On the stabilizability of multivariable systems by minimum order compensation

In this paper, we derive the necessary condition, mp ?? n, for stabilizability by constant gain feedback of the generic degree n, p ?? m system. This follows from another of our main results, which asserts that generic stabilizability is equivalent to generic solvability of a deadbeat control problem, provided mp ?? n. Taken together, these conclusions enable us to make some sharp statements concerning minimum order stabilization. The techniques are primarily drawn from decision algebra and classical algebraic geometry and have additional consequences for problems of stabilizability and pole-assignability. Among these are the decidability (by a Sturm test) of the equivalence of generic pole-assignability and generic stabilizability, the semi-algebraic nature of the minimum order, q, of a stabilizing compensator, and the nonexistence of formulae involving rational operations and extraction of square roots for pole-assigning gains when they exist, answering in the negative a question raised by Anderson, Bose, and Jury.