# zbMATH — the first resource for mathematics

Stability analysis for uncertain neutral systems with discrete and distributed delays. (English) Zbl 1277.93060
Summary: This paper studies the problem of the robustly stability analysis for uncertain neutral systems with discrete and distributed delays, and norm-bounded uncertainties. By constructing an appropriate Lyapunov-Krasovskii functional, some new delay-dependent criteria can be obtained by using the free-weighting matrices approach to estimate the derivative of the Lyapunov functional, which are established in terms of Linear Matrix Inequalities (LMIs). The novelties in this paper are that any bounding technique and any mode transformation method are not utilized, and the decisive variables are much fewer than ones involved in existing literatures. Finally, three numerical examples are presented to illustrate the significant improvement on the conservatism of the delay bound over some reported results in the literature.

##### MSC:
 93D09 Robust stability 34K40 Neutral functional-differential equations
Full Text:
##### References:
 [1] Y. Ariba, F. Gouaisbaut, Delay-dependent stability analysis of linear systems with time-varying delay, in: Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, LA, USA, December 12-14, 2007. · Zbl 1190.93076 [2] Boyd, S.; Ghaoui, L.E.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory, (1994), SIAM Philadelphia · Zbl 0816.93004 [3] Balasubramaniam, P.; Manivanan, A.; Rakkiyappan, R., Exponential stability results for uncertain neutral systems with interval time-varying delays and Markovian jump parameters, Appl. math. comput., 216, 3396-3407, (2010) · Zbl 1197.34160 [4] Balasubramaniam, P.; Krishenasamy, R.; Rakkiyappan, R., Delay-dependent stability of neutral systems with time-varying delays using delay-decomposition approach, Appl. math. model., 36, 2253-2261, (2012) · Zbl 1243.34105 [5] Chen, W.; Zheng, W., Delay-dependent robust stabilization for uncertain neutral systems with distributed delays, Automatica, 43, 95-104, (2007) · Zbl 1140.93466 [6] Fridman, E., New lyapunov – krasovkii functionals for stability of linear retarded and neutral type systems, Syst. control lett., 43, 309-319, (2001) · Zbl 0974.93028 [7] Fridman, E.; Shaked, U., An improved stabilization method for linear time-delay systems, IEEE trans. automat. control, 47, 1931-1937, (2002) · Zbl 1364.93564 [8] Gu, K.; Kharitonov, V.; Chen, J., Stability of time-delay systems, (2003), Birkhauser Boston · Zbl 1039.34067 [9] Gu, K.; Han, Q.; Luo, A.; Niculescu, S., Discretized Lyapunov functional for systems with distributed delay and piecewise constant coefficients, Int. J. control, 74, 737-744, (2001) · Zbl 1015.34061 [10] Han, Q., A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays, Automatica, 40, 1791-1796, (2004) · Zbl 1075.93032 [11] Han, Q., On robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainty, Automatica, 40, 1087-1092, (2004) · Zbl 1073.93043 [12] Han, Q., Improved stability criteria and controller design for linear neutral systems, Automatica, 45, 1948-1952, (2009) · Zbl 1185.93102 [13] He, Y.; Wang, Q.; Lin, C.; Wu, M., Augumented Lyapunov functional and delay-dependent stability criteria for neutral systems, Int. J. robust nonlinear control, 15, 923-933, (2005) · Zbl 1124.34049 [14] He, Y.; Wu, M.; She, J.H.; Liu, G.P., Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays, Syst. control lett., 51, 57-65, (2004) · Zbl 1157.93467 [15] Kwon, O.; Park, J.; Lee, S., On stability criteria for uncertain delay-differential systems of neutral type with time-varying delays, Appl. math. comput., 197, 864-873, (2008) · Zbl 1144.34052 [16] Kwon, O.; Park, J.; Lee, S., On delay-dependent robust stability of uncertain neutral systems with interval time-varying delays, Appl. math. comput., 203, 843-853, (2008) · Zbl 1168.34046 [17] Li, T.; Song, A.; Fei, S.; Wang, T., Delay-derivative-dependent stability for delayed neural networks with unbounded distributed delay, IEEE trans. neural netw., 21, 1365-1371, (2010) [18] Li, T.; Guo, L.; Lin, C., Stability criteria with less LMI variables for neural networks with time-varying delay, IEEE trans. circuits syst. II: express briefs, 55, 1188-1192, (2008) [19] Li, X.; Zhu, X., Stability analysis of neutral systems with distributed delays, Automatica, 44, 2197-2201, (2008) · Zbl 1283.93212 [20] Lien, C.H., Delay-dependent stability criteria for uncertain neutral systems with multiple time-varying delays via LMI approach, IEEE proc. control theory appl., 148, 442-447, (2005) [21] Liu, X.; Wu, M.; Martin, R.; Tang, M., Stability analysis for uncertain systems with mixed delays, J. comput. appl. math., 202, 478-497, (2007) · Zbl 1120.34057 [22] Nian, X.; Pan, H.; Gui, W.; Wang, H., New stability analysis for linear neutral system via state matrix decomposition, Appl. math. comput., 215, 1830-1837, (2009) · Zbl 1201.34119 [23] Park, J.H., A new delay-dependent criterion for neutral systems with multiple delays, J. comput. appl. math., 136, 177-184, (2001) · Zbl 0995.34069 [24] Park, J.H., Simple criterion for asymptotic stability of interval neutral delay-differential systems, Appl. math. lett., 16, 1063-1068, (2003) · Zbl 1058.34094 [25] Park, J.H., Delay-dependent criterion asymptotic stability of a class of neutral equations, Appl. math. lett., 17, 1203-1206, (2004) · Zbl 1122.34339 [26] Park, J.H., LMI optimization approach to asymptotic stability of certain neutral delay differential equations with time-varying coefficients, Appl. math. comput., 160, 355-361, (2005) · Zbl 1062.34084 [27] Park, M.J.; Kwon, O.M.; Park, J.H.; Lee, S.M., A new augmented lyapunov – krasovskii functional approach for stability of linear systems with time-varying delays, Appl. math. comput., 217, 7197-7209, (2011) · Zbl 1219.93106 [28] Park, J.H., Guaranteed cost stabilization of neutral differential systems with parametric uncertainty, J. comput. appl. math., 151, 371-382, (2003) · Zbl 1038.93044 [29] Qian, W.; Liu, J.; Fei, S., Augmented Lyapunov functional approach for stability of neutral systems with mixed delays, Asian J. control, 14, 1-8, (2012) [30] J. Sun, J. Chen, G. Liu, D. Rees, On robust stability of uncertain neutral systems with discrete and distributed delays, in: American Control Conference, Hyatt Regency Riverfront, St. Louis, MO, USA, June 10-12, 2009, pp. 5469-5473. [31] Sun, J.; Liu, G.-P.; Chen, J., Delay-dependent stability and stabilization of neutral time-delay systems, Int. J. robust nonlinear control, 19, 1364-1375, (2009) · Zbl 1169.93399 [32] Wu, M.; He, Y.; She, J., New delay-dependent stability criteria and stabilizing method for neutral systems, IEEE trans. automat. control, 49, 2266-2271, (2004) · Zbl 1365.93358 [33] Wu, Z.-G.; Park, J.; Su, H.; Chu, J., Admissibility and dissipativity analysis for discrete-time singular systems with mixed time-varying delay, Appl. math. comput., 218, 7128-7138, (2012) · Zbl 1243.93066 [34] Wu, Z.-G.; Park, J.; Su, H.; Chu, J., Dissipativity analysis for singular systems with time-varying delays, Appl. math. comput., 218, 4605-4613, (2011) · Zbl 1256.34062 [35] Xiong, L.; Zhong, S.; Tian, J., Novel robust stability criteria of uncertain neutral systems with discrete and distributed delays, Chaos solitons fract, 40, 771-777, (2009) · Zbl 1197.93132 [36] Xu, S.; Lam, J., Improved delay-dependent stability criteria for time-delay systems, IEEE trans. automat. control, 50, 384-387, (2005) · Zbl 1365.93376 [37] Xu, S.; Lam, J., A survey of linear matrix inequality techniques in stability analysis of delay systems, Int. J. syst. sci., 39, 384-387, (2005) [38] Xu, S.; Lam, J.; Zhong, M., New exponential estimates for time-delay systems, IEEE trans. automat. control, 51, 1501-1505, (2006) · Zbl 1366.34101 [39] Xu, S.; Lam, J., On equivalence and efficiency of certain stability criteria for time-delay systems, IEEE trans. automat. control, 52, 95-101, (2007) · Zbl 1366.93451 [40] Yue, D.; Han, Q., A delay-dependent stability criterion of neutral systems and its application to a a partial element equivalent circuit model, IEEE trans. circuit syst., 51, 685-689, (2004) [41] Zhang, B.; Zhou, S.; Xu, S., Delay-dependent $$H_\infty$$ controller design for linear neutral systems with discrete and distributed delays, Int. J. syst. sci., 38, 611-621, (2007) · Zbl 1128.93017 [42] Zhang, L.; Shi, P.; Boukas, E.; Wang, C., $$H_\infty$$ output-feedback control for switched linear discrete-time systems with time-varying delays, Int. J. control, 80, 1354-1365, (2007) · Zbl 1133.93316 [43] Zhang, L.; Shi, P.; Boukas, E.; Wang, C., Robust $$L_2 - L_\infty$$ filtering for switched linear discrete time-delay systems with ploytopic uncertainties, IET control theory appl., 1, 722-730, (2007) [44] Zhang, L.; Boukas, E.; Haidar, A., Delay-range-dependent control synthesis for time-delay systems with actuator saturation, Automatica, 44, 2691-2695, (2008) · Zbl 1155.93350
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.