Optimal control of vortex shedding using low-order models. Part I?open-loop model development

Abstract

An approach to developing active control strategies for separated flows is presented. The methodology proposed is applied to the incompressible unsteady wake flow behind a circular cylinder at a Reynold's number of 100. Control action is achieved via cylinder rotation. Low‐order models which are amenable to control and which incorporate the full non‐linear dynamics are developed by applying the proper orthogonal decomposition technique to data provided by numerical simulation. This process involves extensions to the usual POD approach and the results are therefore assessed for two ‘open‐loop’ test cases. The predictions are found to be satisfactory for control purposes, assuming the model can be reset periodically. The use of these models for optimal control is discussed in a companion paper, Part II. Copyright © 1999 John Wiley & Sons, Ltd.

Reduction procedure for parametrized fluid dynamics problems based on proper orthogonal decomposition and calibration

“…The rest is a linear combination of control inputs (Graham et al, 1999) that are used to remove the inhomogeneous boundary conditions that appear in boundary control problems for PDEs. The control inputs are expressed as a product of a control action parameter g j ðmÞ and a control function r j ðxÞ.…”

“…Examples are the inflow velocity, the position of the cylinder in the flow, etc. Basically, one is free to choose any control function r j ðxÞ, as discussed in Graham et al (1999) and Bergmann et al (2005). However, the choice of r j ðxÞ has an influence on the properties of the reduced model, as will be discussed below.…”

“…The reduced model (13) has a similar form as the model described in Graham et al (1999) with the coefficients a and g collapsed into the vector F.…”

Control of nonlinear systems represented by Galerkin models using adaptation-based linear parameter-varying models

“…This produces a Galerkin model that approximates the original system of nonlinear partial differential equations (PDEs). An approach of this sort has been used, among others, in feedback control of cylinder wakes [12][13][14][15][16][17], control of cavity flow [18][19][20][21][22][23][24], and optimal control of vortex shedding [25,26].…”

An input/output approach to the optimal transition control of a class of distributed chemical reactors

“…A direct use of CFD models for control design or dynamic optimization involves significant computational cost. Although application of advanced model reduction techniques to derive reduced-order models from the detailed partial differential equation (PDE) process models may work well in certain cases and lead to efficient dynamic optimization and control algorithms (see, for example, Armaou and Christofides, 1999Bendersky and Christofides, 2000;Christofides, 2001;Christofides and Daoutidis, 1997;Graham and Kevrekidis, 1996;Graham et al, 1999;Groetsch et al, 2006;Park and Lee, 2000), such a reduced-order model approach might require a huge amount of memory and computational cost when the CFD model consists of millions of grid points needed to accurately describe the process behavior. The reader may also refer to Raja et al (2000) and Varshney and Armaou (2006) for recent applications of model reduction and dynamic optimization to thin film deposition processes described by CFD equations and Balsa-Canto et al (2004 for further recent results on dynamic optimization and control of distributed parameter systems.…”