Decoupling and disturbance rejection

For the linear multivariable system
\dot{x} = Ax + Bu + Er, z_{i} = D_{i}x(i \in k), x \in X
, with disturbance
\Gamma(\cdot)
, it is shown that the decoupling problem and disturbance rejection problem are simultaneously solvable, so as to yield a stable closed-loop system, if and only if
Im(E)\subset V_{g}^{*}
and
R_{i}^{*} + \ker D_{i} = X, i \in k
. Here
V_{g}^{*}
and
R_{i}^{*}
are subspaces uniquely defined by and constructible from the data
A, B, D_{i} i\in k
.