Inverted Pendulum Stabilization Via a Pyragas-Type Controller: Revisiting the Triple Zero Singularity

The use of Pyragas-Type controller proved interest in the stabilization of unstable periodic orbits. The stabilization problem of a balancing inverted pendulum on an horizontally moving cart by the use of such a controller is considered. The main objective of the paper is to propose delayed control law containing only proportional gains able to stabilize the inverted pendulum by avoiding the existence of a triple zero eigenvalue at the origin. We analyze the center dynamics described by a three dimensional system of ordinary differential equations (ODEs) with a codimension-three triple zero bifurcation. Furthermore, the stability analysis of the corresponding linear time invariant system with two delays describing the behavior around the equilibrium is also proposed. This analysis is done in order to characterize the possible local bifurcations. Finally, the proposed control scheme is numerically illustrated and discussed. Time-Delay, Stability, Delayed Feedback, Control, Center Manifold Theorem, Normal Forms, Local Bifurcation, Inverted Pendulum.