Poiseuille Flow Controller Design via the Method of Inequalities

This paper investigates the use of the Method of Inequalities (MoI) to design outputfeedback compensators for the problem of the control of laminar plane Poiseuille flow. In common with many flows, the dynamics of plane Poiseuille flow are very non-normal. Consequently, small perturbations grow rapidly with a large transient that may trigger nonlinearities and lead to turbulence even though such perturbations would, in a linear flow, eventually decay. Such a system can be described as a conditionally linear system. The sensitivity is measured using the maximum transient energy growth, which is widely used in the fluids dynamics community. The paper considers two approaches. In the first, the MoI is used to design low-order proportional and P+D controllers. In the second approach, the MoI is combined with McFarlane and Glover’s H∞ loop-shaping design procedure in a mixed-optimization approach. The results show that the low-order controllers do reduce the maximum transient energy growth but the reduction is not satisfactory. Furthermore, the H∞ approach does not improve the performance.