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Suppose we have a random variable x(t) whose value we want to estimate at certain times t0 ,t1, t2, t3, etc. Also, suppose we know that x(tk) satisfies a linear dynamic equation
x(tk+1) = Fx(tk) + u(k) (the dynamic equation)
In the above equation F is a known number. In order to work through a numerical example let us assume F= 0.9
Kalman assumed that u(k) is a random number selected by picking a number from a hat. Suppose the numbers in the hat are such that the mean of u(k) = 0 and the variance of u(k) is Q. For our numerical example, we will take Q to be 100.
u(k) is called white noise, which means it is not correlated with any other random variables and most especially not correlated with past values of u.
In later lessons we will extend the Kalman filter to cases where the dynamic equation is not linear and where u is not white noise. But for this lesson, the dynamic equation is linear and w is white noise with zero mean.
Now suppose that at time t0 someone came along and told you he thought x(t0) = 1000 but that he might be in error and he thinks the variance of his error is equal to P. Suppose that you had a great deal of confidence in this person and were, therefore, convinced that this was the best possible estimate of x(t0). This is the initial estimate of x. It is sometimes called the a priori estimate.
this explanation is from wikipedia.org:
The Kalman filter is an efficient recursive filter which estimates the state of a dynamic system from a series of incomplete and noisy measurements. An example of an application would be to provide accurate continuously-updated information about the position and velocity of an object given only a sequence of observations about its position, each of which includes some error. It is used in a wide range of engineering applications from radar to computer vision. Kalman filtering is an important topic in control theory and control systems engineering.
For example, in a radar application, where one is interested in tracking a target, information about the location, speed, and acceleration of the target is measured with a great deal of corruption by noise at any time instant. The Kalman filter exploits the dynamics of the target, which govern its time evolution, to remove the effects of the noise and get a good estimate of the location of the target at the present time (filtering), at a future time (prediction), or at a time in the past (interpolation or smoothing).
The filter is named after its inventor, Rudolf E. Kalman, though Peter Swerling actually developed a similar algorithm earlier. Stanley Schmidt is generally credited with developing the first implementation of a Kalman filter. It was during a visit of Kalman to the NASA Ames Research Center that he saw the applicability of his ideas to the problem of trajectory estimation for the Apollo program, leading to its incorporation in the Apollo navigation computer. The filter was developed in papers by Swerling (1958), Kalman (1960) and Kalman and Bucy (1961).
A wide variety of Kalman filters have now been developed, from Kalman's original formulation, now called the simple Kalman filter, to Schmidt's extended filter, the information filter and a variety of square-root filters developed by Bierman, Thornton and many others. Perhaps the most commonly used type of Kalman filter is the phase-locked loop now ubiquitous in radios, computers, and nearly any other type of video or communications equipment.