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D-OSKIL: A New Mechanism o Con olling S ick-Slip Oscilla ions in Oil Well
D ills i ings
A icleinIEEE T ansac ions on Con ol Sys ems Technology · Decembe 2008
DOI: 10.1109/TCST.2008.917873·Sou ce: IEEE Xplo e
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1
DOSKIL: A New Mechanism o Con olling
S ick-Slip Oscilla ions in Oil Well D ills ings
Ca los Canudas-de-Wi
Labo a oi e d’Au oma ique de G enoble, INPG-CNRS, G enoble, FRANCE.
Email: ca los.canuda[email p o ec ed]
F ancisco R. Rubio
Depa men o Au oma ic Con ol, Uni e si y o Se ille, Se ille, SPAIN
Email: [email p o ec ed]
Miguel Angel Co che o
Depa men o Au oma ic Con ol, Uni e si y o Se ille, Se ille, SPAIN
Email: macpe[email p o ec ed]
Abs ac
Limi cycles occu ing in oil well d ills ings esul om he in e ac ion be ween he d ill bi and he ock du ing
d illing ope a ions. In his pape we p opose o use he weigh on he bi (WoB) o ce as an addi ional con ol
a iable o ex inguish limi cycles when hey occu . An app oxima e analysis based on he bias desc ibing unc ion
and comple ed wi h some simula ions, p o ides good e idence ha he o a ional dynamics o he oil well d ills ing
displays such a beha io . In pa icula , we p opose an adap a ion law o he WoB named D-OKILL mechanisms,
which esul s om a a ian o he oscilla ion kille (OSKIL) mechanism s udied in de ail in [6].
In opposi ion o he heu is ic con ol s uc u e p oposed in [7], we show ha he new Weigh on Bi (WoB ) con ol
law esul s in a globally asymp o ically s able closed loop-sys em. Simula ions applying he D-OSKIL mechanism
show ha he s ick-slip oscilla ions can be elimina ed wi hou equi ing a e-design o he eloci y o a y- able con ol.
I. INTRODUCTION
Oil well d ills ings (see Figu e 1) a e sys ems which p esen in e es ing ea u es om he dynamical and con ol
iewpoin s as hey pose many challenging echnological p oblems [26], [34]. The applica ion o dynamic analysis
and con ol echniques in a d illing sys em can lead o conclusions ha allow us o p opose new ecommenda ions
o d illing ope a ions, d ills ing design and con ol algo i hm, which would p oduce economic bene i s h ough a
mix o lowe de elopmen cos s, highe p oduc ion a es and imp o ed eco e y. Pa icula ly, he p esence o s ick-
slip sel -exci ed oscilla ions a he bo om pa o he d ills ings as well as dec easing se ice li e o d ills ings
and downhole equipmen , has d awn he a en ion o he con ol communi y in he las decade. The elimina ion
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Au ho manusc ip , published in "IEEE T ansac ions on Con ol Sys ems Technology 16, 6 (2008) 1177-1191"
2
a) b)
Fig. 1. Oil d illing sys em in he ield (a). Basic scheme o a e ical d illing sys em.(b)
o his kind o oscilla ions is a challenge o d ille s and scien is s since i can p o ide impo an cos sa ings in
d illing ope a ions, in e ms on money and exploi a ion ime [21].
Di e en oscilla ions a ec ing he d ills ing beha io a e una oidable. The occu ence o sel -exci ed s ick-slip
ib a ions (i.e., he op o he d ills ing o a es wi h a cons an o a y speed, whe eas he bi (cu ing de ice) o a y
speed a ies be ween ze o and up o six imes he o a y speed measu ed a he su ace) as a common and damaging
phenomena in d ills ing sys ems has been highly desc ibed and analyzed in ecen yea s, d awing he a en ion o
he con ol communi y. Fo mo e in o ma ion abou d ills ing oscilla ions and s ick-slip phenomenon in oil well
d ills ings, please see [14], [21], [24], [31].
Some causes o s ick-slip oscilla ions a e backlash be ween con ac ing pa s, hys e esis, nonlinea damping
and geome ical impe ec ions which a e e y di icul o model. Howe e , he main cause o such ib a ions in
d ills ings is he ic ion appea ing by con ac wi h he ock o ma ion [3], [17]. Consequen ly, a model desc ibing
he d ills ing beha io should include a bi - ock ic ion o que model adequa e enough o p ope ly ep oduce his
e ec .
Many ways o educing hese ib a ions ha e been p oposed, bo h om p ac ical and heo e ical iewpoin s.
His o ically, he expe ience o d ille s has e ealed ha he manipula ion o di e en d illing pa ame e s (inc eas-
ing he o a y speed, dec easing he weigh -on-bi (WoB), modi ying he d illing mud cha ac e is ics, in oducing
an addi ional ic ion a he bi [25], e c) is an e ec i e s a egy o supp ess s ick-slip mo ion [28]. Howe e , his
s a egy depends oo much on he pe sonal skills o each d illing echnician o be eally e ec i e.
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Usually, d illing sys ems a e eloci y-con olled o make hem o a e a a cons an eloci y, bu no speci ica ions
abou ib a ion supp ession o damping a e conside ed. Ano he con ol op ion would be in oducing new egula ion
me hodologies (ac i e, passi e) in he loop, speci ically aimed o compensa e o d ills ing ib a ions. Among hose,
he ollowing examples can be poin ed ou :
•The so-called So To que Ro a y Sys em (STRS) [13] [28], ha is a o que eedback a he op o he d ills ing
which makes he sys em beha e in a “so e ” way a he han as a ixed hea y lywheel, so ha he o sional
wa es a i ing a he su ace a e abso bed, b eaking he ha m ul cycling mo ion.
•In oducing a ib a ion abso be a he op o he d ills ing [15] which ollows he same app oach gi en in
[13] and [28].
•In oducing a PID con olle s uc u e a he su ace in o de o con ol he o a y speed [1], [23], [24], [25].
•Using obus con olle s, like he linea H∞con ol p oposed in [29], o supp ess s ick-slip mo ion a he bi .
•Using a con olle based on an inpu -s a e eedback linea iza ion o he nonlinea ic ion o que [2].
Howe e , ew wo ks ha e p o ided a o mal s abili y analysis o hei p oposed con ol s a egies. Fo ins ance,
analysis o he dynamical beha io o d ills ing unde ib a ions has been explo ed in [1] and linea app oxima ions
o s abili y o con olled d ills ing has been s udied in [24].
The alue o he sys em weigh measu ed a he bo om pa , called Weigh on Bi (WoB), has been p o ed o
be an impo an pa ame e in he occu ence and possible a oidance o s ick-slip oscilla ions (see [23] and [28]).
E icien d illing ope a ion equi es a ce ain amoun o o ce (WoB) ha may be incompa ible wi h he low o ce
ange which may a oid s ick-slip oscilla ions. This adeo be ween o ce magni udes, p o ides a i s indica ion
ha a egula ion s a egy o WoB seems o be necessa y o main ain a good d illing ope a ion, hus, a oiding such
oscilla ions (see [7] and [24]).
This pape is ocused on he p oblem o s ick-slip oscilla ions p oduced a he bo om-hole assembly (BHA). The
main idea is o use he weigh on he bi (WoB) o ce as an addi ional con ol a iable. In pa icula we adap he
oscilla ion kille (OSKIL) mechanism s udied in [6], o he oil well d ills ing sys ems (named he e D-OSKIL1)
which has been shown o be pa icula ly adap ed o nonlinea sys ems displaying a local s able egion wi h a s able
limi se ou side his local domain.
An app oxima e analysis based on he bias desc ibing unc ion p o ides good e idence ha he o a ional dynamics
o he oil well d ills ing display a simila beha io pa e n. This analysis, al hough app oxima e, also gi es a good
in ui ion in he way ha he WoB needs o be modi ied o supp ess oscilla ions. An impo an p ope y o he
p oposed D-OSKIL mechanism is ha i allows eco e ing he nominal ope a ion condi ion ( he W oB eco e s i s
nominal d illing alue) while oscilla ions a e supp essed.
In opposi ion o he heu is ic con ol s uc u e p oposed in [7], we show ha he new p oposed Weigh on
Bi (WoB) con ol law esul s in a globally asymp o ically s able closed loop-sys em. The e o e, he D-OSKIL
mechanism elimina es he s ick-slip oscilla ions wi hou equi ing a e-design o he eloci y o a y- able con ol.
1D-OSKIL s ands o D illing oscilla ion kille mechanism.
June 6, 2007 DRAFT
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The pape is o ganized as ollows. In Sec ion II he basics abou d ills ing dynamics and ib a ions a e b ie ly
in oduced. In Sec ion III, he d ills ing model used in he pape is p esen ed. The con ol loop used o egula e
o a ional eloci y o he se and he a p io i beha io o he closed loop sys em a e shown in Sec ions IV and V,
espec i ely. In Sec ion VI, a Desc ibing Func ion based analysis is made in o de o ob ain some in o ma ion abou
s ick-slip oscilla ions in he sys em, and nex , in Sec ion VII, he con ol mechanism named D-OSKIL ob ained
om he conclusions o he p e ious analysis is p esen ed. In Sec ion VIII, some simula ions a e shown. Sec ion
IX p oposes an obse ed-based e sion o he same con olle whe e only ield exis ing measu es a e used. In he
same sec ion, we show also some simula ions. And inally, Sec ion X submi s he conclusions and u u e esea ch
lines.
II. BASICS ON DRILLSTRING DYNAMICS AND VIBRATIONS
S anda d o a o y d illing equipmen , as shown in Figu e 1 o depic wha is commonly used by oil companies
o ex ac gas and oil om he ea h su ace, uses a dill-bi (called bi ) o c ush he ock and make he hole in he
g ound. As he hole becomes deepe , some pipe sec ions (called d ill pipes) a e added, lea ing he bi coupled a
he bo om pa o he se . These pipes, oge he wi h he d ill bi , o m he so-called d ills ing. This d ills ing
is mo ed by means o a mo o o sys em o mo o s in he su ace. As i has been shown in he p e ious Sec ion,
ope a ion o he d ills ing looks jus like ha o a household elec ic d ill, whe e a mo o makes he bi o a e, and
enough weigh is applied o main ain he con ac be ween he bi and he objec o be d illed.
In o de o make he s udy o a d illing sys em s uc u e a bi mo e comp ehensi e, he ollowing pa s can be
emphasized:
•Powe Sys em: A se o diesel and elec ic mo o s ha p o ide he necessa y ene gy o pe o m all he asks.
•Suppo ing S uc u e: This is used o mo e he pipes in and ou o he oil well, and so, o a y he weigh
applied du ing he p ocess.
•The Ro a o y Sys em o make he sys em o a e. I is composed by:
– Swi el and Kelly: To connec he Suppo ing and Ro a o y Sys ems.
– Ro a o y Table: Also called u n able. I is a la ge disc-shaped ine ia coupled o he d ills ing ha
d i es he o a ing mo ion using powe om elec ic mo o s.
– D ills ing: As shown be o e, i is a sequence o ubes ha connec he o a o y able and he bi .
– Bi : The cu ing de ice.
•Ci cula ion Sys em: I consis s o a se o pipes and pumps which c ea e a low wi hin he hole by d illing
mud in o i . This subs ance is aimed o lub ica e and e ige a e he con ac be ween he ock and he bi , and
so o li he ock cu ings om he d ill bi o he su ace.
A mo e exhaus i e desc ip ion o he o a o y sys em can be ound, in [21] and [31] among o he pape s.
One o he main p oblems is he appea ance o oscilla o y beha io s (limi cycles), ha cause a dec easing o he
d illing pe o mance om he iewpoin s o di e en pa ame e s ( a e o pene a ion a he su ace, o a ional speed
o he bi , ...) and so p o oking he mechanical ailu e o he d ills ing o he b eakage o any o he elemen s [31].
June 6, 2007 DRAFT
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5
bi -bouncing
whi l
whi l
s ick-slip
Fig. 2. Di e en ypes o ib a ions in d ills ing sys ems.
The ib a ions appea ing in he d ills ing can be di ided in o 3 di e en ca ego ies [14] [10], (see Figu e 2):
•Longi udinal ib a ions a e p oduced in a e ical di ec ion om he d illing owe , causing ebounds o he
bi a he bo om o he oil well, a phenomenon called bi -bouncing.
•La e al ib a ions a e p oduced when he d ills ing’s mass cen e is displaced om he o a ion axis, causing
whi l-like mo emen s and ebounds wi hin he oil well walls, a phenomenon called whi ling.
•To sional ib a ions a e p oduced when he o a ional eloci ies a he su ace and he bo om o he d ills ing
a e di e en , causing s ick-slip mo emen s.
Each oscilla ion phenomenon appea s bo h a di e en imes and di e en equency anges, and so, hey can be
s udied sepa a ely. This wo k is ocused on s ick-slip ib a ions.
The s ick-slip oscilla ions a e gene ally associa ed o ypical d y ic ion p o iles [19], i.e., when he e is no
mo emen , he ic ion o que (s a ic ic ion) is la ge han in non ze o eloci y cases (dynamic ic ion). The
di e ence be ween hose wo magni udes has been shown by many au ho s o be one o he mos ele an a iables
ha cha ac e izes s ick-slip oscilla ions [22].
III. SYSTEM MODELLING
Mul iple kind o models ha e been used in li e a u e o desc ibe d ills ing sys ems (see o example [19] and
[31]). The ype and he complexi y o he model o be used a e closely ela ed o he aim pu sued (modelling,
simula ion, model o con ol, e c). Howe e , lumped pa ame e s models ha e been shown o be alid enough o
p ope ly desc ibe he s ick-slip oscilla ion phenomena and easy enough o make he s udy no oo complex [10].
The p oblem o modelling s ick-slip phenomenon in a d ills ing by means o a lumped-pa ame e model has been
s udied om se e al poin s o iew. Mos o hem conside he d ills ing as a o sional pendulum wi h di e en
deg ees o eedom, o ins ance: [17], [20], [27], [32] p opose single-deg ee-o - eedom models, [1], [5], [22], [24]
June 6, 2007 DRAFT
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6











J
ϕ
d
kc
ϕb
db
Jb
ToB
WoB
Fig. 3. D ills ing wo-coupled masses model.
p opose wo-deg ee-o - eedom models including a linea con olle , and [15], [29] p esen wo-deg ee-o - eedom
models o he mechanical pa o he sys em plus he model o he o a y able elec ic mo o sys em.
As i will be seen in subsequen sec ions, he bi - ock ic ion model is undamen al o p ope ly ep oducing
s ick-slip oscilla ions phenomenon. Many models ha e been p oposed in li e a u e, some o hem summa ized in
[22]. The model used he e (depic ed in Figu e 3) is a wo-deg ee-o - eedom model wi h wo ine ial masses J
and Jb, locally damped by d and db. The ine ias a e coupled wi h each o he by an elas ic sha o s i ness k
and damping c. The a iables ϕ and ϕbs and o he o a y and he bi angle. The o a y o que con ol signal
used o egula e he o a y angula eloci y ˙ϕ . The T oB (To que on Bi ) ep esen s he o al ic ion o que o e
he d ill bi .
The model equa ions a e he ollowing:
J ¨ϕ +c( ˙ϕ −˙ϕb) + k(ϕ −ϕb) + d ˙ϕ = (1)
Jb¨ϕb+c( ˙ϕb−˙ϕ ) + k(ϕb−ϕ ) + db˙ϕb=−T oB (2)
In cons an s abo e, he sub-sc ip ′ ′, and ′b′s ands o o a y and bi , espec i ely.
A sui able model o T oB is essen ial, because he ep oduc ion o s ick-slip ib a ions will s ongly depend on
he pa icula choice o he model o T oB. This o que ep esen s he combined e ec s o eac i e o que on he bi
and nonlinea ic ional o ces along he d ills ing. In ou case, he T oB will be gi en by he p oduc o µ( ˙ϕb, z),
which desc ibes he no malized (dimensionless) o sional bi - ock ic ion (di e en bi - ock ic ion models a e
p esen ed in [22]), and he no mal o ce ucalled Weigh on Bi (WoB), i.e.
T oB =µ( ˙ϕb, z)·u(3)
Se e al o ms o µ( ˙ϕb, z)can be conside ed acco ding he use o he model. Nex , we desc ibe he model o T oB
used o simula ions and o alida ing he con ol law, hen a simpli ied model is in oduced o con ol analysis
June 6, 2007 DRAFT
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7
pu poses.
A. Model o simula ions
The s a e-space ep esen a ion o he la e model is he ollowing:
˙x=Ax +B +Hµ(x, z)u(4)
˙z= (x, z)(5)
wi h
A=




0 1 −1
−k
J
−(d +c)
J
c
J
k
Jb
c
Jb
−(c+db)
Jb





, B =




0
1
J
0





, H =




0
0
−1
Jb





whe e he s a e x= [x1x2x3]Tis de ined as ollows:
x1=ϕ −ϕb
x2= ˙ϕ (6)
x3= ˙ϕb
In his desc ip ion, he s a e z∈R ep esen s he in e nal ic ion s a e, and Equa ion (5) desc ibes he ic ion
dynamics. Va ious ic ion models ha e been shown o wo k p ope ly o cap u e he ypical ic ion phenomena
(s ic ion, S ibeck e ec , e c) which cause s ick-slip oscilla ions ( [12] and [18]). One possible model o Equa ion
(5) is he LuG e ic ion model [8]:
˙z=x3−σ0|x3|
g(x3)z,
g(x3) = µC+ (µS−µC)e−(x3/ s)2(7)
µ(x, z) = σ0z+σ1˙z,
The unc ion g( )i mainly a ec he s eady-s a e cha ac e is ics o he ic ion model. In s eady-s a e, he
model p edic he ollowing ic ion alue, µSS(x3) = g(x3)sgn(x3).In his model, σ0, σ1, s, µC, µSa e posi i e
cons an s cha ac e izing he ic ion physical p ope ies. Also no e ha he o sional linea ic ion a he d ill bi
side is al eady inco po a ed in he Ama ix o he ep esen a ion (4).
B. Model o con ol
No e ha he p e ious model o he µ(x, z)includes an addi ional ic ion dynamics, zwhich is sui ed o desc ibe
mo ion a p e-sliding, and in pa icula o egula ize he di e en ial equa ion desc ibing he sys em dynamics. An
al e na i e is o use s a ic desc ip ion o µ(x)(maps wi hou memo y), which may be simple o con ol analysis.
The di e en be ween bo h models, may no be oo signi ican , as long as compu a ion issues a e s ongly simpli ied.
The model o con ol is hen desc ibed by,
˙x=Ax +B +Hµ(x3)u(8)
June 6, 2007 DRAFT
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8
whe e he e µ(x3)is a s a ic map be ween he bi o a ional eloci y x3= ˙ϕband he no malized ic ion pa ame e
µ, i.e. he s eady-s a e o m o he model (7), o any sui able app oxima ion like he one shown in Figu e 12.
IV. ROTATIONAL VELOCITY REGULATION LOOP
The i s ask o con on is designing a p ope con ol law o mo o o que . This signal will be aimed mainly
a egula ing he o a ional eloci y o he o a o y able ˙ϕ o a ce ain desi ed alue ωd(a ypical alue o ωdis
5 ad/s). As we ha e shown in Sec ion I, many a chi ec u es ha e been p oposed o ha pu pose, om classic PID
s uc u es o a linea H∞ obus con ol, al hough oil well d ills ings usually ope a e wi h educed-o de simple
con ol laws.
A. Ro a y able eloci y con ol loop
In his wo k, he s uc u e o he eloci y con olle is inspi ed by he one p esen ed in [10], as shown by:
=k1+k2
s(ωd−˙ϕ )−k3( ˙ϕ −˙ϕb)(9)
o equi alen
=k1(ωd−x2) + k2x4−k3(x2−x3)
˙x4= (ωd−x2)
hen he closed-loop equa ions ake he o m o he model o simula ion,
˙x=Aclx+Bclωd+Hclµ(x, z)u(10)
˙z= (x, z)(11)
and he ollowing one o he model o con ol analysis,
˙x=Aclx+Bclωd+Hclµ(x3)u(12)
wi h he ob ious obse a ion ha xis now o dimension ou (due o he in oduc ion o an in eg al e m in he
o a y able con ol), i.e. x= [x1, x2, x3, x4], and wi h he Acl,Bcl and Hcl gi en as:
Acl =








0 1 −1 0
−k
J
−(d +c+k1+k3)
J
(c+k3)
J
k2
J
k
Jb
c
Jb
−(c+db)
Jb0
0−1 0 0








, Bcl =








0
k1
J
0
1








, Hcl =








0
0
−1
Jb
0








The s eady-s a e alue o x, conside ing µ∗=µ(x∗
3), is:
x∗
1=u0µ∗+dbωd
k(13)
x∗
2=ωd(14)
x∗
3=ωd(15)
x∗
4=(db+d )ωd+µ∗u0
k2
(16)
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









kek
WoB
WoBmin u0
˜u
a ac i e limi cycle
epellen limi cycle
a
b
b
b
b
Fig. 10. Possible d ills ing sys em ajec o ies.
b) Beha io beyond nominal ope a ions: To supp ess such oscilla ions, he u=WoB mus be educed by
means o he con ol signal ˜u, un il i s ajec o y eaches he bi u ca ion poin , and he local con olle is able o
e u n he sys em ajec o ies o he equilib ium. The ea e , he nominal alue o he WoB mus be eco e ed in
a p ope slow manne , o con inue wi h he d illing ask, i.e. u→u0. Signi ican ly enough, he a ia ion o ˜u
should be es ic ed o a alid domain, and in pa icula es ic ed o a posi i e alues o he han ze o. Wi hou his
es ic ion, i is clea ha d illing may no be e icien , o i will be imp ac ical.
The gene al s uc u e o he a ia ion law o u(o equi alen o ˜u), will be he e o he o m:
˙
˜u=P0
−u0{−σ˜u+ Φ(·)}
whe e σ > 0, can be unde s ood as a ime-cons an o he con olle , and P0
−u0is a p ojec o ope a o ensu ing ha
solu ions o he abo e equa ion makes ˜us ay in he ange (−u0,0], and Φ(·)is a nonlinea unc ion which mus
be designed o ensu e sys em s abili y, ha is:
˙ϕ →ωdand ˜u→0
In o de o make he p esen a ion simple , we will op he explici use o he p ojec ion ope a ion Pin he ollowing
sec ion. Howe e , he eade should keep in mind ha ˜uis a bounded signal in he p esc ibed ange.
Wi h his in mind, he comple e closed-loop equa ions a e:
˙x=Aclx+Bclωd+Hclµ(x3)(u0+ ˜u)(20)
˙
˜u=−σ˜u+ Φ(·)(21)
whe e ma ices Acl, Bcl, Hcl ha e been de ined p e iously.
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16




-
-



Ψ
∆
G(s)
Γ
y
˜u2
˜u1
Fig. 11. E o sys em block diag am.
B. E o equa ions
E o equa ions can be ob ained by applying he change o coo dina es e=x−x∗, and conside ing he s eady-s a e
alues µ(e3)ss =µ∗and ˜uss = 0. This yields2:
˙e=Acle+Hcl [µ(y)˜u+ ˜µ(y)u0](22)
˙
˜u=−σ˜u+ Φ(y)(23)
y=Ce =e3(24)
whe e he e m ˜µ(y)is de ined as ollows:
˜µ(y) = µ(y)−µ∗(25)
and we assume ha he upda e ule o ˜uis designed on he basis o he ou pu y. The e o sys em can be desc ibed
by he block diag am in Figu e 11, wi h he ollowing de ini ions:
G(s) : Ψ 7→ y(26)
Γ : y7→ ˜u1=µ(y)˜u(27)
∆ : y7→ ˜u2= ˜µ(y)u0(28)
whe e Ψ = −(˜u1+ ˜u2).
C. E o equa ion p ope ies
1) PR condi ion on G(s):The map G(s)is:
G(s) = −C(sI −Acl)Hcl (29)
2Wi h an abuse o no a ion, we will use µ(y) o deno e he exp ession o µ(x3), in he shi coo dina e y+ωd, i.e. µ(x3) = µ(y+ωd) = µ(y).
No e ha a y= 0, we ha e µ(y= 0) = µ(ωd) = µ∗.
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17
wi h C= [0 0 1 0], o equi alen ly:
G(s) = s(a2s2+a1s+a0)
b4s4+b3s3+b2s2+b1s+b0
(30)
The e o equa ion and upda ing ule o ˜uha e been designed in such a way ha he esul ing map G(s)has
ela i e deg ee one. This condi ion is necessa y o ob ain PR and SPR unc ions. The e i ica ion ha G(s)is PR
can be done di ec ly on he iple s o ma ices (Acl, Hcl, C)as demons a ed in [30] o SPR unc ion. Adap ing
his esul o PR unc ions and using ou no a ion a hand, esul s in he ollowing elaxed condi ions: conside he
ans e unc ion G(s) = −C(sI −Acl)Hcl.G(s)is PR i and only i : 1) CAclHcl >0,2)Acl is s able, 3) he
ma ix Acl(I−(1/CAclHcl)AclHclC)Acl has no eigen alues on he open nega i e eal axis (−∞,0).
The i s condi ion is easy o compu e in e ms o model pa ame e s. This gi es CAclHcl = (c+db)/J2
b>0,
and always holds om he physics o he sys em. The second condi ion is also e i ied since Acl is designed o be
s able. The las condi ion is mo e in ol ed, bu i can be easily checked nume ically. Fo ypical alues o d ills ing
sys em model pa ame e s and -con ol gains conside ed in his pape , i is possible o show ha condi ion 3) holds.
No e also ha by con inui y o he eigen alues wi h espec o he ma ices pa ame e s, he e will exis s a ce ain
deg ee o obus ness o his condi ion wi h espec he model unce ainly.
Consequen ly we ha e ha G(s)is a Posi i e Real (PR) unc ion, and hence om he Kalman-Yacubo ich-Popo
Lemma [16], he ollowing p ope y holds: ∃P=PT>0, Q =LTL≥0such ha :
AT
clP+P Acl =−Q=−LTL≤0(31)
PHcl =−C(32)
The e o e, as a consequence we ha e he ollowing wo p ope ies o he linea map G(s):
•G(s)is a passi e ela i e o V(e) = eTP e, and
•G(s)has a ini e L2-gain: γ2(G) = supω|G(jω)|<∞
2) Boundedness o signal Ψ( ):F om he de ini ions o ˜u1and ˜u2in Equa ions (27) and (28), oge he wi h he
assump ion ha he adap a ion mechanism yields alues in he ange ˜u∈(−u0,0], i ollows ha bo h signals, ˜u1
and ˜u2, a e bounded, ha is:
||˜u1||∞= sup
≥0|˜u1| ≤ u0<∞,(33)
||˜u2||∞= sup
≥0|˜u2| ≤ 2·u0<∞.(34)
Hence ||Ψ( )||∞≤3·u0.
3) Boundedness o he ou pu y( ):Since G(s) : Ψ 7→ yis a lineal s able map, he ou pu signal yis also
bounded, i.e.
Ψ∈L∞⇒y∈L∞(35)
4) Sec o condi ion on ∆:Wi h ega d o Figu e 11, he ou pu o he map ∆can be seen as a dis u bance
ac ing on he closed loop sys em esul ing om he ope a o s G(s)in eedback connec ion wi h nonlinea ope a o
Γ.
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18





µ( ˙ϕb)
˙ϕb
µS
µC
s




∆
y
−ay
by
a) b)
Fig. 12. No malized ic ion unc ion (a). ∆(y)lies a he in e al [a, b](b).
Figu e 12-(a) shows he memo yless ic ion map used o his s udy. No e ha as he s eady-s a e o a o y bi
speed (ωd) is in gene al much la ge han he S ibeck eloci y s, we can hen assume ha µ∗=µC. The e o e,
aking in o accoun ha y=e3= ˙ϕb−ωd, he ou pu o block ∆will ha e he p o ile shown in Figu e 12-(b).
This ope a o belongs o he cone sec o [a, b]as displayed in he same Figu e (see [33] o u he discussion on
sec o de ini ions). Fo mally his is s a ed as ollows.
The nonlinea ope a o ∆(y)belongs o he sec o [a, b]i he ollowing holds ue:
•∆(0) = 0
•a≤∆(y)
y≤b,∀y≥0, o equi alen ly,
•ay2≤y∆(y)≤by2,∀y∈ ℜ
In ou case, he alues o aand ba e:
a=−µS−µC
ωd
u0(36)
b=µS+µC
ωd
u0(37)
and consequen ly, he map ∆has also ini e L2-gain, which is bounded by:
γ2(∆) ≤max[|a|,|b|](38)
5) Block ans o ma ion: As i can be seen in Figu e 12-(b), he map ∆is almos passi e, since almos he whole
diag am is wi hin he i s and hi d quad an s. This cha ac e is ic is gene ic, as he di e ence be ween b eak-away
and Coulomb ic ion le els is gene ally small (ais small when compa ed o b).
June 6, 2007 DRAFT
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19
Figu e 13 shows a possible block ans o ma ion, whe e he ollowing new ope a o s, ∆∗, and Γ∗a e de ined:
∆∗:y7→ (˜u2+εy)
Γ∗:y7→ (˜u1−εy)








+
+
+
-

-
-
∆
ε
G(s)
Γ
ε
y
∆∗
Γ∗
Ψ
Fig. 13. Modi ied Block Diag am.
Wi h his ans o ma ion, i can be easily p o ed ha he map ∆∗is passi e i he alue o εis aken such ha
ε=|a|, as can be seen in Figu e 14.
ε=|a| ⇒ Z
0
y(˜u2+εy)d ≥0(39)
Wi h his new eedback con igu a ion, he p oblem o designing a s able upda e law o ˜u( )is equi alen o
inding a unc ion Φ(y), and pa ame e condi ions, such ha he ans o med ope a o Γ∗de ines a passi e map.
This design s a egy esul s om well known p ope ies o eedback in e connec ed passi e sys ems.
The nex subsec ion uses such a esul o demons a e he s abili y p ope ies o one possible candida e upda e
ule.
D. D-OSKIL upda ing law
Unde he p emise ha he comple e o m o he upda e law should also include a sui ed p ojec ion ope a o
ensu ing ha he a ia ion o ˜uis limi ed o he admissible pa ame e s ange, he ollowing upda ing ule will be
analyzed.
June 6, 2007 DRAFT
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20



y
∆∗
ε
Fig. 14. P o ile o map ∆∗.
Le us conside he nonlinea unc ion Φ(y):
Φ(y) = λ y sgn(µ(y)) λ≥0(40)
No e ha his choice is condi ioned by he abili y o compu ing he sign o µ(y). Conside ing he o m o he
ic ion model o his s udy, he sign o µ(y) = µ( ˙ϕb−ωd), can be compu ed i ˙ϕbcan be measu ed, o a leas ,
obse ed as shown in simula ions la e on. We p oceed acco ding o his hypo hesis in wha ollows.
E. S abili y analysis
Lemma 1: Le ρ > 0be an a bi a ily posi i e cons an , and λ,σbe such ha he ollowing design inequali y
holds,
λ
σ≥µS
µC−1u0
ωd
+σ
µC
ρ(41)
whe e µS
µC≥1. Then map Γ∗:y7→ (µ(y)˜u−εy)is s ic ly inpu passi e, i.e.
I=Z
0
(µ(y)˜u−εy)y≥ρZ
0
y2−β0(42)
wi h β0=ymax
σ2u0>0.
P oo : Le Ide ine he in eg al o he inpu -ou pu p oduc o he ope a o Γ∗, i.e.
I=Z
0
(µ(y)˜u−εy)y=Z
0
µ(y)˜uy −Z
0
εy2
Subs i u ing ˜u om Equa ion (23) in he abo e exp ession gi es,
I=1
σZ
0
µ(y)Φ(y)y−1
σZ
0
µ(y)y˙
˜u−Z
0
εy2
F om sec ions VII-C.2, and VII-C.3, signals yand µ(y)ha e been shown o be bounded. Le no e hese bounds
as: |y|< ymax,|µ(y)|<1. The e o e,
June 6, 2007 DRAFT
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21
I≥1
σZ
0
µ(y)yΦ(y)−Z
0
εy2−ymax
σZ
0
˙
˜u
Taking now Φ(y) = λ y sgn(µ(y)), gi es
I≥Z
0
(λ
σ|µ(y)|−ε)y2−ymax
σ|˜u( )−˜u(0)|
≥Z
0
(λ
σ|µ(y)|−ε)y2−ymax
σ2u0
≥Z
0
(λ
σµC−|a|)y2−β0
whe e he las inequali y is ob ained by using he lowe bound on |µ(y)|, i.e., |µ(y)| ≥ µC, he de ini ion o
ε=|a|(wi h aas gi en in Equa ion (36)) and he cons an β0=ymax
σ2u0. Finally, in oducing he condi ion (41)
in he abo e exp ession, gi es he ollowing lowe bound on I:
I≥Z
0
(λ
σµC−|a|)y2−β0≥ρZ
0
y2−β0
which p o es he lemma.
Rema k 1: The condi ion (41) exhibi s se e al in e es ing p ac ical ea u es. I ela es he design pa ame e s (σ,
λ) as a unc ion o physical d ills ing sys em cha ac e is ics such as he nominal WoB (u0), he ock ic ion
ea u es (µS−µC), and he desi ed o a ional eloci y (ωd). The pa ame e ρas shown la e , p o ides a measu e
o he con e gence a e o he ou pu y o ze o.
We a e now in a posi ion o es ablish he main s abili y esul .
Theo em 7.1: Conside he closed-loop sys em o Figu e 13 wi h G(s),Γ∗,∆∗holding he ollowing p ope ies:
(i) G(s)is a PR ope a o sa is ying (31)-(32)
(ii) ∆∗is a passi e map sa is ying (39)
(iii) Γ∗is a s ic ly inpu passi e map sa is ying (42), i.e. design pa ame e s a e such ha he condi ion (41) holds.
Then, (e∗,˜u∗) = (0,0) is a globally asymp o ically s able equilib ium o he conside ed closed-loop sys em.
P oo : Le us ake he ollowing scala unc ion:
V(e, ˜u) = 1
2eTPe +Z
0
y(˜u2+εy) + I(˜u, y) + β0−ρZ
0
y2(43)
F om (39) and (42), we ha e ha V(e, ˜u)is semi-posi i e de ini e.
Compu ing he ime-de i a i e o V(e, ˜u), and using he p ope ies (i)−(iii)o he heo em, esul s in:
˙
V(e, ˜u) = −1
2eTLTLe −ρy2≤0,∀e, ˜u(44)
The e o e, om las Equa ion, we ha e ha y→0wi h a a e depending on he alue o ρ. The es o he p oo
ollows om he applica ion o he LaSalle’s in a iance p inciple. F om Equa ion (23), we can see ha i y→0,
June 6, 2007 DRAFT
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22
0 50 100 150
0
10
20
30
40
Time[s]
Ro a o y speed [ ad/s]
Ope a ion unde DOSKIL
Su ace
Downhole
0 50 100 150
0
1
2
3
4x 104
Time[s]
Weig h on Bi [N]
Fig. 15. Simula ion o D-OSKIL scheme wi h σ= 0.3and λ= 2.000.
oge he wi h he ac ha σis a posi i e cons an and Φ(0) = 0, gi es ha ˜u→0. Finally, his implies ha he
wo las e ms in Equa ion (22) also ends o ze o, i.e.
lim
→∞ [µ(y)˜u+ ˜µ(y)u0] = [µ(0)0 + ˜µ(0)u0] = 0
since µ(0) = µ∗=µC, and ˜µ(0) = µ(0) −µ∗=µ∗−µ∗= 0. The e o e, his esul s in
˙e=Acle+Hcl lim
→∞ [·] = Acle
So i can be concluded ha e→0, and hence ha e∗= 0, and ˜u∗= 0 a e a globally asymp o ically s able equilib ia.
VIII. SIMULATION EXAMPLE
In o de o demons a e he beha io o he p oposed adap i e law, simula ions o he d ills ing sys em con olled
unde he D-OSKIL mechanism designed in Sec ion VII-D a e shown in Figu es 15 and 16. The alues3 o sys em
model pa ame e s used in he simula ions a e p esen ed in Table I. In hese Figu es, he ypical p o iles in e ms
on o a o y eloci y, bo h in su ace and downhole, and sys em WoB a e shown.
As i can be obse ed, wi h he nominal weigh u0= 40000N, he sys em is unde a sus ained oscilla ion
egime. The D-OSKIL mechanism is ac i a ed a = 50s, and in bo h cases, he con olle is able o ex inguish
such oscilla ions, al hough he WoB p o iles ob ained a e qui e di e en .
3The nume ical alues o he d illing sys em pa ame e s acco ding o a 2000mlong d ills ing ha e been aken om [29].
June 6, 2007 DRAFT
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0 50 100 150
0
10
20
30
40
Time[s]
Ro a o y speed [ ad/s]
Ope a ion unde DOSKIL
Su ace
Downhole
0 50 100 150
2.5
3
3.5
4
x 104
Time[s]
Weig h on Bi [N]
Fig. 16. Simula ion o D-OSKIL scheme wi h σ= 1.0and λ= 2.500.
In Figu e 15 (wi h con ol pa ame e s λ= 2.000 and σ= 0.3) he D-OSKIl mechanism is able o ex inguish he
oscilla ions wi h a so e olu ion in he WoB con ol signal.
On he o he hand, in Figu e 16 (wi h con ol pa ame e s λ= 2.500 and σ= 1.0), he ansi ion om oscilla ion
egime o s abiliza ion pe iod is as e han he one ob ained in Figu e 15. The s abiliza ion ime o he alue o
WoB is also as e in Figu e 16, bu in his case some oscilla ions occu du ing he ansi ion.
This issue is due o he λ alue, when la ge alues o he λa e chosen a sha p WoB alue ansi ion is ob ained
in he swi ching ins an o ime (in ou simula ions = 50s).
F om a p ac ical poin o iew, and in o de o a oid oscilla ions in he con ol signal, he pa ame e alues
p oposed in Figu e 15 seem o be mo e app op ia e.
IX. OBSERVER-BASED DESIGN
In his sec ion we p esen some ex ensions o he p e ious con ol which has been s udied and designed unde
he hypo hesis o he measu e o he bi o a ional eloci y ˙ϕb. In his sec ion we i s p o ide an al e na i e way
o ge his measu e ough a s a e obse e .
A. Obse e Design
The obse e is designed on he basis o open-loop equa ion (8). Tha is on
˙x=Ax +B + (Hu)·µ(45)
yo=Cox= ˙ϕ (46)
June 6, 2007 DRAFT
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24
Following [35] we p opose o use he ollowing obse e based on he obse a ion o he o a y angula eloci y
yo= ˙ϕ :
˙
ˆx=Aˆx+B + (Hu)·ˆµ+Ko+βΓΓTCT
o(yo−Coˆx)(47)
˙
ˆµ=βΓTCT
o(yo−Coˆx)(48)
˙
Γ = [A−KoCo] Γ + (Hu)(49)
whe e: Γis a (3 ×1) ime- a ying ec o , β > 0is a posi i e scala , ˆx, and ˆµa e he s a e and ic ion coe icien
es ima es espec i ely, yois he measu ed ou pu , Kois a (3 ×1) obse e ec o gain, and Co= [0,1,0]. No e
ha Co6=C.
The obse e no only p o ides an es ima e o he bi o a ional eloci y ˆx3, bu also p o ide an es ima e o he
ic ion coe icien ˆµwhich can be use ul o o he moni o ing pu poses. The obse e , as indica ed in [35], esul s
in a globally exponen ially s able obse e p o iding he ollowing hypo hesis hold:
•µis cons an
•Exis s a ma ix Kosuch ha (A−KoCo)is s ic ly s able ma ix, i.e. The cons an pai (A, Co)is de ec able.
•u( )is pe sis en ly exci ing, i.e. ∃δ, T > 0such ha he ollowing inequali y is sa is ied:
Z +T
Γ(τ)TCT
oCoΓ(τ)dτ > δ > 0
Le commen he p ac ical implica ion o he p e ious hypo hesis.
The i s hypo hesis assumes ha he ic ion coe icien is cons an , o e en ually slow- ime a ian 4˙µ≈0. No e
ha his app oxima ion is o en assumed in he con ex o obse e design wi h unknown inpu s, bu also in he
con ex o adap i e con ol. He e his hypo hesis means ha he a e o a ia ion o he ock ic ion coe icien does
no exhibi subs an ial changes du ing he d ill-ope a ion. E en i he d illed su aces may ha e di e en ic ion
cha ac e is ics, he a e o pene a ion (d illing-speed) emains small.
The second hypo hesis co espond o he necessi y obse a ion p ope y need o build he obse ed. By inspec ing
his condi ion, we can see ha he sys em obse abili y is in a ian wi h espec he ma ic Aand Co.
Finally, he las p ope y is necessa y o he obse e o con e ge. No e ha as he “adap a ion” is done unde
a single pa ame e µ, he equi ed condi ion is weak and will be simple o ul ill. To see his no e ha he pai
(A−KoCo, H)is con ollable and he pai (A−KoCo, Co)is obse able, hen he pe sis en ly exci ing condi ion
is easily e i ied i he WoB o ce is no equal o ze o, i.e. u( )>0. A de ailed jus i ica ion o his can be ound
in [11].
B. Simula ion wi h he obse ed-based con olle
The o iginal con olle has he o m
˙
˜u=−σ˜u+λ(x3−ωd)·sgn(µ(x3−ωd)) (50)
4In his case, i can also be shown ha s abili y (no asymp o ic) ollows is p ese ed
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[31] Spanos, P.D., Che allie , A.M., Poli is N.P. and Payne M.L. Oil well d illing: A ib a ions pe spec i e. The Shock and Vib a ion Diges ,
35(2):81–99, 2003.
[32] an de V ande, B.L., an Campen, D.H., de K ake , A. An App oxima e Analysis o D y- ic ion-induced S ick-slip Vib a ions by a
Smoo hing P ocedu e. Nonlinea Dynamics, 19:157–169, 1999.
[33] Vidyasaga , M. Nonlinea Sys ems Analysis. P en ice-Hall, second edi ion, 1993.
[34] William Ranney, M. O sho e Oil Technology Recen De elopmen . Noyes Da a Co po a ion, 1979.
[35] Zhang, Q. Adap i e obse e o MIMO linea ime a ying sys ems. INRIA epo , No.4111, 2001.
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