A Closed-Form Feedback Controller for Stabilization of Magnetohydrodynamic Channel Flow
R. Vazquez, E. Schuster and M. Krstic
European Control Conference
Kos, Greece, July 2-5, 2007
Abstract
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We present a PDE boundary controller that stabilizes
the velocity, pressure, and electromagnetic fields in a
magnetohydrodynamic (MHD) channel flow, also known as
Hartmann flow, a benchmark model for applications such as
cooling systems, hypersonic flight and propulsion. This flow
is characterized by 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 so-called MHD approximation in a low magnetic Reynolds
number regime, and is unstable for large Reynolds numbers.
Our control design needs actuation of velocity and the electric
potential at only one of the walls. The backstepping method for
stabilization of parabolic PDEs is applied to the velocity field
system written in some appropriate coordinates; this system
is very similar to the Orr-Sommerfeld-Squire system of PDEís
and presents the same difficulties. Thus we use actuation not
only to guarantee stability but also to decouple the system
in order to prevent transients. Control gains are computed
solving linear hyperbolic PDEsóa much simpler task than,
for instance, solving nonlinear Riccati equations. Stabilization
of non-discretized 3-D MHD channel flow has so far been an
open problem.