Modal-Based Nonlinear Model Predictive Control for 3-D Very Flexible Structures

Abstract

In this article a novel nonlinear model predictive control (NMPC) scheme is derived, which is tailored to the underlying structure of the intrinsic description of geometrically exact nonlinear beams (in which velocities and strains are primary variables). This is an important class of partial differential equation (PDE) models whose behavior is fundamental to the performance of flexible structural systems (e.g., wind turbines, high-altitude long-endurance aircraft). Furthermore, this class contains the much-studied Euler–Bernoulli and Timoshenko beam models, but has significant additional complexity (to capture 3-D effects and arbitrarily large displacements) and requires explicit computation of rotations in the PDE dynamics to account for orientation-dependent forces. A challenge presented by this formulation is that uncontrollable modes necessarily appear in any finite dimensional approximation to the PDE dynamics. We show, however, that an NMPC scheme can be constructed in which the error introduced by the uncontrollable modes can be explicitly controlled. It is demonstrated that the asymptotic error can be made insignificant (from a practical perspective) using our NMPC scheme and excellent performance is obtained even when applied to a highly resolved numerical simulation of the PDEs. We also present a generalization of Kelvin–Voigt damping to the intrinsic description of geometrically exact beams. Finally, special emphasis is placed on constructing a framework suitable for real-time NMPC control, where the particular structure of the underlying PDEs is exploited to obtain both efficient finite-dimensional models and numerical schemes.