ContentslistsavailableatScienceDirect
Biomedical
Signal
Processing
and
Control
j ou rn a l h o m e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / b s p c
State-space
analysis
of
fractional-order
respiratory
system
models
Esra
Saatci
∗,
Ertugrul
Saatci
DepartmentofElectricalandElectronicsEngineering,IstanbulKulturUniversity,Bakirkoy,Istanbul,Turkey
a
r
t
i
c
l
e
i
n
f
o
Articlehistory: Received14August2019 Receivedinrevisedform 26November2019 Accepted8December2019 Availableonline21December2019 Keywords:
Fractional-orderrespiratorysystemmodels State-spaceanalysis
Stabilityanalysis Time-delaysystems
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FractionalOrderModels(FOM)oftherespiratorysystemhavebeenusedinthemodel-basedanalysis
oftherespiratorysystem.Althoughtherearestudiesexploringthephysiologicalcorrectnessandfitting
accuracyofthemodels,theyarenotanalyzedintermsofinteractionsbetweenparameters,time-varying
dynamicsandmeasurablesignals.Inthisstudywepurposetousestate-spaceanalysistoyieldthe
time-varyingnatureofthesystemincorporatedbytheparameters,statesandoutput.Wetestedthemodels
forcontrollability,observabilityandstabilitycharacteristicswhileusingtheparametersfoundinthe
literature.Sufficientasymptoticstabilityboundsweredrivenbyusingstabilitytheoryofthediscrete
time-delaysystem.ResultsrevealedthatFOMswithestimatedparametersoffersystemswithdifferent
characteristics.Thus,carefulconsiderationmustbegivenwheninterpretingestimatedparametersin
FOMsduringrespiratorytests.
©2019ElsevierLtd.Allrightsreserved.
1. Introduction
FOMoftherespiratory systemhasattractedagreat interest becausefunctional and structuralproperties of the systemcan be obtained by a few physically motivated parameters. It has beenshown thatwellknownMaxwellandKelvin-Voight mod-elswithintegerorder differentialequations cannot modelthe stressrelaxationofthelungtissue[1],timevaryingratiobetween transpulmonarypressureandtotallungvolume[2]anddistributive self-similarstructureofthesystem[3]verywell.Asitisthecasefor allrespiratorysystemmodellingapproaches,respiratorysystemis excitedbythecombinedsinusoidalpressureatlowfrequenciesand FOMinfrequencydomainisfittothemeasuredimpedancedata. FrequencydomainversionoftheFOMiscalledasConstantPhase Model(CPM)[4]whichisusedfortheimpedancemodellingand systemparameterestimation.
BothFOMandCPMhavebeenevaluatedandreviewedinthe literature.Theaiminthepreviousstudiesweretoestimatethe modelparameterswithlowestfittingerror,toclassifythediseased andhealthyparametersandtoinvestigatethepowerlawproperty oftherespiratorysystem[5],[6],[7],[8],[9].In[10]authors ana-lysedFOMsintermsofstructuralrepresentationoftherespiratory mechanics,whereasapplicationsofthefractionalcalculusinthe biologicalsystemswereinvestigatedin[11].In[12]statistical
anal-∗ Correspondingauthor.Tel:+902124984229.
E-mailaddresses:esra.saatci@iku.edu.tr(E.Saatci),e.saatci@iku.edu.tr
(E.Saatci).
ysisisappliedtocalculateconfidenceintervalsoftheparameters’ estimates.Parameterswereestimatedbyleastsquaresalgorithm fromimpedancedataandparametervariationswereestimatedby sample variancesand sensitivitymatrix. First, in[12],onlyone modelwasfittotheimpedancedataandcombinederrorswere assumedtobeGaussiannoise.Moreimportantly,estimatesofthe parametersinfluencedthesensitivitiesintheproposedmethodand this madetheresultsobliviousof model-parameterassociation. Therefore,weproposeacomprehensiveanalysis,including estab-lishinganassociationbetweeninformationbearingparametersand dynamicsofthesystem,performedonthealreadyproposedand usedFOMsofrespiratorysystem.Here,ourobjectiveistoexplore FOMsincontrolsystemspointofviewandanswerthefollowing questions:
1.HowunmeasurabledynamicsinFOMsarerelatedtothe param-eters?
2.WhichoftheFOMsarecontrollableandobservableforthe esti-matedparametervaluesintheliterature?
3.Istheasymptoticstabilityanissuefortheestimatedparameter valuesinFOMs?
Toanswerthefirstquestion,weseerespiratorysystemasa complexphysicalsystemwhich consistsofmutual interconnec-tions.Thusourstartingpointistoproposeastate-spacemodelling frameworkwherestatesprescribeacertaindynamicalbehavior. Thekeyaspecthereisthat,incontrasttotheCPMs,state-space modelofFOMsneitherdependsonthefrequencyresponseofthe system,noris restrictedtoonlyparameter-systemrelationship. https://doi.org/10.1016/j.bspc.2019.101820
Table1
EstimatedParametersandValuesinFOMs[14].
Healthy FOM1 FOM2 FOM3 FOM4
R 0.22 0.22 0.06 –
L – 0.0007 0.029 0.0374
1/C 0 1.36 3.52 2.02
˛ 0 0 0.48 0.43
ˇ 0.99 0.99 0 0.79
COPD FOM1 FOM2 FOM3 FOM4
R 0.18 0.26 0.27 –
L – 0.0009 0.0021 0.015
1/C 1.73 5.20 8.9 2.94
˛ 0 0 0.87 0.59
ˇ 0.18 0.83 0 0.52
Incontrasttoearliermethods,thismethodleadstoasetof nat-uralstatevariableswhicharedirectlysuitableforaddressingthe parameter-state-systemrelationships.
Controllabilityand observability analysis[13] isproposedto answerthesecondquestion.Inthisstep,ouraimistoinvestigate whethertheparametervaluesestimatedinthepreviousstudiesin FOMsleadtosignificantandmeaningfuldynamicbehaviourin res-piratorysystems.Atthisstage,weshouldexplainthemeaningof controllableandobservablemodels.Ifthemodelisnotcontrollable and/orobservable,eitherrealizationofthemodelisredundant,i.e. somestateshavenorelationshiptotheinputoroutput,or physi-calsystemiscontroldeficient,i.e.dynamicvariablesarenotfully controlledbyphysicalactuators.
Finally, we proposed to explore the stability of FOMs by usingwell-knownasymptoticLyapunovsstabilitytheoremforthe last question. At this stage, discrete-time version of FOMs are utilized, thus the main source of inspiration is to explore the stability in discrete time-delay systems. The effort is made to addressthederivationoftheconditionsthatleadtostability in FOMs.
2. Methods
2.1. ConstantPhaseModelsofRespiratorySystem
CPMoftherespiratorysystemisexpressedasasumofresistive, inductiveandcapacitivetermsas:
ZFOM(s)=R+Ls˛+ 1
Csˇ (1)
whereRrepresentsallresistivepressurelossesintheairwaysand lungparenchyme,LandCaretheinertialeffectsoftheairflowinlow frequenciesandthecomplianceofthelungrespectively.0≤˛,ˇ≤1 arefractionalordersobtainedfromthefractional-orderdifferential equationsoftherespiratorysystemandsistheLaplacevariable.
Based on the estimated parameter values in the literature, fourmostcommonly used version of (1) canbe foundin [14]. Table1summarizestheparametervaluescalculatedbythe real-timesignalsacquiredfrombothhealthyandChronicObstructive PulmonaryDisease(COPD)patients.Theabsenceofsome param-etervalues impliesthattheestimatedvalueistoosmall,hence respectivetermcanbeeliminatedfromthemodel.Electrical anal-ogyofthesemodelscanbeallrealizedbydominoladdercircuit asshowninFig.1,wheretransferimpedanceisZ (s)=Z1(s)+Y1(s)1 andZ1(s) isthesumofresistiveandinertialfractionalimpedance andY1(s) isthefractionaladmittanceduetotheelasticeffects.
Fig.1.Electricalanalogyofconstantphasemodelrealizedbydominoladdercircuit.
Table2
MatricesofState-SpaceModels
Models FOM1 FOM2 FOM3 FOM4
x (t)
X1(t) X2(t) X1(t) X2(t) X1(t) X2(t) X1(t) X2(t) y (t) [Q (t)] [Q (t)] [Q (t)] [Q (t)] u (t) [P (t)] [P (t)] [P (t)] [P (t)] A [0 1 0 −1 RC ] [−10 1 LC −R L ] [−10 1 LC −R L ] [−10 1 LC 0 ] B [ 01 RC ] [ 01 LC ] [ 01 LC ] [ 01 LC ] C 0 C 0 C 0 C 0 C D [0] [0] [0] [0] ˇ ˇ ˇ 1 1 ˛ ˇ ˛2.2. ContinuousTimeState-SpaceModeloftheFOM
If respiratory systemis assumed to becontinuous time lin-eartimeinvariantsystem,FOMoftherespiratorysystemcanbe expressedinthestate-spaceformas[15]:
tx (t)=A
x (t)+Bu (t)y (t)=C
x (t)+Du (t) (2)wherex (t) ∈Rn,y (t) ∈Rm andu (t) ∈Rr arethestate,output and input vectors respectively. A
∈Rn×n is the state tran-sitionmatrix, B ∈Rn×r istheinputmatrix, C∈Rm×n is theoutputmatrixandD ∈Rm×r isthefeedforwardmatrix. Modelparameters,whichareassumedtobetime-invariantover the observationwindow, are expressedby a parameter vector ∈Rp.=1 2 ··· n
Tdenotesthefractionalorder
vec-torandtx (t) isthethorderfractionalderivativeofthestate
vector x (t). Initialconditions in (2) are assumed tobe zero.If 1=2=···=n=q,thesystemiscalledasacommensurateorder system[16].Incaseofordinarylineartimeinvariantsystems, frac-tionalordersareallequaltoone.
State-spacemodelin(2)canbeobtainedfromallversionsof CPMsin(1)withparametersinTable1.Applicationofderivations giveninAppendixAleadtovectorsandmatricesA
,B,C andDwhicharesummarizedinTable2.2.3. DiscreteTimeState-SpaceModelsoftheFOM
State-spaceanalysisofnon-commensurateFOMisperformed onthediscretetimestate-spacemodelof(2),becauseacquired
res-piratorysignalsareindiscreteform.Grünwald-Letnikovdefinition ofthefractionalderivationisoneoftheapproachtocircumvent aproblemofnumericalcalculationsofthefractionalderivations in(2).Thisapproachallowsustocomposediscretematricesfrom thematricesin(2)easilyand”shortmemory”issueswerealready investigatedinpreviousstudies[16].Inthisstudy,discretization stepsof FOMbyGrünwald-Letnikovdefinition ofthefractional derivation,foundin[16]isutilizedtoobtaintime-delayed state-spacemodelofthediscretetimenon-commensurateFOM:
x (k+1)=A0
x (k)+ k j=1 Ajx (k−j)+Bd u (k) y (k)=Cx (k)+Du (k) (3)wherek ∈Z+∪
0isatimeindex.StatetransitionmatrixA j canbeobtainedasfollows:
Letsdefine ¯A
whoseithrowiscalculatedby¯A i,∗= i t
A−In i,∗
wheretisthediscretetimestep(weassumedt=0.01sinthe simulations)andi∈
1,2,...,nisthestatenumber.Inisn×n identitymatrix.Ifadiagonalmatrixiscreatedbyusingnon-integer diagonalcombinations
i j
as ϒj=diag
1 j , 2 j ,··· n j (4) where
j
=
(−1)··· (−j+1) j (j−1)···1 j>0 1 j=0 , then discrete statetransitionmatrixcanbeobtainedas
A0
= ¯A+ϒ1Aj= (−1)jϒj+1
(5) Systemdefinedin(3)–(5)isalsoalineardiscretetime-delay sys-temandinthisworkitisusedforthecontrollability,observability andstabilityanalysis.Howeverfirst,oneshoulddefinethesolution ofthesystemasasumofnaturalandforcedresponsesrespectively. Thecompletesolutionisgivenas
x (k)=Gk
x (0)+ k−1 j=0 Gk−1−jBd u (j) (6)wheretimevaryingtransitionmatrixGk
isgivenby G0 =In Gk =A0 Gk−1+ k−1 j=1 AjGk−1−j (7)Whileproofofthesolutiongivenin(6)and(7)canbefound in[16],thecomplicationsinthemodelandthesolutionshould beexplainedhere.Althoughrespiratorysignalsareprocessedin aMpoint timewindow,theupperlimit ofthesumsin(3) and (6)shouldbelimitedtothepredefinedvalue(K)duetothe com-putationalissues (K<M).Anotherimportantobservationis that transitionmatrixGk
in(7)iscomposedofpowersofdiscrete statetransitionmatrixAj in(5).ThispropertyofGk allows controllabilityandobservabilitytestsbecomputationallyeffective, becauseoriginalcontrollabilityandobservabilitytestsincorporate matrixpowerofstatetransitionmatrix.2.4. ControllabilityandObservabilityAnalysis
Controllabilityandobservabilityaretwoimportantanalysisin thecontrolsystemstotesttheexistenceofthesolutionsforthe giveninputs.Asystemissaidtobecompletelystatecontrollable ifforanykiitispossibletoconstructacontrolvectoru (k) which willtransferanygivenstatex (ki) toanyfinalstatex
kf
inafinite timeintervalki≤k≤kf[17].
Theorem1. Discretetimenon-commensuratefractionalorder
sys-temiscompletelystatecontrollableifandonlyifthereexitsafinite timeKsuchthatthecompositecontrollabilitymatrix,PK
∈Rn×Kr PK =G0 Bd G1 Bd ··· GK−1Bd isofrankn.Asystemissaidtobecompletelyobservableonki≤k≤kfiffor everykiandsomekfeverystatex (ki) canbedeterminedfromthe knowledgeofy (k) onki≤k≤kf[17].
Theorem2. Discretetimenon-commensuratefractionalorder
sys-temiscompletelyobservableifthereexitsafinitetimeKsuchthatthe compositeobservabilitymatrix,OK
∈RKm×n OK =CG0 CG1 ··· CGK−1T isofrankn. [∗]Tdenotesthetransposeofamatrix.Proofsoftheoremscanbefoundin[16]and[17].Theorems1and 2wereappliedtothefractionalorderrespiratorysystemmodels andcontrollabilityandobservabilitymatrixrankswerecompared withthenumber ofstates. Sinceconstant parametervalues() givenintheliteraturewereusedinthecalculationsofthematrices, controllabilityandobservabilitymatricesaretime-invariant matri-ces(seeTable1fortheparametervalues,explanationsandrelated references).
2.4.1. Theeffectofthemaximumtimedelay(K)
Firstweshouldinvestigatetheeffectofthemaximumtimedelay ontherankofcontrollabilityandobservabilityofmatrices.Model equationsplay averyimportantrole toensurethat therankof PK
andOK aren.FirstweobservethateventhoughAis alowrankmatrix(asinFOM1),A0 isfullrank,whichresultsin afullranktimevaryingtransitionmatrixGk in(7).However,it isnotsufficienttohavefullrankGk tosatisfytheTheorems1 and2.IfbothGk Bd andCGk resultinzerorowand columnvectorsforeverykrespectively,regardlessofthemaximum timedelay,augmentedmatricesPK andOK willhavearank lowerthann.2.4.2. Theeffectofthefractionalordervector()
ThereisnodirecteffectoffractionalorderstotherankofPK
andOK ,butsincethefractionalordersinthemodel(˛andˇ) relatedtothemodelstructure,fractionalordervector()affects therankindirectly.2.4.3. Theeffectoftheparameters()
Parametervaluesalsodonothaveadirecteffectontherank ofPK
andOK .However,parametersensitivitiesofthe con-trollabilityand observabilitymatriceswithrespecttonumerical instabilitiesofthematricesshouldbeinvestigatedasafuturestudy. 2.5. StabilityAnalysisEstimated parameters in biological modelsmay not lead to stable model solutions, because fitting models to theacquired
impedancedatawitha minimumfittingerrordoesnot guaran-teethestabledynamicpropertiesofthesystem.Inthiscaseany externalandinternalperturbationsinthestates(pressuresin res-piratorysystem)resultindivergedoutput(airflow).Inotherwords, dynamicpropertiesofthemodelsbecomeunreliabletoindicate physiologicalvariations.Thusweproposetoanalyzethemodels withestimatedparametersintermsofstabilitybeforeany evalua-tionofthephysiologicalmeaningofstates.Stabilityanalysisofthe discretetime-delaysystemin(3)and(5)canbedoneby consider-ingtwodifferenttypeofthestability,namely,asymptoticstability andfinitetimestability.Inthiswork,wefocusonlyonasymptotic stabilityandleavethefinitetimestabilityforthefuturework.
Definition3. Lineardiscretetimesystemin(3)and(5)is
asymp-toticallystablewith respectto
Bx,B0,Bu,K
, Bx,B0,Bu ∈R+, K ∈Z,foreachk≥1,anyinitialcondition
x (0)<B0 andany inputsignalu (k)<Buif x (k)<Bxwhere
x (k)=ni=1xi(k)2isEuclideannormofthestate vec-tor.Basically,therequirementfortheasymptoticallystablesystem isthat[18]: K
j=0 Aj <1 (8)where discrete state transition matrix is defined as in (5) and
Aj = max ATj Aj andmax(•) aretheEuclidean normandmaximumeigenvalueofthematrixrespectively. Condi-tionin(8)canbeobtainedbyconsideringthesolutionin(6)and definitionoftimevaryingtransitionmatrixin(7).Ifabove require-mentisevaluatedbydiscretestatetransitionmatrixin(5),onecan findrestrictivestabilityconditionofdiscretetimefractionalorder system.In[19]theoremgivenbelowwasprovedfortheasymptotic stabilityofcommensurateFOMgivenin(3)and(5).However,for thenon-commensurateFOM,theconditionisslightlymodified.Theorem4. Asystemin(3)–(5)isasymptoticallystableforallstates
attimeinstantkifthefollowingconditionissatisfiedforallfractional orders:
A0 <max i (r (K,i)) wherer (K,i)=1 i=0 i+ (K+2−i) (K+2) (1−i) i>0 iscalleda spec-tralradiusofthematrixA0 .
ProofofTheorem3:Byusingthefactthat
A+B≤A+B, (8)canbewrittenas A0 <1− K j=1 (−1)jϒ j+1Iftheclosedformfortheabovesumisexisted,byusingpartialsums ofgeneralbinomialcoefficientsas
kj=0(−1)ji j
=
k−i k
(i ∈Randk≥i,k ∈Z+∪
0
)andbyreplacingthedefinitionFig.2.r (K,i) inFOM4.
ofϒj+1in(4),theupperbondforthematrixnormofA0
canbe writtenas: A0 <1 −In+diag −1−k+1−1 k+1
,···,−n−
k+1−n k+1
(9)
Finally,whenGammafunctionisreplacedinsteadofthe non-integerbinomialcoefficients,
K+1−i K+1
= (K+2−i) (K+2) (1−i) spectralradiusofthematrixisfoundasinTheorem3
Theorem3isaspecialcaseofoneofthematrixmetricsdefined in[18] (Theorems 3,4 and5)for thenon-commensurateFOM. However,theadvantageofTheorem3shouldbealsonotedthat Theorem3islessrestrictivethanthestabilityrequirementin(8). 2.5.1. Theeffectofthemaximumtimedelay(K)
Spectral radius (max i
(r (K,i))) of the matrix A0
only dependsonthefractionalorderiandmaximumtime delayK. InFig.2r (K,i) isplottedfor1≤K≤200withafixediinFOM4. SimilarfigureswereobtainedinFOM1,FOM2andFOM3.Wefound thatspectralradiusinverselychangesbyKwithadecreasingtrend. InthisstudyweassumedK=100,because(.)functionincreases tothelevelcausingthenumericalinstabilities.2.5.2. Theeffectofthefractionalordervector()
Fractionalordersdirectlyaffectthespectralradiusofthematrix
A0
.Toexploretheeffects,considertheFig.3.Fig.3showsthe spectralradiusversusfractionalorderforK=10andK=100.From thefigure,itis observedthatspectralradiusisminimumwhen 0.2≤i≤0.4anddecreaseswhenmaximumtimedelayincreases. 2.5.3. Theeffectoftheparameters()Itshouldbestatedthat,parametervalueshaveadirecteffect onthenormofA0
,whichisconstructedbythestatetransition matrix.However,theboundonthestability,whichiscalledasa spectralradiushasnoinfluenceontheparameters.Thisfurther meansthatonlymodelstructureandfractionalorderscanchange theupperboundsofthestabilitycriterion.3. Results
Beforetheresults,itshouldbeemphasizedthat,inthisstudy thestate-spaceanalysisisdonebasedontheestimated param-etervalues given in theliterature.Therefore, this analysisonly
Fig.3. r (K,i) for0≤i≤1andK=10K=100.
Table3
ControllabilityandObservabilityResults
Models Controllability rank(PK
) Observability rank(OK ) FOM1 Healthy 2 1 COPD 2 1 FOM2 Healthy 2 2 COPD 2 2 FOM3 Healthy 2 2 COPD 2 2 FOM4 Healthy 2 2 COPD 2 2 Table4 StabilityResultsModels EuclideanNorm SpectralRadius Stability
A0 maxi
(r (K,i))
FOM1 Healthy 0.9795 1.0004 YES
COPD 4.4519 0.6154 NO
FOM2 Healthy 2.0865 1.0004 NO
COPD 1.2990 1 NO
FOM3 Healthy 0.7813 1 YES
COPD 1.1228 1 NO
FOM4 Healthy 0.6474 0.8161 YES
COPD 1.1856 0.6556 NO
requiresparametervalues(Table1)and modelstructuresgiven in(1).Controllability(PK
)andobservability(OK )matrices wereconstructedwiththehelpofTheorems1and2.The control-labilityandobservabilityresultsareshowninTable3.Ascanbe seenfromthetable,resultsprovethatallmodels,exceptFOM1are bothcontrollableandobservable,becausetheranksofPK and OK matchtothenumberofthestatesinthemodels.Asymptoticstabilityanalysiswereperformedaccordingtothe Theorem3.Table4shows spectralradius(max
i
(r (K,i)))ofthe matrixA0
forallmodels.NormofthematrixA0 andstability resultsarealsogiveninthesametable.4. Discussion
InFOMsinput(control)signalisthemouthpressureand out-putsignalisthetotalairflow.HoweverstatesofFOMsdonothave
direct physiological explanations. Allelasticpressure related to thecomplianceparameter(C)andairflowrelatedtotheinertance parameter(L)aredynamicelementsinthemodels.Wecanassume thatX1(t) andX2(t) representthepressureandtheairflowrelated dynamicsinFOMsrespectivelyduetothedefinitionsinthemodels. Table 3 shows theresults of thecontrollability and observ-ability analysis.Results provethatallmodels,exceptFOM1are bothcontrollableandobservable,becausetheranksofPK
and OK matchtothenumberofthestatesinthemodels.Firstof all,in section 2.4.3, it is stated thatcontrollability and observ-abilitydonotdependontheparametervalues,thustheyarenot linkedtothehealthyanddiseasedconditions.Physiological expla-nation for the existence of thecontrollability condition is that mouth pressure can manipulate time-varyingdynamics in res-piratory system. This furtherexplains that as irrelevant to the FOMstructure,pressureand flowsinrespiratorysystemcanbe estimatedfrommouth pressure.On theotherhand,absenceof theobservabilityconditionwasobservedinFOM1.Physiological explanation for this result is that measured airflowalone can-notaffectthetime-varyingdynamicsintherespiratory system, ifitismodelledbyFOM1.Also,itisobservedthatthelossofan inertanceparameter(L)inFOM1resultsinunobservablemodels. Thisfurthermeansthat although structuralstatic propertiesof FOM1isknown,itcannotbedynamicallyidentifiedbythe mea-suredairflowsignal.Theonlyremedyforthismodellingproblem is tochangethemodel structureby includingorexcluding the parameters.Discrete-timeversionofFOMsistheinfinitememorytime-delay systems.ThisistheoneofthereasonsthatFOMsrepresentthe stressrelationofthelungtissue.Controllabilityandobservability analysisalsoindicatedthattheeffectofthemouthpressureand theairflowonthesystemdynamicsisnotafunctionofthememory ofthesystem.Also,parametervaluesandfractionalordersdonot altercontrollabilityandobservabilityresults.Itseemsthatmodel structureplaysanimportantroleondeterminationofthemouth pressureandairfloweffectsinFOMs.
Table4showstheresultsofthestabilityanalysis.Firstly, con-servativeandsufficientdelay-independentstabilitycriterionin(8) yieldsthat thenormof theindefinite statetransitionmatrix is responsibleforthestabilityofFOMs.Thisresultisexpectedand canbeexplainedbythebasicstabilitytheoremindiscrete-time systems.HoweverstabilitybondsfortheFOMsinTheorem3is establishedasanupperboundsforthestabilityrestriction. This meansthatTheorem3alsoboundstheparameterranges.
Fig.2revealstheinterestingrelationshipbetweenthesystem memory,whichisthemaximumtimedelayandstabilityboundof thesystem.InallFOMsstabilityboundsdecreasemoreslowlyafter K=100.Thisisbecauseofthegammafunction.(K)intheratio increasesrapidlyandatthelimitonly(i)determinestheratio. ThiscanbeseenfromFig.3whichshowsthestabilityboundsfor differentfractionalorders.Weobservedthatfractionalordersare noticeablylowerforCOPDdata.ThismaybethereasonthatCOPD patientsdataprovidednon-stablemodels.Onthecontrary,all mod-elsexcept FOM2arestable modelsfortheparametervalues of healthysubjects.Physiologicalexplanationsofthestabilityresults canbedoneaccordingly.Thelinkbetweenmodelparameter val-uesanddiseasedconditions(airwayobstruction,changeintissue elasticityandmorphologicalchangesduetothetissue heterogene-ity)werealreadyestablishedinrecentstudies[6],[7]and[8].Here, weprovedmathematicallythatlowerfractionalordershavesome consequences in thestability ofFOMs. Thisresultdevelops the associationbetweenthecapabilitiesofFOMs inthe representa-tionof therespiratory systemandthefractionalorders.In this stage,weareveryreluctanttoassumearelationshipbetween sta-bilityandpathologyintheunderlyingsystem,sinceitneedsfurther proof.
5. Conclusion
Inthisstudy,controllability,observabilityandstability analy-sisofFOMswereperformedbyusingstate-spacerepresentationof themodelsevaluatedbyreferencevaluesgivenintheliterature. Resultsshowedthatthereisanuncontrollableandunobservable modein FOM1.Furthermore,stability analysisrevealed thatall COPDpatientsdataprovidedunstablemodels.Clinical applicabil-ityoftheproposedapproachinthisstudyisdirectlylinkedtothe applicabilityofthefractionalordermodelstotherespiratory sys-tem.Thisstudyrevealedthatstraightadoptionoftheestimated parametersin theclinicalstudiesmaybemisleadingor subop-timal due to the model behaviour revealed by the state-space analysisofthefractionalorder models.Thus,careful considera-tionmustbegivenwheninterpretingthefindingsinFOMsduring therespiratorytests.Results indicatethat parametershave dif-ferenteffects onthemodels.However,thisneedsfurtherstudy whichmayrevealhowtheimprecisionsintheparametervalues affecttheaccuracyofthesystemdynamics(statesandthe out-put).Finally,resultsand outcomesofthis studyprovidedsome improvementsintheunderstandingofthemodelsinthe litera-ture.
Acknowledgements
Thisstudywas partiallysupported bytheResearch Fund of IstanbulKulturUniversity,GrandNo:IKU-BAP1801.TheStudyhas ethicalapprovalfromEthicalCommitteeofIstanbulKultur Univer-sity,(2018.01anddated1,April2018).
DeclarationofCompetingInterest
Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.
AppendixA.
Firsttransferadmittancein(1)canbewrittenas YFOM(s)= 1 ZFOM(s)= Q (s) P (s) = Csˇ 1+RCsˇ+LCs˛+ˇ (10) where P (s) and Q (s) are the mouth pressure and the airflow throughmouthrespectively.Ifadmittanceisdecomposedintotwo partsbyusingintermediatevariableW (s):
YFOM(s)= Q (s) W (s)
W (s) P (s)
wherepartsareW (s)Q (s) =CsˇandW (s)
P(s) =1+RCsˇ1+LCs˛+ˇ,secondpartis transformedintotimedomainas:
P (t)=W (t)+RCˇtW (t)+LC˛+ˇt W (t) (11) Thentwoequationsintimedomainarebuiltas:
X1(t)=W (t) X2(t)=ˇtW (t)=
ˇ tX1(t)
(12) whereX1(t) andX2(t) arestatesin(2).˛thorderfractional deriva-tiveofX2(t) canbeobtainedfrom(11)and(12)as:
˛tX2(t)=− 1 LCX1(t)− R CX2(t)+ 1 LCP (t) (13)
Next,firstpartistransformedintotimedomainas:
Q (t)=CˇtW (t)=CX2(t) (14)
Finally,from(12),(13)and(14)correspondingstate-space repre-sentationisderived:
⎡
⎣
ˇ tX1(t) ˛ tX2(t)⎤
⎦
=⎡
⎣
0 1 −1 LC −R C⎤
⎦
X 1(t) X2(t)+
⎡
⎣
0 1 LC⎤
⎦
P (t) Q (t)=0 C X 1(t) X2(t)+ [0] P (t) (15) References
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