On Model-Based Feedback Flow Control

Jonathan Epps, Miguel Palaviccini, Louis CattafestaMAE Department, University of FloridaInterdisciplinary Microsystems Group

Florida Center for Advanced Aero-Propulsion

IFFC-2Poitiers, 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 ChoicesRef: Pastoor et al., JFM, 2008

Example: Adaptive Black-Box ModelCavity 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

u= aiui−NA

−1

∑actuationm odes

1 2 3+ u0

baseflow

{ + aiuii=i

NKL

∑PODm odes

1 2 3

standard POD insufficient! Tadm or et al (2010)

1 244 34 4

+ aiuii=NKL+1

NEM

∑non-equilibriumm odes (shift m odes orstability eigenm odes)

1 24 34

daidt

= lij+ lij+

unresolved m odedissipation}

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟aj

j=−NA

NEM

∑

linear term s1 24 4 4 34 4 4

+ qijk + qijk+

non-zeropressure term s}

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟aj

k=−NA

NEM

∑j=−NA

NEM

∑ ak

quadratic term s1 24 4 4 44 34 4 4 4 4

+ gijj=−NA

−1

∑daj

dtboundary control inputs

1 24 34; i =−NAK NEM

Ref: Noack et al., AIAA 2004-2408

Example - Cylinder Wake

Minimal Extended Galerkin POD ModelBased 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= usunstablesteadyN-S sol

{ + a1u1 + a2u2

first 2 POD m odes1 24 34 + a3u3

shift m ode{

us

umean

u3

u2

u1

Example: Cylinder Wake

Resulting Dynamical Systems Model

After projection, phase averaging, & transforming to cylindrical coordinates:

obtain

which has a limit cycle

a1 =rcosΦ, a2 =rsinΦ

&r&Φ&a3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

s r −ba3( )r

w +ga3

ar2 −s 3a3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

a3* =s r

br* =

s 3a3*

a=

s 3s r

ab

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

&r&Φ&a3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

s r −ba3( )r

w +ga3

ar2 −s 3a3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥+

gcos Φ−q( )

−gr

⎛⎝⎜

⎞⎠⎟sin Φ−q( )

0

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

u

Example: Cylinder Wake

Using:

transforms the controlled system to

b1 =rcos Φ−q( ) b2 =rsin Φ−q( )

&b1&b2&a3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥=

s r −ba3( )b1 − w +ga3( )b2w +ga3( )b1 + s r −ba3( )b2

−s 3a3 +a b12 +b2

2( )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥+

g00

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥u

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

&b1&b2&a3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥=

s r −ba3( )b1 − w +ga3( )b2w +ga3( )b1 + s r −ba3( )b2

−s 3a3 +a b12 +b2

2( )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥+

gc

00

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥u

b1 =rcos Φ−q( ) b2 =rsin Φ−q( )

&A&Φ&a3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

s r −ba3( )A

w +ga3

aA2 −s 3a3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥+

gccos Φ−q( )

−gc

r⎛⎝⎜

⎞⎠⎟sin Φ−q( )

0

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

u

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 a1, a2, and a3. 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 reductionUnactuated 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 b1 and b2 “tuned” to match observed transient in simulations w/ sinusoidal forcing at ω

&a1&a2

⎡

⎣⎢⎢

⎤

⎦⎥⎥= s −ar2 −w

w s −ar2⎡⎣⎢⎢

⎤⎦⎥⎥

a1a2

⎡

⎣⎢⎢

⎤

⎦⎥⎥+

b1b2

⎡

⎣⎢⎢

⎤

⎦⎥⎥u

a1 =rcosq, a2 =rsinq

Dynamic Phasor Control

As before, the input is chosen to be u = rc cos(θ – θc) 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 θc and rc 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)

r* =s −ka

Dynamic Phasor Control

Kalman filter, assuming dr/dt = 0, is used to estimate the states

– where η is a p sensor measurement and– L1 > 0, and L2 = ω – L1

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

– Too 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?

0 500 1000 1500 2000 2500 3000 3500138

140

142

144

146

148

150

152

Freqency [Hz]

SP

L [d

B] w

ith P

ref =

20m

Pa

Balanced Truncation(Linear Systems)

Consider a linear (stable) state-space system

Idea for obtaining a reduced-order model:– Change to coordinates in which x1 is “most important” state, x2 “less

important”,…, xn “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: x1 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 influenceon the output

States easily excitedby 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 104 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 oscillationsM=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 inputsx̂

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 #

Mdesign=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 2nd 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. 44th 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. 44th 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.