Comparative Performance Analysis of an SMA-Optimized FOPID Controller for an Inverted Pendulum on a Cart

Öz

This study investigates the tuning of a Fractional-Order Proportional-Integral-Derivative (FOPID) controller for a cart–inverted pendulum system, which is an inherently nonlinear and unstable system requiring simultaneous stabilization of the pendulum angle and regulation of the cart position. Although various optimization methods have been widely employed in the literature for this problem, the performance of the Slime Mould Algorithm (SMA) in the context of inverted pendulum control has not been sufficiently investigated. Motivated by this gap, the five parameters of the controller were determined using SMA, Grey Wolf Optimization (GWO), and Artificial Hummingbird Algorithm (AHA), and all methods were comparatively evaluated under identical simulation conditions based on the Integral Time Absolute Error (ITAE) criterion. The obtained results demonstrate that the SMA-based design yields lower ITAE variability and higher repeatability compared to the other methods. The corresponding closed-loop responses also exhibit less oscillatory angle and position behavior. These findings indicate that SMA provides a more reliable and consistent FOPID tuning performance than GWO and AHA for the considered inverted pendulum problem.

Anahtar Kelimeler

• Inverted Pendulum
• Optimization
• FOPID
• SMA
• AHA
• GWO

Araba Üzerindeki Ters Sarkaç Sistemi için SMA Tabanlı Optimize Edilmiş FOPID Denetleyicisinin Karşılaştırmalı Performans Analizi

Öz

Bu çalışma, doğası gereği doğrusal olmayan ve kararsız bir sistem olan araba–ters sarkaç için Kesir Dereceli Oransal-İntegral-Türev (FOPID) kontrolörünün meta-sezgisel algoritmalar ile ayarlanmasını incelemektedir. Literatürde bu problem için çeşitli optimizasyon yöntemleri yaygın biçimde kullanılmakla birlikte, Cıvık Mantar Algoritması (SMA)'nın ters sarkaç kontrolündeki performansı yeterince araştırılmamıştır. Bu motivasyonla kontrolörün beş parametresi SMA, Gri Kurt Optimizasyonu (GWO) ve Yapay Sinek Kuşu Algoritması (AHA) kullanılarak belirlenmiş; tüm yöntemler özdeş benzetim koşulları altında Integral Zaman Mutlak Hata (ITAE) kriteri temelinde karşılaştırmalı olarak değerlendirilmiştir. Elde edilen sonuçlar, SMA tabanlı tasarımın diğer yöntemlere kıyasla daha düşük ITAE değişkenliği ve daha yüksek tekrarlanabilirlik sunduğunu göstermektedir. Buna karşılık gelen kapalı çevrim tepkileri de daha az salınımlı bir açı ve konum davranışı sergilemektedir. Bu bulgular, SMA'nın söz konusu problem için GWO ve AHA'ya kıyasla daha güvenilir ve tutarlı bir FOPID ayarlama performansı sağladığını ortaya koymaktadır.

Anahtar Kelimeler

• Ters sarkaç
• Optmizasyon
• FOPID
• SMA
• AHA
• GWO

Kaynakça

1. [1] Liu, Y. & Yu, H. (2013). A survey of underactuated mechanical systems. IET Control Theory and Applications, 7(7), 921–935.
2. [2] Furuta, K. (2003). Control of pendulum: From super mechano-system to human adaptive mechatronics. Proceedings of the 42nd IEEE International Conference on Decision and Control, Maui, HI, USA, 149–156.
3. [3] Li, Q. R., Tao, W. H., Sun, N., Zhang, C. Y. & Yao, L. H. (2008). Stabilization control of double inverted pendulum system. Proceedings of the 3rd International Conference on Innovative Computing, Information and Control, Dalian, China, 1–6.
4. [4] Åström, K. J. & Furuta, K. (2000). Swinging up a pendulum by energy control. Automatica, 36(2), 287–295.
5. [5] Yoshida, K. (1999). Swing-up control of an inverted pendulum by energy-based methods. Proceedings of the American Control Conference, San Diego, CA, USA, 4045–4049.
6. [6] Mason, P., Broucke, M. & Piccoli, B. (2008). Time optimal swing-up of the planar pendulum. IEEE Transactions on Automatic Control, 53(8), 1876–1886.
7. [7] Tao, C. W., Taur, J. S., Hsieh, T. W. & Tsai, C. L. (2008). Design of a fuzzy controller with fuzzy swing-up and parallel distributed pole assignment schemes for an inverted pendulum and cart system. IEEE Transactions on Control Systems Technology, 16(6), 1277–1288.
8. [8] Shahnazi, R. & Akbarzadeh-T, M. R. (2008). PI adaptive fuzzy control with large and fast disturbance rejection for a class of uncertain nonlinear systems. IEEE Transactions on Fuzzy Systems, 16(1), 187–197.

9. [9] Bloch, A. M., Leonard, N. E. & Marsden, J. E. (2000). Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem. IEEE Transactions on Automatic Control, 45(12), 2253–2270.
10. [10] Chaturvedi, N. A., McClamroch, N. H. & Bernstein, D. S. (2008). Stabilization of a 3D axially symmetric pendulum. Automatica, 44(9), 2258–2265.
11. [11] Wai, R. J. & Chang, L. J. (2006). Adaptive stabilizing and tracking control for a nonlinear inverted-pendulum system via sliding-mode technique. IEEE Transactions on Industrial Electronics, 53(2), 674–692.
12. [12] Chang, L. H. & Lee, A. C. (2007). Design of nonlinear controller for bi-axial inverted pendulum system. IET Control Theory and Applications, 1(4), 979–986.
13. [13] Wang, H., Dong, H., He, L., Shi, Y. & Zhang, Y. (2010). Design and simulation of LQR controller with the linear inverted pendulum. Proceedings of the International Conference on Electrical and Control Engineering, Wuhan, China, 1–4.
14. [14] Wang, J. J. (2011). Simulation studies of inverted pendulum based on PID controllers. Simulation Modelling Practice and Theory, 19(1), 440–449.
15. [15] Lim, Y. Y., Hoo, C. L. & Wong, Y. M. F. (2018). Stabilising an inverted pendulum with PID controller. MATEC Web of Conferences, 152, 03022.
16. [16] Peker, F. & Kaya, I. (2017). Identification and real-time control of an inverted pendulum using PI–PD controller. Proceedings of the 21st International Conference on System Theory, Control and Computing, Sinaia, Romania, 1–6.
17. [17] Elsayed, B. A., Hassan, M. A. & Mekhilef, S. (2015). Fuzzy swinging-up with sliding mode control for third-order cart–inverted pendulum system. International Journal of Control, Automation and Systems, 13(1), 238–248.
18. [18] Anderson, C. W. (1989). Learning to control an inverted pendulum using neural networks. IEEE Control Systems Magazine, 9(3), 31–37.
19. [19] Ortigueira, M. D. (2011). Fractional Calculus for Scientists and Engineers. Springer, Dordrecht, The Netherlands.
20. [20] Caponetto, R., Dongola, G., Fortuna, L. & Petráš, I. (2010). Fractional Order Systems: Modeling and Control Applications. World Scientific, Singapore.
21. [21] Baleanu, D., Diethelm, K., Scalas, E. & Trujillo, J. J. (2012). Fractional Calculus Models and Numerical Methods. World Scientific, Singapore.
22. [22] Podlubny, I. (1999). Fractional-order systems and PIλDμ controllers. IEEE Transactions on Automatic Control, 44(1), 208–214.
23. [23] Podlubny, I., Petráš, I., Vinagre, B. M., O'Leary, P. & Dorčák, Ľ. (2002). Analogue realizations of fractional-order controllers. Nonlinear Dynamics, 29(1–4), 281–296.
24. [24] Yeroglu, C. & Tan, N. (2011). Note on fractional-order proportional–integral–differential controller design. IET Control Theory and Applications, 5(17), 1978–1989.
25. [25] Maiti, D., Acharya, A., Chakraborty, M., Konar, A. & Janarthanan, R. (2008). Tuning PID and PI/λDδ controllers using the integral time absolute error criterion. Proceedings of the International Conference on Information and Automation for Sustainability, Colombo, Sri Lanka, 1–6.
26. [26] Das, S., Das, S. & Gupta, A. (2011). Fractional order modeling of a PHWR under step-back condition and control of its global power with a robust PIλDμ controller. IEEE Transactions on Nuclear Science, 58(5), 2431–2441.
27. [27] Chen, H., Wang, M., Li, S. & Mirjalili, S. (2019). An opposition-based sine cosine approach with local search for parameter estimation of photovoltaic models. Energy Conversion and Management, 195, 927–942.
28. [28] Chen, H., Xu, Y., Wang, M. & Zhao, X. (2019). A balanced whale optimization algorithm for constrained engineering design problems. Applied Mathematical Modelling, 71, 45–59.
29. [29] Wang, M. & Chen, H. (2020). Chaotic multi-swarm whale optimizer boosted support vector machine for medical diagnosis. Applied Soft Computing, 88, 105946.
30. [30] Wang, M., Chen, H., Li, S. & Mirjalili, S. (2017). Toward an optimal kernel extreme learning machine using a chaotic moth–flame optimization strategy. Neurocomputing, 267, 69–84.
31. [31] Mishra, S. K. & Chandra, D. (2014). Stabilization and tracking control of inverted pendulum using fractional-order PID controllers. Journal of Engineering, 2014, 752918.
32. [32] Lanjewar, A., Khubalkar, S. W. & Junghare, A. S. (2018). Comparative analysis of two-loop integer and fractional-order PID controller for inverted pendulum. Proceedings of the International Conference on Smart Electric Drives and Power Systems, Nagpur, India, 1–6.
33. [33] Erkol, H. O. (2018). Optimal PIλDμ controller design for two-wheeled inverted pendulum. IEEE Access, 6, 75709–75717.
34. [34] Mirjalili, S., Mirjalili, S. M. & Lewis, A. (2014). Grey wolf optimizer. Advances in Engineering Software, 69, 46–61.
35. [35] Zhao, W., Wang, L. & Mirjalili, S. (2022). Artificial hummingbird algorithm: A new bio-inspired optimizer with its engineering applications. Computer Methods in Applied Mechanics and Engineering, 388, 114194.
36. [36] Soylu, S. (2024). Performance analysis of fractional-order controllers for blade pitch angle control of a wind turbine. Proceedings of the 4th International Conference on Electrical, Computer, Communications and Mechatronics Engineering (ICECCME), 1–6.
37. [37] Çelik, M. & Soylu, S. (2023). Parameter estimation study of polymer electrolyte membrane fuel cell using artificial hummingbird algorithm. Proceedings of the Institution of Mechanical Engineers, Part C, 237(8), 1956–1967.
38. [38] Bhoi, S., Rout, N. C., & Rout, B. C. (2017). MGWO meta heuristic algorithm vs. classical tuning method of FOPID controller for inverted pendulum. International Conference on Innovations in Control, Communication and Information Systems, 1–7.
39. [39] Hasan, M. A., Oglah, A. A., & Marie, M. J. (2023). Optimal FOPI-FOPD controller design for rotary inverted pendulum system using grey wolves’ optimization technique. TELKOMNIKA Telecommunication Computing Electronics and Control, 21(3), 657–666.
40. [40] Shah, P. & Agashe, S. (2016). Review of fractional PID controller. Mechatronics, 38, 29–41.
41. [41] Howard, F. L. (1931). The life history of Physarum polycephalum. American Journal of Botany, 18(2), 116–133.
42. [42] Latty, T. & Beekman, M. (2010). Food quality and the risk of light exposure affect patch-choice decisions. Ecology, 91(1), 22–27.
43. [43] Li, S., Chen, H., Wang, M., Heidari, A. A. & Mirjalili, S. (2020). Slime mould algorithm: A new method for stochastic optimization. Future Generation Computer Systems, 111, 300–323.