The Method of Singular Frequencies for Robust Control Design in an Affine Parameter Space

Abstract

The Method of Singular Frequencies is an elegant tool for design of controllers in an affine parameter space. The essential aspects of the method were already introduced in [4], [1] and [2]. In [4] stable PID controlleres are designed by making use of the Theorem of Hermite-Biehler. It was shown that for a fixed proportional-term kp, convex stable polygons result. In [1] it was shown that actually the singular frequencies reside behind this 'mystery', and the method was extended also for sigma stability. In [2] the linear transformation of parameters was introduced to extend the PID design also for circle-stability. In this paper we generalize this approach for controllers in an affine parameter space. The concept of a singular Gamma region is introduced. Particularly, PID-controllers are studied in a systematic approach, as the first step towards the generalization of the method. A symbolic method for detection of stable polygons is introduced. Subsequently robust control of time delay systems is discussed. Finally the method is generalized for design of the robust affine controllers in a three-parameter-space. Gamma-space and gamma-vectors are introduced to derive the set of all singular Gamma regions for a given structure of the affine controller. Presently the method is more suited for the design of controllers, especially for digital control.