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, define differential reachability and observability Gramians. They matrix valued functions of the state trajectory (i.e. initial trajectory), it is difficult to find them as input. The main result paper show that a fixed trajectory, possible compute values these Gramians by using impulse responses variational system. Therefore, doable along without solving partial equations , differently from conventional balancing methods. We further develop approximation method, which only requires trajectories original systems.