Global asymptotic stabilization for controllable systems without drift. (English) Zbl 0760.93067

Summary: This paper proves that the accessibility rank condition on \(\mathbb{R}^ n\backslash\{0\}\) is sufficient to guarantee the existence of a global smooth time-varying (but periodic) feedback stabilizer, for systems without drift. This implies a general result on the smooth stabilization of nonholonomic mechanical systems, which are generically not smoothly stabilizable using time-invariant feedback.


93D20 Asymptotic stability in control theory
93B05 Controllability
Full Text: DOI


[1] R. W. Brockett, Asymptotic stability and feedback stabilization, inDifferential Geometric Control Theory (R. W. Brockett, R. S. Millman, and H. J. Sussmann, eds.), Birkhäuser, Basel, 1983. · Zbl 0528.93051
[2] G. Campion, B. d’Andréa-Novel, and G. Bastin, Modelling and state feedback control of nonholonomic mechanical systems, preprint, LAAS Louvain la Neuve, ENSMP Fontainebleau, 1990.
[3] J.-M. Coron, Linearized control systems and applications to smooth stabilization, preprint, Université Paris-Sud, 1992.
[4] J.-M. Coron and B. d’Andréa-Novel, Smooth stabilizing time-varying control laws for a class of nonlinear systems. Application to mobile robots, preprint, Université Paris-Sud and ENSMP Fontainebleau, 1991.
[5] J.-M. Coron and J.-B. Pomet, A remark on the design of time-varying stabilizing feedback laws for controllable systems without drift, preprint, Université Paris-Sud and ECNantes, 1991.
[6] J. Dieudonné,Foundations of Modern Analysis, Academic Press, New York, 1960.
[7] M. Gromov,Partial Differential Relations, Ergebnisse Mathematik 3, Folge 9, Springer-Verlag, Berlin, 1986. · Zbl 0651.53001
[8] A. Ilchmann, I. Nürnberger, and W. Schmale, Time-varying polynomial matrix systems,Internat. J. Control,40 (1984), 329-362. · Zbl 0545.93045 · doi:10.1080/00207178408933278
[9] J. Kurzweil, On the inversion of Lyapunov’s second theorem on stability of motion,Ann. Math. Soc. Transl., Ser. 2,24 (1956), 19-77.
[10] J.-B. Pomet, Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift,Systems Control Lett., to appear. · Zbl 0744.93084
[11] C. Samson, Velocity and torque feedback control of a wheeled mobile robot: Stability analysis, preprint, INRIA, Sophia-Antipolis, 1990.
[12] R. Sepulchre, private communication, 1991.
[13] L. M. Silverman and H. E. Meadows, Controllability and observability in time variable linear systems,SIAM J. Control,5 (1967), 64-73. · Zbl 0163.11001 · doi:10.1137/0305005
[14] E. D. Sontag, Finite-dimensional open-loop control generators for non-linear systems,Internat. J. Control,47 (1988), 537-556. · Zbl 0641.93035 · doi:10.1080/00207178808906030
[15] E. D. Sontag, Feedback stabilization of nonlinear systems, inRobust Control of Linear Systems and Nonlinear Control (M. A. Kaashoek, J. H. van Schuppen, and A. C. M. Ran, eds.), Birkhäuser, Cambridge, MA, 1990, pp. 61-81. · Zbl 0735.93063
[16] E. D. Sontag,Mathematical Control Theory: Deterministic Finite Dimensional Systems, Springer-Verlag, New York, 1990. · Zbl 0703.93001
[17] E. D. Sontag and H. J. Sussmann, Remarks on continuous feedback, inProceedings of the 19th IEEE Conference on Decision and Control, Albuquerque, NM, 1980, pp. 916-921.
[18] J. Tsinias, Sufficient Lyapunov-like conditions for stabilization,Math. Control Signals Systems,2 (1989), 343-357. · Zbl 0688.93048 · doi:10.1007/BF02551276
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.