We have mentioned that the common filter is a powerful way of expressing the complicated and potentially
computationally expensive process of data fusion down into a more simpler problem that can be easily
solved.
This is done by making a few assumptions about the system dynamics, the estimate, the state, the
error and noise properties.
Mathematically, the common filter is an estimate which solves a linear quadratic estimation problem,
which just means that the estimates, the instantaneous date of a linear dynamic system perturbed by
random white noise, and it does is by taking a series of measurements that are linearly related to
the state but are also are corrupted by white noise.
So we want to calculate the estimated state, which is as close as possible to the true state, which
is unknown, and we do this by using a series of Noisey measurements that are related to the true state.
The estimated state solution is a statistically optimum estimate with regards to the quadratic function
of the estimation error, which just means that it's minimizing the squared main error.
So just like Enlace squares, we want to minimize the cost function, which minimizes the squared main
error of some error measurement.
The cost function that we choose to minimize is the same one as the one that we use in the recursively
square.
We want to minimize the state estimation error.
So that's the difference between the true state and the estimate, the state to give you the state estimation
error.
We know that the state estimation era covariance matrix can be written like so and likewise with the
cost function of the squared mean of the state estimation error.
So just like in recursively squares, minimizing the trace of the covariance matrix also minimizes the
cost function.
We will go into more detail for the Common Feltl later on.
You may have noticed that I didn't mention probability once in this definition, while the previous
videos we talked about representing the state as a probability distribution, but making a few assumptions
about the type of noise and system dynamics, you can cast the probability problem into an optimization
problem and solve it in an appropriate way.
You can look at the common filter from an optimal or geometric or probabilistic point of view, but
you always end up with the same equation solutions in the end, which is pretty amazing.
The common filter is one of the greatest discoveries in the history of estimation and data fusion and
perhaps one of the greatest engineering discoveries of the 20th century.
It has enabled mankind to do and build many things which could not be possible otherwise.
It has immediate application in complicated dynamic systems such as those used in guidance, navigation
and control of cars, ships, aircraft and spacecraft.
It is widely used in robotics and manufacturing and is applicable to pretty much any time domain series
analysis such as used in signal processing, economics, stock market prediction, finance and more.
The common fear is named after its inventor, Rudolph Carmen, who was born in Budapest in the year
1930.
He immigrated into the US during World War Two and eventually ended up at MIT for a bachelor and master's
in electrical engineering.
This eventually led him to role as a lecturer at Columbia University.
The first practical use of the common filter was by the Ames Research Center of NASA, and it was quickly
then used for the Apollo spacecraft mission for the estimation of the trajectory and control of the
spacecraft.