A New Method for Obtaining Non-Autonomous, Reduced-Order Model of Flow Using Proper Orthogonal Decomposition (POD) and Optimal Control of the Resultant Nonlinear Model

Abstract

The focus of this work is on the use of POD and perturbation method to develop non-autonomous, low-dimensional models for the dynamics of the flow evolution and optimal control of the resultant system. The main difficulty of the POD for control purposes is that, control inputs do not show up explicitly in the resulting system making it useless for control purposes. Most of the approaches which are presented to handle this issue make a use of superposition principle and are more appropriate for linear systems. In addition these methods don’t care about time varying nature of the system and consider a constant influence for the inputs on the system. The test bed was selected to be the unsteady, incompressible flow around a circular cylinder. Actuation was performed by two suction-blowing panels on the sides of the cylinder. It is obvious from the physical features of the problem, that the control inputs will not have a constant effect, but a periodic one on the system. The presented method uses perturbation method to account for some nonlinear characteristics of the problem and results in a non-autonomous, time-varying, simply-handled system which tries to capture the periodic influence of the inputs on the system and is expected to predict the Navier-Stokes response to external excitations more precisely. Finally, optimal control theory was employed to design a control law for the nonlinear reduced model, trying to minimize the vorticity content in the fluid domain. Simulations were performed to show the flow’s evolution with the control influence according to the low dimensional model.