This video, we're going to go over the extended comment update, step equations, the appropriate state
assessment, our ex hat with a negative symbol can be updated with the current measurement, innovation,
information.
So our music to form the posterior state estimate are exact, plus using the following equations.
So to recap the definitions of our state vector, again, we have the a priori state vector, so our
that came with negative symbol.
This gives us the best estimate of the current state at TimeStep K without using the current information
from the current measurement.
So this is using all the measurements up to came on this one.
While the posterior state estimate our had for K with the plus symbol is what happens once we fuse in
the current TOMPSETT information.
So we're using all the information up to TimeStep K, including the current measurement.
So this should provide us a better, best estimate for the current timestep state vector compared to
the a priori state estimate, which does not include the current information.
So we can see on our common for the update.
So the questions here, we multiply our innovation vector you by our common field, the game matrix
k so it shows us how much we have to update our previous best estimate to form the better estimate for
the current timestep.
And the common game, OK, Matrix is just calculated with this set of equations here.
So this set of equations is pretty much the same as linear computer, except of using the matrix for
the linear measurement model.
We're now using the Caribbean or the sensor model because we have a nonlinear system.
The poor state Eric Covariance Matrix, Alpay K for negative symbol can also be updated with the current
same information to form the posterior state estimate covariance matrix, our piqué plus symbol using
the following equations.
So we have the common field of covariance update equation shown here.
So, again, looking at our definitions of the covariance matrix for the April and the posterior eye,
we have the covariance matrix for our negative symbol up here is the covariance or the amount of error,
uncertainty, given all the information up to TimeStep K minus one while in a posterior, I can estimate
Alpay Plus is including the current TimeStep information into the system.
So we see that we can modify our previous covariance matrix by this term here to get the updated covariance
matrix.
So you can see that this item here is pretty much the identity matrix.
So this is basically saying one minus this parameter here, which is going to be another matrix.
Looking at that, that should always be less than one because we basically have one minus a value.
So that should be less than one.
So this covariance between this step here up previous careers and the current covariance should always
be smaller.
So this covariance should always be getting smaller.
So as we use more information into the system, our uncertainty inside the system should be shrinking.
So this is the common field covariance update equations.
And again, they're very similar as linear economy filter is that we now using the describing a matrix
for the center model here.
To highlight the differences, we can look at the measurement model update from the state, the linear
common filter is calculated using these equations here, while the extended comma filter is calculated
using these equations here.
And you can see the two equations, that's pretty much exactly the same.
So we have the same fully updated equations here and here.
Except now when we calculate our common field again matrix, we're using the linear approximation of
the nonlinear measurement model here.
While here, we're just using the linear measurement model here.
So they're basically the same is that we're just using a different measurement model.
And if we look at the covering up step, the linear culture that uses this set of equations here, while
the extended comment that it uses is set here, so the linear commenter update uses the linear system
model and our linear measurement model, while the nonlinear system for the extended coming photo,
we use a linear approximation of the measurement model.
So we use that Yukari matrix here instead of our linear matrix here.
So in summary, we can update the common the state estimate using these equations, so we have we first
calculate a common gain and we use the common gain along with the innovation, to work out how much
we need to update our previous best estimate to get the current best estimate.
And we can also update the covariance matrix as we do this by multiplying our previous covariance by
this time here, matrix time here to get our new kind of variance.