  In this work, we present a model-based optimal boundary control design for an
aerial robotic system composed of a quadrotor carrying a flexible cable. The
whole system is modeled by partial differential equations (PDEs) combined with
boundary conditions described by ordinary differential equations (ODEs). The
proper orthogonal decomposition (POD) method is adopted to project the original
infinite-dimensional system on a subspace spanned by orthogonal basis
functions. Based on the reduced order model, nonlinear model predictive control
(NMPC) is implemented online to realize shape trajectory tracking of the
flexible cable in an optimal predictive fashion. The proposed reduced modeling
and optimal control paradigms are numerically verified against an accurate
high-dimensional FDM-based model in different scenarios and the controller's
superior performance is shown compared to an optimally tuned PID controller.
