Table A1.

Summary of the adaptive estimation algorithm.

Initialization

Initialize Ȃ0 and ${\widehat{P}}_1^{-}$, λ, R0

Forward pass

1. For n from 1 to N

2. Construct the data history matrix Hn

3. Compute: one step prediction of observation: ${\stackrel{⌢}{x}}_n=H_n{\stackrel{⌢}{A}}_n^{-}$

4. Compute residuals: rn = xn − x̑n

5. Update the covariance matrix: $R_n=(1-\lambda )R_{n-1}+\lambda r_n{r}_n^T$, and compute $R_n^{\prime }$ according to Eq. (A8)

6. Compute Kalman Gain: $K_n={\stackrel{⌢}{P}}_n^{-}{H}_n^T/\left(H_n{\stackrel{⌢}{P}}_n^{-}{H}_n^T+R_n^{\prime }\right)$

7. Compute state estimate: Ȃn = Ȃn−1 + Knen

8. Compute estimated state covariance: ${\stackrel{⌢}{P}}_n={\stackrel{⌢}{P}}_n^{-}-K_n{H}_n{\stackrel{⌢}{P}}_n^{-}$

9. Update state noise covariance: Wn = λPn

10. Calculate a priori estimated state covariance: ${\stackrel{⌢}{P}}_{n+1}^{-}={\stackrel{⌢}{P}}_n+W_n$

END

Backward pass

For n from N − 1 to 1

Let $S_n=\widehat{P}n/{\stackrel{⌢}{P}}_{n+1}^{-}$

1. Compute smoothed estimated state covariance: $${\widehat{P}}_n^s={\stackrel{⌢}{P}}_n+S_n\left({\widehat{P}}_{n+1}^s-{\widehat{P}}_{n+1}^{-}\right)S_n^T$$

2. Compute smoothed state estimate: ${\stackrel{⌢}{A}}_n^s={\stackrel{⌢}{A}}_{n-}{S}_n\left({\stackrel{⌢}{A}}_{n+1}^s-{\stackrel{⌢}{A}}_n\right)$

END

Note, the initial smoothed estimate at step N for the backward pass is the final state estimate for the forward pass: ${\stackrel{⌢}{A}}_N^s={\stackrel{⌢}{A}}_N$

Similarly, ${\widehat{P}}_{n+1}^s={\stackrel{⌢}{P}}_N$