Identification of observer/Kalman filter Markov parameters - Theory and experiments

This paper discusses an algorithm to compute the Markov parameters of an observer or Kalman filter from experimental input and output data. The Markov parameters can then be used for identification of a state-space representation, with associated Kalman or observer gain, for the purpose of controller design. The algorithm is a nonrecursive matrix version of two recursive algorithms developed in previous works for different purposes, and the relationship between these other algorithms is developed. The new matrix formulation here gives insight into the existence and uniqueness of solutions of certain equations and offers bounds on the proper choice of observer order. It is shown that if one uses data containing noise and seeks the fastest possible deterministic observer, the deadbeat observer, one instead obtains the Kalman filter, which is the fastest possible observer in the stochastic environment. The results of the paper are demonstrated in numerical studies and experiments on the Bubble space telescope.