Necessary and sufficient conditions for existence of decoupling controllers

It is well known that if a linear time-invariant plant is free from coincidences of poles and zeros in the right half-plane, then it can be decoupled with internal stability under unity-feedback configuration. We consider plants for which such coincidences do occur and give necessary and sufficient conditions under which stabilizing decoupling controllers exist. The conditions derived, based on transfer matrices and residues, are simple and straightforward.