Fractional order modeling and control for under-actuated inverted pendulum

Abstract

This paper finds a fractional order model (FOM) of an inverted pendulum system (IPS) based on simulated and experimental data. The traditional integer order model of the IPS is extended to fractional order one in this work. As a preliminary step, the estimation and optimization processes are carried out using the simulated data sets which were obtained from a reference simulated IPS prototype. The coefficients and the fractional differentiation orders of the proposed FOM have been estimated based on Sine Cosine Algorithm (SCA). The accuracy of the FOM is benchmarked against an identified integer order model (IOM). The comparison results show quite good congruence between the output of the FOM and that of the simulated IPS prototype. Practically, to demonstrate the advantages of using FOM in control system design, three fractional order PID (FOPID) controllers are designed according to the identified FOM, the identified IOM and the theoretical nonlinear model (TNM) using the same design scheme for a fair comparison. Furthermore, the FOM-based FOPID (FOPIDFOM) controller is compared to fractional order fuzzy PD controller with grey predictor (FOFPD-GP). Practical results show significantly improved performance using the FOPIDFOM controller; as a strong evidence of the FOM reliability in control system design.