tailieunhanh - Tracking and Kalman filtering made easy P9

INTRODUCTION In Section we developed a recursive least-squares growing memory-filter for the case where the target trajectory is approximated by a polynomial. In this chapter we develop a recursive least-squares growing-memory filter that is not restricted to having the target trajectory approximated by a polynomial [5. pp. 461–482]. The only requirement is that Y nÀi , the measurement vector at time n À i, be linearly related to X nÀi in the error-free situation. The Y nÀi can be made up to multiple measurements obtained at the time n À i as in () instead of a single measurement. | 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 9 GENERAL RECURSIVE MINIMUMVARIANCE GROWING-MEMORY FILTER BAYES AND KALMAN FILTERS WITHOUT TARGET PROCESS NOISE INTRODUCTION In Section we developed a recursive least-squares growing memory-filter for the case where the target trajectory is approximated by a polynomial. In this chapter we develop a recursive least-squares growing-memory filter that is not restricted to having the target trajectory approximated by a polynomial 5. pp. 461-482 . The only requirement is that Tn_i the measurement vector at time n i be linearly related to Xn in the error-free situation. The Yn i can be made up to multiple measurements obtained at the time n i as in instead of a single measurement of a single coordinate as was the case in where Yn_1 yn_1 . The Yn_i could for example be a twodimensional measurement of the target slant range and Doppler velocity. Extensions to other cases such as the measurement of three-dimensional polar coordinates of the target are given in Section and Chapter 17. Assume that at time n we have L 1 observations Yn Yn_1 . Yn L obtained at respectively times n n 1 . n L. These L 1 observations are represented by the matrix Y n of . Next assume that at some later time n 1 we have another observation Yn 1 given by Yn 1 M Xn Nn 1 Assume also that at time n we have a minimum-variance estimate of X n based on the past L 1 measurements represented by Y n . This estimate is given by with Wn given by . In turn the covariance matrix S n is given by . Now to determine the new minimum-variance estimate X 1 n 1 from the set of data consisting of Y n 260 BAYES FILTER 261 and Yn 1 one could again use and with Y n now replaced by Y n i which is Y n of with Yn 1 added to it. Correspondingly the matrices T and R n would then be appropriately changed to