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

aIBISC-EA 4526 laboratory, University of Evry, 40 rue du Pelvoux, Evry 91020, France

bLIM 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-orderhy- 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.

E-mail address: rhouma.mlayeh@ipeit.rnu.tn (R. Mlayeh).

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,thebacksteppingapproachdevelopedforparabolicequa- 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 usetheback- 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).

Thedynamicofthetorsionalvariableϑ(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,I_b 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=^ς_L,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)= ^c_GJ^aL

ω

(^t)−¹_L

GJ I

v

t(^t,1)

,

ι

=

σ

^L

1

IGJ,F(

v

t(^t,0))=−_GJ^LT

1 L

GJ I

v

t(^t,0)

,anda=^LI_I

b. Tolinearizethetipboundarycondition(7),weusethenextform[3]

v

¯

(

^t,x

)

=

ι

^wr

2 x²−F

(

^wr

)

^x+wrt+

v

0 (8)

asareferencetrajectory,suchthatwr=

v

¯t(^t,x)^.

Thenweobtainthenextlinearizedofequationssystem

v

tt

(

^t,x

)

⁼

v

xx

(

^t,x

)

⁻

ι v

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_∂^(w_u^r) 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⁾

where

γ

^{istheoutputinjectiongaintobedesigned.}

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 ¹₂>

>0,

η

≤ −₍²⁺_1−x),suchthatx∈[0,1[,andthenorm^where

²

(

^t

)

=

^wt

²L²(^[0,1])+

^wx

²L²(^[0,1])+

|

^wt

(

^t,0

) |

^2.

Thenn₁²(^t)≤V(^t)≤n₂²(^t)^wheren1=min

{

^e−2^η−^e₄^−η,¹a

}

^andn2=max

{

^e−2^η+^e₄^−η,¹a

}

. Proof.UsingtheCauchy–SchwarzandYoung’sinequalities,weobtain

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−₂^η

^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−₂^η

^wx

²⁺¹_a

|

^wt

(

t,0

) |

²⁺

^e−η

2 1

0

|

^w˜tw˜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

)

^.

Thenn₁²(^t)≤V(^t)≤n₂²(^t)^withn1=min

{

^e−2^η−^e₄^−η,¹a

}

^andn2=max

{

^e−2^η+^e₄^−η,¹a

}

. Now,weareinterestedinthestabilizationoftheobservertargetsystem.

Theorem1. (Observertarget systemstability)Considersystem (15)–(17),withinitial conditionw₀=w(⁰,x)∈L²(^[0,1])^.Then thezeroequilibriumofthesystem(15)–(17)isexponentiallystableinthesenseofthenextnorm

²

(

^t

)

⁼

^wt

²L²([0,1])+

^wx

²L²([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−ηx

(

¹−x

)

^wttwx+1

2

^e−ηx

(

¹−x

)

^wtxwt

dx +1

awtt

(

^t,0

)

^wt

(

^t,0

)

=−

ι

^e−η1

0

w²_t + ₁

0

e^−ηwtxwx+e^−ηwxxwt+1

2

^e−ηx

(

¹−x

)

^wxxwx

+1

2

(

^1−x

)

^wtxwt

dx+1

a

(

^wtt

(

^t,0

)

^wt

(

^t,0

))

⁻

ι

2 ₁

0

e^−ηx

(

^1−x

)

^wtwxdx

=−e^−ηwt

(

^t,0

)

^wx

(

t,0

)

−

4wx

(

t,0

)

²−

4wt

(

^t,0

)

²− 1

0

−

⁻

η (

^1−x

)

2 e^−ηxw²_x

2 dx

−2a

⁺¹

a w²_t

(

^t,0

)

+e^−ηwt

(

^t,0

)

^wx

(

^t,0

)

− 1

0

−

⁻

η (

^1−x

)

2 e^−ηxw²_t

2 dx−

ι

2 1

0

^e−ηx

(

^1−x

)

^wtwxdx−

ι

¹

0

e^−ηw²_tdx

≤ −e^η 1

0

e^−ηw²_x 2 dx−1

aw²_t

(

t,0

)

−

ι

2 1

0

^e−ηx

(

¹−x

)

^wtwxdx−

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⁾

k_ξ

(

⁰,0

)

=ae^−ηk

(

⁰,0

)

+ae^−η

β (

⁰

)

=

β

(

⁰

)

. (26)

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} backsteppingtrans- 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,.

)

²L²(^[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

∞=sup_x∈[0,1]

| β

(^x)

|

,k_∞= max_{(x,ξ )∈}

|

^k(^x,

ξ

)

|

²2,andsoonforl_∞,(p_ξξ)_∞,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,.

)

²L² ≤

(

¹+k_∞+l_∞+

(

^lξ

)

∞+

ι

^p∞

)

v

t

(

^t,.

)

²L²+p_∞

v

x

(

^t,.

)

²L²

+

((

^p_ξξ

)

∞

v (

t,.

)

²L²+

β

∞

|

v

t

(

t,0

) |

²

)

≤a1

²

wherea₁=max

{

¹+k_∞+l_∞+(^l_ξ)∞+

ι

^p∞,(^p_ξξ)∞,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,.

)

²L²+

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 )

^wξ

(

^t,

ξ )

^d

ξ

^.

Asw(^t,0)=w(^t,x)−x

0wy(^t,y)^dy,usingPoincare’sinequality,weobtain

v ₍

t,.

)

²L²≤a₃

(

^wx

(

^t,.

)

²L²+

^wt

(

^t,.

)

²L²

)

,

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,.

)

²L² ≤a4

(

^wt

(

^t,.

)

²L²+

^wx

(

^t,.

)

²L²

)

^,

wherea₄=max

{

¹+e_∞+h_∞+(^h_ξ)∞+

ι

^f∞,f_∞+a₀(^f_ξξ)∞

}

≥0. Also,asw(^t,0)=w(^t,x)−x

0wy(^t,y)^dywefind

v

x

(

^t,.

)

²L² ≤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. TheproofofTheorem2isperformedinthreesteps: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 nonlinearandcom- 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

^˜2_x

(

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

c_a

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

˜_t²

(

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

^˜_t¹

(

^t,0

)

,˜

v

²_t

(

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)andY₀∈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