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

Size: px
Start display at page:

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

Transcription

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

2

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=

The Transport Equation

The Transport Equation The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

More information

Journal Of Business & Economics Research September 2005 Volume 3, Number 9

Journal Of Business & Economics Research September 2005 Volume 3, Number 9 Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo

More information

Multiprocessor Systems-on-Chips

Multiprocessor Systems-on-Chips Par of: Muliprocessor Sysems-on-Chips Edied by: Ahmed Amine Jerraya and Wayne Wolf Morgan Kaufmann Publishers, 2005 2 Modeling Shared Resources Conex swiching implies overhead. On a processing elemen,

More information

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,

More information

The option pricing framework

The option pricing framework Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.

More information

Option Pricing Under Stochastic Interest Rates

Option Pricing Under Stochastic Interest Rates I.J. Engineering and Manufacuring, 0,3, 8-89 ublished Online June 0 in MECS (hp://www.mecs-press.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecs-press.ne/ijem Opion ricing Under Sochasic Ineres

More information

A Generalized Bivariate Ornstein-Uhlenbeck Model for Financial Assets

A Generalized Bivariate Ornstein-Uhlenbeck Model for Financial Assets A Generalized Bivariae Ornsein-Uhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical

More information

Analogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar

Analogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar Analogue and Digial Signal Processing Firs Term Third Year CS Engineering By Dr Mukhiar Ali Unar Recommended Books Haykin S. and Van Veen B.; Signals and Sysems, John Wiley& Sons Inc. ISBN: 0-7-380-7 Ifeachor

More information

TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS

TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.

More information

Monte Carlo Observer for a Stochastic Model of Bioreactors

Monte Carlo Observer for a Stochastic Model of Bioreactors Mone Carlo Observer for a Sochasic Model of Bioreacors Marc Joannides, Irène Larramendy Valverde, and Vivien Rossi 2 Insiu de Mahémaiques e Modélisaion de Monpellier (I3M UMR 549 CNRS Place Eugène Baaillon

More information

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides 7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion

More information

Cointegration: The Engle and Granger approach

Cointegration: The Engle and Granger approach Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be non-saionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require

More information

policies are investigated through the entire product life cycle of a remanufacturable product. Benefiting from the MDP analysis, the optimal or

policies are investigated through the entire product life cycle of a remanufacturable product. Benefiting from the MDP analysis, the optimal or ABSTRACT AHISKA, SEMRA SEBNEM. Invenory Opimizaion in a One Produc Recoverable Manufacuring Sysem. (Under he direcion of Dr. Russell E. King and Dr. Thom J. Hodgson.) Environmenal regulaions or he necessiy

More information

Communication Networks II Contents

Communication Networks II Contents 3 / 1 -- Communicaion Neworks II (Görg) -- www.comnes.uni-bremen.de Communicaion Neworks II Conens 1 Fundamenals of probabiliy heory 2 Traffic in communicaion neworks 3 Sochasic & Markovian Processes (SP

More information

The Application of Multi Shifts and Break Windows in Employees Scheduling

The Application of Multi Shifts and Break Windows in Employees Scheduling The Applicaion of Muli Shifs and Brea Windows in Employees Scheduling Evy Herowai Indusrial Engineering Deparmen, Universiy of Surabaya, Indonesia Absrac. One mehod for increasing company s performance

More information

A UNIFIED APPROACH TO MATHEMATICAL OPTIMIZATION AND LAGRANGE MULTIPLIER THEORY FOR SCIENTISTS AND ENGINEERS

A UNIFIED APPROACH TO MATHEMATICAL OPTIMIZATION AND LAGRANGE MULTIPLIER THEORY FOR SCIENTISTS AND ENGINEERS A UNIFIED APPROACH TO MATHEMATICAL OPTIMIZATION AND LAGRANGE MULTIPLIER THEORY FOR SCIENTISTS AND ENGINEERS RICHARD A. TAPIA Appendix E: Differeniaion in Absrac Spaces I should be no surprise ha he differeniaion

More information

Task is a schedulable entity, i.e., a thread

Task is a schedulable entity, i.e., a thread Real-Time Scheduling Sysem Model Task is a schedulable eniy, i.e., a hread Time consrains of periodic ask T: - s: saring poin - e: processing ime of T - d: deadline of T - p: period of T Periodic ask T

More information

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion

More information

Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average

Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July

More information

Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.

Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations. Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given

More information

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 Introduction to Mathematical Finance Lesson 5 26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

More information

Relationships between Stock Prices and Accounting Information: A Review of the Residual Income and Ohlson Models. Scott Pirie* and Malcolm Smith**

Relationships between Stock Prices and Accounting Information: A Review of the Residual Income and Ohlson Models. Scott Pirie* and Malcolm Smith** Relaionships beween Sock Prices and Accouning Informaion: A Review of he Residual Income and Ohlson Models Sco Pirie* and Malcolm Smih** * Inernaional Graduae School of Managemen, Universiy of Souh Ausralia

More information

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper

More information

A Probability Density Function for Google s stocks

A Probability Density Function for Google s stocks A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural

More information

Term Structure of Prices of Asian Options

Term Structure of Prices of Asian Options Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 1-1-1 Nojihigashi, Kusasu, Shiga 525-8577, Japan E-mail:

More information

INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES

INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES Journal of Applied Analysis 1, 1 (1995), pp. 39 45 INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES A. LUCZAK Absrac. We find exponens of independen marginals

More information

Analysis of Pricing and Efficiency Control Strategy between Internet Retailer and Conventional Retailer

Analysis of Pricing and Efficiency Control Strategy between Internet Retailer and Conventional Retailer Recen Advances in Business Managemen and Markeing Analysis of Pricing and Efficiency Conrol Sraegy beween Inerne Reailer and Convenional Reailer HYUG RAE CHO 1, SUG MOO BAE and JOG HU PARK 3 Deparmen of

More information

3 Runge-Kutta Methods

3 Runge-Kutta Methods 3 Runge-Kua Mehods In conras o he mulisep mehods of he previous secion, Runge-Kua mehods are single-sep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he

More information

Stochastic Optimal Control Problem for Life Insurance

Stochastic Optimal Control Problem for Life Insurance Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian

More information

DETERMINISTIC INVENTORY MODEL FOR ITEMS WITH TIME VARYING DEMAND, WEIBULL DISTRIBUTION DETERIORATION AND SHORTAGES KUN-SHAN WU

DETERMINISTIC INVENTORY MODEL FOR ITEMS WITH TIME VARYING DEMAND, WEIBULL DISTRIBUTION DETERIORATION AND SHORTAGES KUN-SHAN WU Yugoslav Journal of Operaions Research 2 (22), Number, 6-7 DEERMINISIC INVENORY MODEL FOR IEMS WIH IME VARYING DEMAND, WEIBULL DISRIBUION DEERIORAION AND SHORAGES KUN-SHAN WU Deparmen of Bussines Adminisraion

More information

II.1. Debt reduction and fiscal multipliers. dbt da dpbal da dg. bal

II.1. Debt reduction and fiscal multipliers. dbt da dpbal da dg. bal Quarerly Repor on he Euro Area 3/202 II.. Deb reducion and fiscal mulipliers The deerioraion of public finances in he firs years of he crisis has led mos Member Saes o adop sizeable consolidaion packages.

More information

Hybrid System Design for Singularityless Task Level Robot Controllers *

Hybrid System Design for Singularityless Task Level Robot Controllers * Proceedings of he 2000 IEEE Inernaional Conference on Roboics & Auomaion San Francisco, CA April 2000 Hybrid Sysem Design for Singulariyless Task Level Robo Conrollers * Jindong Tan and Ning Xi Deparmen

More information

On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations

On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations On Galerkin Approximaions for he Zakai Equaion wih Diffusive and Poin Process Observaions An der Fakulä für Mahemaik und Informaik der Universiä Leipzig angenommene DISSERTATION zur Erlangung des akademischen

More information

Distributed Echo Cancellation in Multimedia Conferencing System

Distributed Echo Cancellation in Multimedia Conferencing System Disribued Echo Cancellaion in Mulimedia Conferencing Sysem Balan Sinniah 1, Sureswaran Ramadass 2 1 KDU College Sdn.Bhd, A Paramoun Corporaion Company, 32, Jalan Anson, 10400 Penang, Malaysia. sbalan@kdupg.edu.my

More information

On the degrees of irreducible factors of higher order Bernoulli polynomials

On the degrees of irreducible factors of higher order Bernoulli polynomials ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on

More information

AP Calculus AB 2013 Scoring Guidelines

AP Calculus AB 2013 Scoring Guidelines AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a mission-driven no-for-profi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was

More information

A general decomposition formula for derivative prices in stochastic volatility models

A general decomposition formula for derivative prices in stochastic volatility models A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 5-7 85 Barcelona Absrac We see ha he price of an european call opion

More information

USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES

USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES Mehme Nuri GÖMLEKSİZ Absrac Using educaion echnology in classes helps eachers realize a beer and more effecive learning. In his sudy 150 English eachers were

More information

Quality-Of-Service Class Specific Traffic Matrices in IP/MPLS Networks

Quality-Of-Service Class Specific Traffic Matrices in IP/MPLS Networks ualiy-of-service Class Specific Traffic Marices in IP/MPLS Neworks Sefan Schnier Deusche Telekom, T-Sysems D-4 Darmsad +4 sefan.schnier@-sysems.com Franz Harleb Deusche Telekom, T-Sysems D-4 Darmsad +4

More information

Quality-Of-Service Class Specific Traffic Matrices in IP/MPLS Networks

Quality-Of-Service Class Specific Traffic Matrices in IP/MPLS Networks ualiy-of-service Class Specific Traffic Marices in IP/MPLS Neworks Sefan Schnier Deusche Telekom, T-Sysems D-4 Darmsad +4 sefan.schnier@-sysems.com Franz Harleb Deusche Telekom, T-Sysems D-4 Darmsad +4

More information

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

More information

Risk Modelling of Collateralised Lending

Risk Modelling of Collateralised Lending Risk Modelling of Collaeralised Lending Dae: 4-11-2008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies

More information

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

More information

A Bayesian framework with auxiliary particle filter for GMTI based ground vehicle tracking aided by domain knowledge

A Bayesian framework with auxiliary particle filter for GMTI based ground vehicle tracking aided by domain knowledge A Bayesian framework wih auxiliary paricle filer for GMTI based ground vehicle racking aided by domain knowledge Miao Yu a, Cunjia Liu a, Wen-hua Chen a and Jonahon Chambers b a Deparmen of Aeronauical

More information

Optimal Investment and Consumption Decision of Family with Life Insurance

Optimal Investment and Consumption Decision of Family with Life Insurance Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker

More information

Measuring the Effects of Monetary Policy: A Factor-Augmented Vector Autoregressive (FAVAR) Approach * Ben S. Bernanke, Federal Reserve Board

Measuring the Effects of Monetary Policy: A Factor-Augmented Vector Autoregressive (FAVAR) Approach * Ben S. Bernanke, Federal Reserve Board Measuring he Effecs of Moneary Policy: A acor-augmened Vecor Auoregressive (AVAR) Approach * Ben S. Bernanke, ederal Reserve Board Jean Boivin, Columbia Universiy and NBER Pior Eliasz, Princeon Universiy

More information

Working Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619

Working Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619 econsor www.econsor.eu Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;

More information

Performance Center Overview. Performance Center Overview 1

Performance Center Overview. Performance Center Overview 1 Performance Cener Overview Performance Cener Overview 1 ODJFS Performance Cener ce Cener New Performance Cener Model Performance Cener Projec Meeings Performance Cener Execuive Meeings Performance Cener

More information

Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension

Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical

More information

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613. Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised

More information

The naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1

The naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1 Business Condiions & Forecasing Exponenial Smoohing LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecure inroduces ime-series smoohing forecasing mehods. Various models are discussed,

More information

Issues Using OLS with Time Series Data. Time series data NOT randomly sampled in same way as cross sectional each obs not i.i.d

Issues Using OLS with Time Series Data. Time series data NOT randomly sampled in same way as cross sectional each obs not i.i.d These noes largely concern auocorrelaion Issues Using OLS wih Time Series Daa Recall main poins from Chaper 10: Time series daa NOT randomly sampled in same way as cross secional each obs no i.i.d Why?

More information

Why Did the Demand for Cash Decrease Recently in Korea?

Why Did the Demand for Cash Decrease Recently in Korea? Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in

More information

PRECISE positioning/tracking control is being studied

PRECISE positioning/tracking control is being studied Design of High Accuracy Tracking Sysems wih H Preview Conrol Anonio Moran Cardenas, Javier G. Rázuri, Isis Bone, Rahim Rahmani, and David Sundgren Absrac Posiioning and racking conrol sysems are an imporan

More information

Chapter 8: Regression with Lagged Explanatory Variables

Chapter 8: Regression with Lagged Explanatory Variables Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One

More information

Continuous Families of Embedded Solitons in the Third-Order Nonlinear Schrödinger Equation

Continuous Families of Embedded Solitons in the Third-Order Nonlinear Schrödinger Equation Coninuous Families of Embedded Solions in he Third-Order Nonlinear Schrödinger Equaion By J. Yang and T. R. Akylas The nonlinear Schrödinger equaion wih a hird-order dispersive erm is considered. Infinie

More information

Vector Autoregressions (VARs): Operational Perspectives

Vector Autoregressions (VARs): Operational Perspectives Vecor Auoregressions (VARs): Operaional Perspecives Primary Source: Sock, James H., and Mark W. Wason, Vecor Auoregressions, Journal of Economic Perspecives, Vol. 15 No. 4 (Fall 2001), 101-115. Macroeconomericians

More information

Inductance and Transient Circuits

Inductance and Transient Circuits Chaper H Inducance and Transien Circuis Blinn College - Physics 2426 - Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual

More information

AP Calculus BC 2010 Scoring Guidelines

AP Calculus BC 2010 Scoring Guidelines AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board

More information

Research Article Solitary Wave Solutions for a Time-Fraction Generalized Hirota-Satsuma Coupled KdV Equation by a New Analytical Technique

Research Article Solitary Wave Solutions for a Time-Fraction Generalized Hirota-Satsuma Coupled KdV Equation by a New Analytical Technique Hindawi Publishing Corporaion Inernaional Journal of Differenial Equaions Volume, Aricle ID 954674, pages doi:.55//954674 Research Aricle Soliary Wave Soluions for a Time-Fracion Generalized Hiroa-Sasuma

More information

ARCH 2013.1 Proceedings

ARCH 2013.1 Proceedings Aricle from: ARCH 213.1 Proceedings Augus 1-4, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference

More information

4 Convolution. Recommended Problems. x2[n] 1 2[n]

4 Convolution. Recommended Problems. x2[n] 1 2[n] 4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discree-ime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.1-1.

More information

1.2 Goals for Animation Control

1.2 Goals for Animation Control A Direc Manipulaion Inerface for 3D Compuer Animaion Sco Sona Snibbe y Brown Universiy Deparmen of Compuer Science Providence, RI 02912, USA Absrac We presen a new se of inerface echniques for visualizing

More information

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

More information

Analysis of optimal liquidation in limit order books

Analysis of optimal liquidation in limit order books Analysis of opimal liquidaion in limi order books James W. Blair, Paul V. Johnson, & Peer W. Duck Absrac In his paper we sudy he opimal rading sraegy of a passive rader who is rading in he limi order book.

More information

Suggested Reading. Signals and Systems 4-2

Suggested Reading. Signals and Systems 4-2 4 Convoluion In Lecure 3 we inroduced and defined a variey of sysem properies o which we will make frequen reference hroughou he course. Of paricular imporance are he properies of lineariy and ime invariance,

More information

Trends in TCP/IP Retransmissions and Resets

Trends in TCP/IP Retransmissions and Resets Trends in TCP/IP Reransmissions and Reses Absrac Concordia Chen, Mrunal Mangrulkar, Naomi Ramos, and Mahaswea Sarkar {cychen, mkulkarn, msarkar,naramos}@cs.ucsd.edu As he Inerne grows larger, measuring

More information

Distributing Human Resources among Software Development Projects 1

Distributing Human Resources among Software Development Projects 1 Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources

More information

International Journal of Supply and Operations Management

International Journal of Supply and Operations Management Inernaional Journal of Supply and Operaions Managemen IJSOM May 05, Volume, Issue, pp 5-547 ISSN-Prin: 8-59 ISSN-Online: 8-55 wwwijsomcom An EPQ Model wih Increasing Demand and Demand Dependen Producion

More information

INTRODUCTION TO FORECASTING

INTRODUCTION TO FORECASTING INTRODUCTION TO FORECASTING INTRODUCTION: Wha is a forecas? Why do managers need o forecas? A forecas is an esimae of uncerain fuure evens (lierally, o "cas forward" by exrapolaing from pas and curren

More information

Load Prediction Using Hybrid Model for Computational Grid

Load Prediction Using Hybrid Model for Computational Grid Load Predicion Using Hybrid Model for Compuaional Grid Yongwei Wu, Yulai Yuan, Guangwen Yang 3, Weimin Zheng 4 Deparmen of Compuer Science and Technology, Tsinghua Universiy, Beijing 00084, China, 3, 4

More information

SHB Gas Oil. Index Rules v1.3 Version as of 1 January 2013

SHB Gas Oil. Index Rules v1.3 Version as of 1 January 2013 SHB Gas Oil Index Rules v1.3 Version as of 1 January 2013 1. Index Descripions The SHB Gasoil index (he Index ) measures he reurn from changes in he price of fuures conracs, which are rolled on a regular

More information

Idealistic characteristics of Islamic Azad University masters - Islamshahr Branch from Students Perspective

Idealistic characteristics of Islamic Azad University masters - Islamshahr Branch from Students Perspective Available online a www.pelagiaresearchlibrary.com European Journal Experimenal Biology, 202, 2 (5):88789 ISSN: 2248 925 CODEN (USA): EJEBAU Idealisic characerisics Islamic Azad Universiy masers Islamshahr

More information

Strategic Optimization of a Transportation Distribution Network

Strategic Optimization of a Transportation Distribution Network Sraegic Opimizaion of a Transporaion Disribuion Nework K. John Sophabmixay, Sco J. Mason, Manuel D. Rossei Deparmen of Indusrial Engineering Universiy of Arkansas 4207 Bell Engineering Cener Fayeeville,

More information

Information Theoretic Evaluation of Change Prediction Models for Large-Scale Software

Information Theoretic Evaluation of Change Prediction Models for Large-Scale Software Informaion Theoreic Evaluaion of Change Predicion Models for Large-Scale Sofware Mina Askari School of Compuer Science Universiy of Waerloo Waerloo, Canada maskari@uwaerloo.ca Ric Hol School of Compuer

More information

MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR

MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR The firs experimenal publicaion, which summarised pas and expeced fuure developmen of basic economic indicaors, was published by he Minisry

More information

Module 3 Design for Strength. Version 2 ME, IIT Kharagpur

Module 3 Design for Strength. Version 2 ME, IIT Kharagpur Module 3 Design for Srengh Lesson 2 Sress Concenraion Insrucional Objecives A he end of his lesson, he sudens should be able o undersand Sress concenraion and he facors responsible. Deerminaion of sress

More information

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer) Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

More information

Real-time Particle Filters

Real-time Particle Filters Real-ime Paricle Filers Cody Kwok Dieer Fox Marina Meilă Dep. of Compuer Science & Engineering, Dep. of Saisics Universiy of Washingon Seale, WA 9895 ckwok,fox @cs.washingon.edu, mmp@sa.washingon.edu Absrac

More information

Single-machine Scheduling with Periodic Maintenance and both Preemptive and. Non-preemptive jobs in Remanufacturing System 1

Single-machine Scheduling with Periodic Maintenance and both Preemptive and. Non-preemptive jobs in Remanufacturing System 1 Absrac number: 05-0407 Single-machine Scheduling wih Periodic Mainenance and boh Preempive and Non-preempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy

More information

Premium Income of Indian Life Insurance Industry

Premium Income of Indian Life Insurance Industry Premium Income of Indian Life Insurance Indusry A Toal Facor Produciviy Approach Ram Praap Sinha* Subsequen o he passage of he Insurance Regulaory and Developmen Auhoriy (IRDA) Ac, 1999, he life insurance

More information

Dependent Interest and Transition Rates in Life Insurance

Dependent Interest and Transition Rates in Life Insurance Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies

More information

Smooth Priorities for Multi-Product Inventory Control

Smooth Priorities for Multi-Product Inventory Control Smooh rioriies for Muli-roduc Invenory Conrol Francisco José.A.V. Mendonça*. Carlos F. Bispo** *Insiuo Superior Técnico - Universidade Técnica de Lisboa (email:favm@mega.is.ul.p) ** Insiuo de Sisemas e

More information

SkySails Tethered Kites for Ship Propulsion and Power Generation: Modeling and System Identification. Michael Erhard, SkySails GmbH, Hamburg, Germany

SkySails Tethered Kites for Ship Propulsion and Power Generation: Modeling and System Identification. Michael Erhard, SkySails GmbH, Hamburg, Germany SkySails Tehered Kies for Ship Propulsion and Power Generaion: Modeling and Sysem Idenificaion Michael Erhard, SkySails GmbH, Hamburg, Germany Conens Inroducion SkySails Marine and Power Simple Model Sensors

More information

Signal Processing and Linear Systems I

Signal Processing and Linear Systems I Sanford Universiy Summer 214-215 Signal Processing and Linear Sysems I Lecure 5: Time Domain Analysis of Coninuous Time Sysems June 3, 215 EE12A:Signal Processing and Linear Sysems I; Summer 14-15, Gibbons

More information

Dokumentation über die Übernahme von. "GS-R-3" (The Management System for Facilities and Activities) "Sicherheitskriterien für Kernkraftwerke"

Dokumentation über die Übernahme von. GS-R-3 (The Management System for Facilities and Activities) Sicherheitskriterien für Kernkraftwerke Dokumentation über die Übernahme von "GS-R-3" () in die "Sicherheitskriterien für Kernkraftwerke" REVISION D APRIL 2009 1. INTRODUCTION BACKGROUND 1.1 This Safety Requirements publication defines the requirements

More information

Information Theoretic Approaches for Predictive Models: Results and Analysis

Information Theoretic Approaches for Predictive Models: Results and Analysis Informaion Theoreic Approaches for Predicive Models: Resuls and Analysis Monica Dinculescu Supervised by Doina Precup Absrac Learning he inernal represenaion of parially observable environmens has proven

More information

PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM

PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM WILLIAM L. COOPER Deparmen of Mechanical Engineering, Universiy of Minnesoa, 111 Church Sree S.E., Minneapolis, MN 55455 billcoop@me.umn.edu

More information

Bayesian Filtering with Online Gaussian Process Latent Variable Models

Bayesian Filtering with Online Gaussian Process Latent Variable Models Bayesian Filering wih Online Gaussian Process Laen Variable Models Yali Wang Laval Universiy yali.wang.1@ulaval.ca Marcus A. Brubaker TTI Chicago mbrubake@cs.orono.edu Brahim Chaib-draa Laval Universiy

More information

Constant Data Length Retrieval for Video Servers with Variable Bit Rate Streams

Constant Data Length Retrieval for Video Servers with Variable Bit Rate Streams IEEE Inernaional Conference on Mulimedia Compuing & Sysems, June 17-3, 1996, in Hiroshima, Japan, p. 151-155 Consan Lengh Rerieval for Video Servers wih Variable Bi Rae Sreams Erns Biersack, Frédéric Thiesse,

More information

SPEC model selection algorithm for ARCH models: an options pricing evaluation framework

SPEC model selection algorithm for ARCH models: an options pricing evaluation framework Applied Financial Economics Leers, 2008, 4, 419 423 SEC model selecion algorihm for ARCH models: an opions pricing evaluaion framework Savros Degiannakis a, * and Evdokia Xekalaki a,b a Deparmen of Saisics,

More information

Emergence of Fokker-Planck Dynamics within a Closed Finite Spin System

Emergence of Fokker-Planck Dynamics within a Closed Finite Spin System Emergence of Fokker-Planck Dynamics wihin a Closed Finie Spin Sysem H. Niemeyer(*), D. Schmidke(*), J. Gemmer(*), K. Michielsen(**), H. de Raed(**) (*)Universiy of Osnabrück, (**) Supercompuing Cener Juelich

More information

Entropy: From the Boltzmann equation to the Maxwell Boltzmann distribution

Entropy: From the Boltzmann equation to the Maxwell Boltzmann distribution Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are

More information

Time Series Prediction of Web Domain Visits by IF-Inference System

Time Series Prediction of Web Domain Visits by IF-Inference System Time Series Predicion of Web Domain Visis by IF-Inference Sysem VLADIMÍR OLEJ, JANA FILIPOVÁ, PETR HÁJEK Insiue of Sysem Engineering and Informaics Faculy of Economics and Adminisraion Universiy of Pardubice,

More information

Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling

Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se

More information

Chabot College Physics Lab RC Circuits Scott Hildreth

Chabot College Physics Lab RC Circuits Scott Hildreth Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard

More information

Tracking of Multiple Moving Sources Using Recursive EM Algorithm

Tracking of Multiple Moving Sources Using Recursive EM Algorithm EURASIP Journal on Applied Signal Processing 24:18, 1 11 c 24 Hindawi Publishing Corporaion Tracking of Muliple Moving Sources Using Recursive EM Algorihm Pei-Jung Chung Deparmen of Elecrical Engineering

More information

Niche Market or Mass Market?

Niche Market or Mass Market? Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.

More information