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A Schmidt filter is a modification of the Kalman filter that allows to append system parameters as states and considers their uncertainty effect in the filtering process without attempting to estimate such parameters. The states that are only considered but not estimated, are generally known as \textit{consider} or \textit{considered} states. The main contributions of this research are the formulations of a Schmidt-Kalman filter that incorporates the numerical robustness of the well-known square root and factorized filtering forms plus the capacity of actively attempting to update the \textit{considered} states. The filters formulations proposed in this research are a fundamental extension of the Kalman filter. Therefore, the formulations of this work also apply within the Extended Kalman filter framework. More importantly, they are shown to handle nonlinearities, larger initial uncertainties, and poorly conditioned systems better than a typical Extended or Schmidt Kalman filter. Because the new filters are directly based on the Schmidt filter, they offer a novel and straight-forward filtering framework, allowing the use of a more simple filter where a more advanced or elaborated technique could have been needed.