Upload
zared
View
50
Download
0
Embed Size (px)
DESCRIPTION
Anti-Slug Control Experiments Using Nonlinear Observers Esmaeil Jahanshahi, Sigurd Skogestad, Esten I. Grøtli Norwegian University of Science & Technology (NTNU). American Control Conference - June 17 th 2013, Washington, DC. Outline. Introduction Motivation Modeling Observer design - PowerPoint PPT Presentation
Citation preview
1
1
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
2
2
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
Outline
Introduction Motivation Modeling Observer design
• Unscented Kalman Filter (UKF)• High-Gain observe• Fast UKF
State-feedback Experimental results Controllability limitation
3
3
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
Introduction
* figure from Statoil
4
4
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
Slug cycle (stable limit cycle)
Experiments performed by the Multiphase Laboratory, NTNU
5
5
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
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
6
6
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
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
7
7
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
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
8
8
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
Modeling
9
9
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
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
10
10
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
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
11
11
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
Bifurcation diagrams
Top pressure Subsea pressure
Experiment
Gain = slope
12
12
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
Observer Design
13
13
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
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:
14
14
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
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
15
15
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
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
16
16
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
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
17
17
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
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
q
q
ˆ [ ]
[ ]
Tk c k
k Tk m k k
X W YK
Y W Y R
18
18
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
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)
19
19
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
Experimental Results
20
20
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
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
21
21
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
Fast UKF – 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
22
22
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
High-gain observer – subsea pressure
measurement: topside pressurevalve opening: 20 %
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
[%]
top-side valve actual position
23
23
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
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
[%]
top-side valve actual position
24
24
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
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
actualobserverset-point
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
[%]
top-side valve actual position
25
25
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
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
26
26
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
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
27
27
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
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
28
28
E. Jahanshahi, S. Skogestad, E. I. Grøtli | Anti-Slug Control Experiments Using Nonlinear Observers
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!