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 Ž . 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.

Overview of control approaches in Fluid Flow

“…There exists an unique classical solution with some stability properties, different cases, global exponential stability in Lq . James Baker et al [11] presented synthesize nonlinear fmite dimensional output feedback controllers to guarantee stability and enforce the output of the closed loop system to follow the references input asymptotically. The results using this method are better than the linear controller.…”

A Closed-Form Feedback Controller for Stabilization of Linearized Navier-Stokes Equations: The 2D Poisseuille Flow

“…Many efforts in design of feedback controllers for the Navier-Stokes system employ in-domain actuation, using optimal control methods [7] or model reduction techniques [4]. For boundary feedback control, optimal control theory has also been developed [16], and specialized to specific geometries, like cylinder wake [15].…”

A Closed-Form Observer for the 3D Inductionless MHD and Navier-Stokes Channel Flow

“…Frequently cited as a paradigm for transition to turbulence [21], the Poiseuille flow is a benchmark for flow control and turbulence estimation. There are many results in channel flow stabilization, for instance, using optimal control [14], backstepping [27], spectral decomposition/pole placement [7], [26], Lyapunov design/passivity [1], [3], or nonlinear model reduction/indomain actuation [2]. Observer designs are more scarce; apart from the continuum backstepping approach [28], previous works were in the form of an Extended Kalman Filter for the spatially discretized Navier-Stokes equations, employing high-dimensional algebraic Riccati equations for computation of observer gains [11], [13].…”