Chapter 10: The Kalman Filter

Learning Objectives

• Formulate a discrete-time linear Gaussian state-space model and identify its matrices $\mathbf{F}$, $\mathbf{G}$, $\mathbf{H}$, $\mathbf{Q}$, $\mathbf{R}$.
• Derive the Kalman filter recursion from the orthogonality principle and the conditional Gaussian theorem.
• Propagate the prediction/filtering covariance through the matrix Riccati equation and recognize the information-form update.
• Characterize the steady-state filter via the discrete algebraic Riccati equation (DARE) and connect it to the Wiener filter.
• Implement the Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) for mildly nonlinear tracking problems and understand their failure modes.