (1999) · Zbl 0957.34069 |

[16] | 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 |

[17] | Park, P. G.; Ko, J. W.; Jeong, C., Reciprocally convex approach to stability of systems with time-varying delays, Automatica, 47, 1, 235-238, (2011) · Zbl 1209.93076 |

[18] | Park, P. G.; Lee, W. I.; Lee, S. Y., Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems, J. Frankl. Inst., 352, 4, 1378-1396, (2015) · Zbl 1395.93450 |

[19] | Richard, J. P., Time-delay systems: an overview of some recent advances and open problems, Automatica, 39, 10, 1667-1694, (2003) · Zbl 1145.93302 |

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

[21] | Seuret, A.; Gouaisbaut, F., Hierarchy of LMI conditions for the stability analysis of time-delay systems, Syst. Control Lett., 81, 1-7, (2015) · Zbl 1330.93211 |

[22] | Seuret, A.; Gouaisbaut, F.; Fridman, E., Stability of systems with fast-varying delay using improved wirtinger’s inequality, Proceedings of the 52nd IEEE Conference on Decision and Control, Florence, Italy, 946-951, (2013) |

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

[24] | Sun, J.; Liu, G.; Chen, J., Delay-dependent stability and stabilization of neutral time-delay systems, Int. J. Robust Nonlinear Control, 19, 12, 1364-1375, (2009) · Zbl 1169.93399 |

[25] | 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 |

[26] | Xu, S. Y.; Lam, J.; Zhang, B. Y.; Zou, Y., New insight into delay-dependent stability of time-delay systems, Int. J. Robust Nonlinear Control, 25, 7, 961-970, (2015) · Zbl 1312.93077 |

[27] | Yue, D.; Tian, E.; Zhang, Y., A piecewise analysis method to stability analysis of linear continuous/discrete systems with time-varying delay, Int. J. Robust Nonlinear Control, 19, 13, 1493-1518, (2009) · Zbl 1298.93259 |

[28] | Zeng, H. B.; He, Y.; Wu, M.; She, J. H., Free-matrix-based integral inequality for stability analysis of systems with time-varying delay, IEEE Trans. Automatic Control, 60, 10, 2768-2772, (2015) · Zbl 1360.34149 |

[29] | Zeng, H. B.; He, Y.; Wu, M.; She, J. H., New results on stability analysis for systems with discrete distributed delay, Automatica, 60, 189-192, (2015) · Zbl 1331.93166 |

[30] | Zhang, C. K.; He, Y.; Jiang, L.; Wu, M.; Zeng, H. B., Delay-variation-dependent stability of delayed discrete-time systems, IEEE Trans. Automatic Control, 61, 9, 2663-2669, (2016) · Zbl 1359.39009 |

[31] | Zhang, C. K.; He, Y.; Jiang, L.; Wu, M.; Zeng, H. B., Stability analysis of systems with time-varying delay via relaxed integral inequalities, Systems Control Letters, 92, 52-61, (2016) · Zbl 1338.93290 |

[32] | Zhang, C. K.; He, Y.; Jiang, L.; Wu, M.; Zeng, H. B., Summation inequalities to bounded real lemmas of discrete-time systems with time-varying delay, IEEE Trans. Automatic Control, (2016) |

[33] | Zhang, X. M.; Han, Q. L., Robust H_∞ filtering for a class of uncertain linear systems with time-varying delay, Automatica, 44, 1, 157-166, (2008) · Zbl 1138.93058 |

[34] | Zhang, X. M.; Han, Q. L., Event-based H_∞ filtering for sampled-data systems, Automatica, 51, 55-69, (2015) · Zbl 1309.93096 |

[35] | Zhang, X. M.; Wu, M.; She, J. H.; He, Y., Delay-dependent stabilization of linear systems with time-varying state and input delays, Automatica, 41, 8, 1405-1412, (2005) · Zbl 1093.93024 |

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.