CONTROLLABILITY ANALYSIS OF AN

UNSTABLE, NON-MINIMUM PHASE

PROCESS

Espen Storkaas Sigurd Skogestad ,1

Department of Chemical Engineering, Norwegian

University of Science and Technology, Trondheim, Norway

Abstract: The choice of manipulated inputs and measured outputs can be criticalfor the successful design of a control system. This is especially the case whendesigning stabilizing controllers for unstable, non-minimum phase systems, asunstable poles combined with unstable zeros and time delay can render stabilizingcontrol very hard. Since the presence and location of unstable zeros and timedelay usually depends on the choice of measurements and control actuators, athorough analysis prior to the actual controller design is important. In this paper,a controllability analysis is performed on a pipeline-riser system with multiphaseflow, where the design objective is to stabilize an unstable operating point toremove riser slugging. Copyright c2005 IFAC

Keywords: Controllability, Stabilizing Control, Unstable, Non-minimum phase,Feedback control

1. INTRODUCTION

When designing control systems, there are a num-ber of decisions to be made prior to the actualdesign of the controllers. These decisions includewhat variables to measure, which variables tomanipulate and how the manipulated and mea-sured variables should be paired. In addition, in-formation as to whether acceptable performanceis possible without redesigning the process wouldbe valuable at an early stage in a design process.These questions are especially important whendesigning stabilizing controllers for unstable, non-minimum phase systems, where the choice of mea-surements and manipulated variables can be thedeciding factor for a successful design.

A controllability analysis is an investigation intoa systems fundamental performance limitationswith respect to control. Skogestad and Postleth-

1 Author to whom correspondence should be addressed:skoge@chemeng.ntnu.no

waite (1996) defines controllability as a systemsability to achieve acceptable control performance;that is to keep the systems outputs (y) withinspecified bounds or displacement from the refer-ences (r), in spite of unknown but bounded varia-tions, such as disturbances (d) and plant changes,using available inputs (u) and available measure-ments (ym or dm). By performing a controllabil-ity analysis for the system with all the possiblecandidates for measurements and manipulativevariables, the set of variables that results in themost controllable system can be chosen.

In order to illustrate the importance of a control-lability analysis, an example from the offshore oilindustry is studied. Multiphase flow of oil andgas in pipeline-riser system can give rise to anoscillatory flow regime called riser slugging. Riserslugging can be removed by stabilizing an unstableoperating point existing at the same boundaryconditions using feedback control. The pipeline-riser system is very interesting, as it is an unstable

system which, depending on the measured input,contains both unstable zeros and time delay. Inaddition, the unstable poles constitute a complexpair, something which adds additional complexityto the analysis.

The pipeline-riser system is described in section2. In section 3, the system-theoretical backgroundfor the controllability analysis is given, and in sec-tion 4, the results from the controllability analysisis discussed.

2. SYSTEM DESCRIPTION

2.1 Riser Slugging Phenomenon

Most of the newly discovered, smaller oil fieldson the Norwegian continental shelf are beingtied in to existing infrastructure by transportingthe untreated wellstream in multiphase transportpipelines from the wellhead clusters to the pro-duction platform. Riser slugging is a flow patternin multiphase flow characterized by varying orirregular flows and surges of liquid and gas in thepipeline-riser system. Riser slugging is caused byliquid accumulation in the bottom of the riser,which blocks the pipeline cross-section and formsa liquid slug. The slug grows both up into the riserand backwards into the pipeline as more liquid isadded, whilst the gas is trapped upstreams of theslug causing increased pressure. When the pres-sure drop over the riser overcomes the hydrostatichead of the liquid slug, the liquid is transportedout of the riser followed by a burst of gas. Afterthe gas production, the slug cycle starts over. Thiscyclic behavior is illustrated in figure 1. More in-formation on riser slugging can be found in Taitel(1986).

Fig. 1. Graphic illustration of a riser slugging cycle

The highly irregular flow out of the pipelinecaused by riser slugging may lead to severe prob-lems for the receiving facility. Among these prob-

lems are; the large amounts of liquid in the slugcan cause separator overfilling and process tripand in less severe cases poor separation and theirregular gas production causes varying compres-sor loads and unnecessary flaring. Hence, riserslugging must be avoided in pipeline-riser systems.

Recently, feedback control has emerged as thepreferred solution for avoiding riser slugging inpipeline-riser systems. Controller design, experi-ments and industrial implementations has beenreported (Schmidt et al., 1979; Hedne and Linga,1990; Hollenberg et al., 1995; Courbot, 1996; Hen-riot et al., 1999; Havre et al., 2000; Havre andDalsmo, 2002; Kovalev et al., 2003; Skoftelandand Godhavn, 2003; Storkaas et al., 2003). Thesecontrol systems are designed either to keep thesystem away from the riser slugging conditions orto stabilize the unstable operating point existingat the same boundary conditions as riser slugging.

2.2 Case Description

In order to study the dominant dynamic behaviorof a typical, yet simple severe slugging problem,the test case for severe slugging in OLGA isused. OLGA is a commercial multiphase simulatorwidely used in the oil industry. The nomenclatureused for these systems are given in figure 2(a).The case geometry is given in figure 2(b). Thepipe diameter is 0.12 m. The feed into the systemis constant at 9 kg/s, with WL = 8.64 kg/s oiland WG = 0.36 kg/s gas. The pressure behindthe choke valve (P0) is constant at 50 bar. Thisleaves the choke valve opening as the only degreeof freedom in the system.

The stability of the flow is dependent on the top-side choke valve opening Z. It is usually possibleto obtain stable flow (that is, no riser slugging) bya sufficiently low opening Z. This is illustrated infigure 2.2, were the pressure at the inlet PI is showfor simulations with choke valve openings Z= 10,20 and 40%.

To illustrate behavior of the system over the wholeworking range of the choke valve, the bifurcationdiagrams in figure 4 are useful. The thickest linesrepresent the OLGA reference data. The solidlines show the observed pressure as a functionof valve opening. For small valve openings, theflow in the system is stable and the inlet pressurePI in figure 4(a) and the pressure drop over thetopside choke valve DP in figure 4(b) are constantas shown by single solid lines. For large valveopenings there is riser slugging in the system,indicated in the bifurcation diagrams by plottingboth the maximum and minimum pressure in theoscillations. In other words, PI and DP oscillatesbetween the upper and lower solid lines. The

(a) Nomenclature

0 1000 2000 3000 4000 5000

300

200

100

0

Horizontal distanze from inlet [m]

Ver

tical

dep

th [m

]

(b) Case Geometry

Fig. 2. (a) Nomenclature used for the pipeline risersystem and (b) System geometry

0 0.5 1 1.5 2 2.5 3 3.5 4

65

70

75

PI [

Bar

]

Valve opening Z = 10%

0 0.5 1 1.5 2 2.5 3 3.5 4

65

70

75

Time [hrs]

PI [

Bar

]

Valve opening Z = 40%0 0.5 1 1.5 2 2.5 3 3.5 4

65

70

75

PI [

Bar

]

Valve opening Z = 20%

Fig. 3. OLGA simulations for valve openings ofZ= 10, 20 and 40%

dashed line represents the unstable steady-statesolution. This is our desired operating line withclosed-loop operation.

2.3 Model tuning

As multiphase flow in pipelines is very complexand the models describing it are complicated, twodifferent simplified models are used to describethe system. The assumption is that if these twomodels predict the same qualitative controllabilityproperties, the results should be reasonably robusttoward modeling error. The first model is a PDE-based two-fluid model (Storkaas et al., 2001), theother a simplified model with three dynamicalstates tailor-made for controller design and analy-sis (Storkaas et al., 2003).

0 10 20 30 40 50 60 70 80 90 10060

62

64

66

68

70

72

74

76

78

80

Valve opening Z [%]

PI [

bar]

OLGA reference dataThreestate model (Ch. 3)PDE model (Ch. 2)

(a) Inlet Pressure

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

8

9

10

Valve opening Z [%]

DP

[bar

]

OLGA reference dataThreestate model (Ch. 3)PDE model (Ch. 2)

(b) Pressure drop over choke

Fig. 4. Model tuning and verification for (a) Inletpressure PI and (b) Pressure drop over chokevalve DP

The two models are fitted to the reference data,and the resulting responses are shown in figure 4.The figure shows that both models have a correctqualitative behavior, but the pressure drop overthe pipeline (as represented by the inlet pressurePI) for the PDE-based model is about 0.7 bartoo high. There are some deviations for the riserslugging amplitude for both models, but this isof lesser importance as the control system shouldstay away from this solution.

3. CONTROLLABILITY THEORY

3.1 Limitations imposed by unstable zeros

The presence of unstable (RHP) zeros z may causesevere problems for stabilizing control. Youla et al.(1974) prove that for a strictly proper SISO plantG(s) to be stabilized by a stable controller, everyreal RHP-zero in G(s) must lie to the left of aneven number of real RHP-poles in G(s). For pairsof complex conjugate unstable poles, the resultfrom Youla et al. (1974) does not fundamentallylimit the possibility for a stabilizing controller tobe found, irrespective of the possible presence ofRHP-zeros.

Skogestad and Postlethwaite (1996) show thatachievable performance is closely linked to thedistance between unstable poles and unstable ze-ros. For real poles, they derive the approximateperformance requirement z > 4p based on de-manding the maximum peak of the complemen-tary sensitivity function T to be less than 2,MT < 2. Skogestad and Postlethwaite (1996) also

show that complex unstable poles and zeros aremore troublesome the closer they are to the realaxis relative to the imaginary axis.

3.2 Lower bounds on sensitivity peaks

The distance between the unstable poles and zerosis closely related to the lowest achievable peaks insensitivity and complementary functions, denotedMS,min and MT,min, respectively. Based on thework of Chen (2000), the values for MS,min andMT,min can be computed.MS,min andMT,min aregood indications of the theoretically achievableperformance for a system, but to actually achievethe computed peak values may require very com-plicated controllers. Note that the value for thesebounds only depends on the relative location ofthe unstable poles and zeros.

To avoid excessive input usage and input satura-tion, the transfer function KS from measurementnoise n to plant input u should small. By smallwe mean that for a properly scaled system, themaximum amplification of the signal through thetransfer function KS minimized over all possi-ble controllers K, minK KS(s) should be lessthan one in magnitude. Based on the work ofFrancis (1987), an exact value of this norm canbe calculated using equation 1

minK

KS(s) =1

H,u(G(s))(1)

If the bound in equation 1 is less then unity for aproperly scaled system, the plant can theoreticallybe stabilized using the available inputs.

4. CONTROLLABILITY ANALYSIS

The models are linearized around two different un-stable operating points, one easy (slow instabil-ity) corresponding to a valve opening of 17.5% andone difficult (faster instability) corresponding toa valve opening of 30%. The PDE model results ina relatively large linear model (many states), as iteven with a rather coarse discretization results ina set of 50 ODEs. The resulting linear models arescaled as outlined in Skogestad and Postlethwaite(1996).

4.1 Stability - Poles

When the valve opening is increased, the station-ary operating point moves along the single solidline in figure 4, through the bifurcation pointat valve opening Z = 13% and onwards alongthe dashed line for the unstable operating points.At the bifurcation point, there is a pair of poles

(eigenvalues of the state feedback matrix of thelinearized model) that move into the right halfplane as seen from the root-locus plot in figure 5.This indicates that the bifurcation point is a Hopfbifurcation (Thompson and Stewart, 1986), whichis also consistent with the shape of the bifurcationmaps in figure 4

2 0 2 4 6 8 10

x 103

0.015

0.01

0.005

0

0.005

0.01

0.015

Re(p)

Imag

(p)

Z = 13% Z = 5% Z = 100%

Fig. 5. Controllability analysis: Open-loop Root-Locus plot with valve opening Z as indepen-dent parameter. The results are computedusing the PDE-based two-fluid model

4.2 Poles and zeros - Measurement selection

In this work we have considered the followingmeasurement candidates: inlet pressure PI , pres-sure drop over choke valve DP and volumetricflow through the topside choke valve Q. Tables1 and 2 give the location of the smallest RHP-zero, the lower bound on the peaks of the sen-sitivity function S and T (MS,min = MT,min)and the lower bound on the sensitivity functionKS (minK KS) for these measurement can-didates. The stationary process gain |G(0)| is alsoincluded in the analysis to identify measurementcandidates that has small low-frequency gain.

Tables 1 and 2 shows that it is theoretically pos-sible to stabilize the system with all the measure-ment candidates as minK KS < 1 in all cases.However, the unstable zero dynamics (RHP-zeros)associated with y = DP makes it practicallyimpossible to design an effective stabilizing con-troller using this measurement. Thus, the pressuredrop over the topside choke valve (y = DP ) willbe excluded from the following analysis. The con-trollability results from the two different modelsare similar, the differences can all be explained bythe complexity difference between the two models.

4.2.1. Inlet pressure PI The inlet pressure PIseems to be ideally suited as a measurement forstabilizing control of these systems as it has smalllower bounds for all the sensitivity functions, nounstable (RHP) zeros and high stationary gain.This is reassuring since pressure control using the

Table 1. Controllability data for the operating point Z = 17.5%, unstable poles at = 0.0007 0.0073 for simplified model and = 0.0014 0.0085 for PDE-based

two-fluid model.

Measurement Value Smallest RHP-zeroMS,min

=

MT,min

minK KS |G(0)|

PI [bar] (simp) 69.35 - 1 0.01 19

PI [bar] (PDE) 70 99 1.0 0.03 18.9

DP [bar] (simp) 1.91 0.018 1.1 0.02 17.7DP [bar] (PDE) 1.92 0.01 0.01i 1.6 0.04 17.6

Q[m3/s] (simp) 0.0194 - 1 0.01 1.5Q[m3/s] (PDE) 0.0208 - 1 0.02 1.8

Table 2. Controllability data for the operating point Z = 30%, unstable poles at = 0.0038 0.0115i for simplified model and = 0.0045 0.0108 for PDE-based

two-fluid model.

Measurement Value Smallest RHP-zeroMS,min

=

MT,min

minK KS |G(0)|

PI [bar] (simp) 68 - 1 0.11 3.4

PI [bar] (PDE) 68.7 98.1 1.0 0.3 3.3

DP [bar] (simp) 0.68 0.016 1.9 0.25 6.3DP [bar] (PDE) 0.66 0.01 0.01i 4.3 0.62 6.1

Q[m3/s] (simp) 0.0196 - 1 0.09 0.28Q[m3/s] (PDE) 0.0211 - 1 0.117 0.33

inlet or riser base pressure is the most commonindustrial approach to stabilizing these systems(Courbot, 1996; Henriot et al., 1999; Havre etal., 2000; Havre and Dalsmo, 2002; Skoftelandand Godhavn, 2003). A Bode plot for the processtransfer function G(s) from input u = Z tomeasurement y = PI is shown in figure 6. The low-frequency behaviors of the two models are almostidentical except for a slightly higher gain aroundthe unstable frequency for the simplified model.The deviations at higher frequencies are causedby time delay and the dampening effects of thefluid dynamics in the feed section of the pipelinethat is neglected in the simplified model.

The Bode plot indicates that there might be twopotential problems with using the inlet pressurefor stabilizing control; the time delay might betoo long and the high frequency gain might betoo low for effective disturbance rejection. Thetime delay problem can be omitted by usingthe pressure at the riser base instead of theinlet pressure. The problems with disturbancerejections at higher frequencies are not directlyrelated to stabilization, but disturbances at higherfrequencies are considered to be a problem, thesecould be suppressed by adding an extra loop inthe controller with a measurement that has higherhigh-frequency gain (Skofteland and Godhavn,2003).

4.2.2. Topside flow measurements W and Q

The final measurement alternative considered intables 1 and 2 is the topside volumetric flow (Q).From tables 1 and 2, the system is stabilizablewith this measurement as minK KS > 1 forboth operating points and models. However, thestationary gain associated with the volumetric

100

102

104M

agni

tude

(ab

s)

104

102

100

102

720

540

360

180

0

180

360

Frequency (s1)

Pha

se (

deg)

GPI

(3state)

GPI

(twofluid)

Fig. 6. Bode plot for process model from inputu = Z to measurement y = PI

flow Q is low. This can also be seen from the Bodeplot for the process transfer function G(s) frominput u = Z to measurement y = Q in figure 7.The small low-frequency process gain implies thatthere will be a problem achieving low-frequencyperformance (integral action) with this measure-ment as input to a stabilizing controller. Theproblem with the lacking low-frequency processgain is that the process might drift from its desiredoperating point to a operating point which is notstabilizable.

If disturbance rejection at medium to high fre-quencies is important to keep the input away fromsaturation and by that keep the system stable, acomparison between figures 6 and 7 indicates thatit might be better to use a flow measurement asinput to a stabilizing controller. In that case, thepressure measurement could be use in an outerloop in a cascade and by that provide the low-

104

102

100

102

0

90

180

270

360

450

Frequency (s1)

Pha

se (

deg)

101

100

101

102

Mag

nitu

de (

abs)

GQ

(3state)

GQ

(twofluid)

Fig. 7. Bode plot for process model from inputu = Z to measurement y = Q

frequency performance necessary to keep the sys-tem at its desired set point.

5. CONCLUSION

A controllability analysis of a pipeline-riser sys-tem at riser slugging conditions shows that agood control structure for stabilizing the systemis a cascade structure where the flow through thetopside choke valve is controlled in the inner loopand the inlet or riser base pressure is controlled inthe outer loop. A single pressure loop controllingthe inlet or riser base pressure can also a goodsolution, provided that good disturbance rejectionproperties at medium to high frequencies are notneeded. Single loop flow controllers can also sta-bilize the system, but small low-frequency gainmight cause the operation point to drift.

Although the (linear) system is theoretically sta-bilizable using pressure drop over the topsidechoke valve or mixture density at the top ofthe riser as measurement, it is very sensitive touncertainty, nonlinearity and model error due tounstable poles and zeros located relatively close toeach other in the complex right half plane. Thus,in practice, the pressure drop over the topsidechoke valve can not be used for stabilizing control.

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