On the nonuniqueness of balanced nonlinear realizations

The notion of balanced realizations for nonlinear state space model reduction problems was first introduced by Scherpen in 1993. Analogous to the linear case, the so called singular value functions of a system describe the relative importance of each state component from an input-output point of view. In this paper it is shown that the procedure for nonlinear balancing has some interesting ambiguities that do not occur in the linear case. Specifically, it appears that the singular value functions as currently defined are dependent on a particular factorization of the observability function. It is shown by example that in a fixed coordinate frame this factorization is not unique, and thus other distinct sets of the singular value functions and balanced realizations are possible.