Balancing using accelerometers and equations with advanced arguments

The stick balancing problem is considered, where the vertical direction is measured using a single accelerometer attached to the stick. It is shown that the output is a linear combination of the angular position and the angular acceleration of the stick. If this 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 highest derivative, the jerk, appears with delayed argument through the derivative term. Autonomous equations with advanced arguments are typically non-causal and are unstable with infinitely many unstable poles. However, if the sampling effect of the digital controller is modeled, then the argument of the delayed highest derivative term is piecewise constant. In this case, the non-causality does not arise, and the system can also be stabilized by tuning the control parameters properly. In the paper, different models for stick balancing are considered and discussed by analyzing the corresponding stability diagrams.