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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 Symposium on Nonlinear Control Systems – September 4th 2013, Toulouse

2. Outline Introduction Motivation Modeling
Solution 1: State estimation & state feedback
Solution 2: Output linearization
Experimental results
Controllability limitation

3. Introduction
* figure from Statoil

4. Slug cycle (stable limit cycle)
Experiments performed by the Multiphase Laboratory, NTNU

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

6. Motivation Problem 1: NonlinearityPT PC uz
Pt,s
Problem 1: Nonlinearity
Additional Problem 2: Unstable zero dynamics (RHP-zero) for topside pressure

7. Modeling

8. Modeling: Simplified 4-state modelθ 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=0
wL,lp L3
wL,in
wG,in w x2 L1
State equations (mass conservations law):

9. Experiments 3m

10. Bifurcation diagrams Top pressure Subsea pressure Gain = slope
Experiment
Bifurcation diagrams
Top pressure
Subsea pressure
Gain = slope

11. Solution 1: observer & state feedbackPT
Nonlinear observer K
State
variables uc Pt

12. High-Gain Observer

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

14. High-gain observer – top pressure
Experiment
High-gain observer – top pressure
measurement: topside pressure valve opening: 20 %

15. Fundamental limitation – top pressure
Measuring topside pressure we can stabilize the system only in a limited range
RHP-zero dynamics of top pressure
Z = 20%
Z = 40%
Ms,min
2.1
7.0

16. High-gain observer – subsea pressure
Experiment
High-gain observer – subsea pressure
measurement: topside pressure valve opening: 20 %
Not working ??!

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

18. Solution 2: feedback linearizationPT
Nonlinear controller uc
Prt

19. Cascaded system

20. Pipeline subsystem ISS
Hypothesis 1. The Pipeline subsystem with the riser-base pressure, Prb, as its input is “input-to-state stable”.

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 of Isidori (1999).

22. Stabilizing controller for riser subsys.
Riser dynamics:
System equations in y coordinates:

23. Stabilizing controller for riser subsys.
exist and are continuously differentiable.
is a diffeomorphism on

24. Stabilizing controller for riser subsys.
System in normal form:
Feedback controller:
(1) Reduces to
. By choosing
, we get
exponentially stable

25. Stabilizing controller for riser subsys.
If we insert the control law into (2) we get
Since
, K1 is bounded and
exponentially fast,
will remain bounded. This is partial exponential
stabilization of the system with respects to .
(Vorotnikov (1997))

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):
y2 is non-minimum phase
Control signal to valve:

27. CV: riser-base pressure (y1), Z=30%
Experiment
CV: riser-base pressure (y1), Z=30%

28. CV: riser-base pressure (y1), Z=60%
Experiment
CV: riser-base pressure (y1), Z=60%
Gain:

29. CV: topside pressure (y2), Z=20%
Experiment
CV: topside pressure (y2), Z=20%
y2 is non-minimum phase

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!