\begin{eqnarray} X_k &=& F X_{k-1}+ \omega_k \nonumber \\ Z_k &=& H X_k + \nu_k \end{eqnarray}
The first equation is state exploration equation and second one measurement equation. Point to be noted, I dont have $Bu$ term in my state equation unlike common state exploration equation. Here, I was trying to generate the data (forward problem) to Kalman filter application using both the above equations with known process noise $\omega_k$ and measuremenet noise $\nu_k$. First I compute $Xk$ and then plug into second equation to calculate $Z_k$.
During the synthetic data generation, my $X_k$ values are going to NaN as it is recusive. Could some one help me to understand the problem and also any suggestion on Kalman filter examples where the data is generated via these two equations and then a Kalman filter is used to estimate the states $\hat{X}_k$? The main goal is use the known $\omega_k$ and $\nu_k$ in forward problem data generation and then use the same values in Q and R matrix to solve with Kalman filter.
I looked into several examples in the internet and I see data is randomly generated and the it is solved with Kalman filter to estimate the states by randomly creating/tuning Q and R matrix.