Abstract
We recently derived a stability criterion for nonlinear systems, which can be viewed as a dual to Lyapunov’s second theorem. The criterion has a physical interpretation in terms of the stationary density of a substance that is generated in all points of the state space and flows along the system trajectories. If the stationary density is finite everywhere except at a singularity in the origin, then almost all trajectories tend towards the origin.
Here we consider consider state feedback for nonlinear systems and show that the search for a control law and density function that satisfy the convergence criterion can be stated in terms of convex optimization. The method is also applied to the problem of smooth blending of two given control laws.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
Z. Artstein. Stabilization with relaxed controls. Nonlinear Analysis TMA, 7:1163–1173, 1983.
S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory, volume 15 of Studies in Applied Mathematics. SIAM, Philadelphia, 1994.
R.W. Brockett. Asymptotic stability and feedback stabilization. In R.W. Brockett, R.S. Millman, and H.J. Sussmann, editors, Differential Geometric Control Theory, volume 27 of Progress in Mathematics. Birkhauser, Boston, 1983.
L. K. Ford and D. K. Fulkerson. Flows in Networks. Princeton University Press, Princeton, New Jersey, 1962.
W. Hahn. Theory and Applications of Lyapunov's Direct Method. Prentice-Hall, Englewood Cliffs, New Jersey, 1963.
A. Isidori. Nonlinear Control Systems. Springer-Verlag, London, 1995.
M. Krstic, I. Kanellakopoulos, and P. Kokotovich. Nonlinear and Adaptive Control Design. John Wiley & Sons, New York, 1962.
Yuri S. Ledyaev and Eduardo D. Sontag. A Lyapunov characterization of robust stabilization. Nonlinear Analysis, 37:813–840, 1999.
Laurent Praly. Personal communication.
Christophe Prieur and Laurent Praly. Uniting local and global controllers. In Proceedings of IEEE Conference on Decision and Control, pages 1214–1219, Arizona, December 1999.
A. Rantzer. A dual to Lyapunov's second theorem. Submitted for journal publication, March 2000.
A. Rantzer and M. Johansson. Piecewise linear quadratic optimal control. IEEE Trans. on Automatic Control, April 2000.
R. Vinter. Convex duality and nonlinear optimal control. SIAM J. Control and Optimization, 31(2):518–538, March 1993.
J.C. Willems. Dissipative dynamical systems, part I: General theory; part II: Linear systems with quadratic supply rates. Arch. Rational Mechanics and Analysis, 45(5):321–393, 1972.
L. C. Young. Lectures on the Calculus of Variations and Optimal Control Theory. W. B. Saunders Company, Philadelphia, Pa, 1969.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2001 Springer-Verlag London Limited
About this paper
Cite this paper
Rantzer, A. (2001). On convexity in stabilization of nonlinear systems. In: Isidori, A., Lamnabhi-Lagarrigue, F., Respondek, W. (eds) Nonlinear control in the year 2000 volume 2. Lecture Notes in Control and Information Sciences, vol 259. Springer, London. https://doi.org/10.1007/BFb0110311
Download citation
DOI: https://doi.org/10.1007/BFb0110311
Published:
Publisher Name: Springer, London
Print ISBN: 978-1-85233-364-5
Online ISBN: 978-1-84628-569-1
eBook Packages: Springer Book Archive