Parameter Optimization Technique

Abstract

The parameter optimization problem is an optimization problem that specifies first the functional form of the control law and asks the m-parameter vector that minimizes the given criterion function. Since the formulation of a parameter optimization problem differs from that of an optimal control problem, the maximum principle cannot be applied. It may be noted that the parameter optimization problem includes the classical problem of optimal controller setting and the synthesis of an optimal control system whose characteristics are specified partially. Settlement of inaccessible control problem by feeding back the detectable output vector also amounts to solving a parameter optimization problem.
It may be said that only some formulas through the analog computer study have been known under the name of optimal controller setting theory, and that no approach worth going with the modern control theory has yet been tried. The author presents the technique based on the gradient theory, noticing that this kind of problem is a nonlinear programming problem. The technique may be said to be modern in that it presupposes the use of a digital computer. Since the gradient vector for the linear-quadratic problem can be evaluated by solving algebraic equation exclusively, the method is especially suited for the computation with a digital computer. On the other hand, the general nonlinear problem requires integral operations m+1 times to evaluate the gradient vector: once for the differential equation of the controlled system and m times for the differential equation with respect to the first variation of the state vector. Consequently, a considerably large memory capacity is required along with a fairly long computational time. However, unlike the other approaches to the inaccessible control problems, the technique of the paper can deal with a quite general class of control problems. It should be added that the implementation of the optimal control law is quite simple and easy.