Kalman filtering is an optimal state estimation process applied to a dynamic system that involves random perturbations. More precisely, the Kalman filter gives a linear, unbiased, and min­ imum error variance recursive algorithm to optimally estimate the unknown state of a dynamic system from noisy data taken at discrete real-time intervals. It has been widely used in many areas of industrial and government applications such as video and laser tracking systems, satellite navigation, ballistic missile trajectory estimation, radar, and fue control. With the recent development of high-speed computers, the Kalman filter has become more use­ ful even for very complicated real-time applications. lnspite of its importance, the mathematical theory of Kalman filtering and its implications are not well understood even among many applied mathematicians and engineers. In fact, most prac­ titioners are just told what the filtering algorithms are without knowing why they work so well. One of the main objectives of this text is to disclose this mystery by presenting a fairly thor­ ough discussion of its mathematical theory and applications to various elementary real-time problems. A very elementary derivation of the filtering equations is fust presented. By assuming that certain matrices are nonsingular, the advantage of this approach is that the optimality of the Kalman filter can be easily understood. Of course these assump­ tions can be dropped by using the more well known method of orthogonal projection usually known as the innovations approach.

Kalman filtering is an optimal state estimation process applied to a dynamic system that involves random perturbations. More precisely, the Kalman filter gives a linear, unbiased, and min­ imum error variance recursive algorithm to optimally estimate the unknown state of a dynamic system from noisy data taken at discrete real-time intervals. It has been widely used in many areas of industrial and government applications such as video and laser tracking systems, satellite navigation, ballistic missile trajectory estimation, radar, and fue control. With the recent development of high-speed computers, the Kalman filter has become more use­ ful even for very complicated real-time applications. lnspite of its importance, the mathematical theory of Kalman filtering and its implications are not well understood even among many applied mathematicians and engineers. In fact, most prac­ titioners are just told what the filtering algorithms are without knowing why they work so well. One of the main objectives of this text is to disclose this mystery by presenting a fairly thor­ ough discussion of its mathematical theory and applications to various elementary real-time problems. A very elementary derivation of the filtering equations is fust presented. By assuming that certain matrices are nonsingular, the advantage of this approach is that the optimality of the Kalman filter can be easily understood. Of course these assump­ tions can be dropped by using the more well known method of orthogonal projection usually known as the innovations approach.