Normalized coprime factorizations and balancing for unstable nonlinear systems

In this paper we first study the normalized right and left coprime factorizations of a nonlinear system from a state-space point of view. In order to do so, we introduce the notion of inner and co-inner nonlinear systems. Secondly we deal with balancing for unstable nonlinear systems. For linear systems there are several ways to approach this problem, and we generalize the ideas of balancing the normalized coprime factorization and of LQG balancing to the nonlinear case. LQG balancing can be seen as measuring the influence of a state component on a certain future and past energy function. We extend this interpretation to the nonlinear case. Furthermore we use the normalized coprime factorization to give another way of balancing the unstable nonlinear system. We give the relation between these two methods of balancing, which is similar to the relation in the linear case.