Adaptive Reduced-Order Controllers for a Thermal Flow System Using Proper Orthogonal Decomposition

Abstract

An adaptive reduced-order controller design is presented for flow control using proper orthogonal decomposition (POD). In reduced-order controller design, the idea is to start with an ensemble of data obtained from numerical simulation of the underlying partial differential equations (PDEs). POD is then used to obtain a reduced set of basis functions which is then used to derive a reduced-order model of the PDEs via Galerkin projection. This reduced-order model allows us to derive a reduced-order controller. However, it is not clear, a priori, what is the best way to obtain an ensemble of data that would give basis functions that represent the influence of the control action on the system. In this paper we explore an adaptive procedure for reduced-order controller design that improves the reduced-order model by successively updating the ensemble of data during the optimization iterations. We illustrate this method on a control problem in thermal flow system modeled by a thermally coupled Navier--Stokes equations. Numerical results are presented for a vorticity regulation problem in fluid flows using boundary temperature as control mechanism. Through our numerical experiments we demonstrate the feasibility and applicability of the adaptive reduced-order controllers.