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The common filter makes the assumption that all the probability distributions relating to the state

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and measurement can be represented as Gaussian distributions.

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Now, this can dramatically simplify the Bayesian calculations.

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And even though it is an approximate assumption, it can be made to work quite well in real life.

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This has made the comment filter one of the industry's leading methods of data fusion.

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The common filter works by propagating the main and covariance of the Gaussian distributions for the

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estimated state through time using a linear process model.

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This is the prediction step or process update step where the estimates move forward in time when the

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elements are available, which are linear combination of the state, the common filter updates, the

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estimate, the state main and covariance based on the measurements made and covariance of the Gaussian

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distribution.

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This is the update step or corrections step for the common filter.

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These two processes form the prediction corrections step, which is recursively run in the filled up

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with time.

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The estimates are propagated forward in time to the current time.

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If this measurement information available at the current time, it is used to improve the current state

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estimates.

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This is then repeated as time moves forward, since any realization or implementation of the linear

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comma filter is going to be done on a microprocessor or computer, which is inherently discrete, we're

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going to first start off looking at the discrete common filter.

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The discrete filter breaks up the time into discrete blocks and steps forward in chunks of time for

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a smooth and continuous system.

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The system state might change like this curve.

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It is smooth and it is continuous.

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The value of the state can be calculated for any point in time.

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The discrete system, the time is step forward in time, blocks or time steps.

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So if the sample time or time step of the system was point one seconds for each energy step, the system

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steps forward.

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Point one seconds.

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So if K was equal to zero, then T equals zero.

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If K is equal to two and T is equal to two times the timestep apply to the discrete system cannot work

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with fractional steps.

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It needs to be whole integers.

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If we overlay the continuous system responses with the discrete system responses, we should find out

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that they match up exactly at those discrete key point times.

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Suppose we have a linear, discrete time system of the form shown on the slide here.

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It consists of a process model and a measurement model.

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The process model describes how the system evolves with time and is a function of the current state.

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It's a function of control inputs and of noise inputs.

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It also has a measurement model.

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So this measurement model here is a linear combination of the current state of the system is a function

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of the current state and of some of random noise.

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So our F matrix here is a state transition matrix.

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This describes how the state changes with the input of the state.

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The G matrix is the control input matrix.

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It describes how the state changes based on control.

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L is a process model noise sensitivity matrix.

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It describes how the system state changes with noise.

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And down here in the measurement model, we have the matrix, which is the measurement matrix.

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So this describes the linear combination that the measurement is based on the current state.

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And we have this matrix here, which is the measurement model, noise sensitivity matrix.

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So this matrix here describes how the measurement is affected by the noise input.

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And when we're using the common filter, we make a few assumptions about the type of noise in the system.

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We make the assumption that a noise.

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So you get to hear and we want to hear the random variable noise is about Gaussian distribution of zero

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mean with a given covariance matrix.

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So here for W we assume that the random vector comes from a normal distribution zero mean and it has

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covariance.

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Q Over here for the measurement of noise out the vector here, a random vector is from a normal distribution.

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It has zero mean and it has a covariance matrix of our.

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We also make the assumption that the noise variables are not correlated in time.

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So matter what value we get at one instance of time is not going to change the probability of getting

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a different value for the next time period.

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We also make the assumption that the random variable noises between the process of noise and the measurement

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noise are also independent.

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So there's no correlation between the two.

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Our goal is to estimate the state of the system based on our knowledge of the dynamics and availability

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of noise measurements.

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This can be expressed as a posteriorly estimate where the state estimate vector X four times of K is

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the expected value, i.e. the main of the probability distribution of the true state X given all the

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measurement.

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Up to and including the current times to see the ocean for the estimated state and the plastination

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for the indication that this is the posterior, I estimate if we didn't have the current TimeStep measurement,

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said K. But we had all the rest, then we would have the a priori estimate of the state.

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And you can notice that we do know this using the negative symbol on the estimated state.

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Is it expected that once the current TOMPSETT measurement said K is fused into the A4 estimate, then

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the posterior estimate will be the better estimate.

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When we start and initialize a common filter, we let X not be the initial condition and this is going

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to be the expected value of the true state.

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Now, this might just be an assumption.

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We might just assume that the starting state is zero.

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And depending on what we're estimating, you also might need to start the current estimate from the

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first measurement and then initialize the filter.

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After we have this information, we will cover this in a lot more detail later on.

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So ignored is our current best guess at the initial condition of the true state, not the common filter

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also uses the covariance of the estimated error to probabilistically find the updated state estimate.

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The covariance is denoted by the Matrix P with the timestep and estimated type indicated again.

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The a priori covariance matrix is for the apriority state estimate and similarly for the posterior ie.

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The error between the true state and the estimated state can be noted with the tilde symbol.

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This is the estimation error or how accurate the estimate a state is.

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So the covariance P essentially is a measure of how close the estimated state is to the true state of

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probabilistically.

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We will develop the equations of how the common field of calculates the estimated state and covariance

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in a next series of videos.

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We will also develop it step by step, using an example to d track a field of problem.