In this video, we are going to cover the prediction step equations for the extended common filter.
We will cover all the equations in a summary form, and then in later, videos will go back and look
at the derivation of these equations.
So to start off, we begin the estimation step of the extended coming field up with the initial condition
X not which is the best estimate of the current initial state of the system.
We want to probably get Tom Ford to find our estimate at TimeStep one.
So we want to find out estimated state X1, which is just the estimated value of the current system
at timeworn, so we can use the nonlinear motion model of the system to do this.
So using the state transition matrix, if we can substitute in our current state for export our current
control vector, if you want and we can set prices will always be zero because we estimated we want
to estimate the mean to get the predicted machine, which is going to be our X1 here.
We can write this in general as this equation here.
So we basically take our state transition matrix f subject in our current state for time K minus one
substitute in our current control, our UK we set the price is one and zero and we can use this to calculate
our current state estimate ICSC for the current timestep.
So this is the prediction step or the time update step of the extent to come, if we use the process
model of the system, we want to estimate to predict the estimate a state which is just going to be
the main of the Gaussian of the probability density function of the state estimation error.
And we use this to predict it forward in time.
So this equation is the state prediction step for the external karma filter, and it can be simply written
like this.
So exactly the same as linear common.
Philidor, what this equation is doing is taking the current best estimate state for the previous timestep.
So we have the set, we have the A posterior, I estimate our X plus for K minus one, which is also
just a shorthand way of writing our estimate state times of K minus one, given all the information
up to K minus one.
So this is given all the information up to the current timestep, which can be probabilistically written
as the expectation of Raida.
So this parameter here is just the expected value of the state at K minus one, given all the measurements
up to minus one.
So we take this parameter here and we use our process model S to transform this into our apriority state
for the constant timestep K so we can say here we transform our previous timestep into our current timestep,
which is a shorthand way of writing it this way.
Again, sorry to go over here.
We have the estimated state at times of K given all the information up to this one.
So you can say that we've probably gone forward in time said he's came on this one.
This is K, but we haven't included we haven't included any extra information.
So all the measurements are still for K minus one came on this one.
So this is why this is the A.
This is this is why this is the a priori state, because we haven't included any any extra information.
And again, this can be written in the expectation form.
We're looking at the expected value of the state of the system, like given all the information up to
came on this one.
So we're still using the same set of information in the we've only predicted the state forward.
So we've looked at the prediction set for the state of the system or the main of the estimate.
We also want to propagate the covariance forward in time as well.
So to begin with, just like in the linear coming up, we start with the initial covariance matrix peno,
which is the initial Aruca variance for the state estimate ex not
so.
The definition of Pinart again is just going to be the expected value of the estimation error.
So we can say this time, here is the true state minus the estimated state.
So this is the estimation error and then we get the square of that.
So this term here, multiply the transpose of the same time is going to give us the estimation error
squared.
So this is the covariance matrix.
So given Peno, we want to propagate forward in time to find out Paswan so we can do this using the
linnear transform just like in the Linnear midfielder, but said the transformations are going to be
the linear approximation of the non-linear model.
So that gives us this equation shown here.
So this looks very similar to the linear shown below, except now we're going to have these new matrixes
here and replace of the F and L Matrixes.
And these matrices are actually called Jacobean matrixes.
So these are a linear approximation of the nonlinear model.
So this Eucumbene here is the Jacobean of the process model F with respect to the state X, and you
can see again we have this this Jacobean matrix here and it's transposed just like we have no f f transpose
and we have the Jacobean of the process model with respect to the noise.
And again, we have that matrix here and it's transpose over here just like we have the L Matrix and
the Matrix transpose over here.
So basically we've changed the linear equations and we replace the linear matrix with these linear approximations
of the nonlinear model.
So in general, we can write the predictions for the covariance in this form here, so we have the covariance
matrix for the error for the previous timestep, and we use it to transform it forward using our Deucalion
matrix of the process model with respect to the state and to get our predicted state covariance for
TimeStep came and we add onto the contribution for the noise.
So we have our noise covariance matrix here and we use the and we transform it into the state space
by using the Jakov in our process model with respect to the process model noise.
So we use the transformation operator.
So this basically gives us the prediction step for the covariance for the extended common filter.
And these Yukari matrices here can be written in this form here.
So the Jacobean of the process model F respect to X is just the partial derivative of the process model
with respect to X, but we evaluate it at the current best estimate, so we evaluate it at our X, K
minus one best estimate and very similar.
We have our Jacobean for the process model with respect to noise, which is going to be the derivative
of a process model F with respect to our noise vector.
And again, it is going to be evaluated at our best estimate.
So as time steps forward and times changes, we're going to have to keep revalidate reevaluating these
Jacobins because these Jacobins are the linear approximation of the nonlinear system.
As the system changes, we have to keep updating our linear ization point.
So we have to keep updating these matrices here so that you can be Matrix's ethics.
And if you a very similar to the Matrix in the linear coming filter and the matrix in the linear coming
field, they have the same job, except now we're changing the linear model with as time goes on.
So we have to keep updating them using this linear approximation of the nonlinear system.
So we can think of the covariance prediction step in this form here, we start off with the a posterior
eye Erica variance for K minus one.
So we have our covariance matrix for TOMPSETT came on this one, given all the information up to came
on this one, which is another way of writing it out in the expectation of radiophone.
So we have the estimated state error, have the estimated state error over here again.
So it's estimating state R-squared, given all the information or the measurements up to came on this
one.
So this is the kind of going back to the expectation operator.
This is the covariance of the state estimation error.
We can use our Jacobean of the system to transform this process, to transform this covariance matrix
and propagate it forward in time to find the current covariance matrix so we can find out P minus K,
which is going to be various metrics for the current TimeStep, given all the information up to the
previous TimeStep K minus one.
So again, just like in our transformation with me, we've increased the timestep for the prediction.
So we've gone from minus one to carry, but we've kept the same amount of information.
So we're only used in measurements up to Cayman's one K minus one.
So you can say it again in this estimation.
So I can tell you again in the expectation operator again, another way of expressing what the P matrix
is.
We've increased our prediction to K, but we've kept out all the information in the same set that we're
still using.
al-Said came on this one, said K minus one.
So for comparison, when you look at the process model predictions.
So that's the prediction for the state and we can compare it between the Lynnae Common Filter and extend
the common filter.
So Interlinear come and fill it up.
We use the linear system model F and G to propagate and step forward.
So we had the input and we had demesne for the previous timestep.
We put them into this equation here and we use this process model to predict forward to get the prediction
at the current TOMPSETT.
Now when we use the extended common here is very similar.
We have the input for the current timestep.
We have the state estimate for the previous times that we put them into the nonlinear process model
and we use that to predict forward to get a current prediction at times.
Okay, so the only difference between the two steps here is that we're using the full nonlinear system
here.
While up here we were using the linear system.
Now, if we look at the process model predictions that for the government, so looking at the linnear
coming, shouldn't we use this linear transformation here?
We used the previous covariance p we substituted it in using the state transition matrixes and state
noise matrix.
So if an elementary ccis and low as well as our noise covariance matrix, we put it into this equation
here and we transformed from the previous timestep to the current time step to get our covariance matrix
for K the external incoming filter.
Again, very similar, except now we've changed our F and O matrixes to be these giacobbe matrixes.
So Jacobean of the state transition matrix with respect to X and the Jacobean of the state transition
matrix, respect to the noise we take in our previous covariance predictor forward to get the current
covariance.
And every time we do this, every time we step forward in time, we have to re evaluate the psychobilly
matrixes as we've changed our linear ization point.
And Jacobean is a linear approximation, which is only valid at a given state of the system, the linear
regression point.
So we have to keep updating that as we estimate as the time goes on.
So in summary, we have the prediction step equation for the extended common field.
We have the prediction state estimate, which is this equation here.
So we've covered this.
This takes this uses the first model F takes in the previous state and the current control to predict
forward to get the current state.
We also have the predicted covariance equation estimate.
So this takes in the previous covariance uses the Jacobean of the system and the Jacobin of the noise
and predicts forward from the previous timestep K minus one to the current times.
Okay.
And finally, we have our definition of the Jacobins, the actual process.
And what does this mean?
We've got to cover in a little bit here.
But basically the Jakobsson is just the partial derivative of the function or the state transition matrix
respect to the current state evaluated at something linear as Asian Point or the current estimate.
And we have the same thing for the noise giacobbe into the Caribbean of the state transition metrics.
But we respect the noise.
So the difference here is ones with respect to noise.
This one is with respect to the state so that you gravimetric these can be thought of a linear transformation.
So instead of using the only system we can use, this linear approximation of the system, however,
is only valid at the state that it is linear rise about and is only valid at small probation's around
that state.