Dynamics of Simple Balancing Models with State Dependent Switching Control

TL;DR

This paper investigates a simple inverted pendulum model with state-dependent switching control, revealing stable and unstable periodic orbits, bifurcations, and complex dynamics arising from nonsmooth control mechanisms.

Contribution

It provides a detailed analysis of nonsmooth, state-dependent control in a balancing model without small angle approximation, highlighting bifurcations and complex solution behaviors.

Findings

Existence of structurally stable periodic orbits

Bifurcations induced by discontinuities in control

Complex periodic and aperiodic solutions from switching events

Abstract

Time-delayed control in a balancing problem may be a nonsmooth function for a variety of reasons. In this paper we study a simple model of the control of an inverted pendulum by either a connected movable cart or an applied torque for which the control is turned off when the pendulum is located within certain regions of phase space. Without applying a small angle approximation for deviations about the vertical position, we see structurally stable periodic orbits which may be attracting or repelling. Due to the nonsmooth nature of the control, these periodic orbits are born in various discontinuity-induced bifurcations. Also we show that a coincidence of switching events can produce complicated periodic and aperiodic solutions.