A modified narmax model based self-tuner with fault tolerance for unknown nonlinear stochastic hybrid systems with an input–output direct feed-through term

ResearchArticle
A modified NARMAXmodel-basedself-tunerwithfault
toleranceforunknownnonlinearstochastichybrid
systemswithaninput–output directfeed-throughterm
Jason S.-H.Tsai a,n, Wen-TengHsu a, Long-GueiLin a, Shu-MeiGuo b,n, JosephW.Tann c
a Department ofElectricalEngineering,NationalCheng-KungUniversity,Tainan701,Taiwan,ROC
b Department ofComputerScienceandInformationEngineering,NationalCheng-KungUniversity,Tainan701,Taiwan,ROC
c Automation&InstrumentationSystemDevelopmentSectionSteel&IronResearch&DevelopmentDepartment,Kaohsiung81233,Taiwan,ROC
a rticleinfo
Article history:
Received3January2013
Receivedinrevisedform
9 August2013
Accepted10August2013
Availableonline5September2013
This paperwasrecommendedfor
publication byDr.Q.-G.Wang
Keywords:
Self-tuning control
Stochasticsystem
NARMAXmodel
Faulttolerantcontrol
OKID
RELS
Systemidentification
An input–output directtransmissionterm
a b s t r a c t
A modified nonlinearautoregressivemovingaveragewithexogenousinputs(NARMAX)model-basedstate-
spaceself-tunerwithfaulttoleranceisproposedinthispaperfortheunknownnonlinearstochastichybrid
systemwithadirecttransmissionmatrixfrominputtooutput.Throughtheoff-lineobserver/Kalman filter
identification method,onehasagoodinitialguessofmodified NARMAXmodeltoreducetheon-linesystem
identification processtime.Then,basedonthemodified NARMAX-basedsystemidentification, acorrespond-
ing adaptivedigitalcontrolschemeispresentedfortheunknowncontinuous-timenonlinearsystem,withan
input–output directtransmissionterm,whichalsohasmeasurementandsystemnoisesandinaccessible
systemstates.Besides,aneffectivestatespaceself-turnerwithfaulttoleranceschemeispresentedforthe
unknownmultivariablestochasticsystem.Aquantitativecriterionissuggestedbycomparingtheinnovation
processerrorestimatedbytheKalman filterestimationalgorithm,sothataweightingmatrixresetting
techniquebyadjustingandresettingthecovariancematricesofparameterestimateobtainedbytheKalman
filterestimationalgorithmisutilizedtoachievetheparameterestimationforfaultysystemrecovery.
Consequently,theproposedmethodcaneffectivelycope withpartiallyabruptand/orgradualsystemfaults
andinputfailuresbythefaultdetection.
& 2013ISA.PublishedbyElsevierLtd.Allrightsreserved.
1. Introduction
The state-spaceself-tuningcontrolmethods [1,2] havebeen
shown tobeeffectiveindesigningadvancedadaptivecontrollers
for linearmultivariablestochasticsystems [3]. Inthoseapproaches
[1,2], thestandardKalmanstate-estimationalgorithm [4] has been
embedded intoanonlineparameterestimationalgorithm.They
utilize state-spaceself-tunersbasedoninnovationmodels,where
(i) theequivalentinternalstatescanbeestimatedsuccessively;
(ii) thestable/unstableandminimum/nonminimum-phasemulti-
variablesystemscanbecontrolledaccurately;(iii)theself-tuners
aresimple,reliableandrobust;and(iv)theadaptiveKalmangain
can subsequentlybeobtained.
Polynomialexpansionsareusedextensivelyinnonlinearsys-
tem analysis,wherethesystemhasnotheinput–output direct
feed-through term.Iftheresponseofasystemisdominatedby
nonlinear characteristics,itisgeneralnecessarytouseanonlinear
model, andthisimmediatelyraisestheproblemofwhatclassof
models touse.ThetraditionalNARMAXmodel,whichwas first
introducedandrigorouslyderivedby [5], providesaunified
representationforawideclassofnonlinearstochasticsystems
[6]. TheNARMAXmodelisnotrestrictedtopolynomialsystems
and canbeexpandedasarationalmodel [7]. Theadvantageofthe
rationalmodelistheefficiency withwhichitcanseverelydescribe
nonlinear characteristicswithafewparameters.Theseresultscan
be relatedtothemodelsintroducedbySontag [8]. Whentheyare
extendedtotheunknownstochasticcase,thesemodelsprovidea
class ofrationalmodels [7] which canbeusedasthebasisof
parameterestimationalgorithms.
Over thepastdecades,therehasbeenagrowinginterestinthe
singular system.Theapplicationsofsingularsysteminlarge-scale
systems,circuits,powersystems,economics,controltheory,
robots,andotherareas [9,10] are extensively.Thetrackerand
fault tolerancecontrolforthelinearsingularsystemisgivenin
[11]. Actually,thesingularsystemcanbeconvertedintoan
equivalentregularsystemwhichmayhaveadirecttransmission
termfrominputtooutput.Indeed,thesingularsystemwithoutthe
Contents listsavailableat ScienceDirect
journalhomepage: www.elsevier.com/locate/isatrans
ISATransactions
0019-0578/$-seefrontmatter & 2013ISA.PublishedbyElsevierLtd.Allrightsreserved.
http://dx.doi.org/10.1016/j.isatra.2013.08.007
n Corresponding authors.Tel.: þ886 62757575x62630, +886 62757575x62525;
fax: þ886 62345482, +886 62747076.
E-mail addresses: shtsai@mail.ncku.edu.tw (J.-H.Tsai),
guosm@mail.ncku.edu.tw (S.M.Guo).
ISA Transactions53(2014)56–75

impulse modeisjustaspecialclassoftheregularsystemwiththe
direct transmissiontermfrominputtooutput.Totheauthor's
knowledge,theNARMAXmodel-basedstate-spaceoptimaltracker
with faulttolerancefortheregularnonlinearsampled-datasystem
containing thedirecttransmissiontermfrominputtooutputhas
not beenproposedinliterature.
The settingofinitialparametersoftheNARMAXmodelis
important toreducethetimeoftheon-lineidentifyingprocess,so
the observer/Kalman filteridentification (OKID) [12,13] is applied
to estimatetheinitialparametersoftheNARMAXmodelandorder
determination fortheonlinerecursiveextended-least-squares
(RELS)identification inthispaper.Thewell-knownprocessof
on-line systemidentification ofARMA/NARMAXmodel-based
state-space self-tuningcontrolforthesystemwithoutinput–
output feed-throughtermrequitestheone-steppastcontrolinput
and someothermeasurementstodeterminethecurrentcontrol
input.However,forthecaseofthesystemwithinput–output feed-
through term,itrequirestohavethecurrentcontrolinput,which
implies thereisacausalproblem.Toovercomethisproblem,a
modified NARMAXmodel-basedsystemidentification forthe
unknown nonlinearsystemwiththeinput–output feed-through
term willbeproposedinthispaper.TheOKID [12,13] is performed
in off-line,sothereisnocausalproblemtoidentifytheinput–
output feed-throughterm.However,itdoesnotworkfortheon-
line case.Totheauthor'sknowledge,noon-lineOKIDhasbeen
proposed inliterature.Theidentified observerofthestate-space
self-tuning controlisinthestate-spaceinnovationform;however,
the oneidentified bytheOKIDisinthegeneralcoordinateform.
So, thetransformationbetweenthesetwowillbebriefly intro-
duced inthispaper.Then,basedonthemodified NARMAXmodel
and itscorrespondingstatespaceinnovationform,adigital
controller designtodealwiththesystemwithadirecttransmis-
sion term [11] is presented.
One pointmustbenoticedthatthestate-spaceself-tuning
control (STC)schemefornonlinearstochastichybridsystems
proposed byGuoetal. [14] estimatesthesystemparametersat
every samplinginstant,thendesignsanadaptivecontrollerbased
on theestimatedparametersalsoateverysamplinginstant.The
frameworkofthestate-spaceSTCseemstoagreewiththatofthe
activefaulttoleranceinarealtime.Forfaultysystemrecovery,we
use themodified Kalman filter estimationalgorithmbyutilizing
the modified covariancematricesfromestimatederrorsto
improvetheparameterestimation [15], insteadofutilizingthe
estimatedcovariancematriceswhichisobtainedfromtheRELS
algorithm intheconventionalSTCschemeforadaptingparameter
variations.Aboutthefaults,abruptfaultsandgradualfaultsare
considered inthispaper.
This paperisorganizedasfollows.Problemdescriptionand
motivationofthispaperisgivenin Section 2. Section 3 summaries
some preliminaryfortheproposedmethod. Section 4 presents the
modified NARMAX(inpolynomial)model-basedstate-spaceself-
tuner forunknownnonlinearstochastichybridsystemswiththe
input–output feed-throughterm.In Section 5, afaulttolerance
scheme bymodifyingtheconventionalstate-spaceself-tuning
controlapproachfortheunknownmultivariablestochasticsystem
with input–output feed-throughtermisproposed.Finally,an
illustrativeexampleisshownin Section 6.
2. Problemdescriptionandmotivation
Considertheclassofcontinuous-timenonlinearstochasticsystems
as follows:
_xðtÞ ¼ f ðxðtÞÞþgðxðtÞÞuðtÞþw′ðtÞ; ð1aÞ
yðtÞ ¼ hðxðtÞÞþdðxðtÞÞuðtÞþv′ðtÞ; ð1bÞ
where f : ℜn-ℜn, g : ℜn-ℜnm, h : ℜn-ℜp, d : ℜn-ℜpm,
uðkÞAℜm is thecontrolinput, xðkÞAℜn is thestatevector, yðtÞAℜp
isthemeasurableoutputvector, w′ðtÞ and v′ðtÞ areuncorrelatedwhite
noise processes.Theon-linesystemidentification methodologiesof
ARMAXand/orNARMAX(inpolynomialand/orrational)model-based
state-spaceself-tuningcontrolwith/withoutfaulttoleranceforthe
known/unknownlinear/nonlinearsystemwithoutinput–output feed-
throughtermestimatethecurrentsystemparametersandstateat
time index t ¼ kT based oncontrolinputuptotimeindex t ¼ kTT,
uðkTTÞ, andoutputmeasurementsuptoeither t ¼ kTT or t ¼ kT.
Then, determinethecurrentcontrolparameter uðkTÞ basedonthe
estimatedstate ^xoðkTÞ, where ^xoðkTÞ denotestheestimatedcurrent
statefortheconstructedobserverrepresentedintheobserver
canonical form,sothatthesystemoutput yðtÞ canwelltrackthe
desiredreference ΓðtÞ at timeindex t ¼ kTþT, i.e.
yðkTþTÞffiΓðkTþTÞ, butnot yðkTÞffiΓðkTÞ. Theinterpretationofthis
comment isthatthecurrent uðkTÞ is determinedbythecurrentstate
^xoðkTÞ=xðkTÞ, whichimplies yðkTÞ determinedby uðkTTÞ exists
already.So, uðkTÞ cannotaffect yðkTÞ anymore.Theaboveobservation
showsthatoneneeds uðkTTÞ first,thenidentifies thesystem
parameter/state,anddeterminesthecurrentcontrolinput uðkTÞ finally,whichisthewell-knownon-lineprocessofthesystem
identification methodologyforthestate-spaceself-tuningcontrol.
However,whenthesystemhastheinput-outputfeed-through
term,oneneedstohavethecurrentcontrolinput uðkTÞ first forthe
on-line systemidentification, thendeterminesthecontrolinput
uðkTÞ later,whichinducestheso-calledcasualproblem.Toover-
come thisproblem,amodified NARMAmodel-basedsystem
identification fortheunknownnonlinearsystemwiththeinput–
output feed-throughtermwillbeproposedinthispaper.
The structureofthestate-spaceSTCschemeincludesapara-
meter andstateestimatorandacontrollerdesign.Atypicalstate-
space STCstructureisillustratedin Fig. 1.
Fig. 1. Block diagramofatypicalstate-spaceself-tuningcontrol.
J.S.-H. Tsaietal./ISATransactions53(2014)56–75 57

Underthisframework,parametersandstateoftheunknown
model areestimatedfromthecontrolinput ðud
nðkTÞ;
udððk1ÞTÞ; udððk2ÞTÞ;…Þ and thesystemoutput ðyðkTÞ;
yððk1ÞTÞ; yððk2ÞTÞ;…Þ, where ud
nðkTÞ is tobeestimatedbased
on somepredictionconceptpresentedin Section 4. Considering
the complexityofthefaulttolerancecontrol,thispapertakesthe
NARMAXmodelinpolynomialexpansionform,butnotinrational
expansionform.Theextensionofthemodified NARMAXmodel-
based methodologyforthefault-tolerancecontrolfromitspoly-
nomial expansionformtorationalexpansionformcanbecon-
sideredasafutureresearchwork.Basedontheestimated
parameter θðkÞ of themodified NARMAXmodel,anappropriate
controllercanbedesignedin Section 4. Then,thedesigned
adaptivecontrollergeneratesreal-timecontrolactionsforthe
unknown dynamicsystem.Thedetailofthetraditionalalgorithm
to estimatethesystemparameters θðkÞ is describedin Section 2.
Then, theprocesscycleisrepeateduntilthecontrolgoalis
achieved. Noticethatifthecontrolinputispersistentlyexcited,
the convergencetothetruesystemparametersisguaranteedby
Ljung [16].
It iswell-knownthatonecanformulateanARMAXmodel
suitable fortheunknownrealsystem.However,insomecases,the
ARMAX modelforself-tuningcontrolcannotsimulatetheoriginal
true systemaccurately.Inthispaper,wewillproposethemodified
NARMAXmodelinpolynomialexpansionformforthefault
toleranceself-tuningcontrol(STC)scheme.
The expressionofmodified NARMAXmodelforthe m-input p-output
systemisgivenby
~yiðkÞ ¼ Fi½y1ðk1Þy2ðk1Þ…ypðk1Þ…y1ðknnyÞy2ðknnyÞ…ypðknnyÞ
u1
nðkÞun
2ðkÞ⋯un
mðkÞu1ðk1Þu2ðk1Þ…umðk1Þ…u1ðknnuÞ
u2ðknnuÞ…umðknnuÞε1ðk1Þε2ðk1Þ…εpðk1Þ⋯ε1ðknneÞ
ε2ðknneÞ…εpðknneÞ
¼ Σ
j ¼ 1n
θijðkÞϕijðkÞ ¼ θiðkÞTϕiðkÞ; for i ¼ 1; 2;…; p; ð2Þ
where ~yiðkÞ is theestimatedvalueof yiðkÞ, uαðkÞ and yβðkÞ denote
the α th input(α¼1, 2, …,m) andthe β th output(β¼1, 2, …,p)
at time k (k¼0, 1, …). Notation εðkÞ is theresidualscalar, nny, nnu,
nne aretheordersof y, u, ε, respectively, n is theamountofthe
linear/nonlinear variables ðy; u; εÞ of theNARMAXmodel (2), θijðkÞ
denotesthe j-th coefficient oftheNARMAXmodelforthe i-th
estimatedoutput~yiðkÞ. Besides, ϕijðkÞ denotesthe j-th linear/non-
linear variables(y, u, ε) oftheNARMAXmodelforthe i-th
estimatedoutput ~yiðkÞ. Thewholemodelisnonlinear,i.e. FiðdÞ
arenonlinearpolynomialsfor i ¼ 1; 2;⋯; p. Notation n
denotesthe
lags and ϕ
ðkÞ can beanynonlinearfunctionin FiðdÞ.
Each estimatedoutput ~yiðkÞ is identified fromeachclassof
nonlinear ϕiðkÞ as
~yiðkÞ ¼ θi
T ðkÞϕiðkÞ; i ¼ 1; 2;…; p; ð3aÞ
and thestandardRELSalgorithmisappliedby
θiðkÞ ¼ θiðk1Þþ
Siðk1ÞϕiðkÞ
λðkÞþϕi
T ðkÞSiðk1ÞϕiðkÞ
εiðkÞ; ð3bÞ
SiðkÞ ¼
1
λðkÞ
Siðk1Þ
Siðk1ÞϕiðkÞϕiðkÞTSiðk1Þ
λðkÞþϕi
!
; ð3cÞ
where λðkÞ is theforgettingfunctiontodiscounttheoldmeasure-
ments, andcanbedeterminedbythe first-order difference
equation, λðkÞ ¼ λ0λðk1Þþð1λ0Þ, withtheinitialcondition
0:9oλð0Þo1; and theupdatingfactor0oλ0o1. Also, SiðkÞAℜnn
is theparametersestimationerrorcovariancematrixwith
Sið0Þ ¼ αiInn, where αi is thepositivescalar,andtheresidualvector
of eachoutputisgivenby
εiðkÞ ¼ yiðkÞθi
T ðk1ÞϕiðkÞ: ð4Þ
Fordifferentselectionsof FiðUÞ, manyclassesofNARMAXmodels
can bechosen.Tosimplifythewholecontrolschemeforthe
complicateon-linefault-tolerancecontrol,itisdesiredtochoose
some simplestructuresofdynamicnonlinearmodels.Thus,the
self-tuning controlschemewiththeNARMAXmodelfornonlinear
stochasticsystemscanworkmoreprecisely.
3. Preliminary
In thissection,webriefly reviewtheRELS-basedobserver/
Kalman filterstate-spaceformandOKID-basedgeneralcoordinate
form, sincethemodelconversionfromOKID-basedgeneral
coordinatetotheRELS-basedstate-spaceinnovateformisneces-
sary.Oncehavingtheestimatedparameters θiðkÞ from thestan-
dard RELSalgorithm,theNARMAXmodelfortheSTCschemecan
accuratelyapproximatetheresponsesofthenonlinearsystem.
Moreover,theinitialparameters θiðkÞ of NARMAXmodelwillaffect
the convergentspeedofRELSprocess.Inordertogetsuitable
initial parameters θið0Þ toshortenthetransientprocessofRELS,
we applyOKIDtoevaluateithere.
The regressor ϕiðkÞ in ~yiðkÞ ¼ θi
T ðkÞϕiðkÞ is composedof
ðy1ðk1Þ; y1ðk2Þ;…; y1ðknnyÞ; y2ðk1Þ; y2ðk2Þ;…y2ðknnyÞ;…; ypðk1Þ
ypðk2Þ;…; ypðknnyÞ; u1
nðkÞ; u1ðk1Þ;…; u1ðknnuÞ; u2
nðkÞ;
u2ðk1Þ;…u2ðknnuÞ;…um
nðkÞ; umðk1Þ;…; umðknnuÞ; ε1ðk1Þ;
ε1ðk2Þ;…; ε1ðknneÞ; ε2ðk1Þ;
ε2ðk2Þ;⋯; ε2ðknneÞ;⋯; εpðk1Þ; εpðk2Þ;⋯; εpðknneÞÞ:
Components of ϕiðkÞ are notindependentfactors,soitis
difficult todesigndigitalcontrollerdirectlyfromtheSTCscheme
with theNARMAXmodel.Forthisreason,onecouldapplythe
optimallinearizationtotheNARMAXmodeltoconfigure alinear
discrete-timestate-spaceobserverfordesigningthedigitalcon-
trolleroftheSTCscheme.
3.1.RELS-basedobserver/Kalman filter instate-space
innovationform
A preliminarystructureofthediscretestate-spaceobserverof
the linearsystemispresentedin [3]. Considerthefollowinglinear
discretestochasticsystemas
xðkþ1Þ ¼ GxðkÞþHuðkÞþwðkÞ; ð5aÞ
yðkÞ ¼ CxðkÞþDuðkÞþvðkÞ; ð5bÞ
where GAℜnn, HAℜnm, CAℜpn and DAℜpm are systemmatrices,
xAℜn, uAℜm, and yAℜp arestatevector,inputvector,andoutput
vector,respectively, wAℜnand vAℜp arezero-meanwhitenoise
sequenceswithcovariancematricesas
E
wðkÞ
vðkÞ
#
wT ðlÞ vT ðlÞ
( h i)
¼
Q S
S R

δk;l; ð6Þ
QZ0, R40, δk;l ¼1 if k ¼ l, and δk;l ¼0 if kal, k; l ¼ 0; 1; 2;….
System (5) can betransferredintotheblockobservableform,ifthe
rankofthefollowingobservabilitymatrix
Θ ¼ ½ðCGr1
ÞT ; ðCGr2
ÞT…ðCGÞT ; CT
T ð7Þ
is equalto n.
Notethattheobservabilityindexof Θ is ρ ¼ n=p, ifitisan
integer(otherwise,itisundefined). Thisconstraintmeansthatthe
Kroneckerindicesofsystem (5) areallsuchintegers ρs satisfying
n ¼ ρp. Whensystem (5) is blockobservable,itcanbetransformed
J.S.-H. Tsaietal./ISATransactions53(2014)5658 –75

into theblockobservablecompanionformasfollows
xoðkþ1Þ ¼ GoxoðkÞþHouðkÞþwoðkÞ; ð8aÞ
yðkÞ ¼ CoxoðkÞþDouðkÞþvoðkÞ; ð8bÞ
where
To ¼ ½Gr1To1; Gr2To1;…; GTo1; To1; To1 ¼ Θ1CT
o
xoðkÞ ¼ T1
o xðkÞ
Go ¼ T1
o GTo ¼
Go1 Ip 0p ⋯ 0p
Go2 0p Ip ⋯ 0p
⋮ ⋮ ⋮⋱⋮
Goρ1 0p 0p ⋯ Ip
Goρ 0p 0p ⋯ 0p
2
6666664
3
7777775
;
Ho ¼ T1
o H ¼ ½HT
o1;HT
o2;…;HT
oρT
xoðkÞ ¼ ½xT
o1ðkÞ; xT
o2ðkÞ;…; xT
oρðkÞT
Co ¼ CTo ¼ ½Ip; 0p;…; 0p
Do ¼ D
woðkÞ ¼ T1
o wðkÞ; voðkÞ ¼ vðkÞ;
E
woðkÞ
voðkÞ
#
wo
T ðlÞ vo
T ðlÞ
( h i)
¼
T1
o QðT1
o ÞT T1
o S
ðT1
o SÞT R
#
¼
Qo So
So Ro
#
δk;l;
in which Ip and 0p are pp identity andzeromatrices,respectively.
System (8) can berepresentedbyastate-spaceinnovation
model [17] as
^xoðkþ1jkÞ ¼ Go ^xoðkjk1ÞþHouðkÞþKoðkÞeoðkÞ; ð9aÞ
yoðkÞ ¼ Co ^xoðkjk1ÞþDouðkÞ; ð9bÞ
where KoðkÞ is theKalmangain,whichcanbecomputedbythe
following algorithm [18]:
KoðkÞ ¼ ½GoPoðkÞCo
T
þSo½CoPoðkÞCT
o þRo1; ð10aÞ
Poðkþ1Þ ¼ ½GoKoðkÞCoPoðkÞ½GoKoðkÞCoT
þKoðkÞRoKT
o ðkÞSoKT
o ðkÞKoðkÞST
o þQo
¼ Ef~xoðkþ1jkÞ~xoðkþ1jkÞTg ð10bÞ
Poð0Þ ¼ Ef½xoð0Þ^xoð0Þ½xoð0Þ^xoð0ÞT g; ð10cÞ
in which ^xoðkjk1Þ is theoptimalestimateof xoðkÞ by themeasure-
mentdataupto yðk1Þ, i.e., yðiÞ for i ¼ 0; 1;…; k1, ~xoðkjk1Þ ¼ xoðkÞ^xoðkjk1Þ is theestimateerror, eoðkÞ ¼ yðkÞCo ^xoðkjk1Þ DouðkÞ ¼ Co ~xoðkÞþvoðkÞ isthezero-meanwhitenoisesequencewith
Re ¼ EfeoðkÞeT
o ðkÞg ¼ CoPoðkÞCT
o þRo, where eoðkÞ is calledtheinnova-
tion process.
If thepair ðGo; CoÞ is detectableandthepair ðGoSoR1
o Co; ^Q
oÞ,
with ^Q
o
^Q
T
o ¼ QoSoR1
o ST
o , isstable,then PoðkÞ-Po, where Po is the
stationary errorcovariancematrix,sothat KoðkÞ-Ko (the station-
ary Kalmangain)as k-1. Furthermore,theeigenvaluesof
GoKoCo areallinsidetheunitcircle.Let z1 be thebackward
time-shiftoperatorandset Ko ¼ ½KT
o1; KT
o2;…; KT
oρT Aℜnp. Then,the
input–output relationshipofthesteady-stateinnovationrepre-
sentation (9) can berewrittenas
yðkÞ ¼ Co½InGoz11½Hoz1uðkÞþKoz1eoðkÞ
þDouðkÞþeoðkÞ
¼ ½Glðz1Þ1½Hlðz1ÞuðkÞþ½Glðz1Þ1½Kelðz1ÞeoðkÞ; ð11Þ
where
Glðz1Þ ¼ IpþGo1z1þGo2z2þ…þGoρzρ;
Hlðz1Þ ¼ Doþ ~H
o1z1þ ~H
o2z2þ…þ ~H
oρzρ;
~H
oi ¼ HoiþGoiDo;
Kelðz1Þ ¼ IpþKeo1z1þKeo2z2þ…þKeoρzρ
Keoi ¼ GoiþKoi; i ¼ 1; 2;…; ρ:
Notethatallthezerosofdet½Kelðz1Þ must beinsidetheunitcircle
in themultivariableARMAXmodel (11). Iftheparametersmatrices
Goi, Hoi, Coiand Doi, i ¼ 1; 2;…; ρ, in (8) are known,andthe
covariancematricesthereinareavailable,therecursiveestimation
algorithm (10a) can beappliedtodeterminetheKalmangain
KoðkÞ. Thus,thestate xoðkÞ can beoptimallyestimatedusingthe
algorithmin (9). TheestimatedKalmangain, ^K
oiðkÞ, isgivenby
^K
oiðkÞ ¼ ^K
eoiðkÞ ^G
oiðkÞ; i ¼ 1; 2;…; ρ: ð12Þ
and theestimatedstateintheblockobservableformis
xoðkþ1jkÞ ¼ ^G
oðkjk1Þxoðkjk1Þþ ^H
oðkÞuðkÞ
þ ^K
oðkÞeoðkÞ; ð13aÞ
eoðkÞ ¼ yðkÞCoxoðkjk1ÞDouðkÞ; ð13bÞ
where ^G
oðkÞ, ^H
oðkÞ, and ^K
oðkÞ contain theestimatedparameters
matrices ^G
oiðkÞ, ^H
oiðkÞ, and ^K
oiðkÞ, i ¼ 1; 2;…; ρ. Whenallthese
estimatedparametersmatricesconvergetothetruevalues,the
estimated xoðkjk1Þ and eoðkÞ convergetotheoptimalstate
estimate ^xoðkjk1Þ and innovationprocess eoðkÞ. Itisimportant
tonotethataslongasthematrix
Gc ¼ GoKoCo ¼
Keo1 Ip 0p ⋯ 0p
Keo2 0p Ip ⋯ 0p
⋮ ⋮ ⋮⋱⋮
Keoρ1 0p 0p ⋯ Ip
Keoρ 0p 0p ⋯ 0p
2
6666664
3
7777775
is asymptoticallystable,theboundaryofthenoisesequences
implies thattheestimationerrorwillalwaysbebounded.When-
ever KeoiðkÞ ¼ 0p, for i ¼ 1; 2;…; ρ, itdesignatesadead-beat-like
property.
3.2. OKID-basedobserver/Kalman filter ingeneralcoordinateform
Consider thefollowingdiscrete-timestate-spaceequationsofa
multivariablelinearsystem
xðkþ1Þ ¼ GxðkÞþHuðkÞ; ð14aÞ
yðkÞ ¼ CxðkÞþDuðkÞ; ð14bÞ
where GAℜnn, HAℜnm, CAℜpnand DAℜpm aresystemmatrices,
and xðkÞAℜn, yðkÞAℜp, uðkÞAℜm are statevector,outputvector,
inputvector,respectively.WhenthecombinedobserverMarkov
parametersaredetermined,theeigensystemrealizationalgorithm
(ERA) methodisusedtoobtainthedesireddiscretesystem
realization ½ ^G
; ^H
; ^C
; ^D
; F throughsingularvaluedecomposition
(SVD) oftheHankelmatrix [12,13].
The ERAprocessesthefactorizationofthecorresponding
Hankel matrix,usingthesingularvaluedecomposition
^H
ð0Þ ¼ VΣST , wherethecolumnsofmatrices V and S areortho-
normal and Σ is arectangularmatrixoftheformas
Σ¼
Σ~ n 0
0 0

; ð15Þ
where Σ~ n ¼ diag½s1; s2;…; snmin ; snmin þ1;…; s~ n contains monotoni-
cally non-increasingentries s1Zs2Z…ZsnminZsnmin þ1Z
…Zs~ n40. Here,somesingularvalues snmin þ1;…; s~ n are relatively
small ðsnmin þ15snmin Þ and negligibleinthesensethattheycontain
more noiseinformationthansysteminformation.Inorderto
construct theloworderobserverofthesystem,let'sdefine
Σnmin ¼ diag½s1; s2;…; snmin . Inotherwords,thereducedmodel

of order nmin afterdeletingsingularvalues snmin þ1;…; s~ n is then
consideredastherobustlycontrollableandobservablepartofthe
realizedopen-loopsystemwithanacceptableperformance.Simul-
taneous realizationsofthesystemandobserverbytheERAare
givenas
x^ðkÞ ¼ G^ xðk1ÞþH^ uðk1ÞþF½yðk1Þy^ ðk1Þ; ð16aÞ
^yðkÞ ¼ ^CxðkÞþ D^ uðkÞ; ð16bÞ
where
^G
¼Σ1=2
nmin VT
nmin
Hð1ÞSnminΣ1=2
nmin ; ð16cÞ
½ H^ F ¼ First ðmþpÞ columns of Σ1=2
nminST
nmin ; ð16dÞ
^C
¼ FirstprowsofVnminΣ1=2
nmin ; ð16eÞ
^D
¼ Y0: ð16fÞ
The definition of Hð1Þ can bereferredto [12,13].
3.2.1.OKIDformulationforrelationshiptoaKalman filter
Let thesystem (14) be extendedtoincludeprocessand
measurementnoisedescribedas
xðkþ1Þ ¼ GxðkÞþHuðkÞþwðkÞ; ð17aÞ
yðkÞ ¼ CxðkÞþDuðkÞþvðkÞ; ð17bÞ
where wðkÞ is theprocessnoiseassumedtobeGaussian,zero-
mean andwhitewiththecovariancematrix Q and vðkÞ is the
measurementnoisesatisfies thesameassumptionas wðkÞ with a
different covariancematrix R. Thesequences wðkÞ and vðkÞ are
independent ofeachother.Then,atypicalKalman filterforthe
system, (17a) and (17b), canbewrittenas
^xðkþ1Þ ¼ G^xðkÞþHuðkÞþKεrðkÞ; ð18aÞ
^yðkÞ ¼ C^xðkÞþDuðkÞ; ð18bÞ
where ^xðkÞ is theestimatedstate, K is theKalman filtergain,and
εrðkÞ is defined asthedifferencebetweentherealmeasurement
yðkÞ and theestimatedmeasurement ^yðkÞ.
The measurementequationbecomes
yðkÞ ¼ C^xðkÞþDuðkÞþεr ðkÞ: ð19Þ
Systems (16a) and (18a) areidenticalwhen F ¼K and εrðkÞ ¼ 0,
and soareMarkovparameters.Inpractice,anyobserversatisfying
a least-squaressolutionwillproducethesameinput–output map
as aKalman filterdoes,providedthatthedatalengthissufficiently
long andtheorderoftheHankelmatrixissufficiently large,sothat
the truncationerrorisnegligible [12]. Therefore,whentheresidual
εrðkÞ is awhitesequenceoftheKalman filter residual,theobserver
gain F convergestothesteady-stateKalman filtergain K such that
F ¼K.
3.3. Optimallinearization
The optimallinearization [19,20] wasproposedforcontinuous-
time nonlinearsystemsfollowedbystabilizingcontrollerdesign
for uncertainnonlinearsystemsusingfuzzymodels.Theproposed
optimallinearizationattheoperatingstate,notnecessarilythe
equilibriumstate,yieldstheexactlinearmodel.Also,ityieldsthe
optimallinearmodeldefined bysomeconvexconstraintoptimiza-
tion criterioninthevicinityoftheoperatingstate.
Consider theclassofnonlinearsystemsdescribedas
xðkþ1Þ ¼ f ðxðkÞÞþgðxðkÞÞuðkÞ; ð20Þ
where f : Rn-ℜn and g : ℜn-ℜm are nonlinearwithcontinuous
partial derivativeswithrespecttoeachoftheirvariablesatall
steps k, where xðkÞAℜn is thestatevectorattimeindex k, and
uðkÞARm is thecontrolinputvectorattimeindex k. Itisdesiredto
haveanexactlocallinearmodelðAðkÞ; BðkÞÞ at anoperatingstateof
interest, xopðkÞAℜn, intheformof
xðkþ1Þ ¼ AðkÞxðkÞþBðkÞuðkÞ; ð21Þ
where AðkÞ and BðkÞ areconstantmatricesofappropriatedimensions.
Suppose thatwearegivenanoperatingstate xopðkÞa0, which
is notnecessarilyanequilibriumofthegivensystem.Thegoalisto
construct alocalmodel,linearin x and alsolinearin u, thatcan
wellapproximatethedynamicalbehaviorsof (20), inthevicinity
of theoperatingstate xopðkÞ. Inotherwords,onehas
f ðxÞþgðxðkÞÞuðkÞ AðkÞxðkÞþBðkÞuðkÞ; f oranyuðkÞ; ð22aÞ
f ðxÞþgðxðkÞÞuðkÞ ¼ AðkÞxopðkÞþBðkÞuðkÞ; f orany: ð22bÞ
Since thecontrolinput u is tobedesignedanditisarbitrary,one
must have gðxðkÞÞ ¼ BðkÞ, sothat (22a) and (22b) become quite
simple
f ðxðkÞÞ AðkÞxðkÞ ð23Þ
and
f ðxopðkÞÞ ¼ AðkÞxopðkÞ: ð24Þ
Tosatisfythese,let ai
Tdenotethe ith rowofthematrix AðkÞ, and
represent (23) and (24) as
f iðxÞ ai
Tx; i ¼ 1; 2;…; n ð25Þ
and
f iðxopðkÞÞ ¼ ai
TxopðkÞ; i ¼ 1; 2;…; n; ð26Þ
where f i : ℜn-ℜ is the i-th componentof f. Then,expandingthe
left-handsideof (25) about xopðkÞ and neglectingthesecondand
higher orderterms,onehas
f iðxopðkÞÞþ½∇f iðxopðkÞÞT ðxðkÞxopðkÞÞ ai
TxðkÞ; ð27Þ
where ∇f iðxopðkÞÞ : ℜn-ℜn is thegradientcolumnvectorof f i
evaluatedat xðkÞ. Now,using (26), wecanrewrite (27) as
½∇f iðxopðkÞÞT ðxðkÞxopðkÞÞ ai
T ðxðkÞxopðkÞÞ; ð28Þ
in which xðkÞ is arbitrarybutshouldbe “close” to xopðkÞ so thatthe
approximationisgood.Todetermineaconstantvector ai
T, itis “as
close aspossible” to ½∇f iðxopðkÞÞT and alsosatisfies ai
TxopðkÞ ¼ f iðxopðkÞÞ. Then,wemayconsiderthefollowingconstrainedmini-
mization problem:
min E ¼
1
2‖∇f iðxopðkÞÞai‖22
subject to ai
TxopðkÞ ¼ f iðxopðkÞÞ: ð29Þ
Noticethatthisisaconvexconstrainedoptimizationproblem;
therefore,the first ordernecessaryconditionforaminimumof E is
also sufficient, whichis
∇aiEþλ∇ai ðai
TxopðkÞf iðxopðkÞÞÞ ¼ 0; ð30Þ
ai
TxopðkÞ ¼ f iðxopðkÞÞ; ð31Þ
where λ is theLagrangemultiplierandthesubscript ai in ∇ai
indicatesthegradientistakenwithrespectto ai. Itfollowsfrom
(30) that
ai∇f iðxopðkÞÞþλxopðkÞ ¼ 0: ð32Þ
Recallthatwearestudyingthecasewhere xopðkÞa0, sobysolving
(32), weobtain
λ ¼
xT
opðkÞ∇f iðxopðkÞÞf iðxopðkÞÞ
‖xopðkÞ‖22
: ð33Þ
J.S.-H. Tsaietal./ISATransactions53(2014)5660 –75

Substituting (33) into (32) gives
ai ¼ ∇f iðxopðkÞÞþ
f iðxopðkÞÞxT
opðkÞ∇f iðxopðkÞÞ
‖xopðkÞ‖22
xopðkÞ; ð34Þ
where xopðkÞa0. Itiseasilyverified thatwhen xopðkÞ ¼ 0, Eq. (32)
yields
ai ¼ ∇f iðxopðkÞÞ ð35Þ
The controllabilitymatrixforthenonlinearsystemin (20) at
the operatingstate xopðkÞ is derivedfromtheoptimallinearmodel
ðAðkÞ; BðkÞÞ, resultingin
C ¼ BðkÞ AðkÞBðkÞ A2
ðkÞBðkÞ … An1
ðkÞBðkÞ
h i
; ð36Þ
where AðkÞ and BðkÞ areconstructedviathefollowingrule:the j-th
columns ofAðkÞand BðkÞ are settobezerowheneverthe j-th
corresponding componentsof xopðkÞ and uðkÞ arezero,respectively.
4. Modified NARMAXmodel-basedstate-spaceself-tuning
control forunknownnonlinearstochastichybridsystemswith
an input–output directfeed-throughterm
By takingtheproposedNARMAXmodel (2) for theself-tuning
control, thediscrete-timestate-spaceinnovationmodel (9) is
constructed todesignthecontrolinput ud for theunknownreal
system.Sincethemodified NARMAXmodelisnonlinear,the
optimallinearizationmethodin Section 3.3 is presentedto
linearize theNARMAXmodelasalinearARMAXmodelatthe
operating statewithoutanyapproximation.Besides,itisalsothe
optimaloneinthesenseofminimizingoptimizationproblem (29)
in thevicinityofoperatingstate.
In thispaper,weselectanadaptiveclassofthemodified
NARMAXmodelinpolynomialformwith m-inputs, p-outputs
and ρ-time-steps asfollows:
where
BiðyÞ ¼ ½Bi01ðyÞ;…; Bi0mðyÞ; Bi11ðyÞ;…; Bi1mðyÞ; Bi21ðyÞ;…;
Bi2mðyÞ;…; Biρ1ðyÞ;…; BiρmðyÞAR1mρ
KeiðyÞ ¼ ½Kei11ðyÞ;…; Kei1pðyÞ; Kei21ðyÞ;…; Kei2pðyÞ;…;
Keiρ1ðyÞ; :::; KeiρpðyÞAR1pρ
U ¼ ½un
1ðkÞ;…; un
mðkÞ; u1ðk1Þ;…; umðk1Þ; u1ðk2Þ;…;
umðk2Þ;…; u1ðkρÞ;…; umðkρÞT ARmρ1;
E ¼ ½ε1ðk1Þ;…; εpðk1Þ; ε1ðk2Þ;…; εpðk2Þ;…; ε1ðkρÞ;…;
εpðkρÞT ARpρ1
the function FiðUÞ is nonlinear,theoutputs,inputsandresiduals
are yiðkÞ;yiðk1Þ;…;yið0Þ; uj
nðkÞ; ujðk1Þ;ujðk2Þ;…; ujð0Þ and
εiðk1Þ; εiðk2Þ;…; εið0Þ, respectively, i ¼ 1; 2; 3; :::; p and j ¼ 1; 2; 3; :::;m. ThisadaptiveNARMAXmodel (37) has thefeature
that BiðyÞU is linearinallitemsof u and KeiðyÞE is linearinallitems
of ε.
Rewritethemodel (37) to separatelinear/nonlinearvariables
and itscoefficients asfollows:
~yiðkÞ ¼ FiðyÞþBiðyÞUþKeiðyÞE
¼ Σ
j ¼ 1n
θijðkÞϕijðkÞ ð38Þ
~yiðkÞ ¼ θi
T ðkÞϕiðkÞ i ¼ 1; 2; 3;…; p; ð39Þ
where ~yiðkÞ is theestimatedvalueof yiðkÞ, θiðkÞ ¼ ½θi1ðkÞ;…; θinðkÞT
is coefficient matrix, ϕiðkÞ ¼ ½ϕi1ðkÞ;…; ϕinðkÞT is linear/nonlinear
variablematrix.Getthecoefficient matrix θiðkÞ by thestandard
RELSalgorithm (3) to approximatetherealsystem. FiðyÞ only
include theoutput yi's delays,andtheyarenonlinearfunctionsof
yi. Forgettingthecorrespondinglinearmodelofthenonlinear
model (37), thefunctions FiðyÞ must needtobelinearizedbythe
optimallinearization.
Performing theoptimallinearizationapproachon FiðyÞ in (38)
yields
FiðyÞ ¼ AiRYF f ori ¼ 1; 2; 3;…; p; ð40Þ
where
YF ¼ ½y1ðk1Þ…ypðk1Þ; y1ðk2Þ…ypðk2Þ;…; y1ðkρÞ…ypðkρÞT
Aℜ1pρ
and
AiR ¼ ∇FiðYFÞþ
FiðYFÞYTF
ðkÞ∇FiðYF Þ
‖YF‖22
YF ¼ ½ai11R;…; ai1pR; ai21R;…; ai2pR;…; aiρ1R;…; aiρpRAℜ1pρ:
Substitute (40) into (38) to have
~yiðkÞþðAiRÞYF ¼ BiUþKeiE;
i.e.
~yiðkÞþ ~A
iYF ¼ ~B
iUþ ~K
eiE; ð41Þ
where
i ¼ 1; 2; 3;…; p;
~A
i ¼
~A
i11 ⋯ ~A
i1p
~A
i21 ⋯ ~A
i2p ⋯⋯ ~A
iρ1 ⋯ ~A
iρp

¼AiR;
~B
i ¼
~B
i01 ⋯ ~B
i0m ~B
i11 ⋯ ~B
i1m ⋯⋯ ~B
iρ1 ⋯ ~B
iρm
h i
¼ Bi;
~K
ei ¼
~K
ei11 ⋯ ~K
ei1p
~K
ei21 ⋯ ~K
ei2p⋯⋯ ~K
eiρ1 ⋯ ~K
eiρp

¼ Kei:
Let z1 denotethebackwardshiftoperator.Onecouldtransform
(41) to give
~y1ðkÞþ ~A
111z1y1ðkÞþ…þ ~A
11pz1ypðkÞþ…þ ~A
1ρ1zρy1ðkÞ
þ…þ ~A
1ρpzρypðkÞ
¼ ~B
101u1
nðkÞþ…þ~B
10mum
nðkÞþ…þ~B
1ρ1zρu1ðkÞ
þ…þ~B
1ρmzρumðkÞ
þ ~K
e111z1ε1ðkÞþ…þ ~K
e11pz1εpðkÞþ…þ ~K
e1ρ1zρε1ðkÞ
þ…þ ~K
e1ρpzρεpðkÞ; ~y2ðkÞþ ~A
211z1y1ðkÞþ…þ ~A
21pz1ypðkÞ
þ…þ ~A
2ρ1zρy1ðkÞþ…þ ~A
2ρpzρypðkÞ
¼ ~B
201u1
nðkÞþ…þ~B
20mum
nðkÞþ…þ~B
2ρ1zρu1ðkÞ
þ…þ~B
2ρmzρumðkÞ
þ ~K
e211z1ε1ðkÞþ…þ ~K
e21pz1εpðkÞþ…þ ~K
e2ρ1zρε1ðkÞ
þ…þ ~K
e2ρpzρεpðkÞ; ⋮~ypðkÞþ ~A
p11z1y1ðkÞþ…þ ~A
p1pz1ypðkÞ
þ…þ ~A
pρ1zρy1ðkÞþ…þ ~A
pρpzρypðkÞ
¼ ~B
p01u1
nðkÞþ…þ~B
p0mum
nðkÞþ…þ~B
pρ1zρu1ðkÞ
~y1ðkÞ ¼ F1½y1ðk1Þ…ypðk1Þ; y1ðk2Þ…ypðk2Þ;…; y1ðkρÞ…ypðkρÞþB1ðyÞUþKe1ðyÞE;
~y2ðkÞ ¼ F2½y1ðk1Þ…ypðk1Þ; y1ðk2Þ…ypðk2Þ;…; y1ðkρÞ…ypðkρÞþB2ðyÞUþKe2ðyÞE;
⋮
~ypðkÞ ¼ Fp½y1ðk1Þ…ypðk1Þ; y1ðk2Þ…ypðk2Þ;…; y1ðkρÞ…ypðkρÞþBpðyÞUþKepðyÞE;
ð37Þ

þ…þ~B
pρmzρumðkÞ
þ ~K
ep11z1ε1ðkÞþ…þ ~K
ep1pz1εpðkÞþ…þ ~K
epρ1zρε1ðkÞ
þ…þ ~K
epρpzρεpðkÞ: ð42Þ
From (42), onecangetthecorrespondingARMAXmodelas
Glðz1ÞyðkÞ ¼ Hlðz1ÞuðkÞþξðkÞ ¼ Hlðz1ÞuðkÞþKelðz1ÞeoðkÞ; ð43Þ
where
yðkÞ ¼ y1ðkÞ y2ðkÞ … ypðkÞ
h iT
;
uðkÞ ¼ u1ðkÞ u2ðkÞ … umðkÞ
h iT
;
eoðkÞ ¼ eo1ðkÞ eo2ðkÞ … eopðkÞ
h iT
;
Glðz1Þ ¼ IpþGo1z1þ⋯þGoρzρ;
Goiði ¼ 1; 2;…; ρÞAℜpp;
Hlðz1Þ ¼ Ho0þHo1z1þ⋯þHoρzρ;
Hoiði ¼ 0; 1;…; ρÞAℜpm
Kelðz1Þ ¼ IpþKeo1z1þ⋯þKeoρzρ;
Keoiði ¼ 1; 2;…; ρÞAℜpp;
Goi ¼
~A
1i1 ⋯ ~A
1ip
⋮ ⋱ ⋮
~A
pi1 ⋯ ~A
pip
2
664
3
775
; Hoi ¼
~B
1i1 ⋯ ~B
1im
⋮ ⋱ ⋮
~B
pi1 ⋯ ~B
pim
2
664
3
775
; Keoi ¼
~K
e1i1 ⋯ ~K
e1ip
⋮ ⋱ ⋮
~K
epi1 ⋯ ~K
epip
2
664
3
775
:
The specialcharacteristicofthemodified ARMAXmodelisthatit
includes the Ho0 matrix, soit fits thesystemwithadirect
transmissionmatrix.
An alternativerepresentationoftheARMAXmodel (43) is
givenby
yðkÞ ¼ Gl1
ðz1ÞHlðz1ÞuðkÞþGl1
ðz1ÞKelðz1ÞeoðkÞ; ð44Þ
in which (44) is intheleftmatrixfractiondescriptionform(LMFD)
[3]. The first andsecondtermsintheright-handsideof (44) share
the sameleftcharacteristicmatrixpolynomial Gl1
ðz1Þ, which
representstheeffectsofthecontrolandthedisturbances.Once
Gl1
ðz1Þ has beenspecified tocharacterizethedynamicsofthe
plant, theresidualvectormodel Gl1
ðz1ÞDlðz1ÞeoðkÞ presents an
adjustable movingaverageprocessofthenoiseinput Kelðz1ÞeoðkÞ.
UnderthelinearizedNARMAXmodel (44), asysteminan
observableblockcompanionformcanberepresentedinthe
state-space innovationform [21–23] as
^xoðkþ1Þ ¼ Go ^xoðkÞþHouðkÞþKoðkÞeoðkÞ; ð45Þ
eoðkÞ ¼ yðkÞCo ^xoðkÞDouðkÞ; ð46Þ
where
Go ¼
Go1 Ip 0p ⋯ 0p
Go2 0p Ip ⋯ 0p
⋮ ⋮ ⋮⋱⋮
Goρ1 0p 0p ⋯ Ip
Goρ 0p 0p ⋯ 0p
2
6666664
3
7777775
;
Ho ¼
Ho1
Ho2
⋮
Hoρ
2
66664
3
77775
;Hoi ¼ ~H
oiGoiDo; i ¼ 1; 2;⋯; ρ
Co ¼ Ip 0p ⋯ 0p
h i
;
Do ¼ Ho0;
Ko ¼ Ko1 Ko2 ⋯ Koρ
h i
; Koi ¼ KeoiGoi ; i ¼ 1; 2;⋯; ρ
^xoðkÞ ¼ ^xT
o1ðkÞ ^xT
o2ðkÞ ⋯ ^xT
oρðkÞ
h iT
;
^xoiðkÞAℜρ for i ¼ 1; 2;⋯; ρ, 0p is a p p null matrix, ^xoðkÞ is the
estimation ofsystemstate xðkÞ in theobservercoordinates,andthe
initial stateisgivenas ^xoð0Þ ¼ C†y, where C† is thepseudo-inverse
of matrix C and eoðkÞ ¼ yðkÞCo ^xoðkÞDouðkÞ.
However,thezerosof Kelðz1Þ in (44) may notbeallintheunit
circle,sothattheeigenvaluesoftheobservergain KoðkÞ in (45)
may notalllieintheunitcircleeither.InsteadoftheKalmangain
KoðkÞ, onecoulddesignthedigitalestimatorgain LoðkÞ toreplace
KoðkÞ. Thedigitalestimatorgainisindirectlydesignedviathe
discrete-timeobserverdesignbasedon (47), ratherthandirectly
estimatedfromtheidentified parametersofthemodified
NARMAXmodel (44). Therefore,theclosed-loopestimatormatrix
GoðkÞLoðkÞCo has allitseigenvaluesstrictlylyinginsidetheunit
circle.
The observergainis
LoðkÞ ¼ ððCo^P
ðkÞCo
T
þRoÞ1Co^P
ðkÞGo
T
ðkÞÞT ; ð47Þ
where ^P
ðkÞis thesolutionoftheRiccatiequation
GoðkÞ^P
ðkÞGo
T
ðkÞ^P
ðkÞðGoðkÞ^P
ðkÞCo
T
ÞðCo^P
ðkÞCo
T
þRoÞ1ðCo^P
ðkÞGo
T
ðkÞÞ
þQo ¼ 0 ð48Þ
in whichweightingmatrices QoZ0 and Ro40 withappropriate
dimensions. Thenthecorrespondingstate-spaceinnovationform
of (45) is givenas
^xoðkþ1Þ ¼ GoðkÞ^xoðkÞþHoðkÞuðkÞþLoðkÞeoðkÞ; ð49Þ
eoðkÞ ¼ yðkÞCo ^xoðkÞDouðkÞ: ð50Þ
4.1.TheinitialparametersofNARMAXmodelbasedonOKID
The initialparameters θið0Þ of themodified NARMAXmodel
significantly affecttheconvergentspeedofRELSprocess.Inorder
toincreasetheconvergentspeedofRELSalgorithm,onecan
predicttheinitialparameters θið0Þ of themodified NARMAXmodel
by OKID.Forgettingtheinitialparameters θið0Þ of RELSalgorithm
(3), weperformtheoff-linesystemidentification scheme,OKID,in
Section 3.2 to obtainthediscretesystemrealization ^G
, ^H
, ^C
, ^ D, and
F firstly.Then,transferthem(^G
, ^H
, ^C
, ^ D, F) intothecorresponding
observer form(Go, Ho, Co, Do, Ko) in (45) by
To ¼ ½Gρ1To1; Gρ2To1;…; GTo1; To1; To1 ¼ Θ1CT
o ;
Co ¼ CTo ¼ ½Ip; 0p;…; 0p:Do ¼ Ho0;
Go ¼ T1
o GTo;Ho ¼ T1
o H ¼ ½HT
o1;HT
o2;…;HT
oρT :
Ko ¼ T1
o Lo
Based on (43)–(45), wehavethemodified ARMAXmodelas
yðkÞþGo1yðk1ÞþGo2yðk2Þþ⋯þGoρyðkρÞ
¼ Ho0uðkÞnþHo1uðk1Þþ⋯þHoρuðkρÞ
þKe1eðk1ÞþKe2eðk2Þþ⋯þKeρeðkρÞ ð51Þ
where
Goi ¼
Goi11 ⋯ Goi1p
⋮ ⋱ ⋮
Goip1 ⋯ Goipp
2
64
3
75
;
Hoi ¼
Hoi01 ⋯ Hoi0m
⋮ ⋱ ⋮
Hoip1 ⋯ Hoipm
2
64
3
75
;
Keoi ¼
Keoi11 ⋯ Keoi1p
⋮ ⋱ ⋮
Keoip1 ⋯ Keoipp
2
64
3
75
62 J.S.-H. Tsaietal./ISATransactions53(2014)56–75

and i ¼ 1; 2;⋯ρ. From (43) and (51), onehastherelationshipas
Goi ¼
~A
1i1 ⋯ ~A
1ip
⋮ ⋱ ⋮
~A
pi1 ⋯ ~A
pip
2
664
3
775
¼
Goi11 ⋯ Goi1p
⋮ ⋱ ⋮
Goip1 ⋯ Goipp
2
64
3
75
; ð52aÞ
Hoi ¼
~B
1i1 ⋯ ~B
1im
⋮ ⋱ ⋮
~B
pi1 ⋯ ~B
pim
2
664
3
775
¼
Hoi01 ⋯ Hoi0m
⋮ ⋱ ⋮
Hoip1 ⋯ Hoipm
2
64
3
75
; ð52bÞ
Keoi ¼
~K
e1i1 ⋯ ~K
e1ip
⋮ ⋱ ⋮
~K
epi1 ⋯ ~K
epip
2
664
3
775
¼
Keoi11 ⋯ Keoi1p
⋮ ⋱ ⋮
Keoip1 ⋯ Keoipp
2
64
3
75
: ð52cÞ
Then,onehasthecoefficientmatrices ð~A
i; ~B
i; ~K
eiÞ ofthelinearized
NARMAXmodelin (41). Basedontheoptimallinearizationin
Section3.3, wecansolvesimultaneousequations AiR in (40) that
showtherelationshipbetweentheunknowncoefficientsofthe
modified NARMAXmodel (39) andtheknowncoefficientsof
thelinearizedNARMAXmodel (41) togettheparameters θi ofthe
modified NARMAXmodel (39) bythepseudoinverseoperation.
Thus,theparameters θi of themodifiedNARMAXmodelcanbe
utilizedastheinitialparameters θið0Þ ofRELSmethod (3) in STC.
4.2. Thedigitaltrackerforsampled-datalinearsystemwithadirect
transmissionterm
This sectionpresentsadigitalcontrollermethodforthelinear
systemwithadirecttransmissionterm.Consideralineardiscrete-
time systemasfollows
xðkþ1Þ ¼ GxðkÞþHudðkÞ; ð53aÞ
yðkÞ ¼ CxðkÞþDudðkÞ; ð53bÞ
where xðkÞAℜn is thestatevector, udðkÞAℜm is thecontrolinput
vector,and yðkÞAℜp is themeasurableoutputvector.Parameters
G,H,C and D are estimated(orgiven)constantsystemmatricesof
appropriatedimensions.
Define theperformanceindex [24] as
Jd ¼
1
2 Σ k
f
k ¼ 0 ½CðkÞxðkÞþWDðkÞudðkÞΓnðkÞTQ½CðkÞxðkÞþWDðkÞudðkÞ
(
ΓnðkÞþuT
dðkÞRudðkÞ
)
; ð54Þ
where tf ¼kf Ts is the final time,and Ts is thesampletime, Q is the
positivesemi-definite matrix, R is thepositivedefinite matrix,
ΓnðkÞAℜp is thepre-specified referenceinputvector,and W is a
weightingmatrixtoadjustthecontrollergainmatrix.Thisoptimal
controlisgivenby [24]
udðkÞ¼KdðkÞxðkÞþEdðkÞΓnðkÞ; ð55Þ
where
KdðkÞ ¼ ½~R
ðkÞþHT
ðkÞPHðkÞ1½HT
ðkÞPGðkÞþNT
ðkÞ; ð56aÞ
EdðkÞ ¼ ½~R
ðkÞþHT
ðkÞPHðkÞ1fHT
ðkÞ½IðGðkÞHðkÞKdðkÞÞT 1
½WDðkÞKdðkÞCðkÞT þðWDðkÞÞT gQ; ð56bÞ
~R
ðkÞ ¼ RþðWDðkÞÞTQWDðkÞ and NðkÞ ¼ CT
ðkÞQWDðkÞ; ð56cÞ
ΓnðkÞ ¼ ΓðkÞðkþ1Þ for thetrackingpurpose [25,26], and P is the
positivedefinite andsymmetricsolutionofthefollowingRiccati
equation
P ¼ GT
ðkÞPGðkÞþCT
ðkÞQCðkÞðGT
ðkÞPHðkÞ
þNðkÞÞð~R
ðkÞþHT
ðkÞPHðkÞÞ1ðHT
ðkÞPGðkÞþNT
ðkÞÞ: ð57Þ
It iswell-knownthatthehigh-gaincontrollerinducesahigh
qualityperformanceontrajectorytrackingdesignandstate
estimation, anditalsocansuppresssystemuncertaintiessuchas
nonlinear perturbations,parametervariations,modelingerrors
and externaldisturbances.Forthesereasons,thedigitalcontroller
with ahigh-gainpropertyisadoptedinourapproach.Thehigh-
gain propertycontrollercanbeobtainedbychoosingasufficiently
high ratioof Q to R (to beshownin Lemma 1) in (54) so thatthe
systemoutputcancloselytrackthepre-specified trajectory.
Lemma 1. [27] Giventheanalogsysteminthepairofsystem
matrices fA; B; Cg, letapairofweightingmatrices fQ; Rg be givenas
diagonalmatrices Q ¼ qIPbR and R ¼ rIm40. Thereexiststhe
lowerboundofweightingmatrices fQn; Rng, i.e. Qn ¼ qnIp and
Fig. 2. Structureofthehybridstate-spaceself-tuningcontrolwiththemodified NARMAXmodelandOKIDforunknownnonlinearstochastichybridsystem.

Rn ¼ rnIm, determinedby
κn ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
:BTB::CTC:
:ATA:
qn
rn

vuut
;
as longasthepropertyofthehigh-gaincontrolstillholds,thatis,
P244ζP1, for
ζ ¼ κ2=κ1 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð:BTB::CTC:=:ATA:Þðq2=r2Þ
q
=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð:BTB::CTC:=:ATA:Þðq2=r1Þ
q
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðq2=r2Þ
p
=
ffiffiffiffiffiffiffiffiffiffiffiffi
q1=r1
p
and κ24κ1Zκn;
where P2 and P1 are thesymmetricpositive-definite solutionof
the followingRiccatiequations,respectively,
ATP1þP1AP1BR1
1 BTP1þCTQ1C ¼ 0;
ATP2þP2AP2BR1
2 BTP2þCTQ2C ¼ 0:
In thefollowing,weshowthedesignproducefortheclassof
MIMO modelfor ρ42. Theresultscanbeextendedtothegeneral
multivariablecasefor ρ42. Thestructureoftheproposedstate-
space self-tuningcontrolwiththemodified NARMAXmodelis
shown in Fig. 2.
The designprocedureisgivenasfollows:
Step(1)Fortheunknowncontinuous-timenonlinearstochastic
system(1),chooseanappropriatemodified NARMAXmodel
(37) tobeusedtoidentifythissystem.
(i) Performtheoff-linesystemidentification schemein
Section 3.2 to obtainsystemandobserver-gainMarkov
parametersoftheOKIDmodel,thenusetheERAmethod
to obtainthediscretesystemrealization ^G
, ^H
, ^C
, ^ Dand F,
then transferthemtoobserverform Go, Ho, Co, Do and Ko.
(ii) Basedonthestate-spaceinnovationform (45) and the
optimallinearizationoftheNARMAXmodel (38)–(45),
initial parameters θið0Þ of theNARMAXmodelcanbe
reverselyobtainedby Go, Ho, Co, Do and Ko.
Step(2)Whenthemodified NARMAXmodelischosen,perform
the parameteridentification ateachsamplingperiod T.
(i) Setsomereasonableinitialparameterstoperformthe
state-space RELSalgorithmin (3). Letthenumberof θi be
θin. Also,set Sið0Þ ¼ αiI
ðinÞðinÞAℜθinθin40, 0oλ0o1,
0:9oλð0Þo1, ^xoð0Þ, andtheinitialcoefficients matrix
θið0Þ which isobtainedbyOKIDinStep1.
(ii) Predictthecontrolinput un
dð0Þ for theon-linesystem
identification as un
dð0Þ¼Kdð0Þ^xoð0ÞþEdð0ÞΓð1Þ, where
the Kdð0Þ and Edð0Þ are obtainedby (56a) and (56b), and
^xoð0Þ ¼ C þ ydð0Þ:
(iii) Foron-lineidentifyingthegivencontinuous-timenon-
linear stochasticsystem(1)withprecise-constantcontrol
input,utilizetheinformationofinputsandoutputsto
determine theupdatedparameters θiðkÞ at eachsampling
period T by RELSalgorithm,wherethepredictioncontrol
input un
dðkÞ fortheon-linesystemidentificationisdeter-
mined by un
dðkÞ¼Kdðk1ÞxoðkÞþEdðk1ÞΓðkþ1Þ for kZ1.
Step(3)LinearizetheNARMAXmodelbytheoptimallinear-
ization methodandestimatestatesateachsamplingperiod T.
Based ontheestimatedparameters θi, linearize FiðyÞ in (40) by
the optimallinearizationmethodology.Underthislinearized
NARMAXmodel,estimatethepredictedstate ^xoðkþ1jkÞ in (58).
Select appropriate fQo; Rog in (48) tohavethehigh-gain
propertydigitalestimatorgainin (47). Theassociatedstate-
space observer (49), forinstance ρ ¼ 2, isgivenby
^xoðkþ1jkÞ ¼ GoðkÞ^xoðkjk1ÞþHoðkÞuðkÞþLoðkÞeoðkÞ;
^yoðkjk1Þ ¼ Co ^xoðkjk1ÞþDouðkÞ; ð58Þ
where
eoðkÞ ¼ yðkÞCo ^xoðkjk1ÞDouðkÞ;
GoðkÞ ¼
Go1ðkÞ Ip
Go2ðkÞ 0p
#
Aℜ2p2p;
HoðkÞ ¼
Ho1ðkÞ
Ho2ðkÞ
#
Aℜ2pp
Ho1ðkÞ ¼ ~H
o1Go1Do;
Ho2ðkÞ ¼ ~H
o2Go2Do;
DoðkÞ ¼ Ho0ðkÞAℜpp
Co ¼ ½Ip; 0pAℜp2p; ^xoðkjk1ÞAℜ2p
LoðkÞ ¼ fðCo^P
ðkÞCo
T
þRoÞ1Co^P
ðkÞGo
T
ðkÞgT AR2pp
where
GoðkÞ^P
ðkÞGo
T
ðkÞ^P
ðkÞðGoðkÞ^P
ðkÞCo
T
Þ
ðCo^P
ðkÞCo
T
þRoÞ1ðCo^P
ðkÞGo
T
ðkÞÞþQo ¼ 0
Step(4)Generatethedigitalcontrolinputateachsampling
period T:
(i) Selectappropriateweightingmatrices fQd; Rdg in (57)
havethehigh-gainpropertydigitalcontrollawin (55)
and (56b).
(ii) Computethedigitalcontrolgains KdðkÞ and EdðkÞ, bythe
digital controlformulain (56a) and (56b) as follows.
~D
ðkÞ¼WDoðkÞ and Wis aweightingmatrixtoadjustthe
controllergainmatrix,
~R
ðkÞ ¼ Rþ ~D
T
ðkÞQ ~D
ðkÞ;
NðkÞ ¼ CT
oQ ~D
ðkÞ;
KdðkÞ ¼ ½~R
ðkÞþHT
o ðkÞPðkÞHoðkÞ1
½HT
o ðkÞPðkÞGoðkÞþNT
ðkÞ;
EdðkÞ ¼ ½~R
ðkÞþHT
o ðkÞPðkÞHoðkÞ1fHT
o ðkÞ
½IðGoðkÞHoðkÞKdðkÞÞT 1
½~D ðkÞKdðkÞCðkÞT þ ~D
T
ðkÞgQ;
where
PðkÞ ¼ GT
o ðkÞPðkÞGoðkÞþCT
o ðkÞQCoðkÞ
ðGT
o ðkÞPðkÞHoðkÞþNðkÞÞð~R
ðkÞ
þHT
o ðkÞPðkÞHoðkÞÞ1ðHT
o ðkÞPðkÞGoðkÞþNT
ðkÞÞ:
(iii) Set k¼kþ1. GotoStep2-(ii)andcontinuetheadaptive
controlprocess.
5. Self-tuningcontrolwithfaulttolerance
Consider theclassofcontinuous-timenonlinearstochastic
system(1).Ifthesystemstatesorinputsareinpartialfaults,the
Fig. 3. (a) Thegradualfailurefunctionand(b)theabruptfailurefunction.

systemdynamicscanberepresentedby
_xðtÞ ¼ f ðxðtÞÞþgðxðtÞÞuðtÞ
þ Σ Z
ζ ¼ 1
βζ ðtτζ Þ½f ζ ðxÞþgζ ðuÞþwðtÞ; ð59Þ
where f ζ ðxÞþgζ ðuÞ representsthedynamicchangescausedbythe
unknown andunanticipatedfailuremodesofstatesorinputs.
Two typicalfaults,gradualfaultsandabruptfaultsarecon-
sidered online.Theircharacteristicsaredescribedbythetime-
varyingfunction βζ ðUÞ as shownin Fig. 3 by [28], where UðUÞ
denotestheunit-stepfunction. f ζðxÞþgζ ðuÞ, βζ ðUÞ, and τζ are
unknown duetothepossibleoccurrenceofunanticipatedfaults.
If Z ¼ 1, system (59) has asinglefault.If Z ¼ 2; 3; 4; :::, itmeansthe
multiple-fault case
Accordingto [20], anoptimallylinearizedmodelofanonlinear
systemcanbedeterminedandappliedtoawideclassofnonlinear
systems.Asaresult,thenonlinearsystem(1)canbeaccurately
linearized asthefollowinglinearstate-dependencemodel:
_xðtÞ ¼ AðxðtÞÞxðtÞþBðxðtÞÞuðtÞþwðtÞ: ð60Þ
Similarly,thesystemwithfailuredynamicscanalsobeapproxi-
mated asthestate-dependencetime-varyingmodel:
_xðtÞ ¼ AðxðtÞÞxðtÞþBðxðtÞÞuðtÞþ Σ Z
ζ ¼ 1
βζ ðtτζ Þ½Aζ ðxðtÞÞxðtÞ
þBζ ðxðtÞÞuðtÞþwðtÞ_xðtÞ ¼ ½AðxðtÞÞþ Σ Z
ζ ¼ 1
βζ ðtτζ ÞAζðxðtÞÞxðtÞ
þ½BðxðtÞÞþ Σ Z
ζ ¼ 1
βζ ðtτζ ÞBζ ðxðtÞÞuðtÞþwðtÞ: ð61Þ
The systemcouldcontainlargeuncertaintieswhenthefailure
dynamics in (61) arelarge.Underthissituation,thecontrollerhas
to takeanappropriatecontrolactionfortheuncertaintiesoccur-
ring atanytimeinstant τζ . Thisisanadaptivecontrolproblem,in
which controllerparametersareadjustedbasedontheestimated
plant parameters.Themethodbasedonthemodified STCscheme
is proposedtoaccomplishtheFTC.
There arethreeassumptionsoftheproposedmethodthatare
addressed asfollows:
Assumption1. The controlledsystemiscontrollableandobser-
vableeveniffaultsoccur.
Assumption2. The controlinputispersistentlyexcited.
Assumption3. Before thefaultoccurs,thesystemishealthyor
has fullyrecoveredfromthepreviousfault.
TheSTCschemeshouldbemodified tocopewithparameter
variationsduetosystemfaults.Whenapartialfaultoccurs,the
systemparametersvaryaccordingly.Theestimatedstate-depen-
dencetime-varyingparametersobtainedviatheRELSalgorithmin
theconventionalSTCschemewouldgivelargeparametererrorsand
resultinapoorsystemperformance.However,basedontheKalman
filterinterpretationalgorithmofRELSmethod [29], a modified
scheme isproposedtoestimateparametervariations.Theabove
modified state-spaceself-tuningcontrolschemecanbeappliedto
the multivariablestochasticfaulty systemwithoutpriormessageof
systemparametersandnoiseproperties.
In short,inthebeginning,ahealthyandunknownsystemis
welltunedbytheconventionalSTCscheme,andthentheself-
tuning structurewiththeresetcovariancematricesofparameter
estimateismodified toenhancetheparameterestimationand
output responsewhenthesystemand/orinputsarepartially
faulty by [15].
It postulatesthattheestimatedparameterisnotconstantbut
varieslikearandomwalk
θiðkÞ ¼ θiðk1ÞþwiðkÞ; ð62Þ
εiðkÞ ¼ yiðkÞθT
iðk1ÞϕiðkÞ; ð63Þ
E½wiðkÞwi
T
ðkÞ ¼ R1i; ð64Þ
E½εTi
ðkÞεiðkÞ ¼ R2i; ð65Þ
where i ¼ 1; 2;…; p; wiðkÞ is thewhiteGaussiannoisesequence.
The Kalman filterthenstillgivestheconditionalexpectationand
1414.51515.51616.51717.51818.519-0.3-0.2-0.100.10.20.3 Magnitude System output Y1ARMAX identified output Y11414.51515.51616.51717.51818.519-0.2-0.100.10.20.3 Megnitude Time (sec) System output Y1NARMAX identified output Y1
Fig. 4. The trajectoriesofOutput1byRELSmethodwithARMAXmodelandmodified NARMAXmodel.

covarianceof θiðkÞ as
^θ
iðkÞ ¼ ^θ
iðk1ÞþMiðkÞεi
T ðkÞ; ð66Þ
MiðkÞ ¼
Siðk1ÞϕiðkÞ
R2iþϕi
; ð67Þ
SiðkÞ ¼ Siðk1Þ
Siðk1ÞϕiðkÞϕi
T ðkÞSiðk1Þ
R2iþϕi
T ðkÞSiðk1ÞϕiðkÞ þR1i; ð68Þ
with
Sið0Þ ¼ E½^θ
ið0Þθið0Þ ½^θ
ið0Þθið0ÞT : ð69Þ
SiðkÞ is thecovariancematrixoftheparameterestimate ^θ
iðkÞ.
Usually,theestimatedresidualortheinnovationerrorvector
εiðkÞ ¼ yiðkÞ^θ
i
T
ðk1ÞϕiðkÞ will benearwhiteifthemodelwith
parameterestimateisingoodagreementwithitstruesystem.
TomodifyKalman filter interpretationoftheRELSmethod,
some appropriateinitializationsof R1i, R2i, and Sið0Þ in (67)–(69)
areassumedtobepre-specified beforetheparameterestimation
process.Whenunanticipatedsystemfaultsbringtheprocesswith
large parametervariations,theyneedtobereasonablyreset.
TomodifytheconventionalSTCprocesswiththeRELSestimate
algorithmforthefaultysystem,approximate R1i, R2i, and Sið0Þ by
the followingmovingwindow-basedstatisticalquantities:
R2i
1
Nk1
Σ
N1
k ¼ k1
εTi
ðkÞεiðkÞ ð70Þ
051015202530-0.2-0.100.10.20.3 megnitude System output Y1Identified output Y1051015202530-0.2-0.100.10.2 error 1 Time (sec) 051015202530-0.3-0.2-0.100.10.20.3 megnitude System output Y2Identified output Y2051015202530-0.2-0.100.10.2 error 1 Time (sec)
Fig. 5. The outputresponsesofthemodified NARMAXmodelbyRELSmethodwithoutOKID.

and
Sið0Þ diag
1
Nk1
Σ
N1
k ¼ k1½^θ
iðkÞθiðNÞ ½^θ
iðkÞθiðNÞT
( )
; ð71Þ
where k1 is thetimeindexaftertheestimate ^θ
iðkÞ in steady-state,
and N is thelasttimeinstantoftheSTC.Itshouldbenotedthatthe
elements of ^θ
iðkÞ in (71) wouldnotbeindependentwitheach
other,when Nk11 in (71) is notlargeenough.Similarly, R1i can
be approximatedbycomparing (62) with (66) as follows:
R1i ½MiðNÞεi
T ðNÞ ½MiðNÞεi
T ðNÞT ; ð72Þ
where
MiðNÞ ¼
Sið0ÞϕiðNÞ
R2iþϕi
T ðNÞSið0ÞϕiðNÞ
:
The STCwiththealgorithm (66)–(69) and theinitialization
(70)–(72) worksonlyfortheplantwithslowlytime-varying
parameters.Thiscanbeinterpretedbythefactthattheinitialized
R1i obtained from (72) is sosmallwhilethesystemishealthyin
general;hence θiðkÞffiθiðk1Þ in (62). Asaresult,itcannotreflect
the realparametervariationinducedbytheunanticipatedsystem
faults. Therefore, SiðkÞ, R2i, and R1i in (68) need tobeappropriately
reset whenafaultisdetectedattimeinstant kf . Althoughthe
algorithm withanappropriatelyresetforgettingfactor λðkf Þ could
improveestimationsofparametervariationfortheconventional
STC scheme,theresetforgettingfactor λðkf Þ would needsome
trials forvariousfailuremodes.Nevertheless,theresetsof Siðkf Þ,
R1i, andR2i proposed in [15] is asystematicapproachforvarious
failure modes.
Because thefactthattheparametervariationsinducedbyfaults
areunknown,theruleofthumbtoresetthecovariancematricesof
the parameterestimate SiðkÞ in (68) online isgivenasfollows.
When thefaultbedetectedattimeinstant kf , thevariationof
parameterestimationsbeforeandafterthefaultcanbeapproxi-
mated as
δ^θ
iðkfÞ ^θiðkf Þ^θ
iðkhÞ; for khokf ; ð73Þ
where ^θ
iðkhÞ is theparameterestimateofthehealthysystem.Then,
based onthephysicalinterpretationof (69), Siðk1Þ in (68) can be
reasonablyresetas
Siðkf1Þ diagf½^θ
iðkf Þ^θ
iðkhÞ ½^θ
iðkf Þ^θ
iðkhÞT g
δ2diag½^θ
iðkf Þ^θ
iðkf ÞT : ð74Þ
Duetotheadditiveuncertaintiesconsidered,wecanassumethe
averageparametervariationisintherangeofthesameorderof
magnitudeofthefaultsystem.So,itisreasonabletoset δ ¼ 1, which
denotestheworstcaseofthisassumption.Somenumericalexamples
arealsogivenin [15] toshowthesensitivityofvarious δ's tothemean
valueofthetrackingerrortoverifytheeffectivenessofthisruleof
thumb forresettingthecovariancematricesoftheparameterestimate.
Toimprovetheparameterestimationfortheunanticipated
faulty systems, R2i in (70) needs toberesetbyamovingwindow
with theresidual
R2i
1
kfkiþ1 Σ k
f
κ ¼ ki
εi
T ðκÞεiðκÞ; ð75Þ
where 1oðkfkiÞo5 usually.Similarly, R1i in (68) should bereset
by substituting (69) and (70) into (67) and then (66) to obtain
R1i ½Miðkf Þεi
T ðkf Þ ½Miðkf Þεi
T ðkf ÞT : ð76Þ
In theSTCscheme,theestimatedresidualisupdatedforevery
samplingtime.Consequently,itisconvenienttouseitasthe
informationoffaultdetection. Therefore,thetimeinstant kf of
the faultoccurrencecouldbedetectedbyutilizingtheratiowith
theresidual R2i in (75) andtheaveragenormoftheinnovation
vectorsas
R2i
Rf i
4γεi; ð77Þ
where Rf i ¼ ð1=kf1ÞΣkf
k ¼ 1εi
T ðkÞεiðkÞ and γεi isapresetthreshold.
The followingsummarizestheFTCusingthemodified STC
methodology withthefaultdetectionandcovariancematrices
resetting:
1) Applythemodified NARMAXmodel-basedSTCalgorithmin
Section 4 to thehealthysystemuntilitwelltracksthepre-
specified trajectory.
2) SwitchtheconventionalRELSestimationalgorithmin Section 4
Step-(ii)tothemodified Kalman filter estimationalgorithm
(66)–(68) with initialized R1i,R2i, and Sið0Þ via (70)–(72).
3) Performthemodified STCschemeandthefaultdetection.
4) Wheneverafaultisdetectedandtheerrorislargeenoughfor
the ratio ðR2i=Rf iÞ4γεi; reset R1i, R2i, and Siðkf Þ by (74)–(76).
5) Whenthefaulthasrecovered,gotoStep(3),andrepeatthe
modified STCprocess.
6. Anillustrativeexample
6.1.Identification byusingtheRELSmethod
6.1.1.NonlinearNARMAXmodelsystem
Assume thetwo-input-two-outputsystemisunknown,and
choose anappropriatetwo-inputtwo-outputNARMAXmodelfor
the RELSmethod.
Consider thenonlinearNARMAXmodelasfollows
y1ðkÞ¼0:2ð0:2y1ðk1Þþ0:2y2ðk1Þ cos ðy1ðk1ÞÞþy21
ðk2Þ
þ0:4y1ðk1Þu1ðk1Þþu1ðk1Þþ0:8e1ðk1Þ
þ0:7y1ðk2Þe1ðk2Þþu1ðkÞÞþe1ðkÞ; ð78aÞ
y2ðkÞ¼0:2ð0:2y2ðk1Þþ0:3y2ðk1Þ cos ðy2ðk2ÞÞþy22
ðk2Þ
þ0:1y2ðk1Þu2ðk1Þþu2ðk1Þþ0:8e2ðk1Þ
þ0:5y1ðk2Þe2ðk2Þþu2ðkÞÞþe2ðkÞ; ð78bÞ
051015202530-0.4-0.200.20.40.60.811.21.4Time (sec) Parameter
Fig. 6. Estimated parametersin θðkÞ of RELSmethodforthemodified NARMAX
model withoutOKID.

where y1ðkÞ and y2ðkÞ areoutputs, u1ðkÞ and u2ðkÞ are inputs,
e1ðkÞand e2ðkÞ arenoiseswhicharezero-meanGaussiansequences
with variances s2 e1 ¼ s2 e2 ¼ 0:01. BecausethesystemEqs. (78a) and
(78b), isunknown,onlythedataofinputandoutputatsampling
instants areavailabletoidentifythesystem.Thesamplingperiod T
is takenas0.1sathere.
The resultsofsystemidentification throughtheRELSmethod
with themodified ARMAXmodel [11] and theproposedmodified
NARMAXmodelareshownin Fig. 4, respectively.
ThetrajectoriesofOutput2byRELSmethodwithARMAXmodel
and modifiedNARMAXmodelhavethesimilarperformanceofthe
caseforOutput1.Allthesesimulationsshowthattheresultof
systemidentificationthroughtheRELSmethodwiththemodified
NARMAXmodel,ratherthanthemodifiedARMAXmodel,achievesa
betterperformance.
6.1.2.Theinitialparameter θð0Þ of themodified NARMAXmodel
Comparison ofthesystemidentification ofRELSmethodwith
the NARMAXmodelbasedontheintuitiveinitialparameter θð0Þ ¼
½I211 I212 0212 I26T and theinitialparameterobtainedby
the off-lineOKIDisshownasfollows.
1) The NARMAXmodelwiththeintuitiveinitialparameter θð0Þ ¼
½I211 I212 0212 I26T Let thesystem (78) be excitedby
the controlforce uðtÞ ¼ ½u1ðtÞ u2ðtÞT with whitenoisehaving
a zeromeanandcovariance diagðcovðuÞÞ ¼ 0:1I2, wherethe
sampling period T is 0.1s.Theresultsofidentification are
shown in Figs. 5 and 6.
2) The modified NARMAXmodelwiththeinitialparametersobtained
the byOKID. Thesystemandobservergain Go, Ho, Co, Doand Ko
in (58) areobtainedbytheOKID.Basedontheoptimal
051015202530-0.2-0.100.10.20.3 megnitude System output Y1Identified output Y1051015202530-0.2-0.100.10.2 error 1 Time (sec) 051015202530-0.2-0.100.10.2 megnitude System output Y2Identified output Y2051015202530-0.2-0.100.10.2 error 1 Time (sec)
Fig. 7. The outputsresponsesofthemodified NARMAXmodelbyRELSmethodwithOKID.

linearization method,wecanobtaintheinitialparameters θð0Þ
which isclosetotheconvergentvalueof θðkÞ. Aftertheinitial
parameters θð0Þ are obtained,theRELSmethod (3) is appliedto
identify thissystem.Theresultofidentification isshownin
Figs. 7 and 8.
6.2. Activefaulttoleranceusingmodified NARMAXmodel-based
state-space self-tuningcontrol
In fact,thesystemmodelisunknown (78), wehaveonly
information aboutinputandoutputdata.Thesamplingperiod T
is selectedas0.1sfortheoff-lineOKIDandtheon-linestate-space
self-tuning control.First,chooseanappropriatetwo-input-two-
output modified NARMAXmodelviatheoff-lineOKIDfortheself-
tuning controlwithfaulttoleranceusingtheestimationalgorithm
and thefaultdetectionisusedtoadapttothefaulttolerance
control. Noticethattheinitialparameter θð0Þof thedetermined
NARMAXmodelfortheon-lineRELSisobtainedbyoff-lineOKID.
If afaultisdetectedat t ¼ kf , matrices R1i, R2i, and Siðkf Þ with the
reset parameter δ ¼ 1 in (71) is automaticallyresetagain.Thefault
detection thresholdsare γε1 and γε2 are 3.0.Noticethatthis
NARMAXmodelcouldapproximateothertwo-input-two-output
systems,notjustonlyforthisexample.
The proposedtwo-input-two-outputmodified NARMAXmodel
is givenby (79). Inordertoshortentheexpressions,oneassumes
y
10
, ¼ F11ðyÞþB1UþKe1E;
y
20
, ¼ F21ðyÞþB2UþKe2E; ð79Þ
where
y
i j
,¼ yiðkj ,
Þ; u
kl
,¼ ukðkl ,
Þ; ε
mn, ¼ εmðkn,
Þ;
F11ðyÞ ¼ a11y
11
,þa12y
21
,þa13y
12
,þa14y
22
,þa15y2
11
,
þa16y
11
,y
21
,þa17y
11
,y
12
,þa18y
11
,y
22
,
þa19y
12
,y
21
,þa110y2
12
,þa111y
12
,y
22
,;
F21ðyÞ ¼ a21y
11
,þa22y
21
,þa23y
12
,þa24y
22
,þa25y
21
,y
11
,
þa26y2
21
,þa27y
21
,y
12
,þa28y
21
,y
22
,
þa29y
22
,y
11
,þa210y
22
,y
12
,þa211y2
22
,;
BT
1 ¼
b101þb103y
11
,þb105y
12
,
b102þb104y
11
,þb106y
12
,
b11þb15y
11
,þb19y
12
,
b12þb16y
11
,þb110y
12
,
b13þb17y
11
,þb111y
12
,
b14þb18y
11
,þb112y
12
,
2
666666666666664
3
777777777777775
; BT
2 ¼
b201þb203y
21
,þb205y
22
,
b202þb204y
21
,þb206y
22
,
b21þb25y
21
,þb29y
22
,
b22þb26y
21
,þb210y
22
,
b23þb27y
21
,þb211y
22
,
b24þb28y
21
,þb212y
22
,
2
666666666666664
3
777777777777775
KT e1 ¼
d11þd15y
11
,þd19y
12
,
d12þd16y
11
,þd110y
12
,
d13þd17y
11
,þd111y
12
,
d14þd18y
11
,þd112y
12
,
2
66666664
3
77777775
; KT e2 ¼
d21þd25y
21
,þd29y
22
,
d22þd26y
21
,þd210y
22
,
d23þd27y
21
,þd211y
22
,
d24þd28y
21
,þd212y
22
,
2
66666664
3
77777775
U ¼
u
10
,
n
u
20
,
n
u
11
,
u
21
,
u
12
,
u
22
,
2
6666666666664
3
7777777777775
; E ¼
ε
11
,
ε
21
,
ε
12
,
ε
22
,
2
666664
3
777775
:
Simplify themodeltoformalinear-in-the-parametersexpression
~y1ðkÞ ¼ θ1
T ðkÞϕ1ðkÞ;
~y2ðkÞ ¼ θ2
T ðkÞϕ2ðkÞ; ð80Þ
where
θ1ðkÞ ¼ ½a11a12 ⋯ a111 b11b12 ⋯ b112 d11d12 ⋯ d112
b101b102 ⋯ b106T Aℜ411;
θ2ðkÞ ¼ ½a21a22 ⋯ a211 b21b22 ⋯ b212 d21d22 ⋯ d212
b201b202 ⋯ b206T Aℜ411;
θðkÞ ¼ θ1ðkÞ θ2ðkÞ
h i
;
ϕ1ðkÞ and ϕ2ðkÞ are therelatedtermstoeachparameter,
nθ ¼ n ¼ 41 isthenumberofparametersin θ1ðkÞ, soisin θ2ðkÞ.
Estimatetheparameters θ1ðkÞ and θ3ðkÞ using thestandard
recursiveextended-least-squaresalgorithm.Theinitialparameters
θð0Þ is obtainedbyOKID,where λ0 ¼ 0:9, λð0Þ ¼ 0:9 and S1ð0Þ ¼ S2ð0Þ ¼ 1 Inθ .
The optimallinearizationmethod,thelinearmodelsof F11ðyÞ
and F21ðyÞ at samplingtime kT are gottenas
F11ðyÞ ¼ a11Ry
11
,þa12Ry
21
,þa13Ry
12
,þa14Ry
22
,; ð81aÞ
F21ðyÞ ¼ a21Ry
11
,þa22Ry
21
,þa23Ry
12
,þa24Ry
22
,; ð81bÞ
where
a11R
a12R
a13R
a14R
2
6664
3
7775
¼
a11þ2a15y
11
,þa16y
21
,þa17y
12
,þa18y
22
,
a12þa16y
11
,þa19y
12
,
a13þa17y
11
,þa19y
21
,þ2a110y
12
,þa111y
22
,
a14þa16y
11
,þa111y
12
,
2
66666664
3
77777775
þλ1R
y
11
,
y
21
,
y
12
,
y
22
,
2
6666664
3
7777775
;
λ1R ¼
1
‖ðy
11
,; y
21
,; y
12
,; y
22
,Þ‖22
ða15y2
11
,þa16y
11
,y
21
,þa17y
11
,y
12
,

þa18y
11
,y
22
,þa19y
12
,y
21
,þa110y2
12
,þa111y
12
,y
22
,Þ

;
051015202530-1-0.500.51Time (sec) Parameter
Fig. 8. Estimated parametersin θðkÞ of RELSmethodforthemodified NARMAX
model withOKID.

a21R
a22R
a23R
a24R
2
6664
3
7775
¼
a21þa25y
21
,þa29y
22
,
a22þa25y
11
,þ2a26y
21
,þa27y
12
,þa28y
22
,
a23þa27y
21
,þa210y
22
,
a24þa28y
21
,þa29y
11
,þa210y
12
,þ2a211y
22
,
2
66666664
3
77777775
þλ2R
y
11
,
y
21
,
y
12
,
y
22
,
2
6666664
3
7777775
;
λ2R ¼
1
‖ðy
11
,; y
21
,; y
12
,; y
22
,Þ‖22
ða25y
21
,y
11
,þa26y2
21
,þa27y
21
,y
12
,

þa28y
21
,y
22
,þa29y
22
,y
11
,þa210y
22
,y
12
,þa211y2
22
,Þ

;
GoðkÞ and HoðkÞ in theassociatedstate-spaceinnovationform (58)
aredeterminedas
GoðkÞ ¼
Go1ðkÞ I2
Go2ðkÞ 02
#
; HoðkÞ ¼
Ho1ðkÞ
Ho2ðkÞ
#
;
Co ¼ I2 02

; Do ¼ Ho0ðkÞ ð82Þ
012345678910-1-0.500.51 magnitude Y1Reference 1System output Y1012345678910-1-0.500.51Time (sec) magnitude Y2Reference 2System output Y2012345678910-1-0.500.51 magnitude Y1Reference 1System output Y1012345678910-1-0.500.511.5Time (sec) magnitude Y2Reference 2System output Y2012345678910-1-0.500.51 magnitude Y1Reference 1System output Y1012345678910-1-0.500.51Time (sec) magnitude Y2Reference 2System output Y20.10.150.20.250.30.350.40.450.50.550.65678910111213141516weighting factor w the convergence of weighting factor
Fig. 9. The comparisonbetween(i)Output y1ðtÞ and reference r1ðtÞ and (ii)Output y2ðtÞ and reference r2ðtÞ: (a) W ¼ 0:2 I2, (b) W ¼ 0:1 I2, (c) W ¼0:4 I2, and
(d) theconvergenceoferrorsvs.weightingmatrices.

where
Go1 ¼
a11R a12R
a21R a22R
#
;
~H
o1ðkÞ ¼
b11þb15y
11
,þb19y
12
, b12þb16y
11
,þb110y
12
,
b21þb25y
21
,þb29y
22
, b22þb26y
21
,þb210y
22
,
2
64
3
75
;
Go2 ¼
a13R a14R
a23R a24R
#
;
~H
o2ðkÞ ¼
b13þb17y
11
,þb111y
12
, b14þb18y
11
,þb112y
12
,
b23þb27y
21
,þb211y
22
, b24þb28y
21
,þb212y
22
,
2
64
3
75
;
Ho0ðkÞ ¼
b101þb103y
11
,þb105y
12
, b102þb104y
11
,þb106y
12
,
b201þb203y
21
,þb205y
22
, b202þb204y
21
,þb206y
22
,
2
64
3
75
;
Ho1ðkÞ ¼ ~H
o1Go1Do;
Ho2ðkÞ ¼ ~H
o2Go2Do:
05101520253035404550-1-0.500.51 magnitude Reference 1Output Y105101520253035404550-1-0.500.51Time (sec) magnitude Reference 2Output Y2
Fig. 10. The outputresponseswithanabruptinputfaultoccurringatthe20thsecondforthemodified NARMAXmodelwithoutFTC. 05101520253035404550-1-0.500.51 magnitude Reference 1Output Y105101520253035404550-1-0.500.51Time (sec) magnitude Reference 2Output Y2
Fig. 11. The outputresponseswithanabruptinputfaultoccurringatthe20thsecondforthemodified NARMAXmodelwithFTC.

The weightingmatricesare fQ ¼ 108
I2; R ¼ I2g in (54) and
fQo ¼ 108
I4; Ro ¼ I2g in (48).
1) Determination oftheweightingmatrix.
Here, severalcaseswillbegiventodiscusstheeffectsfor
different weightingmatrices.Wecanselectanappropriate
weightingfactortoimprovethetrackingperformanceofthe
on-line systemcontrol.Sometypicalexamplesforthesystem
(78) are illustratedfollows.
From Fig. 9, theweightingfactorischosenas W ¼ 0:2I2 to
minimize theoutputtrackingerrordefined. Then,theFTCwith
an abruptinputfaultandagradualinputfaultisconsideredas
follows
2) Fault scenario1:anabruptinputfault.
Weusethemodified NARMAXmodeltoidentifythenonlinear
system,andswitchittotheFTCatthe5thsecond.Atthe20th
second, theInput1isassumedtobeabruptlyaddedto5times
of itsfunction,andInput2isassumedtobeabruptlyadded
to 0.8timesofitsfunction,thensimulationresultsin Fig. 10.
The simulationresultsusingtheFTCwiththemodified STC
Fig. 12. The outputresponseswithanabruptinputfaultoccurringatthe20thsecondforthemodified ARMAXmodelwithoutFTC. 05101520253035404550-1-0.500.51 magnitude Reference 1Output Y105101520253035404550-1-0.500.51Time (sec) magnitude Reference 2Output Y2
Fig. 13. The outputresponseswithanabruptinputfaultoccurringatthe20thsecondforthemodified ARMAXmodelwithFTC.

methodology areshownin Fig. 11. Forthefaulttolerance,a
comparison ofthenewmethodisgivenin Figs. 10–13, which
showsourproposedapproachismuchsuperiorthan [11].
3) Faultscenario2:agradualinputfault.
Weusethemodified NARMAXmodeltoidentifythenonlinear
system,thenswitchtotheFTCwhichusingthemodified STC
methodology when5thsecond.Afterthe20thsecond,the
Input1isassumedtobeabruptlyaddedto10:8ð1eðt20ÞÞ times ofitsfunctions,thensimulationresultsin Fig. 14. Andthe
simulation resultswiththeFTCwhichusingthemodified STC
methodology isshownin Fig. 15. Forthefaulttolerance,a
comparison ofthenewmethodisgivenin Figs. 14–17, which
showsourproposedapproachismuchsuperiorthan [11].
7. Conclusions
A polynomial-expansion-formmodified NARMAXmodel-based
fault-tolerantstate-spaceself-tunerfortheunknownnonlinear
stochastichybridsystemwithaninput-outputfeed-throughterm
05101520-1-0.500.51 magnitude Y1 Reference 1System output Y105101520-1-0.500.51Time (sec) magnitude Y2 Reference 2System output Y2
Fig. 14. The outputresponses(divergingtoaninfinity valuebefore23s)withagradualinputfaultoccurringatthe20thforthemodified NARMAXmodelsecond
without FTC. 05101520253035404550-1-0.500.51 magnitude Y1Reference 1System output Y105101520253035404550-1-0.500.51Time (sec) magnitude Y2Reference 2System output Y2
Fig. 15. The outputresponseswithagradualinputfaultoccurringatthe20thsecondforthemodified NARMAXmodelwithFTC.

has beenproposedinthispaper.Themaincontrolschemesofthis
paperare(i)initializationfortheon-linepolynomial-expansion-form
NARMAX-basedsystemidentification obtainedbytheoff-lineOKID
is proposedtospeeduptheparameteridentificationprocess,(ii)an
optimal trackerbasedontheidentified modelwithadirecttransmis-
sion termfrominputtooutputisutilizedfortheunknownnonlinear
systemwithadirecttransmissionmatrix,and(iii)whenthe
unknownsystemhasinputfault,thecontrolschemefocuseson
designinganactivefaulttolerancestate-spaceself-tunerusingthe
modified NARMAXmodel-basedsystemidentification.
Totheauthor’s knowledge,theNARMAXmodel-basedstate-
space optimaltrackerwithfaulttolerancefortheregularnonlinear
sampled-data systemcontainingthedirecttransmissionterm
from inputtooutputhasnotbeenproposedinliterature.Con-
sidering thecomplexityofthefaulttolerancecontrol,thispaper
takes theNARMAXmodelinpolynomialexpansionform,butnot
in rationalexpansionform.Theextensionofthemodified NAR-
MAX model-basedmethodologyforthefault-tolerancecontrol
from itspolynomialexpansionformtorationalexpansionform
can beconsideredasafutureresearchwork.
Fig. 16. The outputresponseswithagradualinputfaultoccurringatthe20thsecondforthemodified ARMAXmodelwithoutFTC. 05101520253035404550-1-0.500.51 magnitude Reference 1Output Y105101520253035404550-1-0.500.51Time (sec) magnitude Reference 2Output Y2
Fig. 17. The outputresponseswithagradualinputfaultoccurringatthe20thsecondforthemodified ARMAXmodelwithFTC.

Acknowledgments
This workwassupportedbytheNationalScienceCouncilof
RepublicofChinaundercontractsNSC102-2221-E-208-MY3and
NSC102-2221-E-006-199.
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A modified narmax model based self-tuner with fault tolerance for unknown nonlinear stochastic hybrid systems with an input–output direct feed-through term

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A modified narmax model based self-tuner with fault tolerance for unknown nonlinear stochastic hybrid systems with an input–output direct feed-through term