Controlled synchronization of coupled pendulums by Koopman Model Predictive Control

We propose a method to solve a class of control problems arising from a system of coupled pendulums. The system considered in this work is a one-dimensional array of pendulums pivoting around a single axis with adjacent pendulums coupled through torsion springs. Only a single torque motor attached to one of the two boundary pendulums actuates the system. This setup of coupled pendulums is a mechanical realization of the Frenkel–Kontorova (FK) model – a spatially discrete version of the sine-Gordon equation describing (nonlinear) waves. The main challenges of controlling this system are high order (the number of pendulums can be high), nonlinear and oscillatory dynamics, and only one actuator. The proposed class of problems can be characterized as controlled synchronization – designing a closed-loop controller that synchronizes the motion of the pendulums. Controlled synchronization is a special case of reference tracking, where all pendulums reach a common point or a trajectory. One can formulate many practically motivated problems within this class; here, we identify three problems: a vibration control of a flexible structure, swing-up control of all pendulums in the array, and low-friction sliding of an atomic-scale structure. We show that the presented problems can be dealt with by the Koopman Model Predictive Control (KMPC). The KMPC allows for controlling nonlinear systems by combining the classical linear model predictive control (MPC) with the Koopman operator approach for nonlinear dynamical systems. The main idea of the method is to construct a linear predictor of a nonlinear system in a higher-dimensional, lifted space and use the predictor within the linear MPC. The optimization problem formulation within the KMPC can then be tailored to a specific problem in the class. Simulations and experiments on a hardware platform realizing the FK model show the effectiveness of the method.