Data-Based Model Reduction for Non-Linear Systems Based on Differential Balancing (I)

In this talk we will present an extension of the standard balancing theory for nonlinear systems which is based on an analysis around equilibrium points, to the contraction framework. This extension offers computational advantages. We provide definitions for controllability and observability functions and their differential versions which can be used for simultaneous diagonalization procedures, providing a measure for importance of the states. Generalised balancing methods based on these developments provide a computationally interesting approach. In addition, we propose a data-based model reduction method based on differential balancing for nonlinear systems whose input vector fields are constants by utilizing their variational systems. The difference between controllability and reachability for the variational system is exploited for computational reasons. For a fixed state trajectory, it is possible to compute the values of the differential Gramians by using impulse and initial state responses of the variational system. Therefore, differential balanced truncation is doable along state trajectories without solving nonlinear partial differential equations. Examples illustrate the methods.