Algorithm 2: Adaptive Two-Stage Information Filter (TSIF).
1. Initialization:
${\widehat{x}}_0^{\mathrm{a}}\sim \mathbb{N}(x_0,P_0)$, ${\widehat{x}}_0^{\mathrm{u}}=0$, $F_0=0$, ${\Lambda }_0^{\mathrm{u}}={\omega }^{\mathrm{u}}{I}_p$, ${\widehat{u}}_0^{}=0$, ${\lambda }_k^{\mathrm{u}}\in [0,1]$.	(49)
2. Input: observation $y_k$
3. Recursive computation: For k = 1, 2, 3, …
(1). Information filtering (IF):
${\widehat{x}}_{k|k-1}^{\mathrm{a}}={\Phi }_k{\widehat{x}}_{k-1}^{\mathrm{a}}$,	(50)
$P_{k|k-1}^{}={\Phi }_k{P}_{k-1}^{}{\Phi }_k^T+Q_k$,	(51)
$N_k^{}=(H_k^T{R}_k^{-1}{H}_k+P_{k|k-1}^{-1})$,	(52)
${\widehat{x}}_k^{\mathrm{a}}=N_k^{-1}(H_k^T{R}_k^{-1}{y}_k+P_{k|k-1}^{-1}{\Phi }_k{\widehat{x}}_{k-1}^{\mathrm{a}})$,	(53)
$P_k^{}=N_k^{-1}$.	(54)
(2). Innovation and covariance:
$V_k=y_k-H_k({\Phi }_k{\widehat{x}}_{k-1}^{\mathrm{a}})$,	(55)
${\Sigma }_k=R_k^{}+H_k{P}_{k|k-1}^{}{H}_k^T$.	(56)
${\alpha }_k\approx \{\begin{array}{c}1\begin{array}{cc}& \begin{array}{ccc}& & \Delta {\tilde{v}}_k\le c\end{array}\end{array}\\ \frac{c}{\Delta {\tilde{v}}_k}\begin{array}{cc}& \begin{array}{cc}& \Delta {\tilde{v}}_k\ge c\end{array}\end{array}\end{array}\begin{array}{cc}& \begin{array}{cc}& \end{array}\end{array}$$,\Delta {\tilde{v}}_k={\left(\frac{{(e_j^{-})}^T{e}_j^{-}}{tr({\Sigma }_j)}\right)}^{\frac{1}{2}}$$, c\in [1,2.5]$.	(57)
(3). Correction:
$F_k=N_k^{-1}(P_{k|k-1}^{-1}{\Phi }_k{F}_{k-1}+P_{k|k-1}^{-1}{\Gamma }_k)$,	(58)
$D_k=(H_k{\Phi }_k{F}_{k-1}+H_k{\Gamma }_k)$,	(59)
${\tilde{\Psi }}_k={({\alpha }_k{\Sigma }_k+D_k{\tilde{\Lambda }}_{k-1}^{}{D}_k^T)}^{-1}$,	(60)
${\tilde{G}}_k={\tilde{\Lambda }}_{k-1}^{}{D}_k^T{\tilde{\Psi }}_k$,	(61)
${\tilde{\Lambda }}_k^{}={\alpha }_k^{-1}{\tilde{\Lambda }}_{k-1}^{}-{\alpha }_k^{-1}{\tilde{\Lambda }}_{k-1}^{}{D}_k^T{\tilde{\Psi }}_k{D}_k^{}{\tilde{\Lambda }}_{k-1}^{}$,	(62)
${\widehat{u}}_k={\widehat{u}}_{k-1}+{\tilde{G}}_k(V_k-D_k{\widehat{u}}_{k-1})$,	(63)
${\widehat{x}}_k^{\mathrm{u}}=N_k^{-1}(P_{k|k-1}^{-1}{\Phi }_k{\widehat{x}}_{k-1}^{\mathrm{u}}+P_{k|k-1}^{-1}{\Gamma }_k{\widehat{u}}_{k-1})+F_k({\widehat{u}}_k-{\widehat{u}}_{k-1})$.	(64)
(4). Modified state:
${\widehat{x}}_k^{}={\widehat{x}}_k^{\mathrm{a}}+{\widehat{x}}_k^{\mathrm{u}}$.	(65)
4. Output: ${\widehat{x}}_k^{}$ and ${\widehat{u}}_k$