Let's recap all that we have learned in this course about common filters, starting with the linnear.
Com and filter, the linnear common filter solves the estimation problem of estimating the current state
of the system by minimizing the expected error over all measurements it receives subject to the system
dynamics sharing here.
So here is the estimated error.
We want to minimize the estimated error for the expected error across all the measurements subject to
the system module dynamics here.
So you can see that these models are linear, which is why we're using the linear Matilda.
The filter assumes that all the noise in the system or any of the error in the system follows these
sets of assumptions here.
So it assumes that each of them are uncorrelated and they can be modeled as a Gaussian distribution
with zero meaning and a given covariance.
The actual common of equations used to solve the filter are shown here, so inside the prediction step,
we take the current state, we predict it forward using our process model.
We add on the effects of the controls to predict the step forward in time.
When we do this, we also want to predict the covariance forward in time.
So we use again the process model and the previous experience and predict it forward in time and add
on the uncertainty inside that prediction step.
When we get the measurement, we apply the measurement set of equations here.
So using the current state and the measurement model, we predict what the measurement should be.
We look at the innovation, the difference between the actual measurement and our predicted measurement
to calculate the innovation and then using this innovation, value and outcome and gain value here we
can correct for the estimated state based on the error as we do the correction here, we also want to
correct and update the covariance matrix using this last equation.
So we're not running the common field.
We also want to make sure the filters tuned correctly and one way is to validate it, and that is to
look at the minimum mean squared error, which is the error between the common filter state estimates
and the true states.
And we want to make this as small as possible for the whole run in the filter like we've done in the
simulation.
We want to minimize the error between the estimates and the three states.
So this is how we can validate the filter.
We're not using the filter to make sure our main squared error is as small as possible.
We want to make sure that the measurement innovation is always close to zero.
So it has zero.
So this is one of the operating conditions that the filter assumes.
So we can calculate the innovation for a series of measurements and we want to make sure that is very
close to zero.
I mean, we also want to ensure that the measurement innovation variance is close to the actual innovation
variance.
So inside the common filter, we calculate our innovation and what we want to do is to make sure the
actual innovation covariance matches what the measured innovation covariance is from the series of innovation
measurements over the time period.
So once these two conditions are met, we have a well working and true in common field which should
minimize the main square area over the whole profile.
Now for the extended calm and filter, the extended calm, I feel that solves the same problem as linear
coming theater, except now we're using a non-linear system model, dynamic equations showing here.
So instead of the linear model, we now have these general nonlinear models.
We still use all the same assumptions for the noise and error variances.
However, we've only change these equations over here for the system model.
The actual equations used to implement the common filter are very similar.
Now we use the full nonlinear system to predict forward.
However, when we do the covariance prediction forward, we're using the Jacobean of these nonlinear
functions.
So the linear approximation of the nonlinear function, instead of using the linear system model for
the innovation, we use the full measurement model to calculate the innovation.
But then for the rest of the equations here is very similar as the linear computer swapping out the
linear measurement model with the linear approximation of the Caribbean for the measurement model.
So the equation shown on this slide here, talk about the Jacobean of the different process model and
measurement model.
So the Jacobean is simply just a partial derivative of the output of the process model with respect
to the states.
And this is for the state process model Jacobean.
You can also calculate the Jacobean of the process model with respect to the noise input.
So this term over here, by taking the partial derivative with respect to the noise for the measurement
model, we have the measurement model Jacobean, which is the Jacobean of this function here with respect
to the actual state inputs to this vector over here, the measurement noise Jakobsson model is the Jacobean
of this measurement model.
But respect to the input noise, the over here.
And then lastly, for the unsent to come and filter, we want to take our state prediction augmented
with the process, modernize, to form this augmented vector, and we do the same thing for the covariance
matrix.
We augment the current covariance with the noise covariance.
The forms are going to covariance.
Then we use these two augmented matrixes to calculate the signal points according to this set of equations
here.
So we use the square root of the covariance matrix to work out the deltas of the similar points between
the main vector.
Once we have the signal points, we then use these weightings and we can use them inside the prediction
step.
So we predict forward each sigma point through the process model and then we can and then we can recover
our predicted state by taking the weighted estimate of all the single points.
And we can do the same thing for the covariance matrix.
When it comes to the measurement step, we do the same thing is that we augment the vector now with
the measurement, noise and measurement covariance.
We form the signal points again using our square root of the covariance matrix.
And then we take these augmented signal points and we put them through the nonlinear measurement model.
Once we do this, we can use the weights again to recover the predicted measurement, calculate the
innovation, and then we can form the measurement covariance matrix and across covariance matrix by
taking the weights and the signal points, again, taking the variances between the state and the measurement
or the measurement and the measurement.
Once we have all those information, then we can apply the uncertainty coming through the update step,
which again is just these series of equations here, which are just a slightly different form of the
extended of filter and linear common filter update step equations.
So in summary, the Linnear midfielder is the best linnear estimate of a linear systems, it is stable
for any initial conditions and it's stable for any error probation's.
So this is one advantage of the linnear coming.
If the system is linear, it has some very nice properties.
Now, moving on to non-linear systems, we start to look at the extended midfielder.
Now, unfortunately, this is not the best estimate of a non-linear systems because there is no best
estimate of a non-linear systems.
We can't guarantee stability and the filter can diverge because of the non-linear effects.
The state estimates must always be close to the true state or the field or diverge because we're using
linear approximations of the nonlinear system.
So overall, this is a first order approximate accuracy for the nonlinear state and covariance transformations
due to the Jacobins.
So now looking at a different form for non-linear filters, we're looking at the unscented coming filter.
So the unscented coming has very similar properties as the extended economy of Europe, except now it
is third order approximation accuracy for the nonlinear state and covariance transforms.
So we have a higher order of accuracy here, which is where the advantages of the uncertainty come a
few to come from.
And when we're using the unscented computer, it does not require the calculation of the Jacobean.
We can use a process model and measurement model directly, which can sometimes aid in the computational
complexity or at least aid in the implementation complexity.
Congratulations on completing the course of all the advance common Philadelphia since the Fusion.
I hope you have learned a lot and you have got quite a bit out of this cause.
I hope to see you in one of my other courses.