Stabilizability of Mechanical Systems Subjected to Digital PIDA Control

This paper presents a study on the limits of stabilizability of unstable second-order dynamical systems by means of digital proportional-integral-derivative-acceleration (PIDA) feedback. Four different models are considered, which are all governed by the same dimensionless second-order differential equation. The mathematical model under analysis is a hybrid system involving terms with piecewise constant arguments due to the discrete sampling and actuation of the controller. Closed form formulas are derived for the domain of stability and for the limits of stabilizability as function of the system parameters, the sampling period and the control gains. It is concluded that while the acceleration term extends the limit of stabilizability, the integral term reduces stabilizability properties.