Time Variant Balancing and Nonlinear Balanced Realizations

Abstract

Balancing for linear time varying systems and its application to model reduction via projection of dynamics (POD) are briefly reviewed. We argue that a generalization for balancing nonlinear systems may be expected to be based upon three sound principles: 1) Balancing should be defined with respect to a nominal flow; 2) Only Gramians defined over small time intervals should be used in order 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. The first two principles lead to local balancing, which provides useful information about the dynamics of the system and the topology of the state space. It is shown that an integrability condition generically provides an obstruction towards a notion of a globally balanced realization in the strict sense. By relaxing the conditions of "strict balancing" in various ways useful system approximations may be obtained.