Generalised observations

The observation function ${\bf h}(.)$ in Equation 5.7 does not encapsulate the most general form of observation, since it assumes that the observation vector ${\bf z}$ can be separated as a function of the state ${\bf x}$. It is sometimes therefore necessary to introduce a generalised observation of the form

$${\bf F}({\bf x},{\bf z}-{\bf w}) = {\bf0}$$

where ${\bf w}$ again represents a random noise vector having covariance $N$. However with some manipulation and extra computation we can effectively convert the linearised version of the ${\bf F}$-type function into an ${\bf h}$-type function, allowing it to be incorporated in the same way. We linearise ${\bf F}(.)$ with respect to ${\bf x}$ and ${\bf z}$ around the estimated state $\hat{\bf x}$ and observation ${\bf z}$, assuming that the noise ${\bf w}$ is small:

$${\bf F}({\bf x},{\bf z}_t) = {\bf F}(\hat{\bf x},{\bf z}) + ... ...) - \frac{\partial {\bf F}}{\partial {\bf z}} {\bf w}= {\bf0}$$

where ${\bf x}$ here represents the true value of the state vector, and ${\bf z}_t$ is the true observation vector (as opposed to the actually measured vector ${\bf z}$), so that ${\bf w}= {\bf z}-{\bf z}_t$. We identify the following quantities with their equivalents for an ${\bf h}$-type observation:

The innovation vector is $\mbox{\boldmath$\nu$}=-{\bf F}(\hat{\bf x},{\bf z})$.

The Jacobian matrix is $H=\frac{\partial {\bf F}}{\partial {\bf x}}$.

The noise vector is ${\bf w}' = \frac{\partial {\bf F}}{\partial {\bf z}} {\bf w}$.

The noise covariance matrix is $N' = \frac{\partial {\bf F}}{\partial {\bf z}} N \left(\frac{\partial {\bf F}}{\partial {\bf z}}\right)^{\top}$.

Extra computation is therefore needed to convert the observation covariance from $N$ to $N'$. The innovation vector $\mbox{\boldmath$\nu$}$, Jacobian matrix $H$ and observation covariance $N'$ are substituted into the Levenberg-Marquardt algorithm in place of their equivalents for the ${\bf h}$-type observation. There is no reason why there should not be a robust version of the ${\bf F}$-type observation, but currently it is not implemented.