Swing-Up and Position Control of Inverted Pendulum - Cart Systems Using Optimized Fuzzy Controller
Main Article Content
Abstract
This study proposes a simple approach to optimally design a fuzzy controller in swing-up and position control for inverted pendulum-cart systems. First, the sub-fuzzy controllers to control the pendulum's swing up and the cart's position are designed separately. Each controller includes two state variables to calculate the component control forces. The combination of component control forces determines the final control force through a weight computed from a simple scheme. Parameters of sub-fuzzy controllers and those to determine the weight are optimized to minimize the system's equilibration time. The simulation results show that the proposed controller is simple to set up and optimize, has high control efficiency, is adaptable to the system's state, and is stable and robust for the system's different initial conditions and configurations. When using the proposed controller, the stabilization time of the system is reduced by 14.5%, the maximum control force is reduced by 32.6%,
and the pendulum length is increased by 50% compared to fuzzy controllers in the published studies. The approach of the present work can be applied to control various underactuated systems as well as in the motion control of mobile robot models.
Keywords
Inverted pendulum-cart, swing-up and position control, fuzzy control, sub-controllers, design optimization.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
References
[1] X.-T. Nguyen, T.-L. Bui, H.-A. Bui, D.-A. Pham, N. Miura, Control the movement of mobile robot using fingers gestures based on fuzzy logic, in Proceedings of the International Conference on Advanced Mechanical Engineering, Automation, and Sustainable Development 2021 (AMAS2021), Springer, 2022, pp. 799-804.
https://doi.org/10.1007/978-3-030-99666-6_116
[2] C.A. Ibanez, O.G. Frias, M.S. Castanon, Lyapunov-based controller for the inverted pendulum cart system, Nonlinear Dynamics, vol. 40, no. 4, pp. 367-374, 2005.
https://doi.org/10.1007/s11071-005-7290-y
[3] L. Rapoport, A. Generalov, Control of an inverted pendulum on a wheel, Automation and Remote Control, vol. 83, no. 8, pp. 1151-1171, 2022.
https://doi.org/10.1134/S000511792208001X
[4] A.V. Pesterev, Y.V. Morozov, Stabilization of a cart with inverted pendulum, Automation and Remote Control, vol. 83, no. 1, pp. 78-91, 2022.
https://doi.org/10.1134/S0005117922010064
[5] 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
[6] S.K. Mishra, D. Chandra, Stabilization and tracking control of inverted pendulum using fractional order PID controllers, Journal of Engineering, vol. 2014, no. 1, pp. 752918, 2014.
https://doi.org/10.1155/2014/752918
[7] S. Irfan, A. Mehmood, M.T. Razzaq, J. Iqbal, Advanced sliding mode control techniques for inverted pendulum: Modelling and simulation, Engineering Science and Technology, an International Journal, vol. 21, no. 4, pp. 753-759, 2018.
https://doi.org/10.1016/j.jestch.2018.06.010
[8] S. Hanwate, Y.V. Hote, A. Budhraja, Design and implementation of adaptive control logic for cart-inverted pendulum system, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, vol. 233, no. 2, pp. 164-178, 2019.
https://doi.org/10.1177/0959651818788148
[9] M. Ashok Kumar, S. Kanthalakshmi, H∞ tracking control for an inverted pendulum, Journal of Vibration and Control, vol. 24, no. 16, pp. 3515-3524, 2018.
https://doi.org/10.1177/1077546317750977
[10] M. Abdullah, A.A. Amin, S. Iqbal, K. Mahmood-ul-Hasan, Swing up and stabilization control of rotary inverted pendulum based on energy balance, fuzzy logic, and LQR controllers, Measurement and Control, vol. 54, no. 9-10, pp. 1356-1370, 2021.
https://doi.org/10.1177/0020294
[11] H.-L. Bui, D.-T. Tran, N.-L. Vu, Optimal fuzzy control of an inverted pendulum, Journal of Vibration and Control, vol. 18, no. 14, pp. 2097-2110, 2012.
[12] H.-Y. Zhang, J. Wang, G.-D. Lu, Self-organizing fuzzy optimal control for under-actuated systems, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, vol. 228, no. 8, pp. 578-590, 2014.
https://doi.org/10.1177/0959651814533681
[13] Z. Ping, M. Zhou, C. Liu, Y. Huang, M. Yu, J.-G. Lu, An improved neural network tracking control strategy for linear motor-driven inverted pendulum on a cart and experimental study, Neural Computing and Applications, vol. 34, no. 7, pp. 5161-5168, 2022.
https://doi.org/10.1007/s00521-021-05986-9
[14] U. Riaz, A.A. Amin, M. Tayyeb, Design of active faulttolerant control system for Air-fuel ratio control of internal combustion engines using fuzzy logic controller, Science Progress, vol. 105, no. 2, pp., 2022.
https://doi.org/10.1177/00368504221094723
[15] J. Lin, Y.S. Ding, J. Chang, Balancing and swinging-up control for cart-pendulum-seesaw system by decomposed fuzzy coordination control, Journal of Vibration and Control, vol. 20, no. 6, pp. 925-942, 2014.
https://doi.org/10.1177/1077546312466885
[16] B.A. Elsayed, M.A. Hassan, S. Mekhilef, Fuzzy swinging-up with sliding mode control for third order cart-inverted pendulum system, International Journal of Control, Automation, and Systems, vol. 13, no. 1, pp. 238-248, 2015.
https://doi.org/10.1007/s12555-014-0033-4
[17] A.I. Roose, S. Yahya, H. Al-Rizzo, Fuzzy-logic control of an inverted pendulum on a cart, Computers and Electrical Engineering, vol. 61, no., pp. 31-47, 2017.
https://doi.org/10.1016/j.compeleceng.2017.05.016
[18] T.B. Ivanova, A.A. Kilin, E.N. Pivovarova, Controlled motion of a spherical robot with feedback. II, Journal of Dynamical and Control Systems, vol. 25, no. 1, pp. 1-16, 2019.
https://doi.org/10.1007/s10883-017-9390-7
[19] B. Elkinany, M. Alfidi, R. Chaibi, Z. Chalh, TS fuzzy system controller for stabilizing the double inverted pendulum, Advances in Fuzzy Systems, vol. 2020, no., pp. 1-9, 2020.
https://doi.org/10.1155/2020/8835511
[20] E. Kelekci, T. Yaren, S. Kizir, Design of optimized interval type-2 fuzzy logic controller based on the continuity, monotonicity, and smoothness properties for a cart-pole inverted pendulum system, Transactions of the Institute of Measurement and Control, vol. no., pp. 2291–2307, 2022.
https://doi.org/10.1177/01423312221081309
[21] C.-H. Chiu, Y.-T. Hung, Y.-F. Peng, Design of a decoupling fuzzy control scheme for omnidirectional inverted pendulum real-world control, IEEE Access, vol.9, no., pp. 26083-26092, 2021.
https://doi.org/10.1109/ACCESS.2021.3057658
[22] A. Jain, A. Sharma, V. Jately, B. Azzopardi, S. Choudhury, Real-time swing-up control of nonlinear inverted pendulum using Lyapunov based optimized fuzzy logic control, IEEE Access, vol. 9, no., pp. 50715-50726, 2021.
https://doi.org/10.1109/ACCESS.2021.3058645
[23] A.N.K. Nasir, A.A.A. Razak, Opposition-based spiral dynamic algorithm with an application to optimize type-2 fuzzy control for an inverted pendulum system, Expert Systems with Applications, vol. 195, no., pp. 116661, 2022.
https://doi.org/10.1016/j.eswa.2022.116661
[24] T. Abut, S. Soyguder, Two-loop controller design and implementations for an inverted pendulum system with optimal self-adaptive fuzzy-proportional–integral–derivative control, Transactions of the Institute of Measurement and Control, vol. 44, no. 2, pp. 468-483, 2022.
https://doi.org/10.1177/01423312211040301
[25] H. Choe, K. Jang, K. Ham, C. Kang, A cascade control scheme with TS fuzzy model-assisted linear active disturbance rejection controller for position tracking of cart inverted pendulum, International Journal of Dynamics and Control, vol. 13, no. 2, pp. 1-12, 2025.
https://doi.org/10.1007/s40435-025-01595-9
[26] N. Bhuvaneswary, S.V. Jeevan, G.D. Chowdary, G. Yashwanth, U. Gowtham, B.R. Krishna, Optimizing Inverted Pendulum Stability: Performance Analysis of Self-Tuning Fuzzy PID and PID, in: 2025 5th International Conference on Trends in Material Science and Inventive Materials (ICTMIM), IEEE, 2025, pp. 553-558.
https://doi.org/10.1109/ICTMIM65579.2025.10988396
[27] S.M. Arbatsofla, A.H. Mazinan, M. Mahmoodabadi, An optimal fuzzy fractional-order adaptive robust controller according to feedback linearization for an under-actuated nonlinear inverted pendulum system, International Journal of Dynamics and Control, vol. 13, no. 3, pp. 107,
2025.
https://doi.org/10.1007/s40435-025-01611-y
[28] J. Yi, N. Yubazaki, Stabilization fuzzy control of inverted pendulum systems, Artificial Intelligence in Engineering, vol. 14, no. 2, pp. 153-163, 2000.
https://doi.org/10.1016/S0954-1810(00)00007-8
[29] J.C. Butcher, Numerical Methods For Ordinary Differential Equations, John Wiley & Sons, 2016.
https://doi.org/10.1002/9781119121534
[30] H.-L. Bui, M.-N. Pham, T.-T.-H. Nguyen, Swing-up control of an inverted pendulum cart system using the approach of Hedge-algebras theory, Soft Computing, vol. 26, no. 10, pp. 4613-4627, 2022.
https://doi.org/10.1007/s00500-022-06968-2
[31] T. Le-Duc, Q.-H. Nguyen, H. Nguyen-Xuan, Balancing composite motion optimization, Information Sciences, vol. 520, no., pp. 250-270, 2020.
https://doi.org/10.1016/j.ins.2020.02.013
[32] J. Moreno-Valenzuela, C. Aguilar-Avelar, Motion Control of Underactuated Mechanical Systems, Springer, 2018.
https://doi.org/10.1007/978-3-319-58319-8
https://doi.org/10.1007/978-3-030-99666-6_116
[2] C.A. Ibanez, O.G. Frias, M.S. Castanon, Lyapunov-based controller for the inverted pendulum cart system, Nonlinear Dynamics, vol. 40, no. 4, pp. 367-374, 2005.
https://doi.org/10.1007/s11071-005-7290-y
[3] L. Rapoport, A. Generalov, Control of an inverted pendulum on a wheel, Automation and Remote Control, vol. 83, no. 8, pp. 1151-1171, 2022.
https://doi.org/10.1134/S000511792208001X
[4] A.V. Pesterev, Y.V. Morozov, Stabilization of a cart with inverted pendulum, Automation and Remote Control, vol. 83, no. 1, pp. 78-91, 2022.
https://doi.org/10.1134/S0005117922010064
[5] 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
[6] S.K. Mishra, D. Chandra, Stabilization and tracking control of inverted pendulum using fractional order PID controllers, Journal of Engineering, vol. 2014, no. 1, pp. 752918, 2014.
https://doi.org/10.1155/2014/752918
[7] S. Irfan, A. Mehmood, M.T. Razzaq, J. Iqbal, Advanced sliding mode control techniques for inverted pendulum: Modelling and simulation, Engineering Science and Technology, an International Journal, vol. 21, no. 4, pp. 753-759, 2018.
https://doi.org/10.1016/j.jestch.2018.06.010
[8] S. Hanwate, Y.V. Hote, A. Budhraja, Design and implementation of adaptive control logic for cart-inverted pendulum system, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, vol. 233, no. 2, pp. 164-178, 2019.
https://doi.org/10.1177/0959651818788148
[9] M. Ashok Kumar, S. Kanthalakshmi, H∞ tracking control for an inverted pendulum, Journal of Vibration and Control, vol. 24, no. 16, pp. 3515-3524, 2018.
https://doi.org/10.1177/1077546317750977
[10] M. Abdullah, A.A. Amin, S. Iqbal, K. Mahmood-ul-Hasan, Swing up and stabilization control of rotary inverted pendulum based on energy balance, fuzzy logic, and LQR controllers, Measurement and Control, vol. 54, no. 9-10, pp. 1356-1370, 2021.
https://doi.org/10.1177/0020294
[11] H.-L. Bui, D.-T. Tran, N.-L. Vu, Optimal fuzzy control of an inverted pendulum, Journal of Vibration and Control, vol. 18, no. 14, pp. 2097-2110, 2012.
[12] H.-Y. Zhang, J. Wang, G.-D. Lu, Self-organizing fuzzy optimal control for under-actuated systems, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, vol. 228, no. 8, pp. 578-590, 2014.
https://doi.org/10.1177/0959651814533681
[13] Z. Ping, M. Zhou, C. Liu, Y. Huang, M. Yu, J.-G. Lu, An improved neural network tracking control strategy for linear motor-driven inverted pendulum on a cart and experimental study, Neural Computing and Applications, vol. 34, no. 7, pp. 5161-5168, 2022.
https://doi.org/10.1007/s00521-021-05986-9
[14] U. Riaz, A.A. Amin, M. Tayyeb, Design of active faulttolerant control system for Air-fuel ratio control of internal combustion engines using fuzzy logic controller, Science Progress, vol. 105, no. 2, pp., 2022.
https://doi.org/10.1177/00368504221094723
[15] J. Lin, Y.S. Ding, J. Chang, Balancing and swinging-up control for cart-pendulum-seesaw system by decomposed fuzzy coordination control, Journal of Vibration and Control, vol. 20, no. 6, pp. 925-942, 2014.
https://doi.org/10.1177/1077546312466885
[16] B.A. Elsayed, M.A. Hassan, S. Mekhilef, Fuzzy swinging-up with sliding mode control for third order cart-inverted pendulum system, International Journal of Control, Automation, and Systems, vol. 13, no. 1, pp. 238-248, 2015.
https://doi.org/10.1007/s12555-014-0033-4
[17] A.I. Roose, S. Yahya, H. Al-Rizzo, Fuzzy-logic control of an inverted pendulum on a cart, Computers and Electrical Engineering, vol. 61, no., pp. 31-47, 2017.
https://doi.org/10.1016/j.compeleceng.2017.05.016
[18] T.B. Ivanova, A.A. Kilin, E.N. Pivovarova, Controlled motion of a spherical robot with feedback. II, Journal of Dynamical and Control Systems, vol. 25, no. 1, pp. 1-16, 2019.
https://doi.org/10.1007/s10883-017-9390-7
[19] B. Elkinany, M. Alfidi, R. Chaibi, Z. Chalh, TS fuzzy system controller for stabilizing the double inverted pendulum, Advances in Fuzzy Systems, vol. 2020, no., pp. 1-9, 2020.
https://doi.org/10.1155/2020/8835511
[20] E. Kelekci, T. Yaren, S. Kizir, Design of optimized interval type-2 fuzzy logic controller based on the continuity, monotonicity, and smoothness properties for a cart-pole inverted pendulum system, Transactions of the Institute of Measurement and Control, vol. no., pp. 2291–2307, 2022.
https://doi.org/10.1177/01423312221081309
[21] C.-H. Chiu, Y.-T. Hung, Y.-F. Peng, Design of a decoupling fuzzy control scheme for omnidirectional inverted pendulum real-world control, IEEE Access, vol.9, no., pp. 26083-26092, 2021.
https://doi.org/10.1109/ACCESS.2021.3057658
[22] A. Jain, A. Sharma, V. Jately, B. Azzopardi, S. Choudhury, Real-time swing-up control of nonlinear inverted pendulum using Lyapunov based optimized fuzzy logic control, IEEE Access, vol. 9, no., pp. 50715-50726, 2021.
https://doi.org/10.1109/ACCESS.2021.3058645
[23] A.N.K. Nasir, A.A.A. Razak, Opposition-based spiral dynamic algorithm with an application to optimize type-2 fuzzy control for an inverted pendulum system, Expert Systems with Applications, vol. 195, no., pp. 116661, 2022.
https://doi.org/10.1016/j.eswa.2022.116661
[24] T. Abut, S. Soyguder, Two-loop controller design and implementations for an inverted pendulum system with optimal self-adaptive fuzzy-proportional–integral–derivative control, Transactions of the Institute of Measurement and Control, vol. 44, no. 2, pp. 468-483, 2022.
https://doi.org/10.1177/01423312211040301
[25] H. Choe, K. Jang, K. Ham, C. Kang, A cascade control scheme with TS fuzzy model-assisted linear active disturbance rejection controller for position tracking of cart inverted pendulum, International Journal of Dynamics and Control, vol. 13, no. 2, pp. 1-12, 2025.
https://doi.org/10.1007/s40435-025-01595-9
[26] N. Bhuvaneswary, S.V. Jeevan, G.D. Chowdary, G. Yashwanth, U. Gowtham, B.R. Krishna, Optimizing Inverted Pendulum Stability: Performance Analysis of Self-Tuning Fuzzy PID and PID, in: 2025 5th International Conference on Trends in Material Science and Inventive Materials (ICTMIM), IEEE, 2025, pp. 553-558.
https://doi.org/10.1109/ICTMIM65579.2025.10988396
[27] S.M. Arbatsofla, A.H. Mazinan, M. Mahmoodabadi, An optimal fuzzy fractional-order adaptive robust controller according to feedback linearization for an under-actuated nonlinear inverted pendulum system, International Journal of Dynamics and Control, vol. 13, no. 3, pp. 107,
2025.
https://doi.org/10.1007/s40435-025-01611-y
[28] J. Yi, N. Yubazaki, Stabilization fuzzy control of inverted pendulum systems, Artificial Intelligence in Engineering, vol. 14, no. 2, pp. 153-163, 2000.
https://doi.org/10.1016/S0954-1810(00)00007-8
[29] J.C. Butcher, Numerical Methods For Ordinary Differential Equations, John Wiley & Sons, 2016.
https://doi.org/10.1002/9781119121534
[30] H.-L. Bui, M.-N. Pham, T.-T.-H. Nguyen, Swing-up control of an inverted pendulum cart system using the approach of Hedge-algebras theory, Soft Computing, vol. 26, no. 10, pp. 4613-4627, 2022.
https://doi.org/10.1007/s00500-022-06968-2
[31] T. Le-Duc, Q.-H. Nguyen, H. Nguyen-Xuan, Balancing composite motion optimization, Information Sciences, vol. 520, no., pp. 250-270, 2020.
https://doi.org/10.1016/j.ins.2020.02.013
[32] J. Moreno-Valenzuela, C. Aguilar-Avelar, Motion Control of Underactuated Mechanical Systems, Springer, 2018.
https://doi.org/10.1007/978-3-319-58319-8