1
Nonlinear model-based control of two-phase
flow in risers by feedback linearization
Esmaeil Jahanshahi, Sigurd Skogestad, Esten I. Grøtli
Norwegian University of Science & Technology (NTNU)
9th
IFAC 1
Symposium
on |Nonlinear
Systems
– September
4th 2013,
E. Jahanshahi, S.
Skogestad,
E. I. Grøtli
NonlinearControl
model-based
control
of two-phase
flowToulouse
in risers
2
Outline







Introduction
Motivation
Modeling
Solution 1: State estimation & state feedback
Solution 2: Output linearization
Experimental results
Controllability limitation
2 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
3
Introduction
* figure from Statoil
3 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
4
Slug cycle (stable limit cycle)
Experiments
performed by
the Multiphase
Laboratory,
NTNU
4 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
5
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
5 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
6
Motivation
Pt,s
PC
uz
PT
Problem 1: Nonlinearity
Additional Problem 2: Unstable zero dynamics (RHP-zero) for
topside pressure
6 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
7
Modeling
7 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
8
Modeling: Simplified 4-state model
Choke valve with opening Z
mG1 : mass of gas in the pipeline
mL1 : mass of liquid in the pipeline
mG 2 : mass of gas in the riser
mL 2 : mass of liquid in the riser
x3, P2,VG2, ρG2 , HLT
P0
wmix,out
L3
x1, P1,VG1, ρG1, HL1
x4
wL,in
wG,in
w
L2
h>hc
wG,lp=0
wL,lp
x2
h
L1
θ
hc
State equations (mass conservations law):
mG1
mL1
mG 2
mL 2
 wG ,in  wG ,lp
 wL ,in  wL ,lp
 wG ,lp  wG , out
 wL ,lp  wL ,out
8 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
9
Experiments
P2
Top-side Air to atm.
Valve
Seperator
Riser
3m
P1
safety valve
P3
FT water
Buffer
Tank
FT air
Mixing Point
Pipeline
P4
Subsea Valve
Water
Reservoir
Pump
Water Recycle
9 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
Experiment
10
Bifurcation diagrams
Top pressure
Subsea pressure
Gain = slope
10 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
11
Solution 1: observer & state feedback
uc
K
State
variables
Nonlinear
observer
uc
Pt
PT
11 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
12
High-Gain Observer
zˆ1  f1 ( zˆ )
zˆ2  f 2 ( zˆ )
ˆz  f ( zˆ )  1 ( y  yˆ )
3
3
m

zˆ4  f 4 ( zˆ )
z1 : mass of gas in the pipeline (mgp )
z2 : mass of liquid in the pipeline (mlp )
z3 : pressure at top of the riser (Pr ,t )
z4 : mass of liquid in the riser (mlr )
mgr RTr
Pr ,t 
mLr
M G (Vr 
)
l
12 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
13
State Feedback
t
u (t )   K c ( xˆ (t )  xss )  K i  ( Pˆin ( )  r )d
0
Kc : a linear optimal controller calculated by solving Riccati equation
Ki : a small integral gain (e.g. Ki = 10−3)
13 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
Experiment
14
High-gain observer – top pressure
P1 [kpa gauge]
subsea pressure (estimated by observer)
Open-Loop Stable
40
actual
observer
set-point
30
Open-Loop Unstable
20
0
5
10
15
20
time [min]
25
30
35
P2 [kpa gauge]
top-side pressure (measurement used by observer)
Open-Loop Stable
15
actual
observer
10
5
measurement: topside pressure
valve opening: 20 %
Open-Loop Unstable
0
0
5
10
15
20
time [min]
25
30
35
top-side valve actual position
Zm [%]
60
40
Controller
Off
Controller On
Open-Loop Unstable
20
0
0
Controller Off
Open-Loop Stable
5
10
15
20
time [min]
25
30
35
14 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
15
Fundamental limitation – top pressure
Measuring topside pressure we can stabilize the system only in a limited range
RHP-zero dynamics of top pressure
M S ,min  
i 1
Ms,min
z  pi
RHP-Zeros
RHP-poles
Z=30%
Z=20%
0.1
Z=15%
Z=45%
0.05
z  pi
Z = 20%
Z = 40%
2.1
7.0
Imaginary axis
Np
0.15
Z=95% Z=60%
0
Z=60%
Z=95%
Z=5% Z=95%
-0.05
Z=5%
Z=95%
Z=45%
Z=15%
-0.1
Z=20%
Z=30%
-0.15
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Real axis
0.5
0.6
0.7
0.8
0.9
15 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
Experiment
16
High-gain observer – subsea pressure
P1 [kpa gauge]
subsea pressure (measurement used by observer)
Open-Loop Stable
40
actual
observer
30
Open-Loop Unstable
20
0
2
4
6
8
10
time [min]
P2 [kpa gauge]
top-side pressure (estimated by observer)
Open-Loop Stable
15
actual
observer
10
5
0
0
measurement: topside pressure
valve opening: 20 %
Open-Loop Unstable
2
4
6
8
10
8
10
time [min]
top-side valve actual position
60
Not working ??!
Zm [%]
Controller Off
40
Open-Loop Unstable
20
0
0
Open-Loop Stable
2
4
6
time [min]
16 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
17
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
Pin
f1 ( x)
f 2 ( x)
Prt
17 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
18
Solution 2: feedback linearization
Prt
Nonlinear
controller
uc
PT
PT
18 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
19
Cascaded system
19 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
20
Pipeline subsystem ISS
a) mass of gas in pipeline, mgp
0.032
mgp [kg]
0.0318
0.0316
0.0314
0.0312
0
50
100
150
200
250
300
250
300
250
300
time [sec]
b) mass of liquid in pipeline, mlp
0.832
0.828
0.826
0.824
0
50
100
150
200
time [sec]
c) pressure at riser-base as input, Prb
21
Prb [kPa gauge]
Hypothesis 1. The Pipeline
subsystem with the riser-base
pressure, Prb, as its input is
“input-to-state stable”.
mlp [kg]
0.83
20
19
18
0
50
100
150
200
time [sec]
20 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
21
Stability of cascade system
Riser
Proposition 2. Let hypothesis 1 holds. If the Riser subsystem
becomes globally asymptotically stable under a stabilizing
feedback control, then the pipeline-riser system is globally
asymptotically stable.
Proof : We use conditions for stability of the cascaded
systems as stated by Corollary 10.5.3 of Isidori (1999).
21 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
22
Stabilizing controller for riser subsys.
Riser dynamics:
System equations in y coordinates:
22 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
23
Stabilizing controller for riser subsys.
exist and are continuously differentiable.
is a diffeomorphism on
23 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
24
Stabilizing controller for riser subsys.
System in normal form:
Feedback controller:
(1) Reduces to
. By choosing
, we get
exponentially stable
24 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
25
Stabilizing controller for riser subsys.
If we insert the control law into (2) we get
exponentially fast,
Since
, K1 is bounded and
will remain bounded. This is partial exponential
stabilization of the system with respects to .
(Vorotnikov (1997))
25 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
26
Stabilizing controller for riser subsys.
Final control law by linearizing y1 dynamics, equation (1):
In the same way by linearizing y2 dynamics, equation (2):
Control signal to valve:
y2 is non-minimum phase
26 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
Experiment
27
CV: riser-base pressure (y1), Z=30%
riser-base pressure (controlled variable)
Prb [kPa]
40
open-loop stable
30
set-point
measurement
open-loop unstable
20
10
0
5
10
15
20
t [min]
topside pressure (measurement used by controller)
open-loop stable
Prt [kPa]
15
10
open-loop unstable
5
0
0
5
10
15
20
t [min]
actual valve position (manipulated variable)
100
Zm [%]
open-loop unstable
Controller Off
50
0
Controller On
Controller Off
open-loop stable
0
5
10
15
20
t [min]
27 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
Experiment
28
CV: riser-base pressure (y1), Z=60%
riser-base pressure (controlled variable)
Prb [kPa]
40
open-loop stable
30
open-loop unstable
set-point
measurement
20
10
0
5
10
15
20
t [min]
Prt [kPa]
topside pressure (measurement used by controller)
open-loop stable
15
10
open-loop unstable
5
0
0
5
10
15
Gain:
20
t [min]
actual valve position (manipulated variable)
50
min & max
100
Controller On
Controller Off
50
0
open-loop unstable
open-loop stable
0
5
10
15
Pin [kpa ]
Zm [%]
Controller Off
steady-state
40
30
20
20
10
t [min]
0
10
20
30
40
50
60
70
80
90
100
Z [%]
1
30
28 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
min & max
steady-state
25
Experiment
29
CV: topside pressure (y2), Z=20%
riser-base pressure (measurement used by controller)
Prb [kPa]
40
open-loop stable
30
open-loop unstable
20
0
5
10
15
20
t [min]
Prt [kPa]
topside pressure (controlled variable)
15
open-loop stable
10
open-loop unstable
set-point
measurement
5
0
0
5
10
15
20
t [min]
actual valve position (manipulated variable)
y2 is non-minimum phase
Zm [%]
100
open-loop unstable
Controller Off
50
0
Controller On
Controller Off
open-loop stable
0
5
10
15
20
t [min]
29 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,
30
Conclusions
•
•
•
•
Nonlinear observers work only when measuring topside pressure
This works in a limited range (valve opening of 20%)
Nonlinear observers fail when measuring subsea pressure
New controller (without observer) stabilizes system up to 60%
• World record (on our experiment)! Tie with gain-scheduled IMC
• But cannot bypass fundamental limitations:
-
non-minimum-phase system for topside pressure
Small gain of system for large valve openings
Thank you!
30 E. I. Grøtli | Nonlinear model-based control of two-phase flow in risers
E. Jahanshahi, S. Skogestad,