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Tracking and Kalman filtering made easy P4

LEAST-SQUARES AND MINIMUM– VARIANCE ESTIMATES FOR LINEAR TIME-INVARIANT SYSTEMS 4.1 GENERAL LEAST-SQUARES ESTIMATION RESULTS In Section 2.4 we developed (2.4-3), relating the 1 Â 1 measurement matrix Y n to the 2 Â 1 state vector X n through the 1 Â 2 observation matrix M as given by Y n ¼ MX n þ N n ð4:1-1Þ It was also pointed out in Sections 2.4 and 2.10 that this linear time-invariant equation (i.e., M is independent of time or equivalently n) applies to more general cases that we generalize further here. Specifically we assume Y n is a 1 Â ðr. | 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 4 LEAST-SQUARES AND MINIMUMVARIANCE ESTIMATES FOR LINEAR TIME-INVARIANT SYSTEMS 4.1 GENERAL LEAST-SQUARES ESTIMATION RESULTS In Section 2.4 we developed 2.4-3 relating the 1 x 1 measurement matrix Yn to the 2 x 1 state vector Xn through the 1 x 2 observation matrix M as given by Yn MXn Nn 4.1-1 It was also pointed out in Sections 2.4 and 2.10 that this linear time-invariant equation i.e. M is independent of time or equivalently n applies to more general cases that we generalize further here. Specifically we assume Yn is a 1 x r 1 measurement matrix Xn a 1 x m state matrix and M an r 1 x m observation matrix see 2.4-3a that is Yn y 0 y 1 LjrJ n Xn X 0 t X 1 t -xm 1 t - 4.1-1a 4.1-1b 155 156 LEAST-SQUARES AND MINIMUM-VARIANCE ESTIMATES and in turn v 0 v1 Nn 4.1-1c v r n As in Section 2.4 x0 tn . xm_1 tn are the m different states of the target being tracked. By way of example the states could be the x y z coordinates and their derivatives as given by 2.4-6 . Alternately if we were tracking only a onedimensional coordinate then the states could be the coordinate x itself followed by its m derivatives that is where Xn X tn x Dx Dmx D Xn --- x t dtJ V t tn 4.1-2 4.1-2a n The example of 2.4-1a is such a case with m 1. Let m always designate the number of states of X tn or Xn then for X tn of 4.1-2 m m 1. Another example for m 2 is that of 1.3-1a to 1.3-1c which gives the equations of motion for a target having a constant acceleration. Here 1.3-1a to 1.33-1c can be put into the form of 2.4-1 with Xn Xn x n Xn 4.1-3 and 1 0 0 T 1 0 T 2 2 T 1 4.1-4 Assume that measurements such as given by 4.1-1a were also made at the L preceding times at n 1 . n L. Then the totality of L 1 measurements GENERAL LEAST-SQUARES ESTIMATION RESULTS 157 can be written as 2 Yn 3 2 MXn - Nn - Yn-1 . MXn-1 . . Nn-1 . . . . - Yn-L . -MXn-L. -Nn-L. 4-1-5 .