Page 864 - The Mechatronics Handbook

0066_Frame_C28 Page 4 Wednesday, January 9, 2002 7:19 PM

Generally, it is assumed that the process noise and measurement noise sequences are uncorrelated so that

T = ∀ ∀
E w k v i 0 k, i (28.17)

However, the Kalman filter can be configured to operate in systems where this assumption does not apply [7].

Linear Kalman Filter Error Covariance Propagation
The propagation of the filter and reference states in the LKF were outlined in the previous section in
Eqs. (28.11) and (28.13). However, all Kalman filter formulations must also propagate a confidence metric
of the state estimate in the form of a state error covariance matrix. The state error covariance, P, is defined
as the expectation of the outer product of the estimation error vector

± () ± ±T
P k = E dx k dx k (28.18)

The state error covariance matrix is n × n and symmetric, and must remain positive definite to retain filter
stability. The mechanism for propagating the covariance can be derived by taking the covariance just before
measurement update at time t k+1

− ()
− ()
P k+1 = E dx k+1 dx k+1 (28.19)
− ()T
and substituting the estimation error and propagation definitions in Eqs. (28.4), (28.11), and (28.13) to
yield

T

+ ()
+ ()

− ()
P k+1 = E ΦΦ t k+1 ,t k ) x ˆ k – x k   x ˆ k – x k ΦΦ ΦΦ t k+1 ,t k ) + w k w k T (28.20)
(
ΦΦ(
T


 
Utilizing the definitions of process noise covariance in Eq. (28.16) and state error covariance in Eq.
(28.18) the propagation equation reduces to
− () + () T
(
(
P k+1 = ΦΦ Φ Φ t k+1 ,t k )P k ΦΦ ΦΦ t k+1 ,t k ) + Q k (28.21)
The propagation equation can be interpreted as the sum of the mapping of the previous post-update
error covariance through the system dynamics and the system process noise induced uncertainty. Thus,
process noise acts to increase the state error covariance between measurement updates.
Linear Kalman Filter Update
The linear Kalman filter (LKF) seeks an unbiased, minimum variance solution for the difference state, x k ,
by combining previous state information with available measurements. The state estimate after measure-
ment update is therefore assumed to be a linear combination of the pre-update state and the linearized
measurement information
+ () ∗ ()
−
x ˆ k = K k x ˆ k + K k z k (28.22)
Substituting Eqs. (28.4) and (28.14) into Eq. (28.22) and solving for the estimation error yields

+ () ∗ ∗ − ()
dx k = ( K k + K k H k – I)x k + K k dx k + K k v k (28.23)

©2002 CRC Press LLC