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Control Lyapunov functions for homogeneous “Jurdjevic-Quinn” systems

Published online by Cambridge University Press:  15 August 2002

ludovic faubourg  and
jean-baptiste pomet
ludovic faubourg
Affiliation:
INRIA, BP. 93, 06902 Sophia Antipolis, France; [email protected].
jean-baptiste pomet
Affiliation:
INRIA, BP. 93, 06902 Sophia Antipolis, France; [email protected].
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Abstract

This paper presents a method to design explicit control Lyapunov functions for affine and homogeneous systems that satisfy the so-called “Jurdjevic-Quinn conditions”. For these systems a positive definite function V0 is known that can only be made non increasing by feedback. We describe how a control Lyapunov function can be obtained via a deformation of this “weak” Lyapunov function. Some examples are presented, and the linear quadratic situation is treated as an illustration.

Keywords

Feedback stabilizationcontrol Lyapunov functionLyapunov design.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 5 , 2000 , pp. 293 - 311
Copyright
© EDP Sciences, SMAI, 2000

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References

Aeyels, D., Stabilization of a class of nonlinear systems by smooth feedback control. Systems Control Lett. 5 (1985) 289-294. CrossRef
Artstein, Z., Stabilization with relaxed control. Nonlinear Anal. TMA 7 (1983) 1163-1173. CrossRef
A. Bacciotti, Local stabilizability of nonlinear control systems. World Scientific, Singapore, River Edge, London, Ser. Adv. Math. Appl. Sci. 8 (1992).
R.W. Brockett, Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory, edited by R.W. Brockett, R.S. Millman and H.J. Sussmann. Basel-Boston, Birkäuser (1983) 181-191.
Bupp, R.T., Bernstein, D.S. and Coppola, V.T., A benchmark problem for nonlinear control design. Internat J. Robust Nonlinear Control 8 (1998) 307-310. 3.0.CO;2-7>CrossRef
height 2pt depth -1.6pt width 23pt, Experimental, implementation of integrator back-stepping and passive nonlinear controllers on the RTAC testbed. Internat J. Robust Nonlinear Control 8 (1998) 435-457.
J.-M. Coron, L. Praly and A.R. Teel, Feedback stabilization of nonlinear system: Sufficient conditions and lyapunov and input-output techniques, in Trends in Control, a European Perspective, edited by A. Isidori. Springer-Verlag (1995) 283-348.
L. Faubourg, La déformation de fonctions de Lyapunov, Rapport de DEA d'automatique et informatique industrielle. INRIA-Université de Lille 1 (1997).
L. Faubourg and J.-B. Pomet, Strict control Lyapunov functions for homogeneous Jurdjevic-Quinn type systems, in Nonlinear Control Systems Design Symposium (NOLCOS'98), edited by H. Huijberts, H. Nijmeijer, A. van der Schaft and J. Scherpen. IFAC (1998) 823-829.
L. Faubourg and J.-B. Pomet, Design of control Lyapunov functions for ``Jurdjevic-Quinn'' systems, in Stability and Stabilization of Nonlinear Systems, edited by D. Aeyels et al. Springer-Verlag, Lecture Notes in Contr. & Inform. Sci. (1999) 137-150.
J.-P. Gauthier, Structure des Systèmes non-linéaires. Éditions du CNRS, Paris (1984).
W. Hahn, Stability of Motion. Springer-Verlag, Berlin, New-York, Grundlehren Math. Wiss. 138 (1967).
Jurdjevic, V. and Quinn, J.P., Controllability and stability. J. Differential Equations 28 (1978) 381-389. CrossRef
M. Kawski, Homogeneous stabilizing feedback laws. Control Theory and Adv. Technol. 6 (1990), 497-516.
H.K. Khalil, Nonlinear Systems. MacMillan, New York, Toronto, Singapore (1992).
J. Kurzweil, On the inversion of Ljapunov's second theorem on stability of motion. AMS Trans., Ser. II 24 (1956) 19-77.
LaSalle, J.-P., Stability theory for ordinary differential equations. J. Differential Equations 4 (1968) 57-65. CrossRef
W. Liu, Y. Chitour and E. Sontag, Remarks on finite gain stabilizability of linear systems subject to input saturation, in 32 $^{{\rm th}}$ IEEE Conf. on Decision and Control. San Antonio, USA (1993) 1808-1813.
F. Mazenc, Stabilisation de trajectoires, ajout d'intégration, commandes saturées, Thèse de doctorat. École des Mines de Paris (1989).
Morin, P., Robust stabilization of the angular velocity of a rigid body with two actuators. European J. Control 2 (1996) 51-56. CrossRef
Outbib, R. and Sallet, G., Stabilizability of the angular velocity of a rigid body revisited. Systems Control Lett. 18 (1992) 93-98. CrossRef
G. Sallet, Historique des techniques de Jurdjevic-Quinn (private communication).
R. Sépulchre, M. Jankovic and P.V. Kokotovic, Constructive Nonlinear Control. Springer-Verlag, Comm. Control Engrg. Ser. (1997).
E.D. Sontag, Feedback stabilization of nonlinear systems, in Robust control of linear systems and nonlinear control, Vol. 2 of proceedings of MTNS'89, edited by M.A. Kaashoek, J.H. van Schuppen and A. Ran. Basel-Boston, Birkhäuser (1990) 61-81.
M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1. Publish or Perish, Houston, second Ed. (1979).
Tsinias, J., Remarks on feedback stabilizability of homogeneous systems. Control Theory and Adv. Technol. 6 (1990) 533-542.
Zhao, J. and Kanellakopoulos, I., Flexible back-stepping design for tracking and disturbance attenuation. Internat J. Robust Nonlinear Control 8 (1998) 331-348. 3.0.CO;2-A>CrossRef