Nonlinear stabilization of a thermal convection loop by state feedback

Abstract

A nonlinear feedback control law that achieves global asymptotic stabilization of a 2D thermal convection loop (widely known for its “Lorenz system” approximation) is presented. The loop consists of viscous Newtonian fluid contained in between two concentric cylinders standing in a vertical plane. The lower half of the loop is heated while the upper half is cooled, which makes the no-motion steady state for the uncontrolled case unstable for values of the non-dimensional Rayleigh number R a >1. The objective is to stabilize that steady state using boundary control of velocity and temperature on the outer cylinder. We discretize the original nonlinear PDE model in space using finite difference method and get a high order system of coupled nonlinear ODEs in 2D. Then, using backstepping design, we transform the original coupled system into two uncoupled systems that are asymptotically stable in l 2-norm with homogeneous Dirichlet boundary conditions. The resulting boundary controls actuate velocity and temperature in the original coordinates. The control design is accompanied by an extensive simulation study which shows that the feedback control law designed on a very coarse grid (using just a few measurements of the flow and temperature fields) can successfully stabilize the actual system for a very wide range of the Rayleigh number.