Decoupling and disturbance rejection

Abstract
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 .

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