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Complex Fractional-Order LQIR for Inverted-Pendulum-Type Robotic Mechanisms: Design and Experimental Validation

Saleem, Omer; Abbas, Faisal; Iqbal, Jamshed


Omer Saleem

Faisal Abbas


This article presents a systematic approach to formulate and experimentally validate a novel Complex Fractional Order (CFO) Linear Quadratic Integral Regulator (LQIR) design to enhance the robustness of inverted-pendulum-type robotic mechanisms against bounded exogenous disturbances. The CFO controllers, an enhanced variant of the conventional fractional-order controllers, are realised by assigning pre-calibrated complex numbers to the order of the integral and differential operators in the control law. This arrangement significantly improves the structural flexibility of the control law, and hence, subsequently strengthens its robustness against the parametric uncertainties and nonlinear disturbances encountered by the aforementioned under-actuated system. The proposed control procedure uses the ubiquitous LQIR as the baseline controller that is augmented with CFO differential and integral operators. The fractional complex orders in LQIR are calibrated offline by minimising an objective function that aims at attenuating the position-regulation error while economising the control activity. The effectiveness of the CFO-LQIR is benchmarked against its integer and fractional-order counterparts. The ability of each controller to mitigate the disturbances in inverted-pendulum-type robotic systems is rigorously tested by conducting real-time experiments on Quanser single-link rotary pendulum system. The experimental outcomes validate the superior disturbance rejection capability of the CFO-LQIR by yielding rapid transits and strong damping against disturbances while preserving the control input economy and closed-loop stability of the system.


Saleem, O., Abbas, F., & Iqbal, J. (2023). Complex Fractional-Order LQIR for Inverted-Pendulum-Type Robotic Mechanisms: Design and Experimental Validation. Mathematics, 11(4), Article 913.

Journal Article Type Article
Acceptance Date Feb 7, 2023
Online Publication Date Feb 10, 2023
Publication Date Feb 2, 2023
Deposit Date Mar 10, 2023
Publicly Available Date Mar 13, 2023
Journal Mathematics
Electronic ISSN 2227-7390
Publisher MDPI
Peer Reviewed Peer Reviewed
Volume 11
Issue 4
Article Number 913
Keywords LQIR; Fractional-order control; Complex fractional orders; Differentiation operator; Integration operator; Quanser single-link rotary pendulum
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Copyright Statement
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (

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