Abstract :
The family of characteristic polynomials of a SISO-PID loop with N representative plant operating conditions is Pi = Ai(s)(KI; + Kps + KDs2) + Bi(s), i = 1,2...N. A basic task of robust control design is to find the set of all parameters KI, KP, KD, that simultaneously place the roots of all Pi(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 (KD, KI)-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 KP. 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 Pi(s) with fixed KP can cross the imaginary axis in three ways i) at zero, ii) at infinity, iii) at a finite number of singular frequencies ωk, k = 1,2,3... M. The singular frequencies ωk(Kp) are first determined as roots of a polynomial. For each ωk then a straight line with positive slope ω2k is a boundary in the (KI, KD)-plane, where a pair of roots of Pi(s) crosses the imaginary axis at s = ±ωk. Finally the resulting stable polygons are selected. Computationally the main task is the factorization of a polynomial for finding ωk(KP). This step can be avoided by evaluating the inverse function KP(ωk), 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.
Keywords :
automobiles; control system synthesis; inverse problems; multivariable control systems; polynomials; robust control; set theory; steering systems; three-term control; SISO control system; SISO-PID loop; characteristic polynomials; convex polygons; geometric property; inverse function; nonconvex set; parameter space approach; polynomial factorization; polynomial roots; representative plant operating conditions; robust PID controller design; robustly decoupled car steering control system; simultaneously stabilizing PID controllers; singular frequency; stable polygon; tomographic rendering; Aerospace electronics; Control systems; Polynomials; Robustness; Stability analysis; Tomography; PID control; parameter space; robustness;