Finite Dimensional Optimal Control of Poiseuille Flow

In this paper, we consider linear stabilization of plane, Poiseuille flow using lin-ear quadratic Gaussian optimal control theory. It is shown that we may significantly increase the dissipation rate of perturbation energy, while reducing the required con-trol energy, as compared to that reported using simple, integral compensator control schemes. Poiseuille flow is described by the infinite-dimensional Navier-Stokes equa-tions. Since it is impossible to implement infinite-dimensional controllers, we imple-ment high, but finite-order controllers. We show that this procedure can in theory 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 or-der 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. % c w- / i H + d b &~aA.od S P I J ~ ~ J