Kalman Filter

The Kalman filter is the general solution to the recursive, minimised mean square estimation problem within the class of linear estimators. The Kalman filter gives a linear, unbiased, and minimum error variance recursive algorithm to estimate the unknown states of a dynamic process from noisy data taken at discrete real-time intervals. States, in this context, refer to any quantities of interest involved in the dynamic process, e.g. position velocity, chemical concentration, etc. For Gaussian random variables the Kalman filter is the optimal linear predictor-estimator and for variables of forms other than Gaussian the estimator is the best only within the class of linear estimators. The filter requires a knowledge of the second-order statistics of the noise of process being observed and of the measurement noise in order to provide the solution that minimises the mean square error between the true state and the estimate of state. Kalman filtering provides a convenient means of determining the weightings (denoted as gains) to be given to input measurement data. It also provides an estimate of the estimated state's error statistics through a covariance matrix. Hence the Kalman filter chooses the gain sequence and estimates the estimated state's accuracy in accordance with the variations (in terms of accuracy and update rate) of input data and modelled process dynamics. It should be noted that the quality of the estimation, as described through the error covariance matrix can in many cases be determined a priori, and would therefore be independent of the observations made. The Kalman filter has been used extensively for many diverse applications. For example, Kalman filtering has proved useful in navigational and guidance systems, radar tracking, sonar ranging, and satellite orbit determination. This chapter is mainly concerned with the derivation of the Kalman filter algorithm from the point of view of it being a linear observer, and with how the filter algorithm may be used in practice. As the Kalman filter is generally implemented on digital computers this chapter concerns itself with the discrete time form of the algorithm. A derivation of the Extended Kalman filter, a variation of the Kalman filter applicable to non-linear problems, is described. Two important variations of the Kalman filter are introduced to provide some indication of the its versatility. Finally three simple, but detailed examples of the calculations involved in Kalman filter cycles are presented