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AFRL-RB-WP-TP-2008-3150

A SNAPSHOT DECOMPOSITION METHOD FOR

REDUCED ORDER MODELING AND BOUNDARY

FEEDBACK CONTROL (POSTPRINT)

R.C. Camphouse, James H. Myatt, Ryan F. Schmit, M.N. Glauser, J.M. Ausseur,

M.Y. Andino, and R.D. Wallace

Control Design and Analysis Branch

Control Sciences Division

JUNE 2008

Final Report

Approved for public release; distribution unlimited.

See additional restrictions described on inside pages

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A SNAPSHOT DECOMPOSITION METHOD FOR REDUCED ORDER

MODELING AND BOUNDARY FEEDBACK CONTROL (POSTPRINT)

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R.C. Camphouse (Sandia National Laboratories)

James H. Myatt (AFRL/RBCA)

Ryan F. Schmit (AFRL/RBAI)

M.N. Glauser, J.M. Ausseur, M.Y. Andino, and R.D. Wallace (Syracuse University)

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Conference paper presented at the AIAA 4th Flow Control Conference, June 23 through 26, 2008 in Seattle, WA.

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14. ABSTRACT

In this paper, we develop a reduced basis construction method that allows for separate consideration of baseline and

actuated dynamics in the reduced modeling process. A prototype initial boundary value problem, governed by the two-

dimensional Burgers equation, is formulated to demonstrate the utility of the method in a boundary control setting.

Comparisons are done between reduced and full order solutions under open-loop boundary actuation to illustrate

advantages gained by separate consideration of actuated dynamics. A tracking control problem is specified using a

linear quadratic regulator formulation. Comparisons of feedback control effectiveness are done to demonstrate benefits

in control effectiveness obtained from separate consideration of actuated dynamics during model reduction.

15. SUBJECT TERMS

order reduction, flow control, feedback control, Burgers’ equation, proper orthogonal decomposition, POD, split POD

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Standard Form 298 (Rev. 8-98)

Prescribed by ANSI Std. Z39-18

4th Flow Control Conference AIAA-2008-4195

Seattle, WA June 23 - 26, 2008

A Snapshot Decomposition Method for Reduced Order Modeling

and Boundary Feedback Control

R. C. Camphouse 1

Sandia National Laboratories

Performance Assessment and Decision Analysis Department

J. H. Myatt and R. F. Schmit

Air Force Research Laboratory

Air Vehicles Directorate

M. N. Glauser, J. M. Ausseur, M. Y. Andino, and R. D. Wallace

Syracuse University

Department of Mechanical Engineering

Abstract

In this paper, we develop a reduced basis construction method that allows for separate consideration of baseline and actuated

dynamics in the reduced modeling process. A prototype initial boundary value problem, governed by the two-dimensional

Burgers equation, is formulated to demonstrate the utility of the method in a boundary control setting. Comparisons are

done between reduced and full order solutions under open-loop boundary actuation to illustrate advantages gained by separate

consideration of actuated dynamics. A tracking control problem is speciﬁed using a linear quadratic regulator formulation.

Comparisons of feedback control eﬀectiveness are done to demonstrate beneﬁts in control eﬀectiveness obtained from separate

consideration of actuated dynamics during model reduction.

Nomenclature

Lcovariance matrix

Ssolution snapshot

Ntotal number of snapshots

Tactuated snapshot

NAtotal number of actuated snapshots

φ, α basis mode and temporal coeﬃcient, respectively

Mnumber of basis modes

MB, MAnumber of baseline and actuator modes, respectively

ξ, η baseline and actuator mode, respectively

θ, β baseline and actuator mode temporal coeﬃcient, respectively

λ,veigenvalue and eigenvector, respectively

A, B state and control matrices, respectively

G, F nonlinear and forcing matrices, respectively

Q, R state and control weight matrices, respectively

γcontrol robustness parameter

Ω spatial domain

t time

xspatial coordinate vector

hspatial step-size

Introduction

Reduced order modeling has received signiﬁcant research attention in recent years. For many problems

of practical interest, the order of the system describing the application must be reduced. An illustrative

example where this is required is the development of feedback control laws for ﬂuid ﬂow conﬁgurations. It

is not uncommon for discretized ﬂow models to describe millions of state variables.1Unfortunately, the

development of systematic feedback control laws from systems of such large dimension is a computationally

intractable problem. For example, if one uses a linear quadratic regulator (LQR) control formulation, roughly

1012 Riccati unknowns need to be calculated for a discretized ﬂow model describing 106states. The Riccati

1This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

1

American Institute of Aeronautics and Astronautics

4th Flow Control Conference<br>

23 - 26 June 2008, Seattle, Washington AIAA 2008-4195

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

unknowns are solutions to a nonlinear matrix equation.2Existing computing power and computational

algorithms are not capable of solving an LQR problem of such large dimension. For dynamical models that

are very large scale, such as those describing ﬂuid ﬂow conﬁgurations, it is apparent that the order of the

system must be reduced prior to control law design.3

Several research eﬀorts have been concerned with, or relied upon, the development of order reduction

strategies that provide reduced order models in a form amenable to state-space feedback control law de-

sign.4–14 For the case of boundary control, the development of such models has been an open problem.

For many applications of practical interest, such as feedback control of the air ﬂow over an airplane wing,

boundary actuation is a requirement. These applications require actuation to be located on the surface if

they are to be implemented in hardware in the physical system. Most reduced modeling eﬀorts have either

been concerned with the case of control action via a body force or have approximated boundary actuation

by a body force on the domain interior. Few techniques have been obtained thus far providing reduced order

models with explicit boundary input that can be used for systematic design of feedback control laws. It is

diﬃcult to extract the boundary control input in the reduced model. When proper orthogonal decomposition

(POD) is used in conjunction with Galerkin projection, boundary conditions are absorbed in the process.

An additional step is required to make the action of the control input explicit in the model.

Recent inroads have allowed for accurate implementation of boundary control in reduced order state-space

models. One approach is to capture the inﬂuence of boundary actuation indirectly. Of note is the work of

Serrani and Kasnako˘glu15 where a basis representing the inﬂuence of boundary actuation is captured by solv-

ing a particular optimization problem in L2. The dimensions of reduced order dynamical models constructed

with this method are typically very small, allowing for implementation of nonlinear control techniques where

small system order is critical. Other methods utilize POD in combination with a weak formulation of the

Galerkin projection.16–19 The advantage of a weak formulation is that boundary conditions appear naturally

when the reduced model is written weakly. Posing the system weakly also reduces regularity requirements

of the solution, reducing errors that result from numerical approximation of high order derivatives in the

Galerkin system. In this paper, we utilize a method16 incorporating diﬀerence approximations in the weak

formulation to construct reduced models with boundary control input appearing explicitly.

Finally, order reduction methods must be ﬂexible so that they are amenable to either simulation or

experimental approaches. Each approach has advantages and disadvantages, but the particular application

at hand often dictates the approach taken. Methods solely suitable for a simulation approach will not likely

lend themselves to implementation in an experimental facility. There is currently a need for order reduction

strategies, suitable for practical control conﬁgurations, that can utilize system data generated by simulation

or experiment. It is in this spirit that the techniques presented in this paper are developed. Simulation data

is used to demonstrate methods in the current paper. The application of these methods to an experimental

laser turret conﬁguration is presented in a companion paper.20

Proper Orthogonal Decomposition

Proper orthogonal decomposition is a popular technique used to construct optimal basis functions attrac-

tive for reduced order modeling. In this work, we use a POD algorithm21 based on the snapshot method22

to construct the low order basis needed for the development of the reduced order model. For the sake of

completeness and clarity in what follows, we brieﬂy describe this technique. A data ensemble of snapshots

{Si(x)}N

i=1 is generated for the system via numerical simulation or experiment, where Nis the total number

of snapshots. Each snapshot consists of instantaneous system data. With the snapshot ensemble in hand,

the N×Ncorrelation matrix Ldeﬁned by

Li,j =hSi, Sji(1)

is constructed. In this work, we utilize the standard L2(Ω) inner product

hSi, Sji=ZΩ

SiS∗

jdx,(2)

where S∗

jdenotes the complex conjugate of Sj, in the construction of L.

2

The eigenvalues {λi}N

i=1 of Lare calculated and sorted in descending order. The ratio

100 ÃPM

i=1 λi

PN

i=1 λi!(3)

is used to determine the number of POD basis functions to construct. The quantity in (3) provides a measure

of the ensemble energy that is captured by a POD basis consisting of Mmodes. By requiring a percentage

of the energy contained in the snapshot ensemble be contained in the basis, the smallest value of Mis

calculated such that the quantity in (3) is greater than or equal to that percentage.

The eigenvectors {vi}M

i=1 corresponding to the Meigenvalues of largest magnitude are calculated. Each

eigenvector is normalized so that

kvik2=1

λi

.(4)

The orthonormal POD basis set {φi(x)}M

i=1 is constructed according to

φi(x) =

N

X

j=1

vi,j Sj(x),(5)

where vi,j is the jth component of vi. With the basis in hand, the system solution w(t, x) is approximated

as a linear combination of POD modes, i.e.,

w(t, x)≈

M

X

i=1

αi(t)φi(x).(6)

Split-POD

Using an energy argument based on (3) is a convenient way to determine the number of POD modes

needed for the reduced model. However, blindly applying (3) to determine the necessary number of modes

is problematic in a boundary control setting. In boundary control applications, it is desired that control

input energy be as small as possible while still satisfying the control objective. For example, in ﬂow control

applications where the control is located on a surface, it is desired that small control inputs yield large

changes in the ﬂow ﬁeld behavior.23 In essence, eﬀects of small control inputs are ampliﬁed by the natural

instabilities and comparatively high energy content of the baseline ﬂow ﬁeld. Simply applying (3) to an

ensemble consisting of baseline and actuated data presents the risk of important structures due to control

input being discarded. These structures are typically of much lower energy content than those associated

with the baseline solution. For these reasons, we extend the POD algorithm described above so that baseline

and control input energy are considered separately. We refer to this method as split-POD. The basis resulting

from this method will consist of modes signiﬁcant to the baseline solution as well as those signiﬁcant from

an actuation standpoint.

The basic idea is to decompose each snapshot in the ensemble into a component in the span of a baseline

POD basis and an orthogonal component. This is done by employing useful properties of orthogonal projec-

tions on Hilbert spaces.24 By considering the case of baseline and actuated data separately, the orthogonal

component is constructed such that it contains new information due to the control input.

An ensemble of solution snapshots is generated for the case of zero control input. From this baseline

snapshot ensemble, a set of POD modes is constructed that contains a high percentage of the baseline energy.

For notational convenience, denote this baseline basis by {ξj}¯

MB

j=1 where ¯

MBis the number of modes. By

employing the energy ratio in (3), we choose ¯

MBso that the baseline basis contains most of the energy

contained in the baseline snapshot ensemble.

With a large set of baseline modes in hand, an ensemble of solution snapshots is generated for the case

of nonzero control input. Denote the actuated ensemble by {Ti}NA

i=1 where NAis the number of actuated

snapshots.

For each snapshot in the actuated ensemble, we determine the component that is in the span of the

baseline basis. In particular, deﬁne bij according to

bij =hTi, ξji,1≤i≤NA,1≤j≤¯

MB.(7)

3

Then, bij is the projection of the ith actuated snapshot Tionto the jth baseline POD mode ξj. In other

words, the product bij ξjis the component of Tithat is in the direction of ξj. The linear combination

P¯

MB

j=1 bij ξjis the component of Tiin the span of the baseline basis.

Deﬁne ¯

Tiaccording to

¯

Ti=Ti−

¯

MB

X

j=1

bij ξj.(8)

Then, ¯

Tiis the component of Tinot contained in the span of the high order baseline basis. As Tiis a solution

snapshot for the case of nonzero control input, ¯

Ticonsists of new information due to the control input. A

second set of POD modes is constructed from the data ensemble {¯

Ti}NA

i=1. Denote this set of “actuator

modes” by {ηi}MA

i=1 , where MAis the number of modes. The energy ratio in (3) is used to determine MA

such that the basis of actuator modes contains an arbitrary amount of the additional energy resulting from

the control input. We have employed the high order baseline basis to extricate new information resulting

from the control. Having done that, the baseline basis is now truncated to contain a more reasonable number

of modes for model development. Denote MBas the number of baseline modes retained after truncation

with corresponding basis set {ξj}MB

j=1

To simplify reduced order modeling via Galerkin projection, it is advantageous to use a basis consisting

of orthonormal modes. We now show that the sets of baseline and actuator modes can be combined into an

overall basis set where all modes are orthonormal.

By construction, baseline modes are orthonormal. Similarly, actuator modes are orthonormal. Consider

the inner product of ξkand ¯

Tifor arbitrary i, k. We see that

¯

Ti, ξk®=*Ti−

¯

MB

X

j=1

bij ξj, ξk+(9)

=hTi, ξki −

¯

MB

X

j=1

bij hξj, ξki.(10)

As the baseline modes are orthonormal, hξj, ξki= 0 unless j=k. For j=k,hξj, ξki= 1. As a result,

hTi, ξki −

¯

MB

X

j=1

bij hξj, ξki=hTi, ξki − bik.(11)

By deﬁnition (7), bik =hTi, ξki. Thus, we have

¯

Ti, ξk®= 0,for arbitrary i, k. (12)

From (5), each actuator mode ηiis a linear combination of the snapshots {¯

Ti}NA

i=1. As a result, (12) in

combination with the linearity of the inner product yields

hηi, ξki= 0,for arbitrary i, k. (13)

Therefore, we have

{ηi} ⊥ {ξj}.(14)

This result allows us to combine the baseline and actuator modes into an overall basis set

{φi}MB+MA

i=1 ={ξ1, ξ2, ..., ξMB, η1, η2, ..., ηMA},(15)

where all modes in the basis are orthonormal. The system solution w(t, x) is still approximated as a linear

combination of modes as in (6). Moreover, separate consideration of baseline and actuated energy allows us

to write this linear combination as

w(t, x)≈

MB

X

j=1

θj(t)ξj(x) +

MA

X

i=1

βi(t)ηi(x),(16)

4

where {θj}MB

j=1 and {βi}MA

i=1 are temporal coeﬃcients for the baseline and actuator basis, respectively. This

allows us to consider the system solution as a baseline component and an additional component induced by

the control input.

Model Problem

We now demonstrate the beneﬁt of careful consideration of actuated dynamics on reduced model accuracy

and subsequent boundary control eﬀectiveness. A distributed parameter system is formulated which models

convective ﬂow over an obstacle. Let Ω1⊆R2be the rectangle given by (a, b]×(c, d). Let Ω2⊆Ω1be the

rectangle given by [a1, a2]×[b1, b2] where a < a1< a2< b and c<b1< b2< d. The problem domain, Ω, is

given by Ω = Ω1\Ω2. In this conﬁguration, Ω2is the obstacle. Dirichlet boundary controls are located on

the obstacle bottom and top, denoted by ΓBand ΓT, respectively.

a a1a2b

c

b1

b2

d

48476

43421ΓB

Γin

Γout

ΓT

Ω

Figure 1: Problem Geometry.

The dynamics of the system are described by the two-dimensional Burgers equation

∂

∂t w(t, x, y ) + ∇ · F(w) = 1

Re ∆w(t, x, y) (17)

for t > 0 and (x, y)∈Ω. In (17), F(w) has the form

F(w) = ·C1

w2(t, x, y)

2C2

w2(t, x, y)

2¸T

,(18)

where C1,C2are nonnegative constants. This equation has a convective nonlinearity like that found in

the Navier-Stokes momentum equation modeling ﬂuid ﬂow.25 The quantity Re, a nonnegative constant, is

analogous to the Reynolds number in the Navier-Stokes momentum equation.

To complete the model of the system, boundary conditions must be speciﬁed as well as an initial condition.

For simplicity, boundary controls are assumed to be separable. With this assumption, we specify conditions

on the obstacle bottom and top of the form

w(t, ΓB) = uB(t)ΨB(x),(19)

w(t, ΓT) = uT(t)ΨT(x).(20)

In (19)-(20), uB(t) and uT(t) are the controls on the bottom and top of the obstacle, respectively. The proﬁle

functions ΨB(x) and ΨT(x) describe the spatial inﬂuence of the controls on the boundary. A parabolic inﬂow

condition is speciﬁed of the form

w(t, Γin) = f(y).(21)

At the outﬂow, a Neumann condition is speciﬁed according to

∂

∂x w(t, Γout ) = 0.(22)

For notational convenience, denote the remaining boundary as ΓU. We require that values be ﬁxed at zero

along ΓUas time evolves. The resulting boundary condition is of the form

w(t, ΓU) = 0.(23)

5

The initial condition of the system is given by

w(0, x, y) = w0(x, y)∈L2(Ω).(24)

As developed in previous work,17 the reduced order state-space model for this system with explicit

boundary control input is of the form

˙α=Aα +Bu +G(α) + F. (25)

Projecting the initial condition w0(x, y ) onto the POD basis results in an initial condition for the reduced

order model of the form

α(0) = α0.(26)

Open-Loop Comparisons

We demonstrate the impact of basis construction technique on the ability of the reduced model to

represent dynamics induced by a boundary control input. Instantaneous snapshots are generated for (17),

(19)-(24) via numerical simulation. A positive parabolic proﬁle with unit maximum amplitude is speciﬁed

for the inlet condition in boundary condition (21). In (18), we set C1= 1 and C2= 0 in order to obtain

solutions that convect from left to right for the positive inlet. In addition, we specify that Re = 300. The

problem domain Ω is discretized, resulting in a uniform grid with spatial step-size h. We utilize a ﬁnite

diﬀerence scheme26 to numerically solve the model problem with and without boundary control input. The

resulting discretized system describes roughly 2,000 states.

We construct reduced order models from snapshot ensembles obtained for two scenarios. In the ﬁrst

scenario, snapshots are generated for the baseline solution and for boundary actuation at a ﬁxed frequency.

For the second case, snapshots are generated for the baseline solution and a more complicated chirp input

where the frequency varies with time. For both scenarios, we compare model agreement resulting from

combining baseline and actuated snapshots into an overall lumped snapshot ensemble to that obtained by

decomposing snapshots into their baseline and actuated components and constructing the split-POD basis

as in (15). In the results that follow, basis construction from a lumped snapshot ensemble is referred to

as the “lumped” method. Constructing the basis by considering baseline and actuated energy separately is

referred to as the “split” method.

Scenario 1

Snapshots are generated for the baseline solution and for the solution arising under periodic boundary

actuation. Inputs speciﬁed are of the form

uB(t) = sin(πt)uT(t) = 0,(27)

uB(t) = 0 uT(t) = sin(πt).(28)

In (27), periodic actuation is done on the bottom of the obstacle while values along the obstacle top are held

at zero. In (28), values along the obstacle bottom are held at zero with periodic actuation occurring at the

top. For each control input listed in (27)-(28), snapshots are taken in increments of ∆t= 0.1 starting from

t= 0 and ending at t= 15. The steady baseline solution is used for the initial condition.

With an ensemble of snapshots in hand, ratio (3) is used to determine the number of basis modes to

construct. Requiring that 99.9% of the ensemble energy be contained in the POD basis results in a lumped

basis consisting of 7 modes. Separate consideration of baseline and actuated energy results in a split basis

consisting of 1 baseline mode and 16 actuator modes.

We now employ linear combination (6) to compare boundary condition agreement between the full order

system and reduced models obtained via the lumped and split-POD bases. By specifying characteristic

functions for the control proﬁles ΨB(x) and ΨT(x) in (19)-(20), we see that

M

X

i=1

αi(t)φi(ΓB)≈w(t, ΓB) = uB(t),(29)

M

X

i=1

αi(t)φi(ΓT)≈w(t, ΓT) = uT(t),(30)

6

We construct the linear combinations on the left in (29)-(30) and compare them to the exact boundary

conditions uB(t) and uT(t) speciﬁed in the full order system. We ﬁrst perform comparisons for boundary

inputs explicitly used during ensemble generation. Results obtained for the baseline solution and for the

solution with periodic boundary actuation of the form sin(πt) are shown in Figure 2. In that ﬁgure, dashed

0 5 10

−1

0

1

t

Lumped

0 5 10

−1

0

1

t

Split

0 5 10

−1

0

1

t0 5 10

−1

0

1

t

Figure 2: Boundary Condition Accuracy.

curves denote the linear combination of POD modes restricted to the boundary. Solid curves denote the

exact full order boundary input. Results obtained for the lumped basis method are plotted on the left.

Split-POD basis results are plotted on the right. As seen in Figure 2, both methods result in very good

agreement between the exact boundary conditions and the linear combination of POD modes restricted to

the boundary. Dashed and solid curves are virtually identical.

In a feedback control setting, dynamics induced by the control are typically not known a priori. The

speciﬁcs of the control law and the dynamics induced by it are not known prior to ensemble creation.

Typically, in the closed-loop system, the boundary input resulting from the control law will be diﬀerent

than the inputs used to generate the snapshot ensemble. As a result, it is useful to compare reduced and

full order model agreement for inputs not speciﬁed during ensemble creation. This provides insight into

the suitability of the reduced model for closed-loop control law design. For these reasons, we now compare

boundary condition agreement between the reduced and full order systems for open-loop inputs that were

not used during ensemble creation. Boundary inputs speciﬁed are of the form

uB(t) = min µt

3,1¶,(31)

uT(t) = sin µ3

2πt¶.(32)

Results obtained for the lumped and split basis methods are shown in Figure 3.

As seen in Figure 3, very good agreement is seen between the reduced and full order systems for the

split method, even though the inputs considered were not speciﬁcally included in the snapshot ensemble.

Condition (32) is reconstructed well using the lumped method. However, the reconstruction of the piecewise

linear input in (31) is much less accurate when the lumped basis is used.

To further compare the lumped and split basis methods and their utility for control law design, we project

the full order solution at each time step onto the lumped and split POD bases. The resulting temporal

coeﬃcients are compared to those predicted by the reduced order models. Boundary inputs speciﬁed are

as in (31)-(32). Results obtained for the ﬁrst 5 temporal coeﬃcients of the lumped method are shown in

Figure 4. The ﬁrst 5 temporal coeﬃcients for the split method are shown in Figure 5. In Figures 4-5,

solid curves denote values of temporal coeﬃcients obtained via the projection. Dashed curves denote the

solution of the reduced order model. Overall, the accuracy of the split method is better, particularly for

temporal coeﬃcients with higher frequency content. Separate consideration of actuated energy during basis

7

0 5 10

−1

0

1

t

Lumped

0 5 10

−1

0

1

t

0 5 10

−1

0

1

t

Split

0 5 10

−1

0

1

t

Figure 3: Boundary Condition Accuracy.

0 2 4 6 8 10

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

t

Lumped

Figure 4: Temporal Coeﬃcient Accuracy for the Lumped Method.

construction results in better representation of dynamics induced by boundary inputs not speciﬁed during

ensemble creation. This is advantageous in a boundary feedback control setting where dynamics induced by

the control are typically not known beforehand.

Scenario 2

It is likely that a snapshot ensemble for inputs at a single frequency results in a POD basis that does

not adequately span the dynamics induced by a feedback control. Feedback controls designed from such a

basis are bound to be ineﬀective when implemented in the full order system. In Scenario 2, we compare the

lumped and split basis methods using a snapshot ensemble generated from chirp inputs of the form

uB(t) = sin ³π¡et¢0.3´uT(t) = 0,(33)

uB(t) = 0 uT(t) = sin ³π¡et¢0.3´.(34)

As seen in Figure 6, an input of this form generates the system response over a range of frequencies. The

resulting POD basis is much more likely to suﬃciently span the unknown dynamics generated by a feedback

control.

8

0 2 4 6 8 10

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

t

Split

Figure 5: Temporal Coeﬃcient Accuracy for the Split Method.

0 5 10 15

−1

−0.5

0

0.5

1

Figure 6: The input function u=sin ³π(et)0.3´.

Instantaneous snapshots are generated for the baseline solution as well as for solutions arising from the

inputs in (33)-(34). For each control input listed in (33)-(34), snapshots are taken in increments of ∆t= 0.1

starting from t= 0 and ending at t= 15. The steady baseline solution is used for the initial condition.

Requiring that 99.9% of the ensemble energy be contained in the POD basis results in a lumped basis

consisting of 7 modes. A split basis comprised of 1 baseline mode and 25 actuator modes are needed when

baseline and actuated energy is considered separately.

As in Scenario 1, we compare boundary condition accuracy of the lumped and split methods for boundary

inputs not speciﬁed during ensemble generation. For the sake of comparison, we use the boundary conditions

given by (31)-(32). Results obtained for the lumped and split basis methods are shown in Figure 7. As seen

in that ﬁgure, very good boundary condition agreement is seen between the reduced and full order systems

for both basis methods. In particular, by comparing Figures 3 and 7, we see that the piecewise linear

boundary condition in (31) is represented much better by the lumped method when inputs (33)-(34) are

used to generate the snapshot ensemble.

As before, we now project the full order solution at each time step onto the lumped and split POD bases.

The resulting temporal coeﬃcients are compared to those predicted by the reduced order models. The

results for the lumped method are shown in Figure 8. Split method results are shown in Figure 9. As seen in

those ﬁgures, the accuracy of the split method is better. For temporal coeﬃcients with signiﬁcant frequency

content, values predicted by the reduced model are virtually identical to those obtained by projecting the full

9

0 5 10

0

0.5

1

t

Lumped

0 5 10

−1

0

1

t

0 5 10

0

0.5

1

t

Split

0 5 10

−1

0

1

t

Figure 7: Boundary Condition Accuracy.

solution onto the split basis, even though the boundary inputs speciﬁed are diﬀerent than those used during

ensemble creation. The split method is better suited with regard to feedback control law design as it is more

capable of accurately representing dynamics that are not explicitly included in the snapshot ensemble.

0 2 4 6 8 10

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

t

Lumped

Figure 8: Temporal Coeﬃcient Accuracy for the Lumped Method.

Closed-Loop Results

We now compare the eﬀectiveness of the lumped and split-POD basis methods in a feedback control

setting. The reduced system given by (25), (26) is linearized, yielding a state-space equation of the form

˙α(t) = Aα +Bu, (35)

α(0) = α0.(36)

We consider the tracking problem for (35)-(36). A ﬁxed reference signal wr ef (x) is speciﬁed for the

full order system. Projecting wref (x) onto the POD basis yields tracking coeﬃcients for the reduced order

model, denoted by αref .

To formulate the tracking control problem, we consider the γ-shifted linear quadratic regulator (LQR)

cost function

J(α0, u) = Z∞

0©(α−αref )TQ(α−αref ) + uTRuªe2γ tdt. (37)

10

0 2 4 6 8 10

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

t

Split

Figure 9: Temporal Coeﬃcient Accuracy for the Split Method.

In (37), Qis a diagonal, symmetric, positive semi-deﬁnite matrix of state weights. R is a diagonal, symmetric,

positive deﬁnite matrix of control weights. The quantity γ, a nonnegative constant, is an additional parameter

that provides added robustness in the control.27,28

We use the LQR formulation in (37) to compare closed-loop results obtained from the lumped and split-

POD basis methods. A snapshot ensemble is constructed containing baseline solution data as well as data

Figure 10: Split Basis Modes.

resulting from nonzero boundary actuation. As it is desired that the POD basis spans unknown dynamics

introduced by the LQR feedback control, boundary inputs speciﬁed during ensemble creation are of the form

uB(t) = Csin ³π¡et¢0.3´uT(t) = 0,(38)

uB(t) = 0 uT(t) = Csin ³π¡et¢0.3´(39)

for C= 1,2,3.

For each control input listed in (38)-(39), snapshots are taken in increments of ∆t= 0.1 starting from

t= 0 and ending at t= 15. The steady baseline solution is speciﬁed for the initial condition. The resulting

snapshot ensemble consists of roughly 900 snapshots. Requiring that 99% of the ensemble energy be contained

in the POD basis results in a lumped basis consisting of 5 modes. A split basis comprised of 1 baseline mode

11

and 20 actuator modes is needed when baseline and actuated energy are considered separately. The ﬁrst 9

modes of the split basis are shown in Figure 10. In that ﬁgure, mode 1 is the baseline mode. Modes 2-9 are

actuator modes.

The tracking LQR problem requires the speciﬁcation of the reference signal αref . In the results that

follow, αref is obtained from the unactuated steady solution for the case Re = 50. This solution is projected

0 0.2 0.4 0.6 0.8

0

0.1

0.2

0.3

0.4

x

y

Figure 11: Tracking Reference Function.

onto the lumped and split bases. The temporal values obtained are used as tracking coeﬃcients in the reduced

order control problem. The reference signal obtained by projecting the steady solution at Re = 50 onto the

split basis is shown in Figure 11. The reference function obtained by projecting onto the lumped basis is

similar. To complete the control formulation, each state in the reduced order model is prescribed a weight

of 2,500. The two boundary controls are each given unit weight. The value speciﬁed for γin (37) is 0.25.

The closed-loop solution of the reduced order model constructed with the split POD basis is shown in Figure

12. By comparing the controlled solution of Figure 12 to the reference function in Figure 11, it is apparent

Figure 12: Closed-Loop Split Model.

that the closed-loop reduced order model satisﬁes the control objective quite well. Separate consideration of

actuated energy in the split POD basis method results in satisfactory tracking of the reference signal.

When the energy ratio in (3) is applied to baseline and actuated data lumped together into an overall

snapshot ensemble, the results are much less favorable. Closed-loop solutions of the reduced order model

constructed with the lumped POD basis are shown in Figure 13. As seen in that ﬁgure, virtually no tracking

is achieved by the reduced order control. Adjusting parameters in the control formulation has little eﬀect

on this result. Increasing the state-weights and the parameter γto 10,000 and 0.75, respectively, does not

signiﬁcantly improve the performance of the control. System information relevant from a control standpoint

12

Figure 13: Closed-Loop Lumped Model.

is discarded when an energy argument is applied during order reduction to the lumped snapshot ensemble

containing baseline and actuated data. The resulting reduced order model does not adequately describe

dynamics induced by the control. Consequently designing a feedback control from such a model results in

very ineﬀective response when the control is applied to the system.

Full Order Validation

To validate the eﬀectiveness of the reduced order control obtained via the split basis method, we utilize a

ﬁxed-point projection algorithm16 to incorporate the reduced order boundary control in the full order model.

The closed-loop solution of the full order system is shown in Figure 14. As seen in that ﬁgure, the reduced

Figure 14: Closed-Loop Response with Split Method Feedback Control.

order control eﬀectively drives the full order plant to the target proﬁle. The full order discretized model

is comprised of roughly 2,000 states. The reduced model obtained via the split basis method describes 21

states. As a result, system dimension is reduced by roughly two orders of magnitude with the resulting

reduced order control being quite eﬀective.

Conclusions

In this paper, a reduced basis construction method was developed allowing for separate consideration

of baseline and actuated dynamics in the reduced modeling process. A prototype initial boundary value

13

problem, governed by the two-dimensional Burgers equation, was formulated to demonstrate the utility of

the method. When actuated energy was considered separately, much better agreement was seen between

open-loop solutions of the reduced and full order systems. Separate consideration of energy induced by

the boundary control resulted in eﬀective feedback control for the reduced and full order systems. When

actuated energy was not explicitly accounted for in the reduced modeling process, the resulting feedback

control was completely ineﬀective when applied to the system.

These results demonstrate the need for separate consideration of baseline and actuated energy in the

reduced modeling process when the resulting model is to be used for feedback control law design. Basis

construction relying on an energy argument applied to a lumped snapshot ensemble containing baseline and

actuated data can result in important control information being discarded during order reduction. This is

particularly the case in a boundary control setting where ensemble energy is typically dominated by that in

the baseline data. Feedback controls developed from the resulting reduced model are likely ineﬀective when

applied to the system. Separate consideration of dynamics induced by boundary control input results in

reduced order controllers that are much more eﬀective when applied to the reduced and full order systems.

In a companion paper,20 we apply these methods in an experimental laser turret application with the

control objective of regularizing the unsteady ﬂow over the turret.

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