Thanks jessegrabowski. You understand the system correctly, and yes it can be represented by a linear state-space model with a switching input matrix, or with a time-varying discrete parameter in the input matrix, \varphi_t, as you show (I think the top row in your R_t matrix should be zeros though). Not sure how this helps though.
And yes, it is a hidden Markov model, so I will check out this example you linked to, and get back to you. Thanks very much!
(b.t.w. I didn’t realize you can use latex in this forum, if I’d known that I might have written the question more clearly, sorry)