Empirical differential Gramians for nonlinear model reduction

In this paper, we present an empirical balanced truncation method for nonlinear systems whose input vector fields are constants. First, we define differential reachability and observability Gramians. They are matrix valued functions of the state trajectory (i.e. the initial state and input trajectory), and it is difficult to find them as functions of the initial state and input. The main result of this paper is to show that for a fixed state trajectory, it is possible to compute the values of these Gramians by using impulse and initial state responses of the variational system. Therefore, balanced truncation is doable along the fixed state trajectory without solving nonlinear partial differential equations, differently from conventional nonlinear balancing methods. We further develop an approximation method, which only requires trajectories of the original nonlinear systems. (C) 2021 Elsevier Ltd. All rights reserved.

Automated summary

This paper introduces an empirical balanced truncation method for nonlinear systems with constant input vector fields. By defining differential reachability and observability Gramians, it demonstrates the possibility of computing these values along a fixed state trajectory without solving nonlinear partial differential equations. The development of an approximation method based on trajectories of the original nonlinear systems is also presented.