Feedforward and Feedback Tracking Control of Diffusion Convection Reaction Systems using Summability Methods

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1 Feedforward and Feedback Tracking Conrol of Diffusion Convecion Reacion Sysems using Summabiliy Mehods Von der Fakulä Maschinenbau der Universiä Sugar zur Erlangung der Würde eines Dokor Ingenieurs (Dr. Ing.) genehmige Abhandlung Vorgeleg von Thomas Meurer geboren in Gaildorf Haupbericher: Mibericher: Prof. Dr. Ing. Dr.h.c. M. Zeiz Prof. Dr. P. Rouchon Tag der mündlichen Prüfung: 24. Juni 25 Insiu für Sysemdynamik und Regelungsechnik der Universiä Sugar 25

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3 III Vorwor Die vorliegende Arbei ensand in den Jahren 2 bis 25 während meiner Täigkei als wissenschaflicher Miarbeier am Insiu für Sysemdynamik und Regelungsechnik der Universiä Sugar. Die durchgeführen Unersuchungen wurden u.a. von der Deuschen Forschungsgemeinschaf (DFG) im Rahmen des Projeks Flachheisbasiere Regelung von Sysemen mi vereilen Parameern geförder. Mein besonderer Dank gil Herrn Prof. Dr. Ing. Dr.h.c. M. Zeiz für die ausgezeichnee und engagiere Bereuung bei der Durchführung der Arbei. Ihm gil auch mein herzlicher Dank für die seige und langjährige Förderung sei der Sudienzei, sein allzei offenes Ohr sowie die eingeräumen fachlichen Freiräume. Sehr gefreu habe ich mich über die Übernahme des Miberichs durch Herrn Prof. Dr. P. Rouchon von der École des Mines de Paris, einem der Mibegründer der Theorie der flachen Syseme und deren Erweierung auf den vereil paramerischen Fall. Ihm danke ich für sein Ineresse, die schnelle Durchsich des Manuskrips sowie seine konsrukiven Hinweise. Ausserdem bedanke ich mich bei dem Insiusleier, Herrn Prof. Dr. Ing. Dr.h.c.mul. E.D. Gilles, für die günsigen Arbeisbedingungen in seinem Insiu. Den Kolleginnen und Kollegen am Insiu danke ich für das gue Arbeisklima und see Hilfsbereischaf. Besonders danke ich Mahias Bizer, Knu Graichen und Reo Köhler für die sehr gue Zusammenarbei in der Gruppe von Herrn Prof. Zeiz. Darüberhinaus bedanke ich mich bei Jens Becker, Mahias Bold, Alexander Schaum, Michael Schenk und ganz speziell bei Marc Oliver Wagner, die durch ihre Sudien und Diplomarbeien wervolle Beiräge zu der vorliegenden Themaik geleise haben. Schliesslich, aber nich zulez, danke ich meinen Elern für ihre Unersüzung und Barbara für ihren Rückhal und ihr Versändnis. Sugar, im Juli 25 Thomas Meurer

4 IV Für meine Elern

5 V Conens Lis of Symbols Absrac German summary / Deusche Kurzfassung VIII XI XIII Inroducion. Conrol of sysems governed by PDEs Flaness based mehods for PDEs Goals of he hesis Srucure of he hesis Formal power series parameerizaion of boundary conrolled DCR sysems 8 2. Scalar DCR equaions wih boundary inpu The linear hea equaion The linear diffusion convecion equaion Scalar DCR equaions wih polynomial nonlineariies MIMO DCR equaions wih boundary inpus General DCR equaions A ubular reacor model Formal power series parameerizabiliy, flaness and conrollabiliy of DCR sysems Conclusions Summabiliy mehods and sequence ransformaions Preliminaries Asympoic power series and Gevrey funcions Formal power series on Banach spaces

6 VI CONTENTS Asympoic expansions Gevrey asympoics Summaion of power series General consideraions Summaion in a direcion Towards numerical implemenaion and convergence acceleraion The (N, ξ) approximae k sum Numerical example: he divergen Euler series General sequence ransformaions General consideraions Examples of linear and nonlinear sequence ransformaions Numerical example: he divergen Euler series Conclusions Moion planning and feedforward conrol Enhanced moion planning in Gevrey classes Muliple ransiions beween saionary profiles Muliple ransiions beween arbirary profiles Power series soluions and summabiliy mehods for PDEs Scalar DCR equaions wih boundary inpu Enhanced moion planning for he linear hea equaion Convergence acceleraion for he linear diffusion convecion equaion Summary MIMO DCR equaions wih boundary inpus Feedforward conrol design for general DCR sysems Summaion of series resuling from DCR sysems Choice of summaion parameers in (N, ξ) approximae k sum Feedforward conrol design for he ubular reacor model Conclusions Flaness based feedback boundary racking conrol of DCR sysems Scalar DCR equaions wih boundary inpu Feedback racking conrol design for he linear hea equaion

7 CONTENTS VII 5... Finie dimensional design model via formal power series and summaion mehods Tracking conrol wih observer Sabiliy of he racking conrol scheme Simulaion resuls Nonlinear scalar DCR equaions MIMO DCR equaions wih boundary inpus Feedback racking conrol design for general DCR sysems Finie dimensional design model via formal power series and summaion mehods Flaness based racking conrol wih observer Spaial profile esimaion The case M < K Feedback conrol for he ubular reacor model Nonlinear conroller normal form Simulaion resuls for flaness based racking conrol wih observer Simulaion resuls for feedforward conrol and oupu feedback Conclusions Conclusions and oulook A Mahemaical glossary 3 A. Analysis A.2 Complex analysis B Smooh funcions wih compac suppor 5 C An algorihm for he efficien evaluaion of nonlinear differenial recurrence relaions 7 D Lemmas and parameers for he non isohermal ubular reacor model 9 D. Lemma on recursion D.2 Model and simulaion parameers Bibliography 2 Inernal repors, suden and diploma heses 28

8 VIII Lis of Symbols The following lis only conains symbols ha are used coninuously hroughou he ex. Local symbols are no lised. Variables ζ() u() u d () u() u d () u S x(z, ) x d (z, ) ˆx(z, ) vecor of ime variable funcions scalar inpu from dimensional funcional space U feedforward boundary conrol inpu vecor from M dimensional funcional space U M vecor of feedforward boundary conrols M vecor of saionary inpu values sae from dimensional funcional space X desired sae profile formal power series ˆx(z, ) = n= ˆx n()z n ˆx n () ˆx n () = [ˆx (), ˆx (),..., ˆx n ()] T for n N x(z, ) sae vecor from K dimensional funcional space X [, ] x d (z, ) ˆx(z, ) ˆx 2N 2 K vecor of desired sae profiles formal power series ˆx(z, ) = n= ˆx n()z n sae vecor ˆx 2N 2 = [x, x 2,..., x 2N 2 ] T of he SISO design model in series coefficiens ˆx n () ˆx n () = [ˆx T (), ˆx T 2 (),..., ˆx T n()] T for n N ˆx 2N 2 sae vecor ˆx 2N 2 = [ˆx T, ˆx T 2,..., ˆx T 2N 2] T of he MIMO design model in series coefficiens

9 Lis of Symbols IX x S (z) y() y d () y() y d () y S K vecor of saionary profiles scalar parameerizing funcion desired rajecory for parameerizing funcion y() M vecor of parameerizing funcions M vecor of desired rajecories for parameerizing funcion y() M vecor of saionary values of he parameerizing funcion Scalars K dimension of sae vecor x(z, ) M dimension of inpu vecor u(z, ) p i,n+ q i,n+ parameers for he dynamics of he racking error parameers for he dynamics of he observer error Formal power series and summaion mehods A(Ω, E) se of all funcions x H(Ω, E) having asympoic expansion ˆx(z) AI (q AI) p AI (z) Aiken s 2 formula A (k) (S, E) A α (Ω, E) E E{z} k,d E[[z]] se of all funcions x, holomorphic, of exponenial growh a mos k in S, and coninuous a he origin se of all funcions x H(Ω, E) having asympoic expansion ˆx(z) of order α Banach space equipped wih a norm over he field of complex numbers C se of all k summable power series ˆx(z) in direcion d se of all formal power series ˆx in z C E[[z]] α se of all formal power series ˆx in z C of Gevrey order α E{z} G M,R,α (Ω) H(Ω, E) L k S P se of all convergen power series ˆx in z C Gevrey class of order α in Ω, M, R posiive consans se of all E valued funcions holomorphic in a secorial region Ω Laplace inegral of order k parial summaion

10 X Lis of Symbols S k,d S N,ξ k k summaion in direcion d (N, ξ) approximae k summaion δ (q δ) p δ (ζ) Weniger s δ algorihm ˆx(z) formal power series (in z C) wih coefficiens {ˆx n } n from a Banach space E Abbreviaions Da Le P e 2DOF BC BVP CAS DCR DCRE DPS FPSP GST IBVP IC IVP MIMO MOL ODE PDE SISO I K O K Damköhler number Lewis number Pecle number wo degree of freedom boundary condiion boundary value problem compuer algebra sysem diffusion convecion reacion diffusion convecion reacion equaion disribued parameer sysem formal power series parameerizabiliy generalized sequence ransformaion iniial boundary value problem iniial condiion iniial value problem muliple inpu muliple oupu mehod of lines ordinary differenial equaion parial differenial equaion single inpu single oupu K K uni marix K K zero marix

11 XI Absrac Diffusion convecion reacion (DCR) processes occur in a large variey in chemical and biochemical engineering such as fixed bed ubular reacors for producion or degradaion. These sysems ypically exhibi complex dynamical behavior, which in paricular complicaes conrol design. Thereby, advanced conrol sraegies are required due o he increasing demands on produc qualiy and producion efficiency. Since modeling of hese processes usually leads o disribued parameer sysems (DPSs), conrol design is eiher based on early or lae lumping approaches. In he early lumping approach, he sysem is approximaed firs and conrol design is performed based on he lumped model. This ofen leads o high dimensional and complex feedback conrol srucures. On he oher hand in he lae lumping approach, conrol synhesis is based on he infinie dimensional process model. Alhough heoreically appealing, his approach may lead o non implemenable conrol laws and is mainly resriced o linear sysems. Furhermore, classical early and lae lumping approaches ypically address sabilizaion while neglecing he racking conrol problem. For is soluion, differenial flaness is a well-esablished ool for finie dimensional nonlinear sysems wih recen exensions o moion planning and feedforward conrol design for infinie dimensional sysems governed by parial differenial equaions (PDEs). Here, he propery of parameerizabiliy can be idenified as he consiuive principle wih he ype of PDE deermining he parameerizaion approach. In paricular for linear and cerain nonlinear DCR equaions wih boundary inpus, power series in he spaial coordinae are applied o deermine differenial recursions for he ime variable series coefficiens. The soluion of hese recursions can be expressed in erms of a parameerizing funcion (corresponding o he fla oupu) and is ime derivaives up o infinie order. Once his parameerizaion is obained, he respecive feedforward conrols, i.e. he boundary inpu which ensures racking of an appropriae rajecory for he parameerizing funcion, follows direcly from he evaluaion of he inhomogeneous boundary condiions. Neverheless, he necessary proof of uniform convergence is direcly relaed o he problem of moion planning. Thereby several drawbacks emerge. A firs, uniform convergence resrics moion planning o cerain smooh funcions having compac suppor. Secondly, no informaion on he respecive speed of convergence, which e.g. migh be slow for convecion dominaed sysems, can be exraced from he convergence proof. Finally for nonlinear problems, he resuling convergence condiions relae moion planning and process parameers, which furher consrains he applicabiliy of he approach.

12 XII Absrac In order o overcome hese limiaions, his hesis considers a combinaion of formal power series and suiable summaion mehods, whereby he noion formal denoes he fac ha he radius of convergence migh well be equal o zero. This is in paricular focused on enhanced moion planning and enlarged applicabiliy of he feedforward conrol design using formal power series. Therefore, he noion of formal power series parameerizabiliy (FPSP) is inroduced, which formally allows o overcome he resricion o uniformly convergen power series. The underlying algebraic srucure of he considered space of formal power series is hereby deermined by summabiliy mehods wih he focus on k summabiliy. This in paricular allows o deal wih boh uniformly convergen as well as cerain divergen series. Alhough various resuls on k summabiliy of formal soluions o PDEs are available, he heory is far from complee and is resriced o he Cauchy problem on unbounded domains. Furhermore, nonlinear problems ypically allow only he deerminaion of a finie number of coefficiens for he formal series. Hence, cerain modificaions of he summabiliy approach are required which lead o he consideraion of generalized sequence ransformaions (GSTs). These mehods provide highly accurae approximaions of he sum of a given slowly converging or possibly diverging series based on only a finie number of series coefficiens. Wihin his framework, a varian of k summaion is inroduced, namely he (N, ξ) approximae k summaion, which approximaely combines he advanages of k summaion wih he demands imposed from pracical problems. This novel combinaion of FPSP and GSTs grealy exends he applicabiliy of he formal power series approach for feedforward racking conrol design, as is illusraed in various examples including divergen soluions o he linear hea equaion, slowly converging series in case of he linear diffusion convecion equaion, and he nonlinear model of a ubular reacor wihin several branches of operaion. Since pure feedforward conrol is only applicable for he nominal case wih perfecly known and sable plan, feedback conrol is required o accoun for insabiliy, model errors, and/or exogenous disurbances. Therefore i is shown, ha a re inerpreaion of FPSP allows o deermine a finie dimensional inherenly fla approximaion of he governing infinie dimensional DPS. This in paricular allows o adop sandard echniques from flaness based racking conrol design wih observer. In addiion, i is shown ha he esimaed daa from he observer can be uilized for spaial profile esimaion e.g. for monioring purposes. Thereby, he range of applicabiliy of his feedback conrol approach can be increased by considering he (N, ξ) approximae k summaion, which is illusraed in numerical simulaions for racking conrol of he linear hea equaion and he nonlinear model of a non isohermal ubular reacor. In summary, formal power series in conjuncion wih sophisicaed summaion mehods provide a sysemaic analysis and design approach suiable for numerical evaluaion and compuer aided implemenaion for cerain pracically relevan nonlinear parabolic sysems of second order PDEs wih boundary inpus.

13 XIII Deusche Kurzfassung Seuerung und Folgeregelung von Diffusions Konvekions Reakions Sysemen uner Verwendung von Summaionsmehoden Syseme, die durch Diffusions Konvekions Reakions Gleichungen (DKRGn) beschrieben werden, reen in großer Vielfal beispielsweise im Bereich der Verfahrensechnik auf. Typische Modellprozesse umfassen Rohr und Fesbereakoren mi komplexem dynamischen Verhalen. Die Analyse dieser parabolischen vereil paramerischen Syseme wird durch deren unendlich dimensionalen Charaker besimm, der sich in den ensprechenden Mehoden der reinen, angewanden und numerischen Mahemaik widerspiegel. Aus Ingenieurssich gil im Allgemeinen das spezielle Ineresse der Unersuchung des dynamischen Eingangs /Ausgangsverhalens. Andererseis und wie auch in dieser Arbei gezeig wird, eröffne die Besimmung des inversen Sysems weierführende Einsichen in die Sysemdynamik, die zum modellbasieren Enwurf von Seuerungen und Regelungen genuz werden können. Im Allgemeinen beruh der modellbasiere Regelungsenwurf für Syseme, die durch parielle Differenzialgleichungen (PDGLn) beschrieben werden, auf zwei Konzepen: Enwurf einer Regelung ggf. mi Beobacher basierend auf einer geeigneen Approximaion der Modellgleichungen ( early lumping ) beispielsweise miels Differenzenverfahren, Finie Elemen Mehoden, modalen Ansäzen (Georgakis e al., 977a,b,c; Balas, 978) oder Projekionsverfahren (Ray, 98; Awell and King, 2; Chrisofides, 2). Enwurf einer Regelung ggf. mi Beobacher direk anhand der vereil paramerischen Modelle ( lae lumping ) miels funkionalanalyischer Mehoden (Faorini, 968; Nambu, 979, 984; Lasiecka and Triggiani, 983; Curain and Zwar, 995) und anschliessede Approximaion der unendlich dimensionalen Regelung. Beide Ansäze weisen gewisse Nacheile auf. Im Fall des early lumping sell sich insbesondere die Frage nach der Konvergenz, d.h. sreb die Lösung des reduzieren Modells gegen

14 XIV German summary / Deusche Kurzfassung die Lösung des Originalmodells. Hieraus ergeben sich meis hoch dimensionale Regelgeseze, die die Anwendbarkei deulich einschränken. Andererseis muss der mahemaisch exake, im Allgemeinen auf unendlich dimensionale Regelgeseze führende lae lumping Enwurf zur Realisierung und Implemenierung geeigne approximier werden. Weierhin is diese Mehodik mi wenigen Ausnahmen auf lineare Syseme beschränk. Insbesondere is zu bemerken, dass mi Ausnahme der Arbeien zur opimalen Seuerung und Regelung (Bukovsky, 969; Lions, 97; Ray, 98; Fursikov, 999), deren Ergebnisse jedoch mi einem hohen numerischen Aufwand einhergehen, meis die Sabilisierungsaufgabe jedoch nich das Trajekorien Folgeproblem berache wird. Zu dessen Lösung ha sich bei nichlinearen endlich dimensionalen Sysemen die Eigenschaf der differenziellen Flachhei als eine geeignee Basis zur Trajekorienplanung sowie zum sysemaischen Enwurf von Seuerungen und Folgeregelungen mi Beobacher erwiesen (Fliess e al., 995; Rohfuß e al., 997; Rohfuß, 997). Akuelle Arbeien befassen sich insbesondere mi der Erweierung der flachheisbasieren Mehoden auf Syseme mi vereilen Parameern (SVPn), wobei speziell die Trajekorienplanung und der Seuerungsenwurf im Mielpunk der Unersuchungen sehen (Rudolph, 23a). Hierbei werden bei parabolischen PDGLn Poenzreihenansäze verwende, die die Paramerierung der Zusands und Eingangsgrößen durch eine paramerierende Funkion (ensprechend dem flachen Ausgang) und deren Zeiableiungen bis zur unendlichen Ordnung ermöglichen siehe z.b. Fliess e al. (997); Marin e al. (997); Laroche e al. (998); Fliess e al. (998a,b); Laroche e al. (2); Lynch and Rudolph (22); Rudolph (23a) und dorige Referenzen. In einer gewissen Analogie zum early lumping, zeig sich auch hier das Problem der Konvergenz, da durch eine geeignee Wahl der Trajekorien für die paramerierende Funkion, gleichmäßige Konvergenz des Reihenansazes sichergesell werden muss. Hieraus ergeben sich einige Einschränkungen: Gleichmäßige Konvergenz erzwing die Wahl von Trajekorien aus gewissen Gevrey Klassen, d.h. glaen Funkionen, deren Ableiungen besimmen Wachsumseigenschafen unerliegen. Anderseis zeigen numerische Ergebnisse (Laroche e al., 2), dass diese Einschränkung gelocker werden kann, was jedoch auf punkweise bzw. möglicherweise divergene Reihen führ, die geeigne summier werden müssen. Der Konvergenznachweis liefer keine Informaion über die Konvergenzgeschwindigkei der Reihe, die beispielsweise für konvekionsdominane Syseme wie die lineare Diffusions Konvekions Gleichung sehr klein sein kann (siehe Kapiel 2..2). Für nichlineare DKRGn führ der Konvergenznachweis neben Bedingungen an die Trajekorien der paramerierenden Funkion zu weieren Forderungen an die Sysemparameer, die die Anwendbarkei des Poenzreihenansazes weier einschränken (siehe Kapiel 2..3 und 2.2.2). Diese Beobachungen sellen den Ausgangspunk dieser Arbei dar, in der Lösungsansäze unersuch werden, um die genannen Einschränkungen von Poenzreihen zu überwinden. Au-

15 XV ßerdem wird die weireichende Anwendbarkei von Poenzreihen zum Seuerungs und Folgeregelungsenwurf für nichlineare Diffusions Konvekions Reakions Syseme mi Randeingriffen erläuer. Speziell wird gezeig, dass formale Poenzreihen in Verbindung mi geeigneen Mehoden zur Konvergenzbeschleunigung und Summaion divergener Reihen ein mahemaisches Gerüs zum Seuerungsenwurf für DKRGn mi Randeingriffen darsellen, formale Poenzreihen zum flachheisbasieren Folgeregelungsenwurf mi Beobacher und Profilschäzung geeigne sind, der Seuerungs und Regelungsenwurf basierend auf formalen Poenzreihen rechnergesüz miels Compuer Algebra Sysemen durchgeführ werden kann, komplexe prakische Problemsellungen miels des vorgeschlagenen Ansazes behandel werden können. Zur Illusraion der genannen Punke wird der Einfachhei halber zunächs die lineare Wärmeleiungsgleichung behandel. Mi diesen Ergebnissen wird die Erweierung der Mehodiken anhand des Modells eines nich isohermen Fesbereakors besehend aus zwei gekoppelen nichlinearen DKRGn dargesell. Seuerungsenwurf für DKRGn mi Randeingriff miels Summaionsmehoden Im Hinblick auf eine allgemeine Behandlung von parabolischen SVPn mi Randeingriff werden Poenzreihen in der Orskoordinae mi zeivariablen Koeffizienen angesez. Dies erlaub für eine relaiv große Klasse von DKRGn eine direke Paramerierung von Sysemzusänden und Eingängen in Abhängigkei von einer Basisgröße, der sogenannen paramerierenden Funkion. Das prinzipielle Vorgehen wird im Folgenden zunächs anhand des Seuerungsenwurfs für die lineare Wärmeleiungsgleichung mi Randeingriff vorgesell. Paramerierung der linearen Wärmeleiungsgleichung In der Modellgleichung des beracheen Wärmeleiers werden im Weieren alle Größen der Einfachhei halber als dimensionslos und normier angenommen, d.h. x(z, ) = λ 2 x(z, ) z 2 + βx(z, ), z (, ), > () Formal heiß in diesem Zusammenhang, dass der Konvergenzradius der Poenzreihe idenisch Null sein kann.

16 XVI German summary / Deusche Kurzfassung mi der Anfangsbedingung (AB) und den konsisenen Randbedingungen (RBn) x(z, ) = x (z), z [, ] (2) x (, ) =, z > (3) x p z (, ) + r x(, ) = u(), >. (4) Die normiere Temperaur x(z, ) kann über den Randeingriff u() in (4) beeinfluss werden. Uner der formalen Annahme einer gleichmäßigen Konvergenz der Reihe ˆx(z, ) = ˆx n ()z n (5) n= liefer die Subsiuion von (5) in PDGL () und RB (3) eine Differenzialrekursion 2. Ordnung für die zeiabhängigen Koeffizienen ˆx n (), n 2: ˆx n+2 () = ˆx n () βˆx n () λ(n + 2)(n + ), n N (6) ˆx () =. (7) Zur Lösung der Rekursion (6) is neben (7) eine weiere Sar Bedingung nowendig. Diese kann beispielsweise aus dem Ansaz x(, ) = y() besimm werden, womi sich aus (5) ein weierer Sarwer ergib: ˆx () = y(). (8) Somi is die geschlossene Auswerung der Differenzialrekursion in Abhängigkei von der eingeführen Größe y() und deren Zeiableiungen möglich: ˆx(z, ) = n= z 2n λ n (2n)! n i= ( ) n ( β) n i y (i) (). (9) i Weierhin kann aus dieser Paramerierung durch Differenziaion miels (4) die ensprechende Gleichung der Eingangsgröße angegeben werden: n ( n ) i= i ( β) n i y (i) () n+ ( n+ ) i= i ( β) n+ i y (i) () û() = r + p λ n =: û (2n)! λ n+ n (). () (2n + )! n= n= n= Hieraus is leich ersichlich, dass durch Vorgabe einer geeigneen C Funkion y() die nowendige Seuerung û() besimm werden kann, die die Größe x(, ) enlang von y() führ. Dies sez die gleichmäßige Konvergenz der Reihe ˆx(z, ) voraus. Hierzu kann leich folgender Saz bewiesen werden (siehe auch (Widder, 975, p.5) oder (Taylor, 996, p.225)):

17 XVII u u u u α= α= α= α= y y y y α=.5 Sim. Soll α=2 Sim. Soll α=2.25 Sim. Soll α=2.5 Sim. Soll Bild : Vergleich der numerischen Ergebnisse für die Randseuerung von () (4) mi Parameern λ =, β =, p =, und r = für die Soll Trajekorie y d () = Φ γ,t () ensprechend Glg. (B.) mi der Übergangszei T = bei Variaion von γ bzw. α = + /γ. Links: Seuereingriff u N () aus (2) für N = 2; rechs: Vergleich von Is Trajekorie y() = x(, ) und Soll Trajekorie y d (). Saz. Die Reihe (9) konvergier gleichmäßig für alle z gegen die Lösung von () mi den RBn (3), (4), falls y() eine Gevrey Funkion der Ordnung α < 2 is, d.h. sup y (n) () M R + R n (n!)α, n N, M, R R +. () Offensichlich beding die Realisierung der Seuerfunkion den Abbruch der Reihe () an einer geeigneen Summaionsgrenze N N, so dass anselle von û() die Seuerung durch u N () = N û n () (2) approximier wird. Abbildung zeig Simulaionsergebnisse für die Anwendung der Seuerfunkion (2) auf ein semi diskreisieres Modell des Wärmeleiers bei Variaion der Gevrey Ordnung α. Dabei wird die in Anhang B vorgeselle Funkion Φ γ,t () (B.) der Gevrey Ordnung α = + als Sollrajekorie für y() vorgegeben. Diese Funkion mi kompakem γ n=

18 XVIII German summary / Deusche Kurzfassung Träger erlaub insbesondere die Realisierung von Übergängen innerhalb endlicher Zeiinervalle [, T ]. Wie in Kapiel 4. gezeig wird, kann durch die Vorgabe von y() direk ein Übergang zwischen ensprechenden saionären Profilen erzeug werden. Erwarungsgemäß reen für α 2 keine Abweichungen zwischen Soll und Is Trajekorie auf, wobei mi anseigendem α eine Abnahme der Seuerampliude zu beobachen is. Anderseis zeigen sich an den Rändern des Orsbereichs sörende Oszillaionen bei Überschreien des Schwellenweres von α = 2, deren Inensiä mi zunehmendem α anseig. Für α 2 is ein glaer Seuereingang mi guem Folgeverhalen zu beobachen, obwohl heoreisch divergenes Verhalen aufreen solle. Dies is in diesem Fall durch die Wahl von N = 2 begründe, wobei die Addiion weierer Reihenglieder û n (), n > N, zu einem Verhalen ähnlich dem für α = 2.5 führ. Dieses für einige divergene Reihen ypische Verhalen (Knopp, 964) wird beispielsweise in Laroche e al. (2) in der Anwendung der sogenannen Summaion zum kleinsen Term angewand, bei der Reihenglieder solange addier werden, bis der beragsmäßig kleinse Term erreich wird. Die Anwendbarkei dieser eher heurisischen Mehode 2 is jedoch deulich eingeschränk (Wagner e al., 24). Andererseis exisieren weiaus geeigneere Summaionsmehoden, die Gegensand der weieren Berachungen sind. Der Begriff der formalen Poenzreihenparamerierung Ausgehend von den Ergebnissen für den linearen Wärmeleier kann der Poenzreihenansaz direk auf komplexere PDGLn mi Randeingriff erweier werden. Im Rahmen dieser Arbei werden speziell die lineare Diffusions Konvekions Gleichung, skalare DKRGn mi polynomialen Nichlineariäen sowie Syseme nichlinearer parabolischer PDGLn unersuch siehe Kapiel 2 und das Anwendungsbeispiel am Ende dieser Kurzfassung. Wie einleiend erwähn, sellen der Konvergenznachweis bzw. die sich daraus ergebenden Bedingungen deuliche Einschränkungen der Anwendbarkei des Poenzreihenansazes dar. Hieraus ergib sich die Moivaion, die Berachungen mi sogenannen formalen Poenzreihen forzusezen, deren Konvergenzradius ohne Einschränkung gleich Null sein kann (Balser, 2). Somi sind die Rechenoperaionen im Folgenden als rein formal anzusehen, da sie ypischerweise gleichmäßige Konvergenz der Reihe voraussezen. Hieraus kann eine wesenliche Verallgemeinerung der Paramerierbarkei von DKRGn erreich werden, die zur Definiion der sogenannen formalen Poenzreihenparamerierung (FPRP) führ (Meurer and Zeiz, 24a; Wagner e al., 24; Meurer and Zeiz, 25). Definiion (Formale Poenzreihenparamerierung). Eine K dimensionale Menge von nichlinearen DKRGn 2. Ordnung mi Randeingriffen ( x(z, ) G i, x(z, ), ) x(z, ), 2 x(z, ) =, z (, ), >, i =,..., K (3) z z 2 2 Es is beispielsweise im Allgemeinen nich möglich, den kleinsen Term a priori zu besimmen.

19 XIX ( R,j x(, ), x ) z (, ), u () =, >, j =,..., K (4) ( R,l x(, ), x ) z (, ), u () =, >, l =,..., K (K + K = 2K) (5) x(z, ) = x (z), z [, ] (6) definier auf (z, ) [, ] R + mi dem Zusandsvekor x(z, ) = [x (z, ),..., x K (z, )] T und den Eingangsgrößen u() aus geeigneen Funkionenräumen X [, ] bzw. U, wird formal poenzreihenparamerierbar (FPRP) genann, falls eine formale Poenzreihe ˆx(z, ) = ˆx n () p n (z) (7) n= mi p n (z) einem geeigneen Polynom in z der Ordnung n exisier, die formal (3) (6) erfüll und deren Koeffizienen ˆx n () durch eine paramerierende Funkion ( y() = H x(, ), x ) x (, ), x(, ), z z (, ), u,() (8) mi dim y = dim u + dim u und ihren Zeiableiungen ausgedrück werden können, d.h. ˆx n () = Υ n ( y(), ẏ(),..., y (r x) () ) (9) mi r x N, wobei ggf. r x mi n. Die Eingangsgrößen u, () werden FPRP genann, falls (3) (6) FPRP sind und (4), (5) nach u, () (explizi) auflösbar sind. Diese Definiion drück im Wesenlichen die Paramerierbarkei der Zusands und Eingangsgrößen eines Sysems von DKRGn aus, womi sich eine gewisse Ähnlichkei zum Flachheisbegriff ergib. Insbesondere kann FPRP als eine Ar konsrukiver Seuerbarkeisnachweis inerpreier werden, da direk durch Vorgabe einer Trajekorie für die paramerierende Größe y() formal die zugehörige Seuerung u() ermiel werden kann, die das Sysem im offenen Kreis enlang dieser Trajekorie führ. Speziell für lineare DKRGn kann im Fall gleichmäßiger Reihenkonvergenz gezeig werden, dass FPRP approximaive Seuerbarkei implizier (Laroche, 2, Prop ). Diese verallgemeinere formale Berachungsweise beding, dass aus der paramerieren und möglicherweise divergenen formalen Poenzreihe (7), (9) miels geeigneer mahemaischer Mehoden ein sinnvoller Grenzwer exrahier werden muss. Im Weieren wird gezeig, dass dies über geeignee Summaionsmehoden erreich werden kann. Seuerungsenwurf miels Summaionsmehoden Die gebräuchlichse Summaionsmehode, obwohl nich als solche bezeichne, sell die Grenzwerbildung in der klassischen Parialsummenbildung dar, d.h. S P ( n= ˆx n z n ) = lim N N ˆx n z n. n=

20 XX German summary / Deusche Kurzfassung Hierbei kann S P als lineares Funkional aufgefass werden, das jede konvergene Poenzreihe auf ihre naürliche Summe abbilde. Offensichlich handel es sich hierbei um eine schwache Mehode, da sie nur auf konvergene Reihen anwendbar is, sie zeig jedoch den prinzipiellen Ansaz. Im Folgenden sei S V ein lineares Funkional auf einem linearen Raum X S von Folgen bzw. Reihen und {a n (ξ)} n N eine Folge von Funkionen in ξ. Des Weieren soll die Wirkung von S V auf eine formale Reihe ˆx(z) = n= ˆx nz n mi z C folgendermaßen gegeben sein: S V ( n= ˆx n z n ) = lim ξ n= a n (ξ)ˆx n z n = lim ξ ˆx A (z; ξ). Falls ˆx A (z; ξ) für z < ρ konvergier, so definier dies eine Familie von holomorphen Funkionen in einem Kreis um z = mi Radius ρ. Konvergier der Grenzwer für ξ gleichmäßig in z, so definier x(z) = lim ξ x A (z; ξ) eine holomorphe Funkion in einem Gebie Ω der komplexen Ebene (Ramis, 993). In diesem Fall wird die Reihe ˆx(z) als A summierbar im Gebie Ω bezeichne, und die Funkion x(z) = (S V ˆx)(z) kann als die A Summe von ˆx(z) aufgefass werden. Der Raum X S aller A summierbaren Reihen wird als Summierbarkeisbereich der Summaionsmehode S V bezeichne. Eine allgemeine Diskussion dieser Ergebnisse is beispielsweise in Hardy (949) zu finden. Um eine Summaionsmehode zur Lösung von Differenzialgleichungen einzusezen, muss diese weiere Bedingungen erfüllen (Ramis, 993; Balser, 2). Hierzu muss (i) der Summierbarkeisbereich X S die Srukur einer Differenzialalgebra aufweisen und eine Erweierung des Raums der konvergenen Reihen darsellen; (ii) S V regulär sein, d.h. jede konvergene Reihe auf ihre naürliche Summe aufsummieren; (iii) S V einen linearen Homomorphismus darsellen, der Produke auf Produke sowie Ableiungen auf Ableiungen abbilde; (iv) S V der formalen eine gegebene Differenzialgleichung (DGL) erfüllenden Reihe ˆx(z) eindeuig eine Summe x(z) = (S V ˆx)(z) zuweisen, die ebenfalls die DGL erfüll. Zur Behandlung von PDGLn müssen noch weiere Bedingungen formulier werden, wie insbesondere die gleichmäßige Summierbarkei, um Ableiungen der variablen Reihenkoeffizienen nach der Zei zuzulassen (vgl. Glg. (6)). Eine Summaionsmehode, welche die genannen Forderungen erfüll, is die sogenanne k Summaion, die von Ramis (98) als eine Erweierung der Arbei von Borel (928) zur Summierbarkei von divergenen Reihen eingeführ wird. Im Folgenden wird nur eine der möglichen äquivalenen Definiionen herangezogen (Balser and Braun, 2): x(z) = ˆx(z) = lim ξ n= s n(z) n= ξ n Γ(+ n k ) =: (S ξ n B k ˆx)(z), (2) Γ(+ n k ) wobei s n (z) die n e Parialsumme darsell. Für eine ausführliche Diskussion der Eigenschafen sowie des Nachweises der k Summierbarkei wird auf Ramis (98); Malgrange (995); Balser (994, 2) und deren Referenzen verwiesen. Beispielsweise erlaub die Anwendung von SB k die Summaion der geomerischen Reihe n= zn außerhalb ihres Konvergenzbereiches z <, d.h. in der gesamen komplexen Halbebene Rz <, zur Summe. z

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