Linear feedback control of transient energy growth and control performance limitations in subcritical plane Poiseuille flow

TL;DR

This paper investigates the limitations and design of linear feedback controllers to suppress transient energy growth in subcritical plane Poiseuille flow, demonstrating that certain performance improvements are fundamentally unattainable with linear state feedback.

Contribution

It introduces a Linear Matrix Inequalities-based method for designing full-state feedback controllers targeting transient growth, and evaluates their effectiveness through simulations.

Findings

Linear state-feedback controllers cannot eliminate transient energy growth.

Designed controllers can increase transition thresholds in flow simulations.

Performance varies depending on initial conditions and actuation components.

Abstract

Suppression of the transient energy growth in subcritical plane Poiseuille flow via feedback control is addressed. It is assumed that the time derivative of any of the velocity components can be imposed at the walls as control input, and that full-state information is available. We show that it is impossible to design a linear state-feedback controller that leads to a closed-loop flow system without transient energy growth. In a subsequent step, full-state feedback controllers -- directly targeting the transient growth mechanism -- are designed, using a procedure based on a Linear Matrix Inequalities approach. The performance of such controllers is analyzed first in the linear case, where comparison to previously proposed linear-quadratic optimal controllers is made; further, transition thresholds are evaluated via Direct Numerical Simulations of the controlled three-dimensional Poiseuille flow against different initial conditions of physical interest, employing different velocity components as wall actuation. The present controllers are effective in increasing the transition thresholds in closed loop, with varying degree of performance depending on the initial condition and the actuation component employed.