MixedH2/H∞ obus con olapp oach o
NCSwi h unce ain iesand da ad opou s⋆
I.Ju ado,M.G.O ega and F.R.Rubio
Depa amen o deIngenie ´ıa deSis emasyAu om´a ica
EscuelaSupe io deIngenie os,Uni e sidad deSe illa
Camino delosDescub imien os s/n-41092 Se illa(Spain) (e-mail:
{iju ado,o ega, ubio}@ca uja.us.es).
Abs ac :In hispape ,aRobus Ne wo kedCon olSys em(RNCS)subjec oda alosses
cons ain sisconside ed.Theseda alossesa emodelledasanindependen sequence o i.i.d.
Be noulli andom a iable.This andom a iableis eplaced byanaddi i enoiseplusa gain,
whichisequal o hesuccess ul ansmission p obabili yin he eedbackloop.Also,s uc u al
unce ain iesin hemodelo heplan a e conside ed.
Tocopewi h hisp oblem,amixedH2/H∞con ol echniqueisp oposedin hiswo k.In his
way, heH2app oachisused os abilize heNCS akingin o accoun hep obabili yo da a
d opou s,while heH∞app oachisincha geo making he closed-loopsys em obus enough
agains s uc u alunce ain ieso henominalmodel.
Keywo ds:Con olunde communica ioncons ain s,con oland es ima ionwi h da aloss
and ne wo kedembeddedcon olsys ems.
1.INTRODUCTION
Nowadays,con olsys emswhe einacommunica ion ne -
wo kexi sa ege ingimpo ance mo eand mo e.Usually,
he communica ion ne wo kconnec s some elemen so he
con olsys em.This kind o sys emsand hei cha ac e -
is icsa ewidelydesc ibedinHespanhae al. (2007)and
Zhange al. (2001).
The ea ealo o s udiesin heli e a u eabou hemain
p oblemsassocia edwi hNe wo kedCon olSys ems.
Oneo hesep oblemsis he ela edwi h he a iable
delays in heda a ansmission(see, o example,Guand
Chen(2003)).Oneway o app oach hisissueis o eso
Lyapuno -K aso skii unc ionals(see, o example,Millan
e al. (2010)and Yue e al. (2005)).
Ano he impo an opic obes udiedinNCSsis he
ne wo k-induced da aloss.This kind o p oblemoccu s
when he communica ionchannel isno able o ansmi
heda a and i ge los .Thismayoccu s, o ins ance,
due ocolissionso lowSNR(signal onoise a io).The e
a edi e en sways odealwi h hesekind o NCSs.One
wayis heuseo p edic i e con ol, whichmakespossible
ocalcula e u u emodel-based da a and ouse hem o
compu e he con olac ions.Some exampleso ne wo k
con olbasedonMPC o linea and non-linea sys ems
can be ound inZhange al. (2006),Millane al. (2008)
and inMu˜nozand Ch is o ides(2008).
Adi e en way odealwi hNCSsubjec oda ad opou s
cons ain sconsis sinmodeling hed opou sbymeans
o aswi chedsys em, i.e., aMa ko jumplinea sys em
⋆Theau ho swouldlike o acknowledgeMCyT(G an DPI2010-
19154.),and heEu opeanCommission(EC) (FeedNe BackP ojec ,
g an ag eemen 223866), o unding hiswo k.
(MJLS).Rela edwi h hisapp oach,Ling and Lemmon
(2004)p esen sa esul whichshows ha , o aspeci ic
NCSa chi ec u esubjec oda ad opou scons ain s, he
esul ingMJLSisequi alen o a linea sys emwi han
ex e nalnoisesou ce.Thisnoisehas hepa icula i yo
ha ing a a iance ha isp opo ional o he a iance o
ano he signalwi hin heini ialcon ol loop.This esul is
usedin Sil ae al. (2009) oshow ha he e exi sasecond
o de momen sequi alence be ween he conside edNCS
and anauxilia ycon olsys em.In hisauxilia ycon ol
sys em, heun eliable con olchannelhasbeen eplace by
anaddi i ei.i.d.noise channel ha hasaSignal oNoise
Ra io(SNR)cons ain .In ha pape , hep obabili yo
da alossesisa ixed alue ha isusedin he con ol
syn hesis.Theobjec i ein Sil ae al. (2009)is ominimize
he e o co a iance designing he con olle iaYoula
pa ame iza ion.Howe e , in ha wo konlyape ec
LTInominalmodel isconside ed,and he e o e obus
p ope iesa eno gua an eed.
In hispape ,anNCSwhe einacommunica ionchannel
wi hada ad opou s sou ce exis sisconside ed,aswell
as s uc u alunce ain iesin heplan .The e o e, he
maingoalo hiswo kis o ind a obus con olle o
heplan wi h unce ain iesand wi h da alossesin he
ansmission;also inding ou heminimalp obabili yo
success in he ansmissionsuch ha meansqua es abili y
(MSS)and obus ness p ope iescan begua an eed.A
mixedH2/H∞con olapp oachisp oposedinsuchaway
ha bo hs uc u alunce ain iesin heplan and da a
lossescan be ole a ed.
The emainde o hepape iso ganizedas ollows:In
Sec ion2,ab ie summa yo hemixedH2/H∞con-
olp oblem heo yisexposed.In Sec ion3 he con ol
p oblem obesol edisp esen ed.Sec ion4shows he
P oceedings o he 18 h Wo ld Cong ess
The In e na ional Fede a ion o Au oma ic Con ol
Milano (I aly) Augus 28 - Sep embe 2, 2011
978-3-902661-93-7/11/$20.00 © 2011 IFAC 13269 10.3182/20110828-6-IT-1002.02152
a chi ec u eo he con olscheme.Sec ion5includes he
ob ained esul swi hanexample.Finally,Sec ion6d aws
hemainconclusionso hepape .
2.MIXEDH2/H∞CONTROLPROBLEM
In his sec ion,ab ie mixedH2/H∞con olapp oach
isdesc ibed.Fu he in o ma ioncan be ound inZhou
e al. (1996)and Doyle e al. (1994).The con olsys em
desc ibedinFigu e1isconside ed,whe e hegene alized
plan P(z)and he con olle C(z)a ebo hassumed
obe eal- a ionaland p ope .Thesignalsin ol edin
hediag ama e he ollowing:w′∈Rm1 ep esen s he
dis u bance ec o ,u∈Rm2is he con ol inpu ,z∞∈
Rp1and z2∈Rp2a e he e o ec o s, he i s one o
hemeasu emen o heH∞pe o mance,and hesecond
one o heH2pe o mance.Themeasu emen supplied o
he con olle is ep esen ed bym∈Rp3.
P(z)
C(z)
Z
Z
mu
w ∞
2
Fig.1.MixedH2/H∞syn hesis
Thesyn hesisp oblemconside edin hisapp oachconsis s
in inding a subop imalLTIcon olle C(z) ha minimizes
he ollowingmixedH2/H∞c i e ion:
MinαkT∞k2
∞+βkT2k2
2,(1)
subjec o:
• kT∞k∞<γ0
• kT2k2<ν0
whe eT∞(z)and T2(z)deno e he closed-loop ans e
unc ions omw′ oz∞and z2, espec i ely;and γ0,
ν0∈R+.
Aswill beshown, heminimiza iono kT2k2implies he
minimiza iono helowe bound o hesuccess p obabili y
in heda a ansmission.
Ino de o ind ou acon olle bymeanso hiscon ol
echnique, i isnecessa y opu heo iginalsys emin o
he o mo heblockdiag amshowninFigu e1.Todo
his, heo iginalsys emischangedwi halowe linea
ac ional ans o ma ion.
In hiscase,T∞ischosen o ep esen amixed-sensi i i y
H∞con olp oblem,whichiswidelyexplainedin Sko-
ges adand Pos le hwai e(1996).So woweigh ing unc-
ionsa e chosen:Ws(z) oweigh hesensi i i y unc ion
S(z)and W (z) oweigh he complemen a y sensi i i y
unc ionT(z).Theseweigh ing unc ionsallow ospeci y
he angeo equencieso ele ance o he co esponding
closed-loop ans e ma ix.Asi is known,anapp op i-
a eshaping o T(z)isdesi able o ackingp oblems,
noisea enua ionand o obus s abili ywi h espec o
mul iplica i eou pu unce ain ies.On heo he hand,
acon enien shaping o S(z)will allow oimp o e he
pe o mance o hesys em.So, hisapp oachisuse ul o
ha eanapp op ia epe o mance on ackingp oblems,as
well as o hesys em obus i ica ionagains noisesand
unce ain ies.
3.PROBLEMDEFINITION
Thispape is ocusedonaRNCSwhe ein hemain
p oblemsa e heunce ain iesin hemodelo heplan
and hepacke sd opou s.So, heaimis odesigna
con olle ha s abilize asys emsubjec o hese wo
p oblems oge he .
Theunce ain iesunde conside a ionwill be ep esen ed
by he ollowingequa ion:
G∗(z)=G(z)(I+Wm(z)∆(z)),
whe eG∗(z) ep esen sall hepossibleplan s,G(z)is
henominalplan and Wm(z)∆(z)is hemul iplica i e
unce ain y,wi hk∆(z)k∞<1.
In he ollowing heway odealwi h hein o ma ionlosses
isp esen ed.A e ha ,amo e ealis ic caseisconside ed
including heplan unce ain ies.
Thepacke sd opou simply ha he eisan un eliable
channel in he eedbackpa h.This si ua ionisillus a ed
inFigu e2,whe eG∗(z)is heplan ans e unc ion,
C(z)is he con olle , is he e e ence and yis heplan
ou pu .The ela ion be ween he channel inpu and he
channelou pu wis:
w(k).
=(1−d (k)) (k),∀k∈N0,∀ (k)∈N,(2)
whe ed modelsda alosses,sod (k)∈{0,1}∀k∈N0.
C(z)G(z)
Wm
∆
+
+y
G*(z)
mu
_
1-d
Fig.2.RNCSwi h packe sd opou s
Theno iono s abili yused o his kind o sys emsis
desc ibedin he ollowingde ini ion.
De ini ion1(Meansqua es abili y)Sil ae al.
(2009)Conside asys emdesc ibed byx(k+1)=
(x(k),w(k)),whe ek∈N0, :Rn×Rm→Rn,
x(k)∈Rnis hesys ems a ea imeins an k,x(0)=x0,
whe ex0isasecond o de andom a iable,and heinpu
wisasecond o de wss p ocess independen o x0.The
sys emis said obemeansqua es able(MSS)i and only
i he e exi ini eµ∈Rnand ini eM∈Rn×n,M≥0,
such ha
lim
k→∞
E{x(k)}=µ,
lim
k→∞
E{x(k)x(k)T}=M,(3)
18 h IFAC Wo ld Cong ess (IFAC'11)
Milano (I aly) Augus 28 - Sep embe 2, 2011
13270
ega dless o heini ials a ex0.
C(z)G*(z) y
mu
_
q
p
p
Fig.3.RNCSwi h packe sd opou s
Theunce ain ieswill beincludedin hesys emgi en
by he ollowing heo em.This esul makespossible o
change heo iginalsys emin o ano he equi alen one.
Theo em1.(Equi alence)Sil ae al. (2009),Ling and
Lemmon(2004)Conside he eedbackloopinFigu es2
and 3.I is supossed ha p∈(0,1)and Assump ions1
and 2 omSil ae al. (2009)hold.Then:
(1)I he eedback sys emdepic edinFigu e2isMSS
and he eedback sys eminFigu e3isin e nally
s able, hen hes a iona yPSDso he e o (e.
= −
y)and o all hesignalsin heloopsa e hesamein
bo hsi ua ions.
(2)Thene wo kedsys eminFigu e2isMSS i and only
i he eedbackloopinFigu e3isasymp o ically
s ableand p
1−p>kTp(z)k2
2,(4)
whe eTp(z)is he ans e unc ion omq o pin
Figu e3,namely
Tp(z).
=−pG∗(z)C(z)(1+pG∗(z)C(z))−1.(5)
P oo .Thep oo goesas hesamelinesas hep oo sin
Ling and Lemmon(2004).
Asaconsequence o Theo em1, i is known ha s udying
heMSS o hesys eminFigu e2isequi alen o achie e
hes abili yo hesys eminFigu e3while condi ion(4)
holds.The eby, hep oblemcan beposedas o ind a
con olle C(z) ha s abilizes hesys eminFigu e3 and
sa is ies he equa ion(4), akingin o accoun ha he
plan G(z)is henominalplan modeland ha he closed-
loopsys em mus be obus agains heunce ain iesin he
plan model.
Asbe o emen ioned,s uc u alunce ain iesa egoing o
be conside edin hemodelo heplan G∗(z).Due o
his ac , hemixedsensi i i yapp oachwi hin heH∞
scopeallows oimpose obus pe o mance bymeanso
app op ia edesigno weigh ing unc ions.In pa icula ,
i iswell known ha obus s abili ycan beimposed
byweigh ing he complemen a y sensi i i y unc ioni
s uc u almul iplica i eunce ain yisconside ed(O ega
and Rubio(2004),O ega e al. (2006)),whilepe o mance
can beimposed bymeanso a easonableweigh on he
sensi i i y unc ion.
On heo he hand, i isnecessa y ha condi ion(4)holds.
Then,by sol ing anH2con olp oblemi ispossible o
ind heminimalp obabili yo success in he ansmission
(p).The e o e,bymixing hese wo echniques,amixed
H2/H∞con olp oblemis o mula ed,wi h he ollow-
ingcos unc ion ominimize:αkT∞k2
∞+βkT2k2
2,whe e
kT∞k∞includes someweigh ing unc ions o achie e he
sys em obus i ica ionand kT2k2will bekTp(z)k2, oim-
pose condi ion(4).
P oblem1Conside heRNCSinFigu e2whe e he
plan G(z)hasboundeds uc u almul iplica i eunce -
ain ies.Then, hep oblemconsis sin inding a obus
con olle C(z),using heRNCSinFigu e3, ha achie es
he ollowingcondi ions simul aneously:
•Minimize kT∞k∞ o achie ea good pe o mance
on ackingp oblemsand hesys em obus i ica ion
agains heplan unce ain ies.
•Minimize kT2k2 ocalcula e heminimalsuccess ul
p obabili yo da alossespossible o heNCS, im-
posingcondi ion(4),so hesys emsin heFigu es2
and 3 a e equi alen s.
4.CONTROLLERDESIGN
In his sec ion he con olle syn hesiswill bepe o med by
meanso hedesc ibedmixedH2/H∞con ol echnique.
Someweigh ing ans e unc ionswill bein oducedin
hesys em odealwi h heunce ain ieso heplan
model. Theaugmen edsys emis ep esen edinFigu e4.
Theweigh ing ans e unc ionsWs(z)and W (z)weigh
hesensi i i y unc ion(S(z)) and he complemen a y
sensi i i y unc ion(T(z)), espec i ely.Theou pu so
heseweigh ing ans e unc ionsa e hesignalszsand
z espec i ely,and hey ep esen he componen so he
ec o z∞inFigu e1.
C1(z)G*(z)y
m1 u1
_
q
p
W (z)
Ws(z)
m2
u2
z2
z
zs
C2(z)Vp
_e
Fig.4.RNCSand heweigh ing ans e unc ions
I isimpo an ono e ha hesys emunde conside a-
ion(Figu e3), isanon-uni a y eedback sys em.So, in
o de oelimina e hes eady s a e e o s,a wo-deg ees-
o - eedomcon olle isp oposed.The e o e, he con olle
will be o med by wo ans e unc ions,C1(z)and C2(z).
Also, hesensi i i y unc ion(S(z)) and he complemen-
a y sensi i i y unc ion(T(z)) exp essionswill change.
These exp essionswill be:
S(z)=1+C1(z)G(z) (C2(z)p−1)
1+C1(z)G(z)C2(z)p
18 h IFAC Wo ld Cong ess (IFAC'11)
Milano (I aly) Augus 28 - Sep embe 2, 2011
13271
T(z)=C1(z)C2(z)G(z)p
1+C1(z)C2(z)G(z)p
Thesensi i i y unc ion(S(z)) ep esen s he ans e
unc ion om he e e ence o he e o signal. The comple-
men a y sensi i i y unc ion(T(z)) dependson heopen-
loop ans e unc iono hesys em,whichis:L(z)=
C1(z)C2(z)G(z)p,so he con olsignalu2should be he
inpu o heweigh ing ans e unc ionW (z),asi is
ep esen edinFigu e4.
Theobjec i eso he con olle a e he ollowing:
(1)Minimize heH∞no mo he closedloop om he
exogenousdis u bances ec o o he ec o z∞.
(2)Minimize heH2no mo he closedloopsignal om
ha ec o o hesignalz2.
So,asmen ioned be o e, hemixedH2/H∞con olp ob-
lemwill besol ed o ind asubop imalcon olle which
achie esa ade-o be ween heminimumo he wo
no msunde conside a ion.Toca you hesyn hesis, he
sys eminFigu e4has obe exp essed,bymeanso alowe
linea ac ional ans o ma ion,asinFigu e1.I iseasy o
see ha ,byiden i ying he e ms, he ollowingsequa ions
hold:
z∞=[zsz ]T,w′=[ q]T,
P(z)=
Ws(z)0|−Ws(z)G(z)0
0 0 |0Wks(z)
0 0 |pG(z)0
I0|0−I
0I|pG(z)0
Wi h espec o heminimiza ion p oblemin(1),T∞(z)
and T2(z)a e chosenas ollows:
kT2(z)k2=kTp(z)k2
kT∞(z)k∞=
Ws(z)S(z)
W (z)T(z)
∞
Thepa ame e swill be choseninsuchaway ha he
condi ion(4)holds.Thismeans ha :
ν0=p
1−p
A hispoin , i ’swo hmen ioningsome commen sin
ela ion o he choice o heo he spa ame e s.I isin e -
es ing ono e ha , i hep io i yis o achie e heminimal
possiblep, i isimpo an o ob ainacon olle ha
p o idesanH2no mo T2(z) e yclose oi sminimum.
Then, o hiscase, hepa ame e βshould beg ea e han
α.On he con a y, i hein e es liesonachie ing hebes
pe o mance and obus ness agains noisesand unce ain-
ies, i isbe e ochoose hepa ame e αg ea e han
β.Thismeans ha he esul ingcon olle will p o idea
e y small H∞no mo T∞(z).
Thep obabili yo success in he ansmissionpisassumed
obe ixedin he con olle syn hesis.Thisispossiblei he
ne wo k equi emen sa ewell-known.Inanycase, i he
alueo pchanges, hes abili yo he closed-loopsys em
isgua an eedi pisg ea e han heminimalp obabili y
o success in he ansmissionob ained.
5.NUMERICALRESULTS
Toillus a e heme hodologyp oposedin hispape ,
his sec ionshows heob ained esul swhen he con ol
s a egyisapplied o a pa icula example.In hisexample
he ollowinguns ablenominalplan will be conside ed:
G(z)=z−0.5
z(z−1.1)
Thesampling imewill be m=0.05s.
10−3 10−2 10−1 100101102
−20
−15
−10
−5
0
5
10
15
ecuency ( ad/s)
W , unce ain y (dB)
W as unce ain y supe io limi (dB)
Unce ain y 1
Unce ain y 2
WT
Fig.5.Unce ain iesand W
To akein o accoun heunce ain iesin heplan , wo
non-nominalmodelsha ebeenalsoconside ed.To ob ain
hese wo o he models, he ealplan is supposed oha e
unmodelled dynamics,so,high equencypolesa einclude.
Also a pe cen ageo unce ain yin hemodelgain has
beenconside ed.F om hese wosys emsand henominal
plan , hemul iplica i eunce ain iescan be compu ed.
The equency esponseo heseunce ain iesha ebeen
plo edinFigu e5.
F om hises ima iono heunce ain y, heweigh ing
ans e unc ionW (z) o he complemen a y sensi i i y
unc ionisdesignedinsuchway ha i smodulusmus
beg ea e han hemoduluso heunce ain ies o all
equency.The equency esponseo W (z)hasbeenalso
ep esen edinFigu e5.
By sol ing hemixedH2/H∞con olp oblem o his
caseusingsome unc ionso heµ−AnalysisandSyn hesis
Toolbox o Ma laband conside ing a success p obabili y
p=0.7,a obus con olle isob ainedyielding he
ollowing esul s:
kT∞k∞=0.8441,kT2k2=1.3615
18 h IFAC Wo ld Cong ess (IFAC'11)
Milano (I aly) Augus 28 - Sep embe 2, 2011
13272
10−3 10−2 10−1 100101102
−70
−60
−50
−40
−30
−20
−10
0
10
equency ( ad/s)
gain (dB)
Sensibili y unc ion and i s weigh
Snominal
WS
−1
S1
S2
Fig.6.S(z)o henominalplan modeland Ws(z)
Thismeans ha hesys emcana o dasuccess p obabil-
i ypequal o o g ea e han0.65, o gua an ee MSS and
op ese e hedemanded obus ness p ope ies.
10−3 10−2 10−1 100101102
−80
−70
−60
−50
−40
−30
−20
−10
0
10
20
equency ( ad/s)
gain (dB)
Complemen a y sensibili y unc ion and i s weigh
Tnominal
WT
−1
T1
T2
Fig.7.T(z)o henominalplan modeland W (z)
InFigu e6 hesensi i i y unc ionso henominaland
non-nominalplan smodelsand hein e seo heweigh -
ing ans e unc ionWs(z)a e ep esen ed.Thisg aphic
showshowall hesensi i i y unc ions,o henominalsys-
emand sys emswi h unce ain ies,a ebelow hein e se
o heweigh ing unc ionWs(z).This ac indica es ha
heou pu ycan ollow he e e ence o all heplan
modelsunde conside a ion, ha is,a ackingp oblem
can besol edal hough heplan model isno exac ly
known.
Figu e7 ep esen s he complemen a y sensi i i y unc-
ionso henominaland non-nominalplan smodelsand
hein e seo heweigh ing ans e unc ionW (z).F om
hisg aphici ispossible osee ha all he complemen a y
sensi i i y unc ions,o henominalsys emand sys ems
wi h unce ain ies,a ebelow hein e seo heweigh ing
unc ionW (z),so heob ainedcon olle is obus agains
heunce ain iesin heplan model.
Toco obo a e hese esul s,somesimula ionsha ebeen
ca iedou wi h hep oposedexample.Figu e8shows
how hesys em ollows he e e ence wi hasuccess ul
ansmission p obabili yp=0.7,whichisg ea e han he
minimalp ha can p o ideMSS and obus ness p ope ies
o his sys em.Thisg aphic ep esen s heou pu so
he closed-loopsys emwi h henominalplan ,wi h he
plan wi h heunce ain ies1 and wi h heplan wi h
heunce ain ies2.The esul sa e e y simila because
he obus ness o hesys em.Howe e , he e exis some
di e encesbe ween hedi e en sou pu s.Fo example,
heou pu wi h heunce ain ies1hasano e shoo ha
isg ea e han heo e shoo when henominalmodel is
used.Wi h espec o heou pu wi h heunce ain ies2,
heo e shoo is educedwi h espec heo he cases,bu
hes a iona ype o mance iswo se.
0 5 10 15 20 25 30 35 40
−6
−4
−2
0
2
4
6
Time,
Plan ou pu , y
Re e ence
Nominal plan
Plan 1
Plan 2
Fig.8.Simula ion esul swi h p =0.7
Theou pu so hedi e en sys ems o a alueo p=0.9
a eshowninFigu e9.In hiscase, hep obabili yo suc-
cess in he ansmission hasbeeninc eased,al hough he
con olle usedin hesesimula ionsis heone calcula ed
o p=0.7.Ob iously, he esul sa ebe e hanin he
onesp esen edinFigu e8,bu hedi e encesbe ween
hepe o mance wi h hedi e en sys emsis hesame
asin he caseo p=0.7.Also, he ea es eady s a e
e o sbecause he con olle is he calcula ed o p=0.7
so he eedbackisnon-uni a y.Theses eady s a e e o s
migh bea oid bycalcula ing he con olle usingp=0.9,
bu heobjec i eis ocompa e he esul swi h hesame
con olle ,supposing ha pha e changedin hene wo k.
Finally,Figu e10 p esen s heou pu so all hesys ems
imposingp=0.4,whileusing hesame con olle asin he
p ecedings simula ions.Ob iously, hepe o mancesge
wo se o all hesys ems,and in he caseo heplan wi h
heunce ain ies1 and 2, he closed-loopsys embecomes
uns able.So,wi hp=0.4, he obus s abili yislos .
18 h IFAC Wo ld Cong ess (IFAC'11)
Milano (I aly) Augus 28 - Sep embe 2, 2011
13273
0 5 10 15 20 25 30 35 40
−6
−4
−2
0
2
4
6
Time,
Plan ou pu , y
Re e ence
Nominal plan
Plan 1
Plan 2
Fig.9.Simula ion esul swi h p =0.90
0 5 10 15 20 25 30 35 40
−6
−4
−2
0
2
4
6
Time,
Plan ou pu , y
Re e ence
Nominal plan
Plan 1
Plan 2
Fig.10.Simula ion esul swi h p =0.40
6.CONCLUSIONS
Thepape has ocusedonaNCSsubjec oda ad opou s
cons ain s.In pa icula ,con ol loops o SISOLTI
plan s,whe e he eedbackpa hcomp isesacommuni-
ca ionchannel ha p oducesda alosses,a e conside ed.
This sys emhasbeens udiedasanequi alen onewhe ein
heun eliable channelhasbeen eplaced byanaddi i e
i.i.d.noise channel, plusa gain.
Theobjec i eo hispape hasbeen hesyn hesiso a
con olle ha a oid hemodelunce ain iesand suppo
he ailed ansmissions.Also, helowe bound o he
success p obabili yin he ansmission hasbeen ound.
Tope o m his ask,amixedH2/H∞con olp oblem
hasbeen p oposed.To ob aina obus con olle ,some
unc ionsha ebeenchosen oweigh somesensi i i y unc-
ions.Mo eo e , om hiscon olp oblem, heminimal
success ul ansmission p obabili yisob ainedsuch ha
MSS and obus ness p ope ies o he closed-loopsys em
a egua an eed.
Finally,anexamplehasbeenexposed o ob ainsome
nume ical esul s ha illus a e he closed-loopsys em
pe o mance.Thesesimula ion esul sco obo a ed ha
obus pe o mance isachie edi hesuccess ulp obabil-
i y ansmissionishighe han heminimumcompu ed,
while hedi e en s sys emspe o mancesge wo se,un il
he obus s abili yislos ,as hesuccess ulp obabili y
ansmission dec eases.
ACKNOWLEDGEMENTS
Theau ho swouldlike o acknowledgeMCyT(G an
DPI2010-19154.),and heEu opeanCommission(EC)
(FeedNe BackP ojec ,g an ag eemen 223866), o und-
ing hiswo k.
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