×

zbMATH — the first resource for mathematics

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.

MSC:
93D30 Lyapunov and storage functions
93C55 Discrete-time control/observation systems
93C10 Nonlinear systems in control theory
93C73 Perturbations in control/observation systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[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
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.