ABSTRACT
Karl Gauss postulated that positive and negative errors could be penalized equally if a quadratic measure of error was employed. He also found that the best previous estimate should be updated by weighing current data according to the confidence in the accuracy of the current measurements. This chapter shows how to develop the discrete-time Kalman filter as the optimal state estimator when the estimates of randomly excited states must be made from noisy measurements. This optimality is in the least-squares sense, in that it minimizes the expected errors of the states conditional on the noisy measurements taken to date. The efficient computational technique of eigenvector decomposition is given to solve the stationary Kalman filter gain problem. The chapter examines a technique for computation of the steady-state Kalman filter gains which does not require the forward solution of the matrix difference until stationary values of the covariance matrices are reached.
