Kalman Filtering

This page discusses some work on Kalman filtering.

I am very proud of this piece of work. I consider this to be one of the best pieces of work I have ever done.

Kalman filtering references

Around the time of 1977 I performed a fairly exhaustive literature search on the general subject of Kalman filtering. I poured through journals looking for articles on Kalman filtering. This endeavor lasted an entire summer. The picture above shows a collection of over 1000 index cards containing numerous book and paper references to Kalman filtering. The process of going through journals results in much learning. A very large literature exists regarding Kalman filtering.

Quite a bit of work produced a unified theoretical exposition of the theory of Kalman filtering. It is derived rigorously for the discrete case using rigorous multi-variable Gaussian statistics and for all cases including cases where the Moore-Penrose pseudoinverse must be used. All theorems and results are proved rigorously. The culmination of this work is shown in the pictures below.

An excerpt from page 44, where a critical result is stated and proved, is shown below.

... and so on.

This thesis contains no pictures, graphs, or computer simulations. At that time I knew the correctness based only on theory. Simulation was not necessary to verify it. Of course since that time I have come to rely on digital computers, like everyone.

This thesis was all written by hand, edited by Prof. Higgins, and diligently typed by Mrs. Thompson. She typed this 200+ page thesis for $1 a page, with corrections. I paid for it out of my own funds, which were extremely limited. The whole thesis looks like the p. 44 excerpt shown above.

As a project for an analog computing course, I derived and implemented an extended Kalman filter for satellite tracking. This is based on a set of equations describing orbital dynamics and observations, with implementation and simulation on an analog computer. Ironically the first simulation run resulted in operational amplifier saturation. Funny, coincidentally, the next day there was news of a satellite crash. After fixing a minor error, it worked nicely. The analog computer itself was very interesting: it used +/- 100 volt operational amplifiers, and consisted of all 1% components.

I may conduct some simulations and studies yet, perhaps using Mathematica, when and if I get time.