nonlinear systems gradient descent A control for output tracking of a class of non-minimumphase Technology Journal of Applied Researchand

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Availableonlineatwww.sciencedirect.com

Journal of Applied Research and Technology

www.jart.ccadet.unam.mx JournalofAppliedResearchandTechnology14(2016)383–395

Original

A gradient descent control for output tracking of a class of non-minimum phase nonlinear systems

Khalil Jouili

^∗

, Naceur Benhadj Braiek

LaboratoryofAdvancedSystemsPolytechnicSchoolofTunisia(EPT),B.P.743,2078Marsa,Tunisia Received17March2016;accepted14September2016

Availableonline2December2016

Abstract

Inthispaperwepresentanewapproachtodesigntheinputcontroltotracktheoutputofanon-minimumphasenonlinearsystem.Therefore,a cascadecontrolschemethatcombinesinput–outputfeedbacklinearizationandgradientdescentcontrolmethodisproposed.Therein,input–output feedbacklinearizationformstheinnerloopthatcompensatesthenonlinearitiesintheinput–outputbehavior,andgradientdescentcontrolformsthe outerloopthatisusedtostabilizetheinternaldynamics.Exponentialstabilityofthecascade-controlschemeisprovidedusingsingularperturbation theory.Finally,numericalsimulationresultsarepresentedtoillustratetheeffectivenessoftheproposedcascadecontrolscheme.

Keywords:Input–outputfeedbacklinearization;Non-minimumphasesystem;Singularperturbedsystem;Gradientdescentcontrol

1. Introduction

Thecontrolof nonlinearnon-minimumphasesystems isa challengingproblemincontrol theoryandhasbeenanactive researchareaforthelastfewdecades.Thistechnique,asamatter offact,wassuccessfullyestablishedinvariouspracticalappli- cations (Bahrami, Ebrahimi, &Asadi, 2013; Cannon, Bacic,

&Kouvaritakis, 2006; Charfeddine, Jouili, Jerbi, &Benhadj Braiek,2010;Jouili&BenHadj,2015;Sun,Li,Gao,Yang,&

Zhao,2016).Thissystemcontrolisadelicatetaskowingtothe factthatitisanonlinearsystemwithnon-minimumphase,and thatitisalsocharacterizedbyadynamicpronetotheinstabil- ityofthedynamicsofzero(Jouili&Jerbi,2009;Jouili,Jerbi,

&BenhadjBraiek,2010;Kazantzis,2004;Naiborhu,Firman,

&Mu’tamar,2013).Infactthereexistnogenericmethodsfor controllersynthesisanddesign(Khalil,2002).Severalfunda- mentalmethodsinthe outputtrackingproblemsonnonlinear non-minimumphasesystemshavebeenproposedinthisarea.

∗Correspondingauthor.

E-mailaddress:[email protected](K.Jouili).

PeerReviewundertheresponsibilityofUniversidadNacionalAutónomade México.

Hirschorn andDavis(1998),Isidori (1995),and Hu etal.

(2015)haveproposedthestableinversionmethodtothetracking problemwithunstablezerodynamics.Thismethodtriestofinda stablesolutionforthefullstatespacetrajectorybysteeringfrom theunstablezerodynamicsmanifoldtothestablezerodynamics manifold.

Khalil(2002)hasderivedaminimumphaseapproximation toasingle-inputsingle-outputnonlinear,non-minimumphase system.Aninput–output linearizingcontrollerisdesignedfor thisapproximationandthenappliedtothenon-minimumphase plant.Thisleadstoasystemthatisinternallystable.Naiborhu andShimizu(2000)presentedacontrollerdesignedbasedupon aninternalequilibriummanifoldwherethiscontrollerpushes the state of a nonlinear non-minimum phase system toward that manifold.Thishasafforded approximateoutput tracking for nonlinearnon-minimumphasesystems whilemaintaining internalstability.

Kravaris and Soroush have developed several results on theapproximatelinearizationofnonminimumphasesystems (Kanter,Soroush,&Seider,2001;Kravaris&Daoutidis,1992;

Kravaris, Daoutidis, & Wright, 1994; Soroush & Kravaris, 1996).ForinstanceKanteretal.(2001)andKravarisetal.(1994) investigatedthesystemoutputwhichisdifferentiatedasmany times as the order of the systemwhere the input derivatives

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Nomenclature

x vectorofstatevariables u controlinput

y outputvariable

ξ vectorofslowstatevariables

η vectoroffaststatevariablesoftheinternaldynam- ics

u* localminimalpointofancontrolvariableu y scalaroutput

yref referencetrajectoryfortheoutput Z statevectorofreducedsubsystem η_ref virtualdesiredoutput

uQSS QSScontrolinput uar artificialinput V(x) Lyapunovfunction

Υ(u) performancefunctionofancontrolvariableu ψ(Z) descentfunction

that appear in the control law are set to zerowhen comput- ing the state feedback input. Bortoff (1997) has studied the systeminput–outputfeedbackofthefirstlinearized.Then,the zerodynamicsisfactorizedintostableandunstableparts.The unstable part is approximately linear and independent of the coordinatesofthestablepart.Charfeddine,Jouili,andBenhadj Braiek(2015)dismissedapartofthesystemdynamicsinorder tomaketheapproximatesysteminput-statefeedbacklineariz- able.Theneglectedpartisthenconsideredasaperturbationpart thatvanishesattheorigin.Next,alinearcontrollerisdesigned tocontroltheapproximatesystem.

Moreover, an original technique of control based on an approximationof the methodof exactinput–output linearization, was proposed in the works (Charfeddine, Jouili, Jerbi,

& Benhadj Braiek, 2011; Guardabassi & Savaresi, 2001;

Guemghar, Srinivasan, Mullhaupt, & Bonvin, 2002; Hauser, Sastry,&Kokotovic,1992).Theapproximation (Charfeddine etal.,2011)isusedtoimprovethedesiredcontrolperformance.

Acascade controlschemehas beenconsidered(Charfeddine, Jouili,&BenhadjBraiek,2014;Yakoub,Charfeddine,Jouili,&

BenhadjBraiek,2013)thatcombinestheinput–outputfeedback linearizationandthebacksteppingapproach.

On the otherhand,Firman, Naiborhu,and Saragih(2015) haveappliedthemodifiedsteepestdescentcontrolforthatsys- tem output will be redefined such that the system becomes minimumphasewithrespecttoanewoutput.

Inthispaper,weaddresstheproblemoftrackingcontrolof asingle-inputsingle-output ofnon-minimumphasenonlinear systems. The ideahereis totransform the givensysteminto Byrnes–Isidorinormalform,thentousethesingularperturbed theoryinwhichatime-scaleseparationisartificiallyintroduced through the useof astate feedback witha high-gain for the linearizedpart.Thegradientdescentcontrolmethod(Naiborhu

&Shimizu,2000)isintroducedtogenerateareferencetrajectory forstabilizingtheinternaldynamics.

Thisresultsinacascadecontrolscheme,wheretheouterloop consistsofagradientdescentcontroloftheinternaldynamics, andtheinnerloopistheinput–outputfeedbacklinearization.

The stability analysis of the cascade control scheme is providedusing resultsofsingular-perturbationtheory(Khalil, 2002).

Therestofthispaperisorganizedasfollows.InSection2, somemathematicalpreliminariesarepresented.Theproposed cascade controlschemeandthe stabilityanalysisaregivenin Sections3and4,respectively.InSection5,theeffectivenessof theproposedcontrolschemeisillustratedbynumericalexam- ples. Finally,thispaper willbe closedbyaconclusionanda futureworkspresentation.

2. Theoreticalbackground

Inthispaper,weconsiderasingle-inputsingle-outputnon- linearsystemoftheform:

˙x=f(x)+g(x)u

y=h(x) (1)

wherex∈ ⁿ isthe n-dimensionalstatevariables,u∈ isa scalarmanipulate inputandy∈isascalaroutput.f(·),g(·) andh(·)aresmoothfunctionsdescribingthesystemdynamics.

2.1. Exactinput–outputfeedbacklinearization

Theinputoutputlinearizationisbasedontwoconcepts:the conceptofrelativedegreeandtheconceptofstatetransforma- tion.

Therelativedegreerofthesystem(1)isdefinedasthenumber ofderivationoftheoutputyneededtoappearintheinputu,such as∀x∈ⁿ:

⎧⎨

⎩

L^k_fh(x)=0∀ 1≤k≤r−1

L_gL^(r_f⁻¹⁾h(x)=/ 0 (2)

If r≤n, then system (1) can be feedback linearized into Byrnes–Isidorinormalform(Isidori,1995)usingthefollowing steps:

Step1:Weapplythefollowingcontrollaw

u(x)= v−L^r_fh(x)

L_gL^r_f⁻¹h(x) (3)

withv=y^(r)

This control law compensates the nonlinearities in the input–outputbehavior.

Step 2: First, system (1) is transformed into normal form (Isidori,1995)throughanonlinearchangeofcoordinates:

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T(x)=

⎡

⎢⎢

⎣ h(x) L_fh(x)

... L^r_fh(x)

ξ₁(x) ... ξ_n_−r(x)

⎤

⎥⎥

⎦

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with:

L_gη_i(x)=0, i=1,...,n−r (5) Theresultingsystemwiththetransformedvariables(4)can thenbewrittenas

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

˙ξi=ξ_i₊₁,i=1,...,r−1

˙ξr=L^r_fh(x)+LgL^r_f⁻¹h(x)u

˙η=Q(ξ,η) y=ξ1

(6)

whereηisthestatevectoroftheinternaldynamics.

2.2. Singularperturbedsystem

Asingularlyperturbedsystemisasystemthatexhibitsatwo- timescalebehavior,i.e.ithasslowandfastdynamics,anditis modeledasfollows(Glielmo&Corless,2010):

⎧⎪

⎪⎨

⎪⎪

⎩

ε˙ξ=F₂(ξ,η,u,ε), ξ(0)=ξ₀

˙η=F₁(ξ,η,u,ε), η(0)=η₀ y=h(x)

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whereξ ∈^Pandη∈ ^m arerespectivelytheslowandfast variablesandε>0isasmallpositiveparameter.Thefunctions F1(·)andF2(·)areassumedtobecontinuouslydifferentiable.

ξ0andη0arerespectivelytheinitialconditionsofthevectors ξandη.Ifε→0,thedynamicsofξactsquicklyandleadstoa time-scaleseparation.Suchaseparationcaneitherrepresentthe physicsofthesystemorcanbeartificiallycreatedbytheuseof high-gaincontrollers.

Asε→0,ξ canbeapproximatedbyitsQuasiSteadyState ξ=ϑ(η,u)obtainedbysolving

f₁(η,ξ,0)+g₁(η,ξ,0)u=0 (8) So,thereduced(slow)systemisgivenby:

˙η =f2(η,ϑ(η,u),0)+g2(η,ϑ(η,u),0)u

= ¯F2(η,u) (9)

Notethatthereducedsystem(8)isnotnecessarilyaffinein input.

Inthenexttheorem,weestablishtheexponentialstabilityof thesingularperturbedsystem(7).

Theorem1(Khalil,2002). Assumethatthefollowingcondi- tionsaresatisfied:

• Theoriginisanequilibriumpointfor(7),

• ϑ(η,u)hasauniquesolution,

• Thefunctionsf1,f2,g1,g2,ϑandtheirpartialderivativesup toorder2areboundedforξintheneighborhoodof ¯ξ,

• Theoriginoftheboundary-layersystem(7)isexponentially stableforallη,

• Theoriginofthereducedsystem(9)isexponentiallystable.

Then,thereexistsε^∗>0suchthat,forallε<ε^∗,theoriginof (7)isexponentiallystable.

Theorem2(Khalil,2002). Givensystem(1),ifthereexistsa LyapunovfunctionV(x)andpositiveconstantsχ₁,χ₂ andχ₃ suchthatχ₁x²≤V(x)≤χ₂x²and ˙V(x)≤−χ3x²,then theoriginisexponentiallystable.

2.3. Basicresultsonthetrajectoryfollowingmethod

The trajectory following method (Naiborhu & Shimizu, 2000) is a numerical optimization method based on solving continuousdifferentialequations.

Thebasicideabehinda“trajectoryfollowing”methodisto formasetofdifferentialequationsfromthegradientofthecost function.

Considerfirsttheminimizingproblemoftheform:

minimize Υ(u) (10)

subjecttonoconstraints

whereΥ(u)isaperformancefunctionofacontrolvariableu.

Supposethatweusethelocalminimalpointu*asaninitial conditionforintegratingthedifferentialequation,

˙u=Λ(u) (11)

whereΛ(·)isafunctionatourdisposal,tobedeterminedshortly.

Calculate the time derivative of Υ(u) along the trajectory generatedbythesolutionto(11).Thenatu=u*:

∂Υ

∂t

u^∗

= ∂Υ

∂u

u^∗

Λ(u^∗)≥0 (12)

Sinceweareinterestedinatrajectorythatwillsearchamin- imum,theaboveobservationsuggeststhatweintegrate(11)by choosing

Λ(u)=−

∂Υ

∂u

T

(13) andEq.(12)becomes

dΥ dt = ∂Υ

∂uΛ(u)0, ∀u=/ u^∗ (14)

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Thus,inordertosolve theproblemof minimizingΥ(·)in an unconstrained control space, one need only to choose an appropriateinitialconditionforuandintegrate

˙u=−

∂Υ

∂u

_T

(15) untiltheequilibriumsolutionisachieved.

3. Synthesisoftheproposedcontrolscheme

In this section, a cascade control scheme for the track- ingcontrolproblemofnonlinearnon-minimumphasesystems is proposed. The idea of the control scheme is to use the input–output feedbacklinearization inthe innerloopandthe gradient descent control in the outer loop. The input–output feedbacklinearizationcanbeseenasapre-compensator prior toapplyinggradientdescentcontrol,whereasgradientdescent control canbeviewedas asystematic wayofcontrollingthe internaldynamics.Thefollowingcontrolstructureisproposed (Fig.1).

3.1. Boundarylayersubsystem

Consider the nonlinear system described by (1), then we applythecontrollaw(3)whichisgivenby:

⎧⎪

⎪⎨

⎪⎪

⎩

v =y^(r)

=y^(r)_ref +

r−1

i=0

¯k_i₊₁(yref−y)⁽ⁱ⁾ (16)

with ¯ki+1=k_i₊₁ ε^r⁻ⁱ

where yref is the referencetrajectory for the output, ε→0 a smallpositiveparameter,andki>0,∀i∈{1,2,...,n−1}are thecoefficientsofaHurwitzpolynomial(Isidori,1995)andthe internaldynamicsaregivenby:

˙η=Q(η,y,˙y,...,y^(r⁻¹⁾) (17) Undertheassumptionthatthegains ¯kiarechosenlarge,such asforanychoiceofε>0,theclosedloopisstableandεcanbe usedasasingletuningparameter,thesystem(16)and(17)can bewrittenintheformofasingularperturbedsystem(7).Sothe faststatecanbedefinedby:

ξ_i=εⁱ⁻¹y⁽ⁱ⁻¹⁾, i=1,...,r (18) Ifwereplace(18)by(17),weobtain

˙η=Q(η,ξ)

η(0)=η₀ (19)

andalsoby(16),suchthat

ε˙ξr=ε^ry^(r)_ref +

r−1

i=0

ki+1(ξ(i+1)ref −ξi+1) (20)

with ξ_ref =

y_ref ε˙yref ε²¨yref ... ε^ry^(r_ref⁻¹⁾

T

thus,(17)canbewrittenasfollows:

⎧⎪

⎪⎨

⎪⎪

⎩

ε˙ξi =ξi+1, i=1,...,r−1 ε˙ξ_r =ε^ry_ref^(r) +

r−1

i=0

k_i₊₁

ξ(i+1)ref−ξ_i₊₁

(21)

Gradient descent

control Linear

feedback

Input-output

linearization System

+- ++

y u

uQSS

y

yref

y x

I

II

η

I II

Fig.1.Cascadecontrolschemeusinginput–outputfeedbacklinearizationandgradientdescentcontrol.

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3.2. Reducedsubsystem

Theinternaldynamicsdependsontheoutputyanditsderiva- tives˙y,...,y^(r⁻¹⁾,suchas:

⎧⎪

⎨

⎪⎩

˙η=Q(η,ξ)

=Q

η,˙y,...,y^(r⁻¹⁾ η(0)=η0

(22)

Ingeneral,theinputoutputlinearizationtechniquesdecouple betweentheinput–outputbehavioryandtheinternaldynamics η.Ontheotherhand,theQSSassumptiondecouplesbetween theinternaldynamicsηandtheinputoutputbehaviory.Thus,y doesnothaveanyeffectonη.Therefore,theoutputy^(r)isused for the control of the internaldynamics. Thus, the boundary layersubsystem (21) andthe reducedsubsystem (22) canbe manipulatedseparately.

Firstly,wedefineanovelstatevector:

Z=

η

¯η

=

⎡

⎢⎢

⎢⎣ η1

... η_n_−r

y ... y^(r⁻¹⁾

⎤

⎥⎥

⎥⎦

(23)

Suchasthereducedsubsystem(22)canbewrittenby:

Z˙ = ¯Q(Z,uQSS)

u_QSS =y^(r) (24)

The system (24)will be expressed by the following state equation:

Z˙ =

˙η

˙¯η

=

α(η,ξ)

¯α(¯η,ξ)+β(¯η,ξ)uQSS

(25) where

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ η=

η₁ η₂ ... η_n_−r_T

¯η=

¯η1 ¯η2 ... ¯ηr

T

α(η,ξ)=

α1(η,ξ) α2(η,ξ) ... αn−r(η,ξ)T

¯α(¯η,ξ)=

¯α1(¯η,ξ) ¯α2(¯η,ξ) ... ¯αr(¯η,ξ)T

β(¯η,ξ)=

0 0 ... 1T

Thestabilityoftheinternalstateηisrequiredtoguarantee theoutputsystemytracksthereferenceoutputtrajectoryyref.

Basedonthegradientdescentcontrolalgorithm,weproposea methodtomaketheinternalstateηtendtoηref,η→ηrefift→∞ (internalstateregulation).

Letηbeavirtualoutputofthesystemandη_refbethevirtual desiredoutput.

Then we find γ_η as relative degree of the system if η is the output of the system (1). Weknow that η∈R⁽ⁿ^−r) then γ_η=

γ_η¹ γ_η² ... γ_η⁽ⁿ^−r)

.BasedonZandtheirderivatives, we construct the performanceindex as adescent functionas follows

ψ(Z)=

(n−r) i=1

⎛

⎜⎝

γ_ηⁱ

j=0

bⁱ_j

η^(j)_ref_(i)−η^(j)_i

⎞

⎟⎠

2

+

⎛

⎝^r

j=0

a_j

y^(j)_ref−¯η^(j)

⎞

⎠

2

(26)

wheretheconstantsa₀,a₁...,a_r,bⁱ₀,bⁱ₁,...,bⁱ

γⁱ,_i=1,...,(n−r)will bechosenlater.

Reasonswhywedefinethedescentfunctionasin(26)are:

i. The input uQSS can be designed if there exists an explicit relationship between the input uQSS and the output y. Thus, descent function (26) must be a function of

η^γ^η¹, ... ,η^γ^η⁽ⁿ^−r), y, ... ,y^(r⁻¹⁾

. ii. Weneedthatthedescentfunction:

ψ(Z)=ψ

η₁, ˙η1, ... η^γ

η1

1 , ... η_(n_−r), ˙η(n−r), ... η^γ

(n−r) η

(n−r), y, ˙y, ... y^(r)

(27) beaquadraticformandthatψ(0)beequaltozero.Accord- ingly,the minimumvalueof thedescentfunctionψ₀(Z)is zero.

If ψ(Z) is zero, each term in the right side of Eq. (26) is positive∀t,thereforewehave:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

r j=0

aj

y_ref^(j) −¯η^(j)

=0

r_η¹

j=0

b¹_j

η^(j)_ref

(1)−η^(j)₁

=0

...

γη⁽ⁿ^−r)

j=0

b⁽ⁿ_j^−r)

η^(j)_ref

(n−r)−η^(j)_(n_−r)

=0

(28)

Bychoosingthevaluesofa0,a1...,arsuchthattheEigen valuesofthepolynomial

arS^r+a(r−1)S^(r⁻¹⁾+···+a1S+a0=0 (29)

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are real negative and the values of bⁱ₀,bⁱ₁,...,bⁱ

γⁱ, i= 1,...,(n−r)suchthattheEigenvaluesofthepolynomial bⁱ

γ_ηⁱS^γ^ηⁱ+b_(γi

η−1)S^(γ^ηⁱ⁻¹⁾+···+bⁱ₁S+bⁱ₀=0 (30) arerealnegative,weobtainthat:

η→η_ref

¯η→yref

(31)

ift→∞.

Inotherwordsoutputtrackingandregulationofinternalstate areachievedtogether(Artstein,1983).Ourobjectiveistofind acontrollawuQSSsuchthatthedescentfunctionψ(Z)becomes minimum.Thentheoutputtrackingproblemcanbewrittenas:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

decrease

uQSS

ψ(Z)

subj. to Z=

˙η

˙¯η =

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

˙η1=α1(η,ξ)

˙η2=α2(η,ξ) ...

˙ηn−r =α_n_−r(η,ξ)

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

˙¯η₁= ¯α1(¯η,ξ)

˙¯η₂= ¯α2(¯η,ξ) ...

˙¯η_r₋₁= ¯αr−1(¯η,ξ)

¯ηr = ¯αr(¯η,ξ)+β(¯η,ξ) uQSS

(32)

Inthispaper,theoutputtrackingproblem(32)willbesolved usingthetrajectoryfollowingmethod.

WhenuQSSisascalar, dψ(Z)

du_QSS =−2ar

⎛

⎝^r

j=0

a_i

y^(j)_ref −¯η^(j)

⎞

⎠ ∂¯η^(r)

∂u_QSS

!

−2

n−r

i=1

bⁱ

γηⁱ

⎛

⎜⎝

γ_ηⁱ

j=0

bⁱ_j

η^(j)_ref

(i)−η^(j)_i

⎞

⎟⎠

⎛

⎜⎝∂η

γ_ηⁱ

i

∂u_QSS

⎞

⎟⎠

(33)

Becausetherelativedegreeofthesystemiswelldefined,

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

⎛

⎜⎝∂η

γⁱ_η

i

∂u_QSS

⎞

⎟⎠ /=0

∂¯η^(r)

∂uQSS

!

=/ 0

(34)

Thenecessaryconditionforalocalminimum dψ(Z)

duQSS =0 (35)

issatisfiedifatthesametime:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

r j=0

a_i

y^(j)_ref −¯η^(j)

=0

γ_ηⁱ

j=0

bⁱ_j

η^(j)_ref

(i)−η^(j)_i

=0

(36)

andithappensifthevalueofperformanceindexψ(Z)=0.

In thefollowingweuseanumericaltechniquetosolvethe minimizationproblem(32)and(33)ateverypointofatrajectory.

The ideaof thetrajectoryfollowingmethodistosolvethe numericaloptimizationproblemonlyonce,attheinitialpoint ofthetrajectory.

Forallotherpointsxalongatrajectory,thecontroluQSSis determinedfromthedifferentialequation:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

˙uQSS =−dψ(Z) du_QSS dψ(Z)

duQSS =−2ar

⎛

⎝^r

j=0

a_i

y_ref^(j) −¯η^(j)

⎞

⎠ ∂¯η^(r)

∂uQSS

!

−2

n−r

i=1

bⁱ

γηⁱ

⎛

⎜⎝

γ_ηⁱ

j=0

bⁱ_j

η^(j)_ref

(i)−η^(j)_i

⎞

⎟⎠

⎛

⎜⎝∂η

γ_ηⁱ

i

∂u_QSS

⎞

⎟⎠

(37)

The control law inEq. (37)is called the steepest descent control.

Calculatethetimederivativeofdescentfunction(26)

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

⎧⎨

⎩Z˙ =

⎡

⎣˙η

˙¯η

⎤

⎦ =

⎡

⎣ α(η,ξ)

¯α (¯η,ξ)+β(¯η,ξ) uQSS

⎤

⎦

dψ(Z)

duQSS =−2ar

⎛

⎝^r

j=0

a_i

y_ref^(j) −¯η^(j)

⎞

⎠ ∂¯η^(r)

∂uQSS

!

−2

n−r

i=1

bⁱ

γ_ηⁱ

⎛

⎜⎝

γ_ηⁱ

j=0

bⁱ_j

η^(j)_ref

(j)−η^(j)_i

⎞

⎟⎠

⎛

⎜⎝∂η

γⁱ_η

i

∂u_QSS

⎞

⎟⎠

(38)

wehave ψ(Z)˙ = ∂ψ

∂x ˙x+ ∂ψ

∂u_QSS˙uQSS (39)

Bysubstituting(21)into(38),wehave ψ(Z)˙ = ∂ψ

∂x ˙x− ∂ψ

∂u_QSS

!2

(40) From Eq. (40) we see that the valueof time derivative of descentfunctionalongthetrajectoryof(38)cannotbeguaran- teedtobelessthanzerofort≥0.Considertheextendedsystem

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(38)andtimederivative of descentfunction(40). Wedonot haveavariablewhichcanbeusedtopushthetimederivativeof descentfunction(40)lessthanzero.

Nowwemodifythesteepestdescentcontrol(38)byadding anartificialinputuar.Thentheextendedsystem(38)becomes:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

⎧⎨

⎩Z˙ =

⎡

⎣˙η

˙¯η

⎤

⎦ =

⎡

⎣ α(η,ξ)

¯α (¯η,ξ)+β(¯η,ξ) uQSS

⎤

⎦

˙uQSS =−∂ψ(Z)

∂u_QSS +u_ar

(41)

fromSontag’sformula(Sontag,1989),weget

u_ar=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ 1 dψ(Z) duQSS

⎡

⎣−dψ(Z) dx ˙x−

"

dψ(Z) dx ˙x

!2

+ dψ(Z) du_QSS

!2⎤

⎦ if dψ(Z) du_QSS =/ 0

0 if dψ(Z)

du_QSS =0

(42)

The control law in Eq. (42) is called a modified steepest descentcontrol.Asimilarmethodusingartificialinputcanbe seeninShimizu,Otsuka,andNaiborhu(1999).

4. Stabilityanalysis

Inthissection,weuseTheorem2ofexponentialstabilityof singularperturbedsystemtoanalyzethestabilityoftheclosed loopsystem. Ifboththe reducedandtheboundary layersub- systemsareexponentiallystable,thenthecombinationisalso exponentiallystable.Thefollowingstepswillbeusedtoprove thestabilityoftheproposedapproach.

4.1. Exponentialstabilityoftheboundarylayersubsystem Letusconsidertheerrorvectorgivenby

˜ξ=ξ−ξ_ref (43)

Then,theboundarylayersubsystem(21)becomes:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

ε˙˜ξ =ε˙ξ−ε˙ξref

=

#

˜ξ1 ˜ξ2 ...

r−1

i=0

k_i₊₁˜ξ_i₊₁

$T

(44)

Lettingτ=t/εyields:

d ˜ξ

dτ =A˜ξ (45)

withAisdefinedby

A=

⎡

⎢⎢

⎣

0 1 0 ... 0

0 0 1 ... 0

... ... ... . .. ...

0 0 0 ... 1

−k1 −k2 −k3 ... −kn−r

⎤

⎥⎥

⎦

UsingtheTheorem2,theorigin ˜ξ =0isexponentiallystable, andtheLyapunovfunctionis

V1

˜ξ

= 1

2˜ξ^TP ˜ξ (46)

whereA^TP+PA=−QandQisamatrixdefinedpositive.

4.2. Exponentialstabilityofthereducedsubsystem

Toanalyzethestabilityofthereducedsubsystem,weusethe followingproposition:

Proposition1. Ifthefollowingstatementsaresatisﬁed:

- Thesystem(25)isstabilizableviathechoiceofuQSS

- (Z=0, uQSS=0) corresponds to an equilibrium point if

dψ(Z) du_QSS =0

Assumethatsystem(25)satisfiesthefollowingassumption.

Assumption1. ThereexistsafunctionV2:ⁿ→,V2(0)=0, whichiscontinuous,positivedefiniteandradiallyunbounded such that the unforced dynamic systemof (25), namely ˙Z= Q(Z,¯ 0)isgloballyasymptoticallystable,i.e., ˙V2(Z,uQSS)<

0, Z=/ 0.

Firstwedefinetheperformanceindex

ψ(Z,uQSS)=V2(Z)+u^T_QSSRuQSS (47) whereRisamatrixconstant,R>0.Thenwedeterminethevalue ofuarbySontag’sformula(Sontag,1989)suchthattheextended nonlinearsystem:

⎧⎪

⎨

⎪⎩

Z˙ = ¯Q(Z,u_QSS),Z(0,0)=(0,0)

˙uQSS=−∂ψ(η,¯η)

∂uQSS +u_ar,u_QSS(0)=0 (48) isasymptoticallystableabout(Z,uQSS)=(0,0).Themostimpor- tantthingistoguaranteetheexistenceofuar.

Remark1. Consider(47).IfuQSS=0thenψ(Z,uQSS)=V2(Z).

InotherwordsperformanceindexbecomesLyapunovfunction andso we do notneed todesign control inputuQSS for only stabilizingsystem.

WithuQSS=0wecandonothingtoincreasetherateofconver- gence.However,byaddinguQSStothesystem,wehavefreedom toacceleratetherateofconvergence.

(8)

Fig.2.Schematicdiagramoftheinvertedcart-pendulum.

Remark 2. It would appear that when Z=0 is globally asymptoticallystableasassumedbyAssumption1,theglobal stabilizationofthewholesystemshouldnotbedifficult.

4.3. Globalstability

In order to illustrate the stability of the cascade control scheme, the following exponential stability theorem is introduced.

Theorem3. Forsystem(1),consideracontrollerwhereuQSS

isobtained by solvingthe optimizationproblem (41)and the inputiscomputedusing(3).IfPandRarepositivedeﬁnite,then thereexistsε>0thatwouldexponentiallystabilize(1).

UsingTheorem1,wecanconcludethat thereexistsε*>0 suchthatforallε<ε*,theoriginof(1)isexponentiallystable.

Alltheconditionsoftheorem1aresatisﬁedsuchthat:

- Theorigin((ξ=0),((η,η)=(0,0)))anduQSS=0isanequi- libriumpointforthesubsystems(21)and(25).

- The boundarylayer subsystem(21)resultingfromthe QSS assumptionε=0hasauniquesolutionyref.Also,asaresult ofthetrajectoryfollowingmethodcontrol,uQSSisafunction ofZ.

- Theoriginoftheboundarylayersystem(21)isexponentially stable∀Z.

- Theoriginofthereducedsystem(25)isexponentiallystable.

5. Simulationresults

Inthissection,wewillgivetwoillustrativeexamplestoshow theapplicabilityandefficiencyoftheproposedcascadecontrol scheme.Thefirstexampleisaninvertedcart-pendulumsystem andthesecondoneisaballandbeamsystem.Controllingboth systemshaspracticalimportance.

5.1. Invertedcart-pendulumsystem

Consider the familiar inverted cart-pendulum system (Al- hiddabi,2005),depictedinFigure2.Thecart mustbemoved usingtheforceusothat thependulumremainsintheupright

Table1

Numericalparametersoftheinvertedcart-pendulumsystem.

Notation Description Numericalvalues

Mp Massofthecart 0.455kg

m Massoftherod 0.21kg

L Lengthoftherod 0.355m

G Gravitationalacceleration 9.8m/s²

positionasthecarttracksvaryingpositionsatthedesiredtime.

Thedifferentialequationsdescribingthemotionare(Al-hiddabi, 2005):

(Mp+m)¨yp+ml¨θ cos(θ)+ml˙θ²sin(θ)=u

l¨θ−¨yp cos(θ)−gsin(θ)=0 (49) whereθistheangleofthependulum,ypisthedisplacementof thecart,andu isthecontrolforce,paralleltotherail,applied tothecart.Thenumericalparametersoftheinvertedpendulum systemaregiveninTable1.

Considerypastheoutputandletx=

θ ˙θ yp ˙yp

T

.The invertedcart-pendulumcanbewrittenasthesystem(1).Where yprepresentstheoutput,uistheinput,xisthestate-spacevector.

Hence,onehas:

f(x)=

⎡

⎢⎢

⎣

x2

1

l gsin(x1)−m(lx²₂+gcos(x1))sin(x1) Mp+m(sin(x1))² cos(x1)

!

x3

m(lx²₂+gcos(x1))sin(x1) Mp+m(sin(x1))²

⎤

⎥⎥

⎦

,

g(x)=

⎡

⎢⎢

⎢⎣ 0

cos(x1) M_p+m(sin(x1))² 0

1 M_p+m(sin(x1))²

⎤

⎥⎥

⎥⎦

andh(x)=yp

Therelativedegreeofthesystemisequaltor=2whichis strictlylowerthanthesystemdimensionn=4.

Applyingtheprocedureofinput–outputlinearizationtothe system(49)oftheinvertedcart-pendulum,theboundarylayer systemisgivenby:

ξ₁=x₃

ξ₂=x₄ (50)

itscontrolisgivenby:

u=m(lx²₂+gcos(x1))sin(x1)−M_p+m(sin(x1))²v (51)

with v=¨yref +k2

ε( ˙yref −˙y)+k₁(yref −y)

(9)

andtheinternaldynamicsisgivenby

η(x)=

#η₁(x)

η2(x)

$

=

⎡

⎣ x₃

x₄−cos(x1) l x₂

⎤

⎦ (52)

UndertheQSSassumptionthat ˙θ= ¨θ =v=0andθ→ ∼=0, sin(θ)=θandcos(θ)=1.UsingEq.(25),thereducedsubsystem canbewrittenas:

Z˙ =

⎡

⎢⎢

⎣

˙η1=η₂

˙η2=1

l(gsin(η1)−¨yrefcos(η1))

˙¯η₁= ˙ξ1

˙¯η₂=u_QSS

⎤

⎥⎥

⎦

(53)

Thecontrolobjectiveistomaketheoutputθtrackadesired referencetrajectoryθ_refgiventhatatthesametimethedisplace- mentofthecarttracksthefollowingtrajectory:

y_pref =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

0 t<0

(1−cos(t)) 0≤t≤2π

0 2π≤t≤4π

−2e^0.5(4π^−t) t≥4π

(54)

Thedesireddisplacement(54)hassmoothswitchingatt=0, t=2πandnon-smoothswitchingatt=4π.

Therelativedegreeofη1isequaltoγ_η¹=2andtherelative degreeofη2isequaltoγ_η²=1.

ByusingEq.(26),thedescentfunctionwillbetransformed underthefollowingform:

ψ(Z)=

⎛

⎝²

j=0

b¹_j

η^(j)_1ref−η^(j)₁

⎞

⎠

2

+

⎛

⎝¹

j=0

b²_j

η^(j)_2ref−η^(j)₂

⎞

⎠

2

+

⎛

⎝²

j=0

a_j

y^(j)_ref −¯η^(j)₁

⎞

⎠

2

(55)

with

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ y_pref =

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

0 t<0

(1−cos(t)) 0≤t≤2π

0 2π≤t≤4π

−2e^0.5(4π^−t) t≥4π η1ref = ¨ypref

g η_2ref =0

(56)

0 5 10 15 20 25

–2.5 –2 –1.5 –1 –0.5 0 0.5 1 1.5 2 2.5

Time [s]

Cart displacement

Fig.3.Evolutionofthecartdisplacementypandthereferencetrajectoryypref

(continuousline:ypref,dashedline:yp).

theinput˙uQSS=−_∂u^∂ψ(Z)_QSS + dψ(Z)¹ duQSS

#

−^dψ(Z)_dx ˙x−%

dψ(Z) dx ˙x

2

+

dψ(Z) duQSS

2$

that stabilizes the internaldynamicsisgivenbysolvingtheproblem(25).Insim- ulation, the parametersused inthe input–output linearization are:ε=0.09,k1=3.6,k2=2.9.Forthegradientdescentcontrol algorithm,theemployedparametersare:

R=

⎡

⎢⎢

⎢⎣

100 0 0 0

0 100 0 0

0 0 100 0

0 0 0 100

⎤

⎥⎥

⎥⎦, P =

# 4.15 3.27 10.61 6.89

$

, Q=

#1 0 0 1

$

and a0=5.4, a1= 2.1, a₂=3.4, b¹₀=6, b¹₁=9, b¹₂=1.7, b²₀= 2, b²₁=8.2. The initial conditions are y_p(0)=˙yp(0)=

˙θ(0)=0 and θ(0)=−π, the downward position for the pendulum.

ThesimulationresultsarepresentedbyFigures3–5.Figure3 showstheevolutionofthedisplacementofthecartypcompared tothedesiredoneyref.ThesimulationresultsinFigure3show

0 5 10 15 20 25

–0.2 –0.15 –0.1 –0.05 0 0.05 0.1 0.15

Angle pendulum

Time [s]

Fig.4.Evolutionoftheangleofthependulumθandthereferencetrajectory θref(continuousline:θref,dashedline:θ).

(10)

0 5 10 15 20 25 –1.5

–1 –0.5 0 0.5 1 1.5 2

Time [s]

Control signal

Fig.5.Evolutionofthecontrolsignal.

thatthecontrolschemeprovidesgoodtracking.Inthisfigure, thereisaperfectagreementbetweenthetwotrajectories.

Figure4showstheevolutionofthependulumangle;indeed,it isasmallvariationaroundzero.Theevolutionofthestabilizing control law isshown inFigure5.The dynamics of thiscon- trolsignalisquitesatisfactory.Infact,thereisnounacceptable physicalovershoot.Onecanalsoseethereducedresponsetime inwhichthecontrollawstabilizesthecontrolledvariable.This showsvery interesting resultsgiven bythe proposedcascade schemecontrol.

5.2. Ballandbeamsystem

Inthispart,aballandbeamisconsidered.Theballandbeam systemisanunstablenonlinearsystemwhichiswellknownin automation;therefore,itisregardedasaperfectbenchtestfor the designof control laws for non-minimumphasenonlinear systems.

Theballandbeamsystemiscomposedofarigidbarcarrying aball.Thelatterischaracterizedbyitshorizontalaxisandits momentofinertiaJ.Itsrotationangleθcomparedtothehori- zontaloneiscontrolledbyanenginewithdirectcurrenttowhich itappliesacoupleτ.Aballisplacedonthebeamwhereitis abletomovewithacertainfreedomundertheeffectofgravity, asitisillustratedinFigure6.

Thedynamicmodelgoverningthebehavioroftheballand beamsysteminopenloopcanbe expressedbythe following equations(Hauseretal.,1992):

θ

rb

Fig.6.Synopticdiagramofthe“ballandbeam”system.

Table2

Parametersandnumericalvaluesoftheballandbeamsystem.

Notation Description Numericalvalues

Mb Ballmass 0.05kg

R Ballradius 0.01m

Jb Ballinertia 2×10⁻⁶kg/m²

J Beaminertia 0.02kg/m²

G Accelerationduetogravity 9.81m/s²

Bb Constant 0.7143

⎧⎪

⎨

⎪⎩ J_b R_b²+Mb

!

¨rb+MbGsin(θ)−Mbrb˙θ²=0 (Mbr²_b+J+J_b)¨θ+2Mbr_b˙rb˙θ+M_bGr_b cos(θ)=τ

(57)

Thenumericalparametersof theballandbeamsystemare recapitulatedinTable2.Todefineanewinpututhesystemcan bewritteninstate-spaceformas(Hauseretal.,1992):

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

⎡

⎢⎢

⎢⎣

˙x1

˙x2

˙x3

˙x4

⎤

⎥⎥

⎥⎦

&'( )

˙x

=

⎡

⎢⎢

⎢⎣

x2

B_b(x1x₄²−Gsin(x3)) x₄

0

⎤

⎥⎥

⎥⎦

& '( )

f(x)

+

⎡

⎢⎢

⎢⎣ 0 0 0 1

⎤

⎥⎥

⎥⎦

&'()

g(x)

u

y=x1

(58)

with x=

r_b ˙rb θ ˙θT

, y=h(x) and B_b=

M_b/(Jb/R²_b+M_b).

Therelativedegreeofthesystemisequaltor=3whichis strictlylowerthanthesystemdimensionn=4.

Applyingtheprocedureofinput–outputlinearizationtothe system(58)oftheballandbeam,theboundarylayersystemis givenby:

⎧⎪

⎪⎨

⎪⎪

⎩ ξ1=x1

ξ2=x2

ξ₃=B_bx₁x²₄−B_bGsin(x3)

(59)

itscontrolisgivenby:

u= 1

2Bbx1x4

(BbGx₄ cos(x3)+B_bx₂x²₄+v) (60) with v= តyref +^k_ε³(¨yref −¨y)+k2( ˙yref −˙y)+k1(yref− y)andtheinternaldynamicsisgivenby

η₁(x)=x₂ (61)

UsingEq.(25),thereducedsubsystemcanbewrittenas:

Z˙ =

⎡

⎢⎢

⎢⎣

˙η1= ˙ξ2

˙¯η₁=ξ₂

˙¯η₂=Bb(x1x²₄−Gsin(x3))

˙¯η₃=uQSS

⎤

⎥⎥

⎥⎦

(62)