Abstract
The design problem of delay-dependent robust control for uncertain discrete singular systems with time-varying delay is addressed in this paper. The uncertainty is assumed to be norm-bounded. By establishing a finite sum inequality based on quadratic terms, a new delay-dependent robust stability condition is derived and expressed in terms of linear matrix inequalities (LMIs). A suitable robust state feedback control law is presented, which guarantees that the resultant closed-loop system is regular, causal and stable for all admissible uncertainties. Numerical examples are given to demonstrate the applicability of the proposed method.
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References
Boukas, E.K., Liu, Z.K., 2003. Delay-dependent stability analysis of singular linear continuous-time systems. IEE Proc.-Control Theory Appl., 150(4):325–330. [doi:10.1049/ip-cta:20030635]
Chen, S.H., Chou, J.H., 2003. Stability robustness of linear discrete singular time-delay systems with structured parameter uncertainties. IEE Proc.-Control Theory Appl., 150(3):295–302. [doi:10.1049/ip-cta:20030242]
Chen, S.J., Lin, J.L., 2004. Robust stability of discrete time-delay uncertain singular systems. IEE Proc.-Control Theory Appl., 151(1):45–52. [doi:10.1049/ip-cta:20040060]
Chen, Y., Xue, A., Ge, M., Wang, J., Lu, R., 2007. On exponential stability for systems with state delays. J. Zhejiang Univ. Sci. A, 8(8):1296–1303. [doi:10.1631/jzus.2007.A1296]
Dai, L., 1989. Singular Control Systems. Springer-Verlag, Berlin, Germany.
Fridman, E., 2001. A Lyapunov Based Approach to Stability of Descriptor Systems with Delays. IEEE Proc. 40th Conf. on Decision and Control, p.2850–2855.
Fridman, E., Shaked, U., 2002. A descriptor approach to H ∞ control of linear time-delay systems. IEEE Trans. on Automatic Control, 47(2):253–270. [doi:10.1109/9.983353]
Gao, H., Chen, T., 2007. New results on stability of discrete-time systems with time-varying state delay. IEEE Trans. on Automatic Control, 52(2):328–334. [doi:10.1109/TAC.2006.890320]
Gao, H., Lam, J., Wang, C., Wang, Y., 2004. Delay-dependent output-feedback stabilization of discrete-time systems with time-varying state delay. IEE Proc.-Control Theory Appl., 151(6):691–698. [doi:10.1049/ip-cta:20040822]
Gu, K., Kharitonov, V.L., Chen, J., 2003. Stability of Time-delay Systems. Birkhauser, Boston.
Han, Q.L., 2002a. Robust stability of uncertain delay-differential systems of neutral type. Automatica, 38(4):719–723. [doi:10.1016/S0005-1098(01)00250-3]
Han, Q.L., 2002b. New results for delay-dependent stability of linear systems with time-varying delay. Int. J. Syst. Sci., 33(3):213–228. [doi:10.1080/00207720110092207]
Han, Q.L., 2004a. On robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainty. Automatica, 40(6):1087–1092. [doi:10.1016/j.automatica.2004.01.007]
Han, Q.L., 2004b. A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays. Automatica, 40(10):1791–1796. [doi:10.1016/j.automatica.2004.05.002]
Han, Q.L., 2005a. Stability analysis for a partial element equivalent circuit (PEEC) model of neutral type. Int. J. Circuit Theory Appl., 33(4):321–332. [doi:10.1002/cta.323]
Han, Q.L., 2005b. On stability of linear neutral systems with mixed time-delays: a discretized Lyapunov functional approach. Automatica, 41(7):1209–1218. [doi:10.1016/j.automatica.2005.01.014]
Han, Q.L., 2005c. Absolute stability of time-delay systems with sector-bounded nonlinearity. Automatica, 41(12):2171–2176. [doi:10.1016/j.automatica.2005.08.005]
Han, Q.L., Yu, X., Gu, K., 2004. On computing the maximum time-delay bound for stability of linear neutral systems. IEEE Trans. on Automatic Control, 49(12):2281–2286. [doi:10.1109/TAC.2004.838479]
Ji, X., Su, H., Chu, J., 2006. Delay-dependent Robust Stability of Uncertain Discrete Singular Time-delay Systems. Proc. American Control Conf., p.3843–3848. [doi:10.1109/ACC.2006.1657318]
Jiang, X., Han, Q.L., 2005. On H ∞ control for linear systems with interval time-varying delay. Automatica, 41(12):2099–2106. [doi:10.1016/j.automatica.2005.06.012]
Jiang, X., Han, Q.L., 2006. Delay-dependent robust stability for uncertain linear systems with interval time-varying delay. Automatica, 42:1059–1065. [doi:10.1016/j.automatica.2006.02.019]
Lewis, F.L., 1986. A survey of linear singular systems. Circuits Syst. Signal Process., 5(1):3–36. [doi:10.1007/BF01600184]
Petersen, I.R., 1987. A stabilization algorithm for a class of uncertain linear systems. Syst. Control Lett., 8(4):351–357. [doi:10.1016/0167-6911(87)90102-2]
Wang, H., Xue, A., Lu, R., Xu, Z., Wang, J., 2007. Robust H ∞ control for discrete singular systems with parameter uncertainties. Acta Automatica Sinica, 33:1300–1305 (in Chinese). [doi:10.1360/aas-007-1300]
Wang, H., Xue, A., Guo, Y., Lu, R., 2008. Input-output approach to robust stability and stabilization for uncertain singular systems with time-varying discrete and distributed delays. J. Zhejiang Univ. Sci. A, 9(4):546–551. [doi:10.1631/jzus.A071299]
Wang, R., Liu, Y., 2000. Asymptotic Stability and Robustness for Discrete-time Singular Systems with Multiple Delays. Proc. World Congress in Intelligent Control and Automation, p.3350–3353. [doi:10.1109/WCICA.2000.863156]
Xu, S., Lam, J., 2004. Robust stability and stabilization of discrete singular systems: an equivalent characterization. IEEE Trans. on Automatic Control, 49(4):568–574. [doi:10.1109/TAC.2003.822854]
Xu, S., Dooren, P.V., Stefan, R., Lam, J., 2002. Robust stability and stabilization for singular systems with state delay and parameter uncertainty. IEEE Trans. on Automatic Control, 47(7):1122–1128. [doi:10.1109/TAC.2002.800651]
Yue, D., Han, Q.L., 2005a. Delay-dependent robust H ∞ controller design for uncertain descriptor systems with time-varying discrete and distributed delays. IEE Proc.-Control Theory Appl., 152(6):628–638. [doi:10.1049/ip-cta:20045293]
Yue, D., Han, Q.L., 2005b. Delayed feedback control of uncertain systems with time-varying input delay. Automatica, 41(2):233–240. [doi:10.1016/j.automatica.2004.09.006]
Zhang, X.M., Han, Q.L., 2006. Delay-dependent robust H ∞ filtering for uncertain discrete-time systems with time-varying delay based on a finite sum inequality. IEEE Trans. on Circuits Syst.-II: Expr. Briefs, 53(12):1466–1470. [doi:10.1109/TCSII.2006.884116]
Zhou, S., Lam, L., 2003. Robust stabilization of delayed singular systems with linear fractional parametric uncertainties. Circuits Syst. Signal Process., 22(6):579–588. [doi:10.1007/s00034-003-1218-x]
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Project (Nos. 60434020 and 60604003) supported by the National Natural Science Foundation of China
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Wang, Hj., Zhao, Xd., Xue, Ak. et al. Delay-dependent robust control for uncertain discrete singular systems with time-varying delay. J. Zhejiang Univ. Sci. A 9, 1034–1042 (2008). https://doi.org/10.1631/jzus.A0720008
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DOI: https://doi.org/10.1631/jzus.A0720008