A pendulum can be stabilised in its upright position by proportional-plus-derivative (PD) feedback control only if the latency in the control loop is smaller than a certain critical delay. This critical delay is determined by the presence of a fully symmetric triple-zero eigenvalue singularity, a bifurcation of codimension three. We investigate three possible modifications of the PD scheme with the aim of extending the range of permissible delays. Effectively, these modifications introduce another parameter. This additional parameter can be used to continue the triple-zero singularity in four parameters until it gains a higher-order degeneracy imposing a new limit on the permissible delay. It turns out that the most effective modification is to feed back the value of the position with a small (intentional) additional delay on top of the control loop latency.
Additional information: Submitted to Dynamical Systems
Sponsorship: The research of J.S. is supported by EPSRC grant GR/R72020/01 and that of
B.K. by an EPSRC Advanced Research Fellowship.