Finite-Dimensional Methods for Optimal Control of Autothermal Thermophilic Aerobic Digestion

31 Pages·2012·0.94 MB·English

5

Finite-Dimensional Methods for Optimal Control
of Autothermal Thermophilic Aerobic Digestion
Ellina Grigorieva1, Natalia Bondarenko2,
Evgenii Khailov2 and Andrei Korobeinikov3
1Texas Woman’s University
2Lomonosov Moscow State University
3University of Limerick
1USA
2Russia
3Ireland
1.Introduction
Anactivatedsludgeprocess(ASP)isabiochemicalprocessemployedfortreatingsewageand
industrialwaste-water,whichusesmicroorganismsandair(oroxygen)tobiologicallyoxidize
organicpollutants,producingawastesludge(orfloc)containingtheoxidizedmaterial. The
optimal operationof the biologicaltreatment processesisa challenging task because of the
stringent effluent requirements, the complexityof processesas an object of control and the
needtoreducetheoperationcost.USlawhasstrictrequirementsontheeffluentqualityofthe
ASP;similarstrictrequirementswereadoptedduringthelastdecadeinEuropeandinSouth
Africa(MaineDepartmentofEnvironmentalProtection,Augusta,ME,2010;Tzoneva,2007).
Polluted water contains a wide range of inorganic and organic chemical species and
microorganisms, including pathogens. Filtered concentrated sludge is produced at initial
stagesofthetreatment. Theobjectivesofthekeysubsequentstageareeliminatingpotential
pathogenic micro-organisms and reducing organic chemical content, which might act as a
substrate for further microbial growth, to an acceptable level. Autothermal thermophilic
aerobicdigestion(ATAD)isaprocessthatiswidelyusedtoachievethesegoals.ATADmakes
useofbacterialgrowthwithinthesludgebothtoreduceorganicchemicalcontentandtokill
pathogenic bacteria; aeration of the sludgepromotes the growthof aerobic bacteria, which
feedonandreducetheorganicsubstratesinthesludge.AreviewoftheATADorigin,design
and operationcan be found inBojarskietal. (2010);Capon-Garciaetal. (2010);Graellsetal.
(2010);Layden(2007);Rojas&Zhelev(2009).
1.1Waste-watertreatmentplantoperation
ThissubsectionisextractedfromRojasetal.(2010).
ATADisoperatedasabatchorsemi-batchprocess.Alargereactorcontainingsludgereceives
anadditionalvolumeofuntreatedsludgeatthestartofabatchviaafeedinlet(seeFigures1
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and 2). During the batch reaction air is pumped continuously into the reactor providing
boththeoxygenationrequiredforaerobicbacterialgrowthandthemechanicalmixingofthe
sludge. Digestion of organic substrates proceeds with bacterial growth, predominantly of
thermophilesduetoelevatedtemperature. Atthe endofthebatch period,afractionofthe
treatedvolumeisremovedand isimmediatelyreplacedbythe nextbatch of intakesludge.
Thus the outflow sludge has the same composition as the sludge in the reactor at the end
of the batch reaction time t. Batch outflow and inflow cannot be opened at the same time
topreventthe untreatedinflowsludgemightfrombedrawnoffwithand taintthe outflow
sludge. Thetimebetweensuccessivebatchintakesistypicallyfixedat24hoursforstaffing
Fig.1.Externalviewofareactor.
Fig.2.Schemeofthereactorforwaste-watertreatment.
reasons. In order to achieve the desiredtreatment outcomes, treatment plant designs may
includeasingleATADreactorstage(asisshowninFigure1),ortworeactorsinserieswith
outflowfromthefirstreactorbeinginflowintothesecond.
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1.2ModelsofASPandATAD
InordertoimproveefficiencyofATADprocess,inrecentdecadesvariouscontrolstrategies
for the ASP have been developed by engineers. Simple strategies are typically limited to
themaintenanceofsomedesiredvaluesforeasilydeterminableprocessparameters,suchas
food-microorganismratio, sludgerecycleflowrate, oroxygenconcentration inthe aeration
basin (Holmberg, 1982). More complex models take into consideration that the process
also depends on a number of working conditions, such as air pump power to regulate the
mean oxygen concentration (Rinaldietal., 1979). Optimal working conditions and control
strategies are usually established with a support of mathematical models (Brune, 1985;
Kobouris&Georgakakos, 1991; Moreno, 1999; Morenoetal., 2006; Tzoneva, 2007); relevant
studies and comparison of control strategies have been done by Debuscheretal. (1999);
Fikaretal.(2005);Lindberg(1998);Lukasseetal.(1998);Potteretal.(1996);Qinetal.(1997).
Obviously, aparticularsolutiondependsonthe modelthat was employed; overthe last50
years avariety of modelsof differentlevelof complexitywereproposedforASP. It should
be taken into consideration, however, that there is considerable uncertainty in inferring
both the functional description of processes and the accuracy of parameterizations when
applyinglaboratory-derivedmodelstoareal-lifesystem.Inparticular,manyexistingmodels
of the waste-water treatment are motivated by microbiological and engineering interest
and have been constructed with an intention to incorporate as much as possible of the
apparent understanding of underlyingprocesses. As a result, such models tend to include
toomanyvariables,processesandparameters,whichareknownwithasubstantialdegreeof
uncertainty.
Laboratory scale experiments provided understanding (to varying degrees) of many of
the microbiological and chemical processes involved. By encapsulating much of this
understanding inmathematical models, engineersprovidedtoolsforcomputer simulations
thatcanbeusedinthedesignprocess.
The "Activated Sludge Model 1" (ASM1) (Henzeetal., 2000) has achieved a broad level of
acceptance inthewaste-water treatmentcommunity. BasedonthedominanceoftheASM1
modelinthefield,itsextensiontotheATADprocessassetoutbyGomezetal.(2007)appears
to create a de facto standard for modeling this process. The drawbacks of the model are
also obvious (Rojasetal., 2010). The full-scale ASM1 model incorporates large numbers
of variables and functional responses, and consequently requires many parameters. The
motivationforincludingtheseparticularitiesinthemodelispredominantlymicrobiological,
and seems to arise from the urge to include all possible components about which there is
some knowledge. However, details for many of these functional responses are unknown,
and the accuracy of parameter values is uncertain: they are dependent on the chemical
andmicrobiologicalmake-upoftheparticularsludge,andthereforetypicallyusedasfitting
parameters. Withsuchahighlyparameterizedmodel,setsofparametervaluescanbefound
to fit most data, but it is unclear whether the model remains reliable in such situations.
Applying a parameterized model under different (but physically reasonable) operating
conditionsmaygivespuriousmodelingartifacts.Thismayalsocausenumericaldifficultiesin
attemptinganyoptimization. Moreover,amathematicalmodelisnotveryuseful,ifitcannot
becompletelyinvestigatedusingexistingoptimizationmethods. Mathematicaloptimization
canbeconductedwiththeuseofoptimalcontroltheory.
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1.3Optimalcontrolofnonlinearmodels
Optimal control problem for a system of ordinary differential equations is the problem of
finding an extremum of a functional (objective function) under differential restrictions in
the infinite dimensional functional space (for example, the space of all bounded piecewise
continuous functions). This isachallenging mathematical problem. Thereareanumber of
approachesthatallowtoreducetheproblemtotheanalysisofafinitedimensionalproblem;
one sucheffectiveapproachisthe PontryaginMaximumPrinciple(PMP)(Pontryaginetal.,
1962), which is a necessary condition for optimality in the original infinitely dimensional
problem. It reduces the optimal control problem to the analysis of two point boundary
valueproblemoftheMaximumPrinciple.Thisboundaryvalueproblemisfiniteinthesense
thatitssolutions,phasevectorandadjointvariable,dependeitherontheinitialorterminal
conditions,whichbelongtotheelementsofafinitedimensionalEuclideanspace.Theoptimal
controlisuniquelydeterminedbythe so-calledmaximumcondition. Therefore,adesirable
control depends on the variables selected as the initial or terminal conditions of the phase
vectorandtheadjointvariable.Thus,theconsideredoptimalcontrolproblemcanbereduced
toafiniteproblem,andthewell-knownmethodscanbeappliedtosolveit(Fedorenko,1978).
Ofcourse,theseargumentsexcludefromconsiderationspossibilitiessuchassingulararcsin
the correspondingoptimal control (Bonnard&Chyba, 2003), orsituations when the control
hasaninfinitenumberofswitchingsonafinitetimeinterval(Bressan&Piccoli,2007).
Undertheassumptionsabove,lookattheboundaryvalueproblemoftheMaximumPrinciple
ontheotherside. Forasystemofordinarydifferentialequationswhichislinearincontrols,
the maximumconditionalmosteverywhereuniquelydeterminesthe optimal controlinthe
form of a piecewise constant function with a finite number of switchings. If we take the
moments of switching as the corresponding variables, then the phase vector and adjoint
variablebecomedependentontheseswitchings,andtheconsideredoptimalcontrolproblem
becomesfiniteinthesensethattherequiredcontrolbelongstotheclassofpiecewiseconstant
functionswithanestimatedfinitenumberofswitchingsdeterminedfromtheanalysisofthe
maximumcondition. Grigorieva&Khailov(2010a;b),whichareassociatedwiththeproblem
ofoptimizingthewaste-water treatmentprocess,supportthesearguments. Theconsidered
model, which is due to Brune (1985), is sufficiently simple to be investigated analytically,
while it adequately describes the principal features of the ASP and water cleaning control
processBrune(1985)formulatedanoptimalcontrolproblemfortheminimizationofthewaste
concentration in the ASP and offered the PMP for its solution. However, the analysis of
thecorrespondingboundaryvalueproblemfortheMaximumPrinciplewasnotcompleted;
insteadtheauthorofferedanumericalsolutiontotheproblemfordifferentpiecewiseconstant
controls. Grigorieva&Khailov(2010a;b)dealwiththecompleteanalysisofthismodelfora
different objective functions. Grigorieva&Khailov (2010a) formulated the optimal control
problemofminimizingthepollutionconcentrationsattheterminaltimeandfoundoptimal
solutions forthis. Authors also investigated how these optimal solutionsdepend oninitial
conditions and conducted numerical simulation of the ASP for different parameter values.
Grigorieva&Khailov (2010b) stated an optimal control problem of minimizing the water
pollution concentrations on a given time interval. The optimal solutions for this problem
is found in two stages: firstly, the authors investigated the optimal control problem using
the PMP. Secondly, an approachbased onGreen’sTheorem(Hájek, 1991), was applied. As
a result of this study the authors obtained possible types of optimal solutions. This study
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Finite-DimensionalMethodsforOptimalControlofAutothermalThermophilicAerobicDigestion 5
reviled a possibility of the existence of singular arcs at corresponding optimal trajectories
(Bonnard&Chyba, 2003); this fact makes an analytical study of the problem extremely
difficult.However,theapproachbasedonGreen’sTheoremenabledtheauthorstoprovethe
absenceofsingulararcsatoptimaltrajectories.Inthesametime,thisapproachoverestimates
the number of switchings for optimal controls. Ultimately, a combination of the results
obtainedapplyingthesedifferentapproachesallowedtosolvetheconsideredoptimalcontrol
problemanalytically.
The modelstudiedin these papers is not intended to be the finest orthe mostpreciseASP
model; on the contrary, it is robust and rather basic. However, it includes all the essential
featuresoftheprocess.Furthermore,themodelisnonlinearandassumesaboundedcontrol;
thesefeaturesmakethemodelveryinterestingfromthemathematicalpointofview.
An alternative approach, which also allows a reduction of the original optimal control
problem to a finite dimensional problem, is a constructing of a finite dimensional
parametrization by the moments of switching for some piecewise constant controls of the
corresponding attainable set, which fully characterizes the behavior of the original control
system.Thisapproachcanbeappliedtosolvearangeofproblems,including:
estimatingofcontrolopportunity,thatisinvestigatingwheretrajectoriesofthesystemcan
•
leadunderdifferentadmissiblecontrols;
solvingthecontrollabilityproblem;
•
solving optimal control problems with a terminal functional, such as the problem of
•
minimizingpollutionataterminaltime;
solvingatimeoptimalcontrolproblem(thatis,howtomoveasystemfromaninitialstate
•
totheterminalstateforminimaltime).
Methods for solving these problems, except the last one, with particular application to the
optimalcontrolforATADwillbediscussedinthisChapter. Atimeoptimalcontrolproblem
andoptimalcontrolproblemswithanintegralfunctional,suchastheproblemofminimizing
pollutionforaspecifiedtimeintervalandtheproblemofminimizingenergyconsumptionfor
aerationonagiventimeinterval,aresubjectsoffutureresearch.
1.4Rationalandbasicassumptionsformodeling
The ATAD process is efficient but it can be costly, as it requires continuous aeration that
generallyisenergy-consuming. Optimizingtheaerationcansignificantlyreducethecostof
operation.ExistingATADmodelsarefartoolargeandcomplex,andthiscomplexityprevents
the application of the usual optimization techniques. Our aim is, therefore, to construct a
modelthatretainsthefundamentalpropertiesoftheprocess(aswellastheseofthelargescale
existingmodels)whilebeingsufficientlysimpletoallowtheuseofoptimizationprocedures.
With this intention in mind, we formulate and investigate a simplifiedmodelof the ATAD
reactionbasedontheexistingASM1modelatthermophilictemperatures.
InthisChapterweformulateamodelthat(incontrasttoarathercomplexASP1model)we
are capable of investigating analytically and optimizing numerically. This model includes
the essential mechanisms of the ATAD process while ignoring the issues, which, though
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microbiologicallyknown,arenotmaterialtotheprocesswithinthelikelyrangeofoperating
conditions. It is our expectation that this model will describe the process with a sufficient
degreeofaccuracywhileprovidingasuitablebasisforoptimization.Anumberofsimplifying
assumptions permitus to reduce the complexity of the model while retaining the essential
aspectsoftheprocesses.Theseassumptionsaresummarizedasfollows:
Weassumethatthereactorinwell-stirred(asitisinpractice),andhenceconcentrationsof
•
thereagentsarehomogeneousthroughoutthevolume.
Allthebiologicalactivitytakesplaceonlyinthereactor.
•
The batch outflow and inflow stages are of sufficiently short duration, and the biological
•
activityduringthesestagesisnegligible.
Aerationissufficient,andanaerobicmetabolicactivityisnegligible.
•
We consider this process as a reaction with three reagents, namely: oxygen with
•
concentrationx(t),organicmatterofconcentrationy(t),andbacteriaofconcentrationz(t).
Thereactionisgovernedbythelowofmassaction.
•
Theaerationrateu(t)istheonlycontrol.Thiscontrolfunctionisbounded.
•
2.Mathematicalmodel
Inordertodescribetheprocessofaerobicbiotreatment,weconsiderasimplemathematical
model, which representthe process as a chemical reaction with three reagents, namely the
concentration of oxygen x(t), organicmatter with concentration y(t), and the thermophilic
aerobic bacteria with concentration z(t). It is assumed that the mass in the reactor is well
stirred,andhencethereactantconcentrationsarehomogeneousinthevolume.Thechangesof
concentrationsofthereagentsaredescribedbyathree-dimensionalnonlinearcontrolsystem
x˙(t)= x(t)y(t)z(t)+u(t)(m x(t)), t [0,T],
− − ∈
y˙(t)= x(t)y(t)z(t),
⎧ − (1)
⎪⎪⎨zx˙((t0))==xx(t,)yy((t0))z(=t)y−,bzz((0t)),=z , x (0,m), y >0, z >0.
0 0 0 0 ∈ 0 0
Its nonlinearity⎪⎪⎩is justified by the law of mass action (Krasnovetal., 1995) describing the
dependenceofthe rateofchemicalreactionontheconcentrations ofinitialsubstances. The
firstequationofsystem(1)representstheevolutionofoxygenconcentration: thefirstterm,
x(t)y(t)z(t),describestheprocessofitsabsorptioninthereaction,whereasthesecondterm
−
describesinfluxofoxygen(bypumping)intothereactorfromoutside. Here,u(t)istherate
of aeration, which at the same time is the control function. The second equation describes
adecreaseofthe organicmatter inthe reaction. The thirdequationofsystem(1) showsan
evolutionoftheactivebiomassconcentration;thebacteriamassgrowsattheratex(t)y(t)z(t)
and decays at a rate b. The original system also includes positive initial conditions and a
restrictionontherateofpumpingair.WeconsiderallpossibleLebesguemeasurablefunctions
u(t),whichforalmostallt [0,T]satisfytheinequality
∈
0 u(t) umax, (2)
≤ ≤
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astheadmissiblecontrolsD(T).
Thephasevariablesx,y,zofsystem(1)satisfythefollowingproperties.
Lemma 1. Let u( ) D(T) be an arbitrary control. Then the corresponding solution w(t) =
· ∈
(x(t),y(t),z(t))⊤ofthesystem(1)isdefinedontheinterval[0,T],thecomponentsx(t),y(t),z(t)of
whichsatisfythefollowinginequalities:
0<x(t)<xmax, 0<y(t)<ymax, 0<z(t)<zmax, t [0,T], (3)
∈
where xmax, ymax, zmax - some positive constants depending on initial conditions x0, y0, z0 and
parametersm,b,umax,Toftheoriginalsystem.
Hereandbelow,symbol⊤meanstranspose.
Proof. Let x(t), y(t), z(t) be solutions of system (1), which regarding the Existence and
Uniqueness Theorem for the system of differential equations (Hartman, 1964), are defined
onthebiggestsemi-intervalΔ [0,T].
⊆
Wewillwritethesolutionofthesecondequationofsystem(1)as
y(t)=y0e− 0tx(ξ)z(ξ)dξ. (4)
(cid:6)
Fromthisformulaitfollowsthaty(t)>0forallt Δ.
∈
Next,byanalogy,weobtainthesolutionofthethirdequationoftheconsideredsystem
z(t)=z0e 0t(x(ξ)y(ξ)−b)dξ, (5)
(cid:6)
fromwhichwehavethatz(t)>0forallt Δ.
∈
Finally,wecanwritethesolutionofthefirstequationofsystem(1)as
x(t)=e− 0t(y(ξ)z(ξ)+u(ξ))dξ x0+m te 0s(y(ξ)z(ξ)+u(ξ))dξu(s)ds . (6)
(cid:6) (cid:7) (cid:8)0 (cid:6) (cid:9)
Fromthisformularegardinginequality(2)wefindthatx(t)>0forallt Δ.
∈
Therefore,weobtaintheinequalities:
x(t)>0, y(t)>0, z(t)>0,
thatarevalidforallt Δ.
∈
Further, considering positiveness of solutions x(t) and z(t), from formula (4) we find the
inequalityy(t)<y forallt Δ.
0 ∈
From formula (6), positiveness of solutions y(t), z(t), and inequality (2) we obtain the
relationship
x(t)=x0e− 0t(y(ξ)z(ξ)+u(ξ))dξ+m te− st(y(ξ)z(ξ)+u(ξ))dξu(s)ds<x0+mumaxT.
(cid:6) (cid:8)0 (cid:6)
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Thenwefindtheinequalityx(t)<x0+mumaxTforallt∈Δ.
Using restrictionsonsolutions x(t) and y(t) obtained above, fromformula(5) we have the
relationship
z(t)<z0e 0tx(ξ)y(ξ)dξ<z0ey0T(x0+mumaxT).
Therefore,wefindtheinequalityz(t(cid:6))<z0ey0T(x0+mumaxT)forallt∈Δ.
Thus,solutionsx(t),y(t),z(t)ofsystem(1)onthesemi-intervalΔ [0,T]arebounded,and
⊆
so they cannot go to infinity. Regarding Corollary from Lemma (§14) (Demidovich, 1967),
solutionsx(t),y(t),z(t)ofconsideredsystemaredefinedonentireinterval[0,T]andsatisfy
onitinequalities(3).Theproofiscompleted.
Restrictionsonthefunctionx(t)arespecifiedasfollows.
Lemma2. Foranarbitrarycontrolu( ) D(T)thecorrespondingsolutionx(t)ofthesystem(1)is
· ∈
subjecttotheinequality
0<x(t)<m, t [0,T].
∈
Proof. Letν(t)=m x(t),t [0,T].Then,usingthefirstequationofsystem(1),forfunction
− ∈
ν(t)wehaveCauchyproblem
ν(˙t)= (y(t)z(t)+u(t))ν(t)+my(t)z(t),
ν(0)=−ν =m x >0.
(cid:10) 0 − 0
Using the variation of a parameter method (Hartman, 1964) we find the solution of this
problemas
ν(t)=e− 0t(y(ξ)z(ξ)+u(ξ))dξ ν0+m te 0s(y(ξ)z(ξ)+u(ξ))dξy(s)z(s)ds .
(cid:6) (cid:7) (cid:8)0 (cid:6) (cid:9)
ByLemma1fromthisexpressionwehaverelationshipν(t) >0forallt [0,T],fromwhich
∈
therequiredinequalityfollows.
Lemmas1and2implythatforanycontrolu( ) D(T)solutionsx(t),y(t),z(t)ofthesystem
· ∈
(1)retaintheirphysicalmeaningsforallt [0,T].
∈
3.Attainablesetanditsproperties
Let X(T) ⊂ R3 be the attainable set for system (1) from an initial point w0 = (x0,y0,z0)⊤
at the moment of time T; that is, X(T) is the set of all ends w(T) = (x(T),y(T),z(T))⊤ of
trajectoriesw(t)=(x(t),y(t),z(t))⊤ofsystem(1)underallpossiblecontrolsu( ) D(T).By
· ∈
Lee&Markus(1967)andLemma1,itfollowsthatsetX(T)isacompactsetinR3locatedinto
theregion
w=(x,y,z)⊤ R3 :x>0,y>0,z>0 .
∈
(cid:11) (cid:12)
We denote by ∂Q and intQ the boundary and the interior of the compact set, Q R3,
⊂
respectively.
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Next, we study the boundary of the attainable set X(T). Consider a point w = (x,y,z)⊤,
such that w ∂X(T). It corresponds to control u( ) D(T) and a trajectory w(t) =
∈ · ∈
(x(t),y(t),z(t))⊤, t [0,T], of system (1), such that w = w(T). Then it follows from
∈
Lee&Markus (1967) that there exists a non-trivial solution ψ(t) = (ψ1(t),ψ2(t),ψ3)⊤, t ∈
[0,T]oftheadjointsystem
ψ˙ (t)=u(t)ψ (t)+y(t)z(t)(ψ (t)+ψ (t) ψ (t)),
1 1 1 2 − 3
⎧ψ˙ (t)=x(t)z(t)(ψ (t)+ψ (t) ψ (t)), (7)
⎪⎪⎪⎨ 2 1 2 − 3
ψ˙ (t)=x(t)y(t)(ψ (t)+ψ (t) ψ (t))+bψ (t),
3 1 2 − 3 3
forwhich,byLemm⎪⎪⎪⎩a2,thefollowingrelationshipisvalid:
0, if L(t)<0,
u(t)=⎧ u [0,umax], if L(t)=0, (8)
⎪⎪⎨u∀ma∈x, if L(t)>0.
Here L(t) = ψ (t). Function L⎪⎪⎩(t) is the so-called switching function and its behavior
1
determinesthetypeofcontrolu(t).
Forconvenienceinsubsequentarguments,weintroducethefollowingauxiliaryfunctions:
G(t)=ψ (t)+ψ (t) ψ (t), P(t)= ψ (t), d(t)=y(t)z(t)+z(t)x(t) x(t)y(t), t [0,T].
1 2 − 3 − 3 − ∈
Usingtheadjointsystem(7),wewriteforfunctions L(t), G(t), P(t)thefollowingsystemof
lineardifferentialequations
L˙(t)=u(t)L(t)+y(t)z(t)G(t), t [0,T],
∈
⎧G˙(t)=u(t)L(t)+d(t)G(t)+bP(t), (9)
⎪⎪⎨P˙(t)= x(t)y(t)G(t)+bP(t).
−
⎪⎪⎩
ThevalidityofthefollowingLemmaimmediatelyfollowsfromthenon-trivialityofsolution
ψ(t)=(ψ1(t),ψ2(t),ψ3(t))⊤oftheadjointsystem(7).
Lemma 3. The switching function L(t) and the auxiliary functions G(t) and P(t) are nonzero
solutionsofsystem(9).
Lemma3allowsustorewritetherelationship(8)intheform
0, if L(t)<0,
u(t)= (10)
(cid:13)umax, if L(t)>0.
Atpointsofdiscontinuitywewilldefinefunctionu(t)byitslimitfromtheleft.Consequently,
the control u(t), t [0,T], corresponding to a point w ∂X(T), is a piecewise constant
∈ ∈
function,takingvalues 0,umax .Suchtypeofcontrolisusuallycalledabang-bangcontrol.
{ }
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Next,weestimatethenumberofswitchingsofcontrolfunctionu(t),t [0,T].Itfollowsfrom
∈
(10) that it is sufficientto estimate the number of zeros of the function L(t) on the interval
(0,T).Thefollowingimportantstatementisvalid.
Lemma4. TheswitchingfunctionL(t)hasatmosttwozerosontheinterval[0,T].
Proof. Letusintroduceforsystem(9)newvariables:
r(t)=L(t), v(t)=G(t), μ(t)=P(t)+q (t)L(t)+q (t)G(t),
1 2
wherefunctionsq (t),q (t)mustbedetermined. Usingnewvariablesr(t),v(t),μ(t)system
1 2
(9)hasthefollowingform
r˙(t)=u(t)r(t)+y(t)z(t)v(t),
⎧v˙(t)=(u(t)−bq1(t))r(t)+(d(t)−bq2(t))v(t)+bμ(t),
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨μ˙(t)+=(cid:14)qq˙˙1((tt))++y((ut()tz)(−t)qb)(qt1)(+t)(+d(ut()t)qb2()tq)(−t)bq1b(tq)2q(2t()t)(cid:15)rx((tt))+y(t) v(t)+ (11)
2 1 − 2 − 2 −
Wewillse⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩lectsuc+hb(cid:14)f(u1n+ctiqo2n(st)q)μ((t)t)a.ndq (t)thatinsystem(11)theexpressions(cid:15)insidebrackets
1 2
byr(t)and v(t)arezero. Now, wehave forfunctions q (t), q (t)the systemofdifferential
1 2
equations
q˙ (t)+(u(t) b)q (t)+u(t)q (t) bq (t)q (t)=0,
1 − 1 2 − 1 2 (12)
q˙ (t)+y(t)z(t)q (t)+(d(t) b)q (t) bq2(t) x(t)y(t)=0.
(cid:10) 2 1 − 2 − 2 −
Thensystem(11)willberewrittenas
r˙(t)=u(t)r(t)+y(t)z(t)v(t),
v˙(t)=(u(t) bq (t))r(t)+(d(t) bq (t))v(t)+bμ(t), (13)
⎧ − 1 − 2
⎨μ˙(t)=b(1+q2(t))μ(t).
⎩
Insystem(13)wewillmakethefollowingsubstitutions:
r(t)=r(t), v(t)=v(t)+q (t)r(t), μ(t)=μ(t),
3
whereq (t)isthefunctionthatmustbedetermined.Thensystem(13)willberewrittenas
3 (cid:16) (cid:16) (cid:16)
r˙(t)=(u(t) y(t)z(t)q (t))r(t)+y(t)z(t)v(t),
− 3
⎧v˙(t)= q˙ (t) bq (t)+(u(t) d(t))q (t)+
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨(cid:16)(cid:16) +b(cid:14)q23(t)q−3(t)−1 y(t)z(t)q(cid:16)23−(t)+u(t3) r(t(cid:16))+ (14)
(cid:15)
+(d(t)+y(t)z(t)q (t) bq (t))v(t)+bμ(t),
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩μ˙(t)=b(1+q2(t))μ(t).3 − 2 (cid:16) (cid:16) (cid:16)
(cid:16) (cid:16)
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