Extended differential balancing for nonlinear dynamical systems

In this paper, we construct extended balancing theory for nonlinear systems in the contraction framework. At first, we introduce the concept of the extended differential observability Gramian and inverse of the extended differential controllability Gramian for nonlinear dynamical systems and show their correspondence with generalized differential Gramians. We also provide how extended differential balancing can be utilized for model reduction to get a smaller
apriori error bound in comparison with generalized differential balancing. We illustrate the results with an example of a mass-spring-damper system considering friction.