A feedback control synthesis for the Rayleigh-Bénard convection by means of mode reduction

A state feedback controller is derived to suppress the Rayleigh-Bénard convection in a finite domain by adjusting heat flux profile at the bottom of the system under the constraint of constant heat input. The Karhunen-Loe`ve Galerkin procedure is employed to convert the Boussinesq equation to a reduced order model, from which an extended Kalman filter and the state optimal feedback controller are constructed. The number of measurement locations employed in the Kalman filter determines the number of reliable eigenmodes to be adopted in the feedback controller. With the feedback controller constructed by employing only the reliable dominant eigenmodes, it is found that there exists a threshold Rayleigh number beyond which the suppressed state becomes unstable to evolve to a new stationary convection state. The linear stability analysis is used to investigate the effect of a number of eigenmodes employed in the feedback controller on the threshold Rayleigh number.