# Yousaf A. Alarfaj

A case study of fractional-order control

Lancaster University, Doctor of Philosophy, 2021

Supervisor: C.J. Taylor

## Abstract

This thesis concerns fractional-order (non-integer) methods for control system design. Although fractional-order calculus has a long history in mathematics and engineering, the uptake of relevant fractional-order concepts in control systems research has been relatively slow, and interest in the topic remains comparatively lowâ€”albeit with some important exceptions, as highlighted by the literature review of this thesis.

The first part of the thesis considers fractional-order methods for modelling and control in quite broad terms, before later focusing on one particular approach from the control systems literature, namely Fractional-order Generalised Predictive Control (FGPC). The FGPC approach is of particular interest here because of its relationship with the well-known, conventional control algorithm, namely Generalised Predictive Control (GPC). Both algorithms have a relatively straightforward implementation form, making them attractive to practitioners.

Hence, one contribution of the thesis is to use worked examples in MATLAB as an introduction to GPC and FGPC design methods, in part for tutorial reasons. More significantly, the thesis demonstrates how fractional-order methods are utilised to increase control design flexibility. In this regard, the thesis investigates both conventional GPC and FGPC methods using various simulation examples. The robustness of control systems is investigated via Monte Carlo simulation, with consideration of model mismatch and unmeasured disturbances. These results are utilised to develop recommendations for how to optimise the extra design coefficients introduced in the fractional-order case.

The comparative study is extended to a laboratory example, namely the control of airflow in a 1 m by 2 m by 2 m forced ventilation environmental test chamber. To facilitate further uptake of FGPC methods in the future, the algorithms developed are prepared as a MATLAB toolbox, i.e. a collection of functions that calculate and implement the FGPC approach and subsequently measure the performance of the controller.