Data driven robust control of complex Ginzburg-Landau equations for spatial developing flow

Spatial developing flows may often prove detrimental to engineering objectives and demand effective control, yet their nonlinearity has often primitive control methods infeasible. Nonetheless, in this article, we attempt to analyze and suppress the amplification of disturbances in those flows that could be described by complex Ginzburg-Landaus equation by employing dynamic mode decomposition (DMD) method to capture the dynamics of nonlinearity. We first numerically simulate the entire flow field over time and stream-wise direction, then generate the DMD matrix which provides a linear approximation of the nonlinear characteristics of the equation, and finally apply the aforementioned matrix into $\mathcal{H}_{\infty}$ robust control scheme to obtain the robust controller that minimizes the effects of disturbances. We conclude that 1) DMD can capture the nonlinear dynamics of complex Ginzburg-Landau equation effectively; 2) nonlinear flow is harder to control than linear flow; 3) the maximum of $\mathcal{H}_{\infty}$ control system transfer function is always less than those of other control systems.