Lyapunov conditions for input-to-state stability of impulsive systems. (English) Zbl 1152.93050

Summary: This paper introduces appropriate concepts of input-to-state stability (ISS) and integral-ISS for impulsive systems, i.e., dynamical systems that evolve according to ordinary differential equations most of the time, but occasionally exhibit discontinuities (or impulses). We provide a set of Lyapunov-based sufficient conditions for establishing these ISS properties. When the continuous dynamics are ISS, but the discrete dynamics that govern the impulses are not, the impulses should not occur too frequently, which is formalized in terms of an average dwell-time (ADT) condition. Conversely, when the impulse dynamics are ISS, but the continuous dynamics are not, there must not be overly long intervals between impulses, which is formalized in terms of a novel reverse ADT condition. We also investigate the cases where (i) both the continuous and discrete dynamics are ISS, and (ii) one of these is ISS and the other only marginally stable for the zero input, while sharing a common Lyapunov function. In the former case, we obtain a stronger notion of ISS, for which a necessary and sufficient Lyapunov characterization is available. The use of the tools developed herein is illustrated through examples from a Micro-Electro-Mechanical System (MEMS) oscillator and a problem of remote estimation over a communication network.


93D25 Input-output approaches in control theory
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C15 Control/observation systems governed by ordinary differential equations


Full Text: DOI Link


[1] Angeli, D.; Sontag, E., Forward completeness, unboundedness observability, and their Lyapunov characterizations, Systems and control letters, 38, 209-217, (1999) · Zbl 0986.93036
[2] Angeli, D.; Sontag, E.D.; Wang, Y., A characterization of integral input-to-state stability, IEEE transactions on automatic control, 45, 6, 1082-1097, (2000) · Zbl 0979.93106
[3] Bainov, D.D.; Simeonov, P.S., Systems with impulse effects: stability, theory and applications, (1989), Academic Press New York · Zbl 0676.34035
[4] Cai, C., & Teel, A. R. (2005). Results on input-to-state stability for hybrid systems. In Proc. of the 44th conf. on decision and contr. (pp. 5403-5408)
[5] Cai, C., Teel, A. R., & Goebel, R. (2007). Results on existence of smooth Lyapunov functions for (pre-)asymptotically stable hybrid systems with non-open basins of attraction. In Proc. of the 2007 Amer. contr. conf. (pp. 3456-3461)
[6] Clarke, F.H., Optimization and nonsmooth analysis, (1990), SIAM Philadelphia, PA · Zbl 0727.90045
[7] Haddad, W.M.; Chellaboina, V.S.; Nersesov, S.G., Impulsive and hybrid dynamical systems: stability, dissipativity, and control, (2006), Princeton University Press · Zbl 1114.34001
[8] Hespanha, J.P., Stabilization through hybrid control, ()
[9] Hespanha, J. P., Liberzon, D., & Teel, A. R. (2007). Lyapunov conditions for input-to-state stability of impulsive systems. Technical report. Santa Barbara: Univ. California. Available at: http://www.ece.ucsb.edu/hespanha/techrep.html · Zbl 1152.93050
[10] Hespanha, J. P., & Morse, A. S. (1999). Stability of switched systems with average dwell-time. In Proc. of the 38th conf. on decision and contr. (pp. 2655-2660)
[11] Hespanha, J. P., Ortega, A., & Vasudevan, L. (2002). Towards the control of linear systems with minimum bit-rate. In Proc. of the int. symp. on the mathematical theory of networks and syst
[12] Jiang, Z.P.; Wang, Y., Input-to-state stability for discrete-time nonlinear systems, Automatica, 37, 857-869, (2001) · Zbl 0989.93082
[13] Khalil, H.K., Nonlinear systems, (2002), Prentice-Hall New Jersey · Zbl 0626.34052
[14] Lakshmikantham, V.; Leela, S., ()
[15] Liberzon, D., Switching in systems and control, (2003), Birkhäuser Boston · Zbl 1036.93001
[16] Liberzon, D.; Hespanha, J.P., Stabilization of nonlinear systems with limited information feedback, IEEE transactions on automatic control, 50, 6, 910-915, (2005) · Zbl 1365.93405
[17] Mancilla-Aguilar, J.L.; Garcia, R.A., On converse Lyapunov theorems for ISS and iiss switched nonlinear systems, Systems and control letters, 42, 47-53, (2001) · Zbl 0985.93052
[18] Mehta, A.; Cherian, S.; Hedden, D.; Thundat, T., Manipulation and controlled amplification of Brownian motion of microcantilever sensors, Applied physics letters, 78, 11, 1637-1639, (2001)
[19] Nesic, D.; Teel, A.R., Input-output stability properties of networked control systems, IEEE transactions on automatic control, 49, 10, 1650-1667, (2004) · Zbl 1365.93466
[20] Prajna, S., Papachristodoulou, A., Seiler, P., & Parrilo, P. A. (2004). SOSTOOLS: Sum of squares optimization toolbox for MATLAB
[21] Praly, L.; Wang, Y., Stabilization in spite of matched unmodeled dynamics and an equivalent definition of input-to-state stability, Mathematics of control signals and systems, 9, 1-33, (1996) · Zbl 0869.93040
[22] Sontag, E.D., Smooth stabilization implies coprime factorization, IEEE transactions on automatic control, 34, 4, 435-443, (1989) · Zbl 0682.93045
[23] Sontag, E.D., Comments on integral variants of ISS, Systems and control letters, 34, 93-100, (1998) · Zbl 0902.93062
[24] Sontag, E. D., & Wang, Y. (1995). On characterizations of the input-to-state stability property with respect to compact sets. In Proc. IFAC non-linear control systems design symposium (pp. 226-231)
[25] Tao, Y., ()
[26] Teel, A.R.; Praly, L., On assigning the derivative of a disturbance attenuation control Lyapunov function, Mathematics of control signals and systems, 13, 95-124, (2000) · Zbl 0965.93048
[27] Walsh, G.C.; Beldiman, O.; Bushnell, L.G., Asymptotic behavior of nonlinear networked control systems, IEEE transactions on automatic control, 46, 7, 1093-1097, (2001) · Zbl 1006.93040
[28] Xu, Y., & Hespanha, J. P. (2004). Communication logics for networked control systems. In Proc. of the 2004 Amer. contr. conf. (pp. 572-577)
[29] Zhang, W.; Baskaran, R.; Turner, K.L., Effect of cubic nonlinearity on auto-parametrically amplified resonant MEMS mass sensor, Sensors and actuators A, 102, 139-150, (2002)
[30] Zhang, W., & Turner, K. L. (2004). A mass sensor based on parametric resonance. In Proc. of the solid-state sensor, actuator and microsystems workshop (pp. 49-52)
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.