  We propose and experimentally demonstrate a feedback control method that
allows synchronizing the motion of a chain of several coupled nonlinear
oscillators actuated through one end of the chain. The chain considered in this
work is a one-dimensional array of pendulums pivoting around a single axis and
interacting with adjacent pendulums through torsion springs; the array is
actuated using a single torque motor attached to one of the two boundary
pendulums. This represents a mechanical realization of the Frenkel-Kontorova
model { a spatially discrete version of a sine-Gordon equation describing
(nonlinear) waves. The main challenges of controlling these systems are: high
order (the number of pendulums can be high), nonlinear dynamics, and (as we set
the problem here) only one actuator. The presented problem of synchronization
of motion is a special case of the problem of reference tracking, where all
pendulums reach a common point or a trajectory. In particular, we demonstrate
synchronization to a stable equilibrium (all pendulums downward), unstable
equilibrium (all pendulums upward), and a periodic orbit (all pendulums
revolving). We use the Koopman Model Predictive Control (KMPC) that constructs
a linear predictor of the nonlinear system in a higher-dimensional lifted space
and uses the predictor within a classical linear MPC, thereby maintaining low
computational cost that allows for a real-time implementation, while taking
into account the complex nonlinear dynamics.
