Abstract
In this paper, the minimum-time control problem for rest-to-rest translation of a one-dimensional second-order distributed parameter system by means of two bounded control inputs at the ends is solved. A traveling wave formulation allows the problem to be solved exactly, i.e., without modal truncation. It is found that the minimum-time control is not bang-bang, as it is for systems with a finite number of degrees of freedom. Rather, it is bang-off-bang, where a period of control inactivity in the middle of the control time interval is required for synchronization with waves propagated through the system.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
LaSalle, J. P.,The Time-Optimal Control Problem, Contributions to the Theory of Nonlinear Oscillations, Edited by L. Cesari, J. P. LaSalle, and S. Lefschetz, Princeton University Press, Princeton, New Jersey, Vol. 5, pp. 1–24, 1960.
Egorov, I. V.,Optimal Control in Banach Space, Doklady Akademii Nauk SSSR, Vol. 150, pp. 241–244, 1963 (in Russian).
Fattorini, H. O.,Time-Optimal Control of Solutions of Operational Differential Equations, SIAM Journal on Control, Vol. 2, No. 1, pp. 54–59, 1964.
Friedman, A.,Optimal Control in Banach Spaces, Journal of Mathematical Analysis and Applications, Vol. 19, No. 1, pp. 35–55, 1967.
Balakrishnan, A. V.,Optimal Control Problems in Banach Spaces, SIAM Journal on Control, Vol. 3, No. 1, pp. 152–180, 1965.
Chun, H. M., Turner, J. D., andJuang, J. N.,Frequency-Shaped Large-Angle Maneuvers, Paper No. AIAA-87-0714, AIAA 25th Aerospace Sciences Meeting, Reno, Nevada, 1987.
Chun, H. M., Turner, J. D., andJuang, J. N.,A Frequency-Shaped Programmed-Motion Approach for Flexible Spacecraft Maneuvers, Paper No. AIAA-87-0926-CP, AIAA Dynamics Specialists Conference, Monterey, California, 1987.
Thompson, R. C., Junkins, J. L., andVadali, S. R.,Near-Minimum Time Open-Loop Slewing of Flexible Vehicles, Journal of Guidance, Control and Dynamics, Vol. 12, No. 1, pp. 82–88, 1989.
Meirovitch, L., andQuinn, R. D.,Maneuvering and Vibration Control of Flexible Spacecraft, Journal of the Astronautical Sciences, Vol. 35, No. 3, pp. 301–328, 1987.
Meirovitch, L., andSharony, Y.,Optimal Vibration Control of a Flexible Spacecraft During a Minimum-Time Maneuver, Proceedings of the Sixth VPI&SU/AIAA Symposium on Dynamics and Control of Large Structures, Blacksburg, Virginia, pp. 579–601, 1987.
Meirovitch, L., andKwak, M. K.,A New Approach to the Equations of Motion for the Maneuvering and Control of Flexible Multi-Body Systems, Paper No. AIAA-91-1121-CP, AIAA Structures, Structural Dynamics and Materials Conference, Baltimore, Maryland, 1991.
Lions, J. L.,Exact Controllability, Stabilization, and Perturbations for Distributed Systems, SIAM Review, Vol. 30, No. 1, pp. 1–68, 1988.
Von Flotow, A. H.,Traveling Wave Control for Large Spacecraft Structures, Journal of Guidance, Control, and Dynamics, Vol. 9, No. 4, pp. 462–468, 1986.
Meirovitch, L.,Analytical Methods in Vibrations, Macmillan, New York, New York, pp. 330–331, 1967.
Bennighof, J. K., andBoucher, R. L.,An Investigation of the Time Required for Control of Structures, Journal of Guidance, Control and Dynamics, Vol. 12, No. 6, pp. 851–857, 1989.
Singh, G., Kabamba, P. T., andMcClamroch, N. H.,Planar, Time-Optimal, Rest-to-Rest Slewing Maneuvers of Flexible Spacecraft, Journal of Guidance, Control, and Dynamics, Vol. 12, No. 1, pp. 71–81, 1989.
Author information
Authors and Affiliations
Additional information
Communicated by L. Meirovitch
This research was supported in part by AFOSR Grant No. AFOSR-90-0297. The helpful suggestions of the referees are gratefully acknowledged.
Rights and permissions
About this article
Cite this article
Bennighof, J.K., Boucher, R.L. Exact minimum-time control of a distributed system using a traveling wave formulation. J Optim Theory Appl 73, 149–167 (1992). https://doi.org/10.1007/BF00940083
Issue Date:
DOI: https://doi.org/10.1007/BF00940083