Nonlinear control of incompressible fluid flow: Application to Burgers' equation and 2D channel flow

Abstract

This paper proposes a methodology for the synthesis of nonlinear finite-dimensional feedback controllers for incompressible Newtonian fluid flows described by two-dimensional Navier-Stokes equations. Combination of Galerkin's method with approximate inertial manifolds is employed for the derivation of low-order ordinary differential equation (ODE) systems that accurately describe the dominant dynamics of the flow. These ODE systems are subsequently used as the basis for the synthesis of nonlinear output feedback controllers that guarantee stability and enforce the output of the closed-loop system to follow the reference input asymptotically. The method is successfully used to synthesize nonlinear finite-dimensional output feedback controllers for the Burgers' equation and the two-dimensional channel flow that enhance the convergence rate to the spatially uniform steady-state and the parabolic velocity profile, respectively. The performance of the proposed controllers is successfully tested through simulations and is shown to be superior to the one of linear controllers. © 2000 Academic Press.