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

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 MHD approximation in a low magnetic Reynolds number regime. 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. The observer design for non-discretized 3D MHD or Navier-Stokes channel flow has so far been an open problem