On the Design of Optimal Feedback Control for Systems of Second Order

ABSTRACT

A difficult but important problem in optimal control theory is the design of an optimal feedback control, i.e., the design of an optimal control as function of the phase (state) coordinates [1,2]. This problem can be solved not often. We study here the autonomous nonlinear system of second order in general form. The constraints imposed on the control input can depend on the phase (state) coordinates of the system. The goal of the control is to maximize or minimize one phase coordinate of the considered system while other takes a prescribed in advance value. In the literature, optimal control problems for the systems of second order are most frequently associated with driving both phase coordinates to a prescribed in advance state. In this statement of the problem, the optimal control feedback can be designed only for special kind of systems. In our statement of the problem, an optimal control can be designed as function of the state coordinates for more general kind of the systems. The problem of maximization or minimization of the swing amplitude is considered explicitly as an example. Simulation results are presented.

A difficult but important problem in optimal control theory is the design of an optimal feedback control, i.e., the design of an optimal control as function of the phase (state) coordinates [1,2]. This problem can be solved not often. We study here the autonomous nonlinear system of second order in general form. The constraints imposed on the control input can depend on the phase (state) coordinates of the system. The goal of the control is to maximize or minimize one phase coordinate of the considered system while other takes a prescribed in advance value. In the literature, optimal control problems for the systems of second order are most frequently associated with driving both phase coordinates to a prescribed in advance state. In this statement of the problem, the optimal control feedback can be designed only for special kind of systems. In our statement of the problem, an optimal control can be designed as function of the state coordinates for more general kind of the systems. The problem of maximization or minimization of the swing amplitude is considered explicitly as an example. Simulation results are presented.

KEYWORDS

System of Second Order, Optimal Feedback Control, Design, Swing, Rocking, Damping, Simulation

System of Second Order, Optimal Feedback Control, Design, Swing, Rocking, Damping, Simulation

Cite this paper

nullА. Formalskii, "On the Design of Optimal Feedback Control for Systems of Second Order,"*Applied Mathematics*, Vol. 1 No. 4, 2010, pp. 301-306. doi: 10.4236/am.2010.14039.

nullА. Formalskii, "On the Design of Optimal Feedback Control for Systems of Second Order,"

References

[1] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko, “The Mathematical Theory of Optimal Processes,” Wiley Interscience, New York, 1962.

[2] V. G. Boltyanskii, “Mathematical Methods of Optimal Control,” Holt, Rinehart & Winston, 1971.

[3] A. M. Formalskii, “Controllability and Stability of Systems with Restricted Control Resources,” Nauka, Moscow, 1974 (in Russian).

[4] I. A. Sultanov, “Studying the Control Processes Obeying Equations with Underdefinite Parameters,” Automation and Remote Control, No. 10, October 1980, pp. 30-41.

[5] A. G. Butkovskiy, “Phase Portrait of Control Dynamical Systems,” Kluwer, 1991.

[6] V. V. Alexandrov and V. N. Jermolenko, “On the Absolute Stability of Second-Order Systems,” Bulletin of Moscow University, Series 1, Mathematics and Mechanics, No. 5, October 1972, pp. 102-108.

[7] E. K. Lavrovskii and A. M. Formalskii, “Optimal Control of the Pumping and Damping of a Swing,” Journal of Applied Mathematics and Mechanics, Vol. 57, No. 2, April 1993, pp. 311-320.

[8] K. Magnus, “Schwingungen Eine einfurung in die theoretische behandlung von schwingungsproblemen,” D. G. Teubner Stuttgart, 1976.

[1] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko, “The Mathematical Theory of Optimal Processes,” Wiley Interscience, New York, 1962.

[2] V. G. Boltyanskii, “Mathematical Methods of Optimal Control,” Holt, Rinehart & Winston, 1971.

[3] A. M. Formalskii, “Controllability and Stability of Systems with Restricted Control Resources,” Nauka, Moscow, 1974 (in Russian).

[4] I. A. Sultanov, “Studying the Control Processes Obeying Equations with Underdefinite Parameters,” Automation and Remote Control, No. 10, October 1980, pp. 30-41.

[5] A. G. Butkovskiy, “Phase Portrait of Control Dynamical Systems,” Kluwer, 1991.

[6] V. V. Alexandrov and V. N. Jermolenko, “On the Absolute Stability of Second-Order Systems,” Bulletin of Moscow University, Series 1, Mathematics and Mechanics, No. 5, October 1972, pp. 102-108.

[7] E. K. Lavrovskii and A. M. Formalskii, “Optimal Control of the Pumping and Damping of a Swing,” Journal of Applied Mathematics and Mechanics, Vol. 57, No. 2, April 1993, pp. 311-320.

[8] K. Magnus, “Schwingungen Eine einfurung in die theoretische behandlung von schwingungsproblemen,” D. G. Teubner Stuttgart, 1976.