So in the previous video, we showed you the equations that you have to use for the answer to come a
field update step, however, we want to go into a bit more detail about how we actually get those equations.
So we have seen that the state vector and coverage metrics are updated.
Using these following equations, we calculate a common gain based on this cross covariance term.
Here we use the predictor corrective form to update the current estimates based on how much error we
have in the current system.
And then we can update the uncertainty inside the system that using this equation here.
So the question becomes, where do these equations actually come from?
So this is what we're going to cover inside this video.
So we can start off by expanding the estimated error equation with the innovation correction step.
So this is the estimation error.
So this is how much error there is between the true state of the system and our predicted state of the
system.
So this time here can be broken down into the state estimates before the update as shown here.
Now we can regroup these two terms over here so these two terms can get regrouped to form this equation
here.
And then you should be pretty easy to notice that this is just the estimation error so they can simplify
this down into our exterior.
But this is before we apply the update so I can say this is after the update.
This is how much error we have once we do the correction step.
This is how much error we have before the correction step.
So we basically end up with this recursive relationship here.
Continuing on from that, we can use the definition of the posterior error covariance matrix, so that's
this equation shown here.
So this is the error covariance matrix.
So this is the estimation error squared up for all the measurements up to timestep, including times
up.
Okay, so substituting in our recursive equation that we had on the previous slide, we can form this
equation here.
So we substitute in for Alex Tilde and then we expanded out and we can form this long equation here
with the expectation of right now investigating this equation.
You should be able to see that the few things that we can simplify.
The first thing is that this term over here, this is just equal to the covariance matrix before we
do the update.
So the apriority covariance matrix.
This expectation of right down here is also very common.
This is the innovation conference, so we can just replace this by S..
So once we substitute in these two values here, we end up with this equation here.
So now this is a recursive equation that links the covariance before the update to the covariance,
after the update.
And it's only a function of these key metrics here.
Now, the common filter is a optimal filter.
So we want to find the solution for K that minimizes the trace of the updated covariance matrix.
Then this will give us the optimum solution, which minimizes the conditional mean squared error.
So this is the original problem of the common filter.
And to do this is fairly straightforward.
We want to minimize the trace of this covariance matrix.
We define the trace of the covariance matrix to be this intermediate value L..
And then we want to minimize the value l and to do that, we take the derivative of L with respect to
our common game and we want to set it to zero.
So this is what this is doing here.
Derivative of L with respect to the come again we end up with this is the derivative here.
We want to set it to zero because we want to find the point that minimizes solution.
So it's going to be the minimum minimum point.
And then once we have this equation, here is a fairly simple operation just to rearrange it to find
K.
So that's what this equation here gives us.
So the common game matrix for the uncynical to filter is just the covariance matrix for the cross covariance
between the state and the measurement multiplied by the inverse of the measurement.
Uncertainty is.
So now that we've calculate the common gain, we can use that to update the main for the U.S. economy
filter.
So the solution for the update step is simply the predictive corrective arm.
So we have the measurement innovation here.
We multiply it by the common gain to work out how much we need to update the current state by to get
a better estimate using this current information.
So this equation here is exactly the same as all the other common filters.
However, right now, to calculate the common gain, we're going to use this modified equation here.
So to work out the common gain, we're going to have to calculate this across covariance time as part
of the unscented filter update step.
Now to finalize the update, we want to look at how we can update the covariance, as we've already
seen, that the solution for the covariance update is sharing in this recursive relationship here.
So this is what we use to calculate the common gain.
However, now that we know what the common gain is, we can back into this relationship.
And then this gives us this equation here is basically a simplification.
So now we have the covariance matrix before the update, minus K times as times K transpose.
So this is how much we've minimized the uncertainty inside the system by using in the measurement and
then doing this operation here will give us the updated covariance matrix.
And of course, this updated matrix here is going to be smaller than this matrix here because we've
reduced the uncertainty inside the system.
So the derivation of the uncertainty coming up next, it should be fairly straightforward, is very
similar to the steps we use for the linnear and extending common fields.
However, we have to make a few approximations at a higher level, so we end up with a few more general
equations.
Sorry, the common thought estate update equation is just shown here and the common field covariance
update equation is just shown here.
So these equations are very straightforward compared to the extent the computer is just about how we
go about calculating these terms here and we're going to be using the unscented transform to do that.