Passivity enforcement with relative error control

This paper introduces a new error control strategy in passivity enforcement schemes for linear lumped macromodels. We consider the general class of a posteriori passivity enforcement algorithms based on Hamiltonian matrix perturbation. Standard available formulations preserve the accuracy during passivity enforcement using special matrix norms associated to the controllability Gramian of the macromodel. This procedure leads to absolute error control. On the other hand, it is well known that relative error control in the macromodel is sometimes preferable, especially for structures that are characterized by small coupling coefficients or high dynamic range in their responses. Here, we present a frequency-weighting scheme leading to the definition of a modified Gramian that, when employed during passivity enforcement, effectively leads to relative error control. Several examples illustrate the reliability of the proposed technique.