On the nonuniqueness of singular value functions and balanced nonlinear realizations

The notion of balanced realizations for nonlinear state space model reduction problems was first introduced earlier. 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, distinct sets of singular value functions and balanced realizations are possible

On the Nonuniqueness of Singular Value Functions in Balanced Nonlinear Realizations

The notion of balanced realizations for nonlinear state space model reduction was first introduced 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 usual 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 definitions for the singular value functions and balanced realizations are possible. One method relating singular value functions from different factorizations is presented

State dependent matrices and balanced energy functions for nonlinear systems

The nonlinear extension of the balancing procedure requires the case of state dependent quadratic forms for the energy functions, i.e., the nonlinear extensions of the linear Gramians are state dependent matrices. These extensions have some interesting ambiguities that do not occur in the linear case. Namely, the choice of the state dependent matrix in the semi-quadratic form is not unique, and therefore may result in different eigenvalues. The introduction of so-called null-matrices is useful for the analysis of this problem. Furthermore, the concept of norm-preserving transformations provides further insight on these ambiguities. This paper provides a detailed analysis of this phenomenon and outlines some future directions for research