So let's have a look at the common of prediction step.
So this is the first step we need to implement for the common Fiola.
We begin the estimation step with the initial condition, it's not so this is going to be the best estimate
of the initial state of the system in this step, we want to propagate the time forward to find the
best estimate of the time.
Step one, so we can use the state's best model of the system that we dropped before to predict forward
what the new state of the system is going to be.
So this is going to be the best estimate at time, zero multiplied by our state transition matrix,
plus our control input at TimeStep zero time to control input matrix.
And this is going to propagate the state forward tompsett one.
Or in general we can write the equation out like so.
So the current state estimate is going to be a function of the old state estimate and the old control
input as well as the two state transition matrixes.
So this forms of prediction, step of the time update, step of the common fiodor, we use a process
model of the system we want to estimate to predict state forward.
So this is basically shifting the main of the Gaussian probably doability distributions forward in time.
So the equation here shown is the state prediction equation for the common field.
OK, so now we've looked at the state transition, but we also want to propagate forward the covariance
of the Gaussian probability distribution.
So to do this, we begin with the initial estimate pay not so pay is the initial variance of the state
estimate ignored.
Now, if we have exact knowledge of the initial state of the system, then we can set pay ought to be
equal to zero.
So there's no uncertainty in the estimate.
We know exactly.
Exactly.
If we have no knowledge of X axonal, well, then Peno is going to be turning towards a very large number
to say that a very large uncertainty.
But we usually know an approximate value or good assumption of the value of X not so we pay not to be
the value of the estimated error.
So now that we have Peno, we can propagate this incredible with time using the linear transformation
that we've derived before.
So basically we have our initial covariance.
Pinart we have our state transmission matrix here and we multiply them out by this equation here to
propagate the uncertainty forward in time.
Now, this does not change the amount of uncertainty in the system.
It just shifts it between the states as the dynamics of the system evolve.
So the size of pea before and after the magnitude, the amount of uncertainty in between the two stays
constant.
So if you look at the process model, we can see that this equation up here is going to account for
this function.
This part of the process model is going to take into account the trans state transition, the transition
of the state between tompsett came on this one and Tompsett K or between zero and one vacancy.
Can we also have this other uncertainty in the Saudi system?
So this is where the system process model noise comes in.
We need to account for this new uncertainty in state due to this process model noise.
So to account for this in the system, we can add a new covariance term.
So you can see here this term here now calculates the answer, new uncertainty in the system due to
the process of noise.
So our key metrics, if you remember this, is the covariance of the process model noise el is the process,
model noise, sensitivity matrixes.
Now, usually we assume that the process model noise is an additive, which basically means that these
matrices here are just identity.
If we use this assumption, then we can simplify the equation down to this one chain here.
So the covariance at times that one is going to be the variance that timestep zero propagated by the
state transition matrix plus the additive process model noise que.
So we can sum up the British concept of the common filter by using these equations here.
So this first equation here looks at this transition.
So this is the shifting the main of the galaxy in probability distribution of the estimated state.
This down here is the covariance propagation.
So this shifts the covariance of the probability distribution of the estimated state.
So inside the British of the filter, we update the main.
We update the covariance.