CHAPTER 3. NONLINEAR STATE SPACE ESTIMATION
where matrices Fx(m; k 1) and Hx(m; k) are Jacobians as in the first order EKF, given by Equations (3.12) and (3.13). The matrices F(xxi) (m; k 1) and H(xxi) (m; k) are the Hessian matrices of fi and hi:
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ei = (0 0 1 0 0)T is a unit vector in direction of the |
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is, it has a 1 at position i and 0 at other positions. |
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The prediction and update steps of the second order EKF can be computed in this toolbox with functions ekf_predict2 and ekf_update2, respectively. By taking the second order terms into account, however, doesn’t quarantee, that the results get any better. Depending on problem they might even get worse, as we shall see in the later examples.
3.1.5 The Limitations of EKF
As discussed in, for example, (Julier and Uhlmann, 2004b) the EKF has a few serious drawbacks, which should be kept in mind when it’s used:
1.As we shall see in some of the later demonstrations, the linear and quadratic transformations produces realiable results only when the error propagation can be well approximated by a linear or a quadratic function. If this condition is not met, the performance of the filter can be extremely poor. At worst, its estimates can diverge altogether.
2.The Jacobian matrices (and Hessian matrices with second order filters) need to exist so that the transformation can be applied. However, there are cases, where this isn’t true. For example, the system might be jump-linear, in which the parameters can change abruptly (Julier and Uhlmann, 2004b).
3.In many cases the calculation of Jacobian and Hessian matrices can be a very difficult process, and its also prone to human errors (both derivation and programming). These errors are usually very hard to debug, as its hard to see which parts of the system produces the errors by looking at the estimates, especially as usually we don’t know which kind of performance we should expect. For example, in the last demonstration (Reentry Vehicle Tracking) the first order derivatives were quite troublesome to calcute, even though the equations themselves were relatively simple. The second order derivatives would have even taken many more times of work.
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