Information and Covariance filtering

Information and Covariance filtering

Feature:

Covariance filtering

Information filtering

These KF implementations propagate the state error covariance P, which represents the uncertainty in the state estimate.

These KF implementation propagate the inverse of P (information matrix Y) rather than propagating P. The Y represents the certainty in the state estimate.

Recursively update x : state vector estimate P : state error covariance

d : information state (d = Yx) Y : information matrix (inverse of covariance)

Meaning

If P is 'large', then we have a lot of uncertainty in our state estimate. In the limit as P->0 we have perfect knowledge of x, and as P->infinity we have zero knowledge of x.

If Y is 'large', then we have a lot of confidence in our state estimate. In the limit as P->0 we have zero knowledge of x. As P->infinity we have perfect knowledge of x.

Feasibility

Covariance filtering cannot be used if Information filtering cannot be used if P is P=infinity (i.e. uncertainty is too high) singular.

Numerical robustness (against roundoff)

The observational update of the uncertainty matrix P is less robust against roundoff errors than the temporal update. It is more likely to cause the matrix P to become indefinite, which tends to destabilize the estimator.

The temporal update of the information matrix Y is less robust than the observational update against roundoff errors and is a more likely cause of filter degradation.

Pros & Cons

+ : Problems with perfect Prior knowledge are easy to solve. - : The conventional KF is the Covariance filtering. It is particularly sensitive to roundoff errors (This is due to the observational update of P).

+ : Problems without Prior Information are easy to solve (an information filter starting from the Y0=0 will have absolutely no bias toward the a priori estimate. Covariance filters cannot do this) + : Information filtering offers a possible solution to the roundoff problem when the observational update of P is the culprit. - : The greatest objection to information filtering is the loss of “transparency” of the representation. Although information is a more practical concept than uncertainty for some problems, it can be more difficult to interpret its physical significance and to use it in our thinking. - : Perhaps the greatest impediment to widespread acceptance of information filtering is the loss of physical significance of the associated state vector components (i.e. information state d = Yx)