Abstract
We consider a controlled nonlinear mechanical system described by the Lagrange equations. We determine the control forcesQ 1 and the restrictions for the perturbationsQ 2 acting on the mechanical system which allow to guarantee the asymptotic stability of the program motion of the system. We solve the problem of stabilization by the direct Lyapunov's method and the method of limiting functions and systems. In this case we can use the Lyapunov's functions having nonpositive derivatives. The following examples are considered: stabilization of program motions of mathematical pendulum with moving point of suspension and stabilization of program motions of rigid body with fixed point.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. R. Merkin,Introduction in the theory of stability of motion, Nauka, Moscow, 1987.
Z. Artstein,Topological dynamics of an ordinary equations, J. Differ. Equat.23 (1977), 216–223.
A. S. AndreyevThe asymptotic stability and instability of the zero solution of a nonautonomous system, Prikl. Mat. Mekh. 48 (1984), 225–232.
N. N. Krasovskii,Problem of stabilization of controlled motion, In I. G. Malkin,The theory of motion stability, Nauka, Moscow, 1966, 475–514.
V. V. Rumyantsev,The optimal stabilization of controlled systems, Prikl. Mat. Mekh.34 (1970), 440–456.
E. J. Smirnov, V. J. Pavlikov, P. P. Scherbakov and A. V. Jurkov,The control of motion of mechanical systems, Leningrad, LSU Publisher, 1985.
A. S. Andreyev and S. P. Bezglasnyi,The stabilization of controlled systems with a guaranteed estimate of the control quality, Prikl. Mat. Mekh.61 (1997), 41–47.
A. S. Andreyev,The stability of the equilibrium position of a non-autonomous mechanical system, Prikl. Mat. Mekh.60 (1996), 388–396.
S. P. Bezglasnyi,On stabilization of program motions of controlled mechanical systems, Proceedings of the 22 Yugoslav Congress of theoretical and applied mechanics, Vrnjacka Banja, June 2–7, 1997, pp. 107–112.
V. I. Zubov,The theory of optimal control, Sudostroenie, S.-Peterburg, 1966.
V. V. Krementulo,Stabilization of stationary motions of rigid body through rotating weights, Nauka, Moscow, 1977.
A. S. Andreyev and S. P. Bezglasnyi,To a task of control of rotary motion of rigid body, The theses of the international conference “Space, time, gravitation”, S.-Petersburg, Russia, May 23–28, 1994, pp. 83.
N. Rouche, P. Habets and M. Laloy,Stability theory by Lyapunov's direct method, Springer, New York, 1977.
Author information
Authors and Affiliations
Corresponding author
Additional information
Sergey Bezglasnyi graduated from Moscow Lomonosov State University (MSU) in 1993. He is a Soros postgraduate student (1996), Russian President's grant holder (1997, Institute of Mathematics in Seged, Hungry) and DAAD grant holder (2000, Fern-University in Hagen Germany). He received his PhD at MSU under the supervision of Prof. Alexander Andreev in 1998. From 1993 to 2000 he was a lecture at Ulyanovsk State University. His research interests focus on the control and stability of mechanical systems.
Rights and permissions
About this article
Cite this article
Bezglasnyi, S. The stabilization of program motions of controlled nonlinear mechanical systems. JAMC 14, 251–266 (2004). https://doi.org/10.1007/BF02936112
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02936112