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Input-to-state stability and integral input-to-state stability of nonlinear impulsive systems with delays. (English) Zbl 1166.93370

Summary: This paper is concerned with analyzing Input-to-State Stability (ISS) and Integral-ISS (IISS) for nonlinear impulsive systems with delays. Razumikhin-type theorems are established which guarantee ISS/IISS for delayed impulsive systems with external input affecting both the continuous dynamics and the discrete dynamics. It is shown that when the delayed continuous dynamics are ISS/IISS but the discrete dynamics governing the impulses are not, the ISS/IISS property of the impulsive system can be retained if the length of the impulsive interval is large enough. Conversely, when the delayed continuous dynamics are not ISS/IISS but the discrete dynamics governing the impulses are, the impulsive system can achieve ISS/IISS if the sum of the length of the impulsive interval and the time delay is small enough. In particular, when one of the delayed continuous dynamics and the discrete dynamics are ISS/IISS and the others are stable for the zero input, the impulsive system can keep ISS/IISS no matter how often the impulses occur. Our proposed results are evaluated using two illustrative examples to show their effectiveness.

MSC:

93D25 Input-output approaches in control theory
93C10 Nonlinear systems in control theory
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[1] Angeli, D.; Sontag, E.D.; Wang, Y., A characterization of integral input to state stability, IEEE transactions on automatic control, 45, 1082-1097, (2000) · Zbl 0979.93106
[2] Angeli, D.; Sontag, E.D.; Wang, Y., Further equivalences and semiglobal versions of integral input to state stability, Dynamics and control, 10, 127-149, (2000) · Zbl 0973.93048
[3] 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
[4] Boyd, S.; Ghaoui, E.I.; Feron, L.; Balakrishnan, E., Linear matrix inequalities in systems and control theory, (1994), SIAM Philadephia, PA
[5] Cai, C., & Teel, A. R. (2005). Results on input-to-state stability for hybrid systems. In Proc. 44th IEEE CDC (pp. 5403-5408)
[6] Chaillet, A., & Angeli, D. (2006). Integral input to state stability for cascaded systems. In MTNS 2006 · Zbl 1140.93470
[7] Chen, W.-H.; Wang, J.-G.; Tang, Y.-J.; Lu, X., Robust \(H_\infty\) control of uncertain linear impulsive stochastic systems, International journal of robust and nonlinear control, 18, 1348-1371, (2008) · Zbl 1298.93346
[8] Gu, K.; Kharitonov, V.; Chen, J., Stability of time-delay systems, (2003), Birkhauser Boston, MA · Zbl 1039.34067
[9] Hespanha, J.P.; Liberzon, D.; Teel, A.R., Lyapunov conditions for input-to-state stability of impulsive systems, Automatica, 11, 2735-2744, (2008) · Zbl 1152.93050
[10] Jiang, Z.P.; Teel, A.; Praly, L., Small-gain theorem for ISS systems and applications, Mathematics of control, signals, and systems, 7, 95-120, (1994) · Zbl 0836.93054
[11] Jiang, Z.P.; Wang, Y., Input-to-state stability for discrete-time nonlinear systems, Automatica, 37, 857-869, (2001) · Zbl 0989.93082
[12] Li, Z.G.; Soh, C.B.; Xu, X.H., Stability of impulsive differential systems, Journal of mathematical analysis and application, 216, 644-653, (1997) · Zbl 0899.34010
[13] Li, Z.G.; Wen, C.Y.; Soh, Y.C., Analysis and design of impulsive control systems, IEEE transactions on automatic control, 46, 894-897, (2001) · Zbl 1001.93068
[14] Nes˘ić, D.; Teel, A.R., Input-output stability properties of networked control systems, IEEE transactions on automatic control, 49, 1650-1667, (2004) · Zbl 1365.93466
[15] Nes˘ić, D.; Teel, A.R., Input-to-state stability of networked control systems, Automatica, 40, 2121-2128, (2004) · Zbl 1077.93049
[16] 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
[17] Praly, L.; Jiang, Z.P., Stabilization by output feedback for systems with ISS inverse dynamics, Systems & control letters, 21, 19-33, (1993) · Zbl 0784.93088
[18] 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
[19] Sontag, E.D., Smooth stabilization implies coprime factorization, IEEE transactions on automatic control, 34, 435-443, (1989) · Zbl 0682.93045
[20] Sontag, E.D., Comments on integral variants of input-to-state stability, Systems & control letters, 34, 93-100, (1998) · Zbl 0902.93062
[21] Teel, A.R., Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem, IEEE transactions on automatic control, 43, 960-964, (1998) · Zbl 0952.93121
[22] Vu, L.; Chatterjee, D.; Liberzon, D., Input-to-state stability of switched systems and switching adaptive control, Automatica, 43, 639-646, (2007) · Zbl 1261.93049
[23] Walsh, G.C.; Beldiman, O.; Bushnell, L.G., Asymptotic behaviour of nonlinear networked control systems, IEEE transactions on automatic control, 46, 1093-1097, (2001) · Zbl 1006.93040
[24] Xie, W.X.; Wen, C.Y.; Li, Z.G., Input-to-state stabilization of switched nonlinear systems, IEEE transactions on automatic control, 46, 1111-1116, (2001) · Zbl 1010.93089
[25] Yang, T., Impulsive systems and control: theory and applications, (2001), Nova Science New York
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