Micro-chaotic Behaviour in PD-controlled Systems

The nonlinearity caused by the application of digital control may lead to chaotic behaviour. Our goal is to analyse the problem of computer-controlled stabilization of unstable equilibria, with the application of the PD control law. We consider a linear equation of motion. However, as a consequence of the digital effects, i.e., the sampling and the round-off error, the solutions of the system can be described by a two dimensional piecewise linear map. We show that this system may perform chaotic behaviour. The amplitude of the evolving oscillations is usually very small, this is why these are called micro-chaotic vibrations.