We are now going to have a look at how we can use the measurement innovation inside the update step
for the antenna to come and filter.
So the APRA state estimate, this expert with the bar at the top can be updated with the current measurement
innovation.
Allow me to form the A posterior, I estimate.
So this hat with the plus using the following equations.
So these are the common of update equations, so the top equation is just the normal recursive predictor,
correct characterful.
It takes the current state estimate, it adds on a scaling gain based on the innovation to update the
state.
However, for the answer coming up, we're going to calculate the gain slightly differently.
We're going to still use the inverse of the measurement innovation matrix.
However, now we're going to calculate this covariance matrix.
This is the cross covariance between the measurement and the current state.
And then for recap, the pariah state four times HepC, our expat bar for TimeStep K is the expected
value of the true state, using all the measurements up to the previous time, stepped up to K minus
one while the posteriorly state is this Alleycat plus.
At times Rapkay is the expected value, is the expected value of the true state using all the information
up to the current timestep.
Now the April state error covariance matrix Alpay minus can also be updated with the same information
to form the posteriorly state estimate covariance matrix.
Albay plus using the following equations.
So we have the previous covariance before the update.
We want to subtract off Alcaide times as times k transpose.
So this is how much uncertainty we can reduce the estimates by to form the new covariance matrix.
And again, for recap purposes, the APRA covariance will timestep k is just the covariance of the expected,
just the error covariance between the true state and estimated state using all information up to the
previous timestep so up to K minus one.
While the posteriorly covariance outputs is the expected error between the true state and estimated
dates for when we use all the information or all the measurements up to the current TimeStep K so we
can compare the process between the extended comma filter and the unseen incoming filter for the update
step for the state.
So as you would have previously seen for the extended comma filter, we use the same predictive corrective
form where we multiply the innovation by again Matrix K to get how much we need to update the current
state to get the best estimate for the current information.
And then we also use our Jacobean of the linear approximation of the measurement model to update our
gain matrix.
K Now for the unscented common filter.
The first equation operates in the very same way because it's still a pretty to correct filter how to
calculate the gain matrix.
We're using this crosscourt variance tampoe, which is across covariance between the state and the measurement.
However, we're still using our inverse over here.
So you can think of this term up here is being how much I change measurement will affect the state multiplied
by the uncertainty in the state.
So this is pretty much transforming the uncertainty into the measurement form down here.
The same thing.
This is how much uncertainty there is between the state and the measurement divided by how much error
is inside the measurement.
So this here is basically the same thing, is basically a ratio of how much uncertainty there is in
the measurement compared to how much uncertainty there is in the current state.
Now for the update step for the covariance inside the extent we can use this equation here.
So, again, this is using the Caribbean approximation of the nonlinear model, however, for the extended
that we're going to use this approximation here.
So this equation here can actually be used inside the external and filter as well.
However, inside the extended common filter, since we have this Jakobsson approximation of the system,
it is possible to simplify this equation down to this equation when we're using the Jacobean.
However, since we don't have a Jakobsson in the unscented common filter, we have to keep it at this
more higher level form as shown here.
So in summary for the unscented coming through the update step, we want to use these equations to calculate
how much we may want to change your state by.
So we calculate the common again using this covariance matrix, which is across covariance between the
state and the measurement.
And we have our update step in the correct form based on how much innovation area we have.
And then when we want to update the common field of covariance, we use this equation down here.
So we want to calculate how much we want to shrink the covariance by by how much information the center
is giving us at that time.
Step.