Maximum Power Transfer for Nonlinear State Space Systems
Arjan van der Schaft
TL;DR
This work generalizes the Maximum Power Transfer theorem to nonlinear state-space systems by embedding the problem in a Hamiltonian input-output formalism. It derives an optimal-load characterization from the adjoint variational system and an inverse Hamiltonian IO system $\Sigma^\times$, enabling computation of the load from a given source signal $y_S$. For linear and port-Hamiltonian or gradient families, the adjoint load retains a passive or cyclo-passive structure, yielding global minimization guarantees for the energy transfer problem. The results connect nonlinear impedance modeling with physical load design in Thévenin/Norton settings and point toward network-reduction approaches (e.g., Kron) for broader nonlinear networks.
Abstract
The classical Maximum Power Transfer theorem of linear electrical network theory is generalized to the setting of a nonlinear state space system connected to a source. This yields a state space version of the input-output operator results of Wyatt (1988). Key tool in the analysis is the formulation of a Hamiltonian input-output system, which is closely related to Pontryagin's Maximum principle. The adjoint variational system incorporated in this system defines an optimal load. The structure of such an optimal load is investigated for classes of physical systems.