Controllability of sum linear systems

For two linear systems $S_1$ and $S_2$ described by ordinary matrix difference equations with disjoint sets of solutions $M_1$ and $M_2$ we define the (disjoint) sum system, the set of whose solutions is the sum of $M_1$ and $M_2$. We obtain a controllability criterion for the sum system in the form of conditions on the coefficients of $S_1$ and $S_2$.