Bifurcation analysis of the stability of one dynamical system with a follower force
DOI:
https://doi.org/10.32703/2617-9040-2019-33-2-4Keywords:
dynamical system, inverted double pendulum, follower force, stabilityAbstract
In this paper, for an inverted double pendulum with asymmetric follower force a mathematical model has been constructed in generalized coordinates. The dynamics of the system has been explored through both analytical and numerical approaches. Stability diagrams have been presented. The influence of the follower forces parameters on the dynamical behavior of the pendulum has been analyzed. Numerical simulation has been applied to obtain integral curves and phase portraits. The trajectories of the lower and upper poles of the pendulum have been constructed too.
References
Lundberg K. H., & Barton, T. W. (2010). History of inverted-pendulum systems. IFAC Proceedings Volumes, 42(24), 131-135.
Boubaker, O. (2017). The inverted pendulum: history and survey of open and current problems in control theory and robotics. The Inverted Pendulum in Control Theory and Robotics: From Theory to New Innovations, 111, 1.
Moysis, L. (2016). Balancing a double inverted pendulum using optimal control and Laguerre functions. Aristotle University of Thessaloniki, Greece, 54124.
Mathew, N. J., Rao, K. K., & Sivakumaran, N. (2013). Swing up and stabilization control of a rotary inverted pendulum. IFAC Proceedings Volumes, 46(32), 654-659.
Sultan, G. A., & Farej, Z. K. (2017). Design and Performance Analysis of LQR Controller for Stabilizing Double Inverted Pendulum System.
Nalavade, M. R., Bhagat, M. J., & Patil, V. V. (2014). Balancing Double Inverted Pendulum on A cart by Linearization Technique. International Journal of Recent Technology and Engineering (IJRTE), 3(1), 153-157.
Wu, S. M., Song, J. W., & Zhang, W. Q. (2014). Optimal Control Theory Research on Inverted Pendulum System. In Applied Mechanics and Materials (Vol. 494, pp. 1118-1121). Trans Tech Publications.
Glück, T., Eder, A., & Kugi, A. (2013). Swing-up control of a triple pendulum on a cart with experimental validation. Automatica, 49(3), 801-808.
Sharma, K., & Sahu, V. (2016). Modeling and Stabilization of Cart Triple Link Inverted Pendulum using LQR Controller Incorporating Degree of Stability. International Research Journal of Advanced Engineering and Science, 1(4), 148-152.
Pflüger, A. (2013). Stabilitätsprobleme der Elastostatik. Springer-Verlag.
Ziegler, H. (2013). Principles of structural stability (Vol. 35). Birkhäuser.
Kirillov, O. N. (2010). Structural optimization of the Ziegler's pendulum: singularities and exact optimal solutions. arXiv preprint arXiv:1101.0246.
Kirillov, O. (2011). Re‐visiting structural optimization of the Ziegler pendulum: singularities and exact optimal solutions. PAMM, 11(1), 717-718.
Nikolov, S., & Nedev, V. (2016). Bifurcation analysis and dynamic behavior of an inverted pendulum with bounded control. Journal of Theoretical and Applied Mechanics, 46(1), 17-32.
Kuznetsov, Y. A. (2013). Elements of applied bifurcation theory (Vol. 112). Springer Science & Business Media.
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