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Notes_WRLS - 1 Recursive Weighted Least Square Z k = Hk...

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1 Recursive Weighted Least Square The estimate of θ can be obtained based on measurements available at time k : Z k = H k θ + ǫ k (1.1) by using the least squares technique described earlier in class. Suppose additional measurements are obtained at time k + 1: ζ k +1 = h k +1 θ + ε k +1 . (1.2) In recursive least squares, the problem we consider is how to compute the weighted least squares estimate of θ with all the measurements available at time k + 1, but without recomputing the weighted least squares solution directly, but rather by updating the estimate available at k . We start by examining the weighted least squares estimate of θ with all the measurements available at time k + 1. Let bracketleftbigg Z k ζ k +1 bracketrightbigg = bracketleftbigg H k h k +1 bracketrightbigg θ + bracketleftbigg ǫ k ε k +1 bracketrightbigg (1.3) Z k +1 = H k +1 θ + ǫ k +1 . (1.4) Use weighted least squares to find the new estimate: ˆ θ k +1 = ( H T k +1 W k +1 H k +1 ) - 1 H T k +1 W k +1 Z k +1 , (1.5) where W k +1 = bracketleftbigg W k 0 0 w k +1 bracketrightbigg . (1.6) Therefore H T k +1 W k +1 H k +1 = H T k W k H k + h T k +1 w k +1 h k +1 . (1.7) Let P k +1 = ( H T k +1 W k +1 H k +1 ) - 1 , then P - 1 k +1 = P - 1 k + h T k +1 w k +1 h k +1 . (1.8) ˆ θ k +1 = P k +1 bracketleftbig H T k W k Z k + h T k +1 w k +1 ζ k +1 bracketrightbig (1.9) = P k +1 bracketleftbig P - 1 k P k H T k W k Z k + h T k +1 w k +1 ζ k +1 bracketrightbig . (1.10) Since ˆ θ k = P k H T k W k Z k ,
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