Identity observers design for a class of port-Hamiltonian systems

We study structure-preserving observer design problem for nonlinear port-Hamiltonian systems. We propose an identity observer in the original coordinates with an affine outputinjection term to preserve the explicit port-Hamiltonian representation. Within the convergent-dynamics framework, we derive explicit gain conditions that guarantee tunable exponential rates for state reconstruction. We treat practically relevant subclasses with constant Hessian or structure matrices and extend the results to the two combined cases. We then analyze the structurepreservation and passivity properties of the observer dynamics, the induced error system, and the system-observer cascade, illustrating how this analysis paves the way for observer-based controller design. The results are illustrated on two examples, including rotor-velocity reconstruction for a permanent-magnet synchronous motor from current measurements and an electricalcircuit case study.