Reduced-order-based feedback control of the Kuramoto–Sivashinsky equation

In this paper, we consider the Kuramoto–Sivashinsky equation (KSE), which describes the long-wave motions of a thin film over a vertical plane. Solution procedures for the KSE often yield a large or infinite-dimensional nonlinear system. We first discuss two reduced-order methods, the approximate inertial manifold and the proper orthogonal decomposition, and show that these methods can be used to obtain a reduced-order system that can accurately describe the dynamics of the KSE. Moreover, from this resulting reduced-order system, the feedback controller can readily be designed and synthesized. For our control techniques, we use the linear and nonlinear quadratic regulator methods, which are the first- and second-order approximated solutions of the Hamilton–Jacobi–Bellman equation, respectively. Numerical simulations comparing the performance of the reduced-order-based linear and nonlinear controllers are presented.