Stability Analysis of Unconstrained Receding Horizon Control Schemes

S abili y Analysis o Uncons ained Receding Ho izon Con ol Schemes Von de Uni e si ¨a Bay eu h zu E langung des akademischen G ades eines Dok o de Na u wissenscha en (D . e . na .) genehmig e Abhandlung o geleg on Ka l Wo hmann aus Hanno e 1. Gu ach e : P o . D . La s G ¨une 2. Gu ach e : P o . D . And ew Richa d Teel 3. Gu ach e : P o . D . Hans Jose Pesch Tag de Ein eichung: 15. Dezembe 2011 Tag des Kolloquiums: 27. Ap il 2012 Con en s Deu sche Zusammen assung III Summa y IX 1 Con ol Sys ems, S abili y, and Feedback 1 1.1 Con ol Sys ems and P oblem Fo mula ion . . . . . . . . . . . . . . . . . 1 1.2 Closed Loop Con ol and Asymp o ic S abili y . . . . . . . . . . . . . . 5 1.3 Sampled-Da aSys ems............................ 11 1.4 Ne wo ked Sys ems and Mul is ep Feedback . . . . . . . . . . . . . . . . 19 2 Receding Ho izon Con ol 21 2.1 In oduc ion ................................. 21 2.2 Te minal Equali y Cons ain s . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 Te minal Inequali y Cons ain s and Cos s . . . . . . . . . . . . . . . . . 30 2.4 Feasibili y................................... 35 3 S abili y and Subop imali y o RHC Schemes 39 3.1 Relaxed Lyapuno Inequali y . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Asymp o icS abili y............................. 44 3.3 Linea P og am................................ 46 3.3.1 Auxilia y Resul s . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.4 Ins an aneous Con ol o he Linea Wa e Equa ion . . . . . . . . . . . . 57 3.4.1 Cons uc ing Sui able S age Cos s . . . . . . . . . . . . . . . . 57 3.4.2 Ve i ying Assump ion 3.2 and Closed Loop S abili y . . . . . . 58 3.4.3 Nume ical Resul s . . . . . . . . . . . . . . . . . . . . . . . . . 62 4 Sensi i i y Analysis 65 4.1 In luence o he Op imiza ion Ho izon . . . . . . . . . . . . . . . . . . . 66 4.2 Cha ac e is ics Depending on he Con ol Ho izon . . . . . . . . . . . . 71 4.2.1 P esen ing he Resul s . . . . . . . . . . . . . . . . . . . . . . 71 4.2.2 Symme y Analysis . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2.3 Mono onici y P ope ies . . . . . . . . . . . . . . . . . . . . . 83 4.3 Fu he Resul s................................ 89 4.3.1 Commen s on Assump ion 3.2 . . . . . . . . . . . . . . . . . . 89 4.3.2 Cos Func ional Inco po a ing a Te minal Weigh . . . . . . . 90 4.3.3 Example: Linea In e ed Pendulum . . . . . . . . . . . . . . 91 4.4 Algo i hms .................................. 92 4.4.1 BasicAlgo i hm.......................... 94 4.4.2 Ad anced Algo i hm . . . . . . . . . . . . . . . . . . . . . . . 100 I CONTENTS 5 Sampled-Da a Sys ems and G ow h Condi ion 105 5.1 Disc e iza ion and Sampled-Da a Sys ems . . . . . . . . . . . . . . . . . 110 5.1.1 Auxilia y Resul s o he P oo o Theo em 5.15 . . . . . . . . 117 5.2 Con inuous Time Coun e pa . . . . . . . . . . . . . . . . . . . . . . . . 122 5.2.1 Auxilia y Resul s . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.3 G ow hCondi ion ..............................129 5.3.1 Exponen ial Con ollabili y . . . . . . . . . . . . . . . . . . . . 130 5.3.2 Fini e Time Con ollabili y . . . . . . . . . . . . . . . . . . . . 132 5.3.3 Analy ical Example . . . . . . . . . . . . . . . . . . . . . . . . 134 5.3.4 G ow h Condi ion and Disc e iza ions . . . . . . . . . . . . . . 138 5.3.5 Al e na i e P oo o Theo em 5.31 . . . . . . . . . . . . . . . . 144 5.4 Accumula edBounds.............................147 5.4.1 Reac ion Di usion Equa ion: Impac o Assump ion 5.38 . . . 149 5.4.2 Synch onous Gene a o : A Case S udy . . . . . . . . . . . . . 150 5.5 Compa ison wi h O he App oaches . . . . . . . . . . . . . . . . . . . . 153 5.5.1 A Linea Fini e Dimensional Example . . . . . . . . . . . . . . 156 5.5.2 Synch onous Gene a o . . . . . . . . . . . . . . . . . . . . . . 157 A Supplemen a y Resul s 159 A.1 Fini eEscapeTimes.............................159 A.2 In e edPendulum..............................160 Lis o Tables 163 Lis o Figu es 165 Bibliog aphy 167 II Deu sche Zusammen assung Das Thema diese Disse a ion is die modellp ¨adik i e Regelung (MPC) — im Englischen auch “ eceding ho izon con ol” genann . Typische weise wi d diese Me hodik eingese z , um ein au einem unendlichen Zei ho izon ges ell es Op imals eue ungsp oblem app o- xima i zu l¨osen, beispielsweise um eine gegebene Regels ecke an einem A bei spunk zu s abilisie en. Alle dings sind Op imals eue ungsp obleme mi einem unendlichen Op i- mie ungsho izon im Allgemeinen kaum ode nu mi seh hohem Rechenau wand l¨osba . Deshalb wi d de Zei ho izon abgeschni en und olglich das Ausgangsp oblem du ch eines au einem endlichen Ho izon e se z . In de modellp ¨adik i en Regelung we den die olgenden d ei Sch i e du chge ¨uh : âDas Ve hal en de Regels ecke wi d, ausgehend on einem Modell und de zule z o genommenen Messung, p ¨adizie , um das Op imals eue ungsp oblem zu l¨osen und dami einhe gehend eine Folge on S eue we en zu be echnen. âDas e s e Elemen diese Folge wi d an de S ecke implemen ie . âDe S a zus and des be ach e en Op imals eue ungsp oblems aus dem e s en Sch i wi d ak ualisie . Zudem wi d de Op imie ungsho izon o w¨a s in de Zei e schoben, was den englischen Namen des Ve ah ens e kl¨a . Dieses Vo gehen wi d ad in ini um wiede hol . So wi d eine S eue olge au dem un- endlichen Zei ho izon e zeug . Die modellp ¨adik i e Regelung gene ie also eine Folge on Op imals eue ungsp oblemen mi endlichen Op imie ungsho izon , um die gesuch e L¨osung zu app oximie en. Insbesonde e die M¨oglichkei S eue - und Zus andsbesch ¨ankungen explizi zu be ¨uck- sich igen ha in den le z en Jah zehn en zu e s ¨a k em In e esse an diese Me hodik ge ¨uh . Des Wei e en w¨achs die Anzahl de Indus ieanwendungen s e ig, siehe [33,100]. Ein wei e e Vo eil diese L¨osungss a egie is die inh¨a en e Robus hei eines geschlosse- nen Regelk eises — zum Beispiel gegen¨ube ex e nen S ¨o ein l¨ussen ode Modellie ungs- ehle n, siehe [102]. T o z de wei en Ve b ei ung modellp ¨adik i e Regelungs e ah en in de Anwendung is die zugeh¨o ige S abili ¨a sanalyse nich ein ach. Die e s en Ans¨a ze basie en au (k¨uns lichen) Endbedingungen und -kos en, siehe [17, 66]. Diese du ch die heo e ische Analyse mo i ie en P oblemmodi ika ionen scha en zus¨a zliche Ein lussm¨oglichkei en, um S abili ¨a seigenscha en des geschlossenen Regelk eises zu e besse n. Weil die indus ielle P axis jedoch wei es gehend au den Einsa z diese Hil smi el e zich e , besch¨a igen wi uns mi de so genann en un es ingie en modellp ¨adik i en Regelung, die wede Endbedingungen noch Endkos en in die P oblem o mulie ung au nimm . Diesbez¨uglich kann de in [39] o ges ell e Ansa z als unse Ausgangspunk be ach e we den. In diesem wi d ein Op imie ungsp oblem konzipie , um asymp o ische S abili ¨a des bzw. G¨u eabsch¨a zungen an den mi els modellp ¨adik i e Regelung geschlossenen III DEUTSCHE ZUSAMMENFASSUNG Regelk eis he zulei en. Posi i i ¨a des zugeh¨o igen Subop imali ¨a sg ades is eine no wendige und hin eichende S abili ¨a sbedingung ¨u die Sys emklasse, welche eine o ausgese z e Kon ollie ba kei sbedingung e ¨ull . Gliede ung und eigene Bei ag Diese A bei is in ¨un Kapi el gegliede . Die e s en zwei ¨uh en in g undlegende Konzep e sowie die P oblems ellung ein. Anschließend wi d in Abschni 3.1 die in [39] en wickel e Me hodik ku z zusammenge ass , welche als Ausgangspunk ¨u das wei e e Vo gehen angesehen we den kann. Danach we den eigene Resul a e da ges ell . Diese Gliede ung soll sowohl eine Inhal s¨ube sich bie en als auch den Bei ag de in diese A bei en wickel en Resul a e zu de Analyse un es ingie e modellp ¨adik i e Regelungs e ah en e l¨au e n. +Im e s en Abschni on Kapi el 1 wi d das g undlegende Konzep eines Kon oll- sys ems einge ¨uh . Dabei wi d un e ande em die Zul¨assigkei on Kon oll olgen behandel . Zus¨a zlich wi d die op imale We e unk ion de inie . In Abschni 1.2 wi d die einge ¨uh e Te minologie e wende , um die wesen lichen Un e schiede eines geschlossenen Regelk eises im Ve gleich zu o enen Regelke e he auszua bei en. So e laub de geschlossene Regelk eis beispielsweise au ¨auße e S ¨o ungen ode Meß ehle zu eagie en. In diesem Zusammenhang wi d de Beg i de asymp o- ischen S abili ¨a ben¨o ig , um die allgemeine P oblems ellung zu de inie en. In den le z en beiden Abschni en on Kapi el 1 besch¨a igen wi uns sowohl mi Ab as - als auch mi Ne zwe ksys emen — zwei wich ige Sys emklassen, an denen die E gebnisse de n¨achs en Abschni e demons ie we den. Dabei wi d insbeson- de e gezeig , wie on Di e en ialgleichungen induzie e Sys eme als zei disk e e Sys- eme behandel we den k¨onnen. Zum Abschluss des Kapi els wi d de ¨u diese A bei wich ige Beg i de R¨uckkopplung bzgl. meh e e Ab as in e alle de inie . +In Kapi el 2 wi d die modellp ¨adik i e Regelung — eine Me hodik um Op imal- s eue ungsp obleme au unendlichem Zei ho izon app oxima i zu l¨osen — in ih en e schiedenen Face en be ach e . Beginnend mi de modellp ¨adik i en Regelung in ih e ein achs en Fo m: un es ingie es MPC. Anschließend wi d die gleiche Kon olls a egie um zus¨a zliche Endkos en ode -bedingungen e wei e . Die Be ¨ucksich igung diese k¨uns lich zu den in jedem I e a ionssch i zu l¨osenden Op imals eue ungsp oblemen hinzuge ¨ug en Komponen en ¨uh zu e besse en S abili ¨a seigenscha en des MPC Algo i hmus. De da ¨u zu zahlende P eis is die schwie ige Au gabe, passende Endkos en zu en we en. Genau diese Nach eil is de G und da ¨u , dass in de Indus ie haup s¨achlich un es ingie es MPC zum Einsa z komm . Ein wei e e wich ige Aspek is die Zul¨assigkei modellp ¨adik i e Regelungs e ah en. Dazu we den die wesen lichen Ideen aus [99] in g oben Z¨ugen skizzie . +Am Beginn des olgenden d i en Kapi els wi d die in [39] en wickel e Me hodik ku z o ges ell . Diese e laub es, basie end au eine Kon ollie ba kei sannahme, eine elaxie e Lyapuno -Ungleichung siche zus ellen — ein wesen liches Hil smi el, um S abili ¨a des geschlossenen Regelk eises nachzuweisen. Da ¨ube hinaus lie e de um issene Ansa z einen Subop imali ¨a sindex, de angib , wie gu die mi MPC e ziel e Regelg¨u e im Ve gleich zu bes m¨oglichen is . Im olgenden Abschni 3.2 IV DEUTSCHE ZUSAMMENFASSUNG wi d de en sp echende S abili ¨a sbeweis au zei a ian e Kon ollho izon e e all- gemeine , eine kleine Modi ika ion, die insbesonde e im Ne zwe kkon ex genu z we den kann, um nich e nachl¨assigba e Ve z¨oge ungen sowie Pake aus ¨alle auszu- gleichen, siehe [47, 48]. Zudem wi d sich diese E wei e ung ¨u die He lei ung wei- e e E gebnisse als hil eich e weisen. Um die einge ¨uh e Me hodik anzuwenden, wi d die L¨osung eines linea en P o- g amms ben¨o ig , dessen G ¨oße dem Op imie ungsho izon in de modellp ¨adik i en Regelung en sp ich . In Abschni 3.3 wi d eine L¨osungs o mel ¨u dieses Op- imie ungsp oblem he gelei e , welche eine de Eckp eile ¨u die olgende S a- bili ¨a sanalyse un es ingie e MPC-Schema a is . Um die wesen lichen Beweis- sch i e besse da s ellen zu k¨onnen, wu den einige echnische De ails in einen Hil s- un e abschni ausgegliede . Anschließend wi d die be ei s e w¨ahn e L¨osungs o mel genu z , um zu zeigen, dass MPC das Regelungsp oblem au unendlichem Zei ho i- zon beliebig gu app oximie — o ausgese z de Op imie ungsho izon is hin- eichend g oß, ein Resul a im Einklang mi [32,120]. Im olgenden Abschni we - den die bishe igen E gebnisse anhand de linea en Wellengleichung e anschaulich . Insbesonde e wi d ins an ane Kon ollie ba kei igo os gezeig . Ins an an bedeu e hie , dass de MPC-Algo i hmus mi kleins m¨oglichem Op imie ungsho izon aus- ge ¨uh wi d. Diese Abschni basie au eine Zusammena bei mi Nils Al m¨ulle , siehe [4,5]. Die wich igs en Bei ¨age on Kapi el 3 sind åeine analy ische L¨osungs o mel ¨u das linea e P og amm, åein Beweis ¨u ins an ane Kon ollie ba kei de linea en Wellengleichung und ådie Ve allgemeine ung des S abili ¨a sbeweises aus [39] au den Fall zei a ian e Kon ollho izon e. Einige Resul a e dieses Kapi els wu den be ei s in [45, 46] in eine Vo ab e sion e ¨o en lich . Jedoch wu den insbesonde e die Beweise g ¨undlich ¨ube a bei e , um de en Nach ollziehba kei zu e leich e n. +In Kapi el 4 wi d eine Sensi i i ¨a sanalyse bzgl. de wich igs en Pa ame e du ch- ge ¨uh : Op imie ungs- und Kon ollho izon . Insbesonde e die Bedeu ung des Le z e en soll e man nich un e sch¨a zen. Wi beginnen mi dem Op imie ungs- ho izon . Die in Kapi el 3 he gelei e e Fo mel wi d dazu e wende pa ame e - abh¨angige S abili ¨a sgebie e zu be echnen. Dies e laub R¨uckschl¨usse au den un- e schiedlichen Ein luss des ¨ Ube schwing- und Abkling e hal ens und olglich au den En wu geeigne e S u enkos en ¨u MPC, siehe [6, 39]. Des Wei e en wi d de minimale s abilisie ende Ho izon , also de kleins e Op imie ungsho izon , de asymp o ische S abili ¨a ga an ie , genaue un e such . In diesem Zusammenhang wi d — ¨u passend gew¨ahl e Kon ollho izon e — linea es Wachs um bzgl. de akkumulie en Wachs umssch anken aus de o ausgese z en Kon ollie ba kei s- bedingung gezeig , was eine quali a i en Ve besse ung im Ve gleich zu den Ab- sch¨a zungen aus [120] en sp ich . Im da au olgenden Abschni be ach en wi Kon ollho izon e. Hie we den insbesonde e n¨u zliche Symme ie- und Mono onie- eigenscha en gezeig , welche ¨u die Algo i hmenen wicklung in Abschni 4.4 eine wich ige Rolle spielen. Abschni 4.2 bes eh aus zwei Teilen. Im e s en Teil we den die E gebnisse zusammenge ass w¨ah end im zwei en, de die Un e abschni e 4.2.2 V DEUTSCHE ZUSAMMENFASSUNG und 4.2.3 um ass , die en sp echenden Beweise da ges ell we den. F¨u diese wi d eine ausge eil e Beweis echnik ben¨o ig . Abschni 4.3 is in d ei eigens ¨andige Teile gegliede . Zue s besch¨a igen wi uns mi de o ausgese z en Kon ollie ba kei sbedingung. Danach wi d ein Beispiel eines linea en Pendels au einem Wagen be ach e . Die du chge ¨uh en nume ischen Tes s bes ¨a igen unse e heo e ischen Resul a e bzgl. des Kon ollho- izon s. Als d i es Thema we den Endgewich e und ih e Auswi kungen au den Subop imali ¨a sg ad behandel . In Abschni 4.4 we den Algo i hmen au Basis de du chge ¨uh en Sensi i i ¨a sanalyse en wickel . Weil de Rechenau wand bei wach- sendem Op imie ungsho izon schnell s eig , wi d diese Pa ame e ypische weise als Schl¨usselg ¨oße in MPC au ge ass . Die o ges ell en Algo i hmen nu zen das Konzep des Kon ollho izon s, um Absch¨a zungen ¨u die ga an ie e Regelg¨u e zu e besse n — ohne den Op imie ungsho izon zu e l¨ange n. Zudem wi d de en wickel e G undalgo i hmus wei e ausge eil , um ein e besse es Robus - hei s e hal en zu e zielen. Um die Vo eile de in diesem Abschni en wickel en Algo i hmen besse he auszus eichen, wi d das Beispiel des synch onen Gene a o s eingehend s udie , siehe [28,34,94]. Die Haup esul a e dieses Kapi els sind åSensi i i ¨a sanalyse bez¨uglich des Op imie ungsho izon s asymp o ische Absch¨a zungen ¨u den minimalen s abilisie enden Ho izon , åSensi i i ¨a sanalyse bez¨uglich des Kon ollho izon s Symme ie- und Mono- onieeigenscha en unse e Subop imali ¨a sabsch¨a zungen und åDesign zweie Algo i hmen basie end au den heo e ischen Resul a en, um den ben¨o ig en Op imie ungsho izon und olglich den Rechenau wand zu e- duzie en. +Das le z e Kapi el diese Disse a ionssch i wi d mi eine Falls udie eine Reak ions-Di usions-Gleichung begonnen, um das wei e e Vo gehen zu mo i ie en. In diesem Zusammenhang wi d eine zei kon inuie liche Ve sion unse e Kon ol- lie ba kei sbedingung einge ¨uh . Weil aus abge as e en Di e en ialgleichungen abgelei e e zei disk e e Regels ecken ein Ke nanwendungsgebie on MPC sind, we den E ek e un e such , die mi de Ve wendung eine e Disk e isie ungen e bunden sind. Hie bei we den neben posi i en Auswi kungen auch m¨ogliche Falls icke seh ku ze Ab as a en beleuch e — seh schnelle Ab as ung kann e o de lich sein, um wesen liche Eigenscha en des Ausgangssys em au sein abge- as e es Pendan zu ¨ube agen. Insbesonde e wi d gezeig , dass de Ansa z aus [39] ¨u klassisches MPC in Kombina ion mi beliebig eine Disk e isie ung nich an- wendba is . Beliebig eine Disk e isie ung en sp ich hie eine gegen Null s eben- den Ab as zei . Des Wei e en wi d de G enzwe dieses Disk e isie ungsp ozesses be echne . Diese G enzwe s imm mi seinem zei kon inuie lichen Pendan aus [103,104] ¨ube ein, was kl¨a , wie die Ans¨a ze [39] und [104] zusammenh¨angen. Um die beobach e en P obleme ¨u seh schnelle Ab as ung zu beheben, wi d eine Wachs umsbedingung einge ¨uh . Mi Hil e diese Bedingung k¨onnen zum Beispiel S e igkei seigenscha en, wie sie ypische weise ¨u Ab as sys eme gel en, in un- se e S abili ¨a sanalyse be ¨ucksich ig we den. Dazu wi d die Me hodik aus [39] um diese Annahme e wei e . Anschließend wi d gezeig , dass dieses Vo gehen VI DEUTSCHE ZUSAMMENFASSUNG das beobach e e P oblem l¨os . Zudem we den ein ach nachp ¨u ba e Bedingungen he gelei e , um diese zus¨a zliche Vo ausse zung zu e i izie en. In Abschni 5.4 we den so genann e akkumulie e Sch anken als al e na i e Kon ollie ba kei sannahme einge ¨uh und in unse e Technik zu Bes immung on G¨u eabsch¨a zungen eingebau . Diese akkumulie en Sch anken s ammen aus [120]. Um de en Auswi kungen zu un e suchen, wi d das Beispiel de Reak ions-Di usions- Gleichung wiede au geg i en. Insgesam ¨uh dieses Vo gehen au e besse e G¨u eabsch¨a zungen ¨u den mi els MPC geschlossenen Regelk eis. Im abschließen- den Abschni wi d die in diese Disse a ionssch i en wickel e Me hodik mi al e na i en Ans¨a zen aus [90] sowie [120] e glichen. Dabei we den insbesonde e Un e scheidungsme kmale he ausges ell . Die in [90] einge ¨uh e Me hodik lie e , alls anwendba , die bes en Absch¨a zungen. Alle dings is ih Anwendungsgebie au linea e endlich-dimensionale Sys eme besch ¨ank und e o de zus¨a zliches Wissen ¨ube die op imale We e unk ion — eine es ik i e Zusa zbedingung. Die ande en beiden Ans¨a ze lassen die Behandlung allgemeine nich linea e sowie unendlich-dimensionale Sys eme inklusi e Kon oll- und Zus andsbesch ¨ankungen zu. Obwohl e gleichba e Annahmen ben¨o ig we den, sind die G¨u eabsch¨a zungen aus [120] h¨au ig deu lich konse a i e im Ve gleich zu unse em Ansa z, de olglich ¨ube legen e schein . Die Haup bei ¨age aus Kapi el 5 sind: åUn e suchung de aus de Ve wendung eine e Disk e isie ungen esul ie en- den Auswi kungen au unse e G¨u eabsch¨a zungen sowie die Be echnung des G enzwe es eines en sp echenden Ve eine ungsp ozesses. åAu s ellen eine Wachs umsbedingung, die dazu ¨uh , dass de o ges ell e Ansa z o z seh schnelle Ab as ung gu e E gebnisse lie e . åVe wendung akkumulie e Sch anken, um unse e G¨u eabsch¨a zungen wei e zu e besse n. åVe gleich mi ande en Ans¨a zen. S a eines sepa a en Beispielkapi els we den die he gelei e en Resul a e di ek in ih en jeweiligen Abschni en mi Beispielen e bunden, um ih e Aussagen zu e anschaulichen und so die heo e ischen E gebnisse besse nach ollziehba zu machen. Einige Resul a e diese Disse a ionssch i wu den be ei s in Vo ab e sionen e ¨o en lich , siehe [6,45–47], [41,50], [4,5] und [97]. Danksagung Besonde e Dank gil meinem Dok o a e P o . D . La s G ¨une ¨u seine he o agende Be euung sowie seine we ollen An egungen, ohne die diese A bei in ih e je zigen Fo m nich m¨oglich gewesen w¨a e. Des Wei e en m¨och e ich mich bei P o . D . F ank Lempio und P o . D . Hans Jose Pesch sowie Nils Al m¨ulle , Ma cus on Lossow, D . J¨u gen Pannek, Ma cus Reble, Michael Schamel und D . Ma in Seeha e bedanken. Eben alls besonde e Dank gil meinen El e n Ing id und D . Wilhelm Wo hmann, meinen Geschwis e n Elke und D . Hans Wo hmann sowie Johannes He el, Ekue-sse Si u Tome y und meine Pa ne in Anja Kleinhenz ¨u ih e Un e s ¨u zung in jedwede Hinsich . Zudem m¨och e ich mich bei meinen F eunden bedanken! VII Chap e 1 Con ol Sys ems, S abili y, and Feedback In his chap e he p oblem o mula ion o his hesis is p esen ed. To his end, con ol sys ems, admissible sequences o con ol alues, and an op imal alue unc ion a e de ined in he i s Sec ion 1.1. In he ensuing sec ion he concep o s abili y, which cha ac e izes he long- e m beha io o sys ems e ol ing in ime, is in oduced. The heo y o Lyapuno which allows o igo ously deduce asymp o ic s abili y is o pa icula in e es in his con ex . Fu he mo e, he basic ideas o closed loop con ol a e p esen ed. Then, in Sec ions 1.3 and 1.4 sampled-da a and ne wo ked con ol sys ems a e deal wi h in o de o mo i a e ou disc e e ime se ing as well as he p oposed mul is ep eedback. The se o eal numbe s is deno ed by Rand he se o in ege s by Z. Fu he mo e, Ns ands o he na u al numbe s, i.e. Z>0, as well as N0 o N∪ {0}, i.e. he se o non-nega i e in ege s. We equi e he ollowing de ini ion, c . [106]. De ini ion 1.1 (Me ic space) A me ic space is a se Xwi h a me ic o dis ance unc ion d:X×X→Rsuch ha he ollowing p ope ies a e sa is ied o all x, y, z ∈X: •de ini eness, i.e. d(x, y)≥0and d(x, y)=0i and only i x=y, •symme y d(x, y) = d(y, x), and • iangle inequali y d(x, z)≤d(x, y) + d(y, z). 1.1 Con ol Sys ems and P oblem Fo mula ion In his hesis we a e conce ned wi h con ol sys ems. The s a e o a con ol sys em e ol es depending on i s cu en s a e and a con ol inpu . This inpu pa ame e can be chosen in o de o exe in luence on he sys em. A classical example is he in e ed pendulum on a ca , c . Figu e 1.1 and Sec ion 1.3. He e, he s a e consis s o he angle Φ o he pendulum, he posi ion o he ca and he co esponding eloci ies. The mo emen is de e mined by he cu en s a e and an ex e nal o ce uac ing on he ca . The concep o a con ol sys em is o malized in he ollowing de ini ion. De ini ion 1.2 (Con ol sys em) Le Xand Ube me ic spaces. A con ol sys em is a quad uple Σ = (T, X, U, )consis ing o a ime domain T={Tk |k∈N0},T > 0, a s a e space X, a se o con ol alues U, 1 CONTROL SYSTEMS, STABILITY, AND FEEDBACK u Ф Figu e 1.1: Schema ic illus a ion o he in e ed pendulum on a ca , c . [37]. and a ansi ion map :D →X. The ansi ion map (·,·)is de ined on a subse D o X×U. The s a e space Xneed no sa is y he de ini ion o a linea space, which can be ound, e.g., in [72]. Since con ol sys ems a e de ined o wa d in ime, he ime domain Tis a subse o he posi i e eal axis. In o de o in es iga e he class o con ol sys ems, we ypically conside models which cap u e he dynamical beha io o an unde lying p ocess, c . Sec ion A.2 o a ma he- ma ical model o he in e ed pendulum on a ca . These models a e employed in o de o deduce a sui able ansi ion map. Since he concep o con ol sys ems is used in o de o desc ibe dynamics o p ac ically mo i a ed sys ems, he s a es and con ol alues a e o en es ic ed. Fo ins ance, he se o con ol alues may ha e o be bounded. The ollowing de ini ion allows o inco po a ing cons ain s in ou se ing. De ini ion 1.3 (S a e and con ol cons ain s) Le nonemp y se s X⊆Xand U⊆Udeno e he se o easible s a es and con ols, espec i ely. A sequence u(·) = (u(n))n∈{0,1,...,N−1}∈UN,N∈N, is called admissible o x0∈Xi u(n)∈Uand (xu(n;x0), u(n)) ∈Xholds o all n∈ {0,1, . . . , N −1}. He e, xu(n;x0)is de ined ecu si ely by he sys em dynamics xu(n+ 1; x0) := (xu(n;x0), u(n)) o n∈N0wi h xu(0; x0) := x0.(1.1) UN(x0)deno es he se {u(·)∈UN:u(·)is admissible o x0}and a sequence u(·)=(u(n))n∈N0∈UNis called admissible o x0∈X, i.e. u(·)∈ U∞(x0), i (u(n))n∈{0,1,...,N−1}∈ UN(x0)holds o each N∈N. The abb e ia ions x(n) = xu(n) = xu(n;x0) a e used when he pa ame e s x0and u(·) clea ly ollow om he con ex . Fu he mo e, he s a es x(n), n∈N0, a e enume a ed 2 CONTROL SYSTEMS AND PROBLEM FORMULATION wi hou s a ing he scaling ac o T esul ing om he ime domain Texplici ly. The se X cha ac e izes all easible s a es, e.g. we may choose X=Rnand X={x∈X|h(x)≤0} o h:Rn→Rin o de o model s a e cons ain s. We make he ollowing assump ion which ensu es ha , o each easible s a e x0∈X, an admissible sequence o con ol alues u(·)∈ U∞(x0) exis s on he in ini e ime ho izon, c . [120, Assump ion A3]. Assump ion 1.4 (Con olled o wa d in a iance) Fo each s a e x∈X, le a con ol alue u∈Uexis such ha (x, u)∈Xholds. Assump ion 1.4 is also e med weak o wa d in a iance o iabili y, c . [35]. Suppose ha Assump ion 1.4 does no hold. Then, he s a e cons ain s a e iola ed o a easible s a e x0∈X o all u∈U. Hence, he ask o s ee ing he con ol sys em wi h ini ial alue x0is no well-posed. The sequence o con ol alues u(·) : N0→Uis in e p e ed as an inpu , i.e. u(·) is cons uc ed in o de o sui ably manipula e he beha io o he sys em. In his hesis, ou goal is o s abilize a gi en plan a a desi ed posi ion which is, in gene al, speci ied in ad ance. This ype o p oblem is called se poin s abiliza ion and i s well o he example o he in e ed pendulum on a ca , in which he up igh posi ion is he desi ed s a e. Typically, hese pa icula posi ions a e so called equilib ia x?∈X⊆Xsa is ying (x?, u?) = x?(1.2) o a leas one con ol alue u?∈U, c . [108, Sec ion 5.4]. T ajec o ies emana ing om an equilib ium x?∈Xmay be balanced a his posi ion by a sui ably chosen con ol inpu . We aim a s ee ing he sys em o i s equilib ium x?, a leas asymp o ically. I mo e han one ajec o y con e ges asymp o ically o he desi ed equilib ium, he ansien beha io o he sys em may be aken in o accoun in o de o assess he quali y o he induced beha io o he sys em o be con olled, c . [58, Sec ion 5.5]. To his end, we de ine a cos unc ional which is based on so called s age cos s, c . [7, Subsec ion 1.6.1]. De ini ion 1.5 (Cos unc ional and s age cos s) Le a con ol sys em (T, X, U, )as well as easible se s X⊆Xand U⊆Ube gi en. Then, he cos unc ional J∞:X×UN→R+ 0∪{∞} is de ined by J∞(x0, u(·)) = ∞ X n=0 `(xu(n;x0), u(n)) (1.3) wi h s age ( unning) cos s `:X×U→R+ 0∪{∞} which a e con inuous on X×U. He e, he sys em dynamics a e gi en by (1.1). Hence, ou goal is o minimize he cos unc ional (1.3) and o s abilize he consid- e ed con ol sys em asymp o ically a a gi en se poin x?. In o de o s a e his ask ma hema ically, hese wo objec i es a e coupled by he s age cos s. To his end, he ollowing de ini ion o a compa ison unc ion is equi ed, c . [115, Exe cise 7.3.11], [35,39], and [58, De ini ion 3.2.1]. De ini ion 1.6 (K∞- unc ion) A con inuous unc ion α:R+ 0→R+ 0is said o be o class Ki α(·)is s ic ly inc easing and α(0) = 0. I , addi ionally, he p ope y lim →∞ α( ) = ∞holds, α(·)is said o be o class K∞. 3 CONTROL SYSTEMS, STABILITY, AND FEEDBACK We poin ou ha each unc ion α(·)∈ K∞is in e ible, c . [70]. The ollowing as- sump ion consis s o wo pa s. The i s ensu es ha s aying a he desi ed equilib ium x? o e e a ze o cos is possible. The second, which uses De ini ion 1.6, inco po a es he s abiliza ion ask in he cos unc ional (1.3) because no ending o x?causes in ini e cos s. Assump ion 1.7 Le an equilib ium x?exis which sa is ies: (i) u∈Uwi h (x?, u) = x?and `(x?, u)=0exis s. (ii) K∞- unc ions α1(·),α2(·)exis such ha he inequali ies α1(kxkx?)≤`?(x) := in u∈U: (x,u)∈X`(x, u) = in u∈U1(x0)`(x, u)≤α2(kxkx?) (1.4) hold o each x∈Xwhe e kxkx?:= dX(x, x?). We ema k ha condi ion (ii) can be elaxed in a ious ways, e.g. i could be eplaced by a de ec abili y condi ion simila o he one used in [32]. Howe e , in o de o keep he p esen a ion echnically simple, we wo k wi h Assump ion 1.7(ii). Mo eo e , he equilib ium x?may be eplaced by a closed se Aa which he sys em has o be s abilized, c . [39]. Typical s age cos s a e, e.g. `(x, u) := dX(x, x?)2+λ dU(u, u?)2. He e, λ∈R≥0deno es a egula iza ion pa ame e and dX,dUme ics on X,U, espec i ely. I he me ic space Xexhibi s he s uc u e o a linea space [72], he desi ed equilib ium x?is supposed o be loca ed a he o igin 0Xo his space, c . [38, Rema k 2.4].1The con ibu ion o he egula iza ion pa ame e λis wo old: i s ly, i allows o penalizing he con ol e o which is used in o de o s ee he sys em in he desi ed di ec ion. Secondly, in pa icula o sys ems go e ned by pa ial di e en ial equa ions, i implies some egula i y o he co esponding solu ions, c . [119]. Ou goal is o ind, o a gi en ini ial alue x0∈X, an admissible sequence o con ol alues u(·)∈ U∞(x0) which minimizes a cos unc ional o ype (1.3). In o de o ackle his ask, he op imal alue unc ion is de ined. De ini ion 1.8 (Op imal alue unc ion) Le a con ol sys em (T, X, U, ), a se o easible s a es X⊆X, and a se o easible con ol alues U⊆Ube gi en. Then, o a gi en s a e x0∈X, he op imal alue unc ion V∞(·) : X→R+ 0∪{∞} is de ined by V∞(x0) := in u(·)∈ U∞(x0)J∞(x0, u(·)) (1.5) wi h he se o admissible inpu sequences U∞(x0) om De ini ion 1.3. Subs i u ing he objec i e o s abilizing he plan a a se poin by acking a e e ence signal is possible. To his end, he s age cos s as well as he cos unc ional ha e o explic- i ly depend on he ime, c . [107, Sec ion 3.2]. The esul s o his hesis a e gene alizable o his se ing, c . [44]. Le us suppose ha he op imal alue unc ion is ini e o each easible s a e, i.e. V∞(x0)<∞holds o all x0∈X. O he wise, he conside ed minimiza ion p oblem is 1O en we omi he subsc ip Xand w i e 0 o he o igin o he espec i e (linea ) me ic space. 4 1.2. CLOSED LOOP CONTROL AND ASYMPTOTIC STABILITY ei he no easible o he compu ed con ol causes in ini e cos s and is, hus, no dis in- guishable om an in easible one. In bo h cases he op imiza ion p oblem is no well-posed. Since V∞(x0)<∞on Ximplies he exis ence o an admissible sequence o con ol alues u(·)∈ U∞(x0) o each x0∈X, Assump ion 1.4 is ensu ed. Summa izing, we wan o ind an admissible sequence o con ol alues u(·) which s abilizes he conside ed con ol sys em wi h minimal cos s. The quali a i e goals o s ee ing he sys em easibly and s abilizing i a he desi ed equilib ium a e coupled wi h he quan i a i e objec i e o minimizing a pe o mance c i e ion ia he op imal alue unc ion V∞(·). Since he coupling is done by he s age cos s, modelling hese app op ia ely is an impo an ask. 1.2 Closed Loop Con ol and Asymp o ic S abili y In he p e ious sec ion he basic p oblem o mula ion was gi en. In o de o ske ch he upcoming app oach, he ollowing assump ion is made in o de o a oid echnical di icul ies. Assump ion 1.9 is used only o illus a i e pu poses in he i s chap e o his hesis. Assump ion 1.9 Fo each x0∈X⊆X, le he in imum in De ini ion 1.8 be a minimum, i.e. a sequence o con ol alues u? x0(·)∈ U∞(x0)sa is ies J∞(x0, u? x0(·)) = V∞(x0).(1.6) Le u? x0(·)=(u? x0(n))n∈N0∈ U∞(x0) deno e an admissible sequence o con ol alues depending on he ini ial alue x0∈Xwhich sa is ies (1.6). The co esponding solu ion xu? x0(·;x0) emana ing om x0is called open loop ajec o y. Since model unce ain ies o dis u bances a e ypically p esen while applying he sequence o con ol alues u? x0(·), he gene a ed ajec o y xu? x0(·;x0) migh no be s able - e en o a bi a y small pe u - ba ions, c . [36, Example 5.2]. Hence, in o de o ob ain a solu ion which compensa es a leas o small pe u ba ions, so called closed loop solu ions a e conside ed, c . Figu e 1.2. Applying he i s elemen u? x0(0) o he compu ed open loop con ol, yields he equali y J∞(x0, u? x0(·)) = ∞ X n=0 `(xu? x0(n;x0), u? x0(n)) = `(x0, u? x0(0)) + ∞ X n=1 `(xu? x0(n;x0), u? x0(n)). Fu he mo e, he nex s a e x1:= xu? x0(1; x0) = (x0, u? x0(0)) is de e mined. Then, he ollowing op imiza ion p oblem can be conside ed: Minimize J∞(x1, u(·)) = ∞ X n=0 `(xu(n;x1), u(n)) w. . . u(·)∈ U∞(x1). Le he co esponding solu ion be deno ed by u? x1(·). Conca ena ing u? x0(0) and u? x1(·) yields a con ol sequence ˜u(·)∈ U∞(x0) wi h ˜u(0) = u? x0(0) and ˜u(n) = u? x1(n−1) o n∈N. Since u? x0(·) sa is ies (1.6), J∞(x0, u? x0(·)) ≤J∞(x0,˜u(·)) is known. Now, suppose ha he s ic inequali y J∞(x0, u? x0(·)) < J∞(x0,˜u(·)) holds. Then, J∞(x0,˜u(·)) = ∞ X n=0 `(x˜u(n;x0),˜u(n)) 5 CONTROL SYSTEMS, STABILITY, AND FEEDBACK Con olle Plan x(n+1)= (x(n),u(n)) + Flow o In o ma ion Re e ence signal Va iable o be con olled Obse ed quan i y Figu e 1.2: Scheme o open and closed loop con ol. The dis inc i e ea u es a e d awn in ed: in he closed loop an obse ed quan i y and, hus, in o ma ion abou he cu en s a e is compa ed wi h a e e ence signal, e.g. he dis ance om he desi ed equilib ium, and ansmi ed o he con olle — he con ol loop is closed. Based on his in o ma ion he con ol signal may be upda ed. Wi hou in eg a ing his low o in o ma ion in he con ol loop a eac ion o dis u bances o modelling e o s is no possible. =`(x0, u? x0(0)) + ∞ X n=0 `(xu? x1(n;xu? x0(1; x0)), u? x1(n)) > `(x0, u? x0(0)) + ∞ X n=0 `(xu? x0(1 + n;x0), u? x0(1 + n)) = J∞(x0, u? x0(·)) is ob ained which con adic s he de ini ion o u? x1(·). As a consequence, he op imal alue unc ion V∞(·) sa is ies V∞(x0) = J∞(x0, u? x0(·)) = J∞(x0,˜u(·)) =`(x0, u? x0(0)) + J∞(x1, u? x1(·)) =`(x0, u? x0(0)) + V∞(x1) = `(x0, u? x0(0)) + V∞( (x0, u? x0(0))). The ac ha u? x0(·) depends only on he cu en s a e x0enables us o de ine a s a ic s a e eedback F∞:X→Uby F∞(x0) := u? x0(0). Plugging his de ini ion in o he las chain o equali ies yields V∞(x0) = `(x0, F∞(x0)) + V∞( (x0, F∞(x0))).(1.7) Indeed, (1.7) cha ac e izes an op imal eedback alue o he op imiza ion p oblem o a gi en s a e x0∈Xon he in ini e ime ho izon and allows o an i e a i e compu a ion o an op imal sequence o con ol alues. This echnique is called dynamic p og amming, c . [113] and [81] o i s use as a compu a ional ool. I is based on Bellman’s p inciple o op imali y which s a es ha ails o op imal ajec o ies a e again op imal, c . [9]. Re o mula ing (1.7) p o ides he Lyapuno equa ion V∞( (x0, F∞(x0))) = V∞(x0)−`(x0, F∞(x0)).(1.8) In o de o illus a e he p esen ed ideas, a simple disc e e ime con ol sys em is consid- e ed, which was in oduced in [112] and u he in es iga ed in [39, 90]. No e ha his example does no exhibi any con ol o s a e cons ain s which makes he analysis much easie . 6 CLOSED LOOP CONTROL AND ASYMPTOTIC STABILITY Example 1.10 Le U=U=R,X=X:= R2, and `(x, u) = xTQx+uTRu be gi en. Then, U∞(x0) = UN holds o each x0∈X=X. The ollowing op imal con ol p oblem is conside ed: min u(·)∈UN ∞ X n=0 x(n)TQx(n) + u(n)TRu(n) = min u(·)∈UN ∞ X n=0 x(n)T1 0 0 1 x(n) + u(n)Tu(n) subjec o he linea dynamics x(n+ 1) = Ax(n) + Bu(n) = 1 1.1 −1.1 1 x(n) + 0 1u(n). Fo his example, he op imal alue unc ion is compu able ia V∞(x0) = xT 0Px0whe e P is he unique posi i e de ini e solu ion o he algeb aic Ricca i equa ion (ARE) P=ATPA −ATPB(BTPB +R)−1BTPA +Q. Mo eo e , F∞(x0) = u? x0(0) is gi en by F∞(x0) = −(BTPB +R)−1BTPAx0, c . [74,90] and [8]. He e, his leads app oxima ely o P≈5.09839937 3.210349330 3.21034933 7.406837723 and F∞(x0)≈0.58728054 −1.301110161 T x0. Using his eedback, we ob ain he closed loop sys em x(n+ 1) = Ax(n) + BF∞(x(n)) = (A+BF∞)x(n).(1.9) Hence, o x0= (1 1)T∈X,(1.8) co esponds o 16.416 ≈V∞((A+BF∞)x0) = V∞(x0)−`(x0, F∞(x0)) ≈18.926 −2.510. Supposing ha a s a ic s a e eedback map F:X→Usa is ying F(x)∈Uand (x, F(x)) ∈X o all x∈X(1.10) is gi en, he esul ing closed loop ajec o y xF(·)=(xF(n))n∈N0is gene a ed by xF(n+ 1; x0) = (xF(n;x0), F(xF(n;x0))), n∈N0, wi h xF(0; x0) = x0. The condi ions gi en in (1.10) ensu e ha he co esponding sequence o con ol alues F(xF(·;x0)) = (F(xF(n;x0)))n∈N0is con ained in U∞(x0) o x0∈Xand, hus, admissi- ble. Hence, assuming ha (1.10) holds, sys em dynamics ˜ :X→Xdepending solely on he s a e can be de ined by ˜ (x) := (x, F(x)). This map ˜ de ines a dynamical sys em, c . [53,58,117]. De ini ion 1.11 (Dynamical sys em) A dynamical sys em on Xis a iple (X, T, x)which consis s o he ime domain T:= N0, he s a e space X, and a map x:T×X→Xsuch ha •x(0, x0) = x0 o all x0∈X(consis ency), •x(τ, x( , x0)) = x(τ+ , x0) o all x0∈Xand , τ ∈T(g oup p ope y). 7 CONTROL SYSTEMS, STABILITY, AND FEEDBACK The es ic ion o he ime domain N0is no necessa y bu i s well o ou pu poses. Since he ime domain is con ained in R+ 0, (X, T, x) is said o be a semi dynamical sys em in some e e ences, c . [35]. Nex , we wan o in oduce he concep o asymp o ic s abili y o a dynamical sys em. To his end, compa ison unc ions β∈ KL0a e equi ed, c . [40]. De ini ion 1.12 (KL- and KL0- unc ions) A unc ion β:R+ 0×N0→R+ 0is said o be o class KL i • o each ∈R+ 0,β(·, ) : R+ 0→R+ 0is o class K∞and • o each ≥0,β( , ·) : N0→R+ 0is dec easing wi h lim →∞ β( , ) = 0. Fu he mo e, a unc ion β:R+ 0×N0→R+ 0is said o be o class KL0i • o each ∈R+ 0,β(·, ) : R+ 0→R+ 0is o class K∞o β(·, )≡0and • o each > 0,lim →∞ β( , )=0. Since disc e e ime sys ems a e deal wi h, β(·,·) om De ini ion 1.12 is, in con as o [58, De ini ion 3.2.1], de ined on R+ 0×N0ins ead o R+ 0×R+ 0. Each β(·,·)∈ KL0may be ex ended o a con inuous unc ion on R+ 0×R+ 0, e.g. by linea in e pola ion. Vice e sa, aking a con inuous KL0- unc ion de ined on R+ 0×R+ 0as a s a ing poin allows o de ine a co esponding es ic ion canonically. This mapping is aci ly used in o de o a oid echnical de ails o disc e e ime sys ems o igina ing om con inuous ime ones. Since each con inuous KL0- unc ion β:R+ 0×R+ 0→R+ 0can be o e bounded by a unc ion ˜ β(·,·)∈ KL, e.g. by se ing ˜ β( , ) = supτ≥ β( , τ) + e− , his can also be done o unc ions de ined acco ding o De ini ion 1.12. Two impo an ep esen a i es o class KL0- unc ions β(·,·) a e gi en in he ollowing example. Example 1.13 The i s example is in ac con ained in KL ⊂ KL0. •Le an o e shoo bound C≥1and a decay a e σ∈(0,1) be gi en. Then, exponen- ially decaying unc ions a e de ined by β( , n) = Cσn . (1.11) While he second equi es he mo e gene al class KL0. •A unc ion β(·,·) : R+ 0×N0→R+ 0is linea in i s i s a gumen and equal o ze o o su icien ly la ge second a gumen s i a ini e numbe n0∈N0and a sequence (cn)n∈N0⊂R+ 0sa is ying cn= 0 o all n≥n0exis such ha β( , n) = ·cn o all n∈N0(1.12) holds. Such a unc ion can be de ined by choosing only ini ely many elemen s cn, n∈ {0,1, . . . , n0−1}. No e ha each unc ion o he second class o Example 1.13 may be o e bounded by an exponen ially decaying one. Howe e , using he la ge class KL0allows o employing igh e bounds in o de o es ima e he ac ual beha io o he sys em, c . [39]. The ollowing submul iplica i i y p ope y will be equi ed in his hesis in o de o cha ac e ize he s abili y beha io o a conside ed class o sys ems be e β( , n +m)≤β(β( , n), m)∀n, m ∈N0and ≥0.(1.13) 8 CLOSED LOOP CONTROL AND ASYMPTOTIC STABILITY Fo β( , n +m) = Cσn+m ≤C2σnσm =C·σm(Cσn ) = β(β( , n), m) wi h C≥1, P ope y (1.13) is always sa is ied. While i is sa is ied o he second class i and only i cn+m≤cncmholds. I needed, his p ope y can be assumed wi hou loss o gene ali y by applying Son ag’s KL-Lemma, c . [115]. Fu he commen s on KL0- unc ions can be ound in [39, Sec ion 3]. Using class KL- unc ions β(·,·) allows o de ine asymp o ic s abili y, c . [44]. De ini ion 1.14 (Asymp o ic s abili y) Le a dynamical sys em (X, N0, x), a se X⊆X, and an equilib ium x?be gi en, i.e. x(n, x?) = x? o n∈N0. The equilib ium is said o be asymp o ically s able on X⊆Xi aKL- unc ion βexis s such ha , o each x∈X, he s a e ajec o y x(n;x0),n∈N0, is con ained in Xand, in addi ion, sa is ies he inequali y kx(n;x0)kx?=dX(x(n;x0), x?)≤β(dX(x0;x?), n) = β(kx0kx?, n), n ∈N0.(1.14) De ini ion 1.14 implies wo impo an p ope ies: •s abili y (in he sense o Lyapuno ), i.e. o any ε > 0, δ=δ(ε)>0 exis s such ha x(n;x0)∈Xand dX(x(n;x0), x?)< ε,n∈N0, hold o all x0∈Xsa is ying dX(x0, x?)< δ, i.e. ajec o ies s ay a bi a ily close o he equilib ium x?i hei ini ial s a e is easible and loca ed in a su icien ly small neighbo hood o x?. •a ac ion, i.e. he s a e ajec o y con e ges o x?since dX(x(n;x0), x?) ends o ze o o napp oaching in ini y o all x0∈X. Nex , he concep o Lyapuno unc ions, which will be employed in o de o conclude s abili y o a con ol sys em ope a ed in closed loop, is in oduced, c . [44, De ini ion 2.18]. A Lyapuno unc ion may be seen as an ene gy no m, i.e. i measu es he ene gy p esen in he sys em. Hence, a Lyapuno inequali y ensu es a “loss o ene gy“ and, hus, cha ac e izes he desi ed equilib ium as a s a e o he sys em a which ene gy is anished, c . [115, p.348]. De ini ion 1.15 (Lyapuno unc ion) Le x?= 0 be an equilib ium poin o a dynamical sys em (X, N0, x)and X⊆Xbe a subse o he s a e space. Then, a unc ion V:X→R+ 0is said o be a Lyapuno unc ion on Xi • K∞- unc ions α1(·),α2(·)exis such ha he ollowing condi ion holds α1(kx0kx?)≤V(x0)≤α2(kx0kx?)∀x0∈X(1.15) •and, in addi ion, a K- unc ion W:R+ 0→R+ 0exis s such ha V(x(1; x0)) ≤V(x0)−W(V(x0)) holds o all x0∈Xsa is ying x(1; x0)∈X. Fu he mo e, i X=X, hen V(·)is called global Lyapuno unc ion. Fo ins ance, he i s inequali y in Condi ion (1.15) can be e i ied o a closed loop sys em i he inequali ies α1(kxkx?)≤`?(x)≤`(x, F(x)) ≤V(x)<∞hold o all x∈X⊆X. He e, in con as o [95,115], con inui y o he Lyapuno unc ion V(·) is no assumed which allows o deal, e.g. wi h s a e cons ain s. O en, e en u he egula i y 9 CONTROL SYSTEMS, STABILITY, AND FEEDBACK model o small angula de ia ions because he dynamics can be ea ed sepa a ely o each coo dina e di ec ion, c . [23, pp. 9 – 10]. Hence, we ocus on small angula de ia ions, e.g. |ϕ|is no pe mi ed o exceed one deg ee o a c, c . [23, p.17]. Example 1.22 (Cons ain s o Example 1.21) In o de o ake in o accoun ha he angle is es ic ed o small alues, he s a e cons ain kx3( )k< c o a su icien ly small cons an c∈R>0may be imposed on x( ). Hence, X:= {x∈X:kx3k< c}is chosen as he se o easible s a es. Assuming an unbounded se o easible con ols, e.g. se ing W:= W=R, allows o a bi a ily in luence he angula eloci y x4(·). Hence, o each x0∈X, a con ol alue ux0∈U:= L1([0, T),W)exis s such ha (x0, ux0) = Φ(T;x, ux0)∈Xholds which ensu es ha he imposed s a e cons ain s can be sa is ied. O cou se, his canno be done in he conside ed p ac ical applica ion, i.e. he se o admissible con ol alues Wwill be con ined o some eal in e al, [a, b], a < 0< b. Howe e , since he ini ial alue o he angula eloci y is loca ed in a small neighbo hood o he o igin, he sys em can be s ee ed such ha he o iginal s a e cons ain kx3(·)k< c is sa is ied and, in addi ion, he angula eloci y x4(·) emains su icien ly small. This allows o ensu ing easibili y o he sys em by choosing he con ol inpu app op ia ely. The example o he in e ed pendulum on a ca is conside ed once mo e in o de o in es iga e he impac o using a ze o o de hold eedback. Example 1.23 (Example 1.21 con inued) Le a sampling pe iod T > 0as well as pa ame e s cP= 0,m= 1,l= 1,g= 9.81, M= 0,J= 2,c= 1/10, and β= 0.5be gi en. The sampled-da a sys em wi h ze o o de hold o he linea ized in e ed pendulum on a ca model is conside ed, c . Example 1.21 o a linea ized e sion o he nonlinea pendulum model o Example 1.19. Since a ze o o de hold eedback is assumed, he con ol alue may change only a he ime ins an s 0, T, 2T, . . .. Hence, he cons an con ol unc ion ˜u(·)≡uis iden i ied wi h he co esponding con ol alue u. Then, he ollowing linea sys em dynamics a e ob ained x(n+ 1) = Φ(T;x(n), u(n)) (1.21) =eAT x(n) + u(n)ZT 0 eAs ds. (1.22) A eedback con ol u(n) = Fx(n)is used in o de o ob ain a closed loop sys em. No e ha Fis a linea map ep esen ed by a ma ix. Hence, Fx(n)is w i en ins ead o F(x(n)). Plugging his in o (1.22) yields he closed loop x(n+ 1) = eAT x(n) + ZT 0 eAs ds ·Fx(n) = eAT +ZT 0 eAs dsFx(n). Fu he mo e, le he ollowing s age cos s `:X×U→R+ 0be gi en, c . Example 1.19: `(x, u) := T(xTQx +uTRu) = T(xTx+uTu) = Tkxk2+Tkuk2. The cos unc ional is gi en by J∞(x0, u(·)) = P∞ n=0 `(xu(n;x0), u(n)). Inco po a ing he sampling pe iod Tin he s age cos s, allows o a compa ison o he esul ing closed loops in dependence on he sampling pe iod Tbecause J∞(x0, u(·)) app oxima es he in eg al R∞ 0x( ;x0, u( ))TQx( ;x0, u( )) + u( )TRu( )d . Since using a smalle sampling pe iod implies he possibili y o changing he con ol alue mo e o en, a dec ease o he cos unc ional is expec ed o smalle sampling pe iods. 16 SAMPLED-DATA SYSTEMS We poin ou ha cons ain s a e no conside ed in his example which allows o em- ploy he ma lab ou ine dlq in o de o sol e he co esponding minimiza ion p oblem. The abb e ia ion dlq s ands o disc e e linea quad a ic egula o . This ma lab ou ine p o ides, in addi ion o a ma ix Psa is ying V∞(x0) = xT 0Px0, also a eedback ma ix F. No e ha Pas well as he eedback law ep esen ed by he ma ix Fdepend on he sampling pe iod T. Fo ou nume ical compu a ions, he ini ial alue x0=1 10(1 1 1 1)Tis picked. T ajec o ies o he e y small sampling pe iod T= 2−16 a e d awn in Figu e 1.4. Fo T= 2−i,i∈ {1,2,...,16}, con e gence o he desi ed equilib ium, i.e. xT(n;x0)→x?= 0 is obse ed. In o de o illus a e his, he no m o he solu ion on he in e al [0,6] is compu ed o di e en sampling pe iods T, c . Figu e 1.4 b). a) 0 1 2 3 4 5 6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 xi(⋅) x1 x2 x3 x4 b) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 0 0.5 1 1.5 2 2.5 |xT(⋅)| T = 1 T = 0.5 T = 0.25 T = 0.0625 Figu e 1.4: In a) a g aphical illus a ion o he dynamical beha io o he linea ized in e ed pendulum on a ca is gi en (T= 2−16). In b), he no m o sampled-da a sys ems wi h ze o o de hold is illus a ed o di e en sampling pe iods T. Using smalle Tand, hus, e alua ing he eedback law mo e o en leads o an imp o ed beha io . This obse a ion is subs an ia ed by he o h and i h column o Table 1.1 in which he Euclidean and he in ini y no m a e compu ed a = 6. Fo sampling pe iods T≤2−6, he impac o ze o o de hold seems o be negligible, c . Table 1.1. In he second column o Table 1.1, he op imal alue unc ion VT ∞(x0)is app oxima ed. The op imal alue unc ion g ows o inc easing sampling pe iod T. Choosing T oo la ge leads o a de e io a e dynamical beha io o he esul ing closed loop. As seen in Example 1.17, he eigen alues o he closed loop ansi ion map ha e o be de e mined in o de o ind sui able pa ame e s C≥1(o e shoo ) and σ∈(0,1) (decay a e) o a KL- unc ion β(·,·)which enables us o show asymp o ic s abili y o he closed loop. Howe e , in o de o assess he solu ions based on he di e en sampling pe iods T, only compa ing he eigen alues which a e a ibu ed o he espec i e closed loop is insu icien . Ins ead, he eigen alue is aken o he (T−1)- h powe , e.g λ4 o T= 0.25, c . he hi d column o Table 1.1. This scaling o he eigen alues and, hus, he co esponding decay a es leads o a measu e o he dec ease a e one ime uni , i.e. T−1 imes he sampling pe iod T. Cons an s o he o e shoo bound may be compu ed analogously o Example 1.10. Using he e y small alue T= 2−16 allows o gene a e esul s which can be in e p e ed as a e e ence solu ion which is no a ec ed by he ze o o de hold implemen a ion. Fo his sampling pe iod, Figu e 1.5 shows le el se s o he op imal alue unc ion V∞(x) = xTPx a x3=x4= 0 on he le and x1=x2= 0 on he igh . Taking he ange o alues 17 CONTROL SYSTEMS, STABILITY, AND FEEDBACK Sampling pe iod T xT 0Px0kλ(A+BF)k1/T kxT(·)| =6k2kxT(·)| =6k∞ 1.0000000000000000 19.67513368 0.44677011 0.13877609 0.10871640 0.5000000000000000 10.21853967 0.43498765 0.06912694 0.05567655 0.2500000000000000 8.34728912 0.43168006 0.05627710 0.04564705 0.1250000000000000 7.90776081 0.43082527 0.05331032 0.04331718 0.0625000000000000 7.79907101 0.43060972 0.05258338 0.04274543 0.0312500000000000 7.77169915 0.43055572 0.05240256 0.04260316 0.0156250000000000 7.76470700 0.43054221 0.05235741 0.04256763 0.0078125000000000 7.76288128 0.43053883 0.05234612 0.04255875 0.0039062500000000 7.76238581 0.43053799 0.05234330 0.04255653 0.0019531250000000 7.76224241 0.43053778 0.05234260 0.04255598 0.0009765625000000 7.76219680 0.43053773 0.05234242 0.04255584 0.0004882812500000 7.76218051 0.43053771 0.05234238 0.04255581 0.0002441406250000 7.76217400 0.43053771 0.05234237 0.04255580 0.0001220703125000 7.76217115 0.43053771 0.05234236 0.04255580 0.0000610351562500 7.76216983 0.43053771 0.05234236 0.04255579 0.0000305175781250 7.76216919 0.43053771 0.05234236 0.04255579 0.0000152587890625 7.76216888 0.43053771 0.05234236 0.04255579 Table 1.1: Nume ical esul s o he linea ized in e ed pendulum on a ca in dependence on he sampling pe iod T o ini ial alue x0=1 10(1 1 1 1)T. in o accoun shows ha he op imal alue unc ion is much mo e sensi i e wi h espec o changes in he angle and i s eloci y (x3and x4) han in he posi ion o he ca and i s eloci y (x1and x2). Bo h plo s indica e ha he chosen ini ial alue wi h solely posi i e alues makes he s abiliza ion p oblem mo e di icul . x1 x2 −0.5 0 0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.0625 0.125 0.25 0.5 1 2 0 0.0625 0.125 0.25 x3 x4 −0.5 0 0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.0625 0.25 1 4 16 64 Figu e 1.5: Le el se s o he op imal alue unc ion V∞(·) o he linea ized in e ed pendulum on a ca example o sampling pe iod T= 2−16: on he le he hi d and o h componen o x0a e se o ze o, i.e. VT ∞(x0) o x0,3=x0,4= 0 is depic ed. On he igh , he same is epea ed o x0,1=x0,2= 0. In o de o ensu e ha he dynamical beha io o a sampled-da a sys em wi h ze o o de hold con e ges o he one o he con inuous ime sys em, su icien ly as sampling 18 1.4. NETWORKED SYSTEMS AND MULTISTEP FEEDBACK and, hus, small sampling pe iods Ta e equi ed, c . [91]. Fu he mo e, i is possible o allow o mo e han one con ol alue pe sampling in e al, i.e. mul i a e sampling, c . [63,80]. Tha means he cu en s a e is measu ed, hen a sequence o , le ’s say m, con ol alues is compu ed and applied on he sampling in e al, i.e. he i s is implemen ed on he in e al [0, T/m), he second on [T/m, 2T/m) and so o h. Hence, m(possibly di e en ) con ol alues a e implemen ed du ing one sampling pe iod. 1.4 Ne wo ked Sys ems and Mul is ep Feedback In his sec ion a ne wo ked con ol se ing is in oduced in o de o mo i a e he de ini ion o a mul is ep eedback. This ype o a s a ic s a e eedback will u n ou o be help ul also o o he applica ions, c . Chap e 4. Due o lowe implemen a ion cos s and g ea e in e ope abili y ne wo ked con ol sys- ems (NCS) a e inc easingly used, pa icula ly in he au omo i e and ae onau ical in- dus ies, c . [47]. The si ua ion o a ne wo ked con ol sys em shown in Figu e 1.6 is conside ed. The con olle uses a ne wo k channel a e e y ime ins an n∈Nin o de o ansmi he eedback con ol alue u(n) = µ(x(n)) o he ac ua o . Since, in con as o, e.g. [118,122], no pa icula p o ocol like ound– obin (RR) o y–once–disca d (TOD) is assumed, a packe ei he a i es unpe u bed and wi h negligible delay o e he channel o is ea ed as a d opou . A d opou means ha he con ol alue sen by he con olle does no a i e a he ac ua o . Plan x(n+1)= (x(n),u(n)) Senso Ac ua o Channel Con olle Bu e Figu e 1.6: Scheme o he conside ed ne wo ked con ol sys em. The communica ion be ween he con olle and he ac ua o is ca ied ou ia a channel. In eg a ing his addi ional elemen in he con ol loop may lead o packe d opou s as well as delays. In o de o compensa e o d opou s, we add a bu e de ice in he ac ua o and adap he con olle design: a each ime ins an n, ins ead o a single con ol alue u(n) = µ(x(n)) ∈U, a sequence µ(x(n),0), µ(x(n),1), . . . , µ(x(n), m?−1) o con ol alues is sen . In he ac ua o , he elemen s o his sequence a e bu e ed and used un il he nex sequence a i es. In he ideal case when no packe d opou s occu , he ac ua o applies he con ol sequence µ(x(n),0), µ(x(n+ 1),0), µ(x(n+ 2),0), µ(x(n+ 3),0),.... I , howe e , ansmission is success ul a , e.g. ime nand n+ 3 bu ails a ime n+ 1 and n+ 2, he ac ua o applies µ(x(n),0), µ(x(n),1), µ(x(n),2), µ(x(n+ 3),0),.... 19 CONTROL SYSTEMS, STABILITY, AND FEEDBACK In o de o o malize his idea, we de ine a sequence (mi)i∈N0o con ol ho izons, which coun s he ime ins an s be ween he i- h and he (i+ 1)-s success ul ansmission. Fo hese sequences he ollowing de ini ions a e in oduced. De ini ion 1.24 Le a se M⊆ {1,2, . . . , m?},m?∈N, be gi en. A sequence o con ol ho izons (mi)i∈N0 is said o be admissible i mi∈Mholds o all i∈N0. Fo k, n ∈N0, he ollowing exp essions a e de ined: σ(k) := k−1 X j=0 mi(using he con en ion P−1 j=0 = 0), ϕ(n) := max{σ(k)|k∈N0, σ(k)≤n}. He e σ(k) deno es he k- h success ul ansmission ime while ϕ(n) deno es he la ges success ul ansmission ime be o e o a ime ins an n. No e ha by con en ion, ime n= 0 coincides wi h he i s success ul ansmission. Using his no a ion, he con ol sequence applied by he ac ua o can be exp essed as µ(x(σ(k)),0), . . . , µ(x(σ(k)), mk−1), µ(x(σ(k+ 1)),0), . . . in which mkis unknown a ime σ(k). No e ha his no a ion is a pos e io i and only used in o de o analyze he esul ing scheme a e wa d. Al hough he con ol loop is no closed a each sampling ins an , measu emen s a e used in o de o upda e he sequence o con ol alues which allows o eac o dis u bances o modelling e o s. Ne e heless, in he ne wo ked con ol se ing, we aim a closing he loop as o en as possible in o de o obus i y he closed loop beha io o he conside ed sys em. Hence, mo e han he i s elemen o he open loop sequence o con ol alues is only implemen ed i a packe d opou occu s. Using he p ecompu ed sequence o con ol alues should be a o able in compa ison o using a de aul con ol alue. In o de o be consis en wi h he scheme in oduced abo e, he e m eedback con ol is used in he ollowing gene al sense. De ini ion 1.25 Le m?∈Nand M⊆ {1,2, . . . , m?}be gi en. A mul is ep eedback law is a map µ: X×{0, . . . , m?−1} → Uwhich, o an admissible con ol ho izon sequence (mi)i∈N0⊂M, is applied acco ding o he ule xµ(0) = x0, xµ(n+ 1) = (xµ(n), µ(xµ(ϕ(n)), n −ϕ(n))).(1.23) Fo de ails abou his se ing we e e o [48]. We poin ou ha he concep o mul is ep eedbacks will u n ou o be bene icial also in a se ing wi hou delays and packe d opou s in o de o enhance s abili y p ope ies o closed loop con olled sys ems, c . Sec ion 4.4. 20 Chap e 2 Receding Ho izon Con ol In his chap e we p esen he main idea o eceding ho izon con ol (RHC) which is also called model p edic i e con ol (MPC).1Then, we discuss he s abili y analysis o eced- ing ho izon con ol schemes wi h e minal cons ain s and, i necessa y, e minal cos s. Fu he mo e, a easibili y p oo om [90] o uncons ained RHC schemes is ske ched. In his con ex , uncons ained means ha nei he e minal cons ain s no e minal cos s a e added o he basic eceding ho izon se ing. 2.1 In oduc ion In he las chap e we deal wi h he op imiza ion p oblem min u(·)∈ U∞(x0)J∞(x0;u(·)) = ∞ X n=0 `(xu(n;x0), u(n)) (2.1) subjec o xu(n+ 1; x0) = (xu(n;x0), u(n)) wi h xu(0; x0) = x0∈X,(2.2) U∞(x0) := (u(n))n∈N0u(n)∈U xu(n+ 1; x0)∈X o all n∈N0(2.3) wi h he con en ion V∞(x0) := in u(·)∈ U∞(x0)J∞(x0;u(·)) = ∞when ei he he op imal ajec o y causes cos s summing up o in ini y o he se U=U∞=U∞(x0) o admissible sequences o con ol alues is emp y, i.e. he e does no exis a sequence u(·)=(u(n))n∈N0 o con ol alues sa is ying he con ol cons ain s u(n)∈U,n∈N0, such ha he s a e cons ain s xu(n+ 1; x0)∈Xa e main ained o all n∈N0. Since V∞(x0) = ∞ cha ac e izes he op imal con ol p oblem as no well-de ined, he ollowing assump ion is made in o de o exclude hese scena ios om ou analysis. Assump ion 2.1 Le V∞(x0)<∞hold o each x0∈X. Assump ion 2.1 implies, among o he s, he exis ence o a sequence o con ol alues ux0(·)∈ U∞(x0) such ha V∞(x0)≤J∞(x0;ux0(·)) <∞holds. Since ux0(·)∈ U∞(x0), he s a e cons ain s xux0(n;x0)∈X,n∈N0, a e sa is ied. 1The e ms mo ing o olling ho izon can also be ound in li e a u e. 21 RECEDING HORIZON CONTROL Summa izing, ou goal is o sol e he minimiza ion p oblem (2.1) - (2.3), i.e. o min- imize he cos unc ional subjec o he sys em dynamics and he con ol and s a e con- s ain s. Howe e , sol ing p oblem (2.1) - (2.3) is, in gene al, in ac able because i s solu ion in ol es sol ing a Hamil on-Jacobi di e ence equa ion. In pa icula , his holds o sys ems whose dynamics a e ei he nonlinea o de ined on a space o in ini e dimen- sion. Fo example, con ol sys ems whose dynamics a e go e ned by pa ial di e en ial equa ions belong o he la e ca ego y. Hence, we aim a app oxima ing he desi ed solu ion o , a leas , sol ing he closely ela ed s abiliza ion p oblem, i.e. looking o a sequence o con ol alues u(·)∈ U∞(x0) which s abilizes he sys em a he equilib ium x?. To his end, he desi ed s a e has o be cha ac e ized app op ia ely by he s age cos s `(·,·), i.e. `?(x) = minu∈U1(x)`(x, u) = 0 i and only i x=x?, c . (1.4). I he a o emen- ioned ask may be ul illed by mo e han one sequence o con ol alues, we pick some u(·)∈ U∞(x0) which minimizes he cos unc ional J(x0;·) o , a leas , yields a pe o - mance which does no de ia e oo much om he op imal one. To be mo e p ecise, ou objec i e is ha he compu ed con ol u(·) induces cos s J∞(x0;u(·)) which a e bounded by he op imal cos s V∞(x0) mul iplied by a ce ain ac o 1/α, i.e. J∞(x0;u(·)) ≤1 α·V∞(x0). Fo example, α= 1/2 means ha he cos s associa ed wi h he chosen con ol u(·) a e a mos wice as much as he op imal ones. The op imal alue V∞(x0) coincides wi h he minimal cos s. Ne e heless mo e han one con ol may exis which induces exac ly his amoun o cos s. He e, we aci ly ag ee in picking one o hese whene e we use he e m op imal con ol. Fu he mo e, no e ha such a sequence o con ol alues, o which he in imum in he p oblem o mula ion is a ained, may no exis a all. Be o e we ackle he aised ques ions, he basic ideas o eceding ho izon con ol, which ep esen s a emedy in o de o deal wi h he desc ibed p oblem se ing, a e p esen ed. To his end, we conside he auxilia y p oblem wi h op imiza ion ho izon N∈N: min u(·)∈ UN(x0)JN(x0;u(·)) = N−1 X n=0 `(xu(n;x0), u(n)) (2.4) subjec o xu(n+ 1; x0) = (xu(n;x0), u(n)) wi h xu(0; x0) = x0∈X,(2.5) UN(x0) := (u(n))n∈N0u(n)∈U xu(n+ 1; x0)∈X o 0 ≤n≤N−1.(2.6) The co esponding op imal alue unc ion is gi en by VN(x0) = in u(·)∈ UN(x0)JN(x0;u(·)).(2.7) No e ha his p oblem di e s om p oblem (2.1) - (2.3): he ime ho izon is unca ed, i.e. he cos unc ional e alua es he s age cos s only a he i s N ime ins an s. Mo e- o e , he se UN(x0), which con ains he con ol and s a e cons ain s, is adap ed as well, e.g. he ajec o y only has o be easible un il ime N. In he nex sec ions addi ional e minal cons ain s and cos s a e inco po a ed in his se ing in o de o ensu e easibil- i y o he esul ing eceding ho izon closed loop. In his subsec ion, howe e , we conside he concep ually simples eceding ho izon app oach imposing nei he e minal cos s no 22 INTRODUCTION e minal cons ain s which makes i easie o ca e ou some o he basic p inciples o mo ing ho izon schemes. Fu he mo e, his scheme is p edominan in indus ial appli- ca ions, c . [100], which mo i a es i s analysis in he ollowing chap e s. Fo ques ions conce ning easibili y we e e o Sec ion 2.4. Al eady in he las chap e , we indica ed how o ob ain a closed loop sys em assuming being able o compu e a sequence o con ol alues which sol es he o iginal p oblem, i.e. sa is ies J∞(x0;u(·)) = V∞(x0) in iew o Assump ion 1.9. He e, we p oceed anal- ogously wi h he p oblem posed on he unca ed ime ho izon, i.e. we sol e P oblem (2.4) - (2.6) in o de o ob ain a sequence o Ncon ol alues. This sequence may be ex- ended by conca ena ion wi h he sequence which is iden ically ze o. No e ha he alues u(N), u(N+ 1), . . . do no play a ole o he p oblem on he unca ed ho izon. P oblem (2.4) - (2.6) is, e.g. o dynamics go e ned by a nonlinea o dina y di e en ial equa ion, a nonlinea op imal con ol p oblem. Using he in oduced concep o sampled-da a sys- ems wi h ze o o de hold and, hus, disc e izing he con ol unc ion u(·) ans o ms his op imal con ol p oblem o an op imiza ion p oblem which is compa a i ely easy o sol e, c . [29,84,119]. We poin ou ha his app oach in ol es a p edic ion o he u u e s a es xu(n;x0), n= 1,2, . . . , N, which mo i a es he e m “p edic i e” in model p edic- i e con ol. In addi ion, he me hod is based on a model which is employed in o de o p edic he ajec o y on he in e al [0, NT) in dependence on he con ol u(·). Nex , we implemen he i s m∈ {1, . . . , N −1}elemen s o he compu ed sequence. In o de o s eamline he p esen a ion o he main idea, le us se m= 1, i.e. implemen ing only he i s elemen o he sequence o con ol alues which may be called “classical” MPC. This si ua ion is illus a ed in Figu e 2.1: x0deno es he cu en s a e o he s a e e olu ion which is induced by a sampled-da a sys em, c . Sec ion 1.3. Hence, we use a (mul is ep) eedback con ol acco ding o De ini ion 1.25, e.g. µN(0; x0) = u(0) o m= 1, and implemen his a he plan which yields he new ini ial s a e x0:= xµN(m;x0) o he op imiza ion p oblem (2.4) - (2.6). No e ha xµN(m;x0) may di e om he p edic ed s a e xu(m;x0), e.g. due o modelling e o s. Then, he p ocedu e is epea ed, i.e. an op imiza ion wi h espec o he op imiza ion ho izon Nis ca ied ou which, again, yields ou eceding ho izon eedback, c . Figu e 2.2. This shi ing o he op imiza ion ho izon o wa d in ime explains he e m “ eceding ho izon”. Summa izing, we de ine a mul is ep eedback law µN,m?by picking he i s melemen s o he op imal con ol sequence based on he ini e ho izon op imal alue unc ion VN(x0). This cou se o ac ion is epea ed a e shi ing he ho izon. In o de o o malize his concep , he ollowing de ini ion is gi en. De ini ion 2.2 Fo m≥1and N≥m+1 a mul is ep MPC eedback law is de ined by µN,m(x0, n) = u?(n), whe e u?(·)is a minimizing con ol o p oblem (2.4) -(2.6) wi h ini ial alue x0. Al hough he open loop op imal con ol u?(·) = u? N(·;x0)depends on he ini ial s a e x0and he op imiza ion ho izon N, he subsc ip Nand he co esponding ini ial s a e x0a e o en no lis ed. Using his eedback leads o a eceding ho izon closed loop. No e ha he ollowing is supposed. Rema k 2.3 We assume ha he e is no model plan misma ch and neglec dis u bances. Hence, he ac- ual s a e xµN(m;x0)coincides wi h he p edic ed s a e xu(m;x0), c . [26, p.12]. Supposing his, he esul ing closed loop is in es iga ed wi h espec o so called nominal s abili y. In 23 RECEDING HORIZON CONTROL X u Op imiza ion Ho izon N⋅T N0, x0=u0 u1 u2 u3 u4 uN−2 uN−1 T 0 1 2 N−2 3 4 N−1 N xu⋅; x0 x0 xu1; x0 Figu e 2.1: G aphical illus a ion o he main idea (I/II): he compu ed con ol alues and he co esponding p edic ed ajec o y a e d awn in blue. In classical MPC he i s con ol alue is implemen ed as a eedback wi h espec o he ini ial s a e x0. Hence, xµN(1; x0) = (x0, µN(0; x0)) = xu(1; x0) is ob ained by applying µN(0; x0) = u(0), which is indica ed in ed. o de o emphasize his, he e m “nominal closed loop” is some imes used. Fo obus ness issues we e e o [10], [14, Chap e 8], [85, chap e 8], [109, chap e s 9-11], and [12]. In pa icula , we emphasize ha obus ness may ge los by inco po a ing addi ional e minal cons ain s, c . [31]. Assump ion 1.9, which was used o illus a i e pu poses, is eplaced by he ollowing, weake assump ion. Assump ion 2.4 Le he in imum o P oblem (2.4) -(2.6) be a ained, i.e., o each x0∈X⊆X, le a sequence o con ol alues u?(·)exis such ha JN(x0;u?(·)) = VN(x0)holds. Assump ion 2.4 ensu es ha he in imum o he op imal alue unc ion (2.7) is a minimum. In he ollowing, le us suppose ha an op imiza ion algo i hm is a ou disposal which inds he global minimum. Since he op imize compu es, in gene al, only a local minimum, his is, in pa icula o non-con ex op imiza ion p oblems, no ma e o cou se.2The mo i a ion o his assump ion is mainly o a oid echnical de ails which dis ac he eade om he main ideas o he p esen ed me hodologies. 2The p oblem o no being able o p o ide a global minimum — independen ly o whe he he eason goes back o being s uck in a local minimum o abo ing he compu a ion p ema u ely in o de o educe 24 INTRODUCTION X u Op imiza ion Ho izon N⋅T u'0=u1 u'1=u2 u3 u4 uN−2 uN−1 T 0 1 2 N−2 3 4 N−1 N xu⋅; x0 x0 x0 ':=xu1; x0 u'3 u'2 u'N−3 u'N−2 u'N−1 xu⋅; x0 ' Figu e 2.2: G aphical illus a ion o he main idea (II/II): he i s elemen is al eady implemen ed, c . Figu e 2.1. The cu en s a e is de ined as ou new ini ial s a e and he op imiza ion p oblem (2.4) - (2.6) is sol ed wi h espec o he new ini ial s a e. No e ha he alue u0(N−1) has o be compu ed om sc a ch whe eas he o me compu ed con ol alues may be a sensible ini ial guess o he op imiza ion. The esul ing con ol alues may coincide wi h he ones om he p eceding s ep, cp. u0(0) and u0(1), o di e since he op imiza ion ho izon akes addi ional s a es in o accoun , cp. u0(n), n= 2, . . . , N −2. Hence, he p edic ed ajec o y changes as well. The ollowing ema kable consequence holds o op imal ajec o ies. Rema k 2.5 As men ioned in Sec ion 1.2, ails o op imal ajec o ies a e again op imal o he e- spec i e op imal con ol p oblem. Fo he p oblem on a ini e ime ho izon, his eads as ollows: le u?(·)deno e a sequence o con ol alues sa is ying JN(x0;u?(·)) = VN(x0), hen JN−1(xu?(1; x0), u?(1 + ·)) = VN−1(xu?(1; x0)) holds — he ail u?(1 + ·)o he op imal sequence o con ol alues u?(·)is an op imal con ol o he p oblem on he sho ened ho izon N−1wi h ini ial alue xu?(1; x0), i.e. he s a e a he nex ime ins an o he ajec o y emana ing om x0gene a ed by u?(0), c . [44, Co olla y 3.16] o a p oo . Fu he mo e, he ollowing is poin ed ou o ne wo ked con ol sys ems. he compu a ional e o and, hus, he ime spend o sol ing he co esponding op imiza ion p oblem — is ackled in [43]. 25 TERMINAL INEQUALITY CONSTRAINTS AND COSTS ≥ N−1 X n=0 `(xu? N(n;x0), u? N(n)) + `(xu? N(N;x0),ˆu) + V ( (xu? N(N;x0),ˆu)) =`(x0, u? N(0; x0)) + N−1 X n=0 `(x˜uN(n;xu? N(1; x0)),˜uN(n)) + V ( (xu? N(N;x0),ˆu)) =`(x0, u? N(0; x0)) + J N(xu? N(1; x0),˜uN(·)) ≥`(x0, u? N(0; x0)) + V N(xu? N(1; x0)). Inequali y (2.16) gua an ees ha (xu? N(N;x0),ˆu)∈X and, hus, he admissibili y o ˜u(·) which ensu es he las inequali y. Hence, ecu si e easibili y o he RHC p oblem is a consequence o he assumed ini ial easibili y. S abili y can be deduced by he s anda d Lyapuno a gumen s. As a consequence, ini ial easibili y in combina ion wi h (2.16) gua an ees s abili y o he eceding ho izon closed-loop. The main ad an age o his RHC scheme in compa ison o he one wi h e minal equali y cons ain s is he elaxa ion o he e minal cons ain . No e ha he schemes coincide o X ={x?},V (·)≡0. The scheme based on a e minal egion and a (con ol) Lyapuno unc ion does no equi e exac con ollabili y o he desi ed equilib ium. Rema k 2.12 In pa icula o nonlinea sys ems, inding a sui able e minal egion X which is con- ol o wa d in a ian and sa is ies (2.16) is challenging. Fo sys ems go e ned by ime in a ian o dina y di e en ial equa ion, a linea iza ion a he se poin , i.e. he desi ed equilib ium, o en allows o compu e a locally s abilizing eedback as well as a local (con- ol) Lyapuno unc ion. No e ha (2.16) has o be sa is ied o his eedback K:X →U, i.e. V ( (x, K(x))) + `(x, K(x)) ≤V (x)and (x, K(x)) ∈X ∀x∈X . Hence, one looks o a con ol sequence s ee ing he nonlinea sys em “su icien ly close” o x?. Once, he ajec o y has en e ed he e minal egion X , he con ol inpu may be swi ched o he p ede ined eedback which ensu es he alidi y o he desi ed (con ol) Lyapuno inequali y — his s a egy is also e med dual mode, c . [73, p.8]. Feasibili y o he esul ing closed loop is, as al eady men ioned, ensu ed by supposing ini ial easibili y. In o de o illus a e hese MPC schemes, he example o he nonlinea in e ed pen- dulum on a ca is conside ed as a sampled-da a sys em wi h ze o o de hold. Example 2.13 Ou goal is o s abilize he nonlinea in e ed pendulum on a ca a he o igin, i.e. ou desi ed equilib ium. In o de o apply RHC based on addi ional e minal cos s, a lo- cal (con ol) Lyapuno unc ion has o be speci ied. To his end, he Lyapuno unc ion V (x) = xTPx, which was calcula ed o he linea ized model in Example 1.23, is employed. The s age cos s and pa ame e s a e also aken om his example in o de o ensu e con- sis ency wi h V (·). The e minal egion X is implici ly de ined by {x∈R4:xTPx ≤c}. Fo su icien ly small pa ame e c∈R>0, hese choices heu is ically ensu e he desi ed Lyapuno Inequali y (2.16). This claim is subs an ia ed by ou nume ical esul s, below. Le he ini ial alue x0= (0.1 0.1 0.1 0.1)T, he sampling pa ame e T= 0.0625, and he e minal egion X ={x∈R4:V (x)≤0.1}be gi en. The p edic ed ajec o ies a e compu ed by means o he MATLAB ou ine ode15 which is an implici Runge-Ku a me hod wi h s ep size con ol. Since a cons ained nonlinea minimiza ion p oblem is deal 32 TERMINAL INEQUALITY CONSTRAINTS AND COSTS −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 −5 −4 −3 −2 −1 0 1 2 x1 x2 N = 7 N = 10 N = 16 N = 32 N = 40 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 x3 x4 N = 7 N = 10 N = 16 N = 32 N = 40 Figu e 2.4: T ajec o ies gene a ed by eceding ho izon con ol o a ious op imiza ion ho izons Nwi h ini ial alue x0= 0.1·(1,1,1,1)T. wi h, he ou ine mincon is used in o de o sol e he in ol ed op imiza ion p oblems. The esul ing ajec o ies a e depic ed in Figu e 2.4. Ou nume ical compu a ions show ha N= 7 is he smalles op imiza ion ho izon which allows o an ini ially easible ajec o y, i.e. compu ing a sequence o Ncon- ol alues such ha xu(N;x0)∈ {x∈Rn:xTPx ≤c}holds. Howe e , his leads o J7(x0) = 1628.51369. We poin ou ha he con ibu ion o he addi ional e minal cos is limi ed o c= 0.1and, hus, negligible. Ra he , he la ge alue o he cos unc ional has o be asc ibed o he e minal cons ain xu(N;x0)∈X whose sa is ac ion demands a compa a i ely la ge con ol e o . The ac ual cos s o he co esponding eceding ho izon closed loop sum up o 870.6461.5Inc easing he op imiza ion ho izon, which implici ly enla ges he easible se o he op imiza ion, signi ican ly educes he cos s associa ed wi h he i s 128 s eps, c . Figu e 2.5. Fo he chosen ini ial condi ion, he s a ic s a e eedback compu ed o he linea ized e sion may also be used in o de o s abilize he sys em, howe e , wi hou aking he e minal cons ain in o accoun . In doing so, cos s amoun ing o 60.7659 a e p oduced. RHC ou pe o ms his eedback only o a su icien ly la ge op imiza ion ho izon, e.g. N= 20. Hence, using a e minal egion has a s abilizing e ec bu may sh ink he se U=UN(x0)o admissible con ols u(·)and, as a consequence, may cause highe cos s. A he ex eme, Uequals he emp y se and he op imiza ion p oblem (2.17) - (2.19) becomes in easible, e.g. N≤6. Fo la ge ho izons, he impac o inco po a ing a e minal cons ain in Uis educed, which esul s in an enla ged se Uand lowe cos s on he in ini e ho izon. No e ha RHC wi h smalle op imiza ion ho izons s ee s he closed loop ajec o y, in gene al, as e in o he e minal egion X , c . Table 2.2. The op imal alue unc ion V N(·)dec eases s ic ly along he eceding ho izon closed loop solu ion in ou nume ical compu a ions, c . Figu e 2.5.6The desi ed Lyapuno inequali y is, howe e , only sa is ied o he i s s eps o he RHC solu ion due o ou heu is ic choice o he e minal cos V (·). The pu pose o he inco po a ed local (con ol) Lyapuno unc ion is o app op ia ely 5The closed loop cos s a e only measu ed on he in e al [0,8] ins ead o [0,∞). Howe e , a = 8 he s a e is al eady e y close o he desi ed se poin such ha his unca ion o he ime ho izon does no dis o he nume ical esul s. 6This claim does no hold o he ajec o y gene a ed by he s a ic s a e eedback o N≤32. 33 TERMINAL INEQUALITY CONSTRAINTS AND COSTS N7 10 12 16 24 30 32 40 64 ime 1.125 1.5625 1.9375 2.8750 2.6875 3.0000 3.0625 3.8125 5.4375 Table 2.2: Time elapsed un il he e minal cons ain , i.e. xµN( )∈X , is, depending on he op imiza ion ho izon N, sa is ied. 10 15 20 25 30 35 40 45 50 55 60 101 102 103 Op imiza ion ho izon N J∞ µ 0 1 2 3 4 5 6 7 8 10−6 10−4 10−2 100 102 104 VN (⋅) N = 7 N = 10 N = 16 N = 32 N = 40 Figu e 2.5: On he le , he o e all cos s Jµ ∞and V N(x0) depending on he op imiza- ion ho izon Na e d awn in blue and ed, espec i ely. The cos s associa ed wi h he p ecompu ed eedback a e indica ed by he dashed line. On he igh , he di e ence V N(xµN(n+ 1)) −V N(xµN(n)) is illus a ed o a ious op imiza ion ho izons N. es ima e he cos o go, i.e. o gi e an uppe bound o he emaining cos s which a e needed in o de o ende he sys em asymp o ically s able. O en, such a Lyapuno unc ion, which has o sa is y Inequali y (2.16), is cons uc ed by a linea iza ion a he desi ed se poin , c . Example 2.13. As a consequence, he e minal egion X has o be chosen su icien ly small which makes he e minal cons ain xu(N;x0)∈X mo e es ic i e. Finding a (con ol) Lyapuno unc ion such ha Inequali y (2.16) is sa is ied globally, i.e. o all x∈X, allows o neglec he e minal cons ain en i ely. Howe e , his is, in pa icula o sys ems go e ned by nonlinea o dina y o pa ial di e en ial equa ions, a challenging ask and, in gene al, no possible. Summa izing, a la ge domain o a ac ion equi es, in gene al, a la ge op imiza ion ho izon Nin RHC schemes wi h e minal cons ain s. Fu he mo e, o each ini ial con- di ion x0∈X, an ini ially easible solu ion o (2.17) - (2.19), i.e. a ajec o y emana ing om x0and eaching he e minal egion a e a mos N ime s eps, has o be ound. Hence, he p esumably mos di icul p oblem has o be ackled a he beginning. On he o he hand, easibili y and s abili y o he eceding ho izon closed loop a e gua an eed. In conclusion, inding a e minal egion equipped wi h an app op ia e local (con ol) Lya- puno unc ion and ensu ing ini ial easibili y is demanding and o en oo es ic i e om a p ac ical poin o iew, c . [100] — al hough hese p e equisi es a e al eady easie o e - i y compa ed o he e minal equali y cons ain s om he p e ious sec ion. In addi ion, MPC wi h e minal cons ain s may gi e asymp o ic s abili y wi hou any obus ness, as shown in [31]. Hence, we shi ou ocus o uncons ained RHC schemes. 34 2.4. FEASIBILITY 2.4 Feasibili y In Sec ions 2.2 and 2.3 cons ain s we e in oduced whose sa is ac ion gua an ees easi- bili y and s abili y o he espec i e RHC schemes. Howe e , inding an ini ially easible ajec o y and, i e minal cos s a e used, designing a sui able (con ol) Lyapuno unc- ion, which is used in o de o es ima e he cos o go, is challenging. Fu he mo e, hese app oaches may ende ini ial condi ions in easible o ho izons N o which RHC schemes wi hou e minal cons ain s and cos s s abilize he sys em. The linea wa e equa ion is such an example in which he ini e p opaga ion speed p e en s he sys em om eaching a neighbo hood o he o igin as whe eas so called uncons ained RHC ul ills he p o- posed ask o s abilizing he sys em e en o ex emely sho op imiza ion ho izons N, c . [62]. In his hesis we a e conce ned mainly wi h he s abili y analysis o uncons ained RHC. Howe e , since hese schemes do no gua an ee easibili y o he esul ing RHC closed loop igh om he beginning, he sys em may become in easible al hough a Lya- puno inequali y was sa is ied o he unca ed op imal alue unc ion VN(·) in each o he p eceding s eps. The phenomenon o no being able o de ec easibili y p oblems on ime, is o en e med sho -sigh edness o he eceding ho izon closed loop, c . [2, p.178] and [44, Example 8.1]. In o de o ensu e easibili y, Assump ion 1.4 is supposed which i s in well wi h ou s anda d assump ion ha he op imal alue unc ion is ini e o each s a e x0o he easible se X. He e, a ske ch o a easibili y p oo om [99] is p esen ed which ou lines a way o encoun e he easibili y p oblem wi hou Assump ion 1.4. We poin ou ha he main idea o ende ing a le el se o he alue unc ion VN(·) in a ian wi h espec o he em- ployed eceding ho izon s a egy is also used in o de o ensu e easibili y o an example conside ed in Sec ion 4.4. Since he examples which a e in es iga ed o in ini e dimen- sional sys ems do no exhibi igh s a e cons ain s, we es ic ou sel es mainly o ini e dimensional sys ems. Ne e heless, we emphasize ha he concep s p esen ed in his sec- ion can no be ans e ed o in ini e dimensional sys ems because some conclusions can no be d awn analogously. Fo example he uni sphe e is bounded and closed bu no compac in L2(R,Rn), c . [119] and [110, Co olla y 4.5]. Fu he mo e, we like o poin ou ha [99] only deals wi h sys ems go e ned by linea dynamics. The ideas, howe e , may be gene alized o he nonlinea se ing. Mo e elabo a e echniques in o de o ensu e easibili y o uncons ained RHC schemes a e discussed, e.g. in [44]. A necessa y condi ion o easibili y o uncons ained RHC wi h op imiza ion ho izon Nis easibili y on he in ini e ho izon which is cha ac e ized by a ini e alue o he espec i e op imal alue unc ion V∞(x0). Hence, he i s s ep owa ds a easibili y analysis is aking a close look a his se . The linea se ing is conside ed, i.e. sys em dynamics x(n+ 1) = Ax(n) + Bu(n) wi h a con ollable pai [A, B] and cons ain s gi en by Ex +Fu ≤ψ. Neglec ing he cons ain s, assuming ha [A, B] is a con ollable pai implies ha e e y x0∈Rnis exac ly con ollable o he o igin in a ini e numbe o s eps which is less o equal he dimension n∈No he s a e space, c . [58] o a p ecise de ini ion. Howe e , due o he cons ain which may model simple box cons ain s o he con ol inpu and, hus, excluding unsa u a ed con ols, his does no hold o he whole space. Hence, we de ine he se I0={0}and he se s Ik+1 := {x∈Rn:∃usuch ha Ax +Bu ∈Ikand Ex +Fu ≤ψ}. Thus, I1con ains all poin s which may be s ee ed o he o igin in one s ep wi hou iola ing he imposed cons ain s. Mo eo e , Ik⊆Ik+1 due o he cons uc ion. De ining 35 FEASIBILITY he se I∞:= S∞ k=0 Ikwe ob ain he ollowing esul . Theo em 2.14 Le he pai [A, B]be con ollable and (0,0) be an in e io poin o he cons ain se {(x, u)∈Rn×Rm:Ex +Bu ≤ψ}. Fu he mo e, le he s age cos s sa is y `(x, u)≥ α(kxk) o a K∞- unc ion α:R+ 0→R+ 0, e.g. xTQx +uTRu wi h posi i e de ini e ma ix Qand posi i e semi-de ini e ma ix R. Then he ollowing equi alence holds: x0∈I∞⇐⇒ V∞(x0)<∞. The main ideas o he p oo a e ske ched. Then, possibili ies in o de o gene alize Theo em 2.14 o he nonlinea se ing a e indica ed and b ie ly discussed in Rema k 2.15. Supposing x0∈I∞ensu es he exis ence o an index Ksuch ha x0∈IK. Conse- quen ly, he de ini ion o he se IKallows us o cons uc a sequence o con ol alues (u(k))k∈{0,1,...,K−1}which easibly s ee s he sys em om x0∈IK o I0. Hence, V∞(x0) is bounded by PK−1 n=0 `(xu(n;x0), u(n)) <∞. Le x0/∈I∞be gi en. Since [A, B] is con ollable and (0,0) ∈Rn×Rmis an in e io poin o he cons ain se , deadbea con ol may be ca ied ou . To be mo e p ecise, e e y s a e con ained in a su icien ly small neighbo hood o he o igin may be s ee ed o he o igin in a mos ns eps. Le Bδ(0) ⊂Rn,δ∈R>0, deno e a ball wi h adius δ comple ely con ained in his neighbo hood. Hence, since x0/∈I∞, he e does no exis a sequence o con ol alues s ee ing x0in o Bδ(0). As a consequence, o each easible (u(n))n∈N0 he es ima e `(xu(n;x0), u(n)) ≥α(kxu(n;x0)k)> α(δ/2) >0 holds o all n∈N0. Hence, J∞(x0, u(·)) = P∞ n=0 `(xu(n;x0), u(n)) ≥P∞ n=0 α(δ/2) = ∞ o e e y easible (u(n))n∈N0. Rema k 2.15 Deadbea con ol is a es ic i e assump ion in he nonlinea se ing, cp. RHC wi h e - minal equali y cons ain s in Sec ion 2.2. Hence, his p e equisi e should be weakened, e.g. assuming he exis ence o a neighbo hood o he desi ed equilib ium such ha each poin con ained in his se is s abilizable inducing ini e cos s. This seems o be a eason- able op ion in o de o gene alize he p oposed cha ac e iza ion o he easible se o he p oblem on an in ini e ime ho izon o a nonlinea se ing. Fu he mo e, we emphasize ha RHC wi h addi ional e minal cons ain s and cos s equi es a simila , e en s onge assump ion anyway, cp. Inequali y (2.16). We con inue wi h he main esul conce ning easibili y om [99]. Theo em 2.16 Le he assump ions o Theo em 2.14 be sa is ied and a pa ame e µ∈R>0be gi en. The µsub-le el se Sµo V∞(·)is de ined by {x∈Rn:V∞(x)≤µ}. Then, an op imiza ion ho izon N0∈N≥2exis s such ha Sµis an in a ian se unde any RHC eedback esul ing om he op imiza ion p oblem (2.4) –(2.6) wi h ho izon N≥N0. The p oo , which can be ound in [99, Appendix], consis s o wo pa s and elies essen ially on he mono onici y o he alue unc ion VN(·) wi h espec o he op imiza ion ho izon leng h N. To be mo e p ecise, VN(·) has o be mono onically inc easing in N— a cha ac e is ic which is au oma ically ul illed o uncons ained RHC, c . Sec ion 2.1. We s a by wo auxilia y claims in o de o p epa e he g ound o he ac ual p oo . •Le β∈(0, µ) be chosen such ha {x∈Rn:α(kxk)≤β} ⊆ Sµ. Then, he ollowing calcula ion shows ha xµN(1; x0)∈Sµholds o all x0∈Sβand N∈N≥2: β≥V∞(x0)≥VN(x0)≥VN−1(xµN(1; x0)) ≥`?(xµN(1; x0)) ≥α(kxµN(1; x0)k). 36 FEASIBILITY •Le he se Wbe de ined by {x∈Rn:α(kxk)≤µ}. Then, xµN(1; x0)∈W∩I∞ holds o all x0∈Sµand su icien ly la ge ho izons N. No e ha xµN(1; x0)∈I∞ gua an ees V∞(xµN(1; x0)) <∞. Repea ing he compu a ion used in o de o es ablish he p e ious asse ion wi h β subs i u ed by µyields xµN(1; x0)∈W. In o de o show xµN(1; x0)∈I∞, a line o a gumen s simila o he p oo o Theo em 2.14 is employed: ini ially, o x /∈I∞, a lowe bound o he s age cos s is es ablished in o de o de i e a con adic ion o su icien ly la ge N, c . [99, Lemma 12] o de ails. Taking he i s asse ion in o accoun allows us o ocus on s a es x∈Sµ Sβin o de o p o e Theo em 2.16. Suppose ha a ho izon leng h N0sa is ying he claim o Theo em 2.16 does no exis . Then, o each j∈N, a ho izon leng h Nj≥jand a s a e xj 0∈Sµ Sβ exis such ha xµNj(1; xj 0)∈W Sµ⊂W. Since Wis compac , (xµNj(1; xj 0))j∈Nhas a con e gen subsequence (˜xk)k∈N:= (xµNjk(1; xjk 0))k∈N, (jk)k∈N⊆Nwi h jk+1 > jk o all k∈N, wi h ˜xk→˜x∞ o k ending o in ini y. I ˜x∞is no con ained in I∞,V∞(˜x∞) = ∞holds. O he wise, V∞(˜x∞) = limk→∞ V∞(˜xk)≥µis ensu ed by he second asse ion in iew o Theo em 2.14. Combining hese asse ions, yields V∞(˜x∞)≥µ. Hence, o e e y ε > 0, a ho izon leng h Nexis s such ha VN(˜x∞)> µ −ε/4.(2.20) Nex , we p o e ∞> VN(˜x∞) by con adic ion. To his end, suppose ha ˜x∞is no easible o he op imiza ion p oblem (2.4) - (2.6) wi h ho izon leng h N. The cons ain s speci y a bounded se o any N0, c . [99, Lemma 10], which is sh inking o la ge N. Hence, VN(˜x∞) = ∞implies he exis ence o an open neighbo hood o ˜x∞which is no easible o all N0≥N— a con adic ion o he con e gence ˜xk→˜x∞ o k→ ∞. Choose ε= in x0∈Sµ Sβα(kx0k)≤in x0∈Sµ Sβ`?(x0). Then, N∈Nexis s such ha (2.20) wi h ∞> VN(˜x∞) holds. Since VN(·) is con inuous, picking Njk> N la ge enough ensu es VNjk(˜xk) = VNjk(xµNjk(1; xjk 0)) ≥VN(xµNjk(1; xjk 0)) ≥µ−ε/2.(2.21) Hence, we ob ain he ollowing inequali y which leads o a con adic ion and comple es he p oo o Theo em 2.167 µ≥VNjk(xjk 0) = `(xjk 0, uNjk(0)) + VNjk−1(xµNjk(1; xjk 0)) ≥`?(xjk 0) + VN(xµNjk(1; xjk 0)) ≥ε+µ−ε/2 = µ+ε/2. The main ideas o his p oo a e gene alizable o he nonlinea se ing. Howe e , gene alizing his easibili y esul o he in ini e dimensional se ing may cause addi ional ( echnical) p oblems, e.g. compac ness o he se Wcan no be expec ed. Ins ead one has o use he concep o weak sequen ial compac ness, c . [79, Sec ion 10.2], and, as a consequence, only ob ains a weakly con e gen subsequence. No e ha he di e en compac ness e ms a e equi alen o no med spaces, c . [106, Ebe lein-ˇ Smulian Theo em], and ha , e.g. he uni sphe e is weakly sequen ially compac in e e y e lexi e space, c . [106, Theo em 2.8.2]. Again, we e e o [44] o mo e elabo a e esul s wi h espec o easibili y, in pa icula o a gene aliza ion o he nonlinea case. 7The i s equali y is lawed in [99]. Since using he op imiza ion ho izon Nk+ 1 leads, in gene al, no o xµNk(1; xk 0) as he nex s a e. 37 FEASIBILITY We like o poin ou ha a s anda d assump ion o s abili y esul s in RHC is he ela ion o he unc ion α1(·)∈ K∞and he s age cos s ia (1.4). Hence, he p e equisi es o Theo ems 2.14 and 2.16 a e no oo es ic i e. Summa izing, RHC wi h ei he e minal cons ain s o cos s ensu es easibili y a p io i bu a he expense o assuming an ini ially easible solu ion — independen ly o whe he easibili y issues play a ole o no . Using uncons ained RHC may lead o easibili y p oblems, in pa icula o sho op imiza ion ho izons Ndue o i s “sho sigh edness”, c . [2, p.178]. On he o he hand, neglec ing e minal cons ain s enla ges he se o admissible con ols signi ican ly and, hus, may imp o e he closed loop pe o mance. In his hesis, howe e , we do no ocus on easibili y issues. This mo i a es Assump ion 1.4, which may be weakened. Assump ion 1.4 ensu es, o each ini ial alue x0∈X, he exis ence o a sequence o con ol alues which sa is ies he cons ain s. Ne e heless, RHC may cause in ini e cos s in he long un. 38 Chap e 3 S abili y and Subop imali y o RHC Schemes In his hesis we a e conce ned wi h eceding ho izon schemes which inco po a e nei he e minal cons ain s no addi ional e minal cos s. These schemes exhibi a decisi e ad- an age in con as o hei coun e pa s which ake e minal cos s o cons ain s in o accoun : he op imal alue unc ion VN(·) inc eases, o each easible ini ial alue x0∈X, mono onically in he op imiza ion ho izon N— an inhe en mono onici y p ope y which allows us o exploi Lyapuno ype inequali ies in o de o es ima e, in addi ion o con- cluding s abili y, he pe o mance o he esul ing RHC closed loop. Assump ion 2.1 ensu es boundedness o V∞(·) on Xwhich is a necessa y condi ion o well-posedness o he op imal con ol p oblem on he in ini e ime ho izon because o h- e wise ei he he cons ain s a e ine i ably iola ed o he s age cos s a e no summable. The la e indica es ha he cos unc ional does no p o ide a sui able c i e ion o s a- bilizing he sys em a he desi ed equilib ium and is, hus, inadequa ely designed. Hence, he mono onically inc easing sequence (VN(x0))N∈N≥2is bounded om abo e by V∞(x0). In Sec ion 3.1 a elaxed Lyapuno inequali y is in oduced which o ms he co e o ou s abili y and subop imali y esul s. Based on a con ollabili y condi ion and Bell- man’s p inciple o op imali y, a nonlinea p og am is deduced which gi es us a su icien condi ion in o de o alida e his Lyapuno inequali y. In he ensuing sec ion ou main s abili y heo em is p esen ed. In Sec ion 3.3 he p oposed op imiza ion p oblem is sol ed o an impo an subclass, which p o ides an easily checkable s abili y and pe o mance c i e ion. Then, he in oduced me hodology is demons a ed. To his end, ou key as- sump ion, i.e. Assump ion 3.2, is e i ied o he linea wa e equa ion which allows o ensu e ins an aneous con ollabili y o his hype bolic pa ial di e en ial equa ion igo - ously. 3.1 Relaxed Lyapuno Inequali y In Sec ion 1.4 ne wo ked con ol sys ems we e in oduced and he no a ion o a mul is ep eedback law µ:X×{0,1, . . . , m?−1} → Uwi h m?∈Nwas speci ied. Using a eceding ho izon con olle based on op imiza ion p oblem (2.4) – (2.6) yields a sequence o N inpu alues o a gi en ini ial alue x0. Since we in end o employ hese alues in o de o cons uc he eedback law µN(·,·), he condi ion m?≤N−1 has o be sa is ied. The pa ame e m?de e mines he maximal numbe o con ol alues which may be applied be o e he op imiza ion p oblem has o be sol ed again in o de o upda e — based on a 39 STABILITY AND SUBOPTIMALITY OF RHC SCHEMES measu emen o he cu en s a e — he sequence o con ol alues. Hence, m?limi s he maximal ime he sys em may s ay in open loop. Whe eas he se M⊆ {1,2, . . . , m?} om De ini ion 1.25 mainly places some lexibili y a ou disposal, which migh be con enien o he ne wo ked con ol se ing, e.g. he ne wo k opology may o ce us only o use odd numbe s o elemen s o he compu ed sequence o con ol alues due o ansmission speci ica ions. Ne e heless, one may hink o M={1,2, . . . , m?−1}in he sequel. Le an admissible con ol ho izon sequence (mi)i∈N0be gi en. Then, using he no a ion om De ini ion 1.24, he co esponding cos s on he in ini e ime in e al a e gi en by Vµ,(mi) ∞(x0) := ∞ X n=0 `(xµ(n), µ(xµ(ϕ(n)), n −ϕ(n))). Ou app oach elies on he ollowing esul om elaxed dynamic p og amming [83,101], which is a gene aliza ion o [39, P oposi ion 2.4]. P oposi ion 3.1 Le a mul is ep eedback law ˜µ:X×{0,1, . . . , m?−1} → U, a se M⊆ {1,2, . . . , m?}, and a unc ion e V:X→R+ 0be gi en. Suppose ha , o each x0∈X, he solu ion x˜µ(·) = x˜µ(·;x0)wi h x˜µ(0) = x0sa is ies x˜µ(n;x0)∈X,n∈ {0,1, . . . , N −1}, and e V(x0)≥e V(x˜µ(m)) + α m−1 X k=0 `(x˜µ(k),˜µ(x0, k)) ∀m∈M(3.1) o some α∈(0,1]. Then, o all x0∈Xand all admissible sequences (mi)i∈N0o con ol ho izons, he es ima e αV∞(x0)≤αV ˜µ,(mi) ∞(x0)≤e V(x0) (3.2) holds. P oo : Conside x0∈Xand he ajec o y x˜µ(·) = x˜µ,(mi)(·;x0) gene a ed by he closed loop sys em using he mul is ep eedback ˜µ(·,·) associa ed wi h (mi)i∈N0. Since x˜µ(n;x˜µ(σ(k); x0)) ∈X,n∈ {0,1, . . . , N −1}, implies x˜µ(σ(k+ 1); x0), (3.1) yields α σ(k+1)−1 X n=σ(k) `(x˜µ(n),˜µ(x˜µ(ϕ(n)), n −ϕ(n))) ≤e V(x˜µ(σ(k))) −e V(x˜µ(σ(k+ 1))) o all k∈N0. Summing o e he ansmission imes σ(k), k= 0,1, . . . , k?, yields α σ(k?)−1 X n=0 `(x˜µ(n),˜µ(x˜µ(ϕ(n)), n −ϕ(n))) = α k?−1 X k=0 σ(k+1)−1 X n=σ(k) `(x˜µ(n),˜µ(x˜µ(ϕ(n)), n −ϕ(n))) ≤e V(x(0)) −e V(x(σ(k?)) ≤e V(x(0)). Fo k?→ ∞ his shows ha e V(x0) is an uppe bound o αV ˜µ,(mi) ∞(x0). Since he de ini ion o he op imal alue unc ion V∞(·) ensu es he i s inequali y in (3.2) di ec ly, his comple es he p oo .  40 RELAXED LYAPUNOV INEQUALITY Ou goal consis s o es ablishing (3.1) o e V(·) = VN(·) and he co esponding RHC con olle ˜µ(·,·) = µN(·,·). Then, using he mono onici y o VN(·) in Nyields αV µN,(mi) ∞(x0)≤VN(x0)≤V∞(x0), which gua an ees ha he RHC closed loop p oduces a mos 1/α as much cos s as he op imal eedback on he in ini e ime ho izon, i.e. a subop imali y es ima e. Ou key ing edien in o de o deduce (3.1) is he ollowing con ollabili y assump ion om [39]. The ela ion be ween Assump ion 3.2 and o he con ollabili y condi ions, e.g. he one used in [32], is discussed in Sec ion 5.5, below. Assump ion 3.2 Le a unc ion β(·,·)∈ KL0be gi en. Suppose ha , o each x0∈X, an admissible con ol unc ion ux0(·)∈ U =U∞(x0)⊆UN0exis s, which sa is ies `(xux0(n), ux0(n)) ≤β(`?(x0), n) o all n∈N0.(3.3) Impo an ep esen a i es o class KL0- unc ions lead o exponen ial o ini e ime con ollabili y, c . Rema k 1.13. In addi ion o Assump ion 3.2, he use ul p ope y (1.13) is assumed which ensu es ha any sequence o he o m λn=β( , n), > 0, ul ills λn+m≤β(λn, m), c . Sec ion 1.2. Assump ion 3.2 is e i ied o he disc e e ime sys em om Example 1.10 in o de o illus a e he meaning o Condi ion (3.3). In pa icula , he example shows ha he in ol ed sequence o con ol alues ux0∈ U =U∞(x0) does no need o be op imal. Example 3.3 Example 1.10 is conside ed. The s age cos s a e gi en by `(x, u) = xTQx +uTRu =xT1 0 0 1 x+uTu=kxk2+kuk2. In Example 1.17 we de i ed he es ima e kx(n;x0)k ≤ Cσnkx0k o he s a ic s a e eedback BF∞. Hence, using he eedback BF∞applied o he cu en s a e yields `(x(n), u(n)) = kx(n)k2+kBF∞x(n)k2≤(1 + kBF∞k2)kx(n)k2 ≤(1 + kBF∞k2)C2σ2nkx0k2=˜ C˜σn`?(x0) wi h ˜ C:= (1 + kBF∞k2)C2,˜σ:= σ2, i.e. exponen ial con ollabili y wi h espec o he s age cos s o , equi alen ly, Assump ion 3.2 wi h β( , n) = ˜ C˜σn· . No e ha he KL- unc ion βis linea in i s i s a gumen . Al e na i ely, one may show ha his example is ini e ime con ollable, c . Example 2.7. In iew o hese esul s, we ob ain `(x(0), u(0)) = kx0k2+k21/110 −2x0k2≤60941/12100 ·`?(x0)<5.04 ·`?(x0), `(x(1), u(1)) =    1 1.1 −10/11 −1x01 x02     2 +   221/110 2.21 x01 x02     2 ≤15677961/1210000 ·`?(x0)<12.96 ·`?(x0). Hence, β( , 0) = c0· ,β( , 1) = c1· , and β( , n)=0 o n∈N≥2wi h c0= 5.04 and c1= 12.96. 41 STABILITY AND SUBOPTIMALITY OF RHC SCHEMES 3.10. O he wise, i s op imal alue can s ill be used as a lowe bound o he subop imali y deg ee o he eceding ho izon closed loop. P oblem 3.17 Minimize 1−(γm+1 −ω)λN−1subjec o λ= (λ1, . . . , λN−1)T≥0componen wise and Aλ ≤¯ b, whe e A:=        a1a2. . . aN−2ω d11. . . 1b1 0d2.... . .. . . . . .......1bN−3 0. . . 0dN−2bN−2        and ¯ b:=        γN−1 0 . . . 0 0        wi h aj=γN o j < m 1o he wise bj=ω o j < m γm+1 o he wise dj=1−γN−j o j < m 1−γN−j+mo he wise Theo em 3.18 Le β(·,·)∈ KL0 om Assump ion 3.2 be linea in i s i s a gumen and sa is y (1.13). Then he op imal alue α=αω N,m o P oblem 3.10 o gi en op imiza ion ho izon N, con ol ho izon m, and weigh ωon he inal e m sa is ies αω N,m = 1 i and only i ω≥γm+1. O he wise, we ge αω N,m = 1 − (γm+1 −ω) N Q i=m+2 (γi−1) N Q i=N−m+1 (γi−1) N Q i=m+1 γi−(γm+1 −ω) N Q i=m+2 (γi−1) N Q i=N−m+1 γi− N Q i=N−m+1 (γi−1). (3.21) P oo : We showed ha he linea op imiza ion p oblem s a ed in P oposi ion 3.16 yields he same op imal alue as P oblem 3.10 o KL0- unc ions which a e linea in hei i s a gumen . Technically, his is posed as a minimiza ion p oblem. Taking he es ic ion λN−1≥0 in o accoun , leads o he ques ion, whe he he coe icien o λN−1in he objec i e unc ion is posi i e o no . As a consequence, he aim is ei he minimizing o maximizing λN−1. In he i s case, i.e. γm+1 −ω≤0, choosing λ1=. . . =λN−1= 0 sol es he conside ed ask and p o ides αω N,m = 1. Hence, we suppose λm+1 −ω > 0. In o de o p o e he asse ion, i.e. he s a ed o mula, we sol e he elaxed P oblem 3.17 and show ha i s op imum is also easible o he o iginal p oblem, i.e. P oblem 3.10. The linea sys em o equa ions Aλ =¯ bwi h Aand ¯ b om P oblem 3.17 is sa is ied a he op imum — a c ucial p ope y which is shown by Lemma 3.22. This allows us o deduce exp essions o λN−2, λN−3, . . . , λ1depending (only) on λN−1. Inse ing he ob ained e ms in o A1λ=¯ b1allows o sol ing his equa ion wi h espec o a iable λN−1. Plugging his exp ession o λN−1in o he objec i e unc ion o he op imiza ion p oblem in conside a ion, yields Fo mula (3.18). Suppose N−m≥2. Then λN−j,j= 2,3, . . . , N −mis gi en by λN−j=Qm+j−1 i=m+1 γi Qm+j i=m+2(γi−1) ·λN−1.(3.22) 48 LINEAR PROGRAM We show his claim by induc ion o e j= 2,3, . . . , N −m. Fo j= 2, he asse ion ollows di ec ly om AN−1λ=¯ bN−1= 0. Thus, we con inue wi h he induc ion s ep using Lemma 3.21 wi h m+2, m+j−1 ins ead o m,M. Using −dN−j=−(1−γN−(N−j)+m) = γm+j−1 and bm−j=γm+1 yields λN−j=γm+1λN−1+Pj−1 i=2 λN−i γm+j−1 I.A. = γm+1 hQm+j−1 i=m+2 (γi−1) + Pj−1 i=2 Qm+i−1 k=m+2 γkQm+j−1 k=m+i+1(γk−1)i Qm+j i=m+2(γi−1) ·λN−1 = γm+1 hQm+j−1 i=m+2 (γi−1) + Pm+j−1 i=m+2 Qi−1 k=m+2 γkQm+j−1 k=i+1 (γk−1)i Qm+j i=m+2(γi−1) ·λN−1 (3.32) =Qm+j−1 i=m+1 γi Qm+j i=m+2(γi−1) ·λN−1. Suppose m≥2. Then λm−j,j= 1,2, . . . , m −1 is gi en by λm−j=QN−m+j−1 i=N−m+1 γi QN−m+j i=N−m+1(γi−1) ωλN−1+ N−m X i=2 λN−i!.(3.23) We show (3.23) by induc ion o e j= 1,2, . . . , m −1. Fo an index jchosen om he speci ied ange, −dm−j=γN−m+j−1 and bm−j=ωhold. Hence, conside ing Amλ= ¯ bm= 0 p o ides he asse ion o j= 1. Using Lemma 3.21 wi h N−m+ 1, N−m+j ins ead o m,M, we pe o m he induc ion s ep in o de o show he asse ion: λm−j=ωλN−1+PN−m i=2 λN−i+Pj−1 i=1 λm−i (γN−m+j−1) I.A. = N−m+j−1 Q i=N−m+1 (γi−1) + j−1 P i=1 N−m+i−1 Q k=N−m+1 γk N−m+j−1 Q k=N−m+i+1 (γk−1) QN−m+j i=N−m+1(γi−1) ωλN−1+ N−m X i=2 λN−i! = N−m+j−1 Q i=N−m+1 (γi−1) + N−m+j−1 P i=N−m+1 i−1 Q k=N−m+1 γk N−m+j−1 Q k=i+1 (γk−1) QN−m+j i=N−m+1(γi−1) ωλN−1+ N−m X i=2 λN−i! (3.32) =QN−m+j−1 i=N−m+1 γi QN−m+j i=N−m+1(γi−1) ωλN−1+ N−m X i=2 λN−i!. Be o e we p oceed, we u he in es iga e he second ac o o (3.23). Again, Lemma 3.21 wi h m+ 2, Nins ead o m,Mis o help ul: N−m X j=2 λN−j+ωλN−1 (3.22) = N−m X j=2 Qm+j−1 i=m+1 γi Qm+j i=m+2(γi−1)λN−1+ωλN−1 = ω+PN−m j=2 Qm+j−1 i=m+1 γiQN m+j+1(γi−1) QN i=m+2(γi−1)  λN−1 49 STABILITY AND SUBOPTIMALITY OF RHC SCHEMES = ω+ γm+1 PN j=m+2 Qj−1 i=m+2 γiQN j+1(γi−1) QN i=m+2(γi−1)  λN−1 (3.32) = ω N Q i=m+2 (γi−1) + γm+1 N Q i=m+2 γi− N Q i=m+2 (γi−1) QN i=m+2(γi−1) ·λN−1 = QN i=m+1 γi−(γm+1 −ω)QN i=m+1(γi−1) QN i=m+2(γi−1) !λN−1.(3.24) Now, we p epa ed he g ound in o de o ex ac an explici exp ession o λN−1 om A1λ=¯ b1=γN−1 by applying (3.23). To his end, we conside he le hand side o his equa ion, i.e. A1λ=γN m−1 X i=1 λi+ N−2 X i=m λi+ωλN−1=γN m−1 X j=1 λm−j+ N−m X j=2 λN−j+ωλN−1 (3.23) ="γN m−1 X j=1 QN−m+j−1 i=N−m+1 γi QN−m+j i=N−m+1(γi−1) + 1# ωλN−1+ N−m Y j=2 λN−j!.(3.25) The i s ac o o he le hand side is ew i en by means o Lemma 3.21 applied wi h N−m+ 1, N−1 ins ead o m,M: γN m−1 X j=1 QN−m+j−1 i=N−m+1 γi QN−m+j i=N−m+1(γi−1) + 1 = γNPm−1 j=1 QN−m+j−1 i=N−m+1 γiQN−1 i=N−m+j+1(γi−1) QN−1 i=N−m+1(γi−1) + 1 = γNPN−1 j=N−m+1 Qj−1 i=N−m+1 γiQN−1 i=j+1(γi−1) QN−1 i=N−m+1(γi−1) + 1 (3.32) = γNQN−1 i=N−m+1 γi−QN−1 i=N−m+1(γi−1) QN−1 i=N−m+1(γi−1) + 1 =QN i=N−m+1 γi−QN i=N−m+1(γi−1) QN−1 i=N−m+1(γi−1) .(3.26) Hence, inse ing (3.24) and (3.26) in o (3.25) and sol ing A1λ=γN−1 wi h espec o λN−1yields λN−1=(γN−1) QN−1 i=N−m+1(γi−1) QN i=N−m+1 γi−QN i=N−m+1(γi−1) QN i=m+2(γi−1) QN i=m+1 γi−(γm+1 −ω)QN i=m+1(γi−1). Taking his exp ession o λN−1in o accoun shows ha he op imal alue o P oblem 3.17 is gi en by (3.21). Howe e , he asse ion claims his o be he op imal alue o P oblem 3.10 as well. In o de o p o e his, i is su icien o show ha he op imum o P oblem 3.17 sa is ies (3.19), j=m, . . . , N −2. As a consequence, i sol es he op imiza ion p oblem s a ed in P oposi ion 3.16 which is equi alen o P oblem 3.10. As a byp oduc , his co e s he necessi y o he p e iously conside ed condi ion γm+1 −ω≤0 in o de o ob ain αω N,m = 1. 50 LINEAR PROGRAM To his end, we pe o m a pai wise compa ison o (3.20) and (3.19) o j∈ {m, . . . , N − 2}in o de o show ha he cons ain s gi en by (3.19), j=m, . . . , N −2, a e dispensable. Since (γm+1 −ω)λN−1≥(γN−j+m−γN−j)λjj=m, . . . , N −2 (3.27) ensu es N−2 X n=j λN−γN−jλj+ωλN−1≤ N−2 X n=j λN−γN−j+mλj+γm+1λN−1, i su ices o es ablish (3.27) o he ob ained op imum in o de o show he asse ion. (3.22) cha ac e izes he componen s λj,j=m, . . . , N −2, in he op imum o P oblem 3.17 by means o he equa ion N−j+m Y i=m+2 (γi−1)!λj=γm+1 N−j+m−1 Y i=m+2 γi!λN−1, j =m, . . . , N −2. Using his ep esen a ion o λjwhich (only) depends on λN−1(3.27) is equi alen o (γm+1 −ω) N−j+m Y i=m+2 (γi−1) ≥(γN−j+m−γN−j) N−j+m−1 Y i=m+1 γi, j =m, . . . , N −2. Since he le hand side o his exp ession is equal o (γm+1−ω) N−j+m−1 Y i=m+2 (γi−1)(c0−1)+(γm+1−ω) N−j+m−1 Y i=m+2 (γi−1) "N−j+m−2 X n=1 cn+ωcN−j+m−1#, (c0−1) ≥0, and (γN−j+m−γN−j) = PN−j+m−2 n=N−j−1cn+ωcN−j+m−1−ωcN−j−1, applying Lemma 3.23 wi h k= 1 comple es he p oo .  Rema k 3.19 E en i P ope y (1.13) is no sa is ied, he p oo o Theo em 3.18 shows ha Fo mula (3.21) p o ides he op imal alue o he elaxed P oblem 3.17 and, hus, a lowe bound o P oblem 3.10. Suppose ha Assump ion 3.2 is sa is ied wi h a KL0- unc ion which is linea in i s i s a gumen . Then, he αω N,m- alue o Theo em 3.18 may s ill be used as a lowe bound o he subop imali y deg ee o he eceding ho izon closed loop. Theo em 3.18 allows us o easily compu e pe o mance bounds which a e needed in Theo- em 3.12 in o de o p o e s abili y p o ided β(·,·) is known. Howe e , e en i β(·,·) is no known exac ly, we can deduce aluable in o ma ion. The ollowing co olla y is ob ained by a ca e ul analysis o he ac ion in (3.21). Co olla y 3.20 Le mand ω≥1be gi en. Then, o each summable KL0- unc ion β(·,·)which is linea in i s i s a gumen , i.e. BN( ) = ·γNand limN→∞ γN<∞, he con e gence limN→∞ αω N,m →1holds. 51 STABILITY AND SUBOPTIMALITY OF RHC SCHEMES P oo : Wi hou loss o gene ali y we assume γm+1 −ω > 0. O he wise Theo em 3.18 yields he asse ion o all N≥m+ 1. Hence, we ha e o show ha he sub ahend o he di e ence in o mula (3.21) con e ges o ze o as he op imiza ion ho izon N ends o in ini y. To his end, he conside ed e m is di ided in o he ac o s (γm+1 −ω) N Q i=m+2 (γi−1) N Q i=m+1 γi−(γm+1 −ω) N Q i=m+2 (γi−1)and N Q i=N−m+1 (γi−1) N Q i=N−m+1 γi− N Q i=N−m+1 (γi−1). (3.28) Since β( , n) is linea in i s i s and summable wi h espec o i s second a gumen , (γN)N∈N≥2is a Cauchy sequence. Hence, an index ¯ N=¯ N(ε) exis s such ha ωP∞ n=¯ Ncn≤ ε < 1 and, hus, γ¯ N−(ω−1)c¯ N−1≤γi≤γ¯ N−(ω−1)c¯ N−1+ε≤γ¯ N+ε o all i > ¯ N holds. Fo N≥¯ N+m, his implies N Q i=N−m+1 (γi−1) N Q i=N−m+1 γi− N Q i=N−m+1 (γi−1) ≤m(γ¯ N+ε−1) m[γ¯ N−(ω−1)c¯ N−1−(γ¯ N−(ω−1)c¯ N−1+ε−1)] =γ¯ N+ε−1 1−ε<∞ which ensu es he boundedness o he second quo ien in (3.28) o su icien ly la ge op- imiza ion ho izons N. Hence, showing ha he i s quo ien in (3.28) con e gences o ze o o N ending o in ini y comple es he p oo . To his end, o N > ¯ N, we conside he espec i e ecip ocal N Q i=m+1 γi−(γm+1 −ω) N Q i=m+2 (γi−1) (γm+1 −ω) N Q i=m+2 (γi−1) = ¯ N Q i=m+1 γi (γm+1 −ω) ¯ N Q i=m+2 (γi−1) · N Q i=¯ N+1 γi N Q i=¯ N+1 (γi−1) −1 ≥1·γ¯ N−(ω−1)c¯ N−1 γ¯ N−(ω−1)c¯ N−1+ε−1N−¯ N −1. Since he e m in b acke s is s ic ly g ea e han one, he deduced lowe bound g ows unboundedly o Napp oaching in ini y. Hence, he i s quo ien in (3.28) con e ges o ze o o N→ ∞ which shows he asse ion.  In pa icula , Co olla y 3.20 ensu es, o su icien ly la ge op imiza ion ho izons N, ha he assump ions o Theo em 3.12 hold and, hus, asymp o ic s abili y o he RHC closed loop. Nex , he linea ini e dimensional sys em wi h quad a ic cos unc ion om Examples 1.17, 1.10, 2.7, and 3.3 is conside ed in o de o illus a e he me hodology in oduced in his chap e . No e ha no cons ain s a e p esen in his example. In pa icula , he ole played by he in ol ed KL0- unc ions β(·,·) in ou con ollabili y Assump ion 3.2 is in es iga ed: 52 LINEAR PROGRAM •Using exponen ial con ollabili y acco ding o Example 3.3, i.e. a KL0- unc ion β( , n) = Cσnwi h C≈49.85805 and σ≈0.26288 (3.29) o ype (1.11), p o ides N= 284 o m= 1. Allowing o la ge con ol ho izons educes his es ima e o N= 94 o m= 40, c . Sec ion 4.2 o de ails on imple- men ing mo e han only he i s elemen o he eceding ho izon con ol sequence. •In con as o ha , al eady he easily deduced ini e ime con ollabili y, i.e. a KL0- unc ion o ype (1.12) gi en by c0= 5.04, c1= 12.96, and cn= 0 o n > 1 (3.30) imp o es he esul s ob ained om Theo em 3.18 signi ican ly, i.e. N= 52 (m= 1) and N= 25 (m= 10), espec i ely. These KL0- unc ions we e deduced in o de o demons a e he gene al e i iabili y o Assump ion 3.2 based on asymp o ic s abili y in e ms o he used no m. He e, we aim a cons uc ing a KL0- unc ion which cha ac e izes he s abili y beha io o he conside ed sys em be e and, hus, implies igh e pe o mance bounds. To his end, he known eedback Fp o ided by Example 1.10 is employed in o de es ima e coe icien s cn,n∈N0, o a KL0- unc ion β(·,·) sa is ying Assump ion 3.2 and P ope y (1.13): `(x(n;x0), BFx(n;x0)) = k(A+BF)nx0k2+kBF(A+BF)nx0k2 ≤(k(A+BF)nk2+kBF(A+BF)nk2)kx0k2 = (k(A+BF)nk2+kBF(A+BF)nk2)`?(x0). Hence, Es ima e (3.3) holds wi h KL0- unc ion β( , n) = cn· wi h cn:= k(A+BF)nk2+kBF(A+BF)nk2,n∈N0, (3.31) c . Table 3.1 o nume ically compu ed alues. Using his KL0- unc ion in o de o apply Theo em 3.18 yields α1 N,m >0 o N= 28 (m= 1) and N= 16 (m= 8), espec i ely. Hence, he pe o mance es ima es a e conside ably imp o ed in con as o hose based on he KL0- unc ions β(·,·) om (3.29) and (3.30) which shows ha he in ol ed bounds cn,n∈N0, play an impo an ole o he quali y o he ho izon es ima es. No e ha P ope y (1.13) is no needed in o de o deduce subop imali y bounds bu ensu es ha he p oposed o mula ep esen s he op imal alue o P oblem 3.8, c . Rema k 3.19. In o de o e i y (1.13), he inequali y cncm≥cn+mhas o be ensu ed o all n, m ∈N0. Since c0≥1 holds, his co esponds o checking cn−jcj≥cn,j= 1,2, . . . , n −1, o each n∈N0. Now, we bene i om compu ing he ho izon es ima es i s : since solely coe icien s cn,n < N, a e equi ed in P oblem 3.8, P ope y (1.13) has only o be e i ied o n < 28 — a condi ion which is sa is ied. We poin ou ha he de i ed unc ion β(·,·) is no mono onically dec easing and, hus, does no belong o class KL, c . Table 3.1. We emphasize ha op imali y o he con ol sequence ux0(·) is no assumed — a key ea u e o ou app oach which simpli ies he e i ica ion o Assump ion 3.2 signi ican ly. This allowed us o employ knowledge on he solu ion o he algeb aic Ricca i equa ion in o de o deduce (3.31) and, hus, o igh en he ho izon es ima es, c . Sec ion 5.5.1. In Sec ion 5.5.1 his example is conside ed again and he esul s a e compa ed wi h o he app oaches which can be also used in o de o es ima e he equi ed ho izon leng h in RHC. 53 STABILITY AND SUBOPTIMALITY OF RHC SCHEMES N β(·,·) om (3.29) β(·,·) om (3.30) β(·,·) om (3.31) 0 49.85804850 5.04 3.037786080 1 13.10674606 12.96 5.186783379 2 3.445517772 0.00 2.790245748 3 0.905762015 0.00 0.392116897 4 0.238107850 0.00 0.015203185 5 0.062594089 0.00 0.031327420 6 0.016454812 0.00 0.013169022 7 0.004325662 0.00 0.001422866 8 0.001137135 0.00 0.000105178 9 0.000298932 0.00 0.000179462 10 0.000078584 0.00 0.000059880 Table 3.1: Coe icien s o se e al KL0- unc ions β(·,·) sa is ying Assump ion 3.2 o Ex- ample 1.17. 3.3.1 Auxilia y Resul s In his subsec ion h ee lemma a a e deduced which a e used in o de o p o e Theo em 3.18. The echnical Lemma 3.21 is applied se e al imes in he p oo o Theo em 3.18 as well as needed as a p elimina y esul in o de o p o e he Lemma 3.22. Lemma 3.22 cha ac e izes he op imal solu ion o P oblem 3.17 which is c ucial in o de o show Fo mula (3.21). In conclusion, we p esen Lemma 3.23 which is based on (1.13). Lemma 3.21 Le m, M ∈Zwi h M≥m−1and cons an s γi∈R,i=m, m + 1, . . . , M be gi en. Fu he mo e, he con en ions Qm−1 i=m= 1 and Pm−1 i=m= 0 a e used. Then, he ollowing o mula holds: M Y i=m γi= M Y i=m (γi−1) + M X i=m i−1 Y k=m γk M Y k=i+1 (γk−1)!.(3.32) P oo : We ca y ou an induc ion o e Min o de o p o e (3.32). Since we ha e ag eed on he con en ions wi h espec o he emp y p oduc and emp y sum, he asse ion holds o M=m−1. Hence, we p oceed wi h he induc ion s ep: M+1 Y i=m (γi−1) = (γM+1 −1) M Y i=m (γi−1) I.A. = (γM+1 −1) "M Y i=m γi− M X i=m i−1 Y k=m γk M Y k=i+1 (γk−1)!# = M+1 Y i=m γi− M Y i=m γi− M X i=m i−1 Y k=m γk M+1 Y k=i+1 (γk−1)! = M+1 Y i=m γi− M+1 X i=m i−1 Y k=m γk M+1 Y k=i+1 (γk−1)!.  54 LINEAR PROGRAM The ollowing lemma mainly a gues wi h he signs o he espec i e coe icien s o he ma ix Aand he ec o ¯ b. The condi ion γm+1 −ωis only used in o de o ensu e ha •¯ b1>0, di<0 o i∈ {1,2, . . . , N −2}and ha • he op imiza ion objec i e consis s o maximizing λN−1. Fu he mo e, we poin ou ha we ake he assump ions discussed in Rema k 3.15 wi h espec o he sequence (cn)n∈N0in o accoun in o de o conclude he ollowing lemma. No e ha hese a e based on he linea i y o β(·,·)∈ KL0in i s i s a gumen . Lemma 3.22 Le γm+1 =Pm−1 n=0 cn+ωcmbe s ic ly g ea e han ω. Then he op imal solu ion λo P oblem 3.17 sa is ies Aλ =¯ b,λ > 0componen wise. P oo : γm+1 > ω implies ha he coe icien o λN−1in he objec i e unc ion is nega i e. As a consequence, maximizing λN−1subjec o gi en cons ain s p o ides he op imum o P oblem 3.17, which is deno ed by λ∗= (λ∗ 1, . . . , λ∗ N−1). In o de o p o e he asse ion, we assume he exis ence o an index k∈ {1, . . . , N −1}such ha Akλ∗=PN−1 n=1 Aknλ∗ n<¯ bkand deduce a con adic ion. We begin wi h he case k= 1 and de ine ε:= γN−1−PN−2 i=1 aiλ∗ i−ωλ∗ N−1> 0, i.e. εco esponds o he slack in he i s inequali y, δ:= −maxi=1,...,N−2di, and β:= maxi=1,...,N−2bi. No e ha γm+1 > ω ensu es δ > 0 in iew o Rema k 3.15 o β(·,·)∈ KL0which is linea in i s i s a gumen . Now, we choose ˜ε > 0 such ha ˜ε"ω+β N−2 X i=1 ai (1 + δ)N−2−i δN−1−i#≤ε. Then, we inc ease λN−1by ˜εand λi, i = 1, . . . , N −2, by ˜ε β(1 + δ)N−2−i/δN−1−i. The choice o ˜εensu es he alidi y o he i s inequali y. Since inequali y j∈ {2, . . . , N −1} holds o λ∗ he ollowing compu a ion shows ha i is s ill sa is ied o he modi ied λi, i = 1, . . . , N −1. He e, we use Lemma 3.21 wi h m= 0, M=N−2−jand γi= 1+δ o i∈ {m, m + 1, . . . , M}: dj−1˜εβ (1 + δ)N−1−j δN−j+ N−2 X i=j ˜εβ (1 + δ)N−2−i δN−1−i+ ˜εbj−1 ≤˜ε"−δβ (1 + δ)N−1−j δN−j+ N−2 X i=j β(1 + δ)N−2−i δN−1−i+β# =˜εβ δN−1−j"−(1 + δ)N−1−j+ N−2−j X i=0 (1 + δ)N−2−j−iδi+δN−1−j#(3.32) = 0. Howe e , his con adic s he assumed op imali y o λ∗. Thus, he i s inequali y holds wi h equali y and k > 1 which implies λ∗ k−1>0. This allows us o educe λk−1wi hou iola ing he non-nega i i y condi ion imposed on his a iable. As a consequence, he i s inequali y is no ac i e any mo e while all o he inequali ies emain alid. Hence, epea ing he abo e a gumen a ion w. . . k= 1 leads, again, o a con adic ion and, hus, p o es Aλ∗=¯ b. 55 STABILITY AND SUBOPTIMALITY OF RHC SCHEMES I emains o show ha λ∗ i>0 o all i∈ {1,2, . . . , N −1}. Suppose λ∗ k= 0 o k∈ {1, . . . , N −2}. Then, he (k+ 1)-s inequali y implies λ∗ i= 0 o i∈ {k, k + 1, . . . , N −1}. Since he k- h inequali y is sa is ied wi h equali y, we ob ain λ∗ k−1= 0. I e a i e applica ion o his a gumen shows λ∗≡0. Howe e , since γm+1 > ω and Rema k 3.15 ensu e ¯ b1=γN−1>0, his con adic s A1λ∗=¯ b1. Hence, λ∗>0 holds componen wise which comple es he p oo .  The ollowing lemma is only needed o k= 1. Howe e , we s a e he esul o all k∈N0 since his simpli ies he induc ion s ep signi ican ly. This ick is he main eason o p esen ing his echnical asse ion in a sepa a e lemma. Lemma 3.23 Le N∈N≥2,m∈ {1, . . . , N −2}, and ω≥1be gi en. Fu he mo e, le γi,i∈N≥2, be de ined as Pi−2 n=0 cn+ωci−1, cp. P oposi ion 3.16. In addi ion, le he coe icien s cn,n∈ N0, sa is y (1.13) and use he con en ion Qm+1 m+2 = 1. Then, o j=N−2, N −3, . . . , m, (γm+1 −ω) N−j+m−1 Y i=m+2 (γi−1) "N−j+m+k−3 X n=k cn+ωcN−j+m+k−2# − N−j+m−1 Y i=m+1 γi"N−j+m+k−3 X n=N−j+k−2 cn+ωcN−j+m+k−2−ωcN−j+k−2#≥0∀k∈N. P oo : We ca y ou an induc ion wi h espec o j. The induc ion s a , j=N−2, ollows o a bi a y k∈N om (γm+1 −ω)"m+k−1 X n=k cn+ωcm+k#−γm+1 "m+k−1 X n=k cn+ωcm+k−ωck# =ωckγm+1 −ω"m+k−1 X n=k cn+ωcm+k#=ω"m−1 X n=0 (ckcn−cn+k) + ω(ckcm−cm+k)#(1.13) ≥0. In o de o pe o m he induc ion s ep om j+1 jwe ew i e he conside ed inequali y o a bi a y bu ixed k∈N: (γm+1 −ω) N−j+m−2 Y i=m+2 (γi−1) "ckγN−j+m−1− N−j+m+k−3 X n=k cn−ωcN−j+m+k−2# +γN−j+m−1 (γm+1 −ω) N−j+m−2 Y i=m+2 (γi−1) "N−j+m+k−3 X n=k+1 cn+ωcN−j+m+k−2# − N−j+m−2 Y i=m+1 γi"N−j+m+k−3 X n=N−j+k−2 cn+ωcN−j+m+k−2−ωcN−j+k−2#! ≥0. The posi i i y o his exp ession, which consis s o wo summands, ollows om (1.13) and he induc ion assump ion o j+ 1 and k+ 1.  56 3.4. INSTANTANEOUS CONTROL OF THE LINEAR WAVE EQUATION 3.4 Ins an aneous Con ol o he Linea Wa e Equa- ion In he p e ious sec ion an analy ical o mula was deduced which p o ides he op imal alue o P oblem 3.8. The key assump ion needed in o de o apply he espec i e Theo em 3.18 is he con ollabili y condi ion in oduced in Sec ion 3.1. In his sec ion, Assump ion 3.2 is deduced o he linea wa e equa ion which allows o conclude asymp o ic s abili y o he eceding ho izon closed loop. The one dimensional linea wa e equa ion wi h Di ichle bounda y condi ion and Neu- mann bounda y con ol is conside ed, see (2.9) - (2.11). In Example 2.8 we ackled he ask o s abilizing his hype bolic pa ial di e en ial equa ion a i s unique equilib ium, i.e. he o igin, by eceding ho izon con ol inco po a ing a e minal equali y cons ain . Howe e , he ini e p opaga ion speed implied he need o an ex emely long op imiza- ion ho izon in o de o sa is y he s abilizing e minal cons ain and, hus, o ensu e easibili y as well as s abili y in a sampled-da a se ing wi h sampling pe iod T2L/c, c . Sec ion 2.2. We emphasize ha p ese ing s abili y p ope ies o a con inuous ime sys em ypically equi es su icien ly as sampling, c . [91]. Fo u he esul s ela ed o e minal cons ain s o e minal cos s o in ini e dimensional sys ems, we e e o [64]. He e, in con as o Sec ion 2.2, uncons ained RHC is used. Ra ionale o his ap- p oach a e p o ided by nume ical esul s: he linea wa e equa ion is no only s abilizable bu also pe o ms well using RHC wi h he sho es easible op imiza ion ho izon N= 2, also e med ins an aneous con ol, c . [62].1Ou con ibu ion o his p oblem is he com- ple e heo e ical analysis. In pa icula , we employ Theo em 3.18 in o de o p o e he obse ed s abili y igo ously o sui ably chosen s age cos s. Exploi ing he de i ed o - mula allows us o es ablish his e en o he combina ion o small sampling pe iods and RHC applied wi h he sho es easible op imiza ion ho izon. 3.4.1 Cons uc ing Sui able S age Cos s In Example 2.8 he ma hema ical p oblem o mula ion and he co esponding solu ion space we e al eady in oduced. In addi ion, his con inuous ime sys em was ew i en as a disc e e ime one and he ough shape o app op ia e s age cos s was de ined, c . (2.13). No e ha he unc ion %(·,·) was no exac ly speci ied, which opens up a ce ain deg ee o eedom in o de o design he s age cos s sui ably. Ou goal is o s ee he sys em o he o igin, which is he unique equilib ium. To his end, we conside he cos unc ional JN(y(·,0), u(·)) := N−1 X n=0 1 4ZL 0 %(yx(x, nT), y (x, nT)) dx +λZNT 0 u( )2d which equals i s con inuous ime coun e pa (2.12). Since ou me hodology depends on (3.1), i.e. he elaxed Lyapuno inequali y, sui able s age cos s, which allow o es ab- lishing his es ima e, ha e o be cons uc ed. To his end, (2.9) - (2.11) is nume ically in es iga ed wi h pa ame e s L=c= 1, λ= 10−3, and sampling ime T= 0.025. Le 1In li e a u e, he e m “ins an aneous con ol” is also used in a di e en manne . In [59,60] ins an a- neous con ol means ha he op imiza ion ou ine — which is employed in o de o compu e a sequence o con ol alues u(·)=(u(n))n∈{0,1,2,...,N−1}sa is ying JN(x0, u(·)) = VN(x0) — is s opped p ema u ely in o de o educe he compu a ional e o . 57 Chap e 4 Sensi i i y Analysis We ocus on disc e e ime sys ems which sa is y Assump ion 3.2 wi h a KL0- unc ion linea in i s i s a gumen . Fo his class, he nonlinea op imiza ion P oblem 3.8 (o i s coun e pa P oblem 3.10 which includes an addi ional weigh on he inal e m in he espec i e cos unc ional) becomes a linea p og am, c . Lemma 3.14. Based on his obse a ion, we deduced an explici o mula cha ac e izing he co esponding op imal alue αω N,m which depends on he op imiza ion ho izon N∈N≥2, he con ol ho izon m∈ {1,2, . . . , N −1}, and he e minal weigh ω≥1, c . Theo em 3.18. The eceding ho izon algo i hm yields, in each i e a ion, a sequence o Ncon ol alues. The con ol ho izon de e mines he numbe o elemen s o his sequence o be implemen ed a he plan be o e he RHC p oblem is sol ed again. In his chap e , a sensi i i y analysis is ca ied ou wi h espec o hese pa ame e s: •In Sec ion 3.3 we showed ha a posi i e αω N,m is ob ained o su icien ly long op i- miza ion ho izon Nwhich allows — unde mild echnical condi ions, c . Theo em 3.12 — o conclude asymp o ic s abili y o he eceding ho izon closed loop. In Sec ion 4.1 he impac o he op imiza ion ho izon Nis u he in es iga ed. In pa icula , we aim a deducing asymp o ic bounds on he equi ed ho izon leng h Nin dependence o a gi en KL0- unc ion β(·,·). In his con ex he e m minimal s abilizing ho izon is in oduced which deno es he smalles ho izon Nsuch ha Theo em 3.18 gua an ees a posi i e pe o mance index αω N,m. In addi ion, esul s conce ning he di e en in luence o he o e shoo and he decay a e o exponen- ially con ollable sys ems a e gi en. •In he subsequen Sec ion 4.2 he in luence o he con ol ho izon mis conside ed. In pa icula , Fo mula (3.21) is exploi ed in o de o es ablish symme y and mono- onici y p ope ies which show ha asymp o ic s abili y o he eceding ho izon closed loop wi h ime a ying con ol ho izons holds unde he same condi ions as o classical RHC. This esul is no only essen ial in o de o deal wi h ne wo ked con ol sys ems bu also o ms he co e o he algo i hms in he ensuing sec ion. In Sec ion 4.4 wo algo i hms a e designed — based on he sensi i i y analysis ca ied ou in he p eceding sec ions. The i s algo i hm allows o signi ican ly educe he equi ed op imiza ion ho izon leng h Nin o de o ensu e a desi ed closed loop pe o mance by employing con ol ho izons m > 1. The second, u he de eloped algo i hm deals wi h he loss o obus ness esul ing om s aying in open loop o longe pe iods o ime while main aining he s abili y gua an ees o i s p edecesso . 65 SENSITIVITY ANALYSIS 4.1 In luence o he Op imiza ion Ho izon Co olla y 3.20 ensu es, o su icien ly la ge op imiza ion ho izon N, asymp o ic s abili y — a esul which was al eady shown in [32] unde simila condi ions (see also [65] o an analogous esul in con inuous ime). Addi ionally, Co olla y 3.20 gene alizes his asse ion o a bi a y, bu ixed con ol ho izons m. Using he same a gumen a ion as in he p oo o Theo em 3.12 allows o conclude asymp o ic s abili y o ime a ying con ol ho izons (mi)i∈N0⊆M⊆ {1,2, . . . , m?} o an a bi a y, bu ixed numbe m?∈N. Combining he inequali y α1 N,mVµN,m ∞(·)≤VN(·) om Theo em 3.7 and he inequali y VN(·)≤V∞(·), which is ensu ed by he mono onici y o VN(·) o ω= 1, implies ha he in ini e ho izon cos VµN,m ∞(·) con e ges o he op imal alue V∞(·). In his sec ion, suppose ha a con ol ho izon m∈Nand a e minal weigh ω≥1 a e gi en. A de ailed sensi i i y analysis is ca ied ou in o de o in es iga e he impac o he op imiza ion ho izon N. We a e, in pa icula , in e es ed in so called s abilizing ho izons, i.e. op imiza ion ho izons Ngua an eeing αω N,m ≥0, and, hus, s abili y. In his con ex , wo ques ions a e ackled: (1) Le an op imiza ion ho izon Nbe gi en. Which class KL0- unc ions β(·,·) can be employed in Assump ion 3.2 in o de o conclude αω N,m ≥0 ia Theo em 3.18? He e, β(·,·) is assumed o be o ype (1.11), i.e. β( , n) = Cσn . Fu he mo e, we wan o elabo a e design guidelines. To his end, he in e play o he o e shoo C and he decay a e σis aken in o accoun . (2) Le Assump ion 3.2 be sa is ied wi h a KL0- unc ion linea in i s i s a gumen and γbe de ined as he accumula ed bound P∞ n=0 β( , n)/ =P∞ n=0 cn om he con- ollabili y condi ion (3.3). How does he minimal op imiza ion ho izon Nensu ing s abili y ia Theo em 3.18 depend on his quan i y γ? He e, ou main emphasis is pu on he asymp o ic g ow h o he minimal s abilizing ho izon wi h espec o γ. We poin ou ha Fo mula (3.21) enables us o p o e nume ical obse a ions om [39] igo ously. In o de o answe he i s ques ion, all pa ame e combina ions (C, σ) implying a non- nega i e subop imali y index αω N,m and, hus, s abili y o a gi en op imiza ion ho izon Na e calcula ed , c . Figu e 4.1.1 As expec ed, he s abili y egion g ows wi h inc easing op imiza ion ho izon N. The- o em 3.18 allows us o quan i y he obse ed enla gemen , e.g. doubling N= 2 inc eases he conside ed a ea by 129.4 pe cen . Fu he mo e, we obse e ha o a gi en decay a e σ he e always exis s an o e shoo Csuch ha s abili y is gua an eed. Indeed, Theo em 3.18 enables us o p o e his. To his end, we deal wi h he special case C= 1 which yields a signi ican ly simple exp ession o αω N,m. P oposi ion 4.1 Assume exponen ial con ollabili y wi hou o e shoo , i.e. Assump ion 3.2 wi h a KL0- unc ion o ype (1.11) wi h C= 1. Then, he op imal alue αω N,m o P oblem 3.10 is equal o min{1,1−(1 + σω −ω)σN−1}and s ic ly posi i e, i.e. αω N,m >0. 1The idea o isualizing he pa ame e dependen s abili y egions in his way goes back o [121]. 66 INFLUENCE OF THE OPTIMIZATION HORIZON Figu e 4.1: Illus a ion o he s abili y egion gua an eed by Theo em 3.18 o a ious op imiza ion ho izons Ngi en a KL- unc ion β(·,·) o ype (1.11) o RHC wi h m= 1. P oo : De ining he auxilia y quan i y η:= 1 + σω −ω, we ob ain γi=1−η σi−1 1−σ, γi−1 = σ(1 −η σi−2) 1−σ, γm+1 −ω=η(1 −σm) 1−σ. Hence, he necessa y and su icien condi ion (γm+1 −ω)≤0 om Theo em 3.18 holds i and only i he condi ion η≤0 is sa is ied — an equi alence which is e lec ed by aking he minimum. I emains o conside η > 0 which ensu es ha αω N,m is gi en by (3.21). As a p epa a o y esul , each o he wo ac o s occu ing in he espec i e denomina o is in es iga ed sepa a ely, i.e. N Y i=N−m+1 γi− N Y i=N−m+1 (γi−1) = N Y i=N−m+1 1−η σi−1 1−σ− N Y i=N−m+1 σ(1 −η σi−2) 1−σ =1−η σN−1−(1 −η σN−m−1)σm 1−σ· N Y i=N−m+2 1−η σi−2 1−σ =1−σm 1−σ· N Y i=N−m+2 1−η σi−2 1−σ and, epea ing he same line o a gumen s, N Y i=m+1 γi−(γm+1 −ω) N Y i=m+1 (γi−1) = 1−η σN−m−1 1−σ· N Y i=m+2 1−η σi−2 1−σ. Inse ing hese exp essions in o Fo mula (3.21) yields αω N,m = 1 − η(1−σm) 1−σQN i=m+2 σ(1−η σi−2) 1−σQN i=N−m+1 σ(1−η σi−2) 1−σ 1−η σN−m−1 1−σ·QN i=m+2 1−η σi−2 1−σ1−σm 1−σ·QN i=N−m+2 1−η σi−2 1−σ 67 SENSITIVITY ANALYSIS = 1 − η(1−σm) 1−σ·σN−m−1QN i=m+2 1−η σi−2 1−σ·σmQN i=N−m+1 1−η σi−2 1−σ 1−σm 1−σ·QN i=m+2 1−η σi−2 1−σQN i=N−m+1 1−η σi−2 1−σ = 1 −η σN−1.  Rema k 4.2 No e ha he op imal alue αω N,m, i.e. he solu ion o P oblem 3.10, does no depend on he con ol ho izon m o C= 1. Consequen ly, he con ol ho izon mdoes no play a ole o his special case. P oposi ion 4.1 s a es ha we always ob ain a s ic ly posi i e alue αω N,m o C= 1. Due o con inui y o he in ol ed exp essions his emains ue o C= 1+ε o su icien ly small ε. Hence, o any decay a e σ∈(0,1) and su icien ly small C=C(σ)>1 (depending on N,mand ω)αω N,m >0 is ob ained. Recall ha a posi i e pe o mance index αω N,m is he key ing edien in Theo em 3.12 in o de o deduce asymp o ic s abili y. Howe e , his p ope y does no hold i we exchange he oles o σand C, i.e. o a gi en o e shoo C > 1 s abili y canno in gene al be concluded o a su icien ly small decay a e σ > 0, c . Figu e 4.1. Nex , he in e play o he op imiza ion ho izon Nand γ=P∞ n=0 cnis s udied in o de o ad ess ques ion numbe (2). We aim a de e mining he asymp o ic g ow h a e o he minimal op imiza ion ho izon Ngua an eeing s abili y o a gi en pa ame e γ. To his end, we assume ini e ime con ollabili y in one s ep, i.e. Assump ion 3.2 using a KL0- unc ion o ype (1.12) de ined by c0=γand cn= 0 o all n∈N≥1. Fo gi en γ, his ep esen s, as will be seen in he p oo o Theo em 4.4, he wo s case o e all KL0- unc ions β(·,·) which a e linea in hei i s a gumen s — a leas o he se ing wi hou an addi ional e minal weigh , i.e. ω= 1. This ac will be o pa icula use in o de o p o e Theo em 4.4. No e ha o his p oblem he addi ional condi ion (1.13) is au oma ically sa is ied (since (3.3) ensu es c0≥1). Hence, Theo em 3.18 cha ac e izes he op imal alue o P oblem 3.10 exac ly. We ocus on m= 1, i.e. he smalles possible con ol ho izon, and m=bN/2c, i.e. he con ol ho izon implying – a leas in he exponen ially con ollable and he ini e ime con ollable in a maximum o wo s eps case he la ges αω N,m alue, c . Sec ion 4.2 below. Co olla y 4.3 Le ω≥1be gi en and Assump ion 3.2 hold wi h β( , n) = ·cn,c0=γand ci= 0 o i∈N, i.e. ini e ime con ollabili y in one s ep. Fu he mo e, le he minimal s abilizing ho izon be de ined as ˆ N(γ) := min{N∈N≥2:αω N,m ≥0 o αω N,m gi en by (3.21) based on β( , n)}, i.e. he smalles op imiza ion ho izon Ngua an eeing ha he solu ion αω N,m o he linea p og am gi en by P oblem 3.10 is posi i e. Then, ˆ N(γ)beha es, • o m= 1, asymp o ically like γln γ, i.e. limγ→∞ ˆ N(γ) γln γ= 1, and • o m=bN/2c, asymp o ically like 2 ln 2 ·γ, i.e. limγ→∞ ˆ N(γ) 2 ln 2·γ= 1. 68 INFLUENCE OF THE OPTIMIZATION HORIZON P oo : Since Co olla y 4.3 deals wi h he asymp o ic beha io wi h espec o γ, le γ be s ic ly g ea e han ω≥1. Fu he mo e, no e ha , o ini e ime con ollabili y in one s ep, γi=γholds o all i∈N≥2independen ly o he chosen e minal weigh . Hence, Fo mula (3.21) yields αω N,m = 1 −(γ−ω)(γ−1)N−1 (γN−m−(γ−ω)(γ−1)N−m−1)(γm−(γ−1)m).(4.1) Fo m= 1, we equi e a posi i e op imal alue o P oblem 3.10 in o de o ensu e s abili y. i.e. αω N,1= 1 −(γ−ω)(γ−1)N−1 γN−1−(γ−ω)(γ−1)N−2=γN−1−γ(γ−ω)(γ−1)N−2 γN−1−(γ−ω)(γ−1)N−2≥0 This inequali y holds i and only i he nomina o is posi i e. Since he loga i hm is mono onically inc easing, his is, a e di iding by γ, equi alen o N≥2 + ln(γ−ω) ln γ−ln(γ−1) =: (γ). We show ha (γ) ends o γln γasymp o ically. To his end, we conside lim γ→∞ (γ) γln γ= lim γ→∞ 2 γln γ | {z } =0 + lim γ→∞ ln(γ−ω) ln γ | {z } =1 ·lim γ→∞ 1 γ ln γ−ln(γ−1) = lim γ→∞ γ(γ−1) γ2= 1 whe e we ha e used l’Hˆopi al’s ule, c . [124, Subsec ion 5.4.4]. Clea ly, ounding up he de i ed exp ession o he op imiza ion ho izon Ndoes no change he ob ained esul . Fo m > 1, (4.1) and, hus, Theo em 3.18 yields αω N,m >0 i and only i γN≥γN−m(γ−1)m+ (γ−ω)(γ−1)N−m−1γm. Hence, o m=bN/2cwe ob ain analogously he ollowing lowe bounds o he op i- miza ion ho izon N: N≥   2 ln 2γ−ω−1 γ−1/(ln γ−ln(γ−1)) o e en N ln 2γ−ω γ+ ln 2γ−ω γ−1/(ln γ−ln(γ−1)) o odd N Again in conside a ion o L’Hˆopi al’s ule, he in es iga ed exp ession exhibi s asymp o - ically a beha io like 2 ln 2 ·γ. Since he ob ained app oxima ion 2 ln 2 ·γholds o bo h es ima es co esponding o e en and odd numbe s N o m=bN/2c, he asse ion holds.  Figu e 4.2 illus a es he esul ing ho izon leng hs o gi en γ. We like o poin ou ha hese es ima es coincide wi h he nume ical esul s de i ed in [39, Sec ion 6].2 Co olla y 4.3 deals wi h he asymp o ic g ow h a e o he minimal s abilizing ho i- zon ˆ N o a bi a y, bu ixed e minal weigh ω≥1. We poin ou ha , o ini e ime con ollabili y in one s ep, γi,i∈N≥2, is independen o ω. Hence, he sequence (γi)i∈N≥2is also o ω > 1 non-dec easing — a p ope y which is impo an o he p oo o he ollowing heo em bu canno be assumed o a bi a y KL0- unc ions sa is ying P∞ n=0 cn=γand (1.13). Theo em 4.4 shows ha he es ima es om Co olla y 4.3 ca y o e o a bi a y KL0- unc ions o ω= 1. The asse ion o m= 1 was also deduced in [120] based on simila assump ions. 2Indeed, we de e mined p ecisely he cons an 2 ln 2 o he linea g ow h es ima e in con as o he nume ically obse ed ac o √2. 69 SENSITIVITY ANALYSIS Figu e 4.2: Minimal s abilizing op imiza ion ho izons o one s ep ini e ime con olla- bili y o m= 1 and m=bN/2cin compa ison wi h hei asymp o ic app oxima ions. Theo em 4.4 Le Assump ion 3.2 be sa is ied wi h a KL0- unc ion linea in i s i s a gumen , ω= 1, and de ine γ=P∞ n=0 cn. Then, he asymp o ic g ow h a e o he minimal s abilizing ho izon ˆ Nis bounded by γln γand 2 ln 2 ·γ o m= 1 and m=bN/2c, espec i ely. P oo : In o de o show he asse ion, ake a close look a P oblem 3.8. He e, β(·,·) om Assump ion 3.2 is inco po a ed in he uppe bounds o he cons ain s. Hence, using ini e ime con ollabili y in one s ep elaxes he cons ain s and, hus, enla ges he easible se o he posed minimiza ion p oblem in con as o e e y o he KL0- unc ion summing up o γ. Hence, Theo em 4.4 is a di ec consequence o Co olla y 4.3.  Summa izing, he de i ed es ima es p o ide uppe bounds on he g ow h a e o he min- imal s abilizing ho izon, e.g. o c0=γ=CP∞ n=0 σnwi h C≥1, σ∈(0,1). Mo eo e , o m=bN/2c, Theo em 4.4 exhibi s a linea bound. Hence, he co esponding g ow h a e is linea o e en slowe . Fu he mo e, no e ha he addi ional p ope y (1.13) is no needed in o de o es ablish Theo em 4.4, c . Rema k 3.19. In o de o conclude his sec ion, he ollowing ema k is gi en which deals wi h he se ing based on a ixed γbu allows o a y he e minal weigh ω. Rema k 4.5 Le Assump ion 3.2 be sa is ied wi h a KL0- unc ion o ype (1.12) gi en by c0:= γ and ci= 0,i∈N. Then, choosing he e minal weigh la ge enough always implies γm+1 −ω=γ−ω≤0and, as a consequence, αω N,m = 1. This obse a ion e lec s an impo an p ope y o ini e ime con ollable sys ems: ypically, he op imiza ion ho izon has o be su icien ly la ge in o de o ensu e ha i is p e e able o o e come he obs acle despi e he needed con ol e o ep esen ed by c0=γ. This dilemma can be esol ed by pu ing mo e emphasis on he inal s a e o he p edic ion ho izon. 70 4.2. CHARACTERISTICS DEPENDING ON THE CONTROL HORIZON 4.2 Cha ac e is ics Depending on he Con ol Ho i- zon Delays and packe d opou s, which ypically occu o ne wo ked con ol sys ems, mo i- a ed he in oduc ion o mul is ep eedback laws, c . De ini ion 1.25. Based on hese p elimina y conside a ions Theo em 3.12 was o mula ed o ime a ying con ol ho izons (mi)i∈N0. In o de o check he condi ions o Theo em 3.12, app op ia e solu ions αω N,mi, i∈N0, o P oblem 3.10 a e needed. A i s glance, he condi ions o his heo em appea o be mo e demanding o ime a ying han o ixed con ol ho izon. Howe e , in his sec ion — based on ou s anda d Assump ion 3.2 wi h a KL0- unc ion which exhibi s linea i y in i s i s a gumen — we p o e ha he condi ions coincide wi h hose o m= 1 o a la ge subclass o such KL0- unc ions including exponen ially decaying ones. Summa izing, he desc ibed p oblem o ime a ying con ol ho izons is esol ed. To his end, we ca y ou a sensi i i y analysis wi h espec o he con ol ho izon mwhich de e mines he numbe o elemen s o ou compu ed sequence o con ol alues o be implemen ed a he plan . Pa icula ly, we es ablish symme y and mono onici y p ope ies o αω N,m which may be coun e -in ui i e, e.g. inc easing he con ol ho izon in he in e al [1,bN/2c]⊂Nimp o es he pe o mance bounds om Theo em 3.12. This coincides wi h ou obse a ion om he p e ious sec ion ha he uppe bounds om Theo em 4.4, which connec he needed con ol e o on he in ini e ho izon o he minimal s abilizing ho izon leng h, g ow only linea ly o m=bN/2cins ead o supe - linea ly (γln γ) o m= 1. Fu he mo e, we deduce a symme y p ope y which enables us o handle con ol ho izons m∈ {bN/2c+ 1, . . . , N −2, N −1}. Combining his wi h he de i ed mono onici y, enables us o show ou main esul in his sec ion, namely, ha s abili y o RHC con ol o ime a ying con ol ho izons ia ou main ool Theo em 3.18 can be gua an eed unde he same condi ions as o m= 1. The esul s which a e de i ed in his sec ion o m he basis o he algo i hm de eloped in Sec ion 4.4 which allows o signi ican ly educing he op imiza ion ho izon Nand, hus, demons a es he p ac ical use o hese heo e ical esul s. This sec ion is subdi ided in o wo pa s. We s a , a e p o iding some insigh in o ou mo i a ion, wi h he main esul s which a e discussed di ec ly a e wa d. In he ollowing subsec ions he co esponding p oo s, which a e a he echnical, a e p esen ed. He e, we like o poin ou he elabo a e echnique hough up in o de o deal wi h he exponen ially con ollable case. 4.2.1 P esen ing he Resul s We begin by looking a Figu e 4.3 which depic s pe o mance bounds αω N,m o con ol ho izons m∈ {1,2, . . . , N −1} o an exponen ially decaying unc ion β:R+ 0×N0→R+ 0, β( , n) = Cσn· wi h C= 2 and σ= 0.625,(4.2) and a KL0- unc ion β(·,·) cha ac e izing ini e ime con ollabili y de ined by c0= 1, c1= 5/4, c2= 3/2, c3= 5/4, c4= 1/2, c5= 1/4, c6= 1/16, and cn= 0, n∈N≥7. (4.3) No e ha he la e , which is a KL0- unc ion o ype (1.12) sa is ying (1.13), is no mono onically dec easing. These examples exhibi he key ea u es wi h espec o he co esponding subop imali y indices. 71 SENSITIVITY ANALYSIS a) b) Figu e 4.3: In a) he pe o mance bounds α1 9,m,m= 1,2,...,8, om Theo em 3.18 a e illus a ed o KL0- unc ions gi en by (4.2, ?) and (4.3, ◦). Whe eas in b) a e minal weigh was added, i.e. (ω= 5/4, ?) and (ω= 2, ) o he exponen ially con ollable (4.2) and (ω= 2, ◦) o he ini e ime con ollabili y case (4.3). In Figu e 4.3 a) wo p ope ies can be obse ed o he se ing wi hou an addi ional weigh on he inal e m: •mono onici y, i.e. inc easing he con ol ho izon in he in e al [1,2,...,bN/2c] im- p o es he op imal alue α1 N,m o P oblem 3.8, and •symme y, i.e. α1 N,m =α1 N,N−m,m= 1,2,...,bN/2c, holds o he compu ed sub- op imali y es ima es. The in e play o hese wo p ope ies ensu es α1 N,1≤α1 N,m o each m∈ {1,2, . . . , N −1}. This obse a ion will be essen ial o he p oo o Theo em 4.8. Using e minal weigh s ω > 1 leads — a leas in his example — o a u he imp o emen o he gua an eed s abili y beha io . Bu ins ead o symme y, Figu e 4.3 b) exhibi s αω N,m ≤αω N,N−m, m= 1,2,...,bN/2c. Be o e con inuing ou s udy, we s a e he co esponding esul s conce ning symme y and mono onici y p ope ies o he op imal alue αω N,m o P oblem 3.10 wi h espec o he con ol ho izon m. The ollowing wo p oposi ions – which a e p o en in Subsec ions 4.2.2 and 4.2.3 – do no only pa e he way o answe he encoun e ed ques ion o ne wo ked con ol sys ems and p epa e he g ound in o de o de elop an algo i hm in Sec ion 4.4 bu a e also in e es ing in hei own igh s. P oposi ion 4.6 Le β(·,·)∈ KL0 om Assump ion 3.2 be ei he o ype (1.11) o o ype (1.12) wi h cn= 0 o n≥3sa is ying (1.13). Then, o N∈N≥2and ω≥1,αω N,m om Theo em 3.18 sa is ies he symme ic bound αω N,m ≤αω N,N−m o m∈ {1,2,...,bN/2c}. P oposi ion 4.7 Suppose β(·,·)∈ KL0 om Assump ion 3.2 o be ei he o ype (1.11) wi h e minal weigh 72 CHARACTERISTICS DEPENDING ON THE CONTROL HORIZON ω∈ {1}∪[1/(1 −σ),∞)o o ype (1.12) wi h cn= 0 o n≥2and a bi a y ω≥1. Then, o N∈N≥2,αω N,m om Theo em 3.18 ul ills αω N,m+1 ≥αω N,m o m∈ {1,...,bN/2c−1}. Using he symme ic bound om P oposi ion 4.6 and he mono onici y p ope y om P oposi ion 4.7 he ollowing no ewo hy consequence o ou s abiliza ion p oblem can be concluded. Theo em 4.8 Le β(·,·)∈ KL0 om Assump ion 3.2 be ei he o ype (1.11) wi h e minal weigh ω∈ {1}∪[1/(1 −σ),∞)o o ype (1.12) wi h cn= 0 o n≥2and a bi a y ω≥1. Then, o each N≥2, he s abili y c i e ion om Theo em 3.12 is sa is ied o m?=N−1 i and only i i is sa is ied o m?= 1. P oo : P oposi ion 4.6 and 4.7 imply αω N,m ≥αω N,1 o all m∈M⊆ {1,2, . . . , N −1} which yields he asse ion.  In o he wo ds, o exponen ially con ollable sys ems wi hou o wi h su icien ly la ge e minal weigh and o sys ems which a e ini e ime con ollable in a mos wo s eps, we ob ain s abili y o ou p oposed ne wo ked MPC scheme unde exac ly he same condi ions as o MPC wi h m?= 1. In his con ex we ecall once again ha o m?= 1 he s abili y condi ion o Theo em 3.12 is igh , c . Rema k 3.13. Simila o ou cou se o ac ion in Sec ion 4.1, we in es iga e he s abili y egion o exponen ially con ollable sys ems wi h espec o hei s age cos s, i.e. he se o all pa ame e combina ions o o e shoo C≥1 and decay a e σ∈(0,1) such ha s abili y o he unde lying disc e e ime sys ems is gua an eed by Theo em 3.18. The in es iga ion o he s abili y egion o exponen ially con ollable sys ems in e ms o hei s age cos s is con inued. The s abili y egion con ains all pa ame e combina ions o o e shoo C≥1 and decay a e σ∈(0,1) such ha s abili y o he unde lying disc e e ime sys ems is gua an eed by Theo em 3.18. He e, he ocus is shi ed om he op imiza ion ho izon N, c . Sec ion 4.1, o he con ol ho izon m. Fo simplici y o exposi ion, he case ω= 1 wi hou an addi ional weigh on he inal e m is conside ed. Ha ing in mind he p oposed esul s, in pa icula P oposi ion 4.6 which holds wi h equali y o ω= 1, c . Co olla y 4.10, only con ol ho izons m∈ {1,...,bN/2c} ha e o be deal wi h. Fo ins ance, Figu e 4.4 shows he s abili y egions o N= 7 and N= 11, espec i ely. Appa en ly, inc easing he con ol ho izon enla ges he s abili y egion, e.g. allows o la ge o e shoo s C o gi en decay a es σ. This obse a ion con i ms ou heo e ical esul s, i.e. he mono onici y p ope y claimed in P oposi ion 4.7 is e lec ed. In addi ion, he g ow h o he s abili y egion can be quan i ied, e.g. o op imiza ion ho izon N= 7: he a ea con aining easible (C, σ) pai s is scaled up by 21 (m= 2) and 30 (m= 3) pe cen . Fo longe op imiza ion ho izons (N= 11) inc easing he con ol ho izon enhances he a ainable gain e en u he , e.g. m= 2 and m= 5 enla ge he s abili y egion by 23 and 48 pe cen , espec i ely. In con as o he exponen ially con ollable case, es ic ions ha e o be imposed on class KL0- unc ions sa is ying (1.12) in Theo em 4.8 — al hough (1.13) is sa is ied. S ill, we expended he e o o gi e a comple e cha ac e iza ion e e ing o his se ing, c . 73 SENSITIVITY ANALYSIS Since η∈(0,1), he polynomial p(C) has clea ly m+ 1 s ic ly posi i e oo s and exac ly one nega i e oo . Hence, in o de o apply Lemma 4.14, we ha e o e i y he second and hi d assump ion wi h c= 0. Since P oposi ion 4.1 yields p(1) = q(1) o , in he no a ion o Lemma 4.14, ˜z= 1, we ha e o show ha he posi i e oo −1/(2σmη) + √ξis loca ed in he in e al (0,1), i.e. √ξ < 1+1/(2σmη) o , equi alen ly, ω(1 −σ) σmη(1 −σmη)<1 + 1 σmη which holds since η(1 −σm) + σmη(1 −σmη)>0. Thus, i emains o ensu e condi ion b) o Lemma 4.14 wi h espec o he (m+1)s de i a i e. To his end, we calcula e p(m+1)(·) p(m+1)(C) = (m+ 2)! C+ (m+ 1)! 1 σmη− N X i=m+2 1−σ 1−σi−1η!, and show ha he only oo o his polynomial o deg ee one is s ic ly nega i e. In o de o de e mine he sign o his oo , i is su icien o conside 1 σmη−(1 −σ) N X i=m+2 1 1−σi−1η>1 σmη−(1 −σ) N X i=m+2 1 1−σm+1η =1−σm+1η−m(1 −σ)σmη σmη(1 −σm+1η) >(1 −σ) σm(1 −σm+1η) m X i=0 σi−mσm!>0. Hence, Lemma 4.14 applied wi h c= 0 and ˜z= 1 ensu es (4.8) and, hus, Lemma 4.15 o N= 2m+ 1, i.e. he induc ion s a is ca ied ou . In o de o comple e he p oo we ha e o pe o m he induc ion s ep. Suppose ha he asse ion holds o N≥2m+ 1. Again, (4.5) is aken as ou s a ing poin . Hence, we ha e o show (γN−m+2−ω) N−m+1 Y i=m+1 γi· N+1 Y i=N−m+2 (γi−1)+(γm+1−γN−m+2) N+1 Y i=m+1 γi−(γm+1−ω) N+1 Y i=m+1 (γi−1) ≥0. Using he induc ion assump ion o (γm+1 −ω)QN i=m+1(γi−1), educing he esul ing exp ession by QN−m i=m+1 γi, and combining he summands which ha e he ac o QN i=N−m+1 γi o QN+1 i=N−m+2(γi−1), espec i ely, yields 0≤hγN−m+1(γN−m+2 −ω)−(γN−m+1 −ω)(γN−m+1 −1)i | {z } =γN−m+1(γN−m+2−γN−m+1)+(γN−m+1−ω) N+1 Y i=N−m+2 (γi−1) +hγN+1(γm+1 −γN−m+2)−(γN+1 −1)(γm+1 −γN−m+1)i | {z } =γN+1(γN−m+1−γN−m+2)+(γm+1−γN−m+1) N Y i=N−m+1 γi. Since (γm+1 −γN−m+1) = C(σm−σN−m)(−η)/(1 −σ) and γN−m+2 −γN−m+1 =CσN−mη 80 CHARACTERISTICS DEPENDING ON THE CONTROL HORIZON a e implied by (4.9), his inequali y may be ew i en as4 γN−m+1CσN−mη+(γN−m+1 −ω)N+1 Y N−m+2 (γi−1) ≥γN+1σN−mη+(σm−σN−m)η 1−σC N Y i=N−m+1 γi. (4.10) Analogously o he induc ion s a , he case η≤0 is deal wi h sepa a ely: since he second and o h summand al eady ha e he desi ed signs, showing γN−m+1CσN−m(−η)"N+1 Y i=N−m+2 γi− N+1 Y i=N−m+2 (γi−1)#≥0 yields (4.10) o η≤0. Consequen ly, η > 0 is supposed om now on. No e ha η= 1 −ω(1 −σ)<1 holds. Reducing (4.10) by σN−mη γN+1/C and QN i=N−m+1(γi/C) leads o γN−m+1 γN+1 C2+γN−m+1 −ω σN−mη γN+1/C N Y i=N−m+1 γi+1 −1 γi/C | {z } =:p(C) ≥Cm+1 C+σm−σN−m σN−m(1 −σNη) | {z } =:q(C) . No e ha bo h polynomials ha e deg ee m+ 2 and he coe icien s o Cm+2 a e equal o one, i.e. a e monic. We aim a applying Lemma 4.14 in o de o conclude his inequali y and, hus, he asse ion. To his end, we begin by de e mining he exac loca ion o he espec i e oo s o p(·) and q(·). The polynomial q(·) has exac ly one s ic ly nega i e oo loca ed a −(σm−σN−m)/((1 −ησN)σN−m). Nex , we conside p(·) and, in pa icula , he ac o (γi+1 −1)/(γi/C), i=N−m+ 1, N −m+ 2, . . . , N, mo e closely. Using (4.9) p o ides γi+1 −1 γi/C =C(1 −σiη)−(1 −σ) 1−σi−1η=1−σiη 1−σi−1ηC−1−σ 1−σiη, i.e. a polynomial o deg ee one whose oo is loca ed a (1−σ)/(1−σiη), i.e. in he in e al (0,1) o each i∈ {N−m+ 1, N −m+ 2, . . . , N}. The i s ac o o p(·) s ill needs o be in es iga ed. He e, ex ac ing he ac o γN−m+1/γN+1, which does no depend on C, and using (4.9), yields 1−σN−mη 1−σNηC2+C σN−mη−ω(1 −σ) (1 −σN−mη)σN−mη. Se ing his exp ession equal o ze o and sol ing he esul ing equa ion p o ides he wo emaining oo s o p(·) C=−1 2σN−mη±s1 2σN−mη2 +ω(1 −σ) (1 −σN−mη)σN−mη,(4.11) a posi i e and a nega i e one. Summa izing, p(C) may be ep esen ed by p(C) = Qm+2 i=1 (C−zi) whe e zi,i= 1,2, . . . , m + 2, deno e he de e mined oo s. Mo eo e , P oposi ion 4.1 yields p(1) = q(1). Hence, we ha e o e i y ha he posi i e oo om 4In ac , his inequali y is lawed in [45], i.e. he di iso (1 −σ) is missing. Ne e heless, he ain o hough s used in o de o p o e he asse ion emains subs an ially he same. 81 SENSITIVITY ANALYSIS (4.11) is s ic ly smalle han one and, hus, con ained in he in e al (0,1). To his end, he equi alen inequali y 1 + 1 2σN−mη>s1 2σN−mη2 +ω(1 −σ) (1 −σN−mη)σN−mη is squa ed which leads o ω(1 −σ) (1 −σN−mη)σN−mη<1 + 1 σN−mη=1 + σN−mη σN−mη o , equi alen ly, ω(1 −σ)<(1 + σN−mη)(1 −σN−mη)=1−σ2(N−m)η2. Since η > 0, aking he de ini ion o ηin o accoun shows ha he espec i e oo is loca ed in he in e al (0,1) and, as a consequence, ha he hi d condi ion o Lemma 4.14 is sa is ied wi h ˜z= 1. Hence, he second condi ion has o be e i ied in o de o apply Lemma 4.14 and deduce ha (4.10) holds o C≥1, which comple es he p oo . We calcula e he (m+ 1)s de i a i e o p(C) and q(C) q(m+1)(C) = (m+ 1)! (m+ 2)C+(σm−σN−m) (1 −σNη)σN−m, p(m+1)(C) = (m+ 1)! (m+ 2)C+1 σN−mη− N X i=N−m+1 1−σ 1−σiη!. We ha e o show ha he oo o p(m+1) is s ic ly smalle han i s coun e pa o q(m+1) di ided by m+ 2 ( he deg ee o he polynomial p(·)), i.e. m+ 2 σN−mη−(σm−σN−m) (1 −σNη)σN−m>(m+ 2) N X i=N−m+1 1−σ 1−σiη.(4.12) Since (1 −σiη)−1≤(1 −σN−m+1)−1,i=N−m+ 1, N −m+ 2, . . . , N, i is su icien o es ablish m+ 2 σN−mη−(σm−σN−m) (1 −σN−m+1η)σN−m>(m+ 2) N X i=N−m+1 1−σ 1−σN−m+1η=m(m+ 2)(1 −σ) 1−σN−m+1η o , equi alen ly, (m+ 2)(1 −σN−m+1η)−(σm−σN−m)η > m(m+ 2)(1 −σ)σN−mη in o de o deduce (4.12). Since (1−σN−m+1η)−(σm−σN−m)η > (1−σ)(σm−1−σN−m)η holds, his is ensu ed by (m+ 1) 1−σN−m+1η 1−σ≥(m+ 1) N−m X n=0 σnN−m>m+1 > m(m+ 2)σN−mη.  82 CHARACTERISTICS DEPENDING ON THE CONTROL HORIZON 4.2.3 Mono onici y P ope ies Looking a Figu e 4.3, one obse es — aside om he symme ic bound — a ce ain mono- onici y p ope y, i.e. a mono one g ow h o he pe o mance bounds αω N,m cha ac e izing he op imal alue o P oblem 3.10 un il he con ol ho izon eaches abou hal he leng h o he op imiza ion ho izon. This ea u e is p ecisely s a ed in P oposi ion 4.7. The goal o his subsec ion is o p o e his esul . Combining he espec i e asse ion wi h he symme ic bound de i ed in he p eceding subsec ion, enables us o deduce Theo em 4.8. This heo em, which is based on Theo em 3.18, ensu es ha using ime a ying con ol ho izons does no cause addi ional di icul ies in o de o e i y he assump ions o The- o em 3.12 — a leas o a la ge and impo an subclass o KL0- unc ions linea in hei i s a gumen s. Fo example, ime a ying ho izons a e equi ed in ne wo ked con ol sys ems in o de o compensa e non-negligible delays o packe d opou s. The symme y analysis which was ca ied ou in Subsec ion 4.2.2 exhibi s an especially nice s uc u e o he se ing wi hou an addi ional weigh on he inal e m in he eceding ho izon cos unc ional, c . Co olla y 4.10. Fu he mo e, he es ic ions on he class o ini e ime con ollable sys ems a e necessa y only o e minal weigh s ω > 1, c . Lemma 4.11 and Rema k 4.12. In con as o ha , hese limi a ions (indeed, e en sligh ly igh e ones) a e necessa y o he mono onici y p ope ies deal wi h in his subsec ion — independen ly o whe he a e minal weigh is in ol ed o no . Hence, in o de o bene i om hese heo e ical esul s as demons a ed by Theo em 4.8 and he algo i hms o be de eloped in Sec ion 4.4 he e is no escape om hese modi ica ions. The necessi y o aking hese es ic ions in o accoun is shown by he ollowing coun e example. Example 4.16 Le he con ollabili y beha io o a disc e e ime sys em be cha ac e ized ia Assump ion 3.2 based on one o he ollowing KL0- unc ions o ype (1.12): •β1(·,·)is de ined by c0= 1.24,c1= 1.14,c2= 1.04, and ci= 0 o i∈N≥3, •β2(·,·)is gi en by c0= 1,c1= 1.2,c2= 1.1,c3= 1.1,c4= 1.2,c5= 1,c6= 0.75, c7= 0.25, and ci= 0 o he wise. No e ha bo h unc ions sa is y (1.13). Fu he mo e, β1(·,·)is mono onically dec easing in i s second a gumen . The co esponding op imal alues o P oblem 3.8 depic ed in Fig. 4.6 show ha nei he o β1(,·,·)no o β2(·,·) he desi ed mono onici y p ope y α1 N,m+1 ≥α1 N,m o m= 1,2,...,bN/2−1c is ob ained. Example 4.16 demons a es ha he desi ed mono onici y p ope y does no hold o a bi a y KL0- unc ions β(·,·) o ype (1.12). The emaining pa o his subsec ion deals wi h subclasses o KL0- unc ions mee ing he assump ions o P oposi ion 4.7. I is a anged as ollows: ini ially, KL0- unc ions o ype (1.12) a e add essed. Then, o exponen ially con ollable sys ems, “su icien ly” la ge e minal weigh s a e ea ed sep- a a ely be o e we u n ou a en ion o he mos delica e si ua ion, i.e. ω= 1. He e, we also indica e p oblems occu ing in ex ending P oposi ion 4.7 and, hus, Theo em 4.8 o a bi a y ω≥1. As seen in he p e ious example, ou esul s can no be gene alized o β(·,·) desc ibing ini e ime con ollabili y in mo e han wo s eps. Hence, he ollowing lemma gi es a comple e analysis o KL0- unc ions o ype (1.12). 83 SENSITIVITY ANALYSIS Figu e 4.6: On he le , a isualiza ion o α1 4,m,m∈ {1,2,3} o β1(·,·) om Example 4.16 wi h e minal weigh s ω= 1 (◦) and ω= 1.01 (?) is shown. On he igh , α1 9,m, m= 1,2,...,8. o β2(·,·) o he same example wi h ω= 1 (?) and ω= 4/3 (◦) is illus a ed. Lemma 4.17 Le Assump ion 3.2 be sa is ied wi h a KL0- unc ion β(·,·)o ype (1.12) wi h cn= 0 o all n∈N≥2, i.e. ini e ime con ollabili y in a mos wo s eps. Then, o N∈N≥4,αω N,m om Theo em 3.18 sa is ies αω N,m+1 −αω N,m ≥0 o m∈ {1,2,...,bN/2c−1}.(4.13) P oo : Since cn= 0 o n≥2 ensu es γm+1 ≥γi o all i∈N≥3, he necessa y and su icien condi ion γm+1 −ω≤0 o αω N,m = 1 implies i s alidi y o e e y con ol ho izon la ge o equal o mand, hus, in pa icula γm+2 −ω≤0 holds. Hence, le γm+1 −ω > 0 hold. This allows o plugging (3.21) in (4.13) in o de o show he asse ion which is, as a consequence, equi alen o (γm+1 −ω)(γm+2 −1)"N Y i=m+2 γi−(γm+2 −ω) N Y i=m+3 (γi−1)#" N Y i=N−m γi− N Y i=N−m (γi−1)# ≥(γm+2 −ω)(γN−m−1)"N Y i=m+1 γi−(γm+1 −ω) N Y i=m+2 (γi−1)#" N Y i=N−m+1 γi− N Y i=N−m+1 (γi−1)#. (4.14) No e ha o he conside ed subclass o KL0- unc ions o ype (1.12) γ3=γiholds o all i∈N≥3. Consequen ly, we de ine γ:= γ3=γi,i∈N≥3. Fu he mo e, aking N−m≥m+ 2 ≥3 in o accoun allows us o educe (4.14) by he ac o (γ−1) and leads o ω(γm+1−γ) N Y i=m+2 γ·hγm−(γ−1)mi+(γm+1−ω)(γ−1)mγm"N−m Y i=m+2 γ−(γ−ω) N−m Y i=m+3 (γ−1)#≥0, which shows he desi ed inequali y and, hus, comple es he p oo .  Nex , we aim a deducing he desi ed mono onici y p ope y assuming exponen ial con- ollabili y. To his end, we make a dis inc ion wi h espec o he e minal weigh ω. We 84 CHARACTERISTICS DEPENDING ON THE CONTROL HORIZON begin wi h “su icien ly la ge” ones o , o be mo e p ecise, hose sa is ying ω≥(1 −σ)−1. No e ha his condi ion is always achie able bu may be demanding o decay a es σ close o one. Lemma 4.18 Le Assump ion 3.2 hold wi h a KL0- unc ion o ype (1.11). Fu he mo e, le he e minal weigh ωbe chosen such ha η:= 1 + σω −ω≤0. Then, o N∈N≥4, he asse ion om P oposi ion 4.7, i.e. (4.13) holds. P oo : Taking he asse ion o Lemma 4.13 in o accoun , i su ices o es ablish (4.14) in o de o deduce (4.13), i.e. αω N,m+1 −αω N,m ≥0, m∈ {1,2,...,bN/2c − 1}. Since 2m+ 2 ≤N, expanding he e ms in (4.13) and combining hem app op ia ely yields −a N Y i=m+2 γi+ha+(γm+1−ω)(γm+2−1)iN Y i=N−m+1 (γi−1) N−m Y i=m+2 γi−(γm+1−ω)(γm+2−ω) N Y i=m+2 (γi−1) ≥0 (4.15) wi h a:= −[ωγN−m(γm+1 −γm+2) + γm+1(γm+2 −γN−m) + ω(γN−m−γm+1)] (4.9) =Cη 1−σh(C−1)ωσm+σN−m−1ω(1 −Cησm) + γm+1(σm+1 −σN−m−1)i≤0. Hence, he e m −a N−m Y i=m+2 γi"N Y i=N−m+1 γi− N Y i=N−m+1 (γi−1)# om (4.15) as well as he di e ence o he wo emaining summands in his inequali y is posi i e which ensu es (4.15) and, consequen ly, comple es he p oo .  In o de o p o e he asse ion o P oposi ion 4.7 o ω= 1, le a con ol ho izon m be gi en. Then, an induc ion wi h espec o he op imiza ion ho izon Nis ca ied ou . The asse ion αω N,m+1 ≥αω N,m has o be shown o m∈ {1,2,...,bN/2c − 1}. Hence, N= 2m+ 2 is he smalles op imiza ion ho izon o gi en m. The p oo is di ided in o wo pa s: he induc ion assump ion, i.e. N= 2m+2, is deal wi h sepa a ely in he ollowing lemma. The induc ion s ep is ca ied ou a e wa d in Lemma 4.20. Spli ing up he p oo is mo i a ed by he es ic ions on he e minal weigh ωin P oposi ion 4.7. Lemma 4.19 co e s a bi a y e minal weigh s, i.e. ω∈[1,(1−σ)−1) while he induc ion s ep is only shown o ω= 1.5This app oach allows o indica e p oblems occu ing o e minal weigh s ω∈(1,(1−σ)−1). We conjec u e ha P oposi ion 4.7 holds independen ly o he chosen e minal weigh . Fu he mo e, we poin ou ha he induc ion is ca ied ou wi h espec o he op imiza ion ho izon N. P oceeding he o he way ound, i.e. an induc ion is-`a- is he con ol ho izon m, has no p o en o be ui ul. Lemma 4.19 Le Assump ion 3.2 be sa is ied wi h a KL0- unc ion o ype (1.11). Fu he mo e, le he e minal weigh ωbe chosen such ha η:= 1 + σω −ω > 0. Then, o m∈Nand N= 2m+ 2,(4.14) holds o he op imal alue αω N,m o P oblem 3.10 which is gi en by Theo em 3.18. 5No e ha Lemma 4.18 co e s he asse ion o ω≥(1−σ)−1, i.e. “su icien ly la ge” e minal weigh s. 85 SENSITIVITY ANALYSIS P oo : Since Lemma 4.13 co e s he asse ion o γm+1 −ω≤0, he inequali y γm+1 − ω > 0 is assumed. Hence, showing (4.15) is su icien in o de o show he asse ion. Taking in o accoun N−m−1 = m+1, he e m ain oduced in his inequali y simpli ies o Cσmηω(γm+2 −1). Hence, educing (4.15) by (γm+2 −1) leads o hCσmηωγm+2 +ω(γm+1 −ω)iN Y i=m+3 (γi−1) ≥Cσmηωγm+2 N Y i=m+3 γi. Using he ep esen a ion o γm+2 gi en by (4.9) and N−m−2 = m, p oceeding analogously o he p oo o Lemma 4.15 yields C2+1−σmη σmη(1 −σm+1η)C−ω(1 −σ) σmη(1 −σm+1η)N Y i=m+3 C−1−σ 1−σi−1η | {z } =:p(C) ≥Cm+2 |{z} =:q(C) . We aim a applying Lemma 4.14 in o de o es ablish his inequali y. Since P oposi ion 4.1 ensu es a poin o in e sec ion a C= 1 which is supposed o play he pa o ˜zin he hi d assump ion o Lemma 4.14, he posi i e oo s o he monic polynomial p(·) : R→R ha e o be loca ed in [0,1). S uc u ally, p(·) consis s o wo ac o s. He e, he ac o QN i=m+3[C−(1−σ)/(1−σi−1η)] ep esen s a polynomial o deg ee mwhich is decomposed in linea ac o s and, hus, exhibi s m eal oo s loca ed in he open in e al (0,1). Nex , we de e mine he oo s o he o he ac o in ol ed in he de ini ion o p(·) by comple ing he squa e: C=−(1 −σmη)±p(1 −σmη)2+ 4ω(1 −σ)σmη(1 −σm+1η) 2σmη(1 −σm+1η), i.e. one s ic ly posi i e and one s ic ly nega i e oo . Hence, we comple e ou asse ion wi h espec o he oo s o p(·) by showing ha he posi i e oo o his ac o is s ic ly less han one o , equi alen ly, (1 −σmη+ 2σmη(1 −σm+1η))2>(1 −σmη)2+ 4ω(1 −σ)σmη(1 −σm+1η). Cancelling ou he summand (1 −σmη)2, educing he esul ing exp ession by 4σmη (1 −σm+1η), and using he de ini ion o ηleads o σmη(1 −σm+1η) + (1 −σmη)−ω(1 −σ) = η(1 −σ2m+1η)>0. Consequen ly, i emains o es ablish he second condi ion o Lemma 4.14 which deals wi h he (m+ 1)s de i a i es o he polynomials p(·) and q(·). To his end, we calcula e p(m+1)(C) = (m+ 2)!C+ (m+ 1)! 1−σmη σmη(1 −σm+1η)− N X i=m+3 1−σ 1−σi−1η!. Since q(m+1)(·) has i s espec i e oo a he o igin, we ha e o p o e ha he oo o p(m+1)(C) is s ic ly nega i e. Fo his pu pose, i su ices o es ablish 1−σmη σmη(1 −σm+1η)− N X i=m+3 1−σ 1−σi−1η>0. 86 CHARACTERISTICS DEPENDING ON THE CONTROL HORIZON Taking N−m−2 = min o accoun , his is ensu ed by 1−σmη 1−σ= m−1 X i=0 σi+ωσm> mσm> ησm N X i=m+3 (1 −σm+1η) 1−σi−1η. Hence, all assump ions o Lemma 4.14 a e sa is ied which enables us o conclude he asse ion.  Lemma 4.19, which is used as induc ion s a o he p oo o he ollowing lemma, holds o ω∈[1,(1−σ)−1), i.e. o all e minal weigh s no co e ed by Lemma 4.18. In con as o ha , ω= 1 is assumed in he induc ion s ep, which is ca ied ou in he p oo o Lemma 4.18. Howe e , his es ic ion is no imposed in he beginning o he induc ion s ep in o de indica e and b ie ly discuss p oblems o ex ending Lemma 4.20 o e minal weigh s ω∈(1,(1 −σ)−1). Fu he mo e, we like o poin ou ha Lemma 4.14 was o iginally designed o he ollowing induc ion s ep. Lemma 4.20 Le he KL0- unc ion β(·,·) om Assump ion 3.2 be o ype (1.11) and ω= 1. Then, o N∈N≥4, he op imal alue αN,m =α1 N,m o P oblem 3.8 exhibi s he mono onici y p ope y gi en by P oposi ion 4.7, i.e. αω N,m+1 −αω N,m ≥0 o m∈ {1,...,bN/2c−1}. P oo : Repea ing he line o a gumen s used in Lemma 4.18 yields ha i is su icien o es ablish (4.15). Le a con ol ho izon mbe gi en. Then, he p eceding lemma co e s he asse ion o he smalles possible choice o he op imiza ion ho izon N— ou induc ion assump ion. Hence, ca ying ou he induc ion s ep p o es he claim. Suppose ha (4.15), i.e. (γm+1−ω)(γm+2−ω) N Y i=m+2 (γi−1) ≤ −a N Y i=m+2 γi+ha+(γm+1−ω)(γm+2−1)iN−m Y i=m+2 γi N Y i=N−m+1 (γi−1), holds o N≥2m+ 2. The e m ais gi en by6 a=γN−m(ωγm+2 −ωγm+1 +γm+1 −ω)−γm+1(γm+2 −ω). Ou goal is o show his inequali y o N+ 1, i.e. he induc ion s ep N N+ 1. To his end, we equi e he de ini ion ˜a:= γN+1−m(ωγm+2 −ωγm+1 +γm+1 −ω)−γm+1(γm+2 −ω), i.e. Nis subs i u ed by N+ 1 and, hus, γN−mby γN−m+1 in a. Then — using he induc ion assump ion — he desi ed inequali y is ensu ed by (γN+1 −1) "−a N Y i=m+2 γi+ha+ (γm+1 −ω)(γm+2 −1)iN−m Y i=m+2 γi N Y i=N−m+1 (γi−1)# ≤ −˜a N+1 Y i=m+2 γi+h˜a+ (γm+1 −ω)(γm+2 −1)iN−m+1 Y i=m+2 γi N+1 Y i=N−m+2 (γi−1). 6No e ha , in compa ison o he p oo o Lemma 4.18, we ea anged only he conside ed e m. 87 SENSITIVITY ANALYSIS Since N−m≥m+ 2, di iding his inequali y by QN−m i=m+2 γiand aking ˜a+ (γm+1 −ω)(γm+2 −1) = (γN−m+1 −1)(ωγm+2 −ωγm+1 +γm+1 −ω), a+ (γm+1 −ω)(γm+2 −1) = (γN−m−1)(ωγm+2 −ωγm+1 +γm+1 −ω), in o accoun , leads o (ωγm+2−ωγm+1+γm+1−ω)hγN−m+1−(γN−m−1)iN+1 Y i=N−m+1 (γi−1) ≥h˜aγN+1−a(γN+1−1)iN Y i=N−m+1 γi. Di iding his inequali y by (ωγm+2 −ωγm+1 +γm+1 −ω) and using (4.9) yields (CσN−m−1η+1) N+1 Y i=N−m+1 (γi−1)≥CσN−m−1η+γN−m γN+1 −γm+1(γm+2 −ω) γN+1(ωγm+2−ωγm+1+γm+1−ω)N+1 Y i=N−m+1 γi. Since η= 1 + σω −ω > 0 is assumed, he di iso is posi i e. The quo ien consis ing o he nume a o γm+1(γm+2 −ω) and he denomina o γN+1(ωγm+2 −ωγm+1 +γm+1 −ω) is he mos di icul o handle. He e, he ac o γm+1/γN+1 does no con ain he o e shoo Cand, hus, only con ibu es a cons an . The o he ac o , howe e , has a polynomial o deg ee one in he denomina o — exac ly his p e en s he applicabili y o Lemma 4.14 o ω∈(1,(1 −σ)−1). On he con a y, o e minal weigh ω= 1, (ωγm+2 −ωγm+1 +γm+1 −ω) = (γm+2 −1) holds. Hence, he conside ed ac o cancels ou wi h (γm+2 −ω) = (γm+2 −1). Taking η=σin o accoun and educing he inequali y in conside a ion by σN−mp o ides p(C) := (C+σ−(N−m)) N+1 Y i=N−m+1C−1−σ 1−σi≥Cm+1 C+σm+1 −σN−m (1 −σN+1)σN−m=: q(C). A s aigh o wa d applica ion o Lemma 4.14 ensu es his inequali y and, hus, allows o concluding he asse ion. P oposi ion 4.1 yields he poin o in e sec ion a C= 1, i.e. p(1) = q(1). Fu he mo e, no e ha p(·) has exac ly one nega i e oo and m+ 1 s ic ly posi i e oo s which a e loca ed in he open in e al (0,1). Addi ionally, qcan be ep esen ed as Cm+1(C+c) wi h c > 0. Hence, he only condi ion which has o be e i ied is he one wi h espec o he (m+ 1)s de i a i e. To his end, we calcula e p(m+1)(C)=(m+ 2)!C+ (m+ 1)! σ−(N−m)− N+1 X i=N−m+1 1−σ 1−σi!. Consequen ly, i su ices o es ablish he ollowing inequali y in o de o comple e he p oo : 1−σN−m N+1 X i=N−m+1 1−σ 1−σi>σm+1 −σN−m (1 −σN+1). Since (1 −σi)−1<(1 −σN−m)−1 o all i∈ {N−m+ 1, N −m+ 2, . . . , N + 1}, his is ensu ed by 1−σm+1 1−σ= m X n=0 σn>(m+ 1)σm>(m+ 1)σN−m=σN−m N+1 X i=N−m+1 1 1−σN−m.  88 4.3. FURTHER RESULTS 4.3 Fu he Resul s This sec ion con ains miscellaneous esul s. We begin wi h commen ing on Assump ion 3.2 which may seem o be es ic i e a i s glance. Howe e , since his condi ion is o mula ed in e ms o he s age cos s, i u ns ou ha e en sys ems which a e only asymp o ically bu no exponen ially s able sa is y Assump ion 3.2 wi h a KL- unc ion o ype (1.11) and, hus, exhibi he desi ed linea i y ea u e exploi ed in o de o de- duce he o mula p esen ed in Theo em 3.18. In pa icula , he esul s o his and he p e ious chap e a e applicable. Secondly, we deal wi h he impac o inco po a ing an addi ional e minal weigh in ou se ing which signi ican ly complica ed de i ing esul s on symme y and mono onici y. Then, in o de o conclude his sec ion, he example o he linea ized in e ed pendulum on a ca is conside ed. He e, heo e ically obse ed bu p esumably as onishing p ope ies like he mono onici y in he con ol ho izon a e nume ically esembled. This mo i a es he cons uc ion o algo i hms which employ hese p ope ies in he ollowing Sec ion 4.4. 4.3.1 Commen s on Assump ion 3.2 Assuming linea i y o he KL- unc ion β(·,·) om Assump ion 3.2 in i s i s a gumen seems o be a demanding condi ion. Howe e , since he s age cos s can be used as a design pa ame e , c ., e.g. [39, Sec ion 7] and [6], his e en includes sys ems which a e only asymp o ically con ollable. Fo ins ance, he s age cos s we e manipula ed o espec homogenei y in o de o ge simila p ope ies o sys ems ha a e no exponen ially s abilizable in [32]. In o de o u he subs an ia e his claim, he con ol sys em de ined by x(n+ 1) = x(n) + u(n)x(n)3is conside ed which co esponds o he Eule app oxima ion o he di e en ial equa ion ˙x( ) = u( )x( )3wi h ime s ep T= 1. Fu he mo e, he con ol U= [−1,1] and s a e cons ain s X= (−1,1) ⊂Ra e se .7This sys em is asymp o ically s abilizable wi h con ol unc ion u(·)≡ −1, i.e. x(n+ 1) = x(n)−x(n)3. Bu , aking he cons ain s in o accoun , i is no exponen ially s abilizable. In o de o show he claimed exponen ially con ollabili y in e ms o he con inuous s age cos s, we de ine `(x(n), u(n)) := (e−1 2x(n)2 o kx(n)k ∈ X {0}, 0 o x= 0. No e ha `?(x) = `(x, u) holds o all admissible con ol alues ubecause he con ol e o is no penalized. The KL- unc ion β( , n) = e−n o ype (1.11) wi h pa ame e s C= 1 and σ=e−1is chosen. Hence, we ha e o show he inequali y `(x(n+ 1), u(n+ 1)) = `?(x(n+ 1)) = `?(x(n)(1 −x(n)2)) ≤e−1`?(x(n)) which is, in u n, equi alen o `?(x(n+ 1)) = e−1 2x(n)2(1−x(n)2)2≤e−2x(n)2+1 2x(n)2=e−1e−1 2x(n)2=σ`?(x(n)) Using 1 >1−3x(n)4+ 2x(n)6= 2x(n)2(1 −x(n)2)2+ (1 −x(n)2)2>0 o x∈X= (0,1) ensu es his inequali y and, hus, induc i ely implies exponen ial con ollabili y in e ms o `(·,·). Summa izing, designing he s age cos s `(·,·) sui ably allows o e i ying he needed assump ions in o de o apply he deduced esul s, e en o sys ems which a e no exponen ially con ollable wi h espec o hei no m. 7The s a e and con ol es ic ions a e necessa y o p ese e he cha ac e is ics o he con inuous ime sys em o he Eule app oxima ion. 89 SENSITIVITY ANALYSIS bound may be lowe ed o ˆα. Then, P oposi ion 4.23 s ill allows o gua an eeing he educed bound ˆαbu canno be employed in o de o conclude he o iginal pe o - mance index αany longe . In his hesis, he op imiza ion ho izon Nis chosen such ha condi ion (1)(c) is ensu ed o a leas one con ol ho izon mby ou subop imali y analysis and, hus, (1)(d) is excluded a p io i. Ano he app oach is ou lined in [30]: suppose ha αin (4.17) is s ic ly g ea e han he desi ed pe o mance bound α. Then, a posi i e slack a iable s:= VN(xµN(σ(k)))−VN(xµN(j;xµN(σ(k))))−α j−1 X n=0 `(xµN(n;xµN(σ(k))), µN(n, xµN(σ(k)))) (4.18) is in oduced and added o he nume a o o he igh hand side in (4.17) in he nex i e a ion o he algo i hm. As a consequence αis inc eased and, hus, condi ion (1)(c) is weakened.10 This ea u e can be easily inco po a ed in ou algo i hm and may lead o an imp o emen , c . [97]. Howe e , a iola ion in he i s i e a ion o Algo i hm 4.24 canno be deal wi h. This me hodology can be ex ended such ha a nega i e slack and, hus, a iola ion o he desi ed Lyapuno inequali y, is allowed. Then, he algo i hm may be modi ied in o de o compensa e, i possible, such iola ions in e ms o s abili y o pe o mance a pos e io i, c . [96]. As poin ed ou in Rema k 4.25 we wan o exclude s ep (1)(d). Hence, he algo i hm ensu es he desi ed pe o mance a p io i — a dis inguishing ea u e in compa ison o algo i hms which do only e i y a subop imali y es ima e a pos e io i bu may un in o a dead-end. Rema k 4.26 The in oduced lis S ep esen s a possibili y o implemen he sequence (σ(k))k∈N0. The cu en ime ins an is accessible as ia back(S). In addi ion, he co esponding s a e (xµN(σ(k)))k∈N0may be added o he lis S, whose en ies hen consis o wo elemen s. We emphasize ha VN(·)is employed as a Lyapuno unc ion a xµN(σ(k)), c . Rema k 3.13 (ii). Ano he op ion is o emo e he lis Sand use only he cu en s a e x. Then, nei he he ime ins ances σ(k),k∈N0, no he co esponding s a es xµN(σ(k)),k∈N0, a e sa ed in o de o educe memo y usage. A i s glance, Algo i hm 4.24 seems o inc ease he e o needed in each eceding ho izon s ep. Howe e , he a p io i compu a ion o VN(·) a he nex swi ching ime co - esponds exac ly o he e alua ion o his unc ion a he ensuing ime ins an , which has o be done anyway in o de o sol e he op imal con ol p oblem posed in he eceding ho i- zon o mula ion. Hence, he p oposed algo i hm only p oduces addi ional compu a ional cos i needed. In pa icula , since Algo i hm 4.24 allows us o educe he op imiza ion ho izon Nsigni ican ly and he compu a ional e o g ows apidly wi h espec o N, his expendi u e is, in he majo i y o cases, mo e han compensa ed, c . [97]. In o de o demons a e bene i s o Algo i hm 4.24, ou in es iga ion o Example 1.10 is ca ied on. Based on he KL0- unc ion β(·,·) om (3.31) he minimal op imiza ion ho izon Nensu ing a desi ed pe o mance speci ica ion is de e mined, c . Table 4.1. 10I swas al eady in oduced in a p eceding s ep, he igh hand side o (4.18) co esponds o he change o s. Then, s ep esen s he accumula ed slack. 96 ALGORITHMS Fo ins ance, asymp o ic s abili y is ensu ed o op imiza ion ho izon N= 16 ins ead o N= 28 (m= 1). Hence, o α= 0, using he de eloped algo i hm wi h op imiza ion ho izon N= 16 gua an ees ha no exi s a egy is equi ed. Since he desi ed elaxed Lyapuno inequali y holds o RHC wi h ho izon N≥5, he algo i hm does no employ m > 1, c . [90] and Sec ion 5.5.1 below. RHC wi h m= 1 allowing o m > 1 α N N m 0 28 16 6 0.25 31 17 8 0.5 35 20 7 2/3 39 23 8 0.8 45 26 11 10/11 53 33 11 100/101 77 53 21 Table 4.1: Minimal s abilizing ho izon o RHC wi h m= 1 and o RHC wi h sui ably chosen con ol ho izon m∈ {1,2, . . . , N −1} o he subop imali y bounds α1 N,m om Theo em 3.18 based on Assump ion 3.2 wi h KL0- unc ion β(·,·) om (3.31). Hence, he s a egy exhibi ed in he algo i hm leads o classical RHC sa egua ded by ou heo e ical esul s because he es ima e deduced ia P oblem 3.8 p o ides only conse a i e bounds o his pa icula example. Howe e , in many p ac ical and mo e challenging applica ions m > 1 is indeed a necessa y condi ion which may seem o be coun e -in ui i e. In his connec ion we emphasize ha he elaxed Lyapuno inequali y is checked less o en and, hus, weakened by employing la ge con ol ho izons: ensu ing he inequali y VN(xµN,m (n+k+ 1)) ≤VN(xµN,m (n+k)) + α`(xµN,m (n+k), µN,m(xµN,m (n+k),0)), in each s ep k∈ {0,1, . . . , m −1}implies he desi ed c i e ion VN(xµN,m (n+m)) ≤VN(xµN,m (n)) + α m−1 X k=0 `(xµN,m (n+k), µN,m(xµN,m (n+k),0)) a e ms eps. Bu his implica ion does no hold he o he way ound which explains why using la ge con ol ho izons may ensu e he desi ed c i e ion independen ly o whe he his is accompanied by an ac ual pe o mance imp o emen o no . In o de o in es iga e he p oposed algo i hm mo e ho oughly, we conside he ol- lowing nonlinea example om [28], which was also examined in [34], nume ically. Example 4.27 (Synch onous gene a o ) The sys em dynamics a e gi en by ˙x1( ) = x2( ) ˙x2( ) = −b1x3( ) sin x1( )−b2x2( ) + P ˙x3( ) = b3cos x1( )−b4x3( ) + E+u( ) wi h pa ame e s b1= 34.29,b2= 0.0,b3= 0.149,b4= 0.3341,P= 28.22, and E= 0.2405. Choosing E= 0.2405 ma ches a s essed and, hus, mo e challenging, ope a ing 97 SENSITIVITY ANALYSIS condi ion, c . [28, Subsec ion 6.1]. This example is e o mula ed in a sampled-da a ashion in o de o i in o ou disc e e ime se ing. Le Φ(·;x0,˜u)deno e he solu ion ope a o o he di e en ial equa ion wi h ini ial alue x0and piecewise cons an con ol unc ion ˜u(·) : [0, T)→U, i.e. ˜u( ) = u∈U o ∈[0, T). Hence, he s a e space is gi en by X:= R3and he con ol alue space Umay be iden i ied wi h R. Con ol cons ain s may be easily in eg a ed by adap ing Uapp op ia ely. Fo his example, ou goal is o s ee he sys em o i s s able equilib ium x?≈ (1.124603730,0.0,0.9122974248)T. In pa icula , also i s uns able coun e pa , i.e. he equilib ium ˜x≈(1.170838231,0.0,0.8934977016)T, has o be ende ed o x?, c . [94]. Receding ho izon con ol is employed o he sys em cons uc ed wi h sampling pe iod T= 0.05 and unning cos s `0(x, u) = ZT 0kϕ( ;x, ˜u)−x?k2+λk˜u( )k2d o `1(x, u) = Tkϕ(0; x, ˜u)−x?k2+λk˜u(0)k2=Tkx−x?k2+λkuk2 wi h λ= 10−3. In addi ion, he physically mo i a ed s a e cons ain s om [28] a e aken in o accoun , i.e. X:= x∈R3: 0 ≤x1< π/2 and 0 ≤x3. Le he desi ed pe o mance bound α0:= 0 be speci ied. Since Assump ion 3.2 has o hold on a se o easible s a es x0∈X, le el se s Li:= {x0:V6(x0) = in u∈U 5 X n=0 `i(xu(n), u(n)) ≤0.0196}(4.19) o VN(·)⊆X o op imiza ion ho izon N= 6 a e conside ed. Hence, ensu ing ou elaxed Lyapuno inequali y o each poin con ained in a le el se Li,i∈ {1,2}, gua an ees o be, a e implemen ing he i s mcon ol alues, again in his se , i.e. he le el se s Li, i∈ {1,2}, a e eceding ho izon in a ian and, hus, he s a e cons ain s a e sa is ied a each ansmission ins an σ(k), k∈N0. Fo ou nume ical in es iga ion, a g id Gcon ained in he cube [x? 1−0.25, x? 1+ 0.25] ×[−1,1] ×[x? 3−0.75, x? 3+ 0.75] ⊂X is buil up wi h disc e iza ion accu acy 0.05 in each coo dina e di ec ion. A e emo ing he desi ed se poin x?, his se consis s o 11 ·41 ·31 −1 = 13980 g id poin s. The in e sec ion o his g id Gand he le el se Li,i∈ {1,2}, is a subse o he in oduced cube, c . Figu e 4.10. Then, we compu e, o each x0∈ G ∩Li,i∈ {0,1}, he co esponding subop imali y index α6,1(x0) and dis inguish whe he α6,1≥αis sa is ied o no . I his check ails, he con ol ho izon mis inc eased and he espec i e pe o mance bound is compu ed. Indeed, o each conside ed ini ial s a e, a con ol ho izon m∈ {1,2,3,4,5}exis s such ha α6,m(x0)≥αand, hus, J6(xµ6,m (m, x0)) ∈ Li,i∈ {0,1}holds. Repea ing his line o a gumen s i e a i ely shows ha he p oposed algo i hm may be applied wi hou an exi s a egy in o de o conclude he desi ed s abili y beha io .11 11Since he elaxed Lyapuno inequali y is ensu ed only a each g id poin and no necessa ily o each poin con ained in he espec i e le el se Li,i∈ {0,1}, he a gumen a ion is no igo ous. Ne e heless, he s a ed claim is con i med by ou nume ical compu a ions. 98 ALGORITHMS 0.9 11.1 1.2 1.3 −1 −0.5 0 0.5 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 x1 x2 x3 0.9 11.1 1.2 1.3 −1 −0.5 0 0.5 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 x1 x2 x3 Figu e 4.10: Illus a ion o he le el se s L0(le ) and L1( igh ) om (4.19) by means o he co esponding con ex hulls. L0con ains 3091 g id poin s, whe eas L1consis s o only 1758. The d awn con ex hulls ep esen 23.1% (L0) and 12.9% (L1) o he olume o he cube, espec i ely. Fo i= 0, i.e. inco po a ing he s age cos s based on he de ia ion om x?measu ed along he esul ing ajec o y, 10 g id poin s a e ob ained o which m > 1 is necessa y in o de o ensu e he elaxed Lyapuno inequali y o op imiza ion ho izon N= 6, c . Table 4.2. We poin ou ha , o m= 1, h ee o hese poin s e en equi e an op imiza ion ho izon N= 9 in o de o exhibi he desi ed pe o mance bound. Hence, allowing o la ge con ol ho izons educes he op imiza ion ho izon signi ican ly. Fu he mo e, he hi d abula ed poin is no s abilized o N= 5, m∈ {1,2,3,4}.12 A u he educ ion o he op imiza ion ho izon is, hus, no possible wi hou inco po a ing an “exi s a egy”. `0(·,·) om (4.19) α6,m Minimal N: g id poin x0m= 1 m= 2 m= 3 αN,1≥α +0.9246 -0.1500 +0.9123 -0.0300 -0.0236 +0.0230 7 +0.9246 -0.1000 +0.9123 -0.0730 -0.0096 +0.0420 9 +0.9246 -0.1000 +0.9623 -0.0819 -0.0440 +0.0103 9 +0.9246 -0.0500 +0.9123 -0.0115 +0.0657 - 7 +0.9246 -0.0500 +0.9623 -0.0807 -0.0034 +0.0455 9 +0.9246 -0.0500 +1.0123 -0.0294 -0.0122 +0.0294 7 +0.9746 -0.1000 +0.9123 -0.0305 -0.0133 +0.0299 8 +0.9746 -0.0500 +0.9123 -0.0355 +0.0335 - 8 +0.9746 -0.0500 +0.9623 -0.0597 -0.0214 +0.0240 8 +1.0246 -0.0500 +0.9123 -0.0410 +0.0018 - 8 Table 4.2: G id poin s om L0 iola ing α6,1≥α= 0. Fo each o hese poin s m∈ {1,2,3}exis s such ha α6,m ≥αholds. Fo m= 1, he op imiza ion ho izon has o be inc eased o N= 9 in o de o ensu e he desi ed pe o mance speci ica ion. Simila esul s a e ob ained o `1(·,·), c . Table 4.3. Again, an op imiza ion ho izon o N= 9 u ns ou o be he minimal s abilizing ho izon o RHC wi h m= 1 in o de 12Two poin no con ained in Table 4.2 also equi e an op imiza ion ho izon N > 5. 99 SENSITIVITY ANALYSIS o sa is y he p oposed pe o mance speci ica ion. Fu he mo e, we poin ou ha e en con ol ho izon m= 4 is equi ed in o de o ensu e he subop imali y bound. The second and six h poin abula ed in Table 4.3 a e no s abilizable o N= 5.13 No e ha he gene a ed ajec o ies may lea e he le el se . The algo i hm applied wi h α= 0 ensu es a dec ease only a he ansmission imes. Howe e , since he le el se is loca ed in he in e io o he cube which also exhibi s a sa e y ma gin away om he bounda y o he se o easible s a es Xas well as he small sampling ime in com- bina ion wi h con inui y p ope ies o he conside ed sys em a iola ion o he imposed s a e cons ain s seems o be highly unlikely. `1(·,·) om (4.19) α6,m Minimal N: g id poin x0m= 1 m= 2 m= 3 m= 4 αN,1≥α +0.9246 -0.0500 +0.9123 -0.0594 +0.0069 - - 8 +0.9246 -0.0500 +0.9623 -0.1063 -0.0624 -0.0006 +0.0377 9 +0.9246 +0.0000 +0.9123 -0.0309 +0.1110 - - 8 +0.9246 +0.0000 +0.9623 -0.1036 +0.0190 - - 9 +0.9746 -0.0500 +0.9123 -0.0231 -0.0034 +0.0451 - 7 +0.9746 -0.0500 +0.9623 -0.0213 -0.0465 -0.0092 +0.0260 7 +0.9746 +0.0000 +0.9123 -0.0195 +0.1080 - - 7 +0.9746 +0.0000 +0.9623 -0.0606 +0.0106 - - 8 +1.0246 +0.0000 +0.9123 -0.0096 +0.1047 - - 7 +1.0746 +0.0000 +0.9123 -0.0011 +0.1012 - - 7 Table 4.3: G id poin s om L1 iola ing he pe o mance speci ica ion α6,1≥α= 0. The smalles op imiza ion ho izon Ngua an eeing αN,1≥α o each g id poin is N= 9. Concluding, Algo i hm 4.24 allows o educe he op imiza ion ho izon signi ican ly, i.e. N= 6 ins ead o N= 9 o α= 0. Simila e ec s a e obse able o o he pe o mance bounds, e.g. α= 1/3. He e, applying he p oposed algo i hm enables us o ensu e he desi ed Lyapuno inequali y o N= 13 ins ead o N= 16 o classical RHC o `1(·,·) (o N= 12 ins ead o N= 15 o `0(·,·)). Hence, employing la ge con ol ho izons is no only a o able om a heo e ical poin o iew bu may also be exploi ed in p ac ice. 4.4.2 Ad anced Algo i hm Al hough employing m > 1 is no needed e y o en along he closed loop ajec o y, i may, ne e heless, be ha m ul in e ms o obus ness. Hence, we aim a de eloping he p oposed algo i hm u he in o de o a oid s aying in open loop longe han necessa y. He e, since αN,1< α may occu , s ep 1 o Algo i hm 4.24 seems o be ine i able. Bu we do no know whe he he compu ed sequence o con ol alues is supe io o eceding ho izon con ol wi h m= 1. Indeed, also classical RHC may sa is y he pe o mance speci ica ion a e ms eps. Hence, he main idea consis s o examining whe he he loop can be closed o no wi hou iola ing he imposed pe o mance speci ica ion. Exac ly his issue is ackled by he ollowing algo i hm. 13Two poin which a e no lis ed in Table 4.3 also iola e ou s abili y c i e ion o N= 5, i.e. αN,m <0 o m∈ {1,2, . . . , N −1}. 100 ALGORITHMS Algo i hm 4.28 Le an ini ial s a e x0∈X, a lis S= (0), an op imiza ion ho izon N∈N≥2, and a pe o mance speci ica ion α∈[0,1) be gi en. Se k= 0. Do (1) Ca y ou s ep (1) om Algo i hm 4.24 in o de o ob ain •VN(xµN(σ(k))), •µN(j, xµN(σ(k))), j= 0,1, . . . , mk−1 wi h mk≥1 such ha VN(xµN(σ(k))) −VN(xµN(mk;xµN(σ(k)))) Pmk−1 n=0 `(xµN(n;xµN(σ(k))), µN(n, xµN(σ(k)))) ≥α. (4.20) (2) Se j= 0 and de ine ˆuN(n) := µN(n, xµN(σ(k))), n= 0,1, . . . , mk−1. Do (a) Se j=j+ 1 (b) Implemen ˆuN(j−1) a he plan ˆx(j) := xˆuN(j;xµN(σ(k))). (c) I j < mk: compu e µN(·,ˆx(j)) and VN(xµN(mk−j; ˆx(j))). Check whe he VN(xµN(σ(k))) −VN(xµN(mk−j; ˆx(j))) Pj−1 n=0 `(ˆx(n),ˆuN(n)) + Pmk−1 n=j`(xµN(n−j; ˆx(j)), µN(n−j, ˆx(j))) ≥α (4.21) holds. In case i does: exchange he emaining ail o ˆu, i.e. ˆuN(n) := (ˆuN(n)n < j µN(n−j, ˆx(j)) n≥j. while j < mk (3) Se S:= (S,back(S) + mk), k:= k+ 1, go o (1) while s opping c i e ia no sa is ied. In Algo i hm 4.24 we ensu ed he elaxed Lyapuno inequali y – he key elemen o ou app oach. Howe e , gua an eeing (4.20) may equi e he implemen a ion o mo e han only he i s elemen o he compu ed sequence o con ol alues µN(n, xµN(σ(k))), n= 0,1, . . . , N −1, i.e. s aying in open loop o a longe pe iod o ime. Algo i hm 4.28 p oposes a s a egy o close he esul ing con ol loop mo e o en. In his con ex , we ha e o dis inguish be ween swi ching ins ances, which coincide wi h he ansmission imes, c . Sec ion 1.4, and ime ins ances no con ained in he sequence (σ(k))k∈N0. Acco ding o De ini ion 1.24, he exis ence o such sampling ins ances implies an elemen mk=σ(k+ 1) −σ(k)>1. He e, we decouple he upda e imes, i.e. he sampling ins an s a which he sequence o con ol alues o be implemen ed is modi ied, and he ansmission imes, i.e. sampling ins ances a which he elaxed Lyapuno inequali y has o hold. This leads o he ques ion, which condi ion allows us o upda e he sequence o con ol alues mo e o en and, hus, obus i ies he con ol loop. To his end, Algo i hm 4.28 employs (4.21) which ensu es ha he sequence assembled om he p e iously used and he newly compu ed one also sa is ies he elaxed Lyapuno inequali y. To be mo e 101 SENSITIVITY ANALYSIS p ecise, he condi ion checks whe he a sequence o which we ha e ensu ed ha he elaxed Lyapuno inequali y holds a he nex ansmission ime may be upda ed a a sampling ins an p eceding ha ime ins an . The candida e is conca ena ed by he sequence o which a leas one con ol alue was implemen ed a he plan and he sequence esul ing om applying RHC a he cu en ime ins an . I he conca ena ed sequence also sa is ies he elaxed Lyapuno inequali y a he o hcoming ansmission ins an , he old sequence is eplaced by he newly compu ed one in o de o imp o e obus ness by closing he con ol loop once mo e. Indeed, o ajec o ies emana ing om he poin s iola ing he desi ed Lyapuno inequali y o op imiza ion ho izon N= 6 o he synch onous gene a o his condi ion is ul illed each ime. Hence, Algo i hm 4.28 indeed pe o ms classical RHC bu ensu es — a p io i — asymp o ic s abili y. He e, e i ying he elaxed Lyapuno inequali y o la ge con ol ho izons enabled us o check ou s abili y c i e ion in ad ance. Al hough (4.21) holds o hese ajec o ies he e is no gua an ee ha i always does, i.e. only being able o s ick, i necessa y, o a compu ed con ol sequence o mo e han one sampling ins an yields he desi ed s abili y gua an ee. Hence, Algo i hm 4.28 obus i ies he applied RHC s a egy. Fu he mo e, i smoo hes he esul ing ajec o ies, c . Figu es 4.11 and 4.12. 0.8 1 1.2 −0.06 −0.04 −0.02 00.02 0.04 0.06 0.08 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 0.8 1 1.2 −0.06 −0.04 −0.02 00.02 0.04 0.06 0.08 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 Figu e 4.11: T ajec o ies emana ing om he c ucial poin s om Table 4.3 compu ed wi h he basic (on he le ) and he ad anced algo i hm (on he igh ) based on he s age cos s `1(·,·). The mo e elabo a ed Algo i hm 4.28 upda es each ime and, hus, smoo hes he co esponding ajec o ies, c . also Figu e 4.12. In o de o conclude his subsec ion we commen on some e ec s obse ed o he conside ed example. To his end, we ocus on he in e al be ween σ(0) and σ(1). •Fo s age cos s `1(·,·), he pe o mance es ima e is imp o ed by Algo i hm 4.28 o each upda e, i.e. he le hand side o (4.21) is la ge han he one om (4.20). While he co esponding change in he op imal alue unc ion is non-mono one. •Fo s age cos s `0(·,·), he las upda e, e.g. he second o m= 3, de e io a es he pe o mance es ima e whe eas he p eceding one con ibu es posi i ely. Summing up hese e ec s yields inc eased subop imali y bounds o m= 3 and dec eased es ima es o m= 2. The op imal alue unc ion e alua ed a he nex ansmission ime inc eases by upda ing. Hence, he main bene i o applying he ad anced e sion o he p oposed algo i hm is he concomi an obus i ica ion. The o e all pe o mance o he eceding ho izon closed loop 102 ALGORITHMS is in es iga ed in [96]. Nex , he con ol alues ac ually implemen ed a he plan du ing un ime o Algo i hm 4.28 a e conside ed. The nex con ol alue o be applied a e upda ing he sequence o con ol alues dec eases in no m o each poin om Tables 4.2 and 4.3, c . Table 4.4 o a ypical cou se. Fo ano he upda e c i e ion, which is p e e able om a compu a ional poin o iew, we e e o [97]. ˆu(0) ˆu(1) ˆu(2) ˆu(3) ˆu(4) ˆu(5) ˆu(6) ˆu(7) ˆu(8) ˆx(0) +0.755 +0.596 +0.110 -0.318 -0.775 -0.000 — — — ˆx(1) — +0.458 +0.257 -0.014 -0.336 -0.762 -0.000 — — ˆx(2) — — +0.121 +0.131 -0.038 -0.330 -0.747 -0.000 — ˆx(3) — — — -0.004 +0.104 -0.036 -0.322 -0.735 -0.000 Table 4.4: Compu ed sequences o con ol alues a s a e ˆx(j), j= 0,1,2,3, o he second poin om Table 4.3. An upda e is ca ied ou a each sampling ins an . The applied con ol alues a e w i en in ed. Ou ocus was pu on he con ol ho izon. In pa icula , we poin ed ou ha he heo e ically deduced esul s wi h espec o symme y and mono onici y p ope ies may be exploi ed in such a way ha a posi i e impac no only on ne wo ked con ol sys ems bu also on RHC in gene al is a ainable. In pa icula , he p oposed algo i hms ep esen a me hodology o educe he op imiza ion ho izon Nwhich p edominan ly de e mines he compu a ional e o associa ed wi h sol ing he op imal con ol p oblem in each eceding ho izon s ep. Fu he mo e, he mo e elabo a ed Algo i hm 4.28 ensu es ha he obus ness p ope ies o RHC a e p ese ed. We emphasize once mo e ha bo h algo i hms e i y he desi ed pe o mance es ima es a p io i. 0.05 0.1 0.15 0.2 0.25 0.3 0.35 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 ime x2 0.05 0.1 0.15 0.2 0.25 0.3 0.35 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 ime x2 Figu e 4.12: The x2componen o he ajec o ies which a e gene a ed by Algo i hm 4.24 (solid line) and Algo i hm 4.28 (dashed-do ed line) and emana e om he second, o h, six h, and se en h poin om Table 4.2 (`0(·,·), on he le ) and Table 4.3 (`1(·,·), on he igh ), a e d awn. By cons uc ion he ajec o ies canno de ia e up o he i s sampling ins an T= 0.05. 103 Chap e 5 Sampled-Da a Sys ems and G ow h Condi ion In Chap e 3 we deduced ou key esul s in o de o ensu e asymp o ic s abili y and, in addi ion, ga e es ima es on he pe o mance o he eceding ho izon closed loop p o ided Assump ion 3.2 holds. Fu he mo e, in Sec ion 1.3, sampled-da a sys ems we e in oduced in o de o inco po a e sys ems o iginally de ined con inuously in ime in ou disc e e ime se ing. Typically, sampled-da a sys ems a e induced by an o dina y o pa ial di e en ial equa ion and, hus, he co esponding con ol inpu , which is ac ually implemen ed a he plan , has o be speci ied on he en i e sampling in e al. Allowing o a bi a y me ic spaces in he de ini ion o he admissible se o con ol alues enables us o deal wi h his ac . Howe e , ou s anding Assump ion 3.2 imposes bounds only a he sampling ins ances which i s well o he disc e e ime se ing bu may no ully e lec he s abili y beha io o a sampled-da a sys em, c . he ollowing example o a eac ion di usion equa ion aken om [5]. Example 5.1 (Semi-linea eac ion di usion equa ion) In his example we change he no a ion o be consis en wi h he usual PDE no a ion: x∈Ω⊂Rdis he independen space a iable while he unknown unc ion y(·, ):Ω→R deno es he s a e. Le he open and connec ed se Ωbe a Lipschi z-domain in o de o ensu e well-posedness o he ollowing semi-linea pa abolic pa ial di e en ial equa ion (PDE), c . [119, Subsec ion 2.2.2]. We conside a eac ion di usion equa ion y (x, ) = ∆y(x, )− (y(x, )) + u(x, )on Ω×(0,∞) (5.1) y(x, ) = 0 on ∂Ω×(0,∞) (5.2) wi h homogeneous Di ichle bounda y condi ions, ini ial da a y(x, 0) = y0, dis ibu ed con ol u(·, ) : Ω →R, and con inuously di e en iable non-linea i y :R→R. In addi ion, le (0) = 0 in o de o ensu e ha he o igin is an equilib ium. Fo exis ence and egula i y esul s we e e o [16]. Be o e we con inue ou in es iga ion o Example 5.1, we p esen he ollowing heo- em conce ning he local s abili y beha io o he uncon olled e sion o his semi-linea pa abolic equa ion which is p o en in [16]. Theo em 5.2 Fo each γ∈(0, λ1+ 0(0)), a cons an R=R(γ)exis s such ha o all y0∈ C0(Ω) wi h ky0k ≤ R he solu ion y(·, )o (5.1),(5.2) wi h u(·, )≡0 o all ∈[0,∞)sa is ies ky(·, )k ≤ Mky0ke−γ ∀ ≥0.(5.3) 105 SAMPLED-DATA SYSTEMS AND GROWTH CONDITION Figu e 5.4: The con inuous cu e on he le depic s he bounds induced by Assump ion 5.9 wi h o e shoo C= 2, decay a e µ= ln(2), and sampling pe iod T= 1. The blue line indica es he implied bounds aken in o accoun by Assump ion 3.2 wi h σ=e−µT = 1/2 (). On he igh , we added he espec i e bounds o a mo e accu a e disc e iza ion co esponding o (T2, N2, σ2) = (0.25,20,4 q1 2) (black line). In o de o be able o apply Fo mula (3.21) o a gi en disc e iza ion pa ame e k∈N, he de ini ion o γi om Theo em 3.18 is combined wi h he se ing gi en in his sec ion. Theo em 5.12 Le Assump ion 5.9 be sa is ied wi h decay a e µ > 0and o e shoo C≥1. In addi ion, le an op imiza ion ho izon N∈N≥2, a con ol ho izon m∈ {1,2, . . . , N −1}, and a sampling pe iod T > 0be gi en. Fu he mo e, we de ine σ:= e−µT ∈(0,1),βk( , n) := ·Ck √σn, and γi,k := i−1 X n=0 −1·βk( , n) = C i−1 X n=0 σ1/kn=C(1 −σi/k) 1−σ1/k .(5.10) Then, o each k∈N, Assump ion 3.2 holds o βk( , n), i.e. a KL- unc ion o ype (1.11) wi h o e shoo Cand decay a e k √σ. Fu he mo e, he op imal alue αkN,km(k) o he co esponding op imiza ion p oblem (Pk), i.e. P oblem 3.8 based on βk(·,·)wi h op imiza ion ho izon kN and con ol ho izon km, is gi en by Fo mula (3.21) based on γi,k ins ead o γi, i.e. αkN,km(k)=1−QkN i=km+1(γi,k −1) QkN i=km+1 γi,k −QkN i=km+1(γi,k −1)·QkN i=k(N−m)+1(γi,k −1) QkN i=k(N−m)+1 γi,k −QkN i=k(N−m)+1(γi,k −1). (5.11) P oo : Fo k= 1 e i ying Assump ion 3.2 and showing (5.11) ollows di ec ly om Assump ion 5.9. Fo k∈N≥2, we adap he sampling pe iod, i.e. Tk:= T/k. Hence, he co esponding decay a e e−µT/k equals k √σand, hus, he asse ion is ensu ed by aking he in oduced no a ion in o accoun . 112 DISCRETIZATION AND SAMPLED-DATA SYSTEMS  No e ha (5.10) has o be in e p e ed in he sense ha γi,k does no depend on he i s a gumen o βk( , n). Indeed, γi,k may ha e been de ined di ec ly by he exp ession gi en by he igh hand side o (5.10) bu using he in ol ed KL- unc ion emphasizes i s o iginal backg ound, c . Rema k 3.15. Fu he mo e, he addi ional index kin (5.10) and he addi ional a gumen kin (5.11) clea ly indica e he in ol ed disc e iza ion pa ame e . Theo em 5.12 allows us o begin ou s udy o mo e accu a e disc e iza ions. To his end, he con inuous ime op imiza ion and con ol ho izon a e ixed. P oposi ion 5.13 Le he assump ions o Theo em 5.12 be sa is ied. Fu he mo e, we de ine σ,γi,k, and (Pk) acco ding o Theo em 5.12. Then, o he sequence (kj)j∈N0wi h kj:= 2j, he op imal alues αkjN,kjm(kj)o (Pkj)sa is y αN,m =αk0N,k0m(k0)and αkjN,kjm(kj)≤αkj+1m,kj+1N(kj+1)≤1−σN∀j∈N0, (5.12) i.e. using an i e a i e e inemen as speci ied by (kj)j∈N0o he con ol and he op imiza ion ho izon ensu es mono onically inc easing subop imali y es ima es. P oo : The p oo is subdi ided in o wo pa s. Fi s ly, we show he mono onici y o he sequence (αkjN,kjm(kj))j∈N0. In he second po ion o he p oo we deduce he uppe bound which is independen o he index j. Le kbe an elemen o (kj)j∈N0. Then, using he ep esen a ion gi en by (5.11) yields αkN,km(k)=1−"kN Y i=km+1 γi,k γi,k −1−1#−1  kN Y i=k(N−m)+1 γi,k γi,k −1−1  −1 . Hence, aking kj+1 = 2kjin o accoun , i is su icien o es ablish kN Y i=k%+1 γi,k γi,k −1≤ kN Y i=k%+1 γ2i,2k γ2i,2k−1·γ2i−1,2k γ2i−1,2k−1= 2kN Y i=2k%+1 γi,2k γ2i,k −1 o %∈ {m, N −m}. Since he p oduc s o ei he side o he inequali y sign consis o he same numbe o ac o s, showing he desi ed inequali y componen wise, i.e. γi,k(γ2i,2k−1)(γ2i−1,2k−1) ≤γ2i,2kγ2i−1,2k(γi,k −1) o , equi alen ly, γ2i,2kγ2i−1,2k≤γi,k(γ2i,2k+γ2i−1,2k−1), su ices. No e ha he con- c e e alue o %does no play a ole. In o de o e i y his inequali y, we es ablish γ2i,2kγ2i−1,2k≤γi,k(γ2i,2k+γ2i−1,2k−C) educed by C2, i.e. bea ing (5.10) in mind 1−σi k 1−σ1 2k·1−σ2i−1 2k 1−σ1 2k≤1−σi k 1−σ1 k 1−σi k+ 1 −σ2i−1 2k−1 + σ1 2k 1−σ1 2k! which is, in u n, equi alen o h(1 −σ1 k)−(1 −σ1 2k)i(1 −σ2i−1 2k)≤(1 −σ1 2k)(σ1 2k−σi k). 113 SAMPLED-DATA SYSTEMS AND GROWTH CONDITION Since he le and he igh hand side o his inequali y a e equal o σ1 2k(1−σ1 2k)(1−σ2i−1 2k), his u ns ou o be an equali y. Indeed, s ic g ow h is shown wi h espec o i o C > 1. Fo C= 1, he alue is cons an . I emains o es ablish he uppe bound s a ed in (5.12). To his end, we equi e he es ima e kN Y i=k%+1 γi,k γi,k −1 C≥1 ≤ kN Y i=k%+1 γi,k γi,k −C (5.10) = kN Y i=k%+1 1−σi k (1 −σi−1 k)σ1 k =1−σN (1 −σ%)σN−%. which is independen o he chosen k. Then, combining his bound wi h he abo e ep e- sen a ion o he op imal alue αkN,km(k) p o ides αk 2kN,2km≤1−1−σN (1 −σm)σN−m−1−11−σN (1 −σN−m)σm−1−1 = 1 −(1 −σm)σN−m 1−σN−m(1 −σN−m)σm 1−σm= 1 −σN, i.e. he desi ed uppe bound which is igh o C= 1.  In P oposi ion 5.13 we adap ed bo h he con ol and he op imiza ion ho izon. Hence, e ining he disc e iza ion and, hus, inc easing he disc e e ime op imiza ion ho izon in o de o keep he con inuous one cons an also implied ha he disc e e ime con ol ho i- zon g ows, which is manageable because o ou mul is ep eedback app oach in oduced in Sec ion 1.4. Indeed, P oposi ion 5.13 ensu es enhanced pe o mance es ima es. Example 5.14 We conside he eac ion di usion equa ion om Examples 5.1, 5.3, and 5.6 and in es i- ga e e ec s o an i e a i e e inemen . To his end, he sequence (2j)j∈N0o disc e iza ion pa ame e s is employed, c . Example 5.11 and Figu e 5.4. As shown in Figu e 5.5, he i s e inemen s ep allows o dec ease he minimal s abilizing ho izon by one o m= 1, i.e. αN,1>0holds o N= 9 ins ead o N= 10, c . Figu e 5.2. Ca ying ou a second e inemen s ep yields αN,1>0 o N= 8. A u he educ ion is no possible, c . Sec ion 5.2 below. The imp o emen associa ed o he espec i e e inemen s ep seems o decline such ha he i s e inemen s eps should seem o be he mos impo an ones. Employing la ge con ol ho izons, i.e. m > 1in combina ion wi h a mo e accu a e disc e iza ion does no allow o using smalle op imiza ion ho izons Nin compa ison o he p e iously de i ed esul s, c . Rema k 5.7. Ne e heless, enhanced es ima es a e ob ained. Hence, he ques ion a ises whe he simila esul s a e ob ainable o classical RHC, i.e. m= 1. This co esponds o sho ening he con inuous ime con ol ho izon, e-op imizing mo e o en and, hus, obus i ying he esul ing closed loop. Howe e , in Chap e 4 we obse ed ha using longe con ol ho izons imp o es he deduced subop imali y es ima es, c . Theo em 4.8. He e, i u ns ou ha i e a ing he e inemen p ocess oo o en causes nega i e subop imali y bounds and, hus, makes ou es ima es useless, c . Figu e 5.6. This claim is shown in Theo em 5.15. In o de o p oo his heo em, we need Lemma 5.20 which is based on esul s conce ning he Gamma Γ(·) as well as he Be a B(·,·) unc ion, i.e. he unc ional equa ion o he Gamma unc ion, a o mula which allows o a ansi ion om he one o he o he , and, in pa icula , a mo e sophis ica ed esul which goes back o Bine . Howe e , since his in eg al pa o he ollowing p oo is a he echnical, i is pos poned un il Subsec ion 5.1.1 in o de o s eamline he p esen a ion. 114 DISCRETIZATION AND SAMPLED-DATA SYSTEMS 4 5 6 7 8 9 10 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 op imiza ion ho izon N subop imali y deg ee α k = 0 k = 1 k = 2 k = 3 4 5 6 7 8 9 10 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 op imiza ion ho izon N subop imali y deg ee α k = 0 k = 1 k = 2 k = 3 Figu e 5.5: Impac o using mo e accu a e disc e iza ions o he eac ion di usion equa- ion. Fo classical RHC one e inemen s ep dec eases he equi ed op imiza ion ho izon o N= 9. The second e inemen s ep leads o a u he imp o emen (N= 8). On he igh , one obse es imp o ed es ima es also o con ol ho izon m= 3. He e, he minimal s abilizing ho izon N, howe e , emains he same. Figu e 5.6: The asse ion o Theo em 5.15 is illus a ed o N= 8, C= 2, and σ= 0.5: o a bi a ily as sampling — which co esponds o using a e y ine disc e iza ion — ou subop imali y es ima es become nega i e o m= 1. Hence, nei he asymp o ic s abili y no a pe o mance bound is ensu ed. Theo em 5.15 Le he assump ions o Theo em 5.12 be sa is ied. Fu he mo e, we de ine σ,γi,k, and (Pk) acco ding o Theo em 5.12. Then, o C > 1and he sequence (kj)j∈N0wi h kj:= 2j, he co esponding sequence o op imal alues (αkjN,1(kj))j∈N0di e ges o minus in ini y, i.e. αkjN,1(kj)=1−(γkjN,kj−1) QkjN i=2 (γi,kj−1) QkjN i=2 γi,kj−QkjN i=2 (γi,kj−1) −→ −∞ o j→ ∞. 115 SAMPLED-DATA SYSTEMS AND GROWTH CONDITION P oo : Since QkjN i=2 γi,kj≥QkjN i=2 (γi,kj−1) ≥0 holds, he asse ion ollows om 0≤1 γkjN,kj−1· kjN Y i=2 γi,kj γi,kj−1−→ 0 o j→ ∞.(5.13) In o de o deal wi h (5.13), we i s p o e he auxilia y inequali ies 1 γkjN,kj−1≤1−σ1/kj C1 and kjN Y i=2 γi,kj γi,kj−1≤C0(21/C)j(5.14) wi h C0:= σ−N/C QN i=2 iC iC−1and C1:= C(1 −σN)−1 + σ. No e ha he cons an s C0and C1do no depend on kj. The i s inequali y is di ec ly ensu ed by using (5.10) because σ≤σ1/kjand, hus, he denomina o o he igh hand side is smalle while he nomina o s a e he same. In o de o es ablish he second claim in (5.14), we equi e he auxilia y es ima e γi,kj γi,kj−1=C C−1 + σ1/kj (1 −σi/kj)(C−1 + σ1/kj) (C−1 + σ1/kj)−Cσi/kj≤C C−1 + σ1/kj·iC iC −1(5.15) which is equi alen o iCσi/kj(1−σ1/kj)≤(C−1+σ1/kj)(1−σi/kj) o i∈ {2,3, . . . , kjN}. Di iding his inequali y by (1 −σ1/kj), spli ing up he esul ing le hand side in o he wo ac o s Cσ1/kjand iσ(i−1)/kj, and applying he es ima es Cσ1/kj<(C−1+σ1/kj) and iσ(i−1)/kj≤Pi−1 n=0 σn/kj= (1 −σi/kj)/(1 −σ1/kj) ensu es (5.15). In addi ion, we equi e ano he p elimina y esul , i.e. C C−1 + σ1/kjkjN ≤σ−N/C ,(5.16) in o de o conclude (5.14). Taking he (kjN)- h oo , (5.16) is equi alen o Cσ1/(kjC)≤ (C−1 + σ1/kj). Since σ1/kj∈(0,1), de ining (x) := C−Cx1/C −1 + xand showing (x)≥0 o all x∈[0,1] gua an ees he desi ed inequali y. Since (0) = C−1≥0 and (1) = 0, e i ying ha (·) is mono onically dec easing su ices. Howe e , his is ensu ed because (·) is con inuous on he in e al [0,1], con inuously di e en iable on (0,1), and 0(x) = 1 −(σ1/kj)−(C−1)/C ≤0 o all x∈(0,1). Hence, bea ing in mind ha he ac o C/(C−1 + σ1/kj) is independen o he con ol a iable i, aking (5.15) and (5.16) in o accoun , using kj= 2j, and applying Lemma 5.20 yields kjN Y i=2 γi,kj γi,kj−1< σ−N/C · 2jN Y i=2 iC iC −1=C0 j−1 Y ν=0 2ν+1N Y i=2νN+1 iC iC −1!≤C0(21/C)j o j∈N0, i.e. (5.14). Now, showing (21/C)j(1 −σ1/kj)→0 as japp oaches in ini y is su icien in o de o comple e he p oo . Fo his pu pose, we de ine ηj:= (21/C)j(1−σ1/kj) and show ha he quo ien ηj+1/ηjcon e ges o 21/C/2 o j→ ∞: ηj+1 ηj =1−σ1/2(j+1) 1−σ1/2j21/C =(1 −σ1/2(j+1) )21/C (1 −σ1/2(j+1) )(1 + σ1/2(j+1) )=21/C 1 + σ1/2(j+1) j→∞ −→ 21/C/2. Hence, he e exis s j?such ha he conside ed quo ien ηj+1/ηjis less o equal θ:= (2 + 21/C)/4<1 o all j≥j?, i.e. he quo ien is bounded om abo e by 21/C/2 + ε wi h ε:= (2 −21/C)/4>0. This implies he con e gence o 21/C(1 −σ1/kj) o ze o o j app oaching in ini y, i.e. (5.13) and, hus, he asse ion. 116 DISCRETIZATION AND SAMPLED-DATA SYSTEMS  O en, solu ions o a con ol sys em gene a ed by a di e en ial equa ion a e con inuous which can be exploi ed, e.g. in o de o p ese e s abili y o he co esponding sampled- da a sys em by su icien ly as sampling, c . [91]. Inhe en p ope ies like he men ioned con inui y yield, in pa icula o (su icien ly) small in e als, igh e bounds on he ansien beha io o he conside ed sys em han hose p o ided by ou con ollabili y Assump ion 3.2. Howe e , hey a e no aken in o accoun in he de i a ion o P oblem 3.8 and, hus, in he subop imali y es ima es om Theo em 3.18. Hence, a g ow h condi ion is inco po a ed in ou se ing in o de o e lec , e.g. con inui y p ope ies in he deduced pe o mance bounds. The combina ion o ou con ollabili y assump ion and a g ow h condi ion will esol e he p oblem esul ing om Theo em 5.15 o e y ine disc e iza ion and m= 1, c . Sec ion 5.3. Fu he mo e, he g ow h condi ion o be in oduced will allow o igh en ou pe o mance bounds, c . Sec ion 5.4. 5.1.1 Auxilia y Resul s o he P oo o Theo em 5.15 In his subsec ion, he auxilia y Lemma 5.20 is p esen ed which is needed in o de o show he second inequali y o (5.14), i.e. kjN Y i=2 γi,kj γi,kj−1≤C0(21/C)j(5.17) and, hus, Theo em 5.15. The p oo o his lemma is, as al eady indica ed in Sec ion 5.1, essen ially based on a esul going back o Bine which p o ides a sui able se ies expansion o he be a unc ion B(·,·), c . Lemma 5.19. Fu he mo e, wo al e na i e p oo s o (5.17) and, hus, Theo em 5.15 o he special case C= 2 a e p esen ed a e wa d which a e in e es ing om a ma hema ical poin o iew. In bo h app oaches a ep esen a ion o he analy ic unc ion (z) = cos(z), which is gi en in Lemma 5.21, is applied: •in he i s p oo he asse ion o Lemma 5.20 is deduced wi hou applying Lemma 5.19 which allows o a oid he use o he Be a unc ion B(·,·) en i ely. •While in he second, elemen a y p oo (5.17) is shown independen ly o he auxilia y Lemma a 5.19 and 5.20. In pa icula , nei he he Be a B(·,·) no he Gamma unc ion Γ(·) a e employed. A i s , he Gamma Γ(·) and he be a unc ion B(·,·) a e de ined. Then, some basic p ope ies o hese wo unc ions a e gi en. De ini ion 5.16 Le x, y ∈R>0. Then, we de ine he Eule ian in eg als o i s and second kind by B(x, y) := Z1 0 x−1(1 − )y−1d and Γ(x) := Z∞ 0 x−1e− d . B(·,·)is known as he Be a unc ion, Γ(·)is called Gamma unc ion. Rema k 5.17 The Gamma unc ion Γ(·)is well-de ined on ]0,∞), c . [88, Theo em 6.4.1]. In addi ion, Γ(1) = 1 and Γ(x+ 1) = xΓ(x) (5.18) 117 SAMPLED-DATA SYSTEMS AND GROWTH CONDITION hold, c . [88, Theo em 6.4.4]. This iden i y is said o be he unc ional equa ion o he Gamma unc ion Γ(·)and also known as he educ ion o mula o he di e ence equa ion, c . [123, p.237]. Fo non-nega i e in ege s n∈N, he Gamma unc ion Γ(·) ep esen s he ac o ial, i.e. Γ(n+ 1) = n!as well. Howe e , (5.18) holds o a bi a y eal numbe s. Rema k 5.18 The Be a unc ion B(·,·)is well-de ined on ]0,∞)×]0,∞), c . [124, p.437] and connec ed o he Gamma unc ion Γ(·) ia he o mula B(x, y) = Γ(x)Γ(y) Γ(x+y),(5.19) c . [124, p.442]. Lemma 5.19 Le p > 0,p+s > 0. Then he ollowing equa ion holds o he Be a unc ion B(·,·) B(p, p +s) = B(p, p) 2s1 + s(s−1) 2(2p+ 1) +s(s−1)(s−2)(s−3) 2·4·(2p+ 1) ·(2p+ 3) +. . .. P oo : We p o e only he special case s= 2. Fo s6= 2 we e e o [123, p.262]. Using (5.18) and (5.19) yields B(p, p+2) = Γ(p)Γ(p+ 2) Γ(2p+ 2) =p(p+ 1)Γ(p)Γ(p) 2p(2p+ 1)Γ(2p)= 2−2(2p+ 1) + 1 2p+ 1 =1 221 + 2(2 −1) 2(2p+ 1) and, hus, he asse ion.  Bea ing hese p elimina y esul s in mind allows o ackling Lemma 5.20 which pa es he way in o de o p o e Theo em 5.15. Lemma 5.20 Le N∈N≥2,C≥1, and ν∈Nbe gi en. Then, we ge 2ν+1N Y i=2νN+1 iC iC −1≤21/C =C √2. P oo : In he ollowing, he unc ional equa ion (5.18) o he Gamma unc ion Γ(·), i s in e play wi h he Be a unc ion B(·,·) ia (5.19) and Lemma (5.19) applied wi h s= (C−1)/C ∈[0,1) and p= 2νNa e used in o de o ew i e he e m o be es ima ed 2ν+1N Y i=2νN+1 iC iC −1= 2ν+1N Y i=2νN+1 i i−1 C =(2ν+1N)! (2νN)! 2ν+1N Y i=2νN+1 i−1 C!−1 =Γ(2ν+1N+ 1) Γ(2νN+ 1) ·Γ(2νN+ 1 −1 C) Γ(2ν+1N+ 1 −1 C) =B(2νN, 2νN+C−1 C) B(2νN, 2νN+ 1) = 21/C 1 + s(s−1) 2(2p+ 1) +s(s−1)(s−2)(s−3) 2·4·(2p+ 1) ·(2p+ 3) +. . .. Since s∈[0,1), he e m in b acke s is less o equal o one. Hence, he desi ed inequali y is ob ained. 118 DISCRETIZATION AND SAMPLED-DATA SYSTEMS  Fo he special case C= 2, Lemma 5.19 can be eplaced by he ollowing lemma om [123, §7.5] which allows us o es ablish Lemma 5.20 and, hus, o conclude he asse ion o Theo em 5.15 wi hou employing he Be a unc ion B(·,·). Lemma 5.21 Le :R→Rbe an analy ic unc ion ha ing simple ze os a each elemen o he sequence (ai)i∈N⊂R {0}which sa is ies limn→∞ |an|=∞. Fu he mo e, he e exis s a sequence o ci cles (Cm)m∈Nsa is ying he condi ions desc ibed in [123, §7.4]. Then, (z)may be w i en as an in ini e p oduc o he o m (z) = (0)e 0(0)z/ (0) ∞ Y n=1 1−z anez/an. Lemma 5.21 is, e.g. applicable o sin(z)/z, c . [123, p.137]. P oo : [Al e na i e p oo o Lemma 5.20 o C= 2] Using C= 2 and p oceeding analogously o he p oo o Lemma 5.20 yields 2ν+1N Y i=2νN+1 2i 2i−1=Γ(2ν+1N+ 1) Γ(2νN+ 1) ·Γ(2νN+1 2) Γ(2ν+1N+1 2). Nex , we equi e he duplica ion o mula 22z−1Γ(z)Γ(z+ 1/2) = √πΓ(2z) (5.20) which holds o he Gamma unc ion Γ(·) acco ding o [123, p.240] and goes back o Legen- d e.4Using he iden i y gi en by (5.20) o z= 2νN+1 2in he i s and o z= 2ν+1N+1 2 in he second equa ion, leads o Γ(2ν+1N+ 1) Γ(2νN+ 1) ·Γ(2νN+1 2) Γ(2ν+1N+1 2)=Γ(2ν+1N+ 1) Γ(2νN+ 1) 2√π 22·(2νN+1 2)−1Γ(2ν+1N+1 2) =Γ(2ν+1N+ 1)3 Γ(2νN+ 1)2Γ(2ν+2N+ 1) 42ν+1N 42νN = 42νN·(2ν+1N)! (2ν+1N)! (2ν+1N)! (2νN)! (2νN)! (2ν+2N)! . Using he de ini ion o he ac o ial, enables us o expand his exp ession as a p oduc 42νN·(2ν+1N)! (2ν+1N)! (2ν+1N)! (2νN)! (2νN)! (2ν+2N)! = 2νN Y n=1 4(2n)3(2n−1)3 n24n(4n−1) (4n−2) (4n−3) = 2νN Y n=1 4(2n−1)2 (4n−1)(4n−3). Since each ac o o his p oduc is s ic ly g ea e han one, his e m is s ic ly mono- onically inc easing in ν. In addi ion, we a e in e es ed in deducing an uni o mly uppe 4Indeed, (5.20) may be concluded as a co olla y o he mul iplica ion- heo em o Gauss, c . [123, p.240]. 119 SAMPLED-DATA SYSTEMS AND GROWTH CONDITION bound, i.e. an es ima e which does no depend on ν. Hence, ou goal consis s o calcula ing he ollowing in ini e p oduc 2νN Y n=1 4(2n−1)2 (4n−1)(4n−3) < ∞ Y n=1 (4n−2)2 (4n−2)2−1= ∞ Y n=1 1−(π/4)2 π2(n−1 2)2!−1 .(5.21) Applying Lemma 5.21 o (z) = cos(z) yields ( he espec i e assump ions may easily be checked) cos(z) = cos(0)e−sin(0)z/ cos(0) ∞ Y n=1 1−z π(n−1 2)ez/[π(n−1 2)]1−z π(1 2−n)ez/[π(1 2−n)] = ∞ Y n=1 1−z2 π2(n−1 2)2. Plugging π/4 in his ep esen a ion o cos(·) ensu es, since cos(π/4)−1=√2 = 21/C, he desi ed es ima e o C= 2.  In o de o conclude his subsec ion, an elemen a y p oo o (5.17) and, hus, Theo em 5.15 is, again o he special case C= 2, gi en, which does no make use o he Gamma Γ(·) o he Be a unc ion B(·,·). In pa icula , posi i i y o an auxilia y unc ion is shown by using a gumen s wi h espec o i s de i a i es — p esen ing he espec i e echnique u he mo i a es including he ollowing lemma. In o de o a oid echnical di icul ies we s ick o he no a ion in oduced in Theo em 5.15 and he espec i e p oo . Lemma 5.22 Le N∈N≥2,C= 2,σ∈(0,1), and he sequence (kj)j∈N0⊂Nwi h kj:= 2jbe gi en. In addi ion, le γi,k be gi en by (5.10) and C0be de ined as σ−N/2QN i=2 2i 2i−1. Then, he ollowing inequali y holds kjN Y i=2 γi,kj γi,kj−1≤C0(21/2)j=C0√2j.(5.22) P oo : Since 2σi/kj= 2σ1/2kjσ(2i−1)/2kj≤(1 + σ1/kj)σ(2i−1)/2kjholds, he ac o s o he p oduc om he le hand side o (5.22) can be ew i en as γi,kj γi,kj−1=2(1 −σi/kj) (1 + σ1/kj)−2σi/kj≤2(1 −σi/kj) (1 + σ1/kj)(1 −σ(2i−1)/2kj). Hence, aking C= 2 in o accoun , we ob ain analogously o he p oo o (5.16) 2jN Y i=2 γi,kj γi,kj−1≤σ−N/2 2jN Y i=2 1−σi/kj 1−σ(2i−1)/(2kj).(5.23) The emaining po ion o his p oo is subdi ided in o wo pa s: •Fi s ly, he es ima e σ−N/2· 20N Y i=2 1−σi 1−σ(2i−1)/2=σ−N/2· N Y i=2 1−σi 1−σi−1 2≤σ−N/2· N Y i=2 2i 2i−1=C0(5.24) is shown which co e s he asse ion o j= 0. 120 DISCRETIZATION AND SAMPLED-DATA SYSTEMS •Secondly, he g ow h bound 2j+1N Y i=2 1−σi/kj+1 1−σ(2i−1)/2kj+1 / 2jN Y i=2 1−σi/kj 1−σ(2i−1)/2kj≤√2.(5.25) is deduced which ensu es ha inc emen ing he index jin (5.23) leads a mos o a mul iplica ion o he es ima e o jby a ac o o √2. Combining his g ow h bound wi h he es ima e o j= 0 in o de o es ima e he e m on he igh hand o (5.23) implies he asse ion. In o de o ensu e (5.24), we p o e (x) := 1 −2ixi−1 2+ (2i−1)xi≥0∀x∈(0,1) (5.26) o i∈ {2,3, . . . , N}, which implies, since σ∈(0,1), he inequali y (1 −σi)/(1 −σi−1 2)≤2i/(2i−1) (5.27) o i∈ {2,3, . . . , N}. Since (0) = 1, (1) = 0, and ∈ C1([0,1]) showing 0(x)≤0, i= 2,3, . . . , N, o x∈(0,1) implies (x)≥0 o all x∈[0,1]: 0(x) = xi−3 2i(2i−1)√x−2i(i−1/2)≤xi−3 2[i(2i−1) −2i(i−1/2)] = 0 ∀x∈(0,1). Hence, i emains o e i y he claimed g ow h p ope y (5.25) in o de conclude he asse ion, i.e. o j∈N0, 2j+1N Y i=2 1−σi/kj+1 1−σ(2i−1)/2kj+1 / 2jN Y i=2 1−σi/kj 1−σ(2i−1)/2kj =1−σ2/(kj+1) 1−σ3/(2kj+1)· 2jN Y i=2 (1 −σ2i/kj+1 )(1 −σ(2i−1)/kj+1 )(1 −σ(2i−1)/2kj) (1 −σ(4i−1)/2kj+1 )(1 −σ(4i−3)/2kj+1 )(1 −σi/kj) kj+1=2kj =1−σ2/(kj+1) 1−σ3/(2kj+1)· 2jN Y i=2 (1 −σ(2i−1)/kj+1 )2 (1 −σ(4i−1)/2kj+1 )(1 −σ(4i−3)/2kj+1 )≤√2. Using (5.26) o i= 2 and x=σ1/kj+1 ∈(0,1) ensu es, in analogy o (5.27) wi h σ1/kj+1 ins ead o σ, (1−σ2/(kj+1))/(1−σ3/(2kj+1))≤4/3 = (4ν−2)2/[(4ν−2)2−1] o ν= 1. As a p elimina y goal, we wan o es ablish his inequali y also o he o he ac o s in ol ed in he p oduc in conside a ion, i.e. (1 −σ(2i−1)/kj+1 )2 (1 −σ(4i−1)/2kj+1 )(1 −σ(4i−3)/2kj+1 )≤(4i−2)2 (4i−2)2−1 o i∈ {2,3,...,2jN}.(5.28) Using 2kj+1 =kj+2 and subs i u ing (2i−1) by ν, (5.28) is equi alen o 1−4ν2σ(2ν−1)/kj+2 + 2(4ν2−1)σ2ν/kj+2 −4ν2σ(2ν+1)/kj+2 +σ4ν/kj+2 ≥0 o ν= 3,5,...,2j+1N−1. Ins ead o deducing his inequali y di ec ly, we sub ac he posi i e e m ν2σ2(ν−1)/kj+2 (1 −σ1/kj+2 )4 om he le hand side and p o e ha he esul ing exp ession, i.e. 1−ν2σ(ν−1)/kj+1 + 2(ν2−1)σν/kj+1 −ν2σ(ν+1)/kj+1 +σ2ν/kj+1 o ν∈N≥3,(5.29) 121 SAMPLED-DATA SYSTEMS AND GROWTH CONDITION To his end, we exploi he speci ic o m o he limi s g? ν,l and g? ν,l. In pa icula , Lemma 5.25 p o ides g? ν,lC=σ−1 µ1 1−σν+l/µ = lim j→∞ kj/µ Y i=1 1 + 1−σ1/kj 1−σj+l/µ −1 + σ1/kjand g? ν,lC=σ−1 µ1 1−σν+(l+1)/µ = lim j→∞ kj/µ Y i=1 1 + 1−σ1/kj 1−σj+(l+1)/µ −1 + σ1/kj. This allows o elabo a ing he ollowing chain o inequali ies, which esembles he s uc- u e o (5.37). Again, we use k∈(kj)j∈N0 o a su icien ly la ge con ol index jin o de o a oid echnical di icul ies k/µ Y i=1 1 + 1−σ1/k σ1/k −σν+(l+1)/µ ≤ k/µ Y i=1 1 + 1−σ1/k σ1/k −σν+l/µσi/k < k/µ Y i=1 1 + 1−σ1/k σ1/k −σν+l/µ . Howe e , in con as o (5.37), we a e able o deal wi h he e m ep esen ing he co e o his exp ession using an a gumen simila o hose applied o elescoping se ies k/µ Y i=1 1 + 1−σ1/k σ1/k −σν+l/µσi/k = k/µ Y i=1 1−σν+l/µσi/k σ1/k(1 −σν+l/µσ(i−1)/k)=σ−1/µ ·1−σν+(l+1)/µ 1−σν+l/µ . Hence, using hese p elimina y conside a ions yields G(µ)≤ N−1 Y ν=m µ−1 Y l=0 "σ−1 µ1−σν+l+1 µ 1−σν+l µ#1 C = N−1 Y ν=mσ−11−σν+1 1−σν1 C =σ−(N−m)1−σN 1−σm1 C ≤ G(µ). Since G(µ)≤limµ→∞ G(µ) = limµ→∞ G(µ)≤ G(µ), he espec i e limi s coincide wi h he deduced bound. Summa izing hese compu a ions p o ide σ−(N−m)1−σN 1−σm1 C = lim µ→∞ G(µ)≤lim j→∞ kjN Y i=kjm+1 γi,kj γi,kj−1≤lim µ→∞ G(µ) =σ−(N−m)1−σN 1−σm1 C and, hus, concludes (5.35), i.e. he asse ion.  The ollowing ema k jus i ies he simpli ica ion which was made in he p oo o Lemma 5.26 in o de o s eamline he p esen a ion. Rema k 5.27 In he p oo o Lemma 5.26, he sequence (kj)j∈N0⊂Nwas chosen such ha he condi- ion kj/µ ∈Nholds o su icien ly la ge index jwhich can be assumed o an i e a i e e inemen p ocess wi hou loss o gene ali y. We emphasize ha his assump ion is no necessa y in o de o p o e Lemma 5.26 bu allows he eade o concen a e on he es- sen ial s eps wi hou being dis ac ed by echnical de ails. I his condi ion is iola ed, he swi ching index µ?:= kmod µis de ined. Then, he p oduc µ−1 Y l=0 k/µ Y i=1 1 + 1−σ1 k C1−σν+l µσi k−1 + σ1 k! 128 5.3. GROWTH CONDITION om (5.36) is eplaced by µ?−1 Y l=0 dk/µe Y i=1 1 + 1−σ1 k C1−σν+l µσi k−1 + σ1 k!· µ−1 Y l=µ? bk/µc Y i=1 1 + 1−σ1 k C1−σν+l µσi k−1 + σ1 k!, i.e., he in ol ed ac o s a e dis ibu ed such ha he numbe o ac o s is ei he bk/µc o dk/µeand he o al numbe o ac o s is equal o k. The ollowing chain o inequali- ies emains unchanged, only he co esponding index ange has o be adap ed o he se {1,2,...,dk/µe}. The uppe index o he p oduc in he de ini ions o gν,l(k)and gν,l(k) depends on whe he o no he index lis con ained in [0, µ?−1] o he conside ed a - gumen k. Howe e , since we a e only in e es ed in he limi o kapp oaching in ini y, his dis inc ion does no play a ole: looking a he p oo o Lemma 5.25 shows ha he asse ion also holds o a sequence (ni)i∈N0⊂R+ 0sa is ying ni→ ∞, i he exponen is, o each index i, andomly subs i u ed by ei he bnico dnie. The emaining pa o he p oo o Lemma 5.26 does no equi e u he modi ica ions. 5.3 G ow h Condi ion Al hough he es ima e s a ed in Theo em 3.18 is s ic o he whole class o sys ems sa - is ying he assumed con ollabili y condi ion, c . Rema k 3.13 (i), i may be conse a i e o subse s o his class. Fo ins ance, o sampled-da a sys ems go e ned by an o dina y di e en ial equa ion ˙x( ) = g(x( ),˜u( )) he di e ence be ween x(n+ 1) and x(n) is usu- ally o o de O(T) — a con inui y p ope y which is no e lec ed in Assump ion 3.2 and, hus, in he op imiza ion p oblem cha ac e izing ou subop imali y bounds. Neglec ing his speci ic cha ac e is ic leads o e y pessimis ic es ima es o sampling pe iods T end- ing o ze o, c . Theo em 5.15. In o de o exploi he men ioned con inui y p ope ies, he ollowing g ow h condi ion is in oduced. Assump ion 5.28 (G ow h Condi ion) Fo each x0∈X he e exis s an admissible con ol unc ion ux0∈ U =U∞(x0)sa is ying `(xux0(n), ux0(n)) ≤Ln`?(x0)∀n∈N0.(5.38) He e L≥1deno es he g ow h bound which ypically depends on he sampling pe iod T. This sec ion is subdi ided in o wo pa s: •Fi s ly, he g ow h condi ion Assump ion 5.28 is inco po a ed in P oblem 3.8 and Theo em 3.18 is gene alized acco dingly. The impac on ou pe o mance bounds is in es iga ed o an analy ical example. •Secondly, in Subsec ion 5.3.4, Assump ion 5.28 will be e i ied o sampled-da a sys ems go e ned by o dina y di e en ial equa ions. In pa icula , es ima es on he in ol ed g ow h bound La e deduced which depend explici ly on he sampling pe iod Tand, hus, allow o a e inemen p ocess analogously o Sec ion 5.1. We show ha Assump ion 5.28 p o ides emedy o he p oblem which occu ed o e y as sampling: in Sec ion 5.1 an i e a i e e inemen p ocess was ca ied ou wi hou adap ing he disc e e ime con ol ho izon m= 1. The co esponding sequence o subop imali y es ima es di e ged o minus in ini y and, hus, was no applicable in o de o gua an ee a pe o mance bound o s abili y. The in oduced g ow h bound coun e ac s his phenomenon, c . Theo em 5.37 and he ensuing commen s. 129 SAMPLED-DATA SYSTEMS AND GROWTH CONDITION 5.3.1 Exponen ial Con ollabili y A i s , he exponen ially con ollable case wi hou an addi ional e minal weigh is conside ed, i.e. Assump ion 3.2 is supposed o hold wi h a KL0- unc ion o ype (1.11). To his end, γiis de ined as γi:= min (C· i−1 X n=0 σn, i−1 X n=0 Ln)= min C(1 −σi) 1−σ,1−Li 1−L.(5.39) De ini ion (5.39) e lec s bo h, i.e. he exponen ial con ollabili y and he g ow h condi- ion. Hence, being able o sa is y Assump ion 5.28 and, hus, using (5.39) ins ead o (3.17) yields igh e bounds on he s age cos s in P oblem 3.8 and, hus, allows o cha ac e ize he beha io o he sys em o be in es iga ed be e , c . Figu e 5.9. Figu e 5.9: Visualiza ion o he bounds induced by ou con ollabili y assump ion (dashed- do ed line) and ou g ow h condi ion (solid line) o C= 3, σ= 3/5, and L= 5/4. The minimum is ma ked wi h solid ci cles. Al hough he obse a ion poin ed ou in he ollowing lemma does no seem o be excep ionally ema kable, Lemma 5.29 is e y use ul in o de o p o e Theo ems 5.31 and 5.37. Lemma 5.29 (Swi ching index) Le Assump ions 3.2 and 5.28 based on a KL0- unc ion o ype (1.11) wi h o e shoo C≥1, decay a e σ∈(0,1), and g ow h cons an L≥1hold. I he condi ion 1 + L≤C(1 + σ) (5.40) is sa is ied, exac ly one swi ching index i?∈N≥2exis s such ha γi=(Pi−1 n=0 Ln≤CPi−1 n=0 σn o i≤i?, CPi−1 n=0 σn<Pi−1 n=0 Ln o i>i?. I Condi ion (5.40) is iola ed, no such swi ching index i?∈N≥2exis s. 130 GROWTH CONDITION P oo : I Condi ion (5.40) is iola ed, γ2=C(1 + σ) holds which implies Ln≥L≥C+Cσ −1≥Cσ > Cσn o n∈N≥2 and, hus, Pi−1 n=0 Ln≤CPi−1 n=0 σn. Hence, no swi ching index exis s. Suppose ha Condi ion (5.40) is sa is ied. Then, γ2= 1 + Lholds. Fo each index i sa is ying he inequali y i≥C/(1 −σ) i−1 X n=0 Ln≥ i−1 X n=0 1 = i≥C/(1 −σ) = C ∞ X n=0 σn≥C i−1 X n=0 σn holds. Hence, γi=CPi−1 n=0 σn o each i≥C/(1 −σ) which implies he exis ence o a swi ching index. Le i?∈N≥2deno e he smalles swi ching index. Then, Li?> Cσi?and, hus, Li> Cσi o all i≥i?hold, i.e. he inc emen s Lia e la ge han hei coun e pa s Cσi o i≥i?which shows ha no u he swi ching index exis s and, hus, ha he asse ion holds.  Fo ins ance, he swi ching index i?equals 4 o he pa ame e s C= 3, σ= 0.6, and L= 1.25, c . Figu e 5.9. In o de o gene alize Theo em 3.18 o he se ing inco po a ing he g ow h condi ion, he ollowing de ini ion is equi ed. De ini ion 5.30 (Equi alen sequence o equi alen KL0- unc ion) Le (γi)i∈N≥2⊂R≥1be a mono one sequence and de ine γ1:= 1. Then, o gi en op imi- za ion ho izon N≥N≥2, a KL0- unc ion β:R+ 0×N0→R+ 0o ype (1.12) gi en by c0= 1, cn:= γn+1 −γn,n∈ {1,2, . . . , N −1},and cn= 0 (5.41) is called equi alen sequence o equi alen KL0- unc ion o (γi)i∈N≥2. Equa ion (5.39), Lemma 5.29, and De ini ion 5.30 enable us o ex end Theo em 3.18 o he se ing inco po a ing he g ow h condi ion. Indeed, excep o adap ing γi,i= 2,3, . . . , N, Theo em 3.18 main ains exac ly i s shape. Theo em 5.31 Le Assump ion 3.2 wi h KL0- unc ion o ype (1.11) and Assump ion 5.28 hold. Fu he - mo e, le an op imiza ion ho izon N∈N≥2and a con ol ho izon m∈ {1,2, . . . , N −1} be gi en. Then, he op imal alue αN,m =α1 N,m o P oblem 3.8 wi h γi,i∈ {2,3, . . . , N}, de ined acco ding o (5.39) is gi en by Fo mula (3.21). P oo : I Condi ion (5.40) is no sa is ied, he g ow h condi ion does no change γi, i∈ {2,3, . . . , N}and he asse ion is ensu ed by Theo em 3.18. Hence, Condi ion (5.40) is assumed which implies exac ly one swi ching index i?∈N≥2. No e ha he sequence (γi)i∈N≥2sa is ies he assump ions o De ini ion 5.30. Since he alues cn,n≥N, do no con ibu e o γi,i= 2,3, . . . , N, Condi ion (3.3) om Assump ion 3.2 is no needed o n≥Nin o de o deduce P oblem 3.8 and, consequen ly, Theo em 3.18, c . [51]. Hence, using he equi alen KL0- unc ion om De ini ion 5.30 and, hus, se ing cn= 0 o all n≥Ndoes no change P oblem 3.8 in he se ing wi hou an addi ional e minal weigh . As a consequence, ou goal is o ensu e Condi ion (1.13) o cn,n= 0,1, . . . , N −1, om (5.41) which is su icien in o de o gua an ee ha Theo em 131 SAMPLED-DATA SYSTEMS AND GROWTH CONDITION 3.18 p o ides he op imal alue o P oblem 3.8 wi h γi,i∈ {2,3, . . . , N}, om (5.39) and, hus, he asse ion. An equi alen KL0- unc ion is cons uc ed o he gi en op imiza ion ho izon N. I he swi ching index i?sa is ies i?≥N,cn=Lnholds o all n < N and cn= 0 o he wise which co esponds o a KL0- unc ion o ype (1.12) sa is ying (1.13). Hence, he asse ion is ensu ed by Theo em 3.18. Consequen ly, i?< N is assumed. Taking Lemma 5.29, i s p oo , and (5.41) in o accoun yields cn=     Ln,n∈ {0,1, . . . , i?−1}, CPi? i=0 σi−Pi?−1 i=0 Li,n=i?, Cσn,n∈ {i?+ 1, i?+ 2, . . . , N}. In o de o show cn+m≤cncm, h ee cases a e dis inguished. Since c0= 1 holds, n, m > 0 can be assumed. •n+m<i?: Since max{n, m} ≤ n+mholds, cn+m=Ln+m=LnLm=cncmis implied. •n+m=i?: Since n, m > 0 holds, ci?≤Ln+m=LnLm=cncmis ensu ed by he de ini ion o he swi ching index i?. •n+m > i?: Taking he p oo o Lemma 5.29 in o accoun leads o he inequali y ci?+j=Cσi?+j< Li?+j o all j∈N. Hence, n?:= max{n, m} ≥ i?can be assumed. Fu he mo e, m?:= min{n, m}is de ined. Taking he de ini ion o he swi ching index i?in o accoun yields ci?≥Cσi?and, hus, cn?≥Cσn?. Combining his inequali y wi h cm?≥σm?implies cn+m=cn?+m?=Cσn?+m?≤cm?cn?=cmcn, i.e. he asse ion. Hence, Condi ion (1.13) is ensu ed o he equi alen sequence o De ini ion 5.30 which comple es he p oo .  Theo em 5.31 gene alizes he key esul gi en in Theo em 3.18 o he se ing inco po a ing he g ow h condi ion which allows o e lec ing con inui y p ope ies o a conside ed sys em. The p oo o his heo em shows ha one may easily check whe he Theo em 3.18 is applicable in o de o de e mine he op imal alue o P oblem 3.8 by cons uc ing an equi alen KL0- unc ion o ype (1.12) and e i ying Condi ion (1.13). This applica ion o ini e ime con ollabili y allows us o ans e he esul s wi h espec o KL0- unc ions o ype (1.12) sa is ying (1.13) deduced in he p e ious chap e s o exponen ially con ollable ones which gi es u he eason o he pe o med, comple e symme y and mono onici y analysis o his se ing. Fu he mo e, he cons uc ion o equi alen sequences is no only a heo e ical concep bu may be used in o de o inco po a e u he es ima es in ou con ollabili y assump ion. This will be shown in he ensuing Sec ion 5.4 in de ail. An al e na i e p oo o Theo em 5.31 is gi en in Subsec ion 5.3.5. 5.3.2 Fini e Time Con ollabili y The con ibu ion o his subsec ion is wo old. On he one hand, he coun e pa o Theo em 5.31 is es ablished o con ol sys ems sa is ying Assump ion 3.2 based on a KL0- unc ion o ype (1.12) wi h c0≥1, c2 1≥c2, and cn= 0 o all n∈N≥3, i.e. ini e 132 GROWTH CONDITION ime con ollabili y in a mos h ee s eps such ha (1.13) holds. On he o he hand, we show ha a u he gene aliza ion o a bi a y KL0- unc ions sa is ying (1.13) is no possible and commen on a emedy which wo ks in a majo i y o cases. An example dealing wi h ini e ime con ollabili y will be in es iga ed in he ensuing subsec ion. We begin wi h ex ending ou esul s conce ning he g ow h condi ion o an impo an subclass o ini e ime con ollable sys ems. Theo em 5.32 Le Assump ion 3.2 based on a KL0- unc ion o ype (1.12) sa is ying (1.13) wi h cn= 0 o all n∈N≥3and he g ow h condi ion, i.e. Assump ion 5.28, hold. Fu he mo e, le an op imiza ion ho izon N∈N≥2and a con ol ho izon m∈ {1,2, . . . , N −1}be gi en. Then, he op imal alue αN,m =α1 N,m o P oblem 3.8 wi h γi,i∈ {2,3, . . . , N −1}, de ined acco ding o (5.39) is gi en by Fo mula (3.18). P oo : Since he se ing wi hou inco po a ing an addi ional weigh ωon he inal e m in he eceding ho izon cos Func ional (2.4) is conside ed, he dis ibu ion o γ2on c0 and c1does no play a ole. Hence, c0= 1 and c1=γ2−1 can be assumed because his choice maximizes he ange in which c2has o be loca ed acco ding o (1.13) (c2≤c2 1). Fu he mo e, le , wi hou loss o gene ali y, γ2= 1 + L < c0+c1hold. O he wise, he g ow h condi ion has no impac on γi,i= 2,3, . . . , N −1, and, as a consequence, Theo em 3.18 ensu es he asse ion. Then, since cn= 0 o n∈N≥3, exac ly one swi ching index i?∈N≥2exis s such ha γi=(Pi−1 n=0 Ln≤Pi−1 n=0 cn o i≤i?, Pi−1 n=0 cn<Pi−1 n=0 Ln o i>i? holds, c . Lemma 5.29. This enables us o de ine an equi alen KL0- unc ion o ype (1.12) which exhibi s p ecisely he same γi,i= 2,3, . . . , N, i.e. cn:= Ln o n≤i?−1, ci?=γi?+1 −γi?, and cn= 0 o n > i?. Hence, e i ying (1.13) o he espec i e sequence (cn)n∈N0and applying Theo em 3.18 comple es he p oo . Howe e , since ci?≤Li?holds, (1.13) is gua an eed.  We con inue wi h he men ioned nega i e esul , which shows ha he asse ion o he p e ious heo em is s ic wi h espec o he class o KL0- unc ions conside ed. Example 5.33 Le he KL0- unc ion β(·,·)o ype (1.12) be de ined by c0:= 1,c1:= 10,c2:= 10, c3:= 100, and cn= 0 o all n∈N≥4. No e ha β(·,·)exhibi s linea i y in i s i s a gumen and sa is ies (1.13). Fu he mo e, le Assump ion 3.2 wi h β(·,·)and he g ow h condi ion wi h g ow h bound L= 5 hold. Then, we ob ain γ2= 1 + L= 6 <11 = c0+c1, γ3=c0+c1+c2= 21 <31 = 1 + L+L2, γ4=c0+c1+c2+c3= 121 <156 = 1 + L+L2+L3, and γi=γ4 o i≥5. We wan o es ablish Theo em 5.31 o op imiza ion ho izon N= 5 and m= 1. To his end, we cons uc he equi alen KL0- unc ion o ype (1.12) acco ding o De ini ion 5.30 which is gi en by c0:= 1,c1:= 5,c2:= 15,c3:= 100, and 133 SAMPLED-DATA SYSTEMS AND GROWTH CONDITION cn= 0 o n∈N≥4. Bu , since c1c2= 75 <100 = c3holds, Condi ion (1.13) is iola ed. Hence, an assump ion o Theo em 3.18 is no sa is ied and, hus, his heo em canno be applied in o de o conclude ha he op imal alue o P oblem 3.8 is gi en by Fo mula (3.21). In o de o u he in es iga e his issue, he al e na i e p oo o Theo em 5.31 is conside ed, c . Sec ion 5.3.5 below. This p oo shows ha Condi ion (5.48) has o be sa is ied, i.e. (γ6−j−1) 5−j Y i=2 (γi−1) −(γ6−j−γ5−j) 5−j Y i=2 γi≥0 o j= 1,2,3. E alua ing he le hand side yields 1.440.000 (j= 1), −600 (j= 2), and 10 (j= 3). Hence, his condi ion is iola ed o j= 2. Consequen ly, he solu ion o he o iginal P oblem 3.8 and i s elaxed coun e pa P oblem 3.17 do no coincide. In conclusion, Condi ion (5.48) has o be checked in o de o decide which cons ain s ha e o be aken in o accoun in he co esponding op imiza ion P oblem 3.8. We poin ou ha Theo em 3.18 ne e heless p o ides aluable in o ma ion because he espec i e o mula may s ill be used as a lowe bound o he subop imali y index o P oblem 3.8. In o de o conclude his subsec ion, ano he example iola ing (1.13) and (5.48) o N= 6 and m= 1 is gi en in Figu e 5.10. In pa icula , his example which is based on a non mono one KL0- unc ion sa is ying (1.13) exhibi s mo e han one swi ching index i?, c . Lemma 5.29. Figu e 5.10: Visualiza ion o he bounds induced by Assump ion 3.2 (dash-do ed line) and he g ow h condi ion (solid line) o c0= 1, c1= 4, c2= 8, c3= 5, c4= 10, c5= 40, ci= 0 o i∈N≥6, and g ow h bound L= 2. The minimum is ma ked wi h a solid ci cle each ime. 5.3.3 Analy ical Example He e, we ocus on quan i a i e e ec s caused by inco po a ing Assump ion 5.28 in P oblem 3.10 and, hus, in ou subop imali y analysis. Since he o e shoo Chas been p o en o be 134 GROWTH CONDITION he decisi e pa ame e in his con ex , c . Sec ion 4.1 and [39, sec ion 6], we in es iga e i s sensi i i y o changes in he g ow h bound L. To his end, we ix he decay a e σ= 0.7. Then, ou goal is o de e mine he maximal o e shoo Cwhich allows o gua an ee ou s abili y condi ion αN,1≥0 o he whole class o sys ems sa is ying Assump ion 3.2 o a gi en op imiza ion ho izon N. Table 5.2 shows esul s o wo ex emal alues o L, i.e. comple ely neglec ing ou g ow h condi ion in compa ison o inco po a ing i wi h g ow h cons an L= 1. N C such ha αN,1≥0 (L=∞)Csuch ha αN,1≥0 (L= 1) inc ease (%) 4 1.4028 1.5790 12.56 6 1.6130 2.0397 26.45 8 1.8189 2.5462 39.98 12 2.2208 3.6489 64.30 16 2.6081 4.8128 84.53 24 3.3409 7.1938 115.33 Table 5.2: In his able we gi e he maximal o e shoo Csuch ha he op imal alue αN,1o P oblem 3.8 is ensu ed o be posi i e in dependence on he op imiza ion ho izon N o he se ing wi h and wi hou ou g ow h condi ion. We chose L= 1 in o de o de e mine he maximal inc ease ealizable by Assump ion 5.28. Figu e 5.11 illus a es ha using Assump ion 5.28 allows o signi ican ly la ge alues o C. Fu he mo e, his igu e shows ha hese indings emain basically he same o subop imali y es ima es αN,1> α > 0, i.e. i we aim a ensu ing ce ain pe o mance speci ica ions o ou eceding ho izon eedback. Hence, Assump ion 5.28 allows us o calcula e igh e bounds and, hus, cha ac e izes he beha io o he closed loop mo e accu a ely. In pa icula , we like o poin ou he cu e o N= 8 in Figu e 5.11. He e, he kink ma ks he uppe bounda y o he ange in o which inco po a ing he g ow h condi ion con ibu es posi i ely o posing he op imiza ion p oblem and, hus, o deducing s abili y ma gins. The nex example demons a es he in e play be ween he g ow h condi ion and e - minal weigh s. To his end, he ollowing p oposi ion is needed. P oposi ion 5.34 Le Assump ion 3.2 based on a KL0- unc ion o ype (1.12) sa is ying (1.13) wi h cn= 0 o all n∈N≥2and he g ow h condi ion, i.e. Assump ion 5.28, hold. Fu he mo e, le an op imiza ion ho izon N∈N≥2, a con ol ho izon m∈ {1,2, . . . , N −1}, and a e minal weigh ω≥1be gi en. Then, he op imal alue αω N,m o P oblem 3.8 wi h γi, i∈ {2,3, . . . , N −1}, de ined acco ding o (5.39) is gi en by Fo mula (3.18). P oo : Wi hou loss o gene ali y, 1 + ωL < C +ωCσ is assumed. O he wise he g ow h condi ion does no change γi,i∈ {2,3, . . . , N}, and, hus, P oblem 3.8. Then, an equi alen KL0- unc ion can be cons uc ed such ha γi,i∈ {2,3, . . . , N}, emain unchanged in compa ison o hose esul ing om Assump ions 3.2 and 5.28. Since cn= 0, n∈N≥2, he asse ion can be concluded analogously o he p oo o Theo em 5.32.  135 SAMPLED-DATA SYSTEMS AND GROWTH CONDITION Figu e 5.11: Illus a ion o he co esponding maximal easible o e shoo Cwhich ensu es asymp o ic s abili y in dependence on he g ow h bound L. The la ge he op imiza ion ho izon N, he la ge a e he esul ing bounds. The solid line s ands o N= 8, N= 10 is ep esen ed by he dashed line, and he dash-do ed line illus a es he in e play o he conside ed pa ame e s o N= 12. In o de o simpli y he ollowing calcula ions, we ocus on RHC wi h m= 1. Neglec ing he g ow h condi ion leads o γ2=c0+ωc1and γi=c0+c1 o all i≥3. Hence, o op imiza ion ho izon N≥3, Theo em 3.18 yields αω 2,1= (c0+ωc1)(1 + ω−c0−c1ω)/ω, αω N,1=QN i=2 γi−(γ2−ω)QN i=3(γi−1)γN QN i=2 γi−(γ2−ω)QN i=3(γi−1) N>2 =(c0+ωc1)(c0+c1)N−2−(c0+ωc1−ω)(c0+c1−1)N−2(c0+c1) (c0+ωc1)(c0+c1)N−2−(c0+ωc1−ω)(c0+c1−1)N−2. We choose c0:= 3 and c1:= 2 and de e mine he minimal ho izon which gua an ees ou s abili y condi ion αω N,1≥0 o an app op ia ely chosen inal weigh ω. This is, in u n, equi alen o (c0+ωc1)(c0+c1)N−3≥(c0+ωc1−ω)(c0+c1−1)N−2 o N≥3 and no possible o N= 2 because c0≥1, c1≥1, and c0+c1>2. Inse ing he coe icien s c0,c1in he conside ed inequali y yields he necessa y condi ion N≥3 + ln 12 + 4ω 3+2ω/ln 5 4≥3 + ln 2/ln(5/4) ≈6.106 o all ω≥1. Hence, he minimal s abilizing ho izon esul ing om Theo em 3.18 has o necessa ily sa is y N≥7. Wi hou adding a inal weigh we ob ain N≥9, c . Figu e 5.12 on he le . Fu he mo e, he deduced inequali y allows us o calcula e he minimal addi ional weigh on he inal e m needed in o de o educe his bound on he op imiza ion ho izon 136 GROWTH CONDITION Figu e 5.12: In es iga ion o he in luence o inco po a ing an addi ional weigh on he inal e m in (2.4) on he minimal s abilizing ho izon leng h N. On he igh , he impac o ou g ow h condi ion on his example is depic ed, in addi ion. N. Le Nbe equal o 7. Then, using a inal weigh ω≥1197/226 ≈5.296 leads o αω 7,1≥0 and, hus, ensu es ha he desi ed elaxed Lyapuno inequali y ollows om Theo em 3.18, c . Table 5.3. As al eady men ioned, gua an eeing s abili y ia Theo em 3.18 is no possible o smalle N. minimal ωsuch ha αω N,1≥0 Nwi hou g ow h condi ion wi h g ow h condi ion 3 - - 15/2 7.500 4 - - 55/14 3.929 5 - - 195/86 2.267 6 - - 545/516 1.056 7 1197/226 5.296 1 1.000 8 971/718 1.352 1 1.000 9 1 1.000 1 1.000 Table 5.3: The able shows he inal weigh s needed in o de o ensu e αω N,1 om Theo em 3.18 based on Assump ion 3.2 wi h KL0- unc ion o ype (1.12) gi en by c0:= 3, c1:= 2, and ci= 0 o all i∈N≥2in i s i s wo columns, which con ain he exac and he app oxima ed alues o ω.N= 7 has u ned ou o be he minimal s abilizing ho izon. Taking, in addi ion, Assump ion 5.28 in o accoun and, hus, using Theo em 5.31 allows o gua an eeing s abili y o signi ican ly smalle op imiza ion ho izons N, e.g. choosing he inal weigh ω= 7.5 allows us o educe he ho izon o N= 3, c . he hi d and o h column. We con inue wi h inco po a ing ou g ow h condi ion in he conside ed se ing. To his end, suppose ha Assump ion 5.28 holds wi h g ow h bound L= 1.2. Since he asse ion o Theo em 5.32 also holds o ini e ime con ollabili y in a mos wo s eps in 137