A POD based non-linear observer for unsteady ﬂows · MAB - Université Bordeaux I, France Marcelo Buffoni AFM - University of Southampton, UK Angelo Iollo MAB - Université Bordeaux

POD Workshop

A POD based non-linear observer

for unsteady flows

Edoardo LombardiMAB - Université Bordeaux I, France

Jessie WellerMAB - Université Bordeaux I, France

Marcelo BuffoniAFM - University of Southampton, UK

Angelo IolloMAB - Université Bordeaux I and INRIA Project MC2, France

Bordeaux. April 1th, 2008 – p. 1

POD Workshop

Summary

1. Motivation

2. Low-dimensional modeling of unsteady flows

(a) Low-order model construction

(b) Low-order model with feedback control construction

3. A non-linear state observer for unsteady flows

(a) Non-linear observer

(b) Resultsi. Two-dimensional case with feedback control: Re = 150

ii. Three-dimensional case: Re = 300

4. Analysis of the capabilities with filtering technique

(a) Filtering technique

(b) Results for three-dimensional case: Re = 300


POD Workshop

Motivations

Low-order models gave satisfactory prediction results for laminar 2D flows around bluffbodies and, in particular, for the configuration considered in this work. (Galletti et al.,JFM, 2004)

Typical control tools cannot be applied to Navier-Stokes equations (high number ofdegrees of freedom in their discretization)

Compute control laws by Reduced Order Models

State estimation: recover the entire flow field from a limited number of flowmeasurements


POD Workshop

Low-order model construction

Discrete instantaneous velocity expanded in terms of empirical eingenmodes:

u(x, t) = u(x)+c(t)uc(x) +PNr

n=1an(t)φn(x)

where c(t) is the feedback control law and u(x) and uc(x) are reference velocityfields and chosen such that the snapshots are equal to zero at inflow, outflow and jetboundaries.

Eigenmodes φn(x) are found by proper orthogonal decomposition (POD) using the“snapshots method” of Sirovich (1987).

Limited number of POD modes, Nr , is used in the representation of velocity fields(snapshots) −→ they are the modes giving the main contribution to the flow energy.


POD Workshop


Galerkin projection of the Navier-Stokes equations over the retained POD modesleading to the low-order model:

ar(t) = Ar + Ckrak(t) − Bksrak(t)as(t)−Er c(t) − Frc2(t) + [Gr − Hkrak(t)]c(t)

ar(0) = (u(x, 0) − u(x)−c(0)uc(x), φr)


POD Workshop




ar(0) = (u(x, 0) − u(x)−c(0)uc(x), φr)

Coefficient Bksr derives directly from the Galerkin projection of the non-linear terms inthe Navier-Stokes equations


POD Workshop




ar(0) = (u(x, 0) − u(x)−c(0)uc(x), φr)

Coefficient Bksr derives directly from the Galerkin projection of the non-linear terms inthe Navier-Stokes equations

System matrices A, C, E, F , G and H are calibrated minimizing

J =R T

0

PNrr=1

“

ar(t) − ˙ar(t)”

2

dt +PNr

r=1α

“

Ar − Ar

”

2

+PNr

r=1

PNrk=1

α“

Ckr − Ckr

”

2

+PNr

r=1α

“

Er − Er

”

2

+PNr

r=1α

“

Fr − Fr

”

2

+PNr

r=1α

“

Gr − Gr

”

2

+PNr

r=1

PNrk=1

α“

Hkr − Hkr

”

2

where α << 1.


POD Workshop

Low-order model construction with feedback actuation

��

��

JETS

SENSORS

Control law can be obtained by feedback, using vertical velocity measurements atpoints xS in cylinder wake

c(t) = Kv(t, xS)


POD Workshop


��

��

JETS

SENSORS


c(t) = Kv(t, xS)

Developing the velocity at measurements points

v(t, xS) = v(xS) + c(t)vc(xs) +PNr

n=1an(t)φn(xs)

⇓

v(t, xS) = v(xS) + Kv(t, xS)vc(xs) +PNr

n=1an(t)φn(xs)


POD Workshop


��

��

JETS

SENSORS


c(t) = Kv(t, xS)

Developing the velocity at measurements points

v(t, xS) = v(xS) + c(t)vc(xs) +PNr

n=1an(t)φn(xs)

⇓

v(t, xS) = v(xS) + Kv(t, xS)vc(xs) +PNr

n=1an(t)φn(xs)

⇒ Low-order model with feedback control in compact form:

ar(t) = A∗r + C∗

krak(t) − B∗

ksrak(t)as(t)

where the matrices A∗r , B∗

ksrand C∗

krare functions of K, v(xS),vc(xs) and φn(xs).


POD Workshop

Non-linear observer

Galerkin representation of the velocity field u(x, t) in terms of Nr empiricaleigenfunctions, Φi(x), obtained by Proper Orthogonal Decomposition (POD)

u(x, t) = u(x) + c(t)uc(x) +

NrX

i=1

ai(t)Φi(x)


POD Workshop

Non-linear observer


u(x, t) = u(x) + c(t)uc(x) +

NrX

i=1

ai(t)Φi(x)

Two simple approaches to estimate coefficients ai(t):


POD Workshop

Non-linear observer


u(x, t) = u(x) + c(t)uc(x) +

NrX

i=1

ai(t)Φi(x)


1. LSQ ⇒ approximate flow measurements in a least square sense(Galletti et al. (2004), Venturi & Karniadakis (2004) and Willcox (2006))

aj(τ) =

NsX

k=1

Υkj (Φ(x)) fk (u(x, τ))


POD Workshop

Non-linear observer


u(x, t) = u(x) + c(t)uc(x) +

NrX

i=1

ai(t)Φi(x)



aj(τ) =

NsX

k=1


2. LSE ⇒ assume that a linear correlation exists between the flow measurementsand the value of the POD modal coefficients

aj(τ) =

NsX

k=1

Λkj

“

aj , fk

”

fk (u(x, τ))


POD Workshop

Non-linear observer


u(x, t) = u(x) + c(t)uc(x) +

NrX

i=1

ai(t)Φi(x)



aj(τ) =

NsX

k=1



aj(τ) =

NsX

k=1

Λkj

“

aj , fk

”

fk (u(x, τ))

Problems with linear estimation (LSQ and LSE) when 3D flows with complicatedunsteady patterns are considered


POD Workshop

Non-linear observer


u(x, t) = u(x) + c(t)uc(x) +

NrX

i=1

ai(t)Φi(x)



aj(τ) =

NsX

k=1



aj(τ) =

NsX

k=1

Λkj

“

aj , fk

”

fk (u(x, τ))

Problems with linear estimation (LSQ and LSE) when 3D flows with complicatedunsteady patterns are considered

Contributions in literature aimed to effective sensor placement and extensions of LSE⇒ QSE ( Schmit & Glauser (2005), Cohen et al. (2004), Cohen et al. (2006), Willcox (2006))


POD Workshop

Non-linear observer

Minimize the sum of the residuals

LSQ case ⇒

α(t) = argmina(t)

NmX

m=1

0

@

NrX

r=1

R2r(a(τm)) +

NrX

r=1

(ar(τm) −

NsX

k=1

Υkrfk (u (τm)))2

1

A

LSE case ⇒

α(t) = argmina(t)

NmX

m=1

0

@

NrX

r=1

R2r(a(τm)) +

NrX

r=1

(ar(τm) −

NsX

k=1

Λkrfk (u (τm)))2

1

A

where Rr(a(τm)) is the residual of low-order model

the method represents a non-linear observer of the flow state (K-LSQ and K-LSE)


POD Workshop

DNS : Computational Domain

Dimensions:L = 1

H/L = 8

Lin/L = 12

Lout/L = 20

Lz/L = 0.6, 2D simulations

Lz/L = 6, 3D simulations

Reynolds numbers based on maxi-mum velocity of incoming profile and“L”


POD Workshop

Observer - Results 2D : POD and ROM set-up

Database≈ 30 snapshots shedding cycle

Re = 150 −→ 205 snapshotsFeedback gain k = 0.3

Model:205 snapshots from t = 0.00 to t = 48.46 −→ ∆t = 48.46

20 modes retained −→ E = 99.7% with a new control law

0 10 20 30 40 50−0.05

0

0.05

c(t)

Non−dimensional time

k =0.3


POD Workshop

Results 2D: Modal coefficient predictionsk = 0.3

(a)0 10 20 30 40 50

−2

−1

0

1

2

3

a1


DNS Model

(b)0 10 20 30 40 50

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

a3


DNS Model

(c)0 10 20 30 40 50

−0.05

0

0.05

0.1

0.15

0.2

a7


DNS Model

(d)0 10 20 30 40 50

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

a14


DNS Model

POD modal coefficients a1,a3,a7 and a14. Projection of the fully resolved Navier-Stokes simulations onto PODmodes (continuous line) vs. the integration of the dynamical system inside the calibration interval, obtainedretaining the first 20 POD modes (circles).


POD Workshop

Results 2D: Modal coefficient predictionsk = 1.3

(a)0 10 20 30 40 50

−3

−2

−1

0

1

2

3

a1


DNS Model

(b)0 10 20 30 40 50

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

a3


DNS Model

(c)0 10 20 30 40 50

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

a7


DNS Model

(d)0 10 20 30 40 50

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

a14


DNS Model

POD modal coefficients a1,a3,a7 and a14. Projection of the fully resolved Navier-Stokes simulations onto PODmodes (continuous line) vs. the integration of the dynamical system with a different feedback gain, obtainedretaining the first 20 POD modes (circles).


POD Workshop

Results 2D: Control law reconstructionk = 1.3

0 10 20 30 40 50−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

c(t)


k =1.3

Projection of the actual control law onto POD modes (continuous line) vs. Reconstructed control law using theintegration of the dynamical system with a different feedback gain, obtained retaining the first 20 POD modes(circles).


POD Workshop

Results 2D: KLSQ modal coefficient predictionsk = 1.3

(a)0 10 20 30 40 50

−3

−2

−1

0

1

2

3K−LSQ

t (sec)

a1

DNS K−LSQ

(b)0 10 20 30 40 50

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4K−LSQ

t (sec)

a3

DNS K−LSQ

(c)0 10 20 30 40 50

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6K−LSQ

t (sec)

a7

DNS K−LSQ

(d)0 10 20 30 40 50

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3K−LSQ

t (sec)

a14

DNS K−LSQ

POD modal coefficients a1,a3,a7 and a14. Projection of the fully resolved Navier-Stokes simulations onto PODmodes (continuous line) vs. the estimation with the K-LSQ approach (using only six velocity sensors), obtainedretaining the first 20 POD modes (circles).


POD Workshop

Results 2D: KLSQ reconstructionk = 1.3

0 10 20 30 40 50−0.3

−0.2

−0.1

0

0.1

0.2

c(t)


k =1.3

Projection of the actual control law onto POD modes (continuous line) vs. Reconstructed control law using the theestimation with the K-LSQ approach (using only six velocity sensors),obtained retaining the first 20 POD modes(circles).

Actual Flow vs. reconstruction (video)


POD Workshop

Considered 3D case for low-order modeling: Re = 300


POD Workshop

Results 3D : POD and ROM set-up

Database≈ 23 snapshots shedding cycle

Re = 300 −→ 1980 snapshots

Model:POD : 151 snapshots from t = 360.23 to t = 412.64 −→ ∆t = 52.41

20 modes retained −→ E = 67.6% outside the database

100 200 300 400 500 600 700

−1

−0.5

0

0.5

1


CL

CL numerical simulation Re = 300

Snapshots inside database PODSnapshots outside database POD


POD Workshop

Results 3D: Modal coefficient predictions

495 500 505 510 515 520 525

−10

−5

0

5

10

a1


DNS Model

495 500 505 510 515 520 525

−1

0

1

2

3

4

a3


DNS Model

495 500 505 510 515 520 525

−2

−1

0

1

2

a11


DNS Model

495 500 505 510 515 520 525

−2

−1

0

1

2

a20


DNS Model

495 500 505 510 515 520 525

−5

0

5

a1


DNS LSQ LSE

495 500 505 510 515 520 525

−2

−1

0

1

2

3

a3


DNS LSQ LSE

495 500 505 510 515 520 525

−2

−1

0

1

2

a11


DNS LSQ LSE

495 500 505 510 515 520 525

−1

−0.5

0

0.5

1

a20


DNS LSQ LSE

495 500 505 510 515 520 525−8

−6

−4

−2

0

2

4

6

a1


DNS KLSQ KLSE

495 500 505 510 515 520 525

−1.5

−1

−0.5

0

0.5

1

a3


DNS KLSQ KLSE

495 500 505 510 515 520 525

−1

−0.5

0

0.5

1

a11


DNS KLSQ KLSE

495 500 505 510 515 520 525−1

−0.5

0

0.5

1

1.5

a20


DNS KLSQ KLSE

Some representative modal coefficients estimated vs. DNS projections.

1st line : POD-ROM ; 2nd line : LSQ/LSE ; 3rd line : KLSQ/KLSE (24 velocity sensors).


POD Workshop

Results 3D : Flow field estimation

(a)

(b)

(c)

Isosurfaces of the velocity components of asnapshot outside the database:

u (left): grey = 0.5dark grey = 1.0

v (center): grey = -0.25dark grey = 0.25

w (right): grey = -0.075dark grey = 0.075

(a) actual snapshot (t = 426.6), (b)snapshot projected on the retained PODmodes,(c) reconstructed snapshot usingthe K-LSE technique

Actual Flow vs. Reconstruction (video)


POD Workshop

Filtering Technique

Major limitation is the ability of the POD modes to adequately represent the flow field.

Filtering techinque

Space average filter:

u∗(xj , t) =

P

p∈IjV (Cp)u(xp, t)

P

p∈IjV (Cp)

where Ij is the ensemble of all the vertex of the neighbouring cells of Cj includeditself.


POD Workshop

Filtering Technique

Nr = 20 Space Average Filter - Reconstructed energy inside and outside database

0 1 5 10 2090

91

92

93

94

95

96

97

98

90.49%

92.26%

95.09%

96.57%

97.89%

Filtering Level

Re

co

nstr

ucte

d E

ne

rgy

0 1 5 10 2065

70

75

80

85

90

67.63%

71.89%

79.59%

83.98%

88.36%

Filtering Level

Re

co

nstr

ucte

d E

ne

rgy


POD Workshop

Filtering Technique

Space Average Filter


POD Workshop

Results : Modal coefficients prediction

500 505 510 515 520

−6

−4

−2

0

2

4

6


a1

DNS Model

500 505 510 515 520

−4

−3

−2

−1

0

1


a3

DNS Model

500 505 510 515 520

−1

−0.5

0

0.5

1


a11

DNS Model

500 505 510 515 520

−1

−0.5

0

0.5

1


a20

DNS Model

500 505 510 515 520−15

−10

−5

0

5

10

15


a1

DNS LSQ

500 505 510 515 520

−6

−4

−2

0

2

4


a3

DNS LSQ

500 505 510 515 520

−10

−5

0

5

10


a11

DNS LSQ

500 505 510 515 520

−2

−1

0

1

2

3


a20

DNS LSQ

500 505 510 515 520

−6

−4

−2

0

2

4

6


a1

DNS KLSQ

500 505 510 515 520

−1.5

−1

−0.5

0

0.5

1


a3

DNS KLSQ

500 505 510 515 520

−1

−0.5

0

0.5

1


a11

DNS KLSQ

500 505 510 515 520

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8


a20

DNS KLSQ

Some representative modal coefficients estimated vs. DNS projections.

1st line : POD-ROM ; 2nd line : LSQ ; 3rd line : KLSQ (24 velocity sensors - filtering level 5).


POD Workshop

Reslts : Flow field estimation

Database e(U ′)% e(V ′)% e(W ′)% e (U)% e(V )% e(W )%

No Filt min 57.48 43.41 95.57 8.30 40.15 93.47

KLSQ 64.67 49.77 102.26 9.35 46.02 99.98

Filt 5 min 49.41 33.56 92.37 6.39 30.91 88.77

KLSQ 58.57 46.23 104.27 7.58 42.57 100.27

Filt 10 min 46.61 29.68 90.83 5.66 27.34 86.23

KLSQ 54.26 40.01 104.96 6.59 36.83 99.81

Mean reconstruction error on the U, V, W components for the total and fluctuating fieldat Re = 300: min is the error using the projection of the DNS velocity fields onto 20

POD modes.

Actual Flow (filtering level 5) vs. Reconstruction (video)


Documents

A POD based non-linear observer for unsteady ﬂows · MAB - Université Bordeaux I, France Marcelo Buffoni AFM - University of Southampton, UK Angelo Iollo MAB - Université Bordeaux