Final_1.pdf

Kalman Filtering 55

Next we take a reading of the state of the system after allowing for a
fixed amount of time to elapse, the idea being that the system would now
have transitioned to the new state. The readings can be translated in system-
state terms based on a mathematical model. Similar to the prediction step,
we also estimate the error associated with our observation. The observation
along with an estimate of the error constitutes the observation step.
We now have two estimates for the states involved: one based on our
prediction and the other based on our observation. The natural next step is
then to reconcile the two state estimates, taking into account the magnitude
of the associated errors. Stating it differently, the predicted estimate is cor-
rected based on the observation. This is therefore called the correction step.
This reconciled estimate of the system state from the correction step is the
final estimate of the current system state.
The preceding process is then repeated again for the state at the next
time instance, making the Kalman filter a recursive prediction–correction
method. The preceding steps are also illustrated in the form of a diagram in
Figure 4.1.
The reader is probably now curious as to how the correction to the pre-
dicted value is effected. Let us discuss that briefly. Note that it is possible to
translate the prediction of the next state into a set of expected observations.

FIGURE 4.1 The Kalman Filtering Process.

Observation Step

Take Measurements

Prediction Step

Predict Next State

Estimate Prediction Error

Variance

Estimate Observation

Error Variance

Get Reconciled

State Estimate

+

Variance of

Estimate

Use Reconciled State Estimate for Prediction