Causal polynomial pole assignment of discrete LTI systems by using deadbeat modes

For a SISO discrete LTI process, given by its transfer function in the z-domain, the pole assignment problem is solvable for any desired polynomial, when the numerator and denominator of the process are coprime. However, in general the calculated controller is noncausal, i.e. the control algorithm cannot be executed in real time. The paper provides conditions on the degrees of the given polynomials under which causal solutions exist. For those cases, when no causal controllers exist, a (minimal) number of poles at zero is added to the desired poles, such that we can find causal controllers for the modified problem. This supplement to the originally incorrect problem can be seen as weakest modification yielding causal solutions.