Extended Differential Balancing and Generalized Balancing for Nonlinear Dynamical Systems: (Extended Abstract)

In this work, we introduced extended differential balancing, which is a model reduction approach for nonlinear dynamical systems in the contraction framework. We utilized the solutions of two time-varying LMIs to arrive at a balanced realization of the associated variational system to perform the truncation of less important states of the original system. One of the main contributions of the work is to show a computationally tractable way of performing model reduction providing tighter aprioiri error bounds in comparison with generalized differential balancing. On the other hand, we introduce generalized controllability function and generalized observability function for continuous-time stable nonlinear systems. We also propose a balanced realization for nonlinear port-Hamiltonian systems in which the generalized energy functions are balanced as well as the Hamiltonian of the corresponding system is in diagonal form. Moreover, the reduced order model obtained by truncation preserves the port-Hamiltonian structure.