Chapter Summary

Chapter Summary

Key Points

• 1.

  The discrete-time linear Gaussian state-space model $\mathbf{x}[n+1] = \mathbf{F}\mathbf{x}[n] + \mathbf{G}\mathbf{w}[n]$, $\mathbf{y}[n] = \mathbf{H}\mathbf{x}[n] + \mathbf{v}[n]$ with white Gaussian $\mathbf{w},\mathbf{v}$ admits a recursive closed-form MMSE estimator — the Kalman filter.

• 2.

  The Kalman recursion alternates a prediction step (propagate the mean and covariance through the dynamics) and an update step (incorporate the new observation through the Kalman gain $\mathbf{K}[n] = \mathbf{P}[n|n-1]\mathbf{H}^T(\mathbf{H}\mathbf{P}[n|n-1]\mathbf{H}^T + \mathbf{R})^{-1}$).

• 3.

  The innovation $\mathbf{y}[n] - \mathbf{H}\hat{\mathbf{x}}[n|n-1]$ is a white Gaussian sequence under the model. This is the state-space realization of the innovations representation from the Wiener theory of Chapter 9.

• 4.

  For time-invariant $(\mathbf{F},\mathbf{H},\mathbf{Q},\mathbf{R})$ the prediction covariance converges to the fixed point of the discrete algebraic Riccati equation (DARE) under detectability and stabilizability. The steady-state Kalman filter is LTI and coincides with the causal Wiener filter for the same signal model.

• 5.

  For nonlinear dynamics/observations, the EKF linearizes around the running estimate and the UKF propagates sigma points through the nonlinearity. Both are approximations; they lose optimality and can diverge when the nonlinearity is strong or the state distribution is multimodal. Particle filters provide a Monte Carlo alternative at higher computational cost.