The reference for the below equations can be found in the Link . Note that $k$ is the timestamp and $i$ is the $i^{th}$ entry of a vector or $(i,i)^{th}$ entry of a matrix, $F$ in this case
Equation 1
\begin{equation}
\tilde{x}_{i}(k+1) = a_{i}(k) + b_{i}(k)
\end{equation}
where
\begin{equation}
\begin{aligned}
a_{i}(k) &= \alpha_{i}x_{i}(k) + (1-\alpha_{i})\tilde{x}_{i}(k) \\
b_{i}(k) &= \beta[a_{i}(k) - a_{i}(k-1)] + (1-\beta_{i})b_{i}(k-1) \\
\end{aligned}
\end{equation}
Equation 1 can also be written using the dynamic model as
\begin{equation} \begin{aligned} \tilde{x}_{i}(k+1) &= F_{i}(k)x_{i}(k) + G_{i}(k) \end{aligned} \end{equation}
From the above equations we have
\begin{equation} \begin{aligned} F_{i}(k) &= \alpha_{i}(1+\beta_{i}) \\ G_{i}(k) &= (1+\beta_{i})(1-\alpha_{i})\tilde{x}_{i}(k) - \beta_{i}a_{i}(k-1)+(1-\beta_{i})b_{i}(k-1) \end{aligned} \end{equation}
I've the data, .i.e the x(k) which is time series data (k = 1,2,$\dots$,n) and I want to estimate the values of $\alpha$ and $\beta$. In my case the values of $\alpha$ and $\beta$ are assumed to be constant throughout the interval $n$. I looked online but couldn't find any suitable reference. Please help.