Exponentially Dissipative Nonlinear Dynamical Systems: a Nonlinear Extension of Strict Positive Realness

We extend the notion of dissipative dynamical systems to formalize the concept of the nonlinear analog of strict positive realness and strict bounded realness. In particular, using exponentially weighted system storage functions with appropriate exponentially weighted supply rates, we introduce the concept of exponential dissipativity. The proposed results provide a generalization of the strict positive real lemma and the strict bounded real lemma to nonlinear systems. We also provide a nonlinear analog to the classical passivity and small gain stability theorems for state space nonlinear feedback systems. These results are used to construct globally stabilizing static and dynamic output feedback controllers for nonlinear passive systems that minimize a nonlinear nonquadratic performance criterion.