The Graded Hautus Test of Controllability for 2-D Systems

It is well-known that the Hautus test of controllability of 2-D systems results in two distinct cases of controllability, namely, when the cancelation variety is empty and when it is zero-dimensional. While the case of the empty cancelation variety has a distinctive trajectory level interpretation – i.e., existence of an observable image representation – the case of the zero-dimensional cancelation variety lacks such an interpretation. We show in this letter that the idea of factoring the system’s equations by a linear principal ideal of equations provides this missing trajectory level interpretation.