An Optimal Scaling Method

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.