Continuity properties of balanced realizations

In this paper topological and geometrical properties of pre-balanced and balanced realizations are considered. It is shown that analytic pre-balancing coordinate transformations do exist and that the set of pre-balanced realizations forms an analytic submanifold. Explicit formulas for the tangent spaces and their dimension are derived. Continuity properties of balanced realizations are studied. For systems with more than two inputs and two outputs explicit points of discontinuity of balancing coordinate transformations are described. A certain class of balancing coordinate transformations is shown to be discontinuous, even for distinct and fixed singular values.