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

Stabilityand PerformanceofNetworked ControlSystemswithTime-multiplexed Sensorsand OversampledObserver⋆ L.Orihuela,F.G´omez-Estern,F.R.Rubio Dpto.Ingenier´ıa deSistemasyAutom´atica,Universidad deSevilla. Camino delosDescubrimientos s/n,41092 Seville,Spain {orihuela,fabio,rubio}@us.es Abstract:Inthis paperweanalyze thescenario whereasetofwireless sensorsaretimemultiplexedin ordertoreduce thetrafficloadand energyconsumptionin aNetworkedControl System.Thepossible choicesforschedulingthetransmissionofmeasuresare explored under the conceptofperiodic feedbackpatterns,i.e.repeatedsequencesofdifferentmeasures.The lengthand structureofsuch patternsare chosento guarantee stabilityofthe closedloopand to minimizeapredefined performance index.The controlschemeis enhancedwithacontroller-side observer running ataratehigherthanthesamplingrate,forwhichanovelstabilityproblemis statedand solved. Keywords:NetworkedControlSystems,Wireless sensors,Periodic systems. 1.PROBLEMSTATEMENT InNetworkedControlSystems(NCS)reducingthe energy consumption bylimitingthetransmission data-ratesis crucial. However,less informationmayimply worseperformance or,atworst,causestabilityproblems.Another issuerelatedtofeedbackcontrolsystemwithwireless distributedsensorsliesin thefact thatenergyconsumption maybestrongly unbalanced,requiringfrequentbattery changesforsomesensors,while otherareundermostly idle.Energy-aware controlshould implementenergybalancing algorithmsacross thenetworkand beable to reschedule the completefeedback strategyaccordingtoenergydemandsand availability.Howeverthis rescheduling capabilityrequiresproperknowledgeofthesystemin order torecomputethestabilityconditions. System s1 s2 sm PacketNetwork Observer Controller y1 k y2 k ym k uk uk ˆxk Fig.1.Networkedcontrolscheme Theunderlyingtechnologythatweproposetodealwith scheduledfeedbackis theuseofpartialobservers.Instead ofsendingthe completeoutputvector,only one component travels overthesharedmediumatoneinstant.Onthe receivingsideofthenetwork,theobservermanagesto ⋆Theauthorswouldliketo acknowledgeCICYT(GrantDPI2010- 19154),and theEuropeanCommission(EC) (FeedNetBackProject, grantagreement223866),forfundingthiswork. reconstruct thewhole stateofthesystem,whichwill be usedforcontrolpurposes.Figure1describestheglobal scheme.Duetothe characteristicsofnetworkedsystems, thesystemis describedin discretetime: xk+1=Axk+Buk,yk=Cxk,(1) wherexk∈Rn,yk∈Rmand uk∈Rrarethestate, theoutputand the controlsignal, respectively.A,B,C areknownmatricesofappropriatedimensions.AsFigure1suggests,there existsmsensors,possibly spatially distributed,whichsample eachoneofthe componentsof theoutputvector.Ineachsamplingperiodk,only one sensorcan usethenetworktosend itspacket.Assuming anidealnetwork(withoutnoise,delays ordisturbances), atinstantkthesensorj(j=1,..., m)sendstheoutput yj k=EjCxk,whereEj,j=1,..., m,arematricesof appropriatedimensions(withelementsonesand zeros) usedtodefineanypartialoutput. Inthe controllersideofthe communicationthereis an observertryingtoestimatethestateofthesystemusing theinformationofthereceived partialoutput (see Figure 1).Thedynamicsoftheobserveris, ˆxk+1=Aˆxk+Buk+Ljhyj k−ˆyj ki,(2) ˆyk=Cˆxk,(3) whereˆxk∈Rnand ˆyk∈Rmarethestateand theoutput oftheobserver,respectively.MatricesLj,j=1,..., m, aretheobservationgainswhich,in general, aredifferent depending onthereceivedoutput. Every samplingperiod,the controllerbuildsacontrol signaldefined byuk=Kˆxk,whereKis the controller matrix ofappropriatedimensions. Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011 978-3-902661-93-7/11/$20.00 © 2011 IFAC 9200 10.3182/20110828-6-IT-1002.02435 Finally,the errorbetweenthestateofthesystemand the stateoftheobserveris definedas ek=xk−ˆxk.(4) Thepurposeofthis paperis thedesignoftheobserver (2)-(3)in suchawaythatsomestabilityand optimal propertiesfortheobservationerror(4)were ensured.We proposeaperiodic observer,in whichapattern,defined off-line,is repeatedall time.This patternindicateswhat outputis sentin whatinstant. Thefollowingdefinitionswill beusefulin futurepropositionsand their proofs. Definition1.AmeasurementpatternϕN∈RNis avectorwhose componentsdefinewhichoutputis sent throughthenetwork.Thatis, ϕN(i)≡{ji:ji=1,..., m},i=1,..., N.(5) Hence,ϕN(i)=jiimpliesthatsensorjiusesthenetwork in thei-th positionofthepattern.Withthis definitionit is possible to grantprioritytosomeoutputsovertherest. Definition2.Interval [ϕN(i),ϕN(j)]denotesall the consecutivesamplesbetweenϕN(i)and ϕN(j). 1.1Literaturereviewand openproblems Intheliteraturetherearerelatedworks asRehbinderand Sanfridson(2004)foroptimalperiodic linearquadratic regulator;Zhang and Hristu-Varsakelis (2006);Jiangetal. (2008)forperiodic observers;Hristu-Varsakelis and Zhang (2008)forperiodic Kalman filtersorSinopoli etal. (2004); Guptaetal. (2007)forKalman filtersubject tointermittentobservations. However,noneoftheseworks solvetheproblemoffinding anoptimalpatternwhichmanagesthesentofthe outputs.InZhangetal. (2006);Hristu-Varsakelis and Zhang(2008) the communicationsequencesare chosenin suchawaythat thereachabilityand observabilityofthe systemarepreserved.However,foraconcretesystem may existseveralpatternsthatpreservetheobservabilityand reachability.Howtochoosetheoptimaloneamongthem still remainsanopen problem.Finally,in Luetal. (2003) aperiodic patternis chosenminimizing anH∞norm. Moreover,forthebestofourknowledge,theredoesnot existanyworkin whichtheobserveris running ata frequencyhigherthanthesamplingrate,suitable when the controllawcanchangefasterthanthedata arrival rate.Thoseproblemsarestudiedin Section2. 2.OVERSAMPLEDOBSERVER Bylettingthe controllerand observer run atahigherfrequencythanthetransmissionofpartialoutputs,therewill besamplinginstantswithoutexternalfeedback.Between twoconsecutivetransmissions,thesensorsremain asleep forPoobservation periods.Sothedynamicsoftheobserver differsforthetwotypesofperiods,observation periods (OP)and measurementperiods(MP). (MP):ˆxk+1=Aˆxk+Buk+Liyi k−ˆyi k; (OP):ˆxk+1=Aˆxk+Buk. Figure2showsthe completepatternforacasein which three observation periodshavebeenintroduced. (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.Timeschedulingfortheoversampledobserver Analyzingbothsetsofequations,and definingtheaugmentedstatezT k=xT keT k,thefollowingproposition givesthe complete evolutionofthesystemalong a full period.Theproofis givenin Appendix A. Proposition1.GivenameasurementpatternϕN∈RN, the evolutionofthesystemfromktok+(Po+1)Nis definedas zk+(Po+1)N=  (A+BK)(Po+1)NΞ12 0Y ji∈[ϕN(N),ϕN(1)] [ϑji]ek zk =Ξ(Po+1)Nzk,∀k,(6) whereϑji=(A−LjEjC)APoand Ξ12 is amatrix with complex structure. 2.1Stability Thefollowinglemmaestablishesthestabilityofthesystem. Lemma 2.Thediscrete-timesystemwithevolutiongiven byProposition1is asymptotically stable if and only if the eigenvaluesofthefollowingmatricesareinsidetheunit circle: •Ξ11 =A+BK, •Ξ22 =Qji∈[ϕN(N),ϕN(1)][ϑji]. Theproofis immediateusingfairly extendedeigenvalue propertiesoftriangularmatrices. Example.Considerthediscrete-timesystemfromIshii and Francis (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) observedwith patternϕ2=[1,2]. Assumethat theobservergainsareL1=[0 −2.2−0.8]Tand L2=[0.500]T. UsingLemma 2,thestabilityoftheobservationerror withoutobservation periodsis ensured.The eigenvaluesof Ξ22 areλ(Ξ22)={0,0.4,0.5}.However,withPo=1,the eigenvaluesofΞ22 areλ(Ξ22)={0,−2,0.5},sotheobservationerroris unstable.However,choosing,forexample, L1=[0 −2−0.67]Tand L2=[1 0 0]TwithPo=1yields λ(Ξ22)={0,0.83,0.09}sotheobserveris stable,buta reductionofdatathroughthenetworkis achieved. 18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011 9201 2.2ObserverDesign Inordertoensurethestabilityofthewhole NCS,the eigenvaluesoftwomatricesmustlayinsidetheunit circle. Assumingthat thesystemis stabilizable,wehaveto designtheobserverin suchawaythat the errordynamics becomesasymptotically stable.AnLMI-basedmethodis presented usingideasfromperiodic systemtheory. Ap-periodic discrete-timesystemis described byxk+1= Akxk,wherexkrepresentsthestateand Akarep-periodic matrices,thatis,Ak+p=Ak,∀k. Animportantresult relatedtothesesystemsis thesocalledPeriodic LyapunovLemma,see Bittantietal. (1985),actually anextensionoftheLyapunovLemmafor this class ofsystems. PeriodicLyapunovLemma.Thep-periodicsystem xk+1=AkxkwithAk+p=Ak∀k, isasymptoticallystable if and onlyif there existsa p-periodicmatrixPk>0such that AT kPk+1Ak−Pk<0,∀k∈{1,..., p}.(8) Thedynamicsofthe errorbetweenstateoftheplantand oftheobservercan bedescribedasaperiodic system.The proofofthefollowingpropositionis omitted duetospace restrictions. Proposition3.GivenameasurementpatternϕN∈RN and aconstantPo∈N,thedynamicsofthe errorgivenin Proposition1is equivalent to anN-periodic systemwith periodic matrix ϑk=(A−LkEkC)APo,∀k∈{1,..., N}. ThefollowingtheoremproposesanLMI-based design procedureto obtain theobservationmatrices. Theorem4.Thedynamicsoftheobservationerror,given byProposition1,is asymptotically stable if and only if there existsanN-periodic positivedefinitematrix Pkand anN-periodic matrix Wkofappropriatedimensions such that thefollowingLMIsaresatisfied, Pk−1(APo+1)TPT k−(APo)TCT(Ek)TWT k ∗Pk>0,(9) ∀k∈{1,..., N},whereP0=PN.Then,theobservation matricesaredefinedasLk=P−1 kWk. Proof.ToproveTheorem4weneedto apply Schur complement tothePeriodic LyapunovLemma(8).Then substituteAkbyϑkand defineWk=PkLkto obtain conditions(9).2 Asnoconservatismhasbeenadded,Theorem4canalsobe usedtofind themaximumnumberofobservation periods (Po) thatcan beintroduced betweentwoconsecutive samplings. 2.3Patterndesign In Section2.1,wehadshownthe conditionsthatmustbe verifiedsothat thedynamicsoftheobservationerrorwere asymptotically stable giventheobservationmatricesLj. In Section2.2,aLMI-based designmethodwasproposed to obtain theobservationmatricesgivenameasurement pattern.Thenextstepin this process is,among all possible patterns,whichonestabilizesthesystem? Itis obviousthat thosepatternswhich donotobservethe dynamicsoftheobservationerrorcould bediscard.There existsomeresearchesin theliteratureoncontrollability and observabilityofgeneric periodic system,see Guo and Qiao(2004).However,forthe caseunderstudy,only the outputmatrix EjCis periodic.We canenunciateherean easierobservabilitytest. Lemma 5.System(1)is observable atinstantkif and only if matrix Q:=     C′ C′A′ . . . C′A′n−1     (10) hasrankequalton,whereA′:= ANand C′:=     EkC Ek+1CA . . . Ek+N−1CAN−1     . Theproofis omitted duetospace limitation,butit can bemadebyapplyingCayley-Hamiltontheoremasin the classicaltest.Finally,weintroduce thedefinitionof completeobservability. Definition.System(1)is completely observable if and only if it is observable forall k=1,..., N. Therefore,if apatternachievestheobservabilityofthe system(1),it will beable tostabilizetheobserver.The observabilitytestis clearly easier,computationally spoken, thantheLMI-basedstabilitytest,sowe canstudyoff-line everypossible patternoflengthN. Minimizationofacostindex Ifweneedtocomparethequalityofsomestable patterns, we canestablishanumericalindex.This indexcould be definedtakinginto account theobservationerror. J(i)= i X k=0 eT kQek,(11) beingQapositivedefinitematrix.Asimilarideawas firstly introducedin Havreand Skogestad(2003),butfor optimalfeedback stabilizationofasystemwithonly one unstable pole.Thedependence oftheindex(11)with the chosen patternis verycomplex,butit is possible tocomparethem,atleast,in anumericalway.Inthe following,assumethatPo=0. Proposition6.GivenapatternwithlengthNand a positivedefinitematrix Q,the costindexJ=P∞ k=0eT kQek can be calculatedas J=eT 0"∞ X n=0 (αT N)nΦN(αN)n#e0,(12) where 18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011 9202 αN=ϑNϑN−1...ϑ2ϑ1, ΦN= N X i=1 αT iQαi. Theproofis immediateby substitutingthe evolutionof thesystem(6)withPo=0in the costindex(11). FromProposition6,it turnsout thedependence between theindexand theinitialconditionoftheobservationerror. Itis moreinterestingthesituationin whichtheinitial conditionoftheobservationerroris bounded.Thatis, weknowtheupperbound ofke0k2.Forsuchcase,we needtobound the costindex,employingtherelation: xTAx≤λmax{A}kxk2. So,the costindex(12)is bounded byJ≤λmax{βN}ke0k2, withβN=P∞ n=0(αT N)nΦN(αN)n. Foraperiodic pattern,therearetwocharacteristicswe havetochoose:1) thelengthofthepatternand 2)its structure.Thelengthofthepatternis critical, in such awaythat,if it is notfixeda priorithenumberof combinationsgrowsuntil infinity.Betweenall possible patternsoflengthNwe canchoosetheonewhominimizes themaximumeigenvalueofmatrix βN. Remark.Duetothepatternstability,matrix αNhas all itseigenvaluesinsidetheunit circle.So,theinfinite seriescan bereplaced byafiniteonewithoutincurringin practicalerrors,aswewill see in the examples.Finally, this finiteseriescan be calculated numerically. Example.Optimalpattern design Considerthediscrete-timesystemfromHristu-Varsakelis and Zhang(2008).Inthatpaper,the choiceofthepattern wasmadebasedonstabilitypropertiesonly. 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) There exist twopossible outputstosend throughthe networky1 k=[1 0]ykand y2 k=[0 1]yk.Theweighting matrix is chosenasQ=diag{1,1}.Itis assumedthat e0=x0,thatis,theinitialconditionfortheobserveris exactly zero. Theobjectiveis tochooseand designtheoptimalpattern among all possible ones.We call optimalpatterntheone whichminimizesthe costindex(12).Tocalculatethe cost index,afinitehorizonis needed.Here,this horizonis chosenlargerenoughtoneglect the errors. Figure3depictsthelowermaximumeigenvalueobtained fordifferentlengths.Rememberthat the costis bounded byJ<λMAXeT 0e0. Itcan beseenthatfromsomevalue,thelargerpattern length doesnotimply thelowercosts.Forthis example,a lengthof7is enough.Moreprecisely,theoptimalpattern is [1 1 2 1 2 2 2]. 2 4 6 8 10 12 14 21 22 23 24 25 26 27 28 λMAX Patternlength Fig.3.Maximumeigenvalueforvariouspatternlengths 3.EXPERIMENTALAPPLICATION 3.1Platformdescription,modelingandcontrol The controlstrategydescribedaboveis appliedto a two-degree-of-freedomdirectdriverobot,which hasbeen designedand developed bytheDepartmentofSystems Engineering and AutomationatUniversityofSeville,see Mill´anetal. (2010).Theobjectivewill betomaintain therobotatitsuprightequilibriumpoint,similarasan inverted pendulum. Therobotconfigurationis schematically shownin Figure 4.Thefirstlink(whichis between bothmotors)will be termedasshoulderwhereasthesecond link(whichis betweenthesmallermotorand the edgeoftherobot) is theelbow. l1 lc1 q1 q2 lc2 l2 c.o.g. Shoulder c.o.g. Elbow Fig.4.Two-degree-of-freedomrobotdiagram Theselectedcontrolsoftwareis thexPC TargetenvironmentMostermanetal. (2005)withMATLAB/Simulink. Itis well-knownthat thedynamicsofarobotic manipulatoris extremely non-linear.To apply theresultsofthis paper,therobotwill beoperatedaround theunstable upright equilibriumdefined byqe=[π0],˙qe=[0 0]. Tolinearize thesystemand obtain matrix Aand Bofequation(1) ameansquareiterativeidentification procedurehasbeen followed. Inordertoperformthetestsaninitialcontrolbasedon feedbacklinearizationis applied,whichsteerstherobot fromits stable downward position(bothlinks stopped 18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 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) Position Error (rad) Po=0 Po=1 Po=2 Po=3 Po=4 Po=5 Fig.6.Positionerrorsfortheoversampledobserverwith differentnumberofobservation periods in their lowerpositions) tothesurroundingsofitsunstable uprightequilibrium.This controlleris appliedassumingthestate completely accessible.Once thatpositionis reached,periodic observerpluslinearcontrolleris switched. 3.2Experimentalresults The experimentis thesamein all cases.Theobserveris switched(fromthenonlinearcontrol) fromtheupright equilibriumpoint.Then,anadditivedisturbance in the torqueis appliedatt=10 seconds. Althoughthelinearmodelsuggeststhat thesystemis observable fromeachoftheoutput (classicalobserver), experiments showthat this is not truefortherealrobot. Forinstance,Figures5band 5cshowthatit is notpossible to observethepositionofthe elboworthevelocityof theshoulder.Theonly componentofthestatethatcan beobservedis thepositionoftheshoulder,asFigure5a shows. Tocomparethethroughputoftheoversampledobserver weinsertadditionalobservation periodsand repeat the same experiment.Thepatternis chosenasϕ=[3,2,3,4], thatis,thepositionoftheshoulderis theonly component ofthestatevectorwhichis being observed(theothersare directly measured).Figure6depictsthepositionerrorof theshoulderwhen differentnumberofobservation periods areincluded.Asexpected,asthenumberofobservation periodsgrows,the erroris higher.The constantsteadystate errorareduetothemodelerrors. Finally,weproveadifferentpatternand compareits throughputwiththeoneproposed before.Theobservation erroraredepictedin Figure7. 4.CONCLUSION Wehaveanalyzedin depththedifferentfeedbackpossibilitiesavailable whenasetofwireless networks embedded in acontrollooparetime-multiplexedin orderto avoid collisionsand reduce thetotalnetworktrafficand energy consumption.Finally,asaresult oftheperformance analysis,newcriteria forfeedbackpattern 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) Position Error (rad) [3 2 3 4] [3 2 3 4 3] Fig.7.Comparison between differentpatterns networks havebeenset.Theresultshavebeenillustrated withsimulationsand in areallaboratory2-DOFrobot. Furtherwork should considerthe computationoftheperformance indexofthefeedbackpatternswithout theneed ofnumericalestimations. REFERENCES Bittanti, S., Bolzern,P., and Colaneri, P.(1985).The extended periodic Lyapunovlemma.Automatica,21(5), 603–605. Guo,G.and Qiao,J.F.(2004).Controllabilityofperiodic systems:continuousand discrete.IEE Proceedings: Controltheoryand applications,151(4),439–446. Gupta,V., Hassibi, B., and Murray,R.M.(2007).Optimal LQGcontrolacross packet-droppinglinks.Systemsand ControlLetters,56(6),439–446. Havre,K.and Skogestad,S.(2003).Selectionofvariables forstabilizingcontrolusingpole vectors.IEEE TransactionsonAutomaticControl,48(8),1393–1398. Hristu-Varsakelis,D.and Zhang,L.(2008).LQGcontrol ofnetworkedcontrolsystemswithaccess constraintsand delays.InternationalJournalofControl,81(8),1266– 1280. Ishii, H.and Francis,B.A.(2002).Stabilizationwith controlnetworks.Automatica,38(10),1745–1751. Jiang,C., Zou,D., and Zhang,Q.(2008).Stabilization ofnetworkedcontrolsystems via periodically timevaryinglocalcontroller.InProceedingsofthe7thWorld Congress onIntelligentControlandAutomation,7965– 7969.Chongqing,China. Lu,L., Xie,L., and Fu,M.(2003).Optimalcontrol ofneworkedsystemswithlimitedcommunication:a combined heuristic and convexoptimizationapproach. In42ndConference onDecisionandControl,1194– 1199.Maui, Hawaii, USA. Mill´an,P., Orihuela,L., Vivas,C., and Rubio,F.(2010). AnoptimalcontrolL2-gain disturbance rejection design fornetworkedcontrolsystems.InProceedingsofthe AmericanControlConference.Baltimore,Maryland, USA. Mosterman,P.J., Prabhu,S., Dowd,A., Glass,J., Erkinnen,T., Kluza,J., and Shenoy,R.(2005).Handbook ofNetworked andEmbeddedControlSystems,chapter EmbeddedReal-TimeControlvia MATLAB,Simulink, and xPC Target,419–446.BirkhuserBoston. 18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 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) Position (rad) Shoulder position System Observer (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) Position (rad) Elbow position System Observer (b)ϕ=(1,4,3,4) 0 5 10 15 20 25 −3 −2 −1 0 1 2 3 4 5 6 Time (s) Velocity (rad/s) Shoulder velocity System Observer (c)ϕ=(1,2,1,4) Fig.5.Comparisonofthequalityoftheobserverfordifferentpatterns Rehbinder,H.and Sanfridson,M.(2004).Scheduling of alimitedcommunicationchannelforoptimalcontrol. Automatica,40(3),491–500. Sinopoli, B., Schenato,L., Franceschetti, M., Poolla,K., I., J.M., and Sastry,S.S.(2004).Kalman filtering withintermittentobservations.IEEE Transactionson AutomaticControl,49(9),1453–1464. Zhang,H.S., Duan,G., and Xie,L.(2006).Linear quadratic regulationforlineartimevaryingsystems withmultiple inputdelays.Automatica,42(9),1465– 1476. Zhang,L.and Hristu-Varsakelis,D.(2006).Communicationand controlco-designfornetworkedcontrolsys- tems.Automatica,42(6),953–958. Appendix A.PROOFOFPROPOSITION1 Supposethat therearemsensorsand Noutputs,with N≥m.FurthersupposethatPoobservation periodsareintroduced betweentwoconsecutivesamples. Theordering ofthemeasurementsis defined byϕN≡ {ji:ji∈{1,..., m}and i=1,..., N}. The evolutionoftheaugmentedstateforameasurement periodcan bewrittenas zk+1=hA+BK−BK 0(A−LjiEjiC)izk=Λjizk.(A.1) Foranobservation period,thefollowingrelationshold, zk+1=hA+BK−BK 0Aizk=∆zk. Supposethatfrominstantktok+Po−1,no outputis receivedfromthesystem.AfterPoobservation periods, thefollowingrelationshold, zk+Po=(A+BK)PoΨPo 0APohxk eki=∆Pozk,(A.2) whereΨPo=PPo i=1(A+BK)Po−iBKAi−1.Itis easy to obtain this matrix bymultiplyingmatrix ∆byitself Po times. Supposethat,afterPoobservation periods,the controller receivesameasurement.Hence the complete evolutionof theobservation periodsplusonemeasurementperiodcan beobtained using(A.1)and (A.2). zk+Po+1=(A+BK)Po+1(A+BK)ΨPo−BKAPo 0θjiAPozk =Λji∆Pozk Notethat the element (2,2)ofpreviousmatrix is exactly θjiAPo≡ϑji.This structureis repeatedeveryPoobservation periodsand onemeasurementperiod.So,it is straightforwardto obtain thefollowingrelation: zk+(Po+1)N=(Λji∆Po)Nzk. Finally,wehavetoprovethat thefollowingequalityis true: (Λji∆Po)N=Ξ(Po+1)N,(A.3) whereΞ(Po+1)Nhasbeen defined previously in thehypothesesoftheproposition.Todothat,wewriteΞ(Po+1)N (Ξhereinafter)usingitsblock structure,thatis,Ξ≡ {Ξij:i,j∈{1,2}},and westudythe equivalence between thedifferentblocks. zk+(Po+1)N=  (A+BK)(Po+1)NΞ12 0Y ji∈[ϕN(N),ϕN(1)] [ϑji]ek (A.4) Ξ11:Duetothe zeroblockin position(2,1),whenmultiplyingNtimesΛji∆Po,in position(1,1)weobtain (A+BK)(Po+1)N. Ξ21:BecauseofthestructureofthematricesΛji∆Po,it is obviousthatmultiplyingthem,weobtain azeroblock in position(2,1). Ξ22:Inthis positionwehaveasimilarstructureasin position(1,1),duetothe zeroblock.However,the matricesin block(2,2)ofΞijaredifferent toeachother. So,here,theordering ofthematrix productsmustbe preserved.MultiplyingfromϕN(N)until ϕN(1),we obtain thesameblockthatin matrix Ξ(Po+1)N. Ξ12:This blockhascomplex structure.Althoughit could befound bymultiplyingthematrices,wehavenot doneit becauseit doesnotaffect tostabilityorfurther developments. This completestheproof.2 18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011 9205