Input-to-state stability of impulsive and switching hybrid systems with time-delay. (English) Zbl 1233.93083

Summary: This paper investigates Input-to-State Stability (ISS) and Integral Input-to-State stability (IISS) of impulsive and switching hybrid systems with time-delay, using the method of multiple Lyapunov-Krasovskii functionals. It is shown that, even if all the subsystems governing the continuous dynamics, in the absence of impulses, are not ISS/IISS, impulses can successfully stabilize the system in the ISS/iISS sense, provided that there are no overly long intervals between impulses, i.e., the impulsive and switching signal satisfies a dwell-time upper bound condition. Moreover, these impulsive ISS/IISS stabilization results can be applied to systems with arbitrarily large time-delays. Conversely, in the case when all the subsystems governing the continuous dynamics are ISS/IISS in the absence of impulses, the ISS/IISS properties can be retained if the impulses and switching do not occur too frequently, i.e., the impulsive and switching signal satisfies a dwell-time lower bound condition. Several illustrative examples are presented, with their numerical simulations, to demonstrate the main results.


93D25 Input-output approaches in control theory
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
93C15 Control/observation systems governed by ordinary differential equations
Full Text: DOI


[1] Ballinger, G.; Liu, X., Existence and uniqueness results for impulsive delay differential equations, Dynamics of continuous, discrete & impulsive systems, 5, 579-591, (1999) · Zbl 0955.34068
[2] Ballinger, G.; Liu, X., Practical stability of impulsive delay differential equations and applications to control problems, () · Zbl 0879.34015
[3] Bellman, R., Topics in pharmacokinetics, III: repeated dosage and impulsive control, Mathematical biosciences, 12, 1-5, (1971)
[4] Branicky, M.S., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE transactions on automatic control, 43, 475-482, (1998) · Zbl 0904.93036
[5] Cai, C., & Teel, A.R. (2005). Results on input-to-state stability for hybrid systems. In Proc. of the 44th conf. on decision and control (pp. 5403-5408).
[6] Cai, C.; Teel, A.R., Characterizations of input-to-state stability for hybrid systems, Systems & control letters, 58, 47-53, (2009) · Zbl 1154.93037
[7] Carter, T., Optimal impulsive space trajectories based on linear equations, Journal of optimization theory and applications, 70, 277-297, (1991) · Zbl 0732.49025
[8] Chen, W.-H.; Zheng, W.X., Input-to-state stability and integral input-to-state stability of nonlinear impulsive systems with delays, Automatica, 45, 1481-1488, (2009) · Zbl 1166.93370
[9] Goebel, R.; Sanfelice, R.G.; Teel, A.R., Hybrid dynamical systems: robust stability and control for systems that combine continuous-time and discrete-time dynamics, IEEE control systems magazine, 29, 28-93, (2009) · Zbl 1395.93001
[10] Haddad, W.M.; Chellaboina, V.; Nersesov, S.G., Impulsive and hybrid dynamical systems, (2006), Princeton Univ. Press New Jersey · Zbl 1114.34001
[11] Hale, J.K.; Lunel, S.M.V., Introduction to functional differential equations, (1993), Springer-Verlag New York
[12] Hespanha, J.P., Uniform stability of switched linear systems: extension of lasalle’s invariance principle, IEEE transactions on automatic control, 49, 470-482, (2004) · Zbl 1365.93348
[13] Hespanha, J.P.; Liberzon, D.; Teel, A.R., Lyapunov conditions for input-to-state stability of impulsive systems, Automatica, 44, 2735-2744, (2008) · Zbl 1152.93050
[14] Jiang, Z.P.; Wang, Y., Input-to-state stability for discrete-time nonlinear systems, Automatica, 37, 857-869, (2001) · Zbl 0989.93082
[15] Kolmanovskii, V.; Myshkis, A., Introduction to the theory and applications of functional differential equations, (1999), Kluwer Avademic Publishers Dordrecht · Zbl 0917.34001
[16] Lakshmikantham, V.; Baĭnov, D.; Simeonov, P., Theory of impulsive differential equations, (1989), World Scientific Publishing Teaneck, NJ · Zbl 0719.34002
[17] Liao, Y.C., Switching and impulsive control of a reflected diffusion, Applied mathematics and optimization, 11, 153-159, (1984) · Zbl 0553.93068
[18] Liberzon, D., Switching in systems and control, (2003), Birkhäuser Boston · Zbl 1036.93001
[19] Li, C.; Liao, X.; Yang, X.; Huang, T., Impulsive stabilization and synchronization of a class of chaotic delay systems, Chaos, 15, 043103, (2005), 9 pp · Zbl 1144.37371
[20] Li, Z.; Soh, Y.; Wen, C., ()
[21] Liu, X., Stability of impulsive control systems with time delay, Mathematical and computer modelling, 39, 511-519, (2004) · Zbl 1081.93021
[22] Liu, X., Impulsive stabilization and control of chaotic system, Nonlinear analysis, 47, 1081-1092, (2001) · Zbl 1042.93523
[23] Liu, X., Impulsive stabilization and applications to population growth models, The rocky mountain journal of mathematics, 25, 381-395, (1995) · Zbl 0832.34039
[24] Liu, J.; Liu, X.; Xie, W.-C., Invariance principles for impulsive switched systems, Dynamics of continuous, discrete & impulsive systems. series B. applications & algorithms, 16, 631-654, (2009) · Zbl 1188.93040
[25] Liu, X.; Rohlf, K., Impulsive control of a lotka – volterra system, IMA journal of mathematical control and information, 15, 269-284, (1998) · Zbl 0949.93069
[26] Liu, X.; Wang, Q., Impulsive stabilization of high-order Hopfield-type neural networks with time-varying delays, IEEE transactions on neural networks, 19, 71-79, (2008)
[27] Liu, X.; Wang, Q., The method of Lyapunov functionals and exponential stability of impulsive systems with time delay, Nonlinear analysis, 66, 1465-1484, (2007) · Zbl 1123.34065
[28] Mancilla-Aguilar, J.L.; Garcia, R.A., On converse Lyapunov theorems for ISS and iiss switched nonlinear systems, Systems & control letters, 42, 47-53, (2001) · Zbl 0985.93052
[29] Neuman, C.; Costanza, V., Deterministic impulse control in native forest ecosystems management, Journal of optimization theory and applications, 66, 173-196, (1990) · Zbl 0681.90031
[30] Pepe, P.; Jiang, Z.P., A lyapunov – krasovskii methodology for ISS and iiss of time-delay systems, Systems & control letters, 55, 1006-1014, (2006) · Zbl 1120.93361
[31] Richard, J.-P., Time-delay systems: an overview of some recent advances and open problems, Automatica, 39, 1667-1694, (2003) · Zbl 1145.93302
[32] Shen, J.; Luo, Z.; Liu, X., Impulsive stabilization of functional-differential equations via Lyapunov functionals, Journal of mathematical analysis and applications, 240, 1-15, (1999) · Zbl 0955.34069
[33] Shorten, R.; Wirth, F.; Mason, O.; Wulff, K.; King, C., Stability criteria for switched and hybrid systems, SIAM review, 49, 543-732, (2007)
[34] Sontag, E.D., Smooth stabilization implies coprime factorization, IEEE transactions on automatic control, 34, 435-443, (1989) · Zbl 0682.93045
[35] Sontag, E.D., Comments on integral variants of ISS, Systems & control letters, 34, 93-100, (1998) · Zbl 0902.93062
[36] Teel, A.R., Connection between Razumikhin-type theorems and the ISS nonlinear small gain theorem, IEEE transactions on automatic control, 43, 960-964, (1998) · Zbl 0952.93121
[37] van der Schaft, A.; Schumacher, H., ()
[38] Vu, L.; Chatterjee, D.; Liberzon, D., Input-to-state stability of switched systems and switching adaptive control, Automatica, 42, 639-646, (2007) · Zbl 1261.93049
[39] Walsh, G.C.; Beldiman, O.; Bushnell, L.G., Asymptotic behavior of nonlinear networked control systems, IEEE transactions on automatic control, 46, 1093-1097, (2001) · Zbl 1006.93040
[40] Xie, G.; Wang, L., Necessary and sufficient conditions for controllability and observability of switched impulsive control systems, IEEE transactions on automatic control, 49, 960-966, (2004) · Zbl 1365.93049
[41] Xu, H.; Liu, X.; Teo, K.L., A LMI approach to stability analysis and synthesis of impulsive switched systems with time delays, Nonlinear analysis hybrid systems, 2, 38-50, (2008) · Zbl 1157.93501
[42] Yang, T.; Chua, L., Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication, IEEE transactions on circuits and systems I: fundamental theory and applications, 44, 976-988, (1997)
[43] Yeganefar, N.; Pepe, P.; Dambrine, M., Input-to-state stability of time-delay systems: a link with exponential stability, IEEE transactions on automatic control, 53, 1526-1531, (2008) · Zbl 1367.93284
[44] Zhang, G.; Liu, Z.; Ma, Z., Synchronization of complex dynamical networks via impulsive control, Chaos, 17, 043126, (2007), 9 pp · Zbl 1163.37389
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.