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Research Papers

# Reduced-Order Model-Based Feedback Control of Flow Over an Obstacle Using Center Manifold Methods

[+] Author and Article Information
Coşku Kasnakoğlu1

Department of Electrical and Electronics Engineering, TOBB University of Economics and Technology, Ankara 06560, Turkeykasnakoglu@etu.edu.tr

R. Chris Camphouse

Carlsbad Programs Group, Performance Assessment and Decision Analysis Department, Sandia National Laboratories, Carlsbad, NM 88220

Andrea Serrani

Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43210

Whenever convenient, we will use Einstein notation to omit summation signs and write $q(xy,t)≈q0+ai(t)φi(x,y)$ in place of Eq. 10.

1

Corresponding author. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited.

J. Dyn. Sys., Meas., Control 131(1), 011011 (Dec 08, 2008) (12 pages) doi:10.1115/1.3023122 History: Received December 08, 2007; Revised September 08, 2008; Published December 08, 2008

## Abstract

In this paper, we consider a boundary control problem governed by the two-dimensional Burgers’ equation for a configuration describing convective flow over an obstacle. Flows over obstacles are important as they arise in many practical applications. Burgers’ equations are also significant as they represent a simpler form of the more general Navier–Stokes momentum equation describing fluid flow. The aim of the work is to develop a reduced-order boundary control-oriented model for the system with subsequent nonlinear control law design. The control objective is to drive the full order system to a desired 2D profile. Reduced-order modeling involves the application of an $L2$ optimization based actuation mode expansion technique for input separation, demonstrating how one can obtain a reduced-order Galerkin model in which the control inputs appear as explicit terms. Controller design is based on averaging and center manifold techniques and is validated with full order numerical simulation. Closed-loop results are compared to a standard linear quadratic regulator design based on a linearization of the reduced-order model. The averaging∕center manifold based controller design provides smoother response with less control effort and smaller tracking error.

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## Figures

Figure 13

Reconstruction of the test flow in Fig. 1

Figure 1

Geometry of the problem

Figure 2

Inlet condition on the left boundary of the domain

Figure 3

Flow with no inlet and no excitation

Figure 4

Baseline POD modes

Figure 5

Flow with excitation at the inlet

Figure 6

Flow with excitation at the inlet and at the top

Figure 7

Flow with excitation at the inlet and at the bottom

Figure 8

Excitation signal for system identification

Figure 9

Actuation modes

Figure 10

Excitation signal for model validation

Figure 11

Flow with test excitation signal applied to top and bottom

Figure 12

Response of Galerkin system (blue∕dark) versus modal coefficients from direct projection (green∕light)

Figure 19

Control signals for averaging∕center manifold-based control

Figure 20

Final state, reference profile, and absolute error for averaging∕center manifold-based control

Figure 21

Modal coefficients a(t) of the closed-loop system with LQR control

Figure 22

LQR control applied to full order Burgers’ CFD

Figure 23

Control signals for LQR control

Figure 24

Final state, reference profile, and absolute error for LQR control

Figure 14

ρ̇ versus ρ for the on-manifold dynamics. The star denotes the value of ρ(0) corresponding to the given initial condition.

Figure 15

Lyapunov function V for the off-manifold dynamics. The star denotes the value of ζ̃(0) corresponding to the given initial condition.

Figure 16

Derivative of Lyapunov function V for the off-manifold dynamics for different values of ρ. The star denotes the value of ζ̃(0) corresponding to the given initial condition.

Figure 17

Modal coefficients a(t) of the closed-loop system with control derived from averaging∕center manifold theory

Figure 18

Averaging∕center manifold-based control applied to full order Burgers’ CFD

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