Sufficient condition for minimal realization of incrementally stable nonlinear systems based on differential energy functions

In this paper, we investigate the minimal realization of nonlinear systems in terms of differential controllability and observability functions, which play important roles in constructing a new balancing method in the contraction framework. It is shown that positivity of these functions, in addition to an incremental stability requirement and a few technical conditions, implies minimality. In doing so, connections are established between nonlinear accessibility/observability and recently proposed differential balancing method.