Two nonlinear control schemes contrasted on a hydrodynamiclike model

Abstract

The feasibility of shear flow control using strategies originated by Hübler (H), and by Ott, Grebogi, and Yorke (OGY), has been examined numerically employing fully developed and transitional solutions of the Ginzburg–Landau equation as models for such flows. The general message of this study is that control of nonlinear systems is best obtained by making maximum use possible of the underlying natural dynamics. The effectiveness of both methods in obtaining control of fully developed flows depends strongly upon the ‘‘distance’’ in state space between the uncontrolled flow and goal dynamics. If the goal dynamics is an unstable nonlinear solution of the equation, or a close approximation, and the flow is nearby at the instant control is applied, both methods perform reliably and at low-energy cost in reaching and maintaining this goal. For example, under conditions when the spatially complex and temporally chaotic uncontrolled flow exists on a strange attractor of Lyapunov dimension Dλ≂120, several different spatially periodic limit cycle motions (Dλ=1) were successfully maintained by both methods using less than 1% of the energy of the chaotic motion. However, when the goal dynamics is not nearby the uncontrolled flow state at the instant control is attempted, both methods can fail in their own unique way. In the H method the failure occurs because multiple solutions of the forced equation result from the same forcing. This could be a generic problem in all nonlinear systems, the Navier–Stokes equations included. In some cases this failure may be avoided by defining and controlling to a sequence of intermediate goals that lead to the ultimate dynamics. This staged approach to a goal shows potential of helping the OGY method when it also fails. Here the ‘‘targeting’’ methods useful in low-dimensional systems for moving from uncontrolled states to a goal region fail because of the underlying complexity of the attractor, or because the goal dynamics are not embedded within the chaotic attractor of the uncontrolled system. There are conceptual difficulties in applying the OGY method to transitional, convectively unstable flows, but within certain limitations the H method works well, though at a large cost in energy terms. It works best when the goal dynamics rearranges the energy of the uncontrolled flow into new nonlinear forms, rather than simply trying to suppress it. The performance of both methods degrades due to noise and the spatially discrete nature of realistic forcing, but these effects are not difficult to predict. It is suggested that channel flows may be particularly amenable to control by either the H or OGY methods, while the H method seems more appropriate for boundary layers.