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

We consider an interconnected system S
o
made of linear mulrivariable subsystems which are specified by matrix fractions with elements in a ring of stable scalar transfer functions
H
. Given that the
k
th subsystem is perturbed from
G_{k} = N_{rk}D_{k}^{-1}
to
\tilde{G}_{k} = (N_{rk} + \Delta N_{rk})(D_{k} + \Delta D_{k})^{-1}
and that the system S
o
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.