The effects of matrix scaling on the design and control of multivariable processes

The motivation behind this work was to define matrix scaling so that process conditioning and process interaction applied to steady state process gain matrices would be put on a common basis. Since units are attached to the elements of a steady state gain matrix, it is often difficult to interpret what these elements mean on a physical basis.

In order to lower condition number, both empirical scaling methods and a numerical nonlinear search technique (NLST) were used. Much of this work was based on the scaling techniques used by Prasad (1). For analysis purposes, process steady state gain matrices were arbitrarily classified into three categories depending on the number and position of zeros in the matrix. These three categories were noninteracting, partially interacting, and full.

For noninteracting matrices of dimension 2x2, 3x3, and 4x4, geometric mean scaling and NLST yielded the same minimum condition number. Full matrices of dimension 2x2 had the same minimum condition when scaled either by geometric mean scaling or by a nonlinear least squares technique. By using an analytical method based on Tomlin's algorithm for geometric mean scaling, it can be shown that the minimum condition number for full, 2x2 matrices is the same for row followed by column scaling or for column followed by row scaling. Results of scaling using other empirical techniques such as arithmetic mean or equilibration were mixed. In general, NLST was the more effective method in finding minimum condition number for all types of low order matrices. Much more work is needed to develop a framework for determining interaction for multivariable processes on a common basis.