Optimal Control Functions Described by Known Functions

Abstract

When one seeks the control under the condition that the control has to be completed in some finite time, many control functions which satisfy this condition are found. Utilizing this fact, the control may be determined as a linear combination of functions which are selected in the known groups of functions. In this paper, the method of constructing the control, in other words, the method of deciding the unknown parameters are discussed.
For constructions of controls of lumped parameter systems, in the known groups of functions, which are defined in some finite time, the number of unknown parameters included in a control function is assigned with the order of system differential equation and the number of constraint. In the boundary controls of distributed parameter systems, approximation is often necessary to determine the actual controls. The number of unknown parameters included in the control function is subjected to the number of terms of approximated function and the number of conditions of constraint, where the impulse response of the original system is approximated by some finite number of terms.
In the present method, the control is constructed as a linear combination of known functions, satisfying the constraint conditions of the finite time and others. The present method may be said to lie between the method of ideal optimal control and the method of classical feedback control. The ideal optimal control is obtained mainly seeking the optimal performance index with, from the authour's viewpoint, the state equations as constraints. On the other hand, in the classical design, the controller is constructed by using the compensating transfer functions of known forms, unlike in the persent method.