Estimating Kalman Filter with dummy state variables?

I'm trying to estimate a state space model where the state equations are of the form:

Because of the unusual dependence structure, I'm including dummy variables in the matrix representation. I also need to estimate the variance of the shocks, so following the matlab suggestion https://www.mathworks.com/help/ident/ug/identifying-state-space-models-with-independent-process-and-measurement-noise.html , using greyest with a Kalman filter in the ODE function to estimate K (K the disturbance vector in the innovations form of the state space model, such that ). The page suggests that K can be taken such that .

My question is, how do I enforce that the dummy variables have no disturbance components, and that takes the same shock as that from ? If I don't use greyest, I can estimate the appropriate K, but I need greyest to restrict the covariances. I'm stuck on how to do both.