Nonlinear stabilization of a thermal convection loop by state feedback

A nonlinear feedback control law that achieves global asymptotic stabilizat ion of a 2D thermal convection loop (widely known for its "Lorenz system" a pproximation) is presented. The loop consists of viscous Newtonian fluid co ntained in between two concentric cylinders standing in a vertical plane. T he lower half of the loop is heated while the upper half is cooled, which m akes the no-motion steady state for the uncontrolled case unstable for valu es of the non-dimensional Rayleigh number R-a > 1. The objective is to stab ilize that steady state using boundary control of velocity and temperature on the outer cylinder. We discretize the original nonlinear PDE model in sp ace using finite difference method and get a high order system of coupled n onlinear ODEs in 2D. Then, using backstepping design, we transform the orig inal coupled system into two uncoupled systems that are asymptotically stab le in l(2)-norm with homogeneous Dirichlet boundary conditions. The resulti ng boundary controls actuate velocity and temperature in the original coord inates. The control design is accompanied by an extensive simulation study which shows that the feedback control law designed on a very coarse grid (u sing just a few measurements of the flow and temperature fields) can succes sfully stabilize the actual system for a very wide range of the Rayleigh nu mber. (C) 2001 Published by Elsevier Science Ltd.