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