Fifth Mississippi State Conference on Differential Equations and Computational Simulations,
Electron. J. Diff. Eqns., Conf. 10, 2002, pp. 163-185.

Dynamics of delayed piecewise linear systems

Laszlo E. Kollar, Gabor Stepan, & Janos Turi

In this paper the dynamics of the controlled pendulum is investigated assuming backlash and time delays. The upper equilibrium of the pendulum is stabilized by a piecewise constant control force which is the linear combination of the sampled values of the angle and the angular velocity of the pendulum. The control force is provided by a motor which drives one of the wheels of the cart through an elastic teeth belt. The contact between the teeth of the gear (rigid) and the belt (elastic) introduces a nonlinearity known as "backlash" and causes the oscillation of the controlled pendulum around its upper equilibrium. The processing and sampling delays in the determination of the control force tend to destabilize the controlled system as well. We obtain conditions guaranteeing that the pendulum remains in the neighborhood of the upper equilibrium. Experimental findings obtained on a computer controlled inverted pendulum cart structure are also presented showing good agreement with the simulation results.

Published February 28, 2003.
Subject classifications: 34K35, 37C75.
Key words: Stability analysis, backlash, digitally controlled pendulum..

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Laszlo E. Kollar
Department of Applied Sciences
University of Qu\'ebec at Chicoutimi
555, Boul. de l'Universite, Chicoutimi G7H 2B1 Quebec, Canada
E-mail address:
Gabor Stepan
Department of Applied Mechanics
Budapest University of Technology and Economics
H-1521 Budapest, Hungary
E-mail address:
Janos Turi
Department of Mathematical Sciences
The University of Texas at Dallas
P.O. Box 830688, MS EC 35, Richardson, Texas 75083-0688, USA
E-mail address:

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