The trick is that, even though the augmented state is uncontrollable, you still use it in the predictions, so the MPC algorithm can still compensate for it. Take a look at the before-last graph in the paper, see how that technique improves predictions after learning the real-time disturbances.
That's not what I was concerned about. The subspace that's uncontrollable is the disturbance components of the state, which I don't care to control anyway.
