A note on the structure of dynamic stabilizing controllers for linear time-invariant systems

For a given plant-compensator pair in closed-loop, it is shown that, if a simple matrix equation holds, the closed-loop system behaves like a functional observer of a linear function of the state of the plant. If such a function can be used to augment the plant output, it is possible to use a static output feedback from the augmented output, thus obtaining as a closed-loop system the restriction of the original one to a suitable invariant subspace. A dual problem is also considered.