Minimality and local state decompositions of a nonlinear state space realization using energy functions

In this paper a set of sufficient conditions is developed in terms of controllability and observability functions under which a given state space realization of a formal power series is minimal. Specifically, it is shown that positivity of these functions, in addition to a stability requirement and a few technical conditions, implies minimality. In doing so, connections are established between Hamilton-Jacobi type optimal control theory and the well known necessary and sufficient conditions for minimality in terms of Kalman type rank conditions on the accessibility and observability distributions. Using the nonlinear analogue of the Kalman decomposition, connections are then established between minimality, singular value functions, balanced realizations, and various notions of reachability and observability for nonlinear systems.