Simultaneous decoupling and pole placement without cancelling invariant zeros

In this paper the problem of simultaneous decoupling and pole placement without cancelling invariant zeros is considered. This is an important problem, especially in the case of unstable invariant zeros. It is shown that when the number of inputs and outputs is equal to the number of states, that there are no invariant zeros, thus, classical decoupling pole-placement methods can be used. Furthermore, in the case where the number of inputs and outputs is less than the number of states, a necessary and sufficient condition is derived which results in the cancellation of all invariant zeros, essentially again leading to the classical decoupling pole-placement solution. It is also demonstrated that the general problem of decoupling and pole placement without cancelling the invariants zeros can be solved for some examples, while in other cases no solution exists