We propose a Bias-Aware Kalman Filtering (BKF) algorithm for simultaneous state and parameter estimation in discrete-time, imperfect dynamical systems. Kalman filtering (KF) is a Gaussian recursive method for state estimation based on measurements and a state evolution model. When both state and parameter estimation are required, a classical approach is to combine two Kalman filters, known as dual Kalman filtering [3, 4, 5]. To enhance robustness against model bias (imperfections) and measurement noise, we introduce a new version of this approach, called the Bias-Aware Kalman Filter (BKF). This method integrates a Parameterized Background Data Weak (PBDW)[6] as the new observer [1, 2], which estimates the bias introduced by non-modeled physics. The model parameters are then updated using inverse tracking based on the PBDW observer, while the evolving system state is sequentially estimated through Kalman filtering. We demonstrate the effectiveness of this algorithm in a real-time data assimilation task of optimizing a drone trajectory.