  In this paper, we present an empirical balanced truncation method for
nonlinear systems with linear time-invariant input vector field components.
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) of the original nonlinear system, 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. Our methods are demonstrated by an RL network along a
trajectory.
