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lecture_6 - 2.160 System Identification Estimation and...

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2.160 System Identification, Estimation, and Learning Lecture Notes No. 6 February 24, 2006 4.5.1 The Kalman Gain Consider the error of a posteriori estimate x ı t e t x ı t x t x ı x ı t t K t ( y t H ) x t + = 1 1 1 H t t t x ı x ı t t ) 1 K H ( x t (25) + + x t v t = t t t t t ) K t ε I K H ( + v t = t t t where ε is a priori estimation error, i.e. before assimilating the new measurement t y t . x ı t ε 1 x t (26) t t For the following calculation, let us omit the subscript t for brevity, = t ε [ T ] ε KH ε [ t ] T ε t t T T T ε + ε ε t t 2 T KH K H K K H + K t v t + e t e t v t t t t (27) + T ε ε ε ε 2 T Kv 2 v T T K + ε KH v T T K Kv H K = Let us differentiate the scalar function e t T e t with respect to matrix K by using the following matrix differentiation rules. K 11 K 1 b 1 df f } { b a = = ) ( a 1 T b K b a T f i) a a a = = i j 2 n dK K ij K K b 1 n n (28) ….. Rule 1 T T K × 1 × 1 × b K b R c R K R ii) n g c = , , , = = = c b K T + b c K T dg dK n n n n ∑∑∑ + K c K b K b K b c k m c ik k ij j ij j ik m K i = 1 j = 1 k = 1 j = 1 j = 1 im (29) …... Rule 2 1
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- Using these rules, d T d T T T T T T T ε H e e t = [ H K K ε 2 KH K v ε + Kv K v ] rule 2 dK t dK c T b T + 2 d [ ε T Kv ε KH ε ] rule 1 dK T T T T T T T T T ε = KH εε H + KH εε H 2[ v KH T + Kv ε H ] + 2 Kvv T + 2[ ε v εε H ] (3)0 T he necessary condition for the mean sq uared error of state estimate with respect to the gain matrix K is: J d t = 0 (3 1) dK aking expectation of e e , differentiating it w.r.t. K and setting it to zero yield: T t T t T T T T T T T T T [ ε 2) KH E εε H v KH Kv ε H + Kvv + ε v εε H ] = 0 (3 KH can be factored out, T T T [ T T T T T [ T KHE [ εε ] H KHE [ ε v ] v KE ε ] H + vv KE ] + E [ ε v ] E [ εε ] H = 0 (33 ) Examine the term E [ ε v T ] = E [( x ı x t ) v T ] t t t t 1 t [ ı T [ x E = t t 1 v t ] v x E T ] t t For the first term x ı t t 1 = A t 1 x ı t 1 ı + K t 1 ( y t 1 x H ı t 1 ) x t 1 = x ı t 1 t 2 t 2 H x t 1 + ] [ T v E ε 1 t v Uncorrelated with v t using (26) and (21), A x t 2 + w t 2
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