On Model-Based Feedback Flow Control

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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.