Nonlinear balancing and Mayer-Lie interpolation

Abstract

The notion of balancing for linear systems is extended to the nonlinear realm. The proposed method of 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. Obstruction to the integrability of the Jacobian is generic in dimensions, n > 2. Here we show how to obtain the global balanced realization if the Mayer-Lie conditions are satisfied, and an interpolation method by integrable functions is proposed when this is not the case. The latter thus defines pseudo-balanced realizations.