Nonlinear Oscillations in Delayed Collocated Control of Pendulum on Trolley

Abstract

The paper investigates the nonlinear dynamics of the collocated position control of a trolley that carries a pendulum. Delayed proportional derivative control force is considered, which is based on the position and velocity of the trolley only. Stability charts are constructed for different parameter combinations, which show intricate structures in the plane of the control parameters. To examine the nonlinear behaviour of the system, the Hopf bifurcation calculation is carried out after an infinite dimensional center manifold reduction. This indicates that supercritical Hopf bifurcations always exist at the boundary of the first stable lobe, however, for increasing time delays, the reappearing stable lobes may be bounded with subcritical Hopf bifurcations as well, and even quasi-periodic oscillations may occur.