Empirical Differential Balancing for Nonlinear Systems

In this paper, we consider empirical balancing of a nonlinear system by using its prolonged system, which consists of the original nonlinear system and its variational system. For the prolonged system, we define differential reachability and observability Gramians, which are matrix valued functions of the state trajectory (i.e. the initial state and input trajectory) of the original system. The main result of this paper is showing 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. By using the obtained values of the Gramians, balanced truncation is doable along the fixed state trajectory without solving nonlinear partial differential equations. An example demonstrates our proposed method to compute a reduced order model along a limit cycle of a coupled van der Pol oscillator.