S abili yand Pe o manceo Ne wo ked
Con olSys emswi hTime-mul iplexed
Senso sand O e sampledObse e ⋆
L.O ihuela,F.G´omez-Es e n,F.R.Rubio
Dp o.Ingenie ´ıa deSis emasyAu om´a ica,Uni e sidad deSe illa.
Camino delosDescub imien os s/n,41092 Se ille,Spain
{o ihuela, abio, ubio}@us.es
Abs ac :In his pape weanalyze hescena io whe ease o wi eless senso sa e ime-
mul iplexedin o de o educe he a icloadand ene gyconsump ionin aNe wo kedCon ol
Sys em.Thepossible choices o scheduling he ansmissiono measu esa e explo ed unde
he concep o pe iodic eedbackpa e ns,i.e. epea edsequenceso di e en measu es.The
leng hand s uc u eo such pa e nsa e chosen o gua an ee s abili yo he closedloopand o
minimizeap ede ined pe o mance index.The con olschemeis enhancedwi hacon olle -side
obse e unning a a a ehighe han hesampling a e, o whichano els abili yp oblemis
s a edand sol ed.
Keywo ds:Ne wo kedCon olSys ems,Wi eless senso s,Pe iodic sys ems.
1.PROBLEMSTATEMENT
InNe wo kedCon olSys ems(NCS) educing he ene gy
consump ion bylimi ing he ansmission da a- a esis
c ucial. Howe e ,less in o ma ionmayimply wo sepe -
o mance o ,a wo s ,causes abili yp oblems.Ano he
issue ela ed o eedbackcon olsys emwi hwi eless dis-
ibu edsenso sliesin he ac ha ene gyconsump ion
maybes ongly unbalanced, equi ing equen ba e y
changes o somesenso s,while o he a eunde mos ly
idle.Ene gy-awa e con olshould implemen ene gybal-
ancing algo i hmsac oss hene wo kand beable o
eschedule he comple e eedback s a egyacco ding oen-
e gydemandsand a ailabili y.Howe e his escheduling
capabili y equi esp ope knowledgeo hesys emin o de
o ecompu e hes abili ycondi ions.
Sys em
s1
s2
sm
Packe Ne wo k
Obse e
Con olle
y1
k
y2
k
ym
k
uk
uk
ˆxk
Fig.1.Ne wo kedcon olscheme
Theunde lying echnology ha wep opose odealwi h
scheduled eedbackis heuseo pa ialobse e s.Ins ead
o sending he comple eou pu ec o ,only one componen
a els o e hesha edmediuma oneins an .On he
ecei ingsideo hene wo k, heobse e manages o
⋆Theau ho swouldlike o acknowledgeCICYT(G an DPI2010-
19154),and heEu opeanCommission(EC) (FeedNe BackP ojec ,
g an ag eemen 223866), o unding hiswo k.
econs uc hewhole s a eo hesys em,whichwill be
used o con olpu poses.Figu e1desc ibes heglobal
scheme.Due o he cha ac e is icso ne wo kedsys ems,
hesys emis desc ibedin disc e e ime:
xk+1=Axk+Buk,yk=Cxk,(1)
whe exk∈Rn,yk∈Rmand uk∈R a e hes a e,
heou pu and he con olsignal, espec i ely.A,B,C
a eknownma iceso app op ia edimensions.AsFig-
u e1sugges s, he e exis smsenso s,possibly spa ially
dis ibu ed,whichsample eachoneo he componen so
heou pu ec o .Ineachsamplingpe iodk,only one
senso can use hene wo k osend i spacke .Assuming
anidealne wo k(wi hou noise,delays o dis u bances),
a ins an k hesenso j(j=1,..., m)sends heou pu
yj
k=EjCxk,whe eEj,j=1,..., m,a ema iceso
app op ia edimensions(wi helemen sonesand ze os)
used ode ineanypa ialou pu .
In he con olle sideo he communica ion he eis an
obse e ying oes ima e hes a eo hesys emusing
hein o ma iono he ecei ed pa ialou pu (see Figu e
1).Thedynamicso heobse e is,
ˆxk+1=Aˆxk+Buk+Ljhyj
k−ˆyj
ki,(2)
ˆyk=Cˆxk,(3)
whe eˆxk∈Rnand ˆyk∈Rma e hes a eand heou pu
o heobse e , espec i ely.Ma icesLj,j=1,..., m,
a e heobse a iongainswhich,in gene al, a edi e en
depending on he ecei edou pu .
E e y samplingpe iod, he con olle buildsacon ol
signalde ined byuk=Kˆxk,whe eKis he con olle
ma ix o app op ia edimensions.
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 9200 10.3182/20110828-6-IT-1002.02435
Finally, he e o be ween hes a eo hesys emand he
s a eo heobse e is de inedas
ek=xk−ˆxk.(4)
Thepu poseo his pape is hedesigno heobse e
(2)-(3)in suchaway ha somes abili yand op imal
p ope ies o heobse a ione o (4)we e ensu ed.We
p oposeape iodic obse e ,in whichapa e n,de ined
o -line,is epea edall ime.This pa e nindica eswha
ou pu is sen in wha ins an .
The ollowingde ini ionswill beuse ulin u u ep oposi-
ionsand hei p oo s.
De ini ion1.Ameasu emen pa e nϕN∈RNis
a ec o whose componen sde inewhichou pu is sen
h ough hene wo k.Tha is,
ϕN(i)≡{ji:ji=1,..., m},i=1,..., N.(5)
Hence,ϕN(i)=jiimplies ha senso jiuses hene wo k
in hei- h posi iono hepa e n.Wi h his de ini ioni
is possible o g an p io i y osomeou pu so e he es .
De ini ion2.In e al [ϕN(i),ϕN(j)]deno esall he
consecu i esamplesbe weenϕN(i)and ϕN(j).
1.1Li e a u e e iewand openp oblems
In heli e a u e he ea e ela edwo ks asRehbinde and
San idson(2004) o op imalpe iodic linea quad a ic
egula o ;Zhang and H is u-Va sakelis (2006);Jiange al.
(2008) o pe iodic obse e s;H is u-Va sakelis and Zhang
(2008) o pe iodic Kalman il e so Sinopoli e al. (2004);
Gup ae al. (2007) o Kalman il e subjec oin e mi -
en obse a ions.
Howe e ,noneo hesewo ks sol e hep oblemo ind-
ing anop imalpa e nwhichmanages hesen o he
ou pu s.InZhange al. (2006);H is u-Va sakelis and
Zhang(2008) he communica ionsequencesa e chosenin
suchaway ha he eachabili yand obse abili yo he
sys ema ep ese ed.Howe e , o aconc e esys em may
exis se e alpa e ns ha p ese e heobse abili yand
eachabili y.How ochoose heop imaloneamong hem
s ill emainsanopen p oblem.Finally,in Lue al. (2003)
ape iodic pa e nis chosenminimizing anH∞no m.
Mo eo e , o hebes o ou knowledge, he edoesno
exis anywo kin which heobse e is unning a a
equencyhighe han hesampling a e,sui able when
he con ollawcanchange as e han heda a a i al
a e.Thosep oblemsa es udiedin Sec ion2.
2.OVERSAMPLEDOBSERVER
Byle ing he con olle and obse e un a ahighe e-
quency han he ansmissiono pa ialou pu s, he ewill
besamplingins an swi hou ex e nal eedback.Be ween
woconsecu i e ansmissions, hesenso s emain asleep
o Poobse a ion pe iods.So hedynamicso heobse e
di e s o he wo ypeso pe iods,obse a ion pe iods
(OP)and measu emen pe iods(MP).
(MP):ˆxk+1=Aˆxk+Buk+Liyi
k−ˆyi
k;
(OP):ˆxk+1=Aˆxk+Buk.
Figu e2shows he comple epa e n o acasein which
h ee obse a ion pe iodsha ebeenin oduced.
(OP) (OP)
(OP)
(OP)(OP)(OP)(OP)(OP)
kk+Po+1k+(Po+1)N
ϕN(1)ϕN(1)ϕN(2)ϕN(2)ϕN(N)
Fig.2.Timescheduling o heo e sampledobse e
Analyzingbo hse so equa ions,and de ining heaug-
men eds a ezT
k=xT
keT
k, he ollowingp oposi ion
gi es he comple e e olu iono hesys emalong a ull
pe iod.Thep oo is gi enin Appendix A.
P oposi ion1.Gi enameasu emen pa e nϕN∈RN,
he e olu iono hesys em omk ok+(Po+1)Nis
de inedas
zk+(Po+1)N=

(A+BK)(Po+1)NΞ12
0Y
ji∈[ϕN(N),ϕN(1)]
[ϑji]ek
zk
=Ξ(Po+1)Nzk,∀k,(6)
whe eϑji=(A−LjEjC)APoand Ξ12 is ama ix wi h
complex s uc u e.
2.1S abili y
The ollowinglemmaes ablishes hes abili yo hesys-
em.
Lemma 2.Thedisc e e- imesys emwi he olu iongi en
byP oposi ion1is asymp o ically s able i and only i he
eigen alueso he ollowingma icesa einside heuni
ci cle:
•Ξ11 =A+BK,
•Ξ22 =Qji∈[ϕN(N),ϕN(1)][ϑji].
Thep oo is immedia eusing ai ly ex endedeigen alue
p ope ieso iangula ma ices.
Example.Conside hedisc e e- imesys em omIshii
and F ancis (2002)
xk+1="1 0 0
0−1−3
0 0 −2#xk+"1 0
0 1
0 1 #uk,
yk=0 1 0
1 0 0 xk.(7)
obse edwi h pa e nϕ2=[1,2]. Assume ha heob-
se e gainsa eL1=[0 −2.2−0.8]Tand L2=[0.500]T.
UsingLemma 2, hes abili yo heobse a ione o
wi hou obse a ion pe iodsis ensu ed.The eigen alueso
Ξ22 a eλ(Ξ22)={0,0.4,0.5}.Howe e ,wi hPo=1, he
eigen alueso Ξ22 a eλ(Ξ22)={0,−2,0.5},so heobse -
a ione o is uns able.Howe e ,choosing, o example,
L1=[0 −2−0.67]Tand L2=[1 0 0]Twi hPo=1yields
λ(Ξ22)={0,0.83,0.09}so heobse e is s able,bu a
educ iono da a h ough hene wo kis achie ed.
18 h IFAC Wo ld Cong ess (IFAC'11)
Milano (I aly) Augus 28 - Sep embe 2, 2011
9201
2.2Obse e Design
Ino de oensu e hes abili yo hewhole NCS, he
eigen alueso woma icesmus layinside heuni ci cle.
Assuming ha hesys emis s abilizable,weha e o
design heobse e in suchaway ha he e o dynamics
becomesasymp o ically s able.AnLMI-basedme hodis
p esen ed usingideas ompe iodic sys em heo y.
Ap-pe iodic disc e e- imesys emis desc ibed byxk+1=
Akxk,whe exk ep esen s hes a eand Aka ep-pe iodic
ma ices, ha is,Ak+p=Ak,∀k.
Animpo an esul ela ed o hesesys emsis heso-
calledPe iodic Lyapuno Lemma,see Bi an ie al.
(1985),ac ually anex ensiono heLyapuno Lemma o
his class o sys ems.
Pe iodicLyapuno Lemma.Thep-pe iodicsys em
xk+1=Akxkwi hAk+p=Ak∀k, isasymp o icallys able
i and onlyi he e exis sa p-pe iodicma ixPk>0such
ha
AT
kPk+1Ak−Pk<0,∀k∈{1,..., p}.(8)
Thedynamicso he e o be weens a eo heplan and
o heobse e can bedesc ibedasape iodic sys em.The
p oo o he ollowingp oposi ionis omi ed due ospace
es ic ions.
P oposi ion3.Gi enameasu emen pa e nϕN∈RN
and acons an Po∈N, hedynamicso he e o gi enin
P oposi ion1is equi alen o anN-pe iodic sys emwi h
pe iodic ma ix ϑk=(A−LkEkC)APo,∀k∈{1,..., N}.
The ollowing heo emp oposesanLMI-based design
p ocedu e o ob ain heobse a ionma ices.
Theo em4.Thedynamicso heobse a ione o ,gi en
byP oposi ion1,is asymp o ically s able i and only i
he e exis sanN-pe iodic posi i ede ini ema ix Pkand
anN-pe iodic ma ix Wko app op ia edimensions such
ha he ollowingLMIsa esa is ied,
Pk−1(APo+1)TPT
k−(APo)TCT(Ek)TWT
k
∗Pk>0,(9)
∀k∈{1,..., N},whe eP0=PN.Then, heobse a ion
ma icesa ede inedasLk=P−1
kWk.
P oo .Top o eTheo em4weneed o apply Schu
complemen o hePe iodic Lyapuno Lemma(8).Then
subs i u eAkbyϑkand de ineWk=PkLk o ob ain
condi ions(9).2
Asnoconse a ismhasbeenadded,Theo em4canalsobe
used o ind hemaximumnumbe o obse a ion pe iods
(Po) ha can bein oduced be ween woconsecu i e
samplings.
2.3Pa e ndesign
In Sec ion2.1,wehadshown he condi ions ha mus be
e i iedso ha hedynamicso heobse a ione o we e
asymp o ically s able gi en heobse a ionma icesLj.
In Sec ion2.2,aLMI-based designme hodwasp oposed
o ob ain heobse a ionma icesgi enameasu emen
pa e n.Thenex s epin his p ocess is,among all possible
pa e ns,whichones abilizes hesys em?
I is ob ious ha hosepa e nswhich dono obse e he
dynamicso heobse a ione o could bedisca d.The e
exis some esea chesin heli e a u eoncon ollabili y
and obse abili yo gene ic pe iodic sys em,see Guo and
Qiao(2004).Howe e , o he caseunde s udy,only he
ou pu ma ix EjCis pe iodic.We canenuncia ehe ean
easie obse abili y es .
Lemma 5.Sys em(1)is obse able a ins an ki and
only i ma ix
Q:= 



C′
C′A′
.
.
.
C′A′n−1




(10)
has ankequal on,whe eA′:= ANand
C′:= 



EkC
Ek+1CA
.
.
.
Ek+N−1CAN−1




.
Thep oo is omi ed due ospace limi a ion,bu i can
bemadebyapplyingCayley-Hamil on heo emasin
he classical es .Finally,wein oduce hede ini iono
comple eobse abili y.
De ini ion.Sys em(1)is comple ely obse able i and
only i i is obse able o all k=1,..., N.
The e o e,i apa e nachie es heobse abili yo he
sys em(1),i will beable os abilize heobse e .The
obse abili y es is clea ly easie ,compu a ionally spoken,
han heLMI-baseds abili y es ,sowe cans udyo -line
e e ypossible pa e no leng hN.
Minimiza iono acos index
I weneed ocompa e hequali yo somes able pa e ns,
we canes ablishanume icalindex.This indexcould be
de ined akingin o accoun heobse a ione o .
J(i)=
i
X
k=0
eT
kQek,(11)
beingQaposi i ede ini ema ix.Asimila ideawas
i s ly in oducedin Ha eand Skoges ad(2003),bu o
op imal eedback s abiliza iono asys emwi honly one
uns able pole.Thedependence o heindex(11)wi h
he chosen pa e nis e ycomplex,bu i is possible
ocompa e hem,a leas ,in anume icalway.In he
ollowing,assume ha Po=0.
P oposi ion6.Gi enapa e nwi hleng hNand a
posi i ede ini ema ix Q, he cos indexJ=P∞
k=0eT
kQek
can be calcula edas
J=eT
0"∞
X
n=0
(αT
N)nΦN(αN)n#e0,(12)
whe e
18 h IFAC Wo ld Cong ess (IFAC'11)
Milano (I aly) Augus 28 - Sep embe 2, 2011
9202
αN=ϑNϑN−1...ϑ2ϑ1,
ΦN=
N
X
i=1
αT
iQαi.
Thep oo is immedia eby subs i u ing he e olu iono
hesys em(6)wi hPo=0in he cos index(11).
F omP oposi ion6,i u nsou hedependence be ween
heindexand heini ialcondi iono heobse a ione o .
I is mo ein e es ing hesi ua ionin which heini ial
condi iono heobse a ione o is bounded.Tha is,
weknow heuppe bound o ke0k2.Fo suchcase,we
need obound he cos index,employing he ela ion:
xTAx≤λmax{A}kxk2.
So, he cos index(12)is bounded byJ≤λmax{βN}ke0k2,
wi hβN=P∞
n=0(αT
N)nΦN(αN)n.
Fo ape iodic pa e n, he ea e wocha ac e is icswe
ha e ochoose:1) heleng ho hepa e nand 2)i s
s uc u e.Theleng ho hepa e nis c i ical, in such
away ha ,i i is no ixeda p io i henumbe o
combina ionsg owsun il in ini y.Be weenall possible
pa e nso leng hNwe canchoose heonewhominimizes
hemaximumeigen alueo ma ix βN.
Rema k.Due o hepa e ns abili y,ma ix αNhas
all i seigen aluesinside heuni ci cle.So, hein ini e
se iescan be eplaced bya ini eonewi hou incu ingin
p ac icale o s,aswewill see in he examples.Finally,
his ini ese iescan be calcula ed nume ically.
Example.Op imalpa e n design
Conside hedisc e e- imesys em omH is u-Va sakelis
and Zhang(2008).In ha pape , he choiceo hepa e n
wasmadebasedons abili yp ope iesonly.
xk+1=


1 0.1 0 0
0.1 1.25 0 0
1 0.1 1/6 0.5
0 0 0 1.25


xk+


0 0
1 0
0 0
0 1


uk,
yk=1 1 0 0
0 0 1 1 xk.(13)
The e exis wopossible ou pu s osend h ough he
ne wo ky1
k=[1 0]ykand y2
k=[0 1]yk.Theweigh ing
ma ix is chosenasQ=diag{1,1}.I is assumed ha
e0=x0, ha is, heini ialcondi ion o heobse e is
exac ly ze o.
Theobjec i eis ochooseand design heop imalpa e n
among all possible ones.We call op imalpa e n heone
whichminimizes he cos index(12).Tocalcula e he cos
index,a ini eho izonis needed.He e, his ho izonis
chosenla ge enough oneglec he e o s.
Figu e3depic s helowe maximumeigen alueob ained
o di e en leng hs.Remembe ha he cos is bounded
byJ<λMAXeT
0e0.
I can beseen ha omsome alue, hela ge pa e n
leng h doesno imply helowe cos s.Fo his example,a
leng ho 7is enough.Mo ep ecisely, heop imalpa e n
is [1 1 2 1 2 2 2].
2 4 6 8 10 12 14
21
22
23
24
25
26
27
28
λMAX
Pa e nleng h
Fig.3.Maximumeigen alue o a iouspa e nleng hs
3.EXPERIMENTALAPPLICATION
3.1Pla o mdesc ip ion,modelingandcon ol
The con ols a egydesc ibedabo eis applied o a
wo-deg ee-o - eedomdi ec d i e obo ,which hasbeen
designedand de eloped by heDepa men o Sys ems
Enginee ing and Au oma iona Uni e si yo Se ille,see
Mill´ane al. (2010).Theobjec i ewill be omain ain
he obo a i sup igh equilib iumpoin ,simila asan
in e ed pendulum.
The obo con igu a ionis schema ically shownin Figu e
4.The i s link(whichis be ween bo hmo o s)will be
e medasshoulde whe eas hesecond link(whichis
be ween hesmalle mo o and he edgeo he obo ) is
heelbow.
l1
lc1
q1
q2
lc2
l2
c.o.g. Shoulde
c.o.g. Elbow
Fig.4.Two-deg ee-o - eedom obo diag am
Theselec edcon olso wa eis hexPC Ta ge en i on-
men Mos e mane al. (2005)wi hMATLAB/Simulink.
I is well-known ha hedynamicso a obo ic manipula-
o is ex emely non-linea .To apply he esul so his pa-
pe , he obo will beope a eda ound heuns able up igh
equilib iumde ined byqe=[π0],˙qe=[0 0]. Tolinea ize
hesys emand ob ain ma ix Aand Bo equa ion(1)
ameansqua ei e a i eiden i ica ion p ocedu ehasbeen
ollowed.
Ino de ope o m he es sanini ialcon olbasedon
eedbacklinea iza ionis applied,whichs ee s he obo
omi s s able downwa d posi ion(bo hlinks s opped
18 h IFAC Wo ld Cong ess (IFAC'11)
Milano (I aly) Augus 28 - Sep embe 2, 2011
9203
9.8 10 10.2 10.4 10.6 10.8
−0.02
0
0.02
0.04
0.06
0.08
0.1
Time (s)
Posi ion E o ( ad)
Po=0
Po=1
Po=2
Po=3
Po=4
Po=5
Fig.6.Posi ione o s o heo e sampledobse e wi h
di e en numbe o obse a ion pe iods
in hei lowe posi ions) o hesu oundingso i sun-
s able up igh equilib ium.This con olle is appliedas-
suming hes a e comple ely accessible.Once ha posi-
ionis eached,pe iodic obse e pluslinea con olle is
swi ched.
3.2Expe imen al esul s
The expe imen is hesamein all cases.Theobse e is
swi ched( om henonlinea con ol) om heup igh
equilib iumpoin .Then,anaddi i edis u bance in he
o queis applieda =10 seconds.
Al hough helinea modelsugges s ha hesys emis
obse able omeacho heou pu (classicalobse e ),
expe imen s show ha his is no ue o he eal obo .
Fo ins ance,Figu es5band 5cshow ha i is no possible
o obse e heposi iono he elbowo he eloci yo
heshoulde .Theonly componen o hes a e ha can
beobse edis heposi iono heshoulde ,asFigu e5a
shows.
Tocompa e he h oughpu o heo e sampledobse e
weinse addi ionalobse a ion pe iodsand epea he
same expe imen .Thepa e nis chosenasϕ=[3,2,3,4],
ha is, heposi iono heshoulde is heonly componen
o hes a e ec o whichis being obse ed( heo he sa e
di ec ly measu ed).Figu e6depic s heposi ione o o
heshoulde when di e en numbe o obse a ion pe iods
a eincluded.Asexpec ed,as henumbe o obse a ion
pe iodsg ows, he e o is highe .The cons an s eady-
s a e e o a edue o hemodele o s.
Finally,wep o eadi e en pa e nand compa ei s
h oughpu wi h heonep oposed be o e.Theobse a ion
e o a edepic edin Figu e7.
4.CONCLUSION
Weha eanalyzedin dep h hedi e en eedbackpossibil-
i iesa ailable whenase o wi eless ne wo ks embedded
in acon olloopa e ime-mul iplexedin o de o a oid
collisionsand educe he o alne wo k a icand ene gy
consump ion.Finally,asa esul o hepe o mance anal-
ysis,newc i e ia o eedbackpa e n designin scheduled
9.8 10 10.2 10.4 10.6 10.8 11
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Time (s)
Posi ion E o ( ad)
[3 2 3 4]
[3 2 3 4 3]
Fig.7.Compa ison be ween di e en pa e ns
ne wo ks ha ebeense .The esul sha ebeenillus a ed
wi hsimula ionsand in a eallabo a o y2-DOF obo .
Fu he wo k should conside he compu a iono hepe -
o mance indexo he eedbackpa e nswi hou heneed
o nume icales ima ions.
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18 h IFAC Wo ld Cong ess (IFAC'11)
Milano (I aly) Augus 28 - Sep embe 2, 2011
9204

0 5 10 15 20 25
3.12
3.14
3.16
3.18
3.2
3.22
3.24
3.26
Time (s)
Posi ion ( ad)
Shoulde posi ion
Sys em
Obse e
(a)ϕ=(3,2,3,4)
0 5 10 15 20 25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
Time (s)
Posi ion ( ad)
Elbow posi ion
Sys em
Obse e
(b)ϕ=(1,4,3,4)
0 5 10 15 20 25
−3
−2
−1
0
1
2
3
4
5
6
Time (s)
Veloci y ( ad/s)
Shoulde eloci y
Sys em
Obse e
(c)ϕ=(1,2,1,4)
Fig.5.Compa isono hequali yo heobse e o di e en pa e ns
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Appendix A.PROOFOFPROPOSITION1
Suppose ha he ea emsenso sand Nou pu s,wi h
N≥m.Fu he suppose ha Poobse a ion pe i-
odsa ein oduced be ween woconsecu i esamples.
Theo de ing o hemeasu emen sis de ined byϕN≡
{ji:ji∈{1,..., m}and i=1,..., N}.
The e olu iono heaugmen eds a e o ameasu emen
pe iodcan bew i enas
zk+1=hA+BK−BK
0(A−LjiEjiC)izk=Λjizk.(A.1)
Fo anobse a ion pe iod, he ollowing ela ionshold,
zk+1=hA+BK−BK
0Aizk=∆zk.
Suppose ha omins an k ok+Po−1,no ou pu is
ecei ed om hesys em.A e Poobse a ion pe iods,
he ollowing ela ionshold,
zk+Po=(A+BK)PoΨPo
0APohxk
eki=∆Pozk,(A.2)
whe eΨPo=PPo
i=1(A+BK)Po−iBKAi−1.I is easy o
ob ain his ma ix bymul iplyingma ix ∆byi sel Po
imes.
Suppose ha ,a e Poobse a ion pe iods, he con olle
ecei esameasu emen .Hence he comple e e olu iono
heobse a ion pe iodsplusonemeasu emen pe iodcan
beob ained using(A.1)and (A.2).
zk+Po+1=(A+BK)Po+1(A+BK)ΨPo−BKAPo
0θjiAPozk
=Λji∆Pozk
No e ha he elemen (2,2)o p e iousma ix is exac ly
θjiAPo≡ϑji.This s uc u eis epea ede e yPoob-
se a ion pe iodsand onemeasu emen pe iod.So,i is
s aigh o wa d o ob ain he ollowing ela ion:
zk+(Po+1)N=(Λji∆Po)Nzk.
Finally,weha e op o e ha he ollowingequali yis
ue:
(Λji∆Po)N=Ξ(Po+1)N,(A.3)
whe eΞ(Po+1)Nhasbeen de ined p e iously in hehy-
po heseso hep oposi ion.Todo ha ,wew i eΞ(Po+1)N
(Ξhe eina e )usingi sblock s uc u e, ha is,Ξ≡
{Ξij:i,j∈{1,2}},and wes udy he equi alence be ween
hedi e en blocks.
zk+(Po+1)N=

(A+BK)(Po+1)NΞ12
0Y
ji∈[ϕN(N),ϕN(1)]
[ϑji]ek
(A.4)
Ξ11:Due o he ze oblockin posi ion(2,1),whenmul-
iplyingN imesΛji∆Po,in posi ion(1,1)weob ain
(A+BK)(Po+1)N.
Ξ21:Becauseo hes uc u eo hema icesΛji∆Po,i
is ob ious ha mul iplying hem,weob ain aze oblock
in posi ion(2,1).
Ξ22:In his posi ionweha easimila s uc u easin
posi ion(1,1),due o he ze oblock.Howe e , he
ma icesin block(2,2)o Ξija edi e en oeacho he .
So,he e, heo de ing o hema ix p oduc smus be
p ese ed.Mul iplying omϕN(N)un il ϕN(1),we
ob ain hesameblock ha in ma ix Ξ(Po+1)N.
Ξ12:This blockhascomplex s uc u e.Al houghi could
be ound bymul iplying hema ices,weha eno
donei becausei doesno a ec os abili yo u he
de elopmen s.
This comple es hep oo .2
18 h IFAC Wo ld Cong ess (IFAC'11)
Milano (I aly) Augus 28 - Sep embe 2, 2011
9205