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We have mentioned that the common filter is a powerful way of expressing the complicated and potentially

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computationally expensive process of data fusion down into a more simpler problem that can be easily

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solved.

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This is done by making a few assumptions about the system dynamics, the estimate, the state, the

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error and noise properties.

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Mathematically, the common filter is an estimate which solves a linear quadratic estimation problem,

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which just means that the estimates, the instantaneous date of a linear dynamic system perturbed by

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random white noise, and it does is by taking a series of measurements that are linearly related to

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the state but are also are corrupted by white noise.

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So we want to calculate the estimated state, which is as close as possible to the true state, which

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is unknown, and we do this by using a series of Noisey measurements that are related to the true state.

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The estimated state solution is a statistically optimum estimate with regards to the quadratic function

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of the estimation error, which just means that it's minimizing the squared main error.

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So just like Enlace squares, we want to minimize the cost function, which minimizes the squared main

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error of some error measurement.

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The cost function that we choose to minimize is the same one as the one that we use in the recursively

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square.

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We want to minimize the state estimation error.

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So that's the difference between the true state and the estimate, the state to give you the state estimation

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error.

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We know that the state estimation era covariance matrix can be written like so and likewise with the

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cost function of the squared mean of the state estimation error.

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So just like in recursively squares, minimizing the trace of the covariance matrix also minimizes the

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cost function.

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We will go into more detail for the Common Feltl later on.

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You may have noticed that I didn't mention probability once in this definition, while the previous

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videos we talked about representing the state as a probability distribution, but making a few assumptions

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about the type of noise and system dynamics, you can cast the probability problem into an optimization

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problem and solve it in an appropriate way.

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You can look at the common filter from an optimal or geometric or probabilistic point of view, but

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you always end up with the same equation solutions in the end, which is pretty amazing.

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The common filter is one of the greatest discoveries in the history of estimation and data fusion and

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perhaps one of the greatest engineering discoveries of the 20th century.

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It has enabled mankind to do and build many things which could not be possible otherwise.

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It has immediate application in complicated dynamic systems such as those used in guidance, navigation

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and control of cars, ships, aircraft and spacecraft.

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It is widely used in robotics and manufacturing and is applicable to pretty much any time domain series

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analysis such as used in signal processing, economics, stock market prediction, finance and more.

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The common fear is named after its inventor, Rudolph Carmen, who was born in Budapest in the year

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1930.

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He immigrated into the US during World War Two and eventually ended up at MIT for a bachelor and master's

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in electrical engineering.

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This eventually led him to role as a lecturer at Columbia University.

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The first practical use of the common filter was by the Ames Research Center of NASA, and it was quickly

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then used for the Apollo spacecraft mission for the estimation of the trajectory and control of the

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spacecraft.