Abstract
Hamilton-Jacobi-Bellman theory is shown to provide a unified framework for nonlinear feedback control laws for special classes of nonlinear systems. These classes include Jurdjevic-Quinn type systems, as well as minimum phase systems with relative degree {1, 1, ..., 1}. Several examples are given to illustrate these results. For the controlled Lorenz equation, results obtained by Vincent and Yu are extended. Next, for spacecraft angular velocity stabilization with two torque inputs, a family of nonlinear feedback control laws that globally asymptotically stabilize angular velocity is established. Special cases of this family of control laws include generalizations of the locally stabilizing control laws of Brockett and Aeyels to global stabilization as well as the globally stabilizing control laws of Irving and Crouch and Byrnes and Isidori. Finally, the results are applied to spacecraft angular velocity stabilization with only one torque input. These last results extend control laws given by Outbib and Sallet.
Similar content being viewed by others
References
Aeyels, D., “Stabilization of a class of nonlinear systems by a smooth feedback control,”Sys. Contr. Lett., vol. 5, pp. 289–294, 1985.
Aeyels, D., and Szafranski, M., “Comments on the stabilizability of the angular velocity of a rigid body,”Sys. Contr. Lett., vol. 10, pp. 35–39, 1988.
Bacciotti, A.,Local Stabilizability of Nonlinear Control Systems, Series on Advances in Mathematics for Applied Sciences, vol. 8, World Scientific: Singapore, 1992.
Bass, R. W., and Webber, R. F. “Optimal nonlinear feedback control derived from quartic and higher-order performance criteria,”IEEE Trans. Autom. Contr., vol. AC-11, pp. 448–454, 1966.
Bernstein, D. S. “Nonquadratic cost and nonlinear feedback control,”Int. J. Robust and Nonlinear Control, vol. 3, pp. 211–229, 1993.
Brockett, R. W., “Asymptotic stability and feedback stabilization,” in R. W. Brockett, R. S. Millman and H. J. Sussmann, Eds.,Differential Geometric Control Theory, Progress in Mathematics, vol. 27, pp. 181–191, 1983.
Byrnes, C. I., and Isidori, A., “New results and examples in nonlinear feedback stabilization,”Sys. Contr. Lett., vol. 12, pp. 437–442, 1989.
Byrnes, C. I., and Isidori, A., “Asymptotic stabilization of minimum phase nonlinear systems,”IEEE Trans. Autom. Contr., vol. 36, pp. 1122–1137, 1991.
Irving, M., and Crouch, P. E., “On sufficient conditions for local asymptotic stability of nonlinear systems whose linearization is uncontrollable,”Control Theory Centre Report, No. 114, University of Warwick, 1983.
Isidori, A.,Nonlinear Control Systems, second edition, Springer-Verlag: 1989.
Jurdjevic, V., and Quinn, J. P., “Controllability and stability,”J. of Differential Equations, vol. 28, pp. 381–389, 1978.
Khalil, H. K.,Nonlinear Systems, Macmillan Publishing Co.: New York, 1992.
Massera, J. L., “Contributions to stability theory,”Ann. Math., vol. 64, pp. 182–206, 1956.
Nijmeijer H., and van der Schaft, A.,Nonlinear Dynamical Control Systems, Springer-Verlag: Berlin, 1990.
Outbib, R., and Sallet, G., “Stabilizability of the angular velocity of a rigid body revisited,”Sys. Contr. Lett., vol. 18, pp. 92–98, 1992.
Slotine, J-J. E., and Li, W.,Applied Nonlinear Control, Prentice-Hall: Englewood Cliffs, NJ, 1991.
Sontag, E. D., and Sussmann, H. J., “Further comments on the stabilizability of the angular velocity of a rigid body,”Sys. Contr. Lett., vol. 12, pp. 213–217, 1988.
Vincent, T. L., and Yu, J., “Control of a chaotic system,”Dynamics and Control, vol. 1, pp 35–52, 1991.
Wilson, F. W., Jr., “The structure of the level surfaces of a Lyapunov function,”J. of Differential Equations, vol. 3, pp. 323–329, 1967.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wan, CJ., Bernstein, D.S. Nonlinear feedback control with global stabilization. Dynamics and Control 5, 321–346 (1995). https://doi.org/10.1007/BF01968501
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01968501