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## Generalised observations

The observation function in Equation 5.7 does not encapsulate the most general form of observation, since it assumes that the observation vector can be separated as a function of the state . It is sometimes therefore necessary to introduce a generalised observation of the form

where again represents a random noise vector having covariance . However with some manipulation and extra computation we can effectively convert the linearised version of the -type function into an -type function, allowing it to be incorporated in the same way. We linearise with respect to and around the estimated state and observation , assuming that the noise is small:

where here represents the true value of the state vector, and is the true observation vector (as opposed to the actually measured vector ), so that . We identify the following quantities with their equivalents for an -type observation:
The innovation vector is .
The Jacobian matrix is .
The noise vector is .
The noise covariance matrix is .
Extra computation is therefore needed to convert the observation covariance from to . The innovation vector , Jacobian matrix and observation covariance are substituted into the Levenberg-Marquardt algorithm in place of their equivalents for the -type observation. There is no reason why there should not be a robust version of the -type observation, but currently it is not implemented.

Next: Levenberg-Marquardt software Up: Levenberg-Marquardt minimisation Previous: Robust observations   Contents
2006-03-17