Design of robust PID controllers

The family of characteristic polynomials of a SISO-PID loop with N representative plant operating conditions is P_i = Ai(s)(K_I; + Kps + K_Ds²) + B_i(s), i = 1,2...N. A basic task of robust control design is to find the set of all parameters K_I, K_P, K_D, that simultaneously place the roots of all P_i(a) into a specified region Γ in the complex plane. For Γ in form of the left half plane it is known that the simultaneously stabilizing region in the (K_D, K_I)-plane consists of one or more convex polygons. This fact simplifies the tomographic rendering of the nonconvex set of all simultaneously stabilizing PID controllers by gridding of K_P. A similar result holds if Γ is the shifted left half plane. In the present paper it is shown 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 Γ-region. A parameter space approach shows, that the roots of a polynomial P_i(s) with fixed K_P can cross the imaginary axis in three ways i) at zero, ii) at infinity, iii) at a finite number of singular frequencies ω_k, k = 1,2,3... M. The singular frequencies ω_k(K_p) are first determined as roots of a polynomial. For each ω_k then a straight line with positive slope ω²_k is a boundary in the (K_I, K_D)-plane, where a pair of roots of P_i(s) crosses the imaginary axis at s = ±ω_k. Finally the resulting stable polygons are selected. Computationally the main task is the factorization of a polynomial for finding ω_k(K_P). This step can be avoided by evaluating the inverse function K_P(ω_k), which is explicitely given. - he results are illustrated by the design of an additional PID controller for improved performance of a robustly decoupled car steering control system.