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Kalman Filters on Non-Linear Models

I am doing a comparison, of Kalman Extended Kalaan Unscented(sigma point) Kalman Filters and several varieties of Partile filters on a highly non-linear system, with both gaussian and non-gaussian noise. I know that Kalman Filters aren't going to work so well, but I am a little unclear how to properly set on the Kalman filter so that it gets to give it its best shot.

The first system is modeled like this % x(k+1) = 1 + sin(4e-2*pi*k) + beta*x(k) + v(k); % y(k) = (x(k).^(2))/5 + w(k);

where w(k) and v(k) are noise sources that are gaussian in the first comparison, and then uniform, gamma, TBD in the other cases.

My first thought was ignore the sin(*) term all together, and simulate an additional step input for the constant "1" term. As for the X^2 term in the measurement equation, I am cluless at the moment. What is the fair solution, using the simplest global linearization possible, or redoing the Kalman equations to use the non-linear terms? Any ideas?

Robert Posey

Reply to
Robert Posey
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I am not sure what you really mean here, but in order to apply the Kalman filter, you will have to linearize the model and use the Jacobians to propagate both the means and the covariances. In the Extended Kalman Filter case, you can use the nonlinear system model (without the v(k) term, of course) to propagate the mean, and the linearized model to propagate the covariance. The unscented Kalman filter will not need any linearization of the system and measurement equations, see the works of S. Julier and J. Uhlmann.

Reply to
Andrey Romanenko

course) to propagate the mean, and the linearized model to propagate the covariance.

Hello Andrey,

Do you have know of web sites that discuss any details of Kalman filters (adaptive filers) in details?

If not on a web site, then a good book with example, including sourcecode.

James

Reply to
James

My preferred web site is

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has book references, software links, and online documents. Very comprehensive.

It should be noted that "standard" Kalman filters are not adaptive by default.

Kalman Filtering Theory and Practice by Grewal and Andrews.

Andrey

Reply to
Andrey Romanenko

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