Kalman Filter - Fixed-lag Smoother

Fixed-lag Smoother

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The optimal fixed-lag smoother provides the optimal estimate of for a given fixed-lag using the measurements from to . It can be derived using the previous theory via an augmented state, and the main equation of the filter is the following:

\begin{bmatrix} \hat{\textbf{x}}_{t|t} \\ \hat{\textbf{x}}_{t-1|t} \\ \vdots \\ \hat{\textbf{x}}_{t-N+1|t} \\ \end{bmatrix} = \begin{bmatrix} I \\ 0 \\ \vdots \\ 0 \\ \end{bmatrix} \hat{\textbf{x}}_{t|t-1} + \begin{bmatrix} 0 & \ldots & 0 \\ I & 0 & \vdots \\ \vdots & \ddots & \vdots \\ 0 & \ldots & I \\ \end{bmatrix} \begin{bmatrix} \hat{\textbf{x}}_{t-1|t-1} \\ \hat{\textbf{x}}_{t-2|t-1} \\ \vdots \\ \hat{\textbf{x}}_{t-N+1|t-1} \\ \end{bmatrix} + \begin{bmatrix} K^{(0)} \\ K^{(1)} \\ \vdots \\ K^{(N-1)} \\ \end{bmatrix} y_{t|t-1}

where:

  • is estimated via a standard Kalman filter;
  • is the innovation produced considering the estimate of the standard Kalman filter;
  • the various with are new variables, i.e. they do not appear in the standard Kalman filter;
  • the gains are computed via the following scheme:
K^{(i)} = P^{(i)} H^{T} \left[ H P H^{T} + R \right]^{-1}
and
P^{(i)} = P \left[ \left[ F - K H \right]^{T} \right]^{i}
where and are the prediction error covariance and the gains of the standard Kalman filter (i.e., ).

If the estimation error covariance is defined so that

P_{i} := E \left[ \left( \textbf{x}_{t-i} - \hat{\textbf{x}}_{t-i|t} \right)^{*} \left( \textbf{x}_{t-i} - \hat{\textbf{x}}_{t-i|t} \right) | z_{1} \ldots z_{t} \right],

then we have that the improvement on the estimation of is given by:

P-P_{i} = \sum_{j = 0}^{i} \left[ P^{(j)} H^{T} \left[ H P H^{T} + R \right]^{-1} H \left( P^{(i)} \right)^{T} \right]

Read more about this topic:  Kalman Filter

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