LMI-Based Control of a Double Pendulum Crane

Mustapha Muhammad, Auwalu Muhammad Abdullahi, Amir A. Bature, Salinda Buyamin, M. M. Bello


This paper presents the design of a Linear Matrix Inequality (LMI) based state feedback controller for position tracking, hook and payload oscillations of a double pendulum crane. In this work, a linearised model of the crane was firstly obtained. The idea of formulating the stability condition in the form of LMIs using the linearised model was proposed. Using this approach, the stabilisation conditions constraints the closed-loop poles of the system to be in an LMI region to guarantee the system stability and gives satisfied transient performance. Based on the stabilization conditions formulated, an LMI based state feedback tracking controller for trolley position tracking and swing angles control of the double pendulum crane is proposed.  Mostly, two or more controllers were used for the control of double pendulum crane to control the trolley, hook and payload. However, in this work a single LMI based controller is used to achieve similar performance. This reduces the number of controllers which minimises cost, complexity and size of the control system. The performance of the proposed controller is investigated via simulations. The simulation result shows the proposed controller is able to track the trolley position relatively fast with minimum hook and payload swing angles hence, reduce the main problem of a double pendulum crane. From the performance comparison of the results, the proposed method has improved in terms of reducing the trolley position percentage overshoot, hook oscillation and payload oscillation with 72%, 52% and 65% respectively.


Double Pendulum Crane, Linear Matrix Inequality (LMI), State feedback control, LMI region

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M. Zhang, X. Ma, H. Chai, X. Rong, X. Tian, Y. Li, A novel online motion planning method for double-pendulum overhead cranes. Nonlinear Dyn. 85, 1079–1090, 2016.

Y. Jisup, S. Nation, W. Singhose, J.E. Vaughan, Control of crane payloads that bounce during hoisting, IEEE Transactions on Control Systems Technology. 22 1233-1238, 2014.

N. B. Almutairi, M. Zribi, Sliding mode control of a three-dimensional overhead crane, Journal of Vibration and Control. 15 1679–1730, 2009.

Z. N. Masoud and, M.F. Daqaq, A graphical approach to input-shaping control design for container cranes with hoist, IEEE Transactions on Control Systems Technology. 14 1070-1077, 2006.

J. Smoczek, Fuzzy crane control with sensorless payload deflection feedback for vibration reduction, Mechanical Systems and Signal Processing. 46, pp.70-81, 2014.

N. Sun, Y. Fang, H. Chen, and B. Lu, Amplitude-saturated nonlinear output feedback antiswing control for underactuated cranes with double-pendulum cargo dynamics, IEEE Transactions on Industrial Electronics, 64(3) 2135-2146, 2017.

N. Sun, Y. Fang, H. Chen, and B. Lu. Energy-based control of double pendulum cranes. In The 5th Annual IEEE International Conference on Cyber Technology in Automation, Control and Intelligent Systems, Shenyang, China, (2015) pp. 258–263.

N. Sun, Y. Fang, H. Chen, Y. Fu, Super-twisting-based antiswing control for underactuated double pendulum cranes. In: IEEE International Conference on Advanced Intelligent Mechatronics, Busan, Korea, (2015) pp. 749–754.

L. A. Tuan, S.G. Lee, Sliding mode controls of double-pendulum crane systems. J. Mech. Sci. Technol. 27(6), 1863–1873, 2013.

H. Chen, Y. Fang, N. Sun, Y. Qian, Pseudospectral method-based time optimal trajectory planning for double pendulum cranes. In: Proceedings of the 34th Chinese Control Conference, Hangzhou, China, (2015) pp. 4302–4307

K. A. Alhazza, A. M. Hasan K. A. Alghanim, Z. N. Masoud, An iterative learning control technique for point-to-point maneuvers applied on an overhead crane. Shock Vib. 2014, 1–11, 2014.

W. E. Singhose, S. T. Towell Double-pendulum gantry crane dynamics and control. In Proceedings of the IEEE International Conference Control Applications, Trieste, Italy, (1998) pp. 1205–1209.

W. P. Guo, D. T. Liu, J. Q. Yi, D. B. Zhao, Passivity-based-control for double-pendulum-type overhead cranes. In: Proceedings of the IEEE Region 10 Conference, Chiang Mai, Thailand, pp. 546–549 (2004)

S. Lahres, H. Aschemann, O. Sawodny, E. P. Hofer, Crane automation by decoupling control of a double pendulum using two translational actuators. In: Proceedings of the American Control Conference, Chicago, USA, (2000) pp. 1052–1056.

O’Connor, W., Habibi, H.: Gantry crane control of a double-pendulum, distributed-mass load, using mechanical wave concepts. Mech. Sci. 4, 251–261, 2013.

T. Y. jian and Z. Mohamed, Modelling and Sway Control of a Double-Pendulum Overhead Crane System, Applications of Modelling and Simulation, 1(1), pp. 15-21, 2017.

G. Yang, W. Zhang, Y. Huang, Y. Yu, Simulation research of extension control based on crane-double pendulum system. Comput.. Inf. Sci. 2(1), 103–107, 2009.

M. A., Ahmad, R. M. T. Ismail, M. S. Ramli, N. Hambali, and N. H. Noordin, Active Sway Control of a Laboratory Scale Rotary Crane System. Proceedings of the 2nd International Conference on Computer and Automation Engineering, Febuary 26-28, Singapore, (2010) pp. 408–412.

M. A. Ahmad, M. S. Ramli, M. A. Zawawi and R. M. T. Ismail, Hybrid Collocated PD with Non-Collocated PID for Sway Control of a Lab-Scaled Rotary Crane. Proceedings of the 5th IEEE Conference on Industrial Electronics and Applications, June 15-17, (2010) Taichung, Taiwan pp. 707–711.

M. A Nazemizadeh, PID Tuning Method for Tracking Control of an Underactuated Gantry Crane. Universal Journal of Engineering Mechanics, 2013. 1: 45-49.

M. Zaidi, M. Tumari, S. Latifah, M. Z. Anwar and M. D. Razali, Experimental Investigation on Active Sway Control of a Gantry Crane System using PID Controller. Proceedings of 2nd International Conference on Electrical, Control and Computer Engineering (InECCE2013), August 27-28, 2013. Pahang, Malaysia, 295-299.

S. Y. S. Hussien, H. I. Jaafar, R. Ghazali, and N. R. A. Razif, The Effects of Auto-Tuned Method in PID And PD Control Scheme for Gantry Crane System. International Journal of Soft Computing and Engineering, 4(4): 121-125, 2015.

S. K. Biswas, Optimal control of gantry crane for minimum payload oscillations. Proceedings of 4th International Conference on Dynamic Systems and Applications, May 21-24, 2003. Atlanta, USA, 12-19.

Y. Yoshida and H. Tabata, Visual Feedback Control of an Overhead Crane and its Combination with Time-Optimal Control. Proceedings of IEEE/ASME International Conference on Advanced Intelligent Mechatronics. June 2-5, 2008. Xian, China: 1114-1119.

B. T. Gao, H. J. Chen, X. H. Zhang, and H. Qi , A Practical Optimal Controller for Underactuated Gantry Crane Systems. Proceedings of IEEE International Symposium on Systems and Control in Aerospace and Astronautics. June 19-21, 2006, Harbin, China: 726-730.

B. Wie, R. Sinha, and Q. Liu, Robust Time-Optimal Control of Uncertain Structural Dynamic Systems. Journal of Guidance, Control and Dynamics, 1992. 16: 980-983.

J. Auernig and H. Troger, Time Optimal Control of Overhead Cranes with Hoisting of the Load. Automatica, 23: 437-447, 1987.

K. A. F. Moustafa, K. H. Harib, and F. Omar, Optimal Controller Design of an Overhead Crane: Monte Carlo Versus Pre-Filter-Based Designs. Transactions of the Institute of Measurement and Control, 35(2): 219-226, 2013.

H. I. Jaafar, M. F. Sulaima, Z. Mohamed, and J. J. Jamian, Optimal PID Controller Parameters for Nonlinear Gantry Crane System via MOPSO Technique. Proceedings of IEEE Conference on Sustainable Utilization and Development in Engineering and technology. May 30-June 1, 2013. Selangor, Malaysia: 89-91.

J. Jafari, M. Ghazal, and M. Nazemizadeh, A LQR Optimal Method to Control the Position of an Overhead Crane. International Journal of Robotics and Automation, 3(4): 252-258, 2014.

M. Böck, and A. Kugi, Real-Time Nonlinear Model Predictive Path-Following Control of a Laboratory Tower Crane. IEEE Transactions on Control Systems Technology, 22: 1461–1473, 2014.

P. Pannil, K. Smerpitak, O. V. La, and T. Trisuwannawat, Load Swing Control of an Overhead Crane. Proceedings of the International Conference on Control, Automation and Systems, October 27-30, 2010. Gyeonggi-do, Korea: 1926–1929.

H. Saeidi, M. Naraghi, and A. A. Raie, A Neural Network Self-Tuner based on Input Shapers Behaviour for Anti-Sway System of Gantry Cranes. Journal of Vibration and Control, 19:1936–1949, 2013.

S. Duong, H. Kinjo, E. Uezato, and T. Yamamoto, Particle Swarm Optimisation of a Recurrent Neural Network Control for an Under Actuated Rotary Crane with Particle Filter Based State Estimation. Proceedings of the 24th International Conference on Industrial, Engineering and Other Applications of Applied Intelligent Systems, June 29-July 01, 2011. Syracuse NY, USA: 51–58.

L. Drag, Model of an Artificial Neural Network for Optimisation of Payload Positioning in Sea Waves. Ocean Engineering, 2016. 115: 123–134.

P. Li, Z. Li, and Y. Yang, The Application Research of Anti Colony Optimisation Algorithm for Intelligent Control on Special Crane. Proceedings of the Second International Conference on Instrumentation, Measurement, Computer, Communication and Control, December 8-10, 2012. Heilongjiang, China: 999–1004.

H. C. Cho, J. W. Lee, Y. J. Lee, and K. S. Lee, Lyapunov Theory Based Robust Control of Complicated Nonlinear Mechanical Systems with Uncertainty. Journal of Mechanical Science and Technology, 2008. 22(11): 2142-2150.

Z. M. Chen, W. J. Meng, M. H. Zhao, and J. G. Zhang, Hybrid Robust Control for Gantry Crane System. Applied Mechanics and Mechanical Engineering, 2010. 29: 2082-2088.

Y. Tangwe, Q. Yong. and H. Jianda, Robust Control of Gantry Crane System with Hoisting: A New Solution based on Wave Motion. Journal of Central South University, 42: 288-292, 2011.

J. Smoczek, J. Szpytko, and P. Hyla, Interval Analysis Approach to Prototype the Robust Control of the Laboratory Overhead Crane. Materials Science and Engineering, 65(1): 1-6, 2014.

L. A. Tuan, S. G. Lee, L. C. Nho, and H. M. Cuong, Robust Controls for Ship-Mounted Container Cranes with Viscoelastic Foundation and Flexible Hoisting Cable. Journal of Systems and Control Engineering, 2015. 229: 662-674.

M. Z. Tumari, M. S. Saealal, Y. Abdul Wahab, and M. R. Ghazali, H-Infinity Controller with LMI Region Schemes for a Laboratory Scale Rotary Pendulum Crane System. International Journal of Systems Signal Control and Engineering Application, 1: 14–20, 2012.

G. Hilhorst, G. Pipeleers, and J. Swevers, Reduced-Order Multi-Objective H-∞ Control of an Overhead Crane Test Setup. Proceedings of 52nd IEEE Conference on Decision and Control, December 10-13, 2013: Florence, Italy: 770–775.

M. Z. Tumari, M. S. Saealal, M. R. Ghazali, and Y. Abdul Wahab, H-Infinity Controller with Graphical LMI Region Profile for Gantry Crane System. Proceedings of the 13th International Symposium on Advanced Intelligence Systems, November 20-24, 2012. Kobe, Japan: 1397–1402.

M. K. Bak, M. R. Hansen, and H. R. Karimi, Robust Tool Point Control for Offshore Knuckle Boom Crane. Proceedings of the 18th World Congress of the International Federation of Automatic Control, August 28-September 2, 2011. Milano, Italy: 4594–4599.

F. Castanos, and L. Fridman, Analysis and Design of Integral Sliding Manifolds for System with Unmatched Perturbations. IEEE Transaction on Automatic Control, 2006. 51(5): 853-858.

H. I. Jaafar and Z. Mohamed, "PSO-Tuned PID Controller for a Nonlinear Double-Pendulum Crane System," in Proceedings of the 17th Asia Simulation Conference, AsiaSim 2017, August 27–29, 2017, Melak, Malaysia. pp. 203-215.

M. Chilali and P. Gahinet, “H infinity Design with Pole Placement Constraint: an LMI approach” IEEE Transactions on Automatic Control, 41(3), (1996), 358-366.


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