An example of thermal regulation of a two dimensional non-isothermal incompressible flow

In this paper we consider a tracking/disturbance rejection problem for a nonlinear infinite dimensional control system using boundary control and sensing. The controlled plant consists of a two dimensional Boussinesq approximation of the non-isothermal incompressible Navier-Stokes equations in a box region. The signal to be tracked and disturbance (which enters as a forcing term on the boundary of the region) are time dependent periodic step functions. While the method described in this work is based on the geometric theory of output regulation, the usual assumptions for that theory do not apply. Nevertheless, we show that the regulator equations used to design the control laws for the geometric methodology can be adapted to handle this case, as an extension of a set point tracking problem. The objective in this example is to force the average temperature on an internal boundary to track a prescribed reference signal, while rejecting a disturbance given as a hot surface on a portion of the boundary. The methodology used in this work provides an example of the design and then discretize methodology as opposed to the usual discretize and then design philosophy.