The Kalman filter also assumes every data point conveys a signal of change, however small. That approach inevitably leads to reading tea leaves (interpreting noise as signal).
It is more economical, in my experience, to start from the idea that unless there's been an actual consistent change in the process, any data you're getting reflects noise around the process mean. SPC is designed to tell those two cases apart.
I'm really struggling to follow along. Control theory is nothing but a set of math tools to derive controllers and filters. Just because most people stick to linear systems with gaussian noise doesn't mean you're limited to that. I'm not sure how a quality control methodology has anything to do with control theory in general.
It is more economical, in my experience, to start from the idea that unless there's been an actual consistent change in the process, any data you're getting reflects noise around the process mean. SPC is designed to tell those two cases apart.