Feedback Control of a Thermal Fluid Based on a Reduced Order Observer*

We discuss the problem of designing a feedback control law based on a reduced order observer, which locally stabilizes a two dimensional thermal fluid modeled by the Boussinesq approximation. We consider mixed boundary control for the Boussinesq equations in an open bounded and connected domain. In particular, the controllers are finite dimensional and act on a portion of the boundary through Neumann/Robin boundary conditions. A linear Luenberger observer is constructed based on point observations of the linearized Boussinesq equations. The current setting of the system leads to a problem with unbounded control inputs and outputs. Linear Quadratic Gaussian (LQG) balanced truncation is employed to obtain the reduced order model for the linearized system. The feedback law can be obtained by solving an extended Kalman filter problem. The numerical results show that the nonlinear system coupled with the reduced order observer through the feedback law is locally exponentially stable.