Algebraic theory for robust stability of interconnected systems: Necessary and sufficient conditions

We consider an interconnected system So made of linear multivariable subsystems which are specified by matrix fractions with elements in a ring of stable scalar transfer functions H. Given that the kth sub-system is perturbed from Gk = NrkDk -1 to G??k = (Nrk + ??Nrk)(Dk + ??Dk)-1 and that the system So is H-stable, we derive a computationally efficient necessary and sufficient condition for the H-stability of the perturbed system. These fractional perturbations are more general than the conventional additive and multiplicative perturbations. The result is generalized to handle simultaneous perturbations of two or more subsystems.