On robust stability criterion for dynamic systems with time-varying delays and nonlinear perturbations.

*(English)*Zbl 1168.34354
Appl. Math. Comput. 203, No. 2, 937-942 (2008); erratum ibid. 215, No. 1, 427-430 (2009).

Authors’ abstract: We propose a new delay-dependent stability criterion for dynamic systems with time-varying delays and nonlinear perturbations. Based on the Lyapunov method, a sufficient delay-dependent criterion for asymptotic stability is derived in terms of linear matrix inequality. Numerical examples are given to show the effectiveness of our result.

Reviewer’s comment: On the first glance the setting of the problem is different from that one in the paper of the same authors, published in the same journal and volume [Appl. Math. Comput. 203, No. 2, 843–853 (2008; Zbl 1168.34046)]. Here, the problem is nonlinear, whereas it is linear in the cited paper. However, if one considers the nonlinearities as the uncertainty terms in the cited paper, both settings become less or more the same. So, nothing to wonder that also the approaches and the results are similar. So, one wonders after all, why no one of the two papers contains a reference to the other one.

Reviewer’s comment: On the first glance the setting of the problem is different from that one in the paper of the same authors, published in the same journal and volume [Appl. Math. Comput. 203, No. 2, 843–853 (2008; Zbl 1168.34046)]. Here, the problem is nonlinear, whereas it is linear in the cited paper. However, if one considers the nonlinearities as the uncertainty terms in the cited paper, both settings become less or more the same. So, nothing to wonder that also the approaches and the results are similar. So, one wonders after all, why no one of the two papers contains a reference to the other one.

Reviewer: Ivan Ginchev (Varese)

##### MSC:

34K20 | Stability theory of functional-differential equations |

34K40 | Neutral functional-differential equations |

93D09 | Robust stability |

##### Keywords:

robust stability; neutral systems; Lyapunov method; linear matrix inequality; interval time-vatying delay.##### Software:

LMI toolbox
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\textit{O. M. Kwon} et al., Appl. Math. Comput. 203, No. 2, 937--942 (2008; Zbl 1168.34354)

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##### References:

[1] | Hale, J.; Lunel, S.M.V., Introduction to functional differential equations, (1993), Springer-Verlag New York |

[2] | Kwon, O.; Park, J.H., Matrix inequality approach to novel stability criterion for time-delay systems with nonlinear uncertainties, Journal of optimization theory and applications, 126, 643-656, (2005) · Zbl 1159.34344 |

[3] | Kwon, O.M.; Park, J.H., Exponential stability of uncertain dynamic systems including state delay, Applied mathematics letters, 19, 901-907, (2006) · Zbl 1220.34095 |

[4] | Yue, D.; Won, S., Delay-dependent robust stability of stochastic systems with time delay and nonlinear uncertainties, Electronics letters, 37, 992-993, (2001) · Zbl 1190.93095 |

[5] | Park, J.H., Robust stabilization for dynamic systems with multiple time-varying delays and nonlinear uncertainties, Journal of optimization theory and applications, 108, 155-174, (2001) · Zbl 0981.93069 |

[6] | Park, J.H.; Kwon, O., Novel stability criterion of time delay systems with nonlinear uncertainties, Applied mathematics letters, 18, 683-688, (2005) · Zbl 1089.34549 |

[7] | Cao, Y.Y.; Lam, J., Computation of robust stability bounds for time-delay systems with nonlinear time-varying perturbations, International journal of system science, 31, 359-365, (2000) · Zbl 1080.93519 |

[8] | Zuo, Z.; Wang, Y., New stability criterion for a class of linear systems with time-varying delay and nonlinear perturbations, IEE Proceedings – control theory and applications, 153, 623-626, (2006) |

[9] | Kwon, O.M.; Park, J.H., An improved delay-dependent robust control for uncertain time-delay systems, IEEE transactions on automatic control, 49, 1991-1995, (2004) · Zbl 1365.93370 |

[10] | Kwon, O.M.; Park, J.H., Robust stabilization of uncertain systems with delays in control input, Applied mathematics and computation, 172, 1067-1077, (2006) · Zbl 1137.93406 |

[11] | Zhang, J.; Shi, P.; Qiu, J., Robust stability criteria for uncertain neutral system with time delay and nonlinear uncertainties, Chaos, solitons & fractals, 38, 160-167, (2008) · Zbl 1142.93402 |

[12] | Boyd, S.; El Ghaoui, L.; Feron, E.; balakrishnan, V., Linear matrix inequalities in system and control theory, (1994), SIAM Philadelphia · Zbl 0816.93004 |

[13] | Gahinet, P.; Nemirovskii, A.; Laub, A.; Chilali, M., LMI control toolbox, mathworks, (1995), Natick Massachusetts |

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