Limitations of Linear Control of Thermal Convection in a Porous Medium

The ability of linear controllers to stabilize the conduction (no-motion) state of a saturated porous layer heated from below and cooled from above is studied theoretically. Proportional, suboptimal robust(H∝) and linear quadratic Gaussian (H2) controllers are considered. The proportional controller increases the critical Rayleigh number for the onset of convection by as much as a factor of 2. Both the H2 and H∝ controllers stabilize the linearized system at all Rayleigh numbers. Although all these controllers successfully render negative the real part of the linearized system’s eigenvalues, the linear operator of the controlled system is non-normal and disturbances undergo substantial growth prior to their eventual, asymptotic decay. The dynamics of the nonlinear system are examined as a function of the disturbance’s amplitude when the system is subjected to the "most dangerous disturbances." These computations provide the critical amplitude of the initial conditions above which the system can no longer be stabilized. This critical amplitude decreases as the Rayleigh number increases. To facilitate extensive computations, we examine two-dimensional convection in a box containing a saturated porous medium, heated from below and cooled from above, as a model system. The heating is provided by a large number of individually controlled heaters. The system’s state is estimated by measuring the temperature distribution at the box’s midheight. All the controllers considered here render the linearized, controlled system’s operator non-normal. The transient amplification of disturbances limits the "basin of attraction" of the nonlinear system’s controlled state. By appropriate selection of a controller, one can minimize, but not eliminate, the controlled, linear system’s non-normality.