Tracking oscillations in the presence of delay-induced essential instability

Hybrid testing, which couples physical experiments and computer simulations bidirectionally and in real-time, is a promising experimental technique in engineering. One fundamental problem of this technique are delays in the coupling between simulation and experiment. We discuss this issue for a simple prototype hybrid system: a pendulum that is vertically 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: the linearisation 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. We introduce an approach based on feedback control and Newton iterations that is able to overcome this instability. The basic idea behind our approach consists of two parts. First, we change the bidirectional coupling between experiment and computer simulation to a unidirectional coupling and stabilise the experiment with a feedback loop. Second, we place the modified hybrid system into a Newton iteration scheme. If the iteration converges then the hybrid experiment behaves just as the original emulated system (within the experimental accuracy). We demonstrate, by using a computer simulation for the experimental part, that our approach is able to overcome the essential instability. In combination with path-following methods, it allows us to track oscillations and their bifurcations systematically.