zbMATH — the first resource for mathematics

On the input-to-state stability property. (English) Zbl 1177.93003
Summary: The “input to state stability” (ISS) property provides a natural framework in which to formulate notions of stability with respect to input perturbations. In this expository paper, we review various equivalent definitions expressed in stability, Lyapunov-theoretic, and dissipation terms. We sketch some applications to the stabilization of cascades of systems and of linear systems subject to control saturation.

93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
93Dxx Stability of control systems
93C73 Perturbations in control/observation systems
Full Text: DOI Link
[1] Varaiya, P.P.; Liu, R., ‘bounded-input bounded-output stability of nonlinear time-varying differential systems’, SIAM J. control, 4, 698-704, (1966) · Zbl 0196.46101
[2] Sontag, E.D., ‘smooth stabilization implies coprine factorization’, IEEE transactions on automatic control, 34, 435-443, (1989), AC · Zbl 0682.93045
[3] Willems, J.C., ‘mechanisms for the stability and instability in feedback systems’, Proc. IEEE, 64, 24-35, (1976)
[4] Hill, D.J.; Moylan, P., ‘dissipative dynamical systems: basic input-output and state properties’, J. franklin institute, 5, 327-357, (1980) · Zbl 0451.93007
[5] Hill, D.J., ‘dissipative nonlinear systems: basic properties and stability analysis’, (), 3259-3264
[6] Sontag, E.D.; Wang, Y., ‘on characterizations of the input-to-state stability property’, systems and control letters’, (), 351-359, 24, Dec. 1994, 1994, 3438-3443
[7] Lin, Y.; Sontag, E.D.; Wang, Y., ‘A smooth converse Lyapunov theorem for robust stability’, SIAM J. control and opt, (), 1771-1775, to appear, June
[8] Sontag, E.D.; Lin, Y., ‘stabilization with respect to noncompact sets: Lyapunov characterizations and effect of bounded inputs’, (), 9-14, June
[9] Liu, W.; Chitour, Y.; Sontag, E.D., ‘remarks on finite gain stabilizability of linear systems subject to input saturation’, SIAM J. control and opt, (), 1808-1813, to appear, Dec 1993
[10] Sussmann, H.J.; Sontag, E.D.; Yang, Y., ‘stabilization of linear systems with bounded controls’, IEEE trans. autom. control, 39, 2411-2425,, (1994) · Zbl 0811.93046
[11] Teel AR. ‘A nonlinear small-gain theorem for the analysis of control systems ,vith saturation’, IEEE Trans. Autom. Ctr., to appear.
[12] Jiang, Z.-P.; Teel, A.; Praly, L., ‘small-gain theorem for ISS systems and applications’, Math of control, signals, and systems, 7, 104-130, (1994) · Zbl 0836.93054
[13] Sontag, E.D., Mathematical control theory, Deterministic finite dimensional systems, (1990), SpringerVerlag New York
[14] Tsinias, J., ‘versions of Sontag’s input to state stability condition and the global stabilizability problem’, SIAM journal on control and optimization, 31, 928-941, (1993) · Zbl 0788.93076
[15] Tsinias, J., ‘sontag’s “input to state stability condition” and global stabilization using state detection’, Systems & control letters, 20, 219-226, (1993) · Zbl 0768.93063
[16] Praly, L.; Jiang, Z.-P., ‘stabilization by output feedback for systems with ISS inverse dynamics’, Systems and control letters, 21, 19-34, (1993) · Zbl 0784.93088
[17] Sontag, E.D., “remarks on stabilization and input-tostate stability’, (), 1376-1378, Dec. 1989
[18] Brockett, R.W., (), 181-191
[19] Hahn, W., Stability of motion, (1967), Springer-Verlag New York · Zbl 0189.38503
[20] Praly L, Wang Y. ‘An equivalent definition of input-tostate stability and stabilization in spite of matched unmodelled dynamics’, submitted for publication.
[21] Lin Y. Lyapunov Function Techniques for Stabilization, PhD Thesis, Mathematics Department, Rutgers, The State University of New Jersey, New Brunswick, New Jersey, 1992.
[22] Sontag, E.D.; Wang, Y., ‘characterizing the input-to-state stability property for set stability’, (), June, to appear.
[23] Sontag, E.D.; Teel, A., ‘changing supply functions in input/state stable systems’, IEEE transactions on automatic control, (1995), to appear. · Zbl 0832.93047
[24] Safonov, M.G., Stability and robustness of multivariable feedback systems, (1980), MIT Press Cambridge, MA · Zbl 0552.93002
[25] Mareels, I.M.; Hill, O.J., ‘monotone stability of nonlinear feedback systems’, J. math. sys, Estimation and control, 2, 275-291, (1992) · Zbl 0776.93039
[26] Teel, A.R., ‘global stabilization and restricted tracking for multiple integrators with bounded controls’, Systems and control letters, 18, 165-171, (1992) · Zbl 0752.93053
[27] Byrnes, C.I.; Isidori, A., ‘new results and counterexamples in nonlinear feedback stabilization’, Systems and control letters, 12, 437-442, (1989) · Zbl 0684.93059
[28] Tsinias, J., ‘sufficient lyapunovlike conditions for stabilization’, math. of control, Signals, and systems, 2, 343-357, (1989)
[29] Chitour, Y.; Liu, W.; Sontag, E.D., ‘on the continuity and incremental-gain properties of certain saturated linear feedback loops’, (), 127-132, Robust and Nonlinear Control, to appear. Dec. 1994.
[30] Sontag, E.D.; Sussmann, H.J., ‘nonlinear output feedback design for linear systems with saturating controls’, (), 3414-3416, Dec, 1990
[31] Sontag, E.D., ‘further facts about input to state stabilization’, IEEE transactions on automatic control, 35, 473-476, (1990), AC · Zbl 0704.93056
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.