Step-ahead prediction covariance for a Kalman filter

02 Dec 2022

Given a discrete-time autonomous system

\[\begin{align} x_{k+1} &= A x_k + w_k \\ y_k &= C x_k + v_k \end{align}\]

with

\[\mathcal{E} \left( \begin{array}{c} w \\ v \end{array} \right) \left( \begin{array}{cc} w' & v' \end{array} \right) = \left[ \begin{array}{cc} Q & S \\ S' & R \end{array} \right]\]

and \(\mathcal{E} w = 0\), \(\mathcal{E} v = 0\).

Given a Kalman filter

\[\begin{align} \hat{x}_{k+1} &= A \hat{x}_k + K e_k \\ &= (A-KC) \hat{x}_k + K y_k\\ y_k &= C \hat{x}_k + e_k \\ e_k :&= y_k - C \hat{x}_k, \end{align}\]

the step-ahead state estimate error is

\[\begin{align} x_{k+1} - \hat{x}_{k+1} &= A x_k + w_{k+1} - A \hat{x}_k - K e_k \\ &= A(x_k - \hat{x}_k) + w_{k+1} - Ke_k \end{align}\]

and the output prediction error is

\[\begin{align} e_{k+1} :&= y_{k+1} - C\hat{x}_{k+1} \\ &= C (x_{k+1} - \hat{x}_{k+1}) + v_{k+1} \\ &= CA(x_k - \hat{x}_k) + CKe_k + Cw_{k+1} + v_{k+1} \end{align}\]

Using the \(\delta\)-correlated property of the noise, the covariance of the step-ahead output prediction is

\[\begin{align} \mathcal{E} e_{k+1} e_{k+1}' =& CAQA'C' + CQC' \\ & + R + CKRK'C' \\ & + + CASK'C' + CKS'A'C'. \end{align}\]