Equivalence between proportional integral observer and augmented Kalman filter

The relationship between the proportional integral observer (PIO) and the augmented Kalman filter (AKF) is addressed in this paper. A general PIO is proposed for linear stochastic systems with unknown disturbance affecting both the system state and the output. The proposed PIO is optimal in the minimum variance unbiased sense. We prove that PIO is equivalent to AKF when the disturbance is assumed to be a constant or a stochastic disturbance with known statistics. Stability conditions of the proposed PIO and AKF are also provided.