Optimum sufficient conditions for the stability of linear multivariable systems

By invoking a fixed-point theorem of functional analysis. Freeman has developed a stability criterion for linear multivariable systems. The criterion is derived by using an operator perturbation technique to try to ensure that the operator in the transformed system equation is a contraction. It is clearly desirable to be able to choose the perturbation operator so that the transformed open-loop operator is as small as possible in some sense. In the letter, we determine that perturbation operator, from the class of diagonal linear operators, that gives the smallest contraction for the transformed open-loop system operator.