Document Type : Reseach Article
10.57647/j.mjee.2025.1901.03
Abstract
Controlling a cart-pendulum system involves both swinging the pendulum up and maintaining its upright position. Typically, separate controllers are used for the swing-up and stabilization phases. This paper presents the implementation of a Fractional Order Proportional-Integral-Derivative (FOPID) controller for the cart-pendulum system. A novel metaheuristic method, Class Topper Optimization (CTO), is utilized to design and optimize the FOPID controller’s performance for both the cart and pendulum. The study focuses on the pendulum angle and arm angle as key parameters. Results indicate that the proposed approach outperforms existing methods such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Ant Colony Optimization (ACO).
Keywords
[1] Ahmad Nor Kasruddin Nasir, Mohd. Ashraf Ahmad and mohd. Fuaad Rahmat, “Performance comparison between LQR and PID controllers for an inverted pendulum system,” AIP conference proceedings, Americal Institute of Physics, pp. 124-128, vol. 1052, no. 1, 2008 3 [http://dx.doi.org/10.1063/1.3008655]
[2] Neha kumari Singhal and Akhilesh Swarup, “Application and Comparison of Few Control Methods to Inverted Pendulum Dynamics,” International Conference on Computing and Communication Automation (ICCCA), pp. 1-6, 2018 4 [https://doi.org/10.1109/CCAA.2018.8777474]
[3] Peng Shen, “Application of genetic algorithm optimization LQR weighting matrices control inverted pendulum,” Applied Mechanics and Materials, Trans Tech Publications Ltd., pp.1274-1277, vol. 543, 2014 5 [https://doi.org/10.4028/www.scientific.net/AMM.543-547.1274
[4] Ujjwal, H. P. Singh, N. Verma and A. Jain, “Optimization of PID based Cart-Inverted Pendulum System using GA-LQR," 2021 6th International Conference on Communication and Electronics Systems (ICCES), Coimbatore, India, 2021, pp. 1-5, Doi: 10.1109/ICCES51350.2021.9488955.
[5] S. Dudhe, D. K. Dheer and G. L. Raja, “A Portable Meconium Aspirator with Fractional Augmented Pressure Control System," in IEEE Transactions on Circuits and Systems II [https://doi.org/10.1109/TCSII.2024.3378312]
[6] U. Mehta, P. Aryan and G. L. Raja, "Tri-Parametric Fractional-Order Controller Design for Integrating Systems with Time Delay," in IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 70, no. 11, pp. 4166-4170, Nov. 2023 [DOI: 10.1109/TCSII.2023.3269819]
[7] N. Kumar, P. Aryan, G. L. Raja and U. R. Muduli, "Robust Frequency-Shifting Based Control Amid False Data Injection Attacks for Interconnected Power Systems with Communication Delay," in IEEE Transactions on Industry Applications, vol. 60, no. 2, pp. 3710-3723, March-April 2024 [https://doi.org/10.1109/TIA.2023.3348775]
[8] Kumar, D., Raja, G. L., Arrieta, O., & Vilanova, R. (2022). Fractional-order model identification and indirect internal model controller design for higher-order processes. IFAC-Papers Online, 56(2), 7270-7275 [https://doi.org/10.1016/j.ifacol.2023.10.337]
[9] Mukherjee, D., Raja, G. L., Kundu, P., & Ghosh, A. (2021). Design of Optimal Fractional Order Lyapunov Based Model Reference Adaptive Control Scheme for CSTR. IFAC-Papers Online, 55(1), 436-441 [https://doi.org/10.1016/j.ifacol.2022.04.072].
[10] K. Furuta, T. Ochiai, and N. Ono, "Attitude control of a triple inverted pendulum," International Journal of Control, vol. 39, pp. 1351–1365, 1984 [ Doi: 10.1080/00207178408933251]
[11] L. Messikh, El. H. Guechi, and M. L. Benloucif, "Critically damped stabilization of inverted-pendulum systems using continuous-time cascade linear model predictive control," Journal of the Franklin Institute, vol. 354, no. 16, pp. 7241–7265, 2017 [https://doi.org/10.1016/j.jfranklin.2017.08.039]
[12] K. Furuta, T. Okutani, and H. Sone, "Computer control of a double inverted pendulum," Computer Electrical Engineering, vol. 5, pp. 67- 84, 1978 [https://doi.org/10.1016/0045-7906(78)90018-6].
[13] S.A. Bortoff, "Robust swing up control for a rotational double pendulum," in Proceedings of the Thirteenth World Congress of IFAC, San Francisco, 1996, pp. 413–419 [https://doi.org/10.1016/S1474-6670(17)58102-1]
[14] J. Lee, R. Mukherjee, and H. K. Khalil, "Output feedback stabilization of inverted pendulum on a cart in the presence of uncertainties," Automatica, vol. 54, pp. 146–157, 2015 [https://doi.org/10.1016/j.automatica.2015.01.013]
[15] M. Park and D. Chwa, "Orbital stabilization of inverted-pendulum systems via coupled sliding-mode control," IEEE Transaction Industrial Electronices, vol. 56, no. 9, p. 2009, 3556–3569 [https://doi.org/10.1109/TIE.2009.2021178].
[16] W. D. Chang, R. C. Hwang, and J. G. Hsieh, "A self-tuning PID control for a class of nonlinear systems based on the Lyapunov approach," Journal of Process Control, vol. 12, no. 2, pp. 233–242, 2002 [https://doi.org/10.1016/S0959-1524(01)00041-5].
[17] S. K. Mishra and D. Chandra, "Stabilization and Tracking Control of Inverted Pendulum Using Fractional Order PID Controllers," Journal of Engineering, 2014 [https://doi.org/10.1155/2014/752918].
[18] J. Yi, N. Yubazaki, and K. Hirota, "Upswing and stabilization control of inverted pendulum system based on the SIRMs dynamically connected fuzzy inference model," Fuzzy Sets System, vol. 122, no. 1, pp. 139–152, 2001 [https://doi.org/10.1016/S0165-0114(00)00049-X].
[19] C.W. Anderson, "Learning to control an inverted pendulum using neural networks," IEEE Control System Magazine, vol. 9, no. 3, pp. 31–37, 1989 [https://doi.org/10.1109/37.24809].
[20] J. H. Yang, S. Y. Shim, J. H. Seo, and Y.S. Lee, "Swing-up control for an inverted pendulum with restricted cart raillength," International Journal Control Automation System, vol. 7, no. 4, pp. 674–680, 2009 [https://doi.org/10.1007/s12555-009-0419-x].
[21] M. Park and D. Chwa, "Orbital stabilization of inverted-pendulum systems via coupled sliding-mode control," IEEE Transaction Industrial Electronics, vol. 56, no. 9, pp. 3556–3569, 2009 [https://doi.org/10.1109/TIE.2009.2021178].
[22] M. M. Dalvand, B. Shirinzadeh, and S. Nahavandi, "Swing-Up and Stability Control of Wheeled Acrobot (WAcrobot)," Automatika, vol. 55, pp. 32-40, Jan-Mar 2014 [https://doi.org/10.7305/automatika.2014.01.109].
[23] Y. J. Mon and C. M. Lin, "Double inverted pendulum decoupling control by adaptive terminal sliding-mode recurrent fuzzy neural network," Journal of Intelligent & Fuzzy Systems, vol. 26, pp. 1723- 1729, 2014 [Doi: 10.3233/IFS-130851].
[24] X. Xu, C. Q. Lian, L. Zuo, and H. B. He, "Kernel-Based Approximate Dynamic Programming for Real-Time Online Learning Control: An Experimental Study," IEEE Transactions on Control Systems Technology, vol. 22, pp. 146-156, Jan 2014 [https://doi.org/10.1109/TCST.2013.2246866].
[25] C. L. Wang and Y. Lin, "Output-feedback robust adaptive backstepping control for a class of multivariable nonlinear systems with guaranteed L-infinity tracking performance," International Journal of Robust and Nonlinear Control, vol. 23, pp. 2082-2096, Dec 2013 [https://doi.org/10.1002/rnc.2871].
[26] C. F. Hsu, "Adaptive PI Hermite neural control for MIMO uncertain nonlinear systems," Applied Soft Computing, vol. 13, pp. 2569-2576, May 2013 [https://doi.org/10.1016/j.asoc.2012.11.029].
[27] T. Waegeman, F. Wyffels, and B. Schrauwen, "Feedback Control by Online Learning an Inverse Model," IEEE Transactions on Neural Networks and Learning Systems, vol. 23, pp. 1637-1648, Oct 2012 [https://doi.org/10.1109/TNNLS.2012.2208655].
[28] J. Awrejcewicz, G. Wasilewski, G. Kudra, and S. A. Reshmin, "An experiment with swinging up a double pendulum using feedback control," Journal of Computer and Systems Sciences International, vol. 51, pp. 176-182, Apr 2012 [Doi: 10.1134/S1064230712020037].
[29] I. Hassanzadeh, A. Nejadfard, and M. Zadi, "A Multivariable Adaptive Control Approach for Stabilization of a Cart-Type Double Inverted Pendulum," Mathematical Problems in Engineering, 2011 [https://doi.org/10.1155/2011/970786].
[30] Y. Aoustin, A. Formal'skii, and Y. Martynenko, "Pendubot: combining of energy and intuitive approaches to swing up, stabilization in erected pose," Multibody System Dynamics, vol. 25, pp. 65-80, Jan 2011[https://doi.org/10.1007/s11044-010-9228-5].
[31] P. C. Li, K. P. Song, and F. H. Shang, "Double chains quantum genetic algorithm with application to neuro-fuzzy controller design," Advances in Engineering Software, vol. 42, pp. 875-886, Oct 2011 [https://doi.org/10.1016/j.advengsoft.2011.06.006].
[32] E. V. Kumar and J. Jerome, “Robust lqr controller design for stabilizing and trajectory tracking of inverted pendulum,” Procedia Engineering, vol. 64, pp. 169–178, 2013 [https://doi.org/10.1016/j.proeng.2013.09.088].
[33] J.-J. Wang, “Simulation studies of inverted pendulum based on PID controllers,” Simulation Modelling Practice and Theory, vol. 19, no. 1, pp. 440–449, 2011 [https://doi.org/10.1016/j.simpat.2010.08.003].
[34] R. Feifei, Z. Lei, T. Le, H. Yanhai, and Z. Pengpeng, “Design and research of double closed-loop control strategy for inverted pendulum system,” 3rd International Conference on Intelligent System Design and Engineering Applications (ISDEA), pp. 16–18, January 2013 [https://doi.org/10.1109/ISDEA.2012.134].
[35] D. Chatterjee, A. Patra, and H. K. Joglekar, “Swing-up and stabilization of a cart–pendulum system under restricted cart track length,” Systems & control letters, vol. 47, no. 4, pp. 355–364, 2002 [https://doi.org/10.1016/S0167-6911(02)00229-3].
[36] P. Colaneri, R. Scattolini, and N. Schiavoni, “Stabilization of multirate sampled-data linear systems,” Automatica, vol. 26, no. 2, pp. 377–380, 1990 [https://doi.org/10.1016/0005-1098(90)90132-2]
[37] A. Dey, S. Chakraborty, and S. K. Das, “Stabilization of a cart-inverted pendulum system using a 2-periodic controller: Simulation results,” in 2013 IEEE Conference on Systems, Process & Control (ICSPC). IEEE, 2013, pp. 169–174 [https://doi.org/10.1109/SPC.2013.6735126].
[38] B. Sprenger, L. Kucera, and S. Mourad, “Balancing of an inverted pendulum with a scara robot,” IEEE/ASME transactions on mechatronics, vol. 3, no. 2, pp. 91–97, 1998 [https://doi.org/10.1109/3516.686676].
[39] T. Kaczorek, “Pole placement for linear discrete-time systems by periodic output feedbacks,” Systems & Control Letters, vol. 6, pp. 267–269, October 1985 [https://doi.org/10.1016/0167-6911(85)90078-7].
[40] Ansari, Z. A., & Raja, G. L. (2024). Flow direction optimizer tuned robust FOPID-(1 + TD) cascade controller for oscillation mitigation in multi-area renewable integrated hybrid power system with hybrid electrical energy storage. Journal of Energy Storage, 83, 110616 [https://doi.org/10.1016/j.est.2024.110616].
[41] Kumari, N., Aryan, P., Raja, G. L., & Arya, Y. (2023). Dual degree branched type-2 fuzzy controller optimized with a hybrid algorithm for frequency regulation in a triple-area power system integrated with renewable sources. Protection and Control of Modern Power Systems, 8(1), 1-29[https://doi.org/10.1186/s41601-023-00317-7].
[42] Aryan, P., Raja, G. L., & Vilanova, R. (2023). Experimentally verified optimal bi-loop re-located IMC strategy for unstable and integrating systems with dead time. International Journal of Systems Science, 54(7), 1531–1549 [https://doi.org/10.1080/00207721.2023.2180782].
[43] Anand, A., Aryan, P., Kumari, N., & Raja, G. L. (2022). Type-2 fuzzy-based branched controller tuned using arithmetic optimizer for load frequency control. Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, 44(2), 4575–4596 [doi: 10.1080/15567036.2022.2078444].
[44] Deep Mukherjee; G. Lloyds Raja; Palash Kundu, Apurba Ghosh, “Modified augmented fractional order control schemes for cart inverted pendulum using constrained Luus-Jaakola optimization” International Journal of Modelling, Identification and Control (IJMIC), Vol. 38, No. 3/4, 2021[https://doi.org/10.1504/IJMIC.2021.123361]
[45] D. Izci and S. Ekinci, "An Efficient FOPID Controller Design for Vehicle Cruise Control System Using HHO Algorithm," 2021 3rd International Congress on Human-Computer Interaction, Optimization and Robotic Applications (HORA), Ankara [https://doi.org/10.1109/HORA52670.2021.9461336]
[46] Ö. Can, S. Ekinci and D. Izci, "Honey Badger Algorithm for Adjustment of FOPID Controller Adopted in An Automatic Voltage Regulator System," 2022 Global Energy Conference (GEC), Batman, Turkey, 2022, pp. 262-265, Doi: 10.1109/GEC55014.2022.9986660 [https://doi.org/10.1109/GEC55014.2022.9986660].
[47] Abualigah, L., Ekinci, S., Izci, D., Zitar, R.A., “Modified elite opposition-based artificial hummingbird algorithm for designing FOPID controlled cruise control system.” Intelligent Automation & Soft Computing, 38(2), 169-183, 2023 [Doi: 10.32604/iasc.2023.040291]
[48] Izci, D., & Ekinci, S., “Novel-enhanced metaheuristic algorithm for FOPID-controlled and Bode’s ideal transfer function–based buck converter system.” Transactions of the Institute of Measurement and Control, 2023 [https://doi.org/10.1177/01423312221140671]
[49] Izci, D., & Ekinci, S., “Fractional order controller design via gazelle optimizer for efficient speed regulation of micromotors.” E-Prime - Advances in Electrical Engineering, Electronics and Energy, 6, 2023 [https://doi.org/10.1016/j.prime.2023.100295]
[50] Izci, D., Ekinci, S., Eker, E., & Kayri, M., “A novel modified opposition-based hunger games search algorithm to design fractional order proportional-integral-derivative controller for magnetic ball suspension system. Advanced Control for Applications: Engineering and Industrial Systems, 4(1), e96, 2022 [https://doi.org/10.1002/adc2.96].
[51] Ahangarani Farahani, Alireza, Sayyed Majid Hosseini, and Meysam Delalat. "Adaptive Nonlinear Controller for Quadrotor Altitude Control with Online Control Coefficients Function." Majlesi Journal of Electrical Engineering 17.4 (2023): 45-51[https://doi.org/10.30486/mjee.2023.1986975.1144].
[52] Al-Azzawi, Waleed Khalid, et al. "Optimization of Energy Systems in a Distribution Micro grid: An Application of the Particle Swarm Optimization." Majlesi Journal of Electrical Engineering 17.2 (2023): 41-46 [https://doi.org/10.30486/mjee.2023.1987130.1147].
[53] Verma, Neelam, and Sudarshan K. Valluru. "Comparative Study of Parametric Disturbances Effect on Cart-Inverted Pendulum System Stabilization." Modern Electronics Devices and Communication Systems: Select Proceedings of MEDCOM 2021. Singapore: Springer Nature Singapore, 2023. 217-230 [https://doi.org/10.1007/978-981-19-6383-4_17].
[54] Jain, Anjali, Ashish Mani, and Anwar S. Siddiqui. "Improved Grey Wolf Optimizer with Levy Flight to Solve Dynamic Economic Dispatch Problem with Electric Vehicle Profiles." Optimization Techniques in Engineering: Advances and Applications (2023): 1-35 [https://doi.org/10.1002/9781119906391.ch1].
[55] Mani, Ashish, and Anjali Jain. "Towards realistic mimicking of grey wolves hunting process for bounded single objective optimization." 2020 IEEE Congress on Evolutionary Computation (CEC). IEEE, 2020 [https://doi.org/10.1109/CEC48606.2020.9185535].
[56] Jain, Anjali, Ashish Mani, and Anwar S. Siddiqui. "Dynamic Economic Dispatch of Power System Network Having Thermal Units and Electric Vehicles using IGWO." Majlesi Journal of Electrical Engineering 15.4 (2021): 43-62 [https://doi.org/10.52547/mjee.15.4.43].
[57] Davut Izci, Serdar Ekinci, Erdal Eker, Murat Kayri, “Augmented hunger games search algorithm using logarithmic spiral opposition-based learning for function optimization and controller design,” Journal of King Saud University - Engineering Sciences, 2022 [https://doi.org/10.1016/j.jksues.2022.03.001]
[58] Verma, Neelam, and Sudarshan K. Valluru. "Comparative Analysis of GOA, WCA and PSO Tuned Fractional Order-PID Controller for Cart-Inverted Pendulum." 2022 IEEE International Power and Renewable Energy Conference (IPRECON). IEEE, 2022 [https://doi.org/10.1109/IPRECON55716.2022.10059474].
[59] Verma, Neelam, and Sudarshan K. Valluru. "ANN Based ANFIS controller Design Using Hybrid Meta-Heuristic Tuning Approach for Cart Inverted Pendulum System." Multimedia Tools and Applications (2023): 1-23 [http://dx.doi.org/10.1007/s11042-023-17699-3].
[60] P. Das, D. K. Das and S. Dey, "A New Class Topper Optimization Algorithm with an Application to Data Clustering," in IEEE Transactions on Emerging Topics in Computing, vol. 8, no. 4, pp. 948-959, 1 Oct.-Dec. 2020 [https://doi.org/10.1109/TETC.2018.2812927]
[61] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J. (2003). Fractional Differential Equations: A Emergent Field in Applied and Mathematical Sciences. In: Samko, S., Lebre, A., dos Santos, A.F. (eds) Factorization, Singular Operators and Related Problems. Springer, Dordrecht. [ https://doi.org/10.1007/978-94-017-0227-0_11]
[62] Puneet Mishra, Vineet Kumar, K.P.S. Rana, “A fractional order fuzzy PID controller for binary distillation column control,” Expert Systems with Applications, 2015 [https://doi.org/10.1016/j.eswa.2015.07.008].
[63] Scherer, R., Kalla, S. L., Tang, Y., & Huang, J. (2011). The Grünwald–Letnikov method for fractional differential equations. Computers & Mathematics With Applications, 62(3), 902-917. https://doi.org/10.1016/j.camwa.2011.03.054
[64] Chen, Y., Wang, X., Zhang, B., Xie, F., & Chen, S. (2024). Differences between Riemann–Liouville and Caputo calculus definitions in analyzing fractional boost converter with discontinuous-conduction-mode operation. International Journal of Circuit Theory and Applications, 52(8), 4164-4183. https://doi.org/10.1002/cta.3972
[65] Tang, Y., Cui, M., Hua, C., Li, L. and Yang, Y., Optimum design of fractional order PI^(-λ) D^μ controller for AVR system using chaotic ant swarm, Expert Systems with Applications [https://doi.org/10.1016/j.eswa.2012.01.007]
)