A Projections Approach to Kalman Filtering

Scot Free Kennedy

Abstract:

We will be filtering linear, discrete time control systems disturbed by Gaussian noise. The systems will be represented as difference equations involving random variables, and their output as a Gauss-Markov sequence. ``Gauss'' because we will generally assume the noise to be normally distributed, although the assumption is not neccesary and we will discuss it's lack below. ``Markov'' because the systems are described by ``one-step'' difference equations perturbed by white noise. Note that we assume some a priori knowledge of the system : The transition function, the statistical qualities of the involved random variables, etc. We will build our model in the Hilbert Space of second order Random Variables defined over the state space of our control system. We will characterize our task as the attempt to find the projection of the state onto the subspace generated by the observations. In brief, it's all about recursive decomposition.

(Disclaimer: all integrals are to be assumed Lebesgue, and all inverses, generalized)