Global stabilization of a thermal convection loop

Abstract

A nonlinear feedback control law that achieves global asymptotic stabilization of a 2D thermal convection loop is presented. The loop consists of a 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. Stability analysis of the thermal convection loop shows that the no-motion steady state for the uncontrolled case is unstable for the values of the non-dimensional Rayleigh number R/sub /spl alpha//>1. The objective is to stabilize the unstable no-motion steady state using boundary control of velocity and temperature on the outer cylinder. In our controller design we start by discretizing the original PDE model in space using a finite difference method which gives a high order system of coupled nonlinear ODEs in 2D. Then, using backstepping design, we obtain a discretized coordinate transformation that transforms the original coupled system into two uncoupled systems that are asymptotically stable in 12-norm with homogeneous Dirichlet boundary conditions. Using the property that the discretized coordinate transformation is smoothly invertible for an arbitrary grid choice, we conclude that the discretized version of the original system is globally asymptotically, stable and obtain nonlinear feedback boundary control laws for velocity and temperature in the original set of coordinates. The control design is accompanied by an extensive simulation study. Numerical results show that the feedback control law designed on a very coarse grid can successfully stabilize the system for a very wide range of the Rayleigh number. This means that an excellent closed loop performance is achieved using just a few measurements of the flow and temperature fields implying that the proposed backstepping design has a potential to be successfully applied in a real experimental setting.