In this paper we present a theory which characterizes LTI state-space
realizations of perturbed rational transfer function matrices. Our app
roach is to model system perturbations as sequences in the space of ra
tional matrices. First, we give a definition of convergence in the spa
ce of rational matrices which is motivated by the kinds of parameter u
ncertainties occurring in many robust control problems. A realization
theory is then established under the constraint that the realization o
f any convergent sequence of rational matrices should also be converge
nt. Next, we consider the issue of minimality of realizations and prop
ose a method for calculating the dimension of a minimal realization of
a given transfer matrix sequence. Finally, necessary and sufficient c
onditions are discussed under which a sequence of state-space systems
is a minimal realization and under which minimal realizations of the s
ame transfer function sequence are state-space equivalent. Relationshi
ps with standard algebraic system theoretic results are discussed. (C)
1994 Academic Press, Inc.