Extended Kalman filter for the model x_dot=f(x,u,w)

Question

There is a lot of info about EKF out there but everything I find explains it for the simplified model of the form

x_dot = f(x,u) + w;

i.e. the process noise is a simple additive term.

Please suggest some good reference where the EKF for the model

x_dot = f(x,u,w)

is explained.

An example of such model can be found here (see chapter 6)

http://www.st.ewi.tudelft.nl/~koen/in4073/Resources/MSc_thesis_X-UFO.pdf

thanks!

Answer 1

The 1997 paper here by Julier and Uhlmann seems a good place to start. Their assumed state update model is slightly more general than yours, being the vector time varying model $$ x(k+1)=f_k[x(k),u(k),w(k)]. $$ You can also check the more recent literature citing this paper.

A technique they utilize in 1 seems to be what's called the "unscented kalman filter", which in most cases yields superior estimation accuracy than linearisation for filtering applications.

Of course, in your specific problem the taylor expansion may be such that higher order terms in Taylor expansions of the form $$ \overline{v}=f[\overline{w}]+\frac{1}{2}\nabla^2 f P_{ww}+\cdots $$ applied to transforming the mean $\overline{w}$ and the covariance of the noise can be neglected, and if the noise is Gaussian in which case the first two moments suffice to describe the noise.