raluk

1. Model of system

2. Internal state

3. How is optimal estimation defined

4. Covariance (statistics)

Kalman filter is optimal estimation of internal state and covariance of system based on measurements so far.

Kalman process/filter is mathematical solution to this problem as the system is evolving based on input and observable measurements. Turns out that internal state that includes both estimated value and covariance is all that is needed to fully capture internal state for such model.

It is important to undrstand, that having different model for what is optimum, uncertenty or system model, compared to what Rudolf Kalman presented, gives just different mathematical solution for this problem. Examples of different optimal solutions for different estimation models are nonlinear Kalman filters and Wiener filter.

---

I think that book on this topic from author Alex Becker is great and possibly best introduction into this topic. It has lot of examples and builds requred intuition really well. All I was missing is little more emphasis into mathematical rigor and chapter about LQG regulator, but you can find both of this in original paper by Rudolf Kalman.

For example, you can write sum of numbers less than n as:

  count(uint64_t(0)) 
   | take(n) 
   | sum<uint64_t>();