On Similarity Invariance of Balancing for Nonlinear Systems

A previously obtained balancing method for nonlinear systems is investigated on similarity in variance by generalization of the observations on the similarity invariance of the linear balanced realization theory. For linear systems it is well known that the Hankel singular values are similarity invariants. It is shown that under some additional conditions a similar statement on the similarity in variance holds for nonlinear systems. To be able to do so the concepts of local zero-state observability and local strong accessibility are considered. The local strong accessibility is an additional condition, which is needed to consider a nonlinear generalization of the Kalman decomposition. In the local coordinates that correspond to this decomposition the controllability and observability functions are investigated.