Tracking oscillations in the presence of delay-induced essential instability

Hybrid experiments, which couple mechanical experiments and computer simulations bidirectionally and in real-time, are a promising experimental technique in engineering. A fundamental problem of this technique are delays in the coupling between simulation and experiment. We discuss this issue for a simple prototype hybrid experiment: a mechanical pendulum that is parametrically excited by coupling it to a simulated linear mass-spring-damper system. Under realistic conditions a small delay in the coupling can give rise to an essential instability. Namely, the linearization has infinitely many unstable eigenvalues for arbitrarily small delay. This type of instability is impossible to compensate for with any of the standard compensation techniques known in engineering. We introduce an approach based on feedback control and Newton iterations and show that it is able to overcome the essential instability. The basic idea consists of two parts. First, we change the bidirectional coupling between experiment and computer simulation to a unidirectional coupling and stabilize the experiment with a feedback loop. Second, we place the modified hybrid experiment into a Newton iteration scheme. If the iteration converges then the hybrid experiment behaves just as the original emulated system (within the experimental accuracy). Using path-following, oscillations and their bifurcations can be tracked systematically without knowledge of an underlying model for the experiment.