A converse Lyapunov theorem for discrete-time systems with disturbances. (English) Zbl 0987.93072

Summary: This paper presents a converse Lyapunov theorem for discrete-time systems with disturbances taking values in compact sets. Among several new stability results, it is shown that a smooth Lyapunov function exists for a family of time-varying discrete systems if these systems are robustly globally asymptotically stable.


93D30 Lyapunov and storage functions
93C55 Discrete-time control/observation systems
93C10 Nonlinear systems in control theory
93C73 Perturbations in control/observation systems
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[1] Agarwal, R.P., Difference equations and inequalities: theory, methods and applications, (1992), Marcel Dekker New York · Zbl 0925.39001
[2] Boothby, W.M., An introduction to differentiable manifolds and Riemannian geometry, (1986), Academic Press Orlando, FL · Zbl 0596.53001
[3] Byrnes, C.I.; Lin, W., Losslessness, feedback equivalence and the global stabilization of discrete-time nonlinear systems, IEEE trans. automat. control, 39, 83-98, (1994) · Zbl 0807.93037
[4] Gordon, S.P., On converse to the stability theorems for difference equations, SIAM J. control optim., 10, 76-81, (1972) · Zbl 0244.39001
[5] Z.P. Jiang, Y. Lin, Y. Wang, A local nonlinear small-gain theorem for discrete-time feedback systems, Proceedings of the Third Asian Control Conference, Shanghai, China, 2000, pp. 1227-1232.
[6] Jiang, Z.P.; Wang, Y., Input-to-state stability for discrete-time nonlinear systems, Automatica, 37, 6, 857-869, (2001) · Zbl 0989.93082
[7] Lakshmikantham, V.; Leela, S.; Martynyuk, A.A., Stability analysis of nonlinear systems, (1989), Marcel Dekker New York · Zbl 0676.34003
[8] Lakshmikantham, V.; Trigiante, D., Theory of difference equations: numerical methods and applications, (1988), Academic Press Inc New York · Zbl 0683.39001
[9] LaSalle, J.P., The stability and control of discrete process, (1986), Springer New York · Zbl 0606.93001
[10] Lin, Y.; Sontag, E.D.; Wang, Y., A smooth converse Lyapunov theorem for robust stability, SIAM J. control optim., 34, 124-160, (1996) · Zbl 0856.93070
[11] Mousa, M.S.; Miller, R.K.; Michel, A.N., Stability analysis of hybrid composite dynamical systems: descriptions involving operators and difference equations, IEEE trans. automat. control, 31, 603-615, (1986) · Zbl 0618.93058
[12] Nijmeijer, H.; van der Schaft, A., Nonlinear dynamical control systems, (1990), Springer New York · Zbl 0701.93001
[13] Ortega, J.M., Stability of difference equations and convergence of iterative processes, SIAM J. numer. anal., 10, 268-282, (1973) · Zbl 0253.65054
[14] Rudin, W., Principles of mathematical analysis, (1976), McGraw-Hill New York · Zbl 0148.02903
[15] Sontag, E.D., Comments on integral variants of input-to-state stability, Systems control lett., 34, 93-100, (1998) · Zbl 0902.93062
[16] Sontag, E.D.; Wang, Y., New characterizations of the input to state stability property, IEEE trans. automat. control, 41, 1283-1294, (1996) · Zbl 0862.93051
[17] Stuart, A.; Humphries, A., Dynamical systems and numerical analysis, (1996), Cambridge University Press Cambridge · Zbl 0869.65043
[18] Sugiyama, S., Comparison theorems on difference equations, Bull. sci. eng. res. lab. waseda univ., 47, 77-82, (1970)
[19] Tsinias, J., Versions of Sontag’s “input to state stability conditions” and the global stabilizability problem, SIAM J. control optim., 31, 928-941, (1993) · Zbl 0788.93076
[20] J. Tsinias, S. Kotsios, N. Kalouptsidis, Topological dynamics of discrete-time systems, in: M.A. Kaashoek, J.H. van Schuppen, A.C.M. Ran (Eds.), Robust Control of Linear Systems and Nonlinear Control, Proceedings of International Symposium MTNS-89, Vol. II, Birkhauser, Boston, 1990, pp. 457-463. · Zbl 0735.93068
[21] Xie, L.L.; Guo, L., How much uncertainty can be dealt with by feedback, IEEE trans. automat. control, 45, 2203-2217, (2000) · Zbl 0989.93052
[22] Zubov, V.I., Methods of A.M. Lyapunov and their application (English edition), (1964), P. Noordhoff Groningen, The Netherlands · Zbl 0115.30204
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