Stabilization and tracking control of underactuated ball and beam system using metaheuristic optimization based TID-F and PIDD2–PI control schemes

PLOS ONE RESEARCH ARTICLE Stabilization and tracking control of underactuated ball and beam system using metaheuristic optimization based TID-F and PIDD2–PI control schemes Farhan Zafar1, Suheel Abdullah Malik1, Tayyab Ali1, Amil Daraz ID2*, Abdul Rahman Afzal3, Farkhunda Bhatti4, Irfan Ahmed Khan5 a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 1 Department of Electrical and Computer Engineering, Faculty of Engineering and Technology, International Islamic University Islamabad (IIUI), Islamabad, Pakistan, 2 School of Information Science and Engineering, NingboTech University, Ningbo, China, 3 Department of Industrial Engineering, University of Business and Technology (UBT) University, Jeddah, Saudi Arabia, 4 Department of Electronic Engineering, Mehran University of engineering & Technology Jamshoro, Jamshoro, Pakistan, 5 Department of Electrical Engineering, Faculty of Engineering, Universiti Malaya, Kuala Lumpur, Malaysia * OPEN ACCESS Citation: Zafar F, Malik SA, Ali T, Daraz A, Afzal AR, Bhatti F, et al. (2024) Stabilization and tracking control of underactuated ball and beam system using metaheuristic optimization based TID-F and PIDD2–PI control schemes. PLoS ONE 19(2): e0298624. https://doi.org/10.1371/journal. pone.0298624 Editor: Lalit Chandra Saikia, National Institute of Technology Silchar, India, INDIA Received: December 7, 2023 Accepted: January 26, 2024 Published: February 14, 2024 Copyright: © 2024 Zafar et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All relevant data are within the manuscript and its Supporting information files. Funding: The author(s) received no specific funding for this work.; Abstract In this paper, we propose two different control strategies for the position control of the ball of the ball and beam system (BBS). The first control strategy uses the proportional integral derivative-second derivative with a proportional integrator PIDD2-PI. The second control strategy uses the tilt integral derivative with filter (TID-F). The designed controllers employ two distinct metaheuristic computation techniques: grey wolf optimization (GWO) and whale optimization algorithm (WOA) for the parameter tuning. We evaluated the dynamic and steady-state performance of the proposed control strategies using four performance indices. In addition, to analyze the robustness of proposed control strategies, a comprehensive comparison has been performed with a variety of controllers, including tilt integral-derivative (TID), fractional order proportional integral derivative (FOPID), integral–proportional derivative (I-PD), proportional integral-derivative (PI-D), and proportional integral proportional derivative (PI-PD). By comparing different test cases, including the variation in the parameters of the BBS with disturbance, we examine step response, set point tracking, disturbance rejection analysis, and robustness of proposed control strategies. The comprehensive comparison of results shows that WOA-PIDD2-PI-ISE and GWO-TID-F- ISE perform superior. Moreover, the proposed control strategies yield oscillation-free, stable, and quick response, which confirms the robustness of the proposed control strategies to the disturbance, parameter variation of BBS, and tracking performance. The practical implementation of the proposed controllers can be in the field of under actuated mechanical systems (UMS), robotics and industrial automation. The proposed control strategies are successfully tested in MATLAB simulation. Competing interests: The authors have declared that no competing interests exist Abbreviations: αL(t), Load gear; β, Angle of Beam; θ, Rotation angle with the wheel; d, Offset of the PLOS ONE | https://doi.org/10.1371/journal.pone.0298624 February 14, 2024 1 / 32 PLOS ONE lever arm; Fd, Translational force generated by gravity; Fu, Force from the ball’s inertia; J, Moment of inertia; L, Length of crossbar; m, Ball’s mass; r, Ball’s Radius; r, Torque of the ball; ABC, Artificial Bee Colony technique; AFSMC, Model Reference Adaptive Control; BBS, Ball and Beam System; CDM, Coefficient Diagram Method; EA, Evolutionary Algorithm; FOPID, Fractional Order Proportional Integral Derivative; FSC, FuzzySliding-Control; GWO, Grey Wolf Optimization; IPD, Integral- Proportional Derivative; IAE, Integral Absolute Error; ISE, Integral Square Error; ITAE, Integral Time Absolute Error; ITSE, Integral Time Square Error; LQR, Linear Quadratic Regulator; NN, Neural Network; PI-PD, Proportional Integral– Proportional Derivative; PID, Proportional Integral Derivative; PIDD2-PI, Proportional Integral Derivative-second Derivative with a Proportional Integrator; PSO, Particle Swarm Optimization; SA, Simulated Annealing; SMC, Sliding Mode Controller; TID-F, Tilt Integral Derivative with Filter; TID, Tilt-Integral-Derivative; TORA, Translational Oscillator with Rotational Actuator; UAS, Underactuated Systems; UMS, Underactuated Mechanical Systems; VTOL, Vertical Takeoff and Landing; WOA, Whale Optimization Algorithm. Stabilization and tracking control of UBBS using metaheuristic based TID-F and PIDD2–PI control schemes Introduction Underactuated mechanical systems (UMS) have fewer control actuators than their degree of freedom they possess. Modern science and engineering incorporate these systems in various practical and diverse applications. Diverse fields, including robotics, the aeronautical industry, and aerospace, actively use underactuated systems. Furthermore, researchers find these systems of great interest and importance as prototypes for complex nonlinear systems in addition to their practical applications. In recent years, researchers have focused primarily on underactuated systems control design. As the field of UMS continues to emerge, a fundamental challenge arises: the development of a theoretical framework. Through a theoretical perspective, UMS controllability and stabilization is a significant challenge for the control research community. The utilization of underactuated mechanical systems (UMS) in engineering research and education encompasses various applications, with the ball and beam system (BBS) emerging as a particularly renowned and widely-used benchmark. Using a straightforward yet efficient mechanism, it actively illustrates the fundamental principles of control system engineering, encompassing modeling, identification, analysis, and design. The system consists of a ball that travels along a beam and a sensor that measures the position of the ball. An angle adjustment of the beam controls the position of the ball. Researchers have explored various control strategies, such as Proportional Integral Derivative (PID), Linear Quadratic Regulator (LQR), fuzzy logic, neural networks, adaptive control, and many m (...truncated)