Input-to-state stability for discrete-time nonlinear systems. (English) Zbl 0989.93082

The present work studies the input-to-state stability (ISS) for discrete-time nonlinear systems. The ISS property means that for a bounded control, the state trajectory is bounded and the state becomes small if the input is small independent from the initial state. The authors equivalently characterize the ISS by the existence of a smooth ISS-function and show that many recent results related to ISS for the continuous time case find their analogue in discrete time. In this spirit, the small-gain theorems are presented to treat the ISS-stabilization for interconnected systems. The ISS stabilizability is characterized by the existence of a continuous feedback.


93D25 Input-output approaches in control theory
93C55 Discrete-time control/observation systems
93C10 Nonlinear systems in control theory
Full Text: DOI


[1] Agarwal, R.P., Difference equations and inequalities: theory, methods and applications, (1992), Marcel Dekker New York · Zbl 0925.39001
[2] Byrnes, C.I.; Lin, W., Losslessness, feedback equivalence and the global stabilization of discrete-time nonlinear systems, IEEE transactions on automatic control, 39, 83-98, (1994) · Zbl 0807.93037
[3] Chen, F.-C.; Khalil, H.K., Adaptive control of a class of nonlinear discrete-time systems using neural networks, IEEE transactions on automatic control, 40, 791-801, (1995) · Zbl 0925.93461
[4] Coron, J.-M.; Praly, L.; Teel, A., Feedback stabilization of nonlinear systems: sufficient conditions and Lyapunov and input-output techniques, ()
[5] Desoer, C.; Vidyasagar, M., Feedback systems: input-output properties, (1975), Academic Press New York · Zbl 0327.93009
[6] Guo, L., On critical stability of discrete-time adaptive nonlinear control, IEEE transactions on automatic control, 42, 11, 1488-1499, (1997) · Zbl 0898.93020
[7] Isidori, A., Nonlinear control systems II., (1999), Springer London · Zbl 0924.93038
[8] Jiang, Z. P., Lin, Y., & Wang, Y. (2000). A local nonlinear small-gain theorem for discrete-time feedback systems and its applications. Proceedings of the third Asian control conference (ASCC’ 2000), Shanghai (pp. 1227-1232).
[9] Jiang, Z.P.; Mareels, I.M.Y., A small-gain control method for cascaded nonlinear systems with dynamic uncertainties, IEEE transactions on automatic control, 42, 292-308, (1997) · Zbl 0869.93004
[10] Jiang, Z.P.; Mareels, I.M.Y.; Wang, Y., A Lyapunov formulation of the nonlinear small gain theorem for interconnected ISS systems, Automatica, 32, 1211-1215, (1996) · Zbl 0857.93089
[11] 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
[12] Jiang, Z. P., & Wang, Y. (2001). A converse Lyapunov theorem for discrete time systems with disturbances. Submitted for publication. · Zbl 0987.93072
[13] Kazakos, D.; Tsinias, J., The input to state stability conditions and global stabilization of discrete-time systems, IEEE transactions on automatic control, 39, 2111-3113, (1994) · Zbl 0925.93568
[14] Kotsios, St., & Kalouptsisis, N. (1996). Adaptive control for a certain class of nonlinear systems. preprint.
[15] Krstić, M.; Li, Z., Inverse optimal design of input-to-state stabilizing nonlinear controllers, IEEE transactions on automatic control, 43, 336-350, (1998) · Zbl 0910.93064
[16] Krstić, M.; Kanellakopoulos, I.; Kokotović, P.V., Nonlinear and adaptive control design, (1995), Wiley New York · Zbl 0763.93043
[17] Lakshmikantham, V.; Trigiante, D., Theory of difference equations: numerical methods and applications, (1988), Academic Press New York · Zbl 0683.39001
[18] LaSalle, J.P., The stability and control of discrete process, (1986), Springer New York · Zbl 0606.93001
[19] Lin, Y.; Sontag, E.D.; Wang, Y., A smooth converse Lyapunov theorem for robust stability, SIAM journal on control and optimization, 34, 124-160, (1996) · Zbl 0856.93070
[20] Mareels, I.M.Y.; Hill, D.J., Monotone stability of nonlinear feedback systems, Journal of mathematical systems and estimation control, 2, 275-291, (1992) · Zbl 0776.93039
[21] Nijmeijer, H.; van der Schaft, A., Nonlinear dynamical control systems, (1990), Springer New York · Zbl 0701.93001
[22] 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
[23] 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, (1994) · Zbl 0869.93040
[24] Sontag, E.D., Smooth stabilization implies coprime factorization, IEEE transactions on automatic control, 34, 435-443, (1989) · Zbl 0682.93045
[25] Sontag, E.D., Further facts about input to state stabilization, IEEE transactions on automatic control, 35, 473-476, (1990) · Zbl 0704.93056
[26] Sontag, E.D.; Wang, Y., On characterizations of the input to state stability property, Systems & control letters, 24, 351-359, (1995) · Zbl 0877.93121
[27] Sontag, E.D.; Wang, Y., New characterizations of the input to state stability property, IEEE transactions on automatic control, 41, 1283-1294, (1996) · Zbl 0862.93051
[28] Teel, A.R., A nonlinear small-gain theorem for the analysis of control systems with saturation, IEEE transactions on automatic control, 41, 1256-1270, (1996) · Zbl 0863.93073
[29] Tsinias, J.T., Versions of Sontag’s “input to state stability condition” and the global stabilization problem, SIAM journal on control and optimization, 31, 928-941, (1993) · Zbl 0788.93076
[30] Tsinias, J., Kotsios, S., & Kalouptsidis, N. (1990). Topological dynamics of discrete-time systems. Robust control of linear systems and nonlinear control, Proceedings of international symposium MTNS-89, vol. II (pp. 457-463). Boston: Birkhauser. · Zbl 0735.93068
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