Adaptive Kalman filtering with control input estimation

The application of Kalman filter to a linear stochastic system requires complete information regarding the system's dynamics and observations. A major disadvantage of the Kalman filter is that it is sensitive to erroneous modeling of the system. In practice there are problems associated with unknown control inputs. In this case the solution provided by the Kalman filter is suboptimal. One way to improve the filter's performance is to estimate the unknown control inputs. The estimate is then fed back to the Kalman filter to improve its behavior. When the estimated control inputs are sent back to the Kalman filter, their estimation errors also have the tendency to increase the filter's estimation uncertainty, which, in turn, will further degrade the system's performance. By manipulating the system's observability matrix, one is able to present a new algorithm in which the degraded state estimate generated by the Kalman filter is not involved in the estimation of the unknown control inputs, so that the accumulation of the estimation errors does not have any chance to take place in our adaptive estimation system.