×

zbMATH — the first resource for mathematics

Robust stability of discrete bilinear uncertain time-delay systems. (English) Zbl 1238.93076
Summary: This paper addresses the problem of robust stability for discrete homogeneous bilinear time-delay systems subjected to uncertainties. Two kinds of uncertainties are treated: (1) nonlinear uncertainties and (2) parametric uncertainties. For parametric uncertainties, we also discuss both unstructured uncertainties and interval matrices. By using Lyapunov’s stability theorem associated with some linear algebraic techniques, several delay-independent criteria are developed to guarantee the robust stability of the overall system. One of the features of the newly developed criteria is its independence from the Lyapunov equation, although the Lyapunov approach is adopted. Furthermore, the transient response and the decay rate of the resulting systems are also estimated. In particular, the transient responses for the aforementioned systems with parametric uncertainties also do not involve any Lyapunov equation which remains unsolved. All the results obtained are also applied to analyze the stability of uncertain time-delay systems.

MSC:
93D09 Robust stability
93C55 Discrete-time control/observation systems
93C10 Nonlinear systems in control theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] M. Bacic, M. Cannon, B. Kouvaritakis, Constrained control of SISO bilinear system. IEEE Trans. Autom. Control 48, 1443–1447 (2003) · Zbl 1048.93037 · doi:10.1109/TAC.2003.815042
[2] L. Berrahmoune, Stabilization and decay estimate for distributed bilinear systems. Syst. Control Lett. 36, 167–171 (1999) · Zbl 0914.93055 · doi:10.1016/S0167-6911(98)00065-6
[3] T. Bose, M.Q. Chen, BIBO stability of the discrete bilinear system. Digit. Signal Process. 5, 160–166 (1995) · doi:10.1006/dspr.1995.1016
[4] O. Chabour, J.C. Vivalda, Remark on local and global stabilization of homogeneous bilinear systems. Syst. Control Lett. 41, 141–143 (2000) · Zbl 0985.93047 · doi:10.1016/S0167-6911(00)00045-1
[5] L.K. Chen, R.R. Mohler, Stability analysis of bilinear systems. IEEE Trans. Autom. Control 36, 1310–1315 (1991) · doi:10.1109/9.100945
[6] Y.P. Chen, J.L. Chang, K.M. Lai, Stability analysis and bang-bang sliding control of a class of single-input bilinear systems. IEEE Trans. Autom. Control 45, 2150–2154 (2000) · Zbl 0989.93063 · doi:10.1109/9.887648
[7] J.S. Chiou, F.C. Kung, T.H.S. Li, Robust stabilization of a class of singular perturbed discrete bilinear systems. IEEE Trans. Autom. Control 45, 1187–1191 (2000) · Zbl 0979.93100 · doi:10.1109/9.863604
[8] H.Y. Chung, W.J. Chang, Input and state covariance control for bilinear stochastic discrete systems. Control Theory Adv. Technol. 6, 655–667 (1990)
[9] I. Derese, E. Noldus, Design of linear feedback laws for bilinear systems. Int. J. Control 31, 213–237 (1980) · Zbl 0439.93043 · doi:10.1080/00207178008961039
[10] C. Gounaridis-minaidis, N. Kalouptsidis, Stability of discrete-time bilinear systems with constant inputs. Int. J. Control 43, 663–669 (1986) · Zbl 0581.93052 · doi:10.1080/00207178608933492
[11] J. Guojun, Stability of bilinear time-delay systems. IMA J. Math. Control Inf. 18, 53–60 (2001) · Zbl 0978.93063 · doi:10.1093/imamci/18.1.53
[12] D.W.C. Ho, G. Lu, Y. Zheng, Global stabilization for bilinear systems with time delay. IEE Proc., Control Theory Appl. 149, 89–94 (2002) · doi:10.1049/ip-cta:20020114
[13] M. Jamshidi, A near-optimum controller for cold-rolling mills. Int. J. Control 16, 1137–1154 (1972) · doi:10.1080/00207177208932346
[14] H. Jerbi, Global feedback stabilization of new class of bilinear systems. Syst. Control Lett. 42, 313–320 (2001) · Zbl 1032.93064 · doi:10.1016/S0167-6911(00)00101-8
[15] B.S. Kim, Y.J. Kim, M.T. Lim, Robust H state feeback control methods for bilinear systems. IEE Proc., Control Theory Appl. 152, 553–559 (2005) · doi:10.1049/ip-cta:20050114
[16] S. Kotsios, A note on BIBO stability of bilinear systems. J. Franklin Inst. 332B, 755–760 (1995) · Zbl 0852.93083 · doi:10.1016/0016-0032(95)00072-0
[17] C.H. Lee, Upper and lower matrix bounds of the solution for the discrete Lyapunov equation. IEEE Trans. Autom. Control 41, 1338–1341 (1996) · Zbl 0861.93016 · doi:10.1109/9.536505
[18] C.S. Lee, G. Leitmann, Continuous feedback guaranteeing uniform ultimate boundness for uncertain linear delay systems: An application to river pollution control. Comput. Math. Appl. 16, 929–938 (1983) · Zbl 0673.93052 · doi:10.1016/0898-1221(88)90203-9
[19] C.H. Lee, T.-H.S. Li, F.C. Kung, New stability criteria for discrete time-delay systems with uncertainties. Control Theory Adv. Technol. 10, 1159–1168 (1995)
[20] G. Lu, D.W.C. Ho, Global stabilization controller design for discrete-time bilinear systems with time-delays, in Proceedings of the 4th World Congress on Intelligent Control and Automation (2002), pp. 10–14
[21] G. Lu, D.W.C. Ho, Continuous stabilization controllers for singular bilinear systems: The state feedback case. Automatica 42, 309–314 (2006) · Zbl 1099.93041 · doi:10.1016/j.automatica.2005.09.010
[22] X. Mao, J. Lam, L. Huang, Stabilization of hybrid stochastic differential equations by delay feedback control. Syst. Control Lett. 57(11), 927–935 (2008) · Zbl 1149.93027 · doi:10.1016/j.sysconle.2008.05.002
[23] T. Mori, N. Fukuma, M. Kuwahara, Delay-independent stability criteria for single and composite linear systems with time delays. IEEE Trans. Autom. Control 27, 964–966 (1982) · Zbl 0484.93060 · doi:10.1109/TAC.1982.1103030
[24] H. Schwarz, Stability of discrete-time equivalent homogeneous bilinear systems. Control Theory Adv. Technol. 3, 263–269 (1987)
[25] H.W. Smith, Dynamic control of a two-stand cold mill. Automatica 5, 183–190 (1969) · doi:10.1016/0005-1098(69)90012-0
[26] S.B. Stojanovic, D.Lj. Debeljkovic, Stability of linear discrete time delay systems: Lyapunov-Krasovskii approach, in The 4th IEEE Conference on Industrial Electronics and Applications (ICIEA), Xi’an, China (2009), pp. 2497–2501
[27] C.W. Tao, W.Y. Wang, M.L. Chan, Design of sliding mode controllers for bilinear systems with time varying uncertainties. IEEE Trans. Syst. Man Cybern.. Part B 34, 639–645 (2004) · doi:10.1109/TSMCB.2002.805805
[28] M. Vidyasagar, Nonlinear Systems Analysis, 2nd edn. (Prentice-Hall, Englewood Cliffs, 1993) · Zbl 0900.93132
[29] J. Xing, J. Lam, Stabilization of discrete-time Markovian jump linear systems via delayed controllers. Automatica 42, 747–753 (2006) · Zbl 1137.93421 · doi:10.1016/j.automatica.2005.12.015
[30] S. Xu, G. Feng, Improved robust absolute stability criteria for uncertain time-delay systems. IET Control Theory Appl. 1, 1630–1637 (2007) · doi:10.1049/iet-cta:20060539
[31] S. Xu, J. Lam, A survey of linear matrix inequality techniques in stability analysis of delay systems. Int. J. Syst. Sci. 39, 1095–1113 (2008) · Zbl 1156.93382 · doi:10.1080/00207720802300370
[32] S. Xu, J. Lam, X. Mao, Delay-dependent H control and filtering for uncertain Markovian jump systems with time-varying delays. IEEE Trans. Circuits Syst. I 54, 2070–2077 (2007) · Zbl 1374.93134 · doi:10.1109/TCSI.2007.904640
[33] X. Yang, L.K. Chen, Stability of discrete bilinear systems with time-delayed feedback functions. IEEE Trans. Autom. Control 38, 158–163 (1993) · Zbl 0773.93064 · doi:10.1109/9.186331
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.