A Realization Theory for Perturbed Linear Systems

In this paper we present a theory which characterizes LTI state-space realizations of perturbed rational transfer function matrices. Our approach is to model system perturbations as sequences in the space of rational matrices. First, we give a definition of convergence in the space of rational matrices which is motivated by the kinds of parameter uncertainties occurring in many robust control problems. A realization theory is then established under the constraint that the realization of any convergent sequence of rational matrices should also be convergent. Next, we consider the issue of minimality of realizations and propose a method for calculating the dimension of a minimal realization of a given transfer matrix sequence. Finally, necessary and sufficient conditions are discussed under which a sequence of state-space systems is a minimal realization and under which minimal realizations of the same transfer function sequence are state-space equivalent. Relationships with standard algebraic system theoretic results are discussed.