Backstepping boundary observer based-control for hyperbolic PDE in rotary drilling system

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Backstepping boundary observer based-control for

hyperbolic PDE in rotary drilling system

Rhouma Mlayeh, Samir Toumi, Lotfi Beji

To cite this version:

Rhouma Mlayeh, Samir Toumi, Lotfi Beji. Backstepping boundary observer based-control for

hyper-bolic PDE in rotary drilling system. Applied Mathematics and Computation, Elsevier, 2018, 322,

pp.66–78. �10.1016/j.amc.2017.11.034�. �hal-01674970�

Backstepping

boundary

observer

based-control

for

hyperbolic

PDE

in

rotary

drilling

system

Rhouma

Mlayeh

b,∗

_,

_Samir

_Toumi

a,b

_,

_{Lotfi Beji}

a

a IBISC-EA 4526 laboratory, University of Evry, 40 rue du Pelvoux, Evry 91020, France b LIM laboratory, Polytechnic School of Tunisia, BP 743, La Marsa 2078, Tunisia

Itiswellknownthattorsionalvibrationsinoilwellsystemaffectthedrillingdirections andmaybeinherentfordrillingsystems.Thedrillpipemodelisdescribedbysecondorder hyperbolicPartialDifferentialEquation(PDE)withmixedboundaryconditionsinwhicha sliding velocityisconsideredatthe topend. Inthispaper,weconsider theproblemof boundaryobserverdesign forone-dimensionalPDEwiththe usuallyneglected damping term.Themainpurposeistheconstructionofacontrollawwhichstabilizesthedamped wavePDE,using onlyboundarymeasurements.Fromthe Lyapunovtheory,weshow an exponentially vibration stability of the partially equipped oilwell drilling system. The observer-basedcontrollawisfoundusingthebacksteppingapproachforsecond-order hy-perbolic PDE. Thenumerical simulationsconfirm theeffectivenessof theproposed PDE observerbasedcontroller.

1. Introduction

Acommontype ofinstabilityinoilwell drillingsystemisstick-slip oscillation(moredetails in[1]), causedbyfriction betweenthedrillbit andtherockresultingintorsionalvibrations ofthe drillstring,whichreduce penetrationrates and increasedrillingoperationcosts.Thestick-slipphenomenonisanundesirablelimitcycleofthedrillstringvelocityyielding potentiallysignificantdamagesonoilproductionfacilities.Inthelastcentury,manyresearcheffortontheavoidingtorsional vibrations hasbeen proposed [2–10]. Despite the development ofseveral techniques for eliminating torsional vibrations (stick-sliposcillations),nowadaysmanyproblemsremainsopenfordrillingsystems.Thetorsionaldynamicsofadrillstring aremodeledasadampedwavePDEthatgovernsthedynamicsoftheangulardisplacementofthedrillstring.Basedonthe linearizationofitsdynamics,acontrolmethodforthestabilizationofthedrillinginstabilityispresentedin[11].Theenergy functionisproposedbySaldivaretal.in[6] forthetorsionaldistributedmodelallowstofindacontrollawthatensuresthe energydissipationduringthedrilling.

In[12] theauthorsaredevelopedasimplifiedmodel,wherethereisnodampinginthedomainandthedrillbithasno inertia andhaveproposed anoutput feedbackadaptive controller.The anti-dampingwave equation usedinthe paperby Bresch-PietriandKrstic[12] isonlyanapproximationofthemodelcommonlyusedinourpapertoaccountforthestick-slip phenomenoninwhichafrictionODEisusedastheboundaryconditioninstead.

∗ _{Corresponding author.}

Fig. 1. Drilling system.

In thiswork, we are concerned with the problemof boundary observer stabilizationfor a system ofhyperbolic PDE whichdescribes thedrilling systems.Basically, inour designs we usethe backstepping techniques(more details in [13]) andtheLyapunovtheorytostudythestabilityanalysis.Initially,thebacksteppingapproachdevelopedforparabolic equa-tions,ithasbeenappliedtononlinear PDE,first-orderhyperbolic equations,second-orderhyperbolicequations,fluidflow

[13,14].Historically,in1990,thebacksteppingapproachiswellknowninordinarydifferentialequations(ODE)stability.Itis developedbyKokotovic[15] foranalyzing thestability ofnonlinear ordinarydifferential equation.Ithas theability tocope withthecontrolsynthesis,andaround2000thistechniquebecomesausefultoolintheboundarycontrolofPDE[13].The mainpurposesofthisworkare:first,thedesignofanobserverusingonlyboundaryvelocitymeasurementsatthetopand theconstructionofan observationerrorsystem;second,thedevelopmentofacontrol lawtakingintoaccount in-domain dampingusuallyneglected;andfinally,thewell-posedness problemoftheobservertorsionalvibration.We usethe back-steppingapproachtodesignafull-statefeedbackobserverlawthatmakestheclosed-loopsystemexponentiallystable.The stabilityanalysisisconductedwithinfinite-dimensionalbacksteppingtransformations forthedampedwave PDEstateand byconstructingaLyapunovfunctional.

Thepaperisstructuredasfollows.InSection2,werecallthePDEwiththeboundaryconditionsthatpermitstodescribe thetorsionalvibrationproblem.Anobserverbasedcontrol lawispresentedinthissection. InSection3,we findthe out-putinjectiongain andweprove theconvergenceoftheestimationerrorsystemusingLyapunovtheoryandbackstepping technique.ThesimulationresultsaregiveninSection4.Someconcludingremarksandperspectivesarealsointroduced.

2. Boundaryobserverbasedcontrol

2.1. Distributedparametermodel:dampedwaveequation

Amoreexhaustivedescriptionoftherotarysystemcanbefoundin[1].Oneoftheprincipalproblemsistheappearance ofoscillatory behaviors,that causea decreasingofthe drillingperformance fromtheview pointsofdifferentparameters (rotationalspeedofthe bit,rateofpenetration atthe surface)andso provokingthe mechanicalfailure ofthedrillstring. Somecausesofstick-sliposcillationsarebacklashbetweencontactingparts,nonlineardamping,hysteresis,andgeometrical imperfectionswhich are verydifficult tomodel.However, the maincause ofsuch vibrations indrillstring isthe friction appearingbycontactwiththerockformation[16].Accordingly,amodeldescribingthedrillstringbehaviorshouldinclude abit-rockfrictiontorquemodeladequateenoughtoproperlyreproducethiseffect(Fig.1).

Thedynamicofthetorsionalvariable_ϑ(t,

ς

)alongthedrillpipeisgovernedby[4,6,17]:

GJ

ϑ

_ςς

(

t,

ς

)

− I

ϑ

tt

(

t,

ς

)

−

σϑ

t

(

t,

ς

)

=0 (1)

ς

∈

(

0,L

)

,t∈

(

0,+∞

)

,withtheboundaryconditions

GJ

ϑ

_ς

(

t,0

)

=ca

(

ϑ

t

(

t,0

)

−

ω

(

t

))

(2)

GJ

ϑ

_ς

(

t,L

)

+Ib

ϑ

tt

(

t,L

)

=−T

(

ϑ

t

(

t,L

))

(3) whereListhelength ofthedrillpipe,Iistheinertia,Gtheshearmodulus,Ib ischosentorepresenttheassemblyatthe bottom hole,Jthe geometricalmomentofinertia, ca theslidingtorque coefficient,

σ

thedrillstring damping,and

ω

the controlinput(angularvelocityduetotherotarytable).Theextremity

(

ς

=L

)

,issubjecttoatorqueonthebitT

(

∂ϑ_∂t

(

t,L

))

, whichisafunctionofthebitvelocity[18].

Inordertoimproveclarity,weintroducethenormalizedrodlengthx=ςL,andthenextvariablechange[2]:

v

(

t,x

)

=

ϑ



L



I GJt,L

(

1− x

)



, x∈

(

0,1

)

. (4)

Then,thedynamicofthetorsionalvariablereads

v

tt

(

t,x

)

=

v

xx

(

t,x

)

−

ιv

t

(

t,x

)

(5)

v

x

(

t,1

)

=

(

t

)

(6)

v

tt

(

t,0

)

=a

v

x

(

t,0

)

+aF

(

v

t

(

t,0

))

(7) where

(

t

)

= caL GJ



ω

(

t

)

−1 L



GJ I

v

t

(

t,1

)



,

ι

=

σ

L



1 IGJ,F

(

v

t

(

t,0

))

=− L GJT



1 L



GJ I

v

t

(

t,0

)



,anda=LI I_b. Tolinearizethetipboundarycondition(7),weusethenextform[3]

¯

v

(

t,x

)

=

ι

wr

2 x

2_{− F}

₍

_w

r

)

x+wrt+

v

0 (8)

asareferencetrajectory,suchthatwr=

v

¯t

(

t,x

)

.

Thenweobtainthenextlinearizedofequationssystem

v

tt

(

t,x

)

=

v

xx

(

t,x

)

−

ιv

t

(

t,x

)

(9)

v

x

(

t,1

)

=

(

t

)

(10)

v

tt

(

t,0

)

=a

v

x

(

t,0

)

+ab

v

t

(

t,0

)

(11) whereb= ∂F(wr)

∂u andu

(

t

)

=

v

t

(

t,1

)

.

One ofthe mainchallenge duringdrilling operation liesinthe poorknowledge ofthe downholeconditions (pressure andtemperatureconditions,gasandoilratios).Inthenext,weproposeanapproachtoestimateunknownparameterswhile drillingoilwell.Hence, themainpurposeinthisstudy,isthestabilityanalysisoftheobserverPDEwhichencounteredin andrillingsystem.

In thissection, we designan observerforthe systemgivenabove when one boundary measurement isavailable. We assumethatvelocityatx=1ismeasured(i.e.thetopboundarycondition,meaningthedrillstringhead).

We denote the estimates by a widehat, and we construct system behavior that integrates from an output injection term:



v

tt

(

t,x

)

=

v

xx

(

t,x

)

−

ι



v

t

(

t,x

)

(12)



v

x

(

t,1

)

=

(

t

)

−

γ

(

v

(

t,1

)

−

v

(

t,1

))

(13)



v

tt

(

t,0

)

=a

v

x

(

t,0

)

+ab

v

t

(

t,0

)

(14)

2.2.Observertargetsystemandbacksteppingtransformation

Thissection showstheimportance of observertarget system, backsteppingtechniques,andthe Lyapunovtheory, pro-vidingausefulanalysisforstabilityin oilwell drillingsystem. Here, themain purposeisto finda controllaw



(t) that transforms(12)–(14) toanextdesignedobservertargetsystem,

 wtt

(

t,x

)

=wxx

(

t,x

)

−

ι

wt

(

t,x

)

(15)  wx

(

t,1

)

=0 (16)  wtt

(

t,0

)

=ae−ηwx

(

t,0

)

−

(

2a



+1

)

wt

(

t,0

)

. (17) The

η

and



parameterswillbedefinedbythefollowingLemma.

Lemma1. Letusintroducethefunction

V

(

t

)

= 1 2

1 0



e−η

(

wx

)

2+e−η

(

wt

)

2+



e−ηx

(

1− x

)

wtwx



dx



+1 a

(

wt

(

t,0

))

2_, with 1

2>



>0,

η

≤ −(2+1−x),suchthatx∈[0,1[,andthenorm



where





2

₍

_t

₎

₌

_

_w

t



2L2_([0,1])+



wx



2L2_([0,1])+

|

wt

(

t,0

)

|

2.

Thenn1



2

(

t

)

≤ V

(

t

)

≤ n2



2

(

t

)

wheren1=min

{

e

−η 2 − e−η 4 , 1 a

}

andn2=max

{

e −η 2 + e−η 4 , 1 a

}

.

Proof.UsingtheCauchy–SchwarzandYoung’sinequalities,weobtain

V

(

t

)

= 1₂

1 0



e−η

(

wx

)

2+e−η

(

wt

)

2+



e−ηx

(

1− x

)

wtwx



dx



+1_a

(

wt

(

t,0

))

2 ≥ e−η 2



wt



2₊e−η 2



wx



2₊1 a

|

wt

(

t,0

)

|

2₋



e−η 2 1 0

|

 wtwx

|

dx ≥



e−η 2 −



e−η 4

(



wt



2+



wx



2

)

+ 1 a

|

wt

(

t,0

)

|

2 ≥ min



e−η 2 −



e−η 4 , 1 a







2

₍

_t

₎

_. Ontheotherhand,

V

(

t

)

= 1 2

1 0



e−η

(

wx

)

2+e−η

(

wt

)

2+



e−ηx

(

1− x

)

wtwx



dx+1 a

(

wt

(

t,0

))

2



≤ e−η 2



wt



2₊e−η 2



wx



2₊1 a

|

wt

(

t,0

)

|

2₊



e−η 2 1 0

|

˜ wtw˜x

|

dx ≤



e−η 2 +



e−η 4

(



wt



2+



wx



2

)

+ 1 a

|

wt

(

t,0

)

|

2 ≤ max



e−η 2 +



e−η 4 , 1 a







2

₍

_t

₎

_.

Thenn1



2

(

t

)

≤ V

(

t

)

≤ n2



2

(

t

)

withn1=min

{

e

−η 2 − e−η 4 , 1 a

}

andn2=max

{

e −η 2 + e−η 4 , 1 a

}

.  Now,weareinterestedinthestabilizationoftheobservertargetsystem.

Theorem1. (Observertarget systemstability)Considersystem (15)–(17),withinitial conditionw0=w

(

0,x

)

∈L2

(

[0,1]

)

.Then

thezeroequilibriumofthesystem(15)–(17)isexponentiallystableinthesenseofthenextnorm 



2

₍

_t

₎

₌

_

_w

t



2_L2_([0,1])+



wx



2_L2_([0,1])+

|

wt

(

t,0

)

|

2.

Proof.In order to prove the observer target system stability, let us consider the proposed V(t) asa Lyapunov function candidate, V

(

t

)

= 1 2

1 0



e−η

(

wx

)

2+e−η

(

wt

)

2+



e−ηx

(

1− x

)

wtwx



dx



+1 a

(

wt

(

t,0

))

2

DifferentiatingVwithrespecttotime,weget ˙ V

(

t

)

= 1 0



e−ηwtxwx+e−ηwttwt+1 2



e−η x

₍

_{1− x}

₎

_w ttwx+1 2



e−η x

₍

_{1− x}

₎

_w txwt



dx +1 awtt

(

t,0

)

wt

(

t,0

)

=−

ι

e−η 1 0  w2t + 1 0



e−ηwtxwx+e−ηwxxwt+ 1 2



e −ηx

₍

_{1− x}

₎

_w xxwx +1 2



(

1− x

)

wtxwt



dx+1 a

(

wtt

(

t,0

)

wt

(

t,0

))

−

ι

2 1 0 e−ηx

₍

_{1− x}

₎

_w twxdx =− e−η_wt

₍

_t,0

₎

_wx

₍

_t,0

₎

₋



4wx

(

t,0

)

2₋



4wt

(

t,0

)

2₋ 1 0 −



−

η

(

1− x

)

2 e−η xw2x 2 dx −2a



+1 a w 2 t

(

t,0

)

+e−ηwt

(

t,0

)

wx

(

t,0

)

− 1 0 −



−

η

(

1− x

)

2 e−η xw2t 2 dx−

ι

2 1 0



e−ηx

₍

_{1− x}

₎

_w twxdx−

ι

1 0 e−ηw2 tdx ≤ −eη 1 0 e−ηw 2 x 2 dx− 1 aw 2 t

(

t,0

)

−₂

ι

1 0



e−ηx

₍

_{1− x}

₎

_w twxdx−

ι

1 0 e−ηw2 tdx ≤ − min

(

eη,

ι

,1

)

V

(

t

)

. ByLemma1,wehave n1



2

(

t

)

≤ V

(

t

)

≤ n2



2

(

t

)

. Hencethereexistc>0andk≥ 0suchthat



(

t

)

≤ ce−kt

₍

 ₀

₎

_.

This implies that the observer target system (15)–(17) is exponentially stableat the equilibriumin the sense of the



 norm. 

In orderto convert the observerplant intothe observer target systems (i.e.

v

(

t,x

)

−→w

(

t,x

)

), we consider the next backsteppingtransformation  w

(

t,x

)

=

v

(

t,x

)

− x 0 k

(

x,

ξ

)



v

(

t,

ξ

)

d

ξ

−

β

(

x

)



v

(

t,0

)

− x 0 p

(

x,

ξ

)



v

t

(

t,

ξ

)

d

ξ

− x 0 l

(

x,

ξ

)



v

_ξ

(

t,

ξ

)

d

ξ

. (18)

Pluggingthebacksteppingtransformation(18) intotheobservertargetsystem(15)–(17),integratingbyparts,andusingthe boundaryconditions,weobtain:

• Kernelsurfaceterms(x,

ξ

):

l_ξξ

(

x,

ξ

)

=lxx

(

x,

ξ

)

, (19)

k_ξξ

(

x,

ξ

)

=kxx

(

x,

ξ

)

, (20)

p_ξξ

(

x,

ξ

)

=pxx

(

x,

ξ

)

, (21) • Kerneldiagonalterms(x,x):

lx

(

x,x

)

=0, kx

(

x,x

)

=0, px

(

x,x

)

=0. (22) • Kernelverticalterms(x,0)andpoint-wiseterms(0,0):

l_ξ

(

x,0

)

=k

(

x,0

)

, k_ξ

(

x,0

)

=

β

₍

_x

₎

_{, (23)}

p

(

x,0

)

=0, p_ξ

(

x,0

)

=0 (24)

l

(

x,0

)

=

β

(

x

)

,l

(

0,0

)

=

β

(

0

)

(25)

ThekernelofthebacksteppingtransformationsatisfiesaninterestingsystemofwavePDEwhichiseasilysolvable.These equationsaredefinedonatriangulardomain



=

{

(

x,

ξ

)

∈R2_:0≤

_ξ

_{≤ x≤ 1}

_}

_.

Atthisstep,introducingthebacksteppingtransformation(18) into(16),wededucethenextcontrollaw

(

t

)

= 1 1− l

(

1,1

)



k

(

1,1

)



v

(

t,1

)

+ 1 0 kx

(

1,

ξ

)



v

(

t,

ξ

)

d

ξ

+p

(

1,1

)



v

t

(

t,1

)

+ 1 0 px

(

1,

ξ

)



v

t

(

t,

ξ

)

d

ξ

+ 1 0 lx

(

1,

ξ

)



v

_ξ

(

t,

ξ

)

d

ξ

+

β

(

1

)



v

(

t,0

)



+

γ

(

v

(

t,1

)

−

v

(

t,1

))

. (27)

Itisworthnoticingthat1− l

(

1,1

)

=1−

β

(

0

)

=0andcannotbezerosince

β

(0)=1here.

It remains to study the behavior of the observer plant system from the inverse backstepping transformation (i.e. 

w

(

t,x

)

→

v

(

t,x

)

) andthe stabilityconditionsunder thecontrol law(27).Letusconsider theinverse backstepping trans-formationasfollows 

v

(

t,x

)

=w

(

t,x

)

+ x 0 e

(

x,

ξ

)

w

(

t,

ξ

)

d

ξ

+ x 0 f

(

x,

ξ

)

wt

(

t,

ξ

)

d

ξ

+

π

(

x

)

w

(

t,0

)

+ x 0 h

(

x,

ξ

)

w_ξ

(

t,

ξ

)

d

ξ

. (28)

Introducingtheexpression(28) intotheobserverplantsystem(12)-(14),wefind: • Kernelsurfaceterms(x,

ξ

):

h_ξξ

(

x,

ξ

)

=hxx

(

x,

ξ

)

,

e_ξξ

(

x,

ξ

)

=exx

(

x,

ξ

)

,

f_ξξ

(

x,

ξ

)

= fxx

(

x,

ξ

)

, • Kerneldiagonalterms(x,x):

hx

(

x,x

)

=0, ex

(

x,x

)

=0, fx

(

x,x

)

=0. • Kernelverticalterms(x,0)andpoint-wiseterms(0,0):

e

(

x,0

)

=h_ξ

(

x,0

)

, e_ξ

(

x,0

)

=

π

₍

_x

₎

_,

_π

₍

₀

₎

₌₀

f

(

x,0

)

=0, h

(

x,0

)

=

π

(

x

)

, f_ξ

(

x,0

)

=0

e

(

0,0

)

=h_ξ

(

0,0

)

=0, h

(

0,0

)

=

π

(

0

)

=−1.

Itiseasilytoverifythatthisequationsaredefinedonatriangulardomain



₌

{

(

x_,

ξ

)

_∈R2_:0≤

ξ

_{≤ x≤ 1}

}

_. ThemainresultregardingtheobserverplantsystemstabilityissummarizedinthenextTheorem.

Theorem2. (Observerplantsystemstability) Considersystem (12)–(14) withinitialcondition

v

0∈L2

(

[0,1]

)

,andwithcontrol

law(27)where thekernelsk,p,andlareobtainedfrom(19)–(26).Thenthesystem(12)–(14)isexponentiallystableatthezero equilibriuminthesenseofthenextnorm





2

₍

_t

₎

₌

_

_

_v

₍

_t,.

₎

_

2

L2_([0,1])+





v

t

(

t,.

)



2L2_([0,1])+





v

x

(

t,.

)



2L2_([0,1])+

|



v

t

(

t,0

)

|

2.

Proof.Firstly, wedenotebyL2_=L2

₍

_[0,1]

₎

_{andletintroducethenextnorms(forexample)as:}

_β

∞=supx∈[0,1]

|

β

(

x

)

|

,k∞= max₍x,ξ )∈

|

k

(

x,

ξ

)

|

2

2,andsoonforl∞,(pξξ)∞,p∞,where

|

k

(

x,

ξ

)

|

22denotestheclassicaloperatornorm.Wewillprove thatthereexist

ζ

1>0and

ζ

2>0suchthat

ζ

1

(

 t

)

≤ 

(

t

)

≤

ζ

2

(

 t

)

.

Recallthatp_ξ

(

x,0

)

=0,p

(

x,0

)

=0,px

(

x,x

)

=0,l

(

x,0

)

=

β

(

x

)

.Consequently,wt isrewritteninthisform

 wt

(

t,x

)

=

v

t

(

t,x

)

− x 0 k

(

x,

ξ

)



v

t

(

t,

ξ

)

d

ξ

− p

(

x,x

)



v

x

(

t,x

)

− x 0 p_ξξ

(

x,

ξ

)



v

(

t,

ξ

)

d

ξ

+ x 0

ι

p

(

x,

ξ

)



v

t

(

t,

ξ

)

d

ξ

− l

(

x,x

)



v

t

(

t,x

)

+ x 0 l_ξ

(

x,

ξ

)



v

t

(

t,

ξ

)

d

ξ

−

β

(

x

)



v

t

(

t,0

)

.

UsingCauchy–Schwarz’sinequalities,weprove



wt

(

t,.

)



2_L2 ≤

(

1+k∞+l∞+

(

lξ

)

∞+

ι

p∞

)





v

t

(

t,.

)



2_L2+p∞





v

x

(

t,.

)



2_L2 +

((

p_ξξ

)

_∞





v

(

t,.

)



2

L2+

β

∞

|



v

t

(

t,0

)

|

2

)

wherea1=max

{

1+k∞+l∞+

(

lξ

)

∞+

ι

p∞,

(

pξξ

)

∞,p∞,

β

∞

}

. As

v

(

t,0

)

=

v

(

t,x

)

−x

0

v

y

(

t,y

)

dy,wefind



wx

(

t,.

)



2L2≤ a2

(





v

x

(

t,.

)



2L2+





v

(

t,.

)



2_L2+





v

t

(

t,.

)



2L2

)

wherea2=max

{

1+l∞+

(

lx

)

∞+

β

∞ ,k∞+

(

kx

)

∞+

β

∞ ,p∞+

(

px

)

∞

}

.

Also,wehave

|

wt

(

t,0

)

|

2≤

|



v

t

(

t,0

|

2.Hence,thereexist

ζ

1>0suchthat

ζ

1

(

 t

)

≤ 

(

t

)

. Recallthattheinversebacksteppingtransformationisgivenby



v

(

t,x

)

=w

(

t,x

)

+ x 0 e

(

x,

ξ

)

w

(

t,

ξ

)

d

ξ

+

π

(

x

)

w

(

t,0

)

+ x 0 f

(

x,

ξ

)

wt

(

t,

ξ

)

d

ξ

+ x 0 h

(

x,

ξ

)

w_ξ

(

t,

ξ

)

d

ξ

. Asw

(

t,0

)

=w

(

t,x

)

−x

0wy

(

t,y

)

dy,usingPoincare’sinequality,weobtain





v

(

t,.

)



2 L2≤ a3

(



wx

(

t,.

)



2_L2+



wt

(

t,.

)



2_L2

)

, wherea3=max

{

a0

(

1+e∞

)

+

π

∞

(

1+a0

)

+h∞,f∞

}

>0,a0>0. Besides,as f

(

x,0

)

=0,h

(

x,0

)

=

π

(

x

)

, f_ξ

(

x,0

)

=0, fx

(

x,x

)

=0 weget,





v

t

(

t,.

)



_L22 ≤ a4

(



wt

(

t,.

)



2_L2+



wx

(

t,.

)



2_L2

)

, wherea4=max

{

1+e∞+h∞+

(

hξ

)

∞+

ι

f∞,f∞+a0

(

f_ξξ

)

∞

}

≥ 0. Also,asw

(

t,0

)

=w

(

t,x

)

−x 0wy

(

t,y

)

dywefind





v

x

(

t,.

)



L22 ≤ a5

(



wx

(

t,.

)



2L2+



wt

(

t,.

)



2L2

)

, wherea5=max

{

1+a0e∞+a0

(

ex

)

∞+h∞+

(

hx

)

∞+

π

∞

(

1+a0

)

,f∞+

(

fx

)

∞

}

.

Finally,wehave

|



v

t

(

t,0

)

|

2≤ 4

|

wt

(

t,0

)

|

2.Accordingly,thereexits

ζ

2>0suchthat

(

 t

)

≤

ζ

2

(

 t

)

. Thisimpliesthatthesystem(12)–(14) isexponentiallystableinthesenseofthe



norm. 

Remark1. TheproofofTheorem2 isperformedinthreesteps:first,thestabilityoftheobservertargetsystem;second,the mappingbetweenthe observerplantsystemandthe observertarget,andthe computationoftheobserverbased control law;finally,thestabilityoftheobserverplantsystem.

3. Outputinjectiongain

The first goal of this section is to prove the existence anduniqueness solution using Lumer–Phillips’s theorem. The secondoneisthestabilitystudyoftheestimationerrorsystem.

3.1. Well-posednessproblem

Inthenext,weusesemigrouptheory(Furtherdiscussioninthistheoryin[19])toprovetheexistenceanduniquenessof theproposedobserversolutions.Then,byprovingtheexistenceanduniquenessoftheestimationerrormodel,weconclude theexistence anduniqueness oftheproposed observersystem. Inaddition,duetothe presenceofa nonlinearand com-plexrelationresultingfromthebit-rockinteractionatthetipboundary,thewell-posednessoftheestimationerrorsystem becomesnottrivial.Hence,inthenext,wetreatthiscontributionusingthesemi-grouptheory.

We denote the estimationerror by

v

˜=

v

−

v

.Let T>0, the naturalsolution of the Cauchy problemis written in this form ˜

v

tt

(

t,x

)

=

v

˜xx

(

t,x

)

−

ι

˜

v

t

(

t,x

)

(29) ˜

v

t

(

t,1

)

=−



IGJ ca ˜

v

x

(

t,1

)

−

γ

v

˜

(

t,1

)

(30) ˜

v

tt

(

t,0

)

=a

v

˜x

(

t,0

)

+aF

(

v

˜t

(

t,0

))

(31) ˜

v

(

0,x

)

=

v

˜0

₍

_x

₎

_,

_v

_˜ t

(

0,x

)

=˜

v

1

(

x

)

(32) wherex∈(0,1),t∈(0,T),

v

˜0_∈K:=

_{

_v

_˜∈H1

₍

_0,1

₎

_{; ˜}

_v

0

₍

₀

₎

₌₀

_}

_,and

_v

_˜1_∈L2

₍

_0,1

₎

_.

ThevectorspaceKisequippedwiththescalarproduct



v

˜1

₍

_t,x

₎

_,

_v

_˜2

₍

_t,x

₎



K = 1 0 ˜

v

1 x

(

t,x

)

v

˜2x

(

t,x

)

dx. ItisobviousthatKisaHilbertspace.

LetusintroduceY=

(

v

˜

(

t,x

)

,

v

˜t

(

t,x

)

,

v

˜

(

t,1

)

,˜

v

t

(

t,0

))

T.Eqs.(29)–(32) canbecompactlywrittenas



˙ Y

(

t

)

=AY

(

t

)

+H

(

Y

(

t

))

+f

(

t

)

Y

(

0

)

=Y0 (33) where A=

⎛

⎜

⎜

⎝

0 1 0 0

∂

xx −

ι

0 0 √ IGJ ca



δ

1

(

x

)

,.



−

γ

0 0 0 −a



δ

0

(

x

)

,.



0 0 0

⎞

⎟

⎟

⎠

, H

(

Y

(

t

))

=

⎛

⎜

⎝

0 0 0 aF

(

˜

v

t

(

t,0

))

⎞

⎟

⎠

, and f

(

t

)

=

⎛

⎜

⎜

⎝

0 0 L



I GJ

ω

(

t

)

0

⎞

⎟

⎟

⎠

such as

δ

denotes the Diracfunctionforwhich



δ

₁

(

x

)

_,

v

˜

(

t_,x

)



_=−˜

v

x

(

t,1

)

and



δ

0

(

x

)

,

v

˜

(

t,x

)



=−˜

v

x

(

t,0

)

.

Firstly,letusconsidertheproblem(33) withH

(

Y

)

=0and f

(

t

)

=0,consequentlywehavethenextTheorem.

Theorem3. TheoperatorAgeneratesaC0semigroupS(t),t≥ 0ofcontractionsonE.

Proof.Letusconsiderthefollowingspace

E₌

⎧

⎪

⎨

⎪

⎩

⎛

⎜

⎝

˜

v

(

t,x

)

˜

v

t

(

t,x

)

˜

v

(

t_,1

)

˜

v

t

(

t,0

)

⎞

⎟

⎠

,

v

˜∈K_,

v

˜t∈L2

(

[0,1]

)

,˜

v

(

t,1

)

∈R,˜

v

t

(

t,0

)

∈R

⎫

⎪

⎬

⎪

⎭

. ThisvectorspaceEisequippedwiththeinner-product

!

⎛

⎜

⎝

˜

v

1

₍

_t,x

₎

˜

v

1 t

(

t,x

)

˜

v

1

(

_t,1

)

˜

v

1 t

(

t,0

)

⎞

⎟

⎠

,

⎛

⎜

⎝

˜

v

2

₍

_t,x

₎

˜

v

2 t

(

t,x

)

˜

v

2

(

_t,1

)

˜

v

2 t

(

t,0

)

⎞

⎟

⎠

"

E =



v

˜1_,

_v

_˜2



K+



v

˜t1,

v

˜t2



L2_[0,1]+



v

˜1

(

t,1

)

,

v

˜2

(

t,1

)



_R +



v

˜1 t

(

t,0

)

,˜

v

2t

(

t,0

)



R. Wedenoteby



.



thenorminEassociatedtothisscalarproduct.

LetA:D(A)⊂ E→Ebethelinearoperatordefinedby D

(

A

)

₌

{

⎛

⎜

⎝

˜

v

(

t,x

)

˜

v

t

(

t,x

)

˜

v

(

t_,1

)

˜

v

t

(

t,0

)

⎞

⎟

⎠

∈E_,

v

˜∈H2

₍

_0,1

₎

_,

_v

_˜ t∈K,

v

˜x

(

t,1

)

=˜

v

x

(

t,0

)

=0,

v

˜

(

t,1

)

∈R,

v

˜t

(

t,0

)

∈R

}

. Wehave A

⎛

⎜

⎝

˜

v

(

t,x

)

˜

v

t

(

t,x

)

˜

v

(

t,1

)

˜

v

t

(

t,0

)

⎞

⎟

⎠

=

⎛

⎜

⎝

˜

v

t

(

t,x

)

˜

v

xx

(

t,x

)

−

ι

v

˜t

(

t,x

)

−√IGJ ca

v

˜x

(

t,1

)

−

γ

v

˜

(

t,1

)

a

v

˜x

(

t,0

)

⎞

⎟

⎠

. Moreover



AY,Y



E=−

ι

1 0 ˜

v

2 tdx−

γ

v

˜

(

t,1

)

2≤ 0,

∀

Y∈D

(

A

)

.

Itiseasytoverifythat

∀

y=

⎛

⎜

⎝

f1 f2 f3 f4

⎞

⎟

⎠

∈E,thereexistsw=

⎛

⎜

⎝

w1 w2 w3 w4

⎞

⎟

⎠

∈D

(

A

)

suchthat w− Aw=y.Then,D(A) isdenseinE andAisclosed.Hence,usingtheLumer–Phillipstheorem(TheoremA.4in[20])Aistheinfinitesimalgeneratorofastrongly continuousgroupofisometriesS(t),t≥ 0,onE. 

Now,wearegoingtoprovetheexistenceanduniquenessofthesystem(33) withH(Y)andf(t)aredifferentfromzero.

Theorem4. Letf∈L1_{([0,T],E)andY}

0∈D(A),thentheproblemY˙

(

t

)

=AY

(

t

)

+H

(

Y

)

+f

(

t

)

hasauniquesolution

Y∈C1

(

[0,T],E

)

#C0

(

[0,T],D

(

A

))

givenby: Y

(

t

)

=S

(

t

)

Y

(

0

)

+ t 0 S

(

t− s

)(

H

(

Y

(

s

))

+f

(

s

))

ds

ToprovetheTheorem4,weneedthenextlemmas:

Lemma2. ThenonlinearoperatorH(Y)isdissipativeandlocallyLipschitz.

Proof. RecallthatthenonlinearfunctionFduetothebit-rockcontactisgivenby

F

(

v

˜t

(

t,0

)

)

=− L GJT



1 L



GJ I

v

˜t

(

t,0

)



, =− cb



1 GJI˜

v

t

(

t,0

)

− L GJWobRb

μ



1 L



GJ I

v

˜t

(

t,0

)



sgn



1 L



GJ I

∂

t

v

˜t

(

t,0

)



where

μ

isas[6]

μ



1 L



GJ I ˜

v

t

(

t,0

)



=

μ

cb+

(

μ

sb−

μ

cb

)

e −γb υf| 1 L  GJ I˜vt(t,0)| . Aftercomputing



H

(

Y

(

t

)))

,Y

(

t

)



E =F

(

v

˜t

(

t,0

))

v

˜t

(

t,0

)

=−cb



1 GJI

(

∂

t

v

˜

(

t,0

))

2₋ L GJWobRb



μ

cb+

(

μ

sb−

μ

cb

)

e −γb υf| 1 L  GJ I˜vt(t,0)|



×sgn



1 L



GJ I

v

˜t

(

t,0

)



˜

v

t

(

t,0

)

.

Consequently,



H(Y(t)),Y(t)



E≤ 0as

μ

sb≥

μ

cb.ThisimpliesthattheoperatorH(Y)isdissipative.

It’seasytoverifythat,HislocallyLipschitz.Accordingly,theoperatorH(Y)isdissipativeandlocallyLipschitz. 

Lemma3. For any functionf∈L1_{([0, T],E),and any initialconditionY}

0∈D(A), theproblem (33)has atmostone solutionin

C1

₍

_[0,T],E

₎

$_C0

₍

_[0,T],D

₍

_A

₎₎

_.

Proof. Suppose Y1 and Y2 are two solutions of (33) in the class C1

(

[0,T],E

)

$C0

(

[0,T],D

(

A

))

. Then the difference Y=

Y1− Y2isanelementofC1

(

[0,T],E

)

$C0

(

[0,T],D

(

A

))

whichsatisfiesthenextsystem



˙

Y

(

t

)

=AY

(

t

)

+H

(

Y

(

t

))

Y

(

0

)

=0 . (34)

SinceAandH(Y)aredissipative,weget



_Y˙

₍

_t

₎

_,Y

₍

_t

₎



_E=



_AY

₍

_t

₎

_,Y

₍

_t

₎



_E+



_H

₍

_Y

₍

_t

₎₎

_,Y

₍

_t

₎



_{E≤ 0.} Then 1 2 d dt



Y

(

t

)



E≤ 0⇒



Y

(

t

)



E=0.

Hence,theproblem(34) hasauniquesolutionY

(

t

)

=0foreveryY0∈D(A),thusprovesthatY1=Y2andshowsthat(33) has asolutioninC1

₍

_[0,T],E

₎

$_C0

₍

_[0,T],D

₍

_A

₎₎

_{,thenthisoneisunique. }

Proof. (ofTheorem4)ByapplyingthetwoLemmasgivenaboveandfromresultsgivenin(Theorem4.2in[21],[19,20,22]), itiseasytoprovethatoursystem(33) hasauniquesolution. 

3.2. Stabilityanalysisofestimationerrorsystem

Here,themaincontributionisthestabilizationoftheestimationerrorsystemandthecomputingoftheoutputinjection gain

γ

.Letusrecalltheobserverbasedcontrolsystem



v

tt

(

t,x

)

=

v

xx

(

t,x

)

−

ι



v

t

(

t,x

)



v

x

(

t,1

)

=

(

t

)

−

γ

(

v

(

t,1

)

−

v

(

t,1

))



v

tt

(

t,0

)

=a

v

x

(

t,0

)

+ab

v

t

(

t,0

)

.

Inorderto fructifythe controllaw(27),itisimportanttoidentifythe outputinjectiongain

γ

.The outputinjectiongain

γ

shouldbedesignedusingthebacksteppingprocedure(moredetailsin[13]).Followingtheabovementionedmethod,the output injectionis applied in theboundary aswell asinthe whole spacial domain(

∀

x_∈[0; 1]). The analysistakes into accounttheestimatederror%

v

=

v

−

v

isas

%

v

x

(

t,1

)

=−

γ

%

v

(

t,1

)

. (36)

%

v

tt

(

t,0

)

=a%

v

x

(

t,0

)

+ab%

v

t

(

t,0

)

(37)

Inordertogettheoutput injectiongain

γ

thatensurethat theestimationerrorsystemgotozero,we introducea back-steppingtransformationtoconverttheerrorsystemintothenexterrortargetsystem(i.e.%

v

(

t,x

)

→w%

(

t,x

)

)

% wtt

(

t,x

)

=w%xx

(

t,x

)

−

ι

w%t

(

t,x

)

(38) % wx

(

t,1

)

=0 (39) % wtt

(

t,0

)

=ae−ηw%x

(

t,0

)

−

(

2a



+1

)

w%t

(

t,0

)

. (40) Weproposethefollowingbacksteppingtransformation

%

v

(

t,x

)

=w%

(

t,x

)

− 1 x A

(

x,

ξ

)

w%

(

t,

ξ

)

d

ξ

− 1 x B

(

x,

ξ

)

w%t

(

t,

ξ

)

d

ξ

− 1 x C

(

x,

ξ

)

w%_ξ

(

t,

ξ

)

d

ξ

. (41)

Lemma4. Letusintroducethefunction

V

(

t

)

= 1 2

1 0



e−η

(

w˜x

)

2+e−η

(

w˜t

)

2+



e−ηx

(

1− x

)

w˜tw˜x



dx+1 a

(

w˜t

(

t,0

))

2



with 1

2>



>0,

η

≤ −₍21+−x),inwhichx∈[0,1[,andthe



˜ normwhere

˜



2

₍

_t

₎

₌

_

_w% t



2_L2_([0,1])+



w%x



2_L2_([0,1])+

|

w%t

(

t,0

)

|

2. Then n1



˜2

(

t

)

≤ V

(

t

)

≤ n2



˜2

(

t

)

wheren1=min

{

e−2η−e −η 4 ,21a

}

andn2=max

{

e−2η+e −η 4 ,21a

}

Proof.TheproofissimilartotheoneofLemma1. 

Ourfirstresult,inthissection,onstabilizationisgivenbythefollowingTheorem.

Theorem5. (Estimationerrortargetsystemstability)Considersystem(38)–(40),withinitialconditionw˜0=w˜

(

0,x

)

∈L2

(

[0,1]

)

.

Thenthezeroequilibriumof(38)–(40)isexponentiallystableinthesenseofthenextnorm ˜



2

₍

_t

₎

₌

_

_w%

t



2L2_([0,1])+



w%x



2L2_([0,1])+

|

w%t

(

t,0

)

|

2.

Proof.TheproofissimilartotheoneofTheorem1. 

Togettheoutputinjectiongain

γ

thatguaranteethattheestimationerrordecaystozero,weuseabackstepping trans-formation.Then,plugging(41) into(35)–(37),weobtainthenextkernelPDE

Cxx

(

x,

ξ

)

=C_ξξ

(

x,

ξ

)

Axx

(

x,

ξ

)

=A_ξξ

(

x,

ξ

)

Bxx

(

x,

ξ

)

=B_ξξ

(

x,

ξ

)

withthenextboundaryconditions

Cx

(

x,x

)

=0, Ax

(

x,x

)

=0, Bx

(

x,x

)

=0 A_ξ

(

x,1

)

=B_ξ

(

x,1

)

=C

(

x,1

)

=0, B

(

1,1

)

=B

(

0,0

)

=A

(

0,0

)

=0, A_ξξ

(

0,

ξ

)

=

(

ι

+ab

)

B_ξξ

(

0,

ξ

)

+aAx

(

0,

ξ

)

C_ξξ

(

0,

ξ

)

=aCx

(

0,

ξ

)

, C

(

0,0

)

=−1 B_ξξ

(

0,

ξ

)

=

(

ι

+ab

)

A

(

0,

ξ

)

−

(

ι

+ab

)

C_ξ

(

0,

ξ

)

+aBx

(

0,

ξ

)

−

(

ι

2+

ι

ab

)

B

(

0,

ξ

)

.

Fig. 2. Stabilization at the bottom extremity of  vt(t, 0) .

Plugging(41) into(36),we findtheoutputinjectiongain

γ

=A

(

1,1

)

.Itiseasy toverifythatC(x,

ξ

), A(x,

ξ

) andB(x,

ξ

) satisfiesawavePDEwhichthegeneralsolutionisgivenbyF

(

x,

ξ

)

=

(

x−

ξ

)

+

(

x−

ξ

)

.Then,thisequationsaredefined onatriangulardomain



₌

{

(

x_,

ξ

)

_∈R2_:0≤

_ξ

_{≤ x≤ 1}

_}

_.

Thebacksteppingtransformationshouldbe invertible.Hence,toconvert(35)–(37) into(38)–(40) (i.e.w%

(

t,x

)

→%

v

(

t,x

)

), weintroducethenextinversebacksteppingtransformation:

% w

(

t,x

)

=%

v

(

t,x

)

+ 1 x M

(

x,

ξ

)

%

v

(

t,

ξ

)

d

ξ

+ 1 x N

(

x,

ξ

)

%

v

t

(

t,

ξ

)

d

ξ

+ 1 x J

(

x,

ξ

)

%

v

_ξ

(

t,

ξ

)

d

ξ

. (42)

Plugging(42) into(38)–(40),weobtainthenextkernelPDE

M_ξξ

(

x,

ξ

)

=Mxx

(

x,

ξ

)

N_ξξ

(

x,

ξ

)

=Nxx

(

x,

ξ

)

J_ξξ

(

x,

ξ

)

=Jxx

(

x,

ξ

)

withthenextboundaryconditions

0=Mx

(

x,x

)

, Jx

(

x,x

)

=0, Nx

(

x,x

)

=0 0=M

(

x,1

)

− J_ξ

(

x,1

)

, 0=N

(

x,1

)

=J

(

x,1

)

0=M_ξ

(

x,1

)

, N_ξ

(

x,1

)

=0 1=J

(

0,0

)

,M

(

0,0

)

=0,N

(

0,0

)

=0 0=

(

2a



+1−

ι

)

N_ξξ

(

0,

ξ

)

+M_ξξ

(

0,

ξ

)

− ae−η_Mx

₍

_0,

_ξ

₎

0=−ae−η_Nx

₍

_0,

_ξ

₎

₊

₍

_ι

₋

₍

_2a



₊₁

₎₎

_J ξ

(

0,

ξ

)

+Nξξ

(

0,

ξ

)

+

(

2a



+1−

ι

)

M

(

0,

ξ

)

+

(

ι

2₋

₍

_2a



₊₁

₎

_ι

₎

_N

₍

_0,

_ξ

₎

0=−ae−η_Jx

₍

_0,

_ξ

₎

_+Jξξ

₍

_0,

_ξ

₎

_.

It is easy to findthe existence and uniqueness of thesolution of thekernel PDE. Then, thisequations are definedon a triangulardomain



=

{

(

x,

ξ

)

∈R2_:0≤

_ξ

_{≤ x≤ 1}

_}

_.