Relative-Degree Wall Restricts Passivity-Based Stability Analysis in Inverter-Dominant Grids

What the math gives to ML

The paper identifies a structural limitation of standard passivity analysis: a nonzero passive transfer relation under the usual supply rate must have relative degree no greater than one, so high-fidelity models with electromagnetic or other strictly higher-order input-output delays cannot be certified by ordinary passivity. Its transferable asset is a design test for passive neural state-space and sequence models: inspect the local input-output relative degree before attempting to enforce a passivity certificate. The practical neural-network construction is to expose a direct or first-order input-to-output pathway, while placing additional dynamics in internal states, and to certify the resulting linearization with a storage-function/KYP inequality. A falsifiable prediction is that models whose learned linearization has relative degree two or larger will fail the standard passivity LMI regardless of training, whereas adding a feedthrough or first-order pathway will make certification feasible.