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Let's recap all that we have learned in this course about common filters, starting with the linnear.

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Com and filter, the linnear common filter solves the estimation problem of estimating the current state

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of the system by minimizing the expected error over all measurements it receives subject to the system

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dynamics sharing here.

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So here is the estimated error.

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We want to minimize the estimated error for the expected error across all the measurements subject to

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the system module dynamics here.

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So you can see that these models are linear, which is why we're using the linear Matilda.

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The filter assumes that all the noise in the system or any of the error in the system follows these

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sets of assumptions here.

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So it assumes that each of them are uncorrelated and they can be modeled as a Gaussian distribution

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with zero meaning and a given covariance.

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The actual common of equations used to solve the filter are shown here, so inside the prediction step,

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we take the current state, we predict it forward using our process model.

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We add on the effects of the controls to predict the step forward in time.

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When we do this, we also want to predict the covariance forward in time.

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So we use again the process model and the previous experience and predict it forward in time and add

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on the uncertainty inside that prediction step.

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When we get the measurement, we apply the measurement set of equations here.

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So using the current state and the measurement model, we predict what the measurement should be.

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We look at the innovation, the difference between the actual measurement and our predicted measurement

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to calculate the innovation and then using this innovation, value and outcome and gain value here we

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can correct for the estimated state based on the error as we do the correction here, we also want to

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correct and update the covariance matrix using this last equation.

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So we're not running the common field.

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We also want to make sure the filters tuned correctly and one way is to validate it, and that is to

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look at the minimum mean squared error, which is the error between the common filter state estimates

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and the true states.

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And we want to make this as small as possible for the whole run in the filter like we've done in the

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simulation.

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We want to minimize the error between the estimates and the three states.

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So this is how we can validate the filter.

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We're not using the filter to make sure our main squared error is as small as possible.

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We want to make sure that the measurement innovation is always close to zero.

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So it has zero.

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So this is one of the operating conditions that the filter assumes.

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So we can calculate the innovation for a series of measurements and we want to make sure that is very

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close to zero.

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I mean, we also want to ensure that the measurement innovation variance is close to the actual innovation

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variance.

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So inside the common filter, we calculate our innovation and what we want to do is to make sure the

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actual innovation covariance matches what the measured innovation covariance is from the series of innovation

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measurements over the time period.

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So once these two conditions are met, we have a well working and true in common field which should

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minimize the main square area over the whole profile.

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Now for the extended calm and filter, the extended calm, I feel that solves the same problem as linear

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coming theater, except now we're using a non-linear system model, dynamic equations showing here.

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So instead of the linear model, we now have these general nonlinear models.

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We still use all the same assumptions for the noise and error variances.

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However, we've only change these equations over here for the system model.

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The actual equations used to implement the common filter are very similar.

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Now we use the full nonlinear system to predict forward.

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However, when we do the covariance prediction forward, we're using the Jacobean of these nonlinear

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functions.

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So the linear approximation of the nonlinear function, instead of using the linear system model for

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the innovation, we use the full measurement model to calculate the innovation.

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But then for the rest of the equations here is very similar as the linear computer swapping out the

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linear measurement model with the linear approximation of the Caribbean for the measurement model.

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So the equation shown on this slide here, talk about the Jacobean of the different process model and

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measurement model.

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So the Jacobean is simply just a partial derivative of the output of the process model with respect

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to the states.

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And this is for the state process model Jacobean.

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You can also calculate the Jacobean of the process model with respect to the noise input.

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So this term over here, by taking the partial derivative with respect to the noise for the measurement

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model, we have the measurement model Jacobean, which is the Jacobean of this function here with respect

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to the actual state inputs to this vector over here, the measurement noise Jakobsson model is the Jacobean

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of this measurement model.

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But respect to the input noise, the over here.

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And then lastly, for the unsent to come and filter, we want to take our state prediction augmented

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with the process, modernize, to form this augmented vector, and we do the same thing for the covariance

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matrix.

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We augment the current covariance with the noise covariance.

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The forms are going to covariance.

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Then we use these two augmented matrixes to calculate the signal points according to this set of equations

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here.

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So we use the square root of the covariance matrix to work out the deltas of the similar points between

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the main vector.

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Once we have the signal points, we then use these weightings and we can use them inside the prediction

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step.

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So we predict forward each sigma point through the process model and then we can and then we can recover

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our predicted state by taking the weighted estimate of all the single points.

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And we can do the same thing for the covariance matrix.

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When it comes to the measurement step, we do the same thing is that we augment the vector now with

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the measurement, noise and measurement covariance.

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We form the signal points again using our square root of the covariance matrix.

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And then we take these augmented signal points and we put them through the nonlinear measurement model.

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Once we do this, we can use the weights again to recover the predicted measurement, calculate the

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innovation, and then we can form the measurement covariance matrix and across covariance matrix by

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taking the weights and the signal points, again, taking the variances between the state and the measurement

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or the measurement and the measurement.

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Once we have all those information, then we can apply the uncertainty coming through the update step,

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which again is just these series of equations here, which are just a slightly different form of the

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extended of filter and linear common filter update step equations.

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So in summary, the Linnear midfielder is the best linnear estimate of a linear systems, it is stable

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for any initial conditions and it's stable for any error probation's.

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So this is one advantage of the linnear coming.

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If the system is linear, it has some very nice properties.

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Now, moving on to non-linear systems, we start to look at the extended midfielder.

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Now, unfortunately, this is not the best estimate of a non-linear systems because there is no best

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estimate of a non-linear systems.

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We can't guarantee stability and the filter can diverge because of the non-linear effects.

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The state estimates must always be close to the true state or the field or diverge because we're using

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linear approximations of the nonlinear system.

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So overall, this is a first order approximate accuracy for the nonlinear state and covariance transformations

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due to the Jacobins.

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So now looking at a different form for non-linear filters, we're looking at the unscented coming filter.

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So the unscented coming has very similar properties as the extended economy of Europe, except now it

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is third order approximation accuracy for the nonlinear state and covariance transforms.

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So we have a higher order of accuracy here, which is where the advantages of the uncertainty come a

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few to come from.

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And when we're using the unscented computer, it does not require the calculation of the Jacobean.

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We can use a process model and measurement model directly, which can sometimes aid in the computational

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complexity or at least aid in the implementation complexity.

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Congratulations on completing the course of all the advance common Philadelphia since the Fusion.

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I hope you have learned a lot and you have got quite a bit out of this cause.

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I hope to see you in one of my other courses.