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UNIVERSITATEA TEHNICĂ “GHEORGHE ASACHI” DIN IAŞI Şcoala Doctorală a Facultăţii de Automatică şi Calculatoare OVERALL POWERTRAIN MODELING AND CONTROL BASED ON DRIVELINE SUBSYSTEMS INTEGRATION (Controlul integrat al lanțului de transmisie a puterii) - TEZĂ DE DOCTORAT - Conducător de doctorat: Prof. univ. dr. ing. Corneliu Lazăr Doctorand: Ing. Andreea Elena Bălău IAŞI - 2011
Transcript
  • UNIVERSITATEA TEHNICĂ

    “GHEORGHE ASACHI” DIN IAŞI

    Şcoala Doctorală a Facultăţii de

    Automatică şi Calculatoare

    OVERALL POWERTRAIN MODELING AND CONTROL BASED ON DRIVELINE

    SUBSYSTEMS INTEGRATION

    (Controlul integrat al lanțului de transmisie a puterii)

    - TEZĂ DE DOCTORAT -

    Conducător de doctorat:

    Prof. univ. dr. ing. Corneliu Lazăr

    Doctorand:

    Ing. Andreea Elena Bălău

    IAŞI - 2011

  • UNIUNEA EUROPEANĂ GUVERNUL ROMÂNIEI

    MINISTERUL MUNCII, FAMILIEI ŞI PROTECŢIEI SOCIALE

    AMPOSDRU

    Fondul Social European POSDRU 2007-2013

    Instrumente Structurale 2007-2013

    OIPOSDRU UNIVERSITATEA TEHNICĂ “GHEORGHE ASACHI”

    DIN IAŞI

    UNIVERSITATEA TEHNICĂ “GHEORGHE ASACHI” DIN IAŞI

    Şcoala Doctorală a Facultăţii de Automatică şi Calculatoare

    OVERALL POWERTRAIN MODELING AND CONTROL BASED ON DRIVELINE

    SUBSYSTEMS INTEGRATION (Controlul integrat al lanțului

    de transmisie a puterii) - TEZĂ DE DOCTORAT -

    Conducător de doctorat: Prof. univ. dr. ing. Corneliu Lazăr

    Doctorand: Ing. Andreea Elena Bălău

    IAŞI - 2011

  • UNIUNEA EUROPEANĂ GUVERNUL ROMÂNIEI

    MINISTERUL MUNCII, FAMILIEI ŞI PROTECŢIEI SOCIALE

    AMPOSDRU

    Fondul Social European POSDRU 2007-2013

    Instrumente Structurale 2007-2013

    OIPOSDRU UNIVERSITATEA TEHNICĂ “GHEORGHE ASACHI”

    DIN IAŞI

    Teza de doctorat a fost realizată cu sprijinul financiar al

    proiectului „Burse Doctorale - O Investiţie în Inteligenţă (BRAIN)”.

    Proiectul „Burse Doctorale - O Investiţie în Inteligenţă (BRAIN)”,

    POSDRU/6/1.5/S/9, ID 6681, este un proiect strategic care are ca

    obiectiv general „Îmbunătățirea formării viitorilor cercetători în cadrul

    ciclului 3 al învățământului superior - studiile universitare de doctorat

    - cu impact asupra creșterii atractivității şi motivației pentru cariera în

    cercetare”.

    Proiect finanţat în perioada 2008 - 2011.

    Finanţare proiect: 14.424.856,15 RON

    Beneficiar: Universitatea Tehnică “Gheorghe Asachi” din Iaşi

    Partener: Universitatea “Vasile Alecsandri” din Bacău

    Director proiect: Prof. univ. dr. ing. Carmen TEODOSIU

    Responsabil proiect partener: Prof. univ. dr. ing. Gabriel LAZĂR

  • UNIUNEA EUROPEANĂ GUVERNUL ROMÂNIEI

    MINISTERUL MUNCII, FAMILIEI ŞI PROTECŢIEI SOCIALE

    AMPOSDRU

    Fondul Social European POSDRU 2007-2013

    Instrumente Structurale 2007-2013

    OIPOSDRU UNIVERSITATEA TEHNICĂ “GHEORGHE ASACHI”

    DIN IAŞI

    Motto:

    Learn from yesterday, live for today, hope for tomorrow. The important thing is not to stop questioning.

    Albert Einstein

  • UNIUNEA EUROPEANĂ GUVERNUL ROMÂNIEI

    MINISTERUL MUNCII, FAMILIEI ŞI PROTECŢIEI SOCIALE

    AMPOSDRU

    Fondul Social European POSDRU 2007-2013

    Instrumente Structurale 2007-2013

    OIPOSDRU UNIVERSITATEA TEHNICĂ “GHEORGHE ASACHI”

    DIN IAŞI

  • Acknowledgements

    Looking back, I am surprised and at the same time very grateful for everythingI have received throughout these years. It has certainly shaped me as a personand has led me where I am now.

    Foremost, I would like to express my sincere gratitude to my advisor Prof. Cor-neliu Lazăr, for the continuous support of my Ph.D study and research, for hismotivation, enthusiasm, patience and immense knowledge. His guidance helpedme in all the time of research and writing of this thesis.

    My sincere thanks also goes to Prof. Paul van den Bosch and Asst. Prof. MirceaLazăr, for offering me the opportunity to work in their department, for the de-tailed and constructive comments and for the kind support and guidance thathave been of great value in this study. Also, I would like to thank Dr. ing.Stefano Di Cairano for the constructive discussions and advices.

    I wish to express my warm thanks to Prof. Octavian Păstrăvănu, Prof. MihaelaHanako-Matcovski, Prof. Alexandru Onea, Assoc. Prof. Letiţia Mirea andAssoc. Prof. Lavinia Ferariu, for the extensive discussions around my work,constructive questions and excellent advices. I have to thank Costi for the sti-mulating discussions and for all the times we have worked together on variouspapers, and I also appreciate the short but productive collaboration I have hadwith Cristina.

    It was a pleasure to share doctoral studies and life with wonderful people likeAdrian, Simona, Marius and Alex, my first office mates, and with my Ph.Dcolleagues Alina, Costi, Cosmin, Carlos and Bogdan, who are now my very closefriends. I will never forget Dana’s late night dinners and all the special momentsI have spent with Nicu. I would like to thank all of them for their friendshipand for sharing the glory and sadness of reports and conferences deadlines andday-to-day research, and also for all the fun we have had in the last three years.

    I am forever indebted to my parents Mariana and Gheorghe, who raised me witha love of science and supported me in all my pursuits. I want to thank all of my

  • family for their understanding, their endless patience and encouragement whenit was most required, with a special thanks to my grandmother Paraschiva andmy sister Oana, for everything they have done for me. Finally, I want to dedicatethis thesis to my nephew Rivano, who I most love. He has shown a strong intereston studying when, at the early age of three, he clearly pointed out his interest ofbecoming a Professor Doctor Engineer.

    Andreea BălăuIaşi, 2011

  • Contents

    List of Figures xi

    List of Tables xv

    Glossary xvii

    1 Introduction 11.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.1.1 Driveline Modeling and Control . . . . . . . . . . . . . . . . . . . . . 11.1.1.1 Backlash Nonlinearity . . . . . . . . . . . . . . . . . . . . . 31.1.1.2 Clutch Nonlinearity . . . . . . . . . . . . . . . . . . . . . . 4

    1.2 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    2 Driveline Modeling and Control 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Electro-Hydraulic Valve-Clutch System . . . . . . . . . . . . . . . . . . . . . 122.3 Driveline Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    2.3.1 Drive Shaft Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.2 Flexible Clutch and Drive Shaft Model . . . . . . . . . . . . . . . . . 162.3.3 Continuous Variable Transmission Drive Shaft Model . . . . . . . . . 18

    2.4 Driveline Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.1 PID Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.2 PID Cascade-Based Driveline Control . . . . . . . . . . . . . . . . . . 212.4.3 Explicit MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4.4 Horizon-1 MPC based on Flexible Control Lyapunov Function . . . . 24

    2.4.4.1 Notation and Basic Definitions . . . . . . . . . . . . . . . . 242.4.4.2 Horizon -1 MPC . . . . . . . . . . . . . . . . . . . . . . . . 24

    2.4.5 Delta GPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    vii

  • CONTENTS

    2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    3 Modeling and Control of an Electro-Hydraulic Actuated Wet Clutch 313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Modeling of an Electro-Hydraulic Actuated Wet Clutch as a Subsystem of an

    Automated Manual Transmission . . . . . . . . . . . . . . . . . . . . . . . . 323.2.1 Test Bench Description . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.2 Modeling of an Pressure Reducing Valve . . . . . . . . . . . . . . . . 34

    3.2.2.1 Valve Description . . . . . . . . . . . . . . . . . . . . . . . . 343.2.2.2 Input-Output Model . . . . . . . . . . . . . . . . . . . . . . 363.2.2.3 State-Space model . . . . . . . . . . . . . . . . . . . . . . . 403.2.2.4 Simulators for the Pressure Reducing Valve . . . . . . . . . 41

    3.2.3 Modeling of the Electro-Hydraulic Actuated Wet Clutch System . . . 473.2.3.1 Description of the Valve-Clutch System . . . . . . . . . . . 493.2.3.2 Input-Output Model . . . . . . . . . . . . . . . . . . . . . . 503.2.3.3 State-Space Model . . . . . . . . . . . . . . . . . . . . . . . 513.2.3.4 Simulators for the Electro-Hydraulic Actuated Wet Clutch . 53

    3.3 Control of the Electro-Hydraulic Actuated WetClutch as a Subsystem of an Automated Manual Transmission . . . . . . . . 573.3.1 Generalized Predictive Control . . . . . . . . . . . . . . . . . . . . . . 583.3.2 PID Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

    3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

    4 Two Inertias Driveline Model Including Backlash Nonlinearity 654.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2 Driveline Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

    4.2.1 CVT Driveline Model with Backlash Nonlinearity . . . . . . . . . . . 664.2.1.1 PWA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.2.1.2 Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . . 69

    4.2.2 AMT Driveline Model with Backlash Nonlinearity . . . . . . . . . . . 704.2.2.1 Rigid Driveline Model . . . . . . . . . . . . . . . . . . . . . 704.2.2.2 Flexible Driveline Model . . . . . . . . . . . . . . . . . . . . 724.2.2.3 Flexible Driveline Model with Backlash . . . . . . . . . . . . 73

    4.3 Driveline Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3.1 PID Cascade-Based Driveline Controller . . . . . . . . . . . . . . . . 754.3.2 Horizon -1 MPC Controller . . . . . . . . . . . . . . . . . . . . . . . 78

    4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

    viii

  • CONTENTS

    4.4.1 Simulator for the PWA Model of the CVT Driveline . . . . . . . . . . 804.4.2 Simulator for the Nonlinear Model of the CVT Driveline . . . . . . . 83

    4.5 Real Time Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.5.1 System Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.5.2 Electromechanical Plant Description . . . . . . . . . . . . . . . . . . 884.5.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

    4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

    5 Three Inertias Driveline Model Including Clutch Nonlinearity 995.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.2 Driveline Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

    5.2.1 AMT Affine Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.2.2 AMT Piecewise Affine Model . . . . . . . . . . . . . . . . . . . . . . 1025.2.3 Dual Clutch Transmission Driveline . . . . . . . . . . . . . . . . . . . 105

    5.3 Driveline Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.3.1 Explicit MPC Controller . . . . . . . . . . . . . . . . . . . . . . . . . 1095.3.2 Horizon-1 MPC Controller . . . . . . . . . . . . . . . . . . . . . . . . 1105.3.3 Delta GPC Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 113

    5.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.4.1 Delta GPC for the Affine Model . . . . . . . . . . . . . . . . . . . . . 1155.4.2 Affine Model Versus PWA Model . . . . . . . . . . . . . . . . . . . . 1165.4.3 AMT Driveline Control . . . . . . . . . . . . . . . . . . . . . . . . . . 121

    5.4.3.1 Scenario 1: Acceleration test . . . . . . . . . . . . . . . . . 1225.4.3.2 Scenario 2: Deceleration test . . . . . . . . . . . . . . . . . 1245.4.3.3 Scenario 3: Tip-in tip-out maneuvers . . . . . . . . . . . . . 1275.4.3.4 Scenario 4: Stress test . . . . . . . . . . . . . . . . . . . . . 127

    5.4.4 DCT Driveline Control . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.4.4.1 Up-shift maneuvers . . . . . . . . . . . . . . . . . . . . . . . 1295.4.4.2 Down-shift maneuvers . . . . . . . . . . . . . . . . . . . . . 130

    5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

    6 Conclusions 1356.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

    6.1.1 Modeling and Control of an Electro-Hydraulic Actuated Wet ClutchSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

    6.1.2 Modeling and Control of a Two Inertia Driveline Including BacklashNonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

    ix

  • CONTENTS

    6.1.3 Modeling and Control of a Three Inertia Driveline Including ClutchNonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

    6.2 Suggestion for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . 138

    A 141

    Bibliography 147

    x

  • List of Figures

    2.1 Schematic vehicle structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Driveline subsystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Schematic valve structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Valve plunger subsystem model. . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Drive shaft model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Flexible clutch and drive shaft model. . . . . . . . . . . . . . . . . . . . . . . 172.7 Continuous variable transmission drive shaft model. . . . . . . . . . . . . . . 192.8 PID control structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.9 Cascade based control structure. . . . . . . . . . . . . . . . . . . . . . . . . . 22

    3.1 a) Test bench b) Schematic diagram . . . . . . . . . . . . . . . . . . . . . . . 333.2 a) Section through a real three stage pressure reducing valve; b) Three stage

    valve schematic representation; c) Charging phase of the pressure reducingvalve; d) Discharging phase of the pressure reducing valve. . . . . . . . . . . 35

    3.3 Transfer function block diagram of the pressure reducing valve. . . . . . . . . 393.4 Simulink model with step signal input. . . . . . . . . . . . . . . . . . . . . . 423.5 Simulink transfer functions of the valve model. . . . . . . . . . . . . . . . . . 423.6 Magnetic force and load flow. . . . . . . . . . . . . . . . . . . . . . . . . . . 433.7 Spool displacement and reduced pressure. . . . . . . . . . . . . . . . . . . . . 433.8 Input-output Simulink model. . . . . . . . . . . . . . . . . . . . . . . . . . . 443.9 Current and magnetic force used as input signals. . . . . . . . . . . . . . . . 453.10 Compared spool displacements for input-output model . . . . . . . . . . . . . 463.11 Compared reducing pressures for input-output model. . . . . . . . . . . . . . 463.12 State-space Simulink model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.13 Compared spool displacements for state-space model. . . . . . . . . . . . . . 483.14 Compared reducing pressures for state-space model. . . . . . . . . . . . . . . 483.15 Charging phase of the actuator-clutch system. . . . . . . . . . . . . . . . . . 493.16 Discharging phase of the actuator-clutch system. . . . . . . . . . . . . . . . . 50

    xi

  • LIST OF FIGURES

    3.17 Transfer function block diagram of the actuator-clutch system. . . . . . . . . 513.18 State-space block diagram of the actuator-clutch system. . . . . . . . . . . . 533.19 Input-output Simulink diagram of the actuator-clutch system. . . . . . . . . 543.20 System pressures for the input-output model. . . . . . . . . . . . . . . . . . 543.21 Input-output system simulation. . . . . . . . . . . . . . . . . . . . . . . . . . 553.22 State-space Simulink diagram of the actuator-clutch system. . . . . . . . . . 563.23 System pressures for the state-space model. . . . . . . . . . . . . . . . . . . . 573.24 State-space system simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 583.25 GPC results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.26 PID controller results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

    4.1 Schematic representation of an automotive driveline with backlash. . . . . . 674.2 Rigid driveline model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.3 Flexible driveline model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.4 Nonlinear CVT driveline structure - Simulink representation. . . . . . . . . . 754.5 Validation structure - Simulink representation. . . . . . . . . . . . . . . . . . 764.6 Input command - icvt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.7 Wheel speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.8 PID cascade based control structure - Simulink representation. . . . . . . . . 774.9 Torque controller - Simulink representation. . . . . . . . . . . . . . . . . . . 774.10 Speed controller - Simulink representation. . . . . . . . . . . . . . . . . . . . 784.11 Horizon-1 MPC - Simulink structure. . . . . . . . . . . . . . . . . . . . . . . 814.12 Wheel speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.13 Operating mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.14 Backlash angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.15 Engine torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.16 Optimal fuel-efficiency curve. . . . . . . . . . . . . . . . . . . . . . . . . . . 844.17 Wheel speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.18 Final drive-shaft torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.19 Engine speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.20 CVT ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.21 M220 Industrial plant emulator schematic structure. . . . . . . . . . . . . . . 874.22 Industrial plant emulator M220. . . . . . . . . . . . . . . . . . . . . . . . . . 894.23 Rigid driveline collocated controller - Simulink structure. . . . . . . . . . . . 904.24 Rigid driveline non-collocated controller - Simulink structure. . . . . . . . . . 924.25 Rigid driveline collocated and non-collocated control. . . . . . . . . . . . . . 924.26 Backlash mechanism structure. . . . . . . . . . . . . . . . . . . . . . . . . . 93

    xii

  • LIST OF FIGURES

    4.27 Rigid driveline with 4 degrees backlash angle collocated and non-collocatedcontrol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

    4.28 Rigid driveline with 8 degrees backlash angle collocated and non-collocatedcontrol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

    4.29 Flexible driveline controller - Simulink structure. . . . . . . . . . . . . . . . . 964.30 Flexible driveline with backlash control - engine inertia position. . . . . . . . 964.31 Flexible driveline with backlash control - wheel inertia position. . . . . . . . 97

    5.1 Three inertia driveline model. . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.2 Clutch functionality a) stiffness characteristic; b) clutch springs . . . . . . . 1035.3 AMT clutch switching logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.4 Double clutch transmission driveline model. . . . . . . . . . . . . . . . . . . 1065.5 DCT - Switching logic for the first clutch. . . . . . . . . . . . . . . . . . . . 1075.6 DCT - Switching logic for the second clutch. . . . . . . . . . . . . . . . . . . 1085.7 Simulation results using δ GPC. . . . . . . . . . . . . . . . . . . . . . . . . . 1155.8 Influences of the δ GPC on engine speed, transmission speed and axle wrap. 1165.9 δ GPC simulation results subject to reference changes. . . . . . . . . . . . . 1175.10 Influences of the δ GPC on engine speed, transmission speed and axle wrap,

    subject to reference changes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.11 Vehicle velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.12 Engine speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.13 Axle wrap speed difference. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.14 Engine torque (control signal). . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.15 Clutch mode of operation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.16 Scenario 1: Acceleration test. . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.17 Scenario 1: Clutch mode of operation. . . . . . . . . . . . . . . . . . . . . . 1235.18 Scenario 1: EMPC - Acceleration test. . . . . . . . . . . . . . . . . . . . . . 1255.19 Scenario 1: EMPC - Clutch mode of operation. . . . . . . . . . . . . . . . . 1255.20 Scenario 2: Deceleration test. . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.21 Scenario 3: Tip-in tip-out test. . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.22 Scenario 4: Stress test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.23 Scenario 1: Up-shift maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . 1305.24 MPC - Clutch operation modes for up-shift maneuvers test. . . . . . . . . . 1315.25 PID - Clutch operation modes for up-shift maneuvers test. . . . . . . . . . . 1315.26 Scenario 2: Down-shift maneuvers. . . . . . . . . . . . . . . . . . . . . . . . 1325.27 Clutch operation modes for down-shift maneuvers test. . . . . . . . . . . . . 132

    xiii

  • LIST OF FIGURES

    xiv

  • List of Tables

    A.1 Valve-clutch system parameter values . . . . . . . . . . . . . . . . . . . . . . 142A.2 Vehicle parameter values for two inertia CVT driveline with backlash nonli-

    nearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143A.3 Vehicle parameter values for two inertia AMT driveline with backlash nonli-

    nearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144A.4 Simulation vehicle parameter values for three inertias driveline with clutch

    nonlinearity - 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145A.5 Simulation vehicle parameter values for three inertias driveline with clutch

    nonlinearity -2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

    xv

  • GLOSSARY

    xvi

  • Glossary

    AMT Automated Manual Transmission

    ARX AutoRegressive eXogenous

    CARIMA Controlled AutoRegressive Integrated Moving Average

    CLF Control Lyaponov Function

    CVT Continuous Variable Transmission

    DAC Digital to Analog Converter

    DC Direct Current

    DCT Double clutch Transmission

    DSP Digital Signal Processor

    FCLF Flexible Control Lyapunov Function

    FDS Flexible Drive Shaft

    FRG Final Reduction Gear

    GPC Generalized Predictive Control

    LP Linear Program

    LQ Linear Quadratic

    LQG Linear Quadratic Gaussian

    LQR Linear Quadratic Regulator

    MILP MixtInteger Linear Program

    MPC Model Predictive Control

    xvii

  • GLOSSARY

    MPT Multi-Parametric Toolbox

    PI Proportional-Integrator

    PID Proportional-Integrator-Derivative

    PLC Programmable Logic Controller

    POG Power-Oriented Graphs

    PRBS PseudoRandom Binary Sequence

    PWA PieceWise Affine

    PWL PieceWise Linear

    SR Speed Reduction

    xviii

  • Chapter 1

    Introduction

    Recent studies in automotive engineering explore various engine, transmission and chassismodels and advanced control methods in order to increase overall vehicle performance, fueleconomy, safety and comfort. The goal of this thesis is overall powertrain modeling andcontrol, based on driveline subsystem integration. More complex driveline and drivelinesubsystems models are proposed, and different problems as nonlinearities introduced bybacklash and clutch are addressed, in order to improve vehicle performances.

    1.1 Literature Review

    An automotive powertrain is a system that includes the mechanical components which havethe function of transmitting the engine torque to the driving wheels. In order to transmitthis torque in an efficient way, a proper model of the driveline is needed for controller designpurposes, with the aim of lowering emissions, reducing fuel consumption and increasingcomfort.

    1.1.1 Driveline Modeling and Control

    The automotive driveline is an essential part of the vehicle and its dynamics have beenmodeled differently, according to the driving necessities. The complexity of the numerousmodels reported in the literature varies (Hrovat et al., 2000), but the two masses models aremore commonly used, and this fact is justified in (Pettersson et al., 1997), where it is shownthat this model is able to capture the first torsional vibrational mode. There are also morecomplex three-masses models reported in different research papers, as it will be indicatednext. In (Templin and Egardt, 2009) a simple driveline model with two inertias, one for theengine and the transmission, and one for the wheel and the vehicle mass, was presented.

    1

  • Introduction

    A more complex two-masses model, including a nonlinearity introduced by the backlash,was presented in (Templin, 2008). A mathematical model of a driveline was introduced in(Baumann et al., 2006) and (Bruce et al., 2005) in the form of a third order linear state-spacemodel. A simple model with the pressure in the engine manifold and the engine speed as statevariables and the throttle valve angle as control input was presented in (Saerens et al., 2008).Other two-masses model, with one inertia representing the engine and the other inertia rep-resenting the vehicle (including the clutch, main-shaft and the powertrain), were presented in(Bemporad et al., 2001), (Serrarens et al., 2004), (Larouci et al., 2007), (Song et al., 2010),(Glielmo and Vasca, 2000), (Peterson et al., 2003), (Gao et al., 2009). Two-masses mod-els for automotive driveline with continuous variable transmission (CVT) are presented in(Shen et al., 2001), (Serrarens et al., 2003) and (Liu and Yao, 2008). (Rostalski et al., 2007)presents a piecewise affine (PWA) two masses model for a driveline including a back-lash nonlinearity. In (Grotjahn et al., 2006) a two masses model was presented, with thedriveline main flexibility represented by the drive shafts, as well as a three mass modelto reproduce the behavior of a vehicle with a dual-mass flywheel. Linear and nonlinearthree masses models, in which the clutch flexibility was also considered, were presentedin (Kiencke and Nielsen, 2005). Complex three masses models that includes certain non-linear aspects of the clutch were presented in (Dolcini et al., 2005), (Glielmo et al., 2004),(Liu et al., 2011), (Garafalo et al., 2001), (Crowther et al., 2004), (Lucente et al., 2005),(Van Der Heijden et al., 2007), (Glielmo et al., 2006).

    Concerning the control strategy, different approaches have also been proposed in litera-ture. In (Templin and Egardt, 2009) a linear quadratic regulator (LQR) design that dampsdriveline oscillations by compensating the driver’s engine torque demand was presented. Theperformance cost uses a weighting of the time derivative of the drive shaft torque and the dif-ference between the driver’s torque demand and the actual controller torque demand. LQRcontrollers were also proposed in (David and Natarajan, 2005) and (Dolcini, 2007). Otherlinear quadratic Gaussian controllers designed with loop transfer recovery were presented in(Pettersson et al., 1997),(Fredriksson et al., 2002),(Berriri et al., 2007), (Berriri et al., 2008).Furthermore, (Bruce et al., 2005) proposed the usage of a feed-forward controller in combi-nation with a LQR controller and considering the engine as an actuator to damp power-train oscillations. A robust pole placement strategy was employed in (Richard et al., 1999),(Stewart et al., 2005), (Stewart and Fleming, 2004), an H∞ optimization approach was pre-sented in (Lefebvre et al., 2003), while model predictive control (MPC) strategies were pro-posed in (Lagerberg and Egardt, 2005), (Rostalski et al., 2007), (Baumann et al., 2006),(Falcone et al., 2007). A feedback controller combined with a feed-forward controller is pre-sented in (Adachi et al., 2004) and (Gao et al., 2010) In (Baumann et al., 2006), a model-

    2

  • 1.1 Literature Review

    based approach for anti-jerk control of passenger cars that minimizes driveline oscillationswhile retaining fast acceleration was introduced. The controller was designed with thehelp of the root locus method and an analogy to a classical PI-controller was drawn. In(Rostalski et al., 2007), a constraint was imposed on the difference between the motor speedand the load speed to minimize the driveline oscillations, while reducing the impact offorces between the mechanical parts. A clutch engagement controller based on fuzzy logicis presented in (Wu et al., 2009) a driveline control with torque observer is proposed in(Kim and Choi, 2010).

    In order to improve vehicle overall performances, problems as nonlinearities introducedby backlash and clutch system are modeled, and different control strategies are proposed.

    1.1.1.1 Backlash Nonlinearity

    Backlash is a common problem in powertrain control because it introduces a hard nonlin-earity in the control loop for torque generation and distribution. This phenomenon occurswhenever there is a gap in the transmission link which leads to zero torque transmittedthrough the shaft to the wheels. When the backlash gap is traversed the impact results ina large shaft torque and sudden acceleration of the vehicle. Engine control systems mustcompensate for the backlash with the goal of traversing the backlash as fast as possible.

    In an automotive powertrain, backlash and shaft flexibility results in an angular positiondifference between wheels and engine. The modeling of mechanical systems with backlashnonlinearities is a topic of increasing interest (Lagerberg and Egardt, 2005), (Templin, 2008),(Rostalski et al., 2007), because a backlash can lead to reduced performances and can evendestabilize the control system. Also, it can have as consequence low components reliabilityand shunt and shuffle. In order to model the mechanical system with backlash, two dif-ferent operational modes must be distinguished: backlash mode (when the two mechanicalcomponents are not in contact) and contact mode (when there is a contact between the twomechanical components resulting in a moment transmission).

    New driveline management application and high-powered engines increase the need forstrategies on how to apply the engine torque in an optimal way. (Lagerberg and Egardt, 2002)presents two controllers for a powertrain model including backlash: a standard PID con-troller and a modified switching controller. The concept of PID controller with torquecompensator is presented in (Nakayama et al., 2000) for the backlash. A simple activeswitching controller for a powertrain model including backlash nonlinearities is proposed in(Tao, 1999). In (Setlur et al., 2003) a nonlinear adaptive back-stepping controller is designedin order to ensure asymptotic wheel speed and gear ratio tracking. A nonlinear predictivecontroller is designed in (Saerens et al., 2008) in order to minimize the fuel consumption

    3

  • Introduction

    and to lower emissions. A power management decoupling control strategy is presented in(Barbarisi et al., 2005) with the aim of minimizing fuel consumption and increasing drive-ability. A rule based supervisory control algorithm is designed in (Rotenberg et al., 2008)in order to improve fuel economy. A nonlinear quantitative feedback theory is applied in(Abass and Shenton, 2010), in order to control an automotive driveline with backlash non-linearity.

    1.1.1.2 Clutch Nonlinearity

    In recent years, the use of control systems for automated clutch and transmission actuationhas been constantly increasing, the trend towards higher levels of comfort and driving dy-namics while at the same time minimizing fuel consumption representing a major challenge.

    The basic function of any type of automotive transmission is to transfer the engine torqueto the vehicle with the desired ratio smoothly and efficiently, and the most common controldevices inside the transmission are clutches and actuators. Such clutches can be hydraulicactuated, motor driven or actuated using other means.

    During the last years, the automated actuated clutch systems and different valve typesused as actuators have been actively researched and different models and control strategieshave been developed: physics-based nonlinear model for an exhausting valve (Ma et al., 2008),nonlinear physical model for programmable valves (Liu and Yao, 2008), nonlinear state-space model description of the actuator that is derived based on physical principles andparameter identification (Wang et al., 2002), (Peterson et al., 2003), (Gennaro et al., 2007),(Nemeth, 2004), mathematical model obtained using identification methods for a valve ac-tuation system of an electro-hydraulic engine (Liao et al., 2008), a model for an electro-hydraulic valve used as actuator for a wet clutch (Morselli and Zanasi, 2006), dynamic mod-eling and control of electro-hydraulic wet clutches (Morselli et al., 2003), PID control for awet plate clutch actuated by a pressure reducing valve (Edelaar, 1997), predictive and piece-wise LQ control of a dry clutch engagement (Van Der Heijden et al., 2007), switched controlof an electro-pneumatic clutch actuator (Langjord et al., 2008), Model Predictive Controlof a two stage actuation system using piezoelectric actuators for controllable industrial andautomotive brakes and clutches (Neelakantan, 2008).

    1.2 Outline of the Thesis

    The reminder of this thesis is structured as follows.Chapter 2, entitled Driveline modeling and control presents different driveline models

    and control strategies found in the literature. First, an electro-hydraulic valve-clutch system

    4

  • 1.2 Outline of the Thesis

    is presented, followed by three driveline models: a drive shaft model, a flexible clutch anddrive shaft model, and a continuous variable transmission drive shaft model. Next, a PID,a PID cascade based, an explicit MPC and a horizon-1 MPC based on flexible controlLyapunov function are presented as driveline control strategies. Starting from these models,in what follows, more complex driveline models are developed and also the control strategiespresented in this chapter are applied in order to obtain new controllers able to improveoverall vehicle performances.

    Chapter 3 is entitled Modeling and control of an electro-hydraulic actuated wet clutch.In this chapter, different models for an electro-hydraulic actuated wet clutch system in theautomatic transmission are presented. First, an input-output and a state-space model ofan electro-hydraulic pressure reducing valve are developed and stating from these, an input-output and a state space model of an electro-hydraulic actuated wet clutch is obtained.Simulators for the wet clutch and its actuator were developed and were validated with dataprovided from experiments with the real valve actuator and the clutch on a test bench. Thetest bench was provided by Continental Automotive Romania and it includes the VolkswagenDQ250 wet clutch actuated by the electro-hydraulic valve DQ500. Also, different controlstrategies are applied on the developed models and simulation result are being discussed: aGPC and a PID controller are designed in order to control the output of the electro-hydraulicactuated clutch system, the clutch piston displacement.

    Chapter 4 is entitled Two inertias driveline model including backlash nonlinearity. Inthis chapter, different models for automotive driveline including backlash nonlinearity areproposed. First, a piecewise affine and a nonlinear state-space model for a ContinuousVariable Transmission (CVT) driveline with backlash are proposed. Simulators are developedin Matlab/Simulink for the two driveline models and different control strategies are applied.A horizon-1 MPC controller is designed for the linear model, while a PID cascade basedcontroller is applied for the nonlinear model designed to reduce the fuel consumption by usingthe optimal fuel efficiency curve in the modeling phase. Next, three models are presented foran Automated Manual Transmission (AMT) driveline based on the Industrial plant emulatorM220 : a rigid driveline model, a flexible driveline model and a flexible driveline modelincluding also backlash nonlinearity. Then, real time experiments are conducted on thepresented models in order to test the influences given by drive shaft flexibility and backlashangle, while applying a horizon-1 MPC controller.

    Chapter 5 is entitled Three inertias driveline model including clutch nonlinearity. Thischapter deals with the problem of damping driveline oscillations in order to improve passen-ger comfort. Three driveline models with three inertias are proposed: a state-space affinemodel and a new state-space piecewise affine model of an automated manual transmission

    5

  • Introduction

    (AMT) driveline, and a new state-space piecewise affine model of a double clutch trans-mission (DCT) driveline, all of them taking into consideration the drive shafts as well asthe clutch flexibilities. Three controllers are implemented for the developed models: ex-plicit MPC, delta GPC and horizon-1 MPC, and the experiments showed that the horizon-1MPC control scheme can handle both the performance/physical constraints and the strictlimitations on the computational complexity corresponding to vehicle driveline oscillationsdamping.

    1.3 List of Publications

    This thesis is based on fourteen published articles, divided as follows: one ISI indexed paper(IF=1.762), one Zentralblatt Math indexed paper, three ISI Proceedings papers, four IEEEconference papers, two IFAC conference papers and three papers published at internationalconferences where paper review is conducted.

    Chapter 3 contains results published in:

    • (Balau et al., 2009a) A. E. Balau, C. F. Caruntu, D. I. Patrascu, C. Lazar, M. H.Matcovschi and O. Pastravanu. Modeling of a Pressure Reducing Valve Actuator forAutomotive Applications. In 18th IEEE International Conference on Control Applica-tions, Part of 2009 IEEE Multi-conference on Systems and Control, Saint Petersburg,Russia, 2009.

    • (Balau et al., 2009b) A. E. Balau, C. F. Caruntu, C. Lazar and D. I. Patrascu. NewModel for Predictive Control of an Electro-Hydraulic Actuated Clutch. In The 18thInternational Conference on FUEL ECONOMY, SAFETY and RELIABILITY of MO-TOR VEHICLES (ESFA 2009), Bucharest, Romania, 2009.

    • (Patrascu, Balau et al., 2009) D. I. Patrascu, A. E. Balau, C. F. Caruntu, C. Lazar,M. H. Matcovschi and O. Pastravanu. Modelling of a Solenoid Valve Actuator forAutomotive Control Systems. In The 1tth International Conference on Control Systemsand Computer Science, Bucharest, Romania, 2009.

    • (Caruntu, Matcovschi, Balau et al., 2009) C. F. Caruntu, M. H. Matcovschi, A. E.Balau, D. I. Patrascu, C. Lazar and O. Pastravanu. Modelling of An ElectromagneticValve Actuator. Buletinul Institutului Politehnic din Iasi, vol. Tome LV (LIX), Fasc.2, pages 9–28, 2009.

    6

  • 1.3 List of Publications

    • (Balau et al., 2010) A. E. Balau, C. F. Caruntu and C. Lazar. State-space model of anelectro-hydraulic actuated wet clutch. In IFAC Symposium Advances in AutomotiveControl, Munchen, Germany, 2010.

    • (Balau et al., 2011a) A. E. Balau, C. F. Caruntu and C. Lazar. Simulation and Controlof an Electro-Hydraulic Actuated Clutch. Mechanical Systems and Signal Processing,vol. 25, pages 1911–1922, 2011.

    • (C.Lazar, Caruntu and Balau, 2010) C. Lazar, C. F. Caruntu and A. E. Balau. Mod-elling and Predictive Control of an Electro-Hydraulic Actuated Wet Clutch for Auto-matic Transmission. In IEEE Symposium on Industrial Electronics, Bari, Italy, 2010.

    • (Caruntu, Balau and C.Lazar, 2010a) C. F. Caruntu, A. E. Balau and C. Lazar. Net-worked Predictive Control Strategy for an Electro-Hydraulic Actuated Wet Clutch. InIFAC Symposium Advances in Automotive Control, Munchen, Germany, 2010.

    • (Balau and C.Lazar, 2011a) A. E. Balau and C. Lazar. Predictive control of an electro-hydraulic actuated wet clutch. In The 15th International Conference on System Theory,Control and Computing, Sinaia, Romania, 2011.

    Chapter 4 contains results published in:

    • (Caruntu, Balau and C.Lazar, 2010b) C. F. Caruntu, A. E. Balau and C. Lazar. Cas-cade based Control of a Drivetrain with Backlash. In 12th International Conference onOptimization of Electrical and Electronic Equipment, Brasov, Romania, 2010.

    Chapter 5 contains results published in:

    • (Balau et al., 2011b) A. E. Balau, C. F. Caruntu and C. Lazar. Driveline oscillationsmodeling and control. In The 18th International Conference on Control Systems andComputer Science, Bucharest, Romania, 2011.

    • (Balau and C.Lazar, 2011b) A.E. Balau and C. Lazar. One Step Ahead MPC for anAutomotive Control Application. In The 2nd Eastern European Regional Conferenceon the Engineering of Computer Based Systems, Bratislava, Slovakia, 2011.

    • (Caruntu, Balau et al., 2011) C. F. Caruntu, A. E. Balau, M. Lazar, P. P. J. v. d. Boshand S. Di Cairano. A predictive control solution for driveline oscillations damping. InThe 14th International Conference on Hybrid Systems: Computation and Control,Chicago, USA, 2011.

    7

  • Introduction

    • (Halauca, Balau and C.Lazar, 2011) C. Halauca, A. E. Balau and C. Lazar. StateSpace Delta GPC for Automotive Powertrain Systems. In The16th IEEE InternationalConference on Emerging Technologies and Factory Automation, 2011.

    8

  • Chapter 2

    Driveline Modeling and Control

    An automotive powertrain is a system that includes the mechanical components which havethe function of transmitting the engine torque to the driving wheels. In order to transmitthis torque in an efficient way, a proper model of the driveline is needed for controllerdesign purposes with the aim of lowering emissions, reducing fuel consumption and increasingcomfort. Recent studies in automotive engineering explore various engine, transmission andchassis models and advanced control methods in order to increase overall vehicle performance.

    2.1 Introduction

    The driveline is a fundamental part of a vehicle and its dynamics have been modeled indifferent ways, according to the purpose. The aim of the modeling is to find the mostsignificant physical effects that have as negative result oscillations in the wheel speed. Mostexperiments consider in the modeling phase low gears because the higher torque transmittedto the drive shaft is obtained in the lower gear. Also, the amplitudes of the resonances in thewheel speed are higher for lower gears, because the load and vehicle mass appear reducedby the high conversion ratio.

    The structure of a passenger car consists, in general, of the following parts: engine, clutch,transmission, propeller shaft, final drive, drive shafts and wheels, as it can be seen in Fig. 2.1.In what follows, the fundamental equations of the driveline will be derived by using the gen-eralized Newton’s second law of motion, as described in (Kiencke and Nielsen, 2005). FigureFig. 2.2 shows the labels, the inputs and the outputs of each subsystem of the considereddriveline, and relations between them will be described for each part.

    The output engine torque is given by the driving engine torque Te resulted from thecombustion, the internal engine friction Tfric,e, and the external load from the clutch Tc,

    9

  • Driveline Modeling and Control

    Figure 2.1: Schematic vehicle structure.

    Figure 2.2: Driveline subsystems.

    obtaining the following equation:

    Jeθ̈e = Te−Tfric,e−Tc, (2.1)

    where Je represents the engine moment of inertia, θe is the crankshaft angle, ωe = θ̇e is theengine angular velocity and ω̇e = θ̈e is the engine angular acceleration.

    A friction clutch consists of a clutch disk connecting the flywheel of the engine and thetransmission’s input shaft. When the clutch is engaged, and no internal friction is assumed,then Tc = Tt. The transmitted torque Tt is a function of the angular difference (θe−θc) andthe angular velocity difference (ωe−ωc) over the clutch:

    Tc = Tt = fc (θe− θc, ωe−ωc) , (2.2)

    where θc represents the clutch angle and ωc = θ̇c is the clutch angular velocity.The transmission has a set of gears, each with a different conversion ratio it. The following

    equations between the input and output torque of the transmission is obtained:

    Tp = ft(Tt, Tfric,t, θc− θtit, ωc−ωtit, it

    ), (2.3)

    10

  • 2.1 Introduction

    where Tp is the propeller shaft torque, Tfric,t is the internal friction torque of the trans-mission, θt is the transmission angle and ωt = θ̇t is the corresponding angular velocity. Thereason for considering the angle difference θc−θtit is the possibility of having torsional effectsin the transmission.

    The propeller shaft connects the transmission’s output shaft with the final drive. Nofriction is assumed so Tp = Tf , giving the following equation:

    Tp = Tf = fp (θt− θp, ωt−ωp) , (2.4)

    where Tf is the final drive torque, θp is the propeller shaft angle and ωp = θ̇p is the corre-sponding angular velocity.

    The final drive is characterized by a conversion ratio if in the same way as for thetransmission. The following relation between the input and the output torque holds:

    Td = ff(Tf , Tfric,f , θp− θf if , ωp−ωf if , if

    ), (2.5)

    where Tfric,f is the internal friction torque of the final drive, Td is the drive shaft torque, θfis the final drive angle and ωf = θ̇f is the corresponding angular velocity.

    The drive shafts is the subsystem that connects the wheel to the final drive. Assumingthat θw is the wheel angle, the rotational wheel velocity ωw = θ̇w is the same for both wheelsand neglecting the vehicle dynamics, the rotational equivalent wheel velocity shall be equalto the velocity of the vehicle body’s center of gravity vv:

    ωw =vvrstat

    , (2.6)

    where rstat represents the wheel radius. The shafts are modeled as one shaft and assumingthat no friction exists Tw = Td the following equation for the wheel torque Tw results:

    Tw = Td = fd(θf − θw, ωf −ωw

    ). (2.7)

    Newton’s second law in the longitudinal direction for a vehicle with mass mCoG andspeed vv, gives:

    Fload =mCoGv̇v +Fairdrag +Froll+mCoGg sin(χroad). (2.8)

    The load force Fload is described by the sum of following quantities:

    • Fairdrag, the air drag, is approximated by Fairdrag = 12cairAfρairv2v , where cair is the

    drag coefficient, Af is the maximum vehicle cross section area and ρair is the air density.

    11

  • Driveline Modeling and Control

    • Froll, the rolling resistance, is approximated by Froll = mCoG (cr1 + cr2vv) where cr1and cr2 depend on the tire pressure.

    • mCoG sin(χroad), the gravitational force, where χroad is the road slope.

    The resulting torque Tload is equal with Floadrstat and the equation of motion for thewheel is described by the following relation:

    Jwω̇w = Tw−Floadrstat−TL, (2.9)

    where Jw is the wheel moment of inertia and TL is the friction torque. By including (2.8) in(2.9) gives:(Jw +mCoGr2stat

    )ω̇w = Tw−TL−

    12cairAfρairr

    3statω

    2w− rstatmCoG (cr1 + cr2rstatωw)

    − rstatmCoGg sin(χroad) .(2.10)

    A complete model of the driveline with the clutch engaged is described by equations(2.1) to (2.10). So far functions fc,ft,fp,ff ,fd and the friction torques Tfric,t,Tfric,f ,TL areunknown, and assumption about these can be made, resulting in a series of driveline models,with different complexities.

    2.2 Electro-Hydraulic Valve-Clutch System

    The basic function of any type of automotive transmission is to transfer the engine torqueto the vehicle with the desired ratio smoothly and efficiently and the most common controldevices inside the transmission are clutches and hydraulic pistons. The automatic controlof the clutch engagement plays a crucial role in AMT (Automatic Manual Transmission)vehicles, being seen as an increasingly important enabling technology for the automotiveindustry. It has a major role in automatic gear shifting and traction control for improvedsafety, driveability and comfort and, at the same time, for fuel economy. Recent attention hasfocused on modeling different valve types used as actuators in automotive control systemsand, in what follows, a model found in the literature of an electro hydraulic actuated wetclutch system is presented.

    A new modeling method of automotive control systems, based on power graphs, is pre-sented in (Morselli and Zanasi, 2006), where a system composed of an electro-hydraulic valveand a wet clutch is modeled. The method is called Power-Oriented Graphs (POG) and uti-lizes the power interaction between the subsystems, as a base concept for the modeling phase.The POG technique is suited for modeling various control systems from different energeticdomains.

    12

  • 2.2 Electro-Hydraulic Valve-Clutch System

    Figure 2.3: Schematic valve structure.

    Figure 2.4: Valve plunger subsystem model.

    The valve-clutch system presented in Fig. 2.3 can be divided into four interacting sub-systems: valve plunger, control chamber, user chamber and clutch chamber. In order toillustrate the POG approach, the subsystem corresponding to the valve plunger is repre-sented in Fig. 2.4.

    The plunger mass Mv moves according to the damping coefficient bp, the return springKp(xs) and the pressures PC and PD from the control chamber, and the back chamber,respectively. xs and ẋs represents the displacement and the valve plunger speed, respectively,and Ap is the area of the plunger’s extremities. The nonlinear force Kp(xs) models thereturn spring as well as the contact force between the plunger and the plunger chamber, atthe plunger two extremities. The plunger movement causes the oil flow QC and QD throughthe control chamber and through the back chamber:

    Mvẍs = (PC −PD)Ap− bpẋs−Kp(xs),

    QC =QD = Apẋs.(2.11)

    The pressure from the control chamber PC is obtained by integrating three oil flows:

    13

  • Driveline Modeling and Control

    the flow Q5 from the power source Ps, the flow QC due to the plunger movement, and theflow Qw through the variable discharging orifice. The very small hydraulic capacity CCstores potential energy in terms of oil pressure and it takes into account the small elasticdeformation of the valve case and the oil stiffness:

    CC ṖC =Q5−QC −Qw. (2.12)

    Depending on the plunger position, the output user chamber is connected either to thepower supply through the variable orifice J1 or to the oil tank by the orifice J3. Also, theuser chamber is connected to the back chamber through orifice J4. This orifice plays tworoles: it implements the feedback action since PD becomes a measure of the user pressurePR, and it has the damping effect that avoids plunger oscillations.

    The back chamber and the user chamber are modeled as two small hydraulic capacities,as for the control chamber:

    CDṖD =QD−Q4,

    CRṖR =Q1 +Q4−Q3−QR.(2.13)

    The user chamber is connected to the clutch chamber by means of a pipe with a dynamicthat cannot be neglected and is described by four elements: the user chamber capacityCR, the hydraulic resistance RL, the pipe hydraulic inductance LL and the clutch chambercapacity CL:

    LLQ̇R = Pl−PL = PR−PQR−PL,

    PR−Pl =QR |QR|CL

    =RL(QR),

    CLṖL =QR−ALż.

    (2.14)

    where PL is the clutch pressure.The motion of the pressure plate under the effects of the pressure PL, the elastic force

    KM (z) and the viscous friction bf are given by the following equations:

    Mpz̈ = PLAL− bfxż−KM (z)−Kbcsgn(ż),

    KM (z) =KF (z) +KD(z).(2.15)

    where Mp is the clutch plunger mass, AL is the clutch piston area, KF (z) represents theforce of the return springs and the contact with the gearbox at the two extreme pressureplate positions, and KD(z) is the force generated by the compression of the clutch discs thatdetermines the maximum torque through the clutch.

    This combined equations model the valve-clutch system using the POG technique and thesimulations results are very similar to the experimental data, providing that the modelingapproach is suitable to automotive control systems.

    14

  • 2.3 Driveline Models

    Figure 2.5: Drive shaft model.

    2.3 Driveline Models

    The automotive driveline is an essential part of the vehicle and its dynamics have beenmodeled differently, according to the driving necessities. In this sections, three differentdriveline models reported in literature are presented.

    2.3.1 Drive Shaft Model

    In (Kiencke and Nielsen, 2005) a simplified model of an automotive driveline is presented.The driveline has two inertias and the structure presented in Fig. 2.5 is composed by: internalcombustion engine, transmission, flexible drive shafts and driven wheel. The propeller shaftis considered to be stiff and it is not represented here.

    Starting from the equations (2.1) to (2.10), that describe the complete driveline dynamics,the equation for the lumped engine and transmission inertia is obtained:

    Je+ Jti2t

    + Jfi2t i

    2f

    ω̇e =Te−Tfric,e−dti2t

    + dfi2t i

    2f

    ωe−− kditif

    θei2t i

    2f

    − θw

    − dditif

    ωei2t i

    2f

    −ωw

    ,(2.16)

    where Jt and Jf represents the transmission and the final drive inertias, while dt and dfstands for the corresponding damping coefficients. Also, kd and dd represents the stiffnessand damping coefficients of the drive shaft.

    Also, the equation for the vehicle and wheels inertia is given by:(Jw +mCoGr2stat

    )ω̇w = kd

    θei2t i

    2f

    − θw

    +dd ωei2t i

    2f

    −ωw

    − 12cairAfρar3statω2w−−rstatmCoG (cr1 +g sin(χroad))−

    (dw +mCoGcr2r2stat

    )ωw,

    (2.17)

    15

  • Driveline Modeling and Control

    where dw represents the damping coefficient of the wheel.The drive shaft model is the simplest one considered, and the drive shaft torsion, the

    engine speed and the wheel speed are used as states, according to:

    x1 =θeif it− θw

    x2 = ωex3 = ωw

    . (2.18)

    Also, taking into consideration that:

    J1 = Je+Jti2t

    + Jfi2t i

    2f

    J2 = Jw +mCoGr2stat

    d1 =dti2t

    + dfi2t i

    2f

    d2 = dw +mCoGcr2r2statl = rstatmCoG (cr1 +g sin(χroad))

    , (2.19)

    the following state-space representation is obtain:

    ẋ= Ax+Bu+Hl, (2.20)

    consisting of the system matrices:

    A=

    0 1if it −1

    − kif itJ1−d1+ d

    if it2

    J1d

    if itJ1kJ2

    0 dif itJ2−d+d2J2

    , (2.21)

    B =

    01J10

    ,H = 00−1J2

    . (2.22)2.3.2 Flexible Clutch and Drive Shaft Model

    A more complex model including two torsional flexibilities, the drive shaft and the clutchis also presented in (Kiencke and Nielsen, 2005). The driveline has three inertias like rep-resented in Fig. 2.6, one corresponding to the internal combustion engine, one for thetransmission, and one for the driven wheel.

    The equation that describe the engine dynamics is given by:

    Jeω̇e = Te−Tfric,e−kc (θe− θtit)−dc (ωe−ωtit) , (2.23)

    16

  • 2.3 Driveline Models

    Figure 2.6: Flexible clutch and drive shaft model.

    where kc is the clutch stiffness and dc represents the damping of the clutch.The second equation describe the dynamics of the transmission:Jt+ Jf

    i2f

    ω̇t = Te−Tfric,e− it (kc (θe− θtit) +dc (ωe−ωtit))−−

    dt+ dfi2f

    ωt− 1if

    (kd

    (θtif− θw

    )+dd

    (ωtif−ωw

    )).

    (2.24)

    Also, the equation for the vehicle and wheels inertia is given by:

    (Jw +mCoGr2stat

    )ω̇w = kd

    (θtif− θw

    )+dd

    (ωtif−ωw

    )− 12cairAfρar

    3statω

    2w−

    −rstatmCoG (cr1 +g sin(χroad))− (dw + cr2rstat)ωw.(2.25)

    When studying a clutch in more detail it is seen that the torsional flexibility is a resultof an arrangement with smaller springs in series with springs with much higher stiffness.When the angle difference over the clutch starts from zero and increases, the smaller springswith stiffness kc1 are being compressed. This ends when they are fully compressed at αc1radians. If the angle is increased further, the stiffer springs, with stiffness kc2, are beginningto compress. When αc2 is reached, the clutch hits a mechanical stop. The resulting stiffnessof the clutch is given by:

    kc(x) =

    kc1 if |x|6 αc1kc2 if αc1 < |x|6 αc2∞ otherwise

    . (2.26)

    The flexible clutch and drive shaft model is a more complex one, and the clutch torsion,the drive shaft torsion, the engine speed, the transmission speed and the wheel speed are

    17

  • Driveline Modeling and Control

    used as states, according to:

    x1 = θe− θtit

    x2 =θtif− θw

    x3 = ωex4 = ωtx5 = ωw

    . (2.27)

    The state-space formulation of the linear clutch and drive shaft model consist of thesystem matrices defined next:

    Ac =

    0 0 1 −it 00 0 0 1if −1− kcJ1 0

    −dcJ1

    dcitJ1

    0

    kcitJ2

    − kdifJ2dcitJ2

    −dci2t +d2+ddi2f

    J2ddifJ2

    0 kdJ3 0ddifJ3

    −d3−ddJ3

    , (2.28)

    B =

    001J100

    ,H =

    0000−1J2

    , (2.29)

    where

    J1 = Je

    J2 = Jt+Jfi2f

    J3 = Jw +mCoGr2stat

    d2 = dt+dfi2f

    d3 = dw +mCoGcr2r2stat

    . (2.30)

    2.3.3 Continuous Variable Transmission Drive Shaft Model

    In (Mussaeus, 1997), a nonlinear model for a continuously-variable transmission driveline isdeveloped. The powertrain is represented in Fig. 2.7 and it is composed from the followingcomponents: engine, continuously-variable transmission (CVT), final reduction gear (FRG),flexible drive shaft (FDS) and driving wheel, which can be seen as input-output blocks. Theengine generates a toque which is transmitted towards the wheels through the driveline.

    18

  • 2.3 Driveline Models

    Figure 2.7: Continuous variable transmission drive shaft model.

    A CVT is used to transfer a given amount of torque from the engine to the FRG using acontinuously-variable gear ratio. The final reduction gear has a distributive role inside thepowertrain, efficiently transferring the CVT output torque to the FDS. Obviously, the finaldrive-shaft is not rigid and the torque losses can be very large if a proper mathematical modelis not considered. The FDS transmits the received torque to the wheels and its efficiency isbased on the FRG gear ratio. The driving wheels are the final components of the powertrain,having the aim of moving the vehicle by defending the friction forces with the road surfaceand the aerodynamic drag.

    The internal combustion engine can be seen as an ideal torque source/generator, thefunctionality of the engine being described by the following equations:

    Te = Γ(ωe),

    Jed

    dtωe (t) = Te (t)−T1 (t) ,

    (2.31)

    where T1 is the torque transmitted to the CVT and Γ is chosen to be the optimal fuelefficiency curve. The transmission, described by equations:

    ω2 = iCV Tωe,

    T2 =ηCV TiCV T

    T1,(2.32)

    where ηCV T is the transmission efficiency and iCV T is the CVT ratio. Another gear ratioiFRG is provided by the final reduction gear, which takes the torque from the transmissionand passes it to the flexible drive-shaft of the vehicle:

    ω3 = iFRGω2,

    T3 =ηFRGiFRG

    T2 = r1T1,(2.33)

    19

  • Driveline Modeling and Control

    where r1 = ηF RGηCV TiF RGiCV T and ηFRG is the flexible drive-shaft efficiency. Considering the flexibledrive-shaft speed related to the engine speed and solving 2.32 in 2.33 yields:

    ω3 (t) =ωe (t)r2

    , (2.34)

    where r2 = 1iF RGiCV T .The powertrain flexibility is given by the flexible drive-shaft, which is characterized by

    an elasticity factor kd = Jvπ2 and a damping coefficient dd = 2√kdJv, both used to calculate

    the FDS torque:

    T3 (t) = Tk (t) +Tb (t) , (2.35)

    where we have:

    Tk = kdt∫

    0(ω3−ωw)dσ,

    Tb = dd (ω3−ωw) .

    (2.36)

    The dynamical behavior of the wheel is described by the following equation:

    Jvd

    dtωw (t) = T3 (t)−Tload (t) , (2.37)

    where

    Jv = r2statmCOG,

    Tload (t) = Troll (t) +Tairdrag (t) +Tangle (t) ,

    Tairdrag (t) = c1ω2w (t) ,

    Troll (t) = c2mCOG,

    Tangle (t) = 0.

    (2.38)

    The torque due to hill climbing and all other disturbances are summarized in Tangle,which is assumed to be unknown and might therefore be subject to estimation, Tairdrag isthe load torques due to aerodynamic drag and c1 and c2 are constants.

    The optimized powertrain was designed to reduce the fuel consumption by using theoptimal fuel efficiency curve in the modeling phase.

    20

  • 2.4 Driveline Control Strategies

    2.4 Driveline Control Strategies

    Next step after developing the driveline model, is to find the proper control strategy to obtainthe desired performances. In this section, different control strategies proposed in literaturefor improving overall performances are presented.

    2.4.1 PID Control

    Unlike simple control algorithms, the PID controller is capable of manipulating the processinputs based on the history and rate of change of the signal. This gives a more accurateand stable control method. The basic idea is that the controller reads the system stateby a sensor. Then it subtracts the measurement from a desired reference to generate theerror value. The error will be managed in three ways, to handle the present, through theproportional term, recover from the past, using the integral term, and to anticipate thefuture, through the derivate term.

    Several methods for tuning the PID loop exist. The choice of method will depend largelyon whether the process can be taken off-line for tuning or not. Ziegler-Nichols method is awell-known online tuning strategy. Further tuning of the parameters is often necessary tooptimize the performance of the PID controller. The control structure of the controller ispresented in Fig. 2.8, and the mathematical form is given by:

    u(n) =Kpe(n) +Kin∑k=0

    e(k)−Kd (y (n)−y (n−1)) , (2.39)

    Kp =Kr

    Ki =KpTsTi

    Kd =KpTdTs

    , (2.40)

    where Kr is the controller gain, Ti, and Td denote the time constants of the integral andderivative terms, Ts is the sampling time of the system and Kp, Ki, and Kd represents theproportional, integral, and derivative gains.

    2.4.2 PID Cascade-Based Driveline Control

    The PID controller consists of proportional, integral and derivative elements, being widelyused in feedback control of industrial processes because of its simplicity and robustness.The often variation in parameters and parameter perturbations, which occur in industrialprocesses, can make the system unstable. That is the reason why the PID controller computes

    21

  • Driveline Modeling and Control

    Figure 2.8: PID control structure.

    Figure 2.9: Cascade based control structure.

    an error value as the difference between the output of the system and a desired setpoint.

    Then, the controller attempts to minimize this error by adjusting the control inputs of the

    plant. The PID parameters that are used in the calculation of the control action must be

    tuned according to the nature of the process. The proportional term responds immediately

    to the current error, the integral value yields zero steady-state error in tracking a constant

    setpoint, and the derivative term determines the reaction based on the rate at which the

    error has been changing. The control element uses the weighted sum of these three actions

    in order to adjust the process. A schematic representation of the powertrain control strategy

    is illustrated in Fig. 2.9. The nonlinear state-space powertrain model is represented in the

    Powertrain block and fw (t) represents a function which has as input the wheel speed and

    outputs the load torque. The engine torque is obtained using the optimal fuel efficiency

    curve Γ from the engine speed.

    In order to control the designed powertrain, a PID based cascade controller is imple-

    mented, the most cascade structures still being developed with classical PID controllers due

    to the simplicity of their tuning and good performances. The inner loop controller was

    designed firstly, considering the powertrain model as the plant and then, using the inner

    closed-loop control system as the plant, the external loop controller was designed.

    22

  • 2.4 Driveline Control Strategies

    2.4.3 Explicit MPC

    Traditional control design methods such as PID or LQR cannot explicitly take into ac-count hard constraints. In contrast, a MPC algorithm solves a finite-horizon open-loopoptimization problem on-line, at each sampling instant, while explicitly taking input andstate constraints into account.

    Optimal control of constrained linear and piecewise affine systems has garnered greatinterest in the research community due to the ease with which complex problems can bestated and solved. The Multi-Parametric Toolbox (MPT) provides efficient computationalmeans to obtain feedback controllers for these types of constrained optimal control problemsin a Matlab programming environment. By multi-parametric programming, a linear orquadratic optimization problem is solved off-line. The associated solution takes the formof a PWA state feedback law. In particular, the state-space is partitioned into polyhedralsets and for each of those sets the optimal control law is given as one affine function of thestate. In the online implementation of such controllers, computation of the controller actionreduces to a simple set-membership test, which is one of the reason why this method hasattracted so much interest in the research community (Kvasnica et al., 2006).

    PWA systems are models for describing hybrid systems and the dynamical behavior ofsuch systems is capture by relations of the following form:

    xk+1 = Aixk + Biuk + fiyk = Cixk + Diuk + gi

    , (2.41)

    subject to constraints on outputs, control input, and control input slew rate:

    ymin ≤ yk ≤ ymaxumin ≤ uk ≤ umax

    ∆umin ≤ uk−uk−1 ≤∆umax

    . (2.42)

    The cost function used for the explicit MPC scheme is

    min{uk}k∈Z[0,N−1]

    ‖PNxN‖p+N−1∑k=0‖Qxxk‖p+‖Ruuk‖p

    , (2.43)where u is the vector of manipulated variables over which the optimization is performed, Nis the prediction horizon, p is the linear norm and can be 1 or ∞ for 1- and Infinity-norm,respectively. Also, Qx, Ru and PN represents the weighting matrices imposed on states,manipulated variables and terminal states, respectively.

    23

  • Driveline Modeling and Control

    2.4.4 Horizon-1 MPC based on Flexible Control Lyapunov Func-tion

    Standard MPC techniques require a sufficiently long prediction horizon to guarantee stability,which makes the corresponding optimization problem too complex. Recently, a relaxationof the conventional notion of a Lyapunov function was proposed in (M.Lazar, 2009), whichresulted in a so-called flexible Lyapunov function. A first application of flexible Lyapunovfunctions in automotive control problems was presented in (Hermans et al., 2009). Thereinit was indicated that flexible Lyapunov functions can be used to design stabilizing MPCschemes with a unitary horizon, without introducing conservatism. In what follows, wedemonstrate how the theory introduced in (M.Lazar, 2009) can be employed to design ahorizon-1 MPC controller for the considered application.

    2.4.4.1 Notation and Basic Definitions

    Let R, R+, Z and Z+ denote the field of real numbers, the set of non-negative reals, theset of integer numbers and the set of non-negative integers, respectively. For every c ∈ Rand Π⊆ R define Π≥c := {k ∈ Π | k ≥ c} and similarly Π≤c, RΠ := Π and ZΠ := Z∩Π. Fora vector x ∈ Rn let ‖x‖ denote an arbitrary p-norm and let [x]i, i ∈ Z[1,n], denote the i-thcomponent of x. Let ‖x‖∞ := maxi∈Z[1,n] |[x]i|, where | · | denotes the absolute value. For amatrix Z ∈ Rm×n let ‖Z‖∞ := supx 6=0 ‖Zx‖‖x‖ denote its corresponding induced matrix norm.In ∈ Rn×n denotes the identity matrix. A function ϕ : R+ → R+ belongs to class K if itis continuous, strictly increasing and ϕ(0) = 0. A function ϕ ∈ K belongs to class K∞ iflims→∞ϕ(s) =∞.

    2.4.4.2 Horizon -1 MPC

    Consider the discrete-time constrained nonlinear system

    xk+1 = φ(xk,uk), k ∈ Z+, (2.44)

    where xk ∈ X ⊆ Rn is the state and uk ∈ U ⊆ Rm is the control input at the discrete-timeinstant k. φ : Rn×Rm→Rn is an arbitrary nonlinear, possibly discontinuous, function withφ(0,0) = 0. It is assumed that X and U are bounded sets with 0 ∈ int(X) and 0 ∈ int(U).Next, let α1,α2 ∈K∞ and let ρ ∈ R[0,1).

    Definition 2.4.1 A function V : Rn→ R+ that satisfies

    α1(‖x‖)≤ V (x)≤ α2(‖x‖), ∀x ∈ Rn (2.45)

    24

  • 2.4 Driveline Control Strategies

    and for which there exists a, possibly set-valued, control law π : Rn⇒ U such that

    V (φ(x,u))≤ ρV (x), ∀x ∈ X,∀u ∈ π(x) (2.46)

    is called a control Lyapunov function (CLF) in X for system (2.44).

    Consider the following inequality corresponding to (2.46):

    V (xk+1)≤ ρV (xk) +λk, ∀k ∈ Z+, (2.47)

    where λk is an additional decision variable which allows the radius of the sublevel set {z ∈X |V (z)≤ ρV (xk)+λk} to be flexible, i.e., it can increase if (2.46) is too conservative. Based oninequality (2.47) we can formulate the following optimization problem. Let α3,α4 ∈K∞ andJ : R→ R+ be a function such that α3(|λ|)≤ J(λ)≤ α4(|λ|) for all λ ∈ R and let µ ∈ R[0,1).Let Ω ⊆ X with the origin in its interior be a set where V (·) is a CLF for system (2.44).Such a region can be obtained for the desired application as the region of validity of anexplicit PWA stabilizing state feedback controller obtained for the unconstrained model.More details on how to obtain a local CLF with corresponding PWA state-feedback law formodel (2.44) are given in the next section.

    Problem 2.4.2 Choose the CLF candidate V and the constants ρ ∈ R[0,1), ∆ ∈ R+ andM ∈ Z>0 off-line. At time k ∈ Z+ measure xk and minimize the cost J(λk) over uk,λksubject to the constraints

    uk ∈ U, φ(xk,uk) ∈ X, λk ≥ 0, (2.48a)

    V (φ(xk,uk))≤ ρV (xk) +λk, (2.48b)

    λk ≤ ρ1

    M (λ∗k−1 +ρk−1M ∆), ∀k ∈ Z≥1. (2.48c)

    Above λ∗k denotes the optimum at time k ∈ Z+.Let π(xk) := {uk ∈ Rm | ∃λk ∈ R s.t. (2.48) holds} and let

    φcl(x,π(x)) := {φ(x,u) | u ∈ π(x)}.

    Theorem 2.4.3 Let a CLF V in Ω be known for system (2.44). Suppose that Problem 2.4.2is feasible for all states x in X. Then the difference inclusion

    xk+1 ∈ φcl(xk,π(xk)), k ∈ Z+, (2.49)

    is asymptotically stable in X.

    25

  • Driveline Modeling and Control

    The proof of Theorem 2.4.3 starts from the fact that (2.48c) implies limk→∞λ∗k = 0and then employs standard arguments for proving input-to-state stability and Lyapunovstability. For brevity a complete proof is omitted here and the interested reader is referred to(M.Lazar, 2009) for more details. However, in (M.Lazar, 2009) a more conservative conditionthan (2.48c) was used, which corresponds to setting ∆ = 0 andM = 1. As such, it is necessaryto prove that (2.48c) actually implies limk→∞λ∗k = 0, which is accomplished in the nextlemma.

    Lemma 2.4.4 Let ∆ ∈ R+ be a fixed constant to be chosen a priori and let ρ ∈ R[0,1) andM ∈ Z>0. If

    0≤ λk ≤ ρ1

    M (λ∗k−1 +ρk−1M ∆), ∀k ∈ Z≥1, (2.50)

    then limk→∞λk = 0.

    A complete proof is omitted here and, for more details, the interested reader is referred to(Caruntu, Balau et al., 2011).

    2.4.5 Delta GPC

    The drawback of the classic control techniques are particularly emphasized especially whenprocesses are to be run very fast and involve high sampling frequency. In this context, othercontrol strategies have been proposed to improve both the design and implementation forembedded devices.

    Generalized predictive control (Camacho and Bordons, 1999), (Clarke et al., 1987) is themost popular controller among of all predictive control formulations. At high sampling rates,the conventional GPC suffers from the large number of samples that must be taken intoaccount at each sampling instant. During the last few years some research paid attentionto δ-domain GPC to emphasize the close connection between discrete time and continuoustime theory. Discrete time system analyses is usually done using q forward shift operatorand associated discrete frequency variable z. Although forward shift operator q is the mostcommonly used discrete-time operator, in some applications, the forward shift operator canlead to difficulties (Middleton and Goodwin, 1986). Unfortunately, the discrete domainsare unconnected with the continuous domain; this is because the underlying continuousdomain description cannot be obtained by setting the sample time to zero value. It hasbeen demonstrated that there is a close connection between continuous time result and δrepresentation (Middleton and Goodwin, 1986). In fact, the δ domain description convergesto the continuous time counterpart for sampling period tends to zero.

    26

  • 2.4 Driveline Control Strategies

    The suggestion of connecting the GPC with the advantages offered by a δ parameter-ization has been discussed in (Rostgaard et al., 1997) using an emulator in a state-spaceapproach. The δ domain emulator based GPC has been further investigated in connec-tion to discrete-time GPC in (Sera et al., 2007), using a Diophantine formulation. Thesignificant relationship in fast sampling is the ratio between the dominant time constantof the system and the sample time. For instance, many process systems where GPC is of-ten applied can be considered to be fast-sampled, due to their slowly changing dynamics(Kadirkamanathan et al., 2009).

    The concept of predictive control in δ domain was first associated with GPC algorithmin continuous time domain based on a state space approach, becoming the GPC emulator(Rostgaard et al., 1997). Later, the GPC emulator has been investigated in terms of discreteGPC algorithm designed with Diophantine equations.

    Considering the deterministic case of single input single output, δ domain stat- spacemodel with the known states unaffected by disturbance or noise is:

    δxk=Aδxk+Bδukyk=Cδxk

    , (2.51)

    with xk ∈ Rn,uk ∈ Rm,yk ∈ Rp the state vector, the control vector and the output vector,respectively.

    Proceed from this model, the j-th order δ derivatives state are obtained as follows(Rostgaard et al., 1997):

    δjxk = Aδjxk +j−1∑i=0

    Aδj−i−1Bδδ

    iuk, (2.52)

    with j = 0,Ny, Ny being the prediction horizon. Using the model (2.51) the following δderivative predictors are estimated in the δ domain:

    δjyk = CδAδjxk +min{j,Ny}−1∑

    i=0CδAδ

    j−i−1Bδδiuk. (2.53)

    In a matrix notation the expression of δ derivatives predictors can be written:

    ŷδ= f +Guδ, (2.54)

    where:

    uδ = [uk δuk δ 2uk......δNu−1uk]T ,

    ŷδ = [yk δyk δ 2yk......δNyyk]T .(2.55)

    27

  • Driveline Modeling and Control

    G is the expanded Toeplitz matrix containing the δ based Markov parameters and it hasthe dimension :

    G=

    g(0,0) . . . g(0,Nu−1)

    ... . . . ...g(Ny,0) . . . g(Ny,Nu−1)

    , (2.56)where

    g(j, i) =

    CδAδj−i−1Bδ,06 i6min{k,Nu}−1

    0, otherwise, (2.57)

    and f is the free response:

    f =[CδAδ

    1 . . . CδAδNy]Txk. (2.58)

    The δ GPC controller is implemented following receding horizon strategy and henceonly the first element of control vector needs to be included. Since δ operator offers thesame flexibility and restrictions in modeling as forward shift q operator, it makes possibleto transform q domain control algorithm to the δ domain. The optimal control sequence isobtained by minimizing an objective function, knowing the reference trajectory rk+i:

    J =Ny∑i=N1

    [ŷk+i− rk+i]2 +λNu∑i=1

    [∆uk+i−1]2, (2.59)

    where Nu is control horizon, N1 is minimum costing horizon and λ is the control weightingfactor. In order to obtain the optimal control sequence in δ domain, the set of vectors thatarise in criterion function are obtained from mapping the q domain terms into the δ domainthrough binomial expansion (Kadirkamanathan et al., 2009), Ts being the sampling time.

    2.5 Conclusions

    An automotive driveline is a system that includes the mechanical components which havethe function of transmitting the engine torque to the driving wheels. In order to transmitthis torque in an efficient way, a proper model of the driveline is needed for controller de-sign purposes with the aim of lowering emissions, reducing fuel consumption and increasingcomfort. Next step is to find the proper control strategy to obtain the desired performances.In this chapter, different driveline models and control strategies found in the literature arepresented. First, an electro-hydraulic valve-clutch system is presented, followed by threedriveline models: a drive shaft model, a flexible clutch and drive shaft model, and a con-tinuous variable transmission drive shaft model. Next, a PID, a PID cascade based, an

    28

  • 2.5 Conclusions

    explicit MPC, a horizon-1 MPC controller based on flexible Lyapunov functions and anDelta GPC controller are presented as driveline control strategies. Starting from the modelspresented in this chapter, in what follows, more complex driveline models are developed andalso the control strategies presented in this chapter are applied in order to improve overallperformances.

    29

  • Driveline Modeling and Control

    30

  • Chapter 3

    Modeling and Control of anElectro-Hydraulic Actuated WetClutch

    Transmission is one of the most important subsystem of an automotive powertrain, withthe basic function of transferring the engine torque to the vehicle with the desired ratiosmoothly and efficiently. The most common control devices inside the transmission areclutches and actuators, and considering that the automatic control of the clutch engagementplays a crucial role in AMT vehicles, in this chapter we deal with the problem of modelingand controlling an electro-hydraulic actuated wet clutch. First, new input-output and state-space models of an electro-hydraulic pressure reducing valve are developed and, stating fromthese, an input-output and a state space model of an electro-hydraulic actuated wet clutchare obtained. Simulators of the developed models are implemented in Matlab, and validatedwith data provided from experiments with the real valve actuator on a test bench. The testbench was provided by Continental Automotive Romania and it includes the VolkswagenDQ250 wet clutch actuated by the electro-hydraulic valve DQ500. Also, a GPC controlstrategy and for PID controllers are applied on the develop models and simulation result arebeing discussed.

    3.1 Introduction

    During the last few years, the interest for automated manual transmission (AMT) systemshas increased due to growing demand of driving comfort. Automated clutch actuation makesit easier for the driver, particularly in stop and go traffic, and has especially seen a recent

    31

  • Modeling and Control of an Electro-Hydraulic Actuated Wet Clutch

    growth in the European automotive industry. An AMT system consists of a manual trans-mission through the clutch disc, and an automated actuated clutch during gear shifts. Someof AMT’s largest advantages are low cost, high efficiency, reduced clutch wear and improvedfuel consumption.

    Automotive actuators have become mechatronic systems in which mechanical componentscoexist with electronics and computing devices and because pressure control valves are usedas actuators in many control applications for automotive systems, a proper dynamic modelis necessary. Hydraulic control valves are devices that use mechanical motion to control asource of fluid power and are used as actuators in many control applications for automotivesystems. They vary in arrangement and complexity, depending upon their function. Themany types of valves available are best classified according to their function. Three broadfunctional types can be distinguished: directional control valves, pressure control valves andflow control valves. Pressure control valves act to regulate pressure in a circuit and may besubdivided into pressure relief valves and pressure reducing valves. Pressure relief valves,which are normally closed, open up to establish a maximum pressure and bypass excess flowto maintain the set pressure. Pressure reducing valves, which are normally open, close tomaintain a minimum pressure by restricting flow in the line. Because control valves are themechanical (or electrical) to fluid interface in hydraulic systems, their performance is underscrutiny, especially when system difficulties occurs. Therefore knowledge of the performancecharacteristics of valve is essential.

    3.2 Modeling of an Electro-Hydraulic Actuated WetClutch as a Subsystem of an Automated ManualTransmission

    Control valves are the mechanical (or electrical) to fluid interface in hydraulic systems,and the knowledge of their performance characteristics is essential. Pressure control valvesemploy feedback and may be properly regarded as servo control loops. Because of that, aproper dynamic design is necessary to achieve stability. Starting from equations found in(Merritt, 1967), where a single stage pressure reducing valve is modeled, in this chapter, anew concept of modeling an electro-hydraulic actuated wet clutch is presented. The workis divided into two sub-chapters, first dedicated to the modeling of a three land three waysolenoid valve actuator, and second dedicated to the modeling of the actuator-clutch system.A simulator was created for the developed models, and the results obtained were compared

    32

  • 3.2 Modeling of an Electro-Hydraulic Actuated Wet Clutch as a Subsystem ofan Automated Manual Transmission

    Figure 3.1: a) Test bench b) Schematic diagram

    with data provided from experiments on a real test bench from Continental AutomotiveRomania.

    3.2.1 Test Bench Description

    The STAT-50.100 test-bench can be used for testing the electro-hydraulic equipment used foractuation, assignment and control with the maximum nominal diameter DN10 and maximumpressure of 100 bar. In order to precisely simulate the real working conditions from theinstallations where the equipment will be installed, the test-bench has the possibility tocontrol the three functional parameters (pressure, flow and temperature) to the real fieldconditions. Adjustments can be predefine and are automatically made, with the help of anPLC - Programmable Logic Controller.

    Advantages:

    • Easy working pressure tuning (10−100 bar);

    • Working temperature tuning (20−100 ◦C);

    • Oil flow easy tuning (10−50 l/min);

    • Precise functional parameters measurements.

    33

  • Modeling and Control of an Electro-Hydraulic Actuated Wet Clutch

    The STAT-50.100 test-bench is composed from the following subcomponents: hydraulictank, hydraulic oil equipment, three measurements circuits, cooling/heating oil circuit, elec-trical equipment and electronic automation equipment. Fig. 3.1.a represents the test bench,where the pressure source, the pressure reducing valve (inside of the black box) and thesensors can be easily distinguish. The schematic diagram from Fig. 3.1.b illustrated howthe test bench can be controlled either by computer, throw a software program, or directlyfrom the control panel.

    3.2.2 Modeling of an Pressure Reducing Valve

    Starting from the equations in (Merritt, 1967), where a single stage pressure reducing valveis modeled, in this sub-chapter, a new concept of modeling a three land three way pres-sure reducing valve used as actuator for the clutch system in the automatic transmission ofa Volkswagen vehicle is presented. Two models were developed: a linearized input-outputmodel and a state-space model then implemented in Matlab/Simulink and validated by com-paring the results with data obtained on the test-bench provided by Continental AutomotiveRomania and briefly presented in paragraph 3.2.1.

    3.2.2.1 Valve Description

    Pressure control valves employ feedback and may be properly regarded as servo control loop.Therefore proper dynamic design is necessary to achieve stability. Taking into considerationthat no model and structural description of this valve is found in literature, the electro-hydraulic valve DQ500 was mechanically sectioned in order to be analyzed. Therefore,in Fig. 3.2.a, a section through a real three stage pressure reducing valve is represented.Schematics of the three land three way pressure reducing valve are shown in Fig. 3.2.b.

    A pump produces the line pressure Ps used as input for the electro-hydraulic actuatorrepresented by a pressure reducing valve. This valve releases a pressure depending on thecurrent i in the solenoid, which will have as consequence the magnetic force Fmag exertedon the valve plunger, which moves linearly within a bounded region under the effect of thisforce. Such a force is generated by a solenoid placed at one boundary of the region. Themagnetic force is a function of the solenoid current and the displacement xs, defined by:

    Fmag = f(i,xs) =kai

    2

    2(kb+xs)2; LS

    di

    dt+RSi= v, (3.1)

    where ka and kb are constants, Ls is the solenoid induction, Rs the resistance and v is thesupply voltage.

    34

  • 3.2 Modeling of an Electro-Hydraulic Actuated Wet Clutch as a Subsystem ofan Automated Manual Transmission

    Figure 3.2: a) Section through a real three stage pressure reducing valve; b) Three stage valveschematic representation; c) Cha


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