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The extended common Philidor, or JKF, is a variant of the linnear common photo that we've covered

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before, this is a modified version of the linear common filter that can operate on non-linear systems.

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The nonlinear system that it is operating on is approximated as a linear system, using a first order

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approximation around a given state and covariance.

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This is known as the linear ization point.

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The state and the covariance that this approximation is made about is recursively updated as the state

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and covariance are updated as time goes on and the system evolves.

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This allows us to always maintain a linear system or a linear approximation that is approximating the

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nonlinear system, even though the system is changing.

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So since the filter now works on non-linear systems, we want to look at the discrete time nonlinear

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system models that the system is going to be operating on.

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So the nonlinear process model that we're going to be using in this filter can be represented as a function

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that looks like this.

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So we're going to have a function that calculates to current state.

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So this is the current state vector of time.

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And this function is going to have a number of inputs is going to take in the previous state for time

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K minus one is going to take in the current control input for Time K and the current noise input for

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time K.

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We're also going to have a different model for the measurement model.

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So now we're going to have a nonlinear function and this nonlinear function is going to take in the

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current state tomake the noise input on the measurement at TOMKA and this is going to give us the measurement

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at times.

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So again, this is now going to be a nonlinear model rather than having a linear matrix model.

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We're going to have a nonlinear model to represent anything that we want.

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So the definition of the different matrices and what we call them haven't changed, we still have the

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state vector.

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We still have the control input vector.

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We have the process model noise.

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And we have the measurement model noise.

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And again, we still have the same assumptions that we use in the linear coming filter.

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We know the properties of the noise parameters.

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We know that the noise vector for our presence model is going to be a normal distribution with zero

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main and covariance.

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We know that the measurement model noise out the K is going to be another normal distribution of zero

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amount and has a covariance matrix of OK.

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So these assumptions are the same and we also make the same assumptions about the correlations.

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So we know that these two processes are not correlated with the time and they're independent from each

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other.

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So there's no correlation between the process of noise and the measurement model noise.

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So these assumptions are standard for the linear comment here and there.

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They are as equally applicable for the extended common filter.

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So when we compare the linear material to the extent Carmencita, the linear coming field, it assumes

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linear dynamics.

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So we've seen this equation before.

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This is the process model for the linear common field.

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So we have these process model matrixes, F, G and L that take in the different measurement vectors

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and vectors and calculate the new state for the extended coming.

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We're going to assume a nonlinear dynamics.

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So we're going to have this general function f which can be any function, it could be a linear function

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or it could be a nonlinear function.

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Same thing happens for the measurement model in the linear coming through.

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We've assumed a linear measurement model model, so we assumed that we had this sense of measurement

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matrix, height, cheer and noise model matrix m and that the current measurement is a linear combination

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of the state and the noise to extend the common field.

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We're going to assume that we have a nonlinear measurement model, so we assume that we have nonlinear

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measurements.

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So our measurement said K, is given by a function which can be a nonlinear function of our current

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state and our current noise.

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So just like before the linear common filter and the extent of Kamasutra both make the same assumptions

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about the noise properties of the system.

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We assume that we have our galaxy in noise for our process model and for our measurement model, and

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that they have known variances and they're independent and non correlated with time.

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So we can compare the properties of the two different filters.

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Now, the linear common field, we know from past that it is the best linear estimator for the system.

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However, this cannot be said for the extended incoming filter.

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Now that we're operating on a nonlinear system, it is no longer the best estimate.

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It is, however, the best linear estimate of the system.

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The Linnear coming through had some very nice properties, it was stable for any initial conditions

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and any probation's in state, which is basically saying that no matter where we initialize to filter,

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the filter would always converge to the truth.

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Unfortunately, since we're using linear approximations for the extent to come and filter, this no

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longer can be said.

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So the extend current filter, unfortunately, does not make any guarantees on stability of the system,

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and it can diverge if the current state gets too far away from the true state.

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So what this actually means is that the estimated state must always be close to the true state or the

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filter can diverge, which means the errors can't grow too large and the initial conditions must be

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close to the truth.

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So we can't just set any initial condition.

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We have to have some information about the true state of the system so that we can correctly initialize

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the filter so the filter doesn't diverge.

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Despite its limitations, the IQF has shown itself to be very capable in the real world, especially

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for navigation models or kinetic models such as for inertial navigation or estimating the states of

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dynamic systems.

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This is why the extended common filter is one of the most widely used filters in engineering at the

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moment.

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Now investigate a bit more into the actual definition or mathematical definition of the extent to come

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and filter the extended comment filter is a recursive estimate.

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So is a predictor correct, to form, which means we make predictions and then we use extra information

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to correct that prediction, to minimize the error.

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And this solves the linear quadratic estimation problem using the minimum means squared error method.

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So mathematically, this can be expressed as this equation here.

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Now, this might look complicated, but is actually just explaining a very simple concept.

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So what this equation is calculating.

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It is calculating the state estimate for TimeStep.

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I given all the information up into TimeStep J.

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So this is a state estimate for time.

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I given all the information on measurements up to time J.

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Now I and J don't have the same number.

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They can be the different intervals and the coming for that works is spelled out by a optimization process

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or working out the minimum expected main squared error over here inside the expectation operator.

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We have our EXI.

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So this is the current true state of the system.

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This is the true state and we're subtracting our estimate of here to get the estimation error.

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So what we want to do is we want to make this estimation error as small as possible, which is why this

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is the ARG min.

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So this is working out the value of our state estimate to minimize this error such that as we do this,

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our estimate should go towards our true state.

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This term here is just looking at the estimation error squared.

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So we're just taking the same time here.

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Transposing it is just a fancy way of calculating the estimation error squared over here.

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We want to work out this expectation of of this probabilistic distribution given all these measurements.

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So we want to calculate, given all the information from measurement, said one all the way up to measurement,

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said J.

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So these eye injuries are given over here.

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So this is what the common filter is actually doing, is calculating the state estimate that minimizes

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this equation, which is the expectation of the estimation error squared.

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Given all these measurement information.

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So let's recap the different estimation terms starting off, we have our X, which is the true side

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of the system.

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This is what we don't know, but this is what we want to estimate.

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Our estimate of the true state is going to be our X hat.

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So this is going to give us our estimate of the system and we want to make our estimate as close to

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the true state as possible.

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So if we look at the difference between the true state and estimate the state, we get our estimation

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error.

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And this is going to be indicated by the Tillett or over the top.

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So we know that the common filter expresses our state estimate as a probability density function, it

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has a mean and it has a covariance.

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So when we look at the main of the state estimates, we have our predicted state and we have our update

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of the state.

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So what this means is that our predicted state for TimeStep K or our apre state is going to be the state

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estimate for times, given all the information up to time K minus one.

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So this is where the minus comes in.

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And what this is, is the expected value of our probability density function of our state, given all

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the measurements up to K minus one and of course our updated state estimate our posterior, I estimate

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our X plus is going to be a state estimate of TimeStep K given all the information up to TimeStep K,

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which is also going to be our expected value of the probability distribution of our state estimate,

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given all our measurements up to Tom K.

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So this is the predicted state.

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This includes all the information up to K minus one, while this is the updated state which uses all

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the information up to the current timestep.

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So that gives the main of the probability density function.

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So now we can look at the covariance of the probability distribution function.

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So again, we have our predicted state covariance, our P minus, which is the covariance for Tompsett.

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K, given all the information up to TimeStep K minus one, which is simply the expectation of the probability

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density function of the estimation error squared.

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Given our measurements up to time K minus one hour, Abdellah covariance is simply denoted again by

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this P plus the simply the probability density function covariance for our state estimation error for

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a time.

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Sepak given all the information up to times of K, which is again simply expressed as an expected operator

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for our estimation error, given all the information up to TimeStep K.

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So again, to recap, we can look at what this means in draft form.

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So we have a probability density function, which is a conditional density function of what our state

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estimate is going to be given the set of measurements.

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And we're going to assume that it's a normal distribution.

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So we assume that this can be represented as a normal distribution of main exper and variance sigma

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squared.

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So our estimate, the state is going to be the main of this distribution.

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So this is what these are calculating here.

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So our estimate, the state is going to be the main of this density function, which is the expected

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value of the probability density function of X.

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Given all the measurements said and our covariance of this function here or the spread of the data is

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going to be the variance squared, which is going to be the expected operator on the estimation error

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squared, given all the measurements.

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So to recap, all these terms are exactly the same as the linear common field for the extended common

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photo.

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But we want to make sure that we understand the basic concepts between what they represent and the difference

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between the a priori state and the posterior state, as well as how the states and the covariance map

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to the probability density function of our state estimate.