A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays.

*(English)*Zbl 1075.93032From the text: Based on the descriptor model transformation from [E. Fridman, Syst. Control Lett. 43, No. 4, 309–319 (2001; Zbl 0974.93028)] and a decomposition technique of a discrete-delay term matrix, we investigate the robust stability of uncertain linear neutral systems with discrete and distributed delays. The uncertainties under consideration are norm bounded, and possibly time varying. The proposed stability criteria are formulated in the form of a linear matrix inequality and it is easy to check the robust stability of the considered systems. Numerical examples show that the results obtained in this paper are less conservative than those in Fridman’s paper mentioned above and the others surveyed there. From Examples 1 and 2, one can see that the criteria in this paper and those in [E. Fridman and U. Shaked, Linear Algebra Appl. 351–352, 271–302 (2002; Zbl 1006.93021)] are complementary.

Reviewer: Gunther Reißig (Magdeburg)

##### MSC:

93D09 | Robust stability |

34K40 | Neutral functional-differential equations |

93C23 | Control/observation systems governed by functional-differential equations |

93B17 | Transformations |

15A39 | Linear inequalities of matrices |

##### Keywords:

Stability; Time delay; Discrete delay; Distributed delay; Neutral system; Linear matrix inequality (LMI); Descriptor model; Transformation; Robust stability##### Software:

LMI toolbox
Full Text:
DOI

##### References:

[1] | Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in systems and control theory. Philadelphia: SIAM. · Zbl 0816.93004 |

[2] | Els’golts’, L. E., & Norkin, S. B. (1973). Introduction to the theory and application of differential equations with deviating arguments. Mathematics in Science and Engineering, Vol. 105, New York: Academic Press. |

[3] | Fridman, E., New Lyapunov-krasovskii functionals for stability of linear retarded and neutral type systems, Systems & control letters, 43, 309-319, (2001) · Zbl 0974.93028 |

[4] | Fridman, E.; Shaked, U., A descriptor system approach to H∞ control of linear time-delay systems, IEEE transaction on automatic control, 47, 253-270, (2002) · Zbl 1364.93209 |

[5] | Gahinet, P., Nemirovski, A., Laub, A. J., & Chilali, M. (1995). LMI control toolbox: For use with MATLAB. Math Works: Natick, MA. |

[6] | Goubet-Batholomeus, A.; Dambrine, M.; Richard, J.P., Stability of perturbed systems with time-varying delays, Systems control letters, 31, 155-163, (1997) · Zbl 0901.93047 |

[7] | Gu, K. (2000). An integral inequality in the stability problem of time-delay systems. In Proceedings of the 39th IEEE conference on decision and control, Sydney, Australia, December 2000 (pp. 2805-2810). |

[8] | Gu, K.; Niculescu, S.-I., Further remarks on additional dynamics in various model transformations of linear delay systems, IEEE transactions on automatic control, 46, 497-500, (2001) · Zbl 1056.93511 |

[9] | Hale, J.K.; Lunel, S.M., Introduction to functional differential equations, (1993), Springer New York · Zbl 0787.34002 |

[10] | Han, Q.-L., Robust stability of uncertain delay-differential systems of neutral type, Automatica, 38, 719-723, (2002) · Zbl 1020.93016 |

[11] | Hu, G.Di.; Hu, G.Da., Some simple stability criteria of neutral delay-differential systems, Applied mathematics and computation, 80, 257-271, (1996) · Zbl 0878.34063 |

[12] | Kim, J.-H., Delay and its time-derivative dependent robust stability of timedelayed linear systems with uncertainty, IEEE transactions on automatic control, 46, 789-792, (2001) · Zbl 1008.93056 |

[13] | Kolmanovskii, V.B.; Richard, J.-P., Stability of some linear systems with delays, IEEE transactions on automatic control, 44, 984-989, (1999) · Zbl 0964.34065 |

[14] | Lien, C.-H.; Yu, K.W.; Hsieh, J.G., Stability conditions for a class of neutral systems with multiple time delays, Journal of mathematical analysis and application, 245, 20-27, (2000) · Zbl 0973.34066 |

[15] | Niculescu, S. -I. (2000). Further remarks on delay-dependent stability of linear neutral systems. Proceedings of MTNS 2000, Perpigan, France. |

[16] | Slemrod, M.; Infante, E.F., Asymptotic stability criteria for linear systems of differential equations of neutral type and their discrete analogues, Journal of mathematical analysis and application, 38, 399-415, (1972) · Zbl 0202.10301 |

[17] | Takaba, K.; Morihira, N.; Katayama, T., A generalized Lyapunov theorem for descriptor systems, Systems & control letters, 24, 49-51, (1995) · Zbl 0883.93035 |

[18] | Verriest, E. -I., & Niculescu, S. -I. (1997). Delay-independent stability of linear neutral systems: A Riccati equation approach. In L. Dugard & E.I. Verriest (Eds.), Stability and control of time-delay systems, LNCIS, Vol. 228, London: Springer. (pp. 92-100). · Zbl 0923.93049 |

[19] | Xie, L., Output feedback H∞ control of systems with parameter uncertainty, International journal control, 63, 741-750, (1996) · Zbl 0841.93014 |

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