Iranian Journal of Mechanical Engineering Transactions of the ISME

Iranian Journal of Mechanical Engineering Transactions of the ISME

Enrichment of Approximated Flexible Beam Model for Constrained Control of Vibration Suppression

Document Type : Research Paper

Authors
1 Ph.D. Candidate, Faculty of Mechanical Engineering, Sahand University of Technology, Tabriz, Iran
2 Professor, Faculty of Mechanical Engineering, Sahand University of Technology, Tabriz, Iran
3 Assistant Professor, Faculty of Mechanical Engineering, Sahand University of Technology, Tabriz, Iran
4 B.Sc. Student, Faculty of Mechanical Engineering, Sahand University of Technology, Tabriz, Iran
Abstract
For flexible beams, the application of distributed models with infinite degrees of freedom is complicated to precise control of vibration. Also, the approximated models bring uncertainties that negatively affect the controller performance. This paper aims to enhance the content of an assumed mode model of a cantilever beam by using the measurement information. A new estimation method is developed to estimate the uncertainty of the approximated model by minimizing the output estimation error at the current time. Experimentally tests are conducted to show the performance of the proposed estimation method compared with an extended state observer. Accordingly, a novel controller is developed using the estimated model to reduce the vibration of beam considering the control input limitations. The stability of the constrained second-order control system is mathematically analyzed, and its performance is evaluated through a co-simulation environment via Adams-MATLAB. The obtained results demonstrate that the suggested estimation scheme remarkably improves the accuracy of the approximated beam model. Consequently, the controller that uses the model uncertainty and disturbance information has a great efficiency in suppressing the vibrations of the beam.
Keywords

Subjects


[1] R. Mohsenipour, and G. Liu, "Flexible Single-link Manipulators Control Based on a Full-order Transfer Function Model," IEEE Transactions on Automatic Control, Vol. 69, No. 6, 2024, doi: https://doi.org/10.1109/TAC.2024.3351552.
 
[2] J. K. Viswanadhapalli, V. K. Elumalai, S. Shivram, S. Shah, and D. Mahajan, "Deep Reinforcement Learning with Reward Shaping for Tracking Control and Vibration Suppression of Flexible Link Manipulator," Applied Soft Computing, Vol. 152, pp. 110756, 2024, doi: https://doi.org/10.1016/j.asoc.2023.110756.
 
[3] Y. Wang, W. Wu, X. Lou, and D. Görges, "Iterative Learning Control of an Euler-Bernoulli Beam with Time-varying Boundary Disturbance," Computers & Mathematics with Applications, Vol. 162, pp. 145-154, 2024, doi: https://doi.org/10.1016/j.camwa.2024.03.004.
 
[4] E.-G. Liu, Y.-F. Shao, B. Dou, J.-F. Cui, and H. Ding, "High-order Modal Vibration Control of Timoshenko Beams Based on Nonlinear Energy Sink Cells," Journal of Vibration Engineering & Technologies, Vol. 12, pp. 6809-6819, 2024, doi: https://doi.org/10.1007/s42417-024-01284-5.
 
[5] Q. Yao, M. Luo, and D. Zhang, "Milling Dynamic Model Based on Rotatory Euler–Bernoulli Beam Model under Distributed Load," Applied Mathematical Modelling, Vol. 83, pp. 266-283, 2020, doi: https://doi.org/10.1016/j.apm.2020.02.015.
 
[6] X. Xing, and J. Liu, "PDE Modelling and Vibration Control of Overhead Crane Bridge with unknown Control Directions and Parametric Uncertainties," IET Control Theory & Applications, Vol. 14, No. 1, pp. 116-126, 2020, doi: https://doi.org/10.1007/978-981-16-1532-0_12.
 
[7] Y. Lei, T. Murmu, S. Adhikari, and M. Friswell, "Dynamic Characteristics of Damped Viscoelastic Nonlocal Euler–Bernoulli Beams," European Journal of Mechanics-A/Solids, Vol. 42, pp. 125-136, 2013, doi: https://doi.org/10.1016/j.euromechsol.2013.04.006.
 
[8] H. Wang, and G. Chen, "Asymptotic Locations of Eigenfrequencies of Euler–Bernoulli Beam with Nonhomogeneous Structural and Viscous Damping Coefficients," SIAM Journal on Control and Optimization, Vol. 29, No. 2, pp. 347-367, 1991, doi: https://doi.org/10.1137/0329019.
 
[9] I. Giorgio, and D. Del Vescovo, "Non-linear Lumped-parameter Modeling of Planar Multi-link Manipulators with Highly Flexible Arms," Robotics, Vol. 7, No. 4, pp. 60, 2018, doi: https://doi.org/10.3390/robotics7040060.
 
[10] K. Lochan, B. K. Roy, and B. Subudhi, "Robust Tip Trajectory Synchronisation between Assumed Modes Modelled Two-link Flexible Manipulators using Second-order PID Terminal SMC," Robotics and Autonomous Systems, Vol. 97, pp. 108-124, 2017, doi: https://doi.org/10.1016/j.robot.2017.08.008.
 
[11] R. Vidoni, L. Scalera, and A. Gasparetto, "3-D ERLS Based Dynamic Formulation for Flexible-link Robots: Theoretical and Numerical Comparison between the Finite Element Method and the Component Mode Synthesis Approaches," International Journal of Mechanics and Control, Vol 19, No 1, pp. 39-50, 2018. [Online]. Available: https://bia.unibz.it/esploro/outputs/991005772712201241.
 
[12] S. Grazioso, V. Sonneville, G. Di Gironimo, O. Bauchau, and B. Siciliano, "A Nonlinear Finite Element Formalism for Modelling Flexible and Soft Manipulators," in 2016 IEEE International Conference on Simulation, Modeling, and Programming for Autonomous Robots (SIMPAR), 13-16 December 2016: IEEE, San Francisco, CA, USA,  pp. 185-190, doi: https://doi.org/10.1109/SIMPAR.2016.7862394.
 
[13] N. Mishra, and S. Singh, "Determination of Modes of Vibration for Accurate Modelling of the Flexibility Effects on Dynamics of a Two Link Flexible Manipulator," International Journal of Non-linear Mechanics, Vol. 141, pp. 103943, 2022, doi: https://doi.org/10.1016/j.ijnonlinmec.2022.103943.
 
[14] M. O. Tokhi, and A. K. Azad, "Flexible Robot Manipulators: Modelling, Simulation and Control", First Edition, Institution of Engineering and Technology-IET, London, United Kingdom, 2008, ISBN:9780863414480, 0863414486, doi: https://doi.org/10.1049/PBCE068E.
 
[15] J. Fan, D. Zhang, and H. Shen, "Dynamic Modeling and Simulation of a Rotating Flexible Hub-beam Based on Different Discretization Methods of Deformation Fields," Archive of Applied Mechanics, Vol. 90, pp. 291-304, 2020, doi: https://doi.org/10.1007/s00419-019-01609-x.
 
[16] H. H. Yoo, J. E. Cho, and J. Chung, "Modal Analysis and Shape Optimization of Rotating Cantilever Beams," Journal of Sound and Vibration, Vol. 290, No. 1-2, pp. 223-241, 2006, doi: https://doi.org/10.1016/j.jsv.2005.03.014.
 
[17] M. Raoufi, and H. Delavari, "Experimental Implementation of a Novel Model‐free Adaptive Fractional‐order Sliding Mode Controller for a Flexible‐link Manipulator," International Journal of Adaptive Control and Signal Processing, Vol. 35, No. 10, pp. 1990-2006, 2021, doi: https://doi.org/10.1002/acs.3305.
 
[18] A. Green, and J. Sasiadek, "Robot Manipulator Control for Rigid and Assumed Mode Fexible Dynamics Models," in AIAA Guidance, Navigation, and Control Conference and Exhibit, 11-14 August, 2003, Autin, Texas, pp. 5435, doi: https://doi.org/10.2514/6.2003-5435.
 
[19] T. Bhaskarwar, H. F. Hawari, N. B. Nor, R. H. Chile, D. Waghmare, and S. Aole, "Sliding Mode Controller with Generalized Extended State Observer for Single Link Flexible Manipulator," Applied Sciences, Vol. 12, No. 6, pp. 3079, 2022, doi: https://doi.org/10.3390/app12063079.
 
[20] R. M. Gharamaleki, M. Mirzaei, S. Rafatnia, and B. Alizadeh, "An Analytical Approach to Optimal Control of Nonlinear Systems with Input Constraints," International Journal of Automation and Control, Vol. 14, No. 2, pp. 213-238, 2020, doi: https://doi.org/10.1504/IJAAC.2020.105519.
 
[21] A. Ghiasi, G. Alizadeh, and M. Mirzaei, "Simultaneous Design of Optimal Gait Pattern and Controller for a Bipedal Robot," Multibody System Dynamics, Vol. 23, pp. 401-429, 2010, doi: https://doi.org/10.1007/s11044-009-9185-z.
 
[22] A. Tahouni, M. Mirzaei, and B. Najjari, "Applied Nonlinear Control of Vehicle Stability with Control and State Constraints," Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, Vol. 234, No. 1, pp. 191-211, 2020, doi: https://doi.org/10.1177/0954407019848858.
 
[23] R. Azmi, M. Mirzaei, and A. Habibzadeh-Sharif, "A Novel Optimal Control Strategy for Regenerative Active Suspension System to Enhance Energy Harvesting," Energy Conversion and Management, Vol. 291, pp. 117277, 2023, doi: https://doi.org/10.1016/j.enconman.2023.117277.
 
[24] J. Li, L. Zhang, L. Luo, and S. Li, "Extended State Observer Based Current-constrained Controller for a PMSM System in Presence of Disturbances: Design, Analysis and Experiments," Control Engineering Practice, Vol. 132, pp. 105412, 2023, doi: https://doi.org/10.1016/j.conengprac.2022.105412.
 
[25] F. Pak, M. Mirzaei, S. Rafatnia, and S. Salmani Pour Avval, "Novel Observer-based Input-constrained Control of Nonlinear Second-order Systems with Stability Analysis: Experiment on Lever Arm," Iranian Journal of Science and Technology, Transactions of Electrical Engineering, pp. 1-18, 2024, doi: https://doi.org/10.1007/s40998-024-00713-1.
 
[26] B.-Z. Guo, and Z.-l. Zhao, "On the Convergence of an Extended State Observer for Nonlinear Systems with Uncertainty," Systems & Control Letters, Vol. 60, No. 6, pp. 420-430, 2011, doi: https://doi.org/10.1016/j.sysconle.2011.03.008.
 
[27] M. M. Shahir, M. Mirzaei, M. Farbodi, and S. Rafatnia, "Extended State Observer-Based Feedback Linearization Control of Shape Memory Alloy Actuator: Design and Experiment," International Journal of Dynamics and Control, pp. 1-12, 2023, doi: https://doi.org/10.1007/s40435-023-01362-8.
 
[28] H. Gui, L. Jin, and S. Xu, "Simple Finite-Time Attitude Stabilization Laws for Rigid Spacecraft with Bounded Inputs," Aerospace Science and Technology, Vol. 42, pp. 176-186, 2015, doi: https://doi.org/10.1016/j.ast.2015.01.020.
 
[29] A. M. Zou, A. H. de Ruiter, and K. D. Kumar, "Finite‐Time Attitude Tracking Control for Rigid Spacecraft with Control Input Constraints," IET Control Theory & Applications, Vol. 11, No. 7, pp. 931-940, 2017, doi: https://doi.org/10.1049/iet-cta.2016.1097.
 
[30] J.-H. Tsai, Y.-Y. Du, W.-Z. Zhuang, S.-M. Guo, C.-W. Chen, and L.-S. Shieh, "Optimal Anti-Windup Digital Redesign of Multi-Input Multi-Output Control Systems under Input Constraints," IET Control Theory & Applications, Vol. 5, No. 3, pp. 447-464, 2011, doi: https://doi.org/0.1049/iet-cta.2010.0020.
 
[31] Y. Su, and J. Swevers, "Finite-time Tracking Control for Robot Manipulators with Actuator Saturation," Robotics and Computer-integrated Manufacturing, Vol. 30, No. 2, pp. 91-98, 2014, doi: https://doi.org/10.1016/j.rcim.2013.09.005.
 
[32] Z. Ma, and P. Huang, "Adaptive Neural-network Controller for an Uncertain Rigid Manipulator with Input Saturation and Full-order State Constraint," IEEE Transactions on Cybernetics, Vol. 52, No. 5, pp. 2907-2915, 2020, doi: https://doi.org/10.1109/TCYB.2020.3022084.
[33] D. E. Kirk, "Optimal Control Theory: An Introduction", First Publication, Courier Corporations, Dover Publications, Inc., Mineola, New York, 2004, doi: https://doi.org/10.1002/aic.690170452.
 
[34] R. C. Loxton, K. L. Teo, V. Rehbock, and K. F. C. Yiu, "Optimal Control Problems with a Continuous Inequality Constraint on the State and the Control," Automatica, Vol. 45, No. 10, pp. 2250-2257, 2009, doi: https://doi.org/10.1016/j.automatica.2009.05.029.
 
[35] H. Marzban, and M. Razzaghi, "Rationalized Haar Approach for Nonlinear Constrained Optimal Control Problems," Applied Mathematical Modelling, Vol. 34, No. 1, pp. 174-183, 2010, doi: https://doi.org/10.1016/j.apm.2009.03.036.
 
[36] D. Liu, D. Wang, and X. Yang, "An Iterative Adaptive Dynamic Programming Algorithm for Optimal Control of unknown Discrete-time Nonlinear Systems with Constrained Inputs," Information Sciences, Vol. 220, pp. 331-342, 2013, doi: https://doi.org/10.1016/j.ins.2012.07.006.
 
[37] S. Mashayekhi, Y. Ordokhani, and M. Razzaghi, "Hybrid Functions Approach for Nonlinear Constrained Optimal Control Problems," Communications in Nonlinear Science and Numerical Simulation, Vol. 17, No. 4, pp. 1831-1843, 2012, doi: https://doi.org/10.1016/j.cnsns.2011.09.008.
 
[38] D.-P. Li, D.-J. Li, Y.-J. Liu, S. Tong, and C. P. Chen, "Approximation-based Adaptive Neural Tracking Control of Nonlinear MIMO unknown Time-varying Delay Systems with Full State Constraints," IEEE Transactions on Cybernetics, Vol. 47, No. 10, pp. 3100-3109, 2017, doi: https://doi.org/10.1109/TCYB.2017.2707178.
 
[39] Z. Chen, Z. Li, and C. P. Chen, "Adaptive Neural Control of Uncertain MIMO Nonlinear Systems with State and Input Constraints," IEEE Transactions on Neural Networks and Learning Systems, Vol. 28, No. 6, pp. 1318-1330, 2016, doi: https://doi.org/10.1109/TNNLS.2016.2538779.
 
[40] C. E. Garcia, D. M. Prett, and M. Morari, "Model Predictive Control: Theory and Practice—A Survey," Automatica, Vol. 25, No. 3, pp. 335-348, 1989, doi: https://doi.org/10.1016/0005-1098(89)90002-2.
 
[41] N. F. Silva Jr, C. E. T. Dórea, and A. L. Maitelli, "An Iterative Model Predictive Control Algorithm for Constrained Nonlinear Systems," Asian Journal of Control, Vol. 21, No. 5, pp. 2193-2207, 2019, doi: https://doi.org/10.1002/asjc.1815.
 
[42] S. S. Rao, "Vibration of Continuous Systems", John Wiley & Sons, United Kingdom, 2019, ISBN:9781119424147, 1119424143 , doi: https://doi.org/ 10.1002/9780470117866.
 
[43] Z. A. Sisi, M. Mirzaei, and S. Rafatnia, "Estimation of Vehicle Suspension Dynamics with Data Fusion for Correcting Measurement Errors," Measurement, Vol. 231, pp. 114438, 2024, doi: https://doi.org/10.1016/j.measurement.2024.114438.
 
[44] S. Rafatnia, and M. Mirzaei, "Estimation of Reliable Vehicle Dynamic Model using IMU/GNSS Data Fusion for Stability Controller Design," Mechanical Systems and Signal Processing, Vol. 168, pp. 108593, 2022, doi: https://doi.org/10.1016/j.ymssp.2021.108593.
[45] C.-T. Chen, "Linear System Theory and Design", Saunders College Publishing, Oxford University Press Inc., New York, NY, United States, 1998, [Online] Available: https://dl.acm.org/doi/10.5555/521603.
 
[46] S. Jamshidi, M. Mirzaei, and M. Malekzadeh, "Applied Nonlinear Control of Spacecraft Simulator with Constraints on Torque and Momentum of Reaction Wheels," ISA Transactions, Vol. 138, pp. 705-719, 2023, doi: https://doi.org/10.1016/j.isatra.2023.03.027.
 
[47] B. Najjari, M. Mirzaei, and A. Tahouni, "Decentralized Integration of Constrained Active Steering and Torque Vectoring Systems to Energy-efficient Stability Control of Electric Vehicles," Journal of the Franklin Institute, Vol. 359, No. 16, pp. 8713-8741, 2022, doi: https://doi.org/10.1016/j.jfranklin.2022.08.035.
 
[48] S. K. Samiei, M. Mirzaei, and S. Rafatnia, "Constrained Control of Flexible-joint Lever Arm Based on Uncertainty Estimation with Data Fusion for Correcting Measurement Errors," Nonlinear Dynamics, pp. 1-20, 2024, doi: https://doi.org/10.1007/s11071-024-09637-1.
 
[49] S. J. Wright, "Numerical Optimization", Springer-Verlag New York, Inc., New York, NY, United States, 2006, doi: https://doi.org/10.1007/978-0-387-40065-5.
 
[50] H. K. Khalil, "Control of Nonlinear Systems", Prentice Hall, New York, NY, United States, 2002, Available: https://books.google.com/books/about/Nonlinear_Control.html?id=-WbjoAEACAAJ.