Nonlinear balanced realizations

Properties of sliding interval balancing for linear systems are revisited, as this notion is basic for our extension of balanced realizations to nonlinear systems. Nonlinear balancing is based upon three principles: 1) Balancing should be defined with respect to a nominal flow; 2) Only gramians defined over small time intervals should be used to preserve the accuracy of the linear perturbation model and; 3) Linearization should commute with balancing, in the sense that the linearization of a globally balanced model should correspond to the balanced linearized model in the original coordinates. Whereas it is generically possible to define a balanced framework locally, it is not possible to do so globally, and the notion of balancing was therefore relaxed, using integrating factors, to that of uncorrelatedness. As obstruction to the integrability of the (scaled) Jacobian is generic in dimensions n > 2, we focus our attention on the planar case, for which global balanced or uncorrelated realizations exist As with linear systems, the metric provided by a canonical gramian is shown to provide useful information about the dynamics of the system and the topology of the state space.