Trajectory tracking of cart-pendulum dynamics using multiple time-delayed feedback

A trajectory tracking problem is investigated for a linear time-invariant (LTI) underactuated dynamics using a fixed control law. Differently from the traditional treatment, a possible occurrence of multiple time delays in the feedback control is also considered. It is known that such time-delayed LTI systems may exhibit multiple stable operating zones (pockets) in the space of the delays. The tasks that are undertaken here are the determination of these pockets analytically, the utilisation of them for stabilisation and relevant experimental validation. The practical objective of the study is to accomplish a successful trajectory tracking for an underactuated mechanical system, cart-and-pendulum dynamics, under the presence of multiple and independent time delays. First, the authors model the system as accurately as possible, and select a desirable fixed control law for the non-delayed feedback structure. The authors then investigate the stability map corresponding to this fixed control strategy when unpreventable delays occur in the feedback. This is done using a recent methodology, which is called the cluster treatment of characteristic roots (CTCR). The authors also experimentally verify the analytical findings of CTCR and deploy a counterintuitive control strategy called the ´delay scheduling´ based on the knowledge obtained from CTCR. If there is an unavoidable delay composition which renders unstable behaviour, the ´delay scheduling´ suggests increasing the delays further in order to recover stability, even for underactuated systems.