Many areas of systems deal with sets of nonlinear functions that can be handled more effectively if proper scaling is performed. When working with m functions and n variables, (m + n) scaling factors are used. A variety of approaches to scaling are reviewed. A method for establishing a scaling array for nonlinear functions is given. The array consists of m rows and (n + 1) columns, with the scaling factor for the (n + 1)-st column constrained to be unity. A particular approach to scaling, introduced by Hamming, is generalized to include weighting factors and target values for array entries. The minimum of the resulting scaling performance measure is characterized by sets of linear equations. A numerical procedure for solving for the optimal scaling factors is given for the general case, and closed-form solutions are obtained for a special case. Numerical examples are used to demonstrate benefits of the use of target values and weighting factors.

A trajectory optimization technique for multidimensional nonlinear processes is presented. Problems which are cast in a discrete-time mold are considered. The method is based on dynamic programming and employs a combination of the technique of functional approximation and the method of region-limiting strategies. The cost function at each stage is approximated by a quadratic polynomial in a region which is restricted to be of a size judged appropriate to reduce the error in the approximation. Minimal costs are evaluated at a set of points, called base points. A new control trajectory and an improved state trajectory are then generated within an extrapolation region. The iterative application of this procedure yields an optimal trajectory. Contained in the algorithm is a simple procedure which eliminates matrix inversion to determine the coefficients of the approximating polynomial. The present algorithm is applicable to problems with one bounded control action. It accounts for inequality constraints on state variables in a straightforward manner. The algorithm is applied to solve a number of trajectory optimization problems.

Equations are derived for the optimal design of a frequency divider circuit, and optimal design equations are presented for other typical pulse circuits: a timing circuit, a gating circuit and a clipping circuit. The Lagrange Multiplier Technique used in this paper has also been applied successfully for the optimal design of switching amplifiers [1]. For each of the circuits considered, the reliability and the performance of the circuit are enhanced by adjusting circuit elements so as to maximize or minimize a specified variable of the circuit when the circuit is in one of its several states; when in other of its states, however, the circuit is required to perform in accordance with specified relationships which act as constraints on the optimization procedure.