Finite Dimensional Optimal Control of Poiseuille Flow

In this paper we consider linear stabilization of plane, Poiseuille flow using linear quadratic Gaussian optimal control theory. It is shown that we may significantly increase the dissipation rate of perturbation energy, while reducing the required control energy, as compared with that reported using simple, integral compensator control schemes. Poiseuille flow is described by the infinite dimensional Navier-Stokes equations. Because it is impossible to implement infinite dimensional controllers, we implement high but finite order controllers. We show that this procedure in theory can lead to destabilization of unmodeled dynamics. We then show that this may be avoided using distributed control or, dually, distributed sensing. A problem in high plant order linear quadratic Gaussian controller design is numerical instability in the synthesis equations. We show a linear quadratic Gaussian design that uses an extremely low-order plant model. This low-order controller produces results essentially equivalent to the high-order controller.