Retarded, neutral and advanced differential equation models for balancing using an accelerometer

Stabilization of a pinned pendulum about its upright position via a reaction wheel is considered, where the pendulum’s angular position is measured by a single accelerometer attached directly to the pendulum. The control policy is modeled as a simple PD controller and different feedback mechanisms are investigated. It is shown that depending on the modeling concepts, the governing equations can be a retarded functional differential equation or neutral functional differential equation or even advanced functional differential equation. These types of equations have radically different stability properties. In the retarded and the neutral case the system can be stabilized, but the advanced equations are always unstable with infinitely many unstable characteristic roots. It is shown that slight modeling differences lead to significant qualitative change in the behavior of the system, which is demonstrated by means of the stability diagrams for the different models. It is concluded that digital effects, such as sampling, stabilizes the system independently on the modeling details.