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
D
s
²
) + 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 ω
²
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. The results are illustrated by the design of an additional PID controller for improved performance of a robustly decoupled car steering control system.