Find the best possible state of the dynamic model:
with its modeling errors w(k) subject to a set of observations with the measurement model:
with the measurement uncertainties v(k). The difficulties are not only embedded in nonlinear transformations f and h, but in the fact that x(k), w(k) and v(k) are random variables. The aforementioned basically states the nonlinear filtering problem. There are various approaches one could attack the challange:
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The optimal filter utilizes the conditional pdf to predict the states via the total probability theorem and uses the measurements to update with Baye's rule yielding the Fokker-Plank equation. This approach has several drawbacks: storing and integrating the pdf is in most applications impractical. | |||||||||
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The engineer, of course, thinks of linearizing the nonlinearities and than apply the standard tools. This leads to the so-called Extended Kalman filter (EKF). It has been widely used and after some experience you might be able to find the right process noise. From the view point of statistics the EKF is a work-around and exhibits several shortcomings:
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Based on the probability theory, the engineer would try to approximate the pdf of the random variables and avoids using linearization. If one has all the time in the world (slow processes for example), the engineer would perform a Monte Carlo simulation. This requires thousands (if not more) samples to reach a statistical accurate estimate of the mean and covariance. A better approach is to sample the pdf at a few hundred points, which yields the particle filter. One obtains as a result a fairly good representation of the pdf. |
Following the approach of approximating the pdf, the so-called unscented filter uses judiciously selected samples to represent a set of statistical moments. This ingenious algorithm has been first presented by S. Julier and J. Uhlmann in 1995[1]. For a list of related publications please refer to the references.
This section provides detailed information on state-of-the-art Unscented Transformations. Please select a top from the list below!
| The standard Unscented Transformation (UKF) |
| The scaled Unscented Transformation |
| The minimum sigma-set Unscented Transformation |
| The Higher order Unscented Transformation (HOUF) |
This example illustrates the unscented transformation on a common transformation, where an analytical solution exists. Please follow the link: polar2cart.
