Dynamic Generalized Controllability and Observability functions with applications to model reduction

The Controllability and Observability functions are used in the model reduction problem since they allow to measure states from the input/output energy point of view. For nonlinear systems these functions are the solution of a HJB and of a Lyapunov partial differential equation, respectively. Herein we introduce the notion of Dynamic Generalized Controllability and Observability functions, which are called dynamic and generalized since they make use of additional states and are such that partial differential inequalities are solved in place of equations. The proposed functions are then exploited in the model reduction by balancing for nonlinear systems.