Assigning Frequencies via Determinantal Equations: New Counterexamples and Invariants

The problem of arbitrary pole-zero assignment by fixed structure compensators has been addressed so far in terms of either necessary or sufficient conditions. The strongest existing condition is based on the rank of the differential of a related map on a degenerate controller and holds true generically when the degrees of freedom of the compensator exceed the number of frequencies to be assigned (mp>n for the output feedback problem). The complete (nongeneric) solvability of the problem is still open, even when complex controllers are considered. A simple necessary solvability condition is that the linear map, appearing as a factor of the main determinantal map, defining the assignment problem, is onto. Here we examine the special problem of assignment of matrix pencil zeros via diagonal perturbations and we present a new necessary and sufficient condition for complex solvability in terms of a new invariant involving the minors of this linear map. Based on this result, we demonstrate that for the important case where the degrees of freedom of the controller are equal to the number of frequencies to be assigned, the surjectivity of this linear map although it is necessary, it is not sufficient for the solvability of the problem