Approximation Methods for FO-IMC Controllers for Time Delay Systems

Cristina I Muresan,Robin De Keyser,Isabela R Birs,Ioan Nascu,Ovidiu Prodan

doi:10.1051/e3sconf/201911501003

Abstract

Fractional Order Internal Model Control (FO-IMC) is among the newest trends in extending fractional calculus to the integer order control. Approximation of the FO-IMC is one of the key problems. Apart from this, when dealing with time delay systems, the time delay needs also to be approximated. All these approximations can alter the closed loop performance of the controller. In this paper, FO-IMC controllers will be tested in terms of the approximation accuracy. The case study is a first order system with time delay. Several scenarios will be considered, aiming for a conclusion regarding the choice of the approximation method as a function of the process characteristics, closed loop performance and FO-IMC fractional order. To approximate the time delay, two extensively used techniques will be considered, such as the series and Pade approximations. These will be compared to a novel approximation technique. An analysis of the test cases presented show that the series approximation proves more suitable in a single scenario, whereas the novel approximation method produces better results for the rest of the test cases.

Highlights

Fractional calculus represents the generalization of the integration and differentiation to an arbitrary real or complex order
The Fractional Order Internal Model Control (FO-internal model control (IMC)) controller has the advantage of increasing the robustness of the traditional IMC due to the supplementary tuning parameter involved, the fractional order
The key problem with FO-IMC controllers is represented by the approximation of the FO terms

Summary

Introduction

Fractional calculus represents the generalization of the integration and differentiation to an arbitrary real or complex order. The parameter Ns is a tuning knob This second step produces a vector of frequency response values of the FO discrete time transfer function. To show the effect of the time delay approximation method, as a function of the fractional order α and ratios λ/τ and λ/T, we consider the closed loop response to a unit step reference. The filter time constant has been chosen even smaller compared to the second example Notice that in all cases presented, λ has been chosen smaller than the corresponding process time constant This is due to the fact that a larger value for λ will result in a small closed loop bandwidth. The approximation error of the process time delay at high frequencies becomes of less importance because it occurs at frequencies, which are out of the passband

Pade approximation Novel approximation First order approximation

Conclusions