Optimal trajectories, nonlinear models and constraints in wave energy device control

Abstract

The optimal control problem for a generic, one-degree of freedom Wave Energy Converter (WEC) with dynamical nonlinearities is formulated in the frequency-domain. Previous research, concerning more specifically a heaving point-absorber with nonlinear restoring force, shows that the unconstrained optimal velocity trajectory is influenced neither by the linear inertial terms, nor by the linear or nonlinear static forces. Further to this result, in this paper, we examine the influence of velocity-dependent nonlinear forces on the optimal trajectory, as well as the effect of physical system constraints. In particular, we show that, under state constraints (e.g. position and velocity limitations), the optimal velocity trajectory remains uninfluenced by static forces; but this is no longer true for constraints involving the control force, such as force limitation and passivity constraints. In addition, unlike static terms and linear inertial terms, the velocity-dependent forces, such as viscous drag, significantly influence the optimal velocity trajectory, regardless of constraints, and must be carefully modelled at the control design stage. In any case, even when the optimal velocity trajectory is not affected by some of the forces considered, the optimal control force required to achieve it depends on all the model dynamics (inertial terms, velocity-dependent and static forces). Numerical simulations, in the specific case of a heaving point absorber, are used to validate and illustrate the theoretical results.