Reciprocity of Nonlinear Systems

Reciprocity of linear input-output systems is defined as symmetry of its impulse response or transfer matrix. In the famous 1972 paper by Willems [Dissipative dynamical systems, part II: Linear systems with quadratic supply rates, Arch. Ration. Mech. Anal., 45, pp. 352-393] it was shown how reciprocity can be reflected in the state space realization. Furthermore, it was shown how to combine reciprocity with passivity in order to obtain state space realizations with physically motivated properties, including relaxation systems. The current paper is concerned with the extension of this theory to the nonlinear case. Emphasis is on nonlinear reciprocal systems with a Hessian pseudo-Riemannian metric. The combination of reciprocity with passivity is elucidated from a port-Hamiltonian perspective.