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This paper deals with the practical and theoretical implications of model reduction for aerodynamic flow-based control problems. Various aspects of model reduction are discussed that apply to partial differential equation- (PDE-) based models in general. Specifically, the proper orthogonal decomposition (POD) of a high dimension system as well as frequency domain identification methods are discussed for initial model construction. Projections on the POD basis give a nonlinear Galerkin model. Then, a model reduction method based on empirical balanced truncation is developed and applied to the Galerkin model. The rationale for doing so is that linear subspace approximations to exact submanifolds associated with nonlinear controllability and observability require only standard matrix manipulations utilizing simulation/experimental data. The proposed method uses a chirp signal as input to produce the output in the eigensystem realization algorithm (ERA). This method estimates the system's Markov parameters that accurately reproduce the output. Balanced truncation is used to show that model reduction is still effective on ERA produced approximated systems. The method is applied to a prototype convective flow on obstacle geometry. An

Recently there has been significant
interest in model reduction for the purpose of control design [

The traditional systematic development of feedback control laws for these systems subject to a large number of states is currently an intractable problem. Feedback control strategies offer the possibility to improve performance and reduce control power through the control of unstable structures in the flow field. Reduced models are important for the design of feedback control laws, which rely on models that capture the relevant dynamics of the input-output system and are amenable to control design.

Unfortunately, it is very difficult
to create models that capture the relevant dynamics of the input-output system.
For example, computational fluid dynamics simulations can provide good
solutions to a discretized version of the Navier-Stokes equation [

POD has been extensively investigated
in distributed parameters systems due to its order reduction capability [

In fluid flow configurations it is
not uncommon for discretized flow models to describe thousands to millions of
state variables, for example, if one uses a linear quadratic regulator (LQR) control
formulation, roughly

In the area of fluid mechanics
controls must often be fixed to the boundary of the problem geometry. For
example, control of flow separation over an airfoil requires that actuation and
sensing be done on the airfoil surface [

The
paper is organized as follows. In Section

The specific problem geometry
considered is shown in Figure

Problem geometry.

The system dynamics that act within
the problem domain are described by the two-dimensional (2D) Burgers’ equation [

Dirichlet boundary conditions located
on the obstacle top and bottom are denoted by

The boundary condition on the airflow
intake side is

The
general approach of POD is to construct a series of solution “snapshots.” These
snapshots are generated by numerical simulations of the governing system
equation(s) with a variety of input equations [

Test inputs used to generate the snapshots.

The numerical simulation was
performed to create the ensemble of solution snapshots

The solution to the PDE is assumed be finite
energy, that is, belongs to the Hilbert space

Any basis for

The POD basis function

A spectral
decomposition of the matrix

The resulting orthonormal POD basis

The governing equation is projected
onto the POD basis. The projection is accomplished via a Galerkin type
projection and results in a system of ordinary differential equations (ODEs). The
Galerkin projection results in only a weak solution to the PDE. However, this
weak solution with finite difference approximations of the boundary conditions
eventually leads to a nonlinear temporal model for the temporal or POD
coefficients

Projecting (

First eight POD modes.

To assess the validity of the POD
model the following test inputs, which are different from the inputs used to
generate the snapshots, are applied at the boundary

Boundary control accuracy.

The goal of model reduction is to
construct another nonlinear system [

Several frequency domain
identification techniques are used in practice to identify the model
parameters. One such method is the eigensystem realization algorithm (ERA)
technique [

A
basic relationship between the Markov parameters and the input and output
relationship in discrete time is

Excitation inputs for ERA method.

The dynamics of (finite dimensional)
linear time-invariant (LTI) systems are governed by a state space model of the
form

The first step in applying balanced
truncation is to compute a coordinate transformation

A balanced realization needs a
similarity transformation

Balanced model
reduction requires the knowledge of the controllability and observability
gramians. The latter are obtained by solving Lyapunov equations, which is
prohibitive for large-scale systems. For a system with

The
computation cost to solve large Lyapunov equations for the controllability and observability gramians prompts us to propose a
balanced truncation algorithm, based on empirical gramians constructed from
input-output data measurements. To this end, let us first introduce the

The product of the

The empirical balanced truncation
based on linear systems is applied to the Galerkin model

The reduced-order model is derived as
discussed through the construction of the immersion/projection nonlinear system
pair

Projected and POD model coefficients.

In
Figure

Comparison of Hankel singular values.

In
Figure

Full- and reduced-order models' responses.

An

Closed-loop control system for tracking.

From Figure

Flow initial condition and reference.

Controlled flow.

Empirical
balanced truncation has been considered in conjunction with POD as an approach
for deriving reduced-order models and applied to 2D Burgers’ equation. Like
POD, empirical balanced truncation is based on simulation/experimental data and
can be implemented via standard matrix computations. Improvements to the scheme originally
proposed in [

This work was supported in part by an Air Force Summer Faculty Fellowship Program, Air Force Office of Scientific Research under Contract AF-FA9550-08-1-0450, and National Science Foundation under Contract NSF-CMMI-0825921.