Chaos Control in Gear Transmission System using GPC and SMC ‎Controllers

Document Type : Research Paper

Authors

Department of Electrical Engineering, Imam Khomeini International University, Qazvin, Iran‎

Abstract

Chaos is a phenomenon that occurs in some non-linear systems. Therefore, the output of the system will be heavily dependent on the initial conditions. Since the main characteristic of the chaos is an abnormal behavior of the system output, it should be considered in designing control systems. In this paper, controlling chaos phenomenon in a time-variant non-linear gear transmission system is investigated. To do this, a non-linear model for the system is introduced considering the effective parameters of the system, and then it is shown that chaos appears in the system by plotting phase plane of state-space variables. It should be noted that there is a great difference between random and chaotic behavior. In random cases, the model or input contains uncertainty, and therefore, the system behavior and output are not predictable. However, in chaotic behavior, there is only a brief uncertainty in the system model, input or initial conditions, and designing controller based on output prediction could be achieved. Therefore, model predictive control (MPC) algorithms are used to control the chaos, using the output prediction concept. In many cases, perturbation term also can be considered as uncertainty, and therefore, a robust controller family can be used for eliminating chaos. Both generalized predictive controller (GPC) and sliding mode controller (SMC) are used for chaos control here. The simulation results show the efficiency of the proposed algorithms.

Keywords

Main Subjects

[1] Slotine, J.-.E., Li, W.P., Applied Non-linear Control, 199(1), Englewood Cliffs, NJ: Prentice hall, 1991.
[2] Faieghi, M.R., Delavari, H., Baleanu, B., A novel adaptive controller for two-degree of freedom polar robot with unknown perturbations, Communications in Nonlinear Science and Numerical Simulation, 17(2), 2012, 1021-1030.
[3] Li, J., Li, W., Li, Q., Sliding mode control for uncertain chaotic systems with input nonlinearity, Communications in Nonlinear Science and Numerical Simulation, 17(1), 2012, 341-348.
[4] Yadav, V.K., Shukla, V.K., Das, S., Difference synchronization among three chaotic systems with exponential term and its chaos control, Chaos, Solitons & Fractals, 124, 2019, 36-51.
[5] Mendoza, S.A., et al, Parrondo’s paradox or chaos control in discrete two-dimensional dynamic systems, Chaos, Solitons & Fractals, 106, 2018, 86-93.
[6] Ge, Z-M., et al, Regular and chaotic dynamics of a rotational machine with a centrifugal governor, International Journal of Engineering Science, 37(7), 1999, 921-943.
[7] Zhou, S., et al, Non-linear dynamic response analysis on gear-rotor-bearing transmission system, Journal of Vibration and Control, 24(9), 2018, 1632-1651.
[8] Pan, H., et al, Finite-time stabilization for vehicle active suspension systems with hard constraints, IEEE Transactions on Intelligent Transportation Systems, 16(5), 2015, 2663-2672.
[9] Byrtus, M., Zeman, V., On modeling and vibration of gear drives influenced by non-linear couplings, Mechanism and Machine theory, 46(3), 2011, 375-397.
[10] de Souza, S.L.T., et al, Sudden changes in chaotic attractors and transient basins in a model for rattling in gearboxes, Chaos, Solitons & Fractals, 21(3), 2004, 763-772.
[11] Wang, J., Li, R., Peng, X., Survey of non-linear vibration of gear transmission systems, Applied Mechanics Reviews, 56(3), 2003, 309-329.
[12] Farshidianfar, A., Saghafi, A., Global bifurcation and chaos analysis in non-linear vibration of spur gear systems, Nonlinear Dynamics, 75(4), 2014, 783-806.
[13] Theodossiades, S., Natsiavas, S., Periodic and chaotic dynamics of motor-driven gear-pair systems with backlash. Chaos, Solitons & Fractals, 12(13), 2001, 2427-2440.
[14] Spitas, C., Spitas, V., Coupled multi-DOF dynamic contact analysis model for the simulation of intermittent gear tooth contacts, impacts and rattling considering backlash and variable torque, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 230(7-8), 2016, 1022-1047.
[15] Litak, G., Friswell, M. I., Vibration in gear systems, Chaos, Solitons & Fractals, 16(5), 2003, 795-800.
[16] Sakaridis, E., Spitas, V., Spitas, C., Non-linear modeling of gear drive dynamics incorporating intermittent tooth contact analysis and tooth eigenvibrations, Mechanism and Machine Theory, 136, 2019, 307-333.
[17] Jingyue, W., Lixin, G., Haotion, W., Analysis of bifurcation and non-linear control for chaos in gear transmission system, Research Journal of Applied Sciences, Engineering and Technology, 6(10), 2013, 1818-1824.
[18] Saghafi, A., Farshidianfar, A., An analytical study of controlling chaotic dynamics in a spur gear system, Mechanism and Machine Theory, 96, 2016, 179-191.
[19] Cai-Wan, G.J., Strong nonlinearity analysis for gear-bearing system under non-linear suspension-bifurcation and chaos, Nonlinear Analysis: Real World Applications, 11(3), 2010, 1760-1774.
[20] Sheng, L., et al, Non-linear dynamic analysis and chaos control of multi-freedom semi-direct gear drive system in coal cutters, Mechanical Systems and Signal Processing, 116, 2019, 62-77.
[21] Xie, B., Wang, W., Zuo, Z., Adaptive output feedback control of uncertain gear transmission system with dead zone nonlinearity, 13th IEEE Conference on Industrial Electronics and Applications (ICIEA), 2018, 438-443.
[22] Shen, Y., Yang, S., Liu, X., Non-linear dynamics of a spur gear pair with time-varying stiffness and backlash based on incremental harmonic balance method, International Journal of Mechanical Sciences, 48(11), 2006, 1256-1263.
[23] Clarke, D.W., Mohtadi, C., Properties of generalized predictive control, Automatica, 25(6), 1989, 859-875.
[24] Clarke, D.W., Application of generalized predictive control to industrial processes, IEEE Control Systems Magazine, 8(2), 1988, 49-55.
[25] Tao, J., et al, A generalized predictive control-based path following method for parafoil systems in wind environments, IEEE Access, 7, 2019, 42586-42595.
[26] Yang, Z.Q., Wang, X., Yu, H., Study on generalized predictive control of cement rotary kiln calcining zone temperature, IEEE Advanced Information Management, Communicates, Electronic and Automation Control Conference (IMCEC), 2016, 1653-1658.
[27] Zhang, X., et al, Generalized predictive contour control of the biaxial motion system, IEEE Transactions on Industrial Electronics, 65(11), 2018, 8488-8497.
[28] Valencia, A., et al, Hardware in loop of a generalized predictive controller for a micro grid DC system of renewable energy sources, International Journal of Engineering, 31(8), 2018, 1215-1221.
[29] Wang, G., et al, Design of a model predictive control method for load tracking in nuclear power plants, Progress in Nuclear Energy, 101, 2017, 260-269.
[30] Vajpayee, V., Mukhopadhyay, S., Tiwari, A.P., Data-driven subspace predictive control of a nuclear reactor, IEEE Transactions on Nuclear Science, 65(2), 2017, 666-679.
[31] Azar, A.T., Zhu, Q., Advances and applications in sliding mode control systems, Cham: Springer International Publishing, 2015.
[32] Li, H., et al, Observer-based adaptive sliding mode control for non-linear Markovian jump systems, Automatica, 64, 2016, 133-142.
[33] Wang, Y., et al. "Sliding mode control of fuzzy singularly perturbed systems with application to electric circuit, IEEE Transactions on Systems, Man, and Cybernetics Systems, 48(10), 2017, 1667-1675.
[34] Elmokadem, T., Zribi, M., Youcef-Toumi, K., Trajectory tracking sliding mode control of underactuated AUVs, Nonlinear Dynamics, 84(2), 2016, 1079-1091.