×

Right-invertibility for a class of nonlinear control systems: A geometric approach. (English) Zbl 0622.93027

The paper attempts at clarifying the concept of right-invertibility (functional reproducibility) for analytic affine control systems. The approach adopted goes through the so-called ”invariant subdistribution algorithm” toward expressing conditions for the right-invertibility of a system in terms of its structure at infinity. There are two points in the paper of particular importance. The first is a demonstration that, contrary to the linear case, it only makes sense of speaking about local right-invertibility of nonlinear systems, i.e. an output signal is reproducible only over a definite (small) interval of time. The second point lies in finding necessary and sufficient conditions for the local right-invertibility of a class of affine systems satisfying assumption A1 due to Isidori [cf. A. Isidori, Algebraic and geometric methods in nonlinear control theory, Proc. Conf., Paris 1985, Math. Appl., D. Reidel Publ. Co. 29, 121-146 (1986)]. This being so, the paper answers affirmatively conjecture P4 stated in the reference above.
Reviewer: K.Tchon

MSC:

93C10 Nonlinear systems in control theory
93B27 Geometric methods
93B05 Controllability
93B25 Algebraic methods
93C15 Control/observation systems governed by ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Brockett, R. W.; Mesarovic, M. D., The reproducibility of multivariable systems, J. Math. Anal. Applic., 11, 548-563 (1965) · Zbl 0135.31301
[2] Silverman, L. M., Inversion of multivariable linear systems, IEEE Trans. Automat. Control, 14, 270-276 (1969)
[3] Sain, M. K.; Massey, J. L., Invertibility of linear time-invariant dynamical systems, IEEE Trans. Automat. Control, 14, 141-149 (1969)
[4] Hirschorn, R. M., Invertibility of control systems on Lie groups, SIAM J. Control Optim., 15, 1034-1049 (1977) · Zbl 0402.93011
[5] Hirschorn, R. M., Invertibility of nonlinear control systems, SIAM J. Control Optim., 17, 289-297 (1979) · Zbl 0417.93036
[6] Hirschorn, R. M., Invertibility of multivariable nonlinear systems, IEEE Trans. Automat Control, 24, 855-865 (1979) · Zbl 0427.93020
[7] Hirschorn, R. M., Output tracking in multivariable nonlinear systems, IEEE Trans. Automat. Control, 26, 593-595 (1981) · Zbl 0477.93010
[8] Singh, S. N., Reproducibility in nonlinear systems using dynamic compensation and output feedback, IEEE Trans. Automat. Control, 27, 955-958 (1982) · Zbl 0487.93028
[9] Singh, S. B., Generalized functional reproducibility condition for nonlinear systems, IEEE Trans. Automat. Control, 27, 958-960 (1982) · Zbl 0487.93029
[10] Nijmeijer, H., Invertibility of affine nonlinear control systems: a geometric approach, Systems Control Lett., 2, 163-168 (1982) · Zbl 0505.93030
[11] Nijmeijer, H.; Schumacher, J. M., Les systèmes non lináires à plus d’entrées que de sorties ne sont pas inversible, C. R. Acad. Sci. Paris Série I, 299, 791-794 (1984) · Zbl 0569.93033
[12] Fliess, M., On the inversion of nonlinear multivariable systems, (Mathematical Theory of Networks and Systems. Mathematical Theory of Networks and Systems, Lecture Notes in Control and Information Sciences No. 58 (1984), Springer: Springer Berlin-New York), 323-330
[13] Nijmeijer, H.; Schumacher, J. M., Zeros at infinity for affine nonlinear control systems, IEEE Trans. Automat. Control, 30, 566-573 (1985) · Zbl 0558.93042
[14] Isidori, A.; Krener, A. J.; Gori-Giorgi, C.; Monaco, S., Nonlinear decoupling via feedback: a differential geometric approach, IEEE Trans. Automat. Control, 26, 331-345 (1981) · Zbl 0481.93037
[15] Nijmeijer, H., Controlled invariance for affine control systems, Internat. J. Control, 34, 824-833 (1981) · Zbl 0467.49025
[16] Claude, D., Découplage des systm̀es: du lináire au non-lináire, (Landau, I. D., Outils et Méthodes Mathématiques pour l’Automatique, l’Analyse des Systems et al Théorie du Signal, Vol. III (1983), Editions CNRS: Editions CNRS Paris)
[17] Brc̈ker, Th.; Lander, L., Differential Germs and Catastrophes, London Mathematical Society Lecture Note Series No. 17 (1975)
[18] Isidori, A., Control of nonlinear systems via dynamic state feedback, (Proc. of Conference on Algebraic and Geometric Methods in Nonlinear Control Theory. Proc. of Conference on Algebraic and Geometric Methods in Nonlinear Control Theory, Paris (1985)), to appear · Zbl 0924.93038
[19] Sigh, S. N.; Schy, A. A., Invertibility and robust nonlinear control of robotic systems, (Proceedings of the 27th CDC. Proceedings of the 27th CDC, Las Vegas, NV (1984)) · Zbl 0579.93033
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.