Modern design of classical controllers: Continuous-time first order controllers

This paper presents a new approach to the design of continuous-time first order controllers of the type C(s) = (x₁s + x₂)/(s + x₃) and the special case, Proportional-Integral (PI) controller, when x₃ = 0, C(s) = (x₁s + X₂)/s, respectively. The objective is to determine the x₁, x₂, and x₃ gains to satisfy a desired crossover frequency and closed-loop stability gain and phase margins. The design procedure is based on a simple parametrization consisting of ellipses and straight lines which represent constant magnitude and constant phase loci in the space of controller gains. The intersection of these figures give us the controller gains that satisfy the desired robust performance. It must be ensured that the selected design point is contained in the stabilizing set of controllers, which can be computed using recent methods. A numerical example and a power electronics application are given to illustrate and validate the proposed approach.