Placing Plenty of Poles is Pretty

We discuss the pole placement problem for single-input or multi-input control models of the form _ x = Ax + Bu. This is the problem of determining a linear state feedback of the form u = Fx such that in the closed-loop system _ x = (A + BF)x, the matrix A + BF has a prescribed set of eigenvalues. We analyze the conditioning of this problem and show that it is an intrinsically ill-conditioned problem, and especially so when the system dimension is large. Thus even the best numerical methods for this problem may yield very bad results. On the other hand, we also discuss the question of whether one really needs to solve the pole placement problem. In most circumstances what is really required is stabilization or that the poles are in a speciied region of the complex plane. This related problem may have much better conditioning. We demonstrate this via the example of stabilization.