×

Integrator backstepping for uncertain nonlinear systems with non-smooth dynamics. (English) Zbl 1403.93173

Summary: We solve the problem of global input-to-state stabilization with respect to external disturbances for a class of nonlinear systems with unknown parameters. For this class, the classical backstepping framework is not applicable and needs to be redesigned because of the following obstacles: (a) the systems under consideration are not in strict-feedback form, are not feedback linearizable and their input-output maps are not invertible, and (b) the dynamics is non-smooth and the trajectories starting from an initial point are not necessarily uniquely defined.

MSC:

93D25 Input-output approaches in control theory
93C41 Control/observation systems with incomplete information
93C10 Nonlinear systems in control theory
93B52 Feedback control
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Celikovsky, S., Local stabilization and controllability of nontriangular systems, IEEE Trans. Autom. Control, 45, 1909-1913, (2000) · Zbl 0991.93104
[2] Celikovsky, S.; Arranda-Bricaire, E., Constructive nonsmooth stabilization of triangular systems, Syst. Control Lett., 36, 21-37, (1999) · Zbl 0913.93057
[3] Clarke, F. H.; Ledyaev, Y. S.; Stern, R. J.; Wolenski, P. R., Nonsmooth Analysis and Control Theory, (1998), Springer-Verlag New York, Berlin, Heidelberg · Zbl 1047.49500
[4] Coron, J.-M.; Praly, L., Adding an integrator for the stabilization problem, Syst. Control Lett., 17, 2, 89-104, (1991) · Zbl 0747.93072
[5] Dashkovskiy, S.; Pavlichkov, S., Global uniform input-to-state stabilization of large-scale interconnections of MIMO generalized triangular form systems, Math. Control Signals Syst., 24, 135-168, (2012) · Zbl 1238.93095
[6] Dashkovskiy, S. N.; Pavlichkov, S. S., Robust stabilization of the generalized triangular form nonlinear systems with disturbances, IEEE Trans. Automat. Control, 59, 6, 1577-1582, (2014) · Zbl 1360.93618
[7] Freeman, R. A.; Krstić, M.; Kokotović, P. V., Robustness of adaptive nonlinear control to bounded uncertainties, Automatica, 34, 10, 1227-1230, (1998) · Zbl 0945.93559
[8] Kanellakopoulos, I.; Kokotović, P. V.; Morse, A. S., Systematic design of adaptive controllers for feedback linearizable systems, IEEE Trans. Autom. Control, 36, 11, 1241-1253, (1991) · Zbl 0768.93044
[9] Kojić, A.; Annaswamy, A. M., Adaptive control of nonlinearly parametrized systems with a triangular structure, Automatica, 38, 1, 115-123, (2002) · Zbl 1031.93109
[10] Kokotović, P. V.; Arcak, M., Constructive nonlinear control: a historical perspective, Automatica, 37, 5, 637-662, (2001) · Zbl 1153.93301
[11] Kolmogorov, A.; Fomin, S., Elements of Theory of Functions and Functional Analysis, (1957)
[12] Korobov, V. I., Controllability, stability of certain nonlinear systems, Differencial′nye Uravnenija, 9, 781, 614-619, (1973) · Zbl 0263.93007
[13] Korobov, V. I.; Pavlichkov, S. S., Global properties of the triangular systems in the singular case, J. Math. Anal. Appl., 342, 2, 1426-1439, (2008) · Zbl 1141.93023
[14] Korobov, V. I.; Pavlichkov, S. S.; Schmidt, W. H., Global robust controllability of the triangular integro-differential Volterra systems, J. Math. Anal. Appl., 309, 743-760, (2005) · Zbl 1140.93323
[15] Lee, A. B.; Marcus, L., Foundations of the Optimal Control Theory, (1972), Nauka Moscow · Zbl 0247.49001
[16] Lin, W.; Gong, Q., A remark on partial state feedback stabilization of cascade systems by small-gain theorem, IEEE Trans. Autom. Control, 48, 3, 497-500, (2003) · Zbl 1364.93650
[17] Lin, W.; Qian, C., Adding one power integrator: a tool for global stabilization of high-order lower triangular form systems, Syst. Control Lett., 39, 5, 339-351, (2000) · Zbl 0948.93056
[18] Pavlichkov, S.; Dashkovskiy, S.; Pang, C. K., Uniform stabilization of nonlinear systems with arbitrary switchings and dynamic uncertainties, IEEE Trans. Autom. Control, 62, 5, 2207-2222, (2017) · Zbl 1366.93521
[19] Pavlichkov, S. S.; Ge, S. S., Global stabilization of the generalized MIMO triangular systems with singular input-output links, IEEE Trans. Autom. Control, 54, 8, 1794-1806, (2009) · Zbl 1367.93532
[20] Qian, C.; Lin, W., A continuous feedback approach to global strong stabilization of nonlinear systems, IEEE Trans. Autom. Control, 46, 7, 1061-1079, (2001) · Zbl 1012.93053
[21] Sklyar, G. M.; Sklyar, K. V.; Ignatovich, S. Y., On the extension of the korobov’s class of linearizable triangular systems by nonlinear control systems of the class c^{1}, Syst. Control Lett., 54, 11, 1097-1108, (2005) · Zbl 1129.93350
[22] Sontag, E. D., Smooth stabilization implies coprime factorization, IEEE Trans. Autom. Control, 34, 4, 435-443, (1989) · Zbl 0682.93045
[23] Tsinias, J., Partial-state global stabilization for general triangular systems, Syst. Control Lett., 24, 139-145, (1995) · Zbl 0877.93093
[24] Tsinias, J., Triangular systems: a global extension of the coron-praly theorem on the existence of feedback-integrator stabilizers, Eur. J. Control, 3, 37-46, (1997) · Zbl 0882.93071
[25] Tsinias, J.; Karafyllis, I., ISS property for time-varying systems and application to partial-state feedback stabilization and asymptotic tracking, IEEE Trans. Autom. Control, 44, 11, 2179-2184, (1999) · Zbl 1136.93439
[26] Tzamtzi, M.; Tsinias, J., Explicit formulas of feedback stabilizers for a class of triangular systems with uncontrollable linearization, Syst. Control Lett., 38, 2, 115-126, (1999) · Zbl 1043.93548
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.