Stability and Performance of Networked Control Systems with Time-multiplexed Sensors and Oversampled Observer

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. REFERENCES Bi an i, S., Bolze n,P., and Colane i, P.(1985).The ex ended pe iodic Lyapuno lemma.Au oma ica,21(5), 603–605. 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Sinopoli, B., Schena o,L., F ancesche i, M., Poolla,K., I., J.M., and Sas y,S.S.(2004).Kalman il e ing wi hin e mi en obse a ions.IEEE T ansac ionson Au oma icCon ol,49(9),1453–1464. Zhang,H.S., Duan,G., and Xie,L.(2006).Linea quad a ic egula ion o linea ime a yingsys ems wi hmul iple inpu delays.Au oma ica,42(9),1465– 1476. Zhang,L.and H is u-Va sakelis,D.(2006).Communi- ca ionand con olco-design o ne wo kedcon olsys- ems.Au oma ica,42(6),953–958. 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