Page 865 - The Mechatronics Handbook

0066_Frame_C28 Page 5 Wednesday, January 9, 2002 7:19 PM

− ()
By definition Ε[v k ] = 0 and E[dx k ] = 0 by assumption of unbiased estimation. Therefore, the updated
state estimation error is unbiased
+ () =
E dx k 0 (28.24)

only if

∗
K k + K k H k – I = 0 (28.25)
Substitution of Eq. (28.25) into Eq. (28.22) results in an expression for the updated state estimate

+ () − ()  − () 
x ˆ k = x ˆ k + K k  z k – H k x ˆ k  (28.26)

with estimation error
+ () = ( IK k H k )dx k +
− ()
dx k – K k v k (28.27)
The post-measurement error covariance in Eq. (28.18) may be expanded to
+ () = ( IK k H k )P k ( IK k H k ) + T
− ()
T
P k – – K k R k K k (28.28)
by substitution of Eq. (28.27) and applying the conditions of uncorrelated process and measurement
noise, zero mean measurement noise, and the definition of the pre-measurement state estimation error
covariance. At this point, only the requirement that the Kalman filter be an unbiased estimator has been
satisfied, so now we will select the Kalman gain K k that delivers the minimum summed variance on the
post-measurement state estimation error. In other words, we seek the gain that will minimize

J k = trace P k + () (28.29)

The necessary condition for minimality of J k is that its partial derivative with respect to the Kalman
gain is zero. By employing the following relationship
∂
------- trace ABA )[ ( T ] = 2AB (28.30)
∂A
where B is a symmetric matrix, on the components of Eq. (28.28) with respect to K k results in

− ()
T
( IK k H k )P k H k + K k R k = 0 (28.31)
–
The optimal gain (the Kalman gain) is therefore
− () T − () T – 1
K k = P k H k H k P k H k + R k (28.32)

which is sometimes written as

− () T −1
K k = P k H k W k (28.33)
where the term W k is referred to as the innovations covariance
− () T
H k P k H k + R k (28.34)

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