Algorithm 1: Two-Stage Information Filter (TSIF).
1. Initialization:
${\widehat{x}}_0^{\mathrm{a}}\sim \mathbb{N}(x_0,P_0)$,${\widehat{x}}_0^{\mathrm{u}}=0$,$F_0=0$,${\Lambda }_0^{\mathrm{u}}={\omega }^{\mathrm{u}}{I}_p$,${\widehat{u}}_0^{}=0$.  2. Input: observation $y_k$	(30)
3. Recursive computation: For k = 1, 2, 3, …
(1). Information filtering (IF):
${\widehat{x}}_{k|k-1}^{\mathrm{a}}={\Phi }_k{\widehat{x}}_{k-1}^{\mathrm{a}}$,	(31)
$P_{k|k-1}^{}={\Phi }_k{P}_{k-1}^{}{\Phi }_k^T+Q_k$,	(32)
$N_k^{}=(H_k^T{R}_k^{-1}{H}_k+P_{k|k-1}^{-1})$,	(33)
${\widehat{x}}_k^{\mathrm{a}}=N_k^{-1}(H_k^T{R}_k^{-1}{y}_k+P_{k|k-1}^{-1}{\Phi }_k{\widehat{x}}_{k-1}^{\mathrm{a}})$,	(34)
$P_k^{}=N_k^{-1}$.	(35)
(2). Innovation and covariance:
$V_k=y_k-H_k({\Phi }_k{\widehat{x}}_{k-1}^{\mathrm{a}})$,	(36)
${\Sigma }_k=R_k^{}+H_k{P}_{k|k-1}^{}{H}_k^T$.	(37)
(3). Correction:
$F_k=N_k^{-1}(P_{k|k-1}^{-1}{\Phi }_k{F}_{k-1}+P_{k|k-1}^{-1}{\Gamma }_k)$,	(38)
$D_k=(H_k{\Phi }_k{F}_{k-1}+H_k{\Gamma }_k)$,	(39)
${\Psi }_k={({\Sigma }_k+D_k{\Lambda }_{k-1}^{}{D}_k^T)}^{-1}$,	(40)
$G_k={\Lambda }_{k-1}^{}{D}_k^T{\Psi }_k$,	(41)
${\Lambda }_k^{}={\Lambda }_{k-1}^{}-{\Lambda }_{k-1}^{}{D}_k^T{\Psi }_k{D}_k^{}{\Lambda }_{k-1}^{}$,	(42)
${\widehat{u}}_k={\widehat{u}}_{k-1}+G_k(V_k-D_k{\widehat{u}}_{k-1})$,	(43)
${\widehat{x}}_k^{\mathrm{u}}=N_k^{-1}(P_{k|k-1}^{-1}{\Phi }_k{\widehat{x}}_{k-1}^{\mathrm{u}}+P_{k|k-1}^{-1}{\Gamma }_k{\widehat{u}}_{k-1})+F_k({\widehat{u}}_k-{\widehat{u}}_{k-1})$.	(44)
(4). Modified state:
${\widehat{x}}_k^{}={\widehat{x}}_k^{\mathrm{a}}+{\widehat{x}}_k^{\mathrm{u}}$.	(45)
4. Output: ${\widehat{x}}_k^{}$ and ${\widehat{u}}_k$