I think there’s a way more simple physical processing going on in your data,
Yes, I suspect it can be modeled as a first order or second order plus dead time model like the one found here.
I feel like this is a state space model because every source I have looked at uses K as the gain and u as the inputs and that sounds like a Kalman filter. Other sources mention something about Laplace transformations to make it easier to calculate. I’m just out of my depth with the math.
VARMAX seems to work better than anything I have tried so far. I have yet to try DFA and ETS.