Tài liệu Tracking and Kalman filtering made easy P9 pdf

9 GENERAL RECURSIVE MINIMUM- VARIANCE GROWING-MEMORY FILTER (BAYES AND KALMAN FILTERS WITHOUT TARGET PROCESS NOISE) 9.1 INTRODUCTION In Section 6.3 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 (4.1-1a) instead of a single measurement of a single coordinate, as was the case in (4.1-20), where Y nÀ1 ¼½y nÀ1 . The Y nÀi could, for example, be a two- dimensional 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 16.2 and Chapter 17. Assume that at time n we have L þ 1 observations Y n , Y nÀ1 ; .; Y nÀL obtained at, respectively, times n; n À 1; .; n À L. These L þ 1 observations are represented by the matrix Y ðnÞ of (4.1-11a). Next assume that at some later time n þ 1 we have another observation Y nþ1 given by Y nþ1 ¼ MÈX n þ N nþ1 ð9:1-1Þ Assume also that at time n we have a minimum-variance estimate of X à n;n based on the past L þ 1 measurements represented by Y ðnÞ . This estimate is given by (4.1-30) with W n given by (4.5-4). In turn the covariance matrix S  à n;n is given by (4.5-5). Now to determine the new minimum-variance estimate X à nþ1;nþ1 from the set of data consisting of Y ðnÞ 260 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) and Y nþ1 , one could again use (4.1-30) and (4.5-4) with Y ðnÞ now replaced by Y ðnþ1Þ , which is Y ðnÞ of (4.1-11a) with Y nþ1 added to it. Correspondingly the matrices T and R ðnÞ would then be appropriately changed to account for the increase in Y ðnÞ to include Y nþ1 . This approach, however, has the disadvantage that it does not make use of the extensive computations carried out to compute the previously minimum-variance estimate X à n;n based on the past data Y ðnÞ . Moreover, it turns out that if Y nþ1 is independent of Y ðnÞ , then the minimum- variance estimate of X à nþ1;nþ1 can be obtained directly from Y nþ1 and X à n;n and their respective variances R nþ1 and S à n;n . This is done by obtaining the minimum-variance estimate of X à nþ1;nþ1 using Y nþ1 and X à n;n together with their variances. No use is made of the original data set Y ðnÞ . This says that the estimate X à n;n and its covariance matrix S à n;n contain all the information we need about the previous L þ 1 measurements, that is, about Y ðnÞ . Here, X à n;n and its covariance matrix are sufficient statistics for the information contained in the past measurement vector Y ðnÞ together with its covariance matrix R ðnÞ . (This is similar to the situation where we developed the recursive equations for the growing- and fading-memory filters in Sections 6.3, 7.2, and 1.2.6.) 9.2 BAYES FILTER The recursive form of the minimum variance estimate based on Y nþ1 and X à n;n is given by [5, p. 464] X  à nþ1;nþ1 ¼ X  à nþ1;n þ H  nþ1 ðY nþ1 À MX  à nþ1;n Þð9:2-1Þ where H  nþ1 ¼ S  à nþ1;nþ1 M T R À1 1 ð9:2-1aÞ S  à nþ1;nþ1 ¼½ðS  à nþ1;n Þ À1 þ M T R À1 1 M À1 ð9:2-1bÞ S  à nþ1;n ¼ ÈS  à n;n È T ð9:2-1cÞ X  à nþ1;n ¼ ÈX  à n;n ð9:2-1dÞ The above recursive filter is often referred to in the literature as the Bayes filter (this is because it can also be derived using the Bayes theorem on conditional probabilities [128].) The only requirement needed for the recursive minimum- variance filter to apply is that Y nþ1 be independent of Y ðnÞ . When another measurement Y nþ2 is obtained at a later time n þ 2, which is independent of the previous measurements, then the above equations (indexed up one) can be used again to obtain the estimate X  à nþ2;nþ2 .IfY ðnÞ and Y nþ1 are dependent, the Bayes filter could still be used except that it would not now provide the minimum- variance estimate. If the variates are reasonably uncorrelated though, the estimate should be a good one. BAYES FILTER 261 9.3 KALMAN FILTER (WITHOUT PROCESS NOISE) If we apply the inversion lemma given by (2.6-14) to (9.2-1b), we obtain after some manipulations the following equivalent algebraic equation for the recursive minimum-variance growing-memory filter estimate [5, p. 465]: X  à n;n ¼ X  à n;nÀ1 þ H  n ðY n À MX à n;nÀ1 Þð9:3-1Þ where H  n ¼ S  à n;nÀ1 M T ðR 1 þ MS à n;nÀ1 M T Þ À1 ð9:3-1aÞ S  à n;n ¼ð1 À H n MÞS  à n;nÀ1 ð9:3-1bÞ S  à n;nÀ1 ¼ ÈS  à nÀ1;nÀ1 È T ð9:3-1cÞ X  à n;nÀ1 ¼ ÈX  à nÀ1;nÀ1 ð9:3-1dÞ The preceding Kalman filter equations are the same as given by (2.4-4a) to (2.4-4j) except that the target model dynamic noise (U n or equivalently its covariance matrix Q n ) is not included. Not including the target model dynamic noise in the Kalman filter can lead to computational problems for the Kalman filter [5, Section 12.4]. This form of the Kalman filter is not generally used for this reason, and it is not a form proposed by Kalman. The Kalman filter with the target process noise included is revisited in Chapter 18. 9.4 COMPARISON OF BAYES AND KALMAN FILTERS As discussed in Sections 2.3, 2.5, and 2.6, the recursive minimum-variance growing-memory filter estimate is a weighted sum of the estimates Y nþ1 and X à nþ1;n with the weighting being done according to the importance of the two estimates; see (2.3-1), (2.5-9), and (2.6-7). Specifically, it can be shown that the recursive minimum-variance estimate can be written in the form [5, p. 385] X  à nþ1;nþ1 ¼ S  à nþ1;nþ1 ½ðS  à nþ1;n Þ À1 X  à nþ1;n þ MR À1 1 Y nþ1 ð9:4-1Þ If the covariance matrix of y nþ1 is dependent on n, then R 1 is replaced by R nþ1 . The recursive minimum-variance Bayes and Kalman filter estimates are maximum-likelihood estimates when Y nþ1 and Y ðnÞ are uncorrelated and Gaussian. All the other properties given in Section 4.5 for the minimum- variance estimate also apply. The Kalman filter has the advantage over the Bayes filter of eliminating the need for two matrix inversions in (9.2-1b), which have a size equal to the state vector X à n;n [which can be large, e.g., 10  10 for 262 GENERAL RECURSIVE MINIMUM-VARIANCE GROWING-MEMORY FILTER the example (2.4-6)]. The Kalman filter on the other hand only requires a single matrix inversion in (9.3-1a) of an order equal to the measurement vector Y nþ1 (which has a dimension 4  4) for the example of Section 2.4 where the target is measured in polar coordinates; see (2.4-7)). It is also possible to incorporate these four measurements one at a time if they are independent of each other. In this case no matrix inversion is needed. 9.5 EXTENSION TO MULTIPLE MEASUREMENT CASE In the Bayes and Kalman filters it is not necessary for Y nþ1 to be just a single measurement at time t nþ1 . The term Y nþ1 could be generalized to consist of L þ 1 measurements at L þ 1 times given by Y nþ1 ; Y n ; Y nÀ1 ; .; Y nÀLþ1 ð9:4-2Þ For this more general case we can express the above L þ 1 measurements as a vector given by Y ðnþ1Þ ¼ Y nþ1 ---- Y n ---- . . . ---- Y nÀLþ1 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ð9:4-3Þ Then from (4.1-5) through (4.1-10), (4.1-11) follows. It then immediately follows that (9.2-1) through (9.2-1d) and (9.3-1) through (9.3-1d) apply with M replaced by T of (4.1-11b) and Y nþ1 replaced by Y ðnþ1Þ of (9.4-3). EXTENSION TO MULTIPLE MEASUREMENT CASE 263 . estimate X à nþ1;nþ1 from the set of data consisting of Y ðnÞ 260 Tracking and Kalman Filtering Made Easy. Eli Brookner Copyright # 1998 John Wiley & Sons,. nþ1 and X à n;n and their respective variances R nþ1 and S à n;n . This is done by obtaining the minimum-variance estimate of X à nþ1;nþ1 using Y nþ1 and