Abstract
In this paper, we consider the problem of obtaining optimal controllers which minimize a quadratic cost function for the rotational motion of a rigid body. We are not concerned with the attitude of the body and consider only the evolution of the angular velocity as described by the Euler equations. We obtain conditions which guarantee the existence of linear stabilizing optimal and suboptimal controllers. These controllers have a very simple structure.
Similar content being viewed by others
References
Scrivener, S.L., and Thomson, R. C., Survey of Time-Optimal Attitude Maneuvers, Journal of Guidance, Control, and Dynamics, Vol. 17, pp. 225–233, 1994.
Junkins, J. L., and Turner, J., Optimal Spacecraft Rotational Maneuvers, Elsevier, New York, New York, 1986.
Dixon, M.V., Edelbaum, T., Potter, J.E., and Vandervelde, W. E., Fuel Optimal Reorientation of Axisymmetric Spacecraft, Journal of Spacecraft and Rockets, Vol. 7, pp. 1345–1351, 1970.
Athans, M., Falb, P. L., and Lacoss, R. T., Time, Fuel, and Energy-Optimal Control of Nonlinear Norm-Invariant Systems, IRE Transactions on Automatic Control, Vol. 8, pp. 196–202, 1963.
Debs, A. S., and Athans, M., On the Optimal Angular Velocity Control of Asymmetrical Space Vehicle, IEEE Transactions on Automatic Control, Vol. 14, pp. 80–83, 1969.
Kumar, K. S. P., On the Optimum Stabilization of a Satellite, IEEE Transactions on Aerospace and Electronic Systems, Vol. 1, pp. 82–83, 1965.
Windeknecht, T. G., Optimal Stabilization of Rigid Body Attitude, Journal of Mathematical Analysis and Applications, Vol. 6, pp. 325–335, 1963.
Lu, W. M., and Doyle, J. C., ℋ∞-Control of Nonlinear Systems via Output Feedback: A Class of Controllers, Proceedings of the 32nd Conference on Decision and Control, San Antonio, Texas, pp. 166–171, 1993.
Van der Schaft, A. J., On a State Space Approach to Nonlinear ℋ∞-Control, Systems and Control Letters, Vol. 16, pp. 1–8, 1991.
Byrnes, C. I., Isidori, A., and Willems, J. C., Passivity, Feedback Equivalence, and the Global Stabilization of Minimum Phase Nonlinear Systems, IEEE Transactions on Automatic Control, Vol. 36, pp. 1228–1240, 1991.
Hill, D., and Moylan, P., The Stability of Nonlinear Dissipative Systems, IEEE Transactions on Automatic Control, Vol. 21, pp. 708–711, 1976.
Kailath, T., Linear Systems, Prentice-Hall, Englewood Cliffs, New Jersey, 1980.
Vidyasagar, M., Nonlinear Systems Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1978.
Alberkht, E. G., On the Optimal Stabilization on Nonlinear Systems, Journal of Applied Mathematics and Mechanics, Vol. 25, pp. 1254–1266, 1962.
Lukes, D. L., Optimal Regulation of Nonlinear Dynamical Systems, SIAM Journal on Control and Optimization, Vol. 7, pp. 75–100, 1969.
Willemstein, A. P., Optimal Regulation of Nonlinear Dynamical Systems on a Finite Interval, SIAM Journal on Control and Optimization, Vol. 15, pp. 1050–1069, 1977.
Yoshida, T., and Loparo, K. A., Quadratic Regulatory Theory for Analytic Nonlinear Systems with Additive Controls, Automatica, Vol. 25, pp. 531–544, 1989.
Rotea, M., Tsiotras, P., and Corless, M., Suboptimal Control of Rigid Body Motion with a Quadratic Cost, 3rd IFAC Symposium on Nonlinear Control Systems Design, Tahoe City, California, 1995.
Van der Schaft, A., J., ℒ2-Gain Analysis of Nonlinear Systems and Nonlinear State Feedback ℋ∞-Control, IEEE Transactions on Automatic Control, Vol. 37, pp. 770–784, 1992.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Tsiotras, P., Corless, M. & Rotea, M. Optimal Control of Rigid Body Angular Velocity with Quadratic Cost. Journal of Optimization Theory and Applications 96, 507–532 (1998). https://doi.org/10.1023/A:1022656326640
Issue Date:
DOI: https://doi.org/10.1023/A:1022656326640