Multi-objective optimal tuning and performance comparison of the LQR controller for an underactuated motion control system with GA, ABC, VPS, and BB-BC algorithms

Year 2025, Volume: 5 Issue: 2, 376 - 395, 30.06.2025

Ferhat Kaya , Çağlar Conker , Hasan Hüseyin Bilgiç

https://doi.org/10.53391/mmnsa.1586464

Abstract

The presented study is organized to provide details on the design and performance analysis of multi-objective optimization of LQR controller parameters for an underactuated motion (pendulum type gantry crane) system. The objective of the optimization is to design an LQR controller to eliminate pendulum oscillations caused by the motion of the system. In line with this objective, a new multi-objective function has been designed by considering the important parameters of control responses. The VPS and BB-BC algorithms have been utilized for the first time in the design and development of motion control for single pendulum gantry systems and compared with traditional GA and ABC algorithms. The six different populations or particle size values of GA, ABC, VPS, and BB BC algorithms were examined over 100 iterations to achieve the most successful optimization results. Furthermore, the configurations of the GA, ABC, VPS, and BB-BC algorithms yielding the best control performance were compared amongst themselves and against conventionally designed LQR controllers. Preliminary design findings indicated the elimination of steady-state error in the pendulum cart system, along with a considerable improvement of 51.54\% in settling time. Additionally, a substantial enhancement of up to 67.57\% was achieved in the settling time of the pendulum angle.

Keywords

multi-objective optimization , linear quadratic regulator , genetic algorithm , artificial bee colony , vibrating particles system algorithm , big bang-big crunch algorithm

References

• [1] Conker, C., Kilic, A., Mistikoglu, S., Kapucu, S. and Yavuz, H. An enhanced control technique for the elimination of residual vibrations in flexible-joint manipulators. Strojniški vestnikJournal of Mechanical Engineering, 60(9), 592-599, (2014).
• [2] Abdel-Rahman, E.M., Nayfeh, A.H. and Masoud, Z.N. Dynamics and control of cranes: A review. Journal of Vibration and Control, 9(7), 863-908, (2003).
• [3] Xie, X., Huang, J. and Liang, Z. Using continuous function to generate shaped command for vibration reduction. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 227(6), 523-528, (2013).
• [4] Conker, C., Yavuz, H. and Bilgic, H.H. A review of command shaping techniques for elimination of residual vibrations in flexible-joint manipulators. Journal of Vibroengineering, 18(5), 2947-2958, (2016).
• [5] Masoud, Z.N. and Nayfeh, A.H. Sway reduction on container cranes using delayed feedback controller. Nonlinear Dynamics, 34, 347-358, (2003).
• [6] Burul, I., Kolonic, F. and Matuško, J. The control system design of a gantry crane based on H∞ control theory. In Proceedings, The 33rd International Convention MIPRO, pp. 183-188, Opatija, Croatia, (2010, May).
• [7] Bakhtiari-Nejad, F., Nazemizadeh, M. and Arjmand, H. Tracking control of an underactuated gantry crane using an optimal feedback controller. International Journal of Automotive and Mechanical Engineering, 7, 830-839, (2013).
• [8] Ospina-Henao, P.A. and Lopez-Suspes, F. Dynamic analysis and control PID path of a model type gantry crane. In Proceedings, 5th Colombian Conference of Engineering Physics (V CNIF), pp. 26-30, Medellin, Colombia, (2016, September).
• [9] Fang, Y., Ma, B., Wang, P. and Zhang, X. A motion planning-based adaptive control method for an underactuated crane system. IEEE Transactions on Control Systems Technology, 20(1), 241-248, (2012).
• [10] Tuan, L.A., Moon, S.C., Lee, W.G. and Lee, S.G. Adaptive sliding mode control of overhead cranes with varying cable length. Journal of Mechanical Science and Technology, 27, 885-893, (2013).
• [11] Choi, K. and Lee, J.S. Sliding mode control of overhead crane. International Journal of Modeling and Simulation, 31(3), 203-209, (2011).
• [12] Benhidjeb, A. and Gissinger, G.L. Fuzzy control of an overhead crane performance comparison with classic control. Journal of Engineering Practice, 3(12), 1687-1696, (1995).
• [13] Bilgiç, H.H., Conker, C., Yavuz, H. and ¸Sen, M.A. Sarkaç tipi bir tepe vincinin kontrolüne bulanık yakla¸sım. In Proceedings, 2015 TrC IFToMM Symposium on Theory of Machines and Mechanisms (TrISToMM 2015), pp. 455-461, Izmir, Türkiye, (2015, June).
• [14] Chiu, C.H. and Lin, C.H. Adaptive output recurrent neural network for overhead crane system. In Proceedings, IEEE Proceedings of SICE Annual Conference, pp. 1082-1087, Taipei, Taiwan, (2010, August).
• [15] Kimiaghalam, B., Homaifar, A., Bikdash, M. and Dozier, G. Genetic algorithms solution for unconstrained optimal crane control. In Proceedings, 1999 Congress on Evolutionary ComputationCEC99 (Cat. No. 99TH8406), pp. 2124-2130, Washington, DC, USA, (1999, July).
• [16] Solihin, M.I., Wahyudi and Legowo, A. Fuzzy-tuned PID anti-swing control of automatic gantry crane. Journal of Vibration and Control, 16(1), 127-145, (2009).
• [17] Benhellal, B., Hamerlain, M. and Rahmani, Y. Decoupled adaptive neuro-interval type-2 fuzzy sliding mode control applied in a 3Dcrane system. Arabian Journal for Science and Engineering, 43, 2725-2733, (2018).
• [18] Bilgiç, H.H., Conker, C. and Yavuz, H. Fuzzy logic-based decision support system for selection of optimum input shaping techniques in point-to-point motion systems. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 235(6), 795-808, (2020).
• [19] Ahmad, M.A., Nasir, A.N.K., Hambali, N. and Ishak, H. Hybrid input shaping and PD-type Fuzzy Logic control scheme of a gantry crane system. In Proceedings, 2009 IEEE Control Applications, (CCA) & Intelligent Control, (ISIC), pp. 1051-1056, St. Petersburg, Russia, (2009, July).
• [20] Suh, J., Lee, J., Lee, Y. and Lee, K. Anti-sway position control of an automated transfer crane based on neural network predictive PID controller. Journal of Mechanical Science and Technology, 19, 505-519, (2005).
• [21] Cho, H.C., Fadeli, M.S., Lee, Y.J. and Lee, K.S. Neural robust control for perturbed crane systems. Journal of Mechanical Science and Technology, 20, 591-601, (2006).
• [22] Solihin, M.I., Wahyudi, Kamal, M.A.S. and Legowo, A. Objective function selection of GAbased PID control optimization for automatic gantry crane. In Proceedings, 2008 International Conference on Computer and Communication Engineering, pp. 883-887, Kuala Lumpur, Malaysia, (2008, July).
• [23] Jafari, J., Ghazal, M. and Nazemizadeh, M. A LQR optimal method to control the position of an overhead crane. IAES International Journal of Robotics and Automation, 3(4), 252-258, (2014).
• [24] Yang, B. and Xiong, B. Application of LQR techniques to the anti-sway controller of overhead crane. Advanced Materials Research, 139-141, 1933-1936, (2010).
• [25] He, K.D., Fang, Z.F., Zhu, D.L. and Yang, W.H. The control system model of gantry crane for preventing swing. Applied Mechanics and Materials, 135-136, 1013-1019, (2012).
• [26] Li, M., Jiao, L., Qiao, J. and Ruan, X. Balance control of robot with CMAC based Q-learning. In Proceedings, 2008 Chinese Control and Decision Conference, pp. 2668-2672, Yantai, China, (2008, July).
• [27] Tang, X., Tao, G., Wang, L. and Stankovic, J.A. Robust and adaptive actuator failure compensation designs for a rocket fairing structural-acoustic model. IEEE Transactions on Aerospace and Electronic Systems, 40(4), 1359-1366, (2004).
• [28] Rao, L.V.V.G. and Narayanan, S. Optimal response of half car vehicle model with sky-hook damper using LQR with look ahead preview control. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 42, 471, (2020).
• [29] Petersen, I., Johansen, T.A., Kalkkuhl, J. and Lüdemann, J. Wheel slip control in ABS brakes using gain scheduled constrained LQR. In Proceedings, 2001 European Control Conference (ECC), pp. 606-611, Porto, Portugal, (2001, September).
• [30] Albayrak, G. and Özdemir, I. Yapı projelerinin süre-maliyet optimizasyonunda metasezgisel algoritma kullanımı. Adıyaman Üniversitesi Mühendislik Bilimleri Dergisi, 3(5), 39-49, (2016).
• [31] Blum, C. and Roli, A. Metaheuristics in combinatorial optimization: Overview and conceptual comparison. ACM Computing Surveys (CSUR), 35(3), 268-308, (2003).
• [32] Chen, Y., Wang, D. and Ning, W. Forecasting by TSK general type-2 fuzzy logic systems optimized with genetic algorithms. Optimization and Control Applications and Methods, 39(1), 393–409, (2018).
• [33] Brandão, M.A.L., Oliveira, G.T.S., Saramago, S.F.P. and Doricio, J.L. Using metaheuristics for optimum design of 3R orthogonal manipulators considering their topology. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 37, 1701-1718, (2015).
• [34] Karami, M., Vatankhah, R. and Khosravifard, A. A modified fuzzy-tuned artificial bee algorithm to optimal location of piezoelectric actuators and sensors for active vibration control of isotropic rectangular plates. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 43, 86, (2021).
• [35] Xi, W., Wang, Y., Chen, B. and Wu, H. Iterative learning control of robot based on artificial bee colony algorithm. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 233(9), 1221-1238, (2019).
• [36] Loucif, F., Kechida, S. and Sebbagh, A. Whale optimizer algorithm to tune PID controller for the trajectory tracking control of robot manipulator. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 42, 1, (2020).
• [37] Amini, F. and Ghaderi, P. Optimal locations for MR dampers in civil structures using improved ant colony algorithm. Optimization and Control Applications and Methods, 33(2), 232-248, (2012).
• [38] Eqra, N., Abiri, A.H. and Vatankhah, R. Optimal synthesis of a four-bar linkage for path generation using adaptive PSO. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 40, 469, (2018).
• [39] Bilgiç, H.H., Sen, M.A., Yapici, A., Yavuz, H. and Kalyoncu, M. Meta-heuristic tuning of the LQR weighting matrices using various objective functions on an experimental flexible arm under the effects of disturbance. Arabian Journal for Science and Engineering, 46, 7323-7336, (2021).
• [40] Bilgiç, H.H., Sen, M.A. and Kalyoncu, M. Tuning of LQR controller for an experimental inverted pendulum system based on The Bees Algorithm. Journal of Vibroengineering, 18(6), 3684-3694, (2016).
• [41] Quanser Innovate Educate, Linear Pendulum Gantry Module Datasheets. www.quanser.com
• [42] Holland, J.H. Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence. MIT Press: Cambridge, (1992).
• [43] Karaboga, D. and Basturk, B. A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm. Journal of Global Optimization, 39, 459-471, (2007).
• [44] Kaveh, A. and Ilchi Ghazaan, M. Vibrating particles system algorithm for truss optimization with multiple natural frequency constraints. Acta Mechanica, 228, 307-322, (2017).
• [45] Erol, O.K. and Eksin, I. A new optimization method: Big Bang–Big Crunch. Advances in Engineering Software, 37(2), 106-111, (2006).