anti-slug control experiments using nonlinear observers

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E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers

Anti-Slug Control Experiments Using Nonlinear Observers

Esmaeil Jahanshahi, Sigurd Skogestad, Esten I. Grøtli

Norwegian University of Science & Technology (NTNU)

American Control Conference - June 17th 2013, Washington, DC

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Outline

Introduction Motivation Modeling Observer design

• Unscented Kalman Filter (UKF)• High-Gain observe• Fast UKF

State-feedback Experimental results Controllability limitation

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Introduction

* figure from Statoil

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Slug cycle (stable limit cycle)

Experiments performed by the Multiphase Laboratory, NTNU

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Introduction Anti-slug solutions• Conventional Solutions:

– Choking (reduces the production)

– Design change (costly) : Full separation, Slug catcher

• Automatic control: The aim is non-oscillatory flow regime together with the maximum possible choke opening to have the maximum production

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PT

PC

uz

Pt,s

10-4

10-3

10-2

10-1

100

101

10-2

10-1

100

101

102

[Rad/s]

Sensitivity and Complementary Sensitivity

|S|

|/Wp|

|T|

|/Wt|

,min1

pNi

Si i

z pM

z p

Objective: using topside pressure for control

Problem 1: Nonlinearity

Additional Problem 2: Unstable zero dynamics (RHP-zero)

MS=5.87, MT=6.46

Motivation

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Real axis

Imag

inar

y ax

is

Z=5% Z=95%Z=5% Z=95%

Z=15%

Z=20%

Z=30%

Z=45%

Z=60% Z=95%

Z=15%

Z=20%

Z=30%

Z=45%

Z=60%Z=95%

RHP-Zeros

RHP-poles

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Solution?!

PTNonlinear observer K

Statevariables

uc

uc

Pt

1. Is this solution applicable for anti-slug control?2. Can observer bypass fundamental limitations?3. Which kind of observer is suitable?

Questions:

• Experiments

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Modeling

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Modeling: Simplified 4-state model

State equations (mass conservations law):

θ

h

L2

hc

wmix,out

x1, P1,VG1, ρG1, HL1

x3, P2,VG2, ρG2 , HLT P0

Choke valve with opening Z

x4

h>hc

wG,lp=0wL,lp

L3

wL,in

wG,in

w

x2

L1

1 , ,G G in G lpm w w

1 , ,L L in L lpm w w

2 , ,G G lp G outm w w

2 , ,L L lp L outm w w

1 : mass of gas in the pipelineGm

1 : mass of liquid in the pipelineLm

2 : massof gas in the riserGm

2 : massof liquid in the riserLm

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Experiments

Pump

BufferTank

WaterReservoir

Seperator

Air to atm.

Mixing Point

safety valveP1

Pipeline

Riser

Subsea Valve

Top-sideValve

Water Recycle

FT water

FT air

P3

P4

P2

3m

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Bifurcation diagrams

Top pressure Subsea pressure

Experiment

Gain = slope

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Observer Design

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1. Unscented Kalman Filter

1 1 1( , )

( , )k k k k

k k k k

x f x u v

y h x u w

1 1 1 1 1

1 1

1

[ ] 0

ˆ ( , )

ˆ

ˆ ˆ[ ]

k k k k k

k k k

k k m

Tk k c k k

X x x c P P

X f X u

x X W

P X W X Q

(1) Prediction step:

Nonlinear plant:

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1. Unscented Kalman Filter (UKF)

[ ] 0

( )

[ ]

[ ]

k k k k k

k k

k k m

Tk k m k k

Tk k c k

X x x c P P

Y h X

Y W

S Y W Y R

C X W Y

(2) Update step:

(3) Correction step:

1k k k

k k k k k

Tk k k k k

K C S

x x K y

P P K S K

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2. High-Gain Observer

1 1

2 2

3 3

4 4

ˆ ˆ( )

ˆ ˆ( )

1ˆ ˆˆ( ) ( )

ˆ ˆ( )

m

z f z

z f z

z f z y y

z f z

1 : mass of gas in the pipeline ( )gpz m

2 : mass of liquid in the pipeline ( )lpz m

3 ,: pressure at top of the riser ( )r tz P

4 : mass of liquid in the riser ( )lrz m

,

( )

g r

LrG r

rr t

l

RT

mM

m

VP

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2. High-Gain Observer

,3

, , ,

( ) r t

r t r t r tgr lr

gr lr

dPf z

dtdP P P

m mdt m m

,

,

2( )

r t

gr lr

grr t

lr lr

P a

m b m

amP

m b m

where

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3. Fast UKF

1 1

2 2

3 3

4 4

ˆ ˆ( )

ˆ ˆ( )

ˆ ˆ( )

ˆ ˆ( )

z f z

z f z

z f z

z f z

Nonlinear model with transformed states:

- Large Qk and small Rk increase the UKF gain

UKF gain:

- Scaling of states and measurement in the model

High-gain Strategy:

This is the high-gain observer without the observer term, therefore we do not need to specify the observer gain manually.

min

min

max

min

0 0 0

0 0 0

0 0 0

0 0 0

k

q

qQ

q

q

ˆ [ ]

[ ]

Tk c k

k Tk m k k

X W YK

Y W Y R

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State Feedback

0

ˆˆ( ) ( ( ) ) ( ( ) )t

c ss i inu t K x t x K P r d

Kc : a linear optimal controller calculated by solving Riccati equationKi : a small integral gain (e.g. Ki = 10−3)

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Experimental Results

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High-gain observer – top pressure

measurement: topside pressurevalve opening: 20 %

Experiment

0 5 10 15 20 25 30 35

20

30

40

time [min]

P1 [k

pa g

auge

]

subsea pressure (estimated by observer)

Open-Loop Stable

Open-Loop Unstable

actualobserverset-point

0 5 10 15 20 25 30 350

5

10

15

time [min]

P2 [k

pa g

auge

]

top-side pressure (measurement used by observer)

Open-Loop Stable

Open-Loop Unstable

actualobserver

0 5 10 15 20 25 30 350

20

40

60

ControllerOff

Controller On Controller Off

Open-Loop Stable

Open-Loop Unstable

time [min]

Zm

[%]

top-side valve actual position

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Fast UKF – top pressure


Experiment

0 5 10 15 20 25 30 35

20

30

40

time [min]

P1 [k

pa g

auge

]

subsea pressure (estimated by observer)

Open-Loop Stable

Open-Loop Unstable


0 5 10 15 20 25 30 350

5

10

15

time [min]

P2 [k

pa g

auge

]

top-side pressure (measurement used by observer)

Open-Loop Stable

Open-Loop Unstable

actualobserver

0 5 10 15 20 25 30 350

20

40

60

ControllerOff

Controller On Controller Off

Open-Loop Stable

Open-Loop Unstable

time [min]

Zm

[%]


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High-gain observer – subsea pressure


Experiment

0 2 4 6 8 10

20

30

40 Open-Loop Stable

Open-Loop Unstable

time [min]

P1 [k

pa g

auge

]

subsea pressure (measurement used by observer)

actualobserver

0 2 4 6 8 100

5

10

15 Open-Loop Stable

Open-Loop Unstable

time [min]

P2 [k

pa g

auge

]

top-side pressure (estimated by observer)

actualobserver

0 2 4 6 8 100

20

40

60

Controller Off

Open-Loop Stable

Open-Loop Unstable

time [min]

Zm

[%]


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PI Controller – subsea pressure

measurement: subsea pressurevalve opening: 40 %

Experiment

0 5 10 15 20 25 30 3510

20

30

40

time [min]

P1 [k

pa g

auge

]

subsea pressure (controlled variable)

Open-Loop Stable

Open-Loop Unstable

measurementset-point

0 5 10 15 20 25 30 35-5

0

5

10

15 Open-Loop Stable

Open-Loop Unstable

top-side pressure

time [min]

P2 [k

pa g

auge

]

0 5 10 15 20 25 30 350

20

40

60

80

ControllerOff

Controller On

Controller Off

Open-Loop Stable

Open-Loop Unstable

time [min]

Zm

[%]


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Linear observer (KF) – subsea pressure

measurement: subsea pressurevalve opening: 40 %

Experiment

0 5 10 15 20 25 30 3510

20

30

40

time [min]

P1 [k

pa g

auge

]

subsea pressure (measurement used by observer)

Open-Loop Stable

Open-Loop Unstable


0 5 10 15 20 25 30 35-5

0

5

10

15

time [min]

P2 [k

pa g

auge

]

top-side pressure (estimated by observer)

Open-Loop Stable

Open-Loop Unstable

actualobserver

0 5 10 15 20 25 30 350

20

40

60

80

ControllerOff

Controller On

Controller Off

Open-Loop Stable

Open-Loop Unstable

time [min]

Zm

[%]


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Summary of experiments

Method \ CV Subsea pressure Top Pressure

Linear Controllers (PI, H∞) Working Not Working

Fast Linear Observer Working Not Working

Fast Nonlinear Observer Not Working??!* Working

Slow Nonlinear Observer Not Robust* Not Robust*

Max. Valve 40% 20%

Stabilizing Control

Experiment

* Estimation works (open-loop), but slow* Estimation also not working

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Chain of Integrators

• Fast nonlinear observer using subsea pressure: Not Working??!• Fast nonlinear observer (High-gain) acts like a differentiator• Pipeline-riser system is a chain of integrator• Measuring top pressure and estimating subsea pressure is differentiating• Measuring subsea pressure and estimating top pressure is integrating

2 ( )f x1( )f xrtP

inP

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Controllability limitation – top pressure

,min1

pNi

Si i

z pM

z p

Z = 20% Z = 40%

Ms,min 2.1 7.0

Measuring topside pressure we can stabilize the system only in a limited range

RHP-zero dynamics of top pressure

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Real axis

Imag

inar

y ax

is

Z=5% Z=95%Z=5% Z=95%

Z=15%

Z=20%

Z=30%

Z=45%

Z=60% Z=95%

Z=15%

Z=20%

Z=30%

Z=45%

Z=60%Z=95%

RHP-Zeros

RHP-poles

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Conclusions

• Nonlinear observers work only when measuring topside pressure• This works in a limited range (valve opening)• A fast observer is needed for stabilizing control• Fast nonlinear observers fail when measuring subsea pressure• Observer can counteract nonlinearity• But cannot bypass fundamental limitation (non-minimum-phase system)

Thank you!

anti-slug control experiments using nonlinear observers

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