Simplified models of aperiodic systems for PID control — Setting from an index test

PID control is the simplest and most widely used closed-loop control. It is suitable for aperiodic systems. Correct tuning, however, is a delicate matter, and requires the open-loop system to be modeled and identified. For this purpose, simplified models have been proposed, for example by Broïda or Strejc, which have few parameters that can be obtained from the index response.

We present here two other models applicable to aperiodic systems, whose parameters are obtained from an index response and which quickly lead to the setting of a PID corrector. The two models proposed $F\left(p\right)=\frac{K_0}{\left(1+{\tau }_1\cdot p\right){\left(1+{\tau }_2\cdot p\right)}^2}$ and $F\left(p\right)=\frac{K_0\cdot \left(1+{\tau }_n\cdot p\right)}{\left(1+{\tau }_1\cdot p\right){\left(1+{\tau }_2\cdot p\right)}^3}$ can represent a wide range of processes. They have only three or four parameters with low correlation. Each parameter has a physical meaning, making it easy to rule out aberrant identifications. From these parameters, we deduce the PID corrector setting best suited to the system.

The PID corrector is simplified to $C\left(p\right)=\frac{A\cdot {(1+T\cdot p)}^2}{T\cdot p\cdot \left(1+\frac{T}{r}\cdot p\right)}$ with only two parameters (A and T, since r ≈ 12 varies little). The aim is not to obtain optimal control, but only to achieve fast and satisfactory adjustment of the corrector. The tuning imposes closed-loop stability with...