KALMAN FILTER TWO-STATE KALMAN FILTER Up to now we have used a deterministic description for the target motion. Specifically, we have assumed a target having a constant-velocity motion as given by _ x nþ1 ¼ x n þ T x n _ _ x nþ1 ¼ x n ð1:1-1aÞ ð1:1-1bÞ In the real world the target will not have a constant velocity for all time. There is actually uncertainty in the target trajectory, the target accelerating or turning at any given time. Kalman allowed for this uncertainty in the target motion by adding a random component to the target dynamics [19,. | Tracking and Kalman Filtering Made Easy. Eli Brookner Copyright 1998 John Wiley Sons Inc. ISBNs 0-471-18407-1 Hardback 0-471-22419-7 Electronic 2 KALMAN FILTER TWO-STATE KALMAN FILTER Up to now we have used a deterministic description for the target motion. Specifically we have assumed a target having a constant-velocity motion as given by Xn 1 Xn TX n x n 1 X n In the real world the target will not have a constant velocity for all time. There is actually uncertainty in the target trajectory the target accelerating or turning at any given time. Kalman allowed for this uncertainty in the target motion by adding a random component to the target dynamics 19 20 . For example a random component un could be added to the target velocity as indicated by the following equations for the target dynamics Xn 1 Xn TX n X n 1 X n Un where un is a random change in velocity from time n to time n 1. We assume un is independent from n to n 1 for all n and that it has a variance aU. Physically un represents a random-velocity jump occurring just prior to the n 1 observation. We now have a system dynamics model with some randomness. This model is called the constant-velocity trajectory model with a random-walk velocity. 64 TWO-STATE KALMAN FILTER 65 The random-velocity component un is sized to account for a possible target acceleration or unexpected target turn. The random dynamics model component un in the literature goes by the names process noise 6 30 plant noise 8 29 30 driving noise 5 8 dynamics noise 119 model noise and system noise see Appendix . Let xn 1 represent the true location of the target at time n 1. Let x 1n represent an estimated predicted position of the target at time n 1 based on the measurements made up to and including time n. Kalman addressed the question of finding the optimum estimate among the class of all linear and nonlinear estimates that minimizes the mean square error Xn 1 n - xn 1 After much effort Kalman found that
