Feedback control of chaotic systems using Multiple Shooting Shadowing and application to Kuramoto Sivashinsky equation

Abstract

We propose an iterative method to evaluate thefeedback control kernel of a chaotic system directlyfrom the system’s attractor. Such kernels are currentlycomputed using standard linear optimal controltheory, known as Linear Quadratic Regulator (LQR)theory. This is however applicable only to linearsystems, which are obtained by linearising thesystem governing equations around a target state.In the present paper, we employ the PreconditionedMultiple Shooting Shadowing (PMSS) algorithm tocompute the kernel directly from the non-linear dynamics, thereby bypassing the linear approximation.Using the adjoint version of the PMSS algorithm,we show that we can compute the kernel at any point of the domain in a single computation. The algorithm replaces the standard adjoint equation (that is ill-conditioned for chaotic systems) with a well-conditioned adjoint, producing reliable sensitivities which are used to evaluate the feedback matrix elements. We apply the idea to the Kuramoto Sivashinsky equation. We compare the computed kernel with that produced by the standard LQR algorithm and note similarities and differences. Bothkernels are stabilising, have compact support and similar shape. We explain the shape using two-point spatial correlations that capture the streaky structure of the solution of the uncontrolled system.