Generalized eigenvalue minimization for uncertain first order plus time-delay processes

ResearchArticle
Generalizedeigenvalueminimizationforuncertain
first-orderplustime-delayprocesses
Gongsheng Huang a,n, KeckVoonLing b, XiaoningXu c, YundanLiao a
a Department ofCivilandArchitecturalEngineering,CityUniversityofHongKong,TatCheeRoad,Kowloon,HongKong
b School ofElectricalandElectronicEngineering,NanyangTechnologicalUniversity,50NanyangAvenue,Singapore
c Institute ofCivilEngineering,GuangzhouUniversity,No.230OuterRingRoad,Guanzhou,GuangdongProvince,China
a rticleinfo
Article history:
Received15February2013
Receivedinrevisedform
8 August2013
Accepted6September2013
Availableonline27September2013
Keywords:
Generalized eigenvalueminimization
First-order plustime-delaymodel
Robustcontrol
Linear-matrixinequality
Uncertainty
a b s t r a c t
This papershowshowtoapplygeneralizedeigenvalueminimizationtoprocessesthatcanbedescribed
by a first-orderplustime-delaymodelwithuncertaingain,timeconstantanddelay.Analgorithm
to transformtheuncertain first-order plustimedelaymodelintoastate-spacemodelwithuncertainty
polyhedronis firstly described.Theaccuracyofthetransformationisstudiedusingnumericalexamples.
Then, theuncertaintypolyhedronisrewrittenasalinear-matrix-inequalityconstraintandgeneralized
eigenvalueminimizationisadoptedtocalculateafeedbackcontrollaw.Casestudiesshowthateven
if uncertaintiesassociatedwiththe first-orderplustimedelaymodelaresignificant, astablefeedback
control lawcanbefound.Theproposedcontrolistestedbycomparingwitharobustinternalmodel
control. Itisalsotestedbyapplyingittothetemperaturecontrolofair-handingunits.
& 2013ISA.PublishedbyElsevierLtd.Allrightsreserved.
1. Introduction
Modeluncertaintyisawidelyrecognizedprobleminthecontrol
literature.Itreferstothedifferencesorerrorsbetweenmodelsand
reality [1]. Whenidentifyingamodelofaprocess,modeluncer-
taintyoccurswhen(i)theprocessisill-definedorhasasignificant
amountofvagueness;(ii)alowordermodelisusedbuttheprocess
is ofhighorder;and(iii)amodelwith fixedparametersisusedbut
thedynamicsoftheprocessistimevaryingorchangeswithitsope-
ratingenvironment,forexample,thetimedelayinanetworked
controlsystemisnotalwaysaconstant [2,3]. Althoughmost(non-
robust)controlstrategieshaveinherentrobustnessformodelunce-
rtainties,thecontrolobjectivesthatarepredefinedmightbe
difficulttoachievewhenmodeluncertaintiesaresignificant [4].
Uncertaintiesmayaffectthestabilityofacontrolsystemseriously
[1,5], andtheyshouldbetakenintoaccountdirectlyinthecontrol
designinordertoguaranteecontrolperformance.
Manyindustrialprocessescanbedescribedusinga first-order
plus timedelay(FOPDT)model [6,7]
T
dy
dt þyðtÞ ¼ KuðtτÞ ð1aÞ
where u, y aretheprocessinputandoutput; K, T, τ denotetheprocess
gain, timeconstantanddelay,respectively.Whenuncertainties
associatedwiththismodelareconsidered,theuncertaintiescanbe
specified by
KA½K1; K2; TA½T1; T2; τA½τ1; τ2 ð1bÞ
wherethesubscript ‘1’ denotesthelowerboundand ‘2’ denotesthe
upper bound.TheuncertaintysetsinEq. (1b) can beusedtodescribe
the variationsoftheseparametersaswellastheassociatedvagueness
associatedwiththeidentification ofthemodel (1a).
ThemodelrepresentedbyEqs. (1a) and (1b) canbeusedto
describethedynamicsofmanythermalprocesses,forexample,
in heating,ventilationandairconditioning(HVAC)systems [8–11],
andafewreports,suchas [10,12,13], addresseditsrobustcontrol
issue.However,noneofthestudiesdealtwithalloftheuncertain-
tiesspecified inEq. (1b). Forexample,Underwood [10] andHuang
andWang [12] dealtwithuncertaintiesonlyassociatedwiththe
processgainanddelay;andHuangandWang [13] dealtwith
uncertaintiesonlyassociatedwiththeprocessgainandtime
constant.Notably,Laughlinetal. [14] describedtheuncertainty
setsinEq. (1b) as regionsinthecomplexplaneandemployed
theinternalmodelcontrol(IMC)todesignacontroller,ofwhichthe
robuststabilityandtheperformancewerebalancedbytuningthe
parametersofa filter [14].
Thispapershowshowtousethegeneralizedeigenvaluemini-
mization(GEVM)techniquetodesignarobustcontrollawfor
processesthatcanbedescribedbyEq. (1a) withalltheuncertain-
tiesspecified inEq. (1b). AGEVMproblemistominimizethe
maximumgeneralizedeigenvalueofapairofmatricesthatdepend
affinelyonavariablesubjecttoalinear-matrix-inequality(LMI)
Contents listsavailableat ScienceDirect
journalhomepage: www.elsevier.com/locate/isatrans
ISATransactions
0019-0578/$-seefrontmatter 2013ISA.PublishedbyElsevierLtd.Allrightsreserved.
http://dx.doi.org/10.1016/j.isatra.2013.09.009
n Corresponding author.Tel.: þ852 34427633;fax: þ852 27669716.
E-mail address: gongsheng.huang@cityu.edu.hk (G.Huang).
ISA Transactions53(2014)141–149

constraint [15]. Sinceefficientinteriorpointalgorithmsarenow
availableincomputersoftware(suchasMATLAB)tosolveGEVM
problems [16], therobustcontrollawcanbeeasilycalculated.In
ordertousetheGEVMtechnique,thecontinuousmodel(1)is
sampledtoadiscretemodelandtheuncertaintysetsspecified in
Eq. (1b) aretransformedintoanuncertaintypolyhedron [17,18],
whichcanbeeasilyreformulatedintheformofaLMIconstraint,
suchasin [19,20]. Afterthistransformation,itwillbeshownthat
thecalculationofastandardrobustfeedbackcontrollawcanbe
formulatedasastandardGEVMproblem.
The restofthispaperisorganizedasfollows.Thetransforma-
tion oftheuncertaintysetsspecified inEq. (1b) to anuncertainty
polyhedroniselaboratedin Section 2. Section 3 details the
synthesis ofthecontrollawusingtheGEVMtechnique.The
accuracyofthetransformationisillustratedin Section 4 using
severalnumericalexamples. Section 5 showsthecomparisonof
the proposedrobustcontrolwiththeIMCmethodin [14], andan
application toanair-handlingunit.Concludingremarksaregiven
in Section 6.
2. Uncertaintypolyhedron
In ordertousetheGEVMtechnique,itisnecessarytosample
and transformthedescriptionofEqs. (1a) and (1b) to thatofEqs.
(2a) and (2b), whereEq. (2a) is adiscretestate-spacemodel;while
Eq. (2b) specifies anuncertaintypolyhedron ΩA,B for themodel
coefficient matrices(A,B). Theactualcoefficient matricesarea
linear convexcombinationofthematrices(Az, Bz) attheverticesof
the polyhedron,where N is thenumberofvertices.
xkþ1 ¼ AxkþBΔuk ð2aÞ
ðA; BÞAΩAB; ΩAB : ¼ ðA; BÞ ¼ Σ N
z ¼ 1
γzðAz; BzÞ; 0rγz
r1; Σ N
z ¼ 1
γz ¼ 1

ð2bÞ
The transformationiselaboratedasfollows.
2.1.Discretemodel
The continuousmodel(1a)is firstly sampledasadiscrete
model (3a)usingasamplinginterval h and thezero-order-hold
(ZOH) samplingmethod [21]
ykþ1 ¼ aykþKbdukdþKbdþ1ukd1 ð3aÞ
where k denotesthecurrenttimeinstant; d is thediscretetime-
delay;and
a ¼ eh=T ; bd ¼ 1eðh ~ τÞ=T ; bdþ1 ¼ eðh ~ τÞ=T eh=T ; d ¼ f lðτ=hÞ
ð3bÞ
where fl(x) isafunctiontoround x to thenearestintegertowards
negative,and ~ τ satisfies
τ ¼ d hþ ~ τ and 0r~ τoh ð3cÞ
RewritingEq. (3a) in theincrementalform,itbecomes
ykþ1 ¼ ð1þaÞykayk1þKbdΔukdþKbdþ1Δukd1 ð4Þ
when theoutput y is controlledtotrackaset-point yr , Eq. (4) is
further rewrittenas
M1 : ekþ1 ¼ ð1þaÞekaek1þKbdΔukdþKbdþ1Δukd1
where M1 denotesthesystemdiscretemodel;and ek ¼ ykyr is
the trackingerror.Samplingacontinuous-timesystemwithtime
delay makesthediscrete-timesystem finite-dimensional [21]. The
model parameters a, bd, bdþ1, and d depend onthechoiceofthe
sampling time h. Followingthesuggestionfrom [6,22], h is chosen
tosatisfy hrT1=10, where T1 is theminimumtimeconstant
specified inEq. (1b).
2.2. Uncertaindiscretemodel
Because theparameters K, T and τ in themodel(1a)are
uncertain, theparameters a, bd, bdþ1, and d of themodel M1 are
also uncertain.Sincetheprocessgain K variesintheuncertainty
set [K1, K2], itcanbewrittenasalinearconvexcombinationofthe
lowerandupperbounds K1 and K2:
K ¼ λK;1K1þλK;2K2 ð5aÞ
where λK;1; λK;2 are coefficients, satisfying
0rλK;1; λK;2r1; λK;1þλK;2 ¼ 1 ð5bÞ
when thetimeconstant T variesintheuncertaintyset ½T1; T2, the
parameter a lies alsoinasetdefined as aA½a1; a2 where
a1 ¼ eh=T1 ; a2 ¼ eh=T2 ð6Þ
similarly,theparameter a can bewrittenasalinearconvex
combination ofthelowerandupperbound a1 and a2
a ¼ λa;1a1þλa;2a2 ð7aÞ
where λa; 1; λa;2 arecoefficients, satisfying
0rλa;1; λa;2r1; λa;1þλa;2 ¼ 1 ð7bÞ
when thetimedelay τ variesintheuncertaintyset ½τ1; τ2, the
correspondingdiscretetime-delay d is anintegerandhasitsvalue
in theset ½d; dþ1;…; d1, where
d ¼ f lðτ1=hÞ; d ¼ clðτ2=hÞ ð8Þ
where fl(x) isafunctiondefined asthesameinEq. (3b); and cl(x) is
a functiontoround x to thenearestintegertowardspositive.
When thetimedelay τ varies,theparameters bd and bdþ1 will also
varycorrespondingly(seeEq. (3b)).
Based ontheobservationsthat(i)inthecaseofthetime
constant T being certain,onlytwoneighboring bds arenonzerofor
a given τ, andother bds arezero,asshownin Fig. 1[12]; and(ii)in
the caseofthetimeconstant T being uncertain,therearealsoonly
twoneighboring bds beingnonzeroforeach τ in itsuncertainty
set. Then,asshownin Section 2.3, whenthetimeconstant T has its
valueintheuncertaintyset ½T1; T2 and thetimedelay τ has its
valueintheuncertaintyset ½τ1; τ2, thenonzeroparameter bd and
bdþ1 in M1 canbeapproximatedby
bd ¼ λτ;dλa;1ð1a1Þþλτ;dλa;2ð1a2Þ
bdþ1 ¼ λτ;dþ1λa;1ð1a1Þþλτ;dþ1λa;2ð1a2Þ ð9aÞ
where a1 and a2 are defined inEq. (7a); and λτ,d and λτ,dþ1 are
defined by
λτ;d¼ ςd;1ðT1; T2; ~ τÞ; λτ; dþ1 ¼ 1þςd;1ðT1; T2; ~τÞ ð9bÞ
where ςd;1, whichisancoefficient usedinlinearizing bd and bdþ1
w.r.t.a, isdefined in Section 2.3; and ~τ satisfies Eq. (3c). Numerical
study showsthat λτd and λτ,dþ1 satisfy
0rλτ; d; λτ; dþ1r1; λτ; dþλτ; dþ1 ¼ 1 ð9cÞ
Fig. 1. Variations inthevaluesof bd when timedelaychanges.
G. Huangetal./ISATransactions53(2014)141–149 142

2.3. Twoapproximations
In ordertodeveloptheuncertaintydescriptionfor bds, thetime
delay uncertaintyset ½τ1; τ2 is separatedinto ðddÞ sections as
shown in Fig. 1. Withoutlossofgenerality,considerthesection
τA½d h; ðdþ1Þ hÞ. Accordingto Fig. 1, onlythecoefficients bd
and bdþ1 are nonzeroandtheyaredefined by
bd ¼ 1eðh ~ τÞ=T ; bdþ1 ¼ eðh ~ τÞ=T eh=T ð10Þ
with ~ τ satisfying
τ ¼ d hþ ~ τ and 0r~ τoh ð11Þ
First, considera fixedvalueof ~ τ (i.e. τ is fixedintheset[dh,
(dþ1)h]). AsshownbyHuangandWang [13], if T variesinthe
set ½T1; T2 and h is chosentobemuchsmallerthan T1, then bd and
bdþ1 can beapproximatedby
bd ¼ ςd;1ðT1; T2Þaþςd;2ðT1; T2Þ;
bdþ1 ¼ ςdþ1;1ðT1; T2Þaþςdþ1;2ðT1; T2Þ ð12Þ
where ςðT1; T2Þ denotesthevalueof ς is relatedtotheupperand
lowerboundofthetimeconstant,i.e. T1 and T2, butnottothe
actual valueof T; and
ςdþ1;1ςd;1ðT1; T2Þ ¼ 1ςd;1ςd;1ðT1; T2Þ; ςdþ1;2ςd;1ðT1; T2Þ
¼ 1ςd;2ςd;1ðT1; T2Þ ð13Þ
The approximationdescribedbyEq. (12) is titledas Approx-
imation I.
Second, considerthevariationof ~ τ (i.e. τ variesintheset
½d h; ðdþ1Þ hÞ). Whenthetimedelay τ variesfrom τ¼dh to
τ¼(dþ1)h, theparameter ~ τ will varyfrom ~ τ ¼0 to ~ τ ¼ h. Itis
notedthatthevaluesof ςd;1; ςd;2and ςdþ1;1; ςdþ1;2 will change
when ~ τ changes.Write ςd;1; ςd;2 and ςdþ1;1; ςdþ1;2 as functionsof ~τ
and substituting a by Eq. (6b), andthen bd has theformof
bd ¼ ςd;1ðT1; T2; ~ τÞλa;1a1þςd;1ðT1; T2; ~ τÞλa;2a2þςd;2ðT1; T2; ~τÞ ð14Þ
Because λa;1þλa;2 ¼ 1, theexpressionof bd is furtherwrittenas
bd ¼ ½ςd;1ðT1; T2; ~ τÞλa;1ð1a1Þþ½ςd; 1ðT1; T2; ~ τÞλa;2ð1a2Þ
þςd;1ðT1; T2; ~τÞþςd;2ðT1; T2; ~ τÞ ð15Þ
When thesamplinginterval h is smallenough(suchas
hrT1=10), foreach ~ τA½0; hÞ thereexists
ςd; 1ðT1; T2; ~ τÞþςd; 2ðT1; T2; ~τÞ 0 ð16Þ
This approximationistitledas ApproximationII. Withthis
approximationanddefinition of λτ,d in Eq. (9b), bd can be
formulated astheformdefined inEq. (9a). Numericalstudyshows
that ςd;1ðT1; T2; ~τÞ has avalueinside[1,0].Then λτ,d has avalue
inside [0,1].
Similarly,rewrite bdþ1 as
bdþ1¼ ςdþ1;1ðT1; T2; ~ τÞλa;1ð1a1Þςdþ1;1ðT1; T2; ~τÞλa;2ð1a2Þ
þςdþ1;1ðT1; T2; τ~ Þþςdþ1;2ðT1; T2; τ~ Þ ð17Þ
AccordingtoEq. (13) and the ApproximationII, ςdþ1;1
ðT1; T2; ~τÞþςdþ1;2ðT1; T2; ~τÞ ¼ 0 foreach ~τA½0; hÞ. Then,withthis
approximationanddefinition of λτ,dþ1 in Eq. (9b), bdþ1 becomes
the formdefined inEq. (9a). Since ςd;1ðT1; T2; ~τÞ has avalueinside
[1,0], λτ,dþ1 has avalueinside[0,1]accordingtoitsdefinition.
2.4. Linearcombinationofsub-models
Considering theprocessgainuncertaintyandaccordingtoEqs.
(5a), (7a) and (9a), thediscrete-timemodel M1 canbewrittenasa
linear combinationofthesub-models Mi,j,l
M2 ¼ Σ 2
i ¼ 1
Σ 2
j ¼ 1
Σ
dþ1
l ¼ d
λK;iλa;jλτ;lMi;j;l ð18aÞ
where λK,i, λK,j, and λτ,l aredefined inEqs. (5b), (6b) and (9b), and
Mi;j;l is defined as
Mi; j; l : ekþ1 ¼ ð1þajÞekajek1þKið1ajÞΔukl ð18bÞ
The developmentof M2 isillustratedin Appendix A. Notethat
the discretetimedelay d can beanyvalueintheset ½d; dþ 1;…; d1. Ateachtimedelay,onlytwo bds arenonzerowhile
othersarezero.Hence,onlytwocorresponding λτs arenonzero
and other λτs arezeroforeachtimedelay.Therefore,Eq. (18a) is
written as
M2 ¼ Σ 2
i ¼ 1
Σ 2
j ¼ 1
Σ d
l ¼ d
λK;iλa;jλτ;lMi;j;l ð18cÞ
where thedefinition of Mi;j;l in Eq. (18b) is extendedfor l ¼ d; …; d. AccordingtoEqs. (5b), (7b) and (9c) as wellasthefact
that other λτ s arezero,thecoefficients inEq. (18c) satisfy
Σ 2
i ¼ 1
Σ 2
j ¼ 1
Σ d
l ¼ d
λK;iλa;jλτ;l ¼ 1 ð19Þ
The expressionof M2 inEq. (18c) indicatesthat M1 canbe
written asalinearconvexcombinationofthesub-models Mi,j,l
when theparameters(K, T, τ) areinsidetheuncertaintysets.
It alsoindicatesthattheuncertaintiesassociatedwiththepara-
meters(K, T, τ) aretransformedtotheuncertaintiesassociated
with theparameters(λK,i, λK,j, λτ,l), i.e. K, T, τ varyingintheir
uncertainties setsaretransferredto λK,i, λK,j, λτ,l varyingbetween
0 and1.
2.5. State-spacemodel
Defining astatevectoras
x′k ¼ ðek; ek1;Δukd;…;Δuk1Þ ð20Þ
Then, x′kþ1 has theformof
x′kþ1 ¼ ðekþ1; ek;Δukdþ1;…;ΔukÞ
Since ekþ1 ¼ ð1þaÞekaek1þKbdΔukdþKbdþ1Δukd1 and
d may beanyvalueinsidetheset ½d;⋯d1, ekþ1 can be
rewrittenas
ekþ1 ¼ ½1þa; a; 01;dðdþ1Þ
; Kbdþ1; Kbd; 01;d1xk
where 0 is azeromatrix(orvector)anditssubscriptsindicatethe
dimension. Similarly,theotheritemsof x′
kþ1 has theformof
ek ¼ ½1; 01;dþ1xk
½Δukdþ1;⋯;Δukþ1′ ¼
0d1;3 Id1;d1
01;dþ2
#
xk
It shouldbenotedthat Δuk is thecontrolinputatthetime
instant k. Therefore,themodel M1 isformulatedasastate-space
model intheformofEq. (2a) as follows.When d40, themodel
M1 canbeformulatedasadiscretestate-spacemodelintheform
of Eq. (2a) with
A¼
1þa; a; 01;dðdþ1Þ
; Kbdþ1; Kbd; 01;d1
1 01;dþ1
0d1;3 Id1;d1
01;dþ2
0
BBBBB@
1
CCCCCA
; B ¼ 0dþ1;11

when d¼0, thecorresponding A and B becomes
A ¼
1þa; a; 01;d1; Kbdþ1
1 01;dþ1
0d1;3 Id1;d1
01;dþ2
0
BBBBB@
1
CCCCCA
; B ¼
Kbd
0d;1
1
0
B@
1
CA

where I is theidentitymatrix.Thecoefficient matrices A and B are
uncertain becauseoftheuncertaintiesassociatedwiththepara-
meters K, a, bd, bdþ1, and d.
The sub-models Mi,j,l is alsoformulatedasastate-spacemodelin
theformofEq. (2a), andthecoefficientmatrices Az(i,j,l) and B z(i,j,l) are
givenasbelow.If l40,
Azði;j;lÞ ¼
1þaj aj 01;dl Kið1ajÞ 01;l1
1 01;dþ1
0d1;3 Id1;d1
01;dþ2
0
BBBBB@
1
CCCCCA
; Bzði;j;lÞ ¼
0dþ1;1
1
#
ð21aÞ
If l¼0 (whichoccursinthecaseof d ¼ 0),
Azði;j;0Þ ¼
1þaj aj 01;d
1 01;dþ1
0d1;3 Id1;d1
01;dþ2
0
BBBBB@
1
CCCCCA
; Bzði;j;0Þ ¼
Kið1ajÞ
0d;1
1
2
64
3
75
ð21bÞ
where I is theidentitymatrix; 0 is azeromatrix(orvector);and z
is
z ¼ zði; j; lÞ
¼ ði1Þ 2ðddþ1Þþðj1Þðddþ1Þþldþ1 ð22Þ
Notethat M1 isapproximatedby M2 (seeEq. (18a)) and M2 isa
linear convexcombinationofthesub-models Mi,j,l. Therefore,the
continuous model(1a)associatedwiththesetuncertainties(1b)is
sampled intoandrewrittenasthediscretestate-spacemodel(2a)
associatedwiththeuncertaintypolyhedron(2b).
2.6. Transformationalgorithm
The algorithmforcomputingtherequiredpolyhedronisgiven
below.
Algorithm 1.
Step1:Chooseasamplingratio h that satisfies hrT1=10;
Step2:Calculate a1; a2 by Eq. (6);
Step3:Calculate d and d by Eq. (8);
Step4:For i¼1, 2; j¼1, 2; l ¼ d; …; d, calculate z by Eq. (22),
and thenspecify Azði;j;lÞ
and Bzði;j;lÞ
as Eq. (21a) if l40 orEq. (21b)
if l¼0.
It shouldbenotedthatthenumberoftheverticesofthe
uncertainty polyhedronis2 2 ðddþ1Þ. Thedimensionsof
the coefficients Az and Bz depend onthemaximumdelay(τ) and
the samplinginterval h. Itshouldalsobenotedthattheideaof
transforminganuncertaintysetintoanuncertaintypolyhedron
wasoriginatedfromHuangandWang [12], inwhichonlythe
processgainandtimedelayuncertaintieswereconsidered.
When thetimeconstantuncertaintyistakenintoaccount,the
transformationbecomesmuchmorecomplex.Twoapproxima-
tions areusedinderivingEq. (9a), i.e.theapproximations
representedbyEq. (12) and Eq. (16). Thetwoapproximations
arethekeydevelopmentof Algorithm1. Theaccuracyofthetwo
approximationswillbeinvestigatedusingnumericalexamplesin
Section 4. Sincethemodel M2 willbeusedforcontroldesign
insteadofthemodel M1,theywillbecomparedinboththetime
domain andthefrequencydomain,whichwillalsobeshownin
Section 4.
3. Robustcontrollaw
A robustcontrollawcanbecalculatedusingtheGEVM
techniqueforthemodel(2a)associatedwiththeuncertainty
polyhedron(2b).Toseethis,assumethefeedbackcontrolhas
the formof
Δuk ¼ Fxk ð23Þ
In ordertoachieverobuststabilityforthemodel(2a),allthe
eigenvaluesof(AþBF) shouldbeinsidetheunitcircle.Thefeed-
back law F is thenoptimizedbyminimizingthespectralradiusof
(AþBF), i.e. ρðAþBFÞ, subjecttotheuncertaintiesspecified by
Eq. (2b). Therefore,theoptimizationofthefeedbacklawis
formulatedas
F ¼ argmin
F;P
γ ð24Þ
Subject to(i). P1γ
ðAþBFÞ′P1γ
ðAþBFÞ40; P ¼ Pt40; and(ii).
ðA; BÞAΩAB
Set Q ¼ P 1, andthentheconstraint(i)becomes
Q Q1γ
ðAþBFÞ′Q 11γ
ðAþBFÞQ40;Q ¼ Q′40 ð25Þ
AccordingtoSchurComplements,Eq. (25) is rewrittenas
0 QA′Y′B′
AQ BY 0
!
oγ
Q 0
0 Q
!
ð26aÞ
where
Y ¼ FQ ð26bÞ
Eq. (26a) is alinear-fractionalconstraintinaGEVMproblem
[15]. Because(A,B) isalinearconvexcombinationof ðAz; BzÞ, the
verticesofthepolytope ΩAB, theconstraint(26a)isholdfor
8ðA; BÞAΩAB if andonlyiftheverticesofthepolytope ΩAB satisfy
the constraint(26a),i.e.
0 QA′zY′B′z
AzQ BzY 0
!
oγ
Q 0
0 Q
!
; z ¼ 1;…;N ð27Þ
Therefore,theoptimizationofthefeedbacklawbecomes
min
Q;Y
γ subject totheconstraint(27)with
Q ¼ Q′40 ð28Þ
The optimizationformulatedbyEq. (28) is astandardGEVM
problemunderLMIconstraints [15]. Itcanbesolvedusingthe
GEVP functionintherobustcontroltoolboxinMATLAB [16].When
the optimizationisfeasible,theoptimalfeedbacklawis
F ¼ Q 1Y ð29Þ
Hence, thecontrolsignalatcurrenttimeinstantis uk ¼ uk1þFxk.
If theoptimizedspectralradius γ issmallerthan1,thenthefeedback
controllaw F can stabilizethediscretestatemodel(2a)withthe
uncertainty polyhedron(2b).Whenthesamplingintervalisproperly
selected,therobuststabilityoftheclosed-loopsystemcanbewell
maintained bythecontrollaw F.
4. Numericalstudies
Numericalstudiesareusedtoshowthatwiththeappropriate
selection ofthesamplinginterval h, i.e. hrT1=10, theerrors
introducedbytheapproximationsinEqs. (12) and (16) as well
as thedifferencesbetweenthediscrete-timemodels M1 and M2
areinsignificant.

4.1.Approximationerrors
ApproximationerrorsintroducedbytheapproximationinEq. (12)
were firstlystudied.Aseriesnumericalstudiesshowedthatfora
fixedtimedelay τ andavariabletimeconstant,ifthesampling
intervalischosenas hrT1=10,theapproximationerrorsintroduced
by Eq. (12) areinsignificant.Toshowthis,consideranexample:
TA½20; 500 s, τ¼5 s,whereaverylargeuncertaintysetforthetime
constantischosen.Thesamplingintervalwasselectedas h¼2 s,and
then d¼2 and ~τ ¼ 1 s.Inthisexample,thevariationsof b2 and b3
calculatedbyEq. (3b) areshownin Fig. 2(a)and(b)separately.Using
theapproximationinEq. (12), ς2;1; ς2;2 for b2 and ς3;1; ς3;2 for b3 are
ς2;1¼ 0:5096; ς2;2 ¼ 0:5095; ς3;1¼ 0:4904; ς3;2 ¼ 0:4905 ð30Þ
As shownin Fig. 2c andd,theapproximationerrorsforboth b2
and b3 are smallerthan3.55104 (the maximumerror).
Similar conclusionwasobtainedfortheapproximationerrors
introduced byEq. (16), inwhichthedelayislimitedas τA½d h; ðdþ1Þ hÞ. Toseethis,considerthetime-constantuncertainty
TA½20; 500 s andthesamplinginterval h¼2 s.Assume d¼2, and
thenthetimedelayvariesintheset τA½4; 6Þ s. Thevaluesof ς2;1, ς2;2
and ðς2;1þς2;2Þ areplottedin Fig.3. Itcanbeseenthatthevaluesof
ðς2;1þς2;2Þ areveryclosedtozero.Therefore,theapproximation
errorsintroducedbyEq. (16) arealsoinsignificant.
4.2. ErrorsbetweenthemodelM1andM2
It hasbeenshownin Section 2 that thediscrete-timemodel M1
can beapproximatedby M2 where M2 isalinearconvex
combination ofthesub-models Mi,j,l. Here,thedifferencebetween
M1 and M2 andtheinfluence ofthedifferencewhen M2 isusedto
replace M1 forcontroldesignwerestudied.Considerthefollowing
uncertainties
K1 ¼ 0:4; K2 ¼ 1:0; T1 ¼ 20 s; T2 ¼ 500 s; τ1 ¼ 0:1 s; τ2 ¼ 8 s ð31Þ
0.90.920.940.960.98100.010.020.030.040.05The value of a0.90.920.940.960.98100.010.020.030.040.05The value of a0.90.920.940.960.98101234x 10-4The value of a0.90.920.940.960.98101234x 10-4The value of a
Fig. 2. Errors introducedby ApproximationI when TA[20,500]s, h¼2 sand τ¼5 s. 44.24.44.64.855.25.45.65.86-1.5-1-0.500.511.5time delay (s) 44.24.44.64.855.25.45.65.86-8-6-4-20x 10-5time delay (s)
Fig. 3. The valuesof ς2;1, ς2;2 and ðς2;1þς2;2Þ when τA½2h; 3hÞ.

When thenominalmodelisrandomlychosenas
K ¼ 0:7; T ¼ 100 s; τ ¼ 4:5 s ð32Þ
Using asamplinginterval h¼2 s,theminimumandmaximum
discretetime-delayare d ¼ 0 and d ¼ 4. Bythediscretization(3a)
and (3b),thecorrespondingdiscretemodelor M1 is
ekþ1 ¼ 1:98ek0:98ek1þ1:042 102Δuk2þ0:344 102Δuk3
ð33Þ
Thecoefficients λk;1; λa;1; λτ;d givenbyEqs. (5b), (7b) and (9b) are
λk;1 ¼ 0:5; λa;1 ¼ 0:1734; λτ;2 ¼ 0:757; λτ;3 ¼ 0:2434; λτ;0 ¼ λτ;1 ¼ λτ;4 ¼ 0
the modelcomputedbythelinearconvexcombination,or M2,
is
ekþ1 ¼ 1:98ek0:98ek1þ1:049 102Δuk2þ0:337 102Δuk3
ð34Þ
Comparedwiththemodel(33), b2 of themodel(34)is0.67%
largerwhile b3 is 2.0%smaller.Therefore,thedifferencesbetween
M1 and M2 areinsignificant.
Tostudytheinfluence ofthedifferencesbetween M1 and M2
on controlperformance,Bodediagramandstepresponseareused
as thetoolforanalysis. Fig. 4 illustratestheBodediagramofthe
model (33),denotedas M1,andthemodel(34),denotedas M2.
The frequencyin Fig. 4 starts from1104 rad/sandstopsat
1.57rad/s,wherethestopfrequencyischosentosatisfythe
Nyquist–Shannon samplingtheorem [21]. Theresultsshowthat
the maximumerrorinmagnitudeislessthan1.9%andthe
maximum errorinphaseislessthan0.13%. Fig. 5 compares the
step responsesofthemodel(33)and(34).Themaximumabsolute
error 6.6105 (or relatively0.63%)wasobtainedat t¼508 s.
Once again,theresultsshowthatthemodel(33)candescribethe
dynamics representedbythemodel(34)withacceptableaccuracy
when stepchangesoccur.
4.3. Robuststability
RobuststabilityofthecontrollawdesignedusingtheGEVM
techniquewasalsostudiedusingannumericalexample.Still
considering theexampledetailedinEq. (31), thecontrollawby
GEMV was
F ¼ ½4:3600 4:1008 0:1179 0:1936 0:2356 0:2506 ð35Þ with whichthemaximumofthespectralradiusof(AþBF) is
0.9965. Therefore,thefeedbacklaw Δuk ¼ Fxk can robustlystabi-
lize thesystemwhentheuncertaintiesarespecified byEq. (31).
Fig. 6 illustratesthecontrolinputandthetrackingerrorwhenthe
threemodelparametersvarywiththecontrolinput(butlimited
intotheuncertaintysetsspecified Eq. (31).
KðukÞ ¼ K1þ0:2uk; TðukÞ ¼ T215uk; τðukÞ ¼ τ1þ2uk ð36Þ
Fig. 6 showsthatrobuststabilitywasachievedinspiteofthe
model parametervariationsshownin Fig. 7.
5. Casestudies
5.1.Comparisonwithinternalmodelcontrol(IMC)
The proposedcontroldesignwascomparedwiththerobust
IMC developedbyLaughlinetal. [14]. Theuncertaintiesspecified
by Eq. (31) wereusedastheexample.Theclosed-loopresponses
wereplottedin Fig. 8, where ‘S1’ to ‘S8’ wereprocessmodelsgiven
in Table1 generatedbytakingcombinationsoflower/upper
bounds ofthethreeparameters.TherobustIMCwastuneduntil
oscillations inthetransientof ‘S6’ wereacceptable.Itcanbeseen
that bothmethodscanrobustlystabilizetheseplants.Asshownin
-50-40-30-20-100 Magnitude (dB) 10-410-310-210-1100-540-360-1800 Phase (deg) Bode DiagramFrequency (rad/sec)
Fig. 4. Comparison oftheBodediagramsofthemodelof(33),denotedas M1,and
the modelbyEq. (34), denotedas M2.
020040060080010001200140016001800200000.20.40.60.8System outputTime(second) y 0200400600800100012001400160018002000-1-0.500.51x 10-4Approximation errorTime(second) e
Fig. 5. Comparison ofthestepresponseofthemodel(33),denotedas M1,andthe
model (34),denotedas M2.
05001000150020002500300035004000-20246 control input time (second) 05001000150020002500300035004000-1-0.500.5 tracking error time (second)
Fig. 6. The controlinputandthetrackingerrorachievedbythecontrollawinEq.
(35).

Fig. 8, thetransientsoftheproposedmethodweregenerallyfaster
than thoseoftherobustIMC.ThiswasbecausetherobustIMC
suppressedthetransientresponses,forexample,intheplantsS1
and S2.Theaverageoftheintegraloftheabsolutetrackingerror
(IAE) andthesettlingtime(ST)werelistedandcomparedin
Table2. Inmostcases,theproposedmethodachievedasmaller
trackingerror.Oneexceptionwastheprocess ‘S8’, wherethe
trackingerroroftheproposedmethodwasslightlylarger.Forthe
settling time, Table2 showsthattheproposedmethodachieved
a shortersettlingtimethantherobustIMCinmostcasesandone
exceptionwastheprocess ‘S7’. Therefore,theproposedmethod
had abetterabilityoferrortrackinginthisexample.
5.2. Temperaturecontrolofair-handlingunit
An applicationoftheproposeddesignapproachtoathermal
processofheating,ventilationandairconditioning(HVAC)sys-
temswasstudied.ItisknownthatthermalprocessesinHVAC
systemscanbeadequatelydescribedusinganuncertainFOPTD
model [8–11]. Hereanair-handlingunit(AHU)wasselected
because AHUisanimportantcomponentinair-conditioning
systems [23,24]. ThestructureofAHUsisillustratedin Fig. 9,
where theinletairisconditionedafterpassingthroughtheAHU
and becomesthesupplyair,whichwillbesenttooccupiedspace
for thermalcomfort.Thetemperatureofthesupplyairisadjusted
by thechilledwater flow rateinsidethecoolingcoil,whichis
controlledbyathree-portvalve.Thechilledwaterisprovidedby
a chiller,andthechilledwatersupplytemperatureisusuallyunder
feedbackcontrolandmaintainedaround6 1C.
The manipulatedvariable(MV)oftheprocessistheopenofthe
three-portvalve,denotedby u (which alsoreflects thechilled
water flowrate).Thecontrolledvariable(CV)isthesupplyair
temperature Ta,sup. Thesetpointofthesupplyairtemperatureis
determinedbythethermalconditionoftheoccupiedspace.The
dynamics ofthisprocessaresignificantly affectedbytheair flow
rate m_ a;in and thecooledwater flow rate.Itwillalsobeaffectedby
the inletairtemperature Ta,in, aswellasthechilledwatersupply
temperature.Theuncertaintysetsspecified byEq. (1b) wereused
todescribethedynamicsuncertainties.
A simulatedAHUprocesswasconstructedinSIMULINKusing
SIMBAD toolbox [25], developedbyCSTB,theFrenchcentrefor
building sciences,totestbuildingenergymanagementsystems.
The dynamicsvariationsofthisprocesswereidentified usingstep
responsesoftheprocessatseveralworkingconditionsasshownin
Table3, wheretheinletairtemperatureandcooledwater
temperaturewere fixedat25.5 1C and6.0 1C respectively.Based
on theidentification results,slightlylargeruncertaintysetsforthe
threeparametersweresetas
K1¼ 1 1C; K2¼ 40 1C; T1 ¼ 60 s; T2 ¼ 70 s; τ1 ¼ 15 s; τ2 ¼ 40 s
ð37Þ
Set thesamplingintervalas h¼6.0 s,thecontrollawbythe
optimizationdefined inEq. (28) yielded
F ¼ ½0:174; 0:145; 0:161; 0:221;
0:246; 0:261; 0:269; 0:277; 0:238 ð38Þ
With thiscontrollaw, Fig. 10 demonstratesthecontrolsignals u
and thesupplyairtemperature,whichwascontrolledtotrackthe
set points20 1C (from t¼500sto t¼2400s),12 1C (from t¼2400s
to t¼4800s)and16 1C (from t¼4800sto t¼7200s).Thedis-
turbances shownin Fig. 11 includes thesupplyair flow rate
(a) whichchangesbetween1.2kg/s(thedesignvalue)and
0.48 kg/s(the40%ofthedesignvalue),theinletairtemperature
(b) whichvariesbetween26.5 1C and24.5 1C, andthesupply
watertemperature(c)whichvariesbetween5 1C and7 1C.
It canbeseenthattherobustcontrolapproachsuccessfully
manipulated thesupplyairtemperaturetoitssetpointswhether
the setpointwashigh(lowcoolingloadcondition)orlow(high
cooling loadcondition).Thetransientresponseateachchangeof
0500100015002000250030003500400000.51 variation of K time (second) 05001000150020002500300035004000400450500 variation of T time (second) 050010001500200025003000350040000510 variation of τ time (second)
Fig. 7. The variationsoftheprocessgain(K), timeconstant(T) andtimedelay τ
specified byEq. (36).
05001000150020002500300000.10.20.30.4 Output time(second) 05001000150020002500300000.10.20.30.4 Output time(second) 05001000150020002500300000.10.20.30.4 Output time(second) 05001000150020002500300000.10.20.30.4 Output time(second)
Fig. 8. Comparison oftheclosed-loopresponsesoftheproposeddesign(a)with
those oftherobustIMC(b)whenthereferenceisstepfrom0to0.3.
Table1
Definition oftheprocessmodelsS1toS8.
S1 S2S3S4S5S6S7S8
K 0.4 1.00.41.00.41.00.41.0
T 20 s20s260s260s20s20s260s260s
τ 0.1s0.1s0.1s0.1s8.0s8.0s8.0s8.0s

set pointwassimilarandtheovershoots(undershoots)were
small. Theaveragetrackingerrorwas0.34 1C underthedistur-
bances, andthedisturbancesareshownin Fig. 11. Theproposed
method wascomparedwithawidelyusedanti-windupPIcontrol.
As shownby Fig. 12, ananti-windupPIcontrollermayachievea
good performanceatoneconditionbutabadperformanceatthe
Table2
The averageoftheintegraloftheabsolutetrackingerror(IAE)andthesettlingtimes(ST)(5%).
S1 S2S3S4S5S6S7S8
IAE Theproposedmethod0.00670.00470.03500.02060.00670.00470.03920.0241
RobustIMC0.03240.01300.04230.02190.03240.01300.04280.0223
Comparison (%)21%36%83%94%21%36%92% 108%
ST Theproposedmethod179.0126.51413.8874.6169.6121.91728.7896.1
RobustIMC1383.7718.41590.51020.41373.7710.51631.91023.2
Comparison (%)13%18%89%86%12%17% 106% 88%
Fig. 9. The structureofatypicalair-handlingunit.
Table3
The valuesof(K, T, τ) identified bystepresponsesatseveraloperatingconditions.
StepinputFlowrate
(percentage ofdesignvalue)(%)
(K, T, τ) (1C, s,s)
u : 0:05-0:15 100(17.8,61.4,25.5)
40 (35.7,66.2,39.1)
u : 0:4-0:5 100(13.8,60.8,21.9)
40 (11.1,64.7,30.8)
u : 0:9-1:0 100(2.1,60.8,20.6)
40 (1.2,64.6,29.3) 10002000300040005000600070001015202530time (second) supply air temperature (oC) 100020003000400050006000700000.20.40.60.81time (second) control input
Fig. 10. The variationsofthesupplyairtemperature(top)andthecorresponding
control signals(bottom). 10002000300040005000600070000.40.60.811.21.4time (second) Supply air flow rate (kg/s) 100020003000400050006000700024.52525.52626.5time (second) inlet air temperature (oC) 100020003000400050006000700055.566.57time (second) Chilled water supply temperature (oC)
Fig. 11. Disturbances oftheAHUprocess:supplyair flow rate(a),inletair
temperature(b)andsupplywatertemperature(c). 10002000300040005000600070001015202530outputset point100020003000400050006000700000.20.40.60.81time(second) Supply air temperature Control input (oC)
Fig. 12. The supplyairtemperature(top)andthecorrespondingcontrolsignals
(bottom)underananti-windupPIcontrol.

other conditions.Therefore,theresultsin Fig. 12 illustratedthat
the controllawinEq. (38) achieved agoodrobustnessforthe
closed-loop process.
6. Conclusions
This paperdevelopedamethodofapplyinggeneralizedeigenvalue
minimizationtoprocessesthatcanbedescribedbya first-orderplus
time delaymodelwithuncertaingain,timeconstantanddelay.An
algorithmhasbeendevelopedtotransformtheuncertaintysetsfor
the gain,timeconstantanddelaytouncertaintypolyhedron,whichis
a typeofuncertaintydescription forstate-spacemodel.Withthe
uncertaintytransformation,afeedbackcontrollawcanbedesigned
using thegeneralizedeigenvalueminimizationtechnique.Numerical
exampleshaveshownthatthetransformationcanbeachieved
accurately.Casestudieshaveshownthatthefeedbackcontrollaw
optimizedbythegeneralizedeigenvalueminimizationtechniqueis
abletoachieveabetterperformance comparedwiththatoftherobust
internalmodelcontroldesignandabetterrobustnesscomparedwith
a conventionalPIcontrol.Since first-orderplustimedelaymodelscan
be easilyidentified andarewidelyusedinpractice,theproposed
controldesignapproachoffersadvantagesforpracticalapplications.
Acknowledgements
The workdescribedinthispaperwasfullysupportedbyagrant
from theResearchGrantsCouncil(RGC)oftheHongKongSpecial
AdministrativeRegion,China(ProjectNo.CityU124012)andalso
by aKeyScienceandTechnologyProjectofGuangdongProvince,
China (ProjectNumber:2012A010800004).
Appendix
Appendix A:ThedevelopmentofM2
Substituting K, a, bd and bdþ1 in M1 by
K ¼ λK;1K1þλK;2K2
a ¼ λa;1a1þλa;2a2
bd ¼ λτ;dλa;1ð1a1Þþλτ;dλa;2ð1a2Þ
bdþ1 ¼ λτ;dþ1λa;1ð1a1Þþλτ;dþ1λa;2ð1a2Þ
M1 becomes
ekþ1 ¼ ð1þλa;1a1þλa;2a2Þekðλa;1a1þλa;2a2Þek1
þðλK;1K1þλK;2K2Þ½λτ;dλa;1ð1a1Þþλτ;dλa;2ð1a2ÞΔukd
þðλK;1K1þλK;2K2Þ½λτ;dþ1λa;1ð1a1Þþλτ;dþ1λa;2ð1a2ÞΔukd1
ðA1Þ
Because λK;1þλK;2 ¼ 1 and λa;1þλa;2 ¼ 1, Eq. (A1) is rewrittenas
ekþ1¼ ðλK;1þλK;2Þ½ðλa;1þλa;2þλa;1a1þλa;2a2Þekðλa;1a1þλa;2a2Þek1
þðλK;1K1þλK;2K2Þ½λτ;dλa;1ð1a1Þþλτ;dλa;2ð1a2ÞΔukd
þðλK;1K1þλK;2K2Þ½λτ;dþ1λa;1ð1a1Þþλτ;dþ1λa;2ð1a2ÞΔukd1
ðA2Þ
Eq. (A2) is reformulatedas
ekþ1 ¼ Σ 2
i ¼ 1
Σ 2
j ¼ 1
λK;iλa;j½ð1þajÞekajek1
þλτ;dKið1ajÞΔukdþλτ;dþ1Kið1ajÞΔukd1 ðA3Þ
Because λτ;d þλτ;d þ1 ¼ 1, Eq. (A3) has thefollowingform
ekþ1 ¼ Σ 2
i ¼ 1
Σ 2
j ¼ 1
λK;iλa;jfλτ;d½ð1þajÞekajek1þKið1ajÞΔukd
þλτ;dþ1½ð1þajÞekajek1þKið1ajÞΔukd1g ðA4Þ
With thedefinition of Mi,j,d and Mi,j,dþ1 in Eqs. ((18b) and A4)
can thenberewrittenas M2 inEq. (18a).
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