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A Snapshot Decomposition Method for Reduced Order Modeling and Boundary Feedback Control (Postprint)

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  • Syracuse University, College of Engineering and Computer Science

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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.
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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
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A SNAPSHOT DECOMPOSITION METHOD FOR REDUCED ORDER
MODELING AND BOUNDARY FEEDBACK CONTROL (POSTPRINT)
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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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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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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
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=Z
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,1iNA,1j¯
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 Ω1R2be the rectangle given by (a, b]×(c, d). Let Ω21be 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\2. 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
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(tB(x),(19)
w(t, ΓT) = uT(tT(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=+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)φiB)w(t, ΓB) = uB(t),(29)
M
X
i=1
αi(t)φiT)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
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) = +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©(ααref )TQ(ααref ) + 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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15
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... Meanwhile, an expanding cluster of stabilization and closure algorithms has formed through the past decade in the projection-based ROMs to restore the dynamically important structures that are truncated through the standard POD process [15,16,[31][32][33]. The split-POD method on the other hand, is originated in flow control applications in order to maintain the observable modes that can be used in the control system to excite the desired outputs with a small amount of actuation energy [17,18]. Since the most observable and the most controllable directions are not in general aligned in a fluid system, these observable modes are typically truncated in POD in favor of the most energetic structures. ...
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Technological advances in sensors, actuators, on-board computational capability, modeling and control sciences have offered a possibility of seriously considering closed-loop flow control for practical applications. We can now attempt addressing problems that have over the years not been effectively solved using passive means and/or open-loop techniques. The main strategies to closed-loop control are a model-independent approach, a full-order optimal control approach based on the Navier Stokes equations and a reduced order model strategy. This effort emphasizes the methodology based on the low-dimensional, proper orthogonal decomposition method applied to the problem concerning the suppression of the von Kármán vortex-street in the wake of a circular cylinder. Focus is on the validity of the low-dimensional model, selection of the important modes that need representation, incorporation of ensembles of snapshots that reflect vital transient phenomena, selection of sensor placement and number, and linear stochastic estimation for mapping of sensor data onto modal information. Furthermore, additional issues surveyed include observability, controllability and stability of the closed-loop systems based on low-dimensional models. Examples based on computational and experimental studies on the cylinder wake benchmark are presented to illuminate some of the important issues. Finally, the paper summarizes some of the important lessons learned and notes the open issues that need to be addressed in future work.
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Proper orthogonal decomposition (POD) Galerkin models are typically obtained from a single reference, such as an attractor. The POD model provides a very efficient repre-sentation of the reference but is often incapable to handle transient dynamics and other changes in flow conditions. These shortcomings are detrimental in feedback flow control applications. A novel concept of tuned Galerkin models is suggested, by which the global model interpolates a succession of similar-structure local models. The tuned model covers a controlled transient manifold, compensating for the gradual deformation of dominant flow structures, along such transients. The model is an enabler for both improved tracking performance, as well as for optimized control hardware placement, taking into account the entire dynamic range of interest. These concepts are demonstrated in the benchmark of stabilization of the wake flow behind a circular cylinder.
Conference Paper
A reduced-order model (ROM) is constructed to measure and control aerodynamic forces on au airfoil. Proper orthogonal decomposition (POD) yields a finite series representing the flow velocity. The series is comprised of products of time-invariant basis functions and time-dependent coefficients. Projection of the weak form of the momentum-conservation equation for incompressible flow onto the set of basis functions produces a low-dimensional set of ordinary differential equations, which is the plant with time-dependent POD coefficients as the state variables. Control input consists of surface jet velocity and is represented explicitly in the plant. Control output is aerodynamic force - both pressure-based and viscous. System output is a direct function of the state variables in the measurement equation. Tests of the plant and output measurement compare data directly from CFD simulations with results from the ROM. The test problem consists of flow past a NACA 4412 airfoil at Mach 0.1, eight degrees angle of attack, and a Reynolds number of 1000 (based on chord length). At these conditions, a separation bubble above the downstream one-third of the airfoil produces periodic vortex shedding into the wake and oscillatory aerodynamic forces. The ROM accurately reflects system dynamics with as few as two POD modes - with and without surface jets. Computation of pressure-based force from the state variables involves second derivatives that produce errors in the mean value of the output measurement. The errors do not exist in the viscous component of force, which accurately reflects the evolution of system output.
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In this paper, we consider the problem of controlling a system governed by a two-dimensional nonlinear partial differential equation. Motivation for the problem is the development of control methodologies for fluid flow, where the dynamics of the system are governed by the nonlinear Navier-Stokes equations. An initial boundary value problem described by the two-dimensional Burgers' equation is formulated to model a right-travelling shock over an obstacle. We focus on implementing feedback control via Dirichlet boundary conditions on the obstacle. We formulate a control problem for the system model, and examine two different methods of finding the control. The first method involves obtaining the solution of an algebraic Riccati equation. The second method involves obtaining a steady-state solution of the Chandrasekhar equations. Numerical approximations are developed to numerically simulate solutions of the problem with and without control. Numerical examples are presented to illustrate the efficacy, as well as the shortcomings, of the control method. Additionally, the influence of boundary condition on the functional gains, and the resulting controls, is demonstrated through numerical examples. Avenues of future work are presented.
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A distributed parameter model of a nonlinear two-dimensional convective system is formulated. Proper orthogonal decomposition is used to construct a reduced order state-space model of the system. Open-loop simulatious of the full and reduced models are compared to demonstrate the validity of the reduced model. A linear quadratic regulator control problem is formulated for the system under boundary control. Control effectiveness is demonstrated in reduced and full order simulations.