Designing a Kalman filter when no noise covariance information is available

A problem when designing Kalman filters using first principlesmodels is often that these models lack a description of the noisesthat affect the states and the measurements. In these cases, theKalman filter has to be estimated from data. For this purpose manyalgorithms have been presented in the literature. All methods inthe literature assume that the system under consideration has anobservability matrix that has no small singular values. In thispaper it will be shown that small singular values can lead to poorperformance of estimated Kalman filters. Also a method will beintroduced for estimating the Kalman filter in the case that thesystem has small singular values. This method is able to constructa good filter, even if the first principles model is badlyobservable.