Chapter_11 - CHAPTER 11 11.1 To drive the square-root...

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233 CHAPTER 11 11.1 To drive the square-root information filter, we proceed as follows. Let Taking the Hermitian transpose: Hence, postmultiplying A ( n ) by A H ( n ): This result pertains to the pre-array of the square-root information filter. Consider next the post-array of the square-root information filter. Let Taking the Hermitian transpose: A n () λ 12 K H 2 n -1 λ u n x ˆ H nY n -1 K H 2 n -1 y * n 0 T 1 = A H n λ K n -1 K n -1 x ˆ n -1 0 λ u H n yn 1 = A n A H n λ K 1 n -1 + λ u n u H n λ K 1 n -1 x ˆ n -1 + λ u n y n () λ u n λ x ˆ n -1 K 1 n -1 λ x ˆ H n -1 K 1 n -1 x ˆ n -1 Y n y * n + λ u H n y * n λ u H n 1 = ) ( B n B 11 n b 21 n b 31 n 0 T b 22 n b 32 n =
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234 Hence, pre-multiplying B ( n ) by B H ( n ): Equating terms in A ( n ) A H ( n ) to corresponding terms in B H ( n ) B ( n ), we get, 1. Hence, 2. Hence, 3. Hence, B H n () B 11 H n 0 b 21 H n b 22 * n b 31 H n b 32 * n = B H n B n B 11 H n B 11 n B 11 H n b 21 n B 11 H n b 31 H n b 21 n B 11 n b 21 H n b 21 n b 22 n 2 + b 21 H n b 31 n b 22 * n b 32 n + b 31 H n B 11 n b 31 H n b 31 n b 32 * n b 22 n + b 31 H n b 31 n b 32 n 2 + = B 11 H n B 11 n λ K 1 n -1 λ u n u H n + = K 1 n K H 2 n K 12 n == B 11 H n K H 2 n = B 11 H n b 21 n λ K 1 n -1 x ˆ nY n -1 λ u n yn + = K 1 n x ˆ n +1 Y n = b 21 n K H 2 n x ˆ n +1 Y n = B 11 H n b 31 n λ u n =
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235 4. Hence, That is, 5. Hence, (1) 6. Hence, That is, But, we know that where g ( n ) is the Kalman gain and α ( n ) is the innovation. Therefore, b 31 n () λ 12 K H 2 n u n = b 31 H n b 31 n b 32 n 2 +1 = b 32 n 2 1 λ u H n K n u n = r 1 n = b 32 r n = b 21 H n b 21 n b 22 n 2 + x ˆ H nY n -1 K 1 n -1 x ˆ n -1 yn 2 + = b 22 n 2 x ˆ H n -1 K 1 n -1 x ˆ n -1 2 + = x ˆ n +1 Y n K 1 n x ˆ n +1 Y n b 31 H b b 21 n b 32 * n b 22 n + = r n b 22 n () λ u H n K n K n x ˆ n +1 Y n = b 22 n r n u H n x ˆ n +1 Y n [] = x ˆ n +1 Y n λ 1 –2 x ˆ n -1 α n g n + =
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236 But, Therefore, (2) where we have used r ( n ) = 1 + u H ( n ) K -1 ( n ) u ( n ). Final Check In a laborious way, it can be shown that Eq. (1) is satisfied exactly by the value of b 22 ( n ) defined in Eq. (2).
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Chapter_11 - CHAPTER 11 11.1 To drive the square-root...

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