Improved robust stability criteria of uncertain neutral systems with mixed delays.

*(English)*Zbl 1184.93096Summary: The problem of robust stability for a class of neutral control systems with mixed delays is investigated. Based on Lyapunov stable theory, by constructing a new Lyapunov-Krasovskii function, some new stability criteria are obtained. These criteria are formulated in the forms of linear matrix inequalities. Compared with some previous publications, our results are less conservative. Simulation examples are presented to illustrate the improvement of the main results.

##### MSC:

93D09 | Robust stability |

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

34H05 | Control problems involving ordinary differential equations |

34K40 | Neutral functional-differential equations |

##### Keywords:

robust stability; neutral control systems with mixed delays; Lyapunov stable theory; Lyapunov-Krasovskii function; linear matrix inequalities
PDF
BibTeX
XML
Cite

\textit{Z. Liu} et al., Abstr. Appl. Anal. 2009, Article ID 294845, 18 p. (2009; Zbl 1184.93096)

**OpenURL**

##### References:

[1] | Q.-L. Han, “A new delay-dependent absolute stability criterion for a class of nonlinear neutral systems,” Automatica, vol. 44, no. 1, pp. 272-277, 2008. · Zbl 1138.93039 |

[2] | J. Zhang, P. Shi, and J. Qiu, “Robust stability criteria for uncertain neutral system with time delay and nonlinear uncertainties,” Chaos, Solitons & Fractals, vol. 38, no. 1, pp. 160-167, 2008. · Zbl 1142.93402 |

[3] | D.-Y. Chen and C.-Y. Jin, “Delay-dependent stability criteria for a class of uncertain neutral systems,” Acta Automatica Sinica, vol. 34, no. 8, pp. 989-992, 2008. |

[4] | M. Li and L. Liu, “A delay-dependent stability criterion for linear neutral delay systems,” Journal of the Franklin Institute, vol. 346, no. 1, pp. 33-37, 2009. · Zbl 1298.34153 |

[5] | J. H. Park and S. Won, “A note on stability of neutral delay-differential systems,” Journal of the Franklin Institute, vol. 336, no. 3, pp. 543-548, 1999. · Zbl 0969.34066 |

[6] | J. H. Park and S. Won, “Asymptotic stability of neutral systems with multiple delays,” Journal of Optimization Theory and Applications, vol. 103, no. 1, pp. 183-200, 1999. · Zbl 0947.65088 |

[7] | O. M. Kwon, J. H. Park, and S. M. Lee, “Augmented Lyapunov functional approach to stability of uncertain neutral systems with time-varying delays,” Applied Mathematics and Computation, vol. 207, no. 1, pp. 202-212, 2009. · Zbl 1178.34091 |

[8] | O. M. Kwon, J. H. Park, and S. M. Lee, “On stability criteria for uncertain delay-differential systems of neutral type with time-varying delays,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 864-873, 2008. · Zbl 1144.34052 |

[9] | J. Tian, L. Xiong, J. Liu, and X. Xie, “Novel delay-dependent robust stability criteria for uncertain neutral systems with time-varying delay,” Chaos, Solitons & Fractals, vol. 40, no. 4, pp. 1858-1866, 2009. · Zbl 1198.34165 |

[10] | B. Wang, X. Liu, and S. Zhong, “New stability analysis for uncertain neutral system with time-varying delay,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 457-465, 2008. · Zbl 1145.34046 |

[11] | X. Li and X. Zhu, “Stability analysis of neutral systems with distributed delays,” Automatica, vol. 44, pp. 2197-2201, 2008. · Zbl 1283.93212 |

[12] | D. Liu, S. Zhong, and L. Xiong, “On robust stability of uncertain neutral systems with multiple delays,” Chaos, Solitons & Fractals, vol. 39, no. 5, pp. 2332-2339, 2009. · Zbl 1197.34139 |

[13] | L. Xiong, S. Zhong, and J. Tian, “Novel robust stability criteria of uncertain neutral systems with discrete and distributed delays,” Chaos, Solitons & Fractals, vol. 40, no. 2, pp. 771-777, 2009. · Zbl 1197.93132 |

[14] | J. Gao, H. Su, X. Ji, and J. Chu, “Stability analysis for a class of neutral systems with mixed delays and sector-bounded nonlinearity,” Nonlinear Analysis: Real World Applications, vol. 9, no. 5, pp. 2350-2360, 2008. · Zbl 1156.34345 |

[15] | X. Li, X. Zhu, and A. Cela, “Stability analysis of neutral systems with mixed delays,” Automatica, vol. 11, pp. 2968-2972, 2008. · Zbl 1152.93450 |

[16] | H. Li, H. Li, and S. Zhong, “Stability of neutral type descriptor system with mixed delays,” Chaos, Solitons & Fractals, vol. 33, no. 5, pp. 1796-1800, 2007. · Zbl 1156.34347 |

[17] | Y. Chen and W. Su, “New robust stability of cellular neural networks with time-varying discrete and distributed delays,” International Journal of Innovative Computing, Information and Control, vol. 3, pp. 1549-1556, 2007. |

[18] | Q. Zhang, X. Wei, and J. Xu, “A generalized LMI-based approach to the global asymptotic stability of discrete-time delayed recurrent neural networks,” International Journal of Innovative Computing, Information and Control, vol. 4, pp. 1393-1400, 2008. |

[19] | E. Boukas, “Free-weighting matrices delay-dependent stabilization for systems with time-varying delays,” ICIC Express Letters, vol. 2, pp. 167-173, 2008. |

[20] | L. Xia, M. Xia, and L. Liu, “LMI conditions for global asymptotic stability of neural networks with discrete and distributed delays,” ICIC Express Letters, vol. 2, pp. 257-262, 2008. |

[21] | T. Mori, “Criteria for asymptotic stability of linear time-delay systems,” IEEE Transactions on Automatic Control, vol. 30, no. 2, pp. 158-161, 1985. · Zbl 0557.93058 |

[22] | S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15 of SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1994. · Zbl 0816.93004 |

[23] | L. Xie, “Output feedback H\infty control of systems with parameter uncertainty,” International Journal of Control, vol. 63, no. 4, pp. 741-750, 1996. · Zbl 0841.93014 |

[24] | J. Yan, “Robust stability analysis of uncertain time delay systems with delay-dependence,” Electronics Letters, vol. 37, pp. 135-137, 2001. |

[25] | J. Cao and J. Wang, “Delay-dependent robust stability of uncertain nonlinear systems with time delay,” Applied Mathematics and Computation, vol. 154, no. 1, pp. 289-297, 2004. · Zbl 1060.34041 |

[26] | K.-K. Fan, J.-D. Chen, C.-H. Lien, and J.-G. Hsieh, “Delay-dependent stability criterion for neutral time-delay systems via linear matrix inequality approach,” Journal of Mathematical Analysis and Applications, vol. 273, no. 2, pp. 580-589, 2002. · Zbl 1010.93084 |

[27] | C. Lien and J. Chen, “Discrete-delay-independent and discrete-delay-dependent criteria for a class of neutral systems,” Journal of Dynamic Systems, Measurement, and Control, vol. 125, pp. 33-41, 2003. |

[28] | J. H. Park and O. Kwon, “On new stability criterion for delay-differential systems of neutral type,” Applied Mathematics and Computation, vol. 162, no. 2, pp. 627-637, 2005. · Zbl 1077.34075 |

[29] | C.-H. Lien, K.-W. Yu, and J.-G. Hsieh, “Stability conditions for a class of neutral systems with multiple time delays,” Journal of Mathematical Analysis and Applications, vol. 245, no. 1, pp. 20-27, 2000. · Zbl 0973.34066 |

[30] | S. Neculescu, “Further remarks on delay-dependent stability of linear neutral system,” in Proceedings of the International Symposium on Mathematical Theory of Networks and Systems (MTNS /00), Perpignan, France, 2000. |

[31] | E. Fridman, “New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems,” Systems & Control Letters, vol. 43, no. 4, pp. 309-319, 2001. · Zbl 0974.93028 |

[32] | Y. He, M. Wu, J.-H. She, and G.-P. Liu, “Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays,” Systems & Control Letters, vol. 51, no. 1, pp. 57-65, 2004. · Zbl 1157.93467 |

[33] | Q.-L. Han, “On stability of linear neutral systems with mixed time delays: a discretized Lyapunov functional approach,” Automatica, vol. 41, no. 7, pp. 1209-1218, 2005. · Zbl 1091.34041 |

[34] | J. H. Park, O. M. Kwon, and S. M. Lee, “LMI optimization approach on stability for delayed neural networks of neutral-type,” Applied Mathematics and Computation, vol. 196, no. 1, pp. 236-244, 2008. · Zbl 1157.34056 |

[35] | K. Gu, “A further refinement of discretized Lyapunov functional method for the stability of time-delay systems,” International Journal of Control, vol. 74, no. 10, pp. 967-976, 2001. · Zbl 1015.93053 |

[36] | Q.-L. Han, “Robust stability of uncertain delay-differential systems of neutral type,” Automatica, vol. 38, no. 4, pp. 719-723, 2002. · Zbl 1020.93016 |

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.