A solution to the state estimation problem of systems with unmeasurable non-zero mean inputs/disturbances, which do not satisfy the disturbance decoupling conditions, is given using the Kalman filtering and Bayesian estimation theory. The proposed estimation algorithm, named Supervisory Kalman Filter (SKF), consists of a Kalman filter with an extra update step which is inspired by the particle filtering technique. The extra step, called supervisory layer, numerically solves the measurement equations for the portion of the state vector that cannot be estimated by the Kalman filter. First, it produces N randomly generated state vectors, the particles, which are distributed based on the Kalman filter’s last updated estimate. Then, the estimated measurement vector associated with each particle is compared to the actual measurement vector to identify the particle’s probability to be a solution. Finally, a so-called resampling stage is implemented to refine the particles with higher likelihoods. The effectiveness of the SKF is demonstrated by comparing its estimation performance with that of the standard Kalman Filter and the particle filter for a vehicle state estimation problem. The estimation results confirm that the SKF precisely estimates those states of the vehicle that cannot be estimated by either the Kalman filter or the particle filter, regardless of the unknown disturbances from the road. The filtering methodology offered in the article has a potential to improve performance of the systems presented in the patents WO2011115960, WO2010024751, US8073528, and US20110299730.