ABSTRACT
In the estimation problem considered in Section 12.3, it is obvious that improved estimates of xo and wo could be made by using the measurement z1 . In other words, measurements related to state 1 provide information about the state 0 and about the transition from state 0 to state 1, i.e., about the forcing vector wo . We shall call these estimates smoothing estimates and denote them by x ^ o / 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137667/ebc3ad23-d2ed-4b59-a8e1-8861837bea7d/content/eq3047.tif"/> and w ^ o / 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137667/ebc3ad23-d2ed-4b59-a8e1-8861837bea7d/content/eq3048.tif"/> to distinguish them from the estimates x ^ o https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137667/ebc3ad23-d2ed-4b59-a8e1-8861837bea7d/content/eq3049.tif"/> and w ¯ o https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137667/ebc3ad23-d2ed-4b59-a8e1-8861837bea7d/content/eq3050.tif"/> . To be least-square estimates, the quantities xo/1 and wo/1 must be the values of xo and wo that minimize the quadratic form () j = 1 2 ( x o − x ^ o ) T P o − 1 ( x o − x ^ o ) + 1 2 ( w o − w ¯ o ) T Q o − 1 ( w o − w ¯ o ) + 1 2 ( z 1 − H 1 x 1 ) T R 1 − 1 ( z 1 − H 1 x 1 ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137667/ebc3ad23-d2ed-4b59-a8e1-8861837bea7d/content/eq3051.tif"/>
