The extended common Philidor, or JKF, is a variant of the linnear common photo that we've covered
before, this is a modified version of the linear common filter that can operate on non-linear systems.
The nonlinear system that it is operating on is approximated as a linear system, using a first order
approximation around a given state and covariance.
This is known as the linear ization point.
The state and the covariance that this approximation is made about is recursively updated as the state
and covariance are updated as time goes on and the system evolves.
This allows us to always maintain a linear system or a linear approximation that is approximating the
nonlinear system, even though the system is changing.
So since the filter now works on non-linear systems, we want to look at the discrete time nonlinear
system models that the system is going to be operating on.
So the nonlinear process model that we're going to be using in this filter can be represented as a function
that looks like this.
So we're going to have a function that calculates to current state.
So this is the current state vector of time.
And this function is going to have a number of inputs is going to take in the previous state for time
K minus one is going to take in the current control input for Time K and the current noise input for
time K.
We're also going to have a different model for the measurement model.
So now we're going to have a nonlinear function and this nonlinear function is going to take in the
current state tomake the noise input on the measurement at TOMKA and this is going to give us the measurement
at times.
So again, this is now going to be a nonlinear model rather than having a linear matrix model.
We're going to have a nonlinear model to represent anything that we want.
So the definition of the different matrices and what we call them haven't changed, we still have the
state vector.
We still have the control input vector.
We have the process model noise.
And we have the measurement model noise.
And again, we still have the same assumptions that we use in the linear coming filter.
We know the properties of the noise parameters.
We know that the noise vector for our presence model is going to be a normal distribution with zero
main and covariance.
We know that the measurement model noise out the K is going to be another normal distribution of zero
amount and has a covariance matrix of OK.
So these assumptions are the same and we also make the same assumptions about the correlations.
So we know that these two processes are not correlated with the time and they're independent from each
other.
So there's no correlation between the process of noise and the measurement model noise.
So these assumptions are standard for the linear comment here and there.
They are as equally applicable for the extended common filter.
So when we compare the linear material to the extent Carmencita, the linear coming field, it assumes
linear dynamics.
So we've seen this equation before.
This is the process model for the linear common field.
So we have these process model matrixes, F, G and L that take in the different measurement vectors
and vectors and calculate the new state for the extended coming.
We're going to assume a nonlinear dynamics.
So we're going to have this general function f which can be any function, it could be a linear function
or it could be a nonlinear function.
Same thing happens for the measurement model in the linear coming through.
We've assumed a linear measurement model model, so we assumed that we had this sense of measurement
matrix, height, cheer and noise model matrix m and that the current measurement is a linear combination
of the state and the noise to extend the common field.
We're going to assume that we have a nonlinear measurement model, so we assume that we have nonlinear
measurements.
So our measurement said K, is given by a function which can be a nonlinear function of our current
state and our current noise.
So just like before the linear common filter and the extent of Kamasutra both make the same assumptions
about the noise properties of the system.
We assume that we have our galaxy in noise for our process model and for our measurement model, and
that they have known variances and they're independent and non correlated with time.
So we can compare the properties of the two different filters.
Now, the linear common field, we know from past that it is the best linear estimator for the system.
However, this cannot be said for the extended incoming filter.
Now that we're operating on a nonlinear system, it is no longer the best estimate.
It is, however, the best linear estimate of the system.
The Linnear coming through had some very nice properties, it was stable for any initial conditions
and any probation's in state, which is basically saying that no matter where we initialize to filter,
the filter would always converge to the truth.
Unfortunately, since we're using linear approximations for the extent to come and filter, this no
longer can be said.
So the extend current filter, unfortunately, does not make any guarantees on stability of the system,
and it can diverge if the current state gets too far away from the true state.
So what this actually means is that the estimated state must always be close to the true state or the
filter can diverge, which means the errors can't grow too large and the initial conditions must be
close to the truth.
So we can't just set any initial condition.
We have to have some information about the true state of the system so that we can correctly initialize
the filter so the filter doesn't diverge.
Despite its limitations, the IQF has shown itself to be very capable in the real world, especially
for navigation models or kinetic models such as for inertial navigation or estimating the states of
dynamic systems.
This is why the extended common filter is one of the most widely used filters in engineering at the
moment.
Now investigate a bit more into the actual definition or mathematical definition of the extent to come
and filter the extended comment filter is a recursive estimate.
So is a predictor correct, to form, which means we make predictions and then we use extra information
to correct that prediction, to minimize the error.
And this solves the linear quadratic estimation problem using the minimum means squared error method.
So mathematically, this can be expressed as this equation here.
Now, this might look complicated, but is actually just explaining a very simple concept.
So what this equation is calculating.
It is calculating the state estimate for TimeStep.
I given all the information up into TimeStep J.
So this is a state estimate for time.
I given all the information on measurements up to time J.
Now I and J don't have the same number.
They can be the different intervals and the coming for that works is spelled out by a optimization process
or working out the minimum expected main squared error over here inside the expectation operator.
We have our EXI.
So this is the current true state of the system.
This is the true state and we're subtracting our estimate of here to get the estimation error.
So what we want to do is we want to make this estimation error as small as possible, which is why this
is the ARG min.
So this is working out the value of our state estimate to minimize this error such that as we do this,
our estimate should go towards our true state.
This term here is just looking at the estimation error squared.
So we're just taking the same time here.
Transposing it is just a fancy way of calculating the estimation error squared over here.
We want to work out this expectation of of this probabilistic distribution given all these measurements.
So we want to calculate, given all the information from measurement, said one all the way up to measurement,
said J.
So these eye injuries are given over here.
So this is what the common filter is actually doing, is calculating the state estimate that minimizes
this equation, which is the expectation of the estimation error squared.
Given all these measurement information.
So let's recap the different estimation terms starting off, we have our X, which is the true side
of the system.
This is what we don't know, but this is what we want to estimate.
Our estimate of the true state is going to be our X hat.
So this is going to give us our estimate of the system and we want to make our estimate as close to
the true state as possible.
So if we look at the difference between the true state and estimate the state, we get our estimation
error.
And this is going to be indicated by the Tillett or over the top.
So we know that the common filter expresses our state estimate as a probability density function, it
has a mean and it has a covariance.
So when we look at the main of the state estimates, we have our predicted state and we have our update
of the state.
So what this means is that our predicted state for TimeStep K or our apre state is going to be the state
estimate for times, given all the information up to time K minus one.
So this is where the minus comes in.
And what this is, is the expected value of our probability density function of our state, given all
the measurements up to K minus one and of course our updated state estimate our posterior, I estimate
our X plus is going to be a state estimate of TimeStep K given all the information up to TimeStep K,
which is also going to be our expected value of the probability distribution of our state estimate,
given all our measurements up to Tom K.
So this is the predicted state.
This includes all the information up to K minus one, while this is the updated state which uses all
the information up to the current timestep.
So that gives the main of the probability density function.
So now we can look at the covariance of the probability distribution function.
So again, we have our predicted state covariance, our P minus, which is the covariance for Tompsett.
K, given all the information up to TimeStep K minus one, which is simply the expectation of the probability
density function of the estimation error squared.
Given our measurements up to time K minus one hour, Abdellah covariance is simply denoted again by
this P plus the simply the probability density function covariance for our state estimation error for
a time.
Sepak given all the information up to times of K, which is again simply expressed as an expected operator
for our estimation error, given all the information up to TimeStep K.
So again, to recap, we can look at what this means in draft form.
So we have a probability density function, which is a conditional density function of what our state
estimate is going to be given the set of measurements.
And we're going to assume that it's a normal distribution.
So we assume that this can be represented as a normal distribution of main exper and variance sigma
squared.
So our estimate, the state is going to be the main of this distribution.
So this is what these are calculating here.
So our estimate, the state is going to be the main of this density function, which is the expected
value of the probability density function of X.
Given all the measurements said and our covariance of this function here or the spread of the data is
going to be the variance squared, which is going to be the expected operator on the estimation error
squared, given all the measurements.
So to recap, all these terms are exactly the same as the linear common field for the extended common
photo.
But we want to make sure that we understand the basic concepts between what they represent and the difference
between the a priori state and the posterior state, as well as how the states and the covariance map
to the probability density function of our state estimate.