Page 867 - The Mechatronics Handbook

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

The Continuous–Discrete Extended Kalman Filter
In applications where the reference state may quickly deviate beyond the linear region of the environment
state, the reference may be directly updated at the time of measurement update by adding the LKF filter
state to the reference in an EKF. The EKF is similar to the LKF, in that measurements are processed to
provide an estimate of the difference between the true state and reference state of the spacecraft. Also,
the EKF evaluates dynamics and measurement partials with respect to the reference state in a manner
similar to the LKF. However, the reference state about which these partials are evaluated is modified
through the addition of measurement information

˜ () + () = ˜ () − () + ()
X t k X t k x ˆ t k (28.41)
The reference state dynamics model used in the EKF formulation is given by Eq. (28.35), but the
measurement model is the discrete form given by Eq. (28.1). The filter state representing the estimated
difference between the true state and the reference state is only calculated at the time of measurement
update via dropping the previous estimate information term from Eq. (28.26):

x ˆ k = K k Z k (28.42)

where the innovation is now the actual measurement residual

˜ 
− ()
Z k = Y k – h X k ,ββ ββ,t k   (28.43)

Therefore, in the EKF there is not a separate filter state that needs to be propagated to the time of the
next measurement, as the filter state has been incorporated into the updated reference state.
As before, the error covariance at each measurement is updated by

+ () − () T T
–
–
P k = ( IK k H k )P k ( IK k H k ) + K k R k K k , (28.44)
and the EKF Kalman gain and innovations covariance are analogous to their LKF counterparts
− () −1
K k = P k H k W k (28.45)
− ()
W k = H k P k H k + R k . (28.46)
The difference between EKF operation and LKF operation is illustrated by revisiting the two-dimensional
trajectory illustration in Fig. 28.2. The reference trajectory can now be seen to respond to measurement
information availability and tracks the true environment trajectory.

Estimated/Reference
Trajectory

True Trajectory

Estimation Error

FIGURE 28.2 EKF tracking of a two-dimensional trajectory.

©2002 CRC Press LLC