Starting at the top and working my way down.
Good questions, by the way!
So basically state and output is modelled as gaussian distributions
that have slowly changing means (to be estimated), with additive
zero-mean gaussian noise on top (to be cancelled)?
Yes, that's correct. The KF formulation can work with other distributions, but the standard ...
You are trying to mimic what I did at time $t$. You'll be $y$ for now, say I'm $x$.
Now suppose that our are mimicking my actions with your own modification pattern (called $f$), at the very same time. Then, your location, depending on mine, is:
$$y(t) = f(x(t))\,. $$
Here, you only need the current observation $x(t)$.
Now, suppose that you are ...
This was my recent experience where I found it was absolutely necessary to use PID control versus the PI control I would typically use: I was developing a TEC (thermo-electric cooler) oven control circuit that had extremely low frequency poles (sub Hz). With a typical PI control the response time would be limited by the lowest frequency pole. Such that in my ...