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DOI: https://doi.org/10.20537/vm170310

Abstract: The Ball and Beam system with a nonlinear geometric constraint is considered. Two possible equilibrium positions of this system are found from the complete constraint equation. The structures of the equations of disturbed motion are analyzed in a neighborhood of the equilibrium positions, using equations without Lagrange multipliers in the form of M. F. Shul'gin. The possibility of linearization of the constraint equation is discussed. The stabilization problem is solved for every equilibrium position and two possible variants of the redundant coordinate. Stabilizing control (voltage at the armature of the drive motor) is calculated via solving linear-quadratic problems by N. N. Krasovsky's method for corresponding control subsystems. The coincidence of controls as time functions for the same equilibrium is shown for different choices of the redundant coordinate, and the stabilizing controls are linear functions of different phase variables. The graphs of transient processes in systems closed by the obtained controls are given. The asymptotic stability of both equilibrium positions in a complete nonlinear closed system follows from the previously proved theorem on asymptotic stability in the presence of zero roots of the characteristic equation corresponding to redundant coordinates.

Keywords: geometric constraints, redundant coordinate, M. F. Shul'gin's equations of motion, Ball and Beam, stability, stabilization, equilibrium.

Received: 16.08.2017

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Citation: A. Ya. Krasinskii, A. N. Il'ina, E. M. Krasinskaya, “Modeling of the Ball and Beam system dynamics as a nonlinear mechatronic system with geometric constraint”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 27:3 (2017), 414–430

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This publication is cited in the following articles:

1. A. Ya. Krasinskiy, A. N. Il'ina, È. M. Krasinskaya, “Stabilization of steady motions for systems with redundant coordinates”, Moscow University Mathematics Bulletin, Moscow University Mеchanics Bulletin, 74:1 (2019), 14–19      
2. M. Ding, B. Liu, L. Wang, “Position control for ball and beam system based on active disturbance rejection control”, Syst. Sci. Control Eng., 7:1 (2019), 97–108      
3. Aleksandr Ya Krasinskiy, “On the methods of analytical mechanics for mathematical modeling of the dynamics of non-free systems and some variants of their application to the dynamics of parallel manipulators”, IRATJ, 9:1 (2022), 15  
4. Nguyen Xuan Chiem, Le Tran Thang, Nguyen Cong Dinh, 2023 12th International Conference on Control, Automation and Information Sciences (ICCAIS), 2023, 466  
5. Nur Syazreen Ahmad, “Modeling and Hybrid PSO-WOA-Based Intelligent PID and State-Feedback Control for Ball and Beam Systems”, IEEE Access, 11 (2023), 137866  

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles