Abstract
The work is devoted to studying the dynamics of unstable oscillating systems (in the form of an inverted pendulum) controlled by the action of a hysteretic type. The results for different types of motion of a suspension point are presented, in particular, for vertical and horizontal motion. A mathematical model of the inverted pendulum with an oscillating suspension is considered. For this pendulum the explicit criteria of stability are obtained using the linearized equations of motion. The dependences between the initial conditions and the value of the control parameters providing periodic oscillations of the pendulum are described. A mathematical model of the inverted pendulum with feedback control is given under the conditions of the horizontal motion of the suspension point. The conditions that guarantee the stabilization of the considered system are obtained; the conditions are formulated in terms of constraints on the initial conditions. The solution to the problem of the optimal control of an oscillating system is presented in the sense of minimization of a quadratic goal functional. The stabilization problem for an unstable system with distributed parameters, the flexible inverted pendulum, is also considered, and the stabilization conditions are formulated. Fulfilment of these conditions ensures the boundedness of the phase coordinates in the infinite interval of time. The optimal parameters (in the sense of minimization of a quadratic goal functional) corresponding to stabilization of the distributed system are identified.
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REFERENCES
P. L. Kapitsa, “Pendulum with vibrating suspension,” Usp. Fiz. Nauk 44, 7–20 (1951).
P. L. Kapitsa, “Dynamic stability of the pendulum with an oscillating suspension point,” Zh. Eksp. Teor. Fiz. 21, 588–597 (1951).
A. Stephenson, “On an induced stability,” Phylos. Mag. 15, 233 (1908).
E. I. Butikov, “An improved criterion for Kapitza’s pendulum stability,” J. Phys. A: Math. Theor. 44, 295202 (2011).
E. I. Butikov, “Oscillations of a simple pendulum with extremely large amplitudes,” Eur. J. Phys. 33, 1555–1563 (2012).
Y. V. Mikheev, V. A. Sobolev, and E. M. Fridman, “Asymptotic analysis of digital control systems,” Autom. Remote Control 49, 1175–1180 (1988).
M. E. Semenov, O. O. Reshetova, A. V. Tolkachev, A. M. Solovyov, and P. A. Meleshenko, “Oscillations under hysteretic conditions: From simple oscillator to discrete Sine-Gordon model,” in Topics in Nonlinear Mechanics and Physics: Selected Papers from CSNDD 2018 (Singapore, 2019), pp. 229–253.
M. E. Semenov, M. G. Matveev, P. A. Meleshenko, and A. M. Solov’ev, “Dynamics of a damping device based on ishlinsky material,” Mekhatron., Avtomatiz., Upravl., No. 20, 106–113 (2019).
M. E. Semenov, M. G. Matveev, G. N. Lebedev, and A. M. Solov’ev, “Stabilization of a flexible inverted pendulum with the hysteretic properties,” Mekhatron., Avtomatiz., Upravl., No. 8, 516–525 (2017).
Z. Y. Zhang and X. J. Miao, “Global existence and uniform decay forwave equation with dissipative term and boundary damping,” Comput. Math. Appl. 59, 1003–1018 (2010).
E. I. Butikov, “Subharmonic resonances of the parametrically driven pendulum,” J. Phys. A: Math. Theor. 35, 6209–6231 (2002).
J. Y. Sun, X. C. Huang, and X. T. Liu, “Study on the force transmissibility of vibration isolators with geometric nonlinear,” Nonlin. Dyn. 74, 1103–1112 (2013).
V. I. Ryazhskikh, M. E. Semenov, A. G. Rukavitsyn, O. I. Kanishcheva, A. A. Demchuk, and P. A. Meleshenko, “Stabilization of inverted pendulum on a two-wheeled vehicle,” Vestn. YuUGU, Ser. Mat., Fiz., Mekh. 9 (3), 27–33 (2017).
F. L. Chernous’ko, L. D. Akulenko, and B. N. Sokolov, Swing Control (Nauka, Moscow, 1980) [in Russian.
Lipo Wang and J. Ross, “Synchronous neural networks of nonlinear threshold elements with hysteresis,” Neurobiology 87, 988–992 (1990).
Z. Y. Zhang, Z. H. Liu, X. J. Miao, and Y. Z. Chen, “Global existence and uniform stabilization of a generalized dissipative Klein-Gordon equation type with boundary damping,” Math. Phys. 52, 023502 (2011).
A. M. Solovyov, M. E. Semenov, P. A. Meleshenko, O. O. Reshetova, M. A. Popov, and E. G. Kabulova, “Hysteretic nonlinearity and unbounded solutions in oscillating systems,” Proc. Eng. 201, 578–583 (2017).
M. E. Semenov, A. M. Solovyov, J. M. Balthazar, and P. A. Meleshenko, “Nonlinear damping: From viscous to hysteretic,” in Recent Trends in Applied Nonlinear Mechanics and Physics, Ed. by M. Belhaq, Springer Proc. Phys. 199, 259–275 (2018).
S. A. Reshmin, “Finding the principal bifurcation value of the maximum control torque in the problem of optimal control synthesis for a pendulum,” J. Comput. Syst. Sci. Int. 47, 163 (2008).
S. A. Reshmin and F. L. Chernous’ko, “Time-optimal control of an inverted pendulum in the feedback form,” J. Comput. Syst. Sci. Int. 45, 383 (2006).
N. V. Anokhin, “Bringing a multilink pendulum to the equilibrium position using a single control torque,” J. Comput. Syst. Sci. Int. 52, 717 (2013).
A. M. Formal’skii, “On stabilization of an inverted double pendulum with one control torque,” J. Comput. Syst. Sci. Int. 45, 337 (2006).
S. V. Aranovskii, A. E. Biryuk, E. V. Nikulchev, I. V. Ryadchikov, and D. V. Sokolov, “Observer design for an inverted pendulum with biased position sensors,” J. Comput. Syst. Sci. Int. 58, 297 (2019).
M. S. Osintsev and V. A. Sobolev, “Reduction of dimension of optimal estimation problems for dynamical systems with singular perturbations,” Comput. Math. Math. Phys. 54, 45 (2014).
K. Magnus, Vibrations (Blackie and Son, London, 1965).
R. A. Nelepin, Research Methods for Nonlinear Automatic Control Systems, Ed. by R. A. Nelepin (Nauka, Moscow, 1979) [in Russian].
M. A. Krasnosel’skii and A. V. Pokrovskii, Hysteresis Systems (Nauka, Moscow, 1983) [in Russian].
N. V. Butenin, Yu. I. Neimark, and N. L. Fufaev, Introduction to the Theory of Nonlinear Oscillations (Nauka, Moscow, 1987) [in Russian].
V. A. Pliss, Nonlocal Problems of the Theory of Oscillations (Nauka, Moscow, 1964) [in Russian].
M. A. Krasnosel’skii and A. V. Pokrovskii, “Periodic oscillations in systems with relay nonlinearities,” Dokl. Akad. Nauk SSSR 216, 733–736 (1974).
F. R. Gantmakher, Matrix Theory (Nauka, Moscow, 1966) [in Russian].
Chao Xu and Xin Yu, “Mathematical model of elastic inverted pendulum control system,” J. Control Theory Appl. 3, 281–282 (2004).
M. Dadfarnia, N. Jalili, B. Xian, and D. M. Dawson, “A Lyapunov-based piezoelectric controller for flexible cartesian robot manipulators,” J. Dyn. Syst., Meas. Control 126, 347 (2004).
E. P. Dadios, P. S. Fernandez, and D. J. Williams, “Genetic algorithm on line controller for the flexible inverted pendulum,” J. Adv. Comput. Intell. Intell. Inform. 10 (2) (2006).
Zheng-Hua Luo and Bao-Zhu Guo, “Shear force feedback control of a single-link flexible robot with a revolute joint,” IEEE Trans. Autom. Control 42 (1) (1997).
Guangpu Xia, Tang Zheng, and Yong Li, “Hopfield neural network with hysteresis for maximum cut problem,” Neural Inform. Process. Lett. Rev. 4 (5) (2004).
J. T. Pierce-Shimomura, T. M. Morse, and S. R. Lockery, “The fundamental role of pirouettes in caenorhabditis elegance chemotaxis,” Neuroscience 19, 9557–9569 (1999).
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The work is supported by the Russian Foundation for Basic Research (project no. 19-08-00158) and by the Russian Science Foundation (project no. 19-11-0197).
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Translated by E. Oborin
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Medvedskii, A.L., Meleshenko, P.A., Nesterov, V.A. et al. Unstable Oscillating Systems with Hysteresis: Problems of Stabilization and Control. J. Comput. Syst. Sci. Int. 59, 533–556 (2020). https://doi.org/10.1134/S1064230720030090
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DOI: https://doi.org/10.1134/S1064230720030090