On blocking zeros and strong stabilizability of linear multivariable systems

Al~lract--Motivated by a crucial role blocking zeros play in deciphering the strong stabilizability of a given system, a careful study of blocking zeros is undertaken here. After developing certain properties of blocking zeros and based on the multiplicity structure of invariant zeros, we identify what kind of invariant zeros are blocking zeros. For controllable and observable systems, an invariant zero is a blocking zero if and only if its geometric multiplicity is equal to the normal rank of the transfer function of the given system. This result leads to delineation of the class of controllable and observable time-invariant linear systems into two subclasses, (1) "simply SISO" systems whose normal rank is unity, and (2) "truly MIMO" systems whose normal rank is greater than unity. In a "simply SISO" system, every invariant zero is a blocking zero and hence a "simply SISO" system is not necessarily strongly stabilizable. On the other hand, a "truly MIMO" system with distinct invariant zeros does not have any blocking zeros and hence is always strongly stabilizable. Also, given any "truly MIMO" system, there always exists an arbitrarily small perturbation of its dynamic matrix such that the perturbed system has no blocking zeros and hence is strongly stabilizahle. In this sense, one can say that a MIMO system "almost always" has no blocking zeros and hence is "almost always" strongly stabilizable.