Geodesic
Modeling of the Ball and Beam system dynamics as a nonlinear mechatronic system with geometric constraint
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 27 (2017) no. 3, pp. 414-430

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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. 
@article{VUU_2017_27_3_a9,
     author = {A. Ya. Krasinskii and A. N. Il'ina and E. M. Krasinskaya},
     title = {Modeling of the {Ball} and {Beam} system dynamics as a nonlinear mechatronic system with geometric constraint},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {414--430},
     publisher = {mathdoc},
     volume = {27},
     number = {3},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2017_27_3_a9/}
}
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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. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 27 (2017) no. 3, pp. 414-430. http://geodesic.mathdoc.fr/item/VUU_2017_27_3_a9/