The common filter makes the assumption that all the probability distributions relating to the state
and measurement can be represented as Gaussian distributions.
Now, this can dramatically simplify the Bayesian calculations.
And even though it is an approximate assumption, it can be made to work quite well in real life.
This has made the comment filter one of the industry's leading methods of data fusion.
The common filter works by propagating the main and covariance of the Gaussian distributions for the
estimated state through time using a linear process model.
This is the prediction step or process update step where the estimates move forward in time when the
elements are available, which are linear combination of the state, the common filter updates, the
estimate, the state main and covariance based on the measurements made and covariance of the Gaussian
distribution.
This is the update step or corrections step for the common filter.
These two processes form the prediction corrections step, which is recursively run in the filled up
with time.
The estimates are propagated forward in time to the current time.
If this measurement information available at the current time, it is used to improve the current state
estimates.
This is then repeated as time moves forward, since any realization or implementation of the linear
comma filter is going to be done on a microprocessor or computer, which is inherently discrete, we're
going to first start off looking at the discrete common filter.
The discrete filter breaks up the time into discrete blocks and steps forward in chunks of time for
a smooth and continuous system.
The system state might change like this curve.
It is smooth and it is continuous.
The value of the state can be calculated for any point in time.
The discrete system, the time is step forward in time, blocks or time steps.
So if the sample time or time step of the system was point one seconds for each energy step, the system
steps forward.
Point one seconds.
So if K was equal to zero, then T equals zero.
If K is equal to two and T is equal to two times the timestep apply to the discrete system cannot work
with fractional steps.
It needs to be whole integers.
If we overlay the continuous system responses with the discrete system responses, we should find out
that they match up exactly at those discrete key point times.
Suppose we have a linear, discrete time system of the form shown on the slide here.
It consists of a process model and a measurement model.
The process model describes how the system evolves with time and is a function of the current state.
It's a function of control inputs and of noise inputs.
It also has a measurement model.
So this measurement model here is a linear combination of the current state of the system is a function
of the current state and of some of random noise.
So our F matrix here is a state transition matrix.
This describes how the state changes with the input of the state.
The G matrix is the control input matrix.
It describes how the state changes based on control.
L is a process model noise sensitivity matrix.
It describes how the system state changes with noise.
And down here in the measurement model, we have the matrix, which is the measurement matrix.
So this describes the linear combination that the measurement is based on the current state.
And we have this matrix here, which is the measurement model, noise sensitivity matrix.
So this matrix here describes how the measurement is affected by the noise input.
And when we're using the common filter, we make a few assumptions about the type of noise in the system.
We make the assumption that a noise.
So you get to hear and we want to hear the random variable noise is about Gaussian distribution of zero
mean with a given covariance matrix.
So here for W we assume that the random vector comes from a normal distribution zero mean and it has
covariance.
Q Over here for the measurement of noise out the vector here, a random vector is from a normal distribution.
It has zero mean and it has a covariance matrix of our.
We also make the assumption that the noise variables are not correlated in time.
So matter what value we get at one instance of time is not going to change the probability of getting
a different value for the next time period.
We also make the assumption that the random variable noises between the process of noise and the measurement
noise are also independent.
So there's no correlation between the two.
Our goal is to estimate the state of the system based on our knowledge of the dynamics and availability
of noise measurements.
This can be expressed as a posteriorly estimate where the state estimate vector X four times of K is
the expected value, i.e. the main of the probability distribution of the true state X given all the
measurement.
Up to and including the current times to see the ocean for the estimated state and the plastination
for the indication that this is the posterior, I estimate if we didn't have the current TimeStep measurement,
said K. But we had all the rest, then we would have the a priori estimate of the state.
And you can notice that we do know this using the negative symbol on the estimated state.
Is it expected that once the current TOMPSETT measurement said K is fused into the A4 estimate, then
the posterior estimate will be the better estimate.
When we start and initialize a common filter, we let X not be the initial condition and this is going
to be the expected value of the true state.
Now, this might just be an assumption.
We might just assume that the starting state is zero.
And depending on what we're estimating, you also might need to start the current estimate from the
first measurement and then initialize the filter.
After we have this information, we will cover this in a lot more detail later on.
So ignored is our current best guess at the initial condition of the true state, not the common filter
also uses the covariance of the estimated error to probabilistically find the updated state estimate.
The covariance is denoted by the Matrix P with the timestep and estimated type indicated again.
The a priori covariance matrix is for the apriority state estimate and similarly for the posterior ie.
The error between the true state and the estimated state can be noted with the tilde symbol.
This is the estimation error or how accurate the estimate a state is.
So the covariance P essentially is a measure of how close the estimated state is to the true state of
probabilistically.
We will develop the equations of how the common field of calculates the estimated state and covariance
in a next series of videos.
We will also develop it step by step, using an example to d track a field of problem.