Describing the H∞ set in the controller parameters space

The aim of this paper is to give an instrument for describing the regions in the controller parameter space such that the H_∞-norm of the closed loop transfer function of a given system is less than a prescribed level. In this note we consider a particular family of linear control systems, comprising also plant with PID controller. We show that for such systems the problem can be easily tackled with the analysis of non-homogeneous quadratic forms in the parameters space, so that first the general L_∞ region can be well approximated, and then the H_∞ subregion distinguished. We present a method for both the description of such a region and of its border, and the proposed approach is valid for continuous-time and discrete-time systems. In case of two parameters controller a nice graphical analysis may be performed. Finally some examples are provided in order to explore the technique proposed.