On the compensation of multivariable control systems

The matrix equation AXD - G = 0, where A, D, and G are real, analytic matrix functions of the complex variable s, arises in compensator specification for a multivariable control system. The generalized inverses of A and D may be used to solve for X if the equation has a solution, or to determine that X which minimizes AXD - G if the equation has no solution. Intermediate calculations in the first case involve s, a variable which does not appear in the solution. A nonrealizable compensator is the result in the second case. These undesirable features do not occur if a semi-inverse, other than the generalized inverse, is used. One such semi-inverse is presented.