Research Papers

A Closed-Form Full-State Feedback Controller for Stabilization of 3D Magnetohydrodynamic Channel Flow

[+] Author and Article Information
Rafael Vazquez1

Departamento de Ingeniería Aeroespacial, Universidad de Sevilla, Camino de los Descubrimientos, 41092 Sevilla, Spainrvazquez1@us.es

Eugenio Schuster

Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015-3085schuster@lehigh.edu

Miroslav Krstic

Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411krstic@ucsd.edu

Our approach can be extended to finite periodic channels with only some changes; see, e.g., Ref. 39 for techniques involved.


Corresponding author.

J. Dyn. Sys., Meas., Control 131(4), 041001 (Apr 27, 2009) (10 pages) doi:10.1115/1.3089561 History: Received June 05, 2008; Revised December 15, 2008; Published April 27, 2009

We present a boundary feedback law that stabilizes the velocity, pressure, and electromagnetic fields in a magnetohydrodynamic (MHD) channel flow. The MHD channel flow, also known as Hartmann flow, is a benchmark for applications such as cooling, hypersonic flight, and propulsion. It involves an electrically conducting fluid moving between parallel plates in the presence of an externally imposed transverse magnetic field. The system is described by the inductionless MHD equations, a combination of the Navier–Stokes equations and a Poisson equation for the electric potential under the MHD approximation in a low magnetic Reynolds number regime. This model is unstable for large Reynolds numbers and is stabilized by actuation of velocity and the electric potential at only one of the walls. The backstepping method for stabilization of parabolic partial differential equations (PDEs) is applied to the velocity field system written in appropriate coordinates. Control gains are computed by solving a set of linear hyperbolic PDEs. Stabilization of nondiscretized 3D MHD channel flow has so far been an open problem.

Copyright © 2009 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 2

Streamwise equilibrium velocity Ue(y) (left) and Uye(y) (right) for different values of H. Solid, H=0; dash-dotted, H=10; and dashed, H=50.

Grahic Jump Location
Figure 3

The chain of transformations used to design the control laws. Note that all transformations are invertible.

Grahic Jump Location
Figure 4

A block diagram showing the structure of the controller. (Top) Full-state controller. (Bottom) Output-feedback controller (with measurements on the lower wall).



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