4.2 Driveline Models - Facultatea de Automatică şi Calculatoare

Transcription

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
“GHEORGHE ASACHI”
DIN IAŞI
“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
PROTECŢIEI SOCIALE
AMPOSDRU
POSDRU 2007-2013
2007-2013
OIPOSDRU
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
DIN IAŞI
UNIUNEA EUROPEANĂ
GUVERNUL ROMÂNIEI
PROTECŢIEI SOCIALE
AMPOSDRU
POSDRU 2007-2013
2007-2013
OIPOSDRU
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
PROTECŢIEI SOCIALE
AMPOSDRU
POSDRU 2007-2013
2007-2013
OIPOSDRU
DIN IAŞI
Acknowledgements
Looking back, I am surprised and at the same time very grateful for everything
I have received throughout these years. It has certainly shaped me as a person
and has led me where I am now.
Foremost, I would like to express my sincere gratitude to my advisor Prof. Corneliu Lazăr, for the continuous support of my Ph.D study and research, for his
motivation, enthusiasm, patience and immense knowledge. His guidance helped
me 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. Mircea
Lazăr, for offering me the opportunity to work in their department, for the detailed and constructive comments and for the kind support and guidance that
have 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. Mihaela
Hanako-Matcovski, Prof. Alexandru Onea, Assoc. Prof. Letiţia Mirea and
Assoc. Prof. Lavinia Ferariu, for the extensive discussions around my work,
constructive questions and excellent advices. I have to thank Costi for the stimulating discussions and for all the times we have worked together on various
papers, and I also appreciate the short but productive collaboration I have had
with Cristina.
It was a pleasure to share doctoral studies and life with wonderful people like
Adrian, Simona, Marius and Alex, my first office mates, and with my Ph.D
colleagues Alina, Costi, Cosmin, Carlos and Bogdan, who are now my very close
friends. I will never forget Dana’s late night dinners and all the special moments
I have spent with Nicu. I would like to thank all of them for their friendship
and for sharing the glory and sadness of reports and conferences deadlines and
day-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 with
a 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 when
it was most required, with a special thanks to my grandmother Paraschiva and
my sister Oana, for everything they have done for me. Finally, I want to dedicate
this thesis to my nephew Rivano, who I most love. He has shown a strong interest
on studying when, at the early age of three, he clearly pointed out his interest of
becoming a Professor Doctor Engineer.
Andreea Bălău
Iaşi, 2011
Contents
List of Figures
xi
List of Tables
xv
Glossary
xvii
1 Introduction
1.1
1
Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1
Driveline Modeling and Control . . . . . . . . . . . . . . . . . . . . .
1
1.1.1.1
Backlash Nonlinearity . . . . . . . . . . . . . . . . . . . . .
3
1.1.1.2
Clutch Nonlinearity . . . . . . . . . . . . . . . . . . . . . .
4
1.2
Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2 Driveline Modeling and Control
9
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Electro-Hydraulic Valve-Clutch System . . . . . . . . . . . . . . . . . . . . .
12
2.3
Driveline Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.3.1
Drive Shaft Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.3.2
Flexible Clutch and Drive Shaft Model . . . . . . . . . . . . . . . . .
16
2.3.3
Continuous Variable Transmission Drive Shaft Model . . . . . . . . .
18
Driveline Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.4.1
PID Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.4.2
PID Cascade-Based Driveline Control . . . . . . . . . . . . . . . . . .
21
2.4.3
Explicit MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.4.4
Horizon-1 MPC based on Flexible Control Lyapunov Function . . . .
24
2.4.4.1
Notation and Basic Definitions . . . . . . . . . . . . . . . .
24
2.4.4.2
Horizon -1 MPC . . . . . . . . . . . . . . . . . . . . . . . .
24
Delta GPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.4
2.4.5
vii
CONTENTS
2.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Modeling and Control of an Electro-Hydraulic Actuated Wet Clutch
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Modeling of an Electro-Hydraulic Actuated Wet Clutch as a Subsystem of an
3.4
31
31
Automated Manual Transmission . . . . . . . . . . . . . . . . . . . . . . . .
32
3.2.1
Test Bench Description . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.2.2
Modeling of an Pressure Reducing Valve . . . . . . . . . . . . . . . .
34
3.2.2.1
Valve Description . . . . . . . . . . . . . . . . . . . . . . . .
34
3.2.2.2
Input-Output Model . . . . . . . . . . . . . . . . . . . . . .
36
3.2.2.3
State-Space model . . . . . . . . . . . . . . . . . . . . . . .
40
3.2.2.4
Simulators for the Pressure Reducing Valve . . . . . . . . .
41
Modeling of the Electro-Hydraulic Actuated Wet Clutch System . . .
47
3.2.3.1
Description of the Valve-Clutch System
. . . . . . . . . . .
49
3.2.3.2
Input-Output Model . . . . . . . . . . . . . . . . . . . . . .
50
3.2.3.3
State-Space Model . . . . . . . . . . . . . . . . . . . . . . .
51
3.2.3.4
Simulators for the Electro-Hydraulic Actuated Wet Clutch .
53
3.2.3
3.3
28
Control of the Electro-Hydraulic Actuated Wet
Clutch as a Subsystem of an Automated Manual Transmission . . . . . . . .
57
3.3.1
Generalized Predictive Control . . . . . . . . . . . . . . . . . . . . . .
58
3.3.2
PID Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
4 Two Inertias Driveline Model Including Backlash Nonlinearity
65
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
4.2
Driveline Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
4.2.1
CVT Driveline Model with Backlash Nonlinearity . . . . . . . . . . .
66
4.2.1.1
PWA Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
4.2.1.2
Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . .
69
AMT Driveline Model with Backlash Nonlinearity . . . . . . . . . . .
70
4.2.2.1
Rigid Driveline Model . . . . . . . . . . . . . . . . . . . . .
70
4.2.2.2
Flexible Driveline Model . . . . . . . . . . . . . . . . . . . .
72
4.2.2.3
Flexible Driveline Model with Backlash . . . . . . . . . . . .
73
Driveline Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
4.3.1
PID Cascade-Based Driveline Controller . . . . . . . . . . . . . . . .
75
4.3.2
Horizon -1 MPC Controller . . . . . . . . . . . . . . . . . . . . . . .
78
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
4.2.2
4.3
4.4
viii
CONTENTS
4.5
4.6
4.4.1
Simulator for the PWA Model of the CVT Driveline . . . . . . . . . .
80
4.4.2
Simulator for the Nonlinear Model of the CVT Driveline . . . . . . .
83
Real Time Experiments
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
4.5.1
System Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
4.5.2
Electromechanical Plant Description . . . . . . . . . . . . . . . . . .
88
4.5.3
Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
5 Three Inertias Driveline Model Including Clutch Nonlinearity
99
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Driveline Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3
5.4
5.2.1
AMT Affine Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.2
AMT Piecewise Affine Model . . . . . . . . . . . . . . . . . . . . . . 102
5.2.3
Dual Clutch Transmission Driveline . . . . . . . . . . . . . . . . . . . 105
Driveline Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3.1
Explicit MPC Controller . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3.2
Horizon-1 MPC Controller . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3.3
Delta GPC Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.4.1
Delta GPC for the Affine Model . . . . . . . . . . . . . . . . . . . . . 115
5.4.2
Affine Model Versus PWA Model . . . . . . . . . . . . . . . . . . . . 116
5.4.3
AMT Driveline Control . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.4.4
5.5
5.4.3.1
Scenario 1: Acceleration test
. . . . . . . . . . . . . . . . . 122
5.4.3.2
Scenario 2: Deceleration test . . . . . . . . . . . . . . . . . 124
5.4.3.3
Scenario 3: Tip-in tip-out maneuvers . . . . . . . . . . . . . 127
5.4.3.4
Scenario 4: Stress test . . . . . . . . . . . . . . . . . . . . . 127
DCT Driveline Control . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.4.4.1
Up-shift maneuvers . . . . . . . . . . . . . . . . . . . . . . . 129
5.4.4.2
Down-shift maneuvers . . . . . . . . . . . . . . . . . . . . . 130
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6 Conclusions
6.1
99
135
Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.1.1
Modeling and Control of an Electro-Hydraulic Actuated Wet Clutch
System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.1.2
Modeling and Control of a Two Inertia Driveline Including Backlash
Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
ix
CONTENTS
6.1.3
6.2
Modeling and Control of a Three Inertia Driveline Including Clutch
Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Suggestion for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . 138
A
141
Bibliography
147
x
List of Figures
2.1
Schematic vehicle structure. . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.2
Driveline subsystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.3
Schematic valve structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.4
Valve plunger subsystem model. . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.5
Drive shaft model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.6
Flexible clutch and drive shaft model. . . . . . . . . . . . . . . . . . . . . . .
17
2.7
Continuous variable transmission drive shaft model. . . . . . . . . . . . . . .
19
2.8
PID control structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.9
Cascade based control structure. . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.1
a) Test bench b) Schematic diagram . . . . . . . . . . . . . . . . . . . . . . .
33
3.2
a) Section through a real three stage pressure reducing valve; b) Three stage
valve schematic representation; c) Charging phase of the pressure reducing
valve; d) Discharging phase of the pressure reducing valve. . . . . . . . . . .
35
3.3
Transfer function block diagram of the pressure reducing valve. . . . . . . . .
39
3.4
Simulink model with step signal input. . . . . . . . . . . . . . . . . . . . . .
42
3.5
Simulink transfer functions of the valve model. . . . . . . . . . . . . . . . . .
42
3.6
Magnetic force and load flow. . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.7
Spool displacement and reduced pressure. . . . . . . . . . . . . . . . . . . . .
43
3.8
Input-output Simulink model. . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.9
Current and magnetic force used as input signals. . . . . . . . . . . . . . . .
45
3.10 Compared spool displacements for input-output model . . . . . . . . . . . . .
46
3.11 Compared reducing pressures for input-output model. . . . . . . . . . . . . .
46
3.12 State-space Simulink model. . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
3.13 Compared spool displacements for state-space model. . . . . . . . . . . . . .
48
3.14 Compared reducing pressures for state-space model. . . . . . . . . . . . . . .
48
3.15 Charging phase of the actuator-clutch system. . . . . . . . . . . . . . . . . .
49
3.16 Discharging phase of the actuator-clutch system. . . . . . . . . . . . . . . . .
50
xi
LIST OF FIGURES
3.17 Transfer function block diagram of the actuator-clutch system. . . . . . . . .
51
3.18 State-space block diagram of the actuator-clutch system. . . . . . . . . . . .
53
3.19 Input-output Simulink diagram of the actuator-clutch system. . . . . . . . .
54
3.20 System pressures for the input-output model. . . . . . . . . . . . . . . . . .
54
3.21 Input-output system simulation. . . . . . . . . . . . . . . . . . . . . . . . . .
55
3.22 State-space Simulink diagram of the actuator-clutch system. . . . . . . . . .
56
3.23 System pressures for the state-space model. . . . . . . . . . . . . . . . . . . .
57
3.24 State-space system simulation. . . . . . . . . . . . . . . . . . . . . . . . . . .
58
3.25 GPC results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3.26 PID controller results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
4.1
Schematic representation of an automotive driveline with backlash. . . . . .
67
4.2
Rigid driveline model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
4.3
Flexible driveline model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
4.4
Nonlinear CVT driveline structure - Simulink representation. . . . . . . . . .
75
4.5
Validation structure - Simulink representation. . . . . . . . . . . . . . . . . .
76
4.6
Input command - icvt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
4.7
Wheel speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
4.8
PID cascade based control structure - Simulink representation. . . . . . . . .
77
4.9
Torque controller - Simulink representation. . . . . . . . . . . . . . . . . . .
77
4.10 Speed controller - Simulink representation. . . . . . . . . . . . . . . . . . . .
78
4.11 Horizon-1 MPC - Simulink structure. . . . . . . . . . . . . . . . . . . . . . .
81
4.12 Wheel speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
4.13 Operating mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
4.14 Backlash angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
4.15 Engine torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
4.16 Optimal fuel-efficiency curve.
. . . . . . . . . . . . . . . . . . . . . . . . . .
84
4.17 Wheel speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
4.18 Final drive-shaft torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
4.19 Engine speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
4.20 CVT ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
4.21 M220 Industrial plant emulator schematic structure. . . . . . . . . . . . . . .
87
4.22 Industrial plant emulator M220. . . . . . . . . . . . . . . . . . . . . . . . . .
89
4.23 Rigid driveline collocated controller - Simulink structure. . . . . . . . . . . .
90
4.24 Rigid driveline non-collocated controller - Simulink structure. . . . . . . . . .
92
4.25 Rigid driveline collocated and non-collocated control. . . . . . . . . . . . . .
92
4.26 Backlash mechanism structure. . . . . . . . . . . . . . . . . . . . . . . . . .
93
xii
LIST OF FIGURES
4.27 Rigid driveline with 4 degrees backlash angle collocated and non-collocated
control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.28 Rigid driveline with 8 degrees backlash angle collocated and non-collocated
control.
94
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
4.29 Flexible driveline controller - Simulink structure. . . . . . . . . . . . . . . . .
96
4.30 Flexible driveline with backlash control - engine inertia position. . . . . . . .
4.31 Flexible driveline with backlash control - wheel inertia position. . . . . . . .
96
97
5.1
Three inertia driveline model. . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2
5.3
Clutch functionality a) stiffness characteristic; b) clutch springs . . . . . . . 103
AMT clutch switching logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4
Double clutch transmission driveline model. . . . . . . . . . . . . . . . . . . 106
5.5
DCT - Switching logic for the first clutch. . . . . . . . . . . . . . . . . . . . 107
5.6
5.7
DCT - Switching logic for the second clutch. . . . . . . . . . . . . . . . . . . 108
Simulation results using δ GPC. . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.8
Influences of the δ GPC on engine speed, transmission speed and axle wrap.
5.9
δ GPC simulation results subject to reference changes. . . . . . . . . . . . . 117
116
5.10 Influences of the δ GPC on engine speed, transmission speed and axle wrap,
subject to reference changes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.11 Vehicle velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.12 Engine speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.13 Axle wrap speed difference. . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.14 Engine torque (control signal). . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.15 Clutch mode of operation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.16 Scenario 1: Acceleration test. . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.17 Scenario 1: Clutch mode of operation. . . . . . . . . . . . . . . . . . . . . . 123
5.18 Scenario 1: EMPC - Acceleration test. . . . . . . . . . . . . . . . . . . . . . 125
5.19 Scenario 1: EMPC - Clutch mode of operation. . . . . . . . . . . . . . . . . 125
5.20 Scenario 2: Deceleration test. . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.21 Scenario 3: Tip-in tip-out test. . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.22 Scenario 4: Stress test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.23 Scenario 1: Up-shift maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.24 MPC - Clutch operation modes for up-shift maneuvers test. . . . . . . . . . 131
5.25 PID - Clutch operation modes for up-shift maneuvers test. . . . . . . . . . . 131
5.26 Scenario 2: Down-shift maneuvers. . . . . . . . . . . . . . . . . . . . . . . . 132
5.27 Clutch operation modes for down-shift maneuvers test. . . . . . . . . . . . . 132
xiii
LIST OF FIGURES
xiv
List of Tables
A.1 Valve-clutch system parameter values . . . . . . . . . . . . . . . . . . . . . . 142
A.2 Vehicle parameter values for two inertia CVT driveline with backlash nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A.3 Vehicle parameter values for two inertia AMT driveline with backlash nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
A.4 Simulation vehicle parameter values for three inertias driveline with clutch
nonlinearity - 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
A.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 chassis
models and advanced control methods in order to increase overall vehicle performance, fuel
economy, safety and comfort. The goal of this thesis is overall powertrain modeling and
control, based on driveline subsystem integration. More complex driveline and driveline
subsystems models are proposed, and different problems as nonlinearities introduced by
backlash 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 have
the function of transmitting the engine torque to the driving wheels. In order to transmit
this torque in an efficient way, a proper model of the driveline is needed for controller design
purposes, with the aim of lowering emissions, reducing fuel consumption and increasing
comfort.
1.1.1
Driveline Modeling and Control
The automotive driveline is an essential part of the vehicle and its dynamics have been
modeled differently, according to the driving necessities. The complexity of the numerous
models reported in the literature varies (Hrovat et al., 2000), but the two masses models are
more commonly used, and this fact is justified in (Pettersson et al., 1997), where it is shown
that this model is able to capture the first torsional vibrational mode. There are also more
complex three-masses models reported in different research papers, as it will be indicated
next. In (Templin and Egardt, 2009) a simple driveline model with two inertias, one for the
engine 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-space
model. A simple model with the pressure in the engine manifold and the engine speed as state
variables 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 representing 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 backlash nonlinearity. In (Grotjahn et al., 2006) a two masses model was presented, with the
driveline main flexibility represented by the drive shafts, as well as a three mass model
to reproduce the behavior of a vehicle with a dual-mass flywheel. Linear and nonlinear
three masses models, in which the clutch flexibility was also considered, were presented
in (Kiencke and Nielsen, 2005). Complex three masses models that includes certain nonlinear 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 literature. In (Templin and Egardt, 2009) a linear quadratic regulator (LQR) design that damps
driveline oscillations by compensating the driver’s engine torque demand was presented. The
performance cost uses a weighting of the time derivative of the drive shaft torque and the difference between the driver’s torque demand and the actual controller torque demand. LQR
controllers were also proposed in (David and Natarajan, 2005) and (Dolcini, 2007). Other
linear 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 combination with a LQR controller and considering the engine as an actuator to damp powertrain 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 presented in (Lefebvre et al., 2003), while model predictive control (MPC) strategies were proposed 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 presented in (Adachi et al., 2004) and (Gao et al., 2010) In (Baumann et al., 2006), a model2
1.1 Literature Review
based approach for anti-jerk control of passenger cars that minimizes driveline oscillations
while retaining fast acceleration was introduced. The controller was designed with the
help 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 speed
and the load speed to minimize the driveline oscillations, while reducing the impact of
forces between the mechanical parts. A clutch engagement controller based on fuzzy logic
is 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 introduced
by 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 nonlinearity in the control loop for torque generation and distribution. This phenomenon occurs
whenever there is a gap in the transmission link which leads to zero torque transmitted
through the shaft to the wheels. When the backlash gap is traversed the impact results in
a large shaft torque and sudden acceleration of the vehicle. Engine control systems must
compensate 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 position
difference between wheels and engine. The modeling of mechanical systems with backlash
nonlinearities 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 even
destabilize the control system. Also, it can have as consequence low components reliability
and shunt and shuffle. In order to model the mechanical system with backlash, two different operational modes must be distinguished: backlash mode (when the two mechanical
components are not in contact) and contact mode (when there is a contact between the two
mechanical components resulting in a moment transmission).
New driveline management application and high-powered engines increase the need for
strategies 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 controller and a modified switching controller. The concept of PID controller with torque
compensator is presented in (Nakayama et al., 2000) for the backlash. A simple active
switching 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 designed
in order to ensure asymptotic wheel speed and gear ratio tracking. A nonlinear predictive
controller 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 driveability. 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 nonlinearity.
1.1.1.2
Clutch Nonlinearity
In recent years, the use of control systems for automated clutch and transmission actuation
has been constantly increasing, the trend towards higher levels of comfort and driving dynamics 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 torque
to the vehicle with the desired ratio smoothly and efficiently, and the most common control
devices inside the transmission are clutches and actuators. Such clutches can be hydraulic
actuated, motor driven or actuated using other means.
During the last years, the automated actuated clutch systems and different valve types
used as actuators have been actively researched and different models and control strategies
have 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 statespace model description of the actuator that is derived based on physical principles and
parameter identification (Wang et al., 2002), (Peterson et al., 2003), (Gennaro et al., 2007),
(Nemeth, 2004), mathematical model obtained using identification methods for a valve actuation system of an electro-hydraulic engine (Liao et al., 2008), a model for an electrohydraulic valve used as actuator for a wet clutch (Morselli and Zanasi, 2006), dynamic modeling and control of electro-hydraulic wet clutches (Morselli et al., 2003), PID control for a
wet plate clutch actuated by a pressure reducing valve (Edelaar, 1997), predictive and piecewise LQ control of a dry clutch engagement (Van Der Heijden et al., 2007), switched control
of an electro-pneumatic clutch actuator (Langjord et al., 2008), Model Predictive Control
of a two stage actuation system using piezoelectric actuators for controllable industrial and
automotive 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 and
drive 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 control
Lyapunov function are presented as driveline control strategies. Starting from these models,
in what follows, more complex driveline models are developed and also the control strategies
presented in this chapter are applied in order to obtain new controllers able to improve
overall 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 the
automatic transmission are presented. First, an input-output and a state-space model of
an electro-hydraulic pressure reducing valve are developed and stating from these, an inputoutput 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 data
provided from experiments with the real valve actuator and the clutch on a test bench. The
test bench was provided by Continental Automotive Romania and it includes the Volkswagen
DQ250 wet clutch actuated by the electro-hydraulic valve DQ500. Also, different control
strategies are applied on the developed models and simulation result are being discussed: a
GPC and a PID controller are designed in order to control the output of the electro-hydraulic
actuated clutch system, the clutch piston displacement.
Chapter 4 is entitled Two inertias driveline model including backlash nonlinearity. In
this chapter, different models for automotive driveline including backlash nonlinearity are
proposed. First, a piecewise affine and a nonlinear state-space model for a Continuous
Variable Transmission (CVT) driveline with backlash are proposed. Simulators are developed
in 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 based
controller is applied for the nonlinear model designed to reduce the fuel consumption by using
the optimal fuel efficiency curve in the modeling phase. Next, three models are presented for
an Automated Manual Transmission (AMT) driveline based on the Industrial plant emulator
M220 : a rigid driveline model, a flexible driveline model and a flexible driveline model
including also backlash nonlinearity. Then, real time experiments are conducted on the
presented models in order to test the influences given by drive shaft flexibility and backlash
angle, while applying a horizon-1 MPC controller.
Chapter 5 is entitled Three inertias driveline model including clutch nonlinearity. This
chapter deals with the problem of damping driveline oscillations in order to improve passenger comfort. Three driveline models with three inertias are proposed: a state-space affine
model 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 transmission (DCT) driveline, all of them taking into consideration the drive shafts as well as
the clutch flexibilities. Three controllers are implemented for the developed models: explicit MPC, delta GPC and horizon-1 MPC, and the experiments showed that the horizon-1
MPC control scheme can handle both the performance/physical constraints and the strict
limitations on the computational complexity corresponding to vehicle driveline oscillations
damping.
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 IEEE
conference papers, two IFAC conference papers and three papers published at international
conferences 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 for
Automotive Applications. In 18th IEEE International Conference on Control Applications, 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. New
Model for Predictive Control of an Electro-Hydraulic Actuated Clutch. In The 18th
International Conference on FUEL ECONOMY, SAFETY and RELIABILITY of MOTOR 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 for
Automotive Control Systems. In The 1tth International Conference on Control Systems
and 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 Electromagnetic
Valve 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 an
electro-hydraulic actuated wet clutch. In IFAC Symposium Advances in Automotive
Control, Munchen, Germany, 2010.
• (Balau et al., 2011a) A. E. Balau, C. F. Caruntu and C. Lazar. Simulation and Control
of 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. Modelling and Predictive Control of an Electro-Hydraulic Actuated Wet Clutch for Automatic 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. Networked Predictive Control Strategy for an Electro-Hydraulic Actuated Wet Clutch. In
IFAC Symposium Advances in Automotive Control, Munchen, Germany, 2010.
• (Balau and C.Lazar, 2011a) A. E. Balau and C. Lazar. Predictive control of an electrohydraulic actuated wet clutch. In The 15th International Conference on System Theory,
Control and Computing, Sinaia, Romania, 2011.
• (Caruntu, Balau and C.Lazar, 2010b) C. F. Caruntu, A. E. Balau and C. Lazar. Cascade based Control of a Drivetrain with Backlash. In 12th International Conference on
Optimization of Electrical and Electronic Equipment, Brasov, Romania, 2010.
• (Balau et al., 2011b) A. E. Balau, C. F. Caruntu and C. Lazar. Driveline oscillations
modeling and control. In The 18th International Conference on Control Systems and
Computer Science, Bucharest, Romania, 2011.
• (Balau and C.Lazar, 2011b) A.E. Balau and C. Lazar. One Step Ahead MPC for an
Automotive Control Application. In The 2nd Eastern European Regional Conference
on 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. Bosh
and S. Di Cairano. A predictive control solution for driveline oscillations damping. In
The 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. State
Space Delta GPC for Automotive Powertrain Systems. In The16th IEEE International
Conference on Emerging Technologies and Factory Automation, 2011.
8
Chapter 2
this torque in an efficient way, a proper model of the driveline is needed for controller
design purposes with the aim of lowering emissions, reducing fuel consumption and increasing
comfort. Recent studies in automotive engineering explore various engine, transmission and
chassis 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 in
different ways, according to the purpose. The aim of the modeling is to find the most
significant physical effects that have as negative result oscillations in the wheel speed. Most
experiments consider in the modeling phase low gears because the higher torque transmitted
to the drive shaft is obtained in the lower gear. Also, the amplitudes of the resonances in the
wheel speed are higher for lower gears, because the load and vehicle mass appear reduced
by 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 generalized Newton’s second law of motion, as described in (Kiencke and Nielsen, 2005). Figure
Fig. 2.2 shows the labels, the inputs and the outputs of each subsystem of the considered
driveline, and relations between them will be described for each part.
The output engine torque is given by the driving engine torque Te resulted from the
combustion, the internal engine friction Tf ric,e , and the external load from the clutch Tc ,
9
Figure 2.1: Schematic vehicle structure.
Figure 2.2: Driveline subsystems.
obtaining the following equation:
Je θ̈e = Te − Tf ric,e − Tc ,
(2.1)
where Je represents the engine moment of inertia, θe is the crankshaft angle, ωe = θ̇e is the
engine 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 the
transmission’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 ) and
the 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 , Tf ric,t , θc − θt it , ωc − ωt it , it ,
10
(2.3)
2.1 Introduction
where Tp is the propeller shaft torque, Tf ric,t is the internal friction torque of the transmission, θt is the transmission angle and ωt = θ̇t is the corresponding angular velocity. The
reason for considering the angle difference θc −θt it is the possibility of having torsional effects
in the transmission.
The propeller shaft connects the transmission’s output shaft with the final drive. No
friction 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 corresponding angular velocity.
The final drive is characterized by a conversion ratio if in the same way as for the
transmission. The following relation between the input and the output torque holds:
Td = ff Tf , Tf ric,f , θp − θf if , ωp − ωf if , if ,
(2.5)
where Tf ric,f is the internal friction torque of the final drive, Td is the drive shaft torque, θf
is 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. Assuming
that θw is the wheel angle, the rotational wheel velocity ωw = θ̇w is the same for both wheels
and neglecting the vehicle dynamics, the rotational equivalent wheel velocity shall be equal
to the velocity of the vehicle body’s center of gravity vv :
ωw =
vv
rstat
(2.6)
,
where rstat represents the wheel radius. The shafts are modeled as one shaft and assuming
that 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 and
speed vv , gives:
Fload = mCoG v̇v + Fairdrag + Froll + mCoG g sin (χroad ).
(2.8)
The load force Fload is described by the sum of following quantities:
• Fairdrag , the air drag, is approximated by Fairdrag = 12 cair Af ρair vv2 , where cair is the
drag coefficient, Af is the maximum vehicle cross section area and ρair is the air density.
11
• Froll , the rolling resistance, is approximated by Froll = mCoG (cr1 + cr2 vv ) where cr1
and 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 Fload rstat and the equation of motion for the
wheel is described by the following relation:
Jw ω̇w = Tw − Fload rstat − 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:
1
2
3
2
Jw + mCoG rstat
ω̇w = Tw − TL − cair Af ρair rstat
ωw
− rstat mCoG (cr1 + cr2 rstat ωw )
2
(2.10)
− rstat mCoG g sin (χroad ) .
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 Tf ric,t , Tf ric,f , TL are
unknown, 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 torque
to the vehicle with the desired ratio smoothly and efficiently and the most common control
devices inside the transmission are clutches and hydraulic pistons. The automatic control
of the clutch engagement plays a crucial role in AMT (Automatic Manual Transmission)
vehicles, being seen as an increasingly important enabling technology for the automotive
industry. It has a major role in automatic gear shifting and traction control for improved
safety, driveability and comfort and, at the same time, for fuel economy. Recent attention has
focused on modeling different valve types used as actuators in automotive control systems
and, in what follows, a model found in the literature of an electro hydraulic actuated wet
clutch system is presented.
A new modeling method of automotive control systems, based on power graphs, is presented in (Morselli and Zanasi, 2006), where a system composed of an electro-hydraulic valve
and a wet clutch is modeled. The method is called Power-Oriented Graphs (POG) and utilizes 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 energetic
domains.
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 subsystems: valve plunger, control chamber, user chamber and clutch chamber. In order to
illustrate the POG approach, the subsystem corresponding to the valve plunger is represented in Fig. 2.4.
The plunger mass Mv moves according to the damping coefficient bp , the return spring
Kp (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 the
return spring as well as the contact force between the plunger and the plunger chamber, at
the plunger two extremities. The plunger movement causes the oil flow QC and QD through
the control chamber and through the back chamber:
Mv ẍs = (PC − PD )Ap − bp ẋs − Kp (xs ),
QC = QD = Ap ẋs .
(2.11)
The pressure from the control chamber PC is obtained by integrating three oil flows:
13
the flow Q5 from the power source Ps , the flow QC due to the plunger movement, and the
flow Qw through the variable discharging orifice. The very small hydraulic capacity CC
stores potential energy in terms of oil pressure and it takes into account the small elastic
deformation 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 the
power supply through the variable orifice J1 or to the oil tank by the orifice J3 . Also, the
user chamber is connected to the back chamber through orifice J4 . This orifice plays two
roles: it implements the feedback action since PD becomes a measure of the user pressure
PR , 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:
CD ṖD = QD − Q4 ,
CR ṖR = Q1 + Q4 − Q3 − QR .
(2.13)
The user chamber is connected to the clutch chamber by means of a pipe with a dynamic
that cannot be neglected and is described by four elements: the user chamber capacity
CR , the hydraulic resistance RL , the pipe hydraulic inductance LL and the clutch chamber
capacity CL :
LL Q̇R = Pl − PL = PR − PQR − PL ,
QR |QR |
= RL (QR ),
P R − Pl =
CL
CL ṖL = QR − AL ż.
(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:
Mp z̈ = PL AL − bf xż − KM (z) − Kbc sgn(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 the
force of the return springs and the contact with the gearbox at the two extreme pressure
plate positions, and KD (z) is the force generated by the compression of the clutch discs that
determines the maximum torque through the clutch.
This combined equations model the valve-clutch system using the POG technique and the
simulations results are very similar to the experimental data, providing that the modeling
approach 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 been
modeled differently, according to the driving necessities. In this sections, three different
driveline 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: internal
combustion engine, transmission, flexible drive shafts and driven wheel. The propeller shaft
is 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:




J
d
J
d
Je + t + f  ω̇e =Te − Tf ric,e −  t + f  ωe −
2
2
2
2
it it if
it i2t i2f



(2.16)

kd  θe
dd  ωe
−
− θw  −
− ωw  ,
2
2
it if it if
it if i2t i2f
where Jt and Jf represents the transmission and the final drive inertias, while dt and df
stands for the corresponding damping coefficients. Also, kd and dd represents the stiffness
and damping coefficients of the drive shaft.
Also, the equation for the vehicle and wheels inertia is given by:

2
Jw + mCoG rstat



ωe
1
θe
3
2
ωw
−
ω̇w = kd  2 2 − θw  + dd  2 2 − ωw  − cair Af ρa rstat
it if
it if
2
−rstat mCoG (cr1 + g sin (χroad )) −
15
2
dw + mCoG cr2 rstat
ωw ,
(2.17)
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 =
θe
− θw
if it
(2.18)
.
x2 = ωe
x3 = ωw
Also, taking into consideration that:
J1 = Je +
Jf
Jt
+ 22
2
it it if
2
J2 = Jw + mCoG rstat
df
dt
d1 = 2 + 2 2
it it if
(2.19)
,
2
d2 = dw + mCoG cr2 rstat
l = rstat mCoG (cr1 + g sin (χroad ))
the following state-space representation is obtain:
ẋ = Ax + Bu + Hl,
(2.20)
consisting of the system matrices:



A=


0
− if ikt J1
1
if it
−d1 +
d
if it 2
J1
k
J2
0

0
 1
B=
J1

d
if it J1
d
if it J2


−d+d2
J2


,


(2.21)

0



,H =  0 .
(2.22)
−1
J2
0
2.3.2
−1
Flexible Clutch and Drive Shaft Model
A more complex model including two torsional flexibilities, the drive shaft and the clutch
is also presented in (Kiencke and Nielsen, 2005). The driveline has three inertias like represented in Fig. 2.6, one corresponding to the internal combustion engine, one for the
transmission, and one for the driven wheel.
The equation that describe the engine dynamics is given by:
Je ω̇e = Te − Tf ric,e − kc (θe − θt it ) − dc (ωe − ωt it ) ,
16
(2.23)
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:


J
Jt + f  ω̇t = Te − Tf ric,e − it (kc (θe − θt it ) + dc (ωe − ωt it )) −
i2f


!
df
1
θt
ωt
− dt + 2  ωt −
kd
− θ w + dd
− ωw
if
if
if
if
(2.24)
!!
.
Also, the equation for the vehicle and wheels inertia is given by:
2
Jw + mCoG rstat
!
ω̇w = kd
!
1
θt
ωt
3
2
− θ w + dd
− ωw − cair Af ρa rstat
ωw
−
if
if
2
(2.25)
−rstat mCoG (cr1 + g sin (χroad )) − (dw + cr2 rstat ) ωw .
When studying a clutch in more detail it is seen that the torsional flexibility is a result
of 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 springs
with stiffness kc1 are being compressed. This ends when they are fully compressed at αc1
radians. If the angle is increased further, the stiffer springs, with stiffness kc2 , are beginning
to compress. When αc2 is reached, the clutch hits a mechanical stop. The resulting stiffness
of the clutch is given by:



kc1 if |x| 6 αc1
kc (x) = kc2 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
used as states, according to:
x1 = θe − θt it
θt
x2 = − θw
if
x3 = ωe
(2.27)
.
x4 = ωt
x5 = ωw
The state-space formulation of the linear clutch and drive shaft model consist of the
system matrices defined next:

0
0




k
 − Jc
1
Ac = 


 kc i t
 J
2

0
−it
0
0
1
0
0
−dc
J1
− ifkJd2
dc i t
J2
kd
J3
0

i
 1 

=
 J1  , H


 0 
0
f
J2
dd
if J3


0


 0 
B
1
if
dc it
J1
d
−dc i2t +d2 + 2d



=



0
0
0
0
−1
J2
0
−1
0
dd
if J2
−d3 −dd
J3






,





(2.28)




,



(2.29)
.
(2.30)
where
J1 = Je
J2 = Jt +
Jf
i2f
2
J3 = Jw + mCoG rstat
df
d2 = dt + 2
if
2
d3 = dw + mCoG cr2 rstat
2.3.3
Continuous Variable Transmission Drive Shaft Model
In (Mussaeus, 1997), a nonlinear model for a continuously-variable transmission driveline is
developed. The powertrain is represented in Fig. 2.7 and it is composed from the following
components: engine, continuously-variable transmission (CVT), final reduction gear (FRG),
flexible drive shaft (FDS) and driving wheel, which can be seen as input-output blocks. The
engine generates a toque which is transmitted towards the wheels through the driveline.
18
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 a
continuously-variable gear ratio. The final reduction gear has a distributive role inside the
powertrain, efficiently transferring the CVT output torque to the FDS. Obviously, the final
drive-shaft is not rigid and the torque losses can be very large if a proper mathematical model
is not considered. The FDS transmits the received torque to the wheels and its efficiency is
based 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 surface
and the aerodynamic drag.
The internal combustion engine can be seen as an ideal torque source/generator, the
functionality of the engine being described by the following equations:
Te = Γ(ωe ),
Je
d
ωe (t) = Te (t) − T1 (t) ,
dt
(2.31)
where T1 is the torque transmitted to the CVT and Γ is chosen to be the optimal fuel
efficiency curve. The transmission, described by equations:
ω2 = iCV T ωe ,
ηCV T
T2 =
T1 ,
iCV T
(2.32)
where ηCV T is the transmission efficiency and iCV T is the CVT ratio. Another gear ratio
iF RG is provided by the final reduction gear, which takes the torque from the transmission
and passes it to the flexible drive-shaft of the vehicle:
ω3 = iF RG ω2 ,
T3 =
ηF RG
T2 = r1 T1 ,
iF RG
19
(2.33)
where r1 =
ηF RG ηCV T
iF RG iCV T
and ηF RG is the flexible drive-shaft efficiency. Considering the flexible
drive-shaft speed related to the engine speed and solving 2.32 in 2.33 yields:
ω3 (t) =
where r2 =
1
iF RG iCV T
ωe (t)
,
r2
(2.34)
.
The powertrain flexibility is given by the flexible drive-shaft, which is characterized by
√
an elasticity factor kd = Jv π 2 and a damping coefficient dd = 2 kd Jv , both used to calculate
the FDS torque:
T3 (t) = Tk (t) + Tb (t) ,
(2.35)
where we have:
Tk = kd
Zt
(ω3 − ωw )dσ,
0
(2.36)
Tb = dd (ω3 − ωw ) .
The dynamical behavior of the wheel is described by the following equation:
Jv
d
ωw (t) = T3 (t) − Tload (t) ,
dt
(2.37)
where
2
Jv = rstat
mCOG ,
Tload (t) = Troll (t) + Tairdrag (t) + Tangle (t) ,
2
Tairdrag (t) = c1 ωw
(t) ,
(2.38)
Troll (t) = c2 mCOG ,
Tangle (t) = 0.
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 is
the load torques due to aerodynamic drag and c1 and c2 are constants.
The optimized powertrain was designed to reduce the fuel consumption by using the
optimal 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 obtain
the desired performances. In this section, different control strategies proposed in literature
for improving overall performances are presented.
2.4.1
PID Control
Unlike simple control algorithms, the PID controller is capable of manipulating the process
inputs based on the history and rate of change of the signal. This gives a more accurate
and stable control method. The basic idea is that the controller reads the system state
by a sensor. Then it subtracts the measurement from a desired reference to generate the
error value. The error will be managed in three ways, to handle the present, through the
proportional term, recover from the past, using the integral term, and to anticipate the
future, through the derivate term.
Several methods for tuning the PID loop exist. The choice of method will depend largely
on whether the process can be taken off-line for tuning or not. Ziegler-Nichols method is a
well-known online tuning strategy. Further tuning of the parameters is often necessary to
optimize the performance of the PID controller. The control structure of the controller is
presented in Fig. 2.8, and the mathematical form is given by:
u (n) = Kp e (n) + Ki
n
X
e (k) − Kd (y (n) − y (n − 1)) ,
(2.39)
k=0
Kp = Kr
Kp Ts
Ki =
(2.40)
Ti ,
Kp Td
Kd =
Ts
where Kr is the controller gain, Ti , and Td denote the time constants of the integral and
derivative terms, Ts is the sampling time of the system and Kp , Ki , and Kd represents the
proportional, integral, and derivative gains.
2.4.2
PID Cascade-Based Driveline Control
The PID controller consists of proportional, integral and derivative elements, being widely
used in feedback control of industrial processes because of its simplicity and robustness.
The often variation in parameters and parameter perturbations, which occur in industrial
processes, can make the system unstable. That is the reason why the PID controller computes
21
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 implemented, 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
2.4.3
Explicit MPC
Traditional control design methods such as PID or LQR cannot explicitly take into account hard constraints. In contrast, a MPC algorithm solves a finite-horizon open-loop
optimization problem on-line, at each sampling instant, while explicitly taking input and
state constraints into account.
Optimal control of constrained linear and piecewise affine systems has garnered great
interest in the research community due to the ease with which complex problems can be
stated and solved. The Multi-Parametric Toolbox (MPT) provides efficient computational
means to obtain feedback controllers for these types of constrained optimal control problems
in a Matlab programming environment. By multi-parametric programming, a linear or
quadratic optimization problem is solved off-line. The associated solution takes the form
of a PWA state feedback law. In particular, the state-space is partitioned into polyhedral
sets and for each of those sets the optimal control law is given as one affine function of the
state. In the online implementation of such controllers, computation of the controller action
reduces to a simple set-membership test, which is one of the reason why this method has
attracted so much interest in the research community (Kvasnica et al., 2006).
PWA systems are models for describing hybrid systems and the dynamical behavior of
such systems is capture by relations of the following form:
xk+1 = Ai xk + Bi uk + fi
yk = Ci xk + Di uk + gi
(2.41)
,
subject to constraints on outputs, control input, and control input slew rate:
ymin ≤ yk ≤ ymax
umin ≤ uk ≤ umax
(2.42)
.
∆umin ≤ uk − uk−1 ≤ ∆umax
The cost function used for the explicit MPC scheme is

min
{uk }k∈Z
[0,N −1]
kPN xN kp +
N
−1
X

kQx xk kp + kRu uk kp  ,
(2.43)
k=0
where u is the vector of manipulated variables over which the optimization is performed, N
is 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
2.4.4
Horizon-1 MPC based on Flexible Control Lyapunov Function
Standard MPC techniques require a sufficiently long prediction horizon to guarantee stability,
which makes the corresponding optimization problem too complex. Recently, a relaxation
of the conventional notion of a Lyapunov function was proposed in (M.Lazar, 2009), which
resulted in a so-called flexible Lyapunov function. A first application of flexible Lyapunov
functions in automotive control problems was presented in (Hermans et al., 2009). Therein
it was indicated that flexible Lyapunov functions can be used to design stabilizing MPC
schemes with a unitary horizon, without introducing conservatism. In what follows, we
demonstrate how the theory introduced in (M.Lazar, 2009) can be employed to design a
horizon-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, the
set of integer numbers and the set of non-negative integers, respectively. For every c ∈ R
and Π ⊆ R define Π≥c := {k ∈ Π | k ≥ c} and similarly Π≤c , RΠ := Π and ZΠ := Z ∩ Π. For
a vector x ∈ Rn let kxk denote an arbitrary p-norm and let [x]i , i ∈ Z[1,n] , denote the i-th
component of x. Let kxk∞ := maxi∈Z[1,n] |[x]i |, where | · | denotes the absolute value. For a
matrix Z ∈ Rm×n let kZk∞ := supx6=0 kZxk
kxk denote its corresponding induced matrix norm.
In ∈ Rn×n denotes the identity matrix. A function ϕ : R+ → R+ belongs to class K if it
is continuous, strictly increasing and ϕ(0) = 0. A function ϕ ∈ K belongs to class K∞ if
lims→∞ 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-time
instant k. φ : Rn × Rm → Rn is an arbitrary nonlinear, possibly discontinuous, function with
φ(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 (kxk) ≤ V (x) ≤ α2 (kxk),
24
∀x ∈ Rn
(2.45)
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 on
inequality (2.47) we can formulate the following optimization problem. Let α3 , α4 ∈ K∞ and
J : 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 an
explicit 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 for
model (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+ and
M ∈ Z>0 off-line. At time k ∈ Z+ measure xk and minimize the cost J(λk ) over uk , λk
subject to the constraints
uk ∈ U, φ(xk , uk ) ∈ X, λk ≥ 0,
(2.48a)
V (φ(xk , uk )) ≤ ρV (xk ) + λk ,
(2.48b)
1
M
λk ≤ ρ (λ∗k−1 + ρ
k−1
M
∆),
∀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 2.4.2
is feasible for all states x in X. Then the difference inclusion
xk+1 ∈ φcl (xk , π(xk )),
is asymptotically stable in X.
25
k ∈ Z+ ,
(2.49)
The proof of Theorem 2.4.3 starts from the fact that (2.48c) implies limk→∞ λ∗k = 0
and then employs standard arguments for proving input-to-state stability and Lyapunov
stability. 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 condition
than (2.48c) was used, which corresponds to setting ∆ = 0 and M = 1. As such, it is necessary
to prove that (2.48c) actually implies limk→∞ λ∗k = 0, which is accomplished in the next
lemma.
Lemma 2.4.4 Let ∆ ∈ R+ be a fixed constant to be chosen a priori and let ρ ∈ R[0,1) and
M ∈ Z>0 . If
1
0 ≤ λk ≤ ρ M (λ∗k−1 + ρ
k−1
M
∆),
∀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 when
processes are to be run very fast and involve high sampling frequency. In this context, other
control strategies have been proposed to improve both the design and implementation for
embedded devices.
Generalized predictive control (Camacho and Bordons, 1999), (Clarke et al., 1987) is the
most 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 into
account at each sampling instant. During the last few years some research paid attention
to δ-domain GPC to emphasize the close connection between discrete time and continuous
time theory. Discrete time system analyses is usually done using q forward shift operator
and associated discrete frequency variable z. Although forward shift operator q is the most
commonly used discrete-time operator, in some applications, the forward shift operator can
lead to difficulties (Middleton and Goodwin, 1986). Unfortunately, the discrete domains
are unconnected with the continuous domain; this is because the underlying continuous
domain description cannot be obtained by setting the sample time to zero value. It has
been demonstrated that there is a close connection between continuous time result and δ
representation (Middleton and Goodwin, 1986). In fact, the δ domain description converges
to the continuous time counterpart for sampling period tends to zero.
26
The suggestion of connecting the GPC with the advantages offered by a δ parameterization has been discussed in (Rostgaard et al., 1997) using an emulator in a state-space
approach. The δ domain emulator based GPC has been further investigated in connection to discrete-time GPC in (Sera et al., 2007), using a Diophantine formulation. The
significant relationship in fast sampling is the ratio between the dominant time constant
of the system and the sample time. For instance, many process systems where GPC is often 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 algorithm
in 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 discrete
GPC algorithm designed with Diophantine equations.
Considering the deterministic case of single input single output, δ domain stat- space
model with the known states unaffected by disturbance or noise is:
δxk =Aδ xk +Bδ uk
yk =Cδ xk
,
(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):
δ j xk = A δ j xk +
j−1
X
Aδ j−i−1 Bδ δ i uk ,
(2.52)
i=0
with j = 0, Ny , Ny being the prediction horizon. Using the model (2.51) the following δ
derivative predictors are estimated in the δ domain:
j
j
δ y k = C δ A δ xk +
min{j,N
Xy}−1
Cδ Aδ j−i−1 Bδ δ i uk .
(2.53)
i=0
In a matrix notation the expression of δ derivatives predictors can be written:
ŷδ = f + Guδ ,
(2.54)
where:
uδ = [uk δuk δ 2 uk ......δ Nu −1 uk ]T ,
ŷδ = [yk δyk δ 2 yk ......δ Ny yk ]T .
27
(2.55)
G is the expanded Toeplitz matrix containing the δ based Markov parameters and it has
the dimension :


g(0, 0) . . . g(0, Nu − 1)


..
..
...

,
G=
.
.

g(Ny , 0) . . . g(Ny, Nu − 1)
(2.56)
where
g(j, i) =

 Cδ Aδ j−i−1 Bδ , 0 6 i 6 min{k, Nu } − 1
 0,
,
(2.57)
otherwise
and f is the free response:
h
f = Cδ Aδ 1 . . . Cδ Aδ N y
iT
xk .
(2.58)
The δ GPC controller is implemented following receding horizon strategy and hence
only the first element of control vector needs to be included. Since δ operator offers the
same flexibility and restrictions in modeling as forward shift q operator, it makes possible
to transform q domain control algorithm to the δ domain. The optimal control sequence is
obtained by minimizing an objective function, knowing the reference trajectory rk+i :
J=
Ny
X
2
[ŷk+i − rk+i ] + λ
Nu
X
[∆uk+i−1 ]2 ,
(2.59)
i=1
i=N1
where Nu is control horizon, N1 is minimum costing horizon and λ is the control weighting
factor. In order to obtain the optimal control sequence in δ domain, the set of vectors that
arise in criterion function are obtained from mapping the q domain terms into the δ domain
through 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 have
this torque in an efficient way, a proper model of the driveline is needed for controller design purposes with the aim of lowering emissions, reducing fuel consumption and increasing
comfort. 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 are
presented. First, an electro-hydraulic valve-clutch system is presented, followed by three
driveline models: a drive shaft model, a flexible clutch and drive shaft model, and a continuous 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 an
Delta GPC controller are presented as driveline control strategies. Starting from the models
presented in this chapter, in what follows, more complex driveline models are developed and
also the control strategies presented in this chapter are applied in order to improve overall
performances.
29
30
Chapter 3
Modeling and Control of an
Electro-Hydraulic Actuated Wet
Clutch
Transmission is one of the most important subsystem of an automotive powertrain, with
the basic function of transferring the engine torque to the vehicle with the desired ratio
smoothly and efficiently. The most common control devices inside the transmission are
clutches and actuators, and considering that the automatic control of the clutch engagement
plays a crucial role in AMT vehicles, in this chapter we deal with the problem of modeling
and controlling an electro-hydraulic actuated wet clutch. First, new input-output and statespace models of an 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
are obtained. Simulators of the developed models are implemented in Matlab, and validated
with data provided from experiments with the real valve actuator on a test bench. The test
bench was provided by Continental Automotive Romania and it includes the Volkswagen
DQ250 wet clutch actuated by the electro-hydraulic valve DQ500. Also, a GPC control
strategy and for PID controllers are applied on the develop models and simulation result are
being discussed.
3.1
Introduction
During the last few years, the interest for automated manual transmission (AMT) systems
has increased due to growing demand of driving comfort. Automated clutch actuation makes
it easier for the driver, particularly in stop and go traffic, and has especially seen a recent
31
growth in the European automotive industry. An AMT system consists of a manual transmission through the clutch disc, and an automated actuated clutch during gear shifts. Some
of AMT’s largest advantages are low cost, high efficiency, reduced clutch wear and improved
fuel consumption.
Automotive actuators have become mechatronic systems in which mechanical components
coexist with electronics and computing devices and because pressure control valves are used
as actuators in many control applications for automotive systems, a proper dynamic model
is necessary. Hydraulic control valves are devices that use mechanical motion to control a
source of fluid power and are used as actuators in many control applications for automotive
systems. They vary in arrangement and complexity, depending upon their function. The
many types of valves available are best classified according to their function. Three broad
functional types can be distinguished: directional control valves, pressure control valves and
flow control valves. Pressure control valves act to regulate pressure in a circuit and may be
subdivided 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 flow
to maintain the set pressure. Pressure reducing valves, which are normally open, close to
maintain a minimum pressure by restricting flow in the line. Because control valves are the
mechanical (or electrical) to fluid interface in hydraulic systems, their performance is under
scrutiny, especially when system difficulties occurs. Therefore knowledge of the performance
characteristics of valve is essential.
3.2
Modeling of an Electro-Hydraulic Actuated Wet
Clutch as a Subsystem of an Automated Manual
Transmission
Control valves are the mechanical (or electrical) to fluid interface in hydraulic systems,
and the knowledge of their performance characteristics is essential. Pressure control valves
employ feedback and may be properly regarded as servo control loops. Because of that, a
proper 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, a
new concept of modeling an electro-hydraulic actuated wet clutch is presented. The work
is divided into two sub-chapters, first dedicated to the modeling of a three land three way
solenoid 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 of
an Automated Manual Transmission
Figure 3.1: a) Test bench b) Schematic diagram
with data provided from experiments on a real test bench from Continental Automotive
Romania.
3.2.1
Test Bench Description
The STAT-50.100 test-bench can be used for testing the electro-hydraulic equipment used for
actuation, assignment and control with the maximum nominal diameter DN10 and maximum
pressure of 100 bar. In order to precisely simulate the real working conditions from the
installations where the equipment will be installed, the test-bench has the possibility to
control the three functional parameters (pressure, flow and temperature) to the real field
conditions. Adjustments can be predefine and are automatically made, with the help of an
PLC - 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
The STAT-50.100 test-bench is composed from the following subcomponents: hydraulic
tank, hydraulic oil equipment, three measurements circuits, cooling/heating oil circuit, electrical 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 the
sensors can be easily distinguish. The schematic diagram from Fig. 3.1.b illustrated how
the test bench can be controlled either by computer, throw a software program, or directly
from 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 valve
is modeled, in this sub-chapter, a new concept of modeling a three land three way pressure reducing valve used as actuator for the clutch system in the automatic transmission of
a Volkswagen vehicle is presented. Two models were developed: a linearized input-output
model and a state-space model then implemented in Matlab/Simulink and validated by comparing the results with data obtained on the test-bench provided by Continental Automotive
Romania 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 consideration
that no model and structural description of this valve is found in literature, the electrohydraulic 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 actuator
represented by a pressure reducing valve. This valve releases a pressure depending on the
current i in the solenoid, which will have as consequence the magnetic force Fmag exerted
on the valve plunger, which moves linearly within a bounded region under the effect of this
force. Such a force is generated by a solenoid placed at one boundary of the region. The
magnetic force is a function of the solenoid current and the displacement xs , defined by:
Fmag = f (i, xs ) =
ka i2
2(kb + xs )
2;
LS
di
+ RS i = v,
dt
(3.1)
where ka and kb are constants, Ls is the solenoid induction, Rs the resistance and v is the
supply voltage.
34
Figure 3.2: a) Section through a real three stage pressure reducing valve; b) Three stage valve
schematic representation; c) Charging phase of the pressure reducing valve; d) Discharging
phase of the pressure reducing valve.
35
The pressure to be controlled PR is sensed on the spool end areas C and D and compared
with the magnetic force which actuates on the plunger. The feedback force Ff eed = FC − FD
is the difference between the force applied on the left sensed pressure chamber FC , and the
force applied on the right sensed pressure chamber FD .
The difference in force is used to actuate the spool valve which controls the flow to
maintain the pressure at the set value. In the charging phase, illustrated in Fig. 3.2.c, the
magnetic 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 moved
to 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 describes
the spool motion and the output pressure. This equation of force balance is the same for
both positive and negative displacement of the spool:
Fmag − CPC + DPD = Mv s2 Xs + Ke Xs ,
(3.2)
where PC represents the pressure in the left sensed chamber that acts on the (C) area, PD
represents the pressure in the right sensed chamber that acts on the (D) area, Mv is the
spool mass, Ke = 0.43w(PS0 ˘PR0 ) represents the flow force spring rate calculated for the
nominal pressures PS0 , PR0 , w represents the area gradient of the main orifice, Xs = Xs (s)
is the Laplace transform of the spool displacement and s represents the Laplace operator.
In Fig. 3.2.a, a hydraulic damper that acts to reduce the input pressure spike, which has
negative effects on the output pressure, is also represented.
3.2.2.2
Input-Output Model
The charging phase of the pressure reducing valve has been illustrated in Fig. 3.2.c. A
positive displacement of the spool allows connection between the source and the hydraulic
load, 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 dynamics from the sensed pressure chambers:
QC = K1 (PR − PC ) =
VC
sPC − CsXs ,
βe
(3.3)
QD = K2 (PR − PD ) =
VD
sPD + DsXs ,
βe
(3.4)
36
where K1 , K2 are the flow-pressure coefficients of restrictors, VC , VD are the sensing chamber
volumes and βe represents the effective bulk modulus.
Using the flow through the left and right sensed chambers, the flow through the main
orifice (from the source to the hydraulic load) and the load flow, the linearized continuity
equation at the chamber of the pressure being controlled is:
KC (PS − PR ) − QL − kl PR − K1 (PR − PC ) − K2 (PR − PD ) + Kq Xs =
Vt
sPR ,
βe
(3.5)
where QL is the load flow, KC is the flow-pressure coefficient of main orifice, Kq is the flow
gain of main orifice, kl is the leakage coefficient and Vt represents the total volume of the
chamber where the pressure is being controlled.
These equations define the valve dynamics and combining them into a more useful form,
solving (3.3) and (3.4) w.r.t. PC and PD and substituting into (3.5) yields after some
manipulation:
"
!
1
C
D
s
s
1
(KC PS − QL )
+1
+ 1 + K q Xs 1 +
+
+
−
s+
ω1
ω2
ω1 ω2 Kq Kq
! #
1
C
D
VC ω1 s
2
+
s = PR Kce
+
−
+1 +
ω1 ω2 Kq ω2 Kq ω1
Vt ω3 ω2
s
VC ω1 VD ω2
VD ω2 s
s
s
+1 + 1+
+
+
+1
+1 ,
+
Vt ω3 ω1
ω3 Vt ω3 Vt ω3
ω1
ω2
(3.6)
1
2
where ω1 = βVe K
and ω2 = βVe K
are the break frequency of the left and right sensed chambers,
C
D
ω3 =
βe Kce
Vt
is the break frequency of the main volume and Kce = KC + kl represents the
equivalent flow-pressure coefficient.
Considering that VC Vt and VD Vt , the right side can be factored to give the final
form for the reducing valve model in the charging phase:
"
!
s
1
1
C
D
s
+1
+ 1 + Kq X s 1 +
+
+
−
s+
(KC PS − QL )
ω1
ω2
ω1 ω2 Kq Kq
! #
1
C
D
s
s
s
2
+
+
−
s = PR Kce
+1
+1
+1 .
ω1
ω2
ω3
(3.7)
In the discharging phase, a negative displacement of the pressure reducing valve spool
allows connection between the hydraulic load and the tank, while the channel that connects
the source with the hydraulic load is kept closed.
37
The linearized continuity equations at the sensed pressure chambers for the discharging
phase of the valve, illustrated in Fig. 3.2.d, are:
−QC = K1 (PC − PR ) = −
VC
sPC + CsXs ,
βe
(3.8)
−QD = K2 (PD − PR ) = −
VD
sPD − DsXs .
βe
(3.9)
Using the flow through the left and right sensed chambers, the flow through the main
orifice (from the hydraulic load to the tank) and the load flow, the linearized continuity
equation obtained for the chamber of the pressure being controlled is:
QL + K1 (PC − PR ) + K2 (PD − PR ) − KD (PR − PT ) − kl PR + Kq Xs =
Vt
sPR ,
βe
(3.10)
where KD is the flow-pressure coefficient of main orifice and PT represents the tank pressure.
Combining these equations into a more useful form, solving (3.8) and (3.9) for PC and
PD and substituting into (3.10) yields after some manipulation:
"
!
s
s
1
1
C
D
(KD PT + QL )
+1
+ 1 + K q Xs 1 +
+
+
−
s+
ω1
ω2
ω1 ω2 Kq Kq
! #
1
C
D
VC ω1 s
2
+
+
−
s = PR Kce
+1 +
Vt ω3 ω2
VD ω2 s
s
VC ω1 VD ω2
s
s
+
+1 + 1+
+
+
+1
+1 ,
Vt ω3 ω1
ω3 Vt ω3 Vt ω3
ω1
ω2
(3.11)
In an entire analogue manner, again making the assumption that VC Vt and VD Vt
like for the charging phase model and considering KD = KC the final form for the reducing
valve in the discharging phase was obtained:
"
!
s
s
1
1
C
D
(KD PT + QL )
+1
+ 1 + K q Xs 1 +
+
+
−
s+
ω1
ω2
ω1 ω2 Kq Kq
! #
1
C
D
s
s
s
2
+
+
−
s = PR Kce
+1
+1
+1 .
ω1
ω2
ω3
(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 the
38
Figure 3.3: Transfer function block diagram of the pressure reducing valve.
valve, define the pressure reducing valve dynamics and can be used to construct the transfer
function block diagram represented in Fig. 3.3. Also, the following notation was made:
G(s) =
Kq
Kce
h
1+
1
ω1
1 + ωs2
+ ω12 + KCq − KDq s +
1 + ωs1
1
ω1 ω2
+ KC
− KD
s2
q ω2
q ω1
1 + ωs3
i
.
(3.13)
Considering the resulting force between the magnetic and the feedback force:
F1 = Fmag − CPC + DPD ,
(3.14)
solving PC and PD from the linearized continuity equations (3.3), (3.4) and substituting in
the force balance equation (3.2), the following equation was obtained:
K1 PR + CsXs
K2 PR − DsXs
Fmag − C s
+D s
= Mv s2 Xs + Ke Xs .
ω1 + 1 K1
ω 2 + 1 K2
After some manipulations, where it was considered that ωm =
39
q
Ke
Mv ,
(3.15)
representing the
mechanical natural frequency, and substituting (3.14) into (3.2) yields:

C2
 K1 s F1 −  s
ω1 + 1

+
D2
s
K2 
 Xs
s
ω2 + 1
s2
+ 1 Xs ,
2
ωm
!
= Ke
(3.16)
where

C
+
F1 = Fmag −  s
ω1 + 1

D 
PR ,
s
ω2 + 1
(3.17)
illustrating the closed loop model from Fig. 3.3 for the displacement xs . A switch is used in
order 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 selecting
different disturbances for positive and negative displacement of the spool.
3.2.2.3
State-Space model
Starting from (3.1), (3.2), (3.3), (3.4) and (3.5) for the charging phase of the valve, and
equations (3.1), (3.2), (3.8), (3.9) and (3.10) for the discharging phase of the valve, a statespace model is designed:
ẋ (t) = Ax (t) + Bu (t)
(3.18)
y (t) = Cx (t) + Du (t)
where: x(t) =
h
vs (t) xs (t) PC (t) PD (t) PR (t)
senting the velocity of the spool, y(t) =
h
PS (t) PT (t) QL (t) Fmag (t)





A = βe 




0
1
βe
C
VC
− VDD
0
iT
h
iT
is the state vector with vs (t) repre-
xs (t) PR (t)
iT
is the output vector and u(t) =
is the input vector. The A, C and D matrices are:
− MKv eβe − MCv βe MD
v βe
0
0
0
1
0
−K
0
VC
2
0
0
−K
VD
Kq
Vt
K1
Vt
"
C=
K2
Vt
0
0
K1
VC
K2
VD
1 +K 2 )
− (Kce +K
Vt
0 1 0 0 0
0 0 0 0 1
40





,




#
, D = O2×4 ,
(3.19)
and the matrix B has the B1 expression in the charging phase (for xs > 0) and the B2
expression in the discharging phase (for xs < 0), where:




B1 = βe 



0
0
0
0
KC
Vt
0 0
0 0
0 0
0 0
0 − V1t

1
Mv β e
0
0
0
0



,







B 2 = βe 



0
0
0
0
0
0
0
0
0
0
0
0
0
KC
Vt
1
Vt

1
Mv βe
0
0
0
0



.



(3.20)
This model is more precise because no approximations were used, as for the input-output
model.
3.2.2.4
Simulators for the Pressure Reducing Valve
In this section two simulators that were designed starting from the models previously described are developed in Matlab/Simulink program. Parameter values used for testing in
Simulink are presented in Table A.1. Dimensional parameters were measured directly on
the sectioned valve and the flow coefficients were determined through experiments with the
real valve on a test bench at Continental Automotive Romania. The models were validated
by comparing the results with data obtained on a real test-bench provided by Continental
Automotive Romania.
For testing purposes a Simulink model illustrated in Fig. 3.4 was created, using as input a step signal. The commutation between the charging and the discharging phase was
simulated by a switch that connects different perturbation depending on the value of the
displacement. In Fig. 3.4 two subsystems were used: one noted Model and representing the
transfer functions of the reducing valve model (Fig. 3.5) that were represented as a block
diagram in Fig. 3.3, and one noted as Load Flow representing the load flow.
For a step signal as the magnetic force and a sequence of pulses as the load flow represented in Fig. 3.6, the results obtained for spool displacement and reduced pressure are
presented in Fig. 3.7. For modeling the load flow needed to actuate the clutch, two impulse signals, a positive one and a negative one, for 20 ms with a value of 10−4 m3 /s were
considered, value determined from measurements on the test bench.
The displacement follows the step input behavior while the reducing pressure has almost
the same value like the reference signal. The model shows good performance, being stable
for input signals variations.
1. Input-output model simulation
41
Figure 3.4: Simulink model with step signal input.
Figure 3.5: Simulink transfer functions of the valve model.
42
5
QL
Fmag
Fmg[N], QL [104*m3/s]
4
3
2
1
0
−1
0
0.5
1
1.5
2
2.5
Time [s]
3
3.5
4
4.5
Figure 3.6: Magnetic force and load flow.
7
Displacement
Pressure
Displacement [m], Pressure [bar]
6
5
4
3
2
1
0
−1
0
0.5
1
1.5
2
2.5
Time [s]
3
3.5
4
Figure 3.7: Spool displacement and reduced pressure.
43
4.5
Figure 3.8: Input-output Simulink model.
In order to validate the results obtained for the solenoid valve actuator, a Simulink
model (represented in Fig. 3.8) was created, using a magnetic force as input (Magnet
block). The magnetic force block implements the connection between electric current
trough solenoid and magnetic force generated by the magnetic flux. A force sensor was
utilized to measure the magnetic force and the results were used in a form of a two
dimensional look-up table, designed at Continental Automotive Romania for this type
of valve.
The blocks in the upper part of the model (time_Druck_A, time_Druck_P , time_Strommesszange and time_W eg_M agnet) represent the real data (corresponding to PR , Ps , i and xs respectively) obtained from experiments made on the test bench
at Continental Automotive Romania with this type of valve. The gains in the model
44
8
Fmag
i − curent
7
Fmag [N], curent [A]
6
5
4
3
2
1
0
−1
0
0.2
0.4
0.6
0.8
Time [s]
1
1.2
1.4
1.6
Figure 3.9: Current and magnetic force used as input signals.
are used to transform the values of the parameters that are in international units in
other units used for display (meter to millimeter for the spool displacement and Pascal
to bar for the reduced pressure).
In Fig. 3.8 the saturation block was used to allow only positive values for the magnetic
force and the filter (switching_f ilt) to eliminate the high frequencies caused by the
look-up table. In Fig. 3.9 a real input signal is illustrated, represented either by the
magnetic force or by the current used to obtain the magnetic force through the look-up
table.
The results of the simulations are presented in Figs. 3.10 and 3.11, where the spool
displacement and the reduced pressure were compared with real data obtained from
experiments made on a test bench with the input signal from Fig. 3.9. It can be
seen that the simulated displacement of the spool has even smaller variations than the
measured displacement while the behavior is the same.
Concerning the reduced pressure, the experimental results reveal that the simulated
pressure follows the measured pressure behavior, having in the steady state an irrelevant offset. The amplitude of the simulated pressures variations in steady state is
lower than the amplitude of the measured pressures variations.
2. State-space model simulator
45
1
measured
simulated
0.8
Displacement [mm]
0.6
0.4
0.2
0
−0.2
−0.4
0
0.2
0.4
0.6
0.8
Time [s]
1
1.2
1.4
1.6
Figure 3.10: Compared spool displacements for input-output model .
14
measured
simulated
12
Pressure [bar]
10
8
6
4
2
0
−2
0
0.2
0.4
0.6
0.8
Time [s]
1
1.2
1.4
1.6
Figure 3.11: Compared reducing pressures for input-output model.
46
Figure 3.12: State-space Simulink model.
The state-space model was represented in Simulink as shown in Fig. 3.12, where a
similar switch as in the input-output model was used in order to commutate the reduced
pressure between the charging and the discharging phases. The results obtained for
the spool displacement using the state-space model, are similar to those obtained using
the input-output model and are represented in Fig. 3.13.
Fig. 3.14 illustrates the difference between the simulated and the measured reduced
pressures. It can be seen that the amplitude of the simulated pressures variations in
steady state is lower than the amplitude of the measured pressures variations. Also,
the simulated pressure has in steady state a slight offset.
3.2.3
Modeling of the Electro-Hydraulic Actuated Wet Clutch
System
In previous sub-chapter, two models for an electro-hydraulic actuator were developed: an
input-output model, where simplifications were made in order to obtain a suitable transfer
function to be implemented in Matlab-Simulink, and a state-space model. Starting from the
47
1
measured
simulated
0.8
Displacement [mm]
0.6
0.4
0.2
0
−0.2
−0.4
0
0.2
0.4
0.6
0.8
Time [s]
1
1.2
1.4
1.6
Figure 3.13: Compared spool displacements for state-space model.
14
measured
simulated
12
Pressure [bar]
10
8
6
4
2
0
−2
0
0.2
0.4
0.6
0.8
Time [s]
1
1.2
1.4
1.6
Figure 3.14: Compared reducing pressures for state-space model.
48
Figure 3.15: Charging phase of the actuator-clutch system.
equations that describe the actuator model, an input-output model and a state-space model
for a wet clutch actuated by an electro-hydraulic valve used by Volkswagen for automatic
transmission was developed and it is presented in this sub-chapter.
3.2.3.1
Description of the Valve-Clutch System
Depending on the valve plunger position, there are two phases of the actuator-clutch system:
the charging phase, when the magnetic force is greater than the feedback force and the valve
plunger is moved to the left, connecting the source with the hydraulic actuated clutch (Fig.
3.15), and the discharging phase, when the magnetic force is switched off or has a lower
value than the feedback force so that the valve plunger is moved to the right, connecting the
hydraulic actuated clutch to the tank (Fig. 3.16).
The wet clutch is a chamber with a piston as represented in (Fig. 3.15). In the charging
phase when the valve plunger is moved to the left and the displacement is considered positive,
the oil flows from the source through the valve to the clutch and the piston in the clutch
moves towards the clutch plates compressing them. In the discharging phase, when the valve
plunger is moved to the right and the displacement is negative, the clutch piston moves to
the right and the oil flows from the clutch chamber through the valve to the tank.
For the clutch model, the first equation arises by applying Newton’s second law to the
49
Figure 3.16: Discharging phase of the actuator-clutch system.
forces on the piston, resulting:
AL PL = Mp s2 xp + Kxp ,
(3.21)
where AL is the area of piston, PL the pressure from the piston chamber, xp the piston
displacement, Mp the total mass of the piston and K is the load spring gradient of the
piston.
3.2.3.2
Input-Output Model
Applying the continuity equation to the piston chamber yields:
QL = K3 (PR − PL ) =
VL
sPL + AL sxp ,
βe
(3.22)
for the charging phase of the system, and:
−QL = K3 (PL − PR ) = −
VL
sPL − AL sxp ,
βe
(3.23)
for the discharging phase of the system, where K3 is the flow-pressure coefficient of the pipe
from valve actuator to the clutch and VL is the piston chamber volume.
50
Figure 3.17: Transfer function block diagram of the actuator-clutch system.
Equations (3.22) and (3.23), for the charging phase of the clutch, and (3.22) and (3.24)
for the discharging phase of the clutch, together with the equations that describe the valve
plunger dynamics, define the electro-hydraulic actuated wet clutch system dynamic model.
Starting from these equations, a schematic diagram of the transfer functions for the
actuator-clutch system was created and represented in Fig. 3.17. It can be seen that a switch
block was used in order to commutate between the two phases that describe the functionality
of the actuator, the charging and the discharging phase. The sign of the displacement of the
plunger was used as the switch parameter, thus selecting different perturbations for positive
or negative displacements of the plunger.
3.2.3.3
State-Space Model
Combining the equations (3.1), (3.2), (3.3), (3.4), (3.5), (3.21) and (3.22) that describe the
dynamics of the system in the charging phase, and equations (3.1), (3.2), (3.8), (3.9), (3.10)
, (3.21) and (3.23) that describe the dynamics of the system in the discharging phase, the
state-space model of the electro-hydraulic actuated clutch is design according to:
ẋ (t) = Ax (t) + Bu (t)
y (t) = Cx (t) + Du (t)
51
(3.24)
The state vector is represented by x (t), u (t) is the input vector, and y (t) is considered
to be the output of the system:
iT
u (t) =
h
PS (t) PT (t) Fmag (t)
x (t) =
h
vs (t) xs (t) vp (t) xp (t) PC (t) PD (t) PR (t) PL (t)
y(t) =
h
xs (t) xp (t) PR (t) PL (t).
,
iT
,
(3.25)
iT
Instead of the solenoid current, the magnetic force was used as input because it is a
nonlinear function of the current, the relation between the magnetic force and the current
together with the plunger displacement being implemented in a form of a two dimensional
look-up table designed at Continental Automotive Romania, for this type of valve. The
matrix A is the same for both charging and discharging phase of the actuator-clutch system
and we consider Ksum = Kce + K1 + K2 + K3 in:









A=







0
1
0
0
Cβe
Vc
e
− Dβ
VD
0
0
Ke
−M
v
0
0
0
0
0
0
0
0
1
0
0
Kq βe
Vt
0
AL βe
VL
0
0
0
K
Mt
0
0
0
D
− MCv
Mv
0
0
0
0
0
0
K1 βe
− Vc
0
K2 βe
0
− VD
0
0
K1 βe
Vt
K2 βe
Vt
0
0
0
0
0
0
0
0
AL
Mt
K1 βe
Vc
K2 βe
VD
Ksum βe
− Vt
K3 βe
VL
0
0
0
K3 βe
Vt
− KV3Lβe









,







(3.26)
while different values of the B matrix are used: B1 for the charging phase of the system and
B2 for the discharging phase of the electro-hydraulic actuated clutch:

0
0
0
0
0
0







B1 = 




 K β
 c e
 Vt
0
0
0
0
0
0
0
0
0
1
Mv 

0
0
0
0
0
0
0






 , B2















=







0
0
0
0
0
0
0
0
0
0
0
0
0
0
Kc βe
Vt
0
1
Mv 

0
0
0
0
0
0
0






.







(3.27)
Starting from the equations that illustrates the mathematical model, a block diagram
for the actuator-clutch system was created and represented in Fig. 3.18. It can be seen
that a switch block was used, relative to the sign of the plunger displacement, in order to
commutate between the two phases that describe the functionality of the actuator: B1 for
the charging phase of the system and B2 for the discharging phase.
52
Figure 3.18: State-space block diagram of the actuator-clutch system.
3.2.3.4
Simulators for the Electro-Hydraulic Actuated Wet Clutch
In order to validate the model obtained for the electro-hydraulic actuated clutch, a simulator was designed and developed in Matlab/Simulink starting from the mathematical model
described previously. The parameters used in the model of the valve actuated wet clutch,
summarized in Table A.1, are estimated experimentally at Continental Automotive Romania
using a test-bench, or are already given by the manufacturer. A test-bench which includes
the Volkswagen DQ250 wet clutch actuated by the electro-hydraulic valve DQ500 was provided by Continental Automotive Romania. Experiments made on the test-bench allowed
obtaining the parameters used in simulations for the electro-hydraulic actuator: the volumes
of the actuator chambers, the left and right areas of the valve plunger, the flow-pressure
coefficients. The test-bench also provides measurements for the outputs of the system, represented by the valve plunger displacement and the clutch pressure which are used in order
to validate the implemented simulator.
1. Input-output model simulation
The input-output model of the actuator-clutch system was implemented in Matlab/Simulink like presented in Fig. 3.19 and it can be seen that the switch commutates
between the two phases of the system by connecting different perturbations.
Following experiments made on the test-bench from Continental Automotive Romania,
using as input the solenoid current i, respectively the magnetic force Fmag obtained
through the look-up table, and represented in Fig. 3.9, the real-time clutch pressure
response from Fig. 3.20 was obtained. The input-output model simulation results are
validated due to similar behavior obtained for the pressure in the clutch chamber. Fig.
3.20 shows the comparison between measurements and simulations for the pressure
53
Figure 3.19: Input-output Simulink diagram of the actuator-clutch system.
6
5
Pressure [bar]
4
3
2
simulated PR
simulated PL
measured PL
1
0
−1
0
0.5
1
1.5
2
2.5
3
Time [s]
Figure 3.20: System pressures for the input-output model.
54
3.5
Fmag [N]
5
0
1
1.5
2
2.5
3
2
2.5
3
2
2.5
3
2
2.5
3
Valve
displacement [mm]
Time [s]
1
0
−1
1
1.5
Clutch
displacement [mm]
Time [s]
500
0
1
1.5
Time [s]
Load
3
flow [m /s]
−3
5
x 10
0
−5
1
1.5
Time [s]
Figure 3.21: Input-output system simulation.
obtained in the clutch chamber, along with the simulation results of the reduced pressure. Good agreement between the real-time and simulation results proves that the
model captures the essential dynamics of the system.
The simulation results obtained for the clutch piston displacements are illustrated in
Fig. 3.21, results obtained with the same input signal from Fig. 3.9. It can be seen
that for a positive clutch flow, there are positive displacements both for the valve
plunger and the clutch piston, illustrating the charging phase of the valve when the
clutch chamber is filled with oil, while for a negative clutch flow, there are negative
displacements, illustrating the discharging phase of the valve and the oil going from
the clutch chamber through the valve to the tank.
The value obtained for the valve piston displacement is in the range of [-1,+1] mm,
like it is supposed to be, because the actuator is a closed loop system and the plunger
displacement is restricted by the balance in forces. The clutch piston displacement
55
Figure 3.22: State-space Simulink diagram of the actuator-clutch system.
goes to 400 mm. and it can be seen that it is directly influenced by the value of the
load flow.
2. State-Space Model Simulation
The state-space model of the actuator-clutch system was implemented in Matlab/Simulink
like presented in Fig. 3.22 and it can be seen that the switch commutates between the
two phases of the system by connecting different perturbations.
Like in the case of the input-output model, the state-space model is validated due to
similar behavior obtained for the pressure in the clutch chamber. Fig. 3.23 shows the
comparison between measurements and simulations for the pressure obtained in the
clutch chamber, along with the simulation results of the reduced pressure.
The behavior obtained for the valve plunger displacement and the clutch piston displacement represented in Fig. 3.24, are similar with the behavior obtained for the
input-output model. In the state-space model developed in this paper for the actuator56
3.3 Control of the Electro-Hydraulic Actuated Wet
Clutch as a Subsystem of an Automated Manual Transmission
6
5
Pressure [bar]
4
3
2
simulated PR
simulated PL
measured PL
1
0
−1
0
0.5
1
1.5
2
2.5
3
3.5
Time [s]
Figure 3.23: System pressures for the state-space model.
clutch system, the clutch flow, which is also illustrated in Fig. 3.24, was obtained as a
difference between the pressure from the valve and the clutch pressure.
The value obtained for the valve piston displacement is again in the range of [-1,+1]
mm, because the actuator is a closed loop system and the plunger displacement is
restricted by the balance in forces. On the other hand, the clutch is an open loop
system, with no feedback, resulting a high value of the piston displacement which can
be further limited by designing a proper controller for the electro-hydraulic actuated
wet clutch.
3.3
Control of the Electro-Hydraulic Actuated Wet
Clutch as a Subsystem of an Automated Manual
Transmission
Starting from the electro-hydraulic actuated wet clutch system input-output model presented
in the previous section, an analytical designed PID controller, with the help of the pole
placement method and a GPC controller are presented and the results are compared and
discussed.
57
Fmag [N]
5
0
1
1.5
2
2.5
3
2
2.5
3
2
2.5
3
2
2.5
3
Valve
displacement [mm]
Time [s]
1
0
−1
1
1.5
Clutch
displacement [m]
Time [s]
500
0
1
1.5
Time [s]
Load
flow [m3/s]
−3
5
x 10
0
−5
1
1.5
Time [s]
Figure 3.24: State-space system simulation.
3.3.1
Generalized Predictive Control
Predictive control techniques are of a particular interest from the point of view of both
broad applicability and implementation simplicity, being applied on large scale in industry
processes, having good performances and being robust at the same time.
Consider the plant described by the CARIMA (Controlled AutoRegressive Integrated
Moving Average) model (Camacho and Bordons, 2004):
A z −1 y (k) = B z −1 u (k − 1) +
e (k) C z −1
D (z −1 )
,
(3.28)
where e(k) is white noise with zero mean value, u is the input voltage, y(k) is the clutch
displacement, A z −1 and B z −1 are the system polynomials with the degrees nA and nB ,
and C z −1 = 1 and D z −1 = 1 − z −1 are the disturbances polynomials.
58
The input output model of the electro-hydraulic actuated wet clutch was implemented
in Matlab/Simulink, and, in order to apply a control strategy, the input of the system was
considered to be the v voltage, while the output of the system is the clutch piston displacement. The actuator-clutch system was identified with an ARX equivalent one employing the
simulation model, utilizing as input a PRBS (PseudoRandom Binary Sequence) signal:
A z −1 = 1 − 1.781z −1 + 0.8039z −2 ,
(3.29)
B z −1 = 0.00003312z −1 + 0.0001122z −2 ,
and the disturbances polynomials were considered to be C z −1 = 1 and D z −1 = 1 − z −1
for obtaining a zero steady-state error.
The prediction model is given by
ŷ (k + j|k) = Gj z −1 D z −1 z −1 u (k + j) +
+
Hj z −1 D z −1
C (z −1 )
u (k − 1) +
Fj z −1
C (z −1 )
,
(3.30)
y (k)
with j = hi, hp, where hi is the minimum prediction horizon and hp is the prediction horizon.
u(k + j − 1 |k ), j = 1, hc is the future control, computed at time k and ŷ (k + j|k) are the
predicted values of the output, hc being the control horizon.
The two Diophantine equations presented in (Camacho and Bordons, 2004) are used to
determine the polynomials Fj z −1 , Gj z −1 and Hj z −1 .
Considering as inputs D(z −1 )u(k) and collecting the j-step predictors in a matrix notation, the prediction model can be written as
ŷ = Gud + ŷ0 ,
(3.31)
where ŷ represents the free response and matrix G is given in (Camacho and Bordons, 2004).
The objective function is based on the minimization of the tracking error and on the
minimization of the controller output, the control weighting factor λ being introduced in
order to make a trade-off between these objectives
J = (Gud + ŷ0 − w)T (Gud + ŷ0 − w) + λud T ud ,
(3.32)
subject to D(z −1 )u(k + i) = 0 for i ∈ [hc, hp − 1], where w is the reference trajectory vector
with the components w(k +j |k ), j = hi, hp. Minimizing the objective function (∂J/∂ud = 0),
the optimal control sequence yields as
u∗d = GT G + λIhc
59
−1
GT [w − ŷ0 ] .
(3.33)
−3
x 10
4
reference
displacement
3.5
Displacement [mm]
3
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
Time [s]
0.6
0.7
0.8
0.9
1
Figure 3.25: GPC results.
Using the receding horizon principle and considering that γj , j = hi, hp are the elements
of the first row of the matrix GT G + λIhc
D z
−1
u (k) =
hp
X
−1
GT , the following control algorithm results:
γj [w (k + j |k ) − ŷ0 (k + j |k )].
(3.34)
j=hi
The determined controller of the electro-hydraulic actuated wet clutch system was implemented in Matlab/Simulink. A reference signal was applied for the clutch piston displacement and it was desired that the system tracks the reference signal as fast as possible. The
following figure shows the reference signal and the controlled output of the system.
It can be seen that, when using this predictive control strategy, the system tracks the
reference signal in a very precise way, having no steady state error.
3.3.2
PID Control
A proportional-integral-derivative controller (PID controller) is a generic control loop feedback mechanism (controller) widely used in industrial control systems. The PID controller
calculates an "error" value as the difference between a measured process variable and a desired set-point, and then the controller attempts to minimize the error by adjusting the
process control inputs. By tuning the three parameters in the PID controller algorithm,
the controller can provide control action designed for specific process requirements. The
response of the controller can be described in terms of the responsiveness of the controller
to an error, the degree to which the controller overshoots the set point and the degree of
system oscillation.
There are three different design methods categories: empirical methods, formally known
as the Ziegler-Nichols method, analytical methods, and optimization methods. Starting
60
from the electro-hydraulic actuated wet clutch system model, an analytical designed PID
controller, with the help of the pole placement method is presented.
Having the discrete model of the plant given by:
Gf (z −1 ) =
B(z −1 ) 0.00003312z −1 + 0.0001122z −1
,
=
A(z −1 )
1 − 1.781z −1 + 0.8039z −2
(3.35)
and taking into account the performances imposed to the automated system (ζ = 0.707
and T = 0.08s), we obtain the coefficients α1 = - 1.9001, α2 = 0.9048, α3 = 0.7022 and the
characteristic polynomial in the form of:
Pcd (z) = (z 2 + α1 z + α2 )(z − α3 )(z − α4 ).
(3.36)
Next step is to build the characteristic polynomial of the closed loop system:
Pc0 (z −1 ) = P (z −1 )A(z −1 ) + Q(z −1 )B(z −1 ).
(3.37)
Because m = 2, the system is undetermined and we select the PID controller with filtering
of the derivative component:
GR (z −1 ) =
q0 + q1 z −1 + q2 z −2
Q(z −1 )
.
=
P (z −1 ) (1 − z −1 )(1 − αd z −1 )
(3.38)
The tuning parameters of the discrete PID controller are chosen by solving Pc0 (z −1 ) =
Pcd (z −1 ). This yields:





1
b1 0 0
a1 − 1 b 2 b 1 0
a2 − a1 0 b 2 b 1
−a2
0 0 b2





αd
q0
q1
q2


 
 
=
 
−a1 + 1 − 3.3044
4.0661 − a2 + a1
−2.2074 + a2
0.4461



,

(3.39)
and the following tuning parameters for the discrete PID controller are obtained:





αd
q0
q1
q2


 
 
=
 
- 0.5295
184.8801
- 364.0633
182.0876



.

(3.40)
The PID controller was then implemented in Matlab/Simulink for controlling the electrohydraulic actuated wet clutch system. A reference signal was applied for the clutch piston
displacement and it was desired that the system tracks the reference signal as fast as possible.
Also, different controllers were experimentally tuned using the relay tuning method, and the
61
−3
4
x 10
3.5
Displacement [mm]
3
2.5
2
reference
frequency−response method
indicial response method
relay method
pole placement method
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
Time [s]
0.6
0.7
0.8
0.9
1
Figure 3.26: PID controller results.
Ziegler-Nichols method based on indicial-response and on frequency-response. In Fig. 3.26
the responses obtained with the four methods are presented.
It can be seen that when using the pole placement method to design the PID controller the
set-point response has small overshoot with no steady-state oscillations, but with a downside
concerning the rising time. When using the relay method or the frequency-response method,
the system response has a higher value of the overshoot, and some steady-state oscillations,
but has a faster rising time. A faster response is obtain when using the indicial-response
method, the smallest overshoot and almost no steady-state oscillations.
Comparing the result obtained with the GPC strategy and the PID control strategies,
it can be concluded that the best results are obtained when using the predictive control
strategy because the system precisely tracks the reference signal, and has no overshoot.
3.4
Conclusions
In this chapter a new concept of modeling an electro-hydraulic actuated wet clutch system
is proposed and a GPC and a PID controller are designed in order to control the output of
the electro-hydraulic actuated clutch system: the clutch piston displacement.
First, a new concept of modeling a solenoid valve actuator used in the automotive control
systems is proposed. This pressure reducing valve is not a typical one, and it was not modeled in the literature. Two models were developed: a linearized input-output model, where
simplifications were made in order to obtain a suitable transfer function to be implemented
in Simulink and to obtain an appropriate behavior for the outputs, and a state-space model
with no simplifications. Two simulators are implemented and the models were validated by
comparing the results with data obtained on the real test-bench provided by Continental Automotive Romania. It can be concluded that the simulators have good results illustrated by
62
3.4 Conclusions
the similar behavior obtained for the spool displacement and the reduced pressure compared
with the measured values on the test bench.
Next, starting from the actuator models, two models for an electro-hydraulic actuated
clutch system used in the automotive control systems for automatic transmission were developed: a linearized input-output model and a state-space model. The models were validated
by comparing the results with data obtained on the real test-bench provided by Continental Automotive Romania, which includes a Volkswagen wet clutch actuated by an electrohydraulic valve. Again, it can be concluded that the simulators have good results illustrated
by the similar behavior obtained for clutch pressure compared with the real measured values.
Finally, a GPC and a PID controller were designed in order to control the output of the
electro-hydraulic actuated clutch system: the clutch piston displacement, and the simulation
results of the controllers are presented and discussed. Comparing the result obtained with
the GPC strategy and the PID control strategies, it can be concluded that the best results
are obtained when using the predictive control strategy because the system precisely tracks
the reference signal, with no overshoot.
The results obtained were published as journal papers:
• (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 Electromagnetic
Valve Actuator. Buletinul Institutului Politehnic din Iasi, vol. Tome LV (LIX), Fasc.
2, pages 9–28, 2009.
• (Balau et al., 2011a) A. E. Balau, C. F. Caruntu and C. Lazar. Simulation and Control
of an Electro-Hydraulic Actuated Clutch. Mechanical Systems and Signal Processing,
vol. 25, pages 1911–1922, 2011.
as well as conference papers:
• (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 for
Automotive Applications. In 18th IEEE International Conference on Control Applications, 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. New
Model for Predictive Control of an Electro-Hydraulic Actuated Clutch. In The 18th
International Conference on FUEL ECONOMY, SAFETY and RELIABILITY of MOTOR VEHICLES (ESFA 2009), Bucharest, Romania, 2009.
63
• (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 for
Automotive Control Systems. In The 1tth International Conference on Control Systems
and Computer Science, Bucharest, Romania, 2009.
• (Balau et al., 2010) A. E. Balau, C. F. Caruntu and C. Lazar. State-space model of an
electro-hydraulic actuated wet clutch. In IFAC Symposium Advances in Automotive
Control, Munchen, Germany, 2010.
• (C.Lazar, Caruntu and Balau, 2010) C. Lazar, C. F. Caruntu and A. E. Balau. Modelling and Predictive Control of an Electro-Hydraulic Actuated Wet Clutch for Automatic 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. Networked Predictive Control Strategy for an Electro-Hydraulic Actuated Wet Clutch. In
IFAC Symposium Advances in Automotive Control, Munchen, Germany, 2010.
• (Balau and C.Lazar, 2011a) A. E. Balau, C. F. Caruntu 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.
64
Chapter 4
Two Inertias Driveline Model
Including Backlash Nonlinearity
In this chapter, starting from the Continuous Variable Transmission Drive Shaft model presented in Chapter 2.3.3, two models for automotive driveline including backlash nonlinearity
are proposed. First, a PWA and a nonlinear state-space model for a Continuous Variable
Transmission (CVT) driveline with backlash are proposed. Simulators are developed in Matlab/Simulink for the two driveline models and two control strategies presented in Chapter
2.4 are applied. A horizon-1 MPC is applied on the PWA model while a PID cascade based
controller is applied for the nonlinear model designed to reduce the fuel consumption by using
the optimal fuel efficiency curve in the modeling phase. Next, three models are implemented
for an Automated Manual Transmission (AMT) driveline based on the M220 Industrial plant
emulator: a rigid driveline model, a flexible driveline model and a flexible driveline model
including also backlash nonlinearity. Then, real time experiments are conducted on the
developed models, while applying a horizon-1 MPC controller.
4.1
Introduction
Backlash is a common problem in powertrain control because it introduces a hard nonlinearity in the control loop for torque generation and distribution. This phenomenon occurs
whenever there is a gap in the transmission link which leads to zero torque transmitted
through the shaft to the wheels. When the backlash gap is traversed the impact results in
a large shaft torque and sudden acceleration of the vehicle. Engine control systems must
compensate for the backlash with the goal of traversing the backlash as fast as possible.
65
Two Inertias Driveline Model Including Backlash Nonlinearity
In an automotive powertrain, backlash and shaft flexibility results in an angular position
difference between wheels and engine. The modeling of mechanical systems with backlash
nonlinearities is a topic of increasing interest, because a backlash can lead to reduced performances and can even destabilize the control system. Also, it can have as consequence low
components reliability and shunt and shuffle. In order to model the mechanical system with
backlash, two different operational modes must be distinguished: backlash mode (when the
two mechanical components are not in contact) and contact mode (when there is a contact
between the two mechanical components resulting in a moment transmission).
4.2
Driveline Models
In this section two models for automotive driveline including backlash nonlinearity are proposed for a Continuous Variable Transmission driveline, and then, three models are implemented for an Automated Manual Transmission driveline based on the M220 Industrial plant
emulator.
4.2.1
CVT Driveline Model with Backlash Nonlinearity
Starting from the relations that describe the dynamical behavior for each subsystem of an
automotive conventional powertrain with CVT, presented in Chapter 2.3.3, two models for a
driveline with backlash are developed: a PWA model and a nonlinear state-space model. The
driveline is composed from the same components: engine, continuously-variable transmission, final reduction gear, flexible drive-shaft and driving wheel. In addition to the driveline
components presented in (Mussaeus, 1997), the backlash nonlinearities are considered between the flexible drive-shaft and the wheel. A schematic representation of an automotive
driveline with backlash nonlinearities is illustrated in Fig. 4.1.
The equations that describe the functional operation of the internal combustion engine,
CVT, FRG and FDS are same equations presented in Section 2.3.3 from (2.31) to (2.35).
In order to obtain the new model, it is useful to define the torsion angle of the flexible
drive-shaft θs = θ3 − θ4 and the backlash angle θb = θ4 − θw , resulting the angular velocities:
d
θs (t) = ωs (t)
dt
d
θb (t) = ωb (t) .
dt
66
(4.1)
Figure 4.1: Schematic representation of an automotive driveline with backlash.
Now, because of the backlash, the equation (2.36) becomes:
Tk (t) = kd θs (t) ,
Tb (t) = dd ωs (t) .
(4.2)
The dynamical behavior of the wheel is described by the same equations (2.37) and (2.38).
The developed driveline model is a nonlinear one, with two different operating modes:
contact and non-contact. In the non-contact mode the two mechanical components of the
system are not in contact and the torque is not transmitted from the final reduction gear
through the final drive-shaft to the wheels, while in the contact mode there is a connection
between the mechanical components of the system and the torque is transmitted to the
wheels. The backlash angle is constant whenever the contact mode is active.
The input-output model of the CVT driveline with backlash is now given by the equations:
67
Te = Γ(ωe ),
d
Je ωe (t) = Te (t) − T1 (t) ,
dt
ω2 = iCV T ωe ,
ηCV T
T2 =
T1 ,
iCV T
ω3 = iF RG ω2 ,
ηF RG
T2 = r1 T1 ,
T3 =
iF RG
T3 (t) = Tk (t) + Tb (t) ,
(4.3)
Tk (t) = kd θs (t) ,
Tb (t) = dd ωs (t) ,
d
θs (t) = ωs (t)
dt
d
θb (t) = ωb (t) ,
dt
d
Jv ωw (t) = T3 (t) − Tload (t) .
dt
4.2.1.1
PWA Model
Starting from the input-output model of the CVT driveline with backlash given by (4.3),
and considering a fixed transmission ratio, a new PWA state-space model can be developed:
(
ẋ = Ax + Bu + f
,
y = Cx
(4.4)
where:
x=
h
u=
h
θs ωe ωw θb
i
iT
,
(4.5)
Te .
The model has four states, represented by the drive shaft angle, the engine angular
velocity, the wheel angular velocity and the backlash angle. The input is represented by the
engine torque and the affine term is represented by the load torque. The system outputs are
all the system states.
68
For the contact mode using ωb (t) = 0, yields:


Aco = 


1
r2
0

− Jkedr1 − Jedrd1 r2 − Jdee
kd
Jv
0
dd
Je r1
11
− Jddv − dwJ+c
v
dd
Jv r2
0

−1
0


 1 


Je  , f
 co
 0 
Bco = 



=


0
0

0
0
Troll
Jv
0


,





Cco = 

1
0
0
0
0
1
0
0
0
0
0
0

0
0
1
0
0
0
0
1


,


(4.6)



,

where c11 is an approximation parameter used in order to obtain an linear approximation of
the aerodynamic drag torque Tairdrag = c11 ωw .
In a similar way, the non-contact mode is characterized by transmitting no torque from
the FDS to the wheels, the state-space representation being realized for T3 (t) = 0, with:



Anc = 



− kddd
0
0
de
0 − Je
0
0
0 − dJwv
kd
1
−1
dd
r1
0
0
0
0




,


(4.7)
and matrices Bnc , fnc and Cnc are the same as for the contact mode.
4.2.1.2
Nonlinear Model
If a continuous variable transmission ratio is considered, starting from the input-output
model of the CVT driveline with backlash given by (4.3), a new nonlinear state-space model
can be developed:
ẋ = f (x, u)
y = h (x, u)
(4.8)
,
where
iT
x=
h
ωe ωw θs θb
u=
h
iCV T Te Tload
y=
h
ωe ωw T3
iT
,
iT
,
(4.9)
.
The model has four states, represented by the engine angular velocity, the wheel angular
velocity, the drive shaft angle and the backlash angle. The inputs are represented by the
continuously-variable transmission ratio, the engine torque and by the load torque. The
69
engine angular velocity and the wheel angular velocity, as well as the final drive shaft torque
T3 are the system outputs.
For the contact mode, the equations that describe the dynamical behavior of the driveline
are given by:
!
de
r3 dd
r3 kd
1
iF RG r3 dd 2
iCV T +
ωe +
ωw iCV T −
θs iCV T + Te ,
ω̇e = −
Je
Je
Je
Je
Je
iF RG dd
dd + dw
kd
1
ω̇w =
ωe iCV T −
ωw + θs − Tload ,
Jv
Jv
Jv
Jv
θ̇s = iF RG ωe iCV T − ωw ,
(4.10)
θ̇b = 0,
and for the non-contact mode are given by:
de
1
ωe + Te ,
Je
Je
dw
1
ω̇w = − ωw − Tload ,
Jv
Jv
kd
θ̇s = − θs ,
dd
kd
θ̇b = θs + iF RG ωe iCV T − ωw ,
dd
ω̇e = −
where r3 =
iF RG
ηF RG ηCV T
(4.11)
, and T3 = dd iF RG iCV T ωe − dd ωw + kd θs .
The optimized driveline was designed to reduce the fuel consumption by using the optimal
fuel efficiency curve in the modeling phase.
4.2.2
AMT Driveline Model with Backlash Nonlinearity
An AMT driveline model with backlash nonlinearity is obtained using as plant an electromechanical apparatus that can be transformed into a variety of dynamic configurations which
represent important classes of "real life" systems. The Model 220 apparatus represents many
such physical plants including rigid bodies, flexibility in drive shafts, gearing and belts. Important non-ideal properties such as backlash, drive friction, and disturbances can be easily
introduced and removed. This allows the plants to be characterized in a controlled manner
and facilitates study of control approaches to mitigate their effects.
4.2.2.1
Rigid Driveline Model
Starting from the structure of the driveline illustrated in Fig. 4.2, the equations that describe
the dynamics of a rigid driveline are developed. The structure is composed by two inertias,
70
Figure 4.2: Rigid driveline model.
one corresponding to the engine and one corresponding to the wheels, and a speed reduction
(SR) assembly. No flexibilities are assumed, and the two inertias are rigidly coupled together.
The overall driveline gear ratio between the two inertias is given by itot =
rw rSR1
rSR2 re
so that
θe = itot θw , while the partial gear ratio between the SR assembly and the engine inertia is
given by: ip = rSR1
re , so that θe = ip θSR .
The total inertia reflected to the engine is
−2
Je∗ = Je + Je i−2
p + Jw itot ,
(4.12)
with the total reflected damping coefficient:
d∗e = de + dw i−2
tot .
(4.13)
In a similar way, the total inertia reflected to the wheel is
Jw∗ = Je i2tot + Je (
itot 2
) + Jw ,
ip
(4.14)
with the total reflected damping coefficient:
d∗w = de i2tot + dw .
(4.15)
Finally, the equations of motion are obtained, as reflected at the engine or at the wheel
respectively:
Je∗ θ̈e + d∗e θ̇e = Te ,
Jw∗ θ̈w + d∗w θ̇w = itot Te .
71
(4.16)
Figure 4.3: Flexible driveline model.
Starting from one of the equations (4.16) a state space model of the rigid driveline can
be obtained:
(
ẋ = Ax + Bu
,
y = Cx
(4.17)
where the system input is u = Te and the system states are x = [θe ωe ] when the inertia is
considered reflected at the engine, and x = [θw ωw ] when the inertia is considered reflected
at the wheel.
The system matrices are given by:
"
A=
0
1
∗
0 − Jd ∗
#
"
,B =
0
#
1
J∗
,C =
h
i
1 0 ,
(4.18)
where d∗ and J ∗ stand for the corresponding total damping and inertia reflected to the
engine or the wheels.
4.2.2.2
Flexible Driveline Model
An approximation of the plant with flexibility in the driveline is shown in Fig. 4.3. In
systems where the flexible element contains a significant fraction of the plant damping, it
may be useful to include this damping in the plant model.
By defining the torsional spring constant of the drive shaft:
kd = 2kl rw 2 ,
(4.19)
and also the drive shaft damping constant:
dd = 2dl rw 2 ,
72
(4.20)
the following equations of motion are obtained for the engine and the wheel inertia, respectively:
dd
dd
1
1
)θ̇e −
θ̇e + kd ( 2 θe −
θw ) = Te ,
2
itot
itot
itot
itot
dd
1
Jw θ̈w + (dw + dd )θ̇w −
θ̇w + kd (θw −
θe ) = 0.
itot
itot
∗
θ̈e + (de +
Jep
(4.21)
∗ = J + J i−2 .
The total inertia reflected to the engine was calculated as Jep
e
p p
Starting from these equations, that describe the dynamics of the engine and wheel inertias, a state-space model of the system is obtained:
(
ẋ = Ax + Bu
,
y = Cx
(4.22)
where:
x=
h
u=
h
iT
θe ωe θw ωw
,
(4.23)
i
Te .
The system matrices are given by:
0




A=




B
− J ∗kid2
ep tot
dd
i2
tot
∗
Jep
de +
−
0
0
kd
∗
itot Jep
dd
∗
itot Jep
0
0
0
1
kd
Jw itot
dd
itot Jw
kd
Jw
d
− dwJ+d
w
0
 1
 J∗
ep
=

 0
0
4.2.2.3
1



,C




=


1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1




,



(4.24)



.

Flexible Driveline Model with Backlash
In order to obtain a more accurate and complex model, backlash nonlinearity can be added
to the existing driveline model. Starting from equations (4.21), that describe the dynamics
of a flexible driveline, the mathematical model of the two inertia system including backlash
and drive shaft flexibility was obtained. The equations that describe the engine and the
wheel dynamics are:
dd
dd
1
1
)θ̇e −
θ̇e + Fbklsh (θw − θe )kd ( 2 θe −
θw ) = Te ,
2
itot
itot
itot
itot
dd
1
Jw θ̈w + (dw + dd )θ̇w −
θ̇w + Fbklsh (θw − θe )kd (θw −
θe ) = 0,
itot
itot
∗
Jep
θ̈e + (de +
73
(4.25)
where Fbklsh is the backlash force that equals 0 when the systems is in the non-contact mode,
and equals 1 if the system is in the contact mode.
Starting from these equations, that describe the dynamics of the engine and wheel inertias, a state-space model of the system is obtained:
(
ẋ = Ax + Bu
,
y = Cx
(4.26)
where:
x=
h
u=
h
θe ωe θw ωw
iT
,
(4.27)
i
Te .
The input of the system is represented by the engine torque, and the system states and
outputs are the torsional angle at the engine and wheel inertias, the engine angular velocity
and the wheel angular velocity.
The system has two working modes: the contact mode, when the torque is transmitted
to the wheels, and the non-contact mode, when no torque is transmitted from the engine to
the driven wheels. The system matrices for the contact mode are:
0




Aco = 




1
− J ∗kid2
ep tot
dd
i2
tot
∗
Jep
de +
−
0
0
kd
∗
itot Jep
dd
∗
itot Jep
0
0
0
1
kd
Jw itot
dd
itot Jw
kd
Jw
d
− dwJ+d
w
0
 1
 J∗
ep
Bco = 

0

0




 , Cco




=

1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1




,



(4.28)



,

while for the non-contact mode:



Anc = 



0
− J ∗kid2
ep tot
0
1
0
0
0
kd
Jw itot
0
kd
∗
itot Jep
0
kd
Jw

0

0 

,
1 

0
and matrices Bnc and Cnc are the same as for the contact mode.
74
(4.29)
Figure 4.4: Nonlinear CVT driveline structure - Simulink representation.
4.3
For these driveline models, two control strategies are applied: a PID cascade-based controller
and a horizon-1 MPC based on flexible control Lyapunov functions. The horizon-1 MPC
controller is developed starting from the control strategy that was previously described in
Chapter 2.4.
4.3.1
PID Cascade-Based Driveline Controller
In order to control the nonlinear model of the CVT driveline with backlash, a PID based cascade controller proposed in (Mussaeus, 1997) and presented in Chapter 2.4 is implemented.
First, the input-output model given by equations (4.3) is implemented in Matlab/Simulink,
with separate blocks representing the driveline components, like illustrated in Fig. 4.4.
In order to validate the model, a step signal affected by white noise is given as reference
for the developed driveline model, like illustrated in Fig. 4.5.
The control command applied to the driveline model is represented in Fig. 4.6, while the
resulting wheel speed is illustrated in Fig. 4.7. The input command was applied on time
t = 0 and the sampling time was set to be Ts = 0.1.
75
Figure 4.5: Validation structure - Simulink representation.
1.4
1.2
1
icvt
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
Time [s]
3
3.5
4
4.5
5
4.5
5
Figure 4.6: Input command - icvt .
14
12
Wheel speed [rpm]
10
8
6
4
2
0
−2
0
0.5
1
1.5
2
2.5
Time [s]
3
3.5
Figure 4.7: Wheel speed.
76
4
Figure 4.8: PID cascade based control structure - Simulink representation.
Figure 4.9: Torque controller - Simulink representation.
The design of the cascade structure is presented in Fig. 4.8, and implies that the inner
loop has a faster dynamics than the external loop. The inner loop, which controls the final
drive-shaft (FDS) torque through the continuously-variable transmission (CVT) gear ratio,
has as input a desired torque given by the controller from the external loop. The speed
controller has to bring the wheel speed at a desired value by sending reference values to
the controller for the inner loop, which has to complete the closed-loop performances before
getting another reference value.
The inner loop controller was designed firstly, considering the driveline model as the plant
and it is represented in Fig. 4.9. The control structure involves a PI controller designed to
bring the drive-shaft torque T3 at a desired value, and has the following parameters: P = 0.03
and I = 0.2.
Then, using the inner closed-loop control system as the plant, the external loop controller
was designed, and it is represented in Fig. 4.10. A PID controller is designed to bring the
wheel speed at a desired value, and the control parameters are: P = 50, I = 1 and D = 2.
Also, a feed-forward controller is used in order to compensate the disturbances introduced
by the load torque (aero-dynamical drag and rolling resistance) and it is incorporated in the
Speed controller.
77
Figure 4.10: Speed controller - Simulink representation.
4.3.2
Horizon -1 MPC Controller
The models considered for control are the PWA models proposed in this chapter for the CVT
driveline and for the AMT driveline, and this control strategy based on flexible Lyapunov
functions has the ability to enforce constraints on states, inputs and outputs.
To obtain a discrete-time PWA model, each affine subsystem in (4.4) (for the CVT
driveline) or (4.26) (for the AMT driveline) is discretized with sampling period Ts using the
Euler transform, which yields
m m
m m
m
xm
k+1 = Adi xk + Bd uk + fd
if xk ∈ Ωi ,
(4.30)
m
m
for all k ∈ Z+ , where Am
di and Bd are the corresponding discretized system matrices, fd is
m
the discretized affine term and xm
k , uk are the state and input of the system at time instant
k ∈ Z+ . The active mode i is selected for the discrete-time PWA system and equals 1 for
the non-contact mode and 2 for the contact mode.
Torque rate constraints are important to allow full usage of the airflow to maintain the
torque reserve, so that torque variations can be actuated instantaneously. The engine torque
rate constraint is
∆
−Te∆ ≤ ∆um
k ≤ Te ,
∀k ∈ Z≥1 ,
(4.31)
m
m
∆
where ∆um
k := uk − uk−1 and Te is the maximum allowed increase (decrease) in torque at
each sampling instant.
In what follows, a coordinate transformation is performed in (4.30) to translate the
problem into stabilization of the origin, i.e.,
ss
uk = um
k −u .
ss
xk = xm
k −x ,
(4.32)
The following system description results:
xk+1 = Adi xk + Bdi uk + fdi
78
if xk ∈ Ωi .
(4.33)
Here Adi and Bdi are the discretized and transformed system matrices and fdi are the
discretized and transformed affine terms. Notice that the transformed PWA model (4.33)
has zero as an equilibrium within the contact mode, i.e., fd2 = 0. Also, observe that uss can
be interpreted as the feed-forward component of the control action.
As such, the objective can be formulated as asymptotic stabilization of the desired steadystate point while satisfying the required constraints.
Consider the following cost function to be minimized
J1 (xk , uk , λk ) := JMPC (xk , uk ) + J(λk )
:= kPx xk+1 k∞ + kRuk k∞ + kGλk k∞ ,
(4.34)
subject to constraints:
0 − uss ≤ uk ≤ Temax − uss ,
(4.35)
−Te∆ ≤ ∆uk ≤ Te∆ .
The cost J(·) is chosen as required in Problem 2.4.2 and the matrices Px ∈ Rpx ×n and
R ∈ Rr×n are chosen as full-column rank matrices of appropriate dimensions. Consider the
following infinity-norm based CLF
V (x) = kP xk∞ ,
(4.36)
where P ∈ Rp×n is a full-column rank matrix to be determined, e.g., using techniques from
(M.Lazar, 2006a). This function satisfies (2.45) with α1 (s) =
√σ s,
p
where σ is the smallest
singular value of P , and α2 (s) = kP k∞ s. For xk ∈ Ωi , substituting (4.30) and (4.36) in
(2.48b) yields
kP (Adi xk + bdi uk + fdi )k∞ ≤ ρkP xk k∞ + λk
(4.37)
where xk , P and ρ ∈ R[0,1) are known at k ∈ Z+ . In what follows it is shown that for a unitary
horizon, the above MPC optimization problem can be formulated as a linear program (LP)
via a particular set of equivalent linear inequalities, despite switching dynamics, while for any
other larger horizon it would lead to a mixed integer linear programming (MILP) problem.
By definition of the infinity norm, for kxk∞ ≤ c to be satisfied, it is necessary and sufficient
to require that ±[x]j ≤ c for all j ∈ Z[1,n] . So, for (4.37) to be satisfied, it is necessary and
sufficient to require
±[P (Adi xk + bdi uk + fdi )]j ≤ ρkP xk k∞ + λk
(4.38)
for all j ∈ Z[1,p] . As such, solving Problem 2.4.2, which includes minimizing the cost function
(4.34), can be reformulated as the following problem.
79
Problem 4.3.1 Measure xk , determine the active mode i and
min 1k + 2k + 3k
(4.39)
±[Px (Adi xk + bdi uk + fdi )]j ≤ 1k , ∀j ∈ Z[1,px ] ,
(4.40a)
uk ,λk
subject to (2.48c), (4.35), (4.38) and
±Ruk ≤ 2k ,
(4.40b)
Gλk ≤ 3k .
(4.40c)
Problem 4.3.1 is a linear program, since xk and λ∗k−1 are known at time k ∈ Z≥1 and
thus, all constraints are linear in uk , λk and εlk , l ∈ Z[1,3] . The horizon-1 MPC algorithm is
stated next.
Algorithm 4.3.2
At each sampling instant k ∈ Z+ :
Step 1: Measure the current state xk and obtain the active mode i: non-contact or contact
mode;
Step 2: Solve the LP Problem 4.3.1 and pick any feasible control action, i.e., uf (xk );
Step 3: Implement uk := uf (xk ) as control action.
2
The fact that only a feasible, rather than optimal, solution of Problem 4.3.1 is required in
Algorithm 4.3.2, can reduce the execution time.
4.4
Simulation Results
In this section simulations results for the horizon-1 MPC proposed control strategy are
presented for the CVT driveline with backlash nonlinearity. The models and the controllers
were implemented in Matlab and simulation results are discussed.
4.4.1
Simulator for the PWA Model of the CVT Driveline
In what follows, the simulator for the PWA state-space model for the CVT driveline with
backlash was implemented in Matlab/Simulink and represented in Fig. 4.11 with the aim of
validating the developed model and controlling the wheel speed of the vehicle. A horizon-1
MPC controller is implemented, that has as inputs the wheel speed reference signal, the
system states and the operating mode of the system (contact or non-contact). The controlled output of the horizon-1 MPC controller, represented by the engine torque, goes to
80
4.4 Simulation Results
Figure 4.11: Horizon-1 MPC - Simulink structure.
the driveline model and the system outputs and are obtained. The active working mode and
the evolution of the CLF relaxation variable λ∗k and the corresponding upper bound defined
by (2.48c) are also obtained as outputs of the controller.
The horizon-1 predictive controller uses the following weight matrices of the cost (4.34):
Px = 1.2 · I4 , R = 0.0001 and G = 1. The technique presented in (M.Lazar, 2006a) was used
for the off-line computation of the infinity norm based local CLF V (x) = kP xk∞ for ρ = 0.99
and the PWA model of the driveline in closed-loop with uk := Ki xk if xk ∈ Ωi , i ∈ Z[1,2] . The
following matrices were obtained


P =


0.1241 −0.0004 −0.0945 0.0073
−0.0590 −0.0296 0.0217 −0.0027
0.0039
0.0000 −0.0282 0.2238
0.0079 −0.0000 −0.0615 −0.1651
K1 =
0.0054 0.0001 0.0000 0.0210
K2 =
0.0094 0.2067 0.0002 −0.0000



,

(4.41)
,
The system output represented by the wheel speed is illustrated in Fig. 4.12. It can be
seen how the output of the system reaches the reference speed 20 km/h in 5 seconds, with
no overshoot. It can also be seen that in the first 0.12 seconds, the wheel speed equals zero.
This is because the system is in the non-contact mode as illustrated in Fig. 4.13, and no
torque is transmitted to the wheels. When the system enters contact mode, the torque is
transmitted to the driven wheels and the speed begin to increase.
A switch was used in order to commutate between the two operating modes function of
the backlash angle, the threshold used for the switching between the non-contact mode and
the contact mode being chosen as α = 2 rad.
81
25
Wheel speed [km/h]
20
15
10
5
0
0
1
2
3
Time [s]
4
5
6
5
6
5
6
2
Operaing mode
1.8
1.6
1.4
1.2
1
0
1
2
3
Time [s]
4
Figure 4.13: Operating mode.
2.5
Backlash angle [rad]
2
1.5
1
0.5
0
0
1
2
3
Time [s]
4
Figure 4.14: Backlash angle.
82
600
Engine speed [rpm]
500
400
300
200
100
0
0
1
2
3
Time [s]
4
5
6
Figure 4.15: Engine torque.
The last figure illustrates the engine speed and again, it can be seen that even if in the
first 0.12 seconds there is a high value of the engine speed, it is not transmitted to the wheels
and the vehicle speed remains equal with zero. Only in the contact mode the wheel speed
increases with the engine speed.
4.4.2
Simulator for the Nonlinear Model of the CVT Driveline
In what follows, the simulator for the nonlinear state-space model of the CVT driveline with
backlash was implemented in Matlab/Simulink with the aim of validating the developed
model. Then a PID cascade based control strategy is applied in order to control the wheel
speed of the vehicle.
The parameter values used to implement the driveline model in Simulink, are presented
in Table A.2 in the Appendix. For simulation purposes, the optimal fuel-efficiency curve of
the engine 1.6i ES CVT of a Honda Civic vehicle, as shown in Fig. 4.16, was used. This
figure also illustrates different optimal fuel-efficiency curves corresponding to other vehicles.
The cascade structure controller presented in Fig. 4.8, is applied. The inner loop controls
the final drive-shaft torque and has as input a desired torque given by the controller from the
external loop. The control structure involves a PI controller with the following parameters:
P = 0.03 and I = 0.2. The speed controller has to bring the wheel speed at a desired value by
sending reference values to the controller for the inner loop, and the PID control parameters
are: P = 50, I = 1 and D = 2.
A switch was used in order to commutate between the two operating modes function of
the backlash angle, the threshold used for the switching between the non-contact mode and
the contact mode being chosen as α = 2 rad. Fig. 4.17 illustrates the output of the system,
83
180
160
Engine torque [Nm]
140
120
100
80
60
Honda Civic 1.6i ES CVT
Geo Metro 1.0i
Saturn 1.9i
Toyota Prius 1.5i 1NZ−FXE
Toyota Prius 1.8i 2ZR−FXE
40
20
0
0
100
200
300
400
500
600
Engine speed [rpm]
700
800
900
Figure 4.16: Optimal fuel-efficiency curve.
represented by the wheel speed relative to the input reference signal. It can be seen that
the controllers have good results represented by the similar behavior of the two signals. The
initial difference is due to the backlash nonlinearity and after the contact mode is reached
the wheel speed tracks the desired reference trajectory.
The final drive-shaft torque and the desired reference value, given by the speed controller,
are illustrated in Fig. 4.18, with a detail on the shaft torque when the switching from the
non-contact to the contact mode occurs.
When the system is in the non-contact mode there is no torque transmitted to the driving
wheels because of the backlash. After the system enters the contact mode, there is a large
shaft torque for a small period of time, but after that a normal value for the FDS torque is
reached.
Also, in Fig. 4.19 the engine speed characteristic is illustrated, which is in accordance
with the gear ratio of the continuously-variable transmission represented in Fig. 4.20. It
can be seen that when the wheel speed remains constant, the engine speed increases and the
CVT gear ratio decreases and when the wheel speed grows the engine speed decreases and
the CVT gear ratio increases, in order to maintain the engine characteristics on the optimal
fuel efficiency curve.
Both controllers (the horizon-1 MPC based on FCLF and the cascade-based PID) developed for the CVT driveline with backlash nonlinearity have good results, illustrated by the
84
50
6
Reference speed
Wheel speed
45
5
40
35
Wheel speed [km/h]
Wheel speed [km/h]
4
30
25
20
3
2
15
10
1
5
0
0
10
20
Time [s]
30
0
40
0
0.5
1
Time [s]
1.5
2
600
600
500
500
400
400
Final drive−shaft torque [Nm]
Final drive−shaft torque [Nm]
Desired torque
FDS torque
300
200
200
100
100
0
300
0
5
10
15
20
Time [s]
25
30
35
40
0
0
Figure 4.18: Final drive-shaft torque.
85
0.5
Time [s]
1
4000
3500
2500
2000
1500
1000
500
0
0
5
10
15
20
Time [s]
25
30
35
40
35
40
Figure 4.19: Engine speed.
1.5
CVT gear ratio
Engine speed [rpm]
3000
1
0.5
0
0
5
10
15
20
Time [s]
25
Figure 4.20: CVT ratio.
86
30
4.5 Real Time Experiments
simulation results. The backlash influence is clearly seen in the wheel speed behavior, which
doesn’t increase until the backlash angle reaches the threshold value. After the backlash
angle is passed, the system output follows the desired reference given for the wheel speed
with, having no steady-state error and no overshoot.
4.5
Real Time Experiments
In this sections the real time results obtained on the M220 Industrial plant emulator are
presented. The values of the parameters used in all experiments conducted on the emulator
are given in Table A.3 in the Appendix.
4.5.1
System Overview
The experimental control system is comprised of the three subsystems. The first of these is
the electromechanical plant which is represented in Fig. 4.21 and consists of the emulator
mechanism, its actuator and sensors. The design features brush-less DC servo motors for
both drive and disturbance generation, high resolution encoders, adjustable inertias and
changeable gear ratios. It also has the possibility to introduce coulomb and viscous friction,
driveline flexibility, and backlash (M220, 1995).
Figure 4.21: M220 Industrial plant emulator schematic structure.
The next subsystem is the real-time controller unit which contains the digital signal
processor (DSP) based real-time controller, servo/actuator interfaces, servo amplifiers, and
auxiliary power supplies. The DSP is capable of executing control laws at high sampling
rates allowing the implementation to be modeled as continuous or discrete time. The controller also interprets trajectory commands and supports such functions as data acquisition,
trajectory generation, and system health and safety checks. A logic gate array performs
87
motor commutation and encoder pulse decoding. Two optional auxiliary digital-to-analog
converters (DAC’s) provide for real-time analog signal measurement. This controller is representative of modern industrial control implementation.
The final subsystem is the executive program that runs on a PC under the DOS or
WindowsTM operating system. This program is the user’s interface to the system and supports features as: controller specification, data acquisition, trajectory definition, plotting
and system execution commands.
4.5.2
Electromechanical Plant Description
The electromechanical plant, shown in Fig. 4.22 is designed to emulate a broad range of
typical servo control applications. The Model 220 apparatus consists of a drive motor (servo
actuator) which is coupled via a timing belt to a drive (engine) disk with variable inertia.
Another timing belt connects the drive disk to the speed reduction (SR) assembly while a
third belt completes the drive train to the load (wheel) disk.
The load and drive disks have variable inertia which may be adjusted by moving (or
removing) brass weights and also speed reduction is adjusted by interchangeable belt pulleys
in the SR assembly. Backlash may be introduced through a mechanism incorporated in the
SR assembly, and flexibility may be introduced by an elastic belt between the SR assembly
and the load disk. The drive disk moves one-for-one with the drive motor so that its inertia
may be thought of as being collocated with the motor. The load inertia however will rotate
at a different speed than the drive motor due to the speed reduction. Also, drive flexibility
and/or backlash may exist between it and the drive motor and hence its inertia is considered
to be non-collocated with the motors (M220, 1995).
A disturbance motor connects to the load disk via a 4:1 speed reduction and is used
to emulate viscous friction and disturbances at the plant output. A brake below the load
disk may be used to introduce Coulomb friction. Thus friction, disturbances, backlash, and
flexibility may all be introduced in a controlled manner. These effects represent non-ideal
conditions that are present to some degree in virtually all physically realizable electromechanical systems.
All rotating shafts of the mechanism are supported by precision ball bearings. Needle
bearings in the SR assembly provide low friction backlash motion (when backlash is desired).
High resolution incremental encoders couple directly to the drive (θe ) and load (θw ) disks
providing position (and derived rate) feedback. The drive and disturbance motors are electrically driven by servo amplifiers and power supplies in the Controller Box. The encoders
are routed through the Controller box to interface directly with the DSP board via a gate
array that converts their pulse signals to numerical values.
88
Figure 4.22: Industrial plant emulator M220.
4.5.3
Experimental Results
A horizon-1 MPC scheme described in Chapter 4.3.2 is implemented in Matlab/Simulink and
real time experiments are conducting by used of Real Time Windows Target, that allows
external connection to the M220 Plant Emulator.
First experiments are made considering the rigid driveline model give by equations (4.17)
and (4.18) and results of two different ways of controlling the system are presented: relative to
the engine inertia and relative to the wheel inertia. Controlling of the system relative to the
engine inertia is referred as collocated control, since the sensor and the actuator are rigidly
coupled and hence kinematically lie at the same location. The second way of controlling
the system, relative to the wheel inertia, is referred to as non-collocated control, because
it potentially involves flexibility, backlash and drive nonlinearity between the actuator and
sensor.
The horizon-1 MPC scheme described in Chapter 4.3.2 was implemented in Matlab /
Simulink and illustrated in Fig. 4.23 for the collocated controller and in Fig. 4.24 for the
non-collocated controller.
In the case of the collocated controller, the horizon-1 MPC controller has as inputs, the
position reference for the engine inertia, and the two system states, represented by engine
inertia position and rotational velocity. The output of the horizon-1 MPC controller, the
uctrl command, is represented by the engine torque which, multiplied by the DAC gain kc,
89
Figure 4.23: Rigid driveline collocated controller - Simulink structure.
is the entry of the ECPDSP Driver. The ECPDSP Driver represents the interface with the
M220 Industrial plant emulator and his outputs are multiplied by the controller software
gain ks and by the Encoder gain ks. Because we are referring to the collocated controller,
the ECPDSP Driver first output represents the engine inertia position.
The discrete time system matrices considered for control are given by:
"
Ad =
1 0.0040
0 0.9951
#
"
, Bd =
0
0.2373
#
.
(4.42)
The horizon-1 predictive collocated controller uses the following weight matrices of the
cost (4.34): Px = 0.6 · I2 , R = 0.3 and G = 1. The technique presented in (M.Lazar, 2006a)
was used for the off-line computation of the infinity norm based local CLF V (x) = kP xk∞
for ρ = 0.9977 and the following matrices ware obtained:
−22.5260 −6.4727
3.4248
29.6188
P=
!
,
Z1 =
−2.5188 −3.4977
,
Z2 =
−0.0018 −0.0022
.
(4.43)
In the case of the non-collocated controller, the horizon-1 MPC controller has as inputs,
the position reference for the wheel inertia, and the two system states, represented by wheel
90
inertia position and rotational velocity. The controller strategy is the same as for the collocated controller, with the difference that, in this case, the output of the ECPDSP Driver is
now the wheel position.
"
Ad =
1 0.0040
0 0.9951
#
"
, Bd =
0
0.9494
#
.
(4.44)
The horizon-1 predictive collocated controller uses the following weight matrices of the
cost (4.34): Px = 0.6 · I2 , R = 0.3 and G = 1. The technique presented in (M.Lazar, 2006a)
was used for the off-line computation of the infinity norm based local CLF V (x) = kP xk∞
for ρ = 0.9977 and the following matrices ware obtained:
−24.6153 −4.9667
5.2057
28.5535
P=
!
,
Z1 =
−2.8976 −3.2531
,
Z2 =
−0.0149 −0.0170
.
(4.45)
The collocated controller is designed in order to control the engine inertia position at the
value of 40000 counts. The non-collocated controller is designed in a similar way, in order
to control the wheel inertia position at the value of 10000 counts, which multiplied by the
gear ratio gives 40000 counts at the engine inertia.
Figure 4.25 shows the comparison between the results obtained for the engine inertia,
with blue, and for the wheel inertia, with red. It can be seen that for a rigid driveline the
results obtained by using collocated and non-collocated control are similar.
Backlash flexibility is a common problem in mechanical drives and exist to some extend
in nearly all gear boxes and in many mechanical couplings. The schematic structure of the
backlash mechanism is illustrated in Fig. 4.26. The upper member and the lower member of
the backlash mechanism come together and are coupled through the backlash contact boss.
When backlash influence is desired, by means of the backlash adjust screw, the backlash
angle can be easily adjusted.
Following experiments are conducted in order to observe the influence of the backlash
on the controlled system. First, a 4 degrees backlash angle read at the wheel inertia is
introduced.
Figure 4.27 shows the comparison between the results obtained for the engine inertia,
with blue, and for the wheel inertia multiplied by the gear ratio, with red. In this case, it
can be seen that for a 4 degree backlash angle, the results obtained by using collocated and
non-collocated control are not similar anymore. When using collocated control, the engine
inertia position behaves in a similar way as when no backlash was introduced, while for the
91
Figure 4.24: Rigid driveline non-collocated controller - Simulink structure.
4
x 10
4
3.5
Position [counts]
3
2.5
2
1.5
1
Engine inertia
Wheel inertia
0.5
0
0
1
2
3
4
5
Time [s]
6
7
8
9
10
Figure 4.25: Rigid driveline collocated and non-collocated control.
92
Figure 4.26: Backlash mechanism structure.
non-collocated control, the wheel inertia has a small overshoot and a steady-state error of
300 counts.
Another experiment was conducted, this time by introducing a backlash angle of 8 degrees, read at the wheel inertia. Results showed in Fig. 4.28 illustrates an even more
deteriorate response obtained for the wheel inertia when using non-collocated control, with
a higher overshoot and a steady-state error of 840 counts.
Another influence on the driveline is given by the drive shaft flexibility. In order to
observe this influence on the controlled system, a flexible drive belt was introduced between
the speed reduction assembly and the wheel inertia. Considering the flexible driveline model
given by equations (4.22), (4.23) and (4.46) the horizon-1 MPC controller was designed in
order to control the engine inertia position at the value of 40000 counts, and it is illustrated in
Fig. 4.29. Because the MPC controller has the ability to control all the system outputs, the
wheel inertia position will be also controller at the value of 10000 counts, which multiplied
by the gear ratio gives 40000 counts at the engine inertia. The horizon-1 MPC structure is
the same as the structures previously presented for collocated and non-collocated controllers,
with the difference that both wheel and engine positions are controlled, and are obtained
from the ECPSDP driver.
93
4
4.5
x 10
4
Position [counts]
3.5
3
2.5
2
1.5
1
Engine inertia
Wheel inertia
0.5
0
0
1
2
3
4
5
Time [s]
6
7
8
9
10
Figure 4.27: Rigid driveline with 4 degrees backlash angle collocated and non-collocated
control.
4
5
x 10
4.5
4
3.5
3
2.5
2
1.5
Engine inertia
Wheel inertia
1
0.5
0
0
1
2
3
4
5
6
7
8
9
10
Figure 4.28: Rigid driveline with 8 degrees backlash angle collocated and non-collocated
control.
94
4.6 Conclusions




Ad 1 = 



Bd = 

0.9983
−0.8337
0.0006
0.3094
0
1.5876
0
0
0.0040 0.0067
0.9935 3.3346
0.0000 0.9975
0.0012 −1.2375
0.0000
0.0133
0.0040
0.9877



,

(4.46)



.

Px = 0.6 · I4 , R = 0.3 and G = 1. The technique presented in (M.Lazar, 2006a) was used for
the off-line computation of the infinity norm based local CLF V (x) = kP xk∞ for ρ = 0.999
and the following matrices ware obtained:




P =
Z1 =
Z2 =
−20.7286 −0.1065 −6.2412 −27.8105
52.5911
0.2103 −181.6309 14.0624 

,
3.1011 −0.9612 −23.5795
5.1142 
20.5990
0.0782
25.1895
−0.3502

−1.6346 −0.1370 3.6686 −0.4652
−4.7965 −0.1285 13.5224 −1.3199
(4.47)
,
.
The system response for the engine inertia, when considering the drive shaft flexibility, is
represented with blue in Fig. 4.30. It can be seen that, compared with the system response
of the rigid driveline from Fig. 4.25, a −700 counts steady-state error appear when drive
shaft flexibility are taken into account. Concerning the wheel inertia position represented
with blue in Fig. 4.31, it can be seen that it has a smaller steady state error, of about −80
counts.
A final experiment was conducted by considering the flexible drive shaft and backlash
flexibility together. With the flexible drive belt introduced between the speed reduction
assembly and the wheel inertia, a 4 degrees backlash angle read at the wheel inertia is
introduced. The results obtained for this configuration are presented in red in Fig. 4.30
for the engine inertia, and in Fig. 4.31 for the wheel inertia. System responses for the two
inertias are both slower when backlash is present, and also have the steady-state errors given
by the drive shaft flexibility.
4.6
Conclusions
In this chapter two different driveline structures including backlash nonlinearity are modeled:
a CVT driveline and an AMT driveline.
95
Figure 4.29: Flexible driveline controller - Simulink structure.
4
4
x 10
Engine inertia position [counts]
3.5
3
2.5
2
1.5
1
Flexible driveline model
Flexible driveline model with backlash
0.5
0
0
1
2
3
4
5
Time [s]
6
7
8
9
10
Figure 4.30: Flexible driveline with backlash control - engine inertia position.
96
4.6 Conclusions
11000
10000
Wheel inertia position [counts]
9000
8000
7000
6000
5000
4000
3000
Flexible driveline model
Flexible driveline model with backlash
2000
1000
0
0
1
2
3
4
5
Time [s]
6
7
8
9
10
Figure 4.31: Flexible driveline with backlash control - wheel inertia position.
First, two models for a conventional driveline composed of engine, continuous variable
transmission, final reduction gear, final drive-shaft and driving wheels are developed, including the backlash nonlinearities: a PWA and a nonlinear state-space model.
The PWA model was designed using a fixed transmission ratio and a simulator was
implemented in Matlab. Then, an horizon-1 MPC controller was applied in order to control
the wheel speed of the proposed model.
For the nonlinear model, the optimized driveline was designed to reduce the fuel consumption by using the optimal fuel efficiency curve in the modeling phase. Also, a PID
based cascade controller was implemented in Matlab/Simulink. 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.
The controllers have good performances, illustrated by the simulation result, despite the
nonlinearities introduced by the backlash.
Next, three models were implemented for an Automated Manual Transmission (AMT)
driveline based on the Industrial plant emulator M220: a rigid driveline model, a flexible
driveline model and a flexible driveline model including also backlash nonlinearity. Then, real
time experiments were conducted on the implemented models in order to test the influences
given by drive shaft flexibility and backlash angle, while applying a horizon-1 MPC controller.
It can be seen that, when considering a rigid driveline, the backlash angle influences the
system behavior, and the results obtained by using collocated and non-collocated control
are not similar anymore like in the case when no backlash was considered. When using
collocated control, the engine inertia position behaves in a similar way as when no backlash
was introduced, while for the non-collocated control, the wheel inertia has an small overshoot
and a steady-state error proportional with the backlash angle. Also, drive shaft flexibility has
97
an influence on the the system outputs, resulting in a steady-state error. When considering
the drive shaft flexibility and the backlash together, the system outputs have a steady-state
error given by the drive shaft flexibility and also a slower response given by the backlash
influence.
The results presented in this report were summarized in form of a conference paper:
• (Caruntu, Balau and C.Lazar, 2010b) C. F. Caruntu, A. E. Balau and C. Lazar. Cascade based Control of a Drivetrain with Backlash. In 12th International Conference on
Optimization of Electrical and Electronic Equipment, Brasov, Romania, 2010.
98
Chapter 5
Three Inertias Driveline Model
Including Clutch Nonlinearity
Driveability is one of the most important factors in modern vehicles. This chapter deals
with the problem of damping driveline oscillations in order to improve passenger comfort.
These oscillations, also called “shuffles”, occur during gearshift, while traversing backlash
or when tip-in and tip-out maneuvers are performed. Stating from the models presented
in Chapter 2, two driveline new models with three inertias are proposed: a state-space
piecewise affine model of an automated manual transmission (AMT) driveline and a statespace piecewise affine model of a double clutch transmission (DCT) driveline, all of them
taking into consideration the drive shafts as well as the clutch flexibilities. Also, both
of them consider four working modes in the modeling phase of the clutch: three phases
of the closed mode, and, as a novelty, the opened mode of the clutch. Next, four control
strategies are proposed for the developed models: PID control, explicit MPC control, horizon1 MPC control based on flexible Lyapunov functions and delta GPC control. Simulators
are implemented in Matlab/Simulink, and different test are conducted in order to see if
the proposed control schemes can handle both the performance/physical constraints and
the strict limitations on the computational complexity corresponding to vehicle driveline
oscillations damping.
5.1
Introduction
Recent studies in automotive engineering are exploring various engine, transmission and
chassis models and advanced control methods in order to increase overall vehicle performance,
fuel economy, safety and comfort. Driveability, the ability to quickly respond to drivers
99
Three Inertias Driveline Model Including Clutch Nonlinearity
Figure 5.1: Three inertia driveline model.
action and a high degree of driving comfort are expected in a modern vehicle. Because of
the elastic components of the driveline, mechanical resonance occurs. This phenomenon is
known as driveline oscillations or “shuffles”. When driveline oscillations are induced, the
driveability of the vehicle is reduced, because the oscillations are transmitted via the chassis
to the driver. The objective is to increase the passenger comfort by reducing the oscillations
that occur during gearshift, while traversing backlash or when tip-in and tip-out maneuvers
are performed.
5.2
Driveline Models
In order to develop a controller, an accurate driveline model is required to predict the vehicle’s
response to a torque input. The model can then be used to design and simulate the control
system performance. Starting from the Drive shaft model and the Flexible Clutch and Drive
shaft model presented in Chapter 2.3, two models of an AMT driveline are derived in this
chapter: an affine model and a new piecewise affine model, as well as a new piecewise affine
model of an DCT driveline. All of them have three rotational inertias and consider that the
driveline flexibility is introduced by the drive shafts and also by the clutch.
5.2.1
AMT Affine Model
A three inertias model has been derived from the laws of motion (Kiencke and Nielsen, 2005),
(Grotjahn et al., 2006), (Van Der Heijden et al., 2007), and it takes into consideration the
clutch flexibility together with the driveshaft flexibility. The first inertia corresponds to the
engine, the second one includes the inertia of the gearbox and the inertia of the final drive,
and the last inertia corresponds to the wheel and vehicle mass, as it can be seen in Fig. 5.1.
The driveline fundamental equations are derived by using the generalized Newton’s second
100
law of motion. The equation of motion for the first rotational mass yields:
J1 ω̇e = Te − (kc (θe − θt it ) + dc (ωe − ωt it )) − de ωe .
(5.1)
The engine is described as an ideal torque source Te with a mass moment of inertia
J1 = Je and a viscous friction coefficient de .
The equation of motion of the second body can be derived as:
J2 ω̇t =it (kc (θe − θt it ) + dc (ωe − ωt it ))−
1
θt
ωt
− d2 ωt − (kd ( − θw ) + dd ( − ωw )),
if
if
if
(5.2)
J
where J2 = Jt + if fif is the second inertia, composed by the gearbox inertia Jt and the final
d
drive inertia Jf , with damping d2 = dt + if fif composed by the transmission and final drive
damping.
The last equation of motion corresponds to the wheels and vehicle body and can be
written as:
J3 ω̇w =kd (
ωt
θt
− θw ) + dd ( − ωw ) − dw ωw − Tload ,
if
if
(5.3)
2
where J3 = Jw + mCOG rstat
is the wheel and vehicle.
The load torque is modeled as:
Tload = Troll + Tangle + Tairdrag ,
(5.4)
where Troll is the rolling torque of the tires, Tangle is the torque due to the road slope and
Tairdrag is the aerodynamic drag torque of the vehicle body, which are defined as:
Troll = cr1 mCOG g cos(χroad )rstat ,
Tangle = mCOG g sin(χroad )rstat ,
(5.5)
Tairdrag = 0.5ρair Af cd vv2 rstat .
Because the purpose of the modeling approach is to control the driveline oscillations,
certain physical details may be neglected and some assumptions may be made in order to
reduce the model complexity. In the modeling phase only the terms given by the rolling
torque Troll and by the aerodynamic drag torque Tairdrag are considered, assuming that the
road slope gradient is equal to zero. Also, instead of the nonlinear function that describes the
aerodynamic drag torque, a linear approximation will be used with cr2 as an approximation
parameter
Tairdrag = cr2 ωw .
101
(5.6)
Considering the torsional angle between engine and transmission, the torsional angle between transmission and wheels, the angular speed of the engine, the angular speed of the
transmission and the angular speed of the wheel as state variables, i.e.,
x1 = θe − θt it
θt
x2 = − θw
if
(5.7)
,
x3 = ωe
x4 = ωt
x5 = ωw
the affine state-space model
ẋ(t) = Ac x(t) + bc u(t) + fc ,
(5.8)
consist of the system matrices Ac , bc and the affine term fc , i.e.,

0
0




− Jk1c
Ac = 

 kc it
 J2

0
−it
0
0
1
0
0
−dc −de
J1
dc it
J2
− ifkJd2
kd
J3

0


0


 0 
 1 

bc = 
 J1  , f c


 0 
0
1
if
dc it
J1
−dnot
J2
dd
i f J3



=



0
0
0
0
−Troll
J3
0
−1
0
dd
i f J2
−dw −dd −cr2
J3





,




(5.9)




,



(5.10)
where dnot = dc it 2 + d2 + idd2 . Notice that although there are three angles, only two states
f
are introduced as only the angle difference is relevant. The input of the system is the engine
torque u = Te and the controlled outputs are represented by the engine and wheel angular
speeds.
5.2.2
AMT Piecewise Affine Model
Starting from the affine state-space model, and taking into consideration different working
modes of the clutch, a new piecewise affine model of the driveline is developed. The equations
that describe the driveline dynamics are the same as for the affine model, but the difference
is that distinct values are used for the clutch stiffness and damping, according to the current
mode of operation for the clutch.
102
Figure 5.2: Clutch functionality a) stiffness characteristic; b) clutch springs
When studying a clutch in detail, it is seen that the torsional flexibility is a result of an
arrangement with smaller stiffness springs in series with springs with higher stiffness, like
it can be seen in Fig. 5.2. Fig. 5.2.a) illustrates the clutch stiffness characteristic and the
spring arrangement of the clutch is presented in Fig. 5.2.b). The reason for this arrangement
is vibration insulation. There are two working modes for the clutch: open and closed, and
three different phases of the closed mode. In the open mode, there is no connection between
the engine and the rest of the driveline, so no torque is transmitted from the engine towards
the wheels (kc1 = 0), while in the closed mode the engine torque is transmitted through the
driveline to the wheels. The closed mode is defined by three phases, corresponding to the
stiffness of the springs that are being compressed. In the first phase of the closed mode, the
springs with the smaller stiffness begin to compressed (kc2 ), and the torque is transmitted to
the driveline. In the second phase, the springs with the smaller stiffness are fully compressed,
and the stiffer springs begin to compress (kc3 ). Finally, in the third phase of the closed mode,
the stiffer springs are also fully compressed, and there results a mechanical stop (kc4 ).
Having the same states, input and outputs as for the affine model, the piecewise affine
state-space model is obtained:
ẋ(t) = Aci x(t) + bc u(t) + fc
if x(t) ∈ Ωi ,
(5.11)
where x := (x1 , . . . , x5 )> ∈ R5 and i ∈ I := Z[1,4] . Here i denotes the active mode at time
t ∈ R+ , Aci ∈ R5×5 , bc ∈ R5×1 are the system matrices and fc ∈ R5×1 is the affine term.
The collection of sets {Ωi | i ∈ I} defines a partition of the state-space X ⊆ R5 such that
∪i∈I Ωi = X and int(Ωi ) 6= ∅ for all i ∈ I. The regions are defined as follows:


Ω1




Ω
:= {x ∈ R5 | x3 ≤ ωeclosing },
- open
5
closing
&
|x1 | ≤ θ1 }, - closed I
2 := {x ∈ R | x3 > ωe


Ω3 := {x ∈ R5 | x3 > ωeclosing & θ1 < |x1 | ≤ θ2 }, - closed II




Ω4 := {x ∈ R5 | x3 > ωeclosing & θ2 < |x1 |},
- closed III
103
,
(5.12)
Figure 5.3: AMT clutch switching logic.
where the region Ω1 corresponds to the open mode of the clutch, while regions Ω2 , Ω3 and Ω4
corresponds to the three phases of the closed mode. ωeclosing is the engine closing speed and θ1
and θ2 are threshold values for the torsional angle between the engine and the transmission,
which are used to pass from one working mode of the clutch to another. The switching
logic is illustrated in Fig. 5.3 and it can be seen that the clutch remains in the open mode
while the engine speed doesn’t reach the closing speed value. When this closing speed value
is reached, selection between the three phases of the closed mode is made relative to the
torsional angle between the engine and the transmission. While this angle is smaller than
the threshold value θ1 , the system is in the first phase of the closed mode. When the angle
passes the threshold value θ1 but is still smaller than the second threshold value θ2 , the
system is in the second phased of the closed mode. Finally, when the angle also passes this
second threshold value θ2 , the system enters the third phase of the closed mode.
Note that when a transition from the open mode to the closed mode occurs, the following
reset condition must be imposed:
∀t1 ∈ R+ , ∀t2 ∈ R>t1 , if x(τ ) ∈ Ω1 , ∀τ ∈ R[t1 ,t2 )
and x(t2 ) ∈ Ω2 , set x1 (t2 ) := 0.
(5.13)
As the engine angle θe tends to infinity in the open mode, so the state x1 tends to infinity,
a synchronization of the engine angle and the transmission angle must be attained at the
moment 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 the
104
open mode and the three phases of the closed mode of the clutch, respectively:

0
0




− kJci1
Aci = 

 kci it
 J2

0
0
0
1
0
0
−Dsum1
J1
dci it
J2
− ifkJd2
kd
J3
0
−it
1
if
dci it
J1
−Dsum2
J2
dd
if J3
0
−1
0
dd
i f J2
−dwheel
J3





,




(5.14)
with Dsum1 = dci + de , Dsum2 = dci it 2 + d2 + idd2 , dwheel = dw + dd + cr2 and the corresponding
f
clutch stiffness kci and clutch damping dci .
The novelty of this model consist of the opened working mode of the clutch, that is added
to the three different phases of the closed mode.
5.2.3
Dual Clutch Transmission Driveline
In recent years the driveline oscillation problem has received an increasing interest due
to the introduction of dual-clutch transmission, commonly abbreviated to DCT (sometimes
refereed to as twin-clutch gearbox or double clutch transmission). DCT utilizes two separated
clutches for odd and even gear sets. It can fundamentally be descried as two separate
manual transmissions contained within one housing, and working as one unit. These dry
clutch transmissions offer improved fuel economy, easier packaging and reduced weight with
respect to the standard wet-clutch planetary gear transmissions. Also the torque converter,
which provides a smooth hydrodynamic coupling between the engine and the transmission
and which is present in standard automatic transmissions, can be removed. However, the
absence of the torque converter makes the torque transfer path from the engine to the wheels
entirely mechanical, which means that disturbances, including the inherent reciprocating
behavior of the engine, have more impact on the driveline.
A new piecewise affine model of a driveline complex system including engine, flexible
clutch, Dual Clutch Transmission, flexible shafts and wheels, was developed in this section
taking into account, for each clutch, the regions defined by (5.12), and is represented in Fig.
5.4. The equations that describe the dynamics of the system are the same with the ones
describing the piecewise affine three inertias driveline model including clutch nonlinearity
previously presented, with the difference that there are two clutches: one for the odd gears
and one for the even gears, so the transmission ratio it stands for it1 in 1st gear and for
it2 in 2nd gear. The change of speed ratio in Dual Clutch Transmission can be regarded
as a process of one clutch to be engaged while another being disengaged, process referred
as clutch-to-clutch shifts. The switching between different gears is made relative to engine
speed and two working modes of the clutch are considered: open and closed. Also, three
105
Figure 5.4: Double clutch transmission driveline model.
different phases of the closed mode are modeled, each corresponding to the clutch springs
that are being compressed at that time.
The following PWA state-space model is obtained:
ẋ(t) = Aci x(t) + bc u(t) + fc
if x(t) ∈ Ωi ,
(5.15)
having different switching logics for the two clutches. The regions for the first clutch are
defined as follows:


Ω1




Ω
:= {x ∈ R5 | x3 ≤ ωeclosing1 || x3 ≥ ωeopening1 }, - open
5
closing1 &
|x1 | ≤ θ1 }, - closed I
2 := {x ∈ R | x3 > ωe
5
closing1


Ω := {x ∈ R | x3 > ωe
& θ1 < |x1 | ≤ θ2 }, - closed II

 3


Ω4 := {x ∈ R5 | x3 > ωeclosing1 & θ2 < |x1 |},
- closed III
,
(5.16)
where ωeclosing1 is the engine closing speed and ωeopening1 is the engine opening speed, used
as thresholds for the first gear, and the switching logic is illustrated in Fig. 5.5. It can be
seen that the clutch remains in the open mode while the engine speed doesn’t reach the
closing speed value ωeclosing1 , or is bigger than the opening speed value ωeopening1 . When
the engine speed is situated between this two threshold values the system enters the closed
mode, selection between the three phases of the closed mode being made relative to the
torsional angle between the engine and the transmission. While this angle is smaller than
the threshold value θ1 , the system is in the first phase of the closed mode. When the angle
106
Figure 5.5: DCT - Switching logic for the first clutch.
passes the threshold value θ1 but is still smaller than the second threshold value θ2 , the
system is in the second phased of the closed mode. Finally, when the angle also passes this
second threshold value θ2 , the system enters the third phase of the closed mode.
The regions for the second clutch are defined as follows:


Ω1




Ω
:= {x ∈ R5 | x3 ≤ ωeclosing2 || x3 ≥ ωeopening2 }, - open
5
closing2 &
|x1 | ≤ θ1 }, - closed I
2 := {x ∈ R | x3 > ωe
5
closing2


Ω3 := {x ∈ R | x3 > ωe
& θ1 < |x1 | ≤ θ2 }, - closed II




5
closing2
Ω4 := {x ∈ R | x3 > ωe
& θ2 < |x1 |},
- closed III
,
(5.17)
where ωeclosing2 is the engine closing speed and ωeopening2 is the engine opening speed, used as
thresholds for the second gear. The switching logic of the second clutch is illustrated in Fig.
5.6. It can be seen that the clutch remains in the open mode while the engine speed doesn’t
reach the closing speed value ωeclosing2 , or is bigger than the opening speed value ωeopening2 .
When the engine speed is situated between this two threshold values the system enters the
closed mode, selection between the three phases of the closed mode being made relative to
the torsional angle between the engine and the transmission. While this angle is smaller
than the threshold value θ1 , the system is in the first phase of the closed mode. When the
angle passes the threshold value θ1 but is still smaller than the second threshold value θ2 ,
the system is in the second phased of the closed mode. Finally, when the angle also passes
this second threshold value θ2 , the system enters the third phase of the closed mode.
All developed models have three rotational inertias and consider that the driveline flexibility is introduced by the drive shafts as well as by the clutch. Also, the driving load given
107
Figure 5.6: DCT - Switching logic for the second clutch.
by the airdrag torque, gravity and rolling resistance is taken into consideration resulting into
a more accurate and complex model of the driveline dynamics.
As a novelty, the piecewise affine models of the AMT and DCT driveline include a model
of the clutch with four operating modes, one corresponding to the open mode, and the other
three corresponding to three different phases of the closed mode.
5.3
For the developed driveline models, three predictive control strategies are proposed, with
the aim of reducing driveline oscillations: explicit MPC, horizon-1 MPC based on flexible
control Lyapunov functions and delta GPC.
The PID control strategy is also applied on the piecewise affine three inertias models
developed of both automated and double clutch transmission, in order to compare the performances of the predictive control strategies. The control structure of the controller is
presented in Fig. 2.8, and the mathematical form is given by equation 2.39. Next, an explicit MPC that can impose constraints on inputs, states and outputs is proposed for the
PWA three inertia model of the AMT driveline. A horizon-1 MPC based on flexible control
Lyapunov functions is designed for all three driveline models proposed in this chapter, and,
like the explicit MPC, this control strategy also has the ability to enforce constraints on
states, inputs and outputs.
108
5.3.1
Explicit MPC Controller
The model considered for control is the PWA three inertia model of the AMT driveline
given by (5.11), (5.12) and (5.14). The explicit MPC algorithm solves a finite-horizon openloop optimization problem on-line, at each sampling instant, and has the ability to enforce
constraints on states, inputs and outputs.
The engine torque (i.e., the control input) is restricted by lower and upper bounds and
by a torque rate constraint as follows:
0 ≤ u(t) ≤ Temax ,
∀t ∈ R+ ,
(5.18)
Tem ≤ u̇(t) ≤ TeM ,
∀t ∈ R+ ,
(5.19)
where Temax is the maximum torque that can be generated by the internal combustion engine
and Tem , TeM are torque rate bounds. Furthermore, the engine and wheel speeds are bounded,
i.e.,
ωemin ≤ x3 (t) ≤ ωemax ,
min
max
∀t ∈ R+ , ωw
≤ x5 (t) ≤ ωw
,
∀t ∈ R+ ,
(5.20)
min
where ωemin and ωemax are the idle speed and the engine limit speed, respectively, and ωw
max are the minimum and the maximum speed of the wheels.
and ωw
The control objective is to reach a desired value of the wheel speed as fast as possible and
with minimum overshoot, while damping driveline oscillations. Considering the state-space
system representation (5.11), the problem to solve is to minimize the cost function

min
{u(t)}t∈Z
[0,N −1]
kPN xN k∞ +
N
−1
X

kQx x(t)k∞ + kRu u(t)k∞  ,
(5.21)
t=0
relative to control input, control input slew ant outputs constraints, given by equations
(5.18), (5.30) and (5.20).
Considering the discrete time PWA state-space model obtained from (5.11):
xk+1 = Adi xk + Bdi uk + fdi
yk = Cdi xk + Ddi uk + gdi
,
(5.22)
subject to constraints on outputs, control input, and control input slew rate:
ωemin ≤ yk ≤ ωemax
0 ≤ uk ≤ Temax
Tem ≤ uk − uk−1 ≤ TeM
109
.
(5.23)
In MPT, PWA systems are described by the following fields of the system structure:
sysStruct.A = {Ad1 , ..., Ad4 },
sysStruct.B = {Bd1 , ..., Bd4 },
sysStruct.C = {Cd1 , ..., Cd4 },
sysStruct.D = {Dd1 , ..., Dd4 },
(5.24)
sysStruct.f = {fd1 , ..., fd4 },
sysStruct.g = {gd1 , ..., gd4 },
with the guard-lines:
sysStruct.guardX = {guardX1 , ..., guardX4 },
sysStruct.guardU = {guardU1 , ..., guardU4 },
(5.25)
sysStruct.guardC = {guardC1 , ..., guardC4 },
and system constraints given by:
sysStruct.ymax = {ωemax },
sysStruct.ymin = {ωemin },
sysStruct.umax = {Temax },
sysStruct.umin = {0},
(5.26)
sysStruct.dumax = {TeM },
sysStruct.dumin = {Tem }.
Each dynamics i is active in a polyhedral partition bounded by the so-called guard-lines:
guardXi x(k) + guardUi u(k) 6 guardCi ,
(5.27)
which means that dynamics i will be applied if the above inequality is satisfied.
5.3.2
Horizon-1 MPC Controller
The models considered for control are the three driveline models proposed in this chapter,
and, like the explicit MPC, this control strategy based on flexible Lyapunov functions also
has the ability to enforce constraints on states, inputs and outputs.
To obtain a discrete-time PWA model, each affine subsystem in (5.11) or (5.15) is discretized with sampling period Ts using the Euler transform, which yields
m m
m m
m
xm
k+1 = Adi xk + bd uk + fd
110
if xk ∈ Ωi ,
(5.28)
m
m
for all k ∈ Z+ , where Am
di and bd are the corresponding discretized system matrices, fd is
m
the discretized affine term and xm
k , uk are the state and input of the system at time instant
k ∈ Z+ . The active mode i is selected for the discrete-time PWA system using (5.12) in the
case of the AMT driveline, and using (5.16) and (5.17) in the case of the DCT driveline,
the same as done for the continuous-time PWA system. Letting I51 denote I5 with the first
element on the diagonal equal to zero, the reset condition (5.13) now becomes
1 m
m
m
∀k ∈ Z≥1 , if (xm
k−1 , xk ) ∈ Ω1 × Ω2 , set xk := I5 xk .
(5.29)
The engine torque rate constraint now becomes
∆
−Te∆ ≤ ∆um
k ≤ Te ,
∀k ∈ Z≥1 ,
(5.30)
m
m
∆
where ∆um
k := uk − uk−1 and Te is the maximum allowed increase (decrease) in torque at
each sampling instant. Torque rate constraints are important to allow full usage of the airflow
to maintain the torque reserve, so that torque variations can be actuated instantaneously.
In what follows, for simplicity of exposition, a coordinate transformation is performed in
(5.28) to translate the problem into stabilization of the origin, i.e.,
ss
xk = xm
k −x ,
where xss =
xss
xss
xss
xss
xss
5
4
3
2
1
>
ss
uk = um
k −u ,
(5.31)
, Note that for a desired wheel speed value xss
5 , one
can obtain the corresponding steady-state values of the system states and input, i.e., xss
1 ,
ss
ss
ss
xss
2 , x3 , x4 and u .
The following system description results:
xk+1 = Adi xk + bdi uk + fdi
if xk ∈ Ωi ,
(5.32)
along with the corresponding reset condition (5.29). Here Adi and bdi are the discretized
and transformed system matrices and fdi are the discretized and transformed affine terms.
Notice that the transformed PWA model (5.32) has zero as an equilibrium within region Ω2 ,
i.e., fd2 = 0. Also, observe that uss can be interpreted as the feedforward component of the
control action.
Consider the following cost function to be minimized
J1 (xk , uk , λk ) := JMPC (xk , uk ) + J(λk )
:= kPx xk+1 k∞ + kRuk k∞ + kGλk k∞ ,
(5.33)
subject to constraints:
0 − uss ≤ uk ≤ Temax − uss ,
−Te∆ ≤ ∆uk ≤ Te∆ ,
xmin ≤ H(Adi xk +bdi uk + fdi ) ≤ xmax ,
111
(5.34)
!
!
ωemin − xss
ωemax − xss
3
3
max :=
with
:=
,
x
and H := ( 00 00 10 00 01 ). The cost J(·) is
min − xss
max − xss
ωw
ω
5
w
5
chosen as required in Problem 2.4.2 and the matrices Px ∈ Rpx ×n and R ∈ Rr×n are chosen as
xmin
full-column rank matrices of appropriate dimensions. Consider the following infinity-norm
based CLF
V (x) = kP xk∞ ,
(5.35)
where P ∈ Rp×n is a full-column rank matrix to be determined, e.g., using techniques from
(M.Lazar, 2006a). This function satisfies (2.45) with α1 (s) =
√σ s,
p
where σ is the smallest
singular value of P , and α2 (s) = kP k∞ s. For xk ∈ Ωi , substituting (5.32) and (5.35) in
(2.48b) yields
kP (Adi xk + bdi uk + fdi )k∞ ≤ ρkP xk k∞ + λk
(5.36)
where xk , P and ρ ∈ R[0,1) are known at k ∈ Z+ . In what follows it is shown that for a unitary
horizon, the above MPC optimization problem can be formulated as a linear program (LP)
via a particular set of equivalent linear inequalities, despite switching dynamics, while for any
other larger horizon it would lead to a mixed integer linear programming (MILP) problem.
By definition of the infinity norm, for kxk∞ ≤ c to be satisfied, it is necessary and sufficient
to require that ±[x]j ≤ c for all j ∈ Z[1,n] . So, for (5.36) to be satisfied, it is necessary and
sufficient to require
±[P (Adi xk + bdi uk + fdi )]j ≤ ρkP xk k∞ + λk
(5.37)
for all j ∈ Z[1,p] . As such, solving Problem 2.4.2, which includes minimizing the cost function
(5.33), can be reformulated as the following problem.
Problem 5.3.1 Measure xk , determine the active mode i and
min 1k + 2k + 3k
(5.38)
±[Px (Adi xk + bdi uk + fdi )]j ≤ 1k , ∀j ∈ Z[1,px ] ,
(5.39a)
uk ,λk
subject to (2.48c), (5.34), (5.37) and
±Ruk ≤ 2k ,
(5.39b)
Gλk ≤ 3k .
(5.39c)
Problem 5.3.1 is a linear program, since xk and λ∗k−1 are known at time k ∈ Z≥1 and
thus, all constraints are linear in uk , λk and εlk , l ∈ Z[1,3] . The horizon-1 MPC algorithm is
stated next.
112
Algorithm 5.3.2
At each sampling instant k ∈ Z+ :
Step 1: Measure the current state xk and obtain the active mode i;
Step 2: Solve the LP Problem 5.3.1 and pick any feasible control action, i.e., uf (xk );
2
Step 3: Implement uk := uf (xk ) as control action.
The fact that only a feasible, rather than optimal, solution of Problem 5.3.1 is required in
Algorithm 5.3.2, can reduce the execution time.
The desired objective is to reach a desired value of the wheel speed, i.e., xss
5 , as fast as
possible and with minimum overshoot, while damping driveline oscillations. As such, the
above objective can be formulated as asymptotic stabilization of the desired steady-state
point while satisfying the required constraints.
5.3.3
Delta GPC Controller
A Delta GPC strategy is applied on the affine model of the three inertia driveline, given by
the equations (5.8), (5.9) and (5.10). The state-space model with the affine term is converted
in the δ representation. This model will be used to design and simulate the predictive control
strategy in the δ domain.
The δ-operator can be directly substituted into q-operator from the definition:
δ=
q − 1 esTs − 1
.
=
Ts
Ts
(5.40)
Middleton and Goodwin (Middleton and Goodwin, 1986) suggested the following relations for conversion from s-domain model into the discrete time δ-domain one:
eAc Ts − I
= ΩAc , Bδ = ΩBc ,
Ts
Cδ = Cc , Dδ = Dc ,
Aδ =
(5.41)
with
T
1 Z Ac τ
1
Ω=
e dτ = (eAc Ts − I)Ac −1
Ts
Ts
0
2
=I+
,
(5.42)
2
Ac Ts Ac Ts
+
+ .......
2!
3!
where Ac , Bc , Cc and Dc are continuous-time state-space model matrices and Ts is the
sampling period with q the usual forward-shift operator.
113
The cost function has an important role in designing predictive control strategies and in
δ domain can be expressed similarly, starting from the cost function given by (2.59) :
Jδ = Ky
Ny
X
j
j
2
(δ ŷk − δ wk ) + λQKu
Nu
X
2
(δ j uk ) ,
(5.43)
j=1
j=N1
where Ky , Q and Ku are matrices that allow transforming the system from the q domain to
δ domain, and are given in (Halauca, Balau and C.Lazar, 2011).
The reference vector is assumed to be of the form:
h
wδ = δ 0 w δ 1 w δ 2 w...... δ Ny w]T .,
(5.44)
and the predictor expression from (2.54) is rewritten:
h
i
→
−
ŷδ = f + Gu + Gy Γy −1 ( Γ − Γu ) u1δ ,
(5.45)
→
−
with Gu and Gy being components of the matrix G, while Γy , Γu and Γ are also defined in
(Halauca, Balau and C.Lazar, 2011).
Therefore, the control input is determined by minimizing the desired control criterion
with respect to u1δ :
dJδ
= 0.
du1δ
(5.46)
According with the receding horizon principle only the first element of the control sequence will be applied to the process and at the next step whole algorithm is repeated. The
state-space δ GPC algorithm is remarkable by the fact that the strategy for determining the
optimal control is fully developed in the δ domain. In this way, the numerical calculations
are performed in the δ discrete time representation. Therefore, the rounding errors that
occur in classic q domain are reduced in this case, especially for small sampling periods in
the context of finite number of bits representation (Kadirkamanathan et al., 2009).
5.4
Simulation Results
The developed models and the proposed control strategies were implemented in Matlab/Simulink
and different simulation scenarios were conducted.
114
Engine torque [Nm]
200
150
100
50
0
0
2
4
6
8
10
12
14
8
10
12
14
Wheel speed [Km/h]
Time [s]
60
40
20
0
0
2
4
6
Time [s]
Figure 5.7: Simulation results using δ GPC.
5.4.1
Delta GPC for the Affine Model
This section presents the performances of the proposed GPC strategy designed in the δ
domain, investigated on the vehicle three inertia driveline model given by the equations (5.8),
(5.9) and (5.10), using Matlab software. In the last decade, several experimental studies
encourage the model predictive control to work in practice. For assessing the performance
of predictive control in discrete δ domain applied on automotive transmission system, some
simulation experiments have been performed. The system considered in (5.8), (5.9) and
(5.10), in a state space approach, is transformed from continuous time representation in the
δ domain.
Fig. 5.7 illustrates the performances of the state-space δ GPC algorithm applied on the
automotive transmission system converted in δ domain model (5.41) and (5.42). The engine
torque is represented in the top figure is referred as input signal while the output, the wheel
speed which must follow the reference trajectory, is drawn with dotted line in the bottom
figure.
The predictive control parameters uses in the simulation were set to Nu = 1, Ny = 6 and
weighting factor of 0.0001 in the context of 40 ms sampling period. The Fig. 5.8 shows the
influences of the δ GPC controller on engine speed, transmission speed and axle wrap. The
axles wrap angular speed is represented in the third plot of the Fig. 5.8 as a measure of
powertrain oscillations that appear in the system.
In the Fig. 5.9 are depicted the simulation results when the reference is changed from
40 km/h to 20 km/h, illustrated with dotted line. It is to be mentioned that the control
115
Engine
speed [rpm]
6000
4000
2000
0
0
2
4
6
8
10
12
14
8
10
12
14
8
10
12
14
Time [s]
Transmission
speed [rpm]
2000
1000
0
0
2
4
6
Axle wrap
speed difference
Time [s]
10
5
0
−5
0
2
4
6
Time [s]
Figure 5.8: Influences of the δ GPC on engine speed, transmission speed and axle wrap.
objective is to reach the desired value of the wheel speed as fast as possible with minimum
overshoot.
The wheel speed reaches the desired value in 6 seconds. The results can be clarified
by examination of the Fig. 5.9, which shows that the δ GPC performs better as weighting
factor is smaller, but the control action is increased. The effect of changes λ is that for more
control weighting, the input changes are less active and λ may be used to reduce the power
consumption of the control signal if necessary, for instance to keep the input signal within
lower limits.
Also, Fig. 5.10 shows the influences of the δ GPC controller on engine speed, transmission
speed and axle wrap when tip-in maneuver is performed.
5.4.2
Affine Model Versus PWA Model
The complexity of the numerous models reported in the literature varies from linear two
masses models, to more complex PWA three-masses models. In order to observe the importance of using a more complex driveline model when developing the system controller, two
horizon-1 MPC controllers were designed: one for the affine and one for the piecewise affine
model (MPC-affine and MPC-PWA, respectively), and both control strategies were applied
on the piecewise affine plant.
116
Engine torque [Nm]
200
150
100
50
0
0
2
4
6
8
10
12
14
8
10
12
14
Wheel speed [Km/h]
Time [s]
60
40
20
0
0
2
4
6
Time [s]
Figure 5.9: δ GPC simulation results subject to reference changes.
Engine
speed [rpm]
6000
4000
2000
0
0
2
4
6
8
10
12
14
8
10
12
14
8
10
12
14
Time [s]
Transmission
speed [rpm]
2000
1000
0
0
2
4
6
Axle wrap
speed difference
Time [s]
10
5
0
−5
0
2
4
6
Time[s]
Figure 5.10: Influences of the δ GPC on engine speed, transmission speed and axle wrap,
subject to reference changes.
117
This section presents the validations of the proposed predictive control strategy investigated on the vehicle driveline models using the Matlab/Simulink program. A step signal was
applied as the reference for the vehicle velocity and it was desired that the system tracks the
reference signal as fast as possible, the following figures showing the results obtained in the
simulations for the MPC-affine and for the MPC-PWA. The sampling time of the system
was chosen to be T = 5ms and the value of the parameters that are used in simulations are
given in Table A.4 and Table A.5.
The proposed one step ahead predictive controller uses the following weight matrices of
the cost (5.33): Px = 0.71I5 , R = 0.021 and G = 1 for the PWA model, and Px = 0, R = 0
and G = 1 for the affine model. The technique presented in (M.Lazar, 2006b) was used for
the off-line computation of the infinity norm based local CLF V (x) = kP xk∞ for ρ = 0.99
and the PWA model of the driveline in closed-loop with uk := Ki xk for each active mode
i ∈ Z[1,4] . The following matrices were obtained

P



=



32.63
−42.24
29.85
−151.96
33.98
−12.80 0.15
−10.67 0.16
439.64 0.34
22.31
7.78
360.39 −0.16

−0.45 −2.01

−0.32 10.34 

−0.04 −52.76 
,
−0.63 3.96 

0.06
64.13
K1 =
45.54 −17.58 −1.69 −8.42 46.33
,
K2 =
13.27 −30.15 −5.35 −6.83 94.15
,
K3 =
17.84 −26.83 −7.02 −6.78 88.49
,
K4 =
23.40 −30.89 −6.52 −7.21 31.03
.
(5.47)
The above control law was only employed off-line, to calculate the weight matrix P of
the local CLF V (·), and it was never used for controlling the system.
For the proposed one step ahead MPC scheme, recursive feasibility implies asymptotic
stability. However, recursive feasibility is not a priori guaranteed and hinges mainly on
the constraint (2.48c) on the future evolution of λ∗k . In the case of the MPC-PWA, the
values ∆ = 500 and M = 5 were found through simulations, such that recursive feasibility is
attained. The time needed for computation of the control input for the proposed one step
ahead predictive controller is less than 2ms, so it meets the required timing constraints.
For the MPC-affine feasibility is not guaranteed no matter what the values of ∆ and
M are and even if the weight matrices of the cost were adjusted, so the closed loop system
is not asymptotically stable. In what follows, the results obtained for both controllers are
presented and compared, even if the MPC-affine does not assure stability.
Fig. 5.11 illustrates the reference vehicle velocity value and the response of the system
when the predictive strategy is applied. Initially, the vehicle speed is equal with zero and
118
40
Vehicle velocity [km/h]
35
30
25
20
Reference speed
PWA model
Affine model
15
10
5
0
0
5
10
Time [s]
15
20
Figure 5.11: Vehicle velocity.
5000
PWA model
Affine model
Engine speed [rpm]
4000
3000
2000
1000
0
0
5
10
Time [s]
15
20
Figure 5.12: Engine speed.
does not increase, because the system is in the open mode in the case of the PWA model,
and because the controller is not able to find a feasible solution in the case of the affine
model. It can be seen that, in the case of the MPC-PWA, the system tracks the reference
signal, having no steady state error and no overshoot, while for the MPC-affine there is an
overshoot and a slower response. This can also be seen in Fig. 5.12 where the engine speed
is illustrated.
The axle wrap angular speed is represented in Fig. 5.13 as a measure of driveline oscillations that appear when the clutch switches through the operating modes.
Fig. 5.14 illustrates the engine torque (control signal) for the predictive method for the
MPC-affine as well as for the MPC-PWA. The working modes of the clutch are presented
in Fig. 5.15, where 1, 2, 3 and 4 represents the regions of the system given in (5.12). In
order to put the vehicle in motion, the load torque has to be defeated, so the engine speed
varies around the threshold value (see Fig. 5.12), which results in the switching between the
open and closed mode. It can be seen that the clutch starts from open mode (1) and after
119
100
Axle wrap angular speed [rpm]
PWA model
Affine model
50
0
−50
0
5
10
Time [s]
15
20
Figure 5.13: Axle wrap speed difference.
200
PWA model
Affine model
Engine torque [Nm]
150
100
50
0
0
5
10
Time [s]
15
20
Figure 5.14: Engine torque (control signal).
4
3.5
Clutch mode
3
2.5
2
1.5
1
0
5
10
Time [s]
15
Figure 5.15: Clutch mode of operation.
120
20
it reaches the engine closing speed ωeclosing , the clutch enters the first phase of the closed
mode. Note that the switching between the three phases of the closed mode (2, 3 and 4,
respectively) is made depending on the value of the torsional angle between the engine and
transmission. When this angle value is bigger than the threshold values θ1 , the system enters
the second phase of the closed mode, and when the angle value is bigger than the threshold
values θ2 , the system enters the third phase of the closed mode.
5.4.3
AMT Driveline Control
The continuous-time PWA model (5.11)-(5.13) for the three inertias AMT driveline model,
was implemented in Matlab/Simulink and three different control strategies were applied to
damp driveline oscillations, i.e., the horizon-1 predictive controller, an explicit MPC and a
PID controller. The control objective is to reach a desired speed reference in a short time,
but, at the same time, to increase the passenger comfort by reducing the oscillations that
appear in the driveline. The axle wrap is calculated as the difference between the engine
speed (divided by the total transmission ratio) and the wheel speed, and it is used as a
measure of the driveline oscillations.
A PID controller was designed based on Ziegler-Nichols tuning method (O’Dwyer, 2006)
and it was further manually tuned in order to have a fast response, which yielded the
proportional, integral and derivative terms KR = 30, Ti = 10−3 and Td = 9·10−5 , respectively.
The more common approach to the design of an explicit MPC controller was also applied.
For the considered discrete-time PWA model and operating constraints, using the Multi
Parametric Toolbox for Matlab, a feasible solution to the corresponding mpMILP problem
was only obtained for the prediction horizon equal to 1, but the resulting performance was
substandard. For a prediction horizon larger than 1, despite using a powerful working station
and several robust mpMILP solvers, a solution could not be obtained. This indicates the
non-trivial nature of the considered case study.
The explicit MPC was designed by using the cost function (5.21) with the following
values of weight matrices: PN = 4I5 , Qx = 0.1I5 , where I5 is the unit matrix of size 5, and
Ru = 0.46.
The following paragraph is dedicated analyzing the system performances for each technique. Clearly, no stability guarantee can be obtained for the PWA system in closed-loop
with the PID controller. The closed-loop system that corresponds to the explicit MPC
scheme is a PWA system and as such, stability can be analyzed a posteriori in this case.
However, the stability analysis of the corresponding closed-loop system performed with the
Multi Parametric Toolbox (MPT) for Matlab, which performs a wide variety of tests (e.g.,
piecewise quadratic, linear and even polynomial Lyapunov functions are searched for) did
121
not yield a conclusive result, but ran into numerical errors. For the horizon-1 MPC scheme
developed in this paper, recursive feasibility implies asymptotic stability. However, recursive
feasibility is not a priori guaranteed and hinges mainly on the constraint (2.48c) on the
future evolution of λ∗k . For all simulation scenarios case studies, the values ∆ = 500 and
M = 5 proved to be large enough to guarantee recursive feasibility for the desired operating
scenarios.
Different simulations were conducted, to evaluate the vehicle behavior in response to
acceleration, deceleration, tip-in and tip-out maneuvers and a stress test, which are presented in the following subsections. Note that, although the PID controller does not enforce
constraints on control command, its output was saturated in order to enforce the engine
limitations, i.e., the torque limit Temax .
In what follows, the results obtained for the explicit MPC will be presented separately,
because of the slow response, while a comparison will be made between the FCLF MPC and
the PID controller, which have similar results.
5.4.3.1
Scenario 1: Acceleration test
A first simulation test is performed on an acceleration scenario where the vehicle has to
accelerate from 0 km/h to 30 km/h, so a reference of 30 km/h is given for the wheel speed.
In what follows the comparative performance of the resulting closed-loop systems for the
acceleration scenario is analyzed for the PID and horizon-1 predictive controller, using the
trajectories plotted in Fig. 5.16.
In Fig. 5.16, top right, it can be seen how the horizon-1 predictive controller reaches
the desired reference speed in a shorter time, and with no overshoot, compared with the
PID controller. In both cases, the wheel speed is equal with zero while the clutch is in the
open mode, and starts rising when the clutch enters the closed mode. The amplitude of
the axle wrap is represented in Fig. 5.16, bottom left, and it can be seen how the horizon-1
predictive controller minimizes these oscillations. The engine torque is represented in the
bottom right figure, where the limitation on the input increase is visible for the horizon-1
predictive controller. The evolution of the CLF relaxation variable λ∗k and the corresponding
upper bound defined by (2.48c) for ρ = 0.99, ∆ = 500 and M = 5 is shown in Fig. 5.16, top
left. It can be observed that λ∗k may decrease or even go to 0, after which it is allowed to
increase again, as long as this does not violate the upper bound. However as k → ∞, λ∗k
is forced to converge to 0. In Fig. 5.17 the clutch mode history was represented for the
PID controller and for the horizon-1 predictive controller, to show that in the transient the
closed-loop system frequently switches between the operating modes.
122
40
Lambda
λ∗k
800
upper bound
600
400
200
0
5
10
Time [s]
15
10
0
0
5
10
Time [s]
15
20
200
Engine torque [Nm]
PID
Horizon−1 MPC
50
0
−50
−100
reference
PID
Horizon−1 MPC
20
20
100
0
5
10
Time [s]
15
PID
Horizon−1 MPC
150
100
50
0
20
0
5
10
Time [s]
15
Figure 5.16: Scenario 1: Acceleration test.
4
PID
Horizon−1 MPC
3.5
Clutch mode
0
30
3
2.5
2
1.5
1
0
5
10
Time [s]
15
Figure 5.17: Scenario 1: Clutch mode of operation.
123
20
20
In order to put the vehicle in motion, the load torque has to be defeated, so the engine
speed varies around the threshold value, which results in the switching between the open
and closed mode. It can be seen that the clutch starts from open mode (1) and after it
reaches the engine closing speed ωeclosing = 12000 rad/s, the clutch enters the first phase of
the closed mode (2). When this angle value is bigger than the threshold values θ1 = 0.17
rad, the system enters the second phase of the closed mode (3), and when the angle value
is bigger than the threshold values θ2 = 0.20 rad, the system enters the third phase of the
closed mode (4).
In Fig. 5.18, top, it can be seen how the explicit MPC reaches the desired wheel speed in
almost 20 seconds, which is really slow compared with the other two controllers. This is due
to the behavior of the control signal (the engine torque), represented in Fig. 5.18, bottom,
which has the steady-state value even from the beginning, and does not have a peak like the
PID and the horizon-1 MPC controllers. Even though different design methods were carried
out to improve the performances of the explicit MPC controller, e.g., choosing the weight
matrix PN “larger” than the Px matrix from the horizon-1 predictive controller to make it
more aggressive, or even equal to the matrix P of the local Lyapunov function, these did
not lead to better results. Also, the working modes of the clutch for the explicit MPC are
represented in Fig. 5.19, and like in the case of the PID and horizon-1 MPC, it can be seen
that the wheel speed is equal with zero while the clutch is in the open mode (2), and starts
rising when the clutch enters the first phase of the closed mode (3).
5.4.3.2
Scenario 2: Deceleration test
The second simulation scenario consist of decelerating the vehicle from 30 km/h to 10 km/h.
The mp-MILP problem for the explicit MPC scheme could not be solved even for horizon 1,
using MPT, so only the results obtained for the PID and the horizon-1 predictive controller
are illustrated in Fig. 5.20.
Although both controllers obtained almost the same settling time, the PID controller
produces some undesired axle wrap oscillations, indicating that the horizon-1 predictive
controller has a superior behavior in terms of damping the driveline oscillations. The evolution of the CLF relaxation variable λ∗k and the corresponding upper bound is illustrated
in Fig. 5.20, top left. Although the upper bound starts from 500, due to ∆, only the values
below 50 were plotted. This makes it possible to observe the evolution of λ∗k .
124
Engine torque [Nm] Axle wrap angular speed [rpm]
40
30
20
10
0
0
5
10
Time [s]
15
20
0
5
10
Time [s]
15
20
0
5
10
Time [s]
15
20
100
50
0
−50
200
150
100
50
0
Figure 5.18: Scenario 1: EMPC - Acceleration test.
Clutch mode
4
3
2
1
0
5
10
Time [s]
15
20
Figure 5.19: Scenario 1: EMPC - Clutch mode of operation.
125
50
40
Lambda
40
λ∗
k
upper bound
30
20
10
0
5
10
Time [s]
15
20
20
10
0
5
10
Time [s]
15
20
70
PID
Horizon−1 MPC
10
0
−10
−20
30
0
20
0
5
10
Time [s]
15
Engine torque [Nm]
0
reference
PID
Horizon−1 MPC
50
40
30
20
10
0
20
PID
Horizon−1 MPC
60
0
5
10
Time [s]
15
20
Figure 5.20: Scenario 2: Deceleration test.
50
35
Lambda
40
λ∗
k
upper bound
30
20
10
0
0
20
40
30
25
20
15
10
60
reference
PID
Horizon−1 MPC
0
20
50
60
200
0
PID
Horizon−1 MPC
−50
40
Time [s]
Engine torque [Nm]
Time [s]
0
20
40
100
50
0
60
Time [s]
PID
Horizon−1 MPC
150
0
20
40
Time [s]
Figure 5.21: Scenario 3: Tip-in tip-out test.
126
60
5.4.3.3
Scenario 3: Tip-in tip-out maneuvers
The results of a tip-in, tip-out maneuver simulation, in which the reference vehicle velocity
goes from 30 km/h to 10 km/h and back to 30 km/h, are presented in Fig. 5.21. Again, a
feasible solution was not bound for the explicit MPC scheme even with N = 1. As such, the
results are given only for the PID controller and the proposed horizon-1 MPC controller. It
can be seen that the horizon-1 predictive controller has a slightly faster response with no
overshoot when it approaches the reference velocity. Moreover, the oscillations of the axle
wrap are damped much faster in the acceleration phase. In the deceleration phase, again,
the PID controller produces undesired oscillations of the axle wrap. Note that the controller
performance during deceleration is limited by the actuator authority. For this experiment
the evolution of the CLF relaxation variable λ∗k and the corresponding upper bound defined
by (2.48c) are shown in Fig. 5.21, top left. Due to changing the reference vehicle velocity,
the upper bound of the CLF relaxation variable defined by (2.48c) may become unfeasible,
so whenever a change in the reference vehicle velocity occurs, the value of the upper bound
was re-initialized.
5.4.3.4
Scenario 4: Stress test
The results of a stress test, in which the reference velocity is a square wave that changes
rapidly between 30 km/h and 20 km/h, are presented in Fig. 5.22. The purpose is to check
what happens to the axle wrap speed if it does not have enough time to settle between
two set-point changes, which means that continuous perturbations may occur. The results
illustrate how the horizon-1 predictive controller has again a smaller amplitude for the axle
wrap angular speed, while the PID barely manages to cope with this kind of maneuver.
Whenever a change in the reference vehicle velocity occurs, the value of the upper bound
was re-initialized as done in the previous scenario. This is not visible in Fig. 5.22, top left,
because the upper bound starts again from 500 and it does not reach values below 50 in such
a short amount of time.
5.4.4
DCT Driveline Control
The proposed continuous-time PWA model (5.15) to (5.17) was implemented in Matlab/Simulink
and two different control strategies were applied to damp driveline oscillations, i.e., the
horizon-1 predictive controller and a PID controller. The sampling period of the system was
chosen to be Ts = 5ms. The values of the parameters used in simulations, which relate to a
medium size passenger car, and they are given in Table A.4 and Table A.5 in the Appendix.
The control objective is to reach a desired speed reference in a short time, but, at the same
127
50
40
Lambda
40
λ∗
k
upper bound
30
20
10
0
0
2
4
35
30
25
20
6
reference
PID
Horizon−1 MPC
0
2
50
6
200
PID
Horizon−1 MPC
0
−50
4
Time [s]
Engine torque [Nm]
Time [s]
0
2
4
100
50
0
6
PID
Horizon−1 MPC
150
0
Time [s]
2
4
6
Time [s]
Figure 5.22: Scenario 4: Stress test.
time, to increase the passenger comfort by reducing the oscillations that appear in the driveline. The axle wrap is calculated as the difference between the engine speed (divided by the
total transmission ratio) and the wheel speed, and it is used as a measure of the driveline
oscillations.
A PID controller was designed based on (O’Dwyer, 2006) and it was tuned to have a fast
response, which yielded the proportional, integral and derivative terms KR = 18, Ti = 11.25
and Td = 0.004, respectively.
Px = 1 · I5 , R = 0 and G = 1. The technique presented in (M.Lazar, 2006a) was used for the
off-line computation of the infinity norm based local CLF V (x) = kP xk∞ for ρ = 0.99 and
the PWA model of the driveline in closed-loop with uk := Ki xk if xk ∈ Ωi , i ∈ Z[1,4] . The
128
following matrices were obtained

P



=




−9.96 −25.23 2.78 −0.55 23.24

−58.93 8.35
1.59 −0.35 10.85 

36.06 345.36 −0.06 0.01 −22.16 
,
−83.13 6.69
2.55
0.21
24.01 

25.82 251.51 −0.04 −0.025 41.42
K1 =
59.04 −14.71 −4.71 −6.22 19.50
K2 =
9.00 −20.50 4.39 −8.40 52.74
,
K3 =
1.27 −19.72 5.03 −8.19 51.16
,
K4 =
35.92 −10.39 3.11 −7.98 38.61
,
(5.48)
.
Different simulations were conducted, to evaluate the vehicle behavior in response to
acceleration and deceleration, and are presented in the following subsections. Note that,
although the PID controller does not enforce constraints on control command, its output
was saturated in order to enforce the engine limitations, i.e., the torque limit Temin and Temax .
5.4.4.1
Up-shift maneuvers
A first simulation test is performed on an acceleration scenario where the vehicle has to
accelerate from 0 km/h to 30 km/h. In what follows the performance of the resulting closedloop systems for the acceleration scenario is analyzed using the trajectories plotted in Fig.
5.23.
The input signal (engine torque) is represented in the top left of the figure, and the wheel
speed in the top right. The engine speed is represented in the bottom left of the figure and
it can be seen that, when it reaches a value of 3000 rpm, a gear shift appears and causes
a drop of the engine speed. The amplitude of the axle wrap is represented in Fig. 5.16,
bottom right. It can be seen that, even if the wheel speed of the horizon-1 MPC has a small
overshot, the axle wrap is the same for both controllers at the beginning, but has bigger
oscillations for the PID controller when a gear shift occurs.
In Fig. 5.24 and Fig. 5.25 the mode history for the two clutches was represented for
the PID controller and for the horizon-1 predictive controller, to show that in the transient
phase, the closed-loop system frequently switches between the operating modes. It can also
be clearly seen how, when a gear shift appears, there is a switch between the first and second
clutch.
In order to put the vehicle in motion, the load torque has to be defeated, so the engine
speed varies around the threshold value, which results in the switching between the open
and closed mode. It can be seen that the first clutch starts from open mode (1) and after it
129
Engine torque (control signal) [Nm]
Reference, Wheel speed [km/h]
200
35
30
150
25
20
100
15
PID
Horizon−1 MPC
10
50
5
0
0
2
4
6
8
0
10
Engine speed [rpm]
0
2
4
6
8
10
Axle wrap: Engine speed − Wheel speed [rpm]
100
3500
3000
2500
50
2000
1500
0
1000
500
0
2
4
6
8
−50
10
0
2
4
6
8
10
Figure 5.23: Scenario 1: Up-shift maneuvers
reaches the engine closing speed ωeclosing1 = 1000 rpm, the clutch enters the first phase of the
closed mode. Note that the switching between the three phases of the closed mode (2, 3 and
4, respectively) is made depending on the value of the torsional angle between the engine
and transmission relative to the threshold values θ1 = 0.17 rad and θ2 = 0.20 rad. When the
engine speed passes the opening threshold value of the engine speed ωeclosing1 = 3000 rpm,
the first clutch enters the open mode, while the second clutch closes. The switching between
the three phases of the closed mode for the second clutch, is made, as well, depending on the
value of the torsional angle between the engine and transmission relative to the threshold
values.
5.4.4.2
Down-shift maneuvers
The second simulation scenario consist of decelerating the vehicle from 30 km/h to 10 km/h
and the results obtained are illustrated in Fig. 5.26.
Top right figure illustrates how, with the horizon-1 predictive controller, the system
reaches the reference wheel velocity in approximately 10 seconds, with almost no oscillations, while the PID controller takes a much longer time, 40 seconds, to reach the reference
wheel velocity. The drop in the engine speed is followed by high value of the axle wrap, as
130
First clutch behaviour
4
3
2
1
0
0
2
4
6
8
10
8
10
Second clutch behaviour
3
2.5
2
1.5
1
0.5
0
0
2
4
6
Figure 5.24: MPC - Clutch operation modes for up-shift maneuvers test.
4
3
2
1
0
0
2
4
6
8
10
8
10
4
3
2
1
0
0
2
4
6
Figure 5.25: PID - Clutch operation modes for up-shift maneuvers test.
131
Engine torque (control signal) [Nm]
Reference, Wheel speed [km/h]
100
35
PID
Horizon−1 MPC
30
80
25
60
20
40
15
20
0
10
0
20
40
5
60
Engine speed [rpm]
0
20
40
60
3000
Axle wrap: Engine speed − Wheel speed [rpm]
20
2500
10
2000
0
1500
−10
1000
−20
500
0
20
40
60
0
20
40
Figure 5.26: Scenario 2: Down-shift maneuvers.
2
1.5
1
0.5
0
0
10
20
30
40
50
60
2
PID
Horizon−1 MPC
1.5
1
0.5
0
0
10
20
30
40
50
60
Figure 5.27: Clutch operation modes for down-shift maneuvers test.
132
60
5.5 Conclusions
consequence, indicating that the horizon-1 predictive controller has a superior behavior in
terms of damping the driveline oscillations.
Also, the mode history for the two clutches is represented in Fig. 5.27, for the PID
controller and for the horizon-1 predictive controller. The first clutch starts from open mode
while the second clutch starts from the second phase of the closed mode (2). When the
engine speed passes the opening threshold value of the engine speed ωeopening1 = 1200 rpm,
the second clutch enters the open mode, while the first clutch closes. Then, the switching
between the phases of the closed mode is made relative to the torsional angle between the
engine and transmission with the threshold values θ1 = 0.17 rad and θ2 = 0.20 rad. It can
be seen that, in the deceleration scenario, in the transient phase, the closed-loop system
smoothly switches between the open and closed modes.
The PID controller can be tuned to have a faster response in the decelerations scenario,
but it affects the performances of the system in the acceleration scenario. Regardless what
parameters are used for the PID controller, when considering the overall performances, the
experiments show that the horizon-1 predictive controller has a superior behavior.
5.5
Conclusions
In this chapter the problem of damping driveline oscillations that occur when tip in and tip
out maneuvers are performed is addressed, with the goal of improving drivers comfort. Two
complex models of an automotive driveline are developed and the torsional speed between
the engine and the wheel is used as measure for the driveline oscillations. Both developed
models have three rotational inertias and consider that the driveline flexibility is introduced
by the drive shafts and also by the clutch. Also, the driving load given by the airdrag torque,
gravity and rolling resistance is taken into consideration resulting into a more accurate model
of the driveline dynamics.
First, a state-space affine model of an AMT driveline is presented, including drive shaft
and clutch flexibilities. Then, starting from the affine model, a more complex piecewise
affine model of an AMT driveline is developed. A new modeling approach of the clutch is
introduced, different from the other approaches found in literature, because of the modeling
of the situation when the clutch is opened. The clutch has four operating modes: one
corresponding to the open mode, and the other three corresponding to three different phases
of the closed mode. Finally, a new piecewise affine model of an DCT driveline is developed.
This model also has three rotational inertias with flexibility given by the clutch and drive
shaft, and each clutch has the four operating modes: one for the open mode and the other
ones representing three different phases of the closed mode.
133
MPC is increasingly seen as an attractive technology due to its capability to directly
handle various specifications requirements including the optimization of the cost function
while enforcing constrains on states and control variables. As such, the problem considered
in this chapter is to damp out driveline oscillations by applying predictive control. For
that reason, a recently introduced design method for horizon-1 MPC, which is based on
flexible control Lyapunov functions (M.Lazar, 2009) is used. The algorithm therein has the
potential to satisfy the timing requirements, due to the short horizon, while it can still offer
a non-conservative solution to stabilization due to the flexibility of the Lyapunov function.
Simulators are implemented in Matlab/Simulink for the three proposed driveline models
and different control techniques are applied, beside the horizon-1 MPC. A PID controller is
implemented in order to compare the performances of the predictive control strategies. An
explicit MPC controller is developed for the PWA AMT driveline model, but the resulting
performances were substandard. Also, an delta GPC controller was developed and analyzed
as a solution for real time implementation. Several simulation scenarios validate the proposed
approach and indicate that the proposed scheme (the horizon-1 MPC), besides yielding a
feasible algorithm, outperforms controllers obtained using typical approaches, such as PID
control and explicit model predictive control.
The results obtained were published at different conferences:
• (Balau et al., 2011b) A.E. Balau, C.F. Caruntu and C. Lazar. Driveline oscillations
modeling and control. In The 18th International Conference on Control Systems and
Computer Science, Bucharest, Romania, 2011.
• (Balau and C.Lazar, 2011b) A. E. Balau and C. Lazar. One Step Ahead MPC for an
Automotive Control Application. In The 2nd Eastern European Regional Conference
on 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. Bosh
Chicago, USA, 2011.
• (Halauca, Balau and C.Lazar, 2011) C. Halauca, A. E. Balau and C. Lazar. State
Space Delta GPC for Automotive Powertrain Systems. In The16th IEEE International
Conference on Emerging Technologies and Factory Automation, 2011.
134
Chapter 6
Conclusions
this torque in an efficient way, a proper model of the driveline is needed for controller design
purposes with the aim increasing vehicle overall performances. Different driveline models
and control strategies are proposed, and problems as nonlinearities introduced by backlash
and clutch are addressed in this thesis.
A summary of the main contributions of this thesis and several recommendations for
future research are provided in this chapter.
6.1
Summary of Contributions
The major contribution of this work are related to:
• Modeling and control of and electro-hydraulic actuated wet clutch system
• Modeling and control of a two inertia driveline including backlash nonlinearity
• Modeling and control of a three inertia driveline including clutch nonlinearity
6.1.1
Modeling and Control of an Electro-Hydraulic Actuated Wet
Clutch System
First contribution of this thesis consist of modeling and controlling of an electro-hydraulic
actuated wet clutch system.
First, two new models of a solenoid valve actuator used in the automotive control systems were developed: a linearized input-output model, where simplifications were made in
135
Conclusions
order to obtain a suitable transfer function to be implemented in Simulink and to obtain
an appropriate behavior for the outputs, and a state-space model with no simplifications.
Simulators are implemented for the developed models, in order to validate the proposed
modeling approach. The models were validated by comparing the results with experimental
data obtained on the real test-bench provided by Continental Automotive Romania.
Next, starting from the actuator models, two new models for an electro-hydraulic actuated clutch system used in the automotive control systems for automatic transmission were
developed: a linearized input-output model and a state-space model. Simulators are implemented for the developed models, in order to validate the proposed modeling approach and
to apply different control strategies. The models were validated by comparing the results
with data obtained on the real test-bench provided by Continental Automotive Romania,
which includes a Volkswagen wet clutch actuated by an electro-hydraulic valve.
A GPC controller was designed in order to control the output of the electro-hydraulic
actuated clutch system: the clutch piston displacement, and a PID controller is implemented
in order to compare the simulation results. Analyzing the result obtained with the GPC
strategy and the PID control strategies, it can be concluded that the best results are obtained
when using the predictive control, because the system precisely tracks the reference signal,
with no overshoot.
6.1.2
Modeling and Control of a Two Inertia Driveline Including
Backlash Nonlinearity
The second contribution of this thesis consist of modeling and controlling a two inertia
driveline including backlash nonlinearity.
Two models for a conventional driveline composed of engine, continuous variable transmission, final reduction gear, final drive-shaft and driving wheels are developed, including
the backlash nonlinearities: a PWA and a nonlinear state-space model.
The PWA model was designed using a fixed transmission ratio and a simulator was implemented in Matlab in order to validate the modeling approach and to implement the proposed
control scheme: the horizon-1 MPC controller based on flexible Lyapunov functions.
For the nonlinear model, the optimized driveline was designed to reduce the fuel consumption by using the optimal fuel efficiency curve in the modeling phase. Also, a PID
based cascade controller was implemented in Matlab/Simulink. The inner loop controller
(torque controller) was designed firstly, considering the driveline model as the plant and
then, using the inner closed-loop control system as the plant, the external loop controller
(speed controller) was designed.
136
6.1 Summary of Contributions
Then, three models were implemented for an Automated Manual Transmission (AMT)
driveline based on the Industrial plant emulator M220: a rigid driveline model, a flexible
driveline model and a flexible driveline model including also backlash nonlinearity. Simulator
were implemented in Matlab for the driveline models, in order to implement the horizon-1
MPC control strategy based on flexible Lyapunov functions. The controllers are developed
and implemented, and then, real time experiments are conducted on the industrial plant
emulator in order to test the influences given by drive shaft flexibility and backlash angle.
6.1.3
Modeling and Control of a Three Inertia Driveline Including
Clutch Nonlinearity
The third contribution of this thesis consist of modeling and controlling a three inertia
driveline including clutch nonlinearity.
Two complex models of an automotive driveline were developed and the torsional speed
between the engine and the wheel was used as measure for the driveline oscillations. The
developed models have three rotational inertias and consider that the driveline flexibility
is introduced by the drive shafts and also by the clutch. Also, the driving load given by
the airdrag torque, gravity and rolling resistance is taken into consideration resulting into a
more accurate model of the driveline dynamics.
Starting from the equations that describe the dynamics of an affine model for an AMT
driveline, a more complex piecewise affine model of an AMT driveline was developed, including a new model of the clutch with four operating modes, one corresponding to the
open mode, and the other three corresponding to three different phases of the closed mode.
Taking into account all these factors yields a more accurate model of the driveline dynamics.
The second model is a new piecewise affine model of a driveline complex system including
engine, flexible clutch, Dual Clutch Transmission, flexible shafts and wheels. The change
of speed ratio in Dual Clutch Transmission can be regarded as a process of one clutch to
be engaged while another being disengaged, process referred as clutch-to-clutch shifts. The
switching between different gears is made relative to engine speed and two working modes of
the clutch are considered: open and closed. Also, three different phases of the closed mode
are modeled, each corresponding to the clutch springs that are being compressed at that
time.
Simulators are implemented for the affine AMT driveline model, for the PWA AMT
driveline model as well as for the PWA DCT driveline model, in order to apply different
control strategies and to compare the simulation results.
137
Conclusions
A horizon-1 MPC controller is developed for the AMT driveline model as well as for
the PWA AMT driveline model, in order to compare the results obtain when using more
complex driveline models. Also, an delta GPC controller was developed for the affine AMT
driveline model and it was analyzed as a solution for real time implementation.
An explicit MPC controller is developed for the PWA AMT driveline model, but the
resulting performances were substandard. A horizon-1 MPC controller is developed for the
PWA AMT driveline model as well as for the PWA DCT driveline model, and also a PID
controller is implemented in order to compare the controllers performances.
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
IEEE conference papers, two IFAC conference papers and three papers published at international conferences where paper review is conducted.
6.2
Suggestion for Future Research
Some directions can be formulated starting from the current research work:
• Starting from the three inertias driveline model including clutch flexibility and considering the modeling approach of the backlash nonlinearity from the two inertias model,
a more complex three inertia model can be developed. The model should consider that
the driveline flexibility is introduced by the drive shafts and also by the clutch. Also,
the driving load given by the airdrag torque, gravity and rolling resistance is taken
into consideration resulting into a more accurate model of the driveline dynamics. The
model of the clutch with four operating modes, one corresponding to the open mode,
and the other three corresponding to three different phases of the closed mode must
also be included. In addition, the modeling of the nonlinearity introduced by the backlash would increase the order of the system and would add two more working mode:
contact and non-contact. Taking into account all these factors yields a more accurate
model of the driveline dynamics.
• Another direction would be to introduce the model of the electro-hydraulic actuated
wet clutch system in the three inertia model of the driveline including clutch nonlinearity and after that the backlash nonlinearity can be added as well, in order to obtain
a more detailed model of the driveline.
• Also, more experiments can be conducted on the Industrial plant emulator M220,
taking into consideration coulomb friction and different configurations given by the
138
6.2 Suggestion for Future Research
adjustable inertias and changeable gear ratios. Also, more control strategies can be
implemented and the results can be compared with the ones obtained using the horizon1 MPC controller based on flexible Lyapunov function.
139
Conclusions
140
Appendix A
141
Table A.1: Valve-clutch system parameter values
Symbol
Value
Unit
Ke
1000
[N/m]
K
900
[N/m]
25e-3
[kg]
1.6e+9
[N/m2 ]
KC = KD
7.58e-11
[(m3 /s)/(N/m2 )]
K1
5.50e-10
[(m3 /s)/(N/m2 )]
K2
3.52e-9
[(m3 /s)/(N/m2 )]
K3
1.26e-8
[(m3 /s)/(N/m2 )]
Kq
5.3418
[(m3 /s)/(N/m2 )]
w
3e-3
[m]
PS
1e+6
[N/m2 ]
PT
0
[N/m2 ]
2e-9
[(m3 /s)/(N/m2 )]
VC
7.53e-8
[m3 ]
VD
1.04e-7
[m3 ]
Vt
3.2e-4
[m3 ]
VL
2.51e-5
[m3 ]
AC
3.66e-5
[m2 ]
AD
2.94e-5
[m2 ]
AL
7.75e-4
[m2 ]
Mp
0.5
[kg]
ka
0.005
[Nm2 /A2 ]
kb
0.01
[m]
Ls
0.01
[H]
Rs
0.5
[Ω]
Mv
βe
kl
142
Table A.2: Vehicle parameter values for two inertia CVT driveline with backlash nonlinearity
Symbol
Value
Measure
Description
Unit
0.125
[kg m2 ]
Engine inertia
Jv
88.86
[kg
m2 ]
Vehicle inertia
de
0
[Nms/rad] Engine damping
rstst
0.285
[m]
Wheel radius
mCOG
1094
[kg]
Vehicle mass
0
[Nms/rad] Vehicle damping
Je
dw
iCV T
0.8
CVT gear ratio
iF RG
0.4
Final driveshaft gear ratio
ηF RG
0.985
Final driveshaft efficiency
ηCV T
0.8
Transmission efficiency
Troll
35
[Nm]
Rolling torque
Tagle
0
[Nm]
Resistant torque
Tairdrag
0
[Nm]
Resistant torque
c1
0.0105
Constant
c2
0.032
Constant
Psc
50
Speed controller
Isc
1
Speed controller
Dsc
2
Speed controller
Ptc
0.03
Torque controller
Itc
0.2
Torque controller
0
Torque controller
Dtc
143
Table A.3: Vehicle parameter values for two inertia AMT driveline with backlash nonlinearity
Symbol
Value
Measure Unit
Description
Je
0.025
[kg m2 ]
Engine inertia
Jp
0.000078
[kg m2 ]
Pulley inertia
Jw
0.0271
[kg m2 ]
Wheel inertia
de
0.004
[Nms/rad]
Engine damping constant
dd
0.017
[Nms/rad]
Drive shaft damping constant
dw
0.05
[Nms/rad]
Wheel damping constant
kd
8.45
[Nms/rad]
Drive shaft spring constant
itot
4
Overall gear ratio
ip
2
Partial gear ratio
kc
32.768/10
[DAC counts/V]
DAC gain
ks
1/32
[ref
Controller software gain
input
counts/controller
input counts]
ke
2*pi/16000
[rad/counts]
144
Encoder gain
Table A.4: Simulation vehicle parameter values for three inertias driveline with clutch nonlinearity - 1
Value
Measure Unit
Description
Je
0.17
[kg m2 ]
Engine inertia
Jt
0.014
[kg m2 ]
Transmission inertia
Jf
0.031
[kg m2 ]
Final drive inertia
Symbol
m2 ]
Jw
1
[kg
Wheel inertia
dd
65
[Nms/rad]
Flexible driveshaft damping
kd
5000
[Nm/rad]
Flexible driveshaft stiffness
de
0.159
[Nms/rad]
Engine damping
dt
0.1
[Nms/rad]
Transmission damping
df
0.1
[Nms/rad]
Final drive damping
dw
0.1
[Nms/rad]
Wheel damping
it
3.5
Gearbox ratio (1st gear)
if
3.7
Final drive ratio
mCOG
1400
[kg]
Vehicle mass
rw
0.32
[m]
Wheel radius
cr1
0.01
[Nm/kg]
Rolling coefficient
cr2
0.36
[Nms/rad]
Approximation coefficient
cd
0.3
[rad−2 ]
Airdrag coefficient
1.2
[kg/m3 ]
Air density
Af
2.7
[m2 ]
Frontal area of the vehicle
g
9.8
[m/s2 ]
Gravitational acceleration
0
[rad]
Road slope
θ1
0.1745
[rad]
Clutch switching boundary
θ2
0.2094
[rad]
Clutch switching boundary
dc1
0
[Nms/rad]
Clutch damping (open)
dc2
3
[Nms/rad]
Clutch damping (closed I)
dc3
6
[Nms/rad]
Clutch damping (closed II)
dc4
10
[Nms/rad]
Clutch damping (closed III)
kc1
0
[Nm/rad]
Clutch stiffness (open)
kc2
800
[Nm/rad]
Clutch stiffness (closed I)
kc3
1600
[Nm/rad]
Clutch stiffness (closed II)
kc4
3200
[Nm/rad]
ρair
χroad
Clutch stiffness (closed III)
145
Table A.5: Simulation vehicle parameter values for three inertias driveline with clutch nonlinearity -2
Symbol
Value
Measure
Description
Unit
Temax
Te∆
ωemin
ωeclosing
ωeclosing1
ωeclosing2
ωeopening1
ωeopening2
ωemax
min
ωw
max
ωw
200
[Nm]
Maximum engine torque
3
[Nm]
Maximum engine torque increase/decrease
62.83
[rad/s]
Engine idle speed
104.72
[rad/s]
Engine closing speed
104.72
[rad/s]
Engine closing speed for the first clutch
125.66
[rad/s]
Engine closing speed for the second clutch
314.15
[rad/s]
Engine opening speed for the first clutch
314.15
[rad/s]
Engine opening speed for the second clutch
628.3
[rad/s]
Maximum engine speed
0
[km/h]
Minimum wheel speed
50
[km/h]
Maximum wheel speed
it1
3.5
Gearbox ratio (1st gear)
it2
2.8
Gearbox ratio (2nd gear)
146
Bibliography
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156
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
ciclului
viitorilor
cercetători
în
cadrul
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.
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Beneficiar: Universitatea Tehnică “Gheorghe Asachi” din
Iaşi
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Director proiect: Prof. univ. dr. ing. Carmen TEODOSIU
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4.2 Driveline Models - Facultatea de Automatică şi Calculatoare

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