Hidden Markov Model with control input?

Given that HMMs and Kalman Filters are both state space models with recursive aspects and KFs allow for control inputs to influence the prediction at each step, I'm wondering if any similar methods exist for HMMs.

Let's say I have a 2-state HMM that takes continuous variables A and B as observations, but I know that variable C is associated with transitions. For instance, transitions to S1 (state 1) tend to be preceded by high values of C and transitions to S2 tend to be preceded by low values of C.

So below, the blue line (C) tends to tick up prior to a transition to yellow and ticks down prior to a transition to purple, however, in the long run, there's no difference in the value of C in S1 versus S2. For simplicity, the observation variables A and B are not shown.

If I'm interested in knowing in advance when the transitions will occur, the only simple thing I can think of is to come up with a formula to alter the values of A and B at each step based on the value of C.

In theory, I could make any kind of model that I wanted. If I wanted to use a neural network to generate a transition matrix conditioned on C or other variables, I think that could be done, but something like that is a little over my head at the moment.

So I'm just wondering - are there standard methods for HMMs to address variables that are predictive of transitions rather than simply indicative of states, similar to how control inputs influence the prediction step but not the measurement step with Kalman Filters?