Heart rate control during treadmill exercise using input-sensitivity shaping for disturbance rejection of very-low-frequency heart rate variability

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

Heart

rate

control

during

treadmill

exercise

using

input-sensitivity

shaping

for

disturbance

rejection

of

very-low-frequency

heart

rate

variability

Kenneth

J.

Hunt

∗

,

Simon

E.

Fankhauser

InstituteforRehabilitationandPerformanceTechnology,DivisionofMechanicalEngineering,DepartmentofEngineeringandInformationTechnology,

BernUniversityofAppliedSciences,CH-3400Burgdorf,Switzerland

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received9March2016

Receivedinrevisedform30May2016

Accepted20June2016 Keywords: Feedbackcontrol Heartrate Treadmillexercise Frequencyresponse Physiologicalcontrol

a

b

s

t

r

a

c

t

Background:Automaticandaccuratecontrolofheartrate(HR)duringtreadmillexerciseisimportantfor prescriptionandimplementationoftrainingprotocols.Theprincipaldesignissueforfeedbackcontrol ofHRistoachievedisturbancerejectionofvery-low-frequencyheartratevariability(VLF-HRV)witha levelofcontrolsignalactivity(treadmillspeed)whichissufficientlysmoothandacceptabletotherunner. Thisworkaimedtodevelopanewmethodforfeedbackcontrolofheartrateduringtreadmillexercise basedonshapingoftheinputsensitivityfunction,andtoempiricallyevaluatequantitativeperformance outcomesinanexperimentalstudy.

Methods:Thirtyhealthymalesubjectsparticipated.20subjectswereincludedinaprecedingstudyto determineanapproximate,averagenominalmodelofheartratedynamics,and10werenot.Thedesign methodguaranteesthattheinputsensitivityfunctiongainmonotonicallydecreaseswithfrequency,is thereforedevoidofpeaking,andhasapre-specifiedvalueatachosencriticalfrequency,thusavoiding unwantedamplificationofHRVdisturbancesinthevery-low-frequencyband.Controllersweredesigned usingtheexistingapproximatenominalplantmodelwhichwasnotspecifictoanyofthesubjectstested. Results:Accurate,stableandrobustoverallperformancewasobservedforall30subjects,withamean RMStrackingerrorof2.96beats/minandasmooth,low-powercontrolsignal.Therewerenosignificant differencesintrackingaccuracyorcontrolsignalpowerbetweenthe10subjectswhowerenotinthe precedingidentificationstudyandamatchedsubgroupofsubjectswhowere(respectively:meanRMSE 2.69vs.3.28beats/min,p=0.24;meancontrolsignalpower15.62vs.16.31×10−4m2_/s2_,p=0.37).

Sub-stantialandsignificantreductionsovertimeinRMStrackingerrorandaveragecontrolsignalpowerwere observed.

Conclusions:Theinput-sensitivity-shapingmethodprovidesadirectwaytoaddresstheprincipaldesign challengeforHRcontrol,namelydisturbancerejectioninrelationtoVLF-HRV,anddeliveredrobustand accuratetrackingwithasmooth,low-powercontrolsignal.Issuesofparametricandstructuralplant uncertaintyaresecondarybecauseasimpleapproximateplantmodel,notspecifictoanyofthesubjects tested,wassufficienttoachieveaccurate,stableandrobustheartratecontrolperformance.

©2016TheAuthor(s).PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Theabilitytoautomaticallyandaccuratelycontrolheartrate (HR)duringtreadmillexercisewouldbringimportantbenefitsfor theprescriptionand implementationofexercisetraining proto-cols.Heartrateisusedtodelineatetheexerciseintensityregimes whichformpartofcurrentrecommendationsfordevelopmentand

∗ Correspondingauthor.

E-mailaddresses:[email protected](K.J.Hunt),[email protected]

(S.E.Fankhauser).

maintenanceofcardiorespiratoryfitness[1];these recommenda-tionsaregivenintermsoffrequency,durationandintensity,the lattertypicallylyingintherangeof“moderate”to“vigorous” exer-cise.Exercise intensity,in turn,is described asa percentage of eithermaximalheartrate(HRmax)orofheartratereserve(HRR), whichisthedifferencebetweenmaximalandrestingheartrates: HRRHRmax−HRrest.UsingHRR,moderateandvigorous intensi-tiescorrespondrespectivelytotheranges40–59%and60–89%of HRR[2].

High-intensityintervaltraining(HIT),whichcombinesperiods of vigorous to high-intensity exercise with low or moderate-intensityrecoveryperiods,hasbeenshowntoprovideadditional

http://dx.doi.org/10.1016/j.bspc.2016.06.005

1746-8094/©2016TheAuthor(s).PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).

source:

https://doi.org/10.24451/arbor.5896

Fig.1. Fourprincipalfrequencybandsforheartratevariabilityanalysis:ultra-low

frequency(ULF),very-lowfrequency(VLF),lowfrequency(LF)andhighfrequency

(HF).AmplitudespectrumofHRforsubjectS02(orangetrace,right-handy-axis).

Inputsensitivityfunctionmagnitudes|Uo(jω)|(left-handy-axis)fornaivefeedback

design(bluetrace|Uox|)andforshapedcontrollerC1(redtrace|Uo1|;Eqs.(33)and

(34)).(Forinterpretationofthereferencestocolourinthislegend,thereaderis

referredtothewebversionofthearticle.)

benefits for cardiorespiratory fitness and cardiovascular func-tion when compared toconstant-intensity training (systematic reviews:[3,4]).Thereisconsiderableflexibilityinsettingthe dura-tionsandintensitylevelsofthedifferentregimeswithinHITin healthyadults [3]and in variouspatientgroups, e.g.in cardiac rehabilitation[5].Thismotivatesthedevelopmentofaccurateand robustfeedbackapproachesforautomaticcontrolofarbitraryheart ratereferenceprofiles.Fortreadmill-basedtraining,thefeedback controllerwouldautomaticallyadjustthetreadmillspeedbasedon continuousobservationofthereferenceandactualHRvalues.

Theprimarydesignchallengeforfeedbackcontrolofheartrate is toensure that thecontrol systemmaintains acceptable per-formanceinthefaceofdisturbancestotheheartratecausedby physiologicalheartratevariability(HRV)[6];thisconceptis sup-portedby datapresented inthepreliminaryobservationalcase studybelow. Performanceof heart ratecontrol systemsshould always be quantified in terms of both tracking accuracy (e.g. root-mean-squareheart-ratetrackingerror,RMSE)andthelevel ofactivityof thecontrolsignal(e.g.averagepowerofthe con-trolsignal,i.e.thetreadmillspeedreference).FortheHRcontrol application,theclassicaltrade-offbetweentrackingaccuracyand controlsignalpower–higheraccuracyisusuallyachievedatthe costofincreasedcontrolsignalactivity–isparticularlypronounced andimportant:theHRVdisturbanceenteringthesystemwillbe rejectedtoadegreedefinedbythefrequency-responseofthe sen-sitivityfunction(So,Eq.(11)),withahigherlevelofdisturbance rejectionleadinggenerallytolowertrackingerrorbutrequiring highercontrolsignalpower.Sincethecontrolsignalinthiscase isthetreadmillspeedreference,changesinthisvariabledirectly impactonthehumansubjectrunningonthetreadmill,sothese changesmustbekeptwithinacceptablelimits,evenifsomedegree oftrackingaccuracyhastobesacrificed;hencetheimportanceof theinputsensitivityfunction(Uo,Eq.(12)),whichlinkstheHRV disturbancetothecontrolsignal.

CurrentstandardsformeasurementandinterpretationofHRV identifyfourprincipalfrequencybandsforanalysis[7,8](cf.Fig.1): • ultra-lowfrequency(ULF),withfrequencyf<0.003Hz;

• very-lowfrequency(VLF),where0.003≤f<0.04Hz; • lowfrequency(LF),0.04≤f<0.15Hz;

• highfrequency(HF),0.15≤f≤0.4Hz.

Fordesignofheartratecontrollers,theVLFcomponentisofprimary importancebecausethisbandusuallyincorporatesthecrossover regionofthefeedbackloop;peakingofthesensitivityfunctions canoccurinthecrossoverregion,potentiallyleadingtounwanted powerinthecontrolsignalintheVLFfrequencyband,which man-ifestsaschangesinthetreadmillspeedwhichwouldbestrongly perceptibletotherunner.HRVintheULFband,incontrast, repre-sentsaveryslowdisturbancewhichcanreadilybefullyrejectedby havinghighgaininthecontrollerinthisrange(e.g.byusing inte-gralaction);theresultingvery-slowchangesinthecontrolsignal wouldnotbeperceivedasunpleasantorundesirablebytherunner –theupper-frequencyboundoftheULFrange,0.003Hz, corre-spondstoanoscillationperiodofjustover5min(333.3s).HRVin theLFandHFfrequencybands,ontheotherhand,willtypically lieoutwiththebandwidthofthefeedbackloopandwillhavelittle effectonthecontrolsignalifthecontroller’sfrequencyresponseis appropriatelydesigned;onewaytodothisistoprescribea strictly-propercontrollertransferfunctionsothattheloopgainrollsoff towardszeroabovethecrossoverregion.Thismakesthecontrol signalinsensitivetoHRVdisturbancesintheLFandHFbands.

Theseconsiderationsemphasizethatthefeedbackloop proper-tiesintheVLFbandareparamountandthatdisturbancerejection behaviouristhekeydesignissue;theseconceptsarefurther elu-cidated in thecase presented in Section 2. To directlyaddress thesechallenges,anoveldesignapproach isderived andtested in the present work which is based on shaping the frequency responseoftheinputsensitivityfunction(Uo,Eq.(12):the trans-ferfunctionbetweentheHRVdisturbanceandthecontrolsignal) sothatithasapre-specifiedgainataselectedcriticalfrequency inthecrossoverregionwithintheVLFband.Moreover,thedesign approachisconstrainedtomakethegainoftheinputsensitivity functionmonotonicallydecreasingwithfrequency,sothat peak-ingofthisgaincannotoccur.Finally,therequirementsoftheULF andLF/HFbandsareaddressedrespectively,asalludedtoabove, byincludingintegralactioninthefeedbackcompensatorandby makingitstrictlyproper(i.e.lowpass).

PreviousworkontreadmillHRcontrolhasfocusednotonthe keyissuesofHRVanddisturbancerejection,butratheron paramet-ricandstructuralplantuncertainty[9–12].Afurthernovelelement ofthepresentworkistheassumptionofaverysimpleand approx-imatemodelofheartratedynamics,whichwastheoutcomeofa companionidentificationstudy[13].Thecontroldesignapproach detailedhereusesthissingleapproximatenominalplantmodel, anddoesnotrequireanyinformationon,oridentificationof,heart ratedynamics for individual runners.A similar nominalmodel strategywastakeninrelatedreportsofheartratecontrolduring outdoorrunning[14]andinacomparisonoflinearandnonlinear heartratecontrollers[15].

The aim of the present work was twofold: to set out the input-sensitivity-shaping method for feedback control of heart rateduringtreadmillexercise;andtoempiricallyevaluate quan-titativeperformanceoutcomeswiththeproposedmethodinan experimentalstudywithanumberofsubjectssufficienttoallow statisticallyvalidconclusionstobedrawn.

2. HRV–preliminaryobservationalcasestudy

Theintroductorydiscussionhighlightedtheimportanceofthe VLFbandofHRVforthedesignofHRcontrolsystems,andthat disturbance rejection is the principal design issue. These con-ceptscanbeexemplifiedbyconsideringdatarecordedfromone

ofthesubjectswhoparticipatedinthepresentstudy(subjectS02, Figs.1and2).

Spectralanalysisofthissubject’sHR,obtainedusingFFTanalysis ofrawECGdatarecordedduringa20-minperiodofconstant-speed, vigorous-intensityrunning,showedthatHRVpowerwas concen-tratedprimarilywithintheVLFfrequencyband(Fig.1).Whenthis subject’sHRwascontrolledusinganaivefeedbackdesignwith sub-stantialpeakingoftheinputsensitivityfunction(bluetrace|Uox|in

Fig.1),verystrongandwhollyunacceptablevariationofthe tread-millspeedwasobservedatadominantfrequencycorresponding closelytothefrequencyoftheresonantpeakoftheinputsensitivity functionduringbothreferencetracking(Fig.2(a))andsteady-state regulation(Fig.2(c)):thepeakvalueof|Uox|is0.07(−23dB)ata frequencyof0.008Hz(Fig.1);thecontrolsignal(lowerplotsin Fig.2(a)and(c))canbeseen tobeoscillatingata frequencyof about0.01Hz(e.g.thereareabout10clearly-identifiablepeaksin thetimerange600≤t≤1600sinthelowerplotofFig.2(c)).

Moreover, the ratio of the peak-to-peak amplitudes of the controlsignaloscillationandtheprincipalcomponentofHRV cor-respondscloselytothepeakgainoftheinputsensitivityfunction |Uox|,whichwasidentifiedaboveas0.07(−23dB):consideringthe half-periodfollowing t=820sinFig.2(c),a risein HRofabout

11bpmwasseen,resultinginachangeinspeedofabout0.79m/s, givingagainof0.79/11=0.07.Theseoscillationswereapparently drivenbyboththestepchangesinthereferencesignal(Fig.2(a)) andbytheVLFcomponentofHRV(Fig.2(c));thisisasexpected, sincetheinputsensitivityfunctionlinksboththereferenceandthe disturbancetothecontrolsignal(cf.Eq.(12)).

Whenacontrollerwasemployedwhichwasdesignedusingthe proposedinput-sensitivity-shapingapproach(Section4),denoted ascontrollerC1,wherethegainoftheinputsensitivityfunction atthecriticalfrequency0.01HzwithintheVLFbandwaspinned downtoavaluewhichwasonequarterofthevaluedeterminedfor thenaivecontrollerabove,i.e.0.0174(−35.19dB,redtrace|Uo1| inFig.1),andnopeakingwaspermitted,oscillationofthe con-trolsignalwaseliminated(Fig.2(b)and(d),lowerplots).Forthis example,theLFandHFbandsforHRVliewellabovetheclosed-loop bandwidth(Fig.1).ULFvariability,ontheotherhand,isseentobe eliminatedappropriatelybyintegralaction,apparentthroughthe ultra-slow,decreasingtrendfortreadmillspeedovertimeineach ofthetests.

Withthiscontroller,C1,itcanagainbeseenthattheratioof theobservedpeak-to-peakamplitudesofthecontrolsignalandthe principalcomponentofHRVcorrespondscloselytothechosenpeak

Fig.2. Illustrativeexample–HRcontrolmeasurementswithsubjectS02.Leftcolumn:referencetracking(a)andconstant-referenceregulation(c)usingnaivedesignwith

peakinginputsensitivityfunctionUox.Rightcolumn:referencetracking(b)andconstant-referenceregulation(d)usinginput-sensitivity-shapedcontrollerC1,Uo1.Inthe

upperpartofeachfigure,HR*istheheartratereference,HRnomisthetargetnominalheartrateresponse(simulated),andHRisthemeasuredheartrate.Inthelowergraphs,

Table1 Subjectcharacteristics. Mean(SD) Range Age/(y) 28.4(10.1) 18–57 Bodymass/(kg) 75.3(9.9) 60–101 Height/(m) 1.77(0.07) 1.63–1.93 BMI/(kg/m2_{) 24.0(2.5) 19.6–30.2} n=30,allmale. SD:standarddeviation.

BMI:bodymassindex;BMI=mass/height2_.

gainoftheinputsensitivityfunction,viz.|Uo1|f=0.01=0.0174:the maximumpeak-to-peakchangeinHRduringthetestswithC1was ∼10bpm(Fig.2(b)and(d),upperplots);analysisofthetimeperiod 1625≤t≤1725sduringtheregulationtest(Fig.2(d)),wherethe largestchangeinspeedduringthistestoccurred,gaveachangein speedof0.174m/s.Thustheobservedgainwas0.174/10=0.0174. This magnitude of change in speed in response to the largest observedvariationinheartratearoundthecriticalfrequencyof 0.01Hzwasbarelyperceptibletotherunner.Thecriticalgainand frequencyvaluesgivenby|Uo1|f=0.01=0.0174werethereforetaken forwardforfullexperimentalevaluationofcontrollerC1asdetailed inthesequel.

3. Experimentalmethods

3.1. Subjectsandethics

Thirtyhealthymalesparticipatedinthestudy(details–Table1). Ofthese30subjects,20hadtakenpartinaprecedingstudy[13] todeterminean approximate, averagenominal model of heart ratedynamics,and 10hadnot. Thestudywasapprovedbythe localethicscommittee(EthicsCommitteeoftheSwissCantonof Bern,Ref.KEK-Nr.313/14)andsubjectsprovidedwritten,informed consentduringrecruitment.Theprimaryinclusion/exclusion crite-riawere:healthymalesfrom18to60yearsofage,andabsence ofmuscular,orthopaedicorcardiovascularissues contraindicat-ingvigoroustreadmillexercise.Priortoeachtest,subjectswere requiredtoavoidstrenuousexercise,alcoholandsmoking(all24h inadvance),andalsoheavymeals(4h)andcaffeine(12h). 3.2. Apparatus

Acomputer-controlledtreadmillwasemployed(modelVenus, h/p/cosmosSportsandMedicalGmbH,Germany)connectedbyan RS-232serialcommunicationlinktoaPC(Fig.3).Theheartrate controllerswereimplementedinthePCusingMatlab/Simulinkand theassociatedRealTimeWorkshop(TheMathworks,Inc.,USA).

Controllersweredesignedinthecontinuous-timedomain (Sec-tion4)andimplementeddigitallyusingasampleintervalofTs=5s. Thischoice ofTs was basedon therecommendationof having ∼4–10samplesperrisetimeoftheplant[16]–thenominalplant modelhadatimeconstantof57.6s(Eq.(1),Section3.3),butthis nominaltimeconstantisanaverageobtainedover48individual models,whereindividualvalueswereseentobeaslowas=27.3s [13].

Heart rate was measured using a chest belt (model T34, PolarElectroOy,Finland)andthreeassociatedwirelessreceivers arrangedinatriangleconfigurationwithrespecttothetreadmill surface,andintegratedwiththetreadmillhardwarecontrolunit. HeartratewastransmittedfromthetreadmilltothePCviathe seriallinkandreadintothesimulinkreal-timecontrolmodelat therateof1Hz.Ateachcontrollersampleinstant,i.e.every5s,the 5heartratesamplesobtainedinthepreceding5-ssampleinterval wereaveragedandfedintothecontrolloop.

Fig.3.ExperimentalsetupshowingthePCinwhichthereal-timeheartrate

con-trollerswereimplemented(A),thetreadmillcontrolunit(B)andthetreadmill

(C).

3.3. Approximatenominalplantmodelandcontrolapproach Feedbackcontroldesignutilisedanapproximatenominalmodel Poofthedynamicresponsebetweenthetreadmillspeedreference signal

v

andheartrateHR[13]:

v

→HR: Po(s)= k s+1 =

24.2

57.6s+1, (1)

where k is the steady-state gain and  is the time constant of this first-order model structure. The nominal plant parameter k=24.2bpm/(m/s) and =57.6swere obtained in the previous studyasaveragesof48individualmodelswhichwereempirically identifiedfrom24healthymalesubjectswhoeachran,in sepa-rateidentificationtests,atmoderateandvigorousintensities,thus giving24subjects×2models/subject=48modelsintotal[13].

Thestructureemployedhereforheartratecontrolcompriseda feedbackcompensatorCandareferenceprefilterCpf(Fig.4).The signalHR*istheheartratereference,

v

isthetreadmillspeed refer-ence,whichservesastheplantcontrolinput,anddisadisturbance. randeareintermediatesignalswhichareusedintheanalysisthat follows(Section4).

Thefeedback compensator C wasdesigned using the input-sensitivity-shapingmethodologydetailedinthesequel(Section4),

Fig.4.Controlstructure:CisthefeedbackcompensatorandCpfisareferenceprefilter.TheprincipalsignalsinthisstructurearetheheartratereferenceHR*,theactualheart

rateHR,thetreadmillspeedreferencev(plantcontrolinput),andadisturbancedwhichmainlyrepresentsheartratevariability.

whilethereferenceprefilterCpfwasseparatelydesignedtosetthe dynamicsoftheHRreferencetrackingresponse.

Following the terminology and notation used classically by Kwakernaak [17], theinput sensitivity function, denoted Uo, is definedasthetransferfunctionfromthedisturbancedtotheplant controlinput

v

:

d_→

v

: Uo(s)=

C(s) 1+C(s)Po(s)

. (2)

Inthepresentcontextofheartratecontrol,thedisturbancesignal distakentorepresentnatural,butunpredictable,changesinheart ratestemmingfromphysiologicalheartratevariability.

Thekeyideaoftheinput-sensitivityfeedbackdesignmethod (detailsinSection4)istoshapethefrequencyresponseoftheinput sensitivityfunctionsuchthatitsgain|Uo(jω)|hasaspecified nom-inalvalueatagivencriticalfrequency,and,moreover,suchthat |Uo(jω)|ismonotonicallydecreasingwithfrequencysothat peak-ingcannotoccur.Thesemeasuresareintendedtoensurethatheart ratevariabilitydoesnotleadtoanunacceptablelevelofactivity inthetreadmillspeedaroundthecriticalfrequency.Twodifferent controllersweretested,denotedC1andC2(Section4.5),whereC2 hadagainwhichwas+4dBhigher(i.e.higherbyanabsolutefactor of∼1.6)thanC1atthechosencriticalfrequency.

ThereferenceprefilterCpf wasseparatelydesignedtosetthe dynamicsoftheHRreferencetrackingtoanoverallclosed-loop transferfunctionTcl,whichhadtheformofthestandard second-ordertransferfunction:

HR∗→HR: Tcl(s)= ω2 n s2_+2ωns+ω2 n , (3)

whereistherelativedampingandωnistheundampednatural frequencyofoscillation.

Consideringthecontrolstructureemployed(Fig.4),the desig-natedoverallclosed-looptransferfunctionisobtainedbysetting theprefilterto

Cpf =Tcl·To−1, (4)

whereTo,thecomplementarysensitivityfunction(seeEq.(10)),is thetransferfunctionfromrtoHR.

Forallcontrollertests,wassettothecriticalvalue=1while ωnwasobtainedfromachosen10–90%closed-looprisetimetras ωn=3.35/tr,whichistheapproximationvalidaround=1[18,p.

196].Inalltests,therisetimewassettotr=120s. 3.4. Testingprotocol

Alltestswerecarried outaccordingtoasystematicprotocol (Fig.5):subjectsfirstwarmedupfor10minatamoderateintensity; a10minrestfollowed;thentherewasaformalmeasurementphase lasting35min;finally,therewasa10mincooldownofrunningat lowintensity.Priortothewarmupoftheirfirsttest,restingheart rateHRrestwasobtainedforeachsubjectbymanualpalpationat thewristaftertheyhadlainrestingfor5min.Estimatesofmaximal heartrateweresettoHRmax=220−age(cf.[19]).

Followingeachtest,thequantitativeoutcomemeasures(RMS trackingerror,RMSE,andaveragecontrolsignalpower,Pv,Eqs.(5)

and(6),Section3.5)werecalculatedoveranevaluationperiodof 300≤t≤1800s.Duringtheevaluationperiod,theheartrate refer-enceHR*wasvariedby±10bpmaroundamidlevel.Themidlevel wasindividuallysetforeachsubjectusing70%oftheirownheart ratereserve(HRR)tothevalueHR70%0.7(HRmax−HRrest)+HRrest. Theheartratereferencethus changedina square-waveformat every5min asHR*=HR70%±10 (Fig.5).The cooldownwasat 15bpmbelowtheindividuallowerleveloftheevaluationperiod, viz.HR70%−25.

3.5. Primaryoutcomemeasures

Closed-loopcontrolperformancewasquantitativelyassessed usingtheroot-mean-square trackingerror(RMSE)fortheheart rate,andtheaveragepowerofchangesinthecontrolsignal

v

(Pv),

where

v

isthetreadmillspeedreference.

RMSEwasevaluatedonaninterval[i1,i2]correspondingtothe evaluationperiod300≤t≤1800s(Fig.5),whereidenotethe dis-cretesampleindices,as:

RMSE=









1 N i2



i=i1

(HRnom(i)−HR(i))2 (5)

withN=i2−i1+1.HRnomisthetargetnominalheartrateresponse whichwasobtainedbysimulatingthenominalclosed-looptransfer functionTclforHR*→HR(Eq.(3),Fig.4),i.e.HRnom=TclHR*.

Theintensityofthecontrolsignalwascharacterisedasthe aver-agepowerofchangesin

v

overthesameinterval:

Pv= 1 N−1 i2



i=i1+1 (

v

(i)−

v

(i−1))2, (6)

whichisoftenreferredtointhesequelforsimplicityas“average controlsignalpower.”

3.6. Experimentaldesignandstatisticalanalysis

All30subjectsweretestedwithcontrollerC1;theprimary out-comesRMSE and Pv are presented usingdescriptive statistics

(n=30).

Todeterminewhetherpriorparticipationintheidentification study[13]hadanyinfluenceontheoutcomes–20ofthe30subjects inthepresentstudybelongedtothegroupof24subjectswhose individualmodelswereusedtodeterminetheaveragenominal model–acomparativesub-analysiswascarriedoutwherethe10 subjectsinthepresentstudywhohadnotbeenintheidentification studywerepairedto10ofthe20whohadbeenbymatching accord-ingtoageandBMI.Pairedtwo-sidedhypothesistestswereapplied tothematchedgroupsof10subjects(n=10)todeterminewhether anysignificantdifferencesexistedintheoutcomesRMSEandPv.

Thenullhypothesisforthisanalysiswasthattherewereno differ-encesinoutcomesbetweenthetwogroups,whicharereferredto belowas“C1ID”and“C1noID”.

Asecondcomparativeanalysiswascarriedoutontheoutcomes withcontrollersC1andC2,whereC2hadanadditional+4dBofgain intheinputsensitivityfunctionUoatthecriticalfrequencyωc (Sec-tion3.3),andalso,therefore,ahigherbandwidth(Section4.5).For thisanalysis,10ofthe30subjectsweretestedwithC2inaddition tohavingbeentestedwithC1.Pairedone-sidedhypothesistests (n=10)wereappliedtotheoutcomesRMSEandPv totestthe

expectation,giventhehigherbandwidthandcriticalgain,thatC2 wouldleadtomoreaccurateheartratetracking(i.e.lowerRMSE), butwiththecostofhighercontrolsignalpower(higherPv).The

nullhypothesisherewasthatC2didnotleadtothesedifferences inoutcomes.

Twofurthersub-analyseswerecarriedoutforcontrollerC1to exploreanydependenciesintheoutcomesRMSEandPvontime,

bothforsteady-stateregulationandfordynamicreference track-ing.Forsteady-stateregulation(allsubjects,n=30),theoutcomes werecalculatedandcomparedfortwoperiodsofconstant refer-encetracking:“early”,where300≤t≤600(5–10min);and“late”, where1800≤t≤2100(30–35min);(cf.Fig.5).Pairedtwo-sided hypothesistests(n=30)wereusedtodeterminewhetherany sig-nificantdifferencesexisted intheoutcomesfor thesetwo time periods,thenullhypothesisbeingthatnodifferencesexisted.

Thedatausedintheabovehypothesistestswerecheckfor nor-malityusingaKolmogorov–SmirnovtestwithLillieforssignificance correction.t-testswereappliedwhenthehypothesisofnormality wasnotrejected,andWilcoxonsignedranktestsotherwise.

Thesecondanalysisofpossibletimedependencyevaluatedand compared the outcomes RMSE and Pv for the fourindividual

dynamicstepchangesinheartratereferenceattimest=600,900, 1200and1500,wherebythestepnumber∈{1,2,3,4}servedas aproxytothesinglefactoroftime(Fig.5).Foreachstep,the out-comeswerecalculatedoverthe300-stimeintervalfollowingthe steponset.Thisanalysisusedone-wayrepeated-measures anal-ysisofvariance(ANOVA)withstepnumberasthefactor.When significancewasdetermined,post-hocpairwisecomparisonswith Bonferronicorrectionwerecarriedoutonallstep-numberpairs. Thenull hypothesis wasthatthere werenodifferencesin out-comesbetweentheindividualsteps.Thisindividualstepanalysis wasdonewithn=29becauseonesubject(S22)wasfoundtohave anabnormallyhighRMSEforstep2(cf.Fig.7(c),uppergraph,step 2):thisRMSElayoutwith3standarddeviationsofthemeanRMSE acrossallsubjectsforstep2andwasthereforeconsideredtobean outlier,andS22wasexcludedfromthispartoftheanalysis.

The significance level for all hypothesis tests was setat 5% (˛=0.05).Pairedhypothesistestswerecarriedoutusingthe Mat-labStatisticsandMachineLearningToolbox(TheMathworks,Inc.) andtheANOVAswithSPSSsoftware(IBMCorp.,USA).

4. Inputsensitivityshaping

Theinput-sensitivityshapingapproachdevelopedhereallows thegainoftheinputsensitivityfunctionUotobesettoadesired, pre-specifiedvaluegc atachosen criticalfrequencyωc,i.e.itis requiredthat|Uo(jωc)|=gc.Further,Uoispurposelystructuredso that|Uo(jω)|isamonotonicallydecreasingfunctionoffrequency ωandthereforedevoidofpeaking.Bythismeans,anacceptable nominalamplification/attenuationoftheheartratevariability dis-turbancearoundthecriticalfrequencywithintheVLFbandcanbe setbychoosingtheprimaryfeedbackdesignparametergc.

Thederivationisgivenfirstforthecaseofageneralnominal plantPo=Bo/Ao,i.e.withnorestrictionontheorderofthe sys-tem(Section4.1),andthenspecialisedtothefirst-ordercase,na=1 (Section4.2).

WeconsideracontrolstructurewithfeedbackcompensatorC andreferenceprefilterCpf(Fig.4).Theinput-sensitivityapproach pertainsonlytothefeedbackelementC,whiletheprefilterCpfis designedseparatelytogivetherequiredreferencetracking dynam-ics(Section3.3).

4.1. Generalcase

Theplantisdescribedbythestrictlyproperlineartransfer func-tionPo:

v

→HR: Po(s)= Bo(s) Ao(s)

, nb<na, (7)

wherenaandnbaretherespectivedegreesoftheplant denom-inatorandnumeratorpolynomialsAoandBo.Thestrictlyproper conditionreflectsthelow-passcharacteroftheplanttransfer func-tion,sincenb<na⇔ lim

ω→∞|Po(jω)|=0.

We seek a strictly proper compensator transfer function C, describedinrationalformby

e_→

v

: C(s)₌ G(s)

H(s), ng<nh. (8)

CisassumedhenceforthtobenormalisedsuchthatHismonic. ThestrictlyproperconditionforCisadesignchoicewhichensures high-frequency rolloff in the compensator (because ng<nh⇔

lim

ω→∞|C(jω)|=0)and,therefore,intheinputsensitivityfunctionUo also,thusmakingtheloopinsensitivetohigh-frequency disturb-ancecomponents.Thecompensatorisfurtherconstrainedtohave integralactionbyincludingthefactorsinHviaH(s)=sH(s),giving C(s)= G(s)

sH(s), ng<nh+1, (9)

wheretheintegratorresultsinlim

ω→0|C(jω)|=∞,i.e.thecompensator hasinfinitesteady-stategain.

Inspectionofthefeedbackloop(Fig.4)givesthekeyclosed-loop transferfunctions: Complementarysensitivity, r→HR: To(s)= CPo 1+CPo = _A BoG oH+BoG; (10) Sensitivity, d→HR: So(s)= 1 1+CPo= AoH AoH+BoG; (11) Inputsensitivity, d→

v

, r→

v

: Uo(s)= C 1+CPo = AoG AoH+BoG , (12) fromwhichtheclosed-loopcharacteristic polynomialcanbe identifiedas

=AoH+BoG=AosH+BoG. (13)

Amajoralgebraic-structuralsimplificationcanbeobtainedby usingcompensatorzerostocancelstable,well-dampedplantpoles (cf.[20]);thecancelledplantpolesarethennotshiftedbythe feed-backandbecomepolesoftheclosed-loopsystem.Inthisspirit,we constrainthecompensatornumeratortobeG=AoG.The compen-satortransferfunction(9)thenspecialisesto

C(s)= AoG(s)

sH(s) , na+ng<nh+1 (14)

andthecharacteristicpolynomialto=AosH+BoAoG.Thus,fora solutiontoexist,Aomustbeafactorof,=Ao,showing explic-itlythatthecancelledplantpolesbecomepolesoftheclosed-loop system,andthereducedcharacteristicpolynomialisobtained as

=sH+BoG. (15)

Forauniquealgebraicsolutionbasedonequatingcoefficientson eithersideof(15),thetotalnumberofunknowncoefficientsinG andH,nh+ng+1(recall,Hismonic),mustbesetequaltothe degreeof(15),whichisn=nh+1(thedegreeofsH,because

ofthestrictlyproperconditiononPoandC).Thusnh+ng+1= nh+1,givingthesolutionng=0,thatisGissimplyaconstant, G(s)=g₀.

Thestrict-properconstraintforthecompensatorin(14)allows na+ng=nh asonepossiblesolution,viz.thesolutiongivingH withminimaldegree.Sincefromaboveng=0,nh=nafollows. Thecompensatorstructureistherefore

C(s)= Aog0

sH(s), nh=na (16)

whereg₀ andHareobtainedingeneralastheuniquesolutionof (15)withn=nh+1=na+1,nh=naandng=0.

Attentionnow turnstothestructure oftheinputsensitivity functionUowhich,from(12),generallyhastheform

d→

v

: Uo(s)=AoG

 . (17)

Asnotedabove,cancellationofplantpolesusingG=AoGresultsin =AoandconsequentlyUocanbesimplifiedto

Uo= A 2 oG Ao = AoG  . (18)

Thisformshowsthatanadditionalmajorsimplificationinthe structureofUocanbeobtainedbyplacingafurthersetof closed-looppolesatthelocationoftheopen-looppoles,whichisachieved byincluding Ao as a factorof . Sincen=na+1,  is then allowed to contain just one further pole; setting =(s+p)Ao, wherepisreal,theinputsensitivityfunctionsimplifiesdownto thefollowingfirst-ordertransfer-function:

Uo= G

(s+p). (19)

Since,fromabove,G=g₀ (ng=0),thefinalformforUois Uo=

g₀

(s+p). (20)

Thegainofthisfirst-ordertransferfunction,|Uo(jω)|,is,asdesired, monotonically decreasingwith frequency and cannot therefore haveanypeaking.Ithasabandwidthequaltop[rad/s].The coeffi-cientg₀ ₌Gisobtained,asnotedabove,astheuniquesolutionof (15)with=(s+p)Ao(thusn=na+1)andnh=na.

4.2. Firstordercase

Proceedingontheassumptionofafirst-orderplant Po(s)= Bo(s) Ao(s)= k s+1= k/ s+(1/), (21)

i.e.na=1,weobtainnh=1andthefinalcompensatorstructurecan bewrittenwithng=0,nh=1⇒G=g0,H=s+h0as

C(s)₌AoG(s) sH(s) =

(s+(1/))g₀

s(s+h₀) . (22)

An explicit, unique solution of (15) (sH+BoG=) can be obtained by noting that n=na+1=2, therefore (s)=s2+ ₁s+₀,andwriting s(s+h₀)+k_·g₀ =s2+₁s+₀ (23) togive h₀=₁, g₀ = _k·₀. (24) WithAo=s+(1/),=(s+p)Aois =(s+p)



s+1 



=s2+



p+1 



s+p . (25) Thus, ₁ =p+1 ,   0= p  (26)

and,from(24),thecontrollercoefficientsh₀andg₀ areobtained directlyintermsofthegiven plantparametersk and andthe bandwidthparameterp:

h₀ =₁ =p+1_, g₀ = _k·₀=p_k. (27) Substitutingforh₀andg₀in(22),thecompensatorbecomes C(s)=(s+(1/))·(p/k)

s(s+p+(1/)) (28)

and,using(20),theinputsensitivityfunctionis Uo= g₀ (s+p)= (p/k) (s+p)= (1/k) (1/p)·s+1, (29)

i.e.itisafirst-ordertransferfunctionwithsteady-stategain1/k andbandwidthp.With(28)and(29),thecompensatorand input-sensitivity transfer functions are seen to depend only on the nominalplantparameters k and,and ontheinput-sensitivity bandwidthparameterp.

4.3. Bandwidthselection

Thespecialandsimplestructureoftheinputsensitivityfunction inEq.(29)allowsthebandwidthparameterptobesetaccording toadesiredgainofUoatsomecriticalfrequency.

From(29),themagnitudeofUomaybewritten |Uo(jω)|=

p/k

(ω2_+p2₎1/2. (30)

Thiscanreadilybesolvedtoobtainanexplicitexpressionforp:

p= ω

((1/(k|Uo(jω)|)2)−1)

1/2. (31)

Definingacriticalfrequencyωcandacorrespondingcriticalgain gc,i.e.gc|Uo(jωc)|,thedesiredinput-sensitivitybandwidthpis

p= ωc

((1/(kgc)2)−1)

1/2. (32)

4.4. Algorithmsummary

Thestepsinvolvedincalculationofthecompensatortransfer functioncannowbesummarisedasfollows:

1.Givendata:nominalplantPo=k/(s+1)withsteady-stategaink andtimeconstant,Eq.(21).

2.Choose critical frequency and gain parameters, ωc and gc, and calculate input-sensitivity bandwidth parameter as p₌ ωc/((1/(kgc)2)−1)

1/2

,Eq.(32).

3.Implement feedback compensator as C(s)=(s+(1/))·(p/k)/ (s(s+p+(1/))),Eq.(28).

4.5. Controllercalculation

Twocontrollers,denotedC1andC2,weredesignedandtested basedontheapproximatenominalplantmodelEq.(1):

Po= k s₊1= 24.2 57.6s₊1= 0.420 s₊0.0174.

AsdiscussedinSection2,thecriticalgainandfrequencyvalues givenby|Uo1|f=0.01=0.0174wereselectedforcontrollerC1. Con-trollerC2wasdesignedtohave+4dBofgainatthesamecritical frequency.

Controller1(C1(s)):

1.Given data: the nominalplant, above, has steady-state gain k=24.2bpm/(m/s)andtimeconstant=57.6s.

2.Choosecriticalfrequencyandgainparameters:ωc=0.0628rad/s (=2·0.01≡0.01Hz)andgc1=0.0174(m/s)/bpm(≡−35.19dB). Calculateinput-sensitivitybandwidthparameterasp1=0.0292, Eq.(32).

3.ImplementfeedbackcompensatorasinEq.(28): C1(s)=

0.00121(s+0.0174)

s(s+0.0465) . (33)

Fig.6. Inputsensitivityfunctionmagnitude|Uo(jω)|forcontrollersC1(Uo1)andC2

(Uo2),whereC2wasdesignedtohaveagainof+4dBatthechosencriticalfrequency

fc.Thelatterisshownasfc=0.01Hz(≡ωc=0.0628rad/s).ThecriticalgainsgcforC1

andC2areshownindBasgc1dBandgc2dB,respectively.Bandwidth:theverticalblue

dottedlinemarksthebandwidthofUo1atfB1=0.0046Hz;theverticalreddotted

linemarksthebandwidthofUo2atfB2=0.0090Hz;thehorizontalblackdottedline

markstheassociatedgainreductionof−3dBwithrespecttothesteady-stategain.

(Forinterpretationofthereferencestocolourinthislegend,thereaderisreferred

tothewebversionofthearticle.)

Forthiscompensator,thecorrespondinginputsensitivityfunction is,(29), Uo1= 0.00121 (s+0.0292)= 0.0413 34.3s+1. (34)

This transfer function (Fig. 6) has steady-state gain 1/k= 0.0413(m/s)/bpm (≡−27.7dB) and bandwidth p1=0.0292rad/s (≡fB1=0.0046Hz).

Controller2(C2(s)):

Thesecondcontroller,C2,wasdesignedtohaveasubstantially higherbandwidththanC1:thecriticalfrequencyforUowaskept thesame,butthecriticalgainatthisfrequencywasincreasedby 4dB(i.e.byanabsolutefactorof1.5849).

1.Givendata:thesamenominalplantasabovewithk=24.2and =57.6.

2.Choosecriticalfrequencyandgainparameters:ωc=0.0628rad/s (as above) and gc2=0.0174×1.5849=0.0276(m/s)/bpm (≡−35.19+4=−31.19dB). Calculate input-sensitivity band-widthparameterasp2=0.0563,Eq.(32).

3.ImplementfeedbackcompensatorasinEq.(28): C2(s)=

0.00233(s+0.0174)

s(s+0.0737) . (35)

Theinputsensitivityfunctionisinthiscase,(29), Uo2=

0.00233 (s+0.0563)=

0.0413

17.8s+1. (36)

Thisinput-sensitivityfunction(Fig.6)hasthesamesteady-state gainasforcontroller1,thatistosay1/k=0.0413(−27.7dB),but abandwidthp2=0.0563rad/s(≡fB2=0.0090Hz)whichis approx-imately twice that for controller 1 (cf. Eqs. (34)and (36)), i.e. 0.0563/0.0292=1.93.

Thefrequencyresponsesofthetwoinputsensitivityfunctions, Uo1andUo2,canconvenientlybevisualisedandcomparedinaBode plot(Fig.6).

Table2

PrincipaloutcomemeasuresforoverallevaluationofC1.

Mean(SD) 95%CI Range

RMSE/(bpm) 2.96(0.85) 2.65–3.28 1.40–4.58

Pv/(10−4m2/s2) 16.00(1.41) 15.47–16.52 14.05–19.94

n=30.

SD:standarddeviation.

95%CI:95%confidenceintervalforthemean.

RMSE:root-mean-squaretrackingerror;Pv:averagepowerofchangesinv;bpm:

beatsperminute.

5. Results

ControllerC1 gavehighly accurate,stable and robustoverall performanceforall30 subjectstested,withameanRMS track-ingerrorof2.96bpmanda95%confidenceintervalforthemean RMSEof2.65–3.28bpm(Table2).Thecontrolsignal

v

,i.e.the tread-millspeedreference,wasstableandacceptablysmoothinalltests; theaveragepowerofchangesin

v

was16.00×10−4m2_/s2_(95%CI: 15.47–16.52×10−4m2_/s2_{).Inadditiontothesequantitative} out-comes,thecontrolperformanceforC1canbequalitativelyassessed

byinspectionofheartrateHRandcontrolsignal

v

forthetestswith thelowest,medianandhighestRMSEvalues(Fig.7).

Comparison between the performance of C1 for the non-identifiedsubjectgroup(“C1noID”)andforthematchedsubgroup ofsubjectswhohadparticipatedintheidentificationstudy(“C1 ID”)showednosignificantdifferenceintrackingaccuracy–mean RMSE2.69vs.3.28bpm,C1noIDvs.C1ID,p=0.24—oraverage con-trolsignalpower–meanPv15.62vs.16.31×10−4m2/s2,C1noID vs.C1ID,p=0.37–(Table3,upperrows;C1noIDvs.C1ID;Fig.8). ControllerC2,whichwaspurposelydesignedwith+4dBofgain atthecriticalfrequencyωc,showedsignificantlyhigheraverage controlsignalpower(meanPv 18.54vs.16.00×10−4m2/s2,C2 vs.C1,p=0.0012)andmodestevidenceofmoreaccuratetracking (lowermeanRMSEof2.71vs.2.95bpm,C2vs.C1,p=0.062)than C1(Table3,middlerows;C2vs.C1;Fig.8).

Examinationandcomparisonofthesteady-stateperformance ofC1duringtheearlytimeintervalof300≤t≤600andthelater time interval of 1800≤t≤2100showedsubstantialand signifi-cantimprovementintrackingaccuracy(decreaseinmeanRMSE from2.88to 2.17bpm, p<0.001) and a significantreduction in averagecontrolsignalpower(meanPv2.09vs.1.40×10−4m2/s2,

p=0.0030)overtime(Table3,lowerrows;Fig.8).

Fig.7.ResultswithC1withthelowest(a),median(b)andhighest(c)valuesforRMStrackingerror.Intheupperpartofeachfigure,HR*istheheartratereference,HRnomis

thetargetnominalheartrateresponse(simulated),andHRisthemeasuredheartrate.Inthelowergraphs,visthecontrolsignal,i.e.thetreadmillspeedreference.RMSE:

Table3

Principaloutcomemeasuresforpairedcomparisonsandp-valuesforcomparisonofmeans.

Mean(SD) MD(95%CI) n p-Value

C1noID C1ID RMSE/(bpm) 2.69(0.99) 3.28(0.88) −0.59(−1.66,0.48) 10 0.24 Pv/(10−4m2/s2) 15.62(1.76) 16.31(1.33) −0.69(−2.33,0.95) 10 0.37 C2 C1 RMSE/(bpm) 2.71(0.61) 2.95(0.83) −0.24(−0.55,0.08) 10 0.062 Pv/(10−4m2/s2) 18.54(3.10) 16.00(1.75) 2.55(1.17,3.92) 10 0.0012 C1,300≤t≤600 C1,1800≤t≤2100 RMSE/(bpm) 2.88(0.93) 2.17(1.00) 0.71(0.34,1.07) 30 4.4×10−4 Pv/(10−4m2/s2) 2.09(1.66) 1.40(1.51) 0.69(0.08,1.30) 30 0.0030 SD:standarddeviation. MD:meandifference.

95%CI:95%confidenceintervalforthemeandifference.

p-valuesare:forC1noIDvs.C1ID,pairedtwo-sidedt-test;forC2vs.C1,pairedone-sidedt-test;forC1(300–600)vs.C1(1800–2100),pairedtwo-sidedt-test(RMSE)and

pairedtwo-sidedWilcoxonsignedranktest(Pv).

RMSE:root-mean-squaretrackingerror;Pv:averagepowerofchangesinv;bpm:beatsperminute.

Comparison of tracking error and average control signal powerduringthedynamicchanges ofthefourindividualsteps 1–4showedsignificant differences in RMSE(overall p-value of p=0.0036) butnosignificantdifferenceinP_v (overall,p=0.14) overtime;(Table4,Fig.9).Proceedingtopost-hocpairwise com-parisonsofRMSEfortheindividualsteps,modestevidencewas

Fig.8.Meanvaluesofprincipaloutcomemeasuresforpairedcomparisons(cf.

Table3),withsignificanceindicators:*⇔p<0.05;**⇔p<0.01;***⇔p<0.001.The

overalloutcomesforC1areincludedforreferenceonly.

foundfora reductionbetweensteps1and 2(meanRMSE3.31 vs.2.69bpm,p=0.051),whilesignificantreductionswerefound betweensteps1 and3(3.31vs. 2.77bpm,p=0.017),and 1and 4(3.31vs.2.59bpm,p=0.027).Nootherpairedcomparisonsfor RMSEshowedasignificantdifference.

Fig.9.Meanvaluesofprincipaloutcomemeasuresforindividualsteps1–4(cf.

Table4),withsignificanceindicatorsforpairedcomparisonsonRMSE:*⇔p<0.05.

Table4

OutcomesforindividualstepswithC1.

Mean(SD) p-Value

Stepnumber

1 2 3 4

RMSE/(bpm) 3.31(1.35) 2.69(0.80) 2.77(1.11) 2.59(0.84) 0.0036

Pv/(10−4m2/s2) 19.20(2.60) 20.26(2.63) 18.86(2.67) 19.61(2.45) 0.14

p-valuesforpost-hocpairwisecomparisonsonRMSE:1vs.2,p=0.051;1vs.3,p=0.017;1vs.4,p=0.027;allotherpairedcomparisonsnotsignificant. n=29(S22excluded,seeSection3.6).

Steps1and3:up;steps2and4:down.

ValuesforRMSEandPvcalculatedover300-saftersteponset.

SD:standarddeviation.

RMSE:root-mean-squaretrackingerror;Pv:averagepowerofchangesinv;bpm:beatsperminute.

6. Discussion

This research work had two main objectives: to develop a new method for feedback control of heart rate during tread-millexercisebasedonshapingoftheinputsensitivityfunction, and toempiricallyevaluatequantitativeperformanceoutcomes withtheproposedmethodinanexperimentalstudy.Thecontrol approachwasmotivatedbytheprincipaldesignissueofachieving asuitabledegreeofdisturbancerejectionofvery-low-frequency heartratevariability,whilemaintainingalevelofcontrolsignal activitywhichissufficientlysmoothand acceptabletothe run-ner.

Thecontrol design methodwasfoundmeetthe disturbance rejectionandcontrolsignalrequirements:controllerC1wasrobust acrossall30subjectstested;overall heartratetracking perfor-mancewasaccuratewithameanRMSEof∼3bpm;andthecontrol signal(treadmillspeed)wasverysmoothwithalowaveragepower onatight rangeof∼14–20m2_/s2 _{acrossallsubjects.Theability} oftheinputsensitivityfunctiontoabsorbasubstantialdegreeof inter-subjectdifferencesinHRVismoststrikinglyapparentwhen consideringtheresultswithC1withthelowest,medianandhighest RMStrackingerrors(Fig.7).SubjectS26(lowestRMSEof1.40bpm, Fig.7(a))isclearlyanindividualwithintrinsicallylowHRV,while subjectS22,incontrast,hashigh-amplitudeHRV,leadingtothe highestRMSEof4.58bpm(Fig.7(c)),afactorof∼3.3higherthan thatforS26.DespitethiswiderangeofHRVmagnitudes,the aver-agecontrolsignalpoweranditsqualitativebehaviourremained satisfactorythroughout,withPv ontheproportionatelynarrow

rangeof14.2–17.2×10−4_m2_/s2 _{forthesethreesubjects,namely} thesubjectswiththe“best,”“middle”and“worst”tracking perfor-mance.

Thedesignmethodguaranteesthatthenominalinput sensitiv-ityfunctionUohasagainwhichismonotonicallydecreasingwith frequency,isthereforedevoidofpeaking,andwhichtakesona pre-specifiedanddesirablevalueatachosencriticalfrequency: theonlydesignparameterswhichhavetobechosenarethe criti-calgainandfrequency;calculationofthecontrollerparametersis thenexplicitandstraightforward.Theempiricalcomparisonoftwo differentcontrollers,C1andC2,demonstratedthattuningofthese parameterstomodifythetradeoffbetweenperformanceoutcomes isclearandsimple:controllerC2,with+4dBofgainin|Uo|ata com-moncriticalfrequency,deliveredmoreaccuratetrackingatthecost ofhigheraveragecontrolsignalpower.

It was hypothesised at the outset that parametric and/or structuraluncertaintyintheplantplayonlyasecondaryrolein heart rate control systems:the nominal plant model assumed herewasa lineartime-invariant (LTI)first-order systemwhose parameters were obtained as average values from a previous system identification study [13]; the input-sensitivity-shaping design approach thusused a singleapproximatenominalplant model,anddidnotrequireanyinformationon,oridentificationof, heartratedynamicsforindividualparticipantsinthepresentwork.

Theexperimentalcomparisonoftheidentifiedandnon-identified subjectgroupsshowednosignificantdifferencesinperformance outcomes,thuslendingsupporttotheproposalthatavery approx-imateandsimpleLTIplantmodelissufficienttoachieveaccurate, stableandrobustheartratecontrolperformance.

The analysisof control loop performance over time showed atendencyforbothRMStrackingerrorandaveragecontrol sig-nalpowertodecrease:theearlyvs.late regulationcomparison showedsubstantialandsignificantreductionsinbothRMSEand Pv inthelater time interval;andthedynamic tracking

analy-sisforindividualstepsshowedthatRMSEdecreasedsignificantly over time.Therearetwopossibleexplanationsforthese obser-vations.First,analysisofalltestresultsfromthisstudyindicates that theamplitude of theHRV disturbancegenerally decreases as the exerciseprogresses, presumably due to on-going physi-ological adaptationsinbody temperature,fatiguelevel etc.This reduction in disturbanceamplitudewill naturallylead tomore accuratetrackingandlowercontrolsignalpower.Second,changes in performancewillalsobeattributableinparttovariationsin plantparameters over time. Thepreceding identificationstudy, [13],reporteda significantreductioninsteady-stategainkover time:khadtherelativelyhighvalueof∼33bpm/(m/s)intheearly stageofexercise(after10–15min),butthisvaluewasobservedto reducerapidlytowardstheoverallnominalvalueof∼24bpm/(m/s) with continuing exercise. This pattern of change in plant gain would lead to an increase in the gain of the input sensitivity function(thesteady-stategainofUo is1/k,Eq.(29))thus, theo-retically,givingahigherloopbandwidthand,correspondingly,a reductioninRMStrackingerrorandanincreaseincontrolsignal power.Sinceonlytheformerchangewasseeninthe experimen-talresults(i.e.lowerRMSE), itcanbeconcludedthatreduction in HRVamplitudeover time istheprimary mechanismleading tosimultaneouslylowerRMStrackingerrorandaveragecontrol signalpower;theeffectsofchangesinplantgainareapparently secondary.

7. Conclusions

Theempiricalresultsobtainedinthisstudystronglysupport theconceptthattheprincipaldesignissueforfeedbackcontrolof heartrateisdisturbancerejectionbehaviourinrelationto very-low-frequencyheartratevariability.Thereis astrongdegreeof intra-subjectandinter-subjectinconstancyinobservedheartrate variability:theHRVisitselfhighlyvariable.

The input-sensitivity-shaping method provides a direct and straightforwardwaytoaddressthis designchallenge,leadingto robustandaccuratetrackingandtoasmooth,low-powercontrol signal.

Issuesof parametricand structuralplantuncertainty appear toplay asecondary and indeedminorrole, becausetheresults obtainedherewereachievedusing asimple approximate plant modelwhichwasnotspecifictoanyofthesubjectstested.

Authors’contributions

KHdesignedthestudy.SFdidthedataacquisition.KHandSF contributedtotheanalysisandinterpretationofthedata.KHwrote themanuscript;SFreviseditcriticallyforimportantintellectual content.Bothauthorsreadandapprovedthefinalmanuscript. Competinginterests

Theauthorsdeclarethattheyhavenocompetinginterests. References

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