My understanding of your definition is that there are two independent noises, \epsilon_1 and \epsilon_2, and the R_t matrix determines which one feeds the system (depending on whether \varphi is 0 or 1). But whichever noise is active, it should feed the second state, \eta_t not the a_t. So shouldn’t R_t be:
\begin{bmatrix}
0 & 0 \\
\varphi_t & (1-\varphi_t)
\end{bmatrix}
\begin{bmatrix}
\epsilon_{t,1} \\
\epsilon_{t,2}
\end{bmatrix}
which produces
\begin{bmatrix}
0 \\
\varphi_t \epsilon_{t,1} + (1-\varphi_t) \epsilon_{t,2}
\end{bmatrix}
Thus
\eta_t = \eta_{t-1} + \omega_t = \eta_{t-1} + \varphi_t \epsilon_{t,1} + (1-\varphi_t) \epsilon_{t,2}
Anyway, I might be wrong, it’s not important to the question.
The HMM would have the following transition matrix
\Pi_{i,j} = \begin{bmatrix}
1-p & p\\
1-p & p
\end{bmatrix}
where p is the shock probability, e.g. p=0.01 (I labelled it epsilon in the question).
At the moment I’m having problems running the HMM example linked above. I’m getting errors but trying to figure them out…