Balanced Truncation Model Reduction of a Nonlinear Cable-Mass PDE System with Interior Damping

We consider model order reduction of a nonlinear cable-mass system modeled by a 1D wave equation with interior damping and dynamic boundary conditions. The system is driven by a time dependent forcing input to a linear mass-spring system at one boundary. The goal of the model reduction is to produce a low order model that produces an accurate approximation to the displacement and velocity of the mass in the nonlinear mass-spring system at the opposite boundary. We first prove that the linearized and nonlinear unforced systems are well-posed and exponentially stable under certain conditions on the damping parameters, and then consider a balanced truncation method to generate the reduced order model (ROM) of the nonlinear input-output system. Little is known about model reduction of nonlinear input-output systems, and so we present detailed numerical experiments concerning the performance of the nonlinear ROM. We find that the ROM is accurate for many different combinations of model parameters

An approach to reduced-order modeling and feedback control for wave energy converters

Wave energy holds great promise to be part of the alternative energy portfolio that will provide independence from fossil fuels. As wave energy converter (WEC) technologies mature, designing effective control strategies to extract maximum energy, extend device life, coordinate WEC operation within an array, or mitigate negative impacts of a WEC becomes an increasingly important area of research. However, developing tractable models for the real-time computation of WEC control signals is challenging. This thesis is concerned with developing a model reduction approach for control design that is suitable for application to high fidelity computational fluid-structure interaction. There are many approaches to model reduction; in the last two decades, much attention has been focused on the proper orthogonal decomposition and other singular value decomposition (SVD) type methods. In the control literature, the balanced truncation is an established approach to model reduction. Balanced POD is a computational approach related to the proper orthogonal decomposition in order to compute balanced truncation of a control system. The work presented in this thesis is the investigation into the applicability of a recently developed model reduction technique, Balanced POD, applied to a WEC fluid-structure interaction problem. We first model a one-dimensional fluid-structure interaction model arising in WEC dynamics heuristically, then design two control strategies for the tracking control of the WEC. Finally, we address the problem of estimating the type of information that can be available to the WEC controller and developing estimates of wave heights and forces that are suitable for control design. The work presented here paves the way for further research regarding the suitability of model reduction techniques applied to WEC problem. The simulation results clearly demonstrate that the reduced order models can successfully capture the fundamental nature of WEC dynamics and can be readily used for control design

Model Reduction of a Nonlinear Cable-Mass PDE System with Dynamic Boundary Input

We consider the motion of a flexible cable attached to a mass-spring system at each end. The input to the system is the driving force to the mass-spring system at the left end, and the output of interest is the displacement and velocity of the mass at the right end. We model the system by a 1D damped wave equation coupled to second order oscillators holding on the boundaries. The mass-spring model at the right end includes a nonlinear stiffening force. We prove the linearized system is well-posed and exponentially stable. We perform balanced truncation model reduction of the linearized system, and use the resulting modes to obtain a nonlinear reduced order model. We numerically compare the input-output response of the nonlinear PDE system and the nonlinear reduced order model for various driving forces and model parameters