Algorithm 1 The implementation of the PEKF-VB algorithm

1:Initialization ($k=0$): state estimation $x_0$ and associated error covariance $P_0$, the number of iterations. 2: Compute the predicted state $x_{k|k-1}$ and the associated error covariance $P_{k|k-1}$ $$\begin{array}{c}\hfill x_{k|k-1}=f_k\left(x_{k-1|k-1}\right),P_{k|k-1}=F_k{P}_{k-1|k-1}{F}_k^{\mathrm{T}}+Q_{k-1},\end{array}$$ where $F_k=\frac{\partial f_k\left(x\right)}{\partial x}{|}_{x=x_{k-1|k-1}}$. 3: Compute the filtering grain $K_k$ $$S_k=H_k{P}_{k|k-1}{H}_k^{\mathrm{T}}+R_k,K_k=P_{k|k-1}{H}_k^{\mathrm{T}}{\left(S_k\right)}^{-1},$$ where $H_k=\frac{\partial h_k\left(x\right)}{\partial x}{|}_{x=x_{k|k-1}}$. 4: Update the state estimation $x_{k|k}$ and the associated error covariance $P_{k|k}$ $$\begin{array}{c}\hfill x_{k|k}^{*}=x_{k|k-1}-K_k\left(z_k-h_k\left(x_{k|k-1}\right)\right),P_{k|k}^{*}=P_{k|k-1}-K_k{H}_k{P}_{k|k-1}.\end{array}$$ 5:Let $x_{k|k}^1=x_{k|k}^{*}$, $P_{k|k}^1=P_{k|k}^{*}$, and $i=1$. 6:while not converge do 7: Compute parameters ${\alpha }_k^{i+1}$ and ${\gamma }_k^{i+1}$ by Equations (18) and (19). 8: Compute iterated state estimation $x_{k|k}^{i+1}$ and its error covariance $P_{k|k}^{i+1}$ by Equations (26) and (27). 9: Let $i=i+1$. 10: end while 11:Let $k=k+1$, go back to Step 2.