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

Abstract

In this paper a novel 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 PDE models whose behaviour 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 3D effects and arbitrarily large displacements) and requires explicit computation of rotations in the PDE dynamics to account for orientation-dependent forces such as gravity. 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. Furthermore, in challenging numerical examples exhibiting considerable deformation and nonlinear effects, 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 generalisation 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 finitedimensional models and numerical schemes.

Aeroelastic Control and Estimation with a Minimal Nonlinear Modal Description

“…where 1 ( ), 21 ( ) : R + → R are the temporal coefficients of the expansion (unless otherwise stated, Einstein's summation convention for index = 1, ..., is used). The low-order model is then obtained by a Galerkin projection, in which evolution equations for the expansion weights 1 and 2 are obtained by substituting (15) and ( 16) into ( 1)…”

“…The intrinsic equations are based on strains and velocities, in contrast to conventional displacement and finite-rotation based formulations. Besides, construction of a modal-based nonlinear low-order model of the intrinsic equations, where few modes are often sufficient to capture the main nonlinear geometrical couplings, has been found to provide an appropriate modelling framework to underpin both the estimator and the predictive controller [15]. To overcome the difficulty of rotations not being directly obtained with the intrinsic equations, they are parametrised using quaternions and approximated by a modal-based expansion, which has also been proven to work satisfactorily embedded in the predictive model [15].…”

“…In our particular multiple shooting implementation, piecewise continuous-time systems are used to describe the dynamics at each sub-interval. Hence, continuous-time adjoints can be analytically derived, which leverage the quadratic structure of the resulting reduced order model and show more favourable complexity scaling (with respect to the system size) than automatic-differentiation-based sensitivity analysis [15].…”

Generalized Kelvin–Voigt Damping for Geometrically Nonlinear Beams

“…Another particularity is that rigid-body motions are inherently embedded in the solution space, thanks to expressing the motions through velocities, avoiding the explicit addition of the rigid-body equations, which proves very valuable for modelling free-flying vehicles. It has also been found that this set of equations admits a rather natural modal-based projection which allows constructing low-order models very suitable for control synthesis [21]. In our case, the resulting system is a compact infinite-dimensional 3D description of damped geometrically-exact beam equations, which facilitates theoretical analysis and makes a much needed contribution in the poorly explored large-deflections area, where models are often limited to axial vibrations [22,23] or two-dimensional nonlinear models [24].…”

“…given in [21], however a more detailed version is included here for completeness. Furthermore, the focus in this paper is on the numerical investigation and the practical implications of the novel damped nonlinear intrinsic equations.…”