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OVERALL POWERTRAIN MODELING AND CONTROL BASED ON DRIVELINE SUBSYSTEMS INTEGRATION

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UNIVERSITATEA TEHNIC GHEORGHE ASACHI DIN IAI coala Doctoral a Facultii de Automatic i Calculatoare OVERALL POWERTRAIN MODELING AND CONTROL BASED ON DRIVELINESUBSYSTEMS INTEGRATION (Controlul integrat al lanuluide transmisie a puterii) - TEZ DE DOCTORAT -

Conductor de doctorat: Prof. univ. dr. ing. Corneliu Lazr Doctorand: Ing. Andreea Elena Blu IAI - 2011 UNIUNEA EUROPEANGUVERNUL ROMNIEI MINISTERUL MUNCII, FAMILIEI I PROTECIEI SOCIALE AMPOSDRU Fondul Social European POSDRU 2007-2013 Instrumente Structurale 2007-2013 OIPOSDRUUNIVERSITATEA TEHNICGHEORGHE ASACHIDIN IAI UNIVERSITATEA TEHNICGHEORGHE ASACHI DIN IAI coala Doctoral a Facultii de Automatic i Calculatoare OVERALL POWERTRAIN MODELING AND CONTROL BASED ON DRIVELINESUBSYSTEMS INTEGRATION (Controlul integrat al lanuluide transmisie a puterii) - TEZ DE DOCTORAT -

Conductor de doctorat:Prof. univ. dr. ing. Corneliu Lazr Doctorand: Ing. Andreea Elena Blu IAI - 2011 UNIUNEA EUROPEANGUVERNUL ROMNIEI MINISTERUL MUNCII, FAMILIEI I PROTECIEI SOCIALE AMPOSDRU Fondul Social European POSDRU 2007-2013 Instrumente Structurale 2007-2013 OIPOSDRUUNIVERSITATEA TEHNICGHEORGHE ASACHIDIN IAI Tezadedoctoratafostrealizatcusprijinulfinanciaral proiectului Burse Doctorale - O Investiie n Inteligen (BRAIN). Proiectul Burse Doctorale - O Investiie n Inteligen (BRAIN), POSDRU/6/1.5/S/9,ID6681,esteunproiectstrategiccareareca obiectiv generalmbuntirea formrii viitorilor cercettori n cadrul ciclului 3 al nvmntului superior - studiile universitare de doctorat - cu impact asupra creterii atractivitii i motivaiei pentru cariera n cercetare.Proiect finanat n perioada 2008 - 2011. Finanare proiect: 14.424.856,15 RON Beneficiar: Universitatea Tehnic Gheorghe Asachi din Iai Partener: Universitatea Vasile Alecsandri din Bacu Director proiect: Prof. univ. dr. ing. Carmen TEODOSIU Responsabil proiect partener:Prof. univ. dr. ing. Gabriel LAZR UNIUNEA EUROPEANGUVERNUL ROMNIEI MINISTERUL MUNCII, FAMILIEI I PROTECIEI SOCIALE AMPOSDRU Fondul Social European POSDRU 2007-2013 Instrumente Structurale 2007-2013 OIPOSDRUUNIVERSITATEA TEHNICGHEORGHE ASACHIDIN IAI Motto: Learn from yesterday, live for today, hope for tomorrow. The important thing is not to stop questioning. Albert Einstein UNIUNEA EUROPEANGUVERNUL ROMNIEI MINISTERUL MUNCII, FAMILIEI I PROTECIEI SOCIALE AMPOSDRU Fondul Social European POSDRU 2007-2013 Instrumente Structurale 2007-2013 OIPOSDRUUNIVERSITATEA TEHNICGHEORGHE ASACHIDIN IAI AcknowledgementsLooking 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 Lazr, 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. MirceaLazr, for oering me the opportunity to work in their department, for the de-tailed and constructive comments and for the kind support and guidance thathavebeenofgreatvalueinthisstudy. Also, IwouldliketothankDr. ing.Stefano Di Cairano for the constructive discussions and advices.I wish to express my warm thanks to Prof. Octavian Pstrvnu, Prof. MihaelaHanako-Matcovski, Prof. AlexandruOnea, Assoc. Prof. LetiiaMireaandAssoc. Prof. LaviniaFerariu, fortheextensivediscussionsaroundmywork,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, MariusandAlex, myrstocemates, andwithmyPh.Dcolleagues Alina, Costi, Cosmin, Carlos and Bogdan, who are now my very closefriends. I will never forget Danas 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 myfamily 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 BluIai, 2011ContentsList of Figures xiList of Tables xvGlossary xvii1 Introduction 11.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Driveline Modeling and Control . . . . . . . . . . . . . . . . . . . . . 11.1.1.1 Backlash Nonlinearity . . . . . . . . . . . . . . . . . . . . . 31.1.1.2 Clutch Nonlinearity . . . . . . . . . . . . . . . . . . . . . . 41.2 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 List of Publications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Driveline Modeling and Control 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Electro-Hydraulic Valve-Clutch System. . . . . . . . . . . . . . . . . . . . . 122.3 Driveline Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.1 Drive Shaft Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.2 Flexible Clutch and Drive Shaft Model . . . . . . . . . . . . . . . . . 162.3.3 Continuous Variable Transmission Drive Shaft Model . . . . . . . . . 182.4 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 . . . . 242.4.4.1 Notation and Basic Denitions . . . . . . . . . . . . . . . . 242.4.4.2 Horizon -1 MPC . . . . . . . . . . . . . . . . . . . . . . . . 242.4.5 Delta GPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26viiCONTENTS2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 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 anAutomated Manual Transmission . . . . . . . . . . . . . . . . . . . . . . . . 323.2.1 Test Bench Description . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.2 Modeling of an Pressure Reducing Valve . . . . . . . . . . . . . . . . 343.2.2.1 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 . . . . . . . . . 413.2.3 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. 533.3 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624 Two Inertias Driveline Model Including Backlash Nonlinearity 654.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2 Driveline Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.2.1 CVT Driveline Model with Backlash Nonlinearity . . . . . . . . . . . 664.2.1.1 PWA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.2.1.2 Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . . 694.2.2 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 . . . . . . . . . . . . 734.3 Driveline Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3.1 PID Cascade-Based Driveline Controller . . . . . . . . . . . . . . . . 754.3.2 Horizon -1 MPC Controller . . . . . . . . . . . . . . . . . . . . . . . 784.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80viiiCONTENTS4.4.1 Simulator for the PWA Model of the CVT Driveline . . . . . . . . . . 804.4.2 Simulator for the Nonlinear Model of the CVT Driveline . . . . . . . 834.5 Real Time Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.5.1 System Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.5.2 Electromechanical Plant Description . . . . . . . . . . . . . . . . . . 884.5.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955 Three Inertias Driveline Model Including Clutch Nonlinearity 995.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.2 Driveline Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.2.1 AMT Ane Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.2.2 AMT Piecewise Ane Model . . . . . . . . . . . . . . . . . . . . . . 1025.2.3 Dual Clutch Transmission Driveline. . . . . . . . . . . . . . . . . . . 1055.3 Driveline Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.3.1 Explicit MPC Controller . . . . . . . . . . . . . . . . . . . . . . . . . 1095.3.2 Horizon-1 MPC Controller . . . . . . . . . . . . . . . . . . . . . . . . 1105.3.3 Delta GPC Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.4.1 Delta GPC for the Ane Model . . . . . . . . . . . . . . . . . . . . . 1155.4.2 Ane Model Versus PWA Model . . . . . . . . . . . . . . . . . . . . 1165.4.3 AMT Driveline Control . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.4.3.1 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 . . . . . . . . . . . . . . . . . . . . . 1275.4.4 DCT Driveline Control . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.4.4.1 Up-shift maneuvers . . . . . . . . . . . . . . . . . . . . . . . 1295.4.4.2 Down-shift maneuvers . . . . . . . . . . . . . . . . . . . . . 1305.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336 Conclusions 1356.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.1.1 Modeling and Control of an Electro-Hydraulic Actuated Wet ClutchSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.1.2 Modeling and Control of a Two Inertia Driveline Including BacklashNonlinearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136ixCONTENTS6.1.3 Modeling and Control of a Three Inertia Driveline Including ClutchNonlinearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.2 Suggestion for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . 138A 141Bibliography 147xList of Figures2.1 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. . . . . . . . . . . . . . . . . . . . . . . . . . 223.1 a) Test bench b) Schematic diagram . . . . . . . . . . . . . . . . . . . . . . . 333.2 a) Section through a real three stage pressure reducing valve; b) Three stagevalve schematic representation; c) Charging phase of the pressure reducingvalve; d) Discharging phase of the pressure reducing valve. . . . . . . . . . . 353.3 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 ow. . . . . . . . . . . . . . . . . . . . . . . . . . . 433.7 Spool displacement and reduced pressure. . . . . . . . . . . . . . . . . . . . . 433.8 Input-output Simulink model. . . . . . . . . . . . . . . . . . . . . . . . . . . 443.9 Current and magnetic force used as input signals. . . . . . . . . . . . . . . . 453.10Compared spool displacements for input-output model . . . . . . . . . . . . . 463.11Compared reducing pressures for input-output model. . . . . . . . . . . . . . 463.12State-space Simulink model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.13Compared spool displacements for state-space model. . . . . . . . . . . . . . 483.14Compared reducing pressures for state-space model. . . . . . . . . . . . . . . 483.15Charging phase of the actuator-clutch system. . . . . . . . . . . . . . . . . . 493.16Discharging phase of the actuator-clutch system. . . . . . . . . . . . . . . . . 50xiLIST OF FIGURES3.17Transfer function block diagram of the actuator-clutch system. . . . . . . . . 513.18State-space block diagram of the actuator-clutch system. . . . . . . . . . . . 533.19Input-output Simulink diagram of the actuator-clutch system. . . . . . . . . 543.20System pressures for the input-output model. . . . . . . . . . . . . . . . . . 543.21Input-output system simulation. . . . . . . . . . . . . . . . . . . . . . . . . . 553.22State-space Simulink diagram of the actuator-clutch system. . . . . . . . . . 563.23System pressures for the state-space model. . . . . . . . . . . . . . . . . . . . 573.24State-space system simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 583.25GPC results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.26PID controller results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.1 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. . . . . . . . . . . . . . . . . . . 774.10Speed controller - Simulink representation. . . . . . . . . . . . . . . . . . . . 784.11Horizon-1 MPC - Simulink structure. . . . . . . . . . . . . . . . . . . . . . . 814.12Wheel speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.13Operating mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.14Backlash angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.15Engine torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.16Optimal fuel-eciency curve. . . . . . . . . . . . . . . . . . . . . . . . . . . 844.17Wheel speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.18Final drive-shaft torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.19Engine speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.20CVT ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.21M220 Industrial plant emulator schematic structure. . . . . . . . . . . . . . . 874.22Industrial plant emulator M220. . . . . . . . . . . . . . . . . . . . . . . . . . 894.23Rigid driveline collocated controller - Simulink structure. . . . . . . . . . . . 904.24Rigid driveline non-collocated controller - Simulink structure. . . . . . . . . . 924.25Rigid driveline collocated and non-collocated control. . . . . . . . . . . . . . 924.26Backlash mechanism structure. . . . . . . . . . . . . . . . . . . . . . . . . . 93xiiLIST OF FIGURES4.27Rigid driveline with 4 degrees backlash angle collocated and non-collocatedcontrol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.28Rigid driveline with 8 degrees backlash angle collocated and non-collocatedcontrol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.29Flexible driveline controller - Simulink structure. . . . . . . . . . . . . . . . . 964.30Flexible driveline with backlash control - engine inertia position. . . . . . . . 964.31Flexible driveline with backlash control - wheel inertia position. . . . . . . . 975.1 Three inertia driveline model. . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.2 Clutch functionality a) stiness characteristic; b) clutch springs . . . . . . . 1035.3 AMT clutch switching logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.4 Double clutch transmission driveline model. . . . . . . . . . . . . . . . . . . 1065.5 DCT - Switching logic for the rst clutch. . . . . . . . . . . . . . . . . . . . 1075.6 DCT - Switching logic for the second clutch. . . . . . . . . . . . . . . . . . . 1085.7 Simulation results using GPC. . . . . . . . . . . . . . . . . . . . . . . . . . 1155.8 Inuences of the GPC on engine speed, transmission speed and axle wrap. 1165.9 GPC simulation results subject to reference changes. . . . . . . . . . . . . 1175.10Inuences of the GPC on engine speed, transmission speed and axle wrap,subject to reference changes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.11Vehicle velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.12Engine speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.13Axle wrap speed dierence. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.14Engine torque (control signal). . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.15Clutch mode of operation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.16Scenario 1: Acceleration test. . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.17Scenario 1: Clutch mode of operation. . . . . . . . . . . . . . . . . . . . . . 1235.18Scenario 1: EMPC - Acceleration test. . . . . . . . . . . . . . . . . . . . . . 1255.19Scenario 1: EMPC - Clutch mode of operation. . . . . . . . . . . . . . . . . 1255.20Scenario 2: Deceleration test. . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.21Scenario 3: Tip-in tip-out test. . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.22Scenario 4: Stress test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.23Scenario 1: Up-shift maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . 1305.24MPC - Clutch operation modes for up-shift maneuvers test. . . . . . . . . . 1315.25PID - Clutch operation modes for up-shift maneuvers test. . . . . . . . . . . 1315.26Scenario 2: Down-shift maneuvers. . . . . . . . . . . . . . . . . . . . . . . . 1325.27Clutch operation modes for down-shift maneuvers test. . . . . . . . . . . . . 132xiiiLIST OF FIGURESxivList of TablesA.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 clutchnonlinearity - 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145A.5 Simulation vehicle parameter values for three inertias driveline with clutchnonlinearity -2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146xvGLOSSARYxviGlossaryAMT Automated Manual TransmissionARX AutoRegressive eXogenousCARIMAControlled AutoRegressive Integrated Moving AverageCLF Control Lyaponov FunctionCVT Continuous Variable TransmissionDAC Digital to Analog ConverterDC Direct CurrentDCT Double clutch TransmissionDSP Digital Signal ProcessorFCLF Flexible Control Lyapunov FunctionFDS Flexible Drive ShaftFRG Final Reduction GearGPC Generalized Predictive ControlLP Linear ProgramLQ Linear QuadraticLQG Linear Quadratic GaussianLQR Linear Quadratic RegulatorMILP MixtInteger Linear ProgramMPC Model Predictive ControlxviiGLOSSARYMPT Multi-Parametric ToolboxPI Proportional-IntegratorPID Proportional-Integrator-DerivativePLC Programmable Logic ControllerPOG Power-Oriented GraphsPRBS PseudoRandom Binary SequencePWA PieceWise AnePWL PieceWise LinearSR Speed ReductionxviiiChapter 1IntroductionRecent 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, basedondrivelinesubsystemintegration. Morecomplexdrivelineanddrivelinesubsystemsmodelsareproposed, anddierentproblemsasnonlinearitiesintroducedbybacklash and clutch are addressed, in order to improve vehicle performances.1.1 Literature ReviewAn 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 ecient way, a proper model of the driveline is needed for controller designpurposes, withtheaimof loweringemissions, reducingfuel consumptionandincreasingcomfort.1.1.1 Driveline Modeling and ControlTheautomotivedrivelineisanessential partof thevehicleanditsdynamicshavebeenmodeled dierently, 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 justied in (Pettersson et al., 1997), where it is shownthat this model is able to capture the rst torsional vibrational mode. There are also morecomplex three-masses models reported in dierent 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.1IntroductionA 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 apiecewiseane(PWA) twomasses model for adrivelineincludingaback-lash nonlinearity. In (Grotjahn et al., 2006) a two masses model was presented, with thedrivelinemainexibilityrepresentedbythedriveshafts, aswell asathreemassmodeltoreproducethebehaviorofavehiclewithadual-massywheel. Linearandnonlinearthreemassesmodels, inwhichtheclutchexibilitywasalsoconsidered, werepresentedin (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, dierent 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 drivers 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 drivers 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-21.1 Literature Reviewbased approach for anti-jerk control of passenger cars that minimizes driveline oscillationswhileretainingfastaccelerationwasintroduced. Thecontrollerwasdesignedwiththehelp 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 dierence between the motor speedandtheloadspeedtominimizethedrivelineoscillations, whilereducingtheimpactofforces between the mechanical parts. A clutch engagement controller based on fuzzy logicispresentedin(Wu et al., 2009)adrivelinecontrol withtorqueobserverisproposedin(Kim and Choi, 2010).In order to improve vehicle overall performances, problems as nonlinearities introducedby backlash and clutch system are modeled, and dierent control strategies are proposed.1.1.1.1 Backlash NonlinearityBacklash 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 occurswheneverthereisagapinthetransmissionlinkwhichleadstozerotorquetransmittedthrough 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 exibility results in an angular positiondierence 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 shue. 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-trollerandamodiedswitchingcontroller. Theconceptof PIDcontrollerwithtorquecompensatorispresentedin(Nakayama et al., 2000)forthebacklash. Asimpleactiveswitching 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 consumption3Introductionand 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 NonlinearityIn 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 eciently, 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 dierent valve typesused as actuators have been actively researched and dierent models and control strategieshave been developed: physics-based nonlinear model for an exhausting valve (Ma et al., 2008),nonlinearphysical model forprogrammablevalves(Liu and Yao, 2008), nonlinearstate-spacemodel descriptionoftheactuatorthatisderivedbasedonphysical principlesandparameter identication (Wang et al., 2002), (Peterson et al., 2003), (Gennaro et al., 2007),(Nemeth, 2004), mathematical model obtained using identication methods for a valve ac-tuationsystemof anelectro-hydraulicengine(Liao et al., 2008), amodel foranelectro-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 ThesisThe reminder of this thesis is structured as follows.Chapter 2, entitled Drivelinemodelingandcontrol presents dierent driveline modelsand control strategies found in the literature. First, an electro-hydraulic valve-clutch system41.2 Outline of the Thesisis presented, followed by three driveline models: a drive shaft model, a exible clutch anddrive shaft model, and a continuous variable transmission drive shaft model. Next, a PID,aPIDcascadebased, anexplicitMPCandahorizon-1MPCbasedonexiblecontrolLyapunov function are presented as driveline control strategies. Starting from these models,in what follows, more complex driveline models are developed and also the control strategiespresentedinthischapterareappliedinordertoobtainnewcontrollersabletoimproveoverall vehicle performances.Chapter 3 is entitled Modeling and control of an electro-hydraulic actuated wet clutch.In this chapter, dierent 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-outputandastatespacemodel ofanelectro-hydraulicactuatedwetclutchisobtained.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, dierent 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 modelincluding backlash nonlinearity. Inthis chapter, dierent models for automotive driveline including backlash nonlinearity areproposed. First, apiecewiseaneandanonlinearstate-spacemodel foraContinuousVariable Transmission (CVT) driveline with backlash are proposed. Simulators are developedin Matlab/Simulink for the two driveline models and dierent 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 eciency curve in the modeling phase. Next, three models are presented foran Automated Manual Transmission (AMT) driveline based on the Industrial plant emulatorM220: arigiddrivelinemodel, aexibledrivelinemodel andaexibledrivelinemodelincludingalsobacklashnonlinearity. Then, real timeexperimentsareconductedonthepresented models in order to test the inuences given by drive shaft exibility 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 anemodel and a new state-space piecewise ane model of an automated manual transmission5Introduction(AMT) driveline, and a new state-space piecewise ane model of a double clutch trans-mission (DCT) driveline, all of them taking into consideration the drive shafts as well asthe clutch exibilities. 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 PublicationsThis 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. NewModelfor Predictive Controlof 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 andO. Pastravanu. Model lingof aSolenoidValveActuatorforAutomotive 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. Model ling of An ElectromagneticValve Actuator. Buletinul Institutului Politehnic din Iasi, vol. Tome LV (LIX), Fasc.2, pages 928, 2009.61.3 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 19111922, 2011.(C.Lazar, Caruntu and Balau, 2010) C. Lazar, C. F. Caruntu and A. E. Balau. Mod-el ling 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 oscil lationsmodeling 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 oscil lations damping. InThe14thInternational ConferenceonHybridSystems: ComputationandControl,Chicago, USA, 2011.7Introduction(Halauca, Balau and C.Lazar, 2011)C. Halauca, A. E. BalauandC. Lazar. StateSpace Delta GPC for Automotive Powertrain Systems. In The16th IEEE InternationalConference on Emerging Technologies and Factory Automation, 2011.8Chapter 2Driveline Modeling and ControlAn 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 transmitthistorqueinanecientway, apropermodel of thedrivelineisneededforcontrollerdesign 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 IntroductionThe driveline is a fundamental part of a vehicle and its dynamics have been modeled indierentways, accordingtothepurpose. Theaimof themodelingistondthemostsignicant physical eects 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, nal 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 Newtons 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 outputengine torque isgivenby the driving engine torqueTeresulted fromthecombustion, the internal engine frictionTfric,e, and the external load from the clutchTc,9Driveline Modeling and ControlFigure 2.1: Schematic vehicle structure.Figure 2.2: Driveline subsystems.obtaining the following equation:Jee =TeTfric,eTc, (2.1)whereJe represents the engine moment of inertia,e is the crankshaft angle,e = e is theengine angular velocity and e = e is the engine angular acceleration.A friction clutch consists of a clutch disk connecting the ywheel of the engine and thetransmissions 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 dierence (ec) andthe angular velocity dierence (ec) over the clutch:Tc =Tt =fc(ec,ec), (2.2)wherec represents the clutch angle andc = c is the clutch angular velocity.The transmission has a set of gears, each with a dierent conversion ratio it. The followingequations between the input and output torque of the transmission is obtained:Tp =ft_Tt,Tfric,t,ctit,ctit,it_, (2.3)102.1 IntroductionwhereTpis the propeller shaft torque, Tfric,tis the internal friction torque of the trans-mission,t is the transmission angle andt = t is the corresponding angular velocity. Thereason for considering the angle dierence ctit is the possibility of having torsional eectsin the transmission.The propeller shaft connects the transmissions output shaft with the nal drive. Nofriction is assumed soTp =Tf, giving the following equation:Tp =Tf =fp(tp, tp) , (2.4)whereTfis the nal drive torque, p is the propeller shaft angle andp = p is the corre-sponding angular velocity.Thenal driveischaracterizedbyaconversionratioifinthesamewayasforthetransmission. The following relation between the input and the output torque holds:Td =ff_Tf,Tfric,f,pfif,pfif,if_, (2.5)where Tfric,f is the internal friction torque of the nal drive, Td is the drive shaft torque, fis the nal drive angle andf = fis the corresponding angular velocity.The drive shafts is the subsystem that connects the wheel to the nal drive. Assumingthat w is the wheel angle, the rotational wheel velocity w = w is the same for both wheelsand neglecting the vehicle dynamics, the rotational equivalent wheel velocity shall be equalto the velocity of the vehicle bodys center of gravityvv:w =vvrstat, (2.6)whererstat represents the wheel radius. The shafts are modeled as one shaft and assumingthat no friction existsTw =Td the following equation for the wheel torqueTw results:Tw =Td =fd_fw, fw_. (2.7)Newtonssecondlawinthelongitudinal directionforavehiclewithmass mCoGandspeedvv, gives:Fload =mCoG vv +Fairdrag +Froll+mCoGg sin(road). (2.8)The load forceFload is described by the sum of following quantities:Fairdrag, the air drag, is approximated byFairdrag =12cairAfairv2v, wherecair is thedrag coecient, Af is the maximum vehicle cross section area and air is the air density.11Driveline Modeling and ControlFroll, the rolling resistance, is approximated byFroll = mCoG(cr1 +cr2vv) wherecr1andcr2 depend on the tire pressure.mCoGsin(road), the gravitational force, whereroad is the road slope.The resulting torqueTloadis equal withFloadrstatand the equation of motion for thewheel is described by the following relation:Jw w =TwFloadrstatTL, (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_ w =TwTL12cairAfairr3stat2wrstatmCoG(cr1 +cr2rstatw)rstatmCoGg sin(road).(2.10)A complete modelof 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 dierent complexities.2.2 Electro-Hydraulic Valve-Clutch SystemThe basic function of any type of automotive transmission is to transfer the engine torqueto the vehicle with the desired ratio smoothly and eciently 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 dierent 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 dierent energeticdomains.122.2 Electro-Hydraulic Valve-Clutch SystemFigure 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 massMv moves according to the damping coecientbp, the return springKp(xs) and the pressures PCandPDfrom the control chamber, and the back chamber,respectively. xs and xs represents the displacement and the valve plunger speed, respectively,andApistheareaoftheplungersextremities. ThenonlinearforceKp(xs)modelsthereturn spring as well as the contact force between the plunger and the plunger chamber, atthe plunger two extremities. The plunger movement causes the oil ow QC and QD throughthe control chamber and through the back chamber:Mv xs = (PCPD)Apbp xsKp(xs),QC =QD =Ap xs.(2.11)The pressure from the control chamber PCis obtained by integrating three oil ows:13Driveline Modeling and Controlthe owQ5 from the power sourcePs, the owQCdue to the plunger movement, and theowQwthroughthevariabledischargingorice. Theverysmall hydrauliccapacityCCstores potential energy in terms of oil pressure and it takes into account the small elasticdeformation of the valve case and the oil stiness:CC PC =Q5QCQw. (2.12)Depending on the plunger position, the output user chamber is connected either to thepower supply through the variable oriceJ1 or to the oil tank by the oriceJ3. Also, theuser chamber is connected to the back chamber through oriceJ4. This orice plays tworoles: it implements the feedback action sincePD becomes a measure of the user pressurePR, and it has the damping eect that avoids plunger oscillations.The back chamber and the user chamber are modeled as two small hydraulic capacities,as for the control chamber:CD PD =QDQ4,CR PR =Q1+Q4Q3QR.(2.13)The user chamber is connected to the clutch chamber by means of a pipe with a dynamicthatcannotbeneglectedandisdescribedbyfourelements: theuserchambercapacityCR, the hydraulic resistanceRL, the pipe hydraulic inductanceLL and the clutch chambercapacityCL:LL QR =PlPL =PRPQRPL,PRPl =QR[QR[CL=RL(QR),CL PL =QRAL z.(2.14)wherePL is the clutch pressure.The motion of the pressure plate under the eects of the pressurePL, the elastic forceKM(z) and the viscous frictionbfare given by the following equations:Mp z =PLALbfx z KM(z) Kbcsgn( z),KM(z) = KF(z) +KD(z).(2.15)whereMpis the clutch plunger mass, ALis 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.142.3 Driveline ModelsFigure 2.5: Drive shaft model.2.3 Driveline ModelsTheautomotivedrivelineisanessential partof thevehicleanditsdynamicshavebeenmodeleddierently, accordingtothedrivingnecessities. Inthissections, threedierentdriveline models reported in literature are presented.2.3.1 Drive Shaft ModelIn (Kiencke and Nielsen, 2005) a simplied 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, exible drive shafts and driven wheel. The propeller shaftis considered to be sti 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+Jfi2ti2f__ e =TeTfric,e__dti2t+dfi2ti2f__ekditif__ei2ti2fw__dditif__ei2ti2fw__,(2.16)whereJtandJfrepresents the transmission and the nal drive inertias, whiledtanddfstands for the corresponding damping coecients. Also, kd anddd represents the stinessand damping coecients of the drive shaft.Also, the equation for the vehicle and wheels inertia is given by:_Jw +mCoGr2stat_ w =kd__ei2ti2fw__+dd__ei2ti2fw__12cairAfar3stat2wrstatmCoG(cr1 +g sin(road)) _dw +mCoGcr2r2stat_w,(2.17)15Driveline Modeling and Controlwheredw represents the damping coecient of the wheel.The drive shaft model is the simplest one considered, and the drive shaft torsion, theengine speed and the wheel speed are used as states, according to:x1 =eifitwx2 =ex3 =w. (2.18)Also, taking into consideration that:J1 =Je+ Jti2t+Jfi2ti2fJ2 =Jw +mCoGr2statd1 =dti2t+dfi2ti2fd2 =dw +mCoGcr2r2statl =rstatmCoG(cr1 +g sin(road)), (2.19)the following state-space representation is obtain: x =Ax+Bu+Hl, (2.20)consisting of the system matrices:A =______01ifit1kifitJ1d1+difit2J1difitJ1kJ20difitJ2d+d2J2______, (2.21)B =___01J10___, H =___001J2___. (2.22)2.3.2 Flexible Clutch and Drive Shaft ModelA more complex model including two torsional exibilities, the drive shaft and the clutchis also presented in (Kiencke and Nielsen, 2005). The driveline has three inertias like rep-resentedinFig. 2.6, onecorrespondingtotheinternal combustionengine, oneforthetransmission, and one for the driven wheel.The equation that describe the engine dynamics is given by:Je e =TeTfric,ekc(etit) dc(etit), (2.23)162.3 Driveline ModelsFigure 2.6: Flexible clutch and drive shaft model.wherekc is the clutch stiness anddc represents the damping of the clutch.The second equation describe the dynamics of the transmission:__Jt+ Jfi2f__ t =TeTfric,eit(kc(etit) +dc(etit))__dt+ dfi2f__t 1if_kd_tifw_+dd_tifw__.(2.24)Also, the equation for the vehicle and wheels inertia is given by:_Jw +mCoGr2stat_ w =kd_tifw_+dd_tifw_12cairAfar3stat2wrstatmCoG(cr1 +g sin(road)) (dw +cr2rstat)w.(2.25)When studying a clutch in more detail it is seen that the torsional exibility is a resultof an arrangement with smaller springs in series with springs with much higher stiness.When the angle dierence over the clutch starts from zero and increases, the smaller springswith stinesskc1 are being compressed. This ends when they are fully compressed atc1radians. If the angle is increased further, the stier springs, with stiness kc2, are beginningto compress. When c2 is reached, the clutch hits a mechanical stop. The resulting stinessof the clutch is given by:kc(x) =___kc1if [x[ c1kc2if c1 < [x[ c2 otherwise. (2.26)The exible 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 are17Driveline Modeling and Controlused as states, according to:x1 =etitx2 =tifwx3 =ex4 =tx5 =w. (2.27)Thestate-spaceformulationofthelinearclutchanddriveshaftmodel consistofthesystem matrices dened next:Ac =____________0 0 1 it00 0 01if1kcJ10dcJ1dcitJ10kcitJ2kdifJ2dcitJ2dci2t+d2+ddi2fJ2ddifJ20kdJ30ddifJ3d3ddJ3____________, (2.28)B =________001J100________, H =________00001J2________, (2.29)whereJ1 =JeJ2 =Jt+ Jfi2fJ3 =Jw +mCoGr2statd2 =dt+ dfi2fd3 =dw +mCoGcr2r2stat. (2.30)2.3.3 Continuous Variable Transmission Drive Shaft ModelIn (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), nal reduction gear (FRG),exible drive shaft (FDS) and driving wheel, which can be seen as input-output blocks. Theenginegeneratesa toquewhichistransmittedtowardsthewheelsthroughthedriveline.182.3 Driveline ModelsFigure 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 nal reduction gear has a distributive role inside thepowertrain, eciently transferring the CVT output torque to the FDS. Obviously, the naldrive-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 eciency isbased on the FRG gear ratio. The driving wheels are the nal components of the powertrain,having the aim of moving the vehicle by defending the friction forces with the road surfaceand the aerodynamic drag.Theinternal combustionenginecanbeseenasanideal torquesource/generator, thefunctionality of the engine being described by the following equations:Te = (e),Jeddte(t) =Te(t) T1(t),(2.31)where T1isthetorquetransmittedtotheCVTandischosentobetheoptimal fueleciency curve. The transmission, described by equations:2 =iCV Te,T2 =CV TiCV TT1,(2.32)whereCV Tis the transmission eciency andiCV Tis the CVT ratio. Another gear ratioiFRG is provided by the nal reduction gear, which takes the torque from the transmissionand passes it to the exible drive-shaft of the vehicle:3 =iFRG2,T3 =FRGiFRGT2 =r1T1,(2.33)19Driveline Modeling and Controlwhere r1 =FRGCV TiFRGiCV Tand FRG is the exible drive-shaft eciency. Considering the exibledrive-shaft speed related to the engine speed and solving 2.32 in 2.33 yields:3(t) =e(t)r2, (2.34)wherer2 =1iFRGiCV T.The powertrain exibility is given by the exible drive-shaft, which is characterized byan elasticity factor kd =Jv2and a damping coecient dd = 2kdJv, both used to calculatethe FDS torque:T3(t) = Tk(t) +Tb(t), (2.35)where we have:Tk =kdt_0(3w)d,Tb =dd(3w).(2.36)The dynamical behavior of the wheel is described by the following equation:Jvddtw(t) =T3(t) Tload(t), (2.37)whereJv =r2statmCOG,Tload(t) =Troll(t) +Tairdrag(t) +Tangle(t),Tairdrag(t) = c12w(t),Troll(t) = c2mCOG,Tangle(t) = 0.(2.38)The torque due to hill climbing and all other disturbances are summarized inTangle,which is assumed to be unknown and might therefore be subject to estimation, Tairdragisthe load torques due to aerodynamic drag andc1 andc2 are constants.Theoptimizedpowertrainwasdesignedtoreducethefuel consumptionbyusingtheoptimal fuel eciency curve in the modeling phase.202.4 Driveline Control Strategies2.4 Driveline Control StrategiesNext step after developing the driveline model, is to nd the proper control strategy to obtainthe desired performances. In this section, dierent control strategies proposed in literaturefor improving overall performances are presented.2.4.1 PID ControlUnlike 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 accurateandstablecontrol method. Thebasicideaisthatthecontrollerreadsthesystemstateby 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, recoverfromthepast, usingtheintegral term, andtoanticipatethefuture, 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 o-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=0e(k) Kd(y (n) y (n1)), (2.39)Kp =KrKi =KpTsTiKd =KpTdTs, (2.40)whereKris the controller gain, Ti, andTddenote the time constants of the integral andderivative terms,Ts is the sampling time of the system andKp,Ki, andKd represents theproportional, integral, and derivative gains.2.4.2 PID Cascade-Based Driveline ControlThe PID controller consists of proportional, integral and derivative elements, being widelyusedinfeedbackcontrol ofindustrial processesbecauseofitssimplicityandrobustness.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 computes21Driveline Modeling and ControlFigure 2.8: PID control structure.Figure 2.9: Cascade based control structure.an error value as the dierence between the output of the system and a desired setpoint.Then, the controller attempts to minimize this error by adjusting the control inputs of theplant. The PID parameters that are used in the calculation of the control action must betuned according to the nature of the process. The proportional term responds immediatelyto the current error, the integral value yields zero steady-state error in tracking a constantsetpoint, and the derivative term determines the reaction based on the rate at which theerror has been changing. The control element uses the weighted sum of these three actionsin order to adjust the process. A schematic representation of the powertrain control strategyis illustrated in Fig. 2.9. The nonlinear state-space powertrain model is represented in thePowertrain block andfw(t) represents a function which has as input the wheel speed andoutputs the load torque. The engine torque is obtained using the optimal fuel eciencycurve 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 duetothesimplicityof theirtuningandgoodperformances. Theinnerloopcontrollerwasdesigned rstly, considering the powertrain model as the plant and then, using the innerclosed-loop control system as the plant, the external loop controller was designed.222.4 Driveline Control Strategies2.4.3 Explicit MPCTraditional control designmethodssuchasPIDorLQRcannotexplicitlytakeintoac-counthardconstraints. Incontrast, aMPCalgorithmsolvesanite-horizonopen-loopoptimization problem on-line, at each sampling instant, while explicitly taking input andstate constraints into account.Optimal control of constrained linear and piecewise ane 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 ecient computationalmeans to obtain feedback controllers for these types of constrained optimal control problemsinaMatlabprogrammingenvironment. Bymulti-parametricprogramming, alinearorquadratic optimization problem is solved o-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 ane 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:yminyk ymaxuminuk umaxuminukuk1umax. (2.42)The cost function used for the explicit MPC scheme ismin{uk}kZ[0,N1]__|PNxN|p+N1

k=0|Qxxk|p+|Ruuk|p__, (2.43)whereu 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 Innity-norm,respectively. Also, Qx, RuandPNrepresents the weighting matrices imposed on states,manipulated variables and terminal states, respectively.23Driveline Modeling and Control2.4.4 Horizon-1 MPC based on Flexible Control Lyapunov Func-tionStandard MPC techniques require a suciently 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 exible Lyapunov function. A rst application of exible Lyapunovfunctions in automotive control problems was presented in (Hermans et al., 2009). Thereinit was indicated that exible Lyapunov functions can be used to design stabilizing MPCschemeswithaunitaryhorizon, withoutintroducingconservatism. Inwhatfollows, 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 DenitionsLet R,R+,Z and Z+denote the eld of real numbers, the set of non-negative reals, theset of integer numbers and the set of non-negative integers, respectively. For everyc Rand R dene c := k [ k c and similarly c, R := and Z := Z. Fora vectorx Rnlet |x| denote an arbitraryp-norm and let [x]i, i Z[1,n], denote thei-thcomponent ofx. Let |x| := maxiZ[1,n][[x]i[, where [[ denotes the absolute value. For amatrixZ Rmnlet |Z| := supx=0Zxxdenote its corresponding induced matrix norm.In Rnndenotes the identity matrix. A function : R+ R+belongs to class K if itis continuous, strictly increasing and(0) = 0. A function K belongs to classKiflims(s) =.2.4.4.2 Horizon -1 MPCConsider the discrete-time constrained nonlinear systemxk+1 =(xk, uk), k Z+, (2.44)wherexk X Rnis the state anduk U Rmis the control input at the discrete-timeinstant k. : RnRmRnis an arbitrary nonlinear, possibly discontinuous, function with(0, 0) = 0. It is assumed that X and U are bounded sets with 0 int(X) and 0 int(U).Next, let1, 2 K and let R[0,1).Denition 2.4.1A functionV: RnR+ that satises1(|x|) V (x) 2(|x|), x Rn(2.45)242.4 Driveline Control Strategiesand for which there exists a, possibly set-valued, control law : RnU such thatV ((x, u)) V (x), x X, u (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) V (xk) +k, k Z+, (2.47)where k is an additional decision variable which allows the radius of the sublevel set z X[V (z) V (xk)+k to be exible, i.e., it can increase if (2.46) is too conservative. Based oninequality (2.47) we can formulate the following optimization problem. Let 3, 4K andJ : R R+ be a function such that3([[) J() 4([[) for all R and let R[0,1).Let X with the origin in its interior be a set whereV () is a CLF for system (2.44).Such a region can be obtained for the desired application as the region of validity of anexplicitPWAstabilizingstatefeedbackcontrollerobtainedfortheunconstrainedmodel.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 2.4.2Choose the CLF candidateV and the constants R[0,1), R+andM Z>0o-line. Attimek Z+measurexkandminimizethecost J(k)over uk, ksubject to the constraintsuk U,(xk, uk) X,k 0, (2.48a)V ((xk, uk)) V (xk) +k, (2.48b)k 1M(k1+k1M), k Z1. (2.48c)Abovek denotes the optimum at timek Z+.Let(xk) :=uk Rm[ k R s.t. (2.48) holds and letcl(x, (x)) :=(x, u) [ u (x).Theorem 2.4.3Let a CLF Vin be known for system (2.44). Suppose that Problem 2.4.2is feasible for all statesx in X. Then the dierence inclusionxk+1 cl(xk, (xk)), k Z+, (2.49)is asymptotically stable in X.25Driveline Modeling and ControlTheproof of Theorem2.4.3startsfromthefactthat(2.48c)implieslimkk = 0andthenemploysstandardargumentsforprovinginput-to-statestabilityandLyapunovstability. 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 and M =1. As such, it is necessarytoprovethat(2.48c)actuallyimplieslimkk = 0, whichisaccomplishedinthenextlemma.Lemma 2.4.4Let R+ be a xed constant to be chosen a priori and let R[0,1) andM Z>0. If0 k 1M(k1+k1M), k Z1, (2.50)then limkk = 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 GPCThe 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 suers from the large number of samples that must be taken intoaccount at each sampling instant. During the last few years some research paid attentionto-domain GPC to emphasize the close connection between discrete time and continuoustime theory. Discrete time system analyses is usually done usingq forward shift operatorand associated discrete frequency variablez. Although forward shift operatorq is the mostcommonly used discrete-time operator, in some applications, the forward shift operator canleadtodiculties(Middleton and Goodwin, 1986). Unfortunately, thediscretedomainsareunconnectedwiththecontinuousdomain; thisisbecausetheunderlyingcontinuousdomain 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 andrepresentation (Middleton and Goodwin, 1986). In fact, the domain description convergesto the continuous time counterpart for sampling period tends to zero.262.4 Driveline Control StrategiesThe suggestion of connecting the GPC with the advantages oered by a parameter-izationhasbeendiscussedin(Rostgaard et al., 1997)usinganemulatorinastate-spaceapproach. ThedomainemulatorbasedGPChasbeenfurtherinvestigatedinconnec-tiontodiscrete-timeGPCin(Sera et al., 2007), usingaDiophantineformulation. Thesignicant 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 domain was rst 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, domain stat- spacemodel with the known states unaected by disturbance or noise is:xk=Axk+Bukyk=Cxk, (2.51)withxk Rn, uk Rm, yk Rpthe state vector, the control vector and the output vector,respectively.Proceedfromthis model, the j-thorder derivatives stateareobtainedas follows(Rostgaard et al., 1997):jxk =Ajxk +j1

i=0Aji1Biuk, (2.52)withj = 0, Ny, Nybeing the prediction horizon. Using the model (2.51) the followingderivative predictors are estimated in the domain:jyk =CAjxk +min{j,Ny}1

i=0CAji1Biuk.(2.53)In a matrix notation the expression of derivatives predictors can be written: y=f +Gu, (2.54)where:u = [ukuk2uk......Nu1uk]T, y = [ykyk2yk......Nyyk]T.(2.55)27Driveline Modeling and ControlG is the expanded Toeplitz matrix containing the based Markov parameters and it hasthe dimension :G =__g(0, 0) . . . g(0, Nu1).........g(Ny, 0) . . . g(Ny,Nu1)__, (2.56)whereg(j, i) =___CAji1B, 0 i mink, Nu10, otherwise, (2.57)andf is the free response:f =_CA1. . . CANy_Txk.(2.58)The GPCcontrollerisimplementedfollowingrecedinghorizonstrategyandhenceonly the rst element of control vector needs to be included. Sinceoperator oers thesame exibility and restrictions in modeling as forward shiftq operator, it makes possibleto transformq domain control algorithm to the domain. The optimal control sequence isobtained by minimizing an objective function, knowing the reference trajectoryrk+i:J =Ny

i=N1[ yk+irk+i]2+Nu

i=1[uk+i1]2, (2.59)whereNu is control horizon,N1 is minimum costing horizon and is the control weightingfactor. In order to obtain the optimal control sequence in domain, the set of vectors thatarise in criterion function are obtained from mapping the q domain terms into the domainthrough binomial expansion (Kadirkamanathan et al., 2009),Ts being the sampling time.2.5 ConclusionsAn 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 ecient 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 nd the proper control strategy to obtain the desired performances.In this chapter, dierent 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 exible clutch and drive shaft model, and a con-tinuousvariabletransmissiondriveshaftmodel. Next, aPID, aPIDcascadebased, an282.5 ConclusionsexplicitMPC, ahorizon-1MPCcontrollerbasedonexibleLyapunovfunctionsandanDelta 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.29Driveline Modeling and Control30Chapter 3Modeling and Control of anElectro-Hydraulic Actuated WetClutchTransmission 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 ratiosmoothlyandeciently. Themostcommoncontrol devicesinsidethetransmissionareclutches 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 VolkswagenDQ250wetclutchactuatedbytheelectro-hydraulicvalveDQ500. Also, aGPCcontrolstrategy and for PID controllers are applied on the develop models and simulation result arebeing discussed.3.1 IntroductionDuring 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 trac, and has especially seen a recent31Modeling and Control of an Electro-Hydraulic Actuated Wet Clutchgrowth 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 AMTs largest advantages are low cost, high eciency, 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 uid 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 classied according to their function. Three broadfunctional types can be distinguished: directional control valves, pressure control valves andow 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 owto maintain the set pressure. Pressure reducing valves, which are normally open, close tomaintain a minimum pressure by restricting ow in the line. Because control valves are themechanical (or electrical) to uid interface in hydraulic systems, their performance is underscrutiny, especially when system diculties occurs. Therefore knowledge of the performancecharacteristics of valve is essential.3.2 Modelingof anElectro-HydraulicActuatedWetClutchasaSubsystemofanAutomatedManualTransmissionControl valvesarethemechanical (orelectrical)touidinterfaceinhydraulicsystems,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, rst 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 compared323.2 Modeling of an Electro-Hydraulic Actuated Wet Clutch as a Subsystem ofan Automated Manual TransmissionFigure 3.1: a) Test bench b) Schematic diagramwithdataprovidedfromexperimentsonareal testbenchfromContinental AutomotiveRomania.3.2.1 Test Bench DescriptionThe 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 theinstallationswheretheequipmentwill beinstalled, thetest-benchhasthepossibilitytocontrol the three functional parameters (pressure, ow and temperature) to the real eldconditions. Adjustments can be predene and are automatically made, with the help of anPLC - Programmable Logic Controller.Advantages:Easy working pressure tuning (10100 bar);Working temperature tuning (20100C);Oil ow easy tuning (1050 l/min);Precise functional parameters measurements.33Modeling and Control of an Electro-Hydraulic Actuated Wet ClutchThe 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 ValveStarting 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 briey presented in paragraph 3.2.1.3.2.2.1 Valve DescriptionPressure 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-hydraulicvalveDQ500wasmechanicallysectionedinordertobeanalyzed. 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 pressurePs used as input for the electro-hydraulic actuatorrepresented by a pressure reducing valve. This valve releases a pressure depending on thecurrenti in the solenoid, which will have as consequence the magnetic forceFmagexertedon the valve plunger, which moves linearly within a bounded region under the eect 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 displacementxs, dened by:Fmag =f(i, xs) =kai22(kb+xs)2; LSdidt +RSi =v, (3.1)whereka andkb are constants, Ls is the solenoid induction, Rs the resistance andv is thesupply voltage.343.2 Modeling of an Electro-Hydraulic Actuated Wet Clutch as a Subsystem ofan Automated Manual TransmissionFigure 3.2: a) Section through a real three stage pressure reducing valve; b) Three stage valveschematicrepresentation; c)Charging phase of thepressurereducingvalve; d)Dischargingphase of the pressure reducing valve.35Modeling and Control of an Electro-Hydraulic Actuated Wet ClutchThe pressure to be controlled PR is sensed on the spool end areas C and D and comparedwith the magnetic force which actuates on the plunger. The feedback force Ffeed =FCFDis the dierence between the force applied on the left sensed pressure chamberFC, and theforce applied on the right sensed pressure chamberFD.Thedierenceinforceisusedtoactuatethespool valvewhichcontrolstheowtomaintain the pressure at the set value. In the charging phase, illustrated in Fig. 3.2.c, themagnetic force is greater than the feedback force and moves the plunger to the left (xs >0),connecting the source with the hydraulic load. In the discharging phase, illustrated in Fig.3.2.d, the feedback force becomes grater than the magnetic force and the plunger is movedto the right (xs < 0); the connection between the source and the hydraulic load is closed,the hydraulic load being connected to the tank.Using the magnetic force and the feedback force it results a force balance which describesthe spool motion and the output pressure. This equation of force balance is the same forboth positive and negative displacement of the spool:FmagCPC +DPD =Mvs2Xs+KeXs, (3.2)wherePC represents the pressure in the left sensed chamber that acts on the (C) area,PDrepresents the pressure in the right sensed chamber that acts on the (D) area, Mvis thespool mass, Ke = 0.43w(PS0PR0) represents the ow force spring rate calculated for thenominal pressuresPS0, PR0, w represents the area gradient of the main orice, Xs = Xs(s)is the Laplace transform of the spool displacement ands represents the Laplace operator.In Fig. 3.2.a, a hydraulic damper that acts to reduce the input pressure spike, which hasnegative eects on the output pressure, is also represented.3.2.2.2 Input-Output ModelThe charging phase of the pressure reducing valve has been illustrated in Fig. 3.2.c. Apositive displacement of the spool allows connection between the source and the hydraulicload, while the channel that connects the hydraulic load with the tank is kept closed.The linearized continuity equation from (Merritt, 1967) was used to describe the dynam-ics from the sensed pressure chambers:QC =K1(PRPC) =VCesPCCsXs, (3.3)QD =K2(PRPD) =VDesPD +DsXs, (3.4)363.2 Modeling of an Electro-Hydraulic Actuated Wet Clutch as a Subsystem ofan Automated Manual Transmissionwhere K1, K2 are the ow-pressure coecients of restrictors, VC, VD are the sensing chambervolumes ande represents the eective bulk modulus.Using the ow through the left and right sensed chambers, the ow through the mainorice (from the source to the hydraulic load) and the load ow, the linearized continuityequation at the chamber of the pressure being controlled is:KC (PSPR) QLklPRK1(PRPC) K2(PRPD) +KqXs =VtesPR, (3.5)whereQL is the load ow,KC is the ow-pressure coecient of main orice,Kq is the owgain of main orice, kl is the leakage coecient andVt represents the total volume of thechamber where the pressure is being controlled.These equations dene the valve dynamics and combining them into a more useful form,solving(3.3)and(3.4)w.r.t. PCandPDandsubstitutinginto(3.5)yieldsaftersomemanipulation:(KCPSQL)_s1+1__s2+1_+KqXs_1+_ 11+12+CKqDKq_s++_112+CKq2DKq1_s2_=PRKce_VCVt13_s2+1_++VDVt23_s1+1_+_1+s3+ VCVt13+ VDVt23__s1+1_ _s2+1__,(3.6)where 1 =eK1VCand 2 =eK2VDare the break frequency of the left and right sensed chambers,3 =eKceVtis the break frequency of the main volume andKce = KC +klrepresents theequivalent ow-pressure coecient.Considering thatVC Vt andVD Vt, the right side can be factored to give the nalform for the reducing valve model in the charging phase:(KCPSQL)_s1+1__s2+1_+KqXs_1+_ 11+12+CKqDKq_s++_112+CKq2DKq1_s2_=PRKce_s1+1__s2+1__s3+1_.(3.7)In the discharging phase, a negative displacement of the pressure reducing valve spoolallows connection between the hydraulic load and the tank, while the channel that connectsthe source with the hydraulic load is kept closed.37Modeling and Control of an Electro-Hydraulic Actuated Wet ClutchThe linearized continuity equations at the sensed pressure chambers for the dischargingphase of the valve, illustrated in Fig. 3.2.d, are:QC =K1(PCPR) =VCesPC +CsXs, (3.8)QD =K2(PDPR) =VDesPDDsXs. (3.9)Using the ow through the left and right sensed chambers, the ow through the mainorice (from the hydraulic load to the tank) and the load ow, the linearized continuityequation obtained for the chamber of the pressure being controlled is:QL+K1(PCPR) +K2(PDPR) KD(PRPT) klPR+KqXs =VtesPR, (3.10)where KD is the ow-pressure coecient of main orice and PT represents the tank pressure.Combining these equations into a more useful form, solving (3.8) and (3.9) forPCandPD and substituting into (3.10) yields after some manipulation:(KDPT +QL)_s1+1__s2+1_+KqXs_1+_ 11+12+CKqDKq_s++_112+CKq2DKq1_s2_=PRKce_VCVt13_s2+1_++VDVt23_s1+1_+_1+s3+ VCVt13+ VDVt23__s1+1__s2+1__,(3.11)In an entire analogue manner, again making the assumption thatVC Vt andVD Vtlike for the charging phase model and consideringKD =KC the nal form for the reducingvalve in the discharging phase was obtained:(KDPT +QL)_s1+1__s2+1_+KqXs_1+_ 11+12+CKqDKq_s++_112+CKq2DKq1_s2_=PRKce_s1+1__s2+1__s3+1_.(3.12)Equations (3.1), (3.2), (3.3), (3.4), (3.5) and (3.7) for the charging phase of the valve,and equations (3.1), (3.2), (3.8), (3.9), (3.10) and (3.12) for the discharging phase of the383.2 Modeling of an Electro-Hydraulic Actuated Wet Clutch as a Subsystem ofan Automated Manual TransmissionFigure 3.3: Transfer function block diagram of the pressure reducing valve.valve, dene the pressure reducing valve dynamics and can be used to construct the transferfunction block diagram represented in Fig. 3.3. Also, the following notation was made:G(s) =KqKce_1+_11 +12 +CKqDKq_s +_112 +CKq2 DKq1_s2__1+s1__1+s2__1+s3_ . (3.13)Considering the resulting force between the magnetic and the feedback force:F1 =FmagCPC +DPD, (3.14)solvingPC andPD from the linearized continuity equations (3.3), (3.4) and substituting inthe force balance equation (3.2), the following equation was obtained:FmagCK1PR+CsXs_s1 +1_K1+DK2PRDsXs_s2 +1_K2=Mvs2Xs+KeXs. (3.15)After some manipulations, where it was considered that m = _KeMv, representing the39Modeling and Control of an Electro-Hydraulic Actuated Wet Clutchmechanical natural frequency, and substituting (3.14) into (3.2) yields:F1___C2K1s_s1 +1_+D2K2s_s2 +1____Xs =Ke_s22m+1_Xs, (3.16)whereF1 =Fmag__Cs1 +1 +Ds2 +1__PR, (3.17)illustrating the closed loop model from Fig. 3.3 for the displacement xs. A switch is used inorder to commutate between the two phases of the pressure reducing valve. Like seen in Fig.3.3, switching between the charging and the discharging phase can be realized by selectingdierent disturbances for positive and negative displacement of the spool.3.2.2.3 State-Space modelStarting from (3.1), (3.2), (3.3), (3.4) and (3.5) for the charging phase of the valve, andequations (3.1), (3.2), (3.8), (3.9) and (3.10) for the discharging phase of the valve, a state-space model is designed: x(t) =Ax(t) +Bu(t)y (t) = Cx(t) +Du(t)(3.18)where: x(t) = _vs(t) xs(t) PC(t) PD(t) PR(t) _Tis the state vector withvs(t) repre-senting the velocity of the spool, y(t) = _xs(t) PR(t) _Tis the output vector andu(t) =_PS(t) PT(t) QL(t) Fmag(t) _Tis the input vector. TheA,C andD matrices are:A =e__0 KeMveCMveDMve01e0 0 0 0CVC0 K1VC0K1VCDVD0 0 K2VDK2VD0KqVtK1VtK2Vt(Kce+K1+K2)Vt__,C =_ 0 1 0 0 00 0 0 0 1_, D =O24,(3.19)403.2 Modeling of an Electro-Hydraulic Actuated Wet Clutch as a Subsystem ofan Automated Manual TransmissionandthematrixBhastheB1expressioninthechargingphase(for xs> 0)andtheB2expression in the discharging phase (forxs closinge& [x1[ 1, - closed I3 :=x R5[ x3 >closinge&1 closinge&2 t1, ifx() 1, R[t1,t2)andx(t2) 2, setx1(t2) := 0.(5.13)As the engine anglee tends to innity in the open mode, so the statex1 tends to innity,a synchronization of the engine angle and the transmission angle must be attained at themoment the clutch switches from the open mode to the closed mode.The new model has the following state matrices Ac1, Ac2, Ac3, Ac4, that correspond to the1045.2 Driveline Modelsopen mode and the three phases of the closed mode of the clutch, respectively:Aci =__________0 0 1 it00 0 01if1kciJ10Dsum1J1dciitJ10kciitJ2kdifJ2dciitJ2Dsum2J2ddifJ20kdJ30ddifJ3dwheelJ3__________, (5.14)with Dsum1 =dci+de, Dsum2 =dciit2+d2+ddif2, dwheel =dw+dd+cr2 and the correspondingclutch stinesskci and clutch dampingdci.The novelty of this model consist of the opened working mode of the clutch, that is addedto the three dierent phases of the closed mode.5.2.3 Dual Clutch Transmission DrivelineInrecentyearsthedrivelineoscillationproblemhasreceivedanincreasinginterestdueto the introduction of dual-clutch transmission, commonly abbreviated to DCT (sometimesrefereed to as twin-clutch gearbox or double clutch transmission). DCT utilizes two separatedclutchesforoddandevengearsets. Itcanfundamentallybedescriedastwoseparatemanual transmissions contained within one housing, and working as one unit. These dryclutch transmissions oer improved fuel economy, easier packaging and red

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