A Closed-Form Observer for the Channel Flow Navier-Stokes System

The observer design for non-discretized Navier-Stokes partial differential equations has so far been an open problem. We present a nonlinear PDE observer that estimates the velocity and pressure fields for an infinite channel flow. Also know as the Poisseuille flow, this problem is frequently cited as a paradigm for transition to turbulence, and is a benchmark for flow control and turbulence estimation. Our observer consists of a copy of the nonlinear Navier-Stokes equations, combined with linear injection of output estimation error, with observer gains designed in closed form using backstepping. Pressure and skin friction at one of the walls are the only quantities needed for measurement. For a fully developed channel flow (whether laminar or turbulent), with a Reynolds number possibly above the critical value, but not too far above it, the observer is guaranteed to be convergent to the actual velocity and pressure field, provided its initial estimates are not too far from the actual initial profile. An output feedback result for the linearized infinite channel flow is also presented, combining the observer in this paper with a controller presented in a companion CDC´05 paper. Both results are presented for the 2D case for clarity of exposition, however an extension to 3D is straightforward.