Hybrid Model Reduction for Compressible Flow Controller Design

Xiaoqing Ge* and John T. WenDepartment of Electrical, Computer and Systems Engineering

Rensselaer Polytechnic Institute, Troy, NY, USA

Abstract— Computational fluid dynamics (CFD) has been apowerful simulation tool to gain insight and understanding offluid dynamic systems. However, it is also extremely compu-tationally intensive and thus unsuitable for control design anditeration. Various model reduction schemes have been proposedin the past to approximate the Navier-Stoke equation with alow-dimensional model. There are essentially two approaches:input/output model identification and proper orthogonal de-composition (POD). The former captures mostly the localbehavior near a steady state and the latter is highly dependenton the snap shots of the flow state used to extract the projection.This paper presents a hybrid model reduction approach thatattempts to combine the best features of the two approaches.We first identify an input/output linear model by using thesubspace identification method. We next project the differencebetween CFD response and the identified model response ontoa set of POD basis. This trajectory is then fit to a nonlineardynamical model to augment the input/output linear model. Theresulting hybrid model is then used for control design with thecontroller evaluated with CFD. The proposed methodology hasbeen applied to a 2D compressible flow passing a contractiongeometry. The result indicates that near the steady state usedfor linear system identification, the linear system based designworks well. However, far away from the steady state, the hybridsystem shows much better performance.

I. INTRODUCTION

Efficient numerical techniques are now commonly usedto efficiently simulate physical systems governed by partialdifferential equations, such as fluid, thermal, and structuralsystems. Due to the high dimensionality of such systems, thesimulation is computationally expensive and time consum-ing. For system analysis and control design, low-dimensionalmodels that can capture key attributes of the high ordersystems are needed.

A commonly used tool for complex nonlinear systemmodel reduction is the method of proper orthogonal decom-position (POD) and Galerkin projection [1]–[3]. It providesa systematic way to develop reduced order models froma series of “snapshots” of experimental or computationaldata. Through an orthogonalization procedure, characteristicinformation of the system is extracted from the solutionsnapshots and used to construct a reduced basis set. A low-dimensional approximation is obtained by projecting the fullnonlinear system to this reduced basis. The attraction ofthe POD/Galerkin method is that it can capture nonlinearbehavior of the system (if appropriate snap shots are used),but may lose key system properties such as stability neara stable equilibrium [4]. Some extended approaches have

* Corresponding author: gex@rpi.edu

been proposed based on the POD method. [5] establish aconnection between POD and balanced truncation theory.The method of snapshots is used to obtain low-rank ap-proximations to the system controllability and observabilitygrammians, which are then used to develop a balancedreduced order model. A trust-region POD (TRPOD) methodis proposed in [6]. By embedding the POD based reducedorder modeling technique into a trust-region framework, thisapproach updates the reduced order models to represent theflow dynamics during an optimization process.

For linear time invariant systems, a rich set of tools exist,mostly based on the singular value decomposition (SVD)of some input/output map. In particular, if only input/outputdata are available, such tools, e.g., subspace algorithm, maybe used to construct a low order dynamical model [7]–[9]. This type of tool is attractive as it usually preservessystem stability and has explicit error bounds. Furthermore,it does not require the knowledge of the governing equation.However, it is unable to predict the nonlinear behavior of thesystem nor provide insight into the flow field.

One way of modeling the dynamics of a nonlinear systemis the nonlinear autoregressive moving average model withexogenous inputs (NARMAX) method. In [10], NARMAXis employed to develop a nonlinear model representing thecoupling between jet actuation and measured pressure for aboundary layer flow separation control problem. However,this method only generates an input/output model, thusproviding no information about the internal system dynamics.

This paper presents a reduced order modeling techniquewe refer to as hybrid method, which combines ideas fromsubspace-based system identification and POD method. Itfirst identifies a stable linear dynamical model. Then itrepresents the difference between the full flow field and theidentified model by a POD expansion. Thus, the full systemis decomposed into a dominant linear subsystem and a small-scale nonlinear subsystem. The dynamics of the nonlinearsubsystem can be obtained by projecting the differencebetween linear model and CFD data onto the POD modes.We apply the hybrid approach to a 2D compressible flowthat passes a contraction geometry and compare it with threeother modeling methods. Computation fluid dynamics (CFD)results suggest that the hybrid model accurately predicts thenonlinear behavior of the system. It also performs the bestin terms of feedback control and learning control.

II. CONTRACTION SECTION EXAMPLE

Fig. 1 shows the geometry of the contraction section ina subsonic wind tunnel facility that motivated this research[11]. The contraction, with boundary shaped by a fifth orderpolynomial curvature, settles the flow transitions from thesettling chamber and feeds uniform flow to the test section.In computational simulations, throat Mach number is attainedto precisely match the experimental setup by adjusting themass flux rate at the inlet. The adjustment can be formulatedas a classical control problem: first model the dynamics fromthe inlet mass flux to the throat Mach number, then designcontrol strategies to achieve a desired output.

Fig. 1. Contraction geometry

The current work uses PHASTA, a numerical CFD code,to simulate the flow dynamics on a multiprocessor UNIXcluster. Flow computations are performed using the Stream-line Upwind Petrov-Galerkin (SUPG) stabilized finite ele-ment method [12], which has been proven stable and higherorder accurate. The numerical simulation uses a 2D slabgeometry which approximates the flow in a 40mm slice alongthe span of the contraction. Full flow field data are collectedat a sampling rate of 5× 10−4s, based on a fine 2D meshshown in Fig. 2.

−2 −1.5 −1 −0.5 0

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

x

y

Fig. 2. 2D CFD mesh

III. MODEL REDUCTION

A. POD and Galerkin Projection

The essential idea of POD is to generate a set of basisthat “optimally ” spans a given sample set of data. Consider aHilbert space H with an inner product ⟨·, ·⟩. The goal of PODis, given an ensemble of data y(t) ∈ H, to find an optimalsubspace S, such that the averaged error of the projectionE(∥y−PSy∥) is minimized. Here ∥ · ∥ denotes the norm onH, PS denotes the projection onto the subspace S, and E isan average operator over t. Let {φ j ∈ H| j = 1, · · · ,n} be anorthogonal basis for S, then the projection of y(t) onto S is

PSy(t) =n

∑j=1

⟨y(t),φ j⟩φ j. (1)

The basis functions φ can be obtained by directly from theSVD of the snapshot matrix [y(t1), · · · ,y(tN)].

Suppose the dynamics of a system are described by a highorder ODE (or PDE) of the form:

y = Dx(y), (2)

where Dx is a nonlinear differential operator depending onsome parameter x. We may approximate solutions to (2) byprojecting the equations onto a finite-dimensional subspacesuch as the one spanned by POD modes. Inserting the PODexpansion

y(t) =n

∑j=1

α j(t)φ j, (3)

into (2) and taking the inner product with φk yield a set ofordinary differential equations (ODEs) for αk:

αk = ⟨Dx(y),φk⟩, k = 1, · · · ,n. (4)

B. Hybrid Model Reduction Method

The first step is to use the input/output CFD data (in thecontraction section example, inlet mass flux is the input,throat Mach number is the output) to identify a lineartime invariant system (e.g., using the subspace identificationmethod).

xk+1 = A11xk +B1uk

zk = C1xk. (5)

We next try to relate the state of this (typically low order)system to the full flow field. Denote the kth snapshot of thefull flow field by Yk. A mapping Φ from the reduced statexk to the full state Yk may be obtained by a least-squares fit:

Φ[

x1 . . . xN]=[

Y1 . . . YN]. (6)

The linear model can approximate well the local behavior(near the state state), but has increasing error far awayfrom the steady state. To capture the nonlinear behavior, weapproximate the difference between Yk and Φxk as a linearcombination of POD modes Ψ:

Yk = Φxk +Ψak =n1

∑i=1

φixik +

n2

∑j=1

ψ j a jk, (7)

where Ψ is obtained from the SVD of the difference snapshotmatrix [Y1 −Φx1, · · · ,YN −ΦxN ].

Traditional POD/Galerkin method finds the dynamics ofak by projecting the governing equation onto the POD basis.The projection is computational expensive when there isa large number of sampling points. Moreover, the resultis highly dependent on the choice of inner product forprojection. One needs to take care of different units of flowvariables (pressure, velocity, temperature, etc.), as well asthe weighting ratio of each point if the mesh is nonuniform.Therefore, instead of projecting the governing equation, wedirectly compute ak by projecting the linear system erroronto the POD basis:

ak = ΨT (Yk −Φxk). (8)

Then we use [a1, · · · ,aN ] as a “training” data set to fit anonlinear discrete dynamical system

ak+1 = fa(xk,ak,uk) (9)

to approximate the dynamics of ak. Combining (5), (7) and(9), we have a hybrid dynamical model:

xk+1 = A11xk +B1uk

ak+1 = fa(xk,ak,uk)

zk = g(Yk) = g(Φxk +Ψak) := h(xk,ak) (10)

where zk is the throat Mach number.

C. Application to Contraction Section Example

We have applied the hybrid model reduction method de-scribed above to the flow modeling of the contraction section.Using the input/output data, a second-order linear model ofthe form (5) is identified by using the subspace algorithm. Asshown in Fig. 3, it captures well the input/output relationshipfor the training data set. The linear model is also validatedby a different data set, where the input range is much largerthan the training data. We shall see later that when the inputlevel is far from the training set, the linear model yields apoor prediction of the throat Mach number.

0 0.5 1 1.5 2 2.50.49

0.5

0.51

0.52

0.53

0.54

0.55

0.56

0.57

time (sec)

Mac

h at

thro

at

2nd−order identified linear model (RMS error=3.0537e−4)

CFDlinear model

Fig. 3. Output response comparison of the training set between CFD andlinear model (input range: [0.7, 0.8])

We next use the POD expansion to represent the differencebetween CFD data and the linear model. Fig. 4 plots the

energy captured by the first 10 POD modes. In the expansionwe use the first 4 modes, which capture more than 99% ofthe total energy.

1 2 3 4 5 6 7 8 9 1010

−2

10−1

100

101

102

# POD modes

ener

gy p

erce

ntag

e (%

)

Energy percentage of first 10 POD modes

Fig. 4. Energy captured by the first 10 POD modes

(a) pressure component (b) x velocity component

(c) y velocity component (d) temperature component

Fig. 5. First POD mode

We denote the ith power of a vector yk by (yk)i =

[yik,1, · · · ,yi

k,n]T , where yk, j, j = 1, · · · ,n is the jth element of

yk. The dynamics of ak is fit to a nonlinear function of theform:

ak+1 ≈ fa(xk,ak,uk)

=m1

∑i=1

Γi(xk)i +

m2

∑i=1

Λi(ak)i +

m3

∑i=1

βi(uk)i. (11)

Let na,nx denote the dimension of xk, ak, respectively. ThenΓi is a na × nx matrix and Λi is a na × na matrix. Here βkis a na × 1 vector since uk is a scalar. If fa(·) has a largenumber of terms that increases the computational cost of theapproximation, one can extract dominant basis by performing

SVD of

x1 · · · xN...

. . ....

(x1)m1 · · · (xN)

m1

a1 · · · aN...

. . ....

(a1)m2 · · · (aN)

m2

u1 · · · uN...

. . ....

(u1)m3 · · · (uN)

m3

. (12)

Fig. 6 shows a good approximation of fa(xk,ak,uk) withm1 = 4, m2 = 1, m3 = 1, which yields a hybrid model withstate nonlinearity and output nonlinearity:

xk+1 = A11xk +B1uk

ak+1 = A21xk +A22ak +B2uk +m1

∑i=2

Γi(xk)i

zk = h(xk,ak). (13)

The hybrid model accurately predicts the throat Mach num-ber (Fig. 7) as well as the full Mach number filed (Fig. 8)for a wide input range.

0 0.5 1 1.5 2 2.5 3−5

−4

−3

−2

−1

0

1

2

3

4

5x 10

5

time (sec)

a 1

RMS error = 3884.0574

adirect

apropagated

(a) a1

0 0.5 1 1.5 2 2.5 3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

4

time (sec)

a 2

RMS error = 967.3997

adirect

apropagated

(b) a2

0 0.5 1 1.5 2 2.5 3−1

−0.5

0

0.5

1

1.5x 10

4

time (sec)

a 3

RMS error = 855.9886

adirect

apropagated

(c) a3

0 0.5 1 1.5 2 2.5 3−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

time (sec)

a 4

RMS error = 677.7747

adirect

apropagated

(d) a4

Fig. 6. Comparison between projected ak (solid) and propagated of ak(dash) using fa(xk,ak,uk)

IV. CONTROL DESIGN FOR OUTPUT REGULATION

The goal of obtaining a reduced model is to use it forcontrol design. In this section, we design LQG control andlearning control for the inlet mass flux to regulate the outputMach throat number based on the hybrid model and compareits performance with three other reduced order models:

1) Identified linear model (5).

0 0.5 1 1.5 2 2.5 30.4

0.5

0.6

0.7

0.8

0.9

1

1.1

time (sec)

Mac

h at

thro

at

Comparison of linear model and hybrid model

CFDlinearhybrid

Fig. 7. Comparison of linear and hybrid models (input range: [0.7, 1.5])

(a) Mach number approximated byhybrid model

(b) Error between CFD data and hy-brid model

Fig. 8. Mach number field corresponding to a high input level (inlet massflux =1.5)

2) Semilinear model with input nonlinearity [13]:

xk+1 = Axk +B f (uk)

zk = Cxk. (14)

3) Purely nonlinear model (in the rest of the paper, we callit “nonlinear model” for simplicity): expand Yk = Ψbkand approximate the dynamics of bk by a nonlinearfunction fb(·)

bk+1 = fb(bk,uk)

zk = g(Yk) = g(Ψbk) := p(bk). (15)

This method is very similar to POD method, exceptthat it directly fits the dynamics of bk to a nonlin-ear function, instead of projecting the Navier-Stokesequations to propagate bk. This is also the same as theidentification of the ak dynamics without combining itwith the linear time invariant system.

A. LQG Control

LQG regulator is a combination of a Kalman filter withan Linear-Quadratic-Regulator (LQR). It uses state estimatefrom the Kalman filter and optimized LQR state-feedbackgain to generate the control signal. To eliminate steady-state error, an integrator is added to the LQG regulator. Theextended state space realization of a linear system with an

error integrator is given by

x = Ax+Bu

y = Cx+Du

e = r− y = r−Cx−Du (16)

Let ∆x = x− x∗, ∆u = u− u∗, ∆y = y− y∗, (x∗,y∗,u∗ aresteady state values), then (16) is rewritten as

xe =

[∆xe

]=

[A 0−C 0

][∆xe

]+

[B−D

]∆u

ye =

[∆ye

]=

[C 00 I

][∆xe

]+

[D0

]∆u. (17)

The full control input is

u = u∗+∆u = u∗−Kexe = u∗− [K|Ki]

[xe

], (18)

where x represents the state estimate from a Kalman filter, Keis computed by solving the Riccati differential equation. Kecan be further tuned to improve the system transient responseby adjusting the weights on input, output and state.

0 0.05 0.1 0.15 0.2 0.250.48

0.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

0.66

time (sec)

Mac

h at

thro

at

LQG control using linear model

MATLABCFD

(a) linear

0 0.05 0.1 0.15 0.2 0.250.45

0.5

0.55

0.6

0.65

0.7

time (sec)

Mac

h at

thro

at

LQG control using semilinear model

MATLABCFD

(b) semilinear

0 0.05 0.1 0.15 0.2 0.250.45

0.5

0.55

0.6

0.65

0.7

time (sec)

Mac

h at

thro

at

LQG control using hybrid model

MATLABCFD

(c) hybrid

0 0.05 0.1 0.15 0.2 0.250.45

0.5

0.55

0.6

0.65

0.7

0.75

time (sec)

Mac

h at

thro

at

LQG control using nonlinear model

MATLABCFD

(d) nonlinear

Fig. 9. MATLAB and CFD simulation results (targe Mach number=0.6)

We designed an LQG regulator based on each model (lin-ear, semilinear, hybrid, nonlinear) and tune Ke in MATLABsuch that they have similar settling time. For semilinear,hybrid and nonlinear models, the control design uses lin-earization of the model and the state estimate x is generatedby an extended Kalman filter. The Kalman filter and LQGcontroller are then implemented in CFD simulations.

Fig. 9 and 10 show the performances of four LQG con-trollers which are designed to drive the throat Mach numberto 0.6. Controllers based on linear, hybrid and nonlinearmodels generate a similar output to MATLAB simulation,while the semilinear model yields a very oscillatory output.We also test the four controllers with a larger target Machnumber of 0.9. Although MATLAB simulations (Fig. 12 (a))

0 0.05 0.1 0.15 0.2 0.250.48

0.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

0.66

time (sec)

Mac

h at

thro

at

LQG controller with integrator (MATLAB)

linear (Tset

=0.067s)

semilinear (Tset

=0.054s)

hybrid (Tset

=0.0375s)

nonlinear (Tset

=0.069s)

(a) MATLAB results

0 0.05 0.1 0.15 0.2 0.250.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

time (sec)

Mac

h at

thro

at

LQG controller with integrator (CFD)

linear (Tset

=0.0885s)

semilinearhybrid (T

set=0.0935s)

nonlinear (Tset

=0.0995s)

(b) CFD results

Fig. 10. Comparison of four models (targe Mach number=0.6)

0 0.1 0.2 0.3 0.4 0.50.4

0.5

0.6

0.7

0.8

0.9

1

time (sec)

Mac

h at

thro

at

LQG control using linear model

MATLABCFD

(a) linear

0 0.1 0.2 0.3 0.4 0.50.4

0.5

0.6

0.7

0.8

0.9

1

time (sec)

Mac

h at

thro

at

LQG control using semilinear model

MATLABCFD

(b) semilinear

0 0.1 0.2 0.3 0.4 0.50.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

time (sec)

Mac

h at

thro

at

LQG control using hybrid model

MATLABCFD

(c) hybrid

0 0.1 0.2 0.3 0.4 0.50.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

time (sec)

Mac

h at

thro

at

LQG control using nonlinear model

MATLABCFD

(d) nonlinear

Fig. 11. MATLAB and CFD simulation results (targe Mach number=0.9)

0 0.1 0.2 0.3 0.4 0.50.4

0.5

0.6

0.7

0.8

0.9

1

time (sec)

Mac

h at

thro

at

LQG controller with integrator (MATLAB)

linear (Tset

=0.0575s)

semilinear (Tset

=0.166s)

hybrid (Tset

=0.226s)

nonlinear (Tset

=0.204s)

(a) MATLAB results

0 0.1 0.2 0.3 0.4 0.50.4

0.5

0.6

0.7

0.8

0.9

1

time (sec)

Mac

h at

thro

at

LQG controller with integrator (CFD)

linear (Tset

=0.4365s)

semilinear (Tset

=0.356s)

hybrid (Tset

=0.1115s)

nonlinear (Tset

=0.2785s)

(b) CFD results

Fig. 12. Comparison of four models (targe Mach number=0.9)

suggest that they have similar performances (linear-model-based controller performs even better), only hybrid-model-based controller settles the output within a reasonable timeof 0.146s. The other three controllers have a much longersettling time of more than 0.27s.

To sum up, when the input is small, linear model is goodenough to predict the output as well as to guide feedbackcontrol design. However, it shows inaccuracy at high inputlevel. Semilinear model can not provide a reasonable pre-diction of the output in both cases. Hybrid model is shownto be the most accurate and reliable model for a wide inputrange.

B. Iterative POD Approach

As can be seen from Fig. 9 and 11, there is still a discrep-ancy between MATLAB and CFD trajectories. This suggestsan iterative POD approach to correct the discrepancy, basedon CFD closed loop simulation results. Take the nonlinearmodel in Fig. 11 (d) for example, we extract POD basis fromthe CFD closed loop response (red line) and compute a newset of bk by projecting the full flow field onto the basis. Were-fit a nonlinear function that approximates the dynamicsof bk and used the new nonlinear model for Kalman filterand control design. Fig. 13 shows the result of using onePOD iteration to improve the nonlinear model. We test two

different LQG gains corresponding to a slow trajectory anda fast trajectory, respectively. Note that the LQG gains usedin Fig. 13 are re-tuned and different from that in Fig. 11(d). It can be seen that the new model based on one PODiteration predicts the slow trajectory very well. It also showsimprovement on the fast trajectory, though there is still asmall discrepancy requiring further POD iteration. Fig. 14shows that through a second POD iteration, we obtain anonlinear model that matches the CFD output very well.

0 0.1 0.2 0.3 0.4 0.50.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

time (sec)

Mac

h at

thro

at

Iterative POD method (nonlinear model)

CFD (Tset

=0.1495s)

MATLAB (Tset

=0.155s)

(a) slow trajectory

0 0.1 0.2 0.3 0.4 0.50.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

time (sec)

Mac

h at

thro

at

Iterative POD method (nonlinear model)

CFD (Tset

=0.175s)

MATLAB (Tset

=0.258s)

(b) fast trajectory

Fig. 13. Iterative POD approach applied to the nonlinear model (firstiteration)

C. Learning Control

Iterative learning control (ILC) has gained great successin motion tracking control [14]. We consider the gradientapproach which propagates the output trajectory error to theinput using the gradient map which is just the linearizedsystem [15]:

1) Set u = u0, apply u to the physical system and obtainthe output sequence y;

2) Update u by adding a corrective term

∆u =−αG∗(y− ydes) (19)

where G∗ is the adjoint of G, the linearized systemabout u, and α may be set as a sufficiently smallconstant or found by a line search;

0 0.1 0.2 0.3 0.4 0.50.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

time (sec)

Mac

h at

thro

at

Iterative POD method (nonlinear model)

MATLAB (Tset

=0.196s)

CFD(Tset

=0.241s)

Fig. 14. Iterative POD approach applied to the nonlinear model (seconditeration)

3) Iterate until ∥y− ydes∥ of ∥∆u∥ is sufficiently small.Fig. 15 shows the responses of learning control basedon linear model and hybrid model, respectively. Since thecontrol design uses linearization of the hybrid model, twomodels generate similar gradient and, thus, have similarperformances.

0 0.2 0.4 0.6 0.8 10.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

time (sec)

Mac

h at

thro

at

Comparison of ILC and LQG controls

ILC w/ linear model (Tset

=0.1845s)

ILC w/ hybrid model (Tset

=0.1565s)

LQG w/ hybrid model (Tset

=0.1265s)

Fig. 15. Comparison of learning control and LQG control

V. CONCLUSION

This paper presents a new model reduction approach thatcombines ideas from subspace-based system identificationand POD method. It first identifies a reasonable linear modelusing N4SID, and then introduces nonlinearity to the modelby projecting the difference between CFD data and identifiedmodel onto a set of POD basis. The hybrid model is usedfor control design and shown to have consistently goodperformances, compared to linear, semilinear and nonlinearmodels.

Future directions include applying the hybrid approach to amore complex case (e.g., an inlet duct with settling chamber,

contraction and downstream diffuser), which has a lot moresampling points and parallel computing issue.

VI. ACKNOWLEDGMENT

This work was supported in part by the Center for Au-tomation Technologies and Systems (CATS) under a blockgrant from the New York State Foundation for Science,Technology and Innovation (NYSTAR). The authors wouldalso like to thank Yi Chen, Onkar Sahni, Ken Jansen,Miki Amitay, John Vaccaro for their help and guidance onaerodynamic modeling and computatational fluid dynamics.

REFERENCES

[1] Lumley, J. L., “The Structure of Inhomogeneous Turbulence,” At-mospheric Turbulence and Radio Wave Propagation, edited by Ya-glom, A. M. and Tatarski, V. I., Nauka, Moscow, pp. 166-178, 1967.

[2] Sirovich, L., “Turbulence and the Dynamics of Coherent Structures,”Parts I-III, Quart. Appl. Math. XLV (3), pp. 561-590, 1987.

[3] Rowley, C. W., Colonius, T., and Murray, R. M., “POD Based Modelsof Self-Sustained Oscillations in the Flow Past an Open Cavity,” AIAA2000-1969, 6th AIAA/CEAS Aeroacoustics Conference, 2000.

[4] Smith, T. R. “Low-Dimensional Models of Plane Couette Flow Us-ing the Proper Orthogonal Decomposition,” Ph.D. thesis, PrincetonUniversity, 2003.

[5] Willcox, K. and Peraire, J., “Balanced Model Reduction via the ProperOrthogonal Decomposition,” AIAA Journal, vol. 40, no. 11, pp. 2323-2330, 2002.

[6] Fahl M. “Trust-Region Methods for Flow Control Based on ReducedOrder Modeling,” Ph.D. Thesis, University of Trier, 2000.

[7] Van Overschee, P., and De Moor, B., “N4SID: Subspace Algorithmsfor the Identification of Combined Deterministic-Stochastic Systems,”Automatica, Special Issue on Statistical Signal Processing and Control,30(1): pp. 75-93, 1994.

[8] Van Overschee, P., and De Moor, B., “N4SID: Numerical Algorithmsfor State Space Subspace System Identification,” in Proceedings of the12th IFAC World Congress, vol. 7, pp. 361-364, 1993.

[9] Kim, K., Debiasi, M., Schultz, R., Serrani, A., and Samimy, M., “Dynamic Compensator for a Synthetic-Jet-Like Compression DriverActuator in Closed-Loop Cavity Flow Control,” 45th AIAA AerospaceSciences Meeting and Exhibit, AIAA 2007-880, 2007.

[10] Kim, K., Beskok, A., and Jayasuriya, S., “Nonlinear System Identifi-cation for the Interaction of Synthetic Jets with a Boundary Layer,”in 2005 American Control Conferance, pp. 1313-1318, 2005.

[11] Vaccaro, J. C., Sahni, O., Olles, J., Jansen, K. E., and Amitay, M.,“Experimental and Numerical Investigation of Active Control of InletDucts,” International Journal of Flow Control, vol. 1, pp. 133-154,2009.

[12] Jansen, K. E., “A Stabilized Finite Element Method for ComputingTurbulence,” Computer Methods in Applied Mechanics and Engineer-ing, vol. 174, pp. 299-317, 1999.

[13] Ge, X., Gressick, W., Wen, J. T., Sahni, O., and Jansen, K. E., “ALow-Order Nonlinear Models for Active Flow Control of a Low L/DInlet Duct,” in Proceedings of 2010 American Control Conference,2010, pp. 2989-2994, 2010.

[14] Moore, K. L., “Iterative learning control - An expository overview,”Applied and Computational Control, Signals, and Circuits, 1, pp.151–214, 1999.

[15] Potsaid, B., Wen, J.T., Unrath, M., Watt, D., and Alpay, M., “HighPerformance Motion Tracking Control for Electronic Manufacturing”,Journal of Dynamic Systems, Measurement, and Control, vol. 129, pp.767-776, 2007.