Abstract
We consider local feedback equivalence and local weak feedback equivalence of control systems. The later equivalence is up to local coordinate changes in the state space, local feedback transformations, and state dependent changes of the time scale. We show that, under such equivalence, there are only five nonequivalent local canonical forms (some with parameters) for generic control-affine systems in the plane. For a more general class of planar systems, excluding only a class of infinite codimension, we propose a general normal form. A subclass of such systems, including all systems of codimension 3, can be brought to canonical forms. We examine stabilizability of each of these canonical forms under smooth feedback. As stabilizability under smooth feedback is invariant under weak feedback equivalence, this solves the stabilizability problem for the above mentioned class of systems.
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References
D. Aeyels, “Stabilization of a class of nonlinear systems by smooth feedback control”, Systems & Control Letters 5 (1985), 289–294.
W.M. Boothby, R. Marino, “Feedback stabilization of planar nonlinear systems”, Systems & Control Letters 12 (1989), 87–92.
B. Bonnard, “Quadratic control systems”, preprint.
R.W. Brockett, “Asymptotic stability and feedback stabilization”, in Differential Geometric Control Theory, R.W. Brockett, R.S. Millmann and H.J. Sussmann eds., Progress in Mathematics 27, Birkhäuser, Boston 1983, 181–191.
T. Bröcker, L. Lander, Differential Germs and Catastrophes, London Math. Soc. Lecture Note Series 17, Cambridge University Press, Cambridge 1975.
C.I. Byrnes and A. Isidori, “The analysis and design of nonlinear feedback systems, I,II: Zero Dynamics and Global Normal Forms, preprint.
J. Carr, Applications of Center Manifold Theory, Springer, New York 1981.
W.P. Dayawansa, C.F. Martin, and G. Knowles, Asymptotic stabilization of a class of smooth two dimensional systems, submitted.
R.B. Gardner, “Differential geometric methods interfacing control theory”, in Differential Geometric Control Theory, R.W. Brockett, R.S. Millmann and H.J. Sussmann eds., Progress in Mathematics 27, Birkhäuser, Boston 1983, 181–191.
R.B. Gardner and W.F. Shadwick, “Feedback equivalence of control systems”, System & Control Letters 8 (1987), 463–465.
B. Jakubczyk, “Equivalence and invariants of nonlinear control systems”, in Nonlinear Controllability and Optimal Control, H.J. Sussmann ed., Marcel Decker (to appear).
M. Kawski, “Stabilization of nonlinear systems in the plane”, Systems & Control Letters 12 (1989), 169–175.
R. Marino, High gain stabilization and partial feedback linearization, Proceedings 25-th CDC, Athens 1986, 209–213.
H.J. Sussmann and E.D. Sontag, “Further comments on the stabilizability of the angular velocity of a rigid body”, Systems & Control Letters 12 (1989), 213–217.
K. Tchon, On normal forms of affine systems under feedback, Proc. Conference, Nantes 1988, Lecture Notes in Control and Information Sciences, Springer 1989.
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Jakubczyk, B., Respondek, W. (1990). Feedback Equivalence of Planar Systems and Stabilizability. In: Kaashoek, M.A., van Schuppen, J.H., Ran, A.C.M. (eds) Robust Control of Linear Systems and Nonlinear Control. Progress in Systems and Control Theory, vol 4. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4484-4_43
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DOI: https://doi.org/10.1007/978-1-4612-4484-4_43
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-8839-8
Online ISBN: 978-1-4612-4484-4
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