On Model

Based
Feedback Flow Control
Jonathan Epps, Miguel Palaviccini, Louis Cattafesta
MAE Department, University of Florida
Interdisciplinary Microsystems Group
Florida Center for Advanced Aero

Propulsion
IFFC

2
Poitiers, France
December 8

10, 2010
Supported by AFOSR, NSF, and FCAAP
Outline
Choices for Feedback Flow Control Methods
Desirable Model Features
POD

Galerkin Model Shortcomings
Extensions to POD

Galerkin Models
–
Example I
–
2

D cylinder wake (Re=100)

Nonlinear model and controllers
–
Example II
–
compressible cavity oscillations

Dynamic phasor control
Limit

cycle oscillations
Lightly

damped, linear, stable oscillations?
–
Balanced models and the connection between
theory and experiments
Outlook
Feedback Flow Control Choices
Ref: Pastoor et al., JFM, 2008
Example: Adaptive Black

Box Model
Cavity Oscillations
Control works but lacks physical insight and basis
–
Ref:
Kegerise
et al., JSV, 2007
M=0.275
Overview
Model

based feedback flow control techniques require
an analytically tractable plant model.
–
Motivates development of minimal

order flow models
POD models are useful for flow control but…
–
Exhibit inherent limitations that must be circumvented
Various extensions have been proposed
–
Focus here is on those most amenable to experimental
implementation
POD
–
Limitations & Extensions
for Control
“Standard” POD
method has limitations
–
How
to respect effects of boundary conditions? Pressure gradient?
–
When actuation is introduced, flow structures change.
How do we
account for this?
–
Low

energy features (e.g
.,
acoustic
feedback)
can be important to
the dynamics. How
do we account
for these?
Some recent extensions
–
Traveling POD
: shift reference frame for traveling waves
–
Mode

interpolation techniques
–
Double POD
–
Shift
modes
: add additional
mode(s
)
to capture transient
–
Phenomenological models
: instead of Galerkin projection, base
models on physical
intuition
–
Balanced
truncation
: use adjoint simulations to weight modes
according to dynamical importance
Model Requirements for
Control Synthesis
(a) natural flow (I) as initial condition,
(b) actuated flow (II) not far from the desired controlled flow,
(c) natural transient from (II) to (I) when actuation is turned off,
(d) actuated transient from (I) to (II),
(e) suitability of the model for control design,
(f) possibility of observer design from sensor signals,
(g) implementable in experiments
Ref: Noack et al., AIAA 2004

2408 and JFM 2003.
Model should
describe
dynamics near
I and II
Generalized Galerkin System
Generalized Galerkin approximation
Leads to generalized system model via Galerkin projection
Objective: obtain a minimal Galerkin model suitable for control
Ref: Noack et al
.
, AIAA 2004

2408
Example

Cylinder Wake
Minimal Extended Galerkin POD Model
Based on 3

mode Galerkin POD model with a shift mode for cylinder wake
(Noack et al., 2003, Tadmor et al. 2004).
Galerkin approximation is
u
s
u
mean
u
3
u
2
u
1
Example: Cylinder Wake
Resulting Dynamical Systems Model
After projection, phase averaging, & transforming to cylindrical coordinates:
obtain
which has a limit cycle
Example: Cylinder Wake
Control of the Cylinder Wake (Tadmor et al., 2004; King et al., 2005)
Adding actuation (transverse oscillation velocity w/ appropriate phase) to the
unforced system model yields
Example: Cylinder Wake
Using:
transforms the controlled system to
Example: Cylinder Wake
Key Dynamical Aspects (Tadmor et al., 2004)
Amplitude Dynamics and Inertial Manifold
of the natural system provide insight
(actuation amplitude
g
assumed constant)
Example: Cylinder Wake
Lessons Learned
Qualitatively, the model does a good job predicting the salient
features of the controlled flow, including the existence of globally
stable and unstable limit cycles and a lower bound on the
reduction of fluctuation energy for the given control policy.
The low order model is
quantitatively
accurate near the open

loop
limit cycle, but diverges rapidly as the vortices are suppressed and
the base flow changes.
Techniques suggested for
a
posteriori
corrections to the model
parameters via nonlinear model estimation.
Additional model extensions required.
Nonlinear Control Approaches
King et al. (2005) developed several nonlinear controllers based on
the “Dynamic Phasor” model recast in the following form
–
Also described in Rowley and Juttijudata (2005)
Energy

Based Controller
Dynamics Phasor Models: Cylinder Wake (King et al., 2005)
A simple
energy

based
controller
was developed by
averaging the control influence on
the oscillation amplitude over half
a period when cos(Φ

θ)>0.
This mean influence is inserted
into the dynamics, and the
amplitude A is then forced to
decay with a rate of
–
k
.
In general, the control is:
Note: This assumes we know
a
1
, a
2
, and a
3
. In real

time
experiments, we will need to
estimate these.
Energy

Based Controller
Dynamics Phasor Models: Cylinder Wake (King et al., 2005)
Energy Based Control Results k=0.0075
Other Nonlinear Controllers
Damping controller that preserves the natural oscillation frequency ω
Input

output linearization
Lyapunov controller based on
Backstepping controller
State feedback control based on a linear parametrically varying model
Opposition controller
Have similar
effects as the
energy

based
controller, but the
commanded input
is undesirable.
DNS of the seven controllers were performed
Simplified Galerkin model no longer accurately reflects the flow for
peak suppression and performance suffers
–
Note: passive splitter plate reduces fluctuations by ~60%
DNS of Nonlinear Controllers
Energy

based control
–
max reduction
Unactuated flow

contours of
u
Application to Cavity Oscillations?
Rowley and Juttijudata (2005) postulated a similar, two

state,
dynamical system model for cavity oscillations.
Parameters σ, α, ω “tuned” to match unforced oscillator
–
Parameter estimation methods would be used in experiments
Parameters b
1
and b
2
“tuned” to match observed transient in
simulations w/ sinusoidal forcing at ω
Dynamic Phasor Control
As before, the input is chosen to be
u = r
c
cos(θ
–
θ
c
)
and the
model is averaged over one period (Krylov

Bogoliubov)
–
Assumes r is changing slowly, dθ/dt~ω, and inputs u are
small
An appropriate choice of
θ
c
and
r
c
yields the final model,
which, for
0 < κ < σ
, has a periodic orbit with amplitude
Idea is to reduce amplitude of oscillations but stay within the
range of validity of the model (κ cannot be too large)
Dynamic Phasor Control
Kalman filter, assuming
dr/dt = 0,
is used to estimate the
states
–
where
η
is a p sensor measurement and
–
L
1
> 0
, and
L
2
= ω
–
L
1
2
/ 2ω (chosen for stable, critically
damped observer dynamics)
Both the model and the state observer will only work well
when the oscillations are near
ω,
which depends on the
Mach number
–
Could also estimate ω (as in Pastoor
et al.
2008)
Dynamic Phasor Control
Model

based control works well for the design Mach number
Oscillations are completely eliminated for
1 ≤ κ / σ ≤ 3
–
T
oo high a value of
κ
causes the system to leave the region
of validity of the model, resulting in increased oscillations.
Dynamic Phasor Control
Performance is very sensitive to Mach number (design M=0.6)
M=0.55
M=0.65
M=0.7
Dynamic Phasor Control
Can this approach be applied in experiments?
Issues
Need to account for strong influence of Mach number?
Adaptive parameter estimation and control?
Only applies to limit

cycle oscillation
Cavity oscillations often lightly

damped, stable, linear
What can we do in this case?
Balanced
Truncation
(Linear Systems)
Consider
a linear
(stable) state

space system
Idea for obtaining a reduced

order model:
–
Change to coordinates in which
x
1
is “most important” state,
x
2
“less
important”,…,
x
n
“least important”
–
Then throw out (truncate) the least important states
How to define “most important” states?
–
Two important concepts: controllability and
observability
–
Most controllable states are ones easily excited by an input
–
Most observable states are ones that have a large effect on output
–
Balance these concepts:
x
1
is most controllable and most
observable,
etc.
Typically produces
better control

oriented models
than POD/Galerkin
Overview of balanced truncation
What are you interested
in capturing?
States that have large
influence
on the output
States easily excited
by an input
Hankel
singular
values
Balanced
POD
Can use standard & adjoint simulations to compute approximate
balanced truncation with cost similar to POD (“Balanced POD”)
–
Rowley, Int. J.
Bifurc
. Chaos, 2005
Advantages
–
Explicitly
incorporates effects of
actuators
and
sensors
–
Considering
observability effectively weights the
dynamical
importance
of various modes:
low

energy modes that affect the
dynamics (e.g., acoustic waves) will be strongly observable, and will not be
truncated
–
Guaranteed error bounds for linear systems, close to best
achievable by any model
Disadvantages
–
Works only for linear systems

Extensions available for nonlinear
systems
–
Computation intractable for systems with more than about 10
4
states
Not applicable to experiments!
Application to Experimental Control
of Cavity Oscillations
Limit cycle or lightly damped stable oscillations?
–
Assuming Gaussian input disturbances, then output puff is
Gaussian for a linear system
M=0.45
M=0.34
M=0.34
M=0.45: lightly damped oscillations
M=0.34: self

sustained oscillations
M=0.34: w/ control
Rowley et al. (2006)
Application to Cavity Oscillations

Eigensystem Realization Algorithm
Construct Hankel matrix H(0) from
input/output data
H(0) = observability x controllability
SVD and truncate at order n
Results in balanced model
Calculate A B C D to achieve a
balanced realization
•
Illingworth et al., J Sound Vib, (2010),
doi:10.1016/j.jsv.2010.10.030
•
Cattafesta et al., AIAA

97

1804
Application to Cavity Oscillations

2D DNS
LQG Controller Design AFTER dynamic phasor control!
Includes effects of disturbances and noise
Control feedback law
Quadratic cost function
Use Kalman filter to
estimate unknown
states
Weight matrices used to penalize large
system states and large control inputs
Application to Cavity Oscillations
System identification using ERA compared to spectral analysis
–
140 states nearly matches spectral analysis
–
8 states models frequency range of Rossiter modes
Application to Cavity Oscillations
Results indicate excellent suppression that is robust to Mach #
M
design
=0.6
M=0.5
M=0.7
OL
CL
OL
CL
OL
CL
Present Experiments at UF
Bandpass filter around
Rossiter modes @ M=0.3
ERA
n
=20
Reasonable comparison
versus conventional
frequency response
Next step is estimator
and then LQG
Present Experiments at UF
Outlook
Nonlinear (limit cycle) reduced

order models can be obtained via
extensions to standard POD that are suitable for control design.
–
Dynamic phasor control (appropriate amplitude and phase) via
physically motivated and formal methods can suppress the
oscillations. Lower u is associated with energy

based control.
–
These reduced

order models must “respect” the range of validity of
the model.
–
If the control is too aggressive, the model will no longer be valid and
the control performance will suffer.
–
Suggests additional model extensions and/or adaptive parameter
estimation or…
Dynamic phasor control of nonlinear oscillator produces a stable,
lightly damped oscillator system and then linear control is applied.
–
Robust, linear control approaches possible provided a “balanced”
reduced

order model can be obtained.

Balanced POD for simulations (requires adjoint)

Balanced state

space realization via ERA for experiments
References
B. Noack, K.
Afanasiev
, M.
Morzynski
, G.
Tadmore
, and F. Thiele, “A hierarchy of low

dimensional models
for the transient and post

transient
cyclinder
wake.”
J. Fluid Mech.
, vol. 497, pp. 335

363, 2003.
G. Tadmor, B. Noack, M.
Morzynski
, and S. Siegel, “Low

Dimensional Models For Feedback Flow Control.
Part II: Control Design and Dynamic Estimation.”
Proc. AIAA 2
nd
Flow Control Conference.
, pp. 2004

2409,
2004.
R. King, M.
Seibold
, O. Lehman, B. Noack, M.
Morzynski
, and G. Tadmor, “Nonlinear Flow Control Based on
a Low Dimensional Model of Fluid Flow.” In
Control and Observer Design for Nonlinear Finite and Infinite
Dimensional Systems
(ed. T.
Meurer
et al.
). Lecture Notes in Control and Information Sciences, vol. 322,
pp. 369

386, 2005.
C. Rowley and V.
Juttijudata
, “Model

based Control and Estimation of Cavity Flow Oscillations.”
Proc. 44
th
IEEE Conference on Decision and Control.
, December 2005.
C. Rowley, D. Williams, T.
Colonus
, R.
Muray
, and D.
Macmynowski
V., “Linear Models for Control of Cavity
Oscillations. Journal of Fluid Mechanics, vol. 547, pp.317

330, 2006.
M.
Morzynski
, W.
Stankiewicz
, B. Noack, R. King, F. Thiele, and G. Tadmor, “Continuous Mode Interpolation
for Control

Oriented Models of Fluid Flow.” In
Active Flow Control
(ed. R. King). Notes on Numerical Fluid
Mechanics and Multidisciplinary Design., vol. 95, pp. 260

278, 2007.
S.
Siegal
, K. Cohen, J. Seidel, and T. McLaughlin, “State Estimation of Transient Flow Fields Using Double
Proper Orthogonal Decomposition (DPOD).” In
Active Flow Control
(ed. R. King). Notes on Numerical Fluid
Mechanics and Multidisciplinary Design., vol. 95, pp. 105

118, 2007.
L. Henning and R. King, “Drag Reduction by Closed

Loop Control of a Separated Flow Over a Bluff Body
with a Blunt Trailing Edge.”
Proc. 44
th
IEEE Conference on Decision and Control.
, pp. 494

499, Dec. 2005.
S. Illingworth, A.
Morgans
, and C. Rowley. “Feedback Control of Flow Resonances Using Balanced
Reduced

Order Models.” Journal of Sound and Vibration, to appear.
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