Magnetohydrodynamic State Estimation with Boundary Sensors
R. Vazquez, E. Schuster, M. Krstic
Automatica 44 (2008) 2517-2527
Abstract
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We present a PDE observer that estimates the velocity, pressure, electric potential and current fields in a magnetohydrodynamic
(MHD) channel flow, also known as Hartmann flow. 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 inductionless MHD approximation in a low magnetic Reynolds number regime. We identify physical
quantities (measurable on the wall of the channel) that are sufficient to generate convergent estimates of the velocity, pressure,
and electric potential field away from the walls. Our observer consists of a copy of the linearized MHD equations, combined
with linear injection of output estimation error, with observer gains designed using backstepping. Pressure, skin friction and
current measurements from one of the walls are used for output injection. For zero magnetic field or non-conducting fluid, the
design reduces to an observer for the Navier-Stokes Poiseuille flow, a benchmark for flow control and turbulence estimation.
We show that for the linearized MHD model the estimation error converges to zero in the L2 norm. Despite being a subject of
practical interest, the problem of observer design for non-discretized 3-D MHD or Navier-Stokes channel flow has so far been
an open problem.