KALMAN FILTER REVISITED INTRODUCTION In Section we developed the Kalman filter as the minimization of a quadratic error function. In Chapter 9 we developed the Kalman filter from the minimum variance estimate for the case where there is no driving noise present in the target dynamics model. In this chapter we develop the Kalman filter for more general case [5, pp. 603–618]. The concept of the Kalman filter as a fading-memory filter shall be presented. Also its use for eliminating bias error buildup will be presented. Finally, the use of the Kalman filter driving noise to prevent instabilities in. | 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 18 KALMAN FILTER REVISITED INTRODUCTION In Section we developed the Kalman filter as the minimization of a quadratic error function. In Chapter 9 we developed the Kalman filter from the minimum variance estimate for the case where there is no driving noise present in the target dynamics model. In this chapter we develop the Kalman filter for more general case 5 pp. 603-618 . The concept of the Kalman filter as a fading-memory filter shall be presented. Also its use for eliminating bias error buildup will be presented. Finally the use of the Kalman filter driving noise to prevent instabilities in the filter is discussed. KALMAN FILTER TARGET DYNAMIC MODEL The target model considered by Kalman 19 20 is given by 5 p. 604 d-X t A t X t D t U t where A t is as defined for the time-varying target dynamic model given in D t is a time-varying matrix and U t is a vector consisting of random variables to be defined shortly. The term U t is known as the processnoise or forcing function. Its inclusion has beneficial properties to be indicated later. The matrix D t need not be square and as a result U t need not have the same dimension as X t . The solution to the above linear differential equation is 375 376 KALMAN FILTER REVISITED 5 p. 605 x t t t _1 x t _1 t x D x U x dx where is the transition matrix obtained from the homogeneous part of that is the differential equation without the driving-noise term D t U t which is the random part of the target dynamic model. Consequently satisfies . The time-discrete form of is given by 5 p. 606 X tn tn tn_i X tn_i V tn tn_i where V t tn-1 t X D X U X dX The model process noise U t is white noise that is E U t 0 and E U t U t0 T K t 8 t - t0 where K t is a nonnegative definite matrix dependent on time and 8 t is
