Single-zone cylinder pressure modeling and estimation for heat release analysis of SI engines 9789185831128

Citation preview

Single-Zone Cylinder Pressure Modeling and Estimation for Heat Release Analysis of SI Engines

Markus Klein

Linköping 2007

Linköping Studies in S ien e and Te hnology. Dissertations No. 1124

Single-Zone Cylinder Pressure Modeling and Estimation for Heat Release Analysis of SI Engines

2007 Markus Klein Copyright

markus.s.kleingmail. om http://www.fs.isy.liu.se/ Department of Ele tri al Engineering, Linköping University, SE581 83 Linköping, Sweden. ISBN 978-91-85831-12-8

ISSN 0345-7524

Printed by Liu-Try k, Linköping, Sweden 2007

To Soa, Ebba, Samuel and Nora

Abstra t

Cylinder pressure modeling and heat release analysis are today important and standard tools for engineers and resear hers, when developing and tuning new engines. Being able to a

urately model and extra t information from the ylinder pressure is important for the interpretation and validity of the result. The rst part of the thesis treats single-zone ylinder pressure modeling, where the spe i heat ratio model onstitutes a key part. This model omponent is therefore investigated more thoroughly. For the purpose of referen e, the spe i heat ratio is al ulated for burned and unburned gases, assuming that the unburned mixture is frozen and that the burned mixture is at hemi al equilibrium. Use of the referen e model in heat release analysis is too time onsuming and therefore a set of simpler models, both existing and newly developed, are ompared to the referen e model. A two-zone mean temperature model and the Vibe fun tion are used to parameterize the mass fra tion burned. The mass fra tion burned is used to interpolate the spe i heats for the unburned and burned mixture, and to form the spe i heat ratio, whi h renders a ylinder pressure modeling error in the same order as the measurement noise, and fteen times smaller than the model originally suggested in Gatowski et al. (1984). The omputational time is in reased with 40 % ompared to the original setting, but redu ed by a fa tor 70 ompared to pre omputed tables from the full equilibrium program. The spe i heats for the unburned mixture are aptured within 0.2 % by linear fun tions, i

ii and the spe i heats for the burned mixture are aptured within 1 % by higher-order polynomials for the major operating range of a spark ignited (SI) engine. In the se ond part, four methods for ompression ratio estimation based on ylinder pressure tra es are developed and evaluated for both simulated and experimental y les. Three methods rely upon a model of polytropi ompression for the ylinder pressure. It is shown that they give a good estimate of the ompression ratio at low ompression ratios, although the estimates are biased. A method based on a variable proje tion algorithm with a logarithmi norm of the ylinder pressure yields the smallest onden e intervals and shortest omputational time for these three methods. This method is re ommended when omputational time is an important issue. The polytropi pressure model la ks information about heat transfer and therefore the estimation bias in reases with the ompression ratio. The fourth method in ludes heat transfer, revi e ee ts, and a ommonly used heat release model for ring y les. This method estimates the ompression ratio more a

urately in terms of bias and varian e. The method is more omputationally demanding and thus re ommended when estimation a

ura y is the most important property. In order to estimate the ompression ratio as a

urately as possible, motored y les with as high initial pressure as possible should be used. The obje tive in part 3 is to develop an estimation tool for heat release analysis that is a

urate, systemati and e ient. Two methods that in orporate prior knowledge of the parameter nominal value and un ertainty in a systemati manner are presented and evaluated. Method 1 is based on using a singular value de omposition of the estimated hessian, to redu e the number of estimated parameters oneby-one. Then the suggested number of parameters to use is found as the one minimizing the Akaike nal predi tion error. Method 2 uses a regularization te hnique to in lude the prior knowledge in the riterion fun tion. Method 2 gives more a

urate estimates than method 1. For method 2, prior knowledge with individually set parameter un ertainties yields more a

urate and robust estimates. On e a hoi e of parameter un ertainty has been done, no user intera tion is needed. Method 2 is then formulated for three dierent versions, whi h dier in how they determine how strong the regularization should be. The qui kest version is based on ad-ho tuning and should be used when omputational time is important. Another version is more a

urate and exible to hanging operating onditions, but is more omputationally demanding.

Svenskt referat

Den svenska titeln på avhandlingen är En-zons-modellering o h esti-

1

mering för analys av frigjord värme i bensinmotorer . Förbränningsmotorer har varit den primära maskinen för att generera arbete i fordon i mer än hundra år, o h kommer att vara högintressant även i fortsättningen främst p.g.a. bränslets höga energidensitet. Emissionskrav från främst lagstiftare, prestandakrav såsom eekt o h bränsleförbrukning från potentiella kunder, samt den konkurrens som ges av nya teknologier såsom bränsle eller fortsätter att driva teknikutve klingen av förbränningsmotorer framåt. Teknikutve klingen möjliggörs av att ingenjörer o h forskare har mött dessa krav genom t.ex. grundforskning på förbränningspro essen, nya eller förbättrade komponenter i motorsystemet o h nya teknologier såsom variabla ventiltider o h variabelt kompressionsförhållande. De två sistnämnda är exempel på teknologier som direkt påverkar try kutve klingen i ylindern, där det är viktigt att få noggrann kunskap om hur förbränningspro essen fortlöper o h hur bränslets kemiska energi frigörs som värme o h sedan omvandlas till mekaniskt arbete. Detta kallas analys av frigjord värme o h kopplar direkt till motorns emissioner, eekt o h bränsleförbrukning. En analys av frigjord värme möjliggörs av att man: 1) kan mäta

ylindertry ket under förbränningspro essen; 2) har matematiska modeller för hur ylindertry ket utve klas som funktion av kolvrörelsen o h förbränningspro essen; samt 3) har metodik för att beräkna den

1

Egentligen tändstiftsmotorer iii

iv frigjorda värmen genom att anpassa den valda modellen till ylindertry ksmätningen. Sensorerna för att mäta try ket i ylindern har den senaste tiden blivit både noggrannare o h robustare mot den extrema miljö o h de snabba try k- o h temperaturförändringar som sensorn utsätts för under varje ykel. Idag används ylindertry kssensorer enbart i labb- o h testmiljö, mest beroende på att sensorn är relativt dyr, men det pågår en teknikutve kling av sensorerna som siktar på att de i framtida fordon även skall sitta i produktionsmotorer. Matematiska modeller av ylindertry ksutve klingen o h metoder för att beräkna den frigjorda värmen behandlas i denna avhandling.

Avhandlingens innehåll o h kunskapsbidrag Avhandlingen består av tre delar o h det sammanhållande temat är ylindertry k. Den första delen behandlar en-zons-modellering av ylindertry ksutve klingen, där blandningen av luft o h bränsle i ylindern behandlas som homogen. Detta antagande möjliggör en kort beräkningstid för den matematiska modellen. Ett bidrag i avhandlingen är identiering av den viktigaste modellkomponenten i en allmänt vedertagen en-zons-modell samt en modellförbättring för denna komponent. I den andra delen studeras kompressionsförhållandet, vilket är förhållandet mellan största o h minsta ylindervolym under kolvens rörelse. Denna storhet är direkt kopplad motorns verkningsgrad o h bränsleförbrukning. Avsnittet beskriver o h utvärderar metoder för att bestämma kompressionsförhållandet i en motor utgående från ylindertry ksmätningar. Dessa metoder appli eras o h utvärderas på en av Saab Automobile utve klad prototypmotor med variabelt kompressionsförhållande. När man beräknar parametrar i matematiska modeller där parametrarna har fysikalisk tolkning, såsom temperatur, har användaren ofta förkunskap o h erfarenhet om vilka värden dessa bör anta. Den tredje delen utve klar ett verktyg för hur användaren kan väga in sådan förkunskap när parametrarnas värden beräknas utgående från ylindertry ksmätningar o h den valda matematiska modellen. Särskilt en av metoderna ger en bra kompromiss mellan användarens förkunskap o h mätdata. Detta verktyg kan användas för analys av frigjord värme o h motorkalibrering, som ett diagnosverktyg o h som ett analysverktyg för framtida motordesigner.

A knowledgments

This work has been arried out at the department of Ele tri al Engineering, division of Vehi ular Systems at Linköpings Universitet, Sweden. It was nan ially funded by the Swedish Agen y for Innovation Systems (VINNOVA) through the ex ellen e enter ISIS (Information Systems for Industrial ontrol and Supervision), and by the Swedish Foundation for Strategi Resear h SSF through the resear h enter MOVII. Their support is gratefully a knowledged. My time as a PhD student has broaden my mind and been enjoyable and I would thank all my olleagues, former and present, at Vehi ular Systems sin e your are all jointly the ause of that. I would also like to thank my supervisors Dr. Lars Eriksson and Professor Lars Nielsen for their guidan e and for letting me join the resear h group of Vehi ular Systems. Dr. Eriksson has inspired me greatly through interesting dis ussions and through his true enthusiasm for engine resear h. I have enjoyed working with you. My o-authors Prof. Lars Nielsen, Ylva Nilsson and Dr. Jan Åslund are also a knowledged for fruitful ollaborations. They have, as well as Dr. Ingemar Andersson, Dr. Per Andersson, Dr. Gunnar Cedersund, Dr. Erik Frisk, Martin Gunnarsson, Per Öberg and many others, ontributed to my resear h by insightful dis ussions and reative eorts. Martin Gunnarsson is also a knowledged for keeping the engine lab running, as well as Carolina Fröberg for making administrative stu work smoothly. Dr. Erik Frisk v

vi deserves an extra mention sin e he has onstantly been within range of my questions and he has always helped out without hesitation. Dr. Lars Eriksson has proof-read the entire thesis, while Dr. Erik Frisk, Dr. Jan Åslund and Per Öberg have proof-read parts of it. I am very grateful for the time and eort you have put into my work. Dr. Ingemar Andersson has inspired me to nish the thesis, and Professor Lars Nielsen has helped me to stay fo used during the nal writing of the thesis. To all the people who have hosted me during the middle part of my PhD studies, among those Andrej Perkovi and Maria Asplund, Dan and Tottis Lawesson, Fredrik Gustavsson and Louise Gustavsson-Ristenvik, Pontus Svensson, Mathias and Sara Tyskeng, and espe ially Jan Brugård, I wish to dire t my sin ere gratitude for rooming me but also for making sure I had an enjoyable stay. The year I shared house with Ingemar and Janne is a memory for life. The support from my family, my in-law family and all my friends has been invaluable. This work would never have been a

omplished without the indefatigable love, support and en ouragement of Soa. Finally I would like to thank Nora, Samuel, Ebba and Soa, whom I love more than life itself, for all the joy and adventures you bring along. Linköping, July 2007

Markus Klein

Contents

1

I 2

3

Introdu tion

1

Modeling

9

1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Publi ations . . . . . . . . . . . . . . . . . . . . . . . . .

An overview of single-zone heat-release models

2.1 2.2 2.3 2.4 2.5 2.6 2.7

Model basis and assumptions . . Rassweiler-Withrow model . . . . Apparent heat release model . . Matekunas pressure ratio . . . . Gatowski et al. model . . . . . . Comparison of heat release tra es Summary . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

3.1 Pressure sensor model . . . . . . . . . . . . . . . . . . 3.1.1 Parameter initialization  pressure oset ∆p . . 3.1.2 Parameter initialization  pressure gain Kp . . 3.1.3 Crank angle phasing . . . . . . . . . . . . . . . 3.1.4 Parameter initialization  rank angle oset ∆θ 3.2 Cylinder volume and area models . . . . . . . . . . . .

. . . . . .

Heat-release model omponents

vii

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

3 6 6

11

12 15 18 20 20 24 26

27

27 28 28 29 30 31

viii 3.2.1 Parameter initialization  learan e volume Vc . 3.3 Temperature models . . . . . . . . . . . . . . . . . . . . 3.3.1 Single-zone temperature model . . . . . . . . . . 3.3.2 Parameter initialization  ylinder pressure at IVC pIV C . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Parameter initialization  mean harge temperature at IVC TIV C . . . . . . . . . . . . . . . . . 3.4 Crevi e model . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Parameter initialization  revi e volume Vcr . . 3.4.2 Parameter initialization  ylinder mean wall temperature Tw . . . . . . . . . . . . . . . . . . . . . 3.5 Combustion model . . . . . . . . . . . . . . . . . . . . . 3.5.1 Vibe fun tion . . . . . . . . . . . . . . . . . . . . 3.5.2 Parameter initialization  energy released Qin . . 3.5.3 Parameter initialization  angle-related parameters {θig , ∆θd , ∆θb } . . . . . . . . . . . . . . . . 3.6 Engine heat transfer . . . . . . . . . . . . . . . . . . . . 3.6.1 Parameter initialization  {C1 , C2 } . . . . . . . . 3.7 Thermodynami properties . . . . . . . . . . . . . . . . 3.7.1 Parameter initialization  γ300 and b . . . . . . . 3.8 Summary of single-zone heat-release models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Sensitivity in pressure to parameter initialization . . . .

32 32 33 33 33 35 36 36 37 37 38 39 39 42 42 42 43 45

4 A spe i heat ratio model for single-zone heat release models 49 4.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Chemi al equilibrium . . . . . . . . . . . . . . . . . . . . 4.3 Existing models of γ . . . . . . . . . . . . . . . . . . . . 4.3.1 Linear model in T . . . . . . . . . . . . . . . . . 4.3.2 Segmented linear model in T . . . . . . . . . . . 4.3.3 Polynomial model in p and T . . . . . . . . . . . 4.4 Unburned mixture . . . . . . . . . . . . . . . . . . . . . 4.4.1 Modeling λ-dependen e with xed slope, b . . . . 4.5 Burned mixture . . . . . . . . . . . . . . . . . . . . . . . 4.6 Partially burned mixture . . . . . . . . . . . . . . . . . . 4.6.1 Referen e model . . . . . . . . . . . . . . . . . . 4.6.2 Grouping of γ -models . . . . . . . . . . . . . . . 4.6.3 Evaluation riteria . . . . . . . . . . . . . . . . . 4.6.4 Evaluation overing one operating point . . . . . 4.6.5 Evaluation overing all operating points . . . . . 4.6.6 Inuen e of γ -models on heat release parameters 4.6.7 Inuen e of air-fuel ratio λ . . . . . . . . . . . . 4.6.8 Inuen e of residual gas . . . . . . . . . . . . . . 4.6.9 Summary for partially burned mixture . . . . . .

50 50 52 52 52 53 54 57 58 63 63 64 67 68 72 76 77 79 82

ix 4.7 Summary and on lusions . . . . . . . . . . . . . . . . . II

Compression Ratio Estimation

83 85

5 Compression ratio estimation  with fo us on motored

y les 87 5.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Cylinder pressure modeling . . . . . . . . . . . . . . . . 5.2.1 Polytropi model . . . . . . . . . . . . . . . . . . 5.2.2 Standard model . . . . . . . . . . . . . . . . . . . 5.2.3 Cylinder pressure referen ing . . . . . . . . . . . 5.3 Estimation methods . . . . . . . . . . . . . . . . . . . . 5.3.1 Method 1  Sublinear approa h . . . . . . . . . . 5.3.2 Method 2  Variable proje tion . . . . . . . . . . 5.3.3 Method 3  Levenberg-Marquardt and polytropi model . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Method 4  Levenberg-Marquardt and standard model . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Summary of methods . . . . . . . . . . . . . . . 5.4 Simulation results . . . . . . . . . . . . . . . . . . . . . 5.4.1 Simulated engine data . . . . . . . . . . . . . . . 5.4.2 Results and evaluation for motored y les at OP2 5.4.3 Results and evaluation for motored y les at all OP . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Sensitivity analysis at OP2 . . . . . . . . . . . . 5.4.5 Results and evaluation for red y les at OP2 . . 5.5 Experimental results . . . . . . . . . . . . . . . . . . . . 5.5.1 Experimental engine data . . . . . . . . . . . . . 5.5.2 Results and evaluation for OP2 . . . . . . . . . . 5.5.3 Results and evaluation for all OP . . . . . . . . . 5.6 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . .

III sis

88 89 89 90 90 91 91 92

93 94 94 95 95 96 98 101 103 103 103 105 108 112

Prior Knowledge based Heat Release Analy115

6 Using prior knowledge for single-zone heat release analysis 117 6.1 Outline . . . . . . . . . . . . . . . . 6.2 Cylinder pressure modeling . . . . . 6.2.1 Standard model . . . . . . . . 6.2.2 Cylinder pressure parameters 6.3 Problem illustration . . . . . . . . . 6.4 Estimation methods . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

119 120 120 120 120 123

x 6.4.1 6.4.2 6.4.3 6.4.4

Method 1  SVD-based parameter redu tion . . Method 2  Regularization using prior knowledge How to determine the prior knowledge? . . . . . Summarizing omparison of methods 1 and 2 . .

7 Results and evaluation for motored y les 7.1 Simulation results  motored y les . . . . 7.1.1 Simulated engine data . . . . . . . 7.1.2 Parameter prior knowledge . . . . 7.1.3 Method 1 Results and evaluation 7.1.4 Method 2 Results and evaluation 7.1.5 Summary for simulation results . . 7.2 Experimental results  motored y les . . 7.2.1 Experimental engine data . . . . . 7.2.2 Parameter prior knowledge . . . . 7.2.3 Method 1 Results and evaluation 7.2.4 Method 2 Results and evaluation 7.3 Summary of results for motored y les . .

8 Results and evaluation for red y les

8.1 Simulation results  red y les . . . . . . 8.1.1 Simulated engine data . . . . . . . 8.1.2 Parameter prior knowledge . . . . 8.1.3 Method 1 Results and evaluation 8.1.4 Method 2 Results and evaluation 8.1.5 Summary for simulation results . . 8.2 Experimental results  red y les . . . . 8.2.1 Experimental engine data . . . . . 8.2.2 Parameter prior knowledge . . . . 8.2.3 Method 1 Results and evaluation 8.2.4 Method 2 Results and evaluation 8.3 Summary of results for red y les . . . . 8.4 Future Work . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

124 130 139 141

143 143 143 144 146 157 169 174 174 175 175 176 184

187 187 187 188 189 193 208 209 209 209 211 213 220 221

9 Summary and on lusions

223

Bibliography

229

9.1

A spe i heat ratio model for single-zone heat release models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 9.2 Compression ratio estimation . . . . . . . . . . . . . . . 224 9.3 Prior knowledge based heat release analysis . . . . . . . 226

xi

A A spe i heat ratio model  further details

A.1 Temperature models . . . . . . . . . . . . . . . . . . . . A.1.1 Single-zone temperature model . . . . . . . . . . A.1.2 Two-zone mean temperature model . . . . . . . . A.2 SAAB 2.3L NA  Geometri data . . . . . . . . . . . . . A.3 Parameters in single-zone model . . . . . . . . . . . . . A.4 Crevi e energy term . . . . . . . . . . . . . . . . . . . . A.5 Simple residual gas model . . . . . . . . . . . . . . . . . A.6 Fuel omposition sensitivity of γ . . . . . . . . . . . . . A.6.1 Burned mixture  Hydro arbons . . . . . . . . . A.6.2 Burned mixture  Al ohols . . . . . . . . . . . . A.6.3 Unburned mixtures . . . . . . . . . . . . . . . . . A.6.4 Partially burned mixture  inuen e on ylinder pressure . . . . . . . . . . . . . . . . . . . . . . . A.7 Thermodynami properties for burned mixture . . . . . A.8 Thermodynami properties for partially burned mixture

B Compression ratio estimation  further details B.1 B.2 B.3 B.4

Taylor expansions for sublinear approa h . Variable Proje tion Algorithm . . . . . . . SVC  Geometri data . . . . . . . . . . . Parameters in single-zone model . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

C Prior knowledge approa h  further details

237

237 237 238 239 240 242 245 246 246 247 250 250 253 254

263

263 264 265 265

267

C.1 Levenberg-Marquardt method . . . . . . . . . . . . . . . 267 C.1.1 Minimizing predi tion errors using a lo al optimizer268 C.2 Linear example . . . . . . . . . . . . . . . . . . . . . . . 272 C.2.1 Linear example for methods 1 and 2 . . . . . . . 274 C.3 Motivation for M2:3+ . . . . . . . . . . . . . . . . . . . 275 C.4 L850  Geometri data . . . . . . . . . . . . . . . . . . . 279 C.5 Parameters in single-zone model  motored y les . . . . 279 C.6 Parameters in single-zone model  red y les . . . . . . 279 C.7 Complementary results for prior knowledge approa h . . 282

D Notation

D.1 Parameters . . . . . . . . . . . . . . . . . D.1.1 Heat transfer . . . . . . . . . . . . D.1.2 Engine geometry . . . . . . . . . . D.1.3 Engine y le . . . . . . . . . . . . D.1.4 Thermodynami s and ombustion D.1.5 Parameter estimation . . . . . . . D.2 Abbreviations . . . . . . . . . . . . . . . . D.3 Evaluation riteria . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

289

289 289 290 290 291 292 293 294

xii

1 Introdu tion

Internal ombustion engines have been the primary ma hine for generating work in mobile appli ations for more than a entury, they are also ontinuing to be of high interest due to the high energy density of the fuels and their possibility to give good total fuel onsumption. Continuous improvements and renements are made to meet the in reasing performan e demands from ustomers and legislators, where both emissions and total system e onomy are important. Emission regulations from the legislators provide a hard limit on the designthey must be met. Today the state-of-the-art te hnology for a hieving low emissions from ombustion engines, is the gasoline engine equipped with a three-way atalyst (TWC). Regulations for diesel engines are also ontinuously being made stri ter to rea h those of the gasoline engine with a TWC. Development and ompetition between manufa turers strives to meet the needs of ustomers and deliver produ ts with better performan e both with respe t to power and fuel onsumption. Emerging te hnologies like the gas turbine and now the fuel ell pose possibilities and give a healthy ompetition, whi h also drives the te hnology development of ombustion engines forward. Engineers have met the hallenges posed by stri ter emission regulations through for example fundamental resear h on ombustion, adding new omponents to more omplex systems, as well as optimization of total system performan e. Engine systems are be oming in reasingly

omplex as new te hnologies are developed, but systemati methods are 1

2

CHAPTER 1.

INTRODUCTION

also required to handle and integrate these te hnologies. Some examples of promising te hniques for spark ignited (SI) engines are variable valve a tuation and variable ompression ratio. Both of these exemplify te hnologies that ontrol the development of the in- ylinder pro ess dire tly and where it is of importan e to get a

urate knowledge about the ombustion pro ess. The ombustion pro ess and other in- ylinder pro esses are dire tly ree ted in the measured ylinder pressure, and used as a standard tool for tuning and optimizing engine performan e. This is of ourse also important for onventional engines. In- ylinder pressure modeling and estimation

The in- ylinder pressure is important sin e it ontains information about the work produ tion in the ombustion hamber and thus gives important insight into the ontrol and tuning of the engine. Being able to a

urately model and extra t information from the ylinder pressure is important for the interpretation and validity of the result. Resear hers and engineers strive to extra t as mu h information as possible from the ombustion hamber through the in- ylinder pressure and models of dierent omplexity exist for interpretation of the ylinder pressures. Here the fo us is on single-zone models that treats the in ylinder ontents as a single zone and single uid. These models an des ribe the ylinder pressure well and have a low omputational omplexity, whi h is also an important parameter when analyzing engine data. Due to the short time s ales of the pro ess a sequen e of measurements on an engine gives huge amounts of data. These large sets of data have to be analyzed e iently, systemati ally, and with good a

ura y. in- ylinder pressure analysis, the most e ient models are the single zone models, and the a

ura y of these is the topi of the thesis. The foundation for the analysis of the model is the rst law of thermodynami s where the relation between work, volume, pressure and temperature is des ribed through the ratio of spe i heats. Analyses that have been performed show that the spe i heat ratio is of high importan e for the model and therefore this model omponent is studied in great detail. Therefore, the rst part of the thesis is on single-zone heat release modeling, where the spe i heat ratio model onstitutes a key part. In- ylinder pressure models in general have a number of parameters, that have to be determined. For an a

urate in- ylinder pressure analysis it is ne essary, but not su ient as will be shown later in the thesis, that the dieren e between the measured and modeled data is small. To minimize this dieren e, in a given measure, requires that the parameters are estimated, and this is the estimation problem. Whenever the parameters in a physi al model have a physi al mean-

1.1.

OUTLINE

3

ing, the user usually has an expe tation or prior knowledge of either what values the parameters should have, or at least the range in whi h they should be in. The user might even know whi h parameters that are most ertain. These are examples of information that ome from our prior knowledge. In the third part of the thesis it is shown how su h information an be in orporated in the estimation problem. Compression ratio estimation

The theme in the thesis is ylinder pressure and the se ond part treats

ompression ratio estimation based on measured ylinder pressure data. This parti ular problem is dire tly motivated by the variable ompression engine, where the ompression ratio an be hanged ontinuously to eliminate an important design trade-o made in onventional engines. High ompression ratios give good engine e ien y but at high loads a high ompression ratio an result in engine destru tion through engine kno k. In su h an engine the ompression ratio is hanged ontinuously to get the best performan e from the engine. When the engine is driven at low loads a high ompression ratio is sele ted for good ef ien y and at high loads a low ompression ratio is used to redu e engine kno k. Compression ratio estimation is studied for several reasons where the most important is for diagnosti purposes. A too high

ompression ratio an lead to engine destru tion while a too low ompression ratio gives a too high fuel onsumption. Four dierent methods for ompression ratio estimation are proposed and evaluated. The resear h was motivated by the variable ompression engine, but the methods are generally appli able and an also be used on onventional engines to get a better value of the ompression ratio from experimental data. 1.1

Outline

An outline of the thesis in terms of short summaries of ea h hapter is given below and indi ates the s ope of ea h hapter. The notation used is summarized in appendix D, where the parameters are given in appendix D.1, and the abbreviations are summarized in appendix D.2. In the thesis various evaluation riteria are used, and they are summarized in appendix D.3. Chapter 2: An overview of single-zone heat-release models

This hapter serves as an introdu tion to single-zone heat release modeling. First the basis and assumptions made for single-zone heat release modeling are given. Based on these, four well-known heat release mod-

4

CHAPTER 1.

INTRODUCTION

els are presented. These are ompared with respe t to their omputed heat release tra e given a ylinder pressure tra e.

Chapter 3: Heat-release model omponents The model omponents used in the most des riptive single-zone heat release model in hapter 2, the Gatowski et al. (1984) model, are des ribed. The model omponents of the other three heat release models form a subset of these. For ea h model omponent, a method to initialize the model omponent parameters is given. The sensitivity in

ylinder pressure for ea h of these parameters is then investigated. The

hapter ends with a summary of the equations, parameters, inputs and outputs of the Gatowski et al. model.

Chapter 4: A spe i heat ratio model for single-zone heat release models An a

urate spe i heat ratio model is important for an a

urate heat release analysis. This sin e the spe i heat ratio ouples the systems energy to other thermodynami quantities. This hapter therefore investigates models of the spe i heat ratio for the single-zone heat release model developed by Gatowski et al. (1984). The obje tive is to nd a model a

urate enough to only introdu e a ylinder pressure modeling error in the order of the ylinder pressure measurement noise, while keeping the omputational omplexity at a minimum. Based on assumptions of frozen mixture for the unburned mixture and hemi al equilibrium for the burned mixture, the spe i heat ratio is al ulated using a full equilibrium program for an unburned and a burned air-fuel mixture, and ompared to already existing and newly proposed models of γ . It is assumed that a general single-zone heat release model an be used as a referen e model. The evaluation is performed in terms of modeling error in γ and in

ylinder pressure. The impa t ea h γ -model has on the heat release, in terms of estimated heat release parameters in the Vibe fun tion is illustrated. The inuen e of fuel omposition, air-fuel ratio and residual gas ontent is also investigated. Large parts of the material in this hapter and in appendix A have previously been published in Klein and Eriksson (2004 ) and Klein and Eriksson (2004a). Appendix A ontains further details and argumentation that support the development of the spe i heat ratio models, and gives a ba kground and a thorough explanation of some of the details in the models.

1.1.

OUTLINE

5

Chapter 5: Compression ratio estimation  with fo us on motored y les The purpose of this hapter is to estimate the ompression ratio given a ylinder pressure tra e, in order to diagnose the ompression ratio if it e.g. gets stu k at a too high or too low ratio. Four methods for ompression ratio estimation based on ylinder pressure tra es are developed and evaluated for both simulated and experimental y les. A sensitivity analysis of how the methods perform when subje ted to parameter deviations in rank angle phasing, ylinder pressure bias and heat transfer is also made. In appendix B further details and argumentation on ompression ratio estimation for motored y les are given, and it serves as a omplement to this hapter. Chapter 5 together with appendix B is an edited version of Klein et al. (2006), whi h itself is based on Klein et al. (2004) and Klein and Eriksson (2005b).

Chapter 68: Using prior knowledge for single-zone heat release analysis Two methods that take parameter prior knowledge into a

ount, when performing parameter estimation on the Gatowski et al. model, are presented. The appli ation in mind is a tool for ylinder pressure estimation that is a

urate, systemati and e ient. The methods are des ribed in detail, and it is shown how to in orporate the prior knowledge in a systemati manner. Guidelines of how to determine the prior knowledge for a spe i appli ation are then given. The performan e of the two methods is evaluated for both simulated and experimentally measured ylinder pressure tra es. These evaluations are made in hapter 7: Results and evaluation for motored y les and in hapter 8: Results and evaluation for red y les. Appendix C ontains further details and argumentation that support the development and evaluation of the two parameter estimation methods.

Experimental and simulated engine data During the proje t dierent engines have been available in the Vehi ular Systems engine laboratory. Therefore the simulated and experimental data used in the hapters are from dierent engines. Chapters 24 use a naturally aspirated 2.3L engine from SAAB, and its geometry is given in appendix A.2. In hapter 5 the SAAB Variable Compression (SVC) engine is used, with the geometry given in appendix B.3. The results for hapters 68 are based on data from a turbo harged 2.0L SAAB engine, see appendix C.4.

6

CHAPTER 1.

1.2

INTRODUCTION

Contributions

The following list summarizes the main ontributions of this thesis: • The interrelation between models in the single-zone heat release model family is shown. A method for nding nominal values for all parameters therein is suggested. • It is shown that the spe i heat ratio model is the most important omponent in ylinder pressure modeling. • The importan e of using the ylinder pressure error as a measure of how well a spe i heat ratio model performs is pinpointed. • A new spe i heat ratio model to be used primarily in singlezone heat release models. This model an easily be in orporated with the Gatowski et al.-model, and redu es the modeling error to be of the same order as the ylinder pressure measurement noise. • Four methods for estimating the ompression ratio index, given a ylinder pressure tra e are proposed. One method is re ommended for its a

ura y, while another is preferable when omputational e ien y is important. • Two methods of using prior knowledge applied to the in- ylinder pressure estimation problem are presented and evaluated. For the se ond method, it is shown that prior knowledge with individually set parameter un ertainties yields more a

urate and robust estimates. 1.3

Publi ations

In the resear h work leading to this thesis, the author has published a li entiate thesis and the following papers: Journal papers:

• M. Klein, L. Eriksson, and Y. Nilsson (2003). Compression estimation from simulated and measured ylinder pressure. SAE 2002 Transa tions Journal of Engines, 2002-01-0843, 111(3), 2003. • M. Klein and L. Eriksson (2005a). A spe i heat ratio model for single-zone heat release models. SAE 2004 Transa tions Journal of Engines, 2004-01-1464, 2005. • M. Klein, L. Eriksson and J. Åslund (2006). Compression ratio estimation based on ylinder pressure data. Control Engineering Pra ti e, 14(3):197211.

1.3.

PUBLICATIONS

7

Conferen e papers:

• M. Klein, L. Eriksson and Y. Nilsson (2002). Compression es-

timation from simulated and measured ylinder pressure. Ele SP-1703, SAE Te hni al Paper 2002-010843. SAE World Congress, Detroit, USA, 2002.

troni engine ontrols,

• M. Klein and L. Eriksson (2002). Models, methods and per-

forman e when estimating the ompression ratio based on the

ylinder pressure. Fourth onferen e on Computer S ien e and Systems Engineering in Linköping (CCSSE), 2002.

• M. Klein and L. Eriksson (2004 ). A spe i heat ratio model

for single-zone heat release models. Modeling of SI engines, SP1830, SAE Te hni al Paper 2004-01-1464. SAE World Congress, Detroit, USA, 2004.

• M. Klein, L. Eriksson and J. Åslund (2004). Compression ratio

estimation based on ylinder pressure data. In pro eedings of IFAC symposium on Advan es in Automotive Control, Salerno, Italy, 2004.

• M. Klein (2004). A spe i heat ratio model and ompression

ratio estimation. Li entiate thesis, Vehi ular Systems, Linköping University, 2004. LiU-TEK-LIC-2004:33, Thesis No. 1104.

• M. Klein and L. Eriksson (2004b). A omparison of spe i heat

ratio models for ylinder pressure modeling. Fifth onferen e on Computer S ien e and Systems Engineering in Linköping (CCSSE), 2004.

• M. Klein and L. Eriksson (2005b). Utilizing ylinder pressure

data for ompression ratio estimation. IFAC World Congress, Prague, Cze h Republi , 2005.

The following onferen e papers have also been produ ed by the author during the proje t, but they are not expli itly in luded in the thesis: • M. Klein and L. Nielsen (2000). Evaluating some Gain S hedul-

ing Strategies in Diagnosis of a Tank System. In pro eedings of IFAC symposium on Fault Dete tion, Supervision and Safety for Te hni al Pro esses, Budapest, Hungary, 2000.

• M. Klein and L. Eriksson (2006). Methods for ylinder pres-

sure based ompression ratio estimation. Ele troni Engine Control, SP-2003, SAE Te hni al paper 2006-01-0185. SAE World Congress, Detroit, USA, 2006.

8

CHAPTER 1.

INTRODUCTION

Part I

Modeling

9

2 An overview of single-zone heat-release models

When analyzing the internal ombustion engine the in- ylinder pressure has always been an important experimental diagnosti , due to its dire t relation to the ombustion and work produ ing pro esses (Chun and Heywood, 1987; Cheung and Heywood, 1993). The in- ylinder pressure ree ts the ombustion pro ess, the piston work produ ed on the gas, heat transfer to the hamber walls, as well as mass ows in and out of

revi e regions between the piston, rings and ylinder liner. Thus, when an a

urate knowledge of how the ombustion pro ess propagates through the ombustion hamber is desired, ea h of these pro esses must be related to the ylinder pressure, so the ombustion pro ess an be distinguished. The redu tion of the ee ts of volume

hange, heat transfer, and mass loss on the ylinder pressure is alled heat-release analysis and is done within the framework of the rst law of thermodynami s. In parti ular during the losed part of the engine y le when the intake and exhaust valves are losed. The most

ommon approa h is to regard the ylinder ontents as a single zone, whose thermodynami state and properties are modeled as being uniform throughout the ylinder and represented by average values. No spatial variations are onsidered, and the model is therefore referred to as zero-dimensional. Models for heat transfer and revi e ee ts an easily be in luded in this framework. Another approa h is to do a more detailed thermodynami analysis by using a multi-zone model, where the ylinder is divided into a number of zones, diering in omposition and properties. Ea h zone being uniform in omposition and temper11

12

CHAPTER 2. OVERVIEW OF HEAT-RELEASE MODELS

ature, and the pressure is the same for all zones, see for e.g. (Nilsson and Eriksson, 2001). The goal of this hapter is to show the stru ture of dierent singlezone heat-release model families and how they are derived. The dis ussion of details in the model omponents are postponed to hapter 3, sin e they might distra t the readers attention from the general stru ture of the model family. Chapter 3 gives a more thorough des ription of the model omponents. Single-zone models for analyzing the heat-release rate and simulating the ylinder pressure are losely onne ted; they share the same basi balan e equation and an be interpreted as ea h others inverse. They are both des ribed by a rst order ordinary dierential equation that has to be solved. In heat release models a pressure tra e is given as input and the heat release is the output, while in pressure models a heat release tra e is the input and pressure is the output. For a given heat-release model an equivalent pressure model is obtained by reordering the terms in the ordinary dierential equation. Sin e they are so losely onne ted it is bene ial to dis uss them together. 2.1

Model basis and assumptions

The basis for the majority of the heat-release models is the rst law of thermodynami s; the energy onservation equation. For an open system it an be stated as dU = žQ − žW +

X

hi dmi ,

(2.1)

i

where dU is the hange in internal energy of the mass in the system, žQ is the heat transported to the system, žW is the work produ ed by the P system and i hi dmi is the enthalpy ux a ross the system boundary. Possible mass ows dmi are: 1) ows in and out of the valves; 2) dire t inje tion of fuel into the ylinder; 3) ows in and out of revi e regions; 4) piston ring blow-by. The mass ow dmi is positive for a mass ow into the system and hi is the mass spe i enthalpy of ow i. Note that hi is evaluated at onditions given by the zone the mass element leaves. As mentioned earlier, single-zone models is our fo us at the moment, so we will now look into those in more detail. Some ommonly made assumptions for the single-zone models are: • the ylinder ontents and the state is uniform throughout the

entire hamber.

• the ombustion is modeled as a release of heat.

2.1.

MODEL BASIS AND ASSUMPTIONS

13

Spark plug Burned gas

Qht

Unburned gas

Mcr

Qhr

Wp

Piston

Figure 2.1: S hemati of the ombustion pro ess in the ylinder, that denes the sign onvention used in the pressure and heat-release models. • the heat released from the ombustion o

urs uniformly in the

hamber.

• the gas mixture is an ideal gas.

Consider the ombustion hamber to be an open system (single zone), with the ylinder head, ylinder wall and piston rown as boundary. Figure 2.1 shows a s hemati of the ombustion hamber, where the sign onventions used in pressure and heat-release models are dened. The hange in heat žQ onsists of the released hemi al energy from the fuel žQch , whi h is a heat adding pro ess, and the heat transfer to the hamber walls žQht , whi h is a heat removing pro ess. The heat transport is therefore represented by žQ = žQch − žQht . Note that the heat transfer ools the gases at most times, but in some instan es it heats the air-fuel mixture. The only work onsidered is the work done by the uid on the piston Wp and it is onsidered positive, therefore žW = žWp . The rst law of thermodynami s (2.1) an then be rewritten as žQch = dUs + žWp −

X

hi dmi + žQht .

(2.2)

i

The piston work žWp is assumed to be reversible and an be written as žWp = pdV . For an ideal gas, the hange in sensible energy dUs is a fun tion of mean harge temperature T only, thus: Us = mtot u(T ),

(2.3)

14

CHAPTER 2. OVERVIEW OF HEAT-RELEASE MODELS

whi h in its dierentiated form be omes: dUs = mtot cv (T )dT + u(T )dmtot , 

(2.4)



∂u where mtot is the harge mass, and cv = ∂T is the mass spe i V heat at onstant volume. The mean temperature is found from the , and its dierentiated form is ideal gas law as T = mpV tot R

dT =

1 (V dp + pdV − RT dmtot ), mtot R

(2.5)

assuming R to be onstant. For reading onvenien e, the dependen e of T in cp , cv and γ is often left out in the following equations. Equation (2.2) an now be rewritten as žQch =

X cv cv + R hi dmi + žQht , V dp + p dV + (u − cv T )dmtot − R R i

(2.6) using equations (2.4) and (2.5). The spe i heat ratio is dened as c γ = cpv and with the assumption of an ideal gas the mass spe i gas

onstant R an be written as R = cp −cv , yielding that the mass spe i heat at onstant volume is given by cv =

R . γ−1

(2.7)

The mass spe i heat is the amount of energy that must be added or removed from the mixture to hange its temperature by 1 K at a given temperature and pressure. It relates internal energy with the thermodynami state variables p and T , and is therefore an important part of the heat release modeling. Inserting (2.7) into (2.6) results in žQch =

X γ RT 1 hi dmi + žQht . V dp+ p dV +(u− )dmtot − γ−1 γ−1 γ−1 i

(2.8) From this equation, four dierent single-zone models with various levels of omplexity will be derived. To begin with the isentropi relation is derived, then the polytropi model is formulated and this model forms the basis for al ulating the mass fra tion burned with the RassweilerWithrow method (Rassweiler and Withrow, 1938). Se ondly, a model for omputing the apparent heat release rst proposed in Krieger and Borman (1967) will be derived. Thirdly, the pressure ratio developed by Matekunas (1983) is shortly summarized. Finally, a model in luding heat transfer and revi e ee ts (Gatowski et al., 1984) will be given.

2.2.

RASSWEILER-WITHROW MODEL

15

The isentropi pro ess and isentropi relation In many situations real pro esses are ompared to ideal pro esses and as

omparison the isentropi pro ess is normally used. From the isentropi pro ess an isentropi relation an be found by integrating the rst law of thermodynami s (2.8). The assumptions are: • No mass transfer: Crevi e ee ts and leakages to the rank ase (often alled blow-by) are non-existent, i.e. dmtot = dmi = 0. • Neither heat transfer nor heat release:

- Heat transfer is not expli itly a

ounted for, i.e. žQht = 0, and thus žQ = žQch − žQht = žQch . - Using the fa t that there is no release of hemi al energy during the ompression phase prior to the ombustion or during the expansion phase after the ombustion, therefore žQ = 0 for these regions.

• The spe i heat ratio γ is onstant.

The rst two assumptions yield that (2.8) an be expressed as: dp = −

γp dV. V

(2.9)

From (2.9) and the last assumption above the isentropi relation is found by integrating as pV γ = C = onstant

(2.10)

by noting that γ is onsidered to be onstant. 2.2

Rassweiler-Withrow model

The Rassweiler-Withrow method was originally presented in 1938 and many still use the method for determining the mass fra tion burned, due to its simpli ity and it being omputationally e ient. The mass (θ) is the burned mass mb (θ) normalized by fra tion burned xb (θ) = mmbtot the total harge mass mtot , and it an be seen as a normalized version of the a

umulated heat-release tra e Qch (θ) su h that it assumes values in the interval [0, 1℄. The relation between the mass fra tion burned and the amount of heat released an be justied by noting that the energy released from a system is proportional to the mass of fuel that is burned. The input to the method is a pressure tra e p(θj ) where the

rank angle θ at ea h sample j is known (or equivalently; the volume is known at ea h sample) and the output is the mass fra tion burned tra e xb,RW (θj ).

16

CHAPTER 2. OVERVIEW OF HEAT-RELEASE MODELS

A ornerstone for the method is the fa t that pressure and volume data during ompression and expansion an be approximated by the polytropi relation pV n = onstant. (2.11) This expression omes from the isentropi relation (2.10) but γ is ex hanged for a onstant exponent n ∈ [1.25, 1.35]. This has been shown to give a good t to experimental data for both ompression and expansion pro esses in an engine (Lan aster et al., 1975). The exponent n is termed the polytropi index. It diers from γ sin e some of the ee ts of heat transfer are in luded impli itly in n. It is omparable to the average value of γu for the unburned mixture during the ompression phase, prior to ombustion. But due to heat transfer to the

ylinder walls, index n is greater than γb for the burned mixture during expansion (Heywood, 1988, p.385). When onsidering ombustion where žQ = žQch 6= 0, equation (2.8)

an be rewritten as dp =

n−1 np žQ − dV = dpc + dpv , V V

(2.12)

where dpc is the pressure hange due to ombustion, and dpv is the pressure hange due to volume hange, ompare dp in (2.9). In the Rassweiler-Withrow method (Rassweiler and Withrow, 1938), the a tual pressure hange ∆p = pj+1 −pj during the interval ∆θ = θj+1 −θj , is assumed to be made up of a pressure rise due to ombustion ∆pc , and a pressure rise due to volume hange ∆pv , (2.13)

∆p = ∆pc + ∆pv ,

whi h is justied by (2.12). The pressure hange due to volume hange during the interval ∆θ is approximated by the polytropi relation (2.11), whi h gives ∆pv (j) = pj+1,v − pj = pj



Vj Vj+1

n

 −1 .

(2.14)

Applying ∆θ = θj+1 − θj , (2.13) and (2.14) yields the pressure hange due to ombustion as ∆pc (j) = pj+1 − pj



Vj Vj+1

n

.

(2.15)

By assuming that the pressure rise due to ombustion in the interval ∆θ is proportional to the mass of mixture that burns, the mass fra tion burned at the end of the j 'th interval thus be omes Pj mb (j) k=0 ∆pc (k) xb,RW (j) = , = PM mb (total) k=0 ∆pc (k)

(2.16)

17

RASSWEILER-WITHROW MODEL

Mass fraction burned [−]

Cylinder pressure [MPa]

2.2.

2 1.5 1 0.5 0

−100

−50 0 50 Crank angle [deg ATDC]

100

−100

−50 0 50 Crank angle [deg ATDC]

100

1

0.5

0

Figure 2.2: Top: Fired pressure tra e (solid) and motored pressure tra e (dash-dotted). Bottom: Cal ulated mass fra tion burned prole using the Rassweiler-Withrow method. where M is the total number of rank angle intervals and ∆pc (k) is found from (2.15). The result from a mass fra tion burned analysis is shown in gure 2.2, where the mass fra tion burned prole is plotted together with the orresponding pressure tra e. In the upper plot two

ylinder pressure tra es, one from a red y le (solid) and one from a motored y le (dash-dotted) are displayed. When the pressure rise from the ombustion be omes visible, i.e. it rises above the motored pressure, the mass fra tion burned prole starts to in rease above zero. The mass fra tion burned prole in reases monotonously as the ombustion propagates through the ombustion hamber. Equations (2.15)-(2.16) form the lassi al Rassweiler-Withrow mass fra tion burned method. If instead a heat-release tra e is sought, the pressure hange due to ombustion in (2.12), dpc = n−1 V žQ, an be rewritten and approximated by ∆QRW (j) =

Vj+1/2 ∆pc (j), n−1

(2.17)

where the volume V during interval j is approximated with Vj+1/2 (the volume at the enter of the interval), and ∆pc (j) is found from (2.15). The heat-release tra e is then found by summation. This method will be alled the Rassweiler-Withrow heat release method. The al ulated heat release approximates the released hemi al energy from the fuel minus energy- onsuming pro esses su h as the heat transfer to the

ylinder walls and revi e ee ts. If heat transfer and revi e ee ts

CHAPTER 2. OVERVIEW OF HEAT-RELEASE MODELS

400 200

Mass fraction burned [−]

Heat released Q [J]

18

0 −100

−50 0 50 Crank angle [deg ATDC]

100

−100

−50 0 50 Crank angle [deg ATDC]

100

1

0.5

0

Figure 2.3: Cal ulated heat-release tra e (upper) and mass fra tion burned tra e (lower), using the apparent heat release (solid) and Rassweiler-Withrow (dash-dotted) methods. where non-existent, the heat release would orrespond dire tly to the amount of energy added from the hemi al rea tions. The heat-release tra e for the same data as in gure 2.2 is displayed in the upper plot of gure 2.3 as the dash-dotted line. 2.3

Apparent heat release model

The work by Krieger and Borman (1967) was derived from the rst law of thermodynami s and alled the omputation of apparent heat release. It is also alled the omputation of net heat release. The method takes neither heat transfer nor revi e ee ts into a

ount, thus žQht is lumped into žQ = žQch − žQht and dmtot = dmi = 0 in (2.8). Hen e, the apparent heat release žQ an be expressed as: žQAHR =

γ(T ) 1 V dp + p dV, γ(T ) − 1 γ(T ) − 1

(2.18)

whi h is the same expression as the Rassweiler-Withrow method was based upon (2.12), but assuming that γ(T ) = n. The mass fra tion burned xb,AHR is omputed by integrating (2.18) and then normalizing with the maximum value of the a

umulated heat release QAHR , i.e. R žQAHR dθ QAHR (θ) dθ xb,AHR (θ) = = . max QAHR max QAHR

(2.19)

2.3.

19

APPARENT HEAT RELEASE MODEL

Method Rassweiler-Withrow Apparent heat release

θ10

-6.4 -4.5

θ50

9.8 11.0

θ85

26.9 25.2

∆θb

33.3 29.8

Table 2.1: Crank angle positions for 10 %, 50 % and 85 % mfb as well as the rapid burn angle ∆θb = θ85 − θ10 , all given in degrees ATDC for the mass fra tion burned tra e in gure 2.3. The Rassweiler-Withrow method in (2.16) is a dieren e equation, and this auses an quantization ee t ompared to the ordinary dierential equation given in (2.18). The net heat-release tra e and mass fra tion burned prole from the Krieger and Borman model are similar to those from the Rassweiler-Withrow method, the former being physi ally the more a

urate one. One example is given in gure 2.3, where the upper plot shows the net heat-release tra es and the lower plot shows the mass fra tion burned tra es, from the ylinder pressure in gure 2.2. For this parti ular ase, the Rassweiler-Withrow method yields a slower burn rate ompared to the apparent heat release method for the same data. This is ree ted in the rank angle for 50 % mfb θ50 , whi h is 11.0 [deg ATDC℄ for the apparent heat release method and 9.8 [deg ATDC℄ for the Rassweiler-Withrow method. Table 2.1 summarizes the rankangle positions for 10 %, 50 % and 85 % mfb as well as the rapid burn angle duration ∆θb , and shows that the Rassweiler-Withrow method yields a shorter burn duration for this parti ular ase. The rapid burn angle duration is dened as ∆θb = θ85 − θ10 . The shorter burn duration is also ree ted in the heat release tra e, and the dieren e is due to the assumptions on n and Vj+1/2 in the Rassweiler-Withrow method. The mass fra tion burned prole is al ulated assuming that the mass of burned mixture is proportional to the amount of released hemi al energy.

Pressure simulation An ordinary dierential equation for the pressure an be simulated by solving (2.18) for the pressure dierential dp: dp =

(γ(T ) − 1) žQ − γ(T ) p dV . V

(2.20)

When performing a heat-release analysis the pressure is used as input and the heat release is given as output, and when the pressure tra e is being simulated the heat-release tra e is given as input. Therefore a

ylinder pressure simulation based on (2.20), an be seen as the inverse of the heat release analysis (2.18). The only additional information that is needed for the omputation is the initial value of the pressure.

20 2.4

CHAPTER 2. OVERVIEW OF HEAT-RELEASE MODELS

Matekunas pressure ratio

The pressure ratio on ept was developed by Matekunas (1983) and it is a omputationally e ient method to determine an approximation of the mass fra tion burned tra e. The pressure ratio is dened as the ratio of the ylinder pressure from a red y le p(θ) and the orresponding motored ylinder pressure p0 (θ): P R(θ) =

p(θ) − 1. p0 (θ)

(2.21)

The pressure ratio (2.21) is then normalized by its maximum P RN (θ) =

P R(θ) , max P R(θ)

(2.22)

whi h produ es tra es that are similar to the mass fra tion burned proles. The dieren e between them has been investigated in Eriksson (1999), and for the operating points used, the dieren e in position for P RN (θ) = 0.5 was in the order of 1-2 degrees. This suggests P RN (θ)

an be used as the mass fra tion burned tra e xb,MP R . The ylinder pressure in the upper plot of gure 2.4 yields the pressure ratio PR (2.21) given in the middle plot, and an approximation of the mass fra tion burned in the lower plot. 2.5

Gatowski et al. model

A more omplex model is to in orporate models of heat transfer, revi e ee ts and thermodynami properties of the ylinder harge into the energy onservation equation (2.8). This was done in Gatowski et al. (1984), where a heat-release model was developed and applied to three dierent engine types, among those a spark-ignited engine.

Crevi e ee t model Crevi es are small, narrow volumes onne ted to the ombustion hamber. During ompression some of the harge ows into the revi es, and remain there until the expansion phase, when most of the harge returns to the ombustion hamber and some harge stays in the revi es. Also, a small part of the harge in the ylinder blows by the top piston ring, before it either returns to the ylinder or ends up in the rank ase, a phenomena termed blow-by. Sin e the ame an not propagate into the revi es, the harge residing in the revi es is not ombusted. The temperature in the revi es are assumed to be lose to the ylinder wall temperature, due to that the revi es are narrow (Heywood,

21

GATOWSKI ET AL. MODEL

Pressure [MPa]

2.5.

2 1 0

−100

−50

0

50

100

−100

−50

0

50

100

−100

−50 0 50 Crank angle [deg ATDC]

100

PR [−]

3 2 1

xb,MPR [−]

0 1 0.5 0

Figure 2.4: Top: Fired pressure tra e (solid) and motored pressure tra e (dash-dotted), same as in the upper plot of gure 2.2. Middle: Matekunas pressure ratio P R(θ) (2.21). Bottom: Computed mass fra tion burned prole using (2.22). 1988, p.387). This has the result that during the losed phase a substantial amount of harge ould be trapped in the revi es. A

ording to Gatowski et al. (1984), the revi e volumes onstitute as mu h as 1-2 per ent of the learan e volume in size. It is also shown that due to the temperature dieren e in the ylinder and in the revi es, as mu h as 10 (mass) per ent of the harge ould then be trapped in revi es at peak pressure. The model in Gatowski et al. (1984) assumes that all revi es an be modeled as a single aggregate onstant volume Vcr , and that the

harge in the revi e assumes the wall temperature Tw and has the same pressure as the ombustion hamber. The ideal gas law thus gives the following expression for the mass in the revi e mcr =

Vcr p Vcr =⇒ dmcr = dp, R Tw R Tw

(2.23)

where it is assumed that Tw and R are onstant. Here, we will only onsider spark-ignition engines with a premixed air-fuel harge during the losed part of the engine y le. Blow-by is also negle ted, hen e the only mass ow o

urring is the one in and out of the revi e region. Mass balan e thus yields dmtot = dmi = −dmcr .

(2.24)

With the denition of enthalpy h = RT + u and using (2.23)(2.24),

22

CHAPTER 2. OVERVIEW OF HEAT-RELEASE MODELS

then equation (2.8) an then be rewritten to: žQch = =

1 γ−1 V 1 γ−1 V

dp + dp +

γ γ−1 p dV γ γ−1 p dV

Vcr RT + ( γ−1 + h′ − u) RT dp + žQht w ′ T u −u Vcr ′ + ( γ−1 + T + R ) Tw dp + žQht .

(2.25) To get a ylinder pressure model, equation (2.25) an be solved for the pressure dierential yielding the following expression: dp =

γ p dV − žQht žQch − γ−1 1 γ−1

V +

Vcr Tw



T γ−1

+ T′ +

u′ −u R

(2.26)

.

The enthalpy h′ is evaluated at ylinder onditions when the revi e mass ow is out of the ylinder (dmcr > 0), and at revi e onditions otherwise.

Heat transfer model The heat transfer model relies upon Newton's law of ooling (2.27)

Q˙ ht = hc A ∆T = hc A (T − Tw ),

and Wos hni (1967) found a orrelation between the onve tion heat transfer oe ient hc and some geometri and thermodynami properties1 , hc =

0.013B −0.2p0.8 C1 up + T 0.55

C2 (p−p0 )Tref Vs
0.8 pref Vref

.

(2.28)

Wos hni's heat transfer orrelation model will be further dis ussed in se tion 3.6. Note that when simulating heat transfer in the rank angle domain, 60 žQht žQht dt = = Q˙ ht (2.29) dθ

dt

dθ

2πN

should be used, where the engine speed N [rpm℄ is assumed onstant in the last equality.

Model of thermodynami properties The ratio of spe i heats γ(T ) is modeled as a linear fun tion of temperature γlin (T ) = γ300 + b (T − 300). (2.30) In Gatowski et al. (1984) it is stated that this omponent is important, sin e it aptures how the internal energy varies with temperature. This is an approximation of the thermodynami properties 1 The value of the rst oe ient diers from the one in (Wos hni, 1967), sin e it is re al ulated to t the SI-unit system.

2.5.

23

GATOWSKI ET AL. MODEL

but it is further stated that this approximation is onsistent with the ) other approximations made in the model. Using γ(T ) = ccvp (T (T ) and R(T ) = cp (T ) − cv (T ), together with the linear model of γ(T ) in (2.30), gives the following expression for cv (T ): cv (T ) =

R R = . γ(T ) − 1 γ300 + b(T − 300) − 1

(2.31)

The only thing remaining in (2.25) to obtain a full des ription of the ∂u model, is an expression for u′ − u. Remembering that cv = ( ∂T )V , ′ u −u an be found by integrating cv . This des ribes the sensible energy

hange for a mass that leaves the revi e and enters the ylinder. The integration is performed as: u′ − u = = =

R T′

cv dT T R ′ {ln(γ b  ′300 +b (T γ −1 R lin b ln γlin −1 ,

− 300) − 1) − ln(γ300 + b (T − 300) − 1)}

(2.32)

where equation (2.31) is used. Gross heat-release simulation

Inserting equations (2.23) to (2.32) into (2.8), yields the following expression for the released hemi al energy: žQch = 1 γ R V dp + p dV + žQht + (cv T + R T ′ + ln γ−1 γ−1 b



 Vcr γ′ − 1 ) dp, γ − 1 R Tw

(2.33)

whi h is reformulated as 1 γ V dp + p dV +žQht + γ−1 γ−1 | {z } dQnet   ′ 1 1 γ − 1 Vcr ) dp . T + T ′ + ln ( γ−1 b γ−1 Tw | {z }

žQch =

(2.34)

dQcrevice

This ordinary dierential equation an easily be solved numeri ally for the heat-release tra e, if a ylinder pressure tra e is provided, together with an initial value for the heat release. Given the ylinder pressure in gure 2.2, the heat-release tra e given in gure 2.5 is al ulated. The solid line is the gross heat released, i.e. the hemi al energy released

24

CHAPTER 2. OVERVIEW OF HEAT-RELEASE MODELS

Qcrevice

500

Qgross

450

Q

ht

Heat released Q [J]

400 Qnet

350 300 250 200 150 100 50 0 −100

−50 0 50 Crank angle [deg ATDC]

100

Figure 2.5: Heat-release tra e from the Gatowski model given the ylinder pressure in gure 2.2. during the engine y le. The dash-dotted line shows the heat released if not onsidering the revi e ee t, and the dashed line shows the net heat release, i.e. when neither heat transfer nor revi e ee ts are

onsidered. For this parti ular ase, the heat transfer is about 70 J and the revi e ee t is about 30 J, i.e. approximately 14 and 6 per ent of the total released energy respe tively.

Cylinder pressure simulation Reordering (2.34), gives an expression for the pressure dierential as dp =

γ p dV − žQht žQch − γ−1 1 γ−1

V +

Vcr Tw



T γ−1

 .  ′  −1 + T′ + 1b ln γγ−1

(2.35)

This ordinary dierential equation an easily be solved numeri ally for the ylinder pressure, if a heat-release tra e žQch is provided, together with an initial value for the ylinder pressure. 2.6

Comparison of heat release tra es

The single-zone heat release models presented in the previous se tions all yield dierent heat release tra es for a given ylinder pressure tra e. This is shown in gure 2.6, where the heat release tra es for the Rassweiler-Withrow, apparent heat release and Gatowski models are

25

COMPARISON OF HEAT RELEASE TRACES

400 200

Mass fraction burned [−]

Heat released Q [J]

2.6.

0 −30

−20

−10

0 10 20 30 Crank angle [deg ATDC]

40

50

60

−20

−10

0 10 20 30 Crank angle [deg ATDC]

40

50

60

1

0.5

0 −30

Figure 2.6: Upper: Heat-release tra es from the three methods, Gatowski (solid), apparent heat release (dashed) and RassweilerWithrow (dash-dotted), given the ylinder pressure in gure 2.2. Lower: Mass fra tion burned tra es orresponding to the upper plot with the addition of the Matekunas pressure ratio (dotted). displayed in the upper plot. As expe ted, the a

umulated heat release is higher for the Gatowski model sin e it a

ounts for heat transfer and

revi e ee ts. The mass fra tion burned tra es do not dier as mu h, as displayed in the lower plot of gure 2.6. For this operating point, the apparent heat release model produ es a mass fra tion burned tra e more like the one found by the Gatowski model, as shown by omparing the burn angles given in table 2.2. Note that the heat release tra es from the Rassweiler-Withrow, apparent heat release and Matekunas models are set onstant when they have rea hed their maximum values in gure 2.6. If not, their behavior would be similar to the net heat release tra e Qnet given in gure 2.5. Method Rassweiler-Withrow Apparent heat release Matekunas Gatowski et.al.

θ10

-6.4 -4.5 -6.9 -5.1

θ50

9.8 11.0 9.2 10.4

θ85

26.9 25.2 24.0 24.4

∆θb

33.3 29.8 30.9 29.5

Table 2.2: Crank angle positions for 10 %, 50 % and 85 % mfb as well as the rapid burn angle ∆θb = θ85 − θ10 , all given in degrees ATDC for the mass fra tion burned tra es in the lower plot of gure 2.6.

26 2.7

CHAPTER 2. OVERVIEW OF HEAT-RELEASE MODELS

Summary

A number of single-zone heat release models have been derived starting from the rst law of thermodynami s. The four models des ribed are then ompared and their spe i model assumptions are pointed out. The most elaborate one is the Gatowski et al. model, whi h in ludes heat transfer des ribed by Wos hni's heat transfer orrelation and revi e ee ts. This model also assumes that the spe i heat ratio for the ylinder harge an be des ribed by a linear fun tion in temperature. The other three models, the Rassweiler-Withrow model, the Matekunas pressure ratio and the apparent heat release model, are all more omputationally e ient than the Gatowski et al. model, merely sin e they la k the modeling of heat transfer and revi e ee t, as well as having a onstant spe i heat ratio for the rst two ases. This omputational e ien y of ourse omes to a ost of less des riptive models. The model omponents in the Gatowski et al. model will now be more thoroughly des ribed in hapter 3.

3 Heat-release model

omponents

Single-zone zero-dimensional heat-release models were introdu ed in the previous hapter, where their stru ture and interrelations were dis ussed. Now the attention is turned to the details of the various

omponents given in hapter 2, and espe ially those for the Gatowski et al.-model are treated more fully here. Some of the model omponents have already been introdu ed in se tion 2.5, but all omponents will be more thoroughly explained and ompared to other model omponents in se tions 3.13.7. It is also des ribed how to nd initial values for all parameters. These values are used as initial values when using the single-zone models for parameter estimation. In hapters 58 they will also be used as nominal values for xed parameters, i.e. parameters that are not estimated. The equations that form the omplete Gatowski et al. single-zone heat release model are emphasized by boxes, and the model is summarized in se tion 3.8. In se tion 3.9 the ylinder pressure sensitivity to the initial values of the parameters is briey investigated. 3.1

Pressure sensor model

The in- ylinder pressure is measured using a water- ooled quartz pressure transdu er, a piezoele tri sensor that be omes ele tri ally harged when there is a hange in the for es a ting upon it. Piezoele tri transdu ers rea t to pressure hanges by produ ing a harge proportional to the pressure hange. This harge is then integrated by the harge amplier, that returns a voltage as output proportional to the pressure. 27

28

CHAPTER 3.

HEAT-RELEASE MODEL COMPONENTS

The harge amplier has a slow drift due to harge leakage. It is however assumed that this drift is slow, and a stati model of the pressure sensor an then be used: pm (θ) = Kp p(θ) + ∆p ,

(3.1)

where pm is the measured ylinder pressure and p is the true ylinder pressure. The gain Kp is onsidered to be onstant for the measurement setup, but the oset ∆p hanges during the y le due to leakage in the harge amplier and thermal sho k of the sensor. Due to the assumption of a slow drift in the harge amplier, the pressure oset ∆p is onsidered to be onstant for one y le.

3.1.1 Parameter initialization  pressure oset ∆p The determination of the pressure oset is referred to as pegging the pressure signal, or as ylinder pressure referen ing. The pressure oset

an be estimated with various methods (Randolph, 1990; Brunt and Pond, 1997). It is generally re ommended that pegging is performed on e for every pressure y le. One method is to nd ∆p in the least squares sense using a polytropi model for the ylinder pressure p. Another method is to referen e the measured ylinder pressure pm (θ) to the intake manifold pressure pman before inlet valve losing (IVC), for several samples of pman . This method is often referred to as intake manifold pressure referen ing (IMPR) (Brunt and Pond, 1997). Due to standing waves (tuning) in the intake runners at ertain operating points, see gure 3.1, the referen ing might prove to be insu ient. The referen ing should be done at rank angles where the hange in ylinder pressure is approximately at for all operating points, i.e. where the intake manifold pressure pman and the ylinder pressure p are the same or have a onstant dieren e (Brunt and Pond, 1997). Figure 3.2 shows the ylinder pressure hange for θ ∈ [−200, −160] [deg ATDC℄ for a number of operating points. Using the same approa h as in Brunt and Pond (1997), the referen ing should be done between -167 to -162 CAD. If IMPR proves to be insu ient, ∆p must be estimated from the measured ylinder pressure data during the ompression phase to a hieve a orre t referen ing. If so, referen ing to pman will however still serve as an initial value.

3.1.2 Parameter initialization  pressure gain Kp The gain Kp an be determined in at least three dierent ways, summarized in Johansson (1995): The rst is to determine the gain for ea h

omponent in the measurement hain and multiply them to get Kp ;

3.1.

29

PRESSURE SENSOR MODEL

4

12

x 10

11.5

Cylinder pressure [Pa]

11 10.5 IVC 10 p 9.5

man

9 8.5 8 7.5

p

7 6.5 −240

−220

−200 −180 −160 Crank angle [deg ATDC]

−140

−120

Figure 3.1: Speed dependent ee ts in the intake runners like ram and tuning ee ts, are learly visible between -200 [deg ATDC℄ and IVC. The se ond is to alibrate the total hain by applying a well dened pressure step and measure the result; The third way being to determine the total gain in onjun tion with a thermodynami model. Here the rst method is used, and the gain Kp is determined by using tabulated values from the manufa turer. 3.1.3

Crank angle phasing

The pressure tra e is sampled at ertain events, su h as every rank angle degree. Sin e the mounting an not be performed with innite pre ision, an un ertainty in the exa t rank angle position for the sampling pulses is inherent. Therefore when al ulating the heat-release tra e (2.34), or when simulating the ylinder pressure (2.35) to ompare it with a measurement, the phasing of the pressure tra e relative to the volume tra e will most denitively ae t the out ome. A

ording to Amann (1985); Morishita and Kushiyama (1997); Sta± (2000), this phasing need to be a

urate within 0.1 CAD, in order to a

urately

al ulate the work (imep) from a spe i ylinder. A

ording to Brunt and Emtage (1997) the phasing need to be within 0.2 CAD to nd an imep a

urate within 1 %, sin e typi ally a 1 CAD phase shift indu e a 4 % imep error with gasoline engines and the relationship between imep error and rank angle error is linear (Brunt and Emtage, 1996). In Andersson (2005), models for rank angle oset ∆θ in a multi ylinder engine are developed. The rank angle oset depends on the

ylinder number and the phase of the y le, i.e. ∆θ diers during om-

30

CHAPTER 3.

HEAT-RELEASE MODEL COMPONENTS

8000

Change in pressure [Pa]

6000 4000 2000 0 −2000 −4000 −6000 −8000 −200

1500rpm 0.5bar 1500rpm 1bar 3000rpm 0.5bar 3000rpm 1bar −195

−190

−185 −180 −175 −170 Crank angle [deg ATDC]

−165

−160

Figure 3.2: Pressure oset as fun tion of CAD when referen ing to intake manifold pressure. The pressure hange is approximately at for all operating points between -167 and -162 CAD. pression, ombustion and expansion. This is due to rank-shaft torsion and exibility. Here we will study the ylinder losest to the rank angle en oder, so the torsion will be small. These ee ts are therefore not modeled here. Instead the rank shaft is onsidered to be rigid, and it is assumed that the sampled value at rank angle θi an be modeled as having a onstant oset ∆θ from the true rank angle θi,true , the rank angle phasing is modeled as θi + ∆θ = θi,true .

(3.2)

3.1.4 Parameter initialization  rank angle oset ∆θ The determination of ∆θ is often referred to as TDC determination. An initial value of ∆θ is provided from a number of motored y les, by referen ing the peak pressure position for the measured ylinder pressure with the orresponding position for a simulated pressure tra e, given the same operating onditions. The simulated pressure tra e is

omputed using the Gatowski et al. model.

Other methods of nding the rank angle oset ∆θ The easiest way to nd ∆θ is of ourse to onsider only motored y les, i.e. when there is no ombustion. Then the ylinder pressure would have its maximum at TDC if it were not for heat transfer and

revi e ee ts. Instead, the peak for the ompression pressure o

urs

3.2.

31

CYLINDER VOLUME AND AREA MODELS

before TDC. This dieren e is referred to as thermodynami loss angle (Hohenberg, 1979). Attempts to avoid the problem onne ted with unknown heat transfer have been taken in Sta± (1996); Morishita and Kushiyama (1997); Nilsson and Eriksson (2004), by using the polytropi relation to determine the position of TDC.

3.2

Cylinder volume and area models

The ylinder volume

V (θ, xof f ) onsists of a learan e volume Vc (xof f ) Vid (θ, xof f ), as

and an instantaneous displa ement volume

V (θ, xof f ) = Vc (xof f ) + Vid (θ, xof f ) . The instantaneous displa ed volume

ylinder bore

B,

rank radius

ar ,

Vid

(3.3)

θ, xof f

depends on the rank angle

onne ting rod length l , pin-o

and is given by

Vid (θ, xof f ) =

q πB 2 q ( (l + ar )2 − x2of f − ar cos θ − l2 − (ar sin θ − xof f )2 ) 4

,

(3.4)

for

xof f ∈ [−(l − ar ), (l − ar )].

The pin-o is dened as positive in the

dire tion of the rank angle revolution. Note that for an engine with pin-o, the rank positions for BDC and TDC are ae ted. They are given by:

θT DC = arcsin θBDC = arcsin

xof f , l + ar

(3.5a)

  x of f + π, l − ar

(3.5b)

and are not symmetri , as in the ase without pin-o. The impa t of pin-o on the ylinder volume is investigated in (Klein, 2004, pp. 149). It is found, for the SVC engine with

xof f ∈ [−2.2, 4.7]

mm, that the

relative error in instantaneous ylinder volume an be as large as 3.4 %, when the pin-o is not onsidered.

By interpreting the dieren e in

rank angle position of TDC due to pin-o as a onstant rank angle oset

∆θ,

as in (3.2), the relative error in

V (θ, xof f )

is redu ed to

less than 0.6 % in the worst ase. Not a

ounting for pin-o therefore

ontributes to the problem of TDC determination. Thus if the engine's pin-o is unknown, the dis repan y in omputing the ylinder volume

V (θ, 0)

(3.3) an almost fully be aptured by the rank angle oset

model (3.2). However if the pin-o for not in luding it in of

V (θ, xof f )

V (θ, xof f )

xof f

is known, there is no reason

(3.3), sin e it in reases the a

ura y

at almost no additional omputational ost.

32

CHAPTER 3.

HEAT-RELEASE MODEL COMPONENTS

When simulating either the heat release (2.34) or the ylinder pressure (2.35), it is ne essary to know the dierential of the volume fun of f ) and it is given by tion dV (θ,x dθ (ar sin θ − xof f ) cos θ  πB 2 ar  dV (θ, xof f ) sin θ + p = . dθ 4 l2 − (ar sin θ − xof f )2

(3.6)

When omputing the heat transfer rate (3.28), the instantaneous ombustion hamber surfa e area A(θ, xof f ) through whi h the heat transfer o

urs is omputed as A(θ, xof f ) = Ach + Apc + Alat (θ, xof f ),

(3.7)

where Ach is the ylinder head surfa e area and Apc is the piston rown surfa e area. Here it is assumed that these areas an be approxi2 mated by at surfa es, Apc = Ach = πB4 . The lateral surfa e area Alat (θ, xof f ) is approximated by the lateral surfa e of a ylinder. The instantaneous ombustion hamber surfa e area an then be expressed as A(θ, xof f ) =

q πB 2 (l + ar )2 − x2of f − ar cos θ − + πB 2

q  l2 − (ar sin θ − xof f )2 .

(3.8) The ompression ratio rc is dened as the ratio between the maximum (Vd + Vc ) and minimum (Vc ) ylinder volume: rc =

3.2.1

Vd Vd + Vc =1+ . Vc Vc

(3.9)

Parameter initialization  learan e volume Vc

The learan e volume Vc strongly inuen es the maximum y le temperature and pressure through the ompression ratio, and for heat release and pressure simulations it is therefore of great importan e. Due to geometri un ertainties in manufa turing, a spread of the a tual

learan e volume from engine to engine and ylinder to ylinder is inherent (Amann, 1985). The ompression ratio given from the manufa turer serves well as an initialization. It an also be initialized by using a polytropi relation, an initialization that works better the lower the real ompression ratio is (Klein et al., 2003). This sin e the polytropi relation does not take heat transfer and revi e ee ts into a

ount expli itly. 3.3

Temperature models

Two models for the in- ylinder temperature will be des ribed, the rst is the mean harge single-zone temperature model and it is the one

3.3.

TEMPERATURE MODELS

33

used in the Gatowski et al.-model. The se ond is a two-zone mean temperature model, used to ompute the single-zone thermodynami properties as mean values of the properties in a two-zone model, an approa h that will be introdu ed in the next hapter in se tion 4.6.

3.3.1

Single-zone temperature model

The mean harge temperature T for the single-zone model is found from the state equation pV = mtot RT , assuming the total mass of

harge mtot and the mass spe i gas onstant R to be onstant. These assumptions are reasonable sin e the mole ular weights of the rea tants and the produ ts are essentially the same (Gatowski et al., 1984). If all thermodynami states (pref ,Tref ,Vref ) are known/evaluated at a given referen e ondition ref , su h as IVC, the mean harge temperature T is omputed as T (θ) =

TIV C p(θ)V (θ) . pIV C VIV C

(3.10)

The ylinder volume at IVC is omputed using the ylinder volume given in (3.3) for θIV C and is therefore onsidered to be known. The two other states at IVC (pIV C ,TIV C ) are onsidered unknown and have to be estimated.

3.3.2

Parameter initialization  ylinder pressure at

IVC p

IV C

The parameter pIV C is initialized by the measured ylinder pressure pm in onjun tion with the pressure sensor oset ∆p and gain Kp , the

rank angle position for IVC, and the rank angle oset ∆θ by using equations (3.1) and (3.2). It needs to be pointed out that the position of IVC does not mean when the intake valve tou hes its seat, rather the position where intake mass ow has stopped. Therefore in most ases pIV C > pman . The parameter pIV C is also used as an initial value for the ordinary dierential equation in e.g. (2.35).

3.3.3 Parameter initialization  mean harge temperature at IVC TIV C The mean harge temperature at IVC diers from the gas temperature in the intake manifold Tman . The harge is heated due to both mixing with residual gases that are approximately at 1400 K (Heywood, 1988, p.178), and in- ylinder heat transfer from piston, valves and ylinder walls. On the other hand, fuel evaporation an ool the harge by as mu h as 25 K a

ording to Stone (1999). Altogether these ee ts make TIV C be ome larger than Tman . In Öberg and Eriksson (2006) three

34

CHAPTER 3.

HEAT-RELEASE MODEL COMPONENTS

r (Fox models for omputing the residual gas mass fra tion xr = mmtot et al., 1993; Ponti et al., 2004; Mladek and Onder, 2000) are ompared. The rst model is based on a ow restri tion model, and the other models are based on energy balan e at a referen e point e.g. IVC and require a ylinder pressure measurement. The last two methods also

ompute the residual gas temperature Tr and TIV C . It is found that the model in Ponti et al. (2004) but with the ex lusion of external EGR, is the best hoi e and it is therefore used here. It is des ribed as follows; Compute TIV C using the ideal gas law, the mass of the air-fuel harge maf , and an estimate of xr a

ording to:

mtot =

maf , 1 − xr

(3.11)

where mtot is the total harge mass, and maf = ma (1 +

1 A )s λ( F

(3.12)

),

where λ and ( FA )s are the normalized and stoi hiometri air-fuel ratios respe tively, and ma is al ulated from the measured air mass ow. The spe i gas onstant and temperature at IVC are stated by RIV C = Rb,r (pIV C , Tr ) · xr + Ru,af (Taf ) · (1 − xr ), TIV C =

pIV C VIV C . RIV C mtot

(3.13) (3.14)

Thermodynami properties su h as the spe i gas onstants R and spe i heats cv are evaluated using a hemi al equilibrium program developed by Eriksson (2004), more thoroughly des ribed in se tion 4.2. The new residual gas mass fra tion is omputed using energy balan e at IVC as xr =

cvu,af (Taf ) · (TIV C − Taf ) , cvb,r (pman , Tr ) · (Tr − TIV C ) + cvu,af (Taf ) · (TIV C − Taf )

where the temperature for the air-fuel harge Taf is given by Taf = Tman



pman pIV C

af  1−γ γ af

,

(3.15)

(3.16)

assuming that the fresh harge experien es a polytropi pro ess from manifold onditions to in- ylinder onditions. The residual gas temperature Tr is given by a orrelation model developed in Mladek and Onder (2000) as Tr = −(C1,Tr (mtot N ))C2,Tr + C3,Tr .

(3.17)

3.4.

35

CREVICE MODEL

The parameters in (3.17) have to be tuned using simulations for ea h engine type, as done in Öberg and Eriksson (2006). In order to have a

onverging x-point iteration s heme, xr is updated a

ording to xr,used =

xr,new + xr,old , 2

(3.18)

where xr,new is given by (3.15) and xr,old is the estimate from the previous iteration. The algorithm for omputing xr and TIV C an then be summarized as:  Residual gas mass fra tion (Ponti et al., 2004) Let xr be the initial estimate of the residual gas mass fra tion, and set xr,old = xr . Algorithm 3.1

1.Compute the total mass mtot (3.11). 2.Compute the temperature at IVC, TIV C (3.14). 3.Compute the residual gas mass fra tion, xr,new using (3.15). 4.Update the estimate xr,used a

ording to (3.18). x

−x

r,old | < 1 × 10−4 , if 5.Che k if xr,used has onverged, i.e. if | r,used xr,used not return to step 1 and set xr,old = xr,used .

6.Return xr = xr,used and TIV C .

3.4

Crevi e model

In an engine, gases ow in and out of the revi es onne ted to the

ombustion hamber as the ylinder pressure rises and falls. Crevi es in lude those volumes between piston, rings and liner, any head gasket gap, spark plug threads and spa e around the pressure transdu er. During ompression some of the harge ows into the revi es, and remains there until the expansion phase, when most of the harge returns to the ombustion hamber. The ame an not propagate into the revi es, and therefore some of the harge is not ombusted. A small part of the ylinder harge blows by the piston rings and ends up in rank- ase. Here this part is assumed to be zero and therefore not modeled.

36

CHAPTER 3.

HEAT-RELEASE MODEL COMPONENTS

When modeling the revi e ee t, the temperature in the revi es are assumed to be lose to the ylinder wall temperature, so during the

losed phase a substantial amount of harge ould be trapped in the

revi es. Consider the ratio of total harge mass to the mass in the

revi es: mcr Vcr T (3.19) = . mtot

V Tcr

A

ording to Gatowski et al. (1984), the revi e volume Vcr an onstitute as mu h as 1-2 per ent of the learan e volume Vc in size. The temperature dieren e in the ylinder T and in the revi es Tcr approa hes a fa tor 4-5 at the end of ombustion. As a onsequen e as mu h as 10 (mass) per ent of the harge ould then be trapped in

revi es at peak pressure. The model developed and applied in Gatowski et al. (1984) assumes that all revi es an be modeled as a single aggregate onstant volume Vcr , and that the harge in the revi e assumes the average wall temperature Tw and is at the same pressure as in the ombustion hamber. The ideal gas law thus gives pVcr = mcr RTw =⇒ dmcr =

Vcr dp, R Tw

(3.20)

where it is assumed that Tw and R are onstant. Gatowski et al. (1984) points out that this model is not meant to a

ount for ea h revi e, but rather to a

ount for the overall revi e ee t.

3.4.1

Parameter initialization  revi e volume Vcr

The single aggregate revi e volume Vcr is unknown and is therefore set to 1.5 per ent of the learan e volume Vc , whi h is a reasonable value a

ording to Gatowski et al. (1984). For an engine with varying

learan e volume, su h as the SVC engine, this would yield a revi e volume dependent of the ompression ratio. To avoid this, Vcr is set to 1.5% Vc at rc = 11, i.e. the learan e volume in the mid range of the

ompression ratio is used.

3.4.2 Parameter initialization  ylinder mean wall temperature Tw The ylinder wall temperature Tw is not only used in the revi e model, but also in the heat transfer model des ribed in se tion 3.6. It varies during the engine y le due to heat transfer in the ylinder blo k, but the surfa e temperature u tuations are lo ally relatively small (i.e. 510 K) suggesting that a onstant surfa e temperature an be used (Anatone and Cipollone, 1996). Therefore an area-weighted mean value of

3.5.

37

COMBUSTION MODEL

the temperatures of the exposed ylinder walls, the head and the piston rown for the losed part of the engine y le is used. Here Tw is initialized to a onstant value of 440 K (Brunt and Emtage, 1997; Eriksson, 1998), set only by it being a reasonable value. There exists other methods of estimating the ylinder wall temperature, see e.g. Arsie et al. (1999). 3.5

Combustion model

The ombustion of fuel and air is a very omplex pro ess, and would require extensive modeling to be fully aptured. The approa h here is to use a parameterization of the burn rate of the ombusted harge. The prevailing ombustion model is the Vibe fun tion (Vibe, 1970), whi h in some literature is spelled Wiebe fun tion. 3.5.1

Vibe fun tion

The Vibe fun tion is often used as a parameterization of the mass fra tion burned xb , and it has the following form xb (θ) = 1 − e

 θ−θ  m+1 −a ∆θ ig cd

(3.21)

,

and the burn rate is given by its dierentiated form a (m + 1) dxb (θ) = dθ ∆θcd



θ − θig ∆θcd

m

e

 θ−θ  m+1 −a ∆θ ig cd

,

(3.22)

where θig is the start of the ombustion, a and m are adjustable parameters, and ∆θcd is the total ombustion duration. The Vibe fun tion is over-parameterized in a, m, and ∆θcd , sin e for example the sets [a = 1, ∆θcd = 1, m = 1℄ and [a = 4, ∆θcd = 2, m = 1℄ give identi al fun tion values. To parameterize the mass fra tion burned (mfb) tra e with physi al parameters, two burn rate angles are often used, namely the ame-development angle ∆θd whi h orresponds to the rank angle from 0 % mfb (ignition) to 10 % mfb, and the rapid burn angle ∆θb (10-85 % mfb) (Heywood, 1988), illustrated in gure 3.3. The burn angle parameters have a dire t relation to the parameters in the Vibe fun tion, but due to the over-parameterization in a and ∆θcd , one of them must be spe ied before-hand to get a unique solution. If ∆θcd is spe ied, the Vibe parameters be ome: m=

ln(ln(1 − 0.1) − ln(1 − 0.85)) −1 , ln ∆θd − ln(∆θd + ∆θb )

(3.23a)

38

CHAPTER 3.

HEAT-RELEASE MODEL COMPONENTS

Mass fraction burned profile 1 0.9

Mass fraction burned [−]

0.8

∆θ

b

0.7 0.6 0.5 0.4 0.3 0.2

∆ θd

0.1 0 0

10

20 30 40 50 60 Crank angle degree since ignition

70

80

Figure 3.3: Mass fra tion burned prole with the ame development angle ∆θd and rapid burn angle ∆θb marked.

a = −(

∆θcd m+1 ) ln(1 − 0.1) , ∆θd

(3.23b)

The dierentiated Vibe fun tion (3.22) is used to produ e a mass fra tion burned tra e, i.e. a normalized heat-release tra e. The absolute value of the heat-release rate dQdθch is given by dQch dxb , = Qin dθ dθ

(3.24)

where Qin represents the total energy released from ombustion. Summing up, the ombustion pro ess is des ribed by (3.24) and parameterized by Qin , θig , ∆θd , and ∆θb .

3.5.2 Parameter initialization  energy released Qin The total energy released Qin is inuen ed by a lot of parameters, su h as residual gas fra tion, ombustion e ien y, mass of fuel, fuel heating value, but also the mass fra tion burned rate due to the dependen e of thermodynami properties for the mixture of temperature and pressure. It is here modeled as Qin = mf qHV ηf , (3.25) where mf is the fuel mass, qHV the spe i lower heating value of the fuel and ηf is the ombustion e ien y. There at least two ways

3.6.

39

ENGINE HEAT TRANSFER

of omputing Qin ; If the fuel mass mf is measured, Qin an be found from (3.25) by assuming a ombustion e ien y. If mf is not measured, another approa h must be taken. One su h approa h is to rewrite mf in (3.25) as mf =

maf (1 − xr )mtot = , A 1 + λ( F )s 1 + λ( FA )s

(3.26)

where the total harge mass mtot is found using the ideal gas law at a referen e point during the high-pressure phase, e.g. IVC, yielding pIV C VIV C mtot = R . The fuel is assumed to be iso-o tane, that has IV C TIV C J Ru = 290 kgK (at 300 K), lower heating value qHV = 44.0 MJ kg , and
stoi hiometri air-fuel equivalen e ratio FA s = 14.6 (Heywood, 1988, p.915). The spe i gas onstant RIV C is given by (3.13) and the residual gas fra tion xr by (3.15). So Qin is initialized as Qin = mf ηf qHV =

(1 − xr ) pIV C VIV C ηf qHV , 1 + λ( FA )s RIV C TIV C

(3.27)

where the ombustion e ien y ηf is assumed to be one. To solve the ordinary dierential equation in (2.34) an initial value for the a

umulated heat released at IVC QIV C must be spe ied. It is set to zero.

3.5.3

Parameter initialization  angle-related parameters {θig , ∆θd , ∆θb }

The angle-related parameters {θig , ∆θd , ∆θb } are initialized by the mass fra tion burned tra e found from the Rassweiler-Withrow method mentioned in se tion 2.2. Nominal values are in the intervals (Eriksson, 1999): [-30 , 0℄

θig

[deg ATDC℄

∆θd

[15, 40℄

[deg℄

∆θb

[10, 50℄

[deg℄

To get an unique solution for (3.23), the total ombustion duration ∆θcd is set to 100π 180 rad. 3.6

Engine heat transfer

Typi ally 20-35 % of the fuel energy is passed on by heat transfer to the engine oolant, the upper limit is rea hed for low load onditions (Stone, 1999, p.429). Of the total heat transfer, about half omes from in ylinder heat transfer and the rest from heat transfer in the exhaust port.

40

CHAPTER 3.

HEAT-RELEASE MODEL COMPONENTS

In- ylinder heat transfer

The in- ylinder heat transfer o

urs by both onve tion and radiation, where onve tion onstitutes the major part (S hmidt et al., 1993). In SI engines, up to approximately 20 %, but usually mu h less of the in ylinder heat transfer is due to radiation, but this is in some instan es in luded in the orrelation for onve tive heat transfer (Wos hni, 1967). For CI engines however, the heat transfer originating from radiation an

onstitute a more signi ant part (up to 40 % (Heywood, 1988, p.696)) and has to be a

ounted for expli itly (Annand, 1963). Sin e we are dealing with SI engines, only onve tive heat transfer is modeled, but keeping in mind that the radiative heat transfer is a

ounted for by the

orrelations. The magnitude of the rate of energy transfer by onve tion Q˙ ht , whi h o

urs in a dire tion perpendi ular to the uid surfa e interfa e, is obtained by Newton's law of ooling Q˙ ht = hc A ∆T = hc A (T − Tw ) ,

(3.28)

where A is the surfa e area of the body whi h is in onta t with the uid, ∆T is the appropriate temperature dieren e, and hc is the onve tion heat transfer oe ient. The oe ient hc varies both in time and spa e, and sin e it is a omposite of both mi ros opi and ma ros opi phenomena, many fa tors must be taken into onsideration for a full des ription. A full des ription of hc is needed if for example thermal stress on the ylinder head is to be investigated (Bergstedt, 2002). On the other hand, a position-averaged heat transfer oe ient will be su ient for predi ting the heat ow to the oolant, and this will be the approa h taken here. There exists a number of models for hc , for e.g. (Annand, 1963; Wos hni, 1967; Hohenberg, 1979). Later works (Shayler et al., 1993; Hayes et al., 1993; Pive et al., 1998; Wimmer et al., 2000) experimentally veried the Wos hni orrelation to be the better one for the losed part of the engine y le. Wos hni's heat-transfer orrelation

The form proposed by Wos hni (1967) is: hc = C(pw)0.8 B −0.2 T −0.55 .

(3.29)

Wos hni found that the exponent for T should be -0.53, but this is not onsistent with the prerequisites in the derivation, see (Klein, 2004, pp. 37). Wos hni states that the hara teristi speed w depends on two terms. One is due to piston motion and is modeled as the mean piston speed up = 2aN 60 [m/s℄, where a [m℄ is the rank radius and N [rpm℄ is the engine speed. The other term is due to swirl originating from the

3.6.

41

ENGINE HEAT TRANSFER

ombustion event, whi h is modeled as a fun tion of the pressure rise due to ombustion, i.e. p−p0 where p0 is the motored pressure. Wos hni used the measured motored pressure, but later on Watson and Janota (1982) proposed to use the polytropi pro ess model (2.11) instead: p0 = pref

V

ref

V

n

(3.30)

where n is the polytropi exponent, and (pref , Vref ) are evaluated at any referen e ondition, su h as IVC. The hara teristi speed w an then be expressed as: w = C1 up + C2 (T − T0 ) = C1 up + C2

V TIV C (p − p0 ), (3.31) pIV C VIV C

where the rst term originates from onve tion aused by piston motion and the se ond term from the ombustion itself, where T0 is the motored mean gas temperature. This results in the following expression for the heat transfer oe ient hc 1 :

hc =

 0.013 B −0.2 p0.8 C1 up +

C2 (p−p0 ) TIV C V pIV C VIV C

T 0.55

0.8

,

(3.32)

where

ylinder pressure for red y le

ylinder pressure for motored y le mean gas temperature mean piston speed instantaneous ylinder volume

onstant

onstant at referen e ondition IVC

p p0 T up V C1 C2 (pIV C ,VIV C ,TIV C )

[P a ℄ [P a ℄ [K ℄ [m/s℄ [m 3 ℄ [-℄ [m/(s K)℄

Simulation in rank angle domain Note that Q˙ ht = angle domain,

žQht dt

, thus when simulating heat transfer in the rank

žQht dθ

=

žQht dt dt

dθ

60 = Q˙ ht 2πN

(3.33)

should be used where N [rpm℄ is the engine speed. 1 The numeri al value of the rst oe ient diers from the one in (Wos hni, 1967), sin e it is al ulated to t the SI-unit system.

42

CHAPTER 3.

HEAT-RELEASE MODEL COMPONENTS

3.6.1 Parameter initialization  {C1 , C2} Wos hni found experimentally that during the losed part of the engine

y le, the parameters C1 = 2.28 and C2 = 3.24 · 10−3 gave a good t, and they therefore serve well as initial values. Wos hni also pointed out that the parameters (C1 , C2 ) are engine dependent, and are therefore likely to hange for dierent engine geometries.

3.7

Thermodynami properties

The a

ura y with whi h the energy balan e an be al ulated for a

ombustion hamber depends in part on how a

urately hanges in the internal energy of the ylinder harge are represented. The most important thermodynami property used in al ulating the heat release rates for engines is the ratio of spe i heats, γ = ccpv (Gatowski et al., 1984; Chun and Heywood, 1987; Guezenne and Hamama, 1999). In the Gatowski et al.-model, the spe i heats ratio γ(T ) is modeled as a linear fun tion of temperature, γlin (T ) = γ300 + b (T − 300) .

(3.34)

Gatowski et al. (1984) states that this omponent is important, sin e it

aptures how the internal energy varies with temperature. This is an approximation of the thermodynami properties but it is further stated that this approximation is onsistent with the other approximations made in the model. It will be shown in hapter 4 that the linear model of γ in temperature T introdu es a modeling error in ylinder pressure whi h is 15 times the ylinder pressure measurement noise in mean. A model of γ that introdu es an error in the same order as the noise is also given in hapter 4.

3.7.1

Parameter initialization  γ300 and b

The initial values for the two parameters in the linear model of γ (3.34) are omputed by using a hemi al equilibrium program (Eriksson, 2004). First the spe i heat ratio is omputed for a spe i fuel by assuming that the mixture is burned and at equilibrium at all instan es. Then the two parameters γ300 and b are tted in the least squares sense to the resulting γ . Nominal values are in the intervals; γ300 ∈ [1.35, 1.41] and b ∈ [−8 · 10−5 , −12 · 10−5 ] K −1 .

3.8.

3.8

SUMMARY OF SINGLE-ZONE HEAT-RELEASE MODELS

43

Summary of single-zone heat-release models

The model omponent equations, that are emphasized by boxes in hapter 2 and 3, for the Gatowski et al.-model are summarized here, together with the inputs, outputs and unknown parameters for the model.

Model inputs and outputs Input λ N pexh pman Tman θIV C θEV O

Output p pm Qch

Des ription air-fuel ratio engine speed exhaust manifold pressure intake manifold pressure intake manifold temperature

rank angle degree for IVC

rank angle degree for EVO Des ription

ylinder pressure measured ylinder pressure

hemi al energy released as heat

Unit [-℄ [rpm℄ [Pa℄ [Pa℄ [K℄ [deg ATDC℄ [deg ATDC℄ Unit [Pa℄ [Pa℄ [J℄

In heat release models a pressure tra e is given as input and the heat release is the output, while in pressure models a heat release tra e is the input and pressure is the output.

Model omponent equations Heat release dierential (2.25); žQch =

1 γ−1 V

γ γ−1 p dV

dp +

T + ( γ−1 + T′ +

u′ −u Vcr R ) Tw dp

or ylinder pressure dierential (2.26); dp =

γ p dV − žQht žQch − γ−1 1 γ−1

V +

Pressure sensor model (3.1);

Vcr Tw



T γ−1

+ T′ +

u′ −u R



+ žQht

(3.35) (3.36)

pm = Kp p + ∆p

(3.37)

θi + ∆θ = θi,true

(3.38)

Crank angle phasing (3.2);

44

CHAPTER 3.

HEAT-RELEASE MODEL COMPONENTS

Cylinder volume V (θ, xof f ) (3.3), (3.4) and (3.6); V (θ, xof f ) = Vc (xof f ) + Vid (θ, xof f )

(3.39a)

q p 2 2 2 Vid (θ, xof f ) = πB l2 − (xof f + ar sin θ)2 ) 4 ( (l + ar ) − xof f − ar cos θ −

(xof f + ar sin θ) cos θ  πB 2 ar  dV (θ, xof f ) sin θ + p = dθ 4 l2 − (xof f + ar sin θ)2

(3.39b) (3.39 )

Temperature model (3.10);

T =

TIV C pV pIV C VIV C

(3.40)

Crevi e model (3.20);

Vcr dp R Tw Vibe ombustion model (3.24), (3.22) and (3.23); dmcr =

dxb dQch = Qin dθ dθ  m  θ−θ  m+1 a (m + 1) θ − θig dxb (θ) −a ∆θ ig cd = e dθ ∆θcd ∆θcd

(3.41)

(3.42a) (3.42b)

ln(ln(1 − 0.1) − ln(1 − 0.85)) −1 (3.42 ) ln ∆θd − ln(∆θd + ∆θb ) ∆θcd m+1 ) ln(1 − 0.1) a = −( (3.42d) ∆θd Wos hni's heat transfer orrelation (3.33), (3.28), (3.32) and (3.8); m=

žQht dt 60 žQht = = Q˙ ht dθ dt dθ 2πN

hc = A(θ, xof f ) =

Q˙ ht = hc A ∆T = hc A (T − Tw )  0.8 C (p−p ) T V 0.013 B −0.2 p0.8 C1 up + 2 pref0Vrefref T 0.55

πB 2 2

(3.43a) (3.43b) (3.43 )

q  p + πB (l + ar )2 − x2of f − ar cos θ − l2 − (xof f + ar sin θ)2

(3.43d) To simulate the Gatowski et al.-model, equations (3.36)(3.43) are used together with the linear spe i heat ratio model (3.34); γlin (T ) = γ300 + b (T − 300)

and the orresponding revi e energy term (2.32); R  γ′ − 1  u′ − u = ln b γ−1

(3.44)

(3.45)

3.9. SENSITIVITY PARAMETER INITIALIZATION

Par. γ300 b C1 C2 ∆θ ∆p Kp pIV C TIV C Tw Vc Vcr θig ∆θd ∆θb Qin

Des ription

onstant spe i heat ratio [-℄ slope for spe i heat ratio[K −1℄ heat-transfer parameter [-℄ heat-transfer parameter [m/(s K)℄

rank angle phasing [deg ATDC℄ bias in pressure measurements [kPa℄ pressure measurement gain[-℄

ylinder pressure at IVC [kPa℄ mean harge temperature at IVC [K℄ mean wall temperature [K℄

learan e volume [cm3 ℄ single aggregate revi e volume [% Vc ℄ ignition angle [deg ATDC℄ ame-development angle [deg ATDC℄ rapid-burn angle [deg ATDC℄ released energy from ombustion [J℄

Value 1.3678 −8.13 · 10−5 2.28 3.24 · 10−3 0.4 30 1 100 340 440 62.9 1.5 -20 15 30 1500

45 Equation (3.34) (3.34) (3.32) (3.32) (3.2) (3.1) (3.1) (3.10), (2.26) (3.10) (3.20),(3.28) (3.3) (3.20) (3.22) (3.22) (3.22) (3.24)

Table 3.1: Nominal values for the parameters in the Gatowski et al. single-zone heat release model. For red y les, Tw = 440 K and TIV C = 340 K, and for motored y les, Tw = 400 K and TIV C = 310 K.

Unknown parameters The parameters used in the Gatowski et al. single-zone model and an example of nominal values is summarized in table 3.1.

3.9

Sensitivity in pressure to parameter initialization

The ylinder pressure is simulated for the nominal values in the table 3.2 using (3.36)(3.45), and yields the ylinder pressure given in gure 3.4. The nominal values are the same as in table 3.1. To get an idea of how sensitive the ylinder pressure is to errors in the initialized parameters, a sensitivity analysis is performed by perturbing the parameters one at a time with the realisti perturbations given in table 3.2. In most

ases the perturbation is set to 10 per ent of their nominal value, but when the nominal value is small in omparison to the un ertainty in the parameter this approa h would not give a fair omparison. The perturbation for these parameters are therefore set to reasonable values. The perturbed simulated ylinder pressure is then ompared to the nominal one, in terms of root mean square error (RMSE) and maximum absolute residual value (Max Res), where RMSE is the more important measure when onsidering least squares optimization. The residual is here dened as the dieren e between the nominal and perturbed

46

CHAPTER 3.

Par. γ300 ∆θd θig Vc Kp Tw Qin ∆θb pIV C TIV C Vcr b ∆θ ∆p C2 C1

HEAT-RELEASE MODEL COMPONENTS

Nominal & perturbation value 1.3678 15 -20 62.9 1 440 1500 30 100 340 1.5 -8.13 ·10−5 0.4 30 3.24 ·10−3 2.28

0.137 5 5 6.29 0.1 44 150 5 10 44 1.5 -8.13 ·10−6 0.2 10 3.24 ·10−4 0.228

[-℄ [deg℄ [deg ATDC℄ [cm3 ℄ [-℄ [K℄ [J℄ [deg℄ [kPa℄ [K℄ [% Vc ℄ [K −1 ℄ [deg℄ [kPa℄ [m/(s K)℄ [-℄

RMSE [kPa℄ 522.6 270.5 237.9 210.3 184.9 110.1 101.7 101.6 95.7 63.7 36.3 25.0 10.3 10.0 3.8 1.6

Max Res [kPa℄ 1407.3 1029.2 860.3 603.7 465.0 285.4 263.7 397.4 217.9 176.7 122.3 77.0 31.4 10.0 8.2 2.9

S

[-℄ 5.25 0.58 0.73 1.76 1.55 0.88 0.81 0.45 0.77 0.36 0.03 0.19 0.02 0.02 0.03 0.01

Table 3.2: Nominal and perturbation values, where the perturbations are performed by adding or subtra ting the perturbation from the nominal value. The root mean square error (RMSE), maximal residual (Max Res) and sensitivity fun tion S (3.46) are omputed for the worst ase for ea h parameter. The parameters are sorted in des ending order of their RMSE.

ylinder pressures. The parameter sensitivity is also examined, and it is omputed here as S=

RMSE(p) p¯ |∆x| |x|

,

(3.46)

where x is the nominal parameter value, ∆x is the parameter perturbation and p¯ is the mean ylinder pressure. The results are summarized in table 3.2. When omparing the RMSE for every parameter, the

onstant γ300 in the linear spe i heat ratio model, the burn related angles ∆θd and θig , and the learan e volume Vc show highest sensitivity in the mean and are therefore more in need of a proper initialization than the others. On the other hand, disturban es in the values of the two Wos hni parameters C1 and C2 do not ae t the resulting ylinder pressure signi antly. Note that the model of the ylinder pressure is nonlinear, so the results found from this analysis is only valid lo ally, but it still gives an idea of whi h parameters are the most sensitive ones.

3.9. SENSITIVITY PARAMETER INITIALIZATION

47

Simulated cylinder pressure 5 4.5

Cylinder pressure [MPa]

4 3.5 3 2.5 2 1.5 1 0.5 0

Figure 3.4: table 3.2.

−100

−50 0 50 Crank angle [deg ATDC]

100

Simulated ylinder pressure using the nominal values in

48

CHAPTER 3.

HEAT-RELEASE MODEL COMPONENTS

4 A spe ifi heat ratio model for single-zone heat release models The a

ura y with whi h the energy balan e an be al ulated for a

ombustion hamber depends in part on how a

urately hanges in the internal energy of the ylinder harge are represented. The most important thermodynami property used when al ulating the heat release rates in engines is the ratio of spe i heats, γ(T, p, λ) = ccpv (Gatowski et al., 1984; Chun and Heywood, 1987; Guezenne and Hamama, 1999). Based on the rst law of thermodynami s, Gatowski et al. (1984) developed a single-zone heat release model that has been widely used, where the spe i heat ratio is represented by a linear fun tion in mean

harge temperature T : γlin (T ) = γ300 + b(T − 300).

(4.1)

This allows a riti al examination of the burning pro ess by analysis of the heat release. In order to ompute the heat release orre tly, the parameters in the single-zone model need to be well tuned. These parameters, su h as heat transfer oe ients, γ300 and b in the linear γ -model (4.1) and so on, an be tuned using well known methods. For instan e, Eriksson (1998) uses standard predi tion error methods (Ljung, 1999) to tune the parameters. This is done by minimizing the predi tion error of the measured ylinder pressure, i.e. by minimizing the dieren e between the modeled and measured ylinder pressure. Applying standard parameter identi ation methods usually ends up in non-physi al values of γ300 , as it be omes larger than 1.40, whi h is the value of γ300 for pure air. It has also been shown in table 3.2 that 49

50

CHAPTER 4. A SPECIFIC HEAT RATIO MODEL

...

γ300 is the most sensitive single-zone parameter. But more importantly, the linear approximation of γ (4.1) itself introdu es a model error in the

ylinder pressure whi h has a root mean square error of approximately 30 kPa, for low load engine operating points, and approximately 90 kPa in the mean for operating points overing the entire operating range. These errors are more than four times the error introdu ed by the measurement noise in the former ase, and more than ten times in the latter ase. These errors will ae t the omputed heat release and a better model of γ(T, p, λ) is therefore sought. A orre t model of γ(T, p, λ) is also desirable in order to avoid badly tuned (biased) single-zone model parameters. The obje tive is to investigate models of the spe i heat ratio for the single-zone heat release model, and nd a model a

urate enough to only introdu e a modeling error less than or in the order of the

ylinder pressure measurement noise, while keeping the omputational

omplexity at a minimum. Su h a model would help us to ompute a more a

urate heat release tra e. This work relies upon the single-zone framework and the general single-zone heat release model given by (3.36)(3.43) is used as a referen e model. This is of ourse an approximation, but reasonable sin e single-zone models des ribe the ylinder pressure well. This has been veried experimentally e.g. in Gatowski et al. (1984). 4.1

Outline

In the following se tion three existing γ -models are des ribed. Then based on hemi al equilibrium, a referen e model for the spe i heat ratio is des ribed. Thereafter, the referen e model is al ulated for an unburned and a burned air-fuel mixture respe tively, and ompared to these existing models in the two following se tions. With the knowledge of how to des ribe γ for the unburned and burned mixture respe tively, the fo us is turned to nding a γ -model during the ombustion pro ess, i.e. for a partially burned mixture. This is done in se tion 4.6, where a number of approximative models are proposed. These models are evaluated in terms of the normalized root mean square error related to the referen e γ -model found from hemi al equilibrium, as well as the inuen e the models have on the ylinder pressure, and also in terms of omputational time. 4.2

Chemi al equilibrium

A

ording to Heywood (1988, p.86), it is a good approximation for performan e estimates to onsider the unburned gases as frozen and the

4.2.

51

CHEMICAL EQUILIBRIUM

burned gases as in hemi al equilibrium. Assuming that the unburned air-fuel mixture is frozen and that the burned mixture is at equilibrium at every instant, the spe i heat ratio and other thermodynami properties of various spe ies an be al ulated using the Matlab pa kage CHEPP (Eriksson, 2004). The referen e fuel used is iso-o tane, C8 H18 , whi h rea ts with air a

ording to: 1 C8 H18 + (O2 + 3.773N2 ) −→ λ (8 + 18/4) y1 O + y2 O2 + y3 H + y4 H2 + y5 OH +y6 H2 O + y7 CO + y8 CO2 + y9 N O + y10 N2 ,

(4.2)

where the produ ts given on the right hand side are hosen by the user and λ is the air-fuel ratio (AFR). The oe ients yi are found by CHEPP and they reveal the mole fra tion x˜i = Pyiyi of spe ie i that i the mixture onsists of at a given temperature, pressure and air-fuel ratio. From x˜i , the mass fra tion xi is omputed using the molar mass Mi as xi = x˜i /Mi . The mixture is assumed to obey the Gibbs-Dalton law, whi h states that under the ideal-gas approximation, the properties of a gas in a mixture are not inuen ed by the presen e of other gases, and ea h gas

omponent in the mixture behaves as if it exists alone in the volume at the mixture temperature (Çengel and Boles, 2002, Ch 12). Therefore, the thermodynami properties an be added together as e.g. in: u(T, p, λ) =

X

xi (T, p, λ)ui (T ),

(4.3)

i

where ui is the internal energy from spe ie i and u is the total internal energy. The enthalpy h(T, p, λ) is omputed in the same manner. All thermodynami properties depend on the air-fuel ratio λ, but for notational onvenien e this dependen e is hereafter left out when there  isno ∂u risk of onfusion. The denition of the spe i heat cv is cv = ∂T , V and by using (4.3) it is expressed as cv (T, p) =

X

xi (T, p)cv,i (T ) + ui (T )

i

∂xi (T, p), ∂T

(4.4)

where   the individual spe ies are ideal gases. The spe i heat cp = ∂h is al ulated in the same manner, as ∂T p

cp (T, p) =

X

xi (T, p)cp,i (T ) + hi (T )

i

∂xi (T, p). ∂T

(4.5)

The spe i heat ratio γ is dened as γ(T, p) =

cp (T, p) . cv (T, p)

(4.6)

52

CHAPTER 4. A SPECIFIC HEAT RATIO MODEL

4.3

Existing models of

...

γ

The omputational time involved in repeated use of a full equilibrium program, su h as CHEPP (Eriksson, 2004) or the NASA program (Gordon and M Bride, 1971; Svehla and M Bride, 1973), an be substantial, and therefore simpler models of the thermodynami properties have been developed. Three su h models will now be des ribed. 4.3.1

Linear model in

T

The spe i heat ratio during the losed part of the y le, i.e. when both intake and exhaust valves are losed, is most frequently modeled as either a onstant, or as a linear fun tion of temperature. The latter model is used in (Gatowski et al., 1984), where it is stated that the model approximation is in parity with the other approximations made for this family of single-zone heat-release models. The linear fun tion in T an be written as: γlin (T ) = γ300 + b (T − 300).

(4.7)

Depending on whi h temperature region and what air-fuel ratio λ the model will be used for, the slope b and onstant γ300 in (4.7) have to be adjusted. Con erning the temperature region, this short oming an be avoided by in reasing the omplexity of the model and use a se ond (or higher) order polynomial for γlin (T ). This has been done in for example Brunt et al. (1998). Su h an extension redu es the need for having dierent values of γ300 and b for dierent temperature regions. Later on, γlin (T ) is al ulated in a least squares sense for both burned and unburned mixtures. 4.3.2

Segmented linear model in

T

A

ording to Chun and Heywood (1987), the ommonly made assumption that γ(T ) is onstant or a linear fun tion of mean temperature is not su iently a

urate. Instead, they propose a segmentation of the losed part of the engine y le into three segments; ompression,

ombustion and post- ombustion (expansion). Both the ompression and post- ombustion are modeled by linear fun tions of T , while the

ombustion event is modeled by a onstant γ . They further state that with these assumptions, the one-zone analysis framework will provide a

urate enough predi tions. The model of γ an be written as:  comp  γ300 + bcomp (T − 300) xb < 0.01 γseg (T, xb ) = γ comb 0.01 ≤ xb ≤ 0.99  300 exp γ300 + bexp (T − 300) xb > 0.99

(4.8)

4.3.

EXISTING MODELS OF

53

γ

where the mass fra tion burned xb is used to lassify the three phases. The γ -model proposed by Chun and Heywood (1987) has dis ontinuities when swit hing between the phases ompression, ombustion and post ombustion. 4.3.3

Polynomial model in

p

and

T

The third model is a polynomial model of the internal energy u developed in Krieger and Borman (1967) for ombustion produ ts of Cn H2n . For lean and stoi hiometri mixtures (λ ≥ 1), a single set of equations was stated, whereas dierent sets where found for ea h λ < 1. The model of u for λ ≥ 1 is given by: uKB (T, p, λ) = A(T ) −

B(T ) + ucorr (T, p, λ), λ

(4.9)

given in [kJ/(kg of original air)℄, where A(T ) = a1 T + a2 T 2 + . . . + a5 T 5

(4.10a)

B(T ) = b0 + b1 T + . . . + b4 T 4 .

(4.10b)

The gas onstant was found to be: R(T, p, λ) = 0.287 +

0.020 + Rcorr (T, p, λ), λ

(4.11)

given in [kJ/(kg of original air) K℄. Krieger and Borman suggested that two orre tion terms ucorr and Rcorr should a

ount for disso iation, modeled as non-zero for T > 1450 K and given by: ucorr (T, p, λ) = cu exp (D(λ) + E(T, λ) + F (T, p, λ)) D(λ) = d0 + d1 λ−1 + d3 λ−3 E(T, λ) =

e0 + e1 λ−1 + e3 λ−3 T

(4.12a) (4.12b) (4.12 )

f4 + f5 λ−1 (4.12d) ) ln(f6 p) T   r1 + r2 /T + r3 ln(f6 p) , (4.13) Rcorr (T, p, λ) = cr exp r0 ln λ + λ F (T, p, λ) = (f0 + f1 λ−1 + f3 λ−3 +

where T is given in Kelvin (K) and p in bar. The values of the oef ients are given in table 4.1. For a fuel of omposition Cn H2n , the stoi hiometri fuel-air ratio is 0.0676. Therefore, equations (4.9)-(4.11) should be divided by (1+0.0676λ−1), to get the internal energy per unit mass of produ ts. In general, Krieger and Borman (1967) found that

54

CHAPTER 4. A SPECIFIC HEAT RATIO MODEL

0.692

a1

a2 39.17 · 10−6 b0

0.154226

-14.763

118.27

r0

r1

r2

...

a3 a4 52.9 · 10−9 −228.62 · 10−13 a5 b1 b2 277.58 · 10−17 3049.33 −5.7 · 10−2 −9.5 · 10−5 b3 b4 cu cr 21.53 · 10−9 −200.26 · 10−14 2.32584 4.186 · 10−3 d0 d1 d3 e0 10.41066 7.85125 -3.71257 −15.001 · 103 e1 e3 f0 f1 −15.838 · 103 9.613 · 103 -0.10329 -0.38656 f3 f4 f5 f6

-0.2977

11.98

-25442

14.503 r3

-0.4354

Table 4.1: Coe ient values for Krieger-Borman polynomial given in (4.10)-(4.13). the error in u was less than 2.5 per ent in the pressure and temperature range of interest, where the extreme end states were approximately {2300 K, 0.07 M P a} and {3300 K, 35 M P a}, and less than 1 per ent over most of the range. A model of γ is now given by its denition as cp R γKB = (4.14) =1+ , cv

where R is given by (4.11) and cv = ing (4.9) with respe t to T . 4.4



∂u ∂T

cv 

V

is found by dierentiat-

Unburned mixture

Now the attention is turned to the unburned mixture. First of all, the spe i heat ratio for an unburned frozen mixture of iso-o tane is

omputed using CHEPP in the temperature region T ∈ [300, 1000] K, whi h is valid for the entire losed part of a motored y le. The spe i heat ratio for λ = 1 is shown in gure 4.1 as a fun tion of temperature, together with its linear approximation (4.7) in a least squares sense. u The linear approximation γlin is fairly good for λ = 1. A tually, the spe i heats cp and cv from whi h γ is formed, are fairly well des ribed by linear fun tions of temperature. Table 4.2 summarizes the root mean square error (RMSE), normalized RMSE (NRMSE) and the oe ients of the respe tive linear fun tion for γ , mass-spe i heats cv and cp for u temperature region T ∈ [300, 1000] K and λ = 1. The RMSE of γlin

4.4.

55

UNBURNED MIXTURE

Iso−octane; Unburned mixture @λ=1 1.36 CHEPP Linear

Specific heat ratio − γ [−]

1.34

1.32

1.3

1.28

1.26

1.24 300

400

500

600 700 Temperature [K]

800

900

1000

Figure 4.1: Spe i heat ratio for unburned stoi hiometri mixture using CHEPP and the orresponding linear fun tion of temperature.

Property u γlin [-℄ lin cp,u [J/(kg K)℄ clin v,u [J/(kg K)℄

Constant 1.3488 1051.9 777.0

Slope

−13.0 · 10−5

0.387 0.387

NRMSE 0.19 % 0.15 % 0.20 %

RMSE 0.0024 1.78 1.78

Table 4.2: Coe ients, normalized RMSE and RMSE in linear approximations of γ , mass-spe i cv and cp , for temperature region T ∈ [300, 1000] K and λ = 1.

56

CHAPTER 4. A SPECIFIC HEAT RATIO MODEL

...

Iso−octane; Unburned mixture 1.4

Specific heat ratio − γ [−]

λ=∞

1.35

λ=1.2 1.3

1.25 300

λ=0.975 λ=1 λ=1.025

400

λ=0.8

500

600 700 Temperature [K]

800

900

1000

Figure 4.2: Spe i heat ratio for unburned stoi hiometri mixture using CHEPP for various air fuel ratios λ as fun tions of temperature. λ = ∞ orresponds to pure air. is dened as: v u M u 1 X u (T ))2 , RM SE = t (γ(Tj ) − γlin j M j=1

(4.15)

v u M  u (T ) 2 u 1 X γ(Tj ) − γlin j t . N RM SE = M j=1 γ(Tj )

(4.16)

where M are the number of samples. The NRMSE is then found by normalizing the modeling error in ea h sample j a

ording to:

Besides temperature, the spe i heat ratio also varies with AFR, as shown in gure 4.2 where λ is varied between 0.8 (ri h) and 1.2 (lean). For omparison, γ(T ) is also shown for λ = ∞, i.e. pure air whi h orresponds to fuel ut-o. u (4.7) vary with λ as shown in the two upper The oe ients in γlin plots of gure 4.3. Both the onstant γ300 and the slope b be ome smaller as the air-fuel ratio be omes ri her. From the bottom plot

4.4.

57

UNBURNED MIXTURE

Iso−octane; Linear model of γ for unburned mixture

1.35 1.34 0.8 −4 0.85 x 10

0.9

0.95

1

1.05

1.1

1.15

1.2

0.8 −3 0.85 x 10

0.9

0.95

1

1.05

1.1

1.15

1.2

0.8

0.9

0.95 1 1.05 Air−fuel ratio − λ [−]

1.1

1.15

1.2

Slope b1 [1/K]

γ

300

[−]

1.36

NRMSE [−]

−1.26 −1.28 −1.3 −1.32 −1.34 2.5 2 1.5

0.85

Figure 4.3: Upper: The onstant value γ300 in (4.7) as a fun tion of λ for unburned mixture at equilibrium. Middle: The value of the slope

oe ient b in (4.7) as a fun tion of AFR. Bottom: Normalized root u mean square error (NRMSE) for γlin (T ).

of gure 4.3, whi h shows the NRMSE for dierent AFR:s, it an be u (T ) is better the leaner

on luded that the linear approximation γlin the mixture is, at least for λ ∈ [0.8, 1.2].

4.4.1 Modeling λ-dependen e with xed slope, b Sin e it is always desirable to have as simple models as possible, an important question is: Would it ini t a major dis repan y to x the slope oe ient b and let only γ300 vary with the air-fuel ratio? This is investigated by setting the slope b to the value for λ = 1, and nding the

oe ient γ300 in a least squares sense. The slope is xed at λ = 1, sin e for spark ignited engines this is the region where the engine should be operating most of the time, due to legislations. The results are shown in gure 4.4, where the oe ient γ300 be omes approximately the same as when letting the slope vary. The relative dieren e is less than 0.1 % for λ ∈ [0.8, 1.2]. For the NRMSE an in rease for λ 6= 1 ompared to when b is free is expe ted, but the in rease is not very signi ant at all.

58

CHAPTER 4. A SPECIFIC HEAT RATIO MODEL

...

Iso−octane; Linear model of γ for unburned mixture

γ300 [−]

1.36

Fixed slope Free slope

1.35

1.34 0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

−3

NRMSE [−]

2.5

x 10

Fixed slope Free slope 2 1.5 1

0.8

0.85

0.9

0.95 1 1.05 Air−fuel ratio λ [−]

1.1

1.15

1.2

Figure 4.4: Upper: The onstant value γ300 in (4.7) as a fun tion of λ for unburned mixture at equilibrium with xed and free slope b respe u tively. Bottom: NRMSE for γlin (T ) for xed and free slope oe ient. For λ ∈ [0.8, 1.12] the relative dieren e in RMSE is less than 5 % and for λ ∈ [0.94, 1.06] it is less than 1 %. This suggests that at least for λ ∈ [0.94, 1.06], the linear approximation with xed slope set at λ = 1,

an be used as a model of γ(T ) with good a

ura y for the unburned mixture. The parameter γ300 is then taking are of the λ-dependen e with good a

ura y. 4.5

Burned mixture

The spe i heat ratio γ for a burned mixture of iso-o tane is omputed using CHEPP in temperature region T ∈ [500, 3500] K and pressure region p ∈ [0.25, 100] bar, whi h overs most of the losed part of a ring

y le. The mixture is assumed to be at equilibrium at every instant. The spe i heat ratio depends strongly on mixture temperature T , but γ also on the air-fuel ratio λ and pressure p as shown in gure 4.5 and gure 4.6 respe tively. For the same deviation from λ = 1, ri h mixtures tend to deviate more from the stoi hiometri mixture, than lean mixtures do. The pressure dependen e in γ is only visible for

4.5.

59

BURNED MIXTURE

Iso−octane; Burned mixture @p=7.5 bar 1.4 1.35

Specific heat ratio − γ [−]

λ=1.2

λ=0.8

1.3 1.25

λ=0.8

1.2 λ=1.2 1.15 λ=0.975 λ=1 λ=1.025

1.1 1.05 1 500

1000

1500

2000 2500 Temperature [K]

3000

3500

Figure 4.5: Spe i heat ratio for burned mixture at various air-fuel ratios λ at 7.5 bar using CHEPP.

Region A B C D E

T ∈ [500, 3500] [500, 3000] [500, 2700] [500, 2500] [1200, 3000]

γ300

1.3695 1.3726 1.3678 1.3623 1.4045

b −9.6 · 10−5 −9.9 · 10−5 −9.4 · 10−5 −8.8 · 10−5 −11.4 · 10−5

b Table 4.3: Coe ients in the linear approximation γlin (T ) found in (4.7) for λ = 1 and p = 7.5 bar.

60

CHAPTER 4. A SPECIFIC HEAT RATIO MODEL

...

Iso−octane; Burned mixture @λ=1 1.4 2 bar 7.5 bar 35 bar

Specific heat ratio − γ [−]

1.35 1.3 1.25 1.2 1.15 1.1 1.05 1 500

1000

1500

2000 2500 Temperature [K]

3000

3500

Figure 4.6: Spe i heat ratio for burned stoi hiometri mixture using CHEPP at various pressures. T > 1500 K, and a higher pressure tends to retard the disso iation and yields a higher γ . b (T ) of To model the spe i heat ratio with a linear fun tion γlin temperature, and thereby negle ting the dependen e of pressure, will of ourse introdu e a modeling error. This modeling error depends on whi h temperature (and pressure) region the linear fun tion is tted to, sin e dierent regions will yield dierent oe ient values in (4.7). In gure 4.7 γ is omputed at λ = 1 and p = 7.5bar for T ∈ [500, 3500] K , b and as well as the orresponding linear fun tion γlin (4.7) and the polynomial γKB (4.14) developed in Krieger and Borman (1967). In b table 4.3, the oe ients in γlin are omputed for ve temperature regions. Table 4.4 displays the maximum relative error (MRE) and b b and γKB . The maximum relative error for γlin is NRMSE for γlin dened as M RE = max |

b γ(Tj , p) − γlin (Tj ) |. γ(Tj , p)

(4.17)

b (T ) does not apture the behavior of The linear approximation γlin γ(T ) for λ = 1 very well, as shown in gure 4.7. The oe ients for b the linear model γlin (T ) vary with the spe i temperature region, as

4.5.

61

BURNED MIXTURE

Iso−octane; Burned mixture @λ=1, p=7.5 bar 1.4 CHEPP Linear KB poly

Specific heat ratio − γ [−]

1.35 1.3 1.25 1.2 1.15 1.1 1.05 1 500

1000

1500

2000 2500 Temperature [K]

3000

3500

Figure 4.7: Spe i heat ratio for burned stoi hiometri mixture using b CHEPP, the orresponding linear fun tion γlin and Krieger-Borman polynomial γKB .

Region T ∈ A B C D E

[500, 3500] [500, 3000] [500, 2700] [500, 2500] [1200, 3000]

MRE 2.0 % 1.6 % 1.9 % 2.4 % 1.6 %

b γlin

NRMSE 0.97 % 0.95 % 0.90 % 0.74 % 0.74 %

γKB

MRE 2.0 % 0.7 % 0.3 % 0.3 % 0.7 %

NRMSE 0.56 % 0.20 % 0.17 % 0.17 % 0.21 %

Table 4.4: Maximum relative error (MRE) and normalized root mean square error (NRMSE) for dierent temperature regions at λ = 1 and p = 7.5 bar.

62

CHAPTER 4. A SPECIFIC HEAT RATIO MODEL

Region γKB @λ = 0.975 γKB @λ = 1 MRE NRMSE MRE NRMSE A 1.9 % 0.86 % 2.0 % 0.56 % B 1.8 % 0.73 % 0.7 % 0.20 %

...

γKB @λ = 1.025

MRE NRMSE 2.1 % 0.59 % 0.7 % 0.28 %

Table 4.5: Maximum relative error (MRE) and normalized root mean square error (NRMSE) for dierent temperature regions for γKB (T, p, λ) at p = 7.5 bar and λ = {0.975, 1, 1.025}. displayed for temperature regions A to E in table 4.3. A se ond order polynomial shows the same behavior as the linear ase, but when the order of the polynomial is in reased to three, the model aptures the modes of γ(T ) quite well. By in reasing the omplexity of the model even more, an even better t is found. This has been done in the Krieger-Borman polynomial, and for this example it aptures the behavior of γ(T ) well for temperatures below 2800 K as seen in gure 4.7 and in the right-most (NRMSE) olumn in table 4.4, where the NRMSE value is mu h higher for temperature region A than for the other regions. As expe ted, the Krieger-Borman polynomial is better than the linear approximation in every hosen temperature region, sin e the NRMSE is smaller. Comparing the MRE:s for temperature region A, where the respe tive MRE are approximately the same, one

ould then on lude that the models des ribe γ equally well. However in gure 4.7 it was learly visible that γKB is the better one, whi h is also the on lusion when omparing the respe tive NRMSE. In table 4.5, the NRMSE and MRE for the Krieger-Borman polynomial γKB (T, p, λ) for λ lose to stoi hiometri is displayed. For λ ≥ 1 (lean), γKB ts the equilibrium γ better than for λ < 1, a tenden y whi h is most evident when omparing the NRMSE for temperature region B. For temperature region A the dieren e for dierent λ is less striking, sin e the γKB does not t γ as well for T > 3000 K. Therefore the Krieger-Borman polynomial is preferably only to be used on the lean side. On the ri h side and lose to stoi hiometri (within 2.5 %), the Krieger-Borman polynomial does not introdu e an error larger than the linear approximation given in table 4.4, and γKB should therefore be used in this operating range. Summary for spe ial ase: Linear models

If a linear model of γ is preferred for omputational reasons, the performan e of the linear model ould be enhan ed by proper sele tion of temperature region. However, the MRE does not de rease for every redu tion in interval, as seen when omparing MRE:s for regions D and

4.6.

PARTIALLY BURNED MIXTURE

63

B in table 4.4. Thus, the temperature region should be hosen with

are by using the NRMSE as evaluation riteria: • When using the single-zone temperature T to des ribe the spe i heat ratio of the burned mixture, temperature region B is preferable, sin e during the losed part T ≤ 3000 K.

• When using the burned-zone temperature Tb in a two-zone model, temperature region E is re ommended, sin e for most ases Tb ∈ [1200, 3000]. The temperature limits are found by evaluating

a number of experimental ylinder pressure tra es using (A.1) and (A.7). By hoosing region E instead of region B, the NRMSE is redu ed by 25%.

4.6

Partially burned mixture

The spe i heat ratio γ as a fun tion of mixture temperature T and air-fuel ratio λ for unburned and burned mixture of air and iso-o tane has been investigated in the two previous se tions. During the losed part of a motored engine y le, the previous investigations would be enough sin e the models of the unburned mixture will be valid for the entire region. When onsidering ring y les on the other hand, an assumption of either a purely unburned or a purely burned mixture approa h is not valid for the entire ombustion hamber during the

losed part of the engine y le. To des ribe the spe i heat ratio in the single-zone model for a partially burned mixture, the mass fra tion burned tra e xb is used to interpolate the (mass-)spe i heats of the unburned and burned zones to nd the single-zone spe i heats. The spe i heat ratio is then found as the ratio between the interpolated spe i heats. 4.6.1

Referen e model

The single-zone spe i heats are found from energy balan e between the single-zone and the two-zone model, from whi h the single-zone spe i heat ratio γCE an be stated: cp (T, p, xb ) = xb cp,b (Tb , p) + (1 − xb ) cp,u (Tu )

(4.18a)

cv (T, p, xb ) = xb cv,b (Tb , p) + (1 − xb ) cv,u (Tu )

(4.18b)

γCE (T, p, xb ) =

cp (T, p, xb ) , cv (T, p, xb )

(4.18 )

where the mass fra tion burned xb is used as an interpolation variable. The single-zone (T ), burned zone (Tb ) and unburned zone (Tu ) temperatures are given by the two temperatures models (A.1) and (A.7)

64

CHAPTER 4. A SPECIFIC HEAT RATIO MODEL

...

des ribed in appendix A.1. The rst is the ordinary single-zone temperature model and the se ond is a two-zone mean temperature model presented in Andersson (2002). The mass spe i heats in (4.18) are

omputed using CHEPP (Eriksson, 2004) and γCE then forms the referen e model. To ompute γCE is omputationally heavy. Even when the spe i heats are omputed before-hand at a number of operating points, the

omputational burden is still heavy due to the numerous table lookups and interpolations required. Therefore, a omputationally more e ient model whi h retains a

ura y is sought for. A number of γ models will therefore be des ribed in the following subse tion, where they are divided into three subgroups based upon their modeling assumptions. These γ -models are then ompared to the referen e model γCE (4.18), in terms of four evaluation riteria, spe ied in se tion 4.6.3.

How to nd

xb ?

To ompute the spe i heat ratio γCE (4.18), a mass fra tion burned tra e xb is needed. For simulated pressure data, the mass fra tion burned is onsidered to be known, whi h is the ase in this work. However, if one were to use experimental data to e.g. do heat release analysis, xb an not be onsidered to be known. There are then two ways of determining the mass fra tion burned; The rst is to use a simple and omputationally e ient method to get xb from a given ylinder pressure tra e. Su h methods were des ribed in hapter 2 and in lude the pressure ratio management by Matekunas (1983) des ribed in se tion 2.4. If one does not settle for this, the se ond approa h is to initialize xb using a simple method from the rst approa h, and then iteratively rene the mass fra tion burned tra e xb using the omputed heat release. 4.6.2

Grouping of

γ -models

Thirteen γ -models have been investigated and they are divided into three subgroups based upon their modeling assumptions; The rst group ontains models for burned mixture only. The se ond ontains models based on interpolation of the spe i heat ratios dire tly, and the third group, to whi h referen e model (4.18) belongs, ontains the models based on interpolation of the spe i heats, from whi h the ratio is determined.

Group B : Burned mixture

The rst subgroup represents the in- ylinder mixture as a single zone of burned mixture with single-zone temperature T , omputed by (A.1).

4.6.

65

PARTIALLY BURNED MIXTURE

The rst model, denoted B1 , is the linear approximation in (4.7): b (T ) = γ300 + b (T − 300), B1 : γB1 (T ) = γlin

(4.19)

where the oe ients an be determined in at least two ways; One way is to use the oe ients that are optimized for temperature region T ∈ [500, 3000] (region B in table 4.3) for a burned mixture. This approa h is used in (Gatowski et al., 1984), although the oe ients dier somewhat ompared to the ones given in table 4.3. Another way is to optimize the oe ients from the referen e model (4.18). This approa h will be the one used here, sin e it yields the smallest modeling errors in both γ and ylinder pressure p. The approa h has optimal onditions for the simulations, and will therefore give the best results possible for this model stru ture. The se ond model, denoted B2 , is the Krieger-Borman polynomial des ribed in (4.9) B2 : uB2 = A(T ) −

B(T ) −→ γB2 (T ) = γKB (T ), λ

(4.20)

without the orre tion term for disso iation. The Krieger-Borman polynomial is used in model B3 as well, B(T ) + ucorr (T, p, λ) λ −→ γB3 (T, p) = γKB (T, p),

B3 : uB3 = A(T ) −

(4.21)

with the orre tion term ucorr (T, p, λ) for disso iation in luded. The fourth and simplest model uses a onstant γ : B4 : γB4 = constant.

(4.22)

As for model B1 , the onstant γB4 is determined from the referen e model (4.18).

Group C : Interpolation of spe i heat ratios

The se ond subgroup uses a two-zone model, i.e. a burned and an unburned zone, and al ulates the spe i heat ratio γb (Tb ) and γu (Tu ) for ea h zone respe tively, where the temperatures are given by the two-zone mean temperature model (A.7). The mass fra tion burned tra e xb is then used to nd the single-zone γ by interpolating γb and γu . Note that the relations for determining the thermodynami properties, shown in (4.18), are not fullled for subgroup C during ombustion. It is however fullled prior to ombustion and after the ombustion, i.e. when xb = 0 or xb = 1 and no interpolation is performed.

66

CHAPTER 4. A SPECIFIC HEAT RATIO MODEL

...

The rst model, denoted C1 , interpolates linear approximations of γ for the unburned and burned mixture. The oe ients in the linear fun tions are optimized for temperature region T ∈ [300, 1000] for the unburned mixture, and temperature region T ∈ [1200, 3000] for the burned mixture. The resulting γC1 an therefore be written as: u b (Tb ) + (1 − xb ) γlin (Tu ), C1 : γC1 (T, xb ) = xb γlin

(4.23)

where the oe ients for the linear fun tions are given in table 4.3 and table 4.2 respe tively. The se ond model was proposed in Stone (1999, p.423), here denoted C2 , and is based on interpolation of the internal energy u omputed from the Krieger-Borman polynomial: C2 : uC2 = A(T ) − xb

B(T ) −→ γC2 (T, xb ). λ

(4.24)

This model in ludes neither disso iation nor the internal energy of the unburned mixture. An improvement of model C1 is expe ted when substituting the linear model for the burned mixture with the Krieger-Borman polynomial. This new model is denoted C3 and des ribed by: u (Tu ). C3 : γC3 (T, p, xb ) = xb γKB (Tb , p) + (1 − xb ) γlin

(4.25)

The fourth model interpolates γu (Tu ) and γb (Tb , p) given by CHEPP: C4 : γC4 (T, p, xb ) = xb γb (Tb , p) + (1 − xb ) γu (Tu ),

(4.26)

and this model is denoted C4 . This model will ree t the modeling error introdu ed by interpolating the spe i heat ratios dire tly instead of using the denition through the spe i heats (4.18). The segmented linear model (4.8) developed in Chun and Heywood (1987) is also investigated and here denoted by model C5 :  comp  γ300 + bcomp (T − 300) xb < 0.01 0.01 ≤ xb ≤ 0.99 C5 : γC5 (T, xb ) = γseg (T, xb ) = γ comb  300 exp + bexp (T − 300) xb > 0.99. γ300

(4.27) Model C5 uses the single-zone temperature for ea h phase, and lassies into group C due to that the swit hing used for xb in (4.27) an be seen as a nearest neighbor interpolation. As for model B1 and B4 , the

oe ients in (4.27) are determined from the referen e model (4.18).

Group D : Interpolation of spe i heats

The last subgroup uses a two-zone model, i.e. a burned and an unburned zone, just as the se ond subgroup, and the spe i heats are

4.6.

67

PARTIALLY BURNED MIXTURE

interpolated to get the single-zone spe i heats. The rst model, denoted D1 , uses the Krieger-Borman polynomial for the burned zone to nd cp,b (Tb , p) and cv,b (Tb , p), and the linear approximations of cp,u (Tu ) and cv,u (Tu ) given in table 4.2 for the unburned zone: D1 : γD1 (T, p, xb ) =

lin xb cKB p,b (Tb , p) + (1 − xb ) cp,u (Tu )

KB (T , p) + (1 − x ) clin (T ) xb cv,b b b v,u u

.

(4.28)

An extension of model D1 is to use the unburned spe i heats cp,u (Tu ) and cv,u (Tu ) omputed from CHEPP: D2 : γD2 (T, p, xb ) =

xb cKB p,b (Tb , p) + (1 − xb ) cp,u (Tu )

KB (T , p) + (1 − x ) c xb cv,b b b v,u (Tu )

.

(4.29)

This model is denoted D2 and ree ts the model error introdu ed by using the linear approximation of the unburned mixture spe i heats, when omparing to D1 . When omparing it to D4 , it also shows the error indu ed by using the Krieger-Borman approximation. Model D1 is also extended for the burned mixture, where the spe i heats for the burned mixture cp,b (Tb , p) and cv,b (Tb , p) are omputed using CHEPP. This model is denoted D3 : D3 : γD3 (T, p, xb ) =

xb cp,b (Tb , p) + (1 − xb ) clin p,u (Tu ) , xb cv,b (Tb , p) + (1 − xb ) clin v,u (Tu )

(4.30)

and ree ts the model error introdu ed by using the Krieger-Borman approximation of the spe i heats, when omparing to D1 . The referen e model γCE (4.18) belongs to this group and is denoted D4 : D4 : γD4 (T, p, xb ) = γCE (T, p, xb ). (4.31)

Modeling of revi e energy term

Note that the usage of a γ -model dierent from the linear model used in Gatowski et al. [1984℄, will also ae t the amount of energy left or added to the system when a mass element enters or leaves the revi e volume. This energy term u′ − u is quantied by (2.32) for B1 . It has to be restated for every γ -model at hand ex ept model B1 , and this is done in appendix A.4. 4.6.3

Evaluation riteria

The dierent γ -models given by (4.19)-(4.30) are evaluated in terms of four riteria. The riteria are: 1. Normalized root mean square error (NRMSE) in γ (4.16), whi h gives a measure of the mean error in γ .

68

CHAPTER 4. A SPECIFIC HEAT RATIO MODEL

...

2. Maximum relative error (MRE) for γ (4.17), whi h yields a measure of the maximum error in γ . 3. Root mean square error (RMSE) for the orresponding ylinder pressures, see (4.15) for the RMSE denition. This riterion gives a measure of the impa t that a ertain model error has on the

ylinder pressure and will help to nd a γ -model a

urate enough for the single-zone model. 4. The omputational time. It is measured as the time for one simulation of the ylinder pressure model during the losed part, given a burn rate tra e and a spe i γ -model. The ylinder pressure model used for the simulations is the model developed in Gatowski et al. (1984), summarized in se tion 3.8. The parameters are given in appendix A.3 and the engine geometry is given in appendix A.2. 4.6.4

Evaluation overing one operating point

At rst, only one operating point is onsidered. This operating point is given by the parameter values in table A.1, and orresponds to the

ylinder pressure given in gure 4.8, i.e. a low engine load ondition. The ylinder pressure given in gure 4.8 is used as an example that illustrates the ee t that ea h model has on spe i heat ratio γ and

ylinder pressure. To investigate if the engine operating ondition inuen es the hoi e of model, nine operating points overing most parts of the operating range of an engine are used to do the same evaluations. These operating points are given in table A.2 and their orresponding

Cylinder pressure [MPa]

Simulated cylinder pressure@OP2 2 1.5 1 0.5 0

−100

−50 0 50 Crank angle [deg ATDC]

100

Figure 4.8: Simulated ylinder pressure using Gatowski et al.-model with nominal values in table A.1, and the linear γ -model B1 repla ed by referen e model D4 .

4.6.

PARTIALLY BURNED MIXTURE

69

ylinder pressures are displayed in gure A.6, where operating point 2

orresponds to the ylinder pressure that is used in the rst evaluation and shown in gure 4.8. γ -domain

The γ -models in the three subgroups are ompared to the referen e model γCE (4.18). A summary of the results are given here while a

omplete pi ture is given in appendix A.8, see e.g. gures A.7 and A.8, where γ is plotted as a fun tion of rank angle. The spe i heat ratio for ea h model is also given in gure A.9 and gure A.10 as a fun tion of single-zone temperature T . Table 4.6 summarizes the MRE(γ ) and NRMSE(γ ) for all models. Figure 4.9 ompares the referen e model D4 with the omputed values of γ for a few of these models, namely B1 , B3 , C5 , C4 and D1 . Of these models, model D1 (4.28) gives the best des ription of γ and

aptures the referen e model well. This is onrmed by the MRE(γ ) and NRMSE(γ ) olumns in table 4.6, where only model group D yields errors lower than 1 % for both olumns. The lower plot of gure 4.9 shows that model C4 deviates only during the ombustion, whi h in this

ase o

urs for θ ∈ [−15, 45] deg ATDC. This deviation is enough to yield a NRMSE(γ ) whi h is almost 0.6 %, approximately six times that of D1 . Of the models previously proposed in literature, the linear model B1 (4.19) has the best performan e, although it does not apture the referen e model very well, as seen in the upper plot of gure 4.9. Model B3 (4.21) is only able to apture the referen e model after the ombustion, sin e model B3 is optimized for a burned mixture. Model C5 (4.27) has good behavior before and after the ombustion. But durcomb ing the ombustion, the onstant γ300 does not apture γCE very well. Models B4 and C2 has even worse behavior, as shown in gure A.7. To on lude, model group D yields errors in γ whi h are less than 1 % for this operating point. Of these models, model D3 has the best performan e ompared to the referen e model D4 . Pressure domain

The impa t ea h γ -model has on the orresponding ylinder pressure is shown in gure 4.10 for models B1 , B3 , C5 , C4 and D1 , and for all models in gures A.11 and A.12. The plots show the dieren e between the simulated ylinder pressure for referen e model D4 and the γ -models, i.e. the error in the ylinder pressure that is indu ed by the modeling error in γ . Note that the s aling in the gures are dierent. The RMSE introdu ed in the ylinder pressure is given in table 4.6 for all models. The RMSE of the measurement noise is approximately 6 kPa and

70

CHAPTER 4. A SPECIFIC HEAT RATIO MODEL

...

1.4

γ [−]

B3 1.3 1.2 1.1

Ref B1 B3 C5 −100

C5 −50 0 50 Crank angle [deg ATDC]

100

γ [−]

1.4 C4

1.3 Ref C 4 D1

1.2 1.1

−100

−50 0 50 Crank angle [deg ATDC]

100

Figure 4.9: Upper: Spe i heat ratios for models B1 , B3 and C5 as

ompared to the referen e model D4 . Lower: Spe i heat ratios for models C4 and D1 as ompared to the referen e model D4 . Model B1 B2 B3 B4 C1 C2 C3 C4 C5 D1 D2 D3 D4

(4.19) (4.20) (4.21) (4.22) (4.23) (4.24) (4.25) (4.26) (4.27) (4.28) (4.29) (4.30) (4.18)

MRE: γ [%℄ 4.7 5.9 5.2 7.7 2.3 7.3 2.4 2.3 8.4 0.27 0.26 0.04 0.0

NRMSE: γ [%℄ 1.3 2.7 1.8 4.5 0.69 4.1 0.65 0.58 1.5 0.10 0.09 0.01 0.0

RMSE: p [kPa℄ 52.3 85.8 76.0 62.8 39.8 140.7 25.4 22.8 82.9 2.8 2.6 0.3 0.0

Time [s℄ 3.8 4.1 4.2 3.8 4.7 4.9 5.1 211.1 4.0 5.2 12.3 381.9 384.2

Table 4.6: Evaluation of γ -models, on the single y le shown in gure 4.8.

4.6.

71

PARTIALLY BURNED MIXTURE

Pressure [kPa]

2000 1500 1000 500 0

−100

−50

0

50

100

Pressure error [kPa]

250

B 1 B 3 C5

200 150 100 50 0 −100

−50

0

50

100

Pressure error [kPa]

80

C4 D

60

1

40 20 0 −20

−100

−50 0 50 Crank angle [deg ATDC]

100

Figure 4.10: Upper: Referen e ylinder pressure, the same as given in gure 4.8. Middle: Cylinder pressure error introdu ed by models B1 , B3 and C5 . For onvenien e, the sign for C5 is hanged. Lower: Cylinder pressure error introdu ed by models C4 and D1 . Note that the s aling in the plots are dierent.

72

CHAPTER 4. A SPECIFIC HEAT RATIO MODEL

...

it is only model group D that introdu es a modeling error in the same order as the noise in terms of RMSE. Thus, the other γ -models will introdu e a modeling error whi h is signi antly larger than the measurement noise as seen in table 4.6, and thereby ae t the a

ura y of the parameter estimates. Within group D , models D3 and D4 have the smallest RMSE(p), and therefore yield the highest a

ura y. Model D1 does not introdu e a signi antly larger RMSE(p) than D2 , and therefore the most time e ient one should be used of these two. Altogether this suggests that any model in group D ould be used. The previously proposed γ -models B1 , B2 , B3 , B4 and C5 , des ribed in se tion 4.3, all introdu e modeling errors whi h are at least seven times the measurement noise for this operating point. Clearly, a large error, so none of these models are re ommended. Of these models, B1 indu es the smallest RMSE(p) and should, if any, be the one used of the previously proposed models.

Computational time The right-most olumn of table 4.6 shows the omputational time. The time value given is the mean time for simulating the losed part of one engine y le using Matlab 6.1 on a SunBlade 100, whi h has a 64-bit 500 MHz pro essor. The proposed model D1 is approximately 70 times faster than the referen e model D4 , where the referen e model uses look-up tables with pre omputed values of the spe i heats cp and cv . Introdu ing the model improvement in model D1 of the spe i heat ratio to the Gatowski et al. single-zone heat release model is simple, and it does not in rease the omputational burden immensely ompared to the original setting, i.e. B1 . The in rease in omputational eort is less than 40 % ompared to the linear γ -model when simulating the Gatowski et al. single-zone heat release model. 4.6.5

Evaluation overing all operating points

The same analysis as above has been made for the simulated ylinder pressure from nine dierent operating points, where pIV C ∈ [0.25, 2] bar and TIV C ∈ [325, 372] K. The parameters for ea h y le is given in table A.2 as well as the orresponding ylinder pressures in gure A.6. The operating range in p and T that these y les over is given in gure 4.11, where the upper plot shows the range overed for the unburned mixture, and the lower shows the range overed for single-zone (solid) and burned (dashed) mixture. A

ording to (Heywood, 1988, p.109), the temperature region of interest for an SI engine is 400 to 900 K for the unburned mixture; for the burned mixture, the extreme end states are approximately {1200 K, 0.2 M P a} and {2800 K, 3.5 M P a}. Of

4.6.

73

PARTIALLY BURNED MIXTURE

Pressure [MPa]

10 Unburned zone

5

0 300

400

500

600 700 Temperature [K]

800

900

1000

1500 2000 Temperature [K]

2500

3000

3500

Pressure [MPa]

10 Burned zone Single zone 5

0 0

500

1000

Figure 4.11: Operating range in p and T . Upper: Unburned zone. Single zone (solid) and burned zone (dashed).

Lower:

ourse, not all points in the range are overed but the y les at hand

over the extremes of the range of interest. The results are summarized in terms of MRE for γ in table A.11, NRMSE for γ in table A.12 and RMSE for p in table A.13, where the mean values over the operating points for ea h model as well as the values for ea h y le are given. The mean values for ea h model are also given here as a summary in table 4.7.

Ordering of models When omparing the NRMSE for γ in table 4.7, the ordering of the γ -models, where the best one omes rst, is: D4 ≺ D3 ≺ D2 ≺ D1 ≺ C4 ≺ C3 ≺ C1 ≺ B1 ≺ C5 ≺ B3 ≺ B2 ≺ C2 ≺ B4 .

(4.32) Here B2 ≺ C2 means that model B2 is better than C2 . Comparing RMSE for the ylinder pressure p, the ordering of the γ -models be omes: D4 ≺ D3 ≺ D2 ≺ D1 ≺ C4 ≺ C3 ≺ C1 ≺ B1 ≺ B4 ≺ B3 ≺ B2 ≺ C5 ≺ C2 .

(4.33)

74

CHAPTER 4. A SPECIFIC HEAT RATIO MODEL

Model B1 B2 B3 B4 C1 C2 C3 C4 C5 D1 D2 D3

MRE: γ [%℄ 3.4 5.2 4.5 7.1 1.9 6.6 1.9 1.8 8.3 0.26 0.25 0.044

NRMSE: γ [%℄ 1.2 2.4 1.7 4.2 0.77 3.9 0.53 0.46 1.6 0.097 0.092 0.016

...

RMSE: p [kPa℄ 84.9 153.6 137.3 110.0 56.6 269.2 42.4 36.7 191.9 5.8 5.1 0.7

Table 4.7: Evaluation of γ -models, in terms of the mean values for all operating points in table A.2. This ordering is not the same as in (4.32), but the only dieren e lies in models C5 and B4 that hange their positions between the two orderings. Model C5 has poor performan e in terms of RMSE(p), ompared to NRMSE(γ ). For model B4 , it is the other way around.

Model group D

In terms of NRMSE(γ ) (4.32) and RMSE(p) (4.33) model group D behaves as expe ted, and obeys the rule: the higher the omplexity is, the higher the a

ura y be omes. A

ording to the RMSE(p) olumn in table 4.7, the models in D all introdu e an RMSE(p) whi h is less than that found for the measurement noise. Comparing models D1 (4.28) and D2 (4.29), it is obvious that not mu h is gained in a

ura y by using the unburned spe i heats from CHEPP instead of the linear fun tions. The omputational ost for D2 was more than two times the one for D1 , as shown in table 4.6. This suggests that the unburned spe i heats are su iently well des ribed by the linear approximation. Model D3 (4.30) utilizes the burned spe i heat from CHEPP, and this is an improvement ompared to model D1 whi h uses the KriegerBorman polynomial for cp,b and cv,b . This improvement redu es the RMSE(p) with a fa tor 7, but the ost in omputational time is high, approximately a fa tor 70 a

ording to table 4.6. The omparison also shows that if we want to redu e the impa t on the ylinder pressure, the eort should be to in rease the a

ura y of the Krieger-Borman polynomial for the burned mixture. In gures A.4 and A.5, the spe i

4.6.

PARTIALLY BURNED MIXTURE

75

heats for CHEPP and the Krieger-Borman polynomial are given, and this veries that the polynomial has poorer performan e for higher temperatures. A new polynomial for the burned mixture, valid for a smaller but more relevant region for SI engines ould in rease the a

ura y for high temperatures. A perfe t model of the burned mixture yields the results given by D3 , whi h then poses the lower limit for the a

ura y. The RMSE(p) in table 4.7 for D1 is however onsidered to be small. Therefore nding a better a model for the burned mixture is not pursued here, and model D1 is re ommended as a good ompromise between omputational a

ura y and e ien y.

Model group C

In model group C , model C5 has good performan e when onsidering the NRMSE in γ (4.32), but not as good in RMSE(p) (4.33). This illustrates the importan e of transforming the modeling error in the γ -domain to the ylinder pressure domain. One obje tive was to get a model that gives a good des ription of the ylinder pressure. This motivates why RMSE(p) is the more important model performan e measure of the two. Model C2 (Stone, 1999, p.423) has really bad performan e and would be the last hoi e here. The rest of the models in group C obey the same rule as group D , i.e. C4 ≺ C3 ≺ C1 . When the best model in group C , i.e. C4 , is ompared to all models in group D , and espe ially the referen e model D4 , it is on luded that the spe i heats should be interpolated, and not the spe i heat ratios. This on lusion an be drawn sin e the only dieren e between C4 and D4 is how the interpolation is performed. Model C4 interpolates the spe i heat ratios found from CHEPP dire tly, and model D4 interpolates the spe i heats from CHEPP and then form the spe i heat ratio. Therefore, group D has better performan e than group C . Sin e D1 has higher a

ura y and approximately the same omputational time as all models in group C , there is no point in using any of the models in group C .

Model group B

As expe ted, the models in group B has the worst performan e of them all, if ex luding models C2 and C5 . It is interesting to note that the linear b model γlin (B1 ) performs best in the group, although it introdu es a modeling error in p whi h is at least ten times the measurement noise in the mean. It has better performan e than γKB (B3 ) in the pressure domain, although this is not the ase in the γ -domain. This again points out the ne essity of evaluating the impa t of the γ -model on to the ylinder pressure. Therefore, if the assumption is that the ylinder

ontents should be treated as a burned mixture during the entire losed

76

CHAPTER 4. A SPECIFIC HEAT RATIO MODEL

...

part of the engine y le, B1 is the model to use.

Summary To on lude, the models are ordered by their performan e and with

omputational e ien y in as ending order: D4 ≺ D1 ≺ B1 ≺ B4

(4.34)

Most of the models are ex luded from this list, either due to their low a

ura y, high omputational time, or be ause another model with approximately the same omputational time has higher a

ura y. Of the models given in (4.34), D1 is re ommended as a ompromise between

omputational time and a

ura y. Compared to the original setting in Gatowski et al. (1984), the omputational burden in reases with 40 % and the modeling error is more than ten times smaller in the mean. This also stresses that the γ -model is an important part of the heat release model, sin e it has a large impa t on the ylinder pressure. The fo us is now turned to how the γ -models will ae t the heat release parameters.

4.6.6 Inuen e of γ -models on heat release parameters The next question is: What impa t does ea h of the proposed γ models have on the heat release parameters? This is investigated by using the ylinder pressure for operating point 2, given in gure 4.8, and estimating the three heat release parameters ∆θd , ∆θb and Qin in the Vibe fun tion, introdu ed in se tion 3.5. The ylinder pressure is simulated using referen e model D4 in onjun tion with the Gatowski et al. ylinder pressure model, and this forms the ylinder pressure measurement signal to whi h measurement noise is added. The heat release tra e is then estimated given the measurement from referen e model D4 . The heat release tra e is parameterized by the Vibe fun tion, whi h has the heat release parameters ∆θd , ∆θb and Qin where it is assumed that θig is known a priori. The estimation is performed by minimizing the predi tion error, i.e. by minimizing the dieren e between the measured ylinder pressure and the modeled

ylinder pressure. The Levenberg-Marquardt method des ribed in appendix C.1 is used as optimization algorithm. The heat release parameters are then estimated for ea h of the γ -models using the Gatowski et al.-model, where the γ -model is repla ed in an obvious manner in the equations. In the estimations, only the three heat release parameters are estimated. The other parameters are set to their true values given in table A.1. The results are summarized in table 4.8, whi h displays

4.6.

77

PARTIALLY BURNED MIXTURE

Model B1 B2 B3 B4 C1 C2 C3 C4 C5 D1 D2 D3 D4

∆θd [%℄

RME 5.1 3.1 3.4 6.8 0.074 9.6 0.19 0.14 -8 0.21 0.2 0.22 0.21

RCI 1.7 1.7 1.7 1.7 1.4 2.1 1.4 1.5 1.5 1.5 1.5 1.5 1.5

∆θb [%℄

RME 0.29 0.63 -0.2 -0.11 1.1 -1 0.75 0.64 -2.5 -0.062 -0.08 -0.13 -0.13

RCI 3.1 2.9 2.9 3.2 2.4 3.9 2.4 2.4 2.3 2.4 2.4 2.4 2.4

Qin [%℄

RME -9.2 -7.3 -7.2 -6.2 -2.9 -14 -2.5 -2 27 -0.67 -0.61 -0.48 -0.42

RCI 1.4 1.3 1.3 1.4 1.1 1.7 1.2 1.2 0.92 1.3 1.3 1.3 1.3

RMSE(p) [kPa℄ 9.8 9.1 9.1 10.1 6.5 16.0 6.5 6.7 6.6 6.7 6.7 6.7 6.7

Time [min℄ 3.5 3.8 3.9 3.5 4.4 4.6 4.8 200 3.7 4.9 11 360 360

Table 4.8: Relative mean estimation error (RME) and mean relative 95 % onden e interval (RCI) given in per ent, for heat release parameters using various γ -models at operating point 2. The nominal values for the heat release parameters are: ∆θd = 20 deg, ∆θb = 40 deg and Qin = 760 J. The omputational time and ylinder pressure RMSE are also given. the relative mean estimation error RME (D.4) and the mean relative 95 % onden e interval RCI (D.5) in ∆θd , ∆θb and Qin respe tively for ea h γ -model. The omputational time and RMSE(p) are also given.

Dis ussion The RMSE of the applied measurement noise is approximately 6.7 kPa, whi h is also the RMSE found when using most γ -models. For every γ model used, the rapid burn angle θb is most a

urately estimated of the three parameters, and nearly all of them are a

urate within 1%. On the other hand, only model group D is a

urate within 1 % for all three parameters, and this suggests that any of the D -models an be used, preferably model D1 due to its lower omputational time. Model C5 has the highest deviation in the estimates of them all.

4.6.7 Inuen e of air-fuel ratio λ An investigation is performed here to see how the proposed model D1 behaves for dierent air-fuel ratios λ. The NRMSE(γ, λ|D1 ) and RMSE(p, λ|D1 ) are omputed for model D1 (4.28) ompared to referen e model D4 for the air-fuel ratio region λ ∈ [0.975, 1.025], at operating point 2. It is assumed that the λ- ontroller of the SI engine

78

CHAPTER 4. A SPECIFIC HEAT RATIO MODEL

...

NRMSE(γ | D1) [−]

0.01

0.005

0

0.98

0.99

1

1.01

1.02

0.98

0.99

1 λ [−]

1.01

1.02

RMSE(p | D1) [kPa]

4 3.5 3 2.5

Figure 4.12: Upper: NRMSE(γ, λ|D1 ) for λ ∈ [0.975, 1.025] at OP 2. RMSE(p, λ|D1 ) for λ ∈ [0.975, 1.025] at OP 2.

Lower:

has good performan e, and therefore keeps the variations in λ small. The results are displayed in gure 4.12, where the upper plot shows the NRMSE(γ, λ|D1 ), and the lower plot shows the RMSE(p, λ|D1 ). Lean and stoi hiometri mixtures have the lowest errors in the γ domain, whi h is expe ted sin e the Krieger-Borman polynomial for the burned mixture is estimated for lean mixtures. The error in pressure domain is approximately symmetri around λ = 0.995, and the magnitude is still less than the measurement noise. This assures that for a few per ent deviation in λ from stoi hiometri onditions, the introdu ed error is still small and a

eptable.

Fuel omposition A small, and by no means exhaustive sensitivity analysis is made for fuels su h as methane and two ommer ial fuels in appendix A.6. This in order to see if the results are valid for other fuels than iso-o tane. The hydro arbon ratio for the fuel Ca Hb is given by y = b/a. It is found that if y ∈ [1.69, 2.25], the RMSE(p) introdu ed at OP 2 when using D1 is in reased with less than 20 % ompared to iso-o tane, whi h is a

eptable.

4.6.

79

PARTIALLY BURNED MIXTURE

4.6.8 Inuen e of residual gas The inuen e of the residual gas on the spe i heat ratio has so far been negle ted. Introdu ing the residual gas mass fra tion xr , the single-zone spe i heat ratio γCE in (4.18) is reformulated as: cp (T, p, xb , xr ) = xb cp,b (Tb , p) + (1 − xb ) ((1 − xr )cp,u (Tu ) + xr cp,b (Tu , p))

(4.35a)

cv (T, p, xb , xr ) = xb cv,b (Tb , p) + (1 − xb ) ((1 − xr )cv,u (Tu ) + xr cv,b (Tu , p))

cp (T, p, xb , xr ) γCE (T, p, xb , xr ) = . cv (T, p, xb , xr )

(4.35b) (4.35 )

The model assumptions are: • the residual gas is homogeneously distributed throughout the

ombustion hamber.

• the residual gas is des ribed by a burned mixture at the appro-

priate temperature and pressure.

• a residual gas mass element in the unburned zone assumes the unburned zone temperature Tu . • when a residual gas mass element rosses the ame front, it enters the burned zone and assumes the burned zone temperature Tb .

The pressure is assumed to be homogeneous throughout all zones.

In gure 4.13, the spe i heat ratio γCE is omputed a

ording to (4.35) for residual gas fra tions xr = [0, 0.05, 0.1, 0.15, 0.20] given the ylinder pressure in gure 4.8. It shows that the larger the residual gas fra tion, the larger the γ . The dieren e in γ for xr = [0.05, 0.1, 0.15, 0.20] ompared to xr = 0 is shown in gure 4.14. The dieren e is largest during ompression and ombustion. After the ombustion, the mass spe i heats for the single zone will oin ide with the ones for the burned zone in a

ordan e with the model assumptions, and there is thus no dieren e in γ . Modeling of

xr -dependen e

A simple model of the inuen e of xr on γ is to model the inuen e as a linear fun tion of xr during the losed part, i.e. γ(T, p, xb , xr ) = γCE (T, p, xb ) + bxr xr = γCE (T, p, xb ) + γbias (xr ),

(4.36) where γCE (T, p, xb ) is given by (4.18). Sin e xr is onstant during a

y le, the term bxr xr an be onsidered as a onstant bias γbias (xr )

80

CHAPTER 4. A SPECIFIC HEAT RATIO MODEL

...

Iso−octane; In−cylinder mixture for xr ∈ [0, 0.20] 1.35 xr=0.20 x =0 r

Specific heat ratio − γ [−]

1.3

1.25

1.2

1.15

1.1

−100

−50 0 50 Crank angle [deg ATDC]

100

Figure 4.13: Spe i heat ratio γCE (4.35) for residual gas fra tion xr = [0, 0.05, 0.1, 0.15, 0.20].

that hanges from y le to y le. A better model is gained if the mass fra tion burned xb is used, as des ribed in γ(T, p, xb , xr ) = γCE (T, p, xb ) + (1 − xb )bxr xr = γCE (T, p, xb ) + (1 − xb )γbias (xr ),

(4.37) whi h relies on the fa t that γ(T, p, xb , xr ) oin ides with γCE for every xr when the mixture is fully burned. A more appealing and more physi ally orre t model is to extend model D1 in (4.28) with the Krieger-Borman polynomial for the residual gas fra tion, in the same manner as in (4.35). Thus (4.28) is rewritten as   lin KB cp (T, p, xb , xr ) = xb cKB p,b (Tb , p) + (1 − xb ) (1 − xr )cp,u (Tu ) + xr cp,b (Tu , p)

(4.38a)

cv (T, p, xb , xr ) =

xb cKB v,b (Tb , p)

  KB (T , p) + (1 − xb ) (1 − xr )clin (T ) + x c u u r v,u v,b

cp (T, p, xb , xr ) γD1xr (T, p, xb , xr ) = cv (T, p, xb , xr )

(4.38b) (4.38 )

4.6.

−3

6

Difference in specific heat ratio γ [−]

81

PARTIALLY BURNED MIXTURE

x 10

Iso−octane; In−cylinder mixture for xr ∈ [0, 0.20]

5 xr=0.20 4 xr=0.15 3

xr=0.10

2

1

x =0.05 r

0

−100

−50 0 50 Crank angle [deg ATDC]

100

Figure 4.14: Dieren e in spe i heat ratio γCE (4.35) for residual gas fra tion xr = [0.05, 0.1, 0.15, 0.20] ompared to xr = 0.

to form the spe i heat ratio γD1xr for a partially burned mixture with residual gas mass fra tion xr . In the same spirit as for (4.36) and (4.37) but with model D1 as a base, the following models are formed: γ(T, p, xb , xr ) = γD1 (T, p, xb ) + bxr xr = γD1 (T, p, xb ) + γbias (xr ), (4.39) γ(T, p, xb , xr ) = γD1 (T, p, xb ) + (1 − xb )bxr xr = γD1 (T, p, xb ) + (1 − xb )γbias (xr ).

(4.40)

Evaluation The spe i heat ratio for the six models (4.36)(4.40) and (4.18), i.e. no xr -modeling, are all ompared to the referen e model (4.35) for a given xr . At the operating point in gure 4.14, the NRMSE in γ and the orresponding value of γbias (xr ) for models (4.36), (4.37), (4.39) and (4.40) are given for the xr :s at hand in table 4.9. The NRMSE for all six models are also in luded. Model (4.37) has the best performan e and de reases the NRMSE

82

CHAPTER 4. A SPECIFIC HEAT RATIO MODEL

Model No model (4.36) (4.37) (4.38) (4.39) (4.40) Model (4.36) (4.37) (4.39) (4.40)

...

NRMSE(γ ) [%℄ xr = 0

0 0 0

9.8 · 10−2 9.8 · 10−2 9.8 · 10−2 xr = 0

0 0

−6.3 · 10−5 9.6 · 10−5

xr = 0.05 7.1 · 10−2 4.8 · 10−2 8.2 · 10−3 9.9 · 10−2 1.1 · 10−1 9.5 · 10−2 xr = 0.05 6.5 · 10−4 1.2 · 10−3 5.9 · 10−4 1.3 · 10−3

xr = 0.10 1.4 · 10−1 9.7 · 10−2 1.7 · 10−2 1.0 · 10−1 1.4 · 10−1 9.2 · 10−2 γbias xr = 0.10 1.3 · 10−3 2.5 · 10−3 1.2 · 10−3 2.6 · 10−3

xr = 0.15 2.1 · 10−1 1.5 · 10−1 2.5 · 10−2 1.0 · 10−1 1.8 · 10−1 9.1 · 10−2

xr = 0.20 2.8 · 10−1 1.9 · 10−1 3.4 · 10−2 1.1 · 10−1 2.2 · 10−1 9.0 · 10−2

xr = 0.15 2.0 · 10−3 3.7 · 10−3 1.9 · 10−3 3.8 · 10−3

xr = 0.20 2.6 · 10−3 5.0 · 10−3 2.5 · 10−3 5.0 · 10−3

Table 4.9: Normalized root mean square error (NRMSE) and γbias for xr = [0, 0.05, 0.10, 0.15, 0.20] using approximative models (4.36)(4.40). Model (4.18) orresponds to no xr -modeling. with approximately a fa tor 8, ompared to model (4.18). Note that the NRMSE in table 4.9 are relatively small ompared e.g. to the NRMSE given in table 4.6. When omparing models that are based on D1 the NRMSE(γ ) is in the same order as in the ase of no xr , at least for models (4.38) and (4.40). The values for γbias (xr ) depend almost linearly upon xr , and it therefore seems promising to model γbias (xr ) as a linear fun tion of xr . Espe ially sin e model (4.40) gives a smaller NRMSE than model (4.38). However, the slope bxr in γbias (xr ) = bxr xr will hange for operating onditions other than the one given here. The model used therefore needs to be robust to hanging operating

onditions, a feature the Krieger-Borman polynomial has. Model (4.38) only adds an NRMSE(γ ) of 1.2 % for xr = 0.20 ompared to xr = 0, as shown in table 4.9. Therefore model (4.38) whi h uses the KriegerBorman polynomial is re ommended, although it did not have the best performan e of the xr -models at this operating point. 4.6.9

Summary for partially burned mixture

The results an be summarized as: • The modeling error must be ompared both in terms of how they des ribe γ and the ylinder pressure. • Comparing models C4 and D4 , it is obvious that interpolating the

spe i heat ratios dire tly instead of the spe i heats auses a large pressure error. Interpolation of spe i heat ratios does not fulll the energy equation.

4.7.

SUMMARY AND CONCLUSIONS

83

• The γ -models B1 , B2 , B3 , B4 , C2 and C5 proposed in earlier works,

introdu e a pressure modeling error whi h is at least four times the measurement noise, and at least ten times the measurement noise in the mean in our investigation. Out of these models, model B1 is the best ompromise of omputational time and model a

ura y.

• If only single-zone temperatures are allowed, model B1 is the best

one.

• The omputation times are of the same order for all models ex ept D3 , D4 and C4 .

• The models in group D are required to get a ylinder pressure

RMSE that is of the same order as the measurement noise.

• As a ompromise between a

ura y and omputational time, model D1 is re ommended. Compared to the original setting in Gatowski

et al. (1984), the omputational burden in reases with 40 % and the ylinder pressure modeling error is 15 times smaller in mean.

• For a residual gas mass fra tion xr up to 20 %, model D1 an

be extended with spe i heats for the residual gas (4.38). These spe i heats are modeled by the Krieger-Borman polynomial. This model extension adds a NRMSE(γ ) whi h is less than 1.2 % to the previous modeling error for xr = 0.20.

• Only model group D produ es predi tion error estimates of the

heat release parameters, that are a

urate within 1 % for all three parameters, and this suggests that any of the D -models an be used, preferably model D1 due to its lower omputational time.

4.7

Summary and on lusions

Based on assumptions of frozen mixture for the unburned mixture and

hemi al equilibrium for the burned mixture (Krieger and Borman, 1967), the spe i heat ratio is al ulated, using a full equilibrium program (Eriksson, 2004), for an unburned and a burned air-fuel mixture, and ompared to several previously proposed models of γ . It is shown that the spe i heat ratio and the spe i heats for the unburned mixture is aptured to within 0.25 % by a linear fun tion in mean harge temperature T for λ ∈ [0.8, 1.2]. Furthermore the burned mixture are

aptured to within 1 % by the higher-order polynomial in ylinder pressure p and temperature T developed in Krieger and Borman (1967) for the major operating range of a spark ignited (SI) engine. If a linear model is preferred for omputational reasons for the burned mixture,

84

CHAPTER 4. A SPECIFIC HEAT RATIO MODEL

...

then the temperature region should be hosen with are whi h an redu e the modeling error in γ by 25 %. With the knowledge of how to des ribe γ for the unburned and burned mixture respe tively, the fo us is turned to nding a γ -model during the ombustion pro ess, i.e. for a partially burned mixture. This is done by interpolating the spe i heats for the unburned and burned mixture using the mass fra tion burned xb . The obje tive was to nd a model of γ , whi h results in a ylinder pressure error that is lower than or in the order of the measurement noise. It is found that interpolating the linear spe i heats for the unburned mixture and the higher-order polynomial spe i heats for the burned mixture, and then forming the spe i heat ratio γ(T, p, xb ) =

lin xb cKB cp (T, p, xb ) p,b + (1 − xb ) cp,u = KB + (1 − x ) clin cv (T, p, xb ) xb cv,b b v,u

(4.41)

results in a small enough modeling error in γ . This modeling error results in a ylinder pressure error that is lower than 6 kPa in mean, whi h is in the same order as the ylinder pressure measurement noise. If the residual gas mass fra tion xr is known, it should be in orporated into (4.41) whi h then extends to (4.38). It was also shown that it is important to evaluate the model error in γ to see what impa t it has on the ylinder pressure, sin e a small error in γ an yield a large ylinder pressure error. This also stresses that the γ -model is an important part of the heat release model. Applying the proposed model improvement D1 (4.41) of the spe i heat ratio to the Gatowski et al. (1984) single-zone heat release model is simple, and it does not in rease the omputational burden immensely. Compared to the original setting, the omputational burden in reases with 40 % and the modeling error introdu ed in the ylinder pressure is redu ed by a fa tor 15 in mean.

Part II

Compression Ratio Estimation

85

5 Compression ratio estimation  with fo us on motored y les The ability to vary the ompression ratio opens up new possibilities but if the ompression ratio gets stu k at too high ratios, the risk of engine destru tion by heavy kno k in reases rapidly. On the other hand if the ompression ratio gets stu k at too low ratios, this results in low e ien y, and therefore an unne essary high fuel onsumption. It is therefore vital to monitor and diagnose the ontinuously hanging

ompression ratio. Determination of the ompression ratio is in itself an issue of high importan e (Amann, 1985; Lan aster et al., 1975) sin e it inuen es the analysis and ontrol of the ombustion pro ess. Due to geometri al un ertainties, a spread in ompression ratio among the dierent ylinders is inherent (Amann, 1985), and sin e it is impossible to measure the ompression ratio dire tly it is ne essary to estimate it. Here four

ompression ratio estimation methods are developed and their properties with respe t to 1) a

ura y, 2) onvergen e speed, and 3) over all onvergen e, are investigated. The approa h is to use measured

ylinder pressure tra es ombined with a ylinder pressure model, to estimate the ompression ratio. A desirable property of the estimator is that it should be able to ope with the unknown oset introdu ed by the harge amplier, hanging thermodynami onditions, and possibly also the unknown phasing of the pressure tra e in relation to the rank angle revolution. Two models for the ylinder pressure with dierent omplexity levels are used; a polytropi model and a single-zone zero-dimensional 87

88

CHAPTER 5. CR ESTIM  FOCUS ON MOTORED CYCLES

heat release model. Three dierent optimization algorithms that minimize the predi tion error are utilized to estimate the parameters in the

ylinder pressure models. These are: 1. 2.

3.

. The non-linear problem is formulated as two linear least-squares problems, that are solved alternately.

Linear subproblem method

. It has two steps where the rst step determines only the parameters that appear linearly in the model through the linear least-squares method. In the se ond step a line sear h is performed for the other parameters in the dire tion of the negative gradient. This method is a separable least-squares method. Variable proje tion method

. This is a well known Gauss-Newton method with regularization. Numeri al approximations of the gradient and the Hessian are used.

Levenberg-Marquardt method

Based on these models and optimization algorithms, four dierent ompression ratio estimation methods are formulated. The methods are appli able to both motored and red y les.

5.1

Outline

The two ylinder pressure models that will be used are given in se tion 5.2. They have been derived in hapter 2 and are summarized here for onvenien e. Based on these two models and the three optimization algorithms des ribed above, four methods for ompression ratio estimation are introdu ed in se tion 5.3. Thereafter, the performan e of the four methods is evaluated for simulated ylinder pressure tra es in terms of bias, varian e and omputational time in se tion 5.4. The simulation study is an important and ne essary step for a fair evaluation, due to that the true value of the ompression ratio for an engine is unknown. The simulation evaluation in ludes a sensitivity analysis. In se tion 5.5 the fo us is turned to an evaluation of the methods performan e on experimental data. Data was olle ted from the SAAB Variable Compression (SVC) engine shown in gure 5.1. By tilting the mono-head, the ompression ratio an be ontinuously varied between 8.13 and 14.67. The geometri data for the SVC engine is given in appendix B.3. A dis ussion on ompression ratio diagnosis is given. The on lusions and re ommendations of the hapter are summarized in se tion 5.6.

5.2.

CYLINDER PRESSURE MODELING

89

Figure 5.1: S hemati of engine with variable ompression. With ourtesy of SAAB Automobile AB. 5.2

Cylinder pressure modeling

Two models are used to des ribe the ylinder pressure tra e and they are referred to as the polytropi model and the standard model. 5.2.1

Polytropi model

A simple and e ient model is the polytropi model, presented earlier in se tion 2.2, p(θ)V (θ)n = C, (5.1) where p(θ) is the ylinder pressure as fun tion of rank angle θ, V (θ) is the volume, n is the polytropi exponent and C is a y le-to- y le dependent onstant. Sometimes the volume is written as the following sum, V (θ) = Vid (θ) + Vc , (5.2) where Vid (θ) is the instantaneous volume displa ed by the piston (3.4) and Vc is the learan e volume. The ompression ratio rc is related to these volumes through rc =

max [Vid (θ)] + Vc , Vc

(5.3)

where min [Vid (θ)] = 0. The polytropi model (5.1) des ribes the ompression and expansion phases of the engine y le well, but not the

90

CHAPTER 5. CR ESTIM  FOCUS ON MOTORED CYCLES

ombustion phase (Heywood, 1988). Therefore, for y les with ombustion only data between inlet valve losing (IVC) and start of ombustion (SOC) will be used, while for motored y les all data a quired during the losed part of the y le, i.e. between IVC and exhaust valve opening (EVO), is utilized. 5.2.2

Standard model

Gatowski et al. (1984) develops, tests and applies this model for heat release analysis. This model is from hereon named the standard model. It is based on the rst law of thermodynami s and maintains simpli ity while still in luding the well known ee ts of heat transfer and revi e ows. It has be ome widely used (Heywood, 1988) and is des ribed in se tion 2.5. The pressure is des ribed by the following dierential equation, dp = dθ

γ dQht − γ−1 p dV dθ − dθ   ,  Vcr 1 T ′ + 1 ln γ ′ −1 V + + T γ−1 Tw γ−1 b γ−1 dQch dθ

(5.4)

see Gatowski et al. (1984) for the derivation and se tion 3.8 for details on model omponents and parameters. This orresponds to model B1 in hapter 4. Equation (5.4) is an ordinary dierential equation that

an easily be solved numeri ally, given an initial value for the ylinder ch pressure. The heat release dQ dθ is zero for motored y les and for y les with ombustion it is modeled using the Vibe fun tion xb (Vibe, 1970) in its dierentiated form (5.5b)  θ−θ  m+1 −a ∆θ ig

cd , xb (θ) = 1 − e  m  θ−θ  m+1 a (m + 1) θ − θig dxb (θ) −a ∆θ ig cd = e , dθ ∆θcd ∆θcd

(5.5a) (5.5b)

where xb is the mass fra tion burned, θig is the start of ombustion, ∆θcd is the total ombustion duration, and a and m are adjustable parameters. The heat release is modeled as dQch dxb (θ) = Qin , dθ dθ

(5.6)

where Qin is the total amount of heat released. The standard model (5.4)(5.6) is valid between IVC and EVO. 5.2.3

Cylinder pressure referen ing

Piezoele tri pressure transdu ers are used for measuring the in- ylinder pressure, whi h will ause a drift in the pressure tra e, i.e. the absolute

5.3.

ESTIMATION METHODS

91

level is unknown and it is slowly varying. This issue was introdu ed and dis ussed in se tion 3.1. The modeling assumption was that the pressure oset is onsidered to be onstant during one engine y le. The pressure oset is estimated using intake manifold pressure referen ing as des ribed in se tion 3.1, i.e. by referen ing the measured ylinder pressure pm (θ) to the intake manifold pressure pman just before inlet valve losing (IVC), for several samples of pman . Due to standing waves in the intake runners and ow losses over the valves at ertain operating points, the referen ing might prove to be insu ient. This is investigated by in luding a parameter for ylinder pressure oset in estimation methods 3 and 4, des ribed in the next se tion. 5.3

Estimation methods

Four methods are developed and investigated for ompression ratio estimation. These methods are des ribed below and their relations are summarized at the end of this se tion. All four methods are formulated as least-squares problems in a set of unknown parameters x as min kε(x)k22 , x

(5.7)

where a residual ε(x) is formed as the dieren e between a model and measurement. The dieren es between the methods lie in how the residuals are formed, and in the iterative methods used for solving the resulting problem (5.7). The termination riterion for all methods are the same; If the relative improvement in the residual kε(x)k2 is less than 1 × 10−6 in one iteration, the method terminates. This numeri al value is hosen to ensure onvergen e for all initial values.

5.3.1

Method 1  Sublinear approa h

The rst method uses the polytropi model (5.1) p(θ) (Vid (θ) + Vc )n = C

(5.8)

to estimate the polytropi exponent n, the ompression ratio rc and the onstant C . The method alternates between two problems, one to determine the polytropi exponent n, and the other to determine the

learan e volume Vc (i.e. rc = (max [Vid (θ)] + Vc ) / Vc ). Applying logarithms on (5.8) yields the residual ε1a (C1 , n) = ln p(θ) − (C1 − n ln(Vid (θ) + Vc ))

(5.9)

ε1b (C2 , Vc ) = Vid (θ) − (C2 p(θ)−1/n − Vc )

(5.10)

whi h is linear in the parameters C1 = ln C and n, if Vc is xed. Another residual, that an be derived from (5.8), is

92

CHAPTER 5. CR ESTIM  FOCUS ON MOTORED CYCLES

whi h is linear in the parameters C2 = C 1/n and Vc , if n is xed. The basi idea is to use the two residuals, ε1a and ε1b , alternately to estimate the parameters n, Vc and C by solving two linear least-squares problems. Using a Taylor expansion, see appendix B.1, the following approximate relation between the residuals is obtained ε1a (θ, x) ≈

n ε1b (θ, x). Vid (θ) + Vc

(5.11)

The relation (5.11) must be taken into a

ount and the residual ε1a is therefore multiplied by the weight w(θ) = Vid (θ) + Vc , to obtain

omparable norms in the least-squares problem. To use this weight is of ru ial importan e and without it the algorithm diverges (Klein, 2004, pp.85). Convergen e of the method an however not be proved. If the residuals were equal, i.e. ε1a = ε1b , the problem would be bilinear and the onvergen e linear (Björ k, 1996). Ea h iteration for estimating the three parameters x = [Vc C n] in the algorithm onsists of three steps.  Sublinear approa h Initialize the parameters x = [Vc C n].

Algorithm 5.1

1.Solve the weighted linear least-squares problem min kw ε1a k22

n,C1

with Vc from the previous iteration and C1 = ln C . 2.Solve the linear least-squares problem min kε1b k22

Vc ,C2

with n from step 1 and c2 = C 1/n . 3.Che k the termination riterion, if not fullled return to step 1.

5.3.2 Method 2  Variable proje tion The se ond method also uses the polytropi model (5.1), together with a variable proje tion algorithm. A nonlinear least-squares problem (5.7) is separable if the parameter ve tor an be partitioned x = (y z) su h that min kε(y, z)k22 (5.12) y

5.3.

93

ESTIMATION METHODS

is easy to solve. If ε(y, z) is linear in y , ε(y, z) an be rewritten as ε(y, z) = F (z)y − g(z).

(5.13)

y(z) = [F T (z)F (z)]−1 F (z)T g(z) = F † (z)g(z),

(5.14)

For a given z , this is minimized by

i.e. by using linear least-squares, where F (z) is the pseudo-inverse of F (z). The original problem minkε(x)k22 an then be rewritten as †

x

min kε(y, z)k22 = min kg(z) − F (z)y(z)k22 z

z

(5.15)

and ε(y, z) = g(z) − F (z)y(z) = g(z) − F (z)F † (z)g(z) = (I − PF (z) )g(z), (5.16) where PF (z) is the orthogonal proje tion onto the range of F (z), thus the name variable proje tion method. Rewriting the polytropi model (5.1) as ε2 (C1 , n, Vc ) = ln p(θ) − (C1 − n ln(Vid (θ) + Vc ))

(5.17)

results in an equation that is linear in the parameters C1 = ln C and n, and nonlinear in Vc . It is thus expressed on the form given in (5.13). The residual (5.10) is not suitable, sin e the parameters are oupled for this formulation, see appendix B.2. With this method the three parameters x = [Vc ln C n] are estimated. A omputationally e ient algorithm, based on Björ k (1996, p.352), is summarized in appendix B.2.

5.3.3

Method 3  Levenberg-Marquardt and polytropi model

The third method uses the polytropi model (5.1), as methods 1 and 2 did, with a pressure sensor model (3.1) added a

ording to pm (θ) = Kp p(θ) + ∆p = p(θ) + ∆p

(5.18)

in order to make the pressure referen ing better. The measured ylinder pressure is given by pm (θ), and the additive pressure bias ∆p is

onsidered to be onstant during one y le. The measurement hain is

onsidered to be well alibrated, and therefore the measurement gain Kp is set equal to 1. Furthermore, errors in the rank angle phasing ∆θ between the volume and pressure are also in luded in the polytropi model, whi h then an be written as the following residual: ε3 (Vc , n, C, ∆p, ∆θ) = pm (θ) − ∆p − C (Vid (θ + ∆θ) + Vc )−n . (5.19)

A LevenbergMarquardt method (Gill et al., 1981) is used to solve this nonlinear least-squares problem, that has ve unknown parameters: Vc , n, C , ∆p, ∆θ.

94

CHAPTER 5. CR ESTIM  FOCUS ON MOTORED CYCLES

5.3.4 Method 4  Levenberg-Marquardt and standard model The fourth method uses the single-zone model (5.4) from Gatowski et al. (1984) whi h, in ontrast to the other methods, also in ludes heat transfer and revi e ee ts. The model is given in se tion 3.8 and the parameters used are summarized in appendix B.4. Due to the

omplexity of this model, the sublinear approa h and variable proje tion approa h are not appli able, and therefore only the Levenberg Marquardt method is used. The in reased omplexity also auses identiability problems for some of the parameters, sin e there exist many dependen ies between them. This is the ase for the revi e volume Vcr and the learan e volume Vc , in whi h estimating the two parameters at the same time results in oupled and biased estimates. Therefore one of them is set onstant, in this ase the revi e volume (Klein, 2004, p. 94). The estimation problem is however still hard, as will be shown in hapter 6, and therefore the number of parameters to estimate are redu ed to ve for motored y les, and eight for red y les. The remaining parameters are xed to their initial values. The number of parameters to estimate are determined by omparing the bias and varian e in the parameter values using simulations for dierent settings of parameters. For motored y les the estimated parameters are: Tivc , γ300 , ∆p, ∆θ, and of ourse Vc . For red y les the parameters ∆θd , ∆θb , and Qin are also in luded.

5.3.5 Summary of methods Table 5.1 shows the relations between the methods. For red y les, methods 1, 2 and 3 use ylinder pressure data between IVC and SOC only, in ontrast to method 4 whi h uses data from the entire losed part of the engine y le. For motored y les, all data during the losed part of the y le is utilized by all methods. It is also noteworthy that if the learan e volume Vc is onsidered to be known, then methods 1 and 2 an be reformulated su h that ∆p is estimated instead of Vc . In su h a ase all methods an be used to estimate an additive pressure bias, see Klein (2004, p. 143) for more details. Algorithm Sublinear Variable proje tion Levenberg-Marquardt

Polytropi model Method 1 Method 2 Method 3

Standard model Method 4

Table 5.1: Relation between methods.

5.4.

95

SIMULATION RESULTS

Pressure [bar]

OP2: N=1500 rpm, pman=1.0 bar 30 rc=15 20 10 rc=8 0

−100

−50 0 50 Crank angle [deg ATDC]

100

Figure 5.2: Simulated ylinder pressure at OP2 with the ompression ratio set to integer values from 8 to 15.

5.4

Simulation results

A fundamental problem is that the exa t value of the ompression ratio in an engine is unknown, whi h makes the evaluation of the ompression ratio estimation methods di ult. To over ome this obsta le simulated pressure tra es, with well known ompression ratios, are used. The methods are rst evaluated on data from motored y les in one operating point with respe t to the estimation bias, varian e and the residual. Then they are evaluated on all y les in the generated data set and a sensitivity analysis is performed. Finally the methods are evaluated on red y les. 5.4.1

Simulated engine data

Cylinder pressure tra es were generated by simulating the standard model (5.4) with representative single-zone parameters, given in appendix B.4. Three operating points were sele ted with engine speed N = 1500 rpm and intake manifold pressures pman ∈ {0.5, 1.0, 1.8} bar, whi h dene operating points OP1, OP2 and OP3 respe tively. Pressure tra es were generated for both motored and red y les at ea h operating point for integer ompression ratios between 8 and 15, whi h overs the ompression ratio operating range of the prototype SI engine. To ea h tra e ten realizations of Gaussian noise with zero mean and standard deviation 0.038 bar were added, forming altogether 240 motored and 240 red y les. This noise level was hosen sin e it has the same RMSE as the residual for method 4 in the experimental evaluation, and thus ree ts the appli ation to experimental data. The data was sampled with a resolution of 1 rank-angle degree (CAD).

96

CHAPTER 5. CR ESTIM  FOCUS ON MOTORED CYCLES

In gure 5.2, one motored y le is shown for ea h integer ompression ratio rc at operating point 2 (OP2). 5.4.2

Results and evaluation for motored y les at OP2

The performan e of the estimation methods is rst evaluated for OP2. This operating point has an intake manifold pressure in the midrange of the engine, and a relatively low engine speed for whi h the ee ts of heat transfer and revi es are signi ant. OP2 is therefore hosen as a representable operating point. For the simulated pressure, the parameters in the models are estimated using all methods, for ea h individual y le.

Estimation results For the variable ompression ratio engine, the dieren e in rc between two onse utive y les an be as large as 5 %. Sin e the estimation is performed on ea h y le, the methods should be able to deal with initial values that are 5 % o, in order to ope with ompression ratio transients. For the simulated data, the initial parameter values for the estimations are therefore set to a ±5 % perturbation of the true parameter values. The termination riterion, des ribed in se tion 5.3, is hosen in order to assure that the methods have onverged, i.e. yield the same estimate for every initial parameter set. Figure 5.3 shows a summary of the estimates for all y les at OP2. In the gure the true ompression ratios are the integer values 8 to 15 and for onvenien e, the results for methods 1 to 4 are separated for ea h true value. The estimate should be as lose as possible to the

orresponding dotted horizontal line. Table 5.2 summarizes the results in terms of relative mean error (RME) and mean 95 % relative onden e interval (RCI) in rc , as well as the mean omputational time and mean number of iterations in ompleting the estimate for one y le. The measures RME and RCI are given for the lower (rc = 8) and upper (rc = 15) limits of the ompression ratio, and as a mean for all ompression ratios. The relative mean error RME is dened by RM E =

1 M

PM

j=1 rc rc∗

j

− rc∗

,

(5.20)

where rcj , j = 1, . . . , M is the estimate of rc orresponding to y le j and rc∗ is the true ompression ratio. The mean 95 % relative onden e

5.4.

97

SIMULATION RESULTS

Simulated motored cycles at OP2 16

Estimated compression ratio

15 14

M1 M2 M3 M4 True CR

13 12 11 10 9 8 7 7

8

9

10 11 12 13 True compression ratio

14

15

16

Figure 5.3: Mean value and 95 % onden e interval of the estimated

ompression ratio for motored y les using the four methods, ompared to the true ompression ratio. The estimate should be as lose as possible to the orresponding dotted horizontal line. interval RCI is omputed as M 1 X 1.96σj RCI = , M j=1 rc∗

(5.21)

where σj is omputed using (C.18) and (C.20) in appendix C.1. The mean omputational time and mean number of iterations in table 5.2 are given for the worst ase of the initial parameter sets. The al ulations were made using Matlab 6.1 on a SunBlade 100, whi h has a 64-bit 500 Mhz pro essor.

Analysis of estimation results Figure 5.3 shows that the estimates from methods 1, 2 and 3 be ome poorer the higher the ompression ratio is. It stresses that heat transfer and revi e ee ts, whi h are not onsidered expli itly in the polytropi

98

CHAPTER 5. CR ESTIM  FOCUS ON MOTORED CYCLES

model (5.1), must be taken into a

ount when a better estimate is desired for higher ompression ratios. Method 4 has the orre t model stru ture and therefore yields a better estimate with mu h lower bias and varian e for all ompression ratios. These results are further onrmed by the relative mean errors in table 5.2, where method 4 is the most a

urate but also the most time onsuming method. Method 4 is at the time being not suitable for on-line implementation, unless a bat hing te hnique is used, e.g. where one y le is olle ted and then pro essed until nished. Of the other three methods, method 2 performs best on erning estimation a

ura y and is outstanding regarding onvergen e speed. Method 1 has the potential of low omputational time sin e it solves linear least-squares problems only, however the rate of onvergen e is low whi h makes it unsuitable for pra ti al use.

Residual analysis The residuals orresponding to the ylinder pressures in gure 5.2 are displayed for all four methods in gure 5.4 for rc = 8 and in gure 5.5 for rc = 15, together with their respe tive root mean square error (RMSE). There is a systemati deviation for methods 1 and 2 that be omes more pronoun ed when rc in reases, as seen by omparing gures 5.4 and 5.5. The same systemati deviation o

urs for method 3, although it is smaller. The residuals for method 4 are white noise, whi h is expe ted sin e method 4 has the orre t model stru ture. The systemati deviation for methods 1-3 is due to the polytropi model (5.1), whi h does not expli itly a

ount for heat transfer and revi e ee ts . The systemati deviation for method 3 would be of the same order as for methods 1 and 2, if it were not for the two extra parameters ∆θ and ∆p in (5.19). These two parameters ompensate for the la k of heat transfer and revi e ee t with the penalty of parameter biases. The most pronoun ed hange is the shifted rank angle phasing ∆θ, whi h results in a peak pressure position whi h is loser to top dead enter (TDC). However this model exibility does not improve the ompression ratio estimate for method 3, whi h a

ording to table 5.2 has a bias of -3.6 % in mean. 5.4.3

Results and evaluation for motored y les at all OP

The results for all motored operating points are given in table 5.3, where the relative mean estimation errors and 95 % onden e intervals are omputed as a mean for all ompression ratios. For all operating points, the estimate from method 2 is within 1.5 % and within 0.5 % from method 4. The onden e intervals are smallest for method 4,

5.4.

99

SIMULATION RESULTS

Method vs. True 1 2 3 4

rc = 8

RME [%℄ 2.6 -0.9 -2.8 0.35

RCI [%℄ 4.1 3.5 5.9 0.87

rc = 15

RME [%℄ 2.3 -1.0 -4.4 0.12

RCI [%℄ 6.9 3.8 5.4 0.49

RME [%℄ 2.4 -0.9 -3.6 0.21

RCI [%℄ 5.4 3.6 5.5 0.65

all rc std Time [-℄ [ms℄ 0.093 63 0.072 3 0.19 86 0.0013 2.6 × 105

Iter [-℄ 26.1 3.0 5.0 14.0

Table 5.2: Relative mean error and mean 95 % relative onden e interval (RCI) in the estimated rc for rc = 8, rc = 15 and as a mean for all ompression ratios for simulated data from OP2. The standard deviation, as well as the mean omputational time and mean number of iterations in ompleting the estimation for one engine y le are also given.

RMSE=0.070 bar

RMSE=0.066 bar 0.2 M2: Residual [bar]

M1: Residual [bar]

0.2 0.1 0 −0.1 −0.2

−100

0

0.1 0 −0.1 −0.2

100

RMSE=0.042 bar

100

0.2 M4: Residual [bar]

M3: Residual [bar]

0 RMSE=0.038 bar

0.2 0.1 0 −0.1 −0.2

−100

−100 0 100 Crank angle [deg ATDC]

0.1 0 −0.1 −0.2

−100 0 100 Crank angle [deg ATDC]

Figure 5.4: Dieren e between estimated and simulated ylinder pressure for all methods, given the motored y le at rc = 8 in gure 5.2. The RMSE for the added measurement noise is 0.038 bar.

100 CHAPTER 5. CR ESTIM  FOCUS ON MOTORED CYCLES RMSE=0.194 bar

RMSE=0.176 bar 0.5 M2: Residual [bar]

M1: Residual [bar]

0.5

0

−0.5

−100

0

0

−0.5

100

RMSE=0.069 bar

100

0.5 M4: Residual [bar]

M3: Residual [bar]

0 RMSE=0.038 bar

0.5

0

−0.5

−100

−100 0 100 Crank angle [deg ATDC]

0

−0.5

−100 0 100 Crank angle [deg ATDC]

Figure 5.5: Dieren e between estimated and simulated ylinder pressure for all methods, given the motored y le at rc = 15 in gure 5.2. The RMSE for the added measurement noise is 0.038 bar. followed by method 2. The observations made for OP2 with respe t to bias and varian e are onrmed by the results for all operating points. A trend among the estimates is that a higher intake manifold pressure pman redu es the onden e interval. A higher pressure improves the signal to noise ratio, while the ee ts of heat transfer and revi e ows remain the same. The hoi e of residual is important. This is illustrated by the fa t that the estimates and onden e intervals for methods 1 and 2 dier, for whi h the only dieren e is the formulation of the residual. Furthermore, methods 1 and 2 use a logarithmi residual of the pressure, that weighs the low pressure parts between IVC and EVO relatively higher than method 3 does. For method 3, the samples orresponding to the highest pressure are the most important ones. This however oin ides with the highest mean harge temperature, and thus the highest heat transfer losses. This is where the polytropi pressure model used in methods 1-3 has its largest model error, sin e heat transfer is not

5.4.

101

SIMULATION RESULTS

N pman Method 1 2 3 4

OP1 1500 rpm 0.5 bar RME [%℄ RCI [%℄ 3.6 8.5 -0.35 5.7 -3.8 7.1 0.37 1.2

OP2 1500 rpm 1.0 bar RME [%℄ RCI [%℄ 2.4 5.4 -0.9 3.6 -3.6 5.5 0.21 0.65

OP3 1500 rpm 1.8 bar RME [%℄ RCI [%℄ 0.45 4.3 -1.4 2.7 -3.6 4.7 0.094 0.37

Table 5.3: Relative mean error (RME) and mean 95 % relative onden e interval (RCI) in estimated rc , for simulated motored ylinder pressures from three operating points dened by engine speed N and intake manifold pressure pman . The estimate improves with in reased pressure. Parameter ∆p [kPa℄ ∆θ [deg℄ C1 [-℄ Vcr [% Vc ℄

Nominal value 0 0 2.28 1.5

-5 -0.25 0 0

Deviations -2 2 -0.1 0.1 1.14 3.42 0.75 2.25

5 0.25 4.56 3.0

Table 5.4: Nominal values for the parameters and the deviations used in the sensitivity analysis.

in luded expli itly. This explains why both methods 1 and 2 yield a more a

urate estimate than method 3. 5.4.4

Sensitivity analysis at OP2

The next question asked is:  How sensitive are the methods to hanges in heat transfer, revi e volume, an ina

urate TDC determination or a badly referen ed ylinder pressure? A sensitivity analysis is therefore performed and model parameter values are altered a

ording to table 5.4. The estimates for all four methods are summarized in gure 5.6, where the relative mean error in rc is given as a fun tion of parameter deviation for ea h of the four parameters. A

ording to gure 5.6, methods 3 and 4 are insensitive to an additive pressure bias ∆p, sin e their respe tive models in lude a pressure bias. Methods 1 and 2 whi h do not model a pressure bias, are approximately equally sensitive. This is due to that an additive pressure bias ae ts all pressure samples equally. A parameter deviation of |∆p| ≤2 kPa assures that the estimates from methods 1 and 2 are within the a

ura y of method 3, at least for OP2. Methods 3 and 4 are also insensitive to a rank angle oset ∆θ, for

0.1

0.1

0.05

0.05 RME [−]

RME [−]

102 CHAPTER 5. CR ESTIM  FOCUS ON MOTORED CYCLES

0 −0.05 M1 M2 −0.1 0 5 M3 −5 Pressure bias ∆p [kPa] M4 0.1

−0.05 −0.1

0.05 RME [−]

RME [−]

−0.2 0 0.2 Crank angle offset ∆θ [deg]

0.1

0.05 0 −0.05 −0.1

0

0 −0.05

0

2 4 Woschni C [−] 1

−0.1

0

1 2 3 Crevice volume [% V ] c

Figure 5.6: Relative mean error for parameter deviations in ∆p, ∆θ, normalized C1 and Vcr for methods 1-4 at OP2. Nominal parameter values are given in table 5.4, and orrespond to the third value on the x-axis. the same reasons as for the pressure bias ∆p. Comparing methods 1 and 2, method 1 is far more sensitive to ∆θ than method 2. This is explained by the following: A rank angle oset ae ts the pressure in the vi inity of TDC the most, due to the higher pressure. It is therefore logi al that method 2 whi h uses a logarithmi residual of p, is less sensitive than method 1. For a parameter deviation of |∆θ| ≤0.2 deg, method 2 yields more a

urate estimates of rc than method 3 does. Todays alibration systems have a possibility to determine the rank angle oset with an a

ura y of 0.1 degree. This suggests that the

rank angle oset ∆θ an be left out when using method 2. Methods 1, 2 and 3 are all ae ted, while method 4 is unae ted by

hanges in heat transfer C1 . The heat transfer ae ts the pressure in the vi inity of TDC the most, due to the higher in- ylinder temperature. Therefore method 1 is more sensitive to heat transfer than method 2 for the same reasons as for the rank angle oset ∆θ. All methods are sensitive to deviations in revi e volume Vcr , although method 4 is not as sensitive as the other methods. It is however important to set Vcr onstant, to avoid an even larger bias in learan e

5.5.

EXPERIMENTAL RESULTS

103

volume Vc . For motored y les the learan e volume and the revi e volume are modeled in almost the same way, due to the relatively low in- ylinder temperatures. It is therefore natural that a larger revi e volume auses a larger estimate of the learan e volume, and thereby a smaller ompression ratio. This is the ase in gure 5.6, and it has been seen earlier for motored and red y les in Klein (2004, p. 94). To on lude, the more robust methods 3 and 4 are able to ope with the parameter deviations imposed upon them, due to their high model exibility. Of the other two methods, method 2 has the best performan e although it is equally sensitive to a pressure bias as method 1 is.

5.4.5 Results and evaluation for red y les at OP2 Figure 5.7 displays the results for simulated y les with ombustion, in terms of mean value and 95 % onden e interval at ea h integer

ompression ratio from 8 to 15. The estimates for methods 1-3 are poor, and the relative mean error for all rc is in the order of 10 %. These methods all suer from the fa t that the pressure orresponding to TDC is not in luded, sin e they only use the data between IVC and SOC. This also yields larger onden e intervals ompared to motored y les, ree ting a more un ertain estimate. Method 4 on the other hand uses all data between IVC and EVO, and yields an estimate a

urate within 0.4 % for OP2. Due to the poor performan e of methods 1-3, an experimental evaluation on red data is not pursued. Out of the four proposed methods, method 4 is required to estimate the ompression ratio from a red y le. The on lusions from this investigation of how the methods perform on simulated data is summarized in se tion 5.6. 5.5

Experimental results

The attention is now turned to the issue of evaluating the methods on experimental engine data. As mentioned before, the true value of the

ompression ratio is unknown. Therefore it is important too see if the ee ts and trends from the simulation evaluation are also present when the methods are applied to experimental data. The performan e of the methods is dis ussed using one spe i operating point, and is then followed by an evaluation in luding all operating points.

5.5.1 Experimental engine data Data is olle ted during stationary operation at engine speeds N ∈ {1500, 3000} rpm, intake manifold pressures pman ∈ {0.5, 1.0} bar alto-

104 CHAPTER 5. CR ESTIM  FOCUS ON MOTORED CYCLES Simulated fired cycles at OP2 16 15

Estimated compression ratio

14

M1 M2 M3 M4 True CR

13 12 11 10 9 8 7 6 7

8

9

10 11 12 13 True compression ratio

14

15

16

Figure 5.7: Mean value and 95 % onden e interval of the estimated

ompression ratio for red y les using the four methods, ompared to the true ompression ratio.

gether forming four dierent operating points, dened in the upper part of table 5.6. The measurements are performed for a tuated ompression ratio values from the lower limit 8.13 to the upper limit 14.66, through integer values 9 to 14 in between. With a tuated ompression ratio it is meant the value ommanded from the ele troni ontrol unit (ECU). These values were determined from engine produ tion drawings and implemented in the ECU, but an be ae ted by produ tion toleran es or non-ideal sensors (Amann, 1985), as well as me hani al and thermal deformation during engine operation (Lan aster et al., 1975). For ea h operating point and ompression ratio, 250 onse utive motored y les with the fuel inje tion shut-o were sampled with a

rank-angle resolution of 1 degree, using a Kiestler 6052 ylinder pressure sensor. Figure 5.8 displays one measured y le for ea h rc at operating point 2 (OP2). For a given rc the mono-head of the engine is tilted, see gure 5.1, whi h auses the position for TDC to be advan ed from 0 CAD (Klein et al., 2003). The lower the ompression ratio is,

5.5.

105

EXPERIMENTAL RESULTS

Pressure [bar]

OP2: N=1500 rpm, pim=1.0 bar 30

rc=14.66

20 10 rc=8.13 0

−100

−50 0 50 Crank angle [deg ATDC]

100

Figure 5.8: Experimentally measured ylinder pressures at OP2. The a tuated ompression ratios are 8.13, integer values 9 to 14, and 14.66. the more advan ed the position for TDC be omes. This explains why the peak pressure position for rc = 8.13 in gure 5.8 is advan ed ompared to rc = 14.66, and not the other way around whi h would be the

ase if only heat transfer and revi e ee ts were present. 5.5.2

Results and evaluation for OP2

The performan e of the estimation methods is rst evaluated for operating point OP2, dened in table 5.6. This operating point has an intake manifold pressure in the midrange of the engine, and a relatively low engine speed for whi h the ee ts of heat transfer and revi es are signi ant. OP2 is therefore hosen as a representable operating point.

Estimation results The estimation results are presented in the same manner as for the simulated data. Figure 5.9 displays the mean estimate and the mean 95 % onden e interval for 250 onse utive y les at OP2, where the estimate has been omputed for ea h individual y le. Table 5.5 shows mean omputational time and mean number of iterations, as well as the relative mean error and mean 95 % relative onden e interval. Two examples of residuals are given in gures 5.10 and 5.11.

Analysis of estimation results Figure 5.9 shows that method 3 underestimates and methods 1, 2 and 4 overestimate the ompression ratio. The spread of the estimates between the methods is more pronoun ed than for the simulated data,

106 CHAPTER 5. CR ESTIM  FOCUS ON MOTORED CYCLES Experimental motored cycles at OP2 M1 M2 M3 M4 Act CR

16

Estimated compression ratio

15 14 13 12 11 10 9 8 7

8

9

10 11 12 13 Actuated compression ratio

14

15

Figure 5.9: Mean value and 95 % onden e interval of the estimated

ompression ratio for motored y les using the four methods, ompared to the a tuated ompression ratio. The estimate should be as lose as possible to the dotted horizontal line.

ompare gures 5.9 and 5.3. This spread in reases as the ompression ratio be omes higher, a trend also found for the onden e intervals of the estimates. The interrelation among methods 1-3 are however the same, where method 1 always yields the largest estimates, and method 3 the smallest. The trends and ee ts in the simulation evaluation are also present in the experimental investigation, whi h gives a rst indi ation that the on lusions drawn from the simulation study are valid. All methods have approximately the same onden e intervals for the experimental and simulated data, ompare tables 5.5 and 5.2. Again method 4 yields the smallest onden e intervals followed by method 2. The dieren e is most signi ant for method 4, whi h had the orre t model stru ture in the simulation ase while here it is an approximation of the real engine. This is also seen in the residual, gure 5.11, as a systemati deviation around TDC. This model error thus adds to

5.5.

107

EXPERIMENTAL RESULTS

Method vs. A t. 1 2 3 4

rc = 8.13

RME [%℄ 6.0 5.1 -1.5 3.6

RCI [%℄ 2.9 2.6 6.7 1.6

rc = 14.66

RME [%℄ 5.2 2.7 -6.1 1.0

RCI [%℄ 6.1 3.7 5.8 1.3

RME [%℄ 6.4 3.6 -4.2 2.2

RCI [%℄ 5.3 3.8 6.5 1.3

all rc std Time [-℄ [ms℄ 0.11 89 0.083 5 0.066 128 0.051 6.6 × 105

Iter [-℄ 30.2 3.7 5.7 14.0

Table 5.5: Relative mean error (RME) and mean 95 % relative onden e interval (RCI) of the estimated rc for extreme values rc = 8.13 and rc = 14.66, and as a mean for all ompression ratios for experimental data from OP2. They are omputed relative to the a tuated

ompression ratio. The standard deviation, mean omputational time and mean number of iterations in ompleting the estimation for one engine y le are also given.

the varian e of the estimate. It also hanges the interrelation between methods 1-3 and method 4, e.g. method 4 always gives a smaller estimate on experimental data than method 2, while the onverse o

urs on the simulated data. The mean omputational time and number of iterations are higher for the experimental data, as shown in table 5.5. As for the simulations, method 2 is the most omputationally e ient method of them all.

Residual analysis The residuals orresponding to the ylinder pressures in gure 5.8 are displayed for all four methods in gure 5.10 for rc = 8.13 and in gure 5.11 for rc = 14.66, together with their respe tive root mean square error (RMSE). As for the simulations, there is a systemati deviation for methods 1, 2 and 3. This deviation in reases with rc . The residuals for method 4 have a omparatively small deviation near TDC. This small deviation illustrates that the model stru ture is a

eptable but not perfe t, sin e it is not able to fully apture the measurement data. This is most evident for the higher ompression ratio, rc = 14.66, in gure 5.11. The relative mean error for rc = 14.66 is however small, less than 0.8 %, and method 4 an therefore be onsidered to apture the data well. The residuals from the experimental data have a higher RMSE ompared to the simulations, but they are still of the same order. The dieren es between the methods are due to dierent formulations of the residuals and model simpli ations. These properties give rise to the systemati deviations that are visible in both simulated and experimental data.

108 CHAPTER 5. CR ESTIM  FOCUS ON MOTORED CYCLES RMSE=0.107 bar M2: Residual [bar]

M1: Residual [bar]

RMSE=0.108 bar 0.2 0.1 0 −0.1 −0.2 −100

0

0.2 0.1 0 −0.1 −0.2

100

−100

0.2 0.1 0 −0.1 −0.2 −100 0 100 Crank angle [deg ATDC]

100

RMSE=0.038 bar M4: Residual [bar]

M3: Residual [bar]

RMSE=0.045 bar

0

0.2 0.1 0 −0.1 −0.2 −100 0 100 Crank angle [deg ATDC]

Figure 5.10: Dieren e between estimated and experimental ylinder pressure for all methods, given the motored y le in gure 5.8 at rc = 8.13.

5.5.3

Results and evaluation for all OP

The trends shown for OP2 are also present in the full data set, displayed in table 5.6. This table shows that as the intake manifold pressure pman in reases, the varian e for all methods de reases. A high pman is therefore desirable. However, the inuen e of the engine speed has no

lear trend as two of the methods yield higher varian e and the other two lower varian e, as the engine speed is in reased. For all operating points, method 4 yields the smallest onden e intervals followed by method 2. Therefore the on lusions made in the simulation evaluation with respe t to models, residual formulation, methods, heat transfer and revi e ee ts are the same for the experimental evaluation.

5.5.

109

EXPERIMENTAL RESULTS

RMSE=0.230 bar M2: Residual [bar]

M1: Residual [bar]

RMSE=0.252 bar 0.5 0 −0.5 −100

0

0.5 0 −0.5

100

−100

0.5 0 −0.5

100

RMSE=0.065 bar M4: Residual [bar]

M3: Residual [bar]

RMSE=0.098 bar

0

0.5 0 −0.5

−100 0 100 Crank angle [deg ATDC]

−100 0 100 Crank angle [deg ATDC]

Figure 5.11: Dieren e between estimated and experimental ylinder pressure for all methods, given the motored y le in gure 5.8 at rc = 14.66.

N pman

Method vs. A t. 1 2 3 4

OP1 1500 rpm 0.5 bar RME RCI [%℄ [%℄ 10 6.2 6 4.6 -2.3 8.2 2.8 2.0

OP2 1500 rpm 1.0 bar RME RCI [%℄ [%℄ 6.4 5.3 3.6 3.8 -4.2 6.5 2.2 1.3

OP3 3000 rpm 0.5 bar RME RCI [%℄ [%℄ 8 7.5 1.3 5.2 -3.9 7.3 2.9 1.6

OP4 3000 rpm 1.0 bar RME RCI [%℄ [%℄ 3.6 5.9 -0.083 4 -6.2 5.9 2.4 1.2

Table 5.6: Relative mean error (RME) and mean 95 % relative onden e interval (RCI) in estimated rc , for four dierent operating points dened by engine speed N and intake manifold pressure pman .

110

CHAPTER 5. CR ESTIM  FOCUS ON MOTORED CYCLES

Dis ussion on ompression ratio diagnosis

One appli ation that the proposed methods were developed for was

ompression ratio diagnosis. The appli ability of the methods will be illustrated by a short example. Consider the issue of diagnosing the

ompression ratio. A possible fault mode is that the ompression ratio gets stu k at a too high level, e.g. at rc = 12. A diagnosti test is then used to dete t and alarm when rc gets stu k. This is realized by a hypothesis test, where the null hypothesis is formulated as H 0 : rc not stu k. This is illustrated by the following example whi h uses the diagnosis framework presented in Nyberg (1999) and Nyberg (2001):  Dete ting rc stu k at 12 Consider the task of dete ting when the ompression ratio gets stu k. Only two fault modes are onsidered

Example 5.1

: rc est ∈ [0.9rcact , 1.1rcact ] F : rc est ∈ / [0.9rcact , 1.1rcact ],

NF

(5.22)

i.e. an estimated rcest within 10 per ent of the a tuated rc act is onsidered as no fault (NF). The hypothesis test is then formulated as H0 : N F H 1 : F.

(5.23)

The test quantity T is based on omparing the a tuated rc act and estimated rc est , a

ording to T = |rc act − rc est |.

(5.24)

Then if the ontroller e.g. a tuates rcact = 9 and rc is really stu k at 12, the test should alarm with a ertain level of onden e. This is formalized by the power fun tion β(rc est ) here dened by β(rc est ) = P (reje t H 0 |rcest ) = P (T ≥ J|rc est ),

(5.25)

where J is a threshold designed by the user. The signi an e level of the test is dened by α = max β(rc est |rcest ∈ NF),

(5.26)

whi h orresponds to the worst ase of the false alarm probability. To be able to make the assumption that H 1 is true if H 0 is reje ted, the threshold J must be set to obtain a su iently low false alarm probability. Based on the estimates and onden e intervals for method 2 using the experimental results at OP2 in table 5.5, the power fun tion β is

5.5.

111

EXPERIMENTAL RESULTS

H0: r not stuck c

1 0.9

Power function β [−]

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 8

9

10

11 12 Actuated rc [−]

13

14

15

Figure 5.12: Power fun tion β at OP2 for experimental data when rc is stu k at 12.

omputed and displayed in gure 5.12. In the omputations it is assumed that the true rc is stu k at 12 and that linear interpolation an be used to nd the estimates and onden e intervals at ompression ratios where no a tual measurements have been performed. When designing a diagnosis system, there is generally a ompromise between a low false alarm probability (FAP) and a low missed dete tion probability (MDP), given by (D.6) and (D.7). The (adaptive) threshold is hosen as J = 0.15 rcact , and gure 5.12 shows that this results in a signi an e level of less than 0.2 %, whi h is assumed su iently low. In this ase the false alarm probability omes to the ost of a low missed dete tion probability for rcest lose to but not within the NF fault mode, rcest ∈ [10.8, 13.2]. This is illustrated by the region rc est ∈ [10.5, 10.8[ where M DP = 1 − β ≥ 0.5, i.e. the error will not be dete ted in more than 50 % of the ases. But when rcest ≤ 10 the missed dete tion probability goes to zero, indi ating that we will always dete t if rc gets stu k if a proper ex itation of the a tuated ompression ratio is made.

112 CHAPTER 5. CR ESTIM  FOCUS ON MOTORED CYCLES Example 5.1 illustrates the prin iple of how a diagnosis system an be designed and analyzed using method 2 at operating point OP2. This

an be generalized to a diagnosis system for all operating points, and it is only a matter of design to nd an appropriate threshold for all methods. The diagnosti performan e in terms of false alarm probability, missed dete tion probability and omputational time are of ourse expe ted to be ome best for method 2. Therefore during driving, all methods are able to dete t if the ompression ratio is stu k at a too high or at a too low level, given an appropriate number of y les and an appropriate ex itation of rc . This is su ient both for safety reasons, where the ompression ratio an be too high and engine kno k is the onsequen e, and for fuel e onomi reasons, where a too low ompression ratio will lead to higher fuel onsumption. 5.6

Con lusions

Four methods for ompression ratio estimation based on ylinder pressure tra es have been developed and evaluated for both simulated and experimental y les.

Con lusions from the simulation results The rst three methods rely upon the assumption of a polytropi ompression and expansion. It is shown that this is su ient to get a rough estimate of the ompression ratio rc for motored y les, espe ially for a low rc and by letting the polytropi exponent be ome small. For a high rc it is important to take the heat transfer into a

ount, and then only method 4 is a

urate to within 0.5 % for all operating points. Method 4 is however slow and not suitable for on-line implementation. Method 2 on the other hand is substantially faster and still yields estimates that are within 1.5 %. The formulation of the residual is also important, sin e it inuen es the estimated rc . For red y les, methods 1-3 yield poor estimates and therefore only method 4 is re ommended. A sensitivity analysis, with respe t to rank angle phasing, ylinder pressure bias, revi e volume, and heat transfer, shows that the third and fourth method are more robust. They therefore deal with these parameter deviations better than methods 1 and 2. Of the latter two, method 2 has the best performan e for all parameter deviations ex ept for an additive pressure bias.

Con lusions from the experimental results All methods yield approximately the same onden e intervals for the simulated and experimental data. The onden e intervals resulting

5.6.

CONCLUSIONS

113

from method 4 are smallest of all methods, but it suers from a high

omputational time. Method 2 yields smaller onden e intervals than methods 1 and 3, and is outstanding regarding onvergen e speed. The ee ts and trends shown in the simulation evaluation are also present in the experimental data. Therefore the on lusions made in the simulation evaluation with respe t to models, residuals, methods, heat transfer and revi e ee ts are the same for the experimental evaluation. For diagnosti purposes, all methods are able to dete t if the

ompression ratio is stu k at a too high or too low level.

Con luding re ommendations The a

ura y of the ompression ratio estimate is higher for motored

y les with high initial pressures. Thus if it is possible to hoose the initial pressure, it should be as high as possible. Using motored y les assures that all pressure information available is utilized and the high initial pressure improves the signal-to-noise ratio, while the ee ts of heat transfer and revi e ows remain the same. Two methods are re ommended; If estimation a

ura y has the highest priority, and time is available, method 4 should be used. Method 4 yields the smallest onden e intervals of all investigated methods for both simulated and experimental data. In the simulation ase where the true value of the ompression ratio is known, method 4 gave estimates with smallest bias. If omputational time is the most important property, method 2 is re ommended. It is the most omputationally e ient of all investigated methods, and yields the smallest onden e intervals out of methods 1-3.

114

CHAPTER 5. CR ESTIM  FOCUS ON MOTORED CYCLES

Part III

Prior Knowledge based Heat Release Analysis

115

6 Using prior knowledge for single-zone heat release analysis Internal ombustion engines operate by onverting the hemi al energy in the air-fuel mixture into useful work by raising the ylinder pressure through ombustion. The ylinder pressure data itself is therefore a ree tion of the ombustion pro ess, heat transfer, volume hange et . and thus gives important insight into the ontrol and tuning of the engine. To a

urately model and extra t information from the ylinder pressure is important for the interpretation and validity of the result. Due to the short time s ale of the pro ess, a sequen e of measurements from an engine gives huge amounts of data. These large sets of data have to be analyzed e iently, systemati ally, and with high a

ura y. The obje tive here is therefore to develop a tool for e ient, systemati and a

urate o-line heat release analysis of ylinder pressure data.

Problem outline The fo us is on a single-zone heat release model (Gatowski et al., 1984) that des ribes the ylinder pressure a

urately and has a low omputational omplexity. A low omputational omplexity is an important feature when analyzing large data sets. However, this model in ludes at most 16 unknown parameters, as shown in se tion 3.8, among whi h there are ouplings ausing identiability problems. This is due to that the estimation problem is on the verge of being over-parameterized. An example is the learan e volume and the revi e volume, whi h are hard to identify simultaneously. This is often solved by setting one of these parameters to a onstant xed value. The orre t value of the onstant 117

118

...

CHAPTER 6. USING PRIOR KNOWLEDGE FOR

parameter is however unknown, and is therefore most likely set with an error. This error auses a bias in the other parameter estimates.

Problem solution Two methods for parameter estimation are developed and ompared. Method 1 is alled SVD-based parameter redu tion and is summarized as: 1. Estimate the parameter values and varian es using a onstrained lo al optimizer. 2. Find the most un ertain parameter, i.e. the one that has the highest varian e by using singular value de omposition (SVD) of the estimated hessian, for the given measured ylinder pressure. 3. Set the un ertain parameter from step 2 onstant using prior knowledge of the parameter and return to step 1. The un ertain parameters are set onstant one-by-one in step 3, whi h redu es the varian e for the remaining (e ient) parameters when no modeling error is indu ed. A re ommended number of e ient parameters is then found by minimizing the Akaike nal predi tion error

riterion (Ljung, 1999). Step 3 is motivated by the following; The most un ertain parameters generally drift o from their nominal values. This nominal parameter deviation will ause a bias even in the most ertain parameters. This motivates why the most ertain parameter is not set

onstant. However, if the most un ertain parameters are set xed to their nominal values, it is believed that this will redu e the bias in the e ient parameters. The se ond method uses prior knowledge of all the parameters expli itly in the optimization. The parameters x are estimated by minimizing the riterion fun tion: WN =

N 1 X (p(θi ) − pˆ(θi , x))2 + (x − x# )T δ (x − x# ) N i=1

(6.1)

w.r.t. x, i.e. by minimizing the standard predi tion error ε(θi , x) = p(θi ) − pˆ(θi , x) using a regularization te hnique. Method 2 is therefore

named regularization using prior knowledge. The rst term in (6.1) minimizes the dieren e between the modeled pˆ(θ, x) and measured p(θ) ylinder pressure, while the se ond term pulls the parameters x towards x# , i.e. towards the nominal values obtained from prior knowledge. The matrix δ and ve tor x# a

ount for prior knowledge of the parameters, where δ inuen es how strong the pulling for e should be. This an be interpreted as that all parameters are assigned a prior

6.1.

OUTLINE

119

probability density fun tion (pdf), with a nominal mean value x# . If the pdf is Gaussian, minimizing WN produ es the maximum a posteriori estimate. For example, our prior knowledge of an additive pressure oset (after pegging with the intake manifold pressure) ould be that the parameter is Gaussian distributed with a nominal mean value of 0 kPa, and a standard deviation of 2.5 kPa. It is however only the intergroup relation among the parameters, and not the absolute value of the standard deviation that is important as will be shown later. The nominal ve tor x# is found from appropriate initial values and the size of the matrix δ is tuned using simulations to give a balan ed ompromise between the prior knowledge and the measured ylinder pressure. A fundamental problem is that the true values of the parameters are unknown in a real engine, whi h makes the evaluation of the two methods on experimental data di ult. This is mainly due to two reasons: First of all, the hosen model stru ture is bound to be in orre t sin e not every physi al pro ess inuen ing the measured ylinder pressure is modeled, whi h will ause inexa t parameter values. Se ondly the

riterion fun tion an be small for a given data set, although the parameter estimates are, due to physi al reasons, bad. It is therefore not su ient to minimize the riterion fun tion on validation data to evaluate the two proposed methods. To over ome these obsta les simulated pressure tra es, with well-known parameter values and a well-known model stru ture, are used. In the simulations noise of the same order as for the experimental data is added to the measurement signal. The methods are evaluated on simulated data from motored and red y les with respe t to estimation bias, varian e and residual analysis using both false and orre t prior knowledge. The results are also validated on experimental data from an SI engine. 6.1

Outline

The outline of this hapter and the two following hapters is presented here. One ylinder pressure is des ribed in se tion 6.2. It has been derived in hapters 2 and is summarized here for onvenien e. Then the problem asso iated with this parameter estimation appli ation is illustrated in se tion 6.3 by two simple approa hes, whi h are based purely on either data tting or parameter prior knowledge, and they are both unsu

essful. Therefore two methods, referred to earlier in this hapter as method 1 and 2, that in orporate parameter prior knowledge in the estimation problem are proposed in se tion 6.4. These two methods are then evaluated using simulated and experimental data in hapter 7 for motored y les, and in hapter 8 for red y les. The ompiled

on lusions from both evaluations are given in se tion 9.3. A method of how to express the prior knowledge is also presented in general terms in se tion 6.4, and in hapters 7 and 8 for the spe i ases.

...

120

6.2

CHAPTER 6. USING PRIOR KNOWLEDGE FOR

Cylinder pressure modeling

One single-zone models is used to des ribe the ylinder pressure tra e and it is referred to as the standard model.

Gatowski et al. (1984)

develops, tests and applies a single-zone model for heat release analysis. See se tion 2.5 for the derivation and se tion 3.8 for details on the parameters and model omponents.

The model is based on the rst

law of thermodynami s and maintain simpli ity while still in luding the well known ee ts of heat transfer and revi e ows.

6.2.1

Standard model

The most important thermodynami property when al ulating the heat release rates in engines is the ratio of spe i heats, γ(T, p, λ) = cp cv (Gatowski et al., 1984; Chun and Heywood, 1987; Guezenne and Hamama, 1999). In the standard model γ is represented by a linear fun tion in mean harge temperature

T

γlin (T ) = γ300 + b(T − 300), i.e. model

B1

in hapter 4.

(6.2)

The standard ylinder pressure model is

simulated by using (3.36)(3.45).

6.2.2

Cylinder pressure parameters

The parameters used in the standard model and how to nd its nominal values was given in hapter 3. In the standard model, two parameters (γ300 and

6.3

b)

are estimated for

γ.

Problem illustration

To illustrate the di ulties asso iated with parameter estimation given a ylinder pressure tra e, two simple but dierent approa hes are used. The rst approa h estimates all parameters in the standard ylinder pressure model simultaneously by applying a Levenberg-Marquardt method with onstraints on the parameters, whi h is a lo al optimizer des ribed in appendix C.1. The onstraints are physi ally motivated and are set widely, in order to only inuen e whenever the parameter values be ome physi ally invalid. Two examples of applied onstraints are a positive ylinder pressure at IVC, perature at IVC,

TIV C ,

pIV C ,

and a mean harge tem-

larger than 270 K. Without these onstraints

the approa h yields e.g. a

TIV C -estimate that typi ally is approximately

200 K, i.e. a physi ally invalid value. The se ond approa h tunes the parameters manually one at a time until a satisfa tory pressure tra e is found, an approa h sometimes referred to as heat release playing.

6.3.

121

PROBLEM ILLUSTRATION

OP1: N=1200 rpm, pman=32.3 kPa 2000

Pressure [kPa]

1500 1000 500 0

Pressure [kPa]

−500

−100

−50

−100

−50

0

50

100

0 50 Crank angle [deg ATDC]

100

20 0 −20

Figure 6.1: Filtered experimental ring y le and the modeled ylinder pressure for the two simple approa hes. The residual for approa h 1 (dash-dotted) and approa h 2 (dashed) are also given in both the upper and lower plot. The RMSE:s are 1.3 and 5.8 kPa respe tively. This approa h requires a lot of patien e ombined with prior knowledge of nominal values as well as expert knowledge of what inuen e ea h parameter has on the ylinder pressure, and also ross- ouplings in between the parameters. The results for both approa hes are given in gure 6.1, where the modeled and measured ylinder pressure tra es are displayed, and in table 6.1 where the parameter estimates are given. In fairness of the se ond approa h, it should be pointed out that less than an hour was spent on tuning the parameters, thus the result ould of ourse be better. It however illustrates how time- onsuming this approa h an be. As shown in gure 6.1 approa h 1 has a good residual t, but a

ording to table 6.1 the parameter estimates are bad. The small residual in gure 6.1 also illustrates that the model stru ture is exible enough to des ribe the ylinder pressure, but that the parameters are hard to identify. The bad parameter estimates are exemplied by the temperature at IVC, TIV C , whi h is as low as 270 K and also the lower bound

γ300 b C1 C2 ∆θ ∆p Kp pIV C TIV C Tw Vc Vcr θig ∆θd ∆θb Qin

Par.

0.048 -1.34 0.908 45.7 270.0 563 61.1 0.61 -31.4 24.6 20.1 352

0.043 -1.37 1 41.0 400.0 475 58.8 0.71 -32.6 26.6 20.1 350

4.56

3.24 · 10−3

3.14 · 10−3

2.44

−7.42 · 10−5

Approa h 2 1.335

−2.11 · 10−5

Approa h 1 1.285

0.047 -1.37 1 41.3 388.8 440 58.8 0.58 -32.6 26.6 20.1 350

3.24 · 10−3

2.28

−7.42 · 10−5

Nominal 1.335

Table 6.1: Nominal values and estimates for the two simple approa hes.

Des ription

onstant spe i heat ratio [-℄ slope for spe i heat ratio[K −1℄ heat-transfer parameter [-℄ heat-transfer parameter [-℄

rank angle phasing [deg ATDC℄ bias in pressure measurements [Pa℄ pressure measurement gain[-℄

ylinder pressure at IVC [kPa℄ mean harge temperature at IVC [K℄ mean wall temperature [K℄

learan e volume [cm3 ℄ single aggregate revi e volume [cm3 ℄ ignition angle [deg ATDC℄ ame-development angle [deg ATDC℄ rapid-burn angle [deg ATDC℄ released energy from ombustion [J℄

122 CHAPTER 6. USING PRIOR KNOWLEDGE FOR

...

6.4.

ESTIMATION METHODS

123

hosen for this parameter, learly a too low value. This is ompensated for by having a higher mean ylinder wall temperature, Tw , and a smaller γ300 . Without the onstraints the estimation would render an even smaller TIV C . Typi ally all temperatures drift o to non-physi al values, independent of initial value, when estimating all parameters at the same time. This is of ourse unwanted and yields parameter estimates that are ina

urate and not physi ally valid. The same behavior arise for estimation problems that are either rank-de ient or ill-posed. As an illustration, the behavior for a simple linear example of a rank-de ient problem is given in appendix C.2. For the se ond approa h it is the other way around ompared to the rst approa h. The se ond approa h gives reasonable parameter values, see table 6.1, but the residual t is worse than for the rst approa h as shown in the lower part of gure 6.1. This approa h an therefore be said to give physi ally valid parameter estimates, but not in a time e ient way. In addition the user's presumption also plays a role in the estimation and ould result in a bias. The residual error is however small ompared to the measured ylinder pressure, as shown in the upper part of the gure. However none of these approa hes

omply very well with all our demands on the estimation pro edure; e ient, systemati and a

urate. To enhan e the performan e of these two approa hes, their advantages are ombined by using the systemati minimization of the residual from the rst approa h and the parameter guess approa h based on expert and prior knowledge from the se ond. The idea for doing so is to use prior knowledge or information about the parameters and use it in a entral way to enhan e the estimation pro edure. 6.4

Estimation methods

Two methods that use prior knowledge of the parameters are presented. They both estimate the parameters x by minimizing the dieren e between the measured p(θi ) and modeled ylinder pressure pˆ(θi , x), i.e. by minimizing the predi tion error ε(θi , x) = p(θi ) − pˆ(θi , x). As shown in gure 6.1 the data is well des ribed by a dire t simulation of the standard model in se tion 6.2.1. It is therefore reasonable to use dire t simulation as a predi tor, whi h orresponds to an output-error model assumption. The rst method uses the prior knowledge indire tly by redu ing the number of parameters to estimate. This is done by setting those that are hardest to estimate to xed values, given by the prior knowledge. Method 1 is referred to as SVD-based parameter redu tion. The se ond method in ludes the prior knowledge dire tly into the optimization problem. This is a hieved by in luding a parameter deviation penalty

...

124

CHAPTER 6. USING PRIOR KNOWLEDGE FOR

term in the riterion fun tion, that will help to

regularize

and thereby

ontrol and attra t the estimates to the vi inity of the priorly given nominal values.

This method is therefore alled

prior knowledge.

regularization using

6.4.1 Method 1  SVD-based parameter redu tion The rst method starts by estimating all parameters

VN = w.r.t.

x.

1 N

N X i=1

x ˆ

by minimizing

(p(θi ) − pˆ(θi , x))2

(6.3)

This is done by using a Gauss-Newton method alled the

Levenberg-Marquardt pro edure, see appendix C.1 for details.

The

number of parameters to estimate are then redu ed one by one by setting them xed, starting with the most un ertain parameter. This is motivated by the following; The most un ertain parameters generally drift o from their nominal values. This nominal parameter deviation will ause a bias even in the most ertain parameters, as illustrated for

γ300

in table 6.1 for approa h 1 where all parameters were estimated

simultaneously.

Altogether, this motivates why the most ertain pa-

rameter is not set onstant. However, if the most un ertain parameters are set xed to their nominal values, it is believed that this will redu e the bias in the e ient parameters. Method 1 nds these un ertain (spurious) parameters by studying the SVD of the estimated hessian, whi h ree ts how mu h a ertain parameter ae ts the loss fun tion

VN .

When the most spurious pa-

rameter is found, it is set onstant and the estimation starts over again with one parameter less to estimate. This systemati pro edure of redu ing the number of parameters was presented in (Eriksson, 1998) for motored y les. A re ommended number of parameters was then found by using the ondition number of the hessian. Here all parameter estimates

xˆ

for any given number of parameters are ompared, and the

estimate minimizing the Akaike nal predi tion error (FPE) is the best ˆ∗ . parameter estimate, x

E ient and spurious parameters Method 1 aims at handling the di ulties asso iated with the rst simple approa h in se tion 6.3 that were due to an over-parameterization of the problem.

The over-parameterization auses the ja obian

ψ

to

be ome ill- onditioned or rank de ient. A rank de ient ja obian an o

ur if one or more olumns of

ψ

are parallel or almost parallel. Here

almost means in the order of the working pre ision of the omputer. This ase ree ts sets or ombinations of parameters that do not inuen e the riterion fun tion that mu h. A

ording to (Sjöberg et al.,

6.4.

ESTIMATION METHODS

125

1995; Lindskog, 1996), the parameters an be divided into two sets:

spurious and e ient parameter sets, where the former orresponds to

parameters with small inuen e on the riterion fun tion value. Due to this small inuen e on the riterion fun tion, it is reasonable to treat those parameters as onstants that are not estimated. This parameter

lassi ation an be made in quite a number of ways, as shown in Example 6.1.

Example 6.1  Spurious and e ient parameters Consider the stati model with two parameters y = (x1 + x2 ).

(6.4)

If x2 is set onstant and thereby onsidered as spurious, then x1 is

onsidered to be e ient. The other way around works equally well, and does not ae t the exibility of the model stru ture.

Eigenve tor of hessian HN  nding the spurious parameters Lo ally, the estimated hessian HN (ˆ x) gives insight into how sensitive VN (ˆ x) is to a ertain parameter or ombination of parameters. To be able to estimate a ertain parameter, that parameter has to have a

lear ee t on the output predi tions (Ljung, 1999, p.453). This is ree ted in the hessian, whi h has large values for e ient parameters. Therefore, it is of interest to nd the group of parameters that

orresponds to the smallest ee t, i.e. the spurious parameters, and set these parameters to appropriate onstant values. This is reasonable sin e the spurious parameters are the hardest to estimate with the given model M(x) and the observed data Z N . The observed data is dened as Z N = [y(θ1 ), u(θ1 ), y(θ2 ), u(θ2 ), . . . , y(θN ), u(θN )], given the inputs u(θ) and outputs y(θ). The spurious parameters are lassied by nding the smallest singular value of HN (ˆ x, Z N ), and from the orresponding eigenve tor the group of parameters an be pi ked out. An algorithm for this is given by algorithm 6.1. The algorithm assumes that the parameters that x, Z N ) are the same for inuen e this eigenve tor of the hessian HN (ˆ t the estimate xˆ, as for the true solution x . The use of the algorithm is illustrated in example 6.2. Algorithm 6.1 an in lude an optional preferred ordering of the parameters, that is used in step 7. The ordering tells us whi h parameter is best known a priori, and whi h parameter we therefore prefer to set xed rst. This preferably applies to parameters that are similar in their physi al properties, su h as two temperatures, whi h then an be grouped together and have a relative ordering. This ordering should be

126

...

CHAPTER 6. USING PRIOR KNOWLEDGE FOR

based on prior knowledge of the parameters, and ould be the result of a pre-estimation of the parameters, knowledge of the measurement situation and sensor properties, or by expert knowledge. For example, the ylinder wall temperature Tw and the mean harge temperature at IVC, TIV C , an be ordered as B1 = [Tw TIV C ] if it is believed that Tw is better known than TIV C . This means that if the two temperatures end up in the same group of parameters in the eigenve tor from the hessian and as long as TIV C is not totally dominating, Tw will be set

onstant prior to TIV C . A spe ial ase for the preferred ordering groups, is when only one (large) group B1 ontaining all parameters is used. This ase is supported by the algorithm, but not re ommended sin e it ould result in that one spurious parameter is ex hanged for another although there is no physi al oupling in between them. As mentioned earlier, it is preferable if the parameters in the same ordering group are similar in their physi al properties or oupled in the hosen model stru ture. This speaks in favor of using multiple ordering groups. An example of this are the parameters C1 , Tw and TIV C whi h are oupled by the heat transfer model (3.43b) and the temperature model (3.40). It is assumed in the algorithm that a parameter only o

urs in one ordering group Bi . In the algorithm there are two ad-ho de ision rules, in step 4 and step 6 respe tively. These are motivated after the algorithm. Algorithm 6.1  Determining the spurious parameter using a preferred ordering Let x ˆef f be estimated parameters from the lo al optimizer used for # d e ient parameters and xsp are the spurious parameters.

x, Z N ) (C.8), where xˆ = 1.Compute the estimated hessian HN (ˆ ef f sp T (ˆ x x ) .

2.Compute the singular values ς1 ≥ . . . ≥ ςd# ≥ 0 and the orresponding eigenve tors v1 , . . . , vd# , by using singular value de omposition (SVD). 3.The eigenve tor vd# whi h orresponds to the smallest singular value ςd# , is hosen. Then the element in vd# that has the largest absolute value is named ek and is pi ked out. This element orresponds to the most spurious parameter a

ording to the data. 4.If the element ek is dominating, the preferred ordering is overridden. This is realized as if the absolute value of the element ek is

6.4.

ESTIMATION METHODS

127

greater than 0.9, go to step 8. 5.If no preferred ordering exists, go to step 8. 6.Form a group A of elements from the eigenve tor vd# orrespondFind the elements in ing to ςd# , by using the following strategy: √ vd# that have absolute values greater than 0.1. Sort these elements in a des ending order of their size in group A. 7.Find the ordering group Bi to whi h ek belongs. The element in Bi that has the highest ordering in Bi and at the same time belongs to group A, is the new ek . 8.Return the parameter xk that orresponds to element ek . Steps 13 in algorithm 6.1 are similar to the approa h in prin ipal

omponent analysis, but here a parameter is set xed rather than a dire tion of the eigenve tor. If one parameter dominates the eigenve tor vd# substantially (a

ording to the data), then this parameter is labeled spurious no matter what the preferred ordering, i.e. prior knowledge, says. This assures that the dominating parameter is not ex hanged. This approa h is implemented in step 4 su h that a parameter is lassied as dominating if its absolute value is greater than 0.9, whi h orresponds to more than 81 per ent of the total ve tor length of vd# . In step 6 another ad-ho de ision rule is used. It is based upon the assumption that if a parameter xk is to be ex hanged for another xj , they both have to have an inuen e on the eigenve tor vd# . In step 6 a parameter is lassied as inuen ing if its absolute value is greater √ than 0.1, i.e. if it onstitutes more than 10 per ent of the total length of eigenve tor vd# . This assures that a parameter is not ex hanged for one that only onstitutes a small part of vd# . The numeri al values in steps 4 and 6 are ad-ho hoi es, and an therefore be adjusted a

ording to the spe i requirements of the user. Example 6.2

 Determining the spurious parameter using algorithm 6.1

Consider an estimation problem where there are ve unknown parameters x1 , . . . , x5 . The parameters are ordered relative to ea h other into ordering groups Bi . For this example they are given by B1 = [x1 , x2 ] and B2 = [x3 , x4 , x5 ]. Algorithm 6.1 is used to nd the most spurious parameter in three dierent ases, all having dierent hessians. Following the algorithm, rst the estimated hessian is omputed (step 1),

128

...

CHAPTER 6. USING PRIOR KNOWLEDGE FOR

and from this the singular values are found (step 2). The eigenve tor that orresponds to the smallest singular value is v5 (step 3). Three dierent ases are now illustrated by using dierent v5 ve tors. a)The elements ek in Group Aa are found by   0  1     v5 =   0  ⇒ Aa = {e2 }  0  0

using steps 34. The absolute value of this element is greater than 0.9, and therefore x2 is returned by the algorithm using step 8.

b)The elements ek in Group Ab are found by   0  0.7     v5 =   −0.7  ⇒ Ab = {e2 , e3 }  0.14  0

using steps 36. In step 7 no re-ordering of the elements o

urs sin e the elements e2 and e3 belong to dierent ordering groups Bi , and therefore x2 is returned in step 8.

)The elements ek in Group Ac are found by   −0.4  0.65     v5 =   0.5  ⇒ Ac = {e2 , e3 , e1 }  0.2  −0.2

using steps 36. In step 7 it is found that the largest element e2 belongs to B1 as well as e1 . The ordering tells us that e1 should be pi ked out before e2 , and thus x1 is returned in step 8.

Note that without the preferred ordering Bi , all three ases would return parameter x2 .

Loss fun tion VN  nding the number of e ient parameters x, Z N ) serves not only The value of the estimated loss fun tion VN (ˆ as a measure of the model t to the measured output, but also as a test quantity for over-parameterization. By in reasing the number

6.4.

ESTIMATION METHODS

129

of parameters, and thereby the exibility of a model, more and more features in the measured output data an be explained by the model, and thereby de reasing the value of the loss fun tion. It is therefore reasonable that the loss fun tion is a monotoni ally de reasing fun tion of the number of model parameters. But by adding more and eventually unne essary parameters, the parameters adjust themselves to features in the parti ular noise realization, and will in fa t de rease the loss fun tion even more. This is known as over-t (Ljung, 1999, p.501), and is unwanted sin e the noise realization is bound to hange. To be able to ompare model stru tures with dierent numbers of e ient parameters in a fair way, the Akaike's Final Predi tion Error (FPE) Criterion (Akaike, 1969) is used. In the derivation of the Akaike FPE it is assumed that the true system an be des ribed by our model M(x), and that the parameters are identiable su h that the hessian is invertible. The expe tation value of the loss fun tion VN is then found to be (Ljung, 1999, pp.503): 2d , (6.5) N where λ0 is the noise varian e ( ompare (C.18b)), N is the number of samples and d is the number of parameters for model M(x). The more parameters the model stru ture uses, the smaller the rst term will be. Ea h new parameter ontributes with a varian e penalty of 2λ0 /N , whi h is ree ted by the se ond term. Any parameter that improves the t of VN with less than 2λ0 /N will thus be harmful, and introdu e an over-t. Su h a parameter is termed spurious. From equation (6.5) and an estimate of λ0 (C.18b), the Akaike FPE VNF P E (ˆ x, d) is stated as (Ljung, 1999, pp.503) EVN (ˆ x) ≈ VN (ˆ x, Z N ) + λ0

EVN (ˆ x) ≈

1+ 1−

d N d N

VN (ˆ x, Z N ) =: VNF P E (ˆ x, d).

(6.6)

x) should be modied to The equation shows how the loss fun tion VN (ˆ give a fair measure of the number of model parameters.

Algorithm  Method 1 We are now ready to formulate the algorithm.

Algorithm 6.2  SVD-based parameter redu tion

Let xinit be the initial values for the parameters x ∈ Rd×1 . The pa T # rameter ve tor is partitioned as x = xef f xsp , where xef f ∈ Rd ×1 # are the e ient parameters and xsp ∈ R(d−d )×1 are the spurious parameters. The maximum and minimum number of e ient parameters # # # are given by d# max and dmin respe tively, related by 1 ≤ dmin ≤ dmax ≤ d.

130

...

CHAPTER 6. USING PRIOR KNOWLEDGE FOR

1.Initialization: Set the best parameter estimate xˆ∗ = xinit and the number of e ient parameters d# = d# max . 2.Use the Levenberg-Marquardt pro edure (C.11) to estimate the  T e ient parameters xˆef f in xˆ = xˆef f xsp . The ve tor xinit is used as initial values for xef f and xed values for xsp . # > 3.If there are more than d# min e ient parameters left, i.e. d # sp ef f dmin , ompute the most spurious parameter xnew out of x usef f ing Algorithm 6.1. Then move xsp to xsp and redu e new from x the number of e ient parameters by one, i.e. d# = d# − 1. Return to step 2.

xef f , d# ) from (6.6) for all d# . 4.Compute the Akaike FPE VNF P E (ˆ Find the number of e ient parameters d∗ that minimizes xef f , d# ). The best estimate is then given by VNF P E (ˆ xˆ∗ = x ˆef f (d∗ ).

5.Return the best parameter estimate xˆ∗ . The algorithm an be made more omputationally e ient, either by redu ing the maximum number of e ient parameters d# max in step 1, i.e. by making more parameters spurious from the start, or by in reasing the minimum number of e ient parameters d# min for the riteria in step 4. Both these simpli ations of ourse rely upon some kind of prior knowledge. Another improvement in omputation time an be a hieved if several parameters are made spurious at the same time in step 4. A test for this is to onsider only parameters for whi h the ratio ς12 /ςi2 is larger than the ma hine pre ision. If d# max < d, i.e. the maximum number of e ient parameters are less than the number of parameters, the initial partitioning of the parameter ve tor x into e ient and spurious parameters has to be done by the user. For the sake of ompleteness, the number of e ient parameters in hapters 7 # and 8 are set to d# min = 1 and dmax = d.

6.4.2 Method 2  Regularization using prior knowledge The se ond method uses prior knowledge of the parameters expli itly in the optimization. The drive for this is to regularize the solution of the estimation problem, and this is done by in luding a parameter penalty term in the loss fun tion.

6.4.

ESTIMATION METHODS

131

Regularization through prior knowledge of the parameters

As mentioned earlier, a regularization te hnique an be used to avoid an ill- onditioned ja obian ψ and thereby avoiding a singular or almost singular hessian HN . In the Levenberg-Marquardt pro edure, this was done by adding ν > 0 to (C.10) and thereby a positive denite HN is guaranteed. A regularizing ee t is also imposed by adding a penalty term to the riterion fun tion VN (x, Z N ) (C.2) resulting in PN WN (x, x# , Z N , δ) = N1 i=1 12 ε2 (θi , x) + (x − x# )T δ(x − x# ) = VN (x, Z N ) + δx VNδ (x, x# , δ).

(6.7) It diers from the basi riterion VN by also penalizing the squared distan e between x and x# , where x# are the nominal parameter values. It is divided into two terms, where VN orresponds to the residual error introdu ed by the measurement and VNδ orresponds to the nominal parameter deviation. The s alar term δx is named regularization fa tor and will be returned to later. The main reason for using this riterion is: If the model parameterization ontains many parameters, it may not be possible to estimate several of them a

urately. There are then advantages in pulling the parameters towards a xed point x# (Ljung, 1999, pp.221). This denitively applies to an ill-posed problem. The penalty term VNδ in (6.7) will ee t the parameters with the smallest inuen e on VN , i.e. the spurious ones, the most. This for es the spurious parameters to the vi inity of x# . The regularization matrix δ an be seen as a tuning knob for the number of e ient parameters. A large δ simply means that the number of e ient parameters d# be omes smaller and that more parameters are lo ked to the vi inity of x# . It an also be seen as the weighted ompromise between residual # error ε and nominal parameter deviation ε# x = x − x . A large δ thus

orresponds to a large onden e in the nominal parameter values. The impa t of the prior on the nal estimate will now be illustrated in the following example. Example 6.3

 Impa t of prior x# and δ on nal estimate x

Again, onsider the stati model y = (x1 + x2 ),

(6.8)

whi h formed the base for example 6.1. Here, x1 = x2 = 1 are the true # values of the parameters, the nominal values are given by x# 1 and x2 respe tively, the measurement is y and the regularization matrix δ is given by δ = diag(δ1 , δ2 ). The residual ε is then formed as ε(x) = y − yˆ(x) = y − (x1 + x2 ).

(6.9)

132

...

CHAPTER 6. USING PRIOR KNOWLEDGE FOR

The riterion fun tion WN (x, x# , Z N , δ) (6.7) is thereby given as WN (x, x# , Z N , δ) = (y − x1 − x2 )2 + (x − x# )T δ(x − x# ).

(6.10)

It is minimized analyti ally by dierentiating WN partially w.r.t. ea h parameter and setting the partial derivative to zero, resulting in the following set of equations (1 + 2δ1 )x1 + x2 = y + 2δ1 x# 1 ,

(6.11a)

x1 + (1 + 2δ2 )x2 = y + 2δ2 x# 2 .

(6.11b)

There are two solutions of (6.11), one for δ = 0 and the other for δ 6= 0. For the δ = 0 ase, whi h orresponds to no prior knowledge of the parameters, the solution is given by (6.12)

x1 + x2 = y,

i.e. a parameterized solution whi h has an innite number of solutions. For δ 6= 0, the solution is given by x1 = δ2

# y − (x# 1 + x2 ) + x# 1 , δ1 + δ2 + δ1 δ2

(6.13a)

x2 = δ1

# y − (x# 1 + x2 ) + x# 2 . δ1 + δ2 + δ1 δ2

(6.13b)

In the spe ial ase when the measurement y is fully aptured by the # # model (6.12) and the prior parameter values x# i , i.e. if y = x1 + x2 , # then the estimates xi are given by the nominal values xi . However, when the prior does not exa tly mat h the measurement, i.e. when # y 6= x# 1 + x2 , the estimates xi are biased from their nominal values. Equation 6.13 illustrates that δ an be seen as a tuning knob; the larger the omponents δi are, the more the estimates xi are drawn to the nominal values x# i . In the ase when δ1 ≫ δ2 , x1 is more attra ted # to x# than x is to x 2 1 2 .

Minimizing the riterion fun tion W

N

The regularizing ee t imposed by the penalty term in WN (6.7) requires that the optimization sear h method is reformulated. The expressions for the gradient and the hessian of WN (6.7) are therefore given here. For omparison, referen es to the derivation of a lo al optimizer without regularization (appendix C.1) are given. The gradient VN (x, Z N )′ (C.4) and the hessian approximation HN (C.8) for su h an optimizer have to be rewritten when the parameter penalty term in (6.7) is in luded. It is here assumed that there are no ross- ouplings

6.4.

133

ESTIMATION METHODS

between the parameter prior knowledge, i.e. the regularization matrix

δ is assumed to be diagonal. The gradient is then written as PN WN′ (x, x# , Z N , δ) = N1 t=1 ψ(t, x)ε(t, x) + 2δ(x − x# ) = VN′ (x, Z N ) + 2δ(x − x# ),

(6.14)

d where as before the ja obian ve tor ψ(t, x) = dx ε(t, x) is given by (C.5). ′′ N The hessian WN (x, Z , δ) is then omputed as

WN′′ (x, Z N , δ) = VN′′ (x, Z N ) + 2δ δ ≈ HN (x, Z N ) + 2δ = HN (x, Z N , δ),

(6.15)

and its approximation is given by HNδ (x, Z N , δ) = HN (x, Z N ) + 2δ . The estimate xˆ is found numeri ally by updating the estimate of the minimizing point xˆi iteratively as x ˆi+1 (x# , Z N , δ) δ,i i =x ˆi (x# , Z N , δ) − µiN [HN (ˆ x , Z N , δ)]−1 WN′ (ˆ xi , x# , Z N , δ)

=

x ˆi (x# , Z N , δ)

+

di (ˆ xi , x# , Z N , δ), (6.16)

where i is the ith iterate, in whi h di is the sear h dire tion and HNδ,i is the approximate hessian in (6.15).

Prior distribution of parameters Both x# and δ in (6.7) an represent prior knowledge of the parameters. If a gaussian probability distribution an be assigned as a prior probability density fun tion (pdf) to the parameters x with mean x# and ovarian e matrix 2N1 δ , the maximum a posteriori estimation is made when the noise is gaussian distributed with zero mean (Ljung, 1999, pp.221). However, sin e no restri tions are made onsidering the s aling of the two terms VN and VNδ , they an dier quite substantially in size and an therefore be hard to ompare in a fair way. This is solved by introdu ing a regularization fa tor δx in the regularization matrix δ a

ording to δ = δx diag(δi ), i = 1, . . . , d,

(6.17)

where δi are alled regularization elements. The fa tor δx is then used as a tuning knob to nd a good ompromise between residual error ε and nominal parameter deviation ε# x . In (6.17) no ross-terms in the

ovarian e matrix are onsidered. The methodology ould however be extended to over the ase of non-zero ross-terms, with slight modi ations of the expressions in (6.14), (6.15) and (6.17) by ex hanging the diagonal matrix δ for a symmetri positive semi-denite matrix δ .

134

...

CHAPTER 6. USING PRIOR KNOWLEDGE FOR

The diagonal elements δi in δ are given by δi =

1 , 2N σi 2

(6.18)

where σi is the standard deviation for parameter xi and it is assumed that there are no ross-terms in the ovarian e matrix. From hereon the nominal value x# and the regularization elements δi are referred to as the prior knowledge, and the tuning parameter δx is referred to as the regularization parameter. How to determine the regularization parameter δx ?

The best way to determine δx would be to nd the δx that minimizes the size of the true parameter error, εtx = x − xt . However the true parameter values xt are only available in simulations, and this approa h is therefore not appli able for experimental data. It an however be used for evaluation of simulated data. A systemati method for nding δx that gives a good ompromise between the measures VN and VNδ is therefore sought. In order to nd an unequivo al δx , it is important that VN and VNδ are monotoni . The two measures VN and VNδ are dire tly related to ε and ε# x a

ording to r

1 VN 2 r 1 δ δ # V RMSE(L εx ) = d N

RMSE(ε) =

(6.19a) (6.19b)

where Lδ = diag(δi )1/2 and d is the number of parameters. The two measures in (6.19) are also monotoni whenever VN and VNδ are. Figure 6.2 shows an example of the dependen e between RMSE(ε) and RMSE(Lδ ε# x ). For large values of δx , the regularized estimates xδ are pulled towards the nominal values x# , i.e. a small RMSE(Lδ δ ε# x ). The matrix L serves as a weighting fun tion and determines how strong the pulling for e is for the individual parameters. The form on whi h the prior is given is therefore important for the shape of the resulting urve. The notation xδ is used for the estimate xˆ(δ) from hereon. For small values of δx , the pulling for e is small and the estimates are no longer suppressed by the prior and therefore free to adjust to the measurement data. This results in an RMSE(ε) that is approximately the same as the noise level. These two ases are the extremes

on erning onden e in the prior knowledge and measurement data respe tively. For moderate δx in between the two extremes, there is a sharp transition where the residual error ε falls while the nominal parameter deviation x# remains basi ally onstant. Due to that the upper

6.4.

135

ESTIMATION METHODS

0

10

Small δ

−1

10

x

−2

δ # x

RMSE ( L ε ) [−]

10

−3

10

−4

10

−5

10

Large δ

x

−6

10

−7

10

−2

−1

10

10 RMSE ( ε ) [bar]

0

10

Figure 6.2: Example of a L- urve. The dashed-dotted lines orrespond xt − x# used in the simula-

to the noise level and parameter deviation tion.

part of the graph resembles the letter L, the urve is alled an L-

urve (Hansen and O'Leary, 1993). The noise level and the parameter # t deviation x − x are known in simulations, and are therefore in luded in gure 6.2 as dashed lines.

The ross-over point of these two lines

is lose to the orner of the L. It is therefore natural to expe t that the orresponding

δx

is a good ompromise between data tting and

penalizing the parameter deviation (Engl et al., 1996). Note that the L- urve is omputed in a set of dis rete points that

an be onne ted by e.g. a ubi spline. range of ner.

δx :s

Typi ally there is a wide

orresponding to the points on the L- urve near its or-

Therefore, the lo ation of the orner should be found by some

numeri al optimization routine, and not by visual inspe tion (Hansen and O'Leary, 1993; Engl et al., 1996), espe ially if one wants to automatize the sear h pro edure.

136

...

CHAPTER 6. USING PRIOR KNOWLEDGE FOR

Routines for determining δx Three optimization routines for determining δx are presented here: 1. Millers a priori hoi e rule for δx (Miller, 1970). Consider omputing an estimate xˆ for whi h RM SE(ε(ˆ x, x# , δ, Z N )) ≤ mε , x, x# , δ, Z N )) ≤ mδ . RM SE(Lδ ε# x (ˆ

(6.20a) (6.20b)

Then Miller showed that

δx =

mδ mε

(6.21)

δ yields xδ su h that RM SE(ε(x , x# , δ, Z N )) √ √ a regularized estimate δ # δ # N ≤ 2mε and RM SE(L εx (x , x , δ, Z )) ≤ 2mδ . The result is valid for linear systems. Then if mε and mδ are good estimates of RMSE(η) and RMSE(Lδ ε# x ) respe tively, the regularization parameter δx from (6.21) yields a solution lose to the L- urve's

orner (Hansen, 1998).

2. Morozovs dis repan y prin iple (Morozov, 1984). The regularization parameter δx is the solution to the problem RM SE(ε(x, x# , δ, Z N )) = aε RM SE(η),

(6.22)

where η is the noise level and aε ≥ 1 is a onstant hosen by the user. Typi ally this routine overestimates δx (Hansen, 1994), and therefore gives a solution whi h is regularized to hard. 3. Maximum urvature of Hansen's L- urve (Hansen and O'Leary, 1993). The urvature τ is dened as τ (δx ) =

ψε′′ (δx )ψδ′ (δx ) − ψε′ (δx )ψδ′′ (δx ) 3/2 ,  ψε′ (δx )2 + ψδ′ (δx )2

(6.23a)

(6.23b) ψδ (δx ) = (6.23 ) The L- urve is approximated by a 2D spline urve given the set of dis rete points for whi h it is omputed, and from whi h δx orresponding to the point of maximum urvature τ is determined. ψε (δx ) = log||ε(x, x# , δ, Z N )||2 ,

# N log||Lδ ε# x (x, x , δ, Z )||2 .

The rst two routines require knowledge of the noise level η, while the third does not. The latter is thus ategorized as error-free in Engl et al. (1996). If the noise level is hanged and an approximative value for the new δx is sought for routines 1 and 2, the following example gives an approximation if the new noise level is known.

6.4.

137

ESTIMATION METHODS

Example 6.4  Impa t of measurement noise on regularization param-

eter δx The obje tive is to keep the ratio between the residual loss fun tion VN and the prior loss fun tion VNδ onstant. Assume that measurement noise is white and des ribed by ε1 (t) and that this results in the residual loss fun tion VN (ε1 ) and the prior (or nominal parameter) loss fun tion VNδ (ε1 ). Now assume a se ond noise realization su h that it

an be des ribed by ε2 (t) = aε1 (t), i.e. the rst one is multiplied by a fa tor a. This means that the standard deviation σ is in reased by a fa tor a and yields a residual loss fun tion VN (ε2 ) VN (ε2 ) =

N 1 X (a ε1i )2 = a2 VN (ε1 ). N i=1

(6.24)

Keeping the ratio between VN (ε2 ) and VNδ (ε2 ) onstant results in VN (ε1 ) VN (ε2 ) a2 VN (ε1 ) = = ⇒ VNδ (ε2 ) = a2 VNδ (ε1 ) VNδ (ε1 ) VNδ (ε2 ) VNδ (ε2 )

(6.25)

whi h is realized by setting the prior fa tor δx (ε2 ) = a2 δx (ε1 ). Thus if the standard deviation of the measurement noise is in reased by a fa tor a, the prior fa tor δx should be in reased by a2 as a rst approximation. This is natural sin e the regularization parameter δx is related to the varian e through (6.17)(6.18).

Spe ial ase: Equal δi In the spe ial ase when all elements δi are equal, the matrix Lδ = diag(δi )1/2 is given by the identity matrix times a onstant. This asδ sures that RMSE(ε# x ) is monotoni whenever VN is, and thus the mea)

an be used in the L- urve plots instead of RMSE(Lδ ε# sure RMSE(ε# x x ). The advantage is that RMSE(ε# )

orresponds dire tly to the nominal x parameter deviation. This will be used in hapters 7 and 8 whenever all δi :s are equal.

Algorithms  Method 2 Algorithms for three versions of method 2 are now given. They dier in how they nd the regularization parameter δx based on the three optimization routines des ribed earlier, and are labeled M2:1, M2:2 and M2:3 respe tively. In all ases when the riterion fun tion WN (6.7) is minimized, the initial values xinit are given by the nominal values x# .

138

...

CHAPTER 6. USING PRIOR KNOWLEDGE FOR

Algorithm 6.3

 Millers a priori hoi e rule (M2:1)

d×1 . 1.Assign a prior x# i and δi to ea h of the parameters x ∈ R The regularization matrix is then formed as δ = δx diag(δi ).

2.Compute boundaries mε and mδ in (6.20), that give δx from (6.21). 3.Minimize WN (6.7) w.r.t. x using δx from step 2. 4.Return the estimate xδ,∗ .

Algorithm 6.4 requires an investigation for nding the sear h region for δx in whi h the orner is always in luded. This region is denoted ∆x , and it only needs to be omputed on e for ea h appli ation. Algorithm 6.4

 Morozovs dis repan y prin iple (M2:2)

d×1 . 1.Assign a prior x# i and δi to ea h of the parameters x ∈ R The regularization matrix is then formed as δ = δx diag(δi ).

2.Assign the parameter aε (6.22) a numeri al value slightly larger than 1. 3.Compute the RMSE(η ) by minimizing the riterion fun tion WN (6.7) with δx = 0. 4.Minimize the riterion fun tion WN (6.7) for the dis rete points δx ∈ ∆x , equally spa ed in a logarithmi s ale. 5.Find the δx for whi h (6.22) is fullled, by using a ubi spline interpolation. 6.Minimize WN (6.7) w.r.t. x using δx from step 5. 7.Return the estimate xδ,∗ .

6.4.

ESTIMATION METHODS

139

A qui ker version of algorithm 6.4 is to reuse an estimation of the noise level made earlier in step 3. Another alternative to make the algorithm faster is to start with the lowest value of δx in step 4 and ompute for as long as RMSE(ε(δx )) ≤ aε RMSE(η). A third alternative is to use nearest neighbor interpolation in step 5, in whi h step 6 be omes obsolete. This is an option worth to onsider if relatively many dis rete points are used in ∆x . This will be further investigated in hapters 7 and 8. Algorithm 6.5 also requires an investigation for nding the sear h region ∆x for δx in whi h the orner is always in luded. Algorithm 6.5

 Hansen's L- urve (M2:3)

d×1 1.Assign a prior x# . i and δi to ea h of the parameters x ∈ R The regularization matrix is then formed as δ = δx diag(δi ).

2.Minimize the riterion fun tion WN (6.7) for the dis rete points δx ∈ ∆x , equally spa ed in a logarithmi s ale. 3.Find the δx for whi h (6.22) is fullled, by using a ubi spline interpolation. 4.Minimize WN (6.7) w.r.t. x using δx from step 3. 5.Return the estimate xδ,∗ . An alternative is to use nearest neighbor interpolation in step 3, in whi h step 4 be omes obsolete. This is an option worth to onsider if relatively many dis rete points are used in ∆x . This will be further investigated in hapters 7 and 8. Note that in step 1 for all versions of method 2, the parameters x ∈ Rd×1 ould be assigned a Gaussian distributed prior, as xi ∈ N (x# i , σi ). The matrix δ is then formed as δ = δx diag(δi ), where δi is given by (6.18). As mentioned earlier, this results in the maximum a posteriori estimate. 6.4.3

How to determine the prior knowledge?

The prior knowledge is used for more reasons than to give the best parameter estimates. It should primarily be used to ree t the insight and knowledge the user has of the system at hand, and to give physi ally reasonable parameter values.

140

...

CHAPTER 6. USING PRIOR KNOWLEDGE FOR

Nominal parameter values x

#

The nominal prior parameter values x# an be determined in at least three alternative ways: 1. By using the parameter initialization methods des ribed in hapter 3 for every parameter xi . 2. By using expert prior knowledge of what the numeri al value of x# should be. 3. A ombination of alternative 1 and 2. Depending on the validity of the prior for parameter xi , it has to be updated at ertain events. For the ylinder pressure models des ribed in se tion 6.2 this an be exemplied by the following; Consider the two parameters: pressure sensor oset ∆p and learan e volume Vc . The former is assumed to be onstant during one engine y le a

ording to se tion 3.1, and therefore has to be updated for every engine y le. On the other hand, in se tion 3.2 the latter parameter is assumed to be engine dependent. It therefore only needs to be updated on e a dierent engine is used.

Regularization elements δ

i

The element δi in the regularization matrix δ ree ts the un ertainty of the nominal parameter value xi and is related to the standard deviation σi of the prior pdf, as pointed out earlier. As for x# there are three

orresponding alternatives to determine δi . If the rst alternative is used, the standard deviation of x# i is omputed dire tly from the parameter initializations given in hapter 3. This value does however not ree t how un ertain the model approximation is when omputing x# i . If the nominal value x# is unsure, δi should be hosen small to give a large parameter exibility and a small penalty on the riterion fun tion WN (6.7). On the other hand when x# j is determined with a high

onden e, this should be ree ted in large value of δj ( ompared to δi ). This orresponds to penalizing |xj − x# j | harder in WN (6.7), and thereby exer ising a strong pulling for e on xj towards x# j . The parameter un ertainty is generally dierent for the individual parameters in the ase of ylinder pressure modeling. In this ase it is therefore more suitable to base δi upon the se ond alternative, i.e. expert knowledge of the un ertainty. Also, from (6.17) it an be on luded that it is only the relative intergroup size of δi that is relevant, sin e the regularization parameter δx ompensates for ontingent s alings. It is therefore not vital for the performan e of method 2 that the best value in a εtx -sense for δi is used, as long as the relative size to the other parameters stands. This also

6.4.

ESTIMATION METHODS

141

speaks in favor for the se ond alternative. It also suggests that other probability density fun tions an be assigned to the parameters, as long as they are of the same kind. Note that only the nominal value, and not the un ertainty of x# i that is related to δi , is used in method 1. The prior values x# and δ are determined for ea h spe i appli ation, and this issue is therefore returned to in hapters 7 and 8. 6.4.4

Summarizing omparison of methods 1 and 2

Both methods use prior knowledge of the parameters in the estimation problem, but they dier in how they use it. Method 1 only uses the nominal value x# indire tly when setting the spurious parameters xed, while method 2 is more exible and uses both the nominal value x# and the parameter un ertainty δ dire tly in the riterion fun tion to regularize the solution. If the prior knowledge in method 2 is Gaussian distributed, the estimation method yields the maximum a posteriori estimate (Ljung, 1999, pp.221).

142

...

CHAPTER 6. USING PRIOR KNOWLEDGE FOR

7

Results and evaluation for motored y les Method 1 and method 2 des ribed in se tion 6.4 will now be evaluated for motored ylinder pressure data. The evaluation overs both simulated and experimental data. As mentioned earlier in hapter 6 this is due to that the true value of the parameters are unknown in the experimental data.

The experimental data is olle ted on a turbo- harged

2.0L SAAB engine, whose engine geometry is given in appendix C.4. In this and the following hapter, it is worth noting that all numeri al al ulations have been made for normalized values of the parameters, while the individual parameter values that are given in tables are not normalized if not expli itly stated.

The purpose of the nor-

malization is to yield parameters that are in the order of 1, and the normalization is des ribed in appendix C.1.

7.1 Simulation results  motored y les First the simulated engine data will be des ribed, followed by a dis ussion on the parameter prior knowledge used. The fo us is then turned to evaluating the performan e of method 1, followed by method 2.

7.1.1

Simulated engine data

Cylinder pressure tra es were generated by simulating the standard model from se tion 6.2.1 with representative single-zone parameters, given in appendix C.5.

Eight operating points were sele ted with

143

144

CHAPTER 7. RESULTS AND EVALUATION MOTORED

...

Simulated motored data; OP1−8 Pressure [bar]

25 20

OP8

15 10 5 0

OP1 −100

−50 0 50 Crank angle [deg ATDC]

100

Figure 7.1: Simulated ylinder pressures for the dierent operating points OP18. engine speeds N ∈ {1500, 3000} rpm, mean harge temperatures at IVC TIV C ∈ {310, 370} K and ylinder pressures at IVC pIV C ∈ {50, 100} kPa, where table C.3 denes the individual operating points. For ea h operating point a ylinder pressure tra e was simulated and ten realizations of Gaussian noise with zero mean and standard deviation 3.8 kPa were added, forming altogether 80 motored y les. The

hosen noise level is the same as the one used in hapter 5, and thus ree ts the level seen in experimental data. The data was sampled with a resolution of 1 rank-angle degree (CAD). In gure 7.1, one motored

y le is shown for ea h operating point. The ylinder pressures orresponding to OP1 and OP8 dier the most, and will therefore be the two extremes in the investigation. 7.1.2

Parameter prior knowledge

The simulations are evaluated using simulated pressure tra es with known parameter values and a known model stru ture. To resemble the experimental situation, both a false and a orre t prior knowledge of the nominal parameter values x# will be used. For method 1, a preferred ordering of the parameters an be assigned and this will be done later in se tion 7.1.3. For method 2, the regularization elements δi needs to be assigned. In the simulations a gaussian distributed prior x ∈ N (x# , σ) is assumed, where δi are given by (6.18), i.e. as δi =

1 . 2N σi 2

(7.1)

When (7.1) was stated it was assumed that there are no ross-terms in the ovarian e matrix.

7.1. SIMULATION RESULTS  MOTORED CYCLES

145

Two dierent setups of nominal parameters x# will be used, and for ea h of these setups two ases of regularization elements δi will be investigated for method 2. Two ases (of δi )

In the rst ase, the standard deviation σi for ea h parameter xi is

hosen as σi = 0.01 xti . This hoi e is made for two reasons; As mentioned earlier the absolute value of δi is not interesting in itself, rather it is the relative size of δi ompared to δj that is. This was due to the regularization parameter δx . By hoosing the same δi for all parameters, this allows for a dire t measure of the nominal and true parameter t deviations, ε# x and εx respe tively, in the plots of an L- urve. Se ondly, sin e ea h parameter is assigned the same parameter un ertainty, this results in a parameter guiding of the estimates towards the nominal values but it does not give any joint ordering of the parameters. This

ase an therefore be seen as the rst and simplest approa h. It will from hereon also be referred to as δi = c, where c is a onstant. In the se ond ase the regularization elements are determined by our prior knowledge of the parameter un ertainty, and it illustrates the appli ation of the methods to an experimental situation. It an be seen as the more advan ed ase of the two, and typi ally yields regularization elements that are not the same. This ase is referred to as δi 6= c. For this parti ular appli ation the standard deviation σi for ea h parameter are given in table 7.1. These values are not veried experimentally, they are merely hosen sin e they are physi ally reasonable. To illustrate the prin iple of how the un ertainties are set,

onsider that the nominal value for the revi e volume Vcr is 1 cm3 whi h is approximately 1.5 per ent of the learan e volume Vc . A

ording to se tion 3.4 a reasonable region for Vcr is [1, 2] %Vc . Now if the standard deviation for Vcr is hosen as σi = 0.15 cm3 , this orresponds to that 95 per ent of the values belongs to that region. The two parameters ∆p and ∆θ are set to their expe ted un ertainty, and results in relative mean errors (RME) that are 50 %. Two setups (of x# )

The rst setup uses an equal relative parameter deviation. These nom= inal parameter deviations from the true parameter values, i.e. ε#−t x x# − xt , are hosen to be {0, 1, 2.5, 5} % xt . This will from hereon be referred to as the false prior (FP) at the spe ied level. This setup is

hosen in order to investigate what inuen e the prior knowledge has on the individual parameters. The se ond setup is based upon a relative parameter deviation dire tly related to the relative un ertainty for ea h parameter. This is

146

CHAPTER 7. RESULTS AND EVALUATION MOTORED

xi

Unit

Vc

C1

σi

RME [%℄

[cm3 ℄ 0.5 0.9

[-℄ 0.114 5.0

xi

Tw

Vcr [cm3 ℄

Unit σi

RME [%℄

[K℄ 10 2.5

0.15 15.0

TIV C

[K℄ 10 2.7

∆p

[kPa℄ 2.5 50.0

pIV C

γ300

[kPa℄ 2.5 5.0

[-℄ 0.005 0.4

∆θ

Kp

[deg℄ 0.05 50.0

...

b

[K −1 ℄

1 · 10−5

10.0

[-℄ 0.005 0.5

Table 7.1: Assigned standard deviation σi of the model parameters xi used for se ond ase of δi :s. The relative mean error (RME) orresponding to one standard deviation from the true value (for OP1) is also given. implemented as a nominal parameter deviation from the true values of t one standard deviation of the prior knowledge, i.e. x# i = (1 + σi )x , where the individual standard deviations σi are given in table 7.1.

7.1.3 Method 1 Results and evaluation First method 1 is evaluated for setup 1 using no preferred ordering of the parameters, i.e. the algorithms rely fully upon the measured data and in lude no prior knowledge of the parameters. Se ondly, a preferred ordering is introdu ed and it is investigated how this ee ts the estimation problem. The evaluation starts by determining in whi h order the parameters are set xed. Then the minimizing number of e ient parameters using Akaike's FPE is given, followed by a re ommendation of how many parameters to estimate. After that an investigation of the estimation a

ura y is performed, followed by a residual analysis. Evaluation without a preferred ordering  setup 1 First of all, the order in whi h the parameters are set xed for setup 1 when using algorithm 6.2 without a preferred ordering is investigated. This orresponds to keeping tra k of the spurious parameters xsp new in step 3. In this ase all parameters need to be in luded, and therefore the maximum and minimum number of e ient parameters are given # by d# max = d and dmin = 1, where d = 11 is the number of parameters. The resulting parameter order without a preferred ordering is given by b ≺ pIV C ≺ Vcr ≺ TIV C ≺ ∆θ ≺ ∆p ≺ C1 ≺ Tw ≺ Vc ≺ Kp ≺ γ300 ,

(7.2)

7.1. SIMULATION RESULTS  MOTORED CYCLES OP 8

11

11

10

10

9

9

8

8 No of parameters [−]

No of parameters [−]

OP 1

7 6 5 4 3 2 1 0 0

147

7 6 5 4 3

0 % (5.3) 1 % (5.3) 2.5 % (5.2) 5 % (5.1) 5 Cycle No [−]

2 1 10

0 0

0 % (6) 1 % (6) 2.5 % (6) 5 % (5.9) 5 Cycle No [−]

10

Figure 7.2: Minimizing number of parameters d∗ for method 1, when using no preferred ordering. whi h ree ts a typi al average ase. Here b ≺ pIV C means that b is set xed prior to pIV C . The underlined parameters are invariant in position to the investigated noise realizations, false prior levels and different operating onditions. The other parameters however dier in a few number of ases and out of the three variations, mostly from dierent noise realizations. The dieren es in order appear almost without ex eption as permutations of two groups, namely [C1 , TIV C , Tw ] and [Kp , pIV C ]. This means that pIV C is sometimes repla ed by Kp and that TIV C is at some instan es repla ed by either C1 or Tw . It is also worth to mention that γ300 is the most e ient parameter of them all, a

ording to (7.2). The minimizing number of parameters using algorithm 6.2 are given in gure 7.2 for OP1 and OP8, whi h are the two extremes in the simulation-based investigation. Ten dierent noise realizations orresponding to ten engine y les have been used, as well as four dierent

ases of false prior. The orresponding results for all eight operating points are given in table 7.2, but now as a mean value for the ten y les.

148

CHAPTER 7. RESULTS AND EVALUATION MOTORED

FP [%℄ 0 1 2.5 5 d∗ ∈

1 5.3 5.3 5.2 5.1 [4, 8℄

2 5.4 4.9 4.9 5.0 [4, 8℄

Operating point (OP) 3 4 5 6 6.1 6.1 5.5 5.3 6.1 6.1 5.2 4.9 6.1 6.1 5.2 4.9 6.2 6.0 5.1 5.2 [4, 8℄ [4, 8℄ [4, 8℄ [4, 8℄

7 6.1 6.1 6.1 6.1 [4, 7℄

...

8 6.0 6.0 6.0 5.9 [4, 8℄

Table 7.2: Minimizing number of parameters d∗ for method 1, without a preferred ordering. Four false prior levels are used and the range of d∗ is given for ea h operating point. Table 7.2 shows that the average numbers of d# are between 4.9 and 6.1, but gure 7.2 and table 7.2 indi ate that d# ∈ [4, 8]. The variation in d# depending on the level of false prior FP is omparatively small, but the variation is larger depending on the operating point. This makes it hard to determine how many e ient parameters in (7.2) to use.

Estimation a

ura y Tables 7.3 and 7.4 show the parameter estimates for the entire range of d# , i.e. from 11 to 1 parameter, for one engine y le. For this spe i

y le (OP1, y le1), the Akaike FPE is minimized by four parameters, i.e. d∗ = 4. The false prior level used is 0 % in table 7.3 and 5 % in table 7.4 respe tively. The parameters are set xed using algorithm 6.1, i.e. in a

ordan e with (7.2), and this orresponds to the emphasized values in two tables. For instan e when pIV C is set xed at 50.0 kPa for d# = 9, this is indi ated in the tables by emphasizing the nominal value. The parameter values for pIV C are then xed for d# ∈ [1, 8] as well, but to in rease the readability these values are left out. Table 7.4 shows that the parameter estimates are learly biased when a false prior is present. As expe ted the individual parameter estimates depend upon the number of e ient parameters d# , as well as the false prior level. The estimates of ourse also depend upon what parameters that are lassied as e ient. In order to have a ε¯tx ≤ 1 %, a maximum of three parameters in ombination with a false prior level of 0 % is required, see table 7.3. For F P = 5 % (table 7.4), the same limit on d# is found if the true parameter deviation should be less or equal to the false prior level, i.e. ε¯tx ≤ 5 %. This means that for d# ≥ 4, the parameter estimates deviate more from xt than x# does. However, by onsidering the individual estimates it is notable that the C1 -estimate has the largest normalized bias in general. Until C1 is xed the other parameters, espe ially Kp , TIV C , γ300 and to some

[ ℄ 57.1 57.4 57.4 57.4 57.5 57.5 58.6 58.5 58.1

Vc cm3

[-℄ 1.91 1.88 1.87 1.87 1.96 1.96 1.70

C1

TIV C

pIV C

γ300

b K −1

Tw

Vcr cm3

∆p

∆θ

Kp

ε¯tx

ε¯# x

11 10 9 8 7 6 5 4 3 2 1

[K℄ [kPa℄ [-℄ [ ℄ [K℄ [ ℄ [kPa℄ [deg℄ [-℄ [%℄ [%℄ 402 51.7 1.38 -0.00012 420 1 3.2 0.11 0.99 8.7 8.7 404 51.8 1.38 -0.0001 418 1 3.0 0.10 0.99 6.6 6.6 404 50.0 1.38 418 1 3.0 0.10 1.03 6.7 6.7 404 1.38 418 1 3.0 0.10 1.03 6.7 6.7 370 1.37 381 3.0 0.10 1.03 5.1 5.1 1.37 382 3.0 0.10 1.03 5.0 5.0 1.40 378 5.0 1.00 7.8 7.8 2.28 1.40 446 0.99 3.5 3.5 1.39 400 1.01 0.5 0.5 58.8 1.40 1.00 0.1 0.1 1.40 1.00 0.0 0.0 x# 58.8 2.28 370 50.0 1.40 -0.0001 400 1 5.0 0.10 1.00 5.0 0.0 xt 58.8 2.28 370 50.0 1.40 -0.0001 400 1 5.0 0.10 1.00 0.0 5.0 Table 7.3: Parameter estimates at OP1 for d# ∈ [1,t 11] parameters when using no preferred ordering and a false prior level of 0 %. The true and nominal parameter deviations ε¯x and ε¯#x are given as RMSE:s of their relative values, i.e. ε¯tx = RMSE(εtx ) and ε¯#x = RMSE(ε#x ).

d#

7.1. SIMULATION RESULTS  MOTORED CYCLES

149

℄ 57.1 57.3 57.3 57.3 57.4 57.4 58.5 58.5 57.8

[

Vc cm3

[-℄ 1.91 1.89 1.89 1.89 1.96 1.90 1.63 1.78

C1

TIV C

pIV C

γ300

b K −1

Tw

Vcr cm3

∆p

∆θ

Kp

ε¯tx

ε¯# x

11 10 9 8 7 6 5 4 3 2 1

[K℄ [kPa℄ [-℄ [ ℄ [K℄ [ ℄ [kPa℄ [deg℄ [-℄ [%℄ [%℄ 402 51.7 1.38 -0.00012 420 1 3.2 0.11 0.99 8.7 9.1 405 51.8 1.38 -0.000105 420 1 3.1 0.10 0.99 6.8 8.1 406 52.5 1.38 421 1 3.1 0.10 0.98 6.9 8.2 406 1.38 421 1.1 3.1 0.10 0.98 7.0 8.1 389 1.38 405 3.1 0.10 0.98 5.7 7.3 1.38 400 3.1 0.11 0.98 6.3 7.9 1.40 394 5.3 0.95 9.3 11.0 1.40 420 0.95 7.6 9.0 2.39 1.39 0.96 3.9 3.8 61.7 1.45 0.93 4.6 3.5 1.37 1.05 4.3 2.0 x# 61.7 2.39 389 52.5 1.47 -0.000105 420 1.1 5.3 0.11 1.05 5.0 0.0 xt 58.8 2.28 370 50.0 1.40 -0.0001 400 1 5.0 0.10 1.00 0.0 5.0 Table 7.4: Parameter estimates at OP1 for d# ∈ [1,t 11] parameters when using no preferred ordering and a false prior level of 5 %. The true and nominal parameter deviations ε¯x and ε¯#x are given as RMSE:s of their relative values, i.e. ε¯tx = RMSE(εtx) and ε¯#x = RMSE(ε#x ).

d#

150 CHAPTER 7. RESULTS AND EVALUATION MOTORED

...

7.1. SIMULATION RESULTS  MOTORED CYCLES

151

extent Tw , ompensate for the bad C1 -estimates. This results in biased estimates for these parameters as well. When C1 is xed, the pressure gain Kp still has a onsiderable bias, espe ially in the presen e of a large false prior. It is however believed that Kp is known more a

urately a priori through alibration, ompared to the other parameters. Therefore if a preferred parameter ordering is to be used, it is re ommended that Kp and C1 are set xed at an early stage. As mentioned earlier, the bias in the individual parameters depend upon d# and F P . This is exemplied by the two parameters Vc and γ300 , whi h have their smallest biases for ve parameters for both ases of the false prior level, a

ording to tables 7.3 and 7.4. These two parameters have already been shown to be important in the sensitivity analysis in table 3.2. This illustrates that if ε¯tx or individual parameters are monitored to nd the best d# , this an result in dierent numbers of re ommended parameters, depending on the spe i appli ation. For the spe i appli ation onsidered in hapter 5, where the fo us was on estimating Vc , it is appropriate to settle for ve e ient parameters in method 4. To summarize, method 1 does not yield a unequivo al number of e ient parameters, as it u tuates in between four and eight. Using a mean value of d# is not optimal either sin e it does not orrespond to the best parameter estimates, espe ially in the presen e of a false prior. This is however not unexpe ted, sin e the minimization of VN (6.3) is only based upon the residuals, and is therefore not inuen ed dire tly by the parameter estimates.

Residual analysis In gure 7.3 the residuals for two ases of false prior oset at OP1 are shown. The upper plots show the residual orresponding to minimization of the Akaike riterion, in these ases d∗ =4. The middle plots display the dieren e between the modeled pressure for 4 and 1 parameter, while the lower plots show the dieren e for 4 and 11 parameters. The data is well des ribed by both 11 pars and 4 pars, for both 0 and 5 % false prior oset. However for less than four parameters the residual be omes larger, ree ting that too few parameters are used to des ribe the data. This ee t is more pronoun ed for F P = 5 %, as the dieren e between the modeled pressure for d# =4 and d# =1 be omes larger. The parameter values for d# =4 and d# =11 are however not the same, see table 7.4, and this ree ts that it is not su ient to have a small residual error for an a

urate parameter estimate.

152

CHAPTER 7. RESULTS AND EVALUATION MOTORED

4

Residual [Pa]

2

x 10

*

FP = 0 %, d =4

4

2

1

1

0

0

−1 −100

0

100

p(4) − p (1)

Residual [Pa]

*

FP = 5 %, d =4

−1 −100 4

1000

2

x 10

0

100

p(4) − p (1)

1

0

0 −1000

−2000 −100

−1 0

100

−2 −100

p(11) − p (4) 1000

Residual [Pa]

x 10

...

0

4

2

x 10

0

100

p(11) − p (4)

1 0

−1000

−2000 −100 0 100 Crank angle [deg ATDC]

−1 −2 −100 0 100 Crank angle [deg ATDC]

Figure 7.3: Upper: The residual for four parameters when the false prior is 0 and 5 %. Middle: Dieren e between the ylinder pressure for four parameters and one parameter. Lower: Dieren e between the ylinder pressure for eleven and four parameters. The s ales are dierent for 0 and 5 %. Evaluation with a preferred ordering  setup 1 To in lude parameter prior knowledge and to gain a more unequivo al order in whi h the parameters are set spurious, the aforementioned permutations of the parameters are used in our ase when a preferred ordering is used. The preferred ordering is given by the following ordering groups: B1 = [b, Tw , C1 , TIV C ], B2 = [Kp , pIV C , ∆θ, ∆p], B3 = [Vcr , Vc ] and B4 = [γ300 ]. The order of the parameters in ea h group Bi is set a

ording to what is believed to be known a priori of the parameters. For instan e, for group B1 the slope oe ient b for γ , is well known from the hemi al equilibrium program used in hapter 4 and is therefore set rst in the group. The order given here should by no means be interpreted as the best one, rather as a fully-qualied suggestion. This results in the following parameter order with a preferred or-

7.1. SIMULATION RESULTS  MOTORED CYCLES

153

dering used b ≺ Kp ≺ Vcr ≺ TIV C ≺ ∆θ ≺ ∆p ≺ Tw ≺ C1 ≺ Vc ≺ pIV C ≺ γ300 .

(7.3) The dieren e in orders between (7.2) and (7.3) o

urs for two lo ations; Firstly for Kp and pIV C , and se ondly for C1 and Tw . The permutations however still exist, but are redu ed in their number of o

urren es and would extinguish if the ad-ho de ision rules orresponding to step 4 and step 6 in algorithm 6.1 were set to 1 and 0 respe tively. The parameter TIV C is set xed prior to Tw and C1 although its relative order in B1 , whi h also is an ee t of the ad-ho de ision rules in algorithm 6.1. The minimizing number of parameters d∗ at OP1 and OP8 are given in gure 7.4, and the mean value of d∗ is given in table 7.5 for all operating points and false prior levels. The individual parameter estimates for one engine y le are given in tables 7.6 and 7.7 for FP = 0 % and 5 % respe tively. Compared to the ase with no preferred ordering, the spread of d∗ is smaller, espe ially for OP8 as shown in gure 7.4. This is also ree ted in table 7.5, where the mean values are more fo used. However, as tables 7.6 and 7.7 both indi ate, the estimates are not signi antly improved ompared to tables 7.3 and 7.4. Due to the unsu

essful results of method 1 for setup 1, the evaluation for setup 2 is left out. Summary of method 1 To summarize, using method 1 to estimate the parameters in the presen e of a false prior has not been su

essful. Minimizing the Akaike FPE riterion gives a re ommended number of parameters between 4 and 8, but this number u tuates with both operating point and noise realization. If we instead just onsider the true parameter deviation, a maximum of three parameters an be used in order to yield ε¯tx ≤ 1 % in the ase of no false prior. In the ase of a false prior it gets even worse. Method 1 is therefore in itself not re ommended to use for estimation, and is therefore not investigated further for the simulation part of the

hapter. It an however be used for a spe i appli ation, like ompression ratio estimation in hapter 5. In this ase only the Vc -estimate is monitored when de iding upon the e ient number of parameters to use, and this results in that ve parameters are used. Method 1 has also given valuable insight in whi h parameters that are most e ient. For instan e it has been shown that γ300 is the most e ient parameter, given the stru ture of the standard model in se tion 3.8.

154

CHAPTER 7. RESULTS AND EVALUATION MOTORED

OP 1

OP 8 11

10

10

9

9

8

8 No of parameters [−]

11

No of parameters [−]

...

7 6 5 4 3

7 6 5 4 3

0 % (5.1) 1 % (5.2) 2.5 % (5.1) 5 % (5.7)

2 1 0 0

0 % (5.3) 1 % (5.3) 2.5 % (5.1) 5 % (5.1)

2 1

5 Cycle No [−]

0 0

10

5 Cycle No [−]

Figure 7.4: Minimizing number of parameters

d∗

10

for method 1, when

using a preferred ordering.

Operating point (OP) FP [%℄

1

2

3

4

5

6

7

8

0

5.1

5.6

5.5

5.3

5.5

5.6

5.3

5.3

1

5.2

5.6

5.3

5.3

5.5

5.5

5.3

5.3

2.5

5.1

5.6

5.3

5.3

5.5

5.5

5.1

5.1

5

5.7

5.5

5.1

5.1

5.5

5.5

5.1

5.1

d∗ ∈

[3, 7℄

[4, 7℄

[5, 8℄

[5, 7℄

[4, 6℄

[4, 7℄

[5, 7℄

[5, 7℄

Table 7.5: Minimizing number of parameters using a preferred ordering. range of

d∗

d∗

for method 1, when

Four false prior levels are used and the

is given for ea h operating point.

[ ℄ 57.1 57.4 57.4 57.4 57.5 57.5 58.6 58.6 58.4 58.6

Vc cm3

[-℄ 1.91 1.88 1.88 1.88

C1

TIV C

pIV C

γ300

b K −1

Tw

Vcr cm3

∆p

∆θ

Kp

ε¯tx

ε¯# x

11 10 9 8 7 6 5 4 3 2 1

[K℄ [kPa℄ [-℄ [ ℄ [K℄ [ ℄ [kPa℄ [deg℄ [-℄ [%℄ [%℄ 402 51.7 1.38 -0.00012 420 1 3.2 0.11 0.99 8.7 8.7 404 51.8 1.38 -0.0001 418 1 3.0 0.10 0.99 6.6 6.6 404 51.5 1.38 418 1 3.0 0.10 1.00 6.6 6.6 404 51.5 1.38 418 1 3.0 0.10 6.6 6.6 2.28 352 51.6 1.37 379 2.9 0.08 3.4 3.4 368 51.6 1.37 400 2.9 0.08 2.6 2.6 349 50.0 1.39 4.7 0.07 1.9 1.9 340 1.39 4.5 0.10 2.5 2.5 346 1.39 5.0 2.0 2.0 370 1.40 0.1 0.1 58.8 1.40 0.0 0.0 x# 58.8 2.28 370 50.0 1.40 -0.0001 400 1 5.0 0.10 1.00 5.0 0.0 xt 58.8 2.28 370 50.0 1.40 -0.0001 400 1 5.0 0.10 1.00 0.0 5.0 Table 7.6: Parameter estimates at OP1 for d# ∈ [1, 11] parameters when using a preferred ordering and a false prior level of 0 %. The true and #nominal parameter deviations ε¯tx and ε¯#x are given as RMSE:s of their relative values, i.e. ε¯tx = RMSE(εtx ) # and ε¯x = RMSE(εx ).

d#

7.1. SIMULATION RESULTS  MOTORED CYCLES

155

[ ℄ 57.1 57.3 57.3 57.3 57.5 57.3 54.9 54.9 52.6 51.7

Vc cm3

[-℄ 1.91 1.89 1.86 1.86

C1

TIV C

pIV C

γ300

b K −1

Tw

Vcr cm3

∆p

∆θ

Kp

ε¯tx

ε¯# x

11 10 9 8 7 6 5 4 3 2 1

[K℄ [kPa℄ [-℄ [ ℄ [K℄ [ ℄ [kPa℄ [deg℄ [-℄ [%℄ [%℄ 402 51.7 1.38 -0.00012 420 1 3.2 0.11 0.99 8.7 9.1 405 51.8 1.38 -0.000105 420 1 3.1 0.10 0.99 6.8 8.1 405 49.0 1.38 420 1 3.1 0.10 1.05 7.1 8.5 405 49.0 1.38 420 1.1 3.1 0.10 7.3 8.3 2.39 334 49.2 1.37 364 2.8 0.08 5.6 7.6 376 49.1 1.37 420 2.9 0.07 4.2 4.6 420 52.5 1.33 -1.2 0.10 8.7 8.4 420 1.33 -1.2 0.11 8.7 8.4 565 1.33 5.3 16.7 15.4 389 1.28 6.0 6.5 61.7 1.37 4.3 2.0 x# 61.7 2.39 389 52.5 1.47 -0.000105 420 1.1 5.3 0.11 1.05 5.0 0.0 xt 58.8 2.28 370 50.0 1.40 -0.0001 400 1 5.0 0.10 1.00 0.0 5.0 Table 7.7: Parameter estimates at OP1 for d# ∈ [1, 11] parameters when using a preferred ordering and a false prior level of 5 %. The true and #nominal parameter deviations ε¯tx and ε¯#x are given as RMSE:s of their relative values, i.e. ε¯tx = RMSE(εtx) # and ε¯x = RMSE(εx ).

d#

156 CHAPTER 7. RESULTS AND EVALUATION MOTORED

...

157

7.1. SIMULATION RESULTS  MOTORED CYCLES 10

10

x

δ [−]

0

10

−10

10

−20

10

0

10

20

30 40 Element no. [−]

50

60

70

Figure 7.5: Values of the regularization parameter δx used in the investigations.

7.1.4

Method 2 Results and evaluation

First some important implementation details are given. Then the three versions of method 2 are evaluated using the two setups des ribed in se tion 7.1.2. Implementation details

First the interesting sear h region for the regularization parameter δx is dis ussed, followed by a motivation for the implementation hoi es that an be made for ea h spe i version of method 2. The sear h region ∆x for the regularization parameter δx is hosen as δx ∈ [10−11 , 105 ], whi h is found by testing to assure that the Lshape orresponding to gure 6.2 o

urs for all examined ases. The region is then divided into three intervals, where δx is equally spa ed in a logarithmi s ale for ea h interval. The middle interval is where the

orner of the L- urve is expe ted to appear, and this interval is therefore more densely sampled. The limits of the intervals are 105 , 101 , 10−8 and 10−11 , and the number of samples are 14, 50 and 5 respe tively. The regularization parameter is plotted in gure 7.5. These numeri al values are all based upon that the residual ε is omputed in bars, and should therefore be altered if a dierent unit is used. Millers a priori hoi e rule, see algorithm 6.3, will from hereon be referred to as M2:1. It requires that the onstants mε and mδ from (6.20) are set. The drive for setting these values is twofold; First of all the regularization parameter δx from (6.21) must be assured to be on the horizontal plateau of the L- urve (see gure 6.2), preferably as lose to the leftmost orner as possible. Se ondly, the same equations for mε and mδ should be used for all operating points and setups. The

onstant mε is therefore hosen as mε = 1.01RMSE(η), and orre-

158

CHAPTER 7. RESULTS AND EVALUATION MOTORED

...

sponds to a data tting whi h is 1 % above the estimated noise level. The other onstant is hosen as a nominal parameter deviation of 1 %, δ whi h is then weighted with the un ertainty matrix L . This results in δ mδ = RMSE(0.01 L ), and assures that both requirements are fullled for all the investigated ases. The se ond version M2:2 is alled Morozovs dis repan y prin iple (algorithm 6.4), where the onstant

aε

needs to assigned. Due to the

same reasons as for M2:1, it is hosen as

ln(aε ) = 0.05 ln(RMSE(ε(x# ))) + 0.95 ln(RMSE(η)),

(7.4)

RMSE(ε(x# )) is the RMSE of the residual for the nominal pa# # rameter values x . In luding RMSE(ε(x )), whi h orresponds to δx → ∞, assures the exibility to dierent false prior levels. where

Algorithm 6.5 is the third version and is named Hansen's L- urve or

M2:3. It uses a 2D ubi spline fun tion of order four (Hansen, 1994) to nd the position of maximum urvature. Apart from that, no hoi es are required. An example of an L- urve is given in gure 7.6 for one engine y le FP = 5 %. It also shows the true parameter deviation ε¯tx and the false prior level, as well as the results for the three versions

at OP1 and

of method 2 and the optimal hoi e of δx . The optimal hoi e of δx is ε¯tx = RMSE(εtx ), for the δx :s used in ∗ the omputation. It is hereon referred to as δx .

determined as the one minimizing

Evaluation for setup 1 Method 2 is now evaluated for setup 1, where the nominal values are # given by x = (1 + FP) xt for FP ∈ {0, 1, 2.5, 5} % and the assigned t standard deviation σ = 0.01x is equal for all parameters, whi h yields equal regularization elements δi , i.e. ase 1 of

δi (δi = c). In se tion 6.4.2 # δi :s, ε¯# x = RMSE(εx )

it was shown that in the spe ial ase of equal

ould be used in the L- urve plots. Figure 7.6 shows that as

δx

is de reased,

ε¯tx

be omes smaller until it

rea hes its optimal value. This means that the estimate is a ompromise between the data and the prior knowledge, and it lies in between the true value and the nominal (false) value, whi h is good. The gure also illustrates that all three versions of method 2 nd positions lose to the

orner of the L, at least visually.

Evaluation for OP1 and OP8:

The fo us will now be on evaluating

the performan e of method 2 for the two extreme operating points OP1 ∗ and OP8. Figure 7.7 displays the optimal δx at ten onse utive y les, for whi h only the noise realization diers.

The false prior level is

indi ated by the gure legend, and the orresponding mean value for δx∗ are given within the parentheses. The values for FP = 0 % are left out sin e they all orrespond to the highest value of

δx .

159

7.1. SIMULATION RESULTS  MOTORED CYCLES Method 2; OP1 FP=5%

0

Zoomed version

−1

10

10

−1

10

−2

x

RMSE( ε#) [−]

10

−3

10

−4

10

−5

10

M2:3 M2:2 M2:1 Optimal

−6

10

−7

10

−2

10

−2

−1

10 RMSE( ε ) [bar]

0

10

10

−2

10

−1

0

10 10 RMSE( ε ) [bar]

Figure 7.6: L- urve (solid line) for one engine y le at OP1 with FP=5 % (dotted line) for setup 1. The results for the three versions of method 2 and the optimal hoi e of regularization parameter are indi ated by the legend. The mean true parameter deviation ε¯tx (dashdotted line) is also given. The gure shows that there is a dieren e in optimal δx∗ depending both on the false prior level and the operating point. There is also a smaller dieren e depending on the y le number, whi h is most pronoun ed for OP1 at FP = 5 %. A general trend for δx∗ at a given operating point, is that δx∗ de reases with FP. This is expe ted sin e a δ lower δx is required when ε¯# x is larger, for the two terms VN and δx VN in WN (6.7) to be ome well balan ed. It does however not say anything about the estimation a

ura y. The estimation a

ura y is investigated in gure 7.8, where the true parameter deviations orresponding to the optimal δx :s in gure 7.7 are given. Again, the false prior level is indi ated by the gure legend and the orresponding mean value for ε¯tx are given in per ent within the parentheses. Figure 7.8 illustrates that the resulting ε¯tx is less than the spe ied

160

CHAPTER 7. RESULTS AND EVALUATION MOTORED

OP 1

OP 8

−2

−2

10

10

−3

−3

10

δx [−]

δx [−]

10

−4

10

−5

−4

10

−5

10

10 1 % (0.003) 2.5 % (0.00093) 5 % (0.00022)

−6

10

...

0

5 Cycle No [−]

10

1 % (0.0077) 2.5 % (0.0025) 5 % (0.0013)

−6

10

0

5 Cycle No [−]

10

Figure 7.7: Value of optimal regularization parameter δx∗ in method 2 for ten y les when minimizing ε¯tx for setup 1. OP 8

0.035

0.035

0.03

0.03

RMSE ( ε ) [−]

0.04

0.025

t x

0.025

t

RMSE ( εx ) [−]

OP 1 0.04

0.02 0.015

0.02 0.015

0.01

0.01

0.005

0.005

0 0

5 10 1 % No (0.0078) Cycle [−] 2.5 % (0.019) 5 % (0.037)

0 0

Figure 7.8: Mean true parameter deviation ε¯tx = sponding to gure 7.7 for setup 1.

5 10 1 % No (0.0076) Cycle [−] 2.5 % (0.019) 5 % (0.038)

RMSE(εtx) orre-

161

7.1. SIMULATION RESULTS  MOTORED CYCLES

OP 1 2 3 4 5 6 7 8

0

1 · 105 1 · 105 1 · 105 1 · 105 1 · 105 1 · 105 1 · 105 1 · 105

False prior (FP) [%℄ 1 2.5 5 0.0030 0.00093 0.00022 0.0024 0.00072 2.6 · 10−5 0.0011 0.0044 0.0024 0.0011 0.0024 0.00098 0.0024 0.00098 1.0 · 10−5 0.0024 0.00072 0.00022 0.0011 0.0044 0.0024 0.0077 0.0025 0.0013

Table 7.8: Value of optimal regularization parameter δx∗ , for operating points OP1 to OP8 for setup 1. For FP = 0 %, δx∗ is the highest value used in the omputation.

FP [%℄ 0 1 2.5 5

1

10−7

0.78 1.9 3.7

2

10−7

0.76 1.9 3.7

Operating point (OP) 3 4 5 6

10−7

0.77 1.9 3.8

10−7

0.76 1.9 3.8

10−7

0.77 1.9 3.8

10−7

0.76 1.9 3.8

7

10−7

0.77 1.9 3.8

8

10−7

0.76 1.9 3.8

Table 7.9: Mean true parameter deviation ε¯tx [%] orresponding to optimal δx∗ in table 7.8, for setup 1 and operating points OP1 to OP8. false prior level FP. This renders an estimate in between the true and nominal value, i.e. a ompromise between data and prior knowledge, although the estimate is more biased towards the nominal value. The gure also shows that for a given FP, the true parameter deviation ε¯tx is fairly onstant from y le-to- y le. In the worst ase (OP1, FP = 5 %), the dieren e between the estimates are within 0.2 %. This suggests that it is not vital to nd exa tly the optimal δx∗ , to get good estimates. Evaluation for all OP:s: The fo us is now turned to evaluating method 2 for all operating points. The optimal δx∗ is given in table 7.8 for all operating points and false prior levels used in setup 1. The orresponding true parameter deviations ε¯tx are summarized in table 7.9. The numeri al values in tables 7.8 and 7.9 are given as mean values for ten y les. Table 7.8 shows that the optimal δx∗ diers for the FP level and operating point used, and the ee t is most pronoun ed for hanges in the FP level. This is in a

ordan e with what was pointed out earlier. Table 7.9 shows that the true parameter deviation ε¯tx is less than the false prior level used for FP > 0 %. The table also illustrates that even

162

CHAPTER 7. RESULTS AND EVALUATION MOTORED

OP 1 2 3 4 5 6 7 8 Max Di Mean Di Time [s℄

δx∗

3.7 3.7 3.8 3.8 3.8 3.8 3.8 3.8 0 0 2579

δx (M2:1)

3.8 3.8 3.9 3.8 3.8 3.8 3.9 3.9 0.092 0.040 32

δx (M2:2)

4.1 4.1 3.9 3.9 4.1 4.1 3.9 3.9 0.35 0.19 1333

δx (M2:3)

4.0 4.0 3.9 3.9 4.0 4.0 3.9 3.8 0.31 0.15 2579

...

mv(δx∗ )

3.9 3.9 3.8 3.8 3.9 3.9 3.8 3.8 0.21 0.094 57

Table 7.10: True parameter deviation ε¯tx for the optimal δx∗ in table 7.8, the three versions of method 2 and as mean value for the optimal δx∗

orresponding to table 7.8. The numeri al values are given in per ent and are evaluated for F P = 5 % for setup 1. The mean omputational time for ompleting the estimation for one engine y le is also given.

though the optimal δx∗ varies depending on the operating point at a given FP level, this has almost no ee t on the estimation a

ura y. It therefore seems probable that the estimation a

ura y is relatively insensitive to variations in δx . This is further investigated by omputing the mean value of the optimal δx∗ in table 7.8, dis arding FP = 0 %. This value is from hereon denoted mv(δx∗ ), and is used together with the three versions of method 2 to ompute the true parameter deviation in table 7.10. The ε¯tx -values in table 7.10 are omputed as a mean value of ten y les for all operating points at FP = 5 %. The maximum and mean dieren e are

omputed relative to δx∗ . All of these values are given in per ent. The mean omputational time is also given, where the following assumptions have been made; For δx∗ and algorithm M2:3 all δx in the sear h region ∆x are used, and are therefore in luded in the omputational time. Algorithm M2:2 starts by estimating the noise level η, and then in reases the δx ∈ ∆x until the riterion (6.22) is fullled, see steps 4 and 5 in algorithm 6.4. Thus the omputational time for this algorithm

an dier quite extensively from y le to y le, depending on when a δx resulting in RMSE(ε(δx )) > aε RMSE(η) is found. For algorithm M2:1 and mv(δx∗ ) the omputational time is based on one value of δx , i.e. it is assumed that these values are found priorly. Table 7.10 shows that the estimation a

ura y in terms of ε¯tx is not as good for the approximative methods as for the optimal one, sin e the true parameter deviation ε¯tx in reases for all four approximations of

7.1. SIMULATION RESULTS  MOTORED CYCLES

163

δx . However the mean dieren e is small, espe ially for M2:1 and the mean-value based mv(δx∗ ) as the table indi ates. The latter is however not an option when onsidering experimental data, sin e then δx∗ is not available, and is therefore not in luded in the tables from hereon. An alternative is to use the mean value for δx (M 2 : 3), whi h is available. The performan e of M2:3 itself is however not better than that of M2:1, a

ording to the mean dieren e in table 7.10. It is therefore expe ted that using the mean value of δx (M 2 : 3) instead does not result in better results than for M2:1. There is a disadvantage of M2:1 ompared to M2:3 when hanging operating onditions, in that it might require some ad-ho tuning of one of the parameters mε and mδ to assure good performan e. This has however not been required in the simulations. However if it would be required, one should instead onsider using M2:3 whi h is exible to hanging operating onditions and is the se ond best hoi e with respe t to estimation a

ura y, given that the required omputational time is available. The third hoi e would be M2:2, but like M2:1 it might require some ad-ho tuning, in this ase for the parameter aε . The two algorithms M2:2 and M2:3 both tend to give too high values of δx ompared to δx∗ , as illustrated in gure 7.6. This results in estimates loser to the nominal values, i.e. estimates that are oversmoothed by the regularization. For M2:2, this is due to the restri tive

hoi e made when determining the ad-ho onstant aε in M2:2. This ee t has already been pointed out for M2:2 by Hansen (1994). Con erning omputational time, algorithm M2:1 and mv(δx∗ ) are the fastest and they dier merely due to that they use a dierent number of iterations to minimize the loss fun tion WN . Compared to M2:3, M2:1 is approximately 80 times faster a

ording to table 7.10. Estimation a

ura y for xi : Now the attention is turned to the estimation a

ura y for the individual parameters xi . Tables 7.11 and 7.12 show the individual parameter estimates and the orresponding mean true and nominal parameter deviations ε¯tx and ε¯# x at OP1 and OP8 respe tively for FP = 5 %. These parameter values are omputed as mean values for ten onse utive y les. The tables also entail the relative mean estimation error (RME) in per ent, as well as numeri al values for the true and (emphasized) nominal values. Both tables show that the estimates for C1 , TIV C , Tw , ∆p, ∆θ, and espe ially for Vcr and b, are regularized to the vi inity of their respe tive nominal value x# i , indi ated by an RME lose to 5 %. These seven parameters orresponds fairly well with the order in whi h the parameters are set spurious for method 1 when no preferred ordering is used, ompare (7.2). The dieren e is pIV C , otherwise it orresponds to the seven most spurious parameters. The estimates for the other parameters, i.e. Vc , pIV C , γ300 and Kp ends up lose to their true val-

:

[

℄ 58.4 -0.6 58.8 -0.0 60.6 3.1 60.3 2.6

Vc cm3

C1

TIV C

pIV C

γ300

b K −1

Tw

Vcr cm3

∆p

∆θ

Kp

ε¯tx

ε¯# x

δx∗

[-℄ [K℄ [kPa℄ [-℄ [ ℄ [K℄ [ ℄ [kPa℄ [deg℄ [-℄ [%℄ [%℄ 2.28 366 50.2 1.397 -0.000106 436 1.05 5.1 0.10 0.990 3.7 4.4 RME -0.1 -1.1 0.4 -0.2 6.0 8.9 5.0 1.3 2.9 -1.0 3.7 4.4 δx (M 2 : 1) 2.37 382 50.0 1.403 -0.000105 425 1.05 5.2 0.10 0.998 3.8 3.1 RME 4.0 3.3 0.1 0.2 5.0 6.2 5.0 4.6 4.7 -0.2 3.8 3.1 δx (M 2 : 2) 2.39 389 50.1 1.421 -0.000105 419 1.05 5.2 0.10 1.002 4.1 2.4 4.9 5.1 0.2 1.5 5.0 4.9 5.0 4.8 5.0 0.2 4.1 2.4 RME δx (M 2 : 3) 2.39 388 50.0 1.420 -0.000105 420 1.05 5.2 0.10 1.000 4.0 2.5 RME 4.8 5.0 0.1 1.4 5.0 5.0 5.0 4.8 4.9 0.0 4.0 2.5 x# 61.7 2.39 389 52.5 1.470 -0.000105 420 1.05 5.3 0.11 1.050 5.0 0.0 RME(x# ) 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 0.0 xt 58.8 2.28 370 50.0 1.400 -0.0001 400 1 5.0 0.10 1.000 0.0 5.0 Table 7.11: Parameter estimates and orresponding relative mean estimation error (RME) for OP1 at FP = 5 %, using method 2 and setup 1.

δx

164 CHAPTER 7. RESULTS AND EVALUATION MOTORED

...

:

Vc

[cm3 ℄ 58.8 0.0 58.5 -0.5 59.2 0.7 58.8 0.0

[-℄ 2.38 4.2 2.35 3.2 2.38 4.5 2.38 4.2

C1

[K℄ 320 3.2 313 0.9 323 4.1 320 3.2

TIV C

pIV C

[kPa℄ 100.0 0.0 100.1 0.1 100.0 -0.0 100.0 0.0

[-℄ 1.403 0.2 1.398 -0.1 1.408 0.6 1.403 0.2

γ300

b

[K −1℄ -0.000105 5.0 -0.000105 5.1 -0.000105 4.9 -0.000105 5.0

Tw

[K℄ 425 6.3 432 8.1 423 5.6 425 6.3

Vcr

∆p

∆θ

Kp

ε¯tx

ε¯# x

[cm3 ℄ [kPa℄ [deg℄ [-℄ [%℄ [%℄ ∗ δx 1.05 5.2 0.10 0.998 3.8 3.1 RME 5.0 4.8 4.6 -0.2 3.8 3.1 δx (M 2 : 1) 1.05 5.2 0.10 0.996 3.9 3.6 RME 5.0 4.6 4.0 -0.4 3.9 3.6 δx (M 2 : 2) 1.05 5.2 0.10 0.998 3.9 2.9 RME 5.0 4.9 4.8 -0.2 3.9 2.9 δx (M 2 : 3) 1.05 5.2 0.10 0.998 3.8 3.1 RME 5.0 4.8 4.6 -0.2 3.8 3.1 x# 61.7 2.39 326 105.0 1.470 -0.000105 420 1.05 5.3 0.11 1.050 5.0 0.0 RME(x# ) 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 0.0 xt 58.8 2.28 310 100.0 1.400 -0.0001 400 1 5.0 0.10 1.000 0.0 5.0 Table 7.12: Parameter estimates and orresponding relative mean estimation error (RME) for OP8 at FP = 5 %, using method 2 and setup 1.

δx

7.1. SIMULATION RESULTS  MOTORED CYCLES

165

166

CHAPTER 7. RESULTS AND EVALUATION MOTORED

...

ues espe ially for OP8. This is due to the higher pIV C , whi h results in a higher ylinder pressure and therefore a better ex itation of the parameters and a better signal-to-noise ratio. For both ases the estimates for pIV C has the overall smallest RME (< 0.4 %) losely followed by Kp , while the a

ura y for Vc and γ300 are within 3.1 % and 1.5 % respe tively for all versions. The estimates of the two latter parameters are signi antly better for OP8 ompared to OP1. The estimation a

ura y also depend upon the routine used to determine the regularization parameter δx . From table 7.10 it was found that routine M2:1 gave the smallest overall mean error, followed by M2:3 and M2:2. When onsidering the individual estimates of the four parameters, the same relative ordering is found if both operating points are onsidered. In the ase for M2:1, the a

ura y for the four parameters is high; The estimates are within 0.5 %. The other estimates end up as a ompromise in between the true value and the (false) nominal value, ex ept for Tw . In the ase of the two parameters Vcr and b, these two estimates oin ides with the nominal values. In the presen e of a false prior of 5 %, the estimates for method 2 are equally or more a

urate ompared to method 1 for all number of e ient parameters, as shown by omparing the ε¯tx - olumns in tables 7.4 and 7.11. This illustrates that method 2 is more robust to a false prior than method 1 and this results in better estimates.

Evaluation for setup 2 Method 2 is now evaluated for setup 2, where the nominal values are t given by x# i = (1 + σi )xi , for whi h the standard deviations are given in table 7.1. The assigned regularization elements are given for two 1 1

ases; The rst ase is δi = 2N , i.e. all elements are equal, and (0.01xt )2 i

in the se ond ase δi = 2N1σ2 . Therefore some of the plots and tables i based on the se ond ase from now on use the weighted versions of the nominal parameter deviation RMSE(Lδ ε# x ) and so on. The two ases are abbreviated as δi = c and δi 6= c respe tively, where c is a onstant. The L- urves based on one engine y le for ea h ase at OP1 are given in gures 7.9 and 7.10 respe tively. For ompleteness, the orresponding L- urves at OP8 are given in gures C.5 and C.6 in appendix C.7. The individual parameter estimates are given in tables 7.14 7.17, and orresponds to the form used in table 7.11. Tables 7.14 and 7.15 represent ase 1 (δi = c), while the orresponding tables for

ase 2 (δi 6= c) are given in tables 7.16 and 7.17. Note that the optimal regularization parameter δx∗ now is determined as the one minimizing the weighted true parameter deviation, i.e. RMSE(Lδ εtx ). The nominal and true parameter deviations ε¯tx and ε¯# x for both ases at OP1 and OP8 are then summarized in table 7.13.

167

7.1. SIMULATION RESULTS  MOTORED CYCLES Method 2; OP1

0

Zoomed version

10

−1

10

−1

10 −2

x

RMSE( ε# ) [−]

10

−3

10

−4

10

−2

10

−5

10

M2:3 M2:2 M2:1 Optimal

−6

10

−7

10

−2

10

−1

10 RMSE( ε ) [bar]

0

10

−2

10

−1

10 RMSE( ε ) [bar]

Figure 7.9: L- urve (solid line) for one engine y le at OP1 with false prior RMSE(ε#−t ) (dotted line), when using setup 2 and ase 1 of the δi x (δi = c). The results for the three versions of method 2 and the optimal

hoi e of regularization parameter are indi ated by the legend. The true parameter deviation RMSE(εtx ) (dash-dotted line) is also given. δi = c

δi 6= c

OP1 OP8 OP1 OP8 δx : ε¯tx ε¯# ε¯tx ε¯# ε¯tx ε¯# ε¯tx ε¯# x x x x [%℄ [%℄ [%℄ [%℄ [%℄ [%℄ [%℄ [%℄ δx∗ 14.1 22.8 14.9 18.2 6.1 20.3 6.5 20.1 δx (M 2 : 1) 22.1 1.4 22.1 0.7 6.2 20.7 6.6 20.9 δx (M 2 : 2) 20.6 5.8 20.5 5.7 10.5 27.6 10.7 27.3 δx (M 2 : 3) 22.1 0.9 22.1 0.3 9.4 18.9 9.4 18.4 x# 22.1 0.0 22.1 0.0 22.1 0.0 22.1 0.0 xt 0.0 22.1 0.0 22.1 0.0 22.1 0.0 22.1 Table 7.13: Mean true and nominal parameter deviation for method 2 using two ases of δi at OP1 and OP8 for setup 2.

168

CHAPTER 7. RESULTS AND EVALUATION MOTORED

−1

Method 2; OP1

10

−1

...

Zoomed version

10

−2

−3

10

δ

RMSE( L ε# ) [−] x

10

−4

10

−5

10

−6

10

M2:3 M2:2 M2:1 Optimal

−2

−1

10 RMSE( ε ) [bar]

10

−1

10 RMSE( ε ) [bar]

Figure 7.10: L- urve (solid line) for one engine y le at OP1 with weighted false prior RMSE(Lδ ε#−t ) (dotted line), when using setup 2 x and ase 2 of the δi (δi 6= c). The results for the three versions of method 2 and the optimal hoi e of regularization parameter are indi ated by the legend. The weighted true parameter deviation δ t RMSE(L εx ) (dash-dotted line) is also given. Evaluation for individual estimates xi : When onsidering the four parameters Vc , pIV C , γ300 and Kp , the estimates are again better for the se ond ase as shown in tables 7.147.17. It is also notable that the estimates for the ase of δi 6= c do not ne essarily end up in between the true and nominal value, see for instan e the negative RME:s for pIV C and ∆θ in table 7.16. This ree ts a more exible solution, than the one for the ase of δi = c. Evaluation for all OP:s. Table 7.13 shows that the mean true parameter deviation is smaller when δi 6= c than for δi = c, for all three versions of method 2 and for the optimal hoi e of δx . It is also worth to mention that the mean nominal parameter deviation ε¯# x is relatively small for δi = c and signi antly higher for δi 6= c. This together with the smaller ε¯tx ree ts that the se ond ase is more robust to a false

7.1. SIMULATION RESULTS  MOTORED CYCLES

169

prior, whi h of ourse is preferable. Now the fo us is turned to nding whi h version of method 2 that performs the best. In the ase of δi = c both M2:1 and M2:3 are restri tive and yield estimates lose to the nominal values. This is, as pointed out previously, ree ted by a relatively low value of ε¯# x in table 7.13. M2:2 a tually performs better than these two, but the dieren e is small and the estimation bias is still signi ant as ree ted by the ε¯tx - olumns. In the ase of δi 6= c, all three versions render weighted parameter deviations RMSE(Lδ εtx ) and RMSE(Lδ ε# x ) that are virtually the same, see tables 7.16 and 7.17. The only dieren e is in RMSE(Lδ ε# x ) for M2:2, whi h is due to the relatively large error in ∆θ. However when

onsidering ε¯tx for all investigated operating points, it is lowest for M2:1 and fairly lose to the optimal, followed by M2:3 at most instan es as shown in table 7.13. If omputational time is ru ial and therefore needs to be taken into a

ount, M2:1 is outstanding as shown earlier in table 7.13 and would therefore unequivo ally be the rst hoi e. 7.1.5

Summary for simulation results

Method 1 resulted in parameter estimates that are signi antly biased in the presen e of a false prior, sin e the bias is larger than the false prior level used. It is therefore in itself not re ommended to use for estimation of all parameters in the given formulation. For a spe i appli ation su h as for example ompression ratio estimation it an however serve as a guideline of how many parameters to use. Method 2 outperforms method 1 sin e it is more robust to a false prior level and yields more a

urate parameter estimates. The drive for method 2 was to regularize the solution su h that the parameters that are hard to determine are pulled towards their nominal values, while the e ient parameters are free to t the data. This has shown to be the ase in the simulations. Method 2 is systemati and a

urate, and therefore fullls two of the requirements for the estimation tool. The user an hose between two ases of the parameter un ertainty, namely δi = c and δi 6= c. The former is dire tly appli able on e nominal values of the parameters are determined and yields good estimates. However, they an be too restri ted by the nominal values, as seen in setup 2. Instead it is re ommended to use the se ond ase. It requires more eort to de ide upon the un ertainty for ea h parameter, but pays o in better estimates that are more robust to a false nominal parameter value. Method 2 is given in three versions, and the version M2:1 followed by M2:3 give the most a

urate results for hanging operating onditions, false prior levels and noise realizations. M2:1 is also omputationally e ient and outstanding ompared to the other versions. It

:

[ ℄ 58.8 -0.0 58.7 -0.1 59.2 0.8 59.3 0.9

Vc cm3

[-℄ 1.48 -35.2 2.39 4.9 2.16 -5.3 2.39 5.0

C1

pIV C

γ300

b K −1

Tw

Vcr cm3

[K℄ [kPa℄ [-℄ [ ℄ [K℄ [ ℄ 380 52.0 1.411 -0.000118 382 1.1 2.8 4.0 0.8 17.6 -4.5 9.7 379 50.5 1.404 -0.00011 410 1.15 2.5 0.9 0.3 10.0 2.6 15.0 338 50.5 1.410 -0.000112 436 1.15 -8.7 0.9 0.7 11.8 9.0 15.0 380 51.0 1.398 -0.00011 410 1.15 2.7 2.1 -0.2 10.0 2.5 15.0

TIV C

[kPa℄ 6.1 21.2 7.5 49.9 7.4 47.4 7.5 50.0

∆p

[deg℄ 0.09 -5.5 0.15 49.9 0.14 43.2 0.15 50.0

∆θ

[-℄ 0.950 -5.0 0.989 -1.1 0.964 -3.6 1.001 0.1

Kp

ε¯tx

ε¯# x

δx∗

[%℄ [%℄ 14.1 22.8 RME 14.1 22.8 δx (M 2 : 1) 22.1 1.4 RME 22.1 1.4 δx (M 2 : 2) 20.6 5.8 20.6 5.8 RME δx (M 2 : 3) 22.1 0.9 RME 22.1 0.9 x# 59.3 2.39 380 52.5 1.405 -0.00011 410 1.15 7.5 0.15 1.005 22.1 0.0 RME(x# ) 0.9 5.0 2.7 5.0 0.4 10.0 2.5 15.0 50.0 50.0 0.5 22.1 0.0 xt 58.8 2.28 370 50.0 1.400 -0.0001 400 1 5.0 0.10 1.000 0.0 22.1 Table 7.14: Parameter estimates and orresponding relative mean estimation error (RME) for OP1 when δi = c, using method 2 and setup 2.

δx

170 CHAPTER 7. RESULTS AND EVALUATION MOTORED

...

:

Vc

[cm3℄ 59.1 0.6 58.7 -0.1 59.2 0.8 59.3 0.9

[-℄ 1.56 -31.7 2.39 4.9 2.18 -4.3 2.39 5.0

C1

[K℄ 296 -4.6 319 3.0 283 -8.8 320 3.3

TIV C

pIV C

[kPa℄ 103.6 3.6 100.9 0.9 100.8 0.8 102.0 2.0

[-℄ 1.409 0.7 1.404 0.3 1.410 0.7 1.397 -0.2

γ300

b

[K −1℄ -0.000115 15.3 -0.00011 10.0 -0.000111 11.4 -0.00011 10.0

Tw

[K℄ 404 1.0 411 2.6 438 9.4 410 2.5

Vcr

[cm3℄ 1.12 12.4 1.15 15.0 1.15 15.0 1.15 15.0

∆p

[kPa℄ 6.5 30.2 7.5 49.9 7.4 47.4 7.5 50.0

∆θ

[deg℄ 0.11 8.0 0.15 49.9 0.14 42.9 0.15 50.0

[-℄ 0.949 -5.1 0.989 -1.1 0.964 -3.6 1.000 0.0

Kp

ε¯tx

ε¯# x

[%℄ [%℄ ∗ δx 14.9 18.2 RME 14.9 18.2 δx (M 2 : 1) 22.1 0.7 RME 22.1 0.7 δx (M 2 : 2) 20.5 5.7 RME 20.5 5.7 δx (M 2 : 3) 22.1 0.3 RME 22.1 0.3 x# 59.3 2.39 320 102.5 1.405 -0.00011 410 1.15 7.5 0.15 1.005 22.1 0.0 RME(x# ) 0.9 5.0 3.2 2.5 0.4 10.0 2.5 15.0 50.0 50.0 0.5 22.1 0.0 xt 58.8 2.28 310 100.0 1.400 -0.0001 400 1 5.0 0.10 1.000 0.0 22.1 Table 7.15: Parameter estimates and orresponding relative mean estimation error (RME) for OP8 when δi = c, using method 2 and setup 2.

δx

7.1. SIMULATION RESULTS  MOTORED CYCLES

171

:

[ ℄ 59.0 0.4 59.0 0.4 59.0 0.3 59.1 0.6

Vc cm3

[-℄ 2.37 3.8 2.36 3.7 2.35 3.1 2.36 3.7

C1

pIV C

γ300

b K −1

Tw

Vcr cm3

∆p

∆θ

Kp

[K℄ [kPa℄ [-℄ [ ℄ [K℄ [ ℄ [kPa℄ [deg℄ [-℄ 377 49.9 1.405 -0.000108 411 1.15 5.5 0.10 1.004 2.0 -0.3 0.4 8.5 2.8 15.0 9.0 -3.1 0.4 377 49.8 1.405 -0.000108 411 1.15 5.4 0.10 1.004 2.0 -0.3 0.4 8.5 2.8 15.0 8.8 -4.5 0.4 376 49.8 1.405 -0.000108 412 1.15 5.3 0.07 1.004 1.6 -0.5 0.4 8.4 3.0 15.0 5.3 -29.6 0.4 377 50.0 1.405 -0.000109 412 1.15 6.2 0.09 1.004 1.9 -0.1 0.3 9.4 3.0 15.0 24.2 -7.0 0.4

TIV C

Lδ εtx

δx∗

[%℄ 3.1 RME 3.1 δx (M 2 : 1) 3.1 RME 3.1 δx (M 2 : 2) 3.1 3.1 RME δx (M 2 : 3) 3.3 RME 3.3 x# 59.3 2.39 380 52.5 1.405 -0.00011 410 1.15 7.5 0.15 1.005 4.8 RME(x# ) 0.9 5.0 2.7 5.0 0.4 10.0 2.5 15.0 50.0 50.0 0.5 4.8 xt 58.8 2.28 370 50.0 1.400 -0.0001 400 1 5.0 0.10 1.000 0.0 Table 7.16: Parameter estimates and orresponding relative mean estimation error (RME) for OP1 when δi method 2 and setup 2.

δx

[%℄ 3.3 3.3 3.3 3.3 3.8 3.8 3.1 3.1 0.0 0.0 4.8 6 c, using =

L δ ε# x

172 CHAPTER 7. RESULTS AND EVALUATION MOTORED

...

:

Vc

[cm3℄ 59.0 0.5 59.0 0.5 59.0 0.4 59.1 0.6

[-℄ 2.37 3.9 2.37 3.9 2.36 3.4 2.37 4.0

C1

[K℄ 317 2.3 317 2.2 315 1.8 317 2.3

TIV C

pIV C

[kPa℄ 99.6 -0.4 99.6 -0.4 99.4 -0.6 99.9 -0.1

[-℄ 1.405 0.4 1.405 0.4 1.405 0.4 1.404 0.3

γ300

b

[K −1℄ -0.000109 8.8 -0.000109 8.8 -0.000109 8.7 -0.000109 9.5

Tw

[K℄ 411 2.8 411 2.9 412 3.1 412 2.9

Vcr

[cm3 ℄ 1.15 15.0 1.15 15.0 1.15 15.0 1.15 15.0

∆p

[kPa℄ 5.5 10.6 5.5 10.2 5.4 7.3 6.2 24.3

∆θ

[deg℄ 0.10 -3.8 0.09 -6.7 0.07 -29.6 0.09 -5.4

[-℄ 1.004 0.4 1.004 0.4 1.004 0.4 1.004 0.4

Kp

Lδ εtx

L δ ε# x

[%℄ [%℄ ∗ δx 3.2 2.4 RME 3.2 2.4 δx (M 2 : 1) 3.2 2.5 RME 3.2 2.5 δx (M 2 : 2) 3.2 3.0 RME 3.2 3.0 δx (M 2 : 3) 3.3 2.2 RME 3.3 2.2 x# 59.3 2.39 320 102.5 1.405 -0.00011 410 1.15 7.5 0.15 1.005 4.2 0.0 RME(x# ) 0.9 5.0 3.2 2.5 0.4 10.0 2.5 15.0 50.0 50.0 0.5 4.2 0.0 xt 58.8 2.28 310 100.0 1.400 -0.0001 400 1 5.0 0.10 1.000 0.0 4.2 Table 7.17: Parameter estimates and orresponding relative mean estimation error (RME) for OP8 when δi 6= c, using method 2 and setup 2.

δx

7.1. SIMULATION RESULTS  MOTORED CYCLES

173

174

CHAPTER 7. RESULTS AND EVALUATION MOTORED

...

Experimental motored data; OP1−4 Pressure [bar]

10 OP1 5 OP2 0

−100

−50

0 50 Crank angle [deg ATDC]

100

Figure 7.11: Experimentally measured motored ylinder pressures at operating onditions OP14.

an therefore be stated that the third requirement is also fullled. For the simulated motored y les, M2:1 is therefore re ommended as the best hoi e.

7.2 Experimental results  motored y les The attention is now turned to the issue of evaluating the methods on experimental engine data. As mentioned before, the true model stru ture and parameter values are unknown. Therefore it is important too see if the ee ts and trends from the simulation-based evaluation are also present when the methods are applied to experimental data. First the experimental engine data will be des ribed, followed by a dis ussion on the parameter prior knowledge used. The fo us is then turned to evaluating the performan e of method 1, followed by method 2.

7.2.1

Experimental engine data

Data is olle ted during stationary operation at engine speeds N ∈ [2000, 5000] rpm, intake manifold pressures pman ∈ [32, 43} kPa altogether forming four dierent operating points. These operating points are dened in the upper part of table 7.18. For ea h operating point 101 onse utive motored y les with the fuel inje tion shut-o were sampled for two ylinders with a rank-angle resolution of 1 degree, using an AVL GU21D ylinder pressure sensor. Figure 7.11 displays one measured y le for ea h operating point.

7.2. EXPERIMENTAL RESULTS  MOTORED CYCLES

175

7.2.2 Parameter prior knowledge The parameter prior knowledge used in the experimental evaluation is based on dierent strategies to set the nominal parameter values x# and the regularization elements δi .

Nominal parameter values x

#

The nominal parameter values x# are set based on a ombination of the parameter initial methods des ribed in hapter 3 and expert prior knowledge of what the numeri al value of x# should be. The values for x# for ea h operating point are given as the emphasized values in table 7.19. Of these parameters it is only the nominal value for ∆p that is updated for ea h y le, while the others are updated for ea h operating ondition.

Regularization elements δ

i

Two ases of regularization parameter elements are used in method 2 and they orrespond to the two ases used in the simulation-based evaluation done previously. The rst ase, again denoted δi = c, sets the standard deviation σi for ea h parameter as σi = 0.01¯x# i , where # x ¯# is the mean value of x for one operating point. It is used to i i assure that σi does not u tuate in size from y le-to- y le. In the simulation-based evaluation xt was used instead of x¯# . The se ond ase (δi 6= c) is based on expert knowledge of the un ertainty for ea h parameter and it is therefore subje tively hosen by the user. The standard deviation used here in the experimental evaluation oin ides with the one used for the simulation-based evaluation in table 7.1. The rst ase is updated for ea h operating point, while this is not required for the se ond ase.

7.2.3 Method 1 Results and evaluation Method 1 is evaluated only without a preferred ordering, sin e no gain was found in the simulation-based evaluation by using one. The parameter order for all four operating points is given by C1 ≺ Kp ≺ ∆θ ≺ TIV C ≺ Vcr ≺ ∆p ≺ b ≺ Tw ≺ Vc ≺ pIV C ≺ γ300 ,

(7.5) whi h ree ts the average ase. This parameter order diers somewhat from the orresponding order found in simulations (7.2), but γ300 is still the most e ient parameter. The underlined parameters in (7.5) are invariant in position for the investigated operating onditions y le-to y le variations. The other parameters dier in order as permutations

176

CHAPTER 7. RESULTS AND EVALUATION MOTORED

N [rpm℄ pman [kPa℄ mean(d∗ ) d∗ ∈

OP1 2000 43 9.5 [9, 10]

OP2 3000 32 10.3 [9, 11]

OP3 4000 35 10.4 [9, 11]

...

OP4 5000 37 10.4 [9, 11]

Table 7.18: Mean number and region of d∗ for 101 experimental y les at OP14.

of the three groups [TIV C , Tw ], [∆p, b] and [Kp , C1 ]. These groups are not exa tly the same as the ones for the simulation-based evaluation. The minimizing number of parameters d∗ using algorithm 6.2 are given in table 7.18 as a mean over the 101 y les at all four operating points. The range of d∗ for ea h operating point is also given. The table shows that the mean values of d∗ are larger and the variation is smaller in the experimental ase, ompared to the simulations given in table 7.2. The mean value, standard deviation, and relative mean error for the individual estimates are given in table 7.19. The relative mean error is omputed relative to the nominal parameter value at ea h y le and is given in per ent. All parameters give reasonable values ex ept for C1 , TIV C , and Tw that yield unreasonably large parameter values. In a

ordan e with the simulation-based evaluation in se tion 7.1.3, method 1 is not re ommended to use for parameter estimation in the given formulation. The attention is therefore turned to method 2.

7.2.4

Method 2 Results and evaluation

The same implementation settings, as for the simulation-based evaluation in se tion 7.1.4, are used and are therefore not repeated here. Results for ase 1 (δi = c) and ase 2 (δi 6= c)

The results are rst des ribed for ase 1. An example of an L- urve is given in gure 7.12. A fourth version, M2:3+, is also in luded in the gure. The algorithm is motivated and fully des ribed in appendix C.3. This version is an extension of M2:3, and uses a smaller sear h region ∆x of δx as ompared to the original form. Version M2:3+ uses the regularization parameter δx from M2:1 as a mid-value of the sear h region ∆x , for a ∆x given by 25 samples of δx from gure 7.5. The usage of M2:3+ is only needed for experimental y les and is due to problems o

urring when sear hing for the maximum urvature of the L- urve. For simulated y les this problem has not o

urred, and therefore results in the same value of δx for M2:3 and M2:3+.

mean σest RME x# mean σest RME x# mean σest RME x# mean σest RME x# 5.04 1.63 120.8 2.28

58.8

2.28

58.8

58.1 1.0 -1.1

6.34 1.71 178.1

2.28

58.8

60.8 1.0 3.5

5.94 2.07 160.5

2.28

60.1 1.1 2.3

58.8

314

454 184 44.5

314

403 170 28.3

314

474 187 50.8

314

TIV C [K℄ 588 220 87.1

48.4

43.8 1.7 -23.2

47.3

41.3 2.2 -27.5

42.5

37.4 1.8 -34.4

56.4

pIV C [kPa℄ 54.7 0.4 -4.1

1.400

1.431 0.019 6.1

1.400

1.429 0.010 5.9

1.400

1.422 0.011 5.4

1.400

γ300 [-℄ 1.369 0.007 1.5

b [K −1 ℄ −4.39 · 10−6 5.82 · 10−6 -96.6 −9.3 · 105 −1.11 · 10−5 1.48 · 10−5 -91.5 −9.3 · 10−5 −1.1 · 10−5 1.63 · 10−5 -91.5 −9.3 · 10−5 −5.03 · 10−5 4.39 · 10−5 -61.3 −9.3 · 10−5 400

562 200 33.9

400

526 186 25.3

400

621 209 47.9

400

Tw [K℄ 690 230 64.4

0.882

0.859 0.318 46.1

0.882

0.448 0.352 -23.7

0.882

1.2 0.524 103.9

0.882

Vcr [cm3 ℄ 1.12 0.755 89.8

-1.2

2.2 0.9 -199.0

-0.6

3.4 0.8 -257.1

-1.6

1.7 0.6 -177.2

-1.4

∆p [kPa℄ -0.3 0.5 -88.3

0.47

0.11 0.05 -77.3

0.47

0.13 0.06 -71.7

0.47

0.14 0.08 -70.1

0.47

∆θ [deg℄ 0.32 0.13 -31.7

1.000

1.020 0.032 2.0

1.000

1.041 0.058 4.1

1.000

1.043 0.049 4.3

1.000

Kp [-℄ 1.000 0.000 0.0

ε¯# x [%℄ 92.5 92.5 92.5 0.0 99.8 99.8 99.8 0.0 108.4 108.4 108.4 0.0 88.1 88.1 88.1 0.0

Table 7.19: Mean value, standard deviation σest , relative mean error (RME) and nominal parameter deviation ε¯# x of the estimate for method 1, evaluated at d∗ for 101 experimental y les at OP14.

OP4

OP3

OP2

OP1

C1 [-℄ 5.03 2.98 120.5

Vc [cm3 ℄ 57.8 1.6 -1.6

7.2. EXPERIMENTAL RESULTS  MOTORED CYCLES 177

178

CHAPTER 7. RESULTS AND EVALUATION MOTORED

Experimental OP1: δi=c

2

...

Zoomed version

10

0

10

0

−2

10

−1

x

RMSE(ε# ) [−]

10

10 −4

10

−6

10

M2:3 M2:3+ M2:2 M2:1

−8

10

−3

10

−2

10 −2

−1

10 10 RMSE( ε ) [bar]

0

10

−4

10

−2

0

10 10 RMSE( ε ) [bar]

Figure 7.12: L- urve (solid line) for one engine y le at OP1, when using ase 1 of the δi (δi = c). The results for the four versions of method 2 are indi ated by the legend.

N [rpm℄ pman [kPa℄ δx (M 2 : 1) mean σest δx (M 2 : 2) mean σest δx (M 2 : 3) mean σest δx (M 2 : 3+) mean σest

OP1 2000 43 3.8 · 10−5 6.3 · 10−5 2.3 · 10−8 1.9 · 10−8 4.2 · 10−8 1.5 · 10−8 5.1 · 10−5 1.6 · 10−5

OP2 3000 32 4.1 · 10−5 3.9 · 10−5 7.5 · 10−8 8.1 · 10−8 3.9 · 10−8 3.4 · 10−8 9.2 · 10−5 4.3 · 10−6

OP3 4000 35 6.5 · 10−5 1.5 · 10−6 9.6 · 10−8 6.6 · 10−8 0.4 · 10−8 0.8 · 10−8 9.0 · 10−5 5.8 · 10−5

OP4 5000 37 4.2 · 10−5 4.8 · 10−5 6.3 · 10−8 4.3 · 10−8 2.1 · 10−8 4.1 · 10−8 3.4 · 10−5 3.4 · 10−5

Table 7.20: Mean value and standard deviation (σest ) of δx for the four versions of method 2 using ase 1, evaluated for 101 experimental y les at OP14.

7.2. EXPERIMENTAL RESULTS  MOTORED CYCLES

179

Table 7.20 summarizes the mean value and standard deviation of the omputed regularization parameter δx for 101 experimental y les. This is done for all four operating points and for ea h of the four versions of method 2. The orresponding individual parameter estimates are given as mean values for 101 y les at ea h operating point in table 7.21, as well as the orresponding (emphasized) nominal values x# and the nominal parameter deviation ε¯# x . More detailed results for OP1 are given in table C.6, where also the standard deviation and the relative mean error with respe t to the nominal parameters x# are given. The orresponding gures and tables for ase 2 are given in gure 7.13, table 7.22, table 7.23 that now also in lude the weighted nominal parameter deviation RMSE(Lδ ε# x ), and table C.7 respe tively. Evaluation for ase 1 (δi = c)

Case 1 is onsidered rst. Figure 7.12 illustrates that the L- urve does not have as sharp transition as in the orresponding simulated ase given in gure 7.9. The dieren e is believed to be due to that the model stru ture is not orre t. Figure 7.12 also shows that the four versions yield dierent regularization parameters δx . The two versions M2:2 and M2:3 render approximately the same δx , whi h is onrmed in table 7.20 for all operating points. M2:1 and M2:3+ result in higher δx :s in the mean, and therefore relies more on the prior knowledge

ompared to M2:2 and M2:3. For all four operating points table 7.20 shows that the regularization parameter for the four versions are ordered in size a

ording to δx (M 2 : 3+) > δx (M 2 : 1) > δx (M 2 : 2) > δx (M 2 : 3),

(7.6)

with two ex eptions. For OP1 it is the other way around for M2:2 and M2:3, and the same goes for M2:1 and M2:3+ at OP4. As expe ted the dieren e between δx (M 2 : 1) and δx (M 2 : 3+) is smaller than between δx (M 2 : 1) and δx (M 2 : 3). Unlike M2:1, M2:3+ is assured to nd a

onvex orner of the L- urve. For the investigated operating points, M2:3+ gives a higher value of δx , than for M2:1, M2:2 and M2:3. For M2:3 this orresponds to the high value of the left-most urvature found in gure C.4. Table 7.20 also shows that the mean value of δx depends upon operating ondition, and that for a given version of method 2 δx is in the same order of magnitude for all four operating points. The dierent δx :s result in dierent nominal parameter deviations and individual estimates, as shown in table 7.21 for OP1-4. In general M2:2 and M2:3 yield similar parameter estimates, espe ially for OP1. This is due to that δx is approximately the same. These estimates are however unreasonably large, see e.g. TIV C , Tw and Kp, while b is unreasonably small. All parameters ex ept Vc and γ300 deviate signi antly

:

[ ℄ 58.0 59.0 59.1 57.8 58.8 59.1 60.1 60.9 58.9 58.8 58.2 58.8 60.7 58.2 58.8 57.4 57.7 58.5 57.3

Vc cm3

[-℄ 2.70 6.27 6.54 2.70 2.28 2.67 5.64 5.77 2.61 2.28 2.60 5.96 5.81 2.58 2.28 2.61 4.67 4.82 2.62

C1

pIV C

γ300

[K℄ [kPa℄ [-℄ 358 53.8 1.396 550 34.1 1.370 567 32.8 1.371 357 53.8 1.393 314 56.4 1.400 347 39.6 1.420 502 27.4 1.402 501 27.0 1.412 347 39.8 1.414 314 42.5 1.400 344 43.8 1.420 523 29.1 1.392 500 29.8 1.418 345 43.9 1.418 314 47.3 1.400 350 45.3 1.422 448 36.3 1.406 456 36.1 1.426 352 45.3 1.420

TIV C

[

℄

b K −1 −8.8 · 10−5 −3.0 · 10−7 −2.1 · 10−7 −8.7 · 10−5 −9.3 · 10−5 −7.9 · 10−5 −1.4 · 10−6 −8.4 · 10−7 −8.1 · 10−5 −9.3 · 10−5 −7.6 · 10−5 −3.4 · 10−7 −8.1 · 10−7 −7.6 · 10−5 −9.3 · 10−5 −7.3 · 10−5 −1.3 · 10−5 −2.9 · 10−5 −7.2 · 10−5 −9.3 · 10−5

[K℄ 370 712 741 373 400 374 663 677 372 400 374 696 687 373 400 367 551 578 363

Tw

Vcr cm3

∆p

∆θ

Kp

ε¯# x

OP1

δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x# δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x# δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x# δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x#

[ ℄ [kPa℄ [deg℄ [-℄ [%℄ 0.883 1.1 0.45 0.998 5.8 0.64 -0.4 0.15 1.596 103.9 0.597 -0.3 0.11 1.658 73.5 0.882 1.3 0.47 0.999 0.8 0.882 -1.4 0.47 1.000 0.0 OP2 0.882 1.4 0.39 0.999 3.4 0.909 0.4 0.08 1.460 82.3 0.73 1.0 0.06 1.457 97.6 0.882 1.2 0.47 1.000 0.6 0.882 -1.6 0.47 1.000 0.0 OP3 0.882 2.9 0.37 0.998 2.4 0.782 1.1 0.07 1.534 83.3 0.382 2.7 0.06 1.449 137.1 0.882 2.8 0.47 1.000 2.6 0.882 -0.6 0.47 1.000 0.0 OP4 0.882 1.7 0.25 0.999 3.3 1.03 0.6 0.06 1.272 65.5 0.787 1.6 0.03 1.253 87.0 0.884 1.7 0.47 1.001 11.4 58.8 2.28 314 48.4 1.400 400 0.882 -1.2 0.47 1.000 0.0 Table 7.21: Mean value of the estimate for the four versions of method 2 using ase 1, evaluated for 101 experimental y les at OP1-4. The nominal values x# are also in luded.

δx

180 CHAPTER 7. RESULTS AND EVALUATION MOTORED

...

7.2. EXPERIMENTAL RESULTS  MOTORED CYCLES

181

from their respe tive nominal value. This is ree ted in the relatively large nominal parameter deviation ε¯# x , and is due to a small δx . The individual estimates for M2:1 and M2:3+ are all reasonable, whi h is expe ted sin e they end up lose to their nominal values. The estimates are in general loser to x# for all parameters (ex ept for Vc and γ300 ), than for M2:2 and M2:3. Out of M2:1 and M2:3+, the latter has a smaller nominal parameter deviation ε¯# x for all operating points ex ept OP4. This orresponds to the ordering given in (7.6). The maximum deviation o

urs for the parameters C1 , TIV C and Tw for both versions. To summarize the evaluation for δi = c, using M2:1 or M2:3+ yield estimates that deviate less than 6 and 11 per ent from the nominal values in the mean. Evaluation for ase 2 (δi 6= c)

Now the se ond ase will be evaluated. Figure 7.13 displays that the four versions yield approximately the same δx , whi h is onrmed by table 7.22 for OP1-4. In general, the mean values and standard deviations of the regularization parameter for the three versions M2:1, M2:3 and M2:3+ are basi ally the same for all operating points, as shown in table 7.22. The reason that M2:3 and M2:3+ do not oin ide is that the former nds a orner at a smaller δx in some instan es. The se ond version (M2:2) yields a δx that has a lower mean value than the other three, whi h ree ts a higher onden e in the data. Con erning the individual parameter estimates in table 7.23, all four versions render estimates that are reasonable. An ex eption is the Wos hni heat transfer oe ient C1 , whi h an be onsidered to be too large for M2:2. A trend for all versions is that the estimated pIV C is smaller than the nominal pIV C , whi h is partly ompensated by ∆p and TIV C that are larger than their nominal values. As for the simulation-based evaluation the parameters Vcr , Kp , and b are attra ted to the vi inity of their nominal values, while others like Vc and γ300 are adjusted to the data. Out of the four versions, M2:3+ losely followed by M2:1 yields the smallest weighted nominal parameter deviations for all OP:s, as indi ated by table 7.23. Compared to ase 1, the estimates for M2:2 and M2:3 are mu h

loser to their nominal values, whi h is ree ted by the ε¯# x - olumns in tables 7.21 and 7.23. For M2:1 and M2:3+ it is the other way around. This is mostly due to the ∆p-estimate, whi h yields a large normalized nominal deviation and therefore ontributes very mu h to ε¯# x. To summarize the evaluation for δi 6= c, using M2:1 or M2:3+ yield weighted nominal parameter deviations that are less than 15 per ent in both ases.

182

CHAPTER 7. RESULTS AND EVALUATION MOTORED

Experimental OP1: δi ~= c

2

...

Zoomed version

10

−0.6

10 0

δ # x

RMSE(L ε ) [−]

10

−0.7

10 −2

10

−0.8

10

−4

10

M2:3 M2:3+ M2:2 M2:1

−6

10

−3

10

−0.9

−2

−1

10 10 RMSE( ε ) [bar]

0

10

10

−4

10

−2

0

10 10 RMSE( ε ) [bar]

Figure 7.13: L- urve (solid line) for one engine y le at OP1, when using ase 2 of the δi (δi 6= c). The results for the four versions of method 2 are indi ated by the legend.

N [rpm℄ pman [kPa℄ δx (M 2 : 1) mean σest δx (M 2 : 2) mean σest δx (M 2 : 3) mean σest δx (M 2 : 3+) mean σest

OP1 2000 43 3.3 · 10−5 1.1 · 10−5 7.8 · 10−6 3.7 · 10−5 2.8 · 10−5 2.0 · 10−5 3.4 · 10−5 2.1 · 10−5

OP2 3000 32 3.5 · 10−5 9.6 · 10−6 4.4 · 10−6 2.7 · 10−5 5.9 · 10−5 5.1 · 10−5 6.3 · 10−5 2.1 · 10−5

OP3 4000 35 5.5 · 10−5 2.6 · 10−5 3.0 · 10−5 1.0 · 10−4 5.6 · 10−5 2.7 · 10−5 6.6 · 10−5 2.6 · 10−5

OP4 5000 37 3.6 · 10−5 1.3 · 10−5 2.1 · 10−5 8.1 · 10−5 3.2 · 10−5 2.1 · 10−5 3.5 · 10−5 2.0 · 10−5

Table 7.22: Mean value and standard deviation (σest ) of δx for the four versions of method 2 using ase 2, evaluated for 101 experimental y les at OP14.

δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x# δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x# δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x# δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x#

:

58.8

Vc

[cm3℄ 57.7 57.5 57.7 57.7 58.8 59.1 59.7 58.9 58.9 58.8 58.2 58.4 58.2 58.1 58.8 57.4 57.4 57.4 57.4 2.28

[-℄ 2.73 3.54 2.72 2.72 2.28 2.68 3.87 2.63 2.63 2.28 2.61 3.04 2.60 2.60 2.28 2.63 3.07 2.62 2.62

C1

314

48.4

pIV C

1.400

γ300

[K℄ [kPa℄ [-℄ 359 53.7 1.394 368 53.8 1.389 359 53.7 1.394 359 53.7 1.394 314 56.4 1.400 348 39.5 1.420 353 38.7 1.432 349 39.7 1.416 349 39.7 1.416 314 42.5 1.400 345 43.7 1.420 346 43.4 1.424 345 43.8 1.420 345 43.8 1.419 314 47.3 1.400 351 45.2 1.422 352 45.0 1.425 351 45.2 1.422 351 45.2 1.422

TIV C −8.7 · 10−5 −8.8 · 10−5 −8.7 · 10−5 −8.7 · 10−5 −9.3 · 10−5 −7.9 · 10−5 −7.8 · 10−5 −8 · 10−5 −8 · 10−5 −9.3 · 10−5 −7.5 · 10−5 −7.7 · 10−5 −7.5 · 10−5 −7.5 · 10−5 −9.3 · 10−5 −7.2 · 10−5 −7.7 · 10−5 −7.2 · 10−5 −7.2 · 10−5 −9.3 · 10−5

b

[K −1℄

400

Tw

[K℄ 372 406 371 371 400 374 426 371 371 400 373 392 373 372 400 367 387 366 367

Vcr

∆p

∆θ

Kp

L δ ε# x

ε¯# x

[cm3℄ [kPa℄ [deg℄ [-℄ [%℄ [%℄ 0.882 1.4 0.49 1.000 14.7 34.4 0.882 1.0 0.46 1.002 21.0 36.7 0.882 1.4 0.49 1.000 15.8 35.4 0.882 1.4 0.49 1.000 14.7 34.5 0.882 -1.4 0.47 1.000 0.0 0.0 0.882 1.6 0.41 1.000 13.1 31.6 0.882 2.1 0.33 1.003 23.3 38.3 0.882 1.4 0.42 1.000 13.1 31.1 0.882 1.3 0.42 1.000 12.7 30.9 0.882 -1.6 0.47 1.000 0.0 0.0 0.882 3.0 0.39 1.000 12.7 33.0 0.882 3.2 0.36 1.001 16.0 35.1 0.882 3.0 0.39 1.000 12.9 33.0 0.882 3.0 0.39 1.000 12.6 32.6 0.882 -0.6 0.47 1.000 0.0 0.0 0.882 1.9 0.26 1.000 12.0 25.3 0.882 2.0 0.24 1.001 15.7 28.2 0.882 1.8 0.26 1.000 12.3 25.5 0.882 1.8 0.26 1.000 12.0 25.2 0.882 -1.2 0.47 1.000 0.0 0.0

Table 7.23: Mean value of the estimate for the four versions of method 2 using ase 2, evaluated for 101 experimental y les at OP1-4. The nominal values x# are also in luded.

OP4

OP3

OP2

OP1

δx

7.2. EXPERIMENTAL RESULTS  MOTORED CYCLES

183

184

CHAPTER 7. RESULTS AND EVALUATION MOTORED

...

Summary of method 2 In the simulations the gain in hoosing ase 2 instead of ase 1 was not distin t. However in the experimental situation the gain in using

ase 2 is stronger, espe ially for M2:2 and M2:3 sin e all estimates are physi ally reasonable. It has also been shown that M2:1 and M2:3+ give more reasonable estimates, ompared to M2:2 and M2:3. Version M2:3+ is more exible to hanges in operating ondition than M2:1, sin e it nds a onvex orner of the L- urve. For motored y les, this exibility has not been shown to be ne essary for the examined operating points. To on lude, the argumentation speaks in favor of using version M2:1 and ase 2 of δi for motored y les. 7.3

Summary of results for motored y les

Both the simulation and experimental studies showed that method 1 did not render a

urate estimates. It was even hard to de ide upon the number of e ient parameters to use. However, method 1 allows for the parameters to be ordered in how e ient they are for the given estimation problem and data. It was also shown that γ300 is the most e ient parameter. Method 2 outperforms method 1 sin e it is more robust to a false prior level and yields more a

urate parameter estimates, a

ording to the simulation-based evaluation. In the experimental situation method 2 was found to give reasonable estimates for all parameters. The drive for method 2 was to regularize the solution su h that the parameters that are hard to determine are pulled towards their nominal values, while the e ient parameters are free to t the data. This has shown to be the ase in the simulations. The user an hose between two ases of the parameter un ertainty, namely δi = c and δi 6= c. The former is dire tly appli able on e nominal values of the parameters are determined and yields good estimates, almost as good as for the se ond ase when onsidering only the simulations. However, when also onsidering the experimental evaluation the gain in using the se ond ase is stronger. Thus it is re ommended to use the se ond ase. It requires more eort to de ide upon the un ertainty for ea h parameter, but pays o in better estimates that are more robust to a false nominal parameter value. Method 2 was originally given in three variants, and variant M2:1 gives the most a

urate results for hanges in operating ondition, false prior levels and noise realizations for simulated ylinder pressure data. In the experimental evaluation a fourth variant, M2:3+, was in luded to ope with unreasonably high urvature for low δx that an o

ur for experimental data. It was found that the two variants M2:1 and M2:3+

7.3.

SUMMARY OF RESULTS FOR MOTORED CYCLES

185

both yield reasonable estimates, and that the exibility of M2:3+ for a hange in operating ondition was not ne essary. Sin e M2:1 is also

omputationally e ient and outstanding ompared to the other variants, usage of M2:1 together with δi 6= c is therefore re ommended as the best hoi e for motored y les.

186

CHAPTER 7. RESULTS AND EVALUATION MOTORED

...

8

Results and evaluation for fired y les Method 1 and method 2 des ribed in se tion 6.4 will now be evaluated for red ylinder pressure data. These methods were evaluated for motored y les in hapter 7 and the evaluation in this hapter follows the same stru ture. The evaluation overs both simulated and experimental data. It is worth noting that all numeri al al ulations have been made for normalized values of the parameters, while the individual parameter values that are given in tables are not normalized if not expli itly stated. The purpose of the normalization is to yield parameters that are in the order of 1, and the normalization is des ribed in appendix C.1.

8.1 Simulation results  red y les First the simulated engine data will be des ribed, followed by a dis ussion on the parameter prior knowledge used. The fo us is then turned to evaluating the performan e of method 1, followed by method 2. 8.1.1

Simulated engine data

Cylinder pressure tra es were generated by simulating the standard model from se tion 6.2.1 with representative single-zone parameters, given in appendix C.6. Six operating points were sele ted with engine speeds N ∈ {1500, 3000} rpm, mean harge temperatures at IVC TIV C ∈ {370, 414} K, ylinder pressures at IVC pIV C ∈ {50, 100} kPa 187

188

...

CHAPTER 8. RESULTS AND EVALUATION FIRED

Simulated fired data; OP1−6 Pressure [bar]

50 OP6

40 30 20 10

OP1

0

−100

−50 0 50 Crank angle [deg ATDC]

100

Figure 8.1: Simulated ylinder pressures for the dierent operating points OP16. xi Unit σi RME [%℄ xi Unit σi RME [%℄

Vc [cm3 ℄ 0.5 0.9 ∆p [kPa℄ 2.5 50.0

C1 [-℄ 0.114 5.0 ∆θ [deg℄ 0.05 50.0

TIV C [K℄ 10 2.4 Kp [-℄ 0.005 0.5

pIV C [kPa℄ 2.5 5.0 C2 [m/(s K)℄ 1.62 · 10−4 5.0

γ300 [-℄ 0.005 0.4 Qin [J℄ 25 5.0

b [K −1 ℄ 1 · 10−5 14.3 θig [deg℄ 1 6.7

Tw [K℄ 10 2.1 ∆θd [deg℄ 2 10.0

Vcr [cm3 ℄ 0.15 15.0 ∆θb [deg℄ 2 10.0

Table 8.1: Assigned standard deviation σi of the model parameters xi used for the se ond ase of δi :s. The relative mean error (RME)

orresponding to one standard deviation from the value (for OP1) is also given.

and dierent heat release tra es, where table C.5 denes the individual operating points. For ea h operating point a ylinder pressure tra e was simulated and ten realizations of Gaussian noise with zero mean and standard deviation 3.8 kPa were added, forming altogether 60 red

y les. The hosen noise level is the same as the one used in hapter 5 (and hapter 7), and thus ree ts the level seen in experimental data. The data was sampled with a resolution of 1 rank-angle degree (CAD). In gure 8.1, one red y le is shown for ea h operating point. The

ylinder pressures orresponding to OP1 and OP6 dier the most, and will therefore be the two extremes in the investigation. 8.1.2

Parameter prior knowledge

The setting of parameter prior knowledge for red y les is based on the same prin iples as for motored y les, see se tion 7.1.2, and most

8.1. SIMULATION RESULTS  FIRED CYCLES

189

details are therefore not repeated here. The standard model for red

y les entails 16 parameters, i.e. ve more than for the motored y le. This has onsequen es for the se ond ase of δi , i.e. δi 6= c, where the expe ted parameter un ertainty for the ve extra parameters must be assigned. For this parti ular appli ation the standard deviation σi for ea h parameter are given in table 8.1.

8.1.3

Method 1 Results and evaluation

First of all, the order in whi h the parameters are set xed when using algorithm 6.2 without a preferred ordering is investigated. This orresponds to keeping tra k of the spurious parameters xsp new in step 3. In this ase all parameters need to be in luded, and therefore the maximum and minimum number of e ient parameters are given by # d# max = d and dmin = 1, where d = 16 is the number of parameters. The resulting parameter order without a preferred ordering is given by C2 ≺ C1 ≺ Kp ≺ Tw ≺ ∆θ ≺ ∆p ≺ Vcr ≺ TIV C ≺ b ≺ θig ≺ Vc ≺ pIV C ≺ ∆θb ≺ Qin ≺ ∆θd ≺ γ300 ,

(8.1)

whi h ree ts a typi al ase. As before, the underlined parameter is invariant in position to the investigated noise realizations, false prior levels and dierent operating onditions. The other parameters dier in position, and ompared to the simulated motored ase the permutation groups are larger and as well as the variation in position. At many instan es the dieren e in order o

urs as permutations of the three groups [b, C1 , C2 , TIV C , pIV C , Tw ], [Kp , pIV C , Qin ] and [θig , ∆θd ], but this is not the ase as frequently as in the motored ase. The rst group is expe ted sin e these parameters are oupled to temperature through Wos hni's heat transfer orrelation (3.43) and the linear model of γ (3.44), and orresponds well to the motored ase des ribed in (7.2) ex ept for b whi h is now in luded. All parameters in the se ond group have a multipli ative ee t on the ylinder pressure, although Qin has no ee t on the ompression phase prior to ignition. Comparing the parameter order for motored (7.2) and red (8.1) y les, the order is

hanged for most positions. However γ300 is again the most e ient parameter, given the stru ture of the standard model. The minimizing number of parameters using algorithm 6.2 are given in gure 8.2 for OP1 and OP6, that are the two extremes in the simulation-based investigation. Ten dierent noise realizations orresponding to ten engine y les have been used, as well as four dierent

ases of false prior. The orresponding results for all six operating points are given in table 8.2, but now as a mean value for the ten

y les.

190

...

CHAPTER 8. RESULTS AND EVALUATION FIRED

OP 6 16

14

14

12

12

No of parameters [−]

No of parameters [−]

OP 1 16

10 8 6 4 2 0 0

10 8 6 4 2

5 0 %No (3.6) Cycle [−] 1 % (6.2) 2.5 % (7.4) 5 % (7.6)

10

0 0

5 0 %No(11.8) Cycle [−] 1 % (12.8) 2.5 % (13) 5 % (12.6)

10

Figure 8.2: Minimizing number of parameters d∗ for method 1, when using no preferred ordering. Table 8.2 shows that the number of e ient parameters d# range from 3 to 16, whi h is also illustrated in gure 8.2. The table also illustrates that for a high load, i.e. OP2, OP4 and OP6, the number of e ient parameters d# in reases. Compared to the motored y les the variation in d# depending on the level of false prior FP is larger, but again the variation is larger depending on the operating point. The large range of d# makes it hard to use the Akaike FPE to determine how many of the parameters in (8.1) that are e ient.

Estimation a

ura y Table 8.3 shows the parameter estimates for the entire range of d# , i.e. from 16 to 1 parameter, for one engine y le in the presen e of a false prior level of 5 %. For this spe i y le (OP1, y le1), the Akaike FPE is minimized by seven parameters, i.e. d∗ = 7. The results for a false prior level of 0 % is given in appendix C.7, see table C.8. Table 8.3 shows that the parameter estimates are biased when a

191

8.1. SIMULATION RESULTS  FIRED CYCLES

FP [%℄ 0 1 2.5 5 d∗ ∈

1 3.6 6.2 7.4 7.6 [3, 10℄

Operating point (OP) 2 3 4 5 10.2 4.2 12.6 3.8 15.8 5.6 11.0 5.6 15.6 5.4 13.4 5.6 14.0 6.0 12.2 6.4 [3, 16℄ [3, 8℄ [4, 16℄ [3, 9℄

6 11.8 12.8 13.0 12.6 [3, 16℄

Table 8.2: Minimizing number of parameters d∗ for method 1, without a preferred ordering. Four false prior levels are used and the range of d∗ is given for ea h operating point. false prior is present, whi h was also found for motored y les in table 7.4. As expe ted the individual parameter estimates depend upon the number of e ient parameters d# , the false prior level and what parameters that are lassied as e ient. In order to have a true parameter deviation that is smaller than the false prior level, a maximum of eight parameters should be used a

ording to the ε¯tx - olumn. For the motored ase, this number was as low as three a

ording to table 7.4. The ε¯tx - olumn also shows that ε¯tx is minimized by seven parameters, whi h is also the minimizer of the Akaike FPE for this parti ular y le. This is more of a oin iden e than a pattern. If the individual estimates are onsidered, it is notable that the C1 -estimate has the largest normalized bias in general, whi h was also the ase for the motored y les. Until C1 is xed the other parameters, espe ially TIV C and b, and to some extent Qin , Kp , and Tw , ompensate for the bad C1 -estimates. This results in biased estimates for these parameters as well. When C1 is xed, the pressure gain Kp and the mean wall temperature Tw still have onsiderable biases. The two parameters γ300 and Vc was shown to be important in the sensitivity analysis in table 3.2. If the estimation a

ura y of these two parameters would de ide how many parameters to use, the answer would be eight parameters. This motivates why eight parameters are used for ompression ratio estimation on ring y les in hapter 5. Due to the unsu

essful results of applying method 1 to setup 1 without a preferred ordering of the parameters, as well as the unsu

essful results for a preferred ordering for the motored ase in se tion 7.1.3, the investigations on erning a preferred parameter ordering as well as setup 2 are left out.

Summary of method 1 To summarize, the usage of method 1 for estimating all the parameters in the presen e of a false prior has not been su

essful, whi h onrms

C1

TIV C

pIV C

γ300

[K℄ [kPa℄ [-℄ 561 51.0 1.33 486 48.6 1.34 445 47.7 1.34 423 48.4 1.35 700 50.7 1.32 607 49.8 1.33 531 48.6 1.34 474 47.4 1.36 435 47.2 1.35 47.3 1.35 46.7 1.37 47.1 1.32 52.5 1.32 1.32 61.7 1.32 1.33 61.7 2.39 435 52.5 1.42 58.8 2.28 414 50.0 1.35

Vc

[cm3℄ [-℄ 57.3 3.15 57.5 3.13 57.8 2.39 58.0 57.1 57.2 57.4 58.3 58.6 58.4 59.1 54.8 59.9 61.5 -7e-005

−7.35 · 10−5

−5.42 · 10−5 −6.25 · 10−5 −6.86 · 10−5 −7.21 · 10−5 −4.35 · 10−5 −5.07 · 10−5 −5.78 · 10−5 −6.57 · 10−5 −6.88 · 10−5 −6.83 · 10−5 −7.35 · 10−5

b

[K −1℄

0.62

Vcr

480 0.59

504

Tw

5.0

5.3

5.3

∆p

1.05

Kp

0.10 1.00

0.11

∆θ

· 10−4 3.24 · 10−4

3.40

C2

[K℄ [cm3 ℄ [kPa℄ [deg℄ [-℄ [m/(s K)℄ 552 0.27 2.5 0.08 1.02 2.85 · 10−4 524 0.5 3.5 0.06 1.05 3.40 · 10−4 396 0.5 3.7 0.07 1.07 390 0.5 4.1 0.08 1.05 549 0.12 1.4 0.11 504 0.32 2.4 0.62 3.7

503

528

528

-15.0

-15.8

θig

[J℄ [deg ATDC℄ 544 -16.1 513 -16.0 488 -16.1 493 -16.1 540 -16.1 523 -16.1 506 -16.1 488 -16.2 484 -16.0 485 -15.8 478

Qin

∆θb

[deg℄ 20.0 20.0 20.0 20.0 20.1 20.0 20.0 20.0 20.0 20.0 19.9 19.6 18.9

ε¯tx

[%℄ 20.5 12.0 7.3 6.9 28.6 18.2 9.2 5.7 4.2 3.9 4.3 4.8 4.4 21.0 4.3 4.4 21.0 4.5 21.0 21.0 5.0 20.0 20.0 0.0

∆θd

[deg℄ 21.3 21.2 21.2 21.2 21.4 21.4 21.3 21.4 21.2 20.9 20.9 21.4 20.9 20.0 20.0

ε¯# x

[%℄ 20.7 11.5 9.0 8.6 29.0 18.7 8.9 5.5 4.4 4.5 4.3 4.7 3.3 2.1 2.1 1.6 0.0 5.0

Table 8.3: Method 1: Parameter estimates at OP1 for d# ∈ [1, 16] parameters when using no preferred ordering and a false prior level of 5t %. The# true and nominal parameter deviations ε¯tx and ε¯#x are given as RMSE:s of their relative values, i.e. t # ε¯x = RMSE(εx ) and ε¯x = RMSE(εx ).

x# xt

16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

d#

192 CHAPTER 8. RESULTS AND EVALUATION FIRED

...

8.1. SIMULATION RESULTS  FIRED CYCLES

193

the results found for the motored y les in hapter 7. Minimizing the Akaike FPE riterion gives a re ommended number of parameters between 3 and 16, whi h u tuates with both operating point and noise realization and therefore hardly gives any guidan e at all. If we instead just onsider the true parameter deviation, a maximum of eight parameters an be used in order to yield a true parameter deviation that is less than the applied false prior level. For a spe i appli ation, like ompression ratio estimation in se tion 5.3.4, eight parameters are re ommended to be used. Method 1 has also given valuable insight in whi h parameters that are most e ient. For instan e it has been shown that γ300 is the most e ient parameter, given the stru ture of the standard model in se tion 3.8.

8.1.4

Method 2 Results and evaluation

Now the fo us is turned to evaluating method 2. First the results are presented and then the performan e of the three versions M2:1, M2:2 and M2:3 are evaluated using the two setups des ribed previously in se tion 7.1.2.

Implementation details The implementation details des ribed in se tion 7.1.4 for motored y les apply here as well and are therefore only repeated in a summarized manner. The interesting sear h region ∆x for the regularization parameter δx is again hosen as δx ∈ [10−11 , 105 ], see gure 7.5, to assure that the L-shaped orner is in luded for all examined ases. When it

omes to the three versions of method 2, the onstants mε and mδ for M2:1 and aε for M2:2 are omputed in the same manner for both red and motored y les, see se tion 6.4.2 for details. Version M2:3 requires no hoi es to be made.

Results for setup 1 The results of method 2 are now given for setup 1, where the nominal values are given by x# = (1 + FP) xt for FP ∈ {0, 1, 2.5, 5} % and the assigned standard deviation σ = 0.01xt is equal for all parameters, yielding equal regularization elements δi , i.e. ase 1 of δi (δi = c). An example of an L- urve is given in gure 8.3 for one engine y le at OP1 and FP = 5 %. It also shows the true parameter deviation ε¯tx and the false prior level, as well as the results from the three versions of method 2 and the optimal hoi e of δx . As before, the optimal δx∗ is determined as the one minimizing ε¯tx = RMSE(εtx ), for the δx :s used in the omputation.

194

...

CHAPTER 8. RESULTS AND EVALUATION FIRED

Method 2; OP1 FP=5%

0

Zoomed version

10

−1.3

10 −1

10

−2 −1.4

10

x

RMSE(ε#) [−]

10

−3

10

−1.5

10

−4

10

M2:3 M2:2 M2:1 Optimal

−5

10

−6

10

−2

10

−1.6

10 −1

0

10 10 RMSE( ε ) [bar]

1

10

−2

10

0

2

10 10 RMSE( ε ) [bar]

Figure 8.3: L- urve (solid line) for one engine y le at OP1 with FP=5 % (dotted line) for setup 1. The results for the three versions of method 2 and the optimal hoi e of regularization parameter are indi ated by the legend. The mean true parameter deviation ε¯tx (dashdotted line) is also given. The optimal δx∗ is given in table 8.4 for all operating points and false prior levels used in setup 1. The orresponding true parameter deviations ε¯tx are summarized in table 8.5. The numeri al values in both tables are given as mean values for the ten y les. Table 8.6 summarizes the results for all six operating points in terms of the true parameter deviation ε¯tx in per ent as mean values of ten

y les at FP=5 %. The results are similar for the other false prior levels used. The mean and maximum dieren e ompared to δx∗ are also given, together with the mean omputational time for ompleting the estimation of one engine y le. Table 8.7 shows the individual parameter estimates and the orresponding mean true and nominal parameter deviations ε¯tx and ε¯# x at OP1 for FP = 5 %. The orresponding table for OP6 is given in table 8.8. These parameter values are omputed as mean values for ten

8.1. SIMULATION RESULTS  FIRED CYCLES

195

FP [%℄

1

2

3

4

5

6

0

1 · 105

1 · 105

1 · 105

1 · 105

1 · 105

1 · 105

2.5

0.0062

0.011

0.051

0.0013

0.02

0.0033

5

0.002

0.0044

0.008

0.00098

0.019

0.0044

1

0.085

0.091

0.095

0.56

0.15

0.0081

δx∗ , for setup 1 ∗ %, δx is the highest

Table 8.4: Value of optimal regularization parameter and operating points OP1 to OP6.

For

FP = 0

value used in the omputation.

FP [%℄ 0

1 −7

2 −7

10

10

3 −7

10

4 −7

10

5 −7

10

6 −7

10

1

0.8

0.8

0.8

0.8

0.8

0.7

2.5

1.9

1.9

1.9

1.6

1.8

1.6

5

3.4

3.5

3.4

3.0

3.3

3.2

¯tx [%] orresponding to Table 8.5: Mean true parameter deviation ε ∗ optimal δx in table 8.4, for setup 1 and operating points OP1 to OP6.

onse utive y les. The tables also entail the relative mean estimation error (RME) in per ent, as well as numeri al values for the true and (emphasized) nominal values.

Evaluation for setup 1 Evaluation for OP1:

Figure 8.3 is very similar to gure 7.6 in that

be omes smaller for a de reasing

δx

ε¯tx

until it rea hes its optimal value,

and that all three versions of method 2 visually nd positions lose to the orner of the L. Note that the estimate lies in between the true value and the nominal (false) value, and therefore is a ompromise between the data and the prior knowledge. The pla ement order on the L- urve in gure 8.3 in whi h the versions o

ur is however somewhat altered ompared to the motored ase. As for the motored ase, the approximative versions are a little more ∗

onservative than the optimal δx , resulting in higher values of δx and estimates that are loser to the nominal values. This is however preferable to a situation where

δx

is too low, whi h would result in parameter

estimates that have drifted away from the nominal ones. Just as for the motored ase, the optimal regularization parameter

δx∗

is approximately the same for dierent noise realizations, but de-

pends upon the operating point and false prior level used. Con erning ¯tx is fairly onthe estimation a

ura y, the true parameter deviation ε stant from y le-to- y le and dierent operating points, for a given

FP

196

...

CHAPTER 8. RESULTS AND EVALUATION FIRED

OP 1 2 3 4 5 6 Max Di Mean Di Time [s℄

δx∗

3.4 3.5 3.4 3.0 3.3 3.2 0 0 21275

δx (M2:1)

4.1 4.0 4.1 4.0 4.0 3.9 0.98 0.69 296

δx (M2:2)

4.1 4.0 4.1 4.0 4.0 3.9 1 0.69 9109

δx (M2:3)

4.0 3.9 4.0 4.0 3.9 3.6 0.95 0.58 21275

mv(δx ) 4.0 3.9 4.0 4.0 4.0 3.8 0.95 0.65 358

Table 8.6: True parameter deviation ε¯tx for the optimal δx∗ in table 8.4, the three versions of method 2 and as mean value for the optimal δx∗

orresponding to table 8.4. The numeri al values are given in per ent and are evaluated for F P = 5 % for setup 1. The mean omputational time for ompleting the estimation for one engine y le is also given, where the assumptions made are the same as for the motored y les in se tion 7.1.4.

level. These statements are not shown, but are in a

ordan e with the motored y les and are based on the similarity of gures 7.8 and 7.9 for the motored y les. Evaluation for all OP:s. The fo us is now turned to evaluating method 2 for all operating points. Table 8.4 shows that the optimal δx∗ hanges with the FP level and operating point, and the ee t is most pronoun ed for hanges in the FP level. However the estimation a

ura y is approximately the same although the optimal δx∗ varies depending on the operating point at a given FP level, as shown in table 8.5. The table also shows that the true parameter deviation ε¯tx is less than the false prior level used for FP > 0 %. This is all in a

ordan e with what was on luded earlier for motored y les. Table 8.6 shows that the true parameter deviation is larger for the approximative versions of δx as ompared to the optimal hoi e δx∗ . However all estimates are a ompromise between the true and nominal (false) values, and are somewhat biased towards the nominal ones. As illustrated earlier for gure 8.3, the approximative versions resulted in

onservative δx :s, whi h in turn results in estimates that are lose to the nominal ones, i.e. estimates that are over-smoothed by the regularization. The mean dieren e in ε¯tx ompared to δx∗ is in the order of 0.6-0.7 % as indi ated in the table. The dieren e in between the approximative versions is however relatively small. Compared to the

orresponding motored ase in table 7.10 the estimation a

ura y for δx∗ is higher for red y les, while the approximative versions yield approx-

8.1. SIMULATION RESULTS  FIRED CYCLES

197

imatively the same a

ura y. Algorithm M2:3 however has the overall highest a

ura y of the three versions, and is in the mean 0.1 % more a

urate than M2:1 and M2:2 for all the examined operating points. These two versions yield approximately the same estimate and they

ould both be trimmed to give better estimates, whi h would however require ad-ho tuning. This is therefore not pursued here sin e the goal has been to ompute the values for the onstants mε , mδ and aε in the same way, regardless of operating point. As pointed out in se tion 7.1.4, M2:1 (and M2:2) ould have a disadvantage ompared to M2:3 when hanging operating onditions whi h

ould either result in some loss in estimation a

ura y or requires an ad-ho tuning of the parameters in M2:1. The former ee t is seen in table 8.6 by omparing the olumns for M2:1 and M2:3, espe ially for OP6. For the motored y les this ee t did not show up, whi h ould be due to the simpler model of ylinder pressure, but is more likely due to that the hanges in thermodynami properties are larger when an operating point is hanged for the ring y les. The omputational time given in table 8.6 again shows that algorithm M2:1 and mv(δx∗ ) are the fastest and they dier merely due to that they use a dierent number of iterations to minimize the loss fun tion WN . Compared to M2:3, M2:1 is approximately 70 times faster a

ording to table 8.6, while M2:2 is approximately 30 times slower than M2:1. As expe ted the omputational time in reases signi antly for red y les as ompared to motored, the in rease is approximatively a fa tor eight in the mean. The in rease is due to the more omplex model, more parameters to estimate and more iterations performed before onvergen e of the lo al optimizer for ea h spe i δx . Estimation a

ura y for xi : Now the fo us is turned to the estimation a

ura y for the individual parameters xi . Tables 8.7 and 8.8 show that the estimates for C1 , TIV C , b, Tw , ∆p, and espe ially for Vcr , ∆θ and C2 , are regularized to the vi inity of their respe tive nominal value x# i , indi ated by an RME lose to 5 %. These eight parameters orrespond fairly well with the order in whi h the parameters are set spurious for method 1 when no preferred ordering is used, ompare (8.1). The dieren e is Kp , otherwise it orresponds to the eight most spurious parameters. The estimates for the other parameters, ex ept for θig and Kp , yield estimates that are in between the true and (false) nominal value. For all examined operating points the estimation a

ura y is highest for γ300 , whi h has an overall RME within 1.5 %,

losely followed by ∆θb (1.6 %), pIV C and Qin (within 1.8 %). The a

ura y for Vc is within 3.2 %. The estimation a

ura y also depends upon the routine used to determine the regularization parameter δx . From table 8.6 it was found that routine M2:3 gave the smallest overall mean error, followed by

:

2.39

5.0 5.0 58.8 2.28

61.7

C1

[-℄ 2.40 5.4 2.40 5.3 2.40 5.3 2.40 5.4

Vc

[cm3℄ 59.0 0.4 60.7 3.2 60.5 3.0 59.7 1.6 5.0 414

435

1.418

γ300

5.0 5.0 50.0 1.350

52.5

pIV C

[K℄ [kPa℄ [-℄ 422 50.3 1.356 1.9 0.7 0.5 435 50.9 1.366 5.1 1.8 1.2 435 50.8 1.369 5.0 1.6 1.4 433 50.5 1.369 4.5 1.0 1.4

TIV C

−7 · 10

0.617

Vcr

5.0 5.0

5.3

∆p

1.050

Kp

5.0 5.0 0.10 1.000

0.11

∆θ

3.24 · 10

5.0

−4

3.40 · 10−4

C2

[cm3℄ [kPa℄ [deg℄ [-℄ [m/(s K)℄ 0.613 5.2 0.10 0.984 3.41 · 10−4 4.3 4.9 4.1 -1.6 5.2 0.617 5.2 0.11 0.986 3.40 · 10−4 5.0 4.8 5.0 -1.4 5.1 0.617 5.2 0.11 0.982 3.40 · 10−4 4.9 4.8 5.0 -1.8 5.1 0.617 5.2 0.10 0.975 3.41 · 10−4 4.9 4.9 5.0 -2.5 5.1

5.0 5.0 480 0.588

504

5.0 −5

−7.35 · 10−5

Tw

[K℄ 496 3.2 502 4.7 502 4.6 501 4.5

b

[K −1℄ −7.02 · 10−5 0.2 −7.31 · 10−5 4.5 −7.3 · 10−5 4.3 −7.25 · 10−5 3.5

5.0 503

528

5.0 -15.0

-15.8

θig

[J℄ [deg ATDC℄ 512 -15.7 1.7 4.8 513 -15.9 1.8 6.0 512 -15.9 1.6 5.9 512 -15.9 1.7 5.7

Qin

∆θb

[deg℄ 20.0 -0.1 20.2 1.2 20.2 0.8 20.0 -0.1

ε¯tx

[%℄ 3.3 3.3 4.1 4.1 4.1 4.1 3.9 3.9 21.0 21.0 5.0 5.0 5.0 5.0 20.0 20.0 0.0

∆θd

[deg℄ 20.8 3.9 20.8 3.8 20.8 3.9 20.9 4.6

ε¯# x

[%℄ 3.3 3.3 2.5 2.5 2.6 2.6 2.9 2.9 0.0 0.0 5.0

Table 8.7: Parameter estimates and orresponding relative mean estimation error (RME) for OP1 at FP = 5 %, using method 2, setup 1 and ase 1 (δi = c).

x

RME(t x# )

x#

RME δx (M 2 : 1) RME δx (M 2 : 2) RME δx (M 2 : 3) RME

δx∗

δx

198 CHAPTER 8. RESULTS AND EVALUATION FIRED

...

:

2.39

5.0 5.0 58.8 2.28

61.7

C1

[-℄ 2.40 5.3 2.40 5.4 2.40 5.3 2.41 5.6

Vc

[cm3℄ 59.7 1.5 60.4 2.7 60.4 2.7 60.5 2.9 5.0 370

389

[K℄ 375 1.2 386 4.2 386 4.3 383 3.4

TIV C

1.418

[-℄ 1.358 0.6 1.370 1.5 1.368 1.4 1.369 1.4

γ300

5.0 5.0 100.0 1.350

105.0

pIV C

[kPa℄ 101.3 1.3 101.2 1.2 101.1 1.1 101.3 1.3 −7 · 10

−5

0.617

Vcr

5.0 5.0

5.3

∆p

1.050

Kp

5.0 5.0 0.10 1.000

0.11

∆θ

3.24 · 10

5.0

−4

3.40 · 10−4

C2

[cm3℄ [kPa℄ [deg℄ [-℄ [m/(s K)℄ 0.620 5.3 0.10 0.980 3.40 · 10−4 5.4 5.3 4.9 -2.0 4.8 0.617 5.2 0.10 0.975 3.40 · 10−4 4.9 4.9 4.9 -2.5 5.0 0.617 5.2 0.10 0.978 3.40 · 10−4 4.9 4.9 5.0 -2.2 5.0 0.617 5.3 0.10 0.975 3.40 · 10−4 4.9 5.1 4.9 -2.5 5.1

5.0 5.0 400 0.588

420

5.0

−7.35 · 10−5

Tw

[K℄ 414 3.5 418 4.6 419 4.6 417 4.2

b

[K −1℄ −6.97 · 10−5 -0.4 −7.24 · 10−5 3.4 −7.26 · 10−5 3.7 −7.15 · 10−5 2.2 5.0 1193

1252

5.0 -25.0

-26.3

θig

[J℄ [deg ATDC℄ 1210 -25.2 1.4 0.7 1211 -25.8 1.5 3.3 1214 -25.9 1.8 3.6 1206 -25.6 1.2 2.3

Qin

∆θb

[deg℄ 29.8 -0.6 29.5 -1.6 29.6 -1.2 29.5 -1.6

ε¯tx

[%℄ 3.2 3.2 3.9 3.9 3.9 3.9 3.6 3.6 15.8 31.5 5.0 5.0 5.0 5.0 15.0 30.0 0.0

∆θd

[deg℄ 15.2 1.2 15.8 5.3 15.8 5.4 15.5 3.6

ε¯# x

[%℄ 3.7 3.7 3.1 3.1 3.0 3.0 3.2 3.2 0.0 0.0 5.0

Table 8.8: Parameter estimates and orresponding relative mean estimation error (RME) for OP6 at FP = 5 %, using method 2, setup 1 and ase 1 (δi = c).

x

t

RME(x# )

x#

RME δx (M 2 : 1) RME δx (M 2 : 2) RME δx (M 2 : 3) RME

δx∗

δx

8.1. SIMULATION RESULTS  FIRED CYCLES

199

200

...

CHAPTER 8. RESULTS AND EVALUATION FIRED

M2:1 and M2:2. When onsidering the individual estimates of the ve parameters, the same relative ordering is found if all six operating points are onsidered. In the ase for M2:3 the a

ura y of the estimates for the ve parameters is within 3 %. This value is larger than the 0.5 % whi h was found for the orresponding motored ase, and is probably due to the more omplex model stru ture and harder estimation problem in the red ase. The other estimates end up as a

ompromise in between the true value and the (false) nominal value, ex ept for θig and Kp . In the ase of the two parameters ∆p and C2 , these two estimates oin ide with the nominal values for all ases of δx in table 8.7. For the two parameters Vcr and ∆θ the estimates oin ide with the nominal ones for the approximative versions, but not for the optimal hoi e of δx . As pointed out for the motored ase, in the presen e of a false prior of 5 % the estimates for method 2 are equally or more a

urate ompared to method 1 for all number of, ex ept seven, e ient parameters, as shown by omparing the ε¯tx - olumns in tables 8.3 and 8.7. This again illustrates that method 2 is more robust to a false prior than method 1 and this results in better estimates. The dieren e between method 1 and method 2 are however smaller in the red ase.

Results for setup 2 The results for method 2 is now presented for setup 2, where the nomt inal values are given by x# i = (1 + σi )xi , for whi h the standard deviations are given in table 8.1. The assigned regularization elements are given for the two ases δi = c and δi 6= c. The rst ase is 1 1 δi = 2N (0.01xt )2 , i.e. all elements are equal, and in the se ond ase i

δi = 2N1σ2 . The tables and gures follow the same stru ture as for the i

orresponding motored ase in se tion 7.1.4.

The L- urves based on one engine y le for ea h ase at OP1 are given in gures 8.4 and 8.5 respe tively. The individual parameter estimates are given in tables 8.108.13, and orresponds to the form used in table 8.7. Tables 8.10 and 8.11 represent ase 1 (δi = c) for OP1 and OP6, while the orresponding tables for ase 2 (δi 6= c) are given in tables 8.12 and 8.13. It is worth to mention again that the se ond

ase uses the weighted versions of the nominal parameter deviation RMSE(Lδ ε#x ) and so on in the tables and gures. Therefore the optimal regularization parameter δx∗ is now determined as the one minimizing the weighted true parameter deviation, i.e. RMSE(Lδ εtx ). The nominal and true parameter deviations ε¯tx and ε¯# x for both ases at OP1 and OP6 are then summarized in table 8.9.

8.1. SIMULATION RESULTS  FIRED CYCLES

201

Method 2; OP1

0

Zoomed version

10

−1

10

−2

−1

10

−3

10

x

RMSE(ε# ) [−]

10

−4

10

−5

10

M2:3 M2:2 M2:1 Optimal

−6

10

−7

10

−2

10

−1

10 RMSE( ε ) [bar]

0

10

−2

10

−1

0

10 10 RMSE( ε ) [bar]

Figure 8.4: L- urve (solid line) for one engine y le at OP1 with false prior ε ¯#−t (dotted line), when using setup 2 and ase 1 of the δi (δi = x c). The results for the three versions of method 2 and the optimal

hoi e of regularization parameter are indi ated by the legend. ε¯tx (dash-dotted line) is also given.

The

true parameter deviation

Evaluation for setup 2 Evaluation for OP1:

Figures 8.4 and 8.5 both illustrate the L-shaped

urve whi h was also found for the orresponding motored ase. The three approximative versions of method 2 all yield estimates that are

lose to the orner of the L. This is however not the ase for the optimal regularization parameter. Figure 8.4 (δi = c) shows that the ∗ optimal δx is not as lose to the orner as in the approximative ver∗ sions. This is dire tly ree ted in the parameter estimates for δx that are lose to the true values. They are loser than the estimates from ¯tx - and ε¯# the approximative versions, as seen in the ε x - olumns in ta∗ bles 8.10 and 8.11. For δi 6= c the optimal δx is loser to the orner as shown in gure 8.5, whi h is also ree ted in a better onsisten y be-

tween the optimal and approximative versions. It also ree ts a better

202

...

CHAPTER 8. RESULTS AND EVALUATION FIRED

Method 2; OP1

2

Zoomed version

10

−1.4

10

0

# RMSE( Lδ εx ) [−]

10

−2

10

−1.5

10

−4

10

M2:3 M2:2 M2:1 Optimal

−6

10

−2

10

−1.6

10

−1

10 RMSE( ε ) [bar]

0

10

−2

10

−1

0

10 10 RMSE( ε ) [bar]

Figure 8.5: L- urve (solid line) for one engine y le at OP1 with weighted false prior RMSE(Lδ ε#−t ) (dotted line), when using setup 2 x and ase 2 of the δi (δi 6= c). The results for the three versions of method 2 and the optimal hoi e of regularization parameter are indi atedδ byt the legend. The weighted true parameter deviation RMSE(L εx) (dash-dotted line) is also given. δi = c

δi 6= c

OP1 OP6 OP1 OP6 δx : ε¯tx ε¯# ε¯tx ε¯# ε¯tx ε¯# ε¯tx ε¯# x x x x [%℄ [%℄ [%℄ [%℄ [%℄ [%℄ [%℄ [%℄ δx∗ 6.5 16.6 6.2 17.7 5.8 15.2 11.2 16.0 δx (M 2 : 1) 18.3 3.8 18.3 3.2 11.1 12.7 14.3 11.7 δx (M 2 : 2) 18.3 4.1 18.3 3.3 12.8 10.4 13.8 8.6 δx (M 2 : 3) 18.2 4.3 18.2 3.9 11.4 12.5 12.6 11.9 x# 18.8 0.0 18.7 0.0 18.8 0.0 18.7 0.0 xt 0.0 18.8 0.0 18.7 0.0 18.8 0.0 18.7 Table 8.9: Mean true and nominal parameter deviation for method 2 using two ases of δi at OP1 and OP6 for setup 2.

8.1. SIMULATION RESULTS  FIRED CYCLES

203

(weighted) ompromise between Lδ εtx and Lδ ε# x , as they are equal in size in tables 8.12 and 8.13. This speaks in favor of using δi 6= c. Evaluation for the individual estimates xi : Now onsider the estimation a

ura y of the individual parameters xi , and espe ially the estimates produ e by M2:1, M2:2 and M2:3. Tables 8.10 and 8.11 show that the estimates for the parameters C1 , b, and espe ially for Vcr , ∆p, ∆θ and C2 are regularized to the vi inity of their respe tive nominal value x# i , indi ated by an RME that is lose to the one for the nominal parameter values. Most of the remaining parameters end up in the region between the true and nominal values, ex ept for the estimates of C1 , Vc , γ300 and Kp . These parameters end up just outside of the region with estimates that are within 6 % for the worst ase (Kp ). For δi 6= c, tables 8.12 and 8.13 show that C1 , TIV C , γ300 , Tw , Vcr , ∆θ and C2 end up near their respe tive nominal value, while the burn angle parameters θig , ∆θd and ∆θb , and the released energy Qin yield estimates within 2.3 %. The estimates of the four parameters Vc , γ300 , Qin and ∆θb are again better for δi 6= c than δi = c, while the a

ura y of pIV C is approximately the same. The on lusion from tables 8.10-8.13 is that the regularization is used to pull parameters that are hard to estimate toward their nominal values, while the e ient parameters are free to t the data. This was the drive for using method 2, and it works as intended. Evaluation for all OP:s: Table 8.9 shows that for δi = c the true parameter deviation ε¯tx is almost as large as the false prior level used, while for δi 6= c the true and nominal parameter deviations ε¯tx and ε¯# x are approximately equal in size. This is also illustrated in the weighted true and nominal parameter deviations, Lδ εtx and Lδ ε# x , in tables 8.12 and 8.13. As pointed out earlier, it ree ts a better ompromise between prior knowledge and measurement data for the se ond ase. From table 8.9 it an also be on luded that the true parameter deviation ε¯tx is smaller for the se ond ase for all three versions of method 2, whi h also was the ase for evaluation based upon motored y les, see table 7.13. As for the orresponding motored ase the mean nominal parameter deviation ε¯# x is relatively small for δi = c and signi antly higher for δi 6= c. This together with the smaller ε¯tx ree ts that the se ond ase is more robust to a false prior, whi h of ourse is preferable. Note that the false prior level for the red y les is smaller than for the motored y les, whi h is due to the larger number of parameters and that the RME of the false prior in setup 2 for ∆p and ∆θ are 50 %, see table 8.1. Now the attention is turned to nding whi h version of method 2 that performs the best. In the ase of δi = c all three versions are restri tive and yield estimates lose to the nominal values. This is, as pointed out previously, ree ted by a relatively low value of ε¯# x in

:

2.39

0.9 5.0 58.8 2.28

59.3

C1

[-℄ 2.68 17.5 2.40 5.3 2.40 5.4 2.41 5.6

Vc

[cm3℄ 58.4 -0.6 58.6 -0.3 58.8 -0.0 59.2 0.7 2.4 414

424

1.355

γ300

5.0 0.4 50.0 1.350

52.5

pIV C

[K℄ [kPa℄ [-℄ 436 49.3 1.349 5.2 -1.4 -0.1 422 50.3 1.375 1.9 0.5 1.9 421 50.2 1.379 1.7 0.4 2.1 418 50.4 1.381 0.9 0.8 2.3

TIV C

−7 · 10

−5

0.74

Vcr

[cm3℄ 0.71 21.4 0.74 25.4 0.74 25.5 0.74 25.5

2.1 25.5 480 0.59

490

14.3

−8 · 10−5

Tw

[K℄ 505 5.2 487 1.5 487 1.4 485 1.0

b

[K −1℄ −6.77 · 10−5 -3.2 −7.92 · 10−5 13.1 −7.89 · 10−5 12.7 −7.8 · 10−5 11.5 7.5

0.15

∆θ

[deg℄ 0.11 14.1 0.15 50.0 0.15 50.0 0.15 50.0 1.005

3.24 · 10

10.0

−4

3.56 · 10−4

C2

[-℄ [m/(s K)℄ 1.011 3.49 · 10−4 1.1 7.6 0.956 3.57 · 10−4 -4.4 10.1 0.949 3.57 · 10−4 -5.1 10.1 0.946 3.57 · 10−4 -5.4 10.1

Kp

50.0 50.0 0.5 5.0 0.10 1.000

∆p

[kPa℄ 5.1 1.9 7.5 49.9 7.5 49.9 7.5 49.9

5.0 503

528

-6.7 -15.0

-14.0

θig

[J℄ [deg ATDC℄ 509 -15.5 1.2 3.4 527 -14.9 4.6 -0.5 524 -14.9 4.1 -0.4 520 -15.0 3.4 0.2

Qin

∆θb

[deg℄ 20.0 0.1 20.4 2.0 20.1 0.7 20.0 -0.1

= c

ε¯tx

ε¯# x

[%℄ 16.6 16.6 3.8 3.8 4.1 4.1 4.3 4.3 0.0 0.0 18.8

, using

[%℄ 6.5 6.5 18.3 18.3 18.3 18.3 18.2 18.2 22.0 22.0 18.8 10.0 10.0 18.8 20.0 20.0 0.0

∆θd

[deg℄ 20.6 3.2 20.3 1.3 20.2 1.2 20.3 1.5

Table 8.10: Parameter estimates and orresponding relative mean estimation error (RME) for OP1 when δi method 2 and setup 2.

x

t

RME(x# )

x#

RME δx (M 2 : 1) RME δx (M 2 : 2) RME δx (M 2 : 3) RME

δx∗

δx

204 CHAPTER 8. RESULTS AND EVALUATION FIRED

...

:

0.9 5.0 58.8 2.28

2.7 370

380

2.39

TIV C

59.3

C1

[-℄ [K℄ 2.28 369 -0.2 -0.2 2.40 376 5.4 1.7 2.41 375 5.5 1.3 2.42 369 5.9 -0.3

Vc

[cm3℄ 58.9 0.2 58.1 -1.2 58.7 -0.1 60.1 2.3 1.355

[-℄ 1.352 0.2 1.373 1.7 1.375 1.9 1.377 2.0

γ300

2.5 0.4 100.0 1.350

102.5

pIV C

[kPa℄ 101.8 1.8 98.6 -1.4 99.0 -1.0 99.9 -0.1 −7 · 10

−5

0.74

Vcr

[cm3℄ 0.86 46.1 0.74 25.4 0.74 25.4 0.74 25.5

2.5 25.5 400 0.59

410

14.3

−8 · 10−5

Tw

[K℄ 386 -3.4 408 2.1 407 1.9 405 1.1

b

[K −1℄ −6.91 · 10−5 -1.3 −7.89 · 10−5 12.7 −7.85 · 10−5 12.1 −7.67 · 10−5 9.6 7.5

0.15

∆θ

[deg℄ 0.11 12.8 0.15 50.0 0.15 50.0 0.15 49.9 1.005

3.24 · 10

10.0

−4

3.56 · 10−4

C2

[-℄ [m/(s K)℄ 0.980 3.28 · 10−4 -2.0 1.4 0.970 3.57 · 10−4 -3.0 10.1 0.969 3.57 · 10−4 -3.1 10.1 0.969 3.57 · 10−4 -3.1 10.1

Kp

50.0 50.0 0.5 5.0 0.10 1.000

∆p

[kPa℄ 5.0 -0.9 7.5 50.0 7.5 50.1 7.5 50.3 2.1 1193

1218

-4.0 -25.0

-24.0

θig

[J℄ [deg ATDC℄ 1217 -24.9 2.1 -0.4 1221 -25.5 2.4 2.1 1216 -25.5 2.0 2.1 1203 -25.4 0.8 1.8

Qin

∆θb

[deg℄ 29.9 -0.2 29.9 -0.3 29.8 -0.8 29.6 -1.4

= c

ε¯tx

ε¯# x

[%℄ 17.7 17.7 3.2 3.2 3.3 3.3 3.9 3.9 0.0 0.0 18.7

, using

[%℄ 6.2 6.2 18.3 18.3 18.3 18.3 18.2 18.2 17.0 32.0 18.7 13.3 6.7 18.7 15.0 30.0 0.0

∆θd

[deg℄ 15.0 -0.3 16.1 7.3 16.0 6.6 15.6 4.2

Table 8.11: Parameter estimates and orresponding relative mean estimation error (RME) for OP6 when δi method 2 and setup 2.

x

t

RME(x# )

x#

RME δx (M 2 : 1) RME δx (M 2 : 2) RME δx (M 2 : 3) RME

δx∗

δx

8.1. SIMULATION RESULTS  FIRED CYCLES

205

:

2.39

0.9 5.0 58.8 2.28

59.3

C1

[-℄ 2.38 4.5 2.40 5.3 2.40 5.4 2.40 5.4

Vc

[cm3℄ 58.8 0.1 59.0 0.4 59.2 0.7 59.1 0.5 2.4 414

424

1.355

γ300

5.0 0.4 50.0 1.350

52.5

pIV C

[K℄ [kPa℄ [-℄ 421 49.5 1.352 1.8 -0.9 0.2 423 49.5 1.355 2.1 -1.1 0.4 424 49.7 1.355 2.3 -0.6 0.4 423 49.5 1.355 2.2 -0.9 0.4

TIV C

−7 · 10

−5

0.74

Vcr

7.5

0.15

∆θ

1.005

Kp

50.0 50.0 0.5 5.0 0.10 1.000

∆p

3.24 · 10

10.0

−4

3.56 · 10−4

C2

[cm3℄ [kPa℄ [deg℄ [-℄ [m/(s K)℄ 0.74 5.1 0.12 1.004 3.54 · 10−4 25.5 2.5 20.2 0.4 9.3 0.74 5.3 0.14 1.004 3.58 · 10−4 25.5 5.2 42.5 0.4 10.5 0.74 5.7 0.15 1.004 3.58 · 10−4 25.5 14.0 47.6 0.4 10.5 0.74 5.3 0.14 1.004 3.58 · 10−4 25.5 5.6 43.5 0.4 10.5

2.1 25.5 480 0.59

490

14.3

−8 · 10−5

Tw

[K℄ 491 2.2 490 2.1 490 2.0 490 2.0

b

[K −1℄ −6.89 · 10−5 -1.5 −6.96 · 10−5 -0.6 −7.18 · 10−5 2.5 −6.98 · 10−5 -0.3 5.0 503

528

-6.7 -15.0

-14.0

θig

[J℄ [deg ATDC℄ 504 -15.3 0.2 1.8 503 -15.0 -0.0 -0.0 507 -14.7 0.8 -2.3 504 -14.8 0.1 -1.0

Qin

∆θb

6= c

Lδ εtx

[deg℄ [%℄ 20.0 2.5 0.1 2.5 20.0 2.9 0.1 2.9 19.9 3.1 -0.3 3.1 20.0 3.0 0.1 3.0 22.0 22.0 4.2 10.0 10.0 4.2 20.0 20.0 0.0

∆θd

[deg℄ 20.4 1.9 20.1 0.6 19.8 -0.9 20.0 -0.1

Table 8.12: Parameter estimates and orresponding relative mean estimation error (RME) for OP1 when δi method 2 and setup 2.

x

t

RME(x# )

x#

δx (M 2 : 1)

RME RME δx (M 2 : 2) RME δx (M 2 : 3) RME

δx∗

δx

, using

[%℄ 3.2 3.2 2.9 2.9 2.6 2.6 2.8 2.8 0.0 0.0 4.2

L δ ε# x

206 CHAPTER 8. RESULTS AND EVALUATION FIRED

...

:

2.39

0.9 5.0 58.8 2.28

59.3

C1

[-℄ 2.35 3.0 2.41 5.8 2.41 5.7 2.41 5.8

Vc

[cm3℄ 59.1 0.6 59.2 0.7 59.2 0.7 59.2 0.7 2.7 370

380

[K℄ 381 2.9 380 2.6 380 2.7 380 2.6

TIV C

1.355

[-℄ 1.350 0.0 1.354 0.3 1.354 0.3 1.354 0.3

γ300

2.5 0.4 100.0 1.350

102.5

pIV C

[kPa℄ 99.8 -0.2 99.2 -0.8 99.7 -0.3 99.2 -0.8 −7 · 10

−5

0.74

Vcr

7.5

0.15

∆θ

1.005

Kp

50.0 50.0 0.5 5.0 0.10 1.000

∆p

3.24 · 10

10.0

−4

3.56 · 10−4

C2

[cm3℄ [kPa℄ [deg℄ [-℄ [m/(s K)℄ 0.74 4.6 0.14 1.003 3.38 · 10−4 25.5 -8.1 43.4 0.3 4.3 0.74 5.5 0.15 1.003 3.58 · 10−4 25.5 9.8 54.7 0.3 10.5 0.74 6.2 0.15 1.003 3.57 · 10−4 25.5 23.2 48.3 0.3 10.3 0.74 5.4 0.15 1.003 3.58 · 10−4 25.5 8.8 47.9 0.3 10.4

2.5 25.5 400 0.59

410

14.3

−8 · 10−5

Tw

[K℄ 407 1.8 409 2.3 409 2.3 409 2.3

b

[K −1℄ −6.61 · 10−5 -5.6 −6.75 · 10−5 -3.6 −7 · 10−5 -0.0 −6.75 · 10−5 -3.6 2.1 1193

1218

-4.0 -25.0

-24.0

θig

[J℄ [deg ATDC℄ 1196 -25.0 0.3 -0.2 1193 -25.0 0.1 -0.2 1200 -24.7 0.6 -1.3 1193 -25.0 0.1 -0.1

Qin

32.0

∆θb

[deg℄ 30.0 0.1 30.0 -0.0 29.8 -0.7 30.0 -0.0 13.3 6.7 15.0 30.0

17.0

∆θd

[deg℄ 15.0 0.2 15.1 0.5 14.9 -0.7 15.1 0.6

Table 8.13: Parameter estimates and orresponding relative mean estimation error (RME) for OP6 when δi method 2 and setup 2.

x

t

RME(x# )

x#

δx (M 2 : 1)

RME RME δx (M 2 : 2) RME δx (M 2 : 3) RME

δx∗

δx

6= c

[%℄ 2.4 2.4 3.0 3.0 3.0 3.0 2.9 2.9 4.2 4.2 0.0

Lδ εtx

, using

[%℄ 3.3 3.3 3.0 3.0 2.6 2.6 3.0 3.0 0.0 0.0 4.2

L δ ε# x

8.1. SIMULATION RESULTS  FIRED CYCLES

207

208

...

CHAPTER 8. RESULTS AND EVALUATION FIRED

table 8.9. Out of these three, M2:3 performs best, but the dieren e is small and the estimation bias is still signi ant as ree ted by the ε¯tx - olumns. In the ase of δi 6= c, all three versions render weighted parameter deviations RMSE(Lδ εtx ) and RMSE(Lδ ε# x ) that are approximately the same and fairly lose to the optimal, see tables 8.12 and 8.13. Table 8.9 shows that ε¯tx is smallest for M2:1 at OP1 and for M2:3 at OP6. However when onsidering ε¯tx for all investigated operating points, it is lowest for M2:3 followed by M2:2 and M2:1 for all six operating points ex ept OP1. 8.1.5

Summary for simulation results

Method 1 resulted in parameter estimates that are signi antly biased in the presen e of a false prior, sin e the bias is larger than the false prior level used. It is therefore in itself not re ommended to use for estimation of all parameters in the given formulation. When the true parameter deviation is available, the number of e ient parameters an be determined for a spe i appli ation su h as for example ompression ratio estimation. Method 2 outperforms method 1 sin e it is more robust to a false prior level and yields more a

urate parameter estimates. The drive for method 2 was to regularize the solution su h that the parameters that are hard to determine are pulled towards their nominal values, while the e ient parameters are free to t the data. This has shown to be the ase in the simulations for both red and motored y les. The user an hose between two ases of the parameter un ertainty, namely δi = c and δi 6= c. The former is dire tly appli able on e nominal values of the parameters are determined and yields good estimates. The se ond ase is however more robust to false prior as seen in setup 2, and it is therefore re ommended sin e it pays o in better estimates. Method 2 is given three versions; For the red y les it has been shown that M2:3 yields the best estimates in terms of estimation a

ura y. For motored y les it was shown that M2:1 ould be used in all the examined operating points without any noti eable loss in estimation a

ura y. This has not been the ase for the red y les, where a

hange in operating onditions hanges the thermodynami properties more signi antly and results in larger biases for M2:1 than for M2:3. The dieren e in estimation a

ura y between M2:1 and M2:3 is however small, within 0.3 % and 2 % for setup 1 and 2 respe tively, while the dieren e in omputational time is more signi ant. Thus if time is available the re ommendation is to use M2:3, while if omputational time is an important feature M2:1 is re ommended.

8.2. EXPERIMENTAL RESULTS  FIRED CYCLES

209

8.2 Experimental results  red y les The methods will now be evaluated on experimental engine data. The stru ture of the se tion is similar to the orresponding motored ase, i.e. se tion 7.2. Thus rst the experimental engine data will be des ribed, followed by a dis ussion on the parameter prior knowledge used. The fo us is then turned to evaluating the performan e of method 1, followed by method 2. 8.2.1

Experimental engine data

Data is olle ted during stationary operation at engine speeds N ∈ [1200, 3500] rpm, intake manifold pressures pman ∈ [32, 130} kPa and ignition angle θig ∈ [−33, −5] deg ATDC, altogether forming six dierent operating points. These operating points are dened in the upper part of table 8.14, and are also given in table C.9 in appendix C.7. For ea h operating point 101 onse utive red y les were sampled for two ylinders with a rank-angle resolution of 1 degree, using an AVL GU21D ylinder pressure sensor. But due to the longer omputational time for ring y les as ompared to motored, espe ially for M2:3 as shown in table 8.6, only the rst 40 y les will be onsidered in the evaluation. Figure 8.6 displays one measured y le for ea h operating point. The operating points are numbered in an as ending order of their maximum ylinder pressure. 8.2.2

Parameter prior knowledge

The parameter prior knowledge used in the experimental evaluation is based on dierent strategies to set the nominal parameter values x# and the regularization elements δi .

Nominal parameter values x

#

The nominal parameter values x# are set based on a ombination of the parameter initial methods des ribed in hapter 3 and expert prior knowledge of what the numeri al value of x# should be. The values for x# for ea h operating point are given as the emphasized values in table 8.15. The nominal values are also given in table C.10 in appendix C.7. Of these parameters it is only the nominal values for ∆p, ∆θd and ∆θb that are updated for ea h y le, and therefore for these three, their respe tive mean value is given in the table. The other parameters are updated for ea h operating ondition.

210

...

CHAPTER 8. RESULTS AND EVALUATION FIRED

Experimental fired data; OP1−6 70 OP6

60

Pressure [bar]

50 40 30 20 10 OP1 0

−100

−50

0 50 Crank angle [deg ATDC]

100

Figure 8.6: Experimentally measured red ylinder pressures at operating onditions OP16.

Regularization elements δi Two ases of regularization parameter elements are used for method 2 and they orrespond to the two ases used in the simulation-based evaluation done previously. The rst ase (δi = c) sets the standard x# ¯# deviation σi for ea h parameter as σi = 0.01¯ i , where x i is the mean for one operating point. It is used to assure that σi does not value of x# i u tuate in size from y le-to- y le. In the simulation-based evaluation xt was used instead of x ¯# . The se ond ase (δi 6= c) is based on expert knowledge of the un ertainty for ea h parameter and it is therefore subje tively hosen by the user. The standard deviation used here in the experimental evaluation oin ides with the one used for the simulation-based evaluation in table 8.1. The rst ase is updated for ea h operating point, while this is not required for the se ond ase.

8.2. EXPERIMENTAL RESULTS  FIRED CYCLES

N [rpm℄ pman [kPa℄ θig [deg ATDC℄ mean(d∗ ) d∗ ∈

OP1 1200 32 -33 13.2 [11, 16]

OP2 1500 55 -26 13.2 [12, 16]

OP3 1500 130 -5 13.1 [9, 16]

OP4 2000 123 -12 11.8 [10, 15]

OP5 3000 103 -28 13.6 [11, 16]

211 OP6 3500 120 -24 14.3 [11, 16]

Table 8.14: Mean number and region of d∗ for 40 experimental y les at OP16.

8.2.3 Method 1 Results and evaluation Method 1 is evaluated without a preferred ordering, sin e no gain was found in the simulation-based evaluation by using one in se tion 7.1.3. The parameter order for all six operating points is given by ∆p ≺ C2 ≺ Tw ≺ pIV C ≺ ∆θ ≺ Vcr ≺ C1 ≺ TIV C ≺ ∆θd ≺ b ≺ Qin ≺ Kp ≺ ∆θb ≺ Vc ≺ θig ≺ γ300 ,

(8.2)

whi h ree ts the most ommon ase. This parameter order diers somewhat from the orresponding order found in simulations (8.1), for instan e θig and ∆θd have hanged pla es, but γ300 is still the most e ient parameter and invariant in position for the investigated operating onditions and y le-to- y le variations. The other parameters dier in order at most instan es as permutations of the three groups [b, C1 , C2 , TIV C , pIV C , Tw , Vcr ], [Kp , pIV C , Qin , ∆θb ] and [θig , ∆θd ]. The same groups were found in the simulation-based evaluation in se tion 8.1.3 ex ept for Vcr and ∆θb that are now in luded in the rst and se ond group respe tively. The minimizing number of parameters d∗ using algorithm 6.2 are given in table 8.14 as a mean over the 40 y les at all six operating points. The range of d∗ for ea h operating point is also given. The table shows that the mean values of d∗ are more stable between operating points and that the variation is smaller in the experimental ase,

ompared to the simulation given in table 8.2. The trend of more e ient parameters for high loads seen in table 8.2 does not show up in table 8.14. The mean value, standard deviation, and relative mean error for the individual estimates are given in table 8.15. The relative mean error is omputed relative to the nominal parameter value at ea h y le and is given in per ent. All parameters give reasonable values ex ept for Tw and Vcr that yield unreasonably large parameter values, while TIV C and to some extent also Vc yield small values. The parameter C1 yields estimates that are either too small, reasonable or too large. Espe ially for OP3 and OP4 the C1 -estimate be omes too small, while at the same

mean σest RME x# mean σest RME x# mean σest RME x# mean σest RME x# mean σest RME x# mean σest RME x# 2.00 1.77 -12.2 2.28

58.8

2.28

58.8

54.5 4.7 -7.3

2.16 1.66 -5.2

2.28

58.8

54.2 4.1 -7.8

0.95 1.12 -58.2

2.28

58.8

57.0 3.7 -3.0

0.02 0.02 -99.1

2.28

58.8

54.7 3.8 -6.9

3.24 1.60 42.2

2.28

51.4 1.7 -12.5

58.8

392

338 113 -14.2

400

292 46 -26.8

394

270 0 -31.4

407

335 66 -15.0

425

271 1 -34.7

450

TIV C [K℄ 304 56 -30.6

153.3

142.6 12.4 -1.2

132.2

131.6 3.8 -0.2

152.3

154.9 5.1 1.7

162.5

171.0 9.7 5.1

68.6

68.4 3.3 -0.6

40.8

pIV C [kPa℄ 39.0 4.0 -3.7

1.334

1.284 0.043 -6.9

1.335

1.269 0.034 -7.8

1.336

1.288 0.019 -5.3

1.336

1.293 0.013 -5.5

1.335

1.211 0.017 -9.6

1.336

γ300 [-℄ 1.253 0.050 -6.6

b [K −1 ℄ −3.12 · 10−5 3.18 · 10−5 -57.9 −7.42 · 10−5 −1.13 · 10−5 1.3 · 10−5 -84.7 −7.42 · 10−5 −2.22 · 10−5 0.754 · 10−5 -70.1 −7.42 · 10−5 −2.33 · 10−5 1.32 · 10−5 -68.6 −7.42 · 10−5 −2.22 · 10−5 2.02 · 10−5 -70.1 −7.42 · 10−5 −2.55 · 10−5 1.79 · 10−5 -65.6 −7.42 · 10−5 475

754 304 58.8

475

772 282 62.5

475

631 238 32.8

475

371 85 -22.0

475

677 175 42.5

475

Tw [K℄ 606 203 27.6

0.882

3.55 1.95 302.1

0.882

3.92 2.62 344.3

0.882

2.8 3.32 217.6

0.882

3.12 2.49 254.3

0.882

2.46 2.18 179.1

0.882

Vcr [cm3 ℄ 2.72 2.11 208.8

-2.7

-3.0 1.6 -8.7

-3.8

-3.9 2.6 -16.7

-2.8

-2.8 0.5 -1.0

-2.3

-2.9 1.3 -8.7

0.6

0.2 0.3 -0.1

0.5

∆p [kPa℄ 0.1 1.4 -17.5

0.47

0.06 0.02 -87.7

0.47

0.05 0.00 -89.8

0.47

0.18 0.22 -62.7

0.47

0.09 0.08 -80.4

0.47

0.34 0.39 -28.3

0.47

∆θ [deg℄ 0.06 0.03 -86.7

1.000

1.021 0.062 2.1

1.000

1.011 0.032 1.1

1.000

1.014 0.048 1.4

1.000

1.002 0.058 0.2

1.000

1.037 0.046 3.7

1.000

Kp [-℄ 1.071 0.107 7.1

C2 [m/(s K)℄ 3.13 · 10−4 0.81 · 10−4 -3.2 3.24 · 10−4 3.12 · 10−4 0.41 · 10−4 -3.8 3.24 · 10−4 2.86 · 10−4 0.61 · 10−4 -11.8 3.24 · 10−4 3.16 · 10−4 0.25 · 10−4 -2.6 3.24 · 10−4 3.18 · 10−4 0.09 · 10−4 -1.8 3.24 · 10−4 3.10 · 10−4 0.26 · 10−4 -4.3 3.24 · 10−4 1797

1566 180 -4.7

1502

1400 52 -4.3

1756

1562 100 -9.0

1864

1685 124 -7.7

731

732 60 5.9

373

Qin [J℄ 345 31 -1.6

-24.4

-29.5 5.8 13.9

-28.0

-32.8 5.0 17.2

-12.3

-17.9 3.5 45.2

-5.3

-13.3 4.6 153.4

-26.0

-28.5 5.4 9.7

-32.6

θig [deg ATDC℄ -31.0 3.9 -4.8

21.1

26.2 4.8 19.5

22.0

27.8 4.2 24.0

17.3

24.4 3.2 39.7

15.4

25.3 4.4 63.2

19.9

22.7 5.5 15.4

27.5

∆θd [deg℄ 25.7 3.6 -6.9

19.1

17.7 1.4 -6.6

18.7

17.7 1.2 -5.6

19.1

18.5 1.1 -4.2

19.3

18.3 0.9 -4.5

16.0

15.7 0.8 -2.4

20.3

∆θb [deg℄ 20.3 1.4 -0.9

ε¯# x [%℄ 79.7 79.7 79.7 0.0 74.8 74.8 74.8 0.0 95.7 95.7 95.7 0.0 79.1 79.1 79.1 0.0 102.9 102.9 102.9 0.0 91.8 91.8 91.8 0.0

Table 8.15: Mean value, standard deviation σest , relative mean error (RME) and nominal parameter deviation ε¯# x of the estimate for method 1, evaluated at d∗ for 40 experimental y les at OP16.

OP6

OP5

OP4

OP3

OP2

OP1

C1 [-℄ 3.60 2.08 58.1

Vc [cm3 ℄ 56.7 4.7 -3.5

212 CHAPTER 8. RESULTS AND EVALUATION FIRED

...

8.2. EXPERIMENTAL RESULTS  FIRED CYCLES

213

time a high pIV C , an advan ed ignition angle θig and a prolonged burn delay angle ∆θd as ompared to the nominal values is notable. The la k of a

ura y for the estimates is due to the high mean number of e ient parameters found in table 8.14. These high numbers are probably due to that the Akaike FPE assumes that the system onsidered is overed by the model stru ture used. All models are approximations and this is the ase here, espe ially for ring y les, sin e not every physi al pro ess inuen ing the measured ylinder pressure is modeled. The

ombination of modeling error and the measurement noise makes the Akaike riterion be ome minimized for a high value of d∗ . In a

ordan e with the simulation-based evaluation in se tion 8.1.3, method 1 is not re ommended to use for parameter estimation in the given formulation. The attention is therefore turned to method 2.

8.2.4

Method 2 Results and evaluation

In the experimental evaluation of method 2 for motored y les in se tion 7.2.4, a fourth version alled M2:3+was in luded. The algorithm was motivated and fully des ribed in appendix C.3, and it will also be studied here in the experimental evaluation for ring y les. First the results are presented for the two ases of δi :s, δi = c and δi 6= c, and then the performan e of the four versions M2:1, M2:2, M2:3 and M2:3+ are evaluated. The same implementation settings, as for the simulation-based evaluation in se tion 8.1.4, are used and are therefore not repeated here. Results for ase 1 (δi = c) and ase 2 (δi 6= c)

The results are rst des ribed for ase 1. An example of an L- urve is given in gure 8.7. Table 8.16 summarizes the mean value and standard deviation of the omputed regularization parameter δx for 40 experimental y les. This is done for all six operating points and for ea h of the four versions of method 2. The individual parameter estimates are given as mean values for 40 y les at ea h OP in table 8.18, as well as the orresponding (emphasized) nominal values x# and the nominal parameter deviation ε¯# x. The orresponding gures and tables for ase 2 are given in gure 8.8, table 8.17, table 8.19 that now also in lude the weighted nominal parameter deviation RMSE(Lδ ε# x ). Evaluation for ase 1 (δi = c)

Case 1 is onsidered rst. Figure 8.7 illustrates that the L- urve does not have as sharp transition as in the orresponding simulated ase given in gure 8.4. The dieren e is believed to be due to that the

214

...

CHAPTER 8. RESULTS AND EVALUATION FIRED

Method 2; OP1

0

Zoomed version

10

−1

10

−2

x

RMSE(ε# ) [−]

10

−4

10

−6

10

M2:3 M2:3+ M2:2 M2:1

−8

10

−2

−2

10 −1

10

10 RMSE( ε ) [bar]

−2

10

−1

10 RMSE( ε ) [bar]

Figure 8.7: L- urve (solid line) for one engine y le at OP1, when using

ase 1 of the δi (δi = c). The results for the four versions of method 2 are indi ated by the legend.

δx : δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+)

mean σest mean σest mean σest mean σest

OP1 1.8 · 10−2 2.0 · 10−2 4.9 · 10−7 2.2 · 10−6 2.8 · 10−5 4.9 · 10−4 3.2 · 10−3 3.8 · 10−3

OP2 4.0 · 10−1 8.8 · 10−2 2.9 · 10−7 1.2 · 10−7 7.3 · 10−6 3.3 · 10−5 9.7 · 10−3 2.0 · 10−3

OP3 3.9 · 100 1.4 · 100 9.5 · 10−6 1.0 · 10−5 7.3 · 10−6 1.3 · 10−4 8.8 · 10−2 3.1 · 10−2

OP4 6.7 · 10−1 1.5 · 100 1.1 · 10−6 1.2 · 10−5 6.2 · 10−6 2.6 · 10−5 6.3 · 10−2 1.8 · 10−2

OP5 7.3 · 10−1 7.1 · 10−1 3.4 · 10−6 3.7 · 10−4 8.0 · 10−6 1.5 · 10−4 4.1 · 10−2 1.1 · 10−2

OP6 1.1 · 100 4.2 · 10−1 8.8 · 10−7 2.0 · 10−6 1.5 · 10−6 1.2 · 10−5 4.8 · 10−2 1.1 · 10−2

Table 8.16: Mean value and standard deviation (σest ) of δx for the four versions of method 2 using ase 1, evaluated for 40 experimental y les at OP16.

8.2. EXPERIMENTAL RESULTS  FIRED CYCLES

215

model stru ture is not orre t. Figure 8.7 also shows that the four versions yield dierent regularization parameters δx . Version M2:1 is the most onservative of them all and yields the highest δx , losely followed by M2:3+. Then there is a gap to the two versions M2:2 and M2:3, whi h in turn render approximately the same δx . Both M2:3 and M2:3+ orrespond to orners on the L- urve, although the orner for M2:3 is hardly visible to the eye. For all six operating points table 8.16 shows that the regularization parameter for the four versions are ordered in size a

ording to δx (M 2 : 1) > δx (M 2 : 3+) > δx (M 2 : 3) > δx (M 2 : 2),

(8.3)

ex ept for OP3 where it is the other way around for M2:2 and M2:3. As expe ted the dieren e between δx (M 2 : 1) and δx (M 2 : 3+) is smaller than between δx (M 2 : 1) and δx (M 2 : 3). Version M2:3+ is more exible to hanges in operating ondition than M2:1, sin e it nds a

onvex orner of the L- urve. For the examined operating points, M2:1 yields a higher δx than M2:3+. Using M2:3 in its original form or M2:2 results in lower values of δx . For M2:3 this orresponds to the high value of the urvature found in gure C.2. Table 8.16 also shows that the mean value of δx depends upon operating ondition, and that for a given version of method 2 δx is in the same order of magnitude for all six operating points. The dierent δx :s result in dierent nominal parameter deviations and individual estimates, as shown in table 8.18 for OP1-6. The ε¯# x olumn shows that M2:1 yields the smallest nominal parameter deviation of all versions, losely followed by M2:3+. The nominal deviations ε¯# x are within 2.2 % and 4.4 % respe tively for all operating points onsidered. The other two versions, M2:2 and M2:3, render a signi antly larger ε¯# x , where the parameters b, γ300 and TIV C give unreasonable values. Of these parameters, b is the largest ontributor to the nominal deviations. At some instan es the ignition angle θig is advan ed

ompared to the nominal value, and this is ompensated by a longer ame development angle ∆θd , see e.g. OP2. In the experimental evaluation for motored y les it was found that Kp yielded unreasonable estimates, see table 7.21, but this is not the ase for the ring y les. Of the other parameters pIV C , Kp , C2 , Qin and ∆θb all end up within 5 % of their nominal values for all operating points. The individual estimates for M2:1 and M2:3+ are all reasonable, whi h is expe ted sin e they end up lose to their nominal values. The maximum deviation o

urs for the parameters θig and ∆θd for both versions, the others are all within 5 %. To summarize the evaluation for δi = c, using M2:1 or M2:3+ yield estimates that deviate less than 3 and 5 per ent in the mean.

216

...

CHAPTER 8. RESULTS AND EVALUATION FIRED

Method 2; OP1

0

Zoomed version

10

−1

10

−2

δ # RMSE(L εx ) [−]

10

−3

10

−1

10

−4

10

−5

10

M2:3 M2:3+ M2:2 M2:1

−6

10

−7

10

−2

−1

10

10 RMSE( ε ) [bar]

−2

10

−1

10 RMSE( ε ) [bar]

Figure 8.8: L- urve (solid line) for one engine y le at OP1, when using

ase 2 of the δi (δi 6= c). The results for the four versions of method 2 are indi ated by the legend.

δx : δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+)

mean σest mean σest mean σest mean σest

OP1 7.7 · 10−4 6.1 · 10−4 4.3 · 10−5 3.4 · 10−4 1.5 · 10−6 1.8 · 10−5 2.2 · 10−4 7.9 · 10−4

OP2 1.4 · 10−2 3.0 · 10−3 5.6 · 10−5 3.0 · 10−5 5.8 · 10−7 6.4 · 10−7 4.6 · 10−4 3.0 · 10−4

OP3 1.7 · 10−1 3.8 · 10−2 3.8 · 10−3 2.9 · 10−3 8.5 · 10−5 1.3 · 10−3 1.3 · 100 2.4 · 100

OP4 3.5 · 10−2 5.2 · 10−2 1.4 · 10−4 8.0 · 10−4 7.3 · 10−6 9.5 · 10−5 7.1 · 10−4 1.8 · 10−3

OP5 3.2 · 10−2 2.1 · 10−2 1.9 · 10−3 6.7 · 10−3 1.2 · 10−4 1.9 · 10−3 1.6 · 10−3 1.5 · 10−3

OP6 5.3 · 10−2 2.3 · 10−2 7.1 · 10−4 1.5 · 10−2 8.7 · 10−6 3.6 · 10−4 1.8 · 10−2 1.1 · 10−1

Table 8.17: Mean value and standard deviation (σest ) of δx for the four versions of method 2 using ase 2, evaluated for 40 experimental y les at OP16.

8.2. EXPERIMENTAL RESULTS  FIRED CYCLES

217

Evaluation for ase 2 (δi 6= c)

Now the se ond ase will be evaluated. The L- urve in gure 8.8 is similar to the orresponding L- urve for the simulated ase in gure 8.5, although it is not as sharp in the transition. As for ase 1, both M2:3 and M2:3+ orrespond to orners on the L- urve, although the orner for M2:3 is hardly visible to the eye. Figure 8.8 also displays that the three versions M2:1, M2:2 and M2:3+ yield approximately the same δx for OP1, while M2:3 yields a smaller δx . This observation is onrmed by table 8.17. However the variation in δx for M2:1, M2:2 and M2:3+ is larger for the other ve operating points, and thus ree t that the versions typi ally are a little more spread out on the L- urve as ompared to OP1. Table 8.17 shows that for all six operating points the regularization parameter for the four versions are ordered in size a

ording to δx (M 2 : 1) > δx (M 2 : 3+) > δx (M 2 : 2) > δx (M 2 : 3),

(8.4)

ex ept for OP3 where it is the other way around for M2:1 and M2:3+. Compared to (8.3), the ordering is the same apart from M2:2 and M2:3 whi h now have hanged pla es. As for the rst ase, M2:3 results in low values of δx , due to the same reason. It is also found that the mean value of δx depends upon operating onditions, and that for a given version of method 2 δx is in the same order of magnitude for almost all six operating points. The only ex eption is OP3 for version M2:3+. As expe ted, the dierent δx :s result in dierent nominal parameter deviations and individual estimates, as shown in table 8.19 for OP1-6. The olumn for weighted parameter deviation, Lδ ε# x , shows that M2:1 and then M2:3+ yield the smallest nominal parameter deviations of all versions, ex ept for OP3 where it is the other way around. This

orresponds to the ordering given in (8.4). The other two versions, M2:2 and M2:3, render a signi antly larger Lδ ε# x , mainly due to low estimates of γ300 and TIV C as seen for OP2. Considering the ε¯# x olumn, these values are larger than the orresponding olumn for δi = c, see table 8.18. This is mainly due to the relatively large ontribution of nominal deviation in ∆θ for δi 6= c. The nominal deviations ε¯# x for M2:1 and M2:3+ are within 67 % and 76 % respe tively for all operating points onsidered. Now onsider the individual estimates in table 8.19; Just as for δi = c, the ignition angle θig is advan ed ompared to the nominal value, and this is ompensated by a longer ame development angle ∆θd . Another trend is that ∆θ is larger than the nominal value for all operating points. For M2:2 and M2:3 the parameters pIV C , Vcr , Kp , C2 , Qin and ∆θb all end up within 5 % of their nominal values. These parameters are the same as for δi = c, but now also in ludes Vcr . For M2:3+the estimates of C1 , pIV C , γ300 , Tw , Vcr , Kp, C2 , Qin , ∆θb and

δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x# δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x# δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x# δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x# δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x# δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x#

:

58.8

Vc

[cm3℄ 58.2 58.2 57.9 58.2 58.8 57.8 50.4 50.3 52.1 58.8 59.4 52.7 53.4 59.0 58.8 59.0 53.7 54.9 58.2 58.8 57.9 56.4 56.1 55.8 58.8 58.4 57.9 58.0 56.9 2.28

[-℄ 2.29 2.13 2.42 2.29 2.28 2.28 2.42 2.41 2.29 2.28 2.28 1.80 1.72 2.29 2.28 2.28 1.80 2.17 2.29 2.28 2.28 1.86 2.08 2.29 2.28 2.28 2.13 1.82 2.29

C1

392

[K℄ 449 325 362 449 450 424 270 281 413 425 406 270 270 397 407 390 270 270 384 394 396 290 291 391 400 389 270 270 382

TIV C

153.3

pIV C

[kPa℄ 40.8 42.0 40.8 40.9 40.8 68.6 69.0 68.9 68.1 68.6 162.7 165.9 165.4 161.9 162.5 151.9 155.2 156.4 151.3 152.3 131.5 134.0 134.3 131.3 132.2 152.4 157.4 154.2 152.2 1.335

[-℄ 1.344 1.299 1.296 1.346 1.336 1.349 1.215 1.218 1.326 1.335 1.341 1.231 1.256 1.378 1.339 1.359 1.254 1.262 1.375 1.339 1.358 1.293 1.296 1.356 1.335 1.357 1.295 1.296 1.356

γ300 −7.38 · 10−5 −5.01 · 10−5 −5.22 · 10−5 −7.34 · 10−5 −7.42 · 10−5 −7.37 · 10−5 −3.16 · 10−5 −3.39 · 10−5 −7.12 · 10−5 −7.42 · 10−5 −7.39 · 10−5 −0.0864 · 10−5 −1.83 · 10−5 −7.14 · 10−5 −7.42 · 10−5 −7.31 · 10−5 −1.21 · 10−5 −2.65 · 10−5 −7.14 · 10−5 −7.42 · 10−5 −7.34 · 10−5 −4.37 · 10−5 −5.25 · 10−5 −7.19 · 10−5 −7.42 · 10−5 −7.35 · 10−5 −5.4 · 10−5 −4.95 · 10−5 −7.2 · 10−5 −7.42 · 10−5

b

[K −1℄

475

Tw

[K℄ 474 443 479 473 475 474 478 477 472 475 475 488 418 471 475 474 519 503 471 475 474 566 469 472 475 474 578 606 472 0.882

Vcr

[cm3 ℄ 0.881 1.56 0.916 0.881 0.882 0.881 0.89 0.89 0.879 0.882 0.882 1.07 1.07 0.882 0.882 0.882 1.55 1.05 0.882 0.882 0.881 1.2 1.12 0.88 0.882 0.881 0.96 1.04 0.881 -2.7

∆p

[kPa℄ 0.5 0.4 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.6 -2.2 -2.4 -2.4 -2.2 -2.2 -2.8 -3.4 -3.0 -2.8 -2.8 -3.8 -4.3 -4.1 -3.8 -3.8 -2.7 -2.8 -2.9 -2.7 0.47

∆θ

[deg℄ 0.47 0.81 0.52 0.47 0.47 0.47 0.57 0.56 0.47 0.47 0.47 1.11 1.00 0.47 0.47 0.47 1.61 0.88 0.47 0.47 0.47 0.77 0.74 0.47 0.47 0.47 0.62 0.73 0.47 1.000

3.24 · 10−4

C2

[-℄ [m/(s K)℄ 0.987 3.24 · 10−4 0.978 2.99 · 10−4 1.002 3.23 · 10−4 0.984 3.24 · 10−4 1.000 3.24 · 10−4 0.988 3.24 · 10−4 1.041 3.24 · 10−4 1.041 3.24 · 10−4 0.974 3.24 · 10−4 1.000 3.24 · 10−4 1.001 3.24 · 10−4 1.020 3.07 · 10−4 1.007 3.09 · 10−4 0.963 3.24 · 10−4 1.000 3.24 · 10−4 0.987 3.24 · 10−4 0.995 3.10 · 10−4 0.998 3.21 · 10−4 0.963 3.24 · 10−4 1.000 3.24 · 10−4 0.984 3.24 · 10−4 0.989 2.99 · 10−4 0.992 3.06 · 10−4 0.969 3.24 · 10−4 1.000 3.24 · 10−4 0.991 3.24 · 10−4 0.982 3.24 · 10−4 0.999 3.12 · 10−4 0.979 3.24 · 10−4

Kp

1797

-24.4

θig

[J℄ [deg ATDC℄ 368 -32.2 366 -31.8 364 -31.6 365 -32.1 373 -32.6 721 -25.3 744 -34.8 745 -34.7 715 -24.9 731 -26.0 1864 -5.2 1914 -12.0 1878 -9.8 1802 -5.0 1864 -5.3 1738 -11.8 1827 -20.2 1779 -17.6 1701 -11.6 1756 -12.3 1485 -27.2 1449 -31.1 1467 -32.0 1465 -26.9 1502 -28.0 1790 -23.7 1780 -28.2 1728 -27.5 1771 -23.5

Qin

21.1

∆θd

[deg℄ 27.9 27.2 27.0 27.9 27.5 20.4 30.0 30.0 21.2 19.9 16.0 24.3 22.2 17.7 15.4 18.5 27.6 24.7 19.2 17.3 22.5 26.5 27.5 23.0 22.0 21.6 26.1 25.5 22.1

19.1

∆θb

[deg℄ 20.0 20.0 20.3 19.7 20.3 15.9 15.7 15.6 15.1 16.0 19.2 18.9 18.8 18.6 19.0 18.9 18.9 18.6 18.3 19.1 18.3 18.0 17.9 17.5 18.7 18.7 18.1 18.1 17.8

ε¯# x

[%℄ 1.1 38.9 10.7 1.6 0.0 1.1 24.2 23.4 4.1 0.0 1.2 58.4 46.3 4.4 0.0 2.2 74.8 36.9 3.7 0.0 1.4 33.1 26.8 2.9 0.0 1.3 20.5 28.9 2.8 0.0

Table 8.18: Mean value of the estimate for the four versions of method 2 using ase 1, evaluated for 40 experimental y les at OP1-6. The nominal values x# are also in luded.

OP6

OP5

OP4

OP3

OP2

OP1

δx

218 CHAPTER 8. RESULTS AND EVALUATION FIRED

...

δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x# δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x# δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x# δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x# δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x# δx (M 2 : 1) δx (M 2 : 2) δx (M 2 : 3) δx (M 2 : 3+) x#

:

58.8

Vc

[cm3℄ 58.9 59.4 59.3 58.1 58.8 57.2 55.0 54.6 56.8 58.8 57.8 55.1 53.5 58.1 58.8 57.0 54.6 53.4 55.4 58.8 57.9 57.4 57.7 58.1 58.8 58.2 58.0 59.3 58.3 2.28

[-℄ 2.28 2.21 1.99 2.24 2.28 2.25 1.94 1.92 2.22 2.28 2.29 2.15 2.03 2.26 2.28 2.25 2.12 2.09 2.16 2.28 2.27 2.14 2.09 2.24 2.28 2.27 2.12 1.25 2.25

C1

392

[K℄ 448 390 375 423 450 413 286 273 405 425 405 374 356 399 407 382 352 355 362 394 394 345 362 390 400 386 338 381 382

TIV C

153.3

pIV C

[kPa℄ 40.2 40.3 42.1 41.4 40.8 65.1 69.6 71.1 64.1 68.6 164.4 158.6 157.0 163.6 162.5 151.2 147.1 148.8 147.7 152.3 130.2 133.1 131.1 129.4 132.2 152.4 154.6 151.7 152.6 1.334

[-℄ 1.334 1.305 1.274 1.305 1.336 1.329 1.243 1.229 1.325 1.335 1.333 1.303 1.285 1.332 1.336 1.326 1.296 1.272 1.306 1.336 1.332 1.305 1.317 1.330 1.335 1.333 1.307 1.329 1.331

γ300 −5.91 · 10−5 −4.97 · 10−5 −2.9 · 10−5 −4.96 · 10−5 −7.42 · 10−5 −3.75 · 10−5 −0.742 · 10−5 −0.00404 · 10−5 −3.58 · 10−5 −7.42 · 10−5 −5.02 · 10−5 −3.15 · 10−5 −2.19 · 10−5 −5.89 · 10−5 −7.42 · 10−5 −4.46 · 10−5 −2.83 · 10−5 −1.92 · 10−5 −3.28 · 10−5 −7.42 · 10−5 −4.47 · 10−5 −4.13 · 10−5 −4.18 · 10−5 −4.24 · 10−5 −7.42 · 10−5 −5.05 · 10−5 −4.45 · 10−5 −3.35 · 10−5 −5.17 · 10−5 −7.42 · 10−5

b

[K −1℄

475

Tw

[K℄ 475 558 536 490 475 480 538 549 483 475 476 488 489 478 475 480 490 506 486 475 476 500 482 476 475 476 500 533 478 0.882

Vcr

[cm3℄ 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 0.882 -2.7

∆p

[kPa℄ 1.5 1.0 -1.4 0.0 0.5 3.8 -1.0 -2.9 5.0 0.6 -3.1 4.6 7.3 -1.7 -2.3 -1.2 3.3 0.2 2.5 -2.8 -3.5 -6.4 -4.1 -2.5 -3.8 -2.9 -6.6 -2.4 -3.0 0.47

∆θ

[deg℄ 0.80 0.69 0.57 0.70 0.47 1.64 1.38 1.28 1.78 0.47 0.45 0.98 1.16 0.46 0.47 0.78 1.30 1.38 1.20 0.47 1.08 0.93 1.06 1.18 0.47 0.93 0.97 1.10 0.88 1.000

3.24 · 10−4

C2

[-℄ [m/(s K)℄ 1.000 3.24 · 10−4 0.999 3.24 · 10−4 0.999 3.24 · 10−4 1.000 3.24 · 10−4 1.000 3.24 · 10−4 0.999 3.24 · 10−4 0.999 3.24 · 10−4 1.000 3.24 · 10−4 0.999 3.24 · 10−4 1.000 3.24 · 10−4 0.996 3.24 · 10−4 0.996 3.24 · 10−4 0.998 3.24 · 10−4 1.000 3.24 · 10−4 1.000 3.24 · 10−4 0.996 3.24 · 10−4 0.999 3.24 · 10−4 1.005 3.24 · 10−4 0.998 3.24 · 10−4 1.000 3.24 · 10−4 0.998 3.24 · 10−4 0.999 3.24 · 10−4 0.998 3.24 · 10−4 0.998 3.24 · 10−4 1.000 3.24 · 10−4 0.999 3.24 · 10−4 1.002 3.24 · 10−4 0.992 3.24 · 10−4 0.999 3.24 · 10−4

Kp

1797

-24.4

θig

[J℄ [deg ATDC℄ 366 -32.4 360 -32.3 364 -31.8 375 -32.5 373 -32.6 723 -30.1 745 -33.7 756 -34.0 731 -32.3 731 -26.0 1792 -7.9 1845 -16.2 1885 -17.2 1850 -7.6 1864 -5.3 1715 -17.1 1794 -21.6 1863 -21.5 1770 -21.6 1756 -12.3 1482 -29.4 1488 -31.5 1482 -31.5 1489 -31.3 1502 -28.0 1793 -26.4 1821 -28.4 1749 -28.7 1795 -28.4

Qin

21.1

∆θd

[deg℄ 28.6 27.9 26.7 28.3 27.5 27.6 30.0 30.0 30.0 19.9 20.4 29.0 30.0 20.1 15.4 24.7 29.6 29.5 29.5 17.3 26.0 27.5 27.8 27.9 22.0 25.3 27.1 27.4 27.2

19.1

∆θb

[deg℄ 20.0 20.0 20.1 20.1 20.3 16.1 16.3 16.3 16.2 16.0 18.8 19.3 19.6 18.7 19.3 18.6 19.1 19.2 19.0 19.1 18.1 18.1 18.2 18.2 18.7 18.2 18.3 18.3 18.1

[%℄ 4.2 18.5 35.4 10.8 0.0 9.6 30.1 33.8 12.3 0.0 6.0 18.7 24.4 5.2 0.0 9.1 19.7 25.2 17.1 0.0 6.8 14.8 13.8 9.7 0.0 5.4 13.8 39.1 7.1 0.0

L δ ε# x

ε¯# x

[%℄ 33.5 41.4 47.2 39.0 0.0 66.6 57.7 58.6 76.0 0.0 17.6 74.1 89.9 19.2 0.0 27.2 62.7 69.8 56.6 0.0 35.4 45.2 44.8 42.3 0.0 26.9 45.0 80.2 26.0 0.0

Table 8.19: Mean value of the estimate for the four versions of method 2 using ase 2, evaluated for 40 experimental y les at OP1-6. The nominal values x# are also in luded.

OP6

OP5

OP4

OP3

OP2

OP1

δx

8.2. EXPERIMENTAL RESULTS  FIRED CYCLES

219

220

...

CHAPTER 8. RESULTS AND EVALUATION FIRED

for M2:1 also the parameters Vc and TIV C , are all within 5 % of their nominal value. Compared to δi = c, fewer parameters are within this limit, but as before all the estimates are still reasonable. To summarize the evaluation for δi 6= c, using M2:1 or M2:3+ yield weighted nominal parameter deviations that are less than 10 and 17 per ent respe tively. Summary of method 2 To summarize, ase 1 and 2 yield dierent estimates and for the versions M2:2 and M2:3 almost the same parameters are lose (within 5 %) of their nominal values. For M2:1 and M2:3+ more parameters are within this limit, espe ially for ase 1. It an however not be determined whi h ase that gives the best estimates, sin e the true values are unknown. But it has been shown that the extension of M2:3 to M2:3+ has resulted in a more robust algorithm. It has also been shown that M2:1 and M2:3+ give more reasonable estimates, ompared to M2:2 and M2:3. This is in line with the results from the simulation study. The simulation study also showed that ase 2 (δi 6= c) is to be preferred, and therefore this is the re ommended hoi e. Unlike M2:1, M2:3+ nds a onvex orner of the L- urve. This makes M2:3+ more exible to hanges in operating ondition. Therefore if time is available, M2:3+ is preferable to M2:1.

8.3 Summary of results for red y les For method 1 the same on lusions as for the motored y les in se tion 7.3 an be drawn; Both the simulation and experimental studies showed that method 1 did not render a

urate estimates. It was even hard to de ide upon the number of e ient parameters to use. However, method 1 allows for the parameters to be ordered in how e ient they are for the given estimation problem and data. It was also shown that γ300 is the most e ient parameter. A

ording to the simulation-based evaluation, method 2 is more robust to an introdu ed false prior than method 1 sin e it yields more a

urate parameter estimates. The dieren e in a

ura y is however smaller between method 1 and 2 for ring y les ompared to the motored ase. Just as for the motored ase, the simulation study showed that the regularization pulls some of the parameters towards their nominal values, while the e ient parameters are free to t the data. Method 2 was originally formulated for three variants. For the simulation-based evaluation variant M2:3 is the most a

urate one, followed by M2:2 and M2:1. From the experimental evaluation it was found that an extension of variant M2:3, alled M2:3+, was needed to

8.4.

FUTURE WORK

221

ope with unreasonably high urvature for low δx that an o

ur for experimental data. With this extension, the two variants M2:1 and M2:3+ both give reasonable estimates, and where M2:3+ is more exible to hanges in operating ondition of the two. Considering the two ases of parameter un ertainty, δi = c and δi 6= c, it was found in the simulations that the se ond ase was re ommended sin e it pays o in better estimates that are more robust to a false nominal parameter value. The experimental evaluation an neither onrm or de line this observation, sin e both ases yield reasonable estimates, at least for variants M2:1 and M2:3+. Thus if omputational time is available, variant M2:3+ of method 2 is re ommended due to its a

ura y. However if omputational time is an important feature variant M2:1 is the best hoi e. The hosen variant should preferably be ombined with ase 2 (δi 6= c), i.e. individually set parameter un ertainty. The ompiled on lusions from the evaluations of motored and red data are given in se tion 9.3. 8.4

Future Work

It would be interesting to in lude a multi-zone ylinder pressure model as a referen e model in the simulation study done in se tion 8.1. Su h a study would shed more light on what happens when the hosen model stru ture is not overed by the ylinder pressure data, while the true value of the parameters are known. It would therefore resemble the situation for the experimental ring y les better. The study ould also in lude the model of the spe i heat ratio developed in hapter 4. This extended model, in hapter 4 named model D1 , is simulated by using (3.36)(3.43), (4.28), (A.9) and algorithm A.1. This in lusion would allow for a dire t performan e omparison of the standard and extended ylinder pressure models.

222

...

CHAPTER 8. RESULTS AND EVALUATION FIRED

9

Summary and on lusions

The theme of the thesis is ylinder pressure modeling and estimation. The results from part I are given in se tion 9.1, whi h repeats the

on lusions from se tion 4.7. In the same way, the results for part II are given in se tion 9.2, whi h repeats the on lusions from se tion 5.6. The ompiled on lusions for part III are then given in se tion 9.3.

9.1 A spe i heat ratio model for singlezone heat release models The rst part of the thesis is on single-zone heat release modeling, where the spe i heat ratio model onstitutes a key part. Chapter 2 gives an overview of single-zone heat release models, while hapter 3 gives a more thorough des ription of the model omponents. In hapter 4 various spe i heat ratio models are investigated. The on lusions from hapter 4 are now given here. Based on assumptions of frozen mixture for the unburned mixture and hemi al equilibrium for the burned mixture, the spe i heat ratio is al ulated, using a full equilibrium program, for an unburned and a burned air-fuel mixture, and ompared to several previously proposed models of γ . It is shown that the spe i heat ratio and the spe i heats for the unburned mixture are aptured to within 0.25 % by a linear fun tion in mean harge temperature T for λ ∈ [0.8, 1.2]. Furthermore the burned mixture is aptured to within 1 % by the higher223

224

CHAPTER 9.

SUMMARY AND CONCLUSIONS

order polynomial in ylinder pressure p and temperature T developed in Krieger and Borman (1967) for the major operating range of a spark ignited (SI) engine. If a linear model is preferred for omputational reasons for the burned mixture, then the temperature region should be

hosen with are whi h an redu e the modeling error in γ by 25 %. With the knowledge of how to des ribe γ for the unburned and burned mixture respe tively, the fo us is turned to nding a γ -model during the ombustion pro ess, i.e. for a partially burned mixture. This is done by interpolating the spe i heats for the unburned and burned mixture using the mass fra tion burned xb . The obje tive was to nd a model of γ , whi h results in a ylinder pressure error that is lower than or in the order of the measurement noise. It is found that interpolating the linear spe i heats for the unburned mixture and the higher-order polynomial spe i heats for the burned mixture, and then forming the spe i heat ratio γ(T, p, xb ) =

lin xb cKB cp (T, p, xb ) p,b + (1 − xb ) cp,u = KB + (1 − x ) clin cv (T, p, xb ) xb cv,b b v,u

(9.1)

results in a small enough modeling error in γ . This modeling error results in a ylinder pressure error that is lower than 6 kPa in mean, whi h is in the same order as the ylinder pressure measurement noise. It was also shown that it is important to evaluate the model error in γ to see what impa t it has on the ylinder pressure, sin e a small error in γ an yield a large ylinder pressure error. This also stresses that the γ -model is an important part of the heat release model. Applying the proposed model improvement D1 (9.1) of the spe i heat ratio to the Gatowski et al. (1984) single-zone heat release model is simple, and it does not in rease the omputational burden immensely. Compared to the original setting, the omputational burden in reases with 40 % and the modeling error introdu ed in the ylinder pressure is redu ed by a fa tor 15 in mean. 9.2

Compression ratio estimation

Four methods for ompression ratio estimation based on ylinder pressure tra es are developed and evaluated for both simulated and experimental y les in hapter 5. The on lusions are given here.

Con lusions from the simulation results The rst three methods rely upon the assumption of a polytropi ompression and expansion. It is shown that this is su ient to get a rough estimate of the ompression ratio rc for motored y les, espe ially for a low rc and by letting the polytropi exponent be ome small. For a high

9.2.

COMPRESSION RATIO ESTIMATION

225

rc it is important to take the heat transfer into a

ount, and then only method 4 is a

urate to within 0.5 % for all operating points. Method 4 is however slow and not suitable for on-line implementation. Method 2 on the other hand is substantially faster and still yields estimates that are within 1.5 %. The formulation of the residual is also important, sin e it inuen es the estimated rc . For red y les, methods 1-3 yield poor estimates and therefore only method 4 is re ommended. A sensitivity analysis, with respe t to rank angle phasing, ylinder pressure bias, revi e volume, and heat transfer, shows that the third and fourth method are more robust. They therefore deal with these parameter deviations better than methods 1 and 2. Of the latter two, method 2 has the best performan e for all parameter deviations ex ept for an additive pressure bias.

Con lusions from the experimental results All methods yield approximately the same onden e intervals for the simulated and experimental data. The onden e intervals resulting from method 4 are smallest of all methods, but it suers from a high

omputational time. Method 2 yields smaller onden e intervals than methods 1 and 3, and is outstanding regarding onvergen e speed. The ee ts and trends shown in the simulation evaluation are also present in the experimental data. Therefore the on lusions made in the simulation evaluation with respe t to models, residuals, methods, heat transfer and revi e ee ts are the same for the experimental evaluation. For diagnosti purposes, all methods are able to dete t if the

ompression ratio is stu k at a too high or too low level.

Con luding re ommendations The a

ura y of the ompression ratio estimate is higher for motored

y les with high initial pressures. Thus if it is possible to hoose the initial pressure, it should be as high as possible. Using motored y les assures that all pressure information available is utilized and the high initial pressure improves the signal-to-noise ratio, while the ee ts of heat transfer and revi e ows remain the same. Two methods are re ommended; If estimation a

ura y has the highest priority, and time is available, method 4 should be used. Method 4 yields the smallest onden e intervals of all investigated methods for both simulated and experimental data. In the simulation ase where the true value of the ompression ratio is known, method 4 gave estimates with smallest bias. If omputational time is the most important property, method 2 is re ommended. It is the most omputationally e ient of all investigated methods, and yields the smallest onden e intervals out of methods 1-3.

226 9.3

CHAPTER 9.

SUMMARY AND CONCLUSIONS

Prior knowledge based heat release analysis

The obje tive in part III, as stated in the beginning of hapter 6, was to develop an estimation tool that is a

urate, systemati and e ient. For this purpose two methods that in orporate parameter prior knowledge in a systemati manner are presented in hapter 6. Method 1 is based on using a singular value de omposition (SVD) of the estimated hessian, to redu e the number of estimated parameters one-byone. Then the suggested number of parameters to use is found as the one minimizing the Akaike nal predi tion error. Method 2 uses a regularization te hnique to in lude the prior knowledge in the riterion fun tion. The ompiled on lusions from the evaluation of the methods on motored ( hapter 7) and red ( hapter 8) are given here. Method 1 restrains the estimation problem, sin e not all parameters are estimated simultaneously. More importantly, it introdu es an estimation bias in the e ient parameters sin e the un ertain parameters are almost unavoidably false. Method 2 enables estimation of all parameters simultaneously, whi h is an important feature when a parameter's prior knowledge is false. Compared to method 2, method 1 yields more biased parameter estimates in the ase of a false prior. It has also been shown by using method 1 that, given the Gatowski et al. ylinder pressure model, the onstant γ300 in the linear γ -model is the most important parameter. The drive for method 2 was to regularize the solution su h that the parameters that are hard to determine are pulled towards their nominal values, while the e ient parameters are free to t the data. This has shown to be the ase in the simulations for both red and motored

y les. The user an hose between two ases of the parameter un ertainty, either equal (in a normalized sense) or individually hosen. The former is dire tly appli able on e nominal values of the parameters are determined and yields good estimates in the simulations. For experimental motored y les it has been shown that the se ond ase gives more a

urate and reasonable parameter estimates, while this an not be determined from the ring y les. The on lusion is still to use the se ond ase. It requires more eort to de ide upon the un ertainty for ea h parameter, but pays o in better estimates that are more robust to a false nominal parameter value. On e a hoi e of parameter un ertainty has been done, no user intera tion is needed. Method 2 was originally formulated for three versions. The versions dier in how they determine how strong the regularization should be. For the simulation-based evaluation version M2:1 and M2:3 where found to be the most a

urate one for motored and red y les respe tively.

9.3.

PRIOR KNOWLEDGE BASED HEAT RELEASE ANALYSIS

227

From the experimental evaluation it was found that an extension of version M2:3, alled M2:3+, was needed. With this extension, the two versions M2:1 and M2:3+ both give reasonable estimates, and where M2:3+ is more exible to hanges in operating ondition of the two, a feature whi h is more required for ring y les than for motored y les. Method 2 an therefore be said to be systemati and a

urate, and therefore fullls two of the requirements for the estimation tool. Thus if omputational time is available, version M2:3+ of method 2 is re ommended due to its a

ura y. However if omputational time is an important feature version M2:1 is the best hoi e. The hosen version should preferably be ombined with individually set parameter un ertainties, i.e. ase 2. In this formulation the third requirement is also fullled. To summarize, the proposed tool for heat release analysis is e ient, systemati and a

urate, and an be used for engine alibration, as a diagnosti tool or as an analyzing tool for future engine designs.

228

CHAPTER 9.

SUMMARY AND CONCLUSIONS

Bibliography

H. Akaike. Fitting autoregressive models for predi tion. Stat. Math., (21):243347, 1969.

Ann. Inst.

C.A. Amann. Cylinder pressure measurement and its use in engine resear h. 1985. SAE Te hni al Paper 852067. M. Anatone and R. Cipollone. A ontribution in al ulating the thermal elds in internal ombustion engines omponents. Number 961127 in SAE Te hni al paper series, SAE World Congress, Detroit, USA, 1996. I. Andersson. Cylinder pressure and ionization urrent modeling for spark ignited engines. Te hni al report, Vehi ular Systems, 2002. LiUTEK-LIC-2002:35, Thesis No. 962. I. Andersson.

Ion Current and Torque Modeling for Combustion

. PhD thesis, Chalmers University, De ember 2005. ISBN 91-7291-706-7. Engine Control

W. J. D. Annand. Heat transfer in the ylinders of re ipro ating internal ombustion engines. Pro . Instn. Me hani al Engineers, Vol. 177, 1963. 36. I. Arsie, G. Flauti, C. Pianese, and G. Rizzo. Cylinder thermal state dete tion from pressure y le in SI engine. 4th International Conferen e on Internal Combustion Engines: Experiments and Modeling, Capri, Italy, 1999. 229

230

Bibliography

E. Bergstedt. A sliding mode observer for a nonlinear temperature estimation in a ylinderhead of a diesel engine. Master's thesis, Linköping University, SE-581 83 Linköping, 2002. LiTH-ISY-EX3255-2002. Å. Björ k. Numeri al Methods for Least Squares Problems. Siam, 1996. M.F.J. Brunt and A.L. Emtage. Evaluation of IMEP routines and analysis errors. In Spark-Ignition Engine Combustion and Emissions (SP-1157), 1996. SAE Te hni al Paper 960609. M.F.J. Brunt and A.L. Emtage. Evaluation of burn rate routines and analysis errors. In Spark-Ignition Engine Combustion and Emissions (SP-1267), 1997. SAE Te hni al Paper 970037. M.F.J. Brunt and C.R. Pond. Evaluation of te hniques for absolute

ylinder pressure orre tion. In Spark-Ignition Engine Combustion and Emissions (SP-1267), 1997. SAE Te hni al Paper 970036. M.F.J. Brunt, A. Rai, and A.L. Emtage. The al ulation of heat release energy from engine ylinder pressure data. In SI Engine Combustion, 1998. SAE Te hni al Paper 981052. H.M. Cheung and J.B. Heywood. Evaluation of a one-zone burn-rate analysis pro edure using produ tion SI engine pressure data. 1993. SAE Te hni al Paper 932749. K. M. Chun and J. B. Heywood. Estimating heat-release and mass-ofmixture burned from spark-ignition engine pressure data. Combustion S i. and Te h., Vol. 54:133143, 1987. Y.A. Çengel and M.A. Boles. Thermodynami s: An Engineering ApM Graw-Hill, 4th edition, 2002. http://highered.m grawhill. om.

proa h.

H.W. Engl, M. Hanke, and A.Ñeubauer. Regularization of inverse problems. Kluwer A ademi Publishers, London, 1996. ISBN 0-79234157-0. L. Eriksson. CHEPP a hemi al equilibrium program pa kage for matlab. volume SP-1830, 2004. SAE Te hni al Paper 2004-01-1460. L. Eriksson. Requirements for and a systemati method for identifying heat-release model parameters. Modeling of SI and Diesel Engines, SP-1330:1930, 1998. SAE Te hni al Paper no.980626. L. Eriksson. Spark Advan e Modeling and Control. PhD thesis, Linköping University, May 1999. ISBN 91-7219-479-0, ISSN 03457524.

Bibliography

231

J.W. Fox, W.K. Cheng, and J.B. Heywood. A model for predi ting residual gas fra tion in spark ignition engines. 1993. SAE Te hni al Paper 931025. J.A. Gatowski, E.N. Balles, K.M. Chun, F.E. Nelson, J.A. Ek hian, and J.B. Heywood. Heat release analysis of engine pressure data. 1984. SAE Te hni al Paper 841359. P.E. Gill, W. Murray, and M.H. Wright. Pra ti al Optimization. A ademi Press, 1981. S. Gordon and B.J. M Bride. Computer program for the al ulation of

omplex hemi al equilibrium omposition, ro ket performan e, in ident and ree ted sho ks, and hapman-jouguet detonations. Te hni al Report NASA Publi ation SP-273, NASA Lewis Resear h Center, 1971. Y.G. Guezenne and W. Hamama. Two-zone heat release analysis of

ombustion data and alibration of heat transfer orrelation in an I.C. engine. 1999. SAE Te hni al paper 1999-01-0218. P.C. Hansen. Rank-de ient and dis rete ill-posed problems: numeri al aspe ts of linear inversion. So iety for Industrial and Applied Mathemati s, Philadelphia, PA, USA, 1998. ISBN 0-89871-403-6.

P.C. Hansen. Regularization tools; a matlab pa kage for analysis and solution of dis rete ill-posed problems. Numeri al Algorithms, 6(1): 135, 1994. P.C. Hansen and D.P. O'Leary. The use of the l- urve in the regularization of dis rete ill-posed problems. SIAM Journal on S ienti Computing, 14(6):14871503, 1993. doi: 10.1137/0914086. URL http://link.aip.org/link/?SCE/14/1487/1. T.K. Hayes, R.A. White, and J.E. Peters. Combustion hamber temperature and instantaneous lo al heat ux measurements in a spark ignition engine. 1993. SAE Te hni al Paper 930217. J. B. Heywood. Internal Combustion Engine Fundamentals. M GrawHill series in me hani al engineering. M Graw-Hill, 1988. ISBN 0-07100499-8. G.F. Hohenberg. Advan ed approa hes for heat transfer al ulations. 1979. SAE Te hni al Paper 790825. B. Johansson. On Cy le to Cy le Variations in Spark Ignition Engines. PhD thesis, Lund Institute of Te hnology, 1995.

232

Bibliography

M. Klein. A spe i heat ratio model and ompression ratio estimation. Li entiate thesis, Vehi ular Systems, Linköping University, 2004. LiU-TEK-LIC-2004:33, Thesis No. 1104, http://www.fs.isy.liu.se/Publi ations/. M. Klein and L. Eriksson. A spe i heat ratio model for singlezone heat release models. SAE 2004 Transa tions Journal of Engines, 2004-01-1464, 2005a. M. Klein and L. Eriksson. A omparison of spe i heat ratio models for ylinder pressure modeling. CCSSE, Norrköping, Sweden, O tober 2004a. M. Klein and L. Eriksson. Utilizing ylinder pressure data for ompression ratio estimation. IFAC World Congress, Prague, Cze h Republi , 2005b. M. Klein and L. Eriksson. Models, methods and performan e when estimating the ompression ratio based on the ylinder pressure. Fourth onferen e on Computer S ien e and Systems Engineering in Linköping (CCSSE), 2002. M. Klein and L. Eriksson. A omparison of spe i heat ratio models for ylinder pressure modeling. CCSSE, Norrköping, Sweden, O tober 2004b. M. Klein and L. Eriksson. A spe i heat ratio model for single-zone heat release models. Modeling of SI Engines, SP-1830, 2004 . SAE Te hni al Paper 2004-01-1464. M. Klein and L. Eriksson. Methods for ylinder pressure based ompression ratio estimation. In Ele troni Engine Control, number 200601-0185 in SAE Te hni al paper series SP-2003, SAE World Congress, Detroit, USA, 2006. M. Klein and L. Nielsen. Evaluating some gain s heduling strategies in diagnosis of a tank system. IFAC Fault Dete tion, Supervision and Safety for Te hni al Pro esses, pages 902907, Budapest, Hungary, 2000. M. Klein, L. Eriksson, and Y. Nilsson. Compression estimation from simulated and measured ylinder pressure. In Ele troni Engine Controls, SP-1704, 2002. SAE Te hni al Paper 2002-01-0843. M. Klein, L. Eriksson, and Y. Nilsson. Compression estimation from simulated and measured ylinder pressure. SAE 2002 Transa tions Journal of Engines, 111(3), 2003. SAE Te hni al Paper 2002-01-0843.

Bibliography

233

M. Klein, L. Eriksson, and J. Åslund. Compression ratio estimation based on ylinder pressure data. 2004. In pro eedings of IFAC symposium on Advan es in Automotive Control, Salerno, Italy. M. Klein, L. Eriksson, and J Åslund. Compression ratio estimation based on ylinder pressure data. Control Engineering Pra ti e, 14(3): 197211, 2006. R.B. Krieger and G.L. Borman. The omputation of apparent heat release for internal ombustion engines. ASME, 1967. D. R. Lan aster, R. B. Krieger, and J. H. Lienes h. Measurement and analysis of engine pressure data. 1975. SAE Te hni al Paper 750026. P. Lindskog. Methods, algorithms and tools for system identi ation based on prior knowledge. PhD thesis, Linköping University, 1996. L. Ljung. System Identi ation Theory for the User. Prenti e Hall, 2nd edition, 1999. F.A. Matekunas. Modes and measures of y li ombustion variability. 1983. SAE Te hni al Paper 830337. K. Miller. Least squares methods for ill-posed problems with a pres ribed bound. SIAM Journal on Mathemati al Analysis, 1(1):5274, 1970. doi: 10.1137/0501006. URL http://link.aip.org/link/?SJM/1/52/1. M. Mladek and C.H. Onder. A model for the estimation of indu ted air mass and residual gas fra tion using ylinder pressure measurements. 2000. SAE Te hni al Paper 2000-01-0958. M. Morishita and T. Kushiyama. An improved method for determining the TDC position in a pv-diagram (rst report). SAE Te hni al Paper 970062, 1997. V.A. Morozov. Methods for solving in orre tly posed problems. Springer Verlag, New York, USA, 1984. Y. Nilsson and L. Eriksson. A new formulation of multi-zone ombustion engine models. IFAC Workshop: Advan es in Automotive Control, pages 629634, Karlsruhe, Germany, 2001. Y. Nilsson and L. Eriksson. Determining TDC position using symmetry and other methods. Modeling of SI Engines, SP-1830, 2004. SAE Te hni al Paper 2004-01-1458. M. Nyberg. Model Based Fault Diagnosis: Methods, Theory, and Automotive Engine Appli ations. PhD thesis, Linköpings Universitet, June 1999.

Bibliography

234

M. Nyberg. A general framework for model based diagnosis based on statisti al hypothesis testing (revised version). 12:th International Workshop on Prin iples of Diagnosis, pages 135142, Sansi ario, Via Lattea, Italy, 2001. P. Öberg and L. Eriksson. Control oriented modeling of the gas ex hange pro ess in variable am timing engines. Number SAE Te hni al Paper 2006-01-0660, 2006. R. Pive , T. Sams, and A. Wimmer. Wärmeubergang im ein- und auslassystem. pages 658663, 1998. MTZ Motorte hnis he Zeits hrift 59 (1998) Nr. 10. F. Ponti, J. C. Pianai, and R Suglia. Residual gas model for on-line estimation for inlet and exhaust ontinous VVT engine onguration. In pro eedings of IFAC symposium on Advan es in Automotive Control, Salerno, Italy

, 2004.

A.L. Randolph. Methods of pro essing ylinder-pressure transdu er signals to maximize data a

ura y. 1990. SAE Te hni al Paper 900170. G.M. Rassweiler and L. Withrow. Motion pi tures of engine ames

orrelated with pressure ards. SAE Te hni al Paper 800131, 1938. F. S hmidt, R. Henderson, and C. Wolgemuth. Introdu tion to thermal s ien es. John Wiley & Sons, In ., 2nd edition, 1993. P.J. Shayler, S.A. May, and T. Ma. The determination of heat transfer from the ombustion hambers of SI engines. 1993. SAE Te hni al Paper 931131. J. Sjöberg, Q. Zhang, L. Ljung, A. Benveniste, B. Deylon, P-Y. Glorenne , H. Hjalmarsson, and A. Juditsky. Nonlinear bla k-box modeling in system identi ation: a unied overview. Automati a, 31: 16911724, De 1995. M.J. Sta±. Thermodynami determination of T.D.C. in piston ombustion engines. 1996. SAE Te hni al Paper 960610. M.J. Sta±. A universally appli able thermodynami method for T.D.C. determination. 2000. SAE Te hni al Paper 2000-01-0561. R. Stone. Introdu tion to Internal national, 3rd edition, 1999.

. SAE Inter-

Combustion Engines

R.A. Svehla and B.J. M Bride. Fortran IV omputer programs for

al ulation of thermodynami and transport properties of omplex

hemi al systems. Te hni al Report NASA Te hni al Note TND-7056, NASA Lewis Resear h Center, 1973.

Bibliography

235

I.I. Vibe. Brennverlauf und Kreispro ess von Verbennungsmotoren. VEB Verlag Te hnik Berlin, 1970. German translation of the russian original. N. Watson and M.S. Janota. Turbo harging the Internal Combustion Engine. The Ma millan Press Ltd, 1982. ISBN 0-333-24290-4. A. Wimmer, R. Pive , and T. Sams. Heat transfer to the ombustion

hamber and port walls of IC engines  measurement and predi tion. 2000. SAE Te hni al Paper 2000-01-0568. G. Wos hni. A universally appli able equation for the instantaneous heat transfer oe ient in the internal ombustion engine. 1967. SAE Te hni al Paper No. 670931.

236

Bibliography

A A spe ifi heat ratio model  further details Additional details and argumentation for the results in hapter 4 are given in this appendix. Ea h se tion is referen ed from various se tions in hapter 4, and this appendix is therefore a omplement. A.1

Temperature models

Two models for the in- ylinder temperature will be des ribed, the rst is the mean harge single-zone temperature model. The se ond is a two-zone mean temperature model, used to ompute the single-zone thermodynami properties as mean values of the properties in a twozone model. A.1.1

Single-zone temperature model

The mean harge temperature T for the single-zone model is found from the ideal state equation pV = mtot RT , assuming the total mass of

harge mtot and the mass spe i gas onstant R to be onstant. These assumptions are reasonable sin e the mole ular weights of the rea tants and the produ ts are essentially the same (Gatowski et al., 1984). If all thermodynami states (pref ,Tref ,Vref ) are known/evaluated at a given referen e ondition ref , su h as IVC, the mean harge temperature T is omputed as T =

TIV C pV. pIV C VIV C

This model has already been des ribed in se tion 3.3.1. 237

(A.1)

238

...

APPENDIX A. A SPECIFIC HEAT RATIO MODEL

A.1.2

Two-zone mean temperature model

A two-zone model is divided into two zones; one ontaining the unburned gases and the other ontaining the burned gases, separated by a innitesimal thin divider representing the ame front. Ea h zone is homogeneous onsidering temperature and thermodynami properties, and the pressure is the same throughout all zones, see e.g. (Nilsson and Eriksson, 2001). Here a simple two-zone model will be used to nd the burned zone temperature Tb and the unburned zone temperature Tu , in order to nd a more a

urate value of γ(T ) as an interpolation of the thermodynami properties for burned and unburned mixtures. The model is alled temperature mean value approa h (Andersson, 2002), and is based on a single-zone ombustion model and polytropi ompression of the unburned harge. The single-zone temperature an be seen as a mass-weighted mean value of the two zone temperatures. Prior to start of ombustion (SOC), the unburned zone temperature Tu equals the single-zone temperature T : (A.2)

Tu,SOC = TSOC .

The unburned zone temperature Tu after SOC is then omputed assuming polytropi ompression (2.11) of the unburned harge a

ording to: Tu = Tu,SOC



p pSOC

1−1/n

= TSOC



p pSOC

1−1/n

.

(A.3)

The rank angle position for ignition θig is assumed to oin ide with SOC, and then the unburned zone temperature Tu is given by: Tu (θ) =

(

T (θ) T (θig )



p p(θig )

1−1/n

θ ≤ θig θ > θig .

(A.4)

Energy balan e between the single-zone and the two-zone models yields: (mb + mu )cv T = mb cv,b Tb + mu cv,u Tu .

(A.5)

In order to have a fast omputation it is assumed that cv = cv,b = cv,u , i.e. a alori perfe t gas, whi h ends up in T =

mb T b + mu T u = xb Tb + (1 − xb )Tu , mb + mu

(A.6)

where xb is the mass fra tion burned. The single-zone temperature T

an be seen as the mass-weighted mean temperature of the two zones. If a temperature and pressure dependent model of cv would be used, the weight of Tb in (A.6) would in rease, resulting in a lower value for Tb sin e cv,b > cv,u . From (A.6), Tb is found as Tb =

T − (1 − xb )Tu . xb

(A.7)

A.2. SAAB 2.3L NA  GEOMETRIC DATA

239

The burned zone temperature is sensitive to low values of the mass fra tion burned, xb . Therefore Tb is set to the adiabati ame temperature for xb < 0.01. The adiabati ame temperature Tad for a onstant pressure pro ess is found from: hu (Tu ) = hb (Tad , p)

(A.8)

where hu and hb are the enthalpy for the unburned and burned mixture respe tively. An algorithm for omputing the zone temperatures is summarized as: Algorithm A.1

 Temperature mean value approa h

1.Compute the single-zone temperature T in (A.1). 2.Compute the mass fra tion burned xb by using the Matekunas pressure ratio on ept (2.21)(2.22). 3.Compute the unburned zone temperature Tu using (A.4). 4.If xb ≥ 0.01 then; Compute the burned zone temperature Tb from (A.7). else; Compute the burned zone temperature Tb from (A.8). 5.Return T , Tu and Tb .

In step 2 the Matekunas pressure ratio on ept ould be ex hanged for any of the single-zone heat-release models given in hapter 2, but the Matekunas on ept is used due to its omputational e ien y. As an illustration, the zone temperatures for the ylinder pressure tra e displayed in gure 4.8 are shown in gure A.1.

A.2 SAAB 2.3L NA  Geometri data A SAAB 2.3L NA engine is for simulated and experimental data in

hapters 24. The geometri data for the rank and piston movement are given in the following table:

240

...

APPENDIX A. A SPECIFIC HEAT RATIO MODEL

Temperatures using 2−zone mean value Temperature [K]

3000

T

b

2500 2000

T1zone

1500

T

u

1000

Mass fraction burned [−]

500 −50

0 Crank angle [deg ATDC]

50

0 Crank angle [deg ATDC]

50

1

0.5

0 −50

Figure A.1: Upper: Single-zone temperature T (=T1zone ), unburned Tu and burned Tb zone temperatures for the ylinder pressure given in gure 4.8. Bottom: Corresponding mass fra tion burned tra e al ulated using Matekunas pressure ratio.

Property Bore Stroke Crank radius Conne ting rod No. of ylinders Displa ement volume Clearan e volume Compression ratio

A.3

Abbrev. B S ar = S2 l ncyl Vd Vc rc

Value 90 90 45 147 4 2290 62.9 10.1

Unit [mm℄ [mm℄ [mm℄ [mm℄ [-℄ [ m3 ℄ [ m3 ℄ [-℄

Parameters in single-zone model

The nominal parameters used in the single-zone model (3.36)(3.45) are summarized in table A.1 for operating point 2. The parameter values for Qin , pIV C and TIV C dier for all the nine operating points a

ording to table A.2. All the other parameters remain the same. The dieren e between the γ -models is expe ted to be largest during

A.3.

241

PARAMETERS IN SINGLE-ZONE MODEL

Par. γ300 b C1 C2 ∆θ θig θd θb Vc Vcr ∆p Tw TIV C pIV C Qin Kp

Des ription

onstant ratio of spe i heat [-℄ slope for ratio of spe i heat [K −1 ℄ Wos hni heat transfer parameter [-℄ Wos hni heat transfer parameter [m/(s K)℄

rank angle phasing [deg℄ ignition angle [deg ATDC℄ ame development angle [deg℄ rapid burn angle [deg℄

learan e volume [cm3 ℄ single aggregate revi e volume [% Vc ℄ pressure bias in measurements [kPa℄ mean wall temperature [K℄ mean harge temperature at IVC [K℄

ylinder pressure at IVC [kPa℄ released energy from ombustion [J℄ pressure sensor gain [-℄

Value 1.3678

−8.13 · 10−5

2.28

3.24 · 10−4

0 -15 20 40 62.9 1.5% 0 440 341 50 760 1

Table A.1: Nominal parameter values for OP2 in the single zone model.

OP 1 2 3 4 5 1 6 7 8 9

pIV C [kPa℄

25 50 100 150 200 25 50 100 150 200

TIV C [K℄

372 341 327 326 325 372 372 372 372 372

Qin [J℄

330 760 1620 2440 3260 330 700 1420 2140 2850

Table A.2: Operating points (OP) for the simulated ylinder pressure.

ombustion, therefore a slow burn rate is used to extend the period when the models dier. Compared to table 3.1 the following parameters are hanged; The burn related angles θig , θd and θb are hanged, to ree t a slower burn rate. The osets ∆θ and ∆p are both set to zero sin e they do not have an ee t on the investigation.

242 A.4

...

APPENDIX A. A SPECIFIC HEAT RATIO MODEL

Crevi e energy term

The energy term u′ − u in (2.26) des ribes the energy required to heat up a unit mass that enters the ylinder from the revi e volume. The term depends on whi h γ -model is used and therefore has to be stated for every γ -model ex ept B1 , whi h is already done in (2.32) for the original setting in the Gatowski et al.-model. , the energy term u′ − u in (2.32) is: For model D R T′ 1 ′ u − u = T cv dT R T′ R T ′ lin = xb Tb cKB v,b dT + (1 − xb ) Tu cv,u dT   = xb (uKB (T ′ , p) − uKB (Tb , p)) + (1 − xb ) bRu ln

u γlin (T ′ )−1 u (T )−1 γlin u

, (A.9) in the se ond equal-

where we have used that cv = xb cKB − x )clin v,b + (1  b v,u ∂u for the burned mixture ity, and in the third equality that cv = ∂T V

and equation (2.31) for the linear approximation of the unburned mixture. The rst term in (A.9) is given dire tly by the Krieger-Borman polynomial in its original form (4.9). The se ond term is easily omu and bu for the linear puted when knowing the oe ient values γ300 unburned mixture model, i.e. u u γlin = γ300 + bu (Tu − 300), (A.10)

u where the oe ients γ300 and bu are given in table 4.2. Dierent modeling assumptions in terms of single-zone or two-zone models result in dierent temperatures T ′ . In these ases the temperature T ′ is as follows: For the single-zone model  Tw dmcr < 0 ′ T = (A.11) T dmcr ≥ 0,

for the burned zone

T =



Tw Tb

dmcr < 0 dmcr ≥ 0,

(A.12)

T′ =



Tw Tu

dmcr < 0 dmcr ≥ 0

(A.13)

′

and a

ording to

for the unburned zone. Note that (A.9) is zero whenever T ′ = T , i.e. when the mass ows to the revi e volume. Now the attention is turned to the revi e energy term u′ − u for the γ -models (4.19)(4.31). For C1 , C3 and C4 approximations are made during the ombustion phase. In order to see if a modeling error in u′ −u has a large impa t on the ylinder pressure, a sensitivity analysis will be performed after all models have been des ribed. Model B1 (4.19):   (T ′ )−1 , u′ − u = Rb ln γγlin (A.14) lin (T )−1

A.4.

243

CREVICE ENERGY TERM

a

ording to (2.32). Model B2 (4.20): u′ − u

Model B3 (4.21): Model B4 (4.22):

u′ − u = uB3 (T ′ , p) − uB3 (T, p).

(A.16)

u′ − u = (T ′ − T ) γR . B

(A.17)

4

Model C1 (4.23): u′ − u =

R T′

=

R γ

T

R T ′C1

dT

R b +(1−x )γ u +x bb (T −300)+(1−x )bu (T −300) xb γ300 u b b b b 300 R T x γ b +(1−x )γ u +(xR bb +(1−x )bu )(T −300) dT b 300  b 300 ′ b  b γC1 (T )−1 R ln b u γC1 (Tb ,Tu )−1 , xb b +(1−xb )b

=R ≈

(A.15)

= uB2 (T ′ ) − uB2 (T ).

T

dT

R T′

(A.18) where the approximation is an equality whenever xb = 0, xb = 1 or u dmcr ≥ 0. The oe ients for the unburned mixture γ300 and bu are b and bb are taken from table 4.3 given in table 4.2 and the values for γ300 for temperature region E (T ∈ [1200, 3000] K). Model C2 (4.24): u′ − u = uC (T ′ ) − uC (T ). (A.19) Model C3 (4.25): 2

u′ − u =

R T′ T

R γ

R T ′C3

2

dT

R dT u xb γKB +(1−xb )γlin R T′ R R T′ ≈ R(xb Tb γKB dT + (1 − xb ) Tu γR dT ) u lin   u ′ γ (T )−1 KB ′ KB xb (u (T , p) − u (Tb , p)) + (1 − xb ) bRu ln γ ulin(Tu )−1 ,

=R

T

(A.20) where the approximation is an equality whenever xb = 0, xb = 1 or dmcr ≥ 0. The approximation made for C3 yields the same energy term as for D1 (A.9). For model C4 (4.26) the energy term is approximated in the same way as for C3, i.e. lin

u′ − u

R T′ dT = T γR C4 ≈ xb (ub (T ′ , p) − ub (Tb , p)) + (1 − xb )(uu (T ′ ) − uu (Tu ))

(A.21)

244

...

APPENDIX A. A SPECIFIC HEAT RATIO MODEL

where the internal energies for burned, ub , and unburned mixture, uu , are omputed from CHEPP (4.3), and the approximation is an equality whenever xb = 0, xb = 1 or dmcr ≥ 0. The approximation made for C4 yields the same energy term as for the referen e model D4 (A.25). Model C5 (4.27): u′ − u = =

R T′

T      

R ) dT γC5 (T,xb  γC5 (T ′ ,0)−1 R ln comp b γC5 (T,0)−1 (T ′ − T ) γC R−1   5 ′ γC5 (T ,1)−1 R ln exp b γC5 (T,1)−1

xb < 0.01 0.01 ≤ xb ≤ 0.99

(A.22)

xb > 0.99

The energy term orresponding to xb < 0.01 is at most instan es equal to zero, but not when the start of ombustion o

urs after TDC. Model D2 (4.29): u′ − u

= xb (uKB (T ′ , p) − uKB (Tb , p)) + (1 − xb )(uu (T ′ ) − uu (Tu )).

(A.23)

Model D3 (4.30): u′ − u

= xb (ub (T ′ , p) − ub (Tb , p)) + (1 − xb ) bRu ln

Model D4 (4.31): u′ − u



u γlin (T ′ )−1 u (T )−1 γlin u



.

(A.24)

= xb (ub (T ′ , p) − ub (Tb , p)) + (1 − xb )(uu (T ′ ) − uu (Tu )).

(A.25)

Crevi e term sensitivity An investigation of what impa t a modeling error in the revi e term has on the ylinder pressure is now performed. The same sensitivity analysis as in se tion 3.9 is made, but this time the revi e volume Vcr is set to 0 in the nominal ylinder pressure. The results from the simulations are summarized in table A.3. The investigation shows that the RMSE(p) is higher for all parameters ex ept Tw , TIV C and ∆p when Vcr = 0, suggesting that the revi e volume has a dampening ee t on the ylinder pressure error. The

hange in all the measures RMSE, maximum residual value and sensitivity S are small, when ompared to table 3.2. Although the sensitivity analysis is performed in a spe i operating point and therefore only valid lo ally, this result indi ates that a orre t revi e-term modeling is not ru ial for the single-zone heat-release models. It also indi ates that the revi e ee t has a small impa t on the resulting ylinder pressure. Therefore a modeling error in u′ − u su h as for models C1 , C3 and C4 will not have not a ru ial ee t on the nal result and on lusions

A.5.

245

SIMPLE RESIDUAL GAS MODEL

Par. γ300 ∆θd θig Vc Kp Tw Qin ∆θb pIV C TIV C b ∆θ ∆p C2 C1

Nominal & perturbation value 1.3678 15 -20 62.9 1 440 1500 30 100 340 -8.13 ·10−5 0.4 30 3.24 ·10−3 2.28

0.137 5 5 6.29 0.1 44 150 5 10 44 -8.13 ·10−6 0.2 10 3.24 ·10−4 0.228

[-℄ [deg℄ [deg ATDC℄ [cm3 ℄ [-℄ [K℄ [J℄ [deg℄ [kPa℄ [K℄ [K −1 ℄ [deg℄ [kPa℄ [m/(s K)℄ [-℄

RMSE [kPa℄ 542.0 282.5 248.3 217.5 188.4 108.7 103.6 105.2 97.5 62.8 27.0 10.6 10.0 4.1 1.7

Max Res [kPa℄ 1471.10 1081.2 904.0 632.2 477.2 283.5 271.9 412.7 221.5 175.2 82.7 32.7 10.0 8.8 2.10

S

[-℄ 5.40 0.60 0.74 1.80 1.56 0.86 0.81 0.46 0.77 0.35 0.20 0.2 0.2 0.3 0.1

Table A.3: Nominal and perturbation values, where the perturbations are performed by adding or subtra ting the perturbation from the nominal value. The root mean square error (RMSE), maximal residual (Max Res) and sensitivity fun tion S (3.46) are omputed for the worst ase for ea h parameter. drawn. It is important to note that the γ -model is very important for the ylinder pressure model. This is primarily through its dire t inuen e by oupling the pressure, temperature and volume hanges to ea h other. The revi e ee t is also dire tly dependent on the γ -model, but here γ only has a se ondary ee t on the total ylinder pressure model. A.5

Simple residual gas model

r An approximative model for nding the residual gas fra tion xr = mmtot and temperature Tr ited in Heywood (1988, p.178) is used to nd TIV C . The residual gas mass is given by mr and the total ylinder gas mass by mtot . The residual gas is left behind from the exhaust pro ess and lls the learan e volume Vc at pressure pexh and temperature T6 , where pexh is the exhaust manifold pressure and T6 is the mean harge temperature at θ = 360 [deg ATDC℄, i.e. at the end of the exhaust stroke. The intake manifold ontains a fresh air-fuel harge at pressure pman and temperature Tman . As the intake valve opens, the residual gases expand a

ording to the polytropi relation (2.11) to volume Vr and temperature Tr a

ording to

Vr = Vc

p

exh

pman

1/n

,

(A.26a)

246

...

APPENDIX A. A SPECIFIC HEAT RATIO MODEL

Tr = T6

p

man

pexh

n  n−1

(A.26b)

.

The rest of the ylinder volume Vaf is lled with fresh air-fuel harge, i.e. Vaf = V − Vr . The ideal gas law is then used to ompute the residual gas fra tion xr as xr

=

=

mr = maf + mr 1+

Tr Tman

pman Vr RTr pman Vaf pman Vr RTman + RTr

pman rc − pexh



pman pexh

= ... (n−1)/n !!−1

, (A.27)

and the mean harge temperature at IVC is then omputed as TIV C = Tr rc xr



pman pexh



(A.28)

,

where Tr = 1400 K and (n−1)/n = 0.24 are appropriate average values to use for initial estimates (Heywood, 1988, p.178). A.6

Fuel omposition sensitivity of

γ

So far, the fo us has only been on iso-o tane C8 H18 as the fuel used. Sin e the a tual fuel omposition an dier over both region of ountry and time of year, it is interesting to see what happens with the spe i heat ratios when the fuel omposition is hanged. Consider the general fuel Ca Hb Oc , whi h is ombusted a

ording to 1 Ca Hb Oc + (O2 + 3.773N2 ) −→ λ (a + b/4 − c/2) y1 O + y2 O2 + y3 H + y4 H2 + y5 OH +y6 H2 O + y7 CO + y8 CO2 + y9 N O + y10 N2 ,

(A.29)

where a, b and c are positive integers. First our attention is turned to the properties of hydro arbons and then to a few al ohols, when onsidering burned mixtures. Then a similar investigation is made for unburned mixtures. Finally the properties of partially burned mixtures and their inuen e on the ylinder pressure are examined.

A.6.1 Burned mixture  Hydro arbons Considering hydro arbons Ca Hb only (c = 0), the hydro arbon ratio y = b/a will determine the properties of the air-fuel mixture, sin e the a and b are only relative proportions on a molar basis (Heywood,

A.6.

FUEL COMPOSITION SENSITIVITY OF

γ

247

1988)[p.69℄. The spe i heat ratio is omputed using CHEPP for the fuels given in table A.4. Gasoline 1 and 2 are ommer ial fuels listed in Heywood (1988)[p.133℄. The fuels methane and gasoline 2 are extreme points for the hydro arbon ratio y in the study, a region whi h

overs most hydro arbon fuels. In the upper plot of gure A.2 the spe i heat ratio for the fuels at λ = 1 and p = 7.5 bar are displayed. They are omputed using model D4 (4.31). The dieren e between the fuels is hardly visible. Therefore, the fuels are ompared to iso-o tane, and the dieren e in γ is plotted in the lower part of gure A.2. The dieren e is small, and smallest for the ommer ial gasoline as expe ted, sin e the hydro arbon ratio y is losest to that of iso-o tane. The NRMSE(γ ) are found in table A.4, for p1 = 7.5 and p2 = 35 bar respe tively. Compared to table 4.5, the fuel omposition introdu es a smaller error in γ than the Krieger-Borman polynomial. Therefore the iso-o tane γ an be used as a good approximation for a burned mixture for the hydro arbon fuels used in this study.

A.6.2 Burned mixture  Al ohols Considering more general fuels su h as al ohols, the spe i heat ratio of methanol CH3 OH is omputed and ompared to the ones found for iso-o tane and methane respe tively. The omparison with methane shows what inuen e the extra oxygen atom brings about, and the

omparison with iso-o tane yields the dieren e to the fuel used here as a referen e fuel. The spe i heat ratios are omputed using D4 (4.31). The results are displayed in gure A.3, where the upper plot shows γ for the three fuels listed in table A.5. The lower plot shows the dieren e in γ for methanol when ompared to iso-o tane and methane respe tively. In gure A.3 and table A.5 the results for the fuels are

ompared to methanol instead of iso-o tane due to three reasons; First of all this allows for a dire t omparison of the results for methane and methanol. Se ondly, it allows for a omparison of methanol and iso-o tane. Thirdly γ for methane and iso-o tane have already been

ompared in gure A.2 and table A.4. Surprisingly, the dieren e in γ is smaller between iso-o tane and methanol than between methane and methanol as shown in gure A.3, whi h is also on luded by omparing the NRMSE:s from table A.5. These NRMSE:s are in fa t quite large, whi h is found by omparing them to the ones found in table 4.4. This suggests that the error introdu ed by using iso-o tane γ to des ribe methanol γ is almost as large b . If a better approximation the error introdu ed by the linear model γlin of the methanol γ is needed, one should use CHEPP (Eriksson, 2004) to ompute the thermodynami properties for methanol and then estimate the oe ients in the Krieger-Borman polynomial (4.9) in a least squares sense.

248

...

APPENDIX A. A SPECIFIC HEAT RATIO MODEL

Specific heat ratio − γ [−]

Burned mixture @λ=1, p=7.5 bar 1.4

C H 8 18 CH 4 C7.76H13.1

1.3 1.2 1.1 1 500

1000

1500

2000 2500 Temperature [K]

3000

3500

1500

2000 2500 Temperature [K]

3000

3500

−3

Difference in γ [−]

10

x 10

5

CH 4 C H

7.76 13.1

0 −5 500

1000

Figure A.2: Upper: Spe i heat ratio for various fuels. Lower: Differen e in γ for methane CH4 and gasoline 2 C7.76 H13.1 , ompared to iso-o tane C8 H18 .

Fuel

Ca Hb

Methane CH4 Iso-o tane C8 H18 Gasoline 1 C8.26 H15.5 Gasoline 2 C7.76 H13.1

NRMSE  p1 A B 4 0.19 % 0.17 % 2.25 0 0 1.88 0.06 % 0.05 % 1.69 0.09 % 0.07 % y

NRMSE  p2 A B 0.16 % 0.15 % 0 0 0.05 % 0.04 % 0.07 % 0.07 %

Table A.4: Burned mixtures: Dierent fuels and their hemi al omposition. The NRMSE(γ ) is formed as the dieren e ompared to isoo tane, and evaluated at λ = 1 and temperature regions A and B , for p1 = 7.5 and p2 = 35 bar respe tively.

A.6.

FUEL COMPOSITION SENSITIVITY OF

249

γ

Specific heat ratio − γ [−]

Burned mixture @λ=1, p=7.5 bar 1.4

C H 8 18 CH4 CH OH

1.3

3

1.2 1.1 1 500

Difference in γ [−]

0.015

1000

1500

2000 2500 Temperature [K]

3000

3500

1500

2000 2500 Temperature [K]

3000

3500

C H 8 18 CH 4

0.01

0.005 500

1000

Figure A.3: Upper: Spe i heat ratio for various fuels. Lower: Differen e in γ for methanol CH3 OH ompared to iso-o tane C8 H18 and methane CH4 .

Fuel

Ca Hb Oc NRMSE  p1

NRMSE  p2 A B A B Methanol CH4 O 0 0 0 0 Methane CH4 0.80 % 0.75 % 0.81 % 0.72 % Iso-o tane C8 H18 0.73 % 0.60 % 0.70 % 0.60 % Table A.5: Burned mixtures: Dierent fuels and their hemi al omposition. The NRMSE is formed as the dieren e for methanol ompared to methane and iso-o tane respe tively, and evaluated at λ = 1 and temperature regions A and B , for p1 = 7.5 and p2 = 35 bar respe tively.

250

...

APPENDIX A. A SPECIFIC HEAT RATIO MODEL

A.6.3 Unburned mixtures The spe i heat ratios for both unburned hydro arbons and al ohols are analyzed in a similar manner as for the burned mixtures presented earlier. The results are summarized in table A.6. All fuels but methane are aptured fairly well by the referen e fuel iso-o tane. Comparing gasoline 1 with the linear model of the unburned mixture given in table 4.2, one see that it introdu es an NRMSE whi h is of the same order. A trend in the results shows that for hydro arbons, the spe i heat ratio is more a

urately determined for burned mixtures than unburned. This on lusion an be drawn by omparing tables A.4 and A.6. For the al ohol methanol it is the other way around, ompare tables A.5 and A.6. Fuel Methane Iso-o tane Gasoline 1 Gasoline 2 Methanol

Ca Hb Oc CH4 C8 H18 C8.26 H15.5 C7.76 H13.1 CH3 OH

y = b/a

4 2.25 1.88 1.69 4

NRMSE 2.57 % 0 0.18 % 0.39 % 0.50 %

Table A.6: Unburned mixtures: Dierent fuels and their hemi al

omposition. The NRMSE(γ ) is formed as the dieren e ompared to iso-o tane, and evaluated at λ = 1 for temperature region T ∈ [300, 1000] K.

A.6.4 Partially burned mixture  inuen e on ylinder pressure The ylinder pressure at OP 2 given in gure 4.8 is used to exemplify the impa t a ertain fuel has on the ylinder pressure, given that all the other operating onditions are the same. The referen e model (4.18) is used to model γ for ea h fuel. The impa t is displayed as the RMSE for the pressure in table A.7, as well as the NRMSE and MRE for γ . Iso-o tane is used as the referen e fuel. Compared to table 4.6, the

ylinder pressure impa t (RMSE(p)) of the fuels listed in table A.7 are larger than the impa t of D1 , see the RMSE(p) olumn in table 4.6, for all fuels ex ept gasoline 1. However for all fuels but methane, the RMSE(p) introdu ed is in reased with less than 75 % ompared to iso-o tane, whi h is a

eptable. The goal is to nd a model that approximates the a tual γ well. Model D1 is omputationally e ient and will therefore be used in the evaluation. Table A.8 evaluates the impa t the use of the polynomials in model D1 has on a spe i fuel, in terms of MRE(γ ), NRMSE(γ )

A.7. THERMODYNAMIC PROP. BURNED MIXTURE

Fuel Methane Iso-o tane Gasoline 1 Gasoline 2 Methanol

MRE: γ [%℄ 2.8 0.0 0.20 0.40 0.85

NRMSE: γ [%℄ 2.0 0.0 0.12 0.29 0.63

251

RMSE: p [kPa℄ 36.6 0.0 2.2 5.3 5.1

Table A.7: Evaluation of the impa t on ylinder pressure and spe i heat ratio for various fuels using iso-o tane as referen e fuel, for the simulated ylinder pressure at OP 2 in gure 4.8.

Fuel Methane Iso-o tane Gasoline 1 Gasoline 2 Methanol

MRE: γ [%℄ 2.7 0.27 0.27 0.47 1.1

NRMSE: γ [%℄ 1.9 0.10 0.15 0.29 0.70

RMSE: p [kPa℄ 34.3 2.8 1.5 3.3 5.4

Table A.8: Evaluation of the impa t on ylinder pressure and spe i heat ratio for various fuels by using model D1 , for the simulated ylinder pressure at OP 2 in gure 4.8.

and RMSE(p). In the evaluation model D4 has been used for a spe i fuel to generate the thermodynami properties of the mixture, from whi h a ylinder pressure simulation has been performed. This referen e pressure and spe i heat ratio has then been ompared to the ones given by model D1 for the same operating onditions at OP 2. The table shows that the RMSE(p) for gasolines 1 and 2 are lose to the one found for iso-o tane. In fa t, the RMSE(p) is only in reased by 20 %. It an therefore be on luded that model D1 an be used for fuels that have a hydro arbon ratio lose to 2, at least within [1.69, 2.25] and still have a modeling error in the order of the noise. Note that the loser y is to 2, the smaller RMSE(p) is. This is due to that the hydro arbon ratio for the Krieger-Borman polynomial used in D1 is 2. This also explains why the RMSE(p) are smaller in table A.8 ompared to table A.7.

252

...

APPENDIX A. A SPECIFIC HEAT RATIO MODEL

Iso−octane; Burned mixture @λ=1, p=7.5 bar 6000 CHEPP Linear KB poly

Specific heat − cv [kJ/(kg K)]

5000

4000

3000

2000

1000

0 500

1000

1500

2000 2500 Temperature [K]

3000

3500

Figure A.4: Spe i heat cv,b for burned stoi hiometri mixture using KB CHEPP, the orresponding linear fun tion clin v,b and cv,b found using the Krieger-Borman polynomial.

Region T ∈ A B C D E

[500, 3500] [500, 3000] [500, 2700] [500, 2500] [1200, 3000]

clin v,b

MRE 0.68 % 0.68 % 0.68 % 0.68 % 0.38 %

NRMSE 0.20 % 0.23 % 0.27 % 0.30 % 0.20 %

KB cv,b

MRE 0.42 % 0.09 % 0.04 % 0.04 % 0.09 %

NRMSE 0.21 % 0.03 % 0.02 % 0.02 % 0.04 %

Table A.9: Maximum relative error (MRE) and normalized root mean square error (NRMSE) of spe i heat cv,b for dierent temperature regions at λ = 1 and p = 7.5 bar.

A.7.

THERMODYNAMIC PROPERTIES FOR BURNED MIXTURE

A.7

Thermodynami properties for burned mixture

This se tion entails further details on thermodynami properties for the burned mixture. The fo us is on approximative models for the spe i heats. As mentioned in se tion 4.6, there is a potential of improving the Krieger-Borman polynomial. Here it will be shown why. Figure A.4 displays the referen e spe i heat cv,b as well as the two approximations, i.e. the linear and the Krieger-Borman model respe tively. The linear approximation has poor performan e over the entire temperature region, and does not apture non-linear behavior of the referen e model very well. The Krieger-Borman polynomial ts the referen e model quite well for T < 2800 K, but for higher temperatures the t is worse. This is ree ted in table A.9, whi h displays the maximum relative error (MRE) and normalized root mean square error (NRMSE) for a number of temperature regions. For temperature regions BE, the NRMSE for cKB v,b is immensely lower than for region A, whi h veries that the Krieger-Borman polynomial works well for temperatures below 3000 K. A tually the KriegerBorman polynomial has poorer performan e than the linear model for high temperatures, as seen by omparing the NRMSE:s for temperature region A. This shows that there is a potential of enhan ing the Krieger-Borman polynomial, at least for temperatures above 3000 K. If a better model approximation is sought, one should rst in rease the polynomial order in T of (4.10) with at least 1, to better at h the behavior for T > 3000 K in gure A.4. The thermodynami properties an then be omputed using CHEPP (Eriksson, 2004), and all

oe ients estimated in a least squares sense. The orresponding results for spe i heat cp,b are shown in gure A.5 and table A.10, from whi h the same on lusions as for cv,b an be drawn. Region T ∈ A B C D E

clin p,b MRE NRMSE [500, 3500] 0.51 % 0.17 % [500, 3000] 0.51 % 0.19 % [500, 2700] 0.51 % 0.23 % [500, 2500] 0.51 % 0.25 % [1200, 3000] 0.31 % 0.17 %

cKB p,b MRE NRMSE 0.39 % 0.19 % 0.08 % 0.03 % 0.03 % 0.02 % 0.03 % 0.02 % 0.08 % 0.03 %

Table A.10: Maximum relative error (MRE) and normalized root mean square error (NRMSE) of spe i heat cp,b for dierent temperature regions at λ = 1 and p = 7.5 bar.

253

254

...

APPENDIX A. A SPECIFIC HEAT RATIO MODEL

Iso−octane; Burned mixture @λ=1, p=7.5 bar 6000

Specific heat − cp [kJ/(kg K)]

5000

CHEPP Linear KB poly

4000

3000

2000

1000

0 500

1000

1500

2000 2500 Temperature [K]

3000

3500

Figure A.5: Spe i heat cp,b for burned stoi hiometri mixture using KB CHEPP, the orresponding linear fun tion clin p,b and cp,b found using the Krieger-Borman polynomial.

A.8

Thermodynami properties for partially burned mixture

The operating points (OP) for the simulated ylinder pressure tra es used to evaluate the proposed γ -models are given in table A.2. In operating points 1-9 the mean harge temperature at IVC, TIV C , is omputed as a fun tion of exhaust pressure pexh (A.28), see appendix A.5. The released energy Qin is omputed as in (3.27), where the residual gas ratio xr is determined from (A.27). The ylinder pressure at IVC, pIV C , here ranges from 25 kPa up to 200 kPa, i.e. from low intake pressure to a highly super harged pressure. The values of the parameters in the single-zone heat release model are given in tables A.1 and A.2. The

orresponding ylinder pressures during the losed part of the y le are shown in gure A.6, where the upper gure shows the ylinder pressure for operating points 1-5, and the lower plot displays operating point 1 and 6-9. The results from applying operating points 1-9 to the approximative γ -models are summarized in the following tables and gures; Tables A.11 and A.12 summarizes the maximum relative error and nor-

...

A.8. THERMODYNAMIC PROP. PARTIALLY BURNED

255

Pressure [MPa]

10

5

0

−100

−50

0

50

100

−100

−50 0 50 Crank angle [deg ATDC]

100

Pressure [MPa]

10

5

0

Figure A.6: Upper: Simulated ylinder pressure for operating points 1-5. Lower: Simulated ylinder pressure for operating points 1 and 6-9. malized root mean square error in spe i heat ratio γ , table A.13 summarizes the root mean square error for the ylinder pressure p. Figures A.7 to A.10 display the approximative γ -models and the referen e γ -model as fun tion of rank angle degree and single zone temperature respe tively, for the ylinder pressure tra e given in gure 4.8. In those gures, the referen e model γCE is the dashed line and the solid line

orresponds to ea h spe i model. Figures A.11 and A.12 illustrates the orresponding ylinder pressure errors introdu ed by the model error ea h γ -model brings along.

256 OP 1 2 3 4 5 1 6 7 8 9 Mean

...

APPENDIX A. A SPECIFIC HEAT RATIO MODEL

B1

4.1 4.1 3.6 3.3 3.1 4.1 3.5 3 2.7 2.5 3.4

B2

B3

5.9 5.9 5.5 5.1 4.9 5.9 5.3 4.8 4.5 4.2 5.2

B4

5.2 5.2 4.7 4.4 4.1 5.2 4.6 4.1 3.8 3.6 4.5

C1

7.7 7.8 7.4 7.2 7 7.7 7.2 6.7 6.4 6.2 7.1

2.3 2.3 1.9 1.8 2.1 2.3 1.7 1.4 1.7 2 1.9

C2

7.3 7.3 6.9 6.5 6.3 7.3 6.7 6.2 5.9 5.7 6.6

C3

2.2 2.4 2.3 2.1 2 2.2 1.8 1.5 1.4 1.3 1.9

C4

2.1 2.3 2.1 1.9 1.8 2.1 1.7 1.5 1.3 1.2 1.8

C5

8.1 8.4 8.6 8.6 8.7 8.1 8.1 8.1 8.1 8.1 8.3

D1

0.29 0.27 0.26 0.26 0.26 0.29 0.26 0.24 0.24 0.23 0.26

D2

0.28 0.26 0.25 0.25 0.25 0.28 0.25 0.24 0.23 0.22 0.25

D3

0.049 0.039 0.036 0.036 0.036 0.049 0.049 0.05 0.05 0.05 0.044

Table A.11: Maximum relative error (MRE) [%℄ for γ . OP 1 2 3 4 5 1 6 7 8 9 Mean

B1

1.4 1.3 1.2 1.2 1.1 1.4 1.2 1.1 1.1 1 1.2

B2

2.6 2.7 2.6 2.4 2.4 2.6 2.4 2.3 2.2 2.1 2.4

B3

1.8 1.8 1.7 1.6 1.6 1.8 1.7 1.6 1.5 1.5 1.7

B4

4.2 4.5 4.6 4.5 4.4 4.2 4.1 4 3.9 3.9 4.2

C1

0.62 0.69 0.82 0.93 1 0.62 0.57 0.69 0.81 0.91 0.77

C2

4.1 4.1 4 3.9 3.8 4.1 4 3.8 3.8 3.7 3.9

C3

0.59 0.65 0.62 0.58 0.56 0.59 0.51 0.44 0.4 0.38 0.53

C4

0.52 0.58 0.55 0.5 0.47 0.52 0.44 0.38 0.34 0.32 0.46

C5

1.5 1.5 1.6 1.7 1.7 1.5 1.6 1.6 1.7 1.8 1.6

D1

0.11 0.098 0.095 0.096 0.097 0.11 0.098 0.094 0.092 0.092 0.097

D2

0.1 0.094 0.091 0.091 0.092 0.1 0.092 0.088 0.086 0.086 0.092

D3

0.016 0.014 0.014 0.014 0.014 0.016 0.017 0.017 0.017 0.017 0.016

Table A.12: Normalized root mean square error (NRMSE) [%℄ for γ models. OP 1 2 3 4 5 1 6 7 8 9 Mean

B1

26.1 52.3 98.4 135.1 168.1 28.7 42.0 73.1 100.3 125.1 84.9

B2

37.3 85.8 172.5 248.8 321.7 37.3 70.0 130.9 188.4 243.7 153.6

B3

33.3 76.0 152.9 221.0 286.3 33.3 62.7 117.7 169.8 220.2 137.3

B4

29.5 62.8 125.7 180.3 232.3 29.5 49.6 91.5 130.7 168.4 110.0

C1

17.2 39.8 74.1 98.4 118.9 17.2 28.5 45.3 58.0 68.2 56.6

C2

61.6 140.7 289.4 427.0 561.7 61.6 120.0 233.2 343.9 453.0 269.2

C3

9.7 25.4 53.3 76.7 99.0 9.7 17.5 31.4 44.3 56.6 42.4

C4

8.6 22.8 47.3 67.3 86.0 8.6 15.2 26.9 37.5 47.4 36.7

C5

34.2 82.9 192.3 305.4 422.8 34.2 75.2 163.5 256.4 352.2 191.9

D1

1.3 2.8 5.9 9.2 12.5 1.3 2.5 4.9 7.4 9.9 5.8

D2 D3

1.2 2.6 5.3 8.2 11.2 1.2 2.2 4.3 6.4 8.5 5.1

0.1 0.3 0.7 1.0 1.4 0.1 0.3 0.7 1.1 1.5 0.7

Table A.13: Root mean square error (RMSE) [kPa℄ for ylinder pressure.

...

1.4

1.3

1.3 B : γ [−]

1.4

1.1

1.2 1.1

−100

0

1

100

1.4

1.3

1.3 B : γ [−]

1.4

−100

0

100

−100

0

100

1.2

4

1.2

3

B : γ [−]

1

1.1 1

1.1

−100

0

1

100

1.4

1.3

1.3 C : γ [−]

1.4

1.2

2

1.2

1

C : γ [−]

257

2

1.2

1

B : γ [−]

A.8. THERMODYNAMIC PROP. PARTIALLY BURNED

1.1 1

1.1

−100 0 100 Crank angle [deg ATDC]

1

−100 0 100 Crank angle [deg ATDC]

Figure A.7: Spe i heat ratio γ for models B1 to C2 (solid line) at operating point 2 given in table A.2. The dashed line orresponds to the referen e model D4 found by CHEPP.

258

...

APPENDIX A. A SPECIFIC HEAT RATIO MODEL

1.3

1.3

4

C : γ [−]

1.4

C3: γ [−]

1.4

1.2

1.1

−100

0

1.2

1.1

100

1.3

1.3

1.2

1.1

100

−100

0

−100

0

100

1.2

1.1

100

1.3

1.3

3

D : γ [−]

1.4

D2: γ [−]

1.4

1.2

1.1

0

1

D : γ [−]

1.4

C5: γ [−]

1.4

−100

−100 0 100 Crank angle [deg ATDC]

1.2

1.1

−100 0 100 Crank angle [deg ATDC]

Figure A.8: Spe i heat ratio γ for models C3 to D3 (solid line) at operating point 2 given in table A.2. The dashed line orresponds to the referen e model D4 found by CHEPP.

...

1.4

1.3

1.3

1.2

1.4

1.4

1.3

1.3

1.2

1.1

1.4

1.4

1.3

1.3

1.2

1.1

500 1000 1500 2000 2500 Single−zone temp [K]

500 1000 1500 2000 2500

1.2

1.1

500 1000 1500 2000 2500

259

1.2

1.1

500 1000 1500 2000 2500

B4 γ [−]

B3 γ [−]

1.1

B5 γ [−]

B2 γ [−]

1.4

B6 γ [−]

B1 γ [−]

A.8. THERMODYNAMIC PROP. PARTIALLY BURNED

500 1000 1500 2000 2500

1.2

1.1

500 1000 1500 2000 2500 Single−zone temp [K]

Figure A.9: Spe i heat ratio γ for models B1 to C2 (solid line) at operating point 2 given in table A.2. The dashed line orresponds to the referen e model D4 found by CHEPP.

...

APPENDIX A. A SPECIFIC HEAT RATIO MODEL

1.4

1.3

1.3

1.2

1.2

1.1

500 1000 1500 2000 2500

1.4

1.4

1.3

1.3

D1 γ [−]

C5 γ [−]

1.1

1.2

1.1

D2 γ [−]

C4 γ [−]

1.4

1.4

1.4

1.3

1.3

1.2

1.1

500 1000 1500 2000 2500 Single−zone temp [K]

500 1000 1500 2000 2500

1.2

1.1

500 1000 1500 2000 2500

D3 γ [−]

C3 γ [−]

260

500 1000 1500 2000 2500

1.2

1.1

500 1000 1500 2000 2500 Single−zone temp [K]

Figure A.10: Spe i heat ratio γ for models C3 to D3 (solid line) at operating point 2 given in table A.2. The dashed line orresponds to the referen e model D4 found by CHEPP.

...

250

150

200

100

150

50

100

0 −50

261

B2: p [kPa]

200

1

B : p [kPa]

A.8. THERMODYNAMIC PROP. PARTIALLY BURNED

50 −100

0

0

100

250

−100

0

100

−100

0

100

200

B : p [kPa]

150

100

4

100

3

B : p [kPa]

200

0

50 −100

0

100

−100

400

100

300 C2: p [kPa]

150

50

200

1

C : p [kPa]

0

0 −50

100

−100 0 100 Crank angle [deg ATDC]

0

−100 0 100 Crank angle [deg ATDC]

Figure A.11: Cylinder pressure error for models B1 to C2 (solid line) at operating point 2 given in table A.2, as ompared to the referen e model. The referen e ylinder pressure is given in gure 4.8.

...

262

APPENDIX A. A SPECIFIC HEAT RATIO MODEL

80

80 60 C : p [kPa]

C : p [kPa]

60

20

20

0 −100

0

0

100

8

200

6 D1: p [kPa]

250

150 100

5

C : p [kPa]

−20

40

4

3

40

50 0

−100

0

100

−100

0

100

4 2 0

−100

0

−2

100

8

1

D : p [kPa]

0.5

3

4

2

D : p [kPa]

6

0

2 0

−100 0 100 Crank angle [deg ATDC]

−0.5

−100 0 100 Crank angle [deg ATDC]

Figure A.12: Cylinder pressure error for models

C3

to

D3

(solid line)

at operating point 2 given in table A.2, as ompared to the referen e model. For onvenien e, the sign of the pressure error for model

hanged.

C5

is

B Compression ratio estimation  further details

Further details and argumentation for the results found in hapter 5 are given in this appendix. Ea h se tion is referen ed from various se tions in hapter 5 and this appendix is therefore a omplement.

B.1

Taylor expansions for sublinear approa h

This se tion presents the al ulations that support the algorithm development in se tion 5.3. In parti ular the relation between the two residuals

ε1a (C1 , n) = ln p(θ) − (C1 − n ln(Vid (θ) + Vc ))

(B.1)

ε1b (C2 , Vc ) = Vid (θ) − (C2 p(θ)−1/n − Vc )

(B.2)

and

is investigated. By using that

C1 = ln C

and

C2 = C

1/n

, the following

relation is obtained

ε1a

= n ln p1/n − ln C2 n + n ln(Vid (θ) + Vc )   C2 p(θ)−1/n = −n ln Vid (θ) + Vc   Vid (θ) − (C2 p(θ)−1/n − Vc ) = −n ln 1 − Vid (θ) + Vc   ε1b . = −n ln 1 − Vid (θ) + Vc 263

264

APPENDIX B. COMPRESSION RATIO ESTIMATION

...

If the residuals are small, i.e. ε1b ≪ Vc ≤ Vid (θ) + Vc , then the Taylor expansion gives   n ε1b ≈ −n ln 1 − (B.3) ε1b . Vid (θ) + Vc Vid (θ) + Vc It then follows that ε1a (θ, x) ≈

n ε1b (θ, x), Vid (θ) + Vc

(B.4)

whi h is the sought relation. B.2

Variable Proje tion Algorithm

A omputationally e ient algorithm des ribed in Björ k (1996, p. 352) is summarized here.  Variable proje tion Partition the parameter ve tor x su h that x = (y z)T , where ε(y, z) is linear in y . Rewrite ε(y, z) as Algorithm B.1

ε(y, z) = F (z)y − g(z)

(B.5)

Let xk = (yk , zk ) be the urrent approximation. 1.Solve the linear subproblem min kF (zk )δyk − (g(zk ) − F (zk )yk )k22 δyk

(B.6)

and set xk+1/2 = (yk + δyk , zk ). 2.Compute the Gauss-Newton dire tion pk at xk+1/2 , i.e. solve min kC(xk+1/2 )pk + ε(yk+1/2 , zk )k22 pk

(B.7)

∂ ε(yk+1/2 , zk )) is the Ja obian mawhere C(xk+1/2 ) = (F (zk ), ∂z trix.

3.Set xk+1 = xk+1/2 + αk pk , do a onvergen e test and return to step 1 if the estimate has not onverged. Otherwise return xk+1 . The polytropi model in (5.8) is rewritten as ln p(θ) = C2 − n ln(Vd (θ) + Vc ).

(B.8)

This equation is linear in the parameters C2 = ln C and n and nonlinear in Vc and applies to the form given in (B.5). With the notation from the

265

B.3. SVC  GEOMETRIC DATA

algorithm above, the parameters are x = (C2 n Vc )T , with y = (C2 n)T and z = Vc . The measurement ve tor is formed as g = − ln p and the regression ve tor as F = [−I ln(Vc + Vd (θ))]. Another possibility is to rewrite the polytropi model (5.8)to the following Vd (θ) = C1 p(θ)−1/n − Vc . (B.9)

This equation does also t into the form of (B.5). However this formulation is not as appropriate as (B.8), due to that the parameters C1 = C 1/n and n are oupled.

B.3 SVC  Geometri data The SVC engine is used for simulated and experimental data in hapter 5. The geometri data for the rank and piston movement are given in the following table: Property Bore Stroke Crank radius Conne ting rod No. of ylinders Displa ement volume Maximum ompression ratio Minimum ompression ratio Pin-o Tilting angle

Abbrev. B S ar = S2 l ncyl Vd rcmax rcmin xof f v

Value 68 88 44 158 5 1598 14.66 8.13 [-2.2, 4.7℄ [0, 4℄

Unit [mm℄ [mm℄ [mm℄ [mm℄ [-℄ [ m3 ℄ [-℄ [-℄ [mm℄ [deg℄

B.4 Parameters in single-zone model The nominal parameters used in the single-zone model (5.4) are summarized in the table B.1. For motored y les the numeri al values of Tw and TIV C are 400 K and 310 K respe tively, while for red y les Tw = 440 K and TIV C = 340 K have been used. Compared to table 3.1 the following parameters are hanged; The learan e volume Vc is altered sin e a dierent engine is simulated. The rank angle oset ∆θ and pressure oset ∆p are both set to zero sin e their effe t is investigated expli itly in se tion 5.4.4. The pressure at IVC pIV C ∈ {0.5, 1.0, 1.8} bar depends on the operating point.

266

APPENDIX B. COMPRESSION RATIO ESTIMATION

Par. γ300 b C1 C2 ∆θ θig ∆θd ∆θb Vc Vcr ∆p Tw TIV C pIV C Qin Kp

Des ription

onstant ratio of spe i heat [-℄ slope for ratio of spe i heat [K −1 ℄ Wos hni heat transfer parameter [-℄ Wos hni heat transfer parameter [m/(s K)℄

rank angle phasing [deg℄ ignition angle [deg ATDC℄ ame development angle [deg℄ rapid burn angle [deg℄

learan e volume [cm3 ℄ single aggregate revi e volume [cm3 ℄ pressure bias in measurements [kPa℄ mean wall temperature [K℄ mean harge temperature at IVC [K℄

ylinder pressure at IVC [kPa℄ released energy from ombustion [J℄ pressure sensor gain [-℄

...

Value 1.3678 −8.13 · 10−5 2.28 3.24 · 10−4 0 -20 15 30 22.845.7 1.5% Vc (rc = 11) 0 440 340 100 500 1

Table B.1: Nominal parameter values in the single-zone model.

C Prior knowledge approa h  further details

Further details and argumentation for the results found in hapters 68 are given in this appendix.

C.1

Levenberg-Marquardt method

The parameters x are estimated by minimizing the dieren e between the measured ylinder pressure and the modeled ylinder pressure, i.e. by minimizing the predi tion error. A Gauss-Newton method alled the Levenberg-Marquardt pro edure is used to nd the parameter esˆ for methods 3 and 4 in hapter 5 and for methods 1 and 2 in timate x

hapters 68 for any given number of parameters. A thorough presentation of system identi ation is given in Ljung (1999), from whi h most of the material presented in the subsequent subse tions are from. The material is in luded here for two reasons: First to des ribe how the parameter estimation is performed when using the Levenberg-Marquardt pro edure. Se ondly as a omparison to the expressions given in se tion 6.4.2 for method 2, where a regularization term is in luded in the riterion fun tion. The rst subse tion states the equations used when minimizing the predi tion error for a nonlinear estimation problem. The Levenberg-Marquardt pro edure is then presented as a spe ial ase. The next subse tion on erns issues su h as stopping riteria, lo al minima, s aling of the parameters and asymptoti varian e of the estimate. 267

268 C.1.1

...

APPENDIX C. PRIOR KNOWLEDGE APPROACH

Minimizing predi tion errors using a lo al optimizer

When the estimation problem is non-linear in the parameters, typi ally the minimum of the loss fun tion an not be omputed analyti ally as in the linear ase. Instead, numeri al sear h routines must be used (Björ k, 1996). Given an observed data set Z N = [y(1), u(1), y(2), u(2), . . . , y(N ), u(N )] of inputs u(t) and outputs y(t), a good model M(x) des ribing the data set Z N is found by minimizing the predi tion error ε(t, x) = y(t) − yˆ(t|x),

(C.1)

t = 1, 2, . . . , N

where y(t) is the output of the system and yˆ(t|x) is the predi ted output of the model M(x). The predi tion error is also termed residual. The predi tion error is minimized by using a norm on ε(t, x) and minimize the size of it. A quadrati norm is our hoi e here and it an be written as VN (x, Z N ) =

N 1 X1 2 ε (t, x). N t=1 2

(C.2)

The term VN is a measure of the validity of M(x) and is often alled loss fun tion or riterion fun tion. A problem on this form is known as the nonlinear least-squares problem in numeri al analysis (Ljung, 1999, pp.327) and an be solved by an iterative sear h for minimum, a number of methods are des ribed in e.g. (Björ k, 1996). The estimate x ˆ is dened as the minimizing argument of (C.2): (C.3)

x ˆ(Z N ) = arg min VN (x, Z N ). x

Minimizing the riterion fun tion V

N (x, Z

N

)

To nd the solution to (C.3) numeri al methods are employed and these methods use the gradient and hessian, or approximations of them. The gradient of (C.2) is VN′ (x, Z N ) =

N 1 X ψ(t, x)ε(t, x), N t=1

(C.4)

where ψ(t, x) is the ja obian ve tor given by ψ(t, x) =

 d d ∂ yˆ(t|x) ε(t, x) = − yˆ(t|x) = − dx dx ∂x1

...

−

∂ yˆ(t|x) ∂xd

T

,

(C.5)

C.1.

LEVENBERG-MARQUARDT METHOD

269

where d are the number of parameters. For our problem, the ja obian ψ(t, x) is omputed numeri ally with a forward dieren e approximation, as

∂ε(t, x) ε(t, x + ∆xj ) − ε(t, x) ≈ . ∂xj ∆xj

(C.6)

Computing the entral dieren e approximation instead of the forward one, would double the amount of omputations. Dierentiating the gradient with respe t to x yields the hessian of (C.2) as VN′′ (x, Z N ) =

N N 1 X 1 X ′ ψ(t, x)ψ T (t, x) + ψ (t, x)εT (t, x), N t=1 N t=1

(C.7)

d ˆ(t|x). It is however omputationally heavy to where ψ ′ (t, x) = − dx 2y 2

ompute all d terms in ψ ′ (t, x). An approximation is therefore desirable and it is made reasonable by the following assumption; Assume that at the global minimum x∗ , the predi tion errors are independent. Thus lose to x∗ the se ond sum in (C.7) will be lose to zero, and the following approximation an be made lose to optimum (Ljung, 1999, p.328): 2

VN′′ (x, Z N ) ≈

N 1 X ψ(t, x)ψ T (t, x) = HN (x, Z N ). N t=1

(C.8)

By omitting the se ond sum in (C.7), the estimate HN (x, Z N ) is assured to be positive semidenite, whi h guarantees onvergen e to a stationary point. The estimate xˆ an be found numeri ally by updating the estimate of the minimizing point xˆi iteratively as xˆi+1 (Z N )

i = x ˆi (Z N ) − µiN [RN (ˆ xi , Z N )]−1 VN′ (ˆ xi , Z N ) (C.9) = x ˆi (Z N ) + di (ˆ xi , Z N ),

i where i is the ith iterate, in whi h di is the sear h dire tion and RN is the approximate hessian HN in (C.8). Finding the estimate xˆ in this manner is known as a Gauss-Newton method.

Regularization  Levenberg-Marquardt pro edure If the model M(x) is over-parameterized or the data Z N is not informative enough, this auses an ill- onditioned ja obian whi h results in that the approximative hessian HN (x, Z N ) may be singular or lose to singular. This auses numeri al problems when omputing the iterative estimates in (C.9), when inverting HN . One way to avoid this is the Levenberg-Marquardt pro edure, whi h uses i i RN (ˆ xi , Z N , ν) = HN (ˆ xi , Z N ) + νI

(C.10)

270

...

APPENDIX C. PRIOR KNOWLEDGE APPROACH

to regularize the approximation of the hessian. The iterative parameter estimate xˆi then be omes x ˆi+1 (Z N , ν) = =

i (ˆ xi , Z N , ν)]−1 VN′ (ˆ xi , Z N ) x ˆi (Z N ) − µiN [RN x ˆi (Z N ) + di (ˆ xi , Z N , ν). (C.11)

i For ν > 0, the hessian approximation RN (ˆ xi , Z N , ν) is guaranteed to be positive denite. With ν = 0 this is the Gauss-Newton method and by in reasing ν the step size is redu ed and the sear h dire tion di is turned towards the gradient, resulting in the steepest des ent dire tion as ν → ∞. Generally it an not be guaranteed that di (ˆxi , Z N , ν) in (C.11) is a des ent dire tion. This an happen if the predi tion errors are large or if (C.6) is not a good approximation ( lose to the optimum) (Eriksson, 1998). The approa h here is to start up with a ν > 0, and if VN (ˆ xi+1 , Z N ) > VN (ˆ xi , Z N ) o

urs, ν is in reased and new values of i+1 i+1 d and xˆ are omputed until di+1 is a des ent dire tion.

Stopping riteria A stopping riterion must be stated in order for the optimization pro edure to terminate. In theory this should be done when the gradient VN′ is zero, so an obvious pra ti al test is to terminate on e kVN′ k is small enough. Another useful test is to ompute the relative dieren e in loss fun tion VN between two iterations, and terminate if this dieren e is less than a given toleran e level. The algorithm an also terminate after a given maximum number of iterations.

Lo al minima and initial parameter values Generally, the optimization pro edures onverge to a lo al minimum of

VN (x, Z N ). This is due to that although the stated optimization prob-

lem may only have one lo al minimum, i.e. the global one, the fun tion

VN (x, Z N ) an have several lo al minima due to the noise in the data Z N . To nd the global minimum, there is usually no other way than to

start the iterative optimization routine at dierent feasible initial values xinit and ompare the results (Ljung, 1999, p.338). Therefore, the initial values should be hosen with are by e.g. using prior knowledge. Good initial values generally pays o in fewer iterations and a faster

onvergen e of the optimization pro edure. For instan e the Newtontype methods have good lo al onvergen e rates, but not ne essarily far from optimum.

S aling of parameters The optimization pro edure works best when the size of the unknown parameters are all in the same order (Gill et al., 1981, p.346). From

C.1.

LEVENBERG-MARQUARDT METHOD

271

table 3.1, where the unknown parameters are summarized, it an be

on luded that the nominal parameter values range over 10 de ades. Therefore a s aling in terms of a linear transformation of the parameters is introdu ed, xs = Dx, (C.12) where D is a diagonal matrix with Di,i = 1/xi , xi 6= 0. It is of , importan e for the implementation that the initial guess of xi , xinit i is assured to be non-zero. The gradient of VN (xs , Z N ) in the s aled parameters is given by VN′ (xs , Z N ) = D−T ψ(x)εT (x) = D−T VN′ (x, Z N ),

(C.13)

sin e ψ(t, xs ) = D−T ψ(t, x). The linear transformation matrix D is diagonal and invertible, and therefore D−T = D−1 . The hessian of VN (xs , Z N ) is given by VN′′ (xs , Z N ) = D−T VN′′ (x, Z N )D−1 ,

(C.14)

sin e ψ ′ (t, xs ) = D−T xD−1 .

S aling of parameters  regularized ase This subse tion deals with the issue of s aling the parameters when using method 2 in se tion 6.4.2, i.e. regularization using prior knowledge. Whenever the parameters x are res aled with D from (C.12), the nominal values x# and the regularization matrix must be res aled as well. The res aled nominal values x#,s are dened by (C.15)

x#,s = Dx# ,

where D is a diagonal matrix with Di,i = 1/xi , xi 6= 0. The elements δis in the diagonal s aled regularization matrix δ s is thereby dened by δis =

1 i )2 2N ( Dσi,i

=

2 Di,i . 2N σi 2

(C.16)

Asymptoti varian e and parameter onden e interval Consider the ase when our model M(x) has the orre t model stru ture and is provided with data Z N , su h that the measured output an be predi ted orre tly by the model. This would mean that there is no bias in the parameter estimate xˆ, and thus xˆ → xˆ∗ asymptoti ally as the number of data N goes to innity. It an then be shown (Ljung, 1999, pp.282) that the probability distribution of the random variable √ N (ˆ x − xˆ∗ ) onverges asymptoti ally to a Gaussian distribution with zero mean and ovarian e matrix P . This is formalized as (ˆ x−x ˆ∗ ) ∈ AsN (0,

P ). N

(C.17)

272

...

APPENDIX C. PRIOR KNOWLEDGE APPROACH

For a quadrati predi tion-error riterion the ovarian e matrix P is estimated by ˆN PˆN = λ

!−1 N
−1 1 X T ˆ N HN (ˆ ψ(t, x ˆ)ψ (t, xˆ) =λ x, Z N ) , (C.18a) N t=1 N X ˆN = 1 λ ε2 (t, x ˆ), N t=1

(C.18b)

for the parameter estimate xˆ and N data points, where λˆN is the estimated noise varian e and HN (ˆx, Z N ) is the approximated hessian in (C.8). When using s aled parameters a

ording to (C.12), one has to re-s ale the hessian using (C.14) in a straight forward manner. The result in (C.18) has a natural interpretation. The more data or the less noisier measured output, the more a

urate the estimate. Also, sin e ψ is the gradient of yˆ, the asymptoti a

ura y of a ertain parameter is related to how sensitive the predi tion is with respe t to this parameter. Therefore, the more or less a parameter ae ts the predi tion, the easier or harder respe tively it will be to determine its value (Ljung, 1999, p.284). The asymptoti ovarian e in (C.18) an be used to ompute onden e intervals for the parameter estimates xˆ, and thereby give a reliability measure of a parti ular parameter xˆi . When (C.17) is valid, the (1 − α)- onden e interval for the true parameter xˆ∗i is formed as (Ljung, 1999, p.302) √ Z N exp(−y 2 N/2PˆNii )dy, P (|ˆ x − xˆ | > α) ≈ q ii |y|>α ˆ 2π PN i

∗i

(C.19)

where PˆNii is the i-th diagonal element of PˆN . From this, it an be stated that the true parameter value xˆ∗i lies in the interval around the parameter estimate xˆi with a ertain signi an e 1 − α. The size of the interval is determined by α, and for a 95 % onden e interval the limits for parameter xˆi are x ˆi ± 1.96 C.2

s

PˆNii . N

(C.20)

Linear example

A

ording to Hansen (1998), a rank-de ient or an ill-posed estimation problem an be solved by the same methods. Therefore, the main di ulties with an ill-posed problem is illustrated by the following rankde ient example partly from Hansen (1994);

C.2.

273

LINEAR EXAMPLE

Example C.1

 Linear rank-de ient problem

Consider the following least-squares problem min ||Ax − b||2 , x

where A and b are given by     0.16 0.10 0.27 A =  0.17 0.11  , b =  0.25  . 2.02 1.29 3.33

(C.21)

(C.22)

The true solution is xt = [1 1]T , and the measurement ve tor b is found by adding a small noise perturbation a

ording to:      0.16 0.10  0.01 1.00 b =  0.17 0.11  (C.23) +  −0.03  . 1.00 2.02 1.29 0.02

The A-matrix of this linear problem has a ondition number of 1.1 ·103 , i.e. it is ill- onditioned and thus potentially sensitive to noise. Indeed, solving the ordinary least-squares problem as it is formulated in (C.21) ends up in an estimate xLS = [7.01 − 8.40]T , whi h is far from the true solution xt = [1 1]T . The large ondition number implies that the olumns of A are nearly linearly dependent. One approa h ould therefore be to merge the two parameters into one, and repla e A = [a1 a2 ] with either [a1 0] or [0 a2 ], sin e they are well- onditioned independently. This results in the two solutions xa1 = [1.65 0]T , xa2 = [0 2.58]T , (C.24) whi h are better than xLS but still far from the true solution. Introdu ing onstraints on the parameters results ould result in better estimates, however the di ulty now lies in how to hose the onstraints. When setting the onstraints to |x − xt | ≤ α, for α equal to 0.02, 1 and 10, the estimates be omes x0.02 = [1.02 0.98]T , x1 = [1.65 0.00]T , x10 = [7.01 − 8.40]T = xLS , (C.25) i.e. the solution lies on the boundary as long as xLS is not in luded in the interval.

This example illustrates three main di ulties with ill-posed problems (Hansen, 1994): 1. The ondition number of A is large.

274

...

APPENDIX C. PRIOR KNOWLEDGE APPROACH

2. Repla ing A with a well- onditioned matrix derived from A does not ne essarily lead to a useful solution. 3. Care must be taken when imposing additional onstraints. C.2.1

Linear example for methods 1 and 2

We now return to example C.2 for the two methods presented in se tion 6.4. Example C.2

 Linear rank-de ient problem, ont.

Methods 1 and 2 are now applied to the estimation problem des ribed in Example C.1 by using prior knowledge of the parameters x. The true values are given by xt = [1 1]T , and the nominal values x# are given in four ases; x# = [1 1]T , x# = [1.05 1]T , x# = [1 1.05]T and x# = [1.05 1.05]T . The rst ase orresponds to a true prior knowledge and the remaining three involves a false prior knowledge. The results of the estimations are given in table C.1. For method 1, either x1 or x2 ould be set spurious and therefore both ases are given. For method 2, the estimate is a fun tion of the regularization parameter δx . This is illustrated in gure C.1, where the ompromise between the residual error VN and the nominal parameter error VNδ is obvious. The estimate orresponding to the L- orner of the urve, whi h is the best

ompromise between VN and VNδ , is the one given in table C.1. Nominal x# [1 1]T [1.05 1]T [1 1.05]T [1.05 1.05]T

M1(x1 xed) [1 1.01]T [1.05 0.94]T [1 1.01]T [1.05 0.94]T

M1(x2 xed) [1.01 1]T [1.01 1]T [0.98 1.05]T [0.98 1.05]T

M2 [1.01 1.00]T [1.02 0.98]T [1.01 1.01]T [1.00 1.01]T

Table C.1: Parameter estimates for methods 1 and 2, in four ases of prior knowledge. The true values are [1 1].

Table C.1 shows that when the prior is true, the estimate is lose to the true estimate and approximately the same for methods 1 and 2. This is also the ase for method 1, as long as the prior is true for all spurious parameters, in whi h ase a better estimate is found than for method 2. However, in the presen e of a false prior in the spurious parameters, method 2 yields a smaller estimation bias than method 1. This highlights one of the features with method 2, namely the ompromise between the data tting and the parameter prior knowledge.

C.3. MOTIVATION FOR M2:3+

275

L−curve; Linear example

2

10

1

10

0

#

RMSE(x−x ) [−]

10

5%

−1

10

−2

10

−3

10

−4

10

0%

−0.9

10

−0.8

10

−0.7

10 RMSE(ε) [bar]

−0.6

10

Figure C.1: L- urves for linear example orresponding to true prior (0 %) and false prior (5 %), i.e. ase 1 and 4 in table C.1. The ir les

orrespond to the estimates in the orners of the L-fun tion. Note also that the bias depends upon whi h parameter that has the false prior, as illustrated by omparing rows 2 and 3 for method 2. In this ase the false prior in x2 results in a smaller estimation bias,

ompared to if the false prior is in x1 .

C.3 Motivation for M2:3+

This se tion gives a motivation for why the version M2:3 needs to be extended, or at least handled with are. Note that the fo us of the se tion is on ring y les, while the results and on lusions are valid for the motored y les as well. Consider gure C.2 where the upper plot displays the L- urve orresponding to OP1 for experimental ring y les using method 2 and δi = c, i.e. the same L- urve as in gure 8.7. In the lower plot the orresponding positive, i.e. onvex, urvature τ (6.23) from algorithm 6.5 is given. The orresponding gures for a simulated red y le and an experimental motored y le are given in gures C.3 and C.4. They orrespond to the L- urves presented earlier in gure 8.4 and gure 7.12.

276

...

APPENDIX C. PRIOR KNOWLEDGE APPROACH

0

10

M2:3 M2:3+ M2:2 M2:1

−2

RMSE(ε#x) [−]

10

−4

10

−6

10

−8

10

−2

−1

Curvature τ [−]

10

10

5

10

0

10

−5

10

−2

−1

10

10 RMSE(ε) [bar]

Figure C.2: Upper: L- urve for experimental ring y les using method 2 and δi = c at OP1. The four versions are indi ated by the legend. Lower: Corresponding positive urvature τ from (6.23). 0

−2

10

x

RMSE(ε#) [−]

10

−4

10

−6

10

M2:3 M2:3+ M2:2 M2:1 −1

Curvature τ [−]

10

0

10

10

10

0

10

−10

10

−1

10 RMSE(ε) [bar]

0

10

Figure C.3: Upper: L- urve for simulated ring y les using method 2 and δi = c at OP1. Lower: Corresponding positive urvature τ from (6.23).

C.3. MOTIVATION FOR M2:3+

277

0

x

RMSE(ε#) [−]

10

−5

M2:3 M2:3+ M2:2 M2:1

10

−3

−2

Curvature τ [−]

10

−1

10

0

10

10

5

10

0

10

−5

10

−3

−2

10

−1

10

0

10

10

RMSE(ε) [bar]

Figure C.4:

Upper:

method 2 and

L- urve for experimental motored y les using

δi = c

at OP1.

The four versions are indi ated by

the legend. Lower: Corresponding positive urvature

Figure C.2 shows that the urvature low value of

δx ,

τ

τ

from (6.23).

is maximized for a relatively

as shown for M2:3. The orresponding urvature does

not show up in the simulated ase in gure C.3, and is thus believed to orrespond to a false orner. Su h a orner is believed to be due to that the measurement noise auses a jump in RMSE(ε) between two dierent lo al minima. The same behavior o

urs for ase 2. It will be shown in the evaluation of ase 1 and 2 for experimental red y les, see se tion 8.2.4, that the estimates for experimental y les for M2:3 are less a

urate than the estimates for M2:1. This is shown in tables 8.18 and 8.19 for ase 1 and 2. The orre t orner is instead lose to M2:1, here exemplied by both gure C.2 and gure C.3. Therefore the extension of M2:3 is to use a smaller region

∆x

of

δx .

Sin e M2:1 is both fast and lose to

the orre t orner in the L- urve, the regularization parameter

δx

from

M2:1 is used as a mid-value for this region. The lower and upper limit of this region

∆x

is then hosen as to assure that a onvex urvature is

in luded. This is done by assuring that the urvature at least one sample of

δx ,

τ

is positive for

and that the position for maximum

interior point of the region

∆x .

τ

is an

278

...

APPENDIX C. PRIOR KNOWLEDGE APPROACH

For this appli ation the δx :s are distributed a

ording to the 69 samples in gure 7.5, and it has proved to be su ient to use ±12 samples of δx , starting from δx (M 2 : 1). This typi ally yields a region of δx that overs four orders of magnitude. This hoi e of ∆x also redu es the omputational time approximately by a fa tor 69 25 for M2:3+ as

ompared to M2:3. Applying these modi ations to M2:3 renders the

orner shown for M2:3+ in gure 8.7, and therefore M2:3+ performs as intended for both ase 1 and 2. The algorithm for M2:3+ is given by: Algorithm C.1

 Hansen's L- urve with Miller initialization (M2:3+)

d×1 1.Assign a prior x# . i and δi to ea h of the parameters x ∈ R The regularization matrix is then formed as δ = δx diag(δi ).

2.Compute boundaries mε and mδ in (6.20), that give δx from (6.21). 3.Compute the region ∆x with δx from step 2 as the mid-value, and the upper and lower limit as hosen relative to the same δx . 4.Minimize the riterion fun tion WN (6.7) for the dis rete points δx ∈ ∆x , equally spa ed in a logarithmi s ale. 5.Find the δx for whi h (6.22) is fullled, by using a ubi spline interpolation. 6.Minimize WN (6.7) w.r.t. x using δx from step 5. 7.Return the estimate xδ,∗ . Steps 1 and 2 in algorithm C.1 orrespond to steps 1 and 2 in algorithm 6.3 (M2:1), step 3 omputes the region ∆x , while steps 47

orrespond to steps 25 in algorithm 6.5 (M2:3).

C.4. L850  GEOMETRIC DATA

279

C.4 L850  Geometri data The L850 engine is used for simulated and experimental data in hapters 68. The geometri data for the rank and piston movement are given in the following table: Property Bore Stroke Crank radius Conne ting rod No. of ylinders Displa ement volume Clearan e volume Pin-o Compression ratio

Abbrev. B S ar = S2 l ncyl Vd Vc xof f rc

Value 86 86 43 145.5 4 1998 58.8 0.8 9.5

Unit [mm℄ [mm℄ [mm℄ [mm℄ [-℄ [ m3 ℄ [ m3 ℄ [mm℄ [-℄

C.5 Parameters in single-zone model  motored y les The nominal parameters used in the single-zone model (3.36)(3.45) are summarized in table C.2 for operating point 1. The parameter values for N , pIV C and TIV C diers for all the eight operating points a

ording to table C.3. All the other parameters remain the same. Compared to table 3.1 the following parameters are hanged; The

onstant γ300 is set to 1.40, sin e this is the value for pure air. The slope oe ient b is hanged a

ordingly. The rank angle oset ∆θ is set to 0.1 CAD, a smaller value than in table 3.1, sin e 0.1 CAD is the approximate value for a well- alibrated measurement system. The

learan e volume Vc is altered due to the hanged engine geometry. The revi e volume is now altered to 1 cm3 , whi h is approximately the same Vcr as for table 3.1 in absolute numbers. The pressure oset is now set to 5 kPa.

C.6 Parameters in single-zone model  red

y les The nominal parameters used in the single-zone model (3.36)(3.45) are summarized in table C.4 for operating point 1. The parameter values for Qin , pIV C , TIV C , Tw and the burn related parameters θig , ∆θd and ∆θb diers for all the eight operating points a

ording to table C.5. All the other parameters remain the same.

280

...

APPENDIX C. PRIOR KNOWLEDGE APPROACH

Par. γ300 b C1 ∆θ Vc Vcr ∆p Tw TIV C pIV C Kp

Des ription

onstant ratio of spe i heat [-℄ slope for ratio of spe i heat [K −1 ℄ Wos hni heat transfer parameter [-℄

rank angle phasing [deg℄

learan e volume [cm3 ℄ single aggregate revi e volume [cm3 ℄ pressure sensor bias [kPa℄ mean wall temperature [K℄ mean harge temperature at IVC [K℄

ylinder pressure at IVC [kPa℄ pressure sensor gain [-℄

Value 1.40

−1 · 10−4

2.28 0.1 58.8 1 5 400 370 50 1

Table C.2: Nominal parameter values for motored y les at OP1 in the single-zone model.

OP 1 2 3 4 5 6 7 8

TIV C [K℄

370 310 370 310 370 310 370 310

pIV C [kPa℄

50 50 100 100 50 50 100 100

N [rpm℄

1500 1500 1500 1500 3000 3000 3000 3000

Table C.3: Operating points (OP) for the simulated motored ylinder pressure.

Out of these, the following parameters are hanged ompared to table 3.1; The onstant γ300 and the slope oe ient b are set to lower values, although the dieren e is small. The rank angle oset ∆θ is set to 0.1 CAD, a smaller value than in table 3.1, sin e 0.1 CAD is the approximate value for a well- alibrated measurement system. The

learan e volume Vc is altered due to the hanged engine geometry. The revi e volume is now altered to 0.588 cm3 . The pressure oset is now set to 5 kPa.

281

C.6. SZ MODEL PARAMETERS  FIRED

Par. γ300 b C1 C2 ∆θ θig ∆θd ∆θb Vc Vcr ∆p Tw TIV C pIV C Qin Kp

Des ription

onstant ratio of spe i heat [-℄ slope for ratio of spe i heat [K −1 ℄ Wos hni heat transfer parameter [-℄ Wos hni heat transfer parameter [m/(s K)℄

rank angle phasing [deg℄ ignition angle [deg ATDC℄ ame development angle [deg℄ rapid burn angle [deg℄

learan e volume [cm3 ℄ single aggregate revi e volume [cm3 ℄ pressure sensor bias [kPa℄ mean wall temperature [K℄ mean harge temperature at IVC [K℄

ylinder pressure at IVC [kPa℄ released energy from ombustion [J℄ pressure sensor gain [-℄

Value 1.35

−7 · 10−5

2.28

3.24 · 10−4

0.1 -15 20 20 58.8 0.588 5 480 414 50 503 1

Table C.4: Nominal parameter values for red y les at OP1 in the single-zone model.

Parameter N [rpm℄ pIV C [kPa℄ TIV C [K℄ Tw [K℄ Qin [J℄ θig [deg ATDC℄ ∆θd [deg℄ ∆θb [deg℄

OP1 1500 50 414 480 503 -15 20 20

OP2 3000 100 370 400 1192 -15 20 20

OP3 1500 50 414 480 503 -20 15 30

OP4 3000 100 370 400 1192 -20 15 30

OP5 1500 50 414 480 503 -25 15 30

OP6 3000 100 370 400 1192 -25 15 30

Table C.5: Operating points (OP) for the simulated red ylinder pressure.

282 C.7

...

APPENDIX C. PRIOR KNOWLEDGE APPROACH

Complementary results for prior knowledge approa h

This se tion ontains the omplementary gures and tables that are referen ed from hapters 7 and 8.

Complementary results for simulations  motored y les Figures C.5 and C.6 show the L- urve for one motored y le at operating point OP8 for setup 2 using δi = c and δi 6= c respe tively.

Complementary results for experiments  motored y les Tables C.6 and C.7 show the mean value, standard deviation, and relative mean error for the individual estimates at OP1 for method 2, using δi = c and δi 6= c respe tively. The relative mean error is omputed relative to the nominal parameter value at ea h y le and is given in per ent.

Complementary results for simulations  red y les Table C.8 shows the individual parameter estimates for the entire range of d# , i.e. from 16 to 1 parameter. The numeri al values are given for one engine y le at OP1 in the presen e of a false prior level of 0 %.

Complementary results for experiments  red y les Table C.9 denes the operating points used in experimental evaluation. The orresponding nominal values are given in table C.10.

:

[ ℄ 58.0 0.7 2.0 59.0 0.7 0.3 59.1 0.1 0.5 57.8 0.4 0.6

Vc cm3

[-℄ 2.70 0.11 5.9 6.27 0.32 175.0 6.54 0.09 149.6 2.70 0.11 0.0

C1

pIV C

γ300

[K℄ [kPa℄ [-℄ 358 53.8 1.396 6 0.4 0.009 12.4 -3.2 1.1 550 34.1 1.370 18 1.5 0.002 75.0 -39.6 -2.1 567 32.8 1.371 5 0.4 0.002 64.4 -33.9 -2.0 357 53.8 1.393 11 0.6 0.007 0.3 -1.3 -1.8

TIV C

[

℄

b K −1 −8.8 · 10−5 2.5 · 10−6

[K℄ 370 10 -12.7 712 34 78.1 741 11 68.1 373 8 -0.2

Tw

[ ℄ 0.883 0.00511 1.8 0.64 0.393 -27.4 0.597 0.0528 -25.8 0.882 −5 4.48 · 10 0.0

Vcr cm3

[kPa℄ 1.1 0.9 -0.7 -0.4 0.2 -76.7 -0.3 0.1 -64.3 1.3 0.8 0.0

∆p

[deg℄ 4.48 0.14 -89.9 0.15 0.03 -67.7 0.11 0.01 -78.7 0.47 0.10 -90.0

∆θ

[-℄ 0.998 0.005 -1.6 1.596 0.071 103.9 1.658 0.020 73.5 0.999 0.003 -1.3

Kp

ε¯# x

δx (M 2 : 1) σest

[%℄ 5.8 5.8 -0.6 5.8 RME δx (M 2 : 2) −3.0 · 10−7 103.9 σest 1.2 · 10−7 103.9 RME -99.7 δx (M 2 : 3) −2.1 · 10−7 73.5 σest 3.4 · 10−8 73.5 RME -79.8 δx (M 2 : 3+) −8.7 · 10−5 0.8 σest 2.9 · 10−6 0.8 RME 0.0 0.8 x# 58.8 2.28 314 56.4 1.400 −9.3 · 10−5 400 0.882 -1.4 0.47 1.000 0.0 Table C.6: Mean value, standard deviation (σest ) and RME of the estimate for the four versions of method 2 using ase 1 (δi = c), evaluated for 101 experimental motored y les at OP1.

δx

C.7. COMPLEMENTARY RESULTS

283

:

58.8

[ ℄ 57.7 0.2 -1.8 57.5 0.4 -2.1 57.7 0.3 -1.7 57.7 0.3 -1.8

Vc cm3

2.28

[-℄ 2.73 0.07 19.7 3.54 0.84 55.5 2.72 0.05 23.7 2.72 0.05 19.1

C1

314

56.4

pIV C

1.400

γ300

[K℄ [kPa℄ [-℄ 359 53.7 1.394 5 0.3 0.004 14.2 -4.8 -0.5 368 53.8 1.389 12 0.4 0.007 17.2 -4.6 -0.8 359 53.7 1.394 6 0.4 0.006 14.8 -4.8 -0.5 359 53.7 1.394 5 0.4 0.005 14.2 -4.8 -0.4

TIV C

-6.2

℄

−9.3 · 10−5

-6.5

−8.7 · 10−5 2.6 · 10−6

-9.0

−8.7 · 10−5 2.6 · 10−6

-5.8

−8.8 · 10−5 9.7 · 10−6

[

b K −1 −8.7 · 10−5 2.1 · 10−6

400

[K℄ 372 6 -7.1 406 37 1.4 371 5 -6.0 371 5 -7.2

Tw

0.882

[ ℄ 0.882 4 · 10−7 0.0 0.882 3.61 · 10−6 0.0 0.882 3.6 · 10−7 0.0 0.882 3.73 · 10−7 0.0

Vcr cm3

-1.4

[kPa℄ 1.4 0.4 -414.9 1.0 0.6 -387.6 1.4 0.5 -401.2 1.4 0.5 -403.3

∆p

0.47

[deg℄ 0.49 0.02 5.0 0.46 0.04 -2.8 0.49 0.02 3.6 0.49 0.02 5.0

∆θ

1.000

[-℄ 1.000 0.000 0.0 1.002 0.002 0.2 1.000 0.000 0.0 1.000 0.000 -0.0

Kp

[%℄ 14.7 14.7 14.7 21.0 21.0 21.0 15.8 15.8 15.8 14.7 14.7 14.7 0.0

L δ ε# x

[%℄ 34.4 34.4 34.4 36.7 36.7 36.7 35.4 35.4 35.4 34.5 34.5 34.5 0.0

ε¯# x

Table C.7: Mean value, standard deviation (σest ) and RME of the estimate for the four versions of method 2 using ase 2 (δi 6= c), evaluated for 101 experimental motored y les at OP1.

x#

RME

δx (M 2 : 3+) σest

RME

δx (M 2 : 3) σest

RME

δx (M 2 : 2) σest

RME

δx (M 2 : 1) σest

δx

284 APPENDIX C. PRIOR KNOWLEDGE APPROACH

...

58.8

58.8

58.8

Vc

[cm3 ℄ 57.3 57.5 57.5 57.4 56.8 57.8 58.6 58.8 58.8 58.2 58.4 58.7 58.9 59.0

2.28

2.28

2.28

[-℄ 3.15 3.04 3.45 3.78 4.43

C1

414

414

414

[K℄ 561 499 537

TIV C

50.0

50.0

50.0

pIV C

[kPa℄ 51.0 49.0 52.1 52.5 51.6 51.3 50.4 49.8 49.8 49.9 49.7 49.7

1.35

1.35

[-℄ 1.33 1.34 1.32 1.32 1.32 1.34 1.35 1.35 1.35 1.35 1.35 1.35 1.35 1.35 1.35 1.35

γ300

-7e-005

−7 · 10−5

−5.42 · 10−5 −6.1 · 10−5 −5.6 · 10−5 −7.3 · 10−5 −7.21 · 10−5 −7.2 · 10−5 −6.95 · 10−5 −7.12 · 10−5 −7.05 · 10−5 −6.84 · 10−5 −7 · 10−5

b

[K −1 ℄

480

480

480

Tw

[K℄ 552 525 561 393 476 373

0.59

0.59

0.59

Vcr

[cm3 ℄ 0.27 0.49 0.14 0.2

5.0

5.0

5.0

∆p

[kPa℄ 2.5 3.5 2.4 2.1 2.9 3.4 4.3

0.10

0.10

0.10

∆θ

[deg℄ 0.08 0.09 0.07 0.05 0.05 0.01 0.01 0.06

1.00

1.00

1.00

[-℄ 1.02 1.05

Kp

· 10−4 3.24 · 10−4 3.24

3.24

2.85 · 10−4 · 10−4

C2

[m/(s K)℄

503

503

503

[J℄ 544 516 559 554 556 518 508 503 504 508 506

Qin

-15.0

-15.0

-15.0

θig

[deg ATDC℄ -16.1 -16.1 -16.0 -16.0 -15.9 -15.8 -15.6 -15.8 -15.9

20.0

20.0

20.0

∆θd

[deg℄ 21.3 21.3 21.3 21.2 21.1 20.8 20.7 20.9 21.0 20.1 20.2 20.1 20.1 20.1 20.1

20.0

20.0

20.0

∆θb

[deg℄ 20.0 20.0 20.1 20.0 20.0 20.0 19.9 19.9 20.0 20.0 20.0 20.0 20.0

ε¯tx

[%℄ 20.5 11.8 25.4 24.1 24.0 6.5 2.8 2.2 2.0 0.7 0.3 0.2 0.2 0.1 0.2 0.0 5.0 0.0

ε¯# x

[%℄ 20.5 11.8 25.4 24.1 24.0 6.5 2.8 2.2 2.0 0.7 0.3 0.2 0.2 0.1 0.2 0.0 0.0 5.0

Table C.8: Method 1: Parameter estimates at OP1 (ring y les) for d# ∈ [1, 16] parameters when using no preferred ordering and a false prior level of 0 %. The true and nominal parameter deviations ε¯tx and ε¯# x are given as RMSE:s of their # relative values, i.e. ε¯tx = RMSE(εtx ) and ε¯# x = RMSE(εx ).

x# xt

16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

d#

C.7. COMPLEMENTARY RESULTS

285

286

...

APPENDIX C. PRIOR KNOWLEDGE APPROACH

Method 2; OP8

0

Zoomed version

10

−1

10

−1

10 −2

−3

x

RMSE( ε# ) [−]

10 10

−4

10

−2

10

−5

10

M2:3 M2:2 M2:1 Optimal

−6

10

−7

10

−2

10

−1

0

10 RMSE( ε ) [bar]

10

−2

10

−1

10 RMSE( ε ) [bar]

Figure C.5: L- urve (solid line) for one (motored) engine y le at OP8 with false prior RMSE(ε#−t ) (dotted line), when using setup 2 and x

ase 1 of the δi (δi = c). The results for the three versions of method 2 and the optimal hoi e of regularization parameter are indi ated by the legend. The true parameter deviation RMSE(εtx ) (dash-dotted line) is also given.

N [rpm℄ pman [kPa℄ θig [deg ATDC℄

OP1 1200 32 -33

OP2 1500 55 -26

OP3 1500 130 -5

OP4 2000 123 -12

OP5 3000 103 -28

OP6 3500 120 -24

Table C.9: Operating point onditions for experimental y les at OP1 6.

OP1 OP2 OP3 OP4 OP5 OP6

Vc [cm3 ℄ 58.8 58.8 58.8 58.8 58.8 58.8

TIV C [K℄ 450 425 407 394 400 392

pIV C [kPa℄ 40.8 68.6 162.5 152.3 132.2 153.3

γ300 [-℄ 1.336 1.335 1.339 1.339 1.335 1.335

b [K −1 ℄ −7.42 · 10−5 −7.42 · 10−5 −7.42 · 10−5 −7.42 · 10−5 −7.42 · 10−5 −7.42 · 10−5

Tw [K℄ 475 475 475 475 475 475

Vcr [cm3 ℄ 0.882 0.882 0.882 0.882 0.882 0.882

∆p [kPa℄ 0.5 0.6 -2.2 -2.8 -3.8 -2.7

∆θ [deg℄ 0.47 0.47 0.47 0.47 0.47 0.47

Kp [-℄ 1.000 1.000 1.000 1.000 1.000 1.000

C2 [m/(s K)℄ 3.24 · 10−4 3.24 · 10−4 3.24 · 10−4 3.24 · 10−4 3.24 · 10−4 3.24 · 10−4

Qin [J℄ 373 731 1864 1756 1502 1797

θig [deg ATDC℄ -32.6 -26.0 -5.3 -12.3 -28.0 -24.4

Table C.10: Nominal parameter values for experimental ring y les at operating points OP16.

C1 [-℄ 2.28 2.28 2.28 2.28 2.28 2.28

∆θd [deg℄ 27.5 19.9 15.4 17.3 22.0 21.1

∆θb [deg℄ 20.3 16.0 19.0 19.1 18.7 19.1

C.7. COMPLEMENTARY RESULTS

287

...

288

APPENDIX C. PRIOR KNOWLEDGE APPROACH

0

Method 2; OP8

−1

10

Zoomed version

10

−1

10

δ

RMSE( L ε# ) [−] x

−2

10

−3

10

−4

10

−5

10

−6

M2:3 M2:2 M2:1 Optimal

10

−2

10

−1

10 RMSE( ε ) [bar]

−1

10 RMSE( ε ) [bar]

Figure C.6: L- urve (solid line) for one (motored) engine y le at OP8 with weighted false prior setup 2 and ase 2 of the

) RMSE(Lδ ε#−t x δi (δi 6= c).

(dotted line), when using

The results for the three ver-

sions of method 2 and the optimal hoi e of regularization parameter are indi ated by the legend. The weighted true parameter deviation

RMSE(Lδ εtx) (dash-dotted line) is also given.

D

Notation The parameters are given in appendix D.1, and the abbreviations are summarized in appendix D.2. In the thesis various evaluation riteria are used, and they are summarized in appendix D.3. D.1

Parameters

In system identi ation literature, e.g. (Ljung, 1999), the onvention is to name the parameters by θ. However throughout this thesis, the parameters are instead named x. This in order to assign θ as the rank angle degree, whi h is ommon in engine literature, see e.g. (Heywood, 1988). D.1.1

C1 C2 hc p p0 T T0 Tw up w

Heat transfer

onstant in Wos hni's orrelation

onstant in Wos hni's orrelation

onve tion heat transfer oe ient

ylinder pressure for ring y le

ylinder pressure for motored y le mean gas temperature mean gas temperature for motored y le wall temperature mean piston speed

hara teristi velo ity 289

[-℄ [m/(s K)℄ [W / (m2 K)℄ [Pa℄ [Pa℄ [K℄ [K℄ [K℄ [m/s℄ [m / s℄

290 D.1.2

A Ach Alat Apc ar B l rc S θ V Vc Vcr Vd Vid v xof f

D.1.3

Kp N p ∆p pexh pm pman Tman Tw θ ∆θ θig θppp

APPENDIX D.

Engine geometry

instantaneous surfa e area

ylinder head surfa e area instantaneous lateral surfa e area piston rown surfa e area

rank radius

ylinder bore

onne ting rod length

ompression ratio index piston stroke

rank angle instantaneous ylinder volume

learan e volume aggregate revi e volume displa ement volume instantaneous displa ement volume tilt angle pin-o

Engine y le

pressure measurement gain engine speed

ylinder pressure pressure bias exhaust manifold pressure measured ylinder pressure intake manifold pressure intake manifold temperature mean wall temperature

rank angle

rank angle oset ignition angle peak pressure position

[-℄ [rpm℄ [Pa℄ [Pa℄ [Pa℄ [Pa℄ [Pa℄ [K℄ [K℄ [rad℄ [rad℄ [rad℄ [rad℄

[m2 ℄ [m2 ℄ [m2 ℄ [m2 ℄ [m℄ [m℄ [m℄ [-℄ [m℄ [rad℄ [m3 ℄ [m3 ℄ [m3 ℄ [m 3 ℄ [m3 ℄ [deg℄ [m℄

NOTATION

D.1.

291

PARAMETERS

D.1.4

( FA )s a b cv cp dmi dmcr hi Mi m ma maf mb mf mr mtot n Q Qch Qht Qin qHV R Tb Tr Tu U Vr W xb xi x ˜i xr yi γ γ300 γb γu ηf ∆θb ∆θcd ∆θd λ φ = λ−1

Thermodynami s and ombustion

stoi hiometri air-fuel ratio Vibe parameter slope in linear model of γ (2.30) mass spe i heat at onstant volume mass spe i heat at onstant pressure mass ow into zone i mass ow into revi e region mass spe i enthalpy molar mass of spe ie i Vibe parameter air mass air-fuel harge mass burned harge mass fuel mass residual gas mass total mass of harge polytropi exponent transported heat

hemi al energy released as heat heat transfer to the ylinder walls released energy from ombustion spe i heating value of fuel spe i gas onstant temperature in burned zone residual gas temperature temperature in unburned zone internal energy residual gas volume me hani al work mass fra tion burned mass fra tion of spe ie i mole fra tion of spe ie i residual gas fra tion xr = mr /mc CHEPP oe ient for spe ie i spe i heat ratio

onstant value in linear model of γ (2.30) γ for burned mixture γ for unburned mixture

ombustion e ien y rapid burn angle total ombustion duration ame development angle (gravimetri ) air-fuel ratio (gravimetri ) fuel-air ratio

[-℄ [-℄ [1/K℄ J [ kgK ℄ J [ kgK ℄ kg [s℄ [ kg s ℄ J [ kg ℄ [kg/mole℄ [-℄ [kg℄ [kg℄ [kg℄ [kg℄ [kg℄ [kg℄ [-℄ [J℄ [J℄ [J℄ [J℄ J [ kg ℄ J [ kgK ℄ [K℄ [K℄ [K℄ [J℄ [m 3 ℄ [J=Nm℄ [-℄ [-℄ [-℄ [-℄ [-℄ [-℄ [-℄ [-℄ [-℄ [-℄ [rad℄ [rad℄ [rad℄ [-℄ [-℄

292 D.1.5

APPENDIX D.

NOTATION

Parameter estimation

aε d dk

FP HN Lδ M(x) mδ mε P PˆN VN (x, Z N ) VNδ v WN x xˆ xˆ∗ xδ xδ,∗ xef f xˆef f xi xˆk xinit xs xsp xt x# x#,s y(t) yˆ(t|x) ZN α δ δi δx ∆x ε εtx ε¯tx ε# x ε¯# x

Morozov oe ient (6.22) number of parameters in M(x) des ent dire tion for iteration k false prior level approximative Hessian in (C.8) weighting matrix, Lδ = (δ)1/2 model for x Miller oe ient (6.20b) Miller oe ient (6.20a)

ovarian e matrix estimate of ovarian e at xˆ loss ( riterion) fun tion based on residual error penalty term based on nominal parameter deviation eigenve tor loss fun tion using prior knowledge ve tor used to parameterize model parameter estimate parameter estimate at optimum parameter estimate when using regularization regularized parameter estimate at optimum e ient parameter estimate for the e ient parameters parameter i estimate of parameter k initial values of parameters s aled parameters xs = Dx spurious parameters true value of the parameters nominal value of parameters s aled nominal value of parameters measured output predi ted model output data set [y(1), u(1), y(2), u(2), . . . , y(N ), u(N )] signi an e level regularization matrix diagonal element in δ regularization parameter sear h region for δx predi tion error y(t) − yˆ(t|x) or residual true parameter deviation (D.8) mean true parameter deviation (D.9) nominal parameter deviation (D.10) mean nominal parameter deviation (D.11)

D.2.

ABBREVIATIONS

ε#−t x η λ0 ˆN λ µ ν ς σ τ ψ

D.2

293

nominal parameter deviation from true values (D.12) noise level noise varian e estimated noise varian e step size for optimization algorithm regularization parameter for Levenberg-Marquardt singular value standard deviation

urvature of L- urve Ja obian ve tor, dened in (C.5)

Abbreviations

AFR AHR ATDC CAD CHEPP CI PDF ECU EVO FAP FAR FP FPE GDI IVC MAP MDP MFB MRE NRMSE RCI RE RME RMSE SI SOC SVC SVD TDC TWC

Air-Fuel equivalen e Ratio A

umulated Heat Release After TDC Crank Angle Degree CHemi al Equilibrium Program Pa kage(Eriksson, 2004) Compression Ignited Probability Density Fun tion Ele troni Control Unit Exhaust Valve Opening False Alarm Probability Fuel-Air equivalen e Ratio (level of) False Prior Final Predi tion Error Gasoline Dire t Inje ted Inlet Valve Closing Maximum A Posteriori Missed Dete tion Probability Mass Fra tion Burned Maximum Relative Error Normalized Root Mean Square Error Relative (95 %) Conden e Interval Relative estimation Error Relative Mean estimation Error Root Mean Square Error Spark Ignited Start Of Combustion Saab Variable Compression Singular Value De omposition Top Dead Center, engine rank position at 0 CAD Three-Way Catalyst

294

APPENDIX D.

D.3

NOTATION

Evaluation riteria

The evaluation riteria used in the thesis are summarized here, and given in a general form. Here y t denotes the true value, yˆ denotes the estimated or modeled value, y # is the nominal value and j is the sample number. Root mean square error (RMSE): v u M u 1 X t RM SE = (y t − yˆj )2 . M j=1 j

(D.1)

v u M u 1 X yjt − yˆj )2 . N RM SE = t ( M j=1 yjt

(D.2)

Normalized root mean square error (NRMSE):

Maximum relative error (MRE):

M RE = max | j

yjt − yˆj |. yjt

(D.3)

The relative mean error (RME): RM E =

M 1 X yjt − yˆj . M j=1 yjt

(D.4)

Mean 95 % relative onden e interval (RCI): M 1 X 1.96σj RCI = , M j=1 yjt

(D.5)

where σ is omputed using (C.18) and (C.20) in appendix C.1. False alarm probability (FAP): F AP = P (reje t H 0 |rcest ∈ NF) = P (T ≥ J|rc est ∈ NF).

(D.6)

The variables and nomen lature used are dened in example 5.1. Missed dete tion probability (MDP): M DP = P (not reje t H 0 |rc est ∈ / NF) = P (T < J|rc est ∈ / NF).

The variables and nomen lature used are dened in example 5.1.

(D.7)

D.3.

295

EVALUATION CRITERIA

True parameter deviation

εtx : εtx = yˆ − y t ,

(D.8)

whi h is losely related to the mean true parameter deviation:

v u M u 1 X t t ε¯x = RMSE(εx ) = t (ˆ yj − yjt )2 . M j=1 Nominal parameter deviation

(D.9)

ε# x:

εtx = yˆ − y # ,

(D.10)

whi h is losely related to the mean nominal parameter deviation:

v u M u 1 X # # ε¯x = RMSE(εx ) = t (ˆ yj − yj# )2 . M j=1 Nominal parameter deviation from true values

ε#−t = y# − yt. x

(D.11)

ε#−t : x

(D.12)

296

Notes

Notes

297

298

Notes

Notes

299

300

Linköping Studies in Te hnology and S ien e Department of Ele tri al Engineering Dissertations, Vehi ular Systems

No. 8

Modelling for Fuel Optimal Control of a Variable Compression Engine Ylva Nilsson, Dissertation No. 1119, 2007

No. 7

Fault Isolation in Distributed Embedded Systems Jonas Biteus, Dissertation No. 1074, 2007

No. 6

Design and Analysis of Diagnosis Systems Using Stru tural Methods Mattias Krysander, Dissertation No. 1033, 2006

No. 5

Air harge Estimation in Turbo harged Spark Ignition Engines Per Andersson, Dissertation No. 989, 2005

No. 4

Residual Generation for Fault Diagnosis Erik Frisk, Dissertation No. 716, 2001

No. 3

Model Based Fault Diagnosis: Methods, Theory, and Automotive Engine Appli ations, Mattias Nyberg, Dissertation No. 591, 1999

No. 2

Spark Advan e Modeling and Control Lars Eriksson, Dissertation No 580, 1999

No. 1

Driveline Modeling and Control Magnus Pettersson, Dissertation No. 484, 1997