The next step in the common filter is to work out how we can use measurement information to update the
current state up on the top right here, we have the measurement model that we are going to assume for
the filtering process.
So we have the measurement, said K is a function of the current state and some noise values here.
Our matrix is the measurement model matrix.
Our M matrix is the measurement model, noise sensitivity matrix.
So this is the model that we're going to assume for the linear filter.
This basically says that the measurements that we have are going to be a linear combination of the current
state corrupted by bit of noise.
We now need to derive how to update the current state prediction with the current measurement to form
the updated state prediction so you can see the difference between these two predictions, even though
they're both the same time.
Step one is without the current information from the measurement.
One is with the current information from the measurement.
This prediction includes all the measurement information up to the current time set, while this one
over here includes all the information, including the current TimeStep measurements.
Now, this was done as a way to estimate, based on the error between the measurement said OK and the
prediction measurement, which is calculated from our paychecks or our measurement model, sensitivity
matrix times our current state estimate.
We're going to call this era the innovation, so the innovation is just a difference between the current
measurement that we get from a sensor or some external source and our predicted sensor measurement.
So this is what the sensor should read if we're just using the state prediction.
And the difference between this is the innovation, which we know as why with a over top.
In the coming theater, we should have a good idea of how much noise and uncertainty there are in the
measurements, which is going to be the size or the variance of our vehicle here.
We also have a good idea of the uncertainty in our state prediction, which is just going to be the
matrix.
So what we can do is that we can actually work out how much uncertainty there is in our innovation.
So to do this, we have to calculate the innovation covariance.
And this is just a very simple process.
We know the uncertainty in X is just going to be P, so we have to transfer the uncertainty in our state
space to our innovation space using our standard covariance transformation relationship here.
So this is a convergence of the state.
We transform it from state space into our covariance space using El-Hage times, our transpose.
And we also know the covariance of a measurement that's just going to be the covariance of anois value
up here, which we've already described as just being.
Ah.
So this is going to tell us how much noise there is going to be in this innovation.
So this is going to be the covariance of the innovation.
So how much uncertainty there is in the Gaussian probability density function for our innovation?
The matrix here is how much uncertainty there is in the measurement said.
Now, looking at our measurement model up here, if we if this matrix was just doing this and we just
had additive noise V.K., then I would just be the variance of V.K. But since we have this sensitivity,
Affymetrix m here to handle in this situation, he would have to say the arc is just going to be equal
to M r and transpose.
So overall, we have different definitions here.
I said is going to be the measurement, Victor, I said that is a predicted measurement vector, which
is just going to be times X and Y.
Tilde is going to be the innovation.
So that's going to be the error between our real measurement and predictive measurement and our metrics
is going to be our innovation covariance.
Now, the state is updated based on the size of the innovation using the common game Matrix K., which
itself is formed as a ratio between the innovation uncertainty and the current state uncertainty.
So this is basically saying if a really certain in the current state, then we should trust that more
than the measurement.
However, if we're really uncertain in our measurement, then we should trust that one more than the
current state.
So basically, based on the ratio between the measurement uncertainty and a current state uncertainty,
we work out a weighted estimate to work out how much we need to shift our state to update it.
And this is done by the common game matrix.
So the state estimate can be calculated using this equation.
Here we have our previous state estimate.
This is the one without the measurement information included.
This is our updated state estimate.
And we the amount we update it is based on our common gain of innovation.
So the innovation again, is the error between the measurement and the predicted measurement and the
size of that era multiplied by our common game is going to tell us how much we need to shift our measurement
by to get the updated prediction.
Now, the common game metric can be calculated from this equation here.
The common game metrics pretty much looks at the uncertainty of the state weighted as a ratio of the
uncertainty of the measurement or the innovation.
So looking at this works out how much it needs to trust the measurement or how much it needs to trust
the current state.
Now, these two equations, they should look a bit similar.
These are actually just at least squares, recursive equations that we've calculated before.
Here is the recursively squares equations.
So you can see that the state prediction equation up here is basically exactly the same state update
here.
You can see that we have the previous information, which is updated by a game Matrix K, which is multiplied
by our innovation or error between our measurement and our predicted measurement to get the update for
a current state estimate, the gain is also exactly the same.
You can see that we have these P metrics, you P metric, c, h, transpose, transpose.
And this whole term here is also just the S matrix.
So we have s inverse.
Same thing over here.
So you can see that the common filter up next step is exactly the same equations as the recursively
squares.
It holds the same properties and it's calculated in the same way.
We've worked out how to update the current state estimate.
We also need to work out how to update the current covariance.
So we've worked out the main of the Gaussian distribution for the state estimate.
We also need to work out the Gaussian distribution around that state estimate.
So the covariance.
The whole point of the state update process is to reduce the uncertainty in the estimates and improve
the accuracy so that the covariance of the estimate must change.
So to do this, we can calculate the covariance update, so basically we have the covariance of the
state estimate before the update, before the measurement is applied and we have it what it is afterwards
and the size that we changes is based on the common ghankay.
So this equation here is, again, just straight from the recursively squares equation.
You should also be noted that this term here is always going to be less than one, so we have identity
minus one.
It means that this term, multiplied by pay, is always going to be smaller.
So our updated covariance is always going to be smaller than our current covariance.
If it is not, well, then something's going wrong.
So a summary of the updates that is shown on this slide here, we have the update for the state estimate,
which updates the current state prediction to the updated state estimate based on the common gane and
the error or the innovation between the current measurement and the predicted measurement.
We also have the equation for the updated state covariance.
This is just a function of the current comingling matrix and the predicted state covariance matrix.
The common game matrix is given by this equation here, which is just a ratio of the current uncertainty
inside the current state estimate compared to the current uncertainty inside the measurement, which
is our innovation covariance here, which is calculated from this equation here.