Equations with advanced arguments in stick balancing models

Abstract

A stick balancing problem is considered, where the output for the feedback controller is provided by an accelerometer attached to the stick. This output is a linear combination of the stick's angular displacement and its angular acceleration. If the output is fed back in a PD controller with feedback delay, then the governing equation of motion is an advanced functional differential equation, since the third derivative of the angular displacement (the angular jerk) appears with a delayed argument through the derivative term. Equations with advanced arguments are typically non-causal and are unstable with infinitely many unstable poles. It is shown that the sampling of the controller may still stabilize the system in spite of its advanced nature. In the paper, different models for stick balancing are considered and discussed by analyzing the corresponding stability diagrams. © 2012 Springer-Verlag GmbH Berlin Heidelberg.