On the algebraic geometry of the output feedback pole placement map

In this paper we study the question of pole placement by output feedback. Our approach makes use of the idea that a transfer function can be thought of as a curve in a Grassmannian manifold and the pole placement problem is that of finding a special hypersurface in this manifold (defined by the gain matrix) which intersects the given curve at a prescribed set of points. This is a classical inverse problem in algebraic geometry and, fortunately, near 100 years ago Hermann Schubert considered some enumerative aspects of such problems in connection with his development of a topic in algebraic geometry now known as the Schubert calculus. In this paper we use results in this area to show, for example, that in the case where the number of output feedback gains available (i.e. the product the number of inputs m with the number of outputs p equals the number of poles of the system) there are generically ϕ(m, p) = 1!2!...(p-1)!1!2!...(m-1)!(mp)!/1!2!...(m+p-1)! different (in general complex) gains which yield the same set of poles. This, rather unexpectedly large number emphasizes the nonlinear nature of the pole placement problem and suggests that it is probably rather difficult to solve algorithmically. It turns out that ϕ(m,p) is odd if and only if either min(m,p)=1 or min(m,p)=2 and max(m,p)=2k-1 and in these cases we are able to show that typically there exists at least one real solution to the pole placement problem.