×

Absolute stability criteria for a class of nonlinear singular systems with time delay. (English) Zbl 1168.34048

The paper studies the absolute stability of a class of nonlinear singular systems with constant delay. The uncertainties are norm-bounded. By applying an integral inequality and constructing a Lyapunov-Krasovskii function, they derive some new delay-independent and delay-dependent stability conditions in terms of linear matrix inequality which can be solved numerically using the effective interior-point algorithm. Two numerical examples are given to illustrate the feasibility and effectiveness of the developed technique.

MSC:

34K20 Stability theory of functional-differential equations
93D09 Robust stability
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lur’e, A. I., Some Nonlinear Problem in the Theory of Automatic Control (1957), H.M. Stationary Office: H.M. Stationary Office London
[2] Bliman, P.-A., Lyapunov-Krasovskii functionals and frequency domain: Delay-independent absolute stability criteria for delay systems, International Journal of Robust and Nonlinear Control, 11, 771-788 (2001) · Zbl 0992.93069
[3] Gan, Z. X.; Ge, W. G., Lyapunov functional for multiple delay general Lur’e control systems with multiple nonlinearities, Journal of Mathematics Analysis and Applications, 256, 596-608 (2001) · Zbl 0995.93041
[4] He, Y.; Wu, M., Absolute stability for multiple delay general Lur’e control systems with multiple nonlinearities, Journal of Computational and Applied Mathematics, 159, 241-248 (2003) · Zbl 1032.93062
[5] L. Yu, Q.-L. Han, S. Yu, F. Gao, Delay-dependent conditions for robust absolute stability of uncertain time-delay systems, in: Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, USA, 2003, pp. 6033-6037; L. Yu, Q.-L. Han, S. Yu, F. Gao, Delay-dependent conditions for robust absolute stability of uncertain time-delay systems, in: Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, USA, 2003, pp. 6033-6037
[6] Han, Q.-L., Absolute stability of time-delay systems with sector-bounded nonlinearity, Automatica, 41, 2171-2176 (2005) · Zbl 1100.93519
[7] Han, Q.-L.; Yue, D., Absolute stability of Lur’e systems with time-varying delay, IET Control Theory Applications, 1, 854-859 (2007)
[8] Han, Q.-L., A new delay-dependent absolute stability criterion for a class of nonlinear neutral systems, Automatica, 41 (2007)
[9] Q.-L. Han, A. Xue, S. Liu, X. Yu, Robust absolute stability criteria for uncertain Lur’e systems of neutral type, International Journal of Robust and Nonlinear Control, doi.10.1002/rnc.1219; Q.-L. Han, A. Xue, S. Liu, X. Yu, Robust absolute stability criteria for uncertain Lur’e systems of neutral type, International Journal of Robust and Nonlinear Control, doi.10.1002/rnc.1219 · Zbl 1284.93178
[10] Y. Wang, Z. Zuo, G. Zhang, New absolute stability condition for time-delay systems with sector-bounded nonlinearity, in: Proceedings of the 2007 American Control Conference, New York City, USA, 2007, pp. 6027-6030; Y. Wang, Z. Zuo, G. Zhang, New absolute stability condition for time-delay systems with sector-bounded nonlinearity, in: Proceedings of the 2007 American Control Conference, New York City, USA, 2007, pp. 6027-6030
[11] He, Y.; Wu, M.; She, J.-H.; Liu, G.-P., Robust stability for delay Lurie control systems with multiple nonlinearities, Journal of Computational and Applied Mathematics, 176, 371-380 (2005) · Zbl 1076.93036
[12] He, Y.; Wang, Q.-G.; Xie, L.; Lin, C., Further improvement of free-weighting matrices technique for systems with time-varying delay, IEEE Transactions on Automatic Control, 52, 293-299 (2007) · Zbl 1366.34097
[13] J. Gao, H. Su, X. Ji, J. Chu, Stability analysis for a class of neutral systems with mixed delays and sector-bounded nonlinearity, Nonlinear Analysis: Real World Applications (2007), in press (doi:10.1016/j.nonrwa.2007.07.003); J. Gao, H. Su, X. Ji, J. Chu, Stability analysis for a class of neutral systems with mixed delays and sector-bounded nonlinearity, Nonlinear Analysis: Real World Applications (2007), in press (doi:10.1016/j.nonrwa.2007.07.003) · Zbl 1156.34345
[14] Xu, S.; Lam, J., On equivalence and efficiency of certain stability criteria for time-delay systems, IEEE Transactions on Automatic Control, 52, 95-101 (2007) · Zbl 1366.93451
[15] Dai, L., Singular Control Systems (1989), Springer-Verlag: Springer-Verlag Berlin, Germany · Zbl 0669.93034
[16] Lewis, F. L., A survey of linear singular systems, Circuits, Systems Signal Processing, 5, 3-36 (1986) · Zbl 0613.93029
[17] Xu, S.; Dooren, P. V.; Stefan, R.; Lam, J., Robust stability and stabilization for singular systems with state delay and parameter uncertainty, IEEE Transactions on Automatic Control, 47, 1122-1128 (2002) · Zbl 1364.93723
[18] Wang, H.; Xue, A.; Lu, R., Robust \(H_\infty\) control for discrete singular systems with parameter uncertainties, Acta Automatica Sinica, 33, 1299-1304 (2007)
[19] Zhou, S.; Lam, J., Robust stabilization of delayed singular systems with linear fractional parametric uncertainties, Circuits, Systems Signal Processing, 6, 579-588 (2003) · Zbl 1045.93042
[20] Lee, L.; Chen, J. L., Strict positive real lemma and absolute stability for discrete-time descriptor systems, IEEE Transactions on Circuits and Systems—I, 50, 788-794 (2003) · Zbl 1368.93596
[21] Yang, C.; Zhang, Q.; Zhou, L., Generalised absolute stability analysis and synthesis for Lur’e-type descriptor systems, IET Control Theory Applications, 1, 617-623 (2007)
[22] Gu, K.; Niculescu, S.-I., Additional dynamics in transformed time-delay systems, IEEE Transactions on Automatic Control, 45, 572-575 (2000) · Zbl 0986.34066
[23] Khalil, H. K., Nonlinear Systems (1996), Prentice Hall: Prentice Hall Upper Saddle River, NJ · Zbl 0626.34052
[24] Kristic, M.; Deng, H., Stabilization of Nonlinear Uncertain Systems (1998), Springer-Verlag: Springer-Verlag London, UK · Zbl 0906.93001
[25] Boyd, S.; Ghaoui, L. E., Linear Matrix Inequalities in System and Control Theory (1994), SIAM: SIAM Philadelphia · Zbl 0816.93004
[26] Petersen, I. R., A stabilization algorithm for a class of uncertain linear systems, Systems & Control Letters, 8, 351-357 (1987) · Zbl 0618.93056
[27] Han, Q.-L.; Yu, X.; Gu, K., On computing the maximum time-delay bound for stability of linear neutral systems, IEEE Transactions on Automatic Control, 49, 2281-2286 (2004) · Zbl 1365.93224
[28] Han, Q.-L., On stability of linear neutral systems with mixed time-delays: A discretized Lyapunov functional approach, Automatica, 41, 1209-1218 (2005) · Zbl 1091.34041
[29] Han, Q.-L., On designing time-varying delay feedback controller for master-slave synchronization of Lur’e systems, IEEE Transactions on Circuits and Systems—I: Regular Papers, 54, 1573-1583 (2007) · Zbl 1374.93299
[30] Han, Q.-L., New delay-dependent synchronization criteria for Lur’e systems using time delay feedback control, Physics Letters A, 360, 563-569 (2007) · Zbl 1236.93072
[31] S.Q. Zhu, Z.L. Chen, J. Feng, Delay-dependent robust stability criterion and robust stabilization for uncertain singular time-delay systems, in: Proceeding of American Control Conference, Portland, USA, 2005, pp. 2839-2844; S.Q. Zhu, Z.L. Chen, J. Feng, Delay-dependent robust stability criterion and robust stabilization for uncertain singular time-delay systems, in: Proceeding of American Control Conference, Portland, USA, 2005, pp. 2839-2844
[32] Yang, F.; Zhang, Q., Delay-dependent \(H_\infty\) control for linear descriptor systems with delay in state, Journal of Control Theory and Application, 3, 76-84 (2005)
[33] Zhong, R.; Yang, Z., Delay-dependent robust control of descriptor systems with time delay, Asian Journal of Control, 8, 36-44 (2006)
[34] Gao, H. L.; Zhu, S. Q.; Chen, Z. L.; Xu, B. G., Delay-dependent state feedback guaranteed cost control uncertain singular time-delay systems, (Proceeding of IEEE Conference on Decision and Control, and European Control Conference (2003), IEEE), 4354-4359
[35] Fridman, E.; Shaked, U., \(H_\infty\) control of linear state-delay descriptor systems: An LMI approach, Linear Algebra and its Applications, 351, 271-302 (2002) · Zbl 1006.93021
[36] Wu, Z. G.; Zhou, W. N., Delay-dependent robust stabilization for uncertain singular systems with state delay, Acta Automatica Sinica, 33, 714-718 (2007) · Zbl 1164.93407
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.