Design of Robust PID Controllers

Abstract

The family of characteristic polynomials of a SISO-PID loop with N representative plant operating conditions is Pi = Ai(s) (K1 + Kps+KDs2) + Bi(s), i=1,2..N. A basic task of robust control design is to find the set of all parameters K1, Kp, KD, that simultaneously place the roots of all Pi(s) into a specified region in the complex plane. For in form of the left half plane it is know that the simultaneously stabilizing region in the (KD,KI)-plane consists of one or more convex polygons. This fact simplifies the tomograhic rendering of the nonconvex set of all simultaneously stabilizing PID controllers by gridding of Kp. A similar result holds if is the shifted left half plane. In the present paper it is show that the nice geometric property also holds for circles with arbitrary real center and radius. It is further shown, that it cannot hold for any other region.