MODELLING AND CONTROL OF A TURBOCHARGED DIESEL ENGINE
By
Tao Zeng

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mechanical Engineering - Doctor of Philosophy
2017

ABSTRACT
MODELLING AND CONTROL OF A TURBOCHARGED DIESEL ENGINE
By
Tao Zeng
The diesel engine is known for its high efficiency, performance, and durability. With
stringent fuel economy and emission regulations, diesel engines face increasing challenges. To
accommodate emission regulations, fuel economy and performance requirements, modern diesel
engines are equipped with the variable geometry turbocharger (VGT) and exhaust gas
recirculation (EGR) system. VGT extracts energy from exhaust gas to drive the compressor to
improve transient response, steady-state performance, and fuel efficiency under wide range of
engine flow conditions. Meanwhile, EGR dilutes fresh air with exhaust gas to reduce the
formation of mono-nitrogen oxides NO and NO2 (NOx). The VGT and EGR control design is
complicated due to the natural coupling between VGT and EGR, and high nonlinearity of diesel
engine air-path system. The extra assisted power and regenerative power on the turbocharger
shaft further increase the control system complexity. In this dissertation, new approaches for
turbocharger system modelling and multivariable control design for the coordinated actuation of
the VGT-EGR system are investigated. The control design is further extended to hydraulic
regenerative assisted turbocharger system.
New modelling approaches for turbocharger system are proposed based on turbomachinery
physics. Proposed turbine and compressor models eliminate the interpolation error, and
especially, allow smooth extrapolation outside the mapped region. A high fidelity reduced order
mean value model of a diesel engine for automotive application is developed based on developed

turbocharger model. Further, new models for high-speed hydraulic turbines and centrifugal
pumps are developed for hydraulic assisted and regenerative turbochargers.
A regenerative hydraulic assisted turbocharger (RHAT) system is investigated in this
dissertation. A system level approach based on 1-D simulations is used to understand the assist
benefits and design trade-offs. Simulation results show that 3-5% fuel economy improvement for
FTP 75 driving cycle, depending on different sub-component sizing. The study also identifies
technical challenges for optimal design and control of RHAT systems.
A linear controller design approach is proposed in this dissertation for regulating both boost
pressure and EGR mass flow rate of the VGT-EGR system. The linear quadratic control with
integral action is designed based on the linearized system. Local controllers are scheduled based
on engine operational parameter: engine speed and fuel injection quantity. The gain scheduled
liner controller is validated against baseline controller based on the nonlinear plant. Results show
that designed multi-input and multi-output (MIMO) controller can well manage the trade-offs
between boost pressure tracking and EGR mass flow tracking, compared to baseline controller
(two single input single output (SISO) controllers). A novel approach is proposed for closed-loop
control design with respect to engine performance and engine emission trade-offs.

The

controller design is further extended to assisted and regenerative turbocharger system with VGT
and EGR. The results show that emission reduction, engine performance and fuel economy
improvement can be achieved at the same time with external power applied to the turbocharger
shaft.

To my grandparents, my parents, my wife Mengyan and my son Franklin

iv

ACKNOWLEDGMENTS

Without the help and encouragement from the people around, it would be impossible to make
through one of the most important learning and experience in my life journey. Foremost, I would
like to express the deepest appreciation to my advisor, Prof. Guoming (George) Zhu for his
persistent support and advising for my study and research. When I started of this new journey at
2012 to pursuit my engineering dream, it was my great honor to choose you as my advisor since
then. Few professors have the similar theoretical and practice background as you, who always
would like provide the student the most comprehensive, trustable guidance and extensive
engineering experience. You have been a tremendous mentor for me. I would like to thank you
for encouraging my research and for allowing me to grow as a research scientist. Your advice on
both research as well as on my career have been priceless.

My sincere thanks also go to my committee members, Dr. Hassan Khalil, Dr. Harold Schock and
Dr. Ranjan Mukherjee. Their insight discussions and constructive comments over the years were
of great help in providing valuable feedback for my research.

I have spent almost three years at Ford Research and Innovative Center (Dearborn,MI) for my
Ph.D. study and research. Without the support from the supervisors/mentors and colleagues at
Ford, the work won’t be such smooth and successful. I would like to express my gratitude to Dr.
Harold Sun, Dr. Devesh Upadhyay, Dr. Satheesh Makkapati, Dr. Eric Curtis, Dr. James Yi, Dr.

v

Michiel Van Nieuwstadt, Vladimir Kokotovic, Dr. Liangjun Hu, Dr. Xiaowu Zhang, Dr. Kang
Song, and Dr. Ben Zhao.

I would like also to extend my thanks to the fellow graduate students, research assistants in our
lab: Dr. Jie Yang, Peng Xu, Yingxu Wang, Yifan Men and Ruitao Song. They were always
willing to help and provide their best suggestions.

Finally, special thanks to Andrew Kim, who encouraged me at my early study stage to pursuit
the art of engineering.

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TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................................... x
LIST OF FIGURES ....................................................................................................................... xii
KEY TO SYMBOLS AND ABBREVIATIONS ......................................................................... xix
CHAPTER 1: INTRODUCTION ................................................................................................... 1
1.1 Motivation ............................................................................................................................. 1
1.2 Research Overview ............................................................................................................... 2
1.2.1 Turbine modelling .......................................................................................................... 2
1.2.2 Compressor modelling ................................................................................................... 3
1.2.3 Hydraulic assisted turbocharger modelling for controller design .................................. 4
1.2.4 Hydraulic regenerative and assisted turbocharger system level investigation............... 4
1.2.5 Gain-scheduling controller design for diesel engines with EGR-VGT system ............. 5
1.3 Dissertation contributions ..................................................................................................... 6
1.4 Dissertation outline ............................................................................................................... 8
CHAPTER 2: CONTROL ORIENTED VGT TURBINE POWER MODELS FOR
TURBOCHARGED ENGINE ...................................................................................................... 10
2.1 Abstract ............................................................................................................................... 10
2.2 Introduction ......................................................................................................................... 10
2.3 VGT Turbine Power Model-based on Euler Turbine Equation .......................................... 16
2.3.1 Euler turbine equation .................................................................................................. 16
2.3.2 VGT turbine power model as a function of vane angle ............................................... 17
2.3.3 Turbocharge friction model ......................................................................................... 20
2.4 Model Validation and Mechanical Loss Identification ....................................................... 22
2.4.1 Model validation using steady-state engine test data and mechanical loss estimation 22
2.4.2 Turbine model validation against standard hot gas flow bench test data .................... 29
2.4.3 Turbine power model validation using GT-Power transient simulation data .............. 31
2.4.4 Model validation using vehicle test data ...................................................................... 35
2.5 Conclusion .......................................................................................................................... 39
CHAPTER 3: A REDUCED COMPLEXITY MODEL FOR THE COMPRESSOR POWER OF
AN AUTOMOTIVE TURBOCHARGER ................................................................................... 40
3.1 Abstract ............................................................................................................................... 40
3.2 Introduction ......................................................................................................................... 41
3.3 Compressor Power Modelling ............................................................................................ 45
3.3.1 Compressor power model-based on the Euler equations ............................................. 45
3.3.2 Investigation of slip factor models for flows over centrifugal compressor ................. 49
3.3.3 Compatibility with corrected mass flow rates ............................................................. 52
3.4 Model Identification............................................................................................................ 53
3.4.1 Model identification using standard hot gas flow bench test data ............................... 53
3.4.2 Model identification with ‘Supercharger Standard Test’............................................. 58

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3.5 Model Validation ................................................................................................................ 60
3.5.1 Model validation based on steady state engine dynamometer test data....................... 60
3.5.2 Model validation for US06 transient cycle based on GT-Power cycle simulations .... 64
3.5.3 Model validation over FTP cycle based on engine dynamometer tests. ...................... 65
3.6 Conclusion .......................................................................................................................... 67
CHAPTER 4: MODELLING OF HYDRAULIC ASSISTED AND REGENERATIVE
TURBOCHARGED D ENGINE .................................................................................................. 68
4.1 Abstract ............................................................................................................................... 68
4.2 Regenerative Hydraulic Assisted Turbocharger With VGT-EGR Overview ..................... 68
4.2.1 Engine air-path modelling overview ............................................................................ 70
4.2.2 Regenerative hydraulic assisted turbocharged engine modelling ................................ 73
4.3 Engine Modelling and Validation ....................................................................................... 76
4.3.1 Engine intake and exhaust mass flow rate ................................................................... 77
4.3.2 Exhaust manifold temperature ..................................................................................... 79
4.3.4 Intake manifold temperature ........................................................................................ 80
4.3.3 EGR mass flow rate modelling .................................................................................... 81
4.4 Variable Geometry Turbocharger modelling and validation .............................................. 83
4.4.1 VGT turbine model ...................................................................................................... 84
4.4.1.1 Turbine power ........................................................................................................ 84
4.4.1.2 Turbine mass flow rate........................................................................................... 85
4.4.2 Compressor modelling ................................................................................................. 86
4.4.2.1 Compressor power ................................................................................................. 86
4.4.2.2 Compressor mass flow rate .................................................................................... 87
4.5 Hydraulic System Modelling .............................................................................................. 94
4.5.1 System overview .......................................................................................................... 94
4.5.2 Hydraulic centrifugal pump modelling ........................................................................ 96
4.5.2.1 Hydraulic centrifugal pump power model ............................................................. 96
4.5.2.1 Centrifugal pump flow rate model ......................................................................... 98
4.5.3 Hydraulic turbine modelling ...................................................................................... 100
4.5.3.1 Turbine power model ........................................................................................... 100
4.5.3.2 Turbine mass flow rate model.............................................................................. 102
4.5.3 Valve model ............................................................................................................... 103
4.5.4 Modelling for piston accumulator .............................................................................. 104
4.6 Model Validation and Plant Investigation ........................................................................ 105
4.6.1 Model validation ........................................................................................................ 105
4.6.2 Plant behavior investigation ....................................................................................... 108
4.7 Conclusion ........................................................................................................................ 116
CHAPTER 5: SYSTEM ANALYSIS FOR HYDRAULIC ASSISTED TURBOCHAED
DIESEL ENGINE THROUGH 1-D SIMULATION ................................................................. 117
5.1 Abstract ............................................................................................................................. 117
5.2 Introduction ....................................................................................................................... 118
5.2.1 Assisted turbocharger................................................................................................. 118
5.2.2 Hydraulic assisted and regenerative turbocharger ..................................................... 121
5.2.3 System causality......................................................................................................... 128
5.3 Vehicle Level Integrated Simulation in GT-Power / Simulink ........................................ 131

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5.3.1 Simulation platform and control algorithm................................................................ 131
5.3.2 Hydraulic components ............................................................................................... 135
5.4 Simulation results and discussion ..................................................................................... 138
5.4.1 Engine alone steady state investigation for feedforward calibration ......................... 138
5.4.2 Transient response improvement investigation ......................................................... 146
5.4.2.1 Engine transient performance improvement ........................................................ 146
5.4.2.2 Turbocharger transient performance improvement ............................................. 148
5.4.2.3 Hydraulic components transient performance ..................................................... 150
5.4.3 Design trade-offs for fuel benefit through driving cycle simulation ......................... 151
5.5 Conclusion ........................................................................................................................ 160
CHAPTER 6: LINEAR QUADRATIC CONTROLLER DESIGN FOR VGT-EGR DIESEL
ENGINE AIR-PATH .................................................................................................................. 162
6.1 Abstract ............................................................................................................................. 162
6.2 Introduction ....................................................................................................................... 163
6.3 Controller design ............................................................................................................... 167
6.3.1 Control objective and problem formulation ............................................................... 167
6.3.2 Linearization of Nonlinear System ............................................................................ 169
6.3.3 Model linearization for diesel engine air-path system ............................................... 170
6.3.4 Augmented with actuator dynamics........................................................................... 172
6.3.6 Linear quadratic regulator with integral action (LQI) ............................................... 174
6.3.7 Observability and controllability analysis ................................................................. 177
6.3.8 Plant scaling ............................................................................................................... 179
6.4 Controller Design Validation ............................................................................................ 180
6.4.1 Baseline controller for control development.............................................................. 180
6.4.2 LQI controller design for different design target ....................................................... 185
6.4.3 Gain-scheduling for linear controllers ....................................................................... 192
6.4.4 Extended controller design for assisted and regenerative turbocharger .................... 202
6.5 Conclusion ........................................................................................................................ 212
CHAPTER 7. CONCLUSIONS AND FUTURE WORK .......................................................... 213
7.1 Conclusions ....................................................................................................................... 213
7.2 Future work ....................................................................................................................... 215
REFERENCES ........................................................................................................................... 216

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LIST OF TABLES

Table 1. Friction model candidates ............................................................................................... 21
Table 2. Model validation results using steady-state engine test data for turbocharger 1 ............ 26
Table 3. Model validation results using steady-state flow bench test data for turbocharger 2 ..... 30
Table 4. Averaged Error between proposed model predicted turbine power and GT simulated
turbine power ................................................................................................................................ 33
Table 5. Average error for different models ................................................................................. 37
Table 6. Comparison among different efficiency modelling approaches (Assuming constant inlet
condition) ...................................................................................................................................... 44
Table 7. Model coefficients for various slip models..................................................................... 50
Table 8. Identified model coefficients for 3 model variants and 3 compressor design variants ... 55
Table 9. Fitted model coefficients identified through supercharging test data ............................. 60
Table 10. Mass flow rate governing equation ............................................................................... 76
Table 11. Load step simulation inputs ........................................................................................ 108
Table 12. Operating points for model linearization .................................................................... 111
Table 13. A Survey for current electric motor/ generator design space ..................................... 124
Table 14. Vehicle information .................................................................................................... 133
Table 15. Vehicle fuel benefits and tank energy......................................................................... 154
Table 16. Controllability and observability analysis for VGT-EGR system .............................. 178
Table 17. Parameters numeric range ........................................................................................... 180
Table 18. C matrix reformulation ............................................................................................... 180
Table 19. Normalized range for Q matrix ................................................................................... 187
Table 20. Controller designs for different engine operation point.............................................. 192

x

Table 21. Control input value range and physical interpretation ................................................ 203
Table 22. Benchmarking with baseline controller ...................................................................... 212

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LIST OF FIGURES

Figure 1. Dissertation outline .......................................................................................................... 9
Figure 2. Normalized turbine steady-state and transient operational ranges vs. the hot gas flow
bench test range............................................................................................................................. 12
Figure 3. Turbocharger system structure ...................................................................................... 13
Figure 4. Simplified turbine model [27] ....................................................................................... 16
Figure 5. Variable geometry turbocharger layout and speed triangle [1, 20] ............................... 18
Figure 6.

 m tan 1

with respect to the VGT position ...................................................................... 24

Figure 7.

 m tan 1

with respect to the VGT position and turbocharger speed ............................... 25

Figure 8. VGT actuation mechanism ............................................................................................ 26
Figure 9. Predicted power loss, vane angle and turbine power at SS (steady-state)..................... 29
Figure 10. Turbocharger 2 test range ............................................................................................ 30
Figure 11. Turbocharger 2 predicted turbine power ..................................................................... 31
Figure 12. Vane angle and the VGT map position with GT simulation ....................................... 33
Figure 13. Model prediction results vs GT simulation results ...................................................... 33
Figure 14. Modelling error for FTP-75 in GT simulation ............................................................ 34
Figure 15. Modelling results for US_06 driving cycle ................................................................. 34
Figure 16. Model validation with transient vehicle test data ........................................................ 38
Figure 17. Operating range deficit between mapped and desired engine operating range. .......... 42
Figure 18. Velocity triangles of a centrifugal compressor at the rotor inlet and outlet ................ 45
Figure 19. Identification Results for the generalized Compressor-Power model for Compressors1,2,3............................................................................................................................................... 56

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Figure 20. Modelling error (27) for compressor power models ................................................... 58
Figure 21. Identification Results for the generalized Compressor-Power model for Compressors4,5,6............................................................................................................................................... 59
Figure 22. Comparison of Cpower model performance against calculated values based on flow
bench data and engine steady state dynamometer test data for Compressor-1 ............................. 61
Figure 23. Model predicted power vs Measured power under 30K rpm TC speed ...................... 62
Figure 24. Model validation over US-06 GT-Power transient simulation.................................... 64
Figure 25. Normalized measurements for mass flow rate, TC speed and Compressor downstream
temperature for a load step ............................................................................................................ 65
Figure 26. Model validation against transient engine test data for a FTP 75 cycle ...................... 66
Figure 27. Diesel engine air-path system with hydraulic actuation system .................................. 69
Figure 28. Diesel engine control volume lay out [52] .................................................................. 71
Figure 29. System modelling architecture and calculating loop. .................................................. 74
Figure 30. System inputs and outs for control perspective ........................................................... 75
Figure 31. Model identification results for volumetric efficiency ................................................ 78
Figure 32. Model identification results for exhaust temperature .................................................. 80
Figure 33. Model identification results for temperature after compressor ................................... 81
Figure 34. Identified EGR effective area ...................................................................................... 82
Figure 35. Model identification results for EGR mass flow rate .................................................. 83
Figure 36. Turbine modelling layout ............................................................................................ 84
Figure 37. Turbine mass flow rate model identification ............................................................... 86
Figure 38. Compressor model layout ............................................................................................ 86
Figure 39. Automotive compressor layout.................................................................................... 87
Figure 40. Hammer-Wiener Model............................................................................................... 89

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Figure 41. Compressor mass flow rate static nonlinear model ..................................................... 90
Figure 42. Nonlinearity for H-W compressor mass flow rate model ........................................... 91
Figure 43. Zeros and poles for linear system in H-W model ........................................................ 92
Figure 44. Identification results for H-W compressor mass flow rate model ............................... 93
Figure 45. Comparison between physics based mass flow rate model and H-W compressor mass
flow rate ........................................................................................................................................ 94
Figure 46. Layout of hydraulic actuation system.......................................................................... 95
Figure 47. Hydraulic pump efficiency (y-axis is pump power) .................................................... 97
Figure 48. Identified pump model and test data ........................................................................... 98
Figure 49. Identified mass flow rate through modified flow coefficient and modified head
coefficient ................................................................................................................................... 100
Figure 50. Hydraulic turbine efficiency (y-axis is turbine power) ............................................. 100
Figure 51. Identified hydraulic turbine model and test data ....................................................... 102
Figure 52. Identified turbine mass flow rate model .................................................................... 103
Figure 53. Control valve flow factor........................................................................................... 104
Figure 54. Engine system modeling in Simulink ........................................................................ 105
Figure 55. Open loop simulation with test inputs. ...................................................................... 106
Figure 56. Model validation results for load step test ................................................................. 107
Figure 57. Model validation results for FTP 75 driving cycle .................................................... 107
Figure 58. Modelling error for FTP 75 driving cycle ................................................................. 108
Figure 59. Engine air-path response respect to VGT position with step change ........................ 112
Figure 60. Engine air-path response with respect to EGR valve with step change .................... 113
Figure 61. Engine air-path response with respect to assisted and regenerative power with step
change ......................................................................................................................................... 114
Figure 62. Frequency analysis for different engine operating points ......................................... 115

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Figure 63. Instantaneous engine power lug curve based on a light duty diesel load step test .... 119
Figure 64. Instantaneous Engine EGR Rate, Soot and NOx Emissions during d light duty diesel
FTP transient test ........................................................................................................................ 120
Figure 65. System layout of diesel engine with hydraulic assisted and regenerative turbocharger
..................................................................................................................................................... 126
Figure 66. Hydraulic assisted turbocharger layout ..................................................................... 126
Figure 67. With assisted turbocharger, turbocharger can be regulated to its higher efficiency
operation range............................................................................................................................ 127
Figure 68. The Hydraulic turbo pump can recover exhaust energy and avoid tip-out surge. ..... 128
Figure 69. System causality for regenerative hydraulic assisted turbocharger ........................... 129
Figure 70. Simulation platform and control structure ................................................................. 131
Figure 71. Modeling layout in GT-suites .................................................................................... 132
Figure 72. Engine air-path controller overview .......................................................................... 133
Figure 73. Vehicle model validation through FTP_75 driving cycle ......................................... 134
Figure 74. Hydraulic turbine efficiency ...................................................................................... 137
Figure 75. Hydraulic pump efficiency ........................................................................................ 137
Figure 76. Driveline pump efficiency ......................................................................................... 137
Figure 77. Torque comparison between electric motor and hydraulic components ................... 138
Figure 78. VGT impact on turbine power and mass flow rate.................................................... 140
Figure 79. Engine BSFC under assist and regeneration ............................................................. 141
Figure 80. Compressor and turbine power distribution under assist and regeneration ............... 142
Figure 81. Air fuel ratio under assist and regeneration ............................................................... 143
Figure 82. Max engine power with 19 kW assist with same AFR ............................................. 144
Figure 83. Loading power on TC shaft for engine BSFC impact (+: regeneration,-: assist) ...... 145
Figure 84. Engine brake power under different assist ................................................................ 147

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Figure 85. Air fuel ratio with different assist .............................................................................. 148
Figure 86. Turbocharger speed transient profile with different hydraulic assisted power ......... 149
Figure 87. TC turbine efficiency transient profile with different hydraulic assisted power. ...... 149
Figure 88. Compressor efficiency during 10.5 s to 11.2 s with different assisted power ........... 150
Figure 89. Hydraulic turbine output power transient profile ...................................................... 151
Figure 90. VGT position distribution for traditional VGT control over FTP 75 cycle. ............. 153
Figure 91. Fuel benefit for driving cycle with different FGT turbine ........................................ 154
Figure 92. Pumping loss reduction with hydraulic assisted power ............................................. 156
Figure 93. Improved engine efficiency during transient ............................................................. 157
Figure 94. Engine brake efficiency [%] distribution .................................................................. 158
Figure 95. Energy usage and recovery for hydraulic components.............................................. 159
Figure 96. Design space for hydraulic components with normalized tank size.......................... 159
Figure 97. Diesel engine with VGT and EGR system ................................................................ 164
Figure 98. Set-point for diesel engine air-path control ............................................................... 168
Figure 99. Tracking reference generation in baseline controller ................................................ 168
Figure 100. Proposed linear quadratic regulator for Engine EGR-VGT air-path system ........... 177
Figure 101. Baseline controller for control development ........................................................... 181
Figure 102. Simulation results with baseline controller ............................................................. 183
Figure 103. Tip-in investigation for baseline controller ............................................................. 184
Figure 104. Tip-out investigation for baseline controller ........................................................... 184
Figure 105. A load step test profile for engine operating at 800 RPM ....................................... 187
Figure 106. Weighting selection for different controller design index for 800 rpm engine speed /
20 mg/cc fuel injection................................................................................................................ 188

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Figure 107. Trade-off between performance, emission and fuel efficiency for controller design
for 800 rpm engine speed / 20 mg/cc fuel injection.................................................................... 189
Figure 108. Trade-off between performance, emission and fuel efficiency for controller design
for 800 rpm engine speed / 20 mg/cc fuel injection.................................................................... 189
Figure 109. Comparison controller design with baseline controller. .......................................... 191
Figure 110. Pressure difference across EGR valve for three different controllers ..................... 191
Figure 111. Gain scheduling for local linear controllers ............................................................ 193
Figure 112. Controller based on gain scheduling ....................................................................... 194
Figure 113. Gain-scheduling for VGT control (target for emission) .......................................... 195
Figure 114. Gain-scheduling for EGR controller (target for emission) ...................................... 195
Figure 115. Gain-scheduling for VGT (target for performance) ................................................ 196
Figure 116. Gain-scheduling for EGR (target for performance) ................................................ 196
Figure 117. Gain scheduling route for controller validation....................................................... 197
Figure 118. Gain scheduling controller validation with baseline controller ............................... 199
Figure 119. Pressure difference across EGR valve ..................................................................... 200
Figure 120. Gain-scheduling for VGT-EGR .............................................................................. 201
Figure 121. Benchmarking with baseline controller ................................................................... 201
Figure 122. Regenerative hydraulic assisted turbocharged diesel engine .................................. 203
Figure 123. Controller design for VGT-EGR-RHAT system ..................................................... 204
Figure 124. Normalized indexes with normalized Q matrix for controller design ..................... 205
Figure 125. Simulation results for different controller design .................................................... 208
Figure 126. Hydraulic actuation inputs for VGT-EGR-RHAT .................................................. 208
Figure 127. Pressure difference across EGR valve ..................................................................... 209
Figure 128. EGR mass fraction, EGR mass flow rate and EGR valve position for different
control design .............................................................................................................................. 210

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Figure 129. Benchmarking with baseline controller ................................................................... 211
Figure 130. Comparison of different control designs ................................................................. 211

xviii

KEY TO SYMBOLS AND ABBREVIATIONS

𝑃0 :

Environment pressure (𝑃𝑎𝑖𝑟 )

𝑃1 :

Compressor inlet pressure

𝑃2 :

Intake manifold pressure

𝑃3 :

Exhaust manifold pressure

𝑃4 :

Turbine downstream pressure

𝜔:

Turbocharger speed

𝑃𝑝 :

Pipe pressure between pump and pump valve

𝑃𝑡 :

Pipe pressure between hydraulic turbine and turbine valve

𝑃𝐴𝑐𝑐 :

High pressure accumulator pressure

𝜒:

Piston position

𝜐:

Piston velocity

𝑃𝑟𝑒𝑡𝑢𝑟𝑛 :

Hydraulic tank return pressure

𝓂:

Mass of piston

𝑇3 :

Engine exhaust manifold temperature

𝑇2 :

Engine intake manifold temperature

𝜂𝑣𝑜𝑙 :

Volumetric efficiency

𝑚̇𝑐 :

Compressor mass flow rate

𝑚̇𝑡 :

Turbine mass flow rate

𝑚̇𝑖𝑛 :
𝑚̇𝑜𝑢𝑡 :

Engine intake mass flow rate
Engine exhaust mass flow rate

xix

𝑚̇𝑒𝑔𝑟 :

Exhaust gas recirculation mass flow rate

𝑚̇𝑡𝑢𝑟𝑏𝑖𝑛𝑒 :

Hydraulic turbine mass flow rate

𝑚̇𝑣𝑎𝑙𝑣𝑒_𝑡 :

Hydraulic turbine vale mass flow rate

𝑚̇𝑝𝑢𝑚𝑝 :
𝑚̇𝑣𝑎𝑙𝑣𝑒_𝑝 :
𝜅:
𝜌𝑓𝑙𝑢𝑖𝑑 :
𝐹𝑓 :

Hydraulic pump mass flow rate
Hydraulic pump valve mass flow rate
Spring coefficient
Hydraulic fluid density
Piston friction force

𝑊̇𝑇 :

Turbine power

𝑊̇𝐶 :

Compressor power

𝑊̇𝐿𝑜𝑠𝑠 :
Wturbine :
W pump

:

Turbocharger mechanical loss power
Hydraulic turbine power
Hydraulic pump power

𝑉0 :

Initial tank volume

𝛽:

Hydraulic fluid bulk modules

𝑉𝑡 :

Pipe volume between hydraulic turbine and turbine valve

𝛢:

Piston area

𝑉𝑝 :

Pipe volume between hydraulic pump and pump valve

𝐹0 :

Spring preload

𝑐:
𝑐𝑝𝑎𝑖𝑟 :
𝜂:
𝛾𝑎𝑖𝑟 :

Damping of friction coefficient associated with the piston
Air constant pressure specific heat [J/ kgK ]
Efficiency
Isentropic index for air

xx

ℎ:

Specific flow enthalpy [J]

𝑚̇:

Mass flow rate [kg/sec]

𝜏:

Torque [Nm]

𝜔:

TC angular velocity [rad/sec]

𝑁𝑇𝐶 :
𝐽𝑇𝐶 :

TC rotational speed [RPM]
TC shaft inertia [kg/m2]

𝛼:

VGT vane angle [radians]

𝜌:

Gas density[kg/m3]

ℜ:

Universal gas constant

A:

Geometric area[m2]

VGT:

Variable geometry turbocharger

EGR

Exhaust gas recirculation

RHAT

Regenerative hydraulic assisted turbocharger

REAT

Regenerative electric assisted turbocharger

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CHAPTER 1: INTRODUCTION
1.1 Motivation
Modern diesel engines are widely equipped with variable geometry turbocharger (VGT) and
exhaust gas recirculation (EGR). A control system for a diesel engine must meet driver’s torque
demand, meanwhile satisfying constraints on emission and fuel economy. Two devices (VGT
and EGR) can provide diesel engine with fresh air as well as required exhaust gas fraction. This
helps to meet the requirement of engine emission standard, fuel economy and transient
performance requirements. However, due to the natural coupling between VGT and EGR
systems, and high nonlinearity of diesel engine air-path, designing a robust and optimal
controller is challenging. Especially, regenerative and assisted turbocharger system adds another
actuator on turbocharger (TC) shaft, leading to additional complexity for controller design.
Model-based control is popular with the automotive research community. Not only it
reduces the design duration but also allows control engineers to gain insight into plant behaviour
without running engine in the test cell. For turbocharged diesel engine modelling, turbocharger
system is one of the critical subsystem models. Most of the previous modelling methods for
turbocharger utilized manufacturer based maps, which leads to non-physical extrapolation and
modelling errors. Especially for modelling variable geometry turbocharger, it involves 3dimensional map extrapolation based on traditional method. To mitigate the requirement of
accurate representation for the complex of physics system and simple structured control-oriented
modelling, physics-based turbocharger modelling is needed for diesel engine air-path system.
Considering hydraulic assisted and regenerative power on the turbocharger shaft, it provides
extra control input, which dramatically changes plant performance and controller design. In order

1

to understand the benefit and challenging for hydraulic assisted and regenerative turbocharger
(RHAT), coordinated control of the VGT-EGR-RHAT system is needed for improving engine
performance, reducing emissions and improving fuel economy. To design closed-loop controller,
system level analysis for RHAT system is needed, beforehand.
For the traditional diesel engine controller development, first, a static map is developed that
provides the desired steady fuel command as a function of engine speed and driver’s pedal
position. By analysing the emission, performance and fuel economy trade-offs, optimal setpoints for air-path actuators, such as VGT vane and EGR valve positions, are obtained with
respected to engine fuel injection and engine speed. During transient operations, it is clear that
the coupling effect between VGT and EGR is ignored, and the VGT vane position is used to
control boost pressure and EGR valve position is utilized to control EGR mass flow rate. To
achieve better tracking performance, coordinated control for VGT and EGR has enjoyed
increased interest. However, most of the current multivariable controls fail to take account of the
design target directly during the controller design process. There is no systematic approach of
designing controller for different performance targets, namely, handling trade-offs between
performance, emissions and fuel economy for closed-loop control design.

Hence, a new

coordinated control design approach is needed for diesel engine air-path control. The proposed
approach in this thesis can also be easily extended to VGT-EGR-RHAT systems.
1.2 Research Overview
1.2.1 Turbine modelling
Control-oriented VGT models are required for model-based control design. Typically, the
VGT turbine power is modelled using a fixed or a map-based turbocharger mechanical efficiency
with isentropic assumptions. However, the fixed efficiency approach could be over simplified,

2

leading to large modelling errors; and on the other hand, the map-based approach may suffer the
interpolation error between two VGT vane positions, especially with large extrapolation errors
when the turbine is operated outside the mapped region. In this thesis, physics-based models of
the turbine power and its power loss are modelled as functions of both VGT vane position and
turbine shaft speed, where the mechanical efficiency is defined as a function of the vane position.
As a result, the proposed model eliminates the interpolation errors, especially with smooth
extrapolation outside the mapped region. The proposed model is validated against test data sets
under both steady-state and transient operational conditions.
1.2.2 Compressor modelling
Control-oriented models for automotive turbocharger compressors typically describe the
compressor power, assuming an isentropic thermodynamic process with fixed isentropic and
mechanical efficiencies for power transmission between the turbine and compressor. Although
these simplifications make the control-oriented model tractable, they also introduce additional
errors due to un-modelled dynamics. This is especially true for map-based approaches since the
manufacture-provided maps tend to be sparse and often incomplete at the operational boundaries,
especially at operational conditions with low mass flow rate at low speed. Extrapolation scheme
is often used when the compressor is operated outside the mapped region, which introduces
additional errors. Furthermore, the manufacture-provided compressor maps, obtained using
steady-flow bench test data, could be quite different from these under pulsating engine flow
conditions. In this dissertation, a physics-based model of compressor power is developed using
Euler equations for turbo-machinery, where the mass flow rate and compressor rotational speed
are used as model inputs. Two new coefficients, speed and power coefficients, are defined. As a
result, this makes it possible to directly estimate the compressor power over the entire

3

compressor operational range based on a single analytic function. The proposed modelling
approach is validated against test data from standard turbocharger flow bench tests, standard
supercharger tests, steady-state and transient engine dynamometer tests. Model validation results
show that the proposed model has acceptable accuracy for model-based control design and also
reduces the dimension of the parameter space typically needed to model compressor dynamics.

1.2.3 Hydraulic assisted turbocharger modelling for controller design
A systematic modelling approach for engine air-path system with hydraulic assisted and
regenerative turbocharger system are proposed and presented in this thesis. Newly developed
turbocharger sub-models are integrated with engine air-path and EGR systems. Furthermore,
new modelling approaches for high speed hydraulic turbine and hydraulic centrifugal turbopump are proposed. This modelling approach also reduces the complexity of high speed
hydraulic turbomachinery. System level model integration and plant behavior investigation are
carried out for engine air-path system with hydraulic assisted turbocharger. The results show
proposed reduced order engine model has adequate accuracy and can be used for model-based
analysis and control design.
1.2.4 Hydraulic regenerative and assisted turbocharger system level investigation.
A regenerative hydraulic assisted turbocharger (RHAT) system is investigated in this
dissertation. For this new system, a hydraulic turbine is used to spin the turbocharger shaft via
high pressure supply hydraulic fluid; a turbo pump is used to absorb excessive power from the
turbocharger shaft while pressurizing the fluid and pumping it back into the supply tank. A
driveline pump is also used to recover vehicle kinetic energy during vehicle deceleration
operations and pump the fluid into the high pressure supply tank. Both hydraulic turbine and the
turbo pump are packaged inside the central house of the turbocharger accembly. Compared to

4

traditional electric assisted and regenerative turbocharger, RHAT has a much higher assist and
regenerative capability due to its high power density, and is more durable and cost effective. The
abundance hydraulic energy that is recovered during vehicle deceleration can be used to assist
the turbine so that VGT can operate at most efficient position with large opening rather than at
small opening (low efficient position) to meet the compressor power demand. With two
additional actuators on turbocharger shaft, VGT could be potentially replaced by the fixed
geometry turbocharger for reduced cost and improved durability with the same efficiency. A 1-D
medium duty turbocharged diesel engine model for a production vehicle was used in the
investigation. The hydraulic turbine, turbo pump, and driveline pump maps were obtained using
the 1-D hydraulic model provided by suppliers, respectively. A baseline controller was
developed and coupled with the 1-D model to control the engagement and disengagement of
RHAT system and manage the energy stored in the hydraulic supply tank. The 1-D simulation
demonstrates that the proposed RHAT turbocharger system can significantly improve engine
transient responses. The 1-D vehicle level simulation shows that 3-5% fuel economy
improvement for the FTP 75 driving cycle, depending on different sub-component sizes. The
study also identifies technical challenges for optimal design and operation of RHAT, as well as
additional fuel economy improvement opportunities from the RHAT system.

1.2.5 Gain-scheduling controller design for diesel engines with EGR-VGT system
Control design for a diesel engine air-path system equipped with variable geometry
turbocharger (VGT) and exhaust gas recirculation (EGR) is critical for engine performance
enhancement, emission reduction, and fuel efficiency improvement. The challenges for control
design in this class of dynamic system lies in the inherent coupling between VGT and EGR

5

systems and high nonlinearity of engine air-path system. A linear controller design approach is
proposed in this dissertation for simultaneously regulating boost pressure and EGR mass flow
rate. Linear quadratic controllers with integral action are designed based on the linearized
systems over the engine operational map. Controller is scheduled based on engine operational
parameters: engine speed and fuel injection amount. The gain-scheduling linear controller is
validated against the baseline controller using the nonlinear plant. Results show that designed
MIMO gain-scheduling controller can manage the tracking trade-offs between boost pressure
and EGR mass flow over the baseline controller (two individual loop SISO controllers). This
validates the proposed novel approach for designing controllers with trade-offs between engine
performance and emissions. This controller design approach is further extended to assisted and
regenerative turbocharger system with VGT and EGR.
1.3 Dissertation contributions
The following is a list of major contributions:
1. A new control-oriented turbine power model based on the turbine vane position, using
only turbine downstream conditions, is proposed, where the VGT vane position is a direct
input for turbine power. A generalized approach of identifying friction loss for the
proposed turbine model is proposed. The turbine power and its mechanical efficiency
models are suitable for the model-based VGT control due to the analytic nature of the
proposed models. The proposed model has adequate accuracy due to the introduction of
turbocharger dynamic friction.
2. A new physics-based model of compressor power is developed using Euler equations for
turbo-machinery, where the mass flow rate and compressor rotational speed are used as
model inputs. Two new coefficients, speed and power coefficients, are defined. As a

6

result, this makes it possible to directly estimate the compressor power over the entire
compressor operational range based on a single analytic relationship. Proposed model
largely reduces the number of parameters to be identified, compared to traditional
modelling approach. The reduced-order and reduced-complexity model is especially
useful for the control applications.
3. A system modelling approach for a reduced-order diesel engine air-path system with a
regenerative hydraulic assisted turbocharger is proposed. Newly developed turbocharger
sub-models are integrated with engine air-path and EGR systems. Furthermore, new
modelling approaches for high speed hydraulic turbine and hydraulic centrifugal turbopump are proposed. The proposed model can be used for model-based control design for
VGT-EGR system as well as VGT-EGR-RHAT system.
4. A system level investigating for hydraulic assisted and regenerative turbocharger systems
shows the benefits and challenges for the proposed system. The preliminary 1-D
simulation results demonstrate that the proposed RHAT turbocharger system can
significantly improve engine transient responses. The vehicle level simulations show that
3-5% fuel economy improvement for the FTP 75 driving cycle, depending on different
sub-component sizes. The study also identifies technical challenges for optimal design
and operation of RHAT systems, as well as additional fuel economy improvement
opportunities.
5. A linear control design scheme for diesel engine air-path system is proposed to handle

engine performance and emission trade-offs for closed-loop controller design. It can not
only regulate the boost pressure and EGR mass flow rate to their desired values with the
proposed coordinated control for EGR-VGT system but also design the closed-loop

7

controller to achieve different design targets, such as transient response performance and
emission target by LQ weighting selection. Gain-scheduling based on engine speed and
fuel injection quantity is used to extended linear controller design to the nonlinear engine
plant. The simulation validation results show that the designed controller has high
flexibility for different performance targets, compared to the baseline controller. The
proposed control design process has potential to significantly reduce efforts for early
prototyping control design and calibrations. Furthermore, a coordinated VGT-EGRRHAT controller is designed for diesel air-path system based on the same scheme.
1.4 Dissertation outline
The organization of dissertation is shown in Error! Reference source not found.. The
control-oriented VGT turbine model and the associated friction identification are investigated in
Chapter 2; and the control-oriented compressor model is investigated in Chapter 3.

The

turbocharged diesel engine model with the regenerative hydraulic assisted turbocharger (RHAT)
is developed in Chapter 4; and its system level simulation investigation results using the high
fidelity 1-D engine and vehicle model is presented in Chapter 5. Finally, a multivariable gainscheduling control design approach is developed for both traditional VGT-EGR and proposed
VGT-EGR-RHAT systems in Chapter 6. The last Chapter adds some conclusions and future
work.

8

Figure 1. Dissertation outline

9

CHAPTER 2: CONTROL ORIENTED VGT TURBINE POWER MODELS FOR
TURBOCHARGED ENGINE
2.1 Abstract
Control-oriented models of Variable Geometry Turbochargers (VGT) are required for modelbased VGT control. Typically, the VGT turbine power is modelled using a fixed or a map-based
turbocharger mechanical efficiency with isentropic assumptions. However, the fixed efficiency
approach could be over simplified, leading to large modelling error; and on the other hand, the
map-based approach may suffer the interpolation error between two VGT vane positions, and
especially the extrapolation inaccuracy when the turbine is operated outside the mapped region.
In this chapter physics-based models of the turbine power and its power loss are modeled using
both the VGT vane position and turbine shaft speed as inputs, where the mechanical efficiency is
defined as a function of the vane position. As a result, the proposed model eliminates the
interpolation error and especially allows smooth extrapolation outside the mapped region. The
proposed model is validated against a few test data sets under both steady-state and transient
operational conditions.
2.2 Introduction
Variable Geometry Turbocharger (VGT) is common in modern diesel engines. Benefits of
using VGT over traditional Fixed Geometry Turbocharger (FGT) in diesel engines have also
been long established [1]-[4]. Federally mandated emission standards on Nitrogen Oxides (NOx)
have forced the utilization of the exhaust gas recirculation (EGR) in diesel engines. This
introduces an additional coupling between the VGT vane position and EGR and it has been a
topic of active research within the engine control community [5]-[8] for the last two decades.

10

Most of control-oriented turbocharger models describe using turbocharger efficiency maps;
the associated air-path models of VGT-EGR diesel engines calculate the turbine power based
upon the ideal isentropic expansion assumption; and the overall turbocharger (TC) efficiency,
used to obtain the power available to the compressor, is based on a family of manufacture
provided maps (one map corresponding to one VGT vane position). However, TC performance
maps, provided by the manufactures, are often sparse or not available (see Figure 2) at light
engine load. A comparison of turbine operational ranges obtained from turbine flow bench,
engine steady-state and transient dynamometer tests is shown in Figure 2. It is clear that the
operational range of the flow bench steady-state tests does not match with the actual steady-state
and transient operational ranges. As a result, operating the TC outside its performance map
requires extrapolation under the assumption of a convex hull using the base map, leading to
several investigations in extrapolation methods; see [8]-[10],[16] and [18]. Usually, empirical
fitted models (usually 2nd or 3rd order) as a function of blade speed ratio (BSR) are used for
extrapolation when the turbocharger is operated outside the provided map. This could lead to
physically impossible values and extremely large modelling errors. Note that the typical
performance map or the data set, used to calibrate and validate the model, is provided by the TC
manufacture and based on the data from hot gas flow bench tests. Since these tests are performed
under steady flow conditions, the mapped data cannot match the pulsating flow operational
conditions when the TC is coupled to an internal combustion (IC) engine [1],[2],[14],[15].

11

a. Turbine pressure ratio range

b. Turbine mass flow rate range

Figure 2. Normalized turbine steady-state and transient operational ranges vs. the hot gas flow
bench test range

Also, the turbine efficiency maps provided by turbine manufactures combine the turbine
efficiency with mechanical loss one. Note that it is difficult to measure mechanical loss
[17],[29],[29] and it is not required to be measured (see the test code described in [26]) in the
standard flow bench tests. As a result, enthalpy change across the compressor is calculated as the
turbine output power. This makes the calculated turbine efficiency depend on the compressor
characteristics and the measured turbine efficiency is not the actual turbine efficiency since it is a
combination of both turbine and mechanical efficiencies. This makes turbine performance map
obtained from flow bench different from the turbine characteristics when it is coupled with
engine as show in Figure 2. In the automotive turbocharger system, turbine extracts energy from
engine exhaust to drive compressor for increased boost pressure. Note that in this application the
available turbine power due to the exhaust gas expansion process needs to overcome both
bearing mechanical and heat transfer losses to drive the compressor; see Figure 3. Friction loss is
due to both journal and thrust bearings in radial and axial directions; and heat transfer loss also

12

drives turbocharger operation away from the adiabatic behavior. Both losses affect turbine
efficiency. However, under transient engine operations, friction loss dominates the turbocharger
behaviors, and especially, under fast transient operations, thrust friction loss is determined by the
balanced thrust force between turbine and compressor and it is not available from TC
manufactures. Hence, turbine efficiency map provided by the manufacture is not suitable for
transient operations. Most of the existing control-oriented turbocharger model assumes constant
mechanical loss efficiency for simplicity [2],[5],[7]-[10]. Note that the axial friction due to the
thrust bearing load and radial friction due to journal bearing cannot be directly measured on the
hot gas flow bench, which makes it challenge to model the mechanical loss directly.

Figure 3. Turbocharger system structure

Another issue for map-based model is that the required interpolation and extrapolation of the
manufacture TC performance map to obtain turbine efficiency at the current operational
condition leads to multiple dimensional lookup tables with at least three inputs (pressure ratio of
upstream and downstream, reduced mass flow rate, and VGT vane position). This could
introduce additional computational burden for real-time control. Further issues with map based

13

traditional isentropic modelling; it needs both upstream and downstream conditions, either for
pressure or temperature. This needs extra dynamics states for engine air-path modelling or extra
sensors for real time application.
Finally, map-based model is not applicable to the assisted and regenerative turbocharger
since the operation ranges of turbine and compressor are quite different from the turbocharger
without assisted power. With the assisted power on the TC shaft, the turbine could operate at the
condition with much lower pressure ratio and higher shaft speed, leading to operating the turbine
operation outside of its traditional operation range. Also with assisted power on TC shaft, the
thrust friction torque could change dramatically since the thrust force direction could vary during
the transient operations. This is due to the fact that thrust force balance between turbine and
compressor is a function of the assisted or regenerative load on the TC shaft. This results in
different thrust forces for thrust bearing, and therefore the mechanical loss is different from the
non-assisted turbocharger.
Hence, turbocharger modelling for turbine, compressor and mechanical loss need to be
separated based on their own physics operation principles. In open literatures, some researchers
[23] proposed a fluid dynamics based approach to model turbine power rather supplier’s map.
The advantage of this model is compactness of its form. It needs only upstream conditions rather
than map based approach, which needs both upstream and downstream conditions. However, this
model needs rotor inlet conditions (after vane nozzle), which are difficult for measurements.
Further, this model also needs mechanical loss model to fulfill mechanical efficiency modelling.
No one has investigated the turbine downstream conditions for turbine modelling. Since, turbine
downstream conditions are shared with both turbine and after-treatment system. Hence, turbine

14

downstream conditions would be a good model inputs candidate for turbine modelling, as well as
serving as model inputs of after-treatment for model-based control.
With the turbine modelling issues addressed above, this chapter proposes to develop a
physics-based control-oriented model for the turbine power based on turbine downstream
conditions and a general method to identify turbocharger mechanical loss to accurately model the
compressor power under both steady-state and transient conditions. The proposed modelling
approach can be directly applied to power-assisted turbocharger. The fundamental Euler turbine
equation is used to model the turbine power [1],[2],[4],[19],[20] by incorporating flow and rotordynamics. Three different shaft mechanical loss models (only friction loss in this study) are also
evaluated. In order to improve the shaft speed transient dynamic model, shaft mechanical loss is
modeled based on both thrust and axial friction torques. The turbine power and mechanical loss
models are validated with given compressor power and the compressor power calculation is
based on the standard isentropic compression assumption with measured compressor upstream
and downstream conditions (temperature, pressure, compressor mass flow rate). Turbocharger
rotor dynamic equation is used to couple the compressor, turbine, and mechanical loss powers
during steady-state and transient operations. The main contributions of this dissertation are twofold: a new control-oriented turbine power model as a function of input turbine vane position
based on only turbine downstream conditions and a generalized approach of identifying friction
loss for the proposed turbine model.
The rest of this chapter is organized as follows. Section II discusses the turbine modelling
using the Euler equation with three mechanical loss candidate models and Section III provides
both steady-state, transient validation results. The last section adds some conclusions.

15

2.3 VGT Turbine Power Model-based on Euler Turbine Equation
2.3.1 Euler turbine equation
The Euler turbine equation bridges the power added to (or removed from) the flow and the
characteristics of rotating blades as described in literature [1],[19],[21],[27]. The Euler equation
is based on the concepts of conservation of angular momentum and conservation of energy and it
can be illustrated in Figure 4.
uout
u in

wout

m in

w in

m out

vout

v in
rin



rout

Figure 4. Simplified turbine model [27]
Applying conservation of angular momentum, it is noted that the turbine torque τ should
equal to the rate of change of angular momentum in a stream tube that flows through the turbine:
τ = 𝑚̇(𝑣𝑖𝑛 𝑟𝑖𝑛 − 𝑣𝑜𝑢𝑡 𝑟𝑜𝑢𝑡 )

(2.1)

Where, vin and vout are the inlet and outlet turbine flow velocities, respectively; rin and
rout are the radius of the inlet and outlet flow with respect to the rotation axis, respectively; and
ṁ is the mass flow rate. The work per unit time Ẇ, or power can be defined as

𝑊̇ = 𝜔𝜏 = 𝜔𝑚̇(𝑣𝑖𝑛 𝑟𝑖𝑛 − 𝑣𝑜𝑢𝑡 𝑟𝑜𝑢𝑡 )

(2.2)

where 𝜔 is the turbine speed. Now consider the steady flow energy equation
𝑄̇ − 𝑊̇ = 𝑚̇∆ℎ 𝑇

(2.3)

where∆ℎ𝑇 is the enthalpy change across the turbine, and 𝑄̇ is turbine heat transfer rate. Assuming
an adiabatic process for the gas expansion process inside turbine [27] with 𝑄̇ = 0 leads to

16

𝑊̇ = 𝑚̇(ℎ𝑖𝑛 − ℎ𝑜𝑢𝑡 )

(2.4)

Combining the above equation with the conservation (6) yields
𝑊̇ = 𝑚̇(ℎ𝑖𝑛 − ℎ𝑜𝑢𝑡 ) = 𝑚̇𝜔(𝑣𝑖𝑛 𝑟𝑖𝑛 − 𝑣𝑜𝑢𝑡 𝑟𝑜𝑢𝑡 )

(2.5)

Since under the ideal gas assumption that the constant pressure specific heat value 𝑐𝑝 for the
flow is a constant [22], we have
𝑊̇ = 𝑚̇𝑐𝑝 (𝑇𝑇1 − 𝑇𝑇2 ) = 𝑚̇𝜔(𝑣𝑖𝑛 𝑟𝑖𝑛 − 𝑣𝑜𝑢𝑡 𝑟𝑜𝑢𝑡 )

(2.6)

Note that equation (2.6) is the well-known Euler turbine equation, that links the temperature ratio
(and hence the pressure ratio) across a turbine to the rotational speed and the change of
momentum per unit mass.

2.3.2 VGT turbine power model as a function of vane angle
For completeness the following turbine power equation is from [1] and [20]. A typical
variable geometry turbine is shown in Figure 5 along with their gas flow velocity triangles at
the turbine rotor inlet and outlet, see [1] and [21] for details. Using the Euler turbine equation
yields:
𝑊̇ = 𝜔𝜏 = 𝜔𝑚̇(𝑣𝑖𝑛 𝑟𝑖𝑛 − 𝑣𝑜𝑢𝑡 𝑟𝑜𝑢𝑡 ) = 𝑚̇(𝑣𝑖𝑛 𝜔𝑟𝑖𝑛 − 𝑣𝑜𝑢𝑡 𝜔𝑟𝑜𝑢𝑡 ) = 𝑚̇(𝑢1 𝑐𝜃1 − 𝑢2 𝑐𝜃2 )

(2.7)

where 𝜔 is the turbine rotational speed in radian per second; 𝜏is the turbine torque; and 𝑚̇ is
the turbine mass flow rate. Note that the swirl at the turbine exit [21] can be ignored under full
energy recovery assumption within the turbine, which leads to 𝑐𝜃2 = 0 the following power
equation:
𝑊̇ = 𝑚̇𝑢1 𝑐𝜃1

17

(2.8)

Figure 5. Variable geometry turbocharger layout and speed triangle [1, 20]

Note that the turbine is normally designed such that its radial velocity at the vane nozzle
outlet (𝑐𝑟1 in Figure 5) is the same as the axial velocity at the rotor exit ( 𝑐𝑟2in Figure 5), i.e.,
𝑐𝑟1 = 𝑐𝑎2 = 𝑐1 𝑐𝑜𝑠𝑎1. This assumption leads to the following velocity relationship:
cθ1 = 𝑐1 ∙ sin 𝛼1 = 𝑐𝛼2 ∙

sin 𝛼1
cos 𝛼1

= 𝑐𝛼2 ∙ tan 𝛼1

(2.9)

where Îą1 is the gas entry angle to the rotor and is determined by the vane guide blade angular
position controlled by the VGT position actuator. Since the mass flow rate at the rotor outlet is a
function of the exit geometric area𝐴2 , the gas velocity at the turbine outlet can be defined as:
cÎą2 =

𝑚̇
𝐴2 𝜌

=

𝑚̇ ℜ𝑇𝑇2
𝐴2 𝑝𝑇2

(2.10)

where 𝑇𝑇2 and 𝑃𝑇2 are turbine outlet temperature and pressure (see Figure 5). Note that the
turbine outlet area of gas flow 𝐴2 can be calculated using the turbine outlet and nut diameters
𝐷𝑡2 and 𝐷𝑡𝑛 as follows,

18

A2 =

π 2
(D − D2tn )
4 t2

(2.11)

and the blade tip speed 𝑢1 in equation (8) can be calculated using the turbine wheel
diameter𝐷𝑡1 below.
𝑢1 = 𝜔

𝐷𝑡1
2

(2.12)

Combining equations (8) to (12), the turbine power can be defined as

𝑊̇ 𝑇 = 𝑚̇𝜔

𝐷𝑡1
𝑚̇
ℜ𝑇𝑇2
𝐷1
ℜ𝑇𝑇2
𝑡𝑎𝑛 𝛼1 = 2𝜔𝑚̇2
𝑡𝑎𝑛 𝛼1
2
2
2 𝜋 (𝐷 2 − 𝐷 2 ) 𝑃𝑇2
𝜋(𝐷
−
𝐷
)
𝑡𝑛 𝑃𝑇2
𝑡2
𝑡𝑛
4 𝑡2

(2.13)

𝑇𝐶
Now substituting the TC shaft speed, 𝜔 = 2𝜋𝑁
, exhaust mass flow rate, 𝑚̇ = 𝑚̇𝑒𝑥 ,, the power
60

and torque produced by the turbine can be expressed as:

𝑊̇𝑇 =

1
𝐷𝑡1
ℜ𝑇𝑇2
𝑁𝑇𝐶 (𝑚̇𝑒𝑥 )2 2
𝑡𝑎𝑛 𝛼1
2
15
𝐷𝑡2 − 𝐷𝑡𝑛 𝑃𝑇2

(2.14)

2
𝐷𝑡1
ℜ𝑇𝑇2
(𝑚̇𝑒𝑥 )2 2
tan 𝛼1
2
𝜋
𝐷𝑡2 − 𝐷𝑡𝑛 𝑃𝑇2

(2.15)

τT =

Using equations (14) and (15), turbine power and torque can be easily obtained. This model
can be directly applied to turbocharger shaft rotor dynamics equation or turbocharger shaft speed
differential equation. This finding is in line with model proposed in [23] as shown in (4).
𝑊̇ 𝑇 =

1
1 ℜ𝑇 ∗
𝑁𝑇𝐶 (𝑚̇𝑒𝑥 )2
tan 𝛼1
60
𝐵 𝑃∗

(2.16)

where B is the turbine inlet clearance as shown in Figure 5; and 𝑇 ∗ ,and 𝑃∗ , are temperature
and pressure at rotor inlet (location 1) in Figure 5, respectively. The advantage of equations (14)
and (15) is that it only depends on the downstream conditions of turbine. Note that, for
traditional turbine efficiency model, at least one pair of upstream and downstream conditions are

19

(temperature or pressure) required for turbine efficiency model. However, rotor inlet conditions
are often not available for measurement. Furthermore, when turbocharger is coupled with the
after-treatment system, downstream conditions are critical for modelling and control of both
turbocharger and after-treatment system. Hence, when the downstream measurements are
available, equations (14) or (15) are more practical than (16) for both turbocharger and aftertreatment system. In this case, turbine power is a direct function of control input-VGT vane
position. This reduces the complexity for VGT turbocharger controller design.
2.3.3 Turbocharge friction model
Power transmission between turbine and compressor is coupled with turbocharger
mechanical loss that is a combination of both heat transfer and friction losses. In this study, only
friction loss is considered. The turbocharger friction consists of mainly two parts: radial and
axial direction ones associated with radial and thrust bearings, respectively. The bearing system
and the forces on the rotor are shown in Figure 3 . Therefore, the associated mechanical loss
𝑊̇𝑙𝑜𝑠𝑠 (also called the shaft loss) can be expressed as
𝑊̇𝑙𝑜𝑠𝑠 = 𝑊̇𝑗𝑜𝑢𝑟𝑛𝑎𝑙 + 𝑊̇𝑡ℎ𝑟𝑢𝑠𝑡

(2.17)

where, 𝑊̇𝑗𝑜𝑢𝑟𝑛𝑎𝑙 and 𝑊̇𝑡ℎ𝑟𝑢𝑠𝑡 are the losses associated with the journal and thrust bearings,
respectively. Note that journal bearing friction depends on shaft angular speed, oil film thickness
as well as oil viscosity; and thrust bearing friction depends on both thrust force from axial
direction and shaft angular speed. Thrust force is due to the pressure difference between the
compressor inlet pressure 𝑃𝐶1 and turbine outlet pressure 𝑃𝑇2 (see Figure 3 ). Note that the
impulse force in the axial direction due to the axial direction flow inside both compressor and
turbine wheels is normally ignored in the model. The compressor wheel action force 𝐹𝐶 due to

20

𝑃𝐶1 and turbine wheel action force𝐹𝑇 due to 𝑃𝑇2 (see Figure 3 ) will be used to determine the
thrust bearing load. There are three existing modelling approaches from literature shown in Table
1, where Model 1 and Model 3 lump the thrust and journal bearing friction losses into a
polynomial of turbocharger shaft speed; and Model 2 separates the thrust and journal bearing
friction losses by considering the turbine upstream pressure (𝑃𝑇1 ) and compressor downstream
pressure (𝑃𝐶2 ) (see Figure 3 ). However, in order to capture turbocharger transient behavior
more accurate, in this study, another friction model is proposed for transient operations as in
Model 4. A transient dynamic compensation term c3 |ṄTC | for dynamic operation is added to
Model 2. This term is set as a tuning parameter for unmodeled transient dynamics. For steady
state operation, Model 2 and Model 4 are identical, since ṄTC = 0. All the four friction models
will be investigated in next section for steady state and transient operation. For simplicity, all
friction model coefficients are assumed to be constant in this study. The heat transfer loss,
bearing oil film thickness, lubrication oil viscosity as well as oil temperature variations are not
considered in this study.
Table 1. Friction model candidates
Model 1 [2]

𝑊̇𝐿𝑜𝑠𝑠 = 𝑐1 (𝑁𝑇𝐶 )2 + 𝑐2 𝑁𝑇𝐶 + 𝑐3

Model

𝑊̇𝐿𝑜𝑠𝑠 = 𝑊̇𝑗𝑜𝑢𝑟𝑛𝑎𝑙 + 𝑊̇𝑡ℎ𝑟𝑢𝑠𝑡 , where 𝑊̇𝑗𝑜𝑢𝑟𝑛𝑎𝑙 = 𝑐1 (𝑁𝑇𝐶 )2 and

Model 2 [29]
Model 3 [23]
Model 4 (this work)

𝑊̇𝑡ℎ𝑟𝑢𝑠𝑡 = 𝑐2 3√𝑃𝑇1 − 𝑃𝐶2 (𝑁𝑇𝐶 )2
𝑊̇𝐿𝑜𝑠𝑠 = 𝑐1 (𝑁𝑇𝐶 )2
𝑊̇𝐿𝑜𝑠𝑠 = 𝑐1 (𝑁𝑇𝐶 )2 + 𝑐2 3√𝑃𝑇1 − 𝑃𝐶2 (𝑁𝑇𝐶 )2 + 𝑐3 |𝑁̇𝑇𝐶 |

21

2.4

Model Validation and Mechanical Loss Identification

2.4.1 Model validation using steady-state engine test data and mechanical loss estimation
After the turbine power and friction models are addressed, the fraction of the turbine power
made available to the compressor can be determined. Note that under normal operating
conditions the power balance between the turbine and compressor of a turbocharger is
determined by the following governing equation, which includes shaft kinetic energy.

ωJTc

𝑑𝜔
= 𝜂𝑚 𝑊̇𝑇 − 𝑊̇𝐶
𝑑𝑡

(2.18)

The mechanical loss is typically defined as a fraction of the source power (or turbine power)
and modeled by the source efficiency 𝜂𝑚 , so the governing equation can be expressed as
ωJTc

𝑑𝜔
𝑑𝑡

= 𝑊̇𝑇 − 𝑊̇𝐿𝑜𝑠𝑠 − 𝑊̇𝐶 , where 𝑊̇𝐿𝑜𝑠𝑠 = (1 − 𝜂𝑚 )𝑊̇𝑇

Under steady-state operation

𝑑𝜔
𝑑𝑡

(2.19)

= 0 leads to
𝑊̇𝐶 = 𝑊̇𝑇 − 𝑊̇𝐿𝑜𝑠𝑠 = 𝜂𝑚 𝑊̇𝑇

(2.20)

In this study, compressor power is assumed to be known by solving stand isentropic power
equation using its upstream condition and downstream conditions. Standard compressor power
(isentropic power corrected for isentropic efficiency) [1] can be expressed by
𝛾𝑎𝑖𝑟 −1
𝛾𝑎𝑖𝑟

1

𝑃𝐶2
𝑊̇𝑐 = 𝑖𝑠 𝑚̇𝑎𝑖𝑟 𝑇𝐶1 𝑐𝑝𝑎𝑖𝑟 (( )
𝑃𝐶1
𝜂𝐶

− 1)

(2.21)

Substituting (24) and (18) into (23) yields:
𝜂𝑚 𝑡𝑎𝑛 𝛼1 =

1

𝑃𝐶2

𝜂𝐶

𝑃𝐶1

𝑎𝑖𝑟
((
𝑖𝑠 𝑚̇𝑎𝑖𝑟 𝑇𝐶1 𝑐𝑝

)

𝛾𝑎𝑖𝑟 −1
𝛾𝑎𝑖𝑟

− 1) 𝛤 −1 , where, 𝛤 =

1
15

𝑁𝑇𝐶 (𝑚̇𝑒𝑥 )2

𝐷𝑡1

ℜ𝑇𝑇2

2 −𝐷2 𝑃
𝐷𝑡2
𝑡𝑛 𝑇2

(2.22)

From equation (22) it can be found that the mechanical efficiency, 𝜂𝑚 , depends on the
turbocharger operational condition. It is not a constant and can be identified by varying VGT

22

vane position 𝛼1 , assuming that all parameters on the right hand side (RHS) of equation (22) are
available. This also indicates that the turbocharger mechanical loss can be solved analytically
using the proposed physics-based model. This could potentially separate actual turbine efficiency
from the measured turbine efficiency using based on the supplied turbine map. The next step is
to find an empirical relationship between𝜂𝑚 and other turbocharger operational parameters so
that the turbine mechanical efficiency can be predicted. Note that the left hand side (LHS) term
𝜂𝑚 𝑡𝑎𝑛 𝛼1

is a function varying VGT vane positions. A heavy-duty turbocharged diesel engine is

used as an example to study the relationship, where the engine parameters and the experimental
set-up can be found in [24] and [25]. The entire engine operating range is mapped via 195
steady-state speed and load points that provide enough data for conduction turbocharger power
balancing study. The ideal gas parameters (such as isentropic index of air, constant pressure
specific heat of air, and ideal gas constant) are from [26]. Vane position is based on turbocharger
control signal between 0% to 100%. Figure 6 plots the 195 values (𝜂𝑚 𝑡𝑎𝑛 𝛼1 ) calculated using the
RHS term of equation (24) as a function of the VGT position𝛼1 , where 0% is associated with the
fully opened position. Each point on this plot represents a unique steady-state operational
condition for the power balanced turbocharger. Since 𝑡𝑎𝑛 𝛼1 is a constant for a given vane
position, the variations observed in Figure 6 must be from 𝜂𝑚 . This confirms that the
turbocharger mechanical efficiency is not a constant, which is discussed in the introduction
section.

23

 *tan
m

1

15

m*tan 1

10

5

0

0

20
u

vgt

Figure 6.

 m tan 1

40
60
80
(VGT closing precentage)

100

with respect to the VGT position

From Figure 6, it is also clear that for vane positions in the range between fully open (or 0%
closed) to about 60% open (or 40% closed) the variation in𝜂𝑚 is small. This is an important
observation since it indicates that with large vane openings (with unrestricted flow across turbine)
the steady state mechanical efficiency of the TC (or the overall efficiency) is a constant for a
given vane position regardless of the operational condition of the VGT-engine system. Note that
constant 𝜂𝑚 was used by several researchers in literature [5], [7]. However, as the vane position
is gradually closed to less than 60% open, flow restriction is introduced and as a result the TC
mechanical efficiency shows large variability. Figure 6 shows large variability in 𝜂𝑚 with varying
TC speed for a fixed vane position. Unfortunately this also clearly indicates that for more
restricted flows the constant 𝜂𝑚 assumption may introduce large error. Therefore, for the region
of restricted flow and high TC shaft speed, the loss is a function of shaft speed and a new model
for the mechanical loss should be developed to include other sensitivities. Based on the results in
Figure 7, where an obvious correlation between shaft speed and the variability can be seen, this
is in line with findings in friction models shown in Table 1.

24

 *tan
m

1

15

m*tan 1

10

5

0
0
5
x 10

4

100
10

50
15

0
u

Shaft Speed [RPM]

Figure 7.

 m tan 1

vgt

(VGT closing precentage)

with respect to the VGT position and turbocharger speed

To determine the coefficients in both turbine and friction model (power loss) models defined
in Table 1 a cost function is proposed in (23).
2
J = ∑𝑛𝑖=1|𝑊̇𝑇_𝑐𝑎𝑙𝑐 (𝑖) − (𝑊̇𝑐 (𝑖) + 𝑊̇𝑙𝑜𝑠𝑠 (𝑖))|

(2.23)

where 𝑊̇𝑇_𝑐𝑎𝑙𝑐 is the calculated turbine power defined in (14) or (16); 𝑊̇𝑐 and 𝑊̇𝑙𝑜𝑠𝑠 are
defined in equation (21) and Table 1, respectively; and n is the number of turbine testing points.
The test data set used for optimization covers the entire VGT vane position. Ideally, the
exact relationship between vane angle 𝛼1 and the VGT actuator position can be obtained for a
given turbocharger; see Figure 5. That is, 𝑡𝑎𝑛 𝛼1 should be known for each given VGT vane
position. However, for this study, the VGT vane position signal is not available and the VGT
actuation duty cycle is used for estimating 𝛼1 directly. Based on the kinematic model of the
linkage mechanism from the VGT actuator to its vane (See Figure 8 for a typical VGT actuation
mechanism), the following polynomial relationship(2.24) from the vane position control dutycycle to the actual vane angle 𝛼1 is used.

25

Figure 8. VGT actuation mechanism
α1 = 𝑓(𝑢𝑣𝑔𝑡 ) = 𝑐4 𝑢3𝑣𝑔𝑡 + 𝑐5 𝑢2𝑣𝑔𝑡 + 𝑐6 𝑢𝑣𝑔𝑡 + 𝑐7

(2.24)

Therefore, the maximal number of optimization parameters is up to six, where up to three ( 𝑐1
to 𝑐3 from the friction model (see Table 1) and four (𝑐4 to 𝑐7 ) from vane position model in (24). A
Least-Squares optimization method is used to find the optimal coefficients (𝑐1 to 𝑐7 ) to minimize
the cost function defined in (23). Since the vane position coefficients (𝑐4 to 𝑐7 ) are independent of
these for the friction model defined in Table 1, friction Model 1 is used to obtain optimal
coefficients (𝑐1 to 𝑐7 ) and the corresponding vane position coefficients (𝑐4 to 𝑐7 ) will be used for
initial conditions for all other three friction models. For steady state, derivative of TC speed is
zero. This makes Model 2 and Model 4 identical for steady state operation. The optimal value of
𝑐4 to 𝑐7

are shown in Table 2.

Table 2. Model validation results using steady-state engine test data for turbocharger 1

Friction model
coefficients

Vane angle coefficients

c1

Model 1
6.957×10-10

Model 2/Model 4
4.507×10-10

Model 3
6.958×10-10

c2

2.357×10-14

5.909×10-11

-

c3

7.5023×10-12

0.00

-

c4

1.011

0.961

1.003

c5

0.171

0.206

0.175

c6

0.349
0.572
0.572
1.478
4.38

0.344
0.572
0.572
1.471
4.23

0.349
0.556
0.556
1.460
4.38

c7

Vane angle[rad]

Min
Max

Model error[%]

-

26

Based on the calibrated vane angle model in Table 2, the VGT fully opened vane angle is
0.556-0.572 rad (31.85 degree to 32.77) that is corresponding to the zero position control
command (𝑢𝑣𝑔𝑡 = 0), and the fully closed vane angle is 1.460-1.478 rad (83.65degree to 84.68
degree) associated with the maximal (100 percent) control signal (𝑢𝑣𝑔𝑡 = 0.8). These calibrated
model coefficients agree well with the designed ones. The TC mechanical loss is plotted in
Figure 9, along with the corresponding VGT vane position. It is clear that the power loss
increases as the TC speed does for all three friction models and the maximal mechanical loss
over the entire engine operational range is 8.5 kW, which matches the test data in [29],[29]. The
deviation between Models 1 and 3 are not significant for steady-state operations since the
quadratic term in both models dominates friction loss. Since Model 2/Model 4 accounts for the
pressure difference, the friction power variation between 40k and 100k rpm is due to the thrust
load change. Note that, the friction variation is only for steady state operation for this validation.
For transient operation, thrust friction will increase due to unbalanced thrust force from both
turbine and compressor side.
As shown in Figure 2, max turbine pressure ratio for engine steady state is much smaller than
max of that for transient operation. Thus, during transient operation, thrust forces would be
higher due to increased pressure difference between turbine and compressor. Thus Model
2/Model 4 would have higher thrust friction due to higher pressure difference across
turbocharger. Dynamics compensation term in Model 4 will be further investigated through
transient test data in section D. Figure 9. Predicted turbine power subtracted by mechanical loss
and compressor power at steady-state (power balanced conditions) is shown in the third plot of
Figure 9. The error function is defined in (25). Admittedly, there are certain errors in this
modelling approach due to unknown relationship between VGT position and vane angle as well

27

as unmolded physics with simplification. From Table 2, it shows that averaged errors for
proposed models are about 4% error for steady state operation.
𝑛

𝑛

Error = (∑|𝑊̇𝑇_𝑐𝑎𝑙𝑐 (𝑖) − (𝑊̇𝑐 (𝑖) + 𝑊̇𝑙𝑜𝑠𝑠 (𝑖))|) (∑|𝑊̇𝐶 (𝑖)|)
𝑖−1

−1

(2.25)

𝑖=1

where n stands for total sample number. Model 2/ model 4 have smallest error. Model 1 and
Model 3 are equivalent for model accuracy. For the selection of mechanical loss model to model
or calculate friction loss, Model 2/ Model 4 should be used to capture accurate rotor dynamics
related with thrust friction. However, Model 2/Model 4 needs the two extra two inputs compared
to Model 1 and Model 3, which are turbine input pressure and compressor downstream pressure.
Without these pressure measurement inputs or dynamics state models, Model 3 should be chosen
for its simple structure. In summary, this section provides validation process of the proposed new
turbine power model, as well as a general procedure for identifying mechanical loss for steady
state engine operation.

28

Predicted vane angle

Vane angle [rad]

2
Friction model 1
Friction model 2
Friction model 3

1.5

Close

1
0.5
0

Open
0

0.1

0.2

0.3

0.4
0.5
VGT position

0.6

0.7

0.8

Predicted mechanical loss
10
Friction model 1
Friction model 2
Friction model 3

Power [kW]

8
6
4
2
0

2

3

4

5

6
7
TC Speed [rpm]

8

9

10

11
x 10

4

Predicted turbine power-friction loss
70
Friction model 1
Friction model 2
Friction model 3

Power [kW]

60
50

y=x

40
30
20
10
0

0

10

20

30
40
50
Compressor power [kW]

60

70

Figure 9. Predicted power loss, vane angle and turbine power at SS (steady-state)

2.4.2 Turbine model validation against standard hot gas flow bench test data
Secondly, the test data from the standard hot gas flow bench of a different turbocharger,
called turbocharger 2, is used to validate the turbine model. In this case, the turbine inlet is under
steady-state flow, which is quite different from the engine dynamometer test data under pulsation
flow. The detailed test setup can be found in [26]. The test points over the VGT operational
range are shown in Figure 10. The speed range for turbocharger 2 is between 5k and 150k rpm
and the pressure is between 1.1 and 5.0 bar. The turbine model described in subsection III.A is
used. Friction Model 1-3 were chosen for this study.

29

Turbine pressure ratio

5

4

3

2

1
1
0.8

15
0.6

10

0.4
5

0.2
VGT position

0

0

x 10

4

Turbocharger speed [rpm]

Figure 10. Turbocharger 2 test range
Table 3. Model validation results using steady-state flow bench test data for turbocharger 2
Model 1

Model 2/Model 4

6.988×10-10

6.513×10-10

c2

1.800×10-10

-4.8812×10-14

Model 3
6.990×1010
-

c3

1.000×10-10

0.00

-

c4

0.8765

1.107

1.085

c5

0.002476

-0.2495

-0.2191

c6

0.1581
0.2541
0.2541
0.972
3.23

0.2169
0.2479
0.2479
0.986
3.20

0.0121
0.2508
0.2508
0.995
3.33

c1

Friction model
coefficients

Vane angle
coefficients

c7

Vane angle[rad]
Error [%]

Min
Max
-

Note that for this study the vane angle equation (24) is used as a function of the percentage
VGT control duty cycle. The optimized mechanical loss and vane angle coefficients are shown in
Table 3 for four models, and the predicted mechanical loss and the predicted vane angle as
functions of speed and VGT position are shown in Figure 11. Since this turbocharger has a
maximum operational speed of 150k rpm, high friction loss (peaked at 16 kW) is expected. Error
function is defined in (25). The results are comparable with these presented in the previous
subsection. Is shows that the proposed turbine model is also suitable for the data set obtained

30

from steady-state flow bench tests and can be used to extrapolate turbine efficiency based on the
given turbine efficiency map.

Predicted vane angle

Vane angle [rad]

1.2
Friction model 1
Friction model 2
Friction model 3

1
0.8

Close

0.6
0.4
0.2

Open
0

0.2

0.4
0.6
VGT position

0.8

1

Predicted friction Loss

Power [kW]

20
Friction model 1
Friction model 2
Friction model 3

15
10
5
0
0.4

0.6

0.8

1
1.2
TC Speed [rpm]

1.4

1.6
x 10

5

Predicted turbine power-friction loss

Power [kW]

80
Friction model 1
Friction model 2
Friction model 3

60

y=x

40
20
0

0

10

20

30
40
50
Compressor power [kW]

60

70

80

Figure 11. Turbocharger 2 predicted turbine power
2.4.3 Turbine power model validation using GT-Power transient simulation data
In order to understand the model characteristics during transient operations, the proposed
model is calibrated using the transient responses obtained from the GT-Power model in this
subsection. The GT-Power model is developed for the engine and turbocharger 1 described in
subsection III.A, and the detailed GT-Power model and its simulation setup can be found in [25].
In the transient simulations, the engine follows the desired torque based acceleration pedal
position. The VGT feedback control is used to track the target calibrated boost pressure. The
purpose of this study is to study the model behavior under transient operational conditions.

31

First, the developed turbine model is calibrated using the FTP 75 simulation data to obtain
the vane angle model with the VGT control duty cycle as input, and then, the calibrated model is
used to predict the turbine power for US 06 driving cycle. Since the turbine map in GT-Power
simulation is based on the steady-state flow bench map, friction loss is not modeled in the GTPower model and the turbine power is an output in GT-Power simulations by interpolating or
extrapolating the manufacture provided efficiency map [28] . The turbine vane angle can be
calculated with equation (26):
tan 𝛼1 =

2𝜋

𝐷

𝑡1
(15 𝑁𝑇𝐶 (𝑚̇ 𝑒𝑥 )2 𝐷2 −𝐷
2
𝑡2

ℜ𝑇𝑇2

𝑡𝑛 𝑃𝑇2

−1

) 𝑊̇ 𝑠𝑖𝑚_𝐺𝑇

(2.26)

where, 𝑊̇𝑠𝑖𝑚_𝐺𝑇 is GT-Power simulated turbine power. The relationship between vane angle
and VGT control duty cycle is shown in Figure 12 and the following vane angle fitting is
obtained
𝛼1 = −3.358𝑢3𝑣𝑔𝑡 + 5.881𝑢2𝑣𝑔𝑡 − 1.594𝑢𝑣𝑔𝑡 + 0.58

(2.27)

where 𝑢𝑣𝑔𝑡 is VGT map position. Using the proposed turbine model equation (14) and fitted
vane relationship equation (27), the predicted turbine power can be obtained.
In order to show the consistence of the calibrated model, the model calibrated using the FTP
75 cycle is used to simulate turbine power under the US 06 driving cycle for the same engine.
The simulation results and the associated errors are shown in Figure 13 and Table 4, where the
power model error is defined in equation (28). Note that the error could be due to the map
extrapolation inaccuracy based on the empirical equation used in the GT-Power model; and the
unmodeled friction loss in the GT-Power model could also contribute to the error.
The above study shows that the proposed turbine power model can be used for modelling the
turbine transient operations.

32

𝑛

−1

𝑛

Error = (∑|𝑊̇𝑠𝑖𝑚_𝐺𝑇 (𝑖) − (𝑊̇𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 (𝑖))|) (∑|𝑊̇𝑠𝑖𝑚_𝐺𝑇 (𝑖)|)
𝑖=1

(2.28)

𝑖=1

Table 4. Averaged Error between proposed model predicted turbine power and GT simulated
turbine power
Driving cycle

Value

FTP 75

US 06

6.6

8.2

Error
[%]

1.8

1.6

Vane angle [Rad]

1.4

1.2

1

0.8

0.6

0.4

0.2

0.3

0.4

0.5
0.6
VGT map position

0.7

0.8

0.9

Figure 12. Vane angle and the VGT map position with GT simulation
70
FTP 75
US 06

Model Predicted Turbine Power [kW]

60

y=x+5

y=x

50
40

y=x-5

30
20
10
0
-10
-10

0

10
20
30
40
50
GT Simulated Turbine Power [kW ]

60

70

Figure 13. Model prediction results vs GT simulation results

33

Figure 14. Modelling error for FTP-75 in GT simulation

Figure 15. Modelling results for US_06 driving cycle

34

2.4.4 Model validation using vehicle test data
At last, vehicle test data is used to investigate the transient performance of the proposed
turbine and friction models. The calibrated turbine and friction loss models of Turbocharge 1 in
subsection III.A is used, where the turbine power is calculated based on (2.14) and the
mechanical loss and the vane angle models are from (2.25) and Table 2. Note that for transient
operations the turbocharger energy balance is defined in (2.18) and it is used to study the
proposed model accuracy related to the mechanical loss and the relationship between VGT
actuator position and the associated vane angle. For the model verification purpose, equation
(2.18) is reorganized below in equation (2.29).
𝑊̇ 𝑇 = 𝜔𝐽𝑇𝑐

𝑑𝜔
𝑑𝑡

+ 𝑊̇ 𝐶 + 𝑊̇ 𝐿𝑜𝑠𝑠

(2.29)

The turbine power defined in equation (2.29) is compared with that calculated using the
proposed model in equation (2.14). Note that the compressor power, 𝑊̇𝐶 , in equations (2.29) can
be calculated based on change in flow enthalpy (see equation (2.18)) that can be measured
directly, and four different mechanical loss (𝑊̇𝐿𝑜𝑠𝑠 ) models defined in Table 1 are used with the
calibrations shown in Table 2, where the transient compensation coefficient 𝑐4 in Model 4 is set
to 0.001 to improve transient operation characteristics. In order to compare the performance of
the proposed turbine model with the traditional map based one, the turbine model-based on map
interpolation and extrapolation in [8] is used as the baseline in this study and the comparison
results is shown in the top plot of Figure 16. Good agreements among turbine power calculated
from the physics-based model (2.14) and the turbine powers calculated using equation (2.29)
based on the experimental data with four friction models, while the turbine power from mapbased model [8] deviates significantly from these calculated based on (2.29). This indicates that

35

the proposed turbine power and TC mechanical loss models are adequate. From the top plot of
Figure 16, it clearly shows that the proposed physics-based turbine power model captures the TC
transient operations better than the map-based model, and also note that the map-based model is
reasonable when it operates within the map provided by flow bench test data and transient
operation data shown in Figure 2. However, turbine transient operation is quite different from
flow bench test range. Since the mapped data is not available at high TC speed with low pressure
ratio and at low TC speed with light load, large extrapolation error over both regions leads fairly
large model error (see the top plot of Figure 16), where the map-based turbine efficiency is
significantly lower than the actual one. The study also shows that the proposed model is able to
capture fast transient characteristics during tip-out and this is due to the fact that the proposed
turbine and friction loss models depend on the TC speed.
From the fourth plot (from top) of Figure 16, the results also agree with previous discussion
that the thrust friction increases as the pressure difference across turbine and compressor goes up.
The modelling errors of the four friction models are similar when the turbocharger operates
within the envelop of engine steady-state operations (see the second and third plots of Figure 16
for speed and pressure ratio); and the thrust friction increases significantly with high pressure
difference across turbine and compressor during transient operations, which leads to about 10%
mechanical efficiency. Also, the mechanical efficiency decreases similarly during the transient
tip-outs, leading to increased compressor modelling error; see Table 5. The error for the physicsbased model is defined in (2.30) and the error for the map-based model is the same as that in
(2.30) by setting 𝑊̇𝐿𝑜𝑠𝑠 (𝑖) = 0 since the mapped based model lumped the friction loss 𝑊̇𝐿𝑜𝑠𝑠
together with turbine power𝑊̇𝑇 . Note that large error of map-based model in Table 5 is due to the
map extrapolation error. Although the physics-based model is able to capture the turbocharger

36

dynamics over the entire operating range, certain errors exists that are due to measurement errors
(such as temperature sensor time constant effect for the transient measurement, flow
measurement error under light compressor load) and unmolded physics (such as heat transfer,
bearing friction loss due to oil viscosity variation).
−1
Error = (∑𝑛𝑖−1|𝑊̇𝑇 (𝑖) − (𝑊̇𝐶 (𝑖) + 𝑊̇𝑙𝑜𝑠𝑠 (𝑖) + 𝑊̇𝑘𝑖𝑛𝑒𝑡𝑖𝑐 )|)(∑𝑛𝑖=1|𝑊̇𝐶 (𝑖)|)

(2.30)

From Table 5 it can be observed that with the thrust load model (friction Models 2 and 4) the
turbine power accuracy can be improved by 5% and 6% for Models 2 and 4 under the transient
operations, respectively. Note that the steady-state operation study in Table 2 and Table 3 shows
no significant error difference. This shows the importance of including the thrust friction in the
turbine power model during transient operations. Even though the proposed turbine model is
derived based on a power balanced (steady-state) TC, with the help of the friction model
(especially Models 2 and 4) is can be also used under transient operations since the modelling
error is reasonable (10%) under transient operations. As a conclusion, the proposed physicsbased model with friction Model 4 reduce the model error under both steady-state and transient
operations, comparing to the conventional map-based model.
Table 5. Average error for different models

Transient
Error [%]

Mapbased
turbine
model

Physicsbased
turbine
model +
friction
Model 1

Physicsbased
turbine
model +
friction
Model 2

Physicsbased
turbine
model +
friction
Model 3

Physicsbased
turbine
model +
friction
Model 4

22.8

15.9

11.1

16.3

10.1

37

Figure 16. Model validation with transient vehicle test data

38

2.5 Conclusion
A physics-based turbine power model of a variable geometry turbocharger (VGT) is
proposed in this chapter along with the thrust friction model. The turbine power model is
derived based on the Euler turbine equation with the VGT vane position as the control parameter.
Three existing friction loss models and one newly proposed model are also investigated. All four
friction models have the potential of including the oil viscosity and heat transfer effect in the
model. The proposed turbine power model, along with the friction models are investigated
against two steady-state data sets (engine dynamometer test and flow bench test) and two
transient data sets (1-D GT-Power transient simulations and vehicle transient test data). The
steady-state data study shows that the proposed model is fairly accurate (with less than 4.5%
modelling error) and the four friction models provides similar modelling accuracy; and for the
transient data investigation, the proposed turbine power and acceleration based thrust friction
models are able to reduce the transient modelling error from 22.8% (conventional map-based
model) down to 10.1%. This indicates that thrust friction is a key to have an accurate transient
model. Note that the proposed the turbine power and its mechanical efficiency models are
suitable for the model-based VGT control due to the analytic nature of the proposed models as a
function of the VGT vane angle.

39

CHAPTER 3: A REDUCED COMPLEXITY MODEL FOR THE COMPRESSOR POWER OF
AN AUTOMOTIVE TURBOCHARGER
3.1 Abstract
Control-oriented models for automotive turbocharger compressors typically describe the
compressor power assuming an isentropic thermodynamic process with fixed isentropic and
mechanical efficiencies for power transmission between the turbine and compressor. Although
these simplifications make the control-oriented model tractable, they also introduce additional
errors due to un-modeled dynamics. This is especially true for map-based approaches since the
manufacture-provided maps tend to be sparse and often incomplete at the operational boundaries,
especially at operational conditions with low mass flow rate and low speed. Extrapolation
scheme is often used when the compressor is operated outside the mapped regions, which
introduces additional errors. Furthermore, the manufacture-provided compressor maps, based on
steady-flow bench tests, could be quite different from these under pulsating engine flow. In this
chapter, a physics-based model of compressor power is developed using Euler equations for
turbo-machinery, where the mass flow rate and compressor rotational speed are used as model
inputs. Two new coefficients, speed and power coefficients, are defined. As a result, this makes
it possible to directly estimate the compressor power over the entire compressor operational
range based on a single analytic relationship. The proposed modelling approach is validated
against test data from standard turbocharger flow bench tests, standard supercharger tests,
steady-state and transient engine dynamometer tests. Model validation results show that the
proposed model has acceptable accuracy for model-based control design and also reduces the
dimension of the parameter space typically needed to model compressor dynamics.

40

3.2 Introduction
It is common for plant models, used in the air-path control of turbocharged diesel engines, to
assume that the ideal power consumed by the compressor is defined by an isentropic
thermodynamic process. The actual power is then derived from either a compressor isentropic
efficiency map [1] or an empirically fitted isentropic efficiency map [8]. A common alternative
approach is to define the compressor power as a first order dynamics using an ad-hoc time
constant with the turbine power as the input [5], [7]. On the other hand, map-based compressor
power models are relied on the overall Turbo-Charger (TC) system efficiency maps as a function
of the vane positions [32] applied to the calculated turbine power. Empirically fitted compressor
efficiencies are typically 2nd or 3rd order polynomials of the Blade Speed Ratio (BSR). The
polynomial coefficients are often dependent on the shaft speed [8]-[12] and are identified against
the populated regions of the manufacturer-supplied maps. These polynomial models are,
therefore, also subject to extrapolation when used under operational conditions outside the
manufacturer-supplied test points. Note that the typical manufacturer-supplied compressor
performance map is based on hot-gas flow-bench test data under steady flow conditions. Hence,
actual maps could deviate from the manufacture-supplied ones under pulsating flow when the
compressor is coupled to an Internal Combustion Engine (ICE) [1], [2], [33]. Another issue,
often encountered when operating at light load conditions, is the sparsity of the manufacturersupplied compressor performance maps in these operational areas, indicated in Figure 17 for a
sample turbocharged engine. When operating the compressor outside the mapped region,
extrapolation under some smoothness constraint becomes necessary. Several investigations into
extrapolation based on map extension are reported in open literature; see [16], [18].

41

Figure 17. Operating range deficit between mapped and desired engine operating range.
The traditional actual compressor power is computed from the isentropic efficiency in equation
(3.1).
̇ 𝛾𝑎𝑖𝑟 −1
𝛾𝑎𝑖𝑟
1
𝑃
𝑜𝑢𝑡
𝑊̇𝑐 = 𝑖𝑠 𝑚̇𝑐 𝑇𝑖𝑛 𝑐𝑝𝑎𝑖𝑟 ((
)
− 1)
𝜂𝑐
𝑃𝑖𝑛

(3.1)

This compressor power model (𝑊̇𝑐 ) relies on a number of measured inputs: the compressor
inlet temperature (𝑇𝑖𝑛 ), the mass flow rate (𝑚̇𝑐 ), and the inlet and outlet pressures (𝑃𝑖𝑛 and 𝑃𝑜𝑢𝑡 ),
respectively. Additionally, the air properties, such as specific heat ( 𝑐𝑝𝑎𝑖𝑟 ) and isentropic index
(𝛾𝑎𝑖𝑟 ), are assumed to be fixed. The isentropic efficiency (𝜂𝑐𝑖𝑠 ) used in (1) is either an empirical
model or is available from mapping data based flow bench tests. In a review of existing literature,
there are various empirical models used to describe the isentropic efficiency. In [10], the
efficiency is expressed as a third-order polynomial function of the non-dimensional mass flow
rate, φc , that is a function of the mass flow rate and the blade tip speed. The polynomial
coefficients ai are three individual functions of inlet Mach number. In [37], the efficiency is
modeled using an elliptical fit based on mass flow rate and pressure ratio. This model depends on

42

the maximum efficiency in terms of corrected mass flow rate and compressor pressure ratio. In
[34], the efficiency model from [37] is further modified for pressure ratio variations. In [36],
compressor work is defined as a polynomial function of corrected mass flow rate and the model
is further fitted with experimental data. Table 6 summarizes these approaches and presents
results from a previous performance assessment [39] of these models inside the manufacturesupplied maps. However, it has been well established that these models do not necessarily
behave well under extrapolation, especially under light load conditions [11].
In this study, an alternative approach is investigated such that the model parameters are
derived from compressor physics and therefore have clear physical interpretation. Further, a
reduced-order model is proposed. Prediction capability of the proposed reduced-order model is
investigated and is shown to be adequate for control design. Additionally this model also allows
smooth extension to operational conditions beyond the typically mapped operational range. In
the approach adopted in this study, the Euler equations for turbomachinery are used for
developing a model for predicting compressor power. The proposed model uses the compressor
mass flow rate and compressor angular velocity as inputs. Two new parameters, the speed and
power coefficients, are defined. It is found that using a quadratic analytic function to model these
two coefficients is adequate to characterize the compressor performance over the entire
operational range. The model is validated using hot gas flow bench test data, steady-state engine
dynamometer test data as well as transient simulation and test data. The validation results
indicate that the proposed reduced-order model is suitable for both steady-state and transient
operations under realistic pulsating exhaust flow conditions.
The rest of the chapter is organized as follows. Section II discusses the development of the
proposed compressor model-based on the Euler equations with different slip factors. Section III

43

provides results from model identification based on flow bench tests. Section IV discusses results
from model validation against transient data from GT Power simulations as well as from engine
dynamometer tests. The last section adds some conclusions

Table 6. Comparison among different efficiency modelling approaches (Assuming constant inlet
condition)

Reference

Jensen et al.
[10]

[Number of fitting

[R2]{ Error

coefficients]

Mean

{number of

Deviation

inputs}

(EMD)}

[9]{2}

[0.971] {6.1}

Compressor efficiency model

S𝜂𝐶 = 𝑎1𝜑𝑐2 + 𝑎2 𝜑𝑐 + 𝑎3 ; 𝑎𝑖 =

𝑎𝑖1 +𝑎𝑖2 𝑀

𝑖 = 1,2,3

𝑎𝑖3 −𝑀

Where, M is inlet Mach number
𝜂𝑐 = 𝜂𝑐,𝑚𝑎𝑥 − 𝜒 𝑇 𝑄𝜒
𝜒 𝑇 = [𝜑𝑐𝑜𝑟𝑟 − 𝜑𝑐𝑜𝑟𝑟,𝜂𝑚𝑎𝑥 , 𝜋𝑐 − 𝜋𝑐,𝑚𝑎𝑥 ]

Guzzella-

𝑎11
𝑄 = [𝑎
21

Amstutz [37]

𝑎12
𝑎22 ]

[4]{3}

[0.927]{ 10.9}

φcorr,ηmax and πc,max are the corrected mass flow parameter 𝜑𝑐𝑜𝑟𝑟 and the compression
ratio 𝜋𝑐 corresponding to the maximum efficiency ηc,max

Andersson
[34]

𝜂𝑐 = 𝜂𝑐,𝑚𝑎𝑥 − 𝜒 𝑇 𝑄𝜒; 𝜒 𝑇 = [𝜑𝑐𝑜𝑟𝑟 − 𝜑𝑐𝑜𝑟𝑟,𝜂𝑚𝑎𝑥
𝑎11
1 + √𝜋𝑐 − 1 − 𝜋𝑐,𝑚𝑎𝑥 ]; 𝑄 = [𝑎
21
𝐺𝑐 𝐶𝑝 𝑇𝑖𝑛 (𝜋𝑐

Canova et al.
[36]

𝜂𝑐 =

𝑘−1
𝑘

− 1) (

𝑎12
𝑎22 ]

[4]{3}

[0.790]{22.2}

𝑃𝑟𝑒𝑓 𝑇𝑟𝑒𝑓
)√
𝑃𝑖𝑛
𝑇𝑖𝑛

𝑊𝑐𝑜𝑟𝑟

2
3
𝑊𝑐𝑜𝑟𝑟 = 𝐴1 + 𝐴2𝜑𝑐𝑜𝑟𝑟 + 𝐴3 𝜑𝑐𝑜𝑟𝑟
+ 𝐴4𝜑𝑐𝑜𝑟𝑟

[4]{3}

[0.947]{4.9}

Where 𝐴𝑖 (𝑖 = 1,2,3,4) are linear functions of corrected speed:
𝜔𝑐𝑜𝑟𝑟 = 𝜔√𝑇𝑟𝑒𝑓 /𝑇𝑖𝑛

Sieros et al.:
Simple linear

𝑌 = 𝑎1 + 𝑎2 𝑋 ;𝑌 = (𝜋𝑐 + 𝜂𝑐 )2
𝑋 = (𝜋𝑐 + 𝜂𝑐 + 1/𝜋𝑐 )(𝜋𝑐 + 𝜂𝑐 )

[2]{3}

[0.919]{11.6}

[38]
Sieros et al.:

𝜂𝑐 = 𝐴1 + 𝐴2𝜋𝑐 + 𝐴3 𝜋𝑐2

Generalized

2
𝐴1 = 𝑎1 + 𝑎2 𝜔𝑛𝑜𝑟 + 𝑎3 𝜔𝑐𝑜𝑟𝑟

linear 1 [38]

𝐴2 = 𝑎4 + 𝑎5 𝜔𝑛𝑜𝑟 ; 𝐴3 = 𝑎6

[6]{3}

[0.978]{5.3}

Sieros et al.:
Generalized

𝜂𝑐 = 𝐴1 + 𝐴2 𝜋𝑐 + 𝐴3 𝑙𝑜𝑔(𝜋𝑐 )
SS𝐴1 = 𝑎1 ;𝐴2 = 𝑎2 + 𝑎3 𝜔𝑛𝑜𝑟 ;𝐴3 = 𝑎4 + 𝑎5 𝜔𝑛𝑜𝑟

linear 2 [38]

44

[5]{2}

[0.980]{4.8}

3.3 Compressor Power Modelling
3.3.1 Compressor power model-based on the Euler equations
The proposed model is derived, in part, using Euler equations of Turbomachinery. In dealing
with the Euler equations, one must rely on the compressor geometry and velocity triangles
associated with the gas flow at the impeller inlet and outlet. For completeness, figures of the
compressor geometry and velocity triangle are reproduced from [1] and [21]. A typical
centrifugal compressor geometry layout and the velocity triangles of the gas flow at the
compressor impeller inlet and outlet are shown in Figure 18.

Figure 18. Velocity triangles of a centrifugal compressor at the rotor inlet and outlet

The Euler equation provides the energy transfer to the fluid as a product of the angular
velocity and torque shown in (3.2):
𝑊̇𝑐 = 𝜔𝜏 = 𝜔𝑚̇𝑐 (𝑣𝑜𝑢𝑡 𝑟𝑜𝑢𝑡 − 𝑣𝑖𝑛 𝑟𝑖𝑛 ) = 𝑚̇𝑐 (𝑈2 𝐶𝜃2 − 𝑈1 𝐶𝜃1 )

(3.2)

For a centrifugal impeller, it is assumed that the air enters the impeller eye in the axial
direction so that the initial angular momentum of the air at the inlet of the compressor can be

45

assumed to be zero ( 𝑈1 𝐶𝜃1 ≈ 0 ) [1], [21]. The ideal compressor power equation can then be
reduced to:
𝑊̇𝑐_𝑖𝑑𝑒𝑎𝑙 = 𝑚̇𝑐 𝑈2 𝐶𝜃2

(3.3)

𝑊̇𝑐_𝑖𝑑𝑒𝑎𝑙 = 𝑚̇𝑐 𝜔𝑅2 𝐶𝜃2

(3.4)

or equivalently:

where, U2 = ωR 2 . Under nominal operation the flow exiting the blades will deviate from
the ideal blade back-sweep angle 𝛽, and exit at some angle𝛽2′ . This deviation from the ideal is
referred to as slip. Under the influence of slip, the corrected absolute flow velocity can be
expressed as:
′
𝐶𝜃2
= 𝑈2 − 𝐶𝑟2 tan 𝛽2′

where

Cr 2 is

(3.5)

′
the impeller outlet flow radial velocity. The ratio of 𝐶𝜃2 to 𝐶𝜃2
is defined as slip

factor  :

𝜎=

𝐶𝜃2
′
𝐶𝜃2

(3.6)

The slip factor depends on a number of factors such as the number of impeller blades, the
passage geometry, the ratio of impeller eye tip to impeller exit diameters, the mass flow rate and
the compressor speed [21]. Introducing the slip factor in (3.4) and the expression for𝐶𝜃2 , the ideal
compressor power is written as:
′
𝑊̇𝑐_𝑖𝑑𝑒𝑎𝑙 = 𝑚̇𝑐 𝜔𝑅2 𝐶𝜃2
𝜎 = 𝑚̇𝑐 𝜔𝑅2 (𝑈2 − 𝐶𝑟2 tan 𝛽2′ )𝜎

Using the conservation of mass flow rate yields:

46

(3.7)

𝑚̇𝑐 = 𝐶𝑟2 𝜌2 𝐴2 = 𝐶𝑟1 𝜌1 𝐴1

Further assuming
𝑚̇𝑐
)
𝜌2 𝐴2

𝑚̇𝑐
).
1 𝐴1

≈ (𝐶𝑟1 = 𝜌

C r 2  C r1

(3.8)

(uniform radial flow with no radial flow losses) leads to (𝐶𝑟2 =
𝑚̇𝑐
1 𝐴1

Substituting for 𝐶𝑟2 = 𝜌

in (3.7), the compressor power can be re-

formulated as:
′

𝜎𝑅 tan 𝛽2
𝑊̇𝑐_𝑖𝑑𝑒𝑎𝑙 = 𝜎𝑅22 𝑚̇𝑐 𝜔2 − 2
𝜔𝑚̇𝑐2
𝜌1 𝐴1

(3.9)

In order to account for flow and ‘windage’ losses, the actual required input work must be
greater than the theoretical value necessary to achieve the target flow rate [45]. To account for
this, a power loss factor 𝜓 is used to modify the ideal power and a friction loss power term 𝑊̇𝑓 is
introduced into power model to define the actual, loss-compensated, power required to achieve a
desired mass flow rate as:
𝑊̇𝑐 = 𝜓𝑊̇𝑐_𝑖𝑑𝑒𝑎𝑙 + 𝑊̇𝑓

(3.10)

In this study, the factor 𝜓 is assumed to be a design parameter and is therefore assumed fixed
for a given compressor design. Ideally, this parameter must be established experimentally or
identified through standard techniques (this work). In order to define the friction loss term, the
loss models proposed in [21] and [46] are used here. The friction loss is modeled as a sum of the
losses over the impeller and the diffuser and is defined as cubic functions of the mass flow rate:
𝑊̇𝑓 = 𝑘𝑓 𝑚̇𝑐3 = (𝑘𝑓𝑖 + 𝑘𝑓𝑑 )𝑚̇𝑐3

(3.11)

where 𝑘𝑓𝑖 and 𝑘𝑓𝑑 are friction coefficients for the impeller and diffuser, respectively. Hence,
the actual compressor power needed to achieve a desired mass flow rate can be written in its
expanded form with all the loss modifiers as:

47

𝑊̇𝑐 = 𝜓𝜎𝑅22 𝑚̇𝑐 𝜔2 −

𝜓𝜎𝑅2 tan 𝛽2′
𝜔𝑚̇𝑐2 + 𝑘𝑓 𝑚̇𝑐3
𝜌1 𝐴1

(3.12)

Two new parameters are defined: the power coefficient 𝐶𝑝𝑜𝑤𝑒𝑟 and speed coefficient 𝐶𝑠𝑝𝑒𝑒𝑑 , as
follows:
𝐶𝑝𝑜𝑤𝑒𝑟 =

𝑊̇𝑐
𝑚̇𝑐3

(3.13)

𝐶𝑠𝑝𝑒𝑒𝑑 =

𝜔
𝑚̇𝑐

(3.14)

Using this notation, the compressor power in (3.12) can be rearranged in terms of the power
and speed coefficients as follows:
2

𝐶𝑃𝑜𝑤𝑒𝑟 = 𝜓𝜎𝑅22 (𝐶𝑆𝑝𝑒𝑒𝑑 ) −

𝜓𝜎𝑅2 tan 𝛽2′
(𝐶𝑆𝑝𝑒𝑒𝑑 ) + 𝑘𝑓
𝜌1 𝐴1

(3.15)

It is clear from (3.15) that, 𝐶𝑃𝑜𝑤𝑒𝑟 varies not only with compressor operating condition (such
as 𝐶𝑠𝑝𝑒𝑒𝑑 ) but also with compressor designs parameters. Different compressor designs could also
impact the power law (3.15) through compressor specific geometry, power loss factor, and slip
factor. A generalized power coefficient model can, however, be written for a specific compressor
design. Since for a given compressor design, the parameters, 𝜓, 𝑘𝑓 , 𝑅2 tan 𝛽2′ , 𝐴1 are constant. This
allows the power coefficient to be expressed compactly as a function of the operating parameters,
speed coefficient and the slip factor for a given operating condition i as:
𝐶𝑃𝑜𝑤𝑒𝑟_𝑖 = 𝑓(𝐶𝑆𝑝𝑒𝑒𝑑𝑖 , 𝜎𝑖 ), 𝑖 = 1,2,∙∙∙, 𝑛

(3.16)

One of the goals of this work was to derive a reduced order power model and investigate its
prediction capability. The power and speed coefficients in conjunction with an appropriate slip
factor model may be such a candidate model. It may be possible to further reduce the model

48

order by making the power coefficient a function of the speed coefficient only as indicated in
(3.17).
𝐶𝑃𝑜𝑤𝑒𝑟 = 𝑓(𝐶𝑆𝑝𝑒𝑒𝑑 )

(3.17)

It is clear that in order to achieve the form shown in (3.17) the slip factor must be defined
either as a parameter fixed by design or defined in terms of the speed and/or power coefficient to
maintain the homogeneity of the compressor power model with respect to 𝐶𝑃𝑜𝑤𝑒𝑟 and 𝐶𝑠𝑝𝑒𝑒𝑑 . This
leads us into an investigation of slip factor models.
3.3.2 Investigation of slip factor models for flows over centrifugal compressor
Several slip factors models for centrifugal compressors are readily available in the open
literature [17, 22, 29-32, 34]. The slip factor typically depends on the compressor design
parameters as well as the operating conditions, such as: compressor rotational speed and mass
flow rate. In order to preserve the impact of flow variations on slip, the slip factor models, as
proposed in [27], [41], and [44] are investigated in this work:
Slip 1: 𝜎 = 1 +

𝑚̇ tan 𝛽2′
𝜌1 𝐴1 𝜔𝑅2

− 0.5 ∗ (1 − 𝑒

−2𝜋
cos 𝛽2′
𝑍

) (0 < 𝜎 < 1) (Reffstrup)

(3.18)

𝜋

Slip 2: 𝜎 = 1 −

(𝑍)𝜔𝑅2 cos 𝛽2′

Slip 3: 𝜎 = 1 − 𝑎

where

a

𝑚̇
tan 𝛽2′
𝜌1 𝐴1

𝜔𝑅2 −
𝑚̇

𝜔𝑅23

(0 < 𝜎 < 1) (Stodola)

(0 < 𝜎 < 1) (Stahler)

(3.19)
(3.20)

is a design parameter in (3.20) related to the flow exit angle and is a constant for a

given impeller design, and Z is the number of impeller blades. Integrating each of these models
into the compressor power model (3.15), a generalized compressor power model can be
established with the following structure:

49

𝑊̇𝑐
𝜔 2
𝜔
2
3 = 𝜀1 (𝑚̇ ) − 𝜀2 (𝑚̇ ) + 𝜀3 ⇒ 𝐶𝑃𝑜𝑤𝑒𝑟 = 𝜀1 (𝐶𝑆𝑝𝑒𝑒𝑑 ) − 𝜀2 (𝐶𝑆𝑝𝑒𝑒𝑑 ) + 𝜀3
𝑚̇𝑐
𝑐
𝑐

(3.21)

where, the coefficients 𝜀1 , 𝜀2, and 𝜀3depend on the selected slip factor model and are defined in
Table 7.The derivations as below:
Table 7. Model coefficients for various slip models
Slip model 1

𝜀1

𝜀2

𝜀3

𝜓𝑅22 (0.5 + 0.5𝑒

−2𝜋
cos 𝛽2′
𝑍

)

−2𝜋
𝜓𝑅2 tan 𝛽2′
′
(0.5𝑒 𝑍 cos 𝛽2 − 0.5)
𝜌1 𝐴1

𝑘𝑓 − 𝜓 (

tan 𝛽2′
)
𝜌1 𝐴1

Slip model 2

Slip model 3

𝜓𝑅22 (1 − (𝜋/𝑍) cos 𝛽2′ )

𝜓𝑅22

𝜓

𝑅2 tan 𝛽2′
𝜌1 𝐴1

𝜓(

2

𝑘𝑓

𝜓

𝑎
𝑅2

+

tan 𝛽2′ 𝑅2

tan 𝛽2′ 𝑎
𝜌1 𝐴1 𝑅2

𝜌1 𝐴1

)

+ 𝑘𝑓

Introduce slip factor model 1 into compressor power equation:
-2
-2
2





cos  2  
m tan  2
1


  R 2      1  m tan  2 - 0.5 * 1 - e Z cos  2   R

W c   1 
- 0.5 * 1 - e Z



  2  m 

  2  A tan  2  m   k f
1 A1 R2
1 A1 R2



1 1






-2
2


cos  2  


  R 2      R2 tan  2
  1 - 0.5 * 1 - e Z


2

   m 
1 A1





-2
2


cos  2  

  R 2     R2 tan  2
  1 - 0.5 * 1 - e Z


   m 
1 A1






-2


cos  2  



  R2 tan  2
    1 - 0.5 * 1 - e Z
   A

m

 

1 1




-2


 0.5e Z cos  2  0.5     k    tan  2
f
  A

 m 
 1 1



Introduce slip factor model 2 into compressor power equation:

50






 tan  2

    

m
 
 1 A1
2

2


 kf



W c  m R U 2  C r 2 tan  2 



 / Z R cos  2 )
m
 m R R 
tan  2 (1 
m
1 A1


R 
tan  2
1 A1


m
 m R R 
tan  2   m R  / Z R cos  2
1 A1


2

m R
 m R 2 
tan  2   / Z cos  2 m R 2  k f m 3
1 A1
R tan  2 2
 (1   / Z cos  2 ) R22 m  2  2
m   k f m 3
1 A1

Introduce slip factor model 3 into compressor power equation:
W c  m R U 2  C r 2 tan  2   k f


m
m 
 m R R 
tan  2 1  a
  k f

A
R 3 
1 1



m
m
m
m 
kf
 m R R  a 3 R 
tan  2 
tan  2 a
1 A1
1 A1
R
R 3 


m
m
1 m 2 
 m R R  a 2 
tan  2  tan  2 a
kf

1 A1
1 A1 R 3 
R

m 2
m 2
1 m 3
 m R 2  a

tan  2 R  tan  2 a
kf
R
1 A1
1 A1 R 2
 a tan  2 R  2
tan  2 a 3
m  
 m R 2   
m  k f

2
R

A

1
1


1 A1 R

The model structure in (3.21) is referred to as the generalized compressor power model in the
rest of this chapter. Note that the three coefficients in (3.21) take constant values and are fixed by
compressor design under the assumption of constant or slowly varying inlet conditions. Once the
relationship between the power and speed coefficients has been identified, the compressor power
for a given operating point defined the pair < 𝜔 , 𝑚̇𝑐 > , can be established as in (3.22).
𝜔 2
𝜔
𝑊̇𝑐 = 𝑚̇𝑐3 (𝜀1 ( ) − 𝜀2 ( ) + 𝜀3 )
𝑚̇𝑐
𝑚̇𝑐

(3.22)

This approach shows that the compressor operation can be represented via an analytic
quadratic function rather. The proposed model has only two inputs (the compressor mass flow
rate and TC rotational speed) and three parameters to be identified. In practical applications,

51

while the mass flow rate is typically available as a measurement, the TC speed is not a standard
measurement. In the absence of a TC speed measurement, the observed value of the TC speed
via an appropriate observer design may be used. Alternately TC speed can also be obtained by
solving the turbocharger rotor dynamic differential equations as in [3]-[7], and the compressor
mass flow rate can be modeled with shaft speed and pressure ratio as inputs [10]. In this study,
the compressor power model is validated under the former assumption that two parameters, mass
flow rate and TC rotational speed, are available as measurements.
3.3.3 Compatibility with corrected mass flow rates
1
Corrected compressor mass flow rates 𝑚̇𝑐𝑜𝑟𝑟𝑒𝑐𝑡 = 𝑚̇ √𝑇
and corrected TC angular velocity
𝑃
1

𝜔𝑐𝑜𝑟𝑟𝑒𝑐𝑡 = 𝜔/√𝑇1

are typically used in compressor performance maps. So it was natural to

investigate the structure of the power coefficient based compressor power model when using
these corrected terms. It is found that the power coefficient with the corrected variables has the
1
same structure as before and is scaled by the modifying term √𝑇
and 1/√𝑇1 as shown below. To
𝑃
1

show this, the corrected power coefficient is defined as: 𝐶𝑝𝑜𝑤𝑒𝑟_𝑐𝑜𝑟𝑟𝑒𝑐𝑡 = (𝑚̇

𝑊̇ 𝑐

3
𝑐𝑜𝑟𝑟𝑒𝑐𝑡 )

; and the

𝜔𝑐𝑜𝑟𝑟𝑒𝑐𝑡
corrected speed coefficient can be defined as 𝐶𝑠𝑝𝑒𝑒𝑑_𝑐𝑜𝑟𝑟𝑒𝑐𝑡 = 𝑚
. Introducing these terms
̇
𝑐𝑜𝑟𝑟𝑒𝑐𝑡

directly into the power coefficient model as in (3.23), the proposed model with corrected power
coefficient is shown in (3.24):
𝑊̇𝑐
𝜔 2
𝜔
= 𝜀1 ( ) − 𝜀2 ( ) + 𝜀3
3
𝑚̇
𝑚̇
𝑚̇
3

𝑊̇𝑐 =

1
√𝑇1
( 𝑃1 )

(

𝑚̇√𝑇1
)
𝑃1

3

2

𝑇1
𝜀1 ( )
𝑃1
(

𝜔
√𝑇1
𝑚̇√𝑇1
( 𝑃1 )

2

(3.23)
2

𝑇1
𝜔𝑐𝑜𝑟𝑟𝑒𝑐𝑡
− 𝜀2 ( ) (
) + 𝜀3
𝑃1
𝑚̇𝑐𝑜𝑟𝑟𝑒𝑐𝑡
)

52

3

𝑊̇𝑐 = (

𝑇1 2 𝜔𝑐𝑜𝑟𝑟𝑒𝑐𝑡 2
𝑇1 2 𝜔𝑐𝑜𝑟𝑟𝑒𝑐𝑡
) (𝑚̇𝑐𝑜𝑟𝑟𝑒𝑐𝑡 )3 (𝜀1 ( ) (
) − 𝜀2 ( ) (
) + 𝜀3 )
𝑃1
𝑚̇𝑐𝑜𝑟𝑟𝑒𝑐𝑡
𝑃1
𝑚̇𝑐𝑜𝑟𝑟𝑒𝑐𝑡
√𝑇
𝑃
1

3

𝑊̇𝑐
(𝑚̇𝑐𝑜𝑟𝑟𝑒𝑐𝑡 )3

=

1
√𝑇1
( 𝑃1 )

𝑇1 2 𝜔𝑐𝑜𝑟𝑟𝑒𝑐𝑡 2
𝑇1 2 𝜔𝑐𝑜𝑟𝑟𝑒𝑐𝑡
(𝜀1 ( ) (
) − 𝜀2 ( ) (
) + 𝜀3 )
𝑃1
𝑚̇𝑐𝑜𝑟𝑟𝑒𝑐𝑡
𝑃1
𝑚̇𝑐𝑜𝑟𝑟𝑒𝑐𝑡

(3.24)

3

𝐶𝑝𝑜𝑤𝑒𝑟_𝑐𝑜𝑟𝑟𝑒𝑐𝑡 =

1
√𝑇1
( 𝑃1 )

𝑇1 2
𝑇1 2
2
(𝜀1 ( ) (𝐶𝑠𝑝𝑒𝑒𝑑_𝑐𝑜𝑟𝑟𝑒𝑐𝑡 ) − 𝜀2 ( ) (𝐶𝑠𝑝𝑒𝑒𝑑𝑐𝑜𝑟𝑟𝑒𝑐𝑡 ) + 𝜀3 )
𝑃1
𝑃1

Eq. (3.24) means that the model needs to be calibrated by reference temperature and pressure
when compressor operates at different inlet conditions. In the rest of this chapter, all derivations
and results are based on the model (3.22) without the corrected parameters, since all the
measurements are under the atmospheric pressure.
3.4 Model Identification
3.4.1 Model identification using standard hot gas flow bench test data
Test data from standard hot gas flow bench tests was used to identify the three model
parameters, 𝜀1 , 𝜀2 , and 𝜀3 for the generalized compressor model as in (3.21) for three different
compressor designs, compressor-1, 2, 3. The inlet conditions of compressors are assumed fixed
in line with standard test protocol [26]. The minimum test speed for Compressor-1,2 was 30k
RPM, while the lowest speed for Compressor-3 was 46k RPM. Model parameters for the power
coefficient model as in (3.21) are identified using a Least Squares optimization based on a model
error cost function as in (3.25):
𝑛

𝐽 = ∑ [𝑊̇𝑐

𝑚𝑜𝑑𝑒𝑙

(𝑖) − 𝑊̇𝑐

𝑚𝑒𝑠

𝑖=1

53

(𝑖)]

2

(3.25)

In (3.25), n = 322, is the number of steady state compressor operating test points for the hot
gas flow bench tests. 𝑊̇𝑐

𝑚𝑜𝑑𝑒𝑙

is the calculated compressor power using (3.22); and 𝑊̇𝑐

𝑚𝑒𝑠

is the

standard compressor power calculated from measured inputs as (3.26):

𝑚𝑒𝑠
𝑚𝑒𝑠
)
𝑊̇𝑐𝑚𝑒𝑠 = 𝑚̇𝑐𝑚𝑒𝑠 𝑐𝑝 (𝑇𝑜𝑢𝑡
− 𝑇𝑖𝑛

(3.26)

Since the geometric design parameters 𝑅2 , 𝐴1 and 𝛽2′ for Compressor-1 were available.
Compressor-1 was selected as a candidate for identifying the parameters, 𝜓, 𝛽2′ and 𝑎, for all three
slip models using the model structure (3.15). Note that the parameter “a” is relevant only for the
Slip Model-3. The values of the identified parameters are shown in Table 8.
It is interesting to note that identified value of the power loss factor 𝜓, is greater than unity,
indicating that the compressor power necessary to achieve the desired flow rate must be larger
than the ideal power calculated in (3.4) in order to account for losses. The order of magnitude of
the identified parameter 𝑎 (Slip Model-3) agrees with the result in [41]. Given the value for the
friction coefficient 𝑘𝑓 , the maximum friction loss power identified was around 2.38kW over the
entire compressor operating range. Modelling errors are investigated using four error metrics: the
coefficient of determination 𝑅2 , error mean deviation (EMD), and 𝑃𝐸𝐵±5% , and 𝑃𝐸𝐵±10% that are
the percentage of total data points within the error bound Âą5% andÂą10%, respectively. The
error evaluation parameters used in these calculations are defined in (3.27), (3.28) and (3.29):

𝐸𝑟𝑟𝑜𝑟(𝑖) =

𝐸𝑀𝐷 =

𝑊̇𝑐𝑚𝑜𝑑𝑒𝑙 (𝑖) − 𝑊̇𝑐𝑚𝑒𝑠 (𝑖)
𝑊̇𝑐𝑚𝑒𝑠 (𝑖)

𝐸𝑟𝑟𝑜𝑟 − 𝑚𝑒𝑎𝑛(𝐸𝑟𝑟𝑜𝑟)
𝑛

54

(3.27)

(3.28)

𝑃𝐸𝐵±𝑘% =

𝑚
𝑛

, 𝑚 = ∑𝑛𝑖=1 𝑁𝑘 (𝑖), where |𝐸𝑟𝑟𝑜𝑟(𝑁𝑘 )| ≤ 𝑘%

(3.29)

where, n is the total sample number. Based on model fitting error, as reported in Table 8 , the
Slip Model-3 has the best fit (least error) for Compressor-1. Since compressor design parameters
were not available for Compressors-2, 3, we could only identify the 3 lumped parameters 𝜀1 , 𝜀2,
and 𝜀3 for the generalized power model (3.21) for these two compressor designs. The identified
values for the coefficients of the generalized power coefficient are shown for all three
compressor designs in Table 8.

Table 8. Identified model coefficients for 3 model variants and 3 compressor design variants
Model (21)
Compressor
-1

Model (22)

Slip model 1
Slip model 2
Slip model 3
Compressor
1
Compressor
2
Compressor
3



kf

a

1.295
1.323
1.351
1

19040
6374
11390
2

NA
NA
1.334

3.52e-8

3

R2
0.968
0.968
0.989
R2

EMD*
4.42
4.56
3.24
EMD

PEB** (Âą5%)
58.9%
66.44%
79.15%
PEB (Âą5%)

PEB (Âą10%)
83.52%
84.00%
92.90%
PEB (Âą10%)

0.0123

1752

0.992

3.24

79.15%

92.90%

2.98e-8

0.0162

3547

0.995

3.73

68.31%

90.22%

1.68e-8

0.0086

1987

0.996

2.61

81.34%

91.69%

2

p
q
NA
R
EMD
PEB (Âą5%)
Compressor
2.439
-10.25
NA
0.994
5.97
70.07%
1
Model(30)
Compressor
2.423
-10.31
NA
0.989
3.78
58.01%
2
Compressor
2.262
-9.551
NA
0.996
2.37
86.63%
3
*EMD: Error Mean Deviation **PEB: Percentage of data points within Error Bound

55

PEB (Âą10%)
86.11%
85.83%
94.61%

Compresspr mass flow rage [kg/s]

(a) flow bench test range for three compressors
0.7
Compressor 1
Compressor 2
Compressor 3

0.6
0.5
0.4
0.3
0.2
0.1
0

7

2

4

x 10

4

8
10
Turbo Speed [RPM]

12

14

16
x 10

4

(b) model identification under linear scale
Compressor
Compressor
Compressor
Compressor
Compressor
Compressor

6
5

Power coeff

6

4

1 test data
1 model (21)
2 test data
2 model (22)
3 test data
3 model (22)

3
mass flow rate decreasing
2
1
0

mass flow rate decreasing
0

0.5

1

1.5

2

2.5

Speed coeff

x 10

6

(c) model identification under log scale
5.5
Compressor
Compressor
Compressor
Compressor
Compressor
Compressor

log10(Power coeff)

5
4.5
4

1 test data
1 model (30)
2 test data
2 model (30)
3 test data
3 model (30)

mass flow rate decreasing
3.5
3
2.5
2
5.2

mass flow rate decreasing
5.3

5.4

5.5

5.6

5.7
5.8
5.9
log (Speed coeff)

6

6.1

6.2

10

Figure 19. Identification Results for the generalized Compressor-Power model for Compressors1,2,3.
Results from the fitted models for the three different compressor designs are shown in Figure
19 (b) the log-log plot is shown in Figure 19 (c) that each compressor design has a unique
characteristic curve. This is expected and reflects the design differences among the different
compressors. In fact, such plots allow a quick and easy comparison of different compressor
designs. As an example, it is clear that for a given mass flow rate and TC shaft speed the
compressor power follows the trend, Power1 > Power2 > Power3, indicating that Design-1
perhaps have a larger loss relative to the other designs. Also note that it is clear that Compressor-

56

3 may offer the widest operating range of the three designs considered in this study. The Log10scale plots offer new insights. It is easy to see that the Log10-scale representation is a linear
transformation of quadratic curves; that is, a power law relationship exists between the power
and speed coefficients. The Log model is shown in (3.30).
log10 (𝐶𝑃𝑜𝑤𝑒𝑟 ) = 𝑝 log10 (𝐶𝑆𝑝𝑒𝑒𝑑 ) + 𝑞

(3.30)

In (30), 𝑝 is the slope of each characteristic line and 𝑞 is the y- intercept for each design.
From (30), we get the power law model, derived from a log10 plot as:
𝑝

𝐶𝑃𝑜𝑤𝑒𝑟 = 10𝑞 (𝐶𝑆𝑝𝑒𝑒𝑑 ) ⇒

𝑊̇𝑐
𝑚̇𝑐3

(3.31)

The 𝑝 and 𝑞 values for the power-law plots were also identified and are noted in TABLE 3.
The power law representation implies a scaling invariance and therefore provides a good
measure of the sensitivity of power coefficient to changes in the speed coefficient. One important
advantage of this is that (3.31) can, theoretically, be obtained from only two compressor
operating conditions that span the high and low load operational conditions. This is in line with
the two-point fitting method used for determining the power law exponent. Hence this method
may significantly reduce the experimental burden during the early stages of compressor
development.
In summary, model fitting results confirm that the generalized compressor power model
proposed in this work is able to reproduce the compressor behavior. Proposed models only need
two or three parameters to be identified. These results also demonstrate that for a centrifugal
compressor the operating characteristics can be reduced to an analytic quadratic function in
linear scale and a straight line in logarithmic scale. A summary of the relative errors of

57

prediction, for the various compressor designs and the model variants, is shown in Figure 20. It
is clear that prediction error is well contained within a Âą 5% relative error band.
120

Compressor-1 model (22)
Compressor-2 model (22)
Compressor-3 model (22)
Compressor-1 model (30)
Compressor-2 model (30)
Compressor-3 model (30)
+10% error
+5% error
- 5% error
-10% error

Model predicted compressor power [kW]

100

80

60

40

20

0

0

20

40
60
80
Measured compressor power [kW ]

100

120

Figure 20. Modelling error (27) for compressor power models

3.4.2 Model identification with ‘Supercharger Standard Test’
Supercharging Standard Tests (SST) are performed with an electric motor driving the
compressor as opposed to the turbine. Given the wider speed control of electric motor, it is
possible to perform more extremely light load (low speed, low mass flow rates) tests relative to
standard hot-gas flow bench testes. Three different compressor designs, Compressor-4, 5, 6 were
tested under this test protocol. The lowest compressor angular velocity achieved was 20k RPM
compared to 30k RPM in the flow bench tests. The minimum mass flow rate was 0.01 kg/s in the
compressor dynamometer, compared to 0.02 kg/s in the flow bench test.

58

Following the same procedure as for the previous set of compressors, we identified the three
lumped parameters

i

for the generalized power coefficient model as in (3.22). The identified

values are indicated in Table 9. The behavior of the fitted model is compared against SST data in
Figure 21 . Results indicate, as before, that the proposed model is able to reproduce the observed
data. The Log scale plots also reproduce a similar power law behavior observed for the previous
set of compressors. The parameters for the power law model are included in Table 9.

Compressor mass flow rate [kg/s]

(a) supercharger test ranges for three compressors
0.5
Compressor 4
Compressor 5
Compressor 6

0.4
0.3
light load

0.2
0.1
0

2

x 10

2.5

4

4

12

14

16
x 10

4

(b) model identification under linear scale
Compressor
Compressor
Compressor
Compressor
Compressor
Compressor

2

Power coeff

6
8
10
Compressor speed [RPM]

1.5

4 test data
4 model (22)
5 test data
5 model (22)
6 test data
6 model (22)

1
mass flow rate decreasing
0.5
mass flow rate decreasing
0

0

1

2

3

4

5
6
Speed coeff

7

8

9

10

11
x 10

5

(c) model identification under log scale
4.5
Compressor
Compressor
Compressor
Compressor
Compressor
Compressor

log10(Power coeff)

4

3.5

4 test data
4 model (30)
5 test data
5 model (30)
6 test data
6 model (30)

mass flow rate decreasing

3
mass flow rate decreasing
2.5
5.3

5.4

5.5

5.6
5.7
5.8
log (Speed coeff)

5.9

6

6.1

10

Figure 21. Identification Results for the generalized Compressor-Power model for Compressors4,5,6.

59

Table 9. Fitted model coefficients identified through supercharging test data
Model (22)
Compressor 4
Compressor 5
Compressor 6
Model (29)
Compressor 4
Compressor 5
Compressor 6

1
1.928 e-06
1.965e-06
1.967e-06
p
2.379
2.378
2.378

2
0.0618
0.0618
0.0564
q
-10.09
-10.07
-10.06

3

858.1
858.1
718.3
NA
NA
NA
NA

R2
0.995
0.996
0.995
R2
0.995
0.989
0.996

3.5 Model Validation
3.5.1 Model validation based on steady state engine dynamometer test data
The previously identified model for, 𝐶𝑝𝑜𝑤𝑒𝑟 = 𝑓(𝐶𝑠𝑝𝑒𝑒𝑑 ) , was validated against a, more realistic,
data set from engine dynamometer steady-state tests. These tests were conducted on a heavy duty
diesel engine. The turbocharger on this engine used a compressor equivalent to the design variant
Compressor-1. Test details can be found from previously published work [24] and [25]. The
steady-state test covers the entire engine operating range (185 testing points). The compressor
operating range is defined by a mass flow rate range between 0.01 and 0.45 kg/s and compressor
speed range between 5.9k and 109k RPM. Note that these tests achieve lighter operational
conditions relative to the flow bench tests used for model identification for which the TC speed
was limited, at the low end to, 30k rpm for the hot-gas tests, and 20k rpm for the SST tests. The
engine operational range drives compressor operation beyond the flow-bench data and provides
an opportunity to test the range extension properties of the proposed model. In Figure 22 (a), we
show the engine test grid overlaid on the compressor mapping points (from hot-gas flow-bench
tests). It is clear that the engine operation at light load conditions forces the compressor to
operate in regions not covered by the flow-bench data.

60

Compressor mass flow rate [kg/s]

(a) engine test range and flow bench test range
0.7
TC Flow bench test data
Engine steady state dyno(TC speed>30000 RPM)
Engine steady state dyno(TC speed<30000 RPM)

0.6
0.5
0.4
0.3
0.2
0.1
0

0

x 10

10

2

4

6
8
TC speed [rpm]

10

12

14
x 10

4

(b) comparsion under linear scale
TC Flow bench test data
Engine steady state dyno(TC speed>30000 RPM)
Engine steady state dyno(TC speed<30000 RPM)
Compressor-1 model (21)

8

Power coeff

4

6
mass flow rate decreasing
4
mass flow rate decreasing
2
0

1

2

3

4

5

6
7
Speed coeff

8

9

10

11
x 10

5

(c) comparsion under log scale
7
TC Flow bench test data
Engine steady state dyno(TC speed>30000 RPM)
Engine steady state dyno(TC speed<30000 RPM)
Compressor-1 model (30)

log10(Power coeff)

6

5
mass flow rate decreasing
4

3
mass flow rate decreasing
2
5.2

5.3

5.4

5.5

5.6
5.7
log10(Speed coeff)

5.8

5.9

6

Figure 22. Comparison of Cpower model performance against calculated values based on flow
bench data and engine steady state dynamometer test data for Compressor-1

The 𝐶𝑝𝑜𝑤𝑒𝑟 model, previously identified for Compressor-1, was used to reproduce the power–
coefficient vs. speed-coefficient relationship for the engine test data. In Figure 22 (a), the
predicted 𝐶𝑝𝑜𝑤𝑒𝑟 is compared against the calculated values for 𝐶𝑝𝑜𝑤𝑒𝑟 for both the flow-bench tests
as well as the engine steady–state tests. Recall that the calculated values for 𝐶𝑝𝑜𝑤𝑒𝑟 , are based on
(26) and use measured inputs. Since the engine data set extends to lower load conditions (TC
speed < 30k RPM), the characteristic curves for the engine data are plotted in two sets to span

61

operational regimes above and below the 30k RPM TC speed threshold. This was done primarily
to assess the quality of the model prediction for light conditions that extend beyond the standard
mapping domain. It is clear from Figure 22 (b) and (c), that, while the model adequately
replicates the calculated values for 𝐶𝑝𝑜𝑤𝑒𝑟 for TC speeds > 30K RPM, there is a significant error
under operational conditions of TC speeds < 30kRPM, where the calculated values for
𝐶𝑝𝑜𝑤𝑒𝑟 lose

monotonicity and appear to be quite random. This obviously anomalous behavior may

have sources other than flow irregularities from extremely low flows. . The actual compressor

Model predicted compressor power [kW]

power (below 30K rpm) is investigated as show in
1.5
y=x

1
y=0.9952*x-0.16

0.5

0

0

0.5
1
Measured compressor power [kW ]

1.5

0.24
0.22

Power [kW]

0.2
0.18
0.16
0.14
0.12
0.1
0.08
Measured compressor power-Model predicted compressor power
0.06

0

0.5
1
Measured compressor power [kW ]

1.5

Figure 23. Model predicted power vs Measured power under 30K rpm TC speed

The compressor power error between model predicted value and measured value are between
0.1kW and 0.23 kW with an averaged value of 0.16 kW as shown in Figure 23. Measured

62

compressor power is higher than model predicted compressor power. In order to identify the
error sources for both increased power coefficient and increased compressor power, the
sensitivity analysis is conducted for both power coefficient 𝐶𝑝𝑜𝑤𝑒𝑟 and compressor power 𝑊̇𝑐 as
shown in Table 10. Sensitivity gain ranges are based on the value range from measurements.

Table 10. Sensitivity analysis for compressor power and power coefficient with TC speed
below 30k RPM
C power

W c
Sensitivity
gain

Measurement (26)
W c
W c
W c
W c 
m c 
Tout 
Tin
m c
Tout
Tin


 C p (Tout  Tin )mc  C p mc Tout  C p m c Tin

Model (31)
C power 

C power
C power
m c 

m c


Measurement (26)
C power 

 p 
 p 1 
m  10 q p 

 10 q p
 m p 1  c
 m p 
 c

 c 





m c (kg/s)

C p (Tout  Tin ) ∈{10.1 20.2}

 p 
 10q p p 1 
 m

 c

∈{-6.96e5, -1.05e5}

 (rpm)

NA

  p 1 
 ∈{0.18,1.51}
10 q p
 m p 
 c 

Tout (k)

C p m c ∈{0.012 0.069}

NA

Tin (k)

C p m c ∈{-0.069,-0.012 }

NA

C power
C power
C power
m c 
Tout 
Tin
m c
Tout
Tin

C p (Tout  Tin )
m c 3

m c 

C p (Tout  Tin )
m c 3

Cp
m c 2

Tout 

Cp
m c 2

Tin

∈{-4.81e6, -6.47e4}

NA

Cp
m c 2


Cp
m c 2

∈{14.51,78.90}
∈{-78.90, -14.51}

The error in the measured power-coefficient is amplified by artificially low mass flow rates
reported by the production mass flow sensor. Mass flow rate sensors are known to suffer from
nonlinearity and loss of accuracy at low flow rates as observed for these tests. This is confirmed
from the model behavior for the SST tests that used laboratory grade sensors and for which the
𝐶𝑝𝑜𝑤𝑒𝑟 model

was able to reproduce the measured values quite adequately. An additional source of

error may be attributed to a positive bias in the temperature downstream of the compressor due to
heat transfer from the hot-end (turbine). This effect of heat transfer on the calculated compressor
under light load operating conditions was also verified experimentally by the authors in [47][48].
However, this effect may not completely describe the error in the 𝐶𝑝𝑜𝑤𝑒𝑟 model observed here
since the successive low load test conditions would result in a progressively cooler turbine

63

housing leading to reduced heat transfer effects. In summary, since the error in the predicted
power is less than 0.23kW in light load range and the model showed good predictive capability
for the SST experiments for light load operation it is safe to project that the proposed model is
adequate for light load extrapolation.
3.5.2 Model validation for US06 transient cycle based on GT-Power cycle simulations
Next, the model behavior is verified against transient data from GT-Power simulations for a
US06 cycle. The TC speed varied between 14k and 105k RPM during the test. The model
selected for evaluation was based on Compressor-1 design and Slip Model-3. Model prediction
for 𝐶𝑝𝑜𝑤𝑒𝑟 is compared against calculated values as before. Figure 24 shows that the model
prediction for 𝐶𝑝𝑜𝑤𝑒𝑟 agrees quite well with the expected values as calculated values over the test
cycle.
12

x 10

4

Turbocharger Speed [RPM]

Turbocharger speed
10
8
6
4
2
0

0

100

200

300
Time [s]

400

500

600

15000

Power coeff

Compressor-1 GT simulation
Compressor-1 model (22)
10000

5000

0

2

3

4

5
6
Speed coeff

7

8

9
x 10

4

Figure 24. Model validation over US-06 GT-Power transient simulation

64

3.5.3 Model validation over FTP cycle based on engine dynamometer tests.
Model prediction capability over an FTP test cycle with engine dynamometer data was
investigated. The model candidate used is based on Compressor-1 design and slip Model-3.
Since the calculated 𝐶𝑝𝑜𝑤𝑒𝑟 (for comparison) relies on the compressor downstream temperature,
we assessed the signal behavior of the relevant sensor. This is because the location of the post
compressor temperature sensor, in the test engine configuration, was expected to introduce
measurement error from slow dynamics and delay as well as heat transfer losses. In Figure 25, it
is clear that during this load step the mass flow rate and turbocharger speed has a much faster
response compared to the temperature sensor. In order to address this, we corrected the measured
temperature signal using a lead-lag filter. The lead-lag filter was designed to match the expected
temperature profile for the same load step by inverting the compressor power model (3.26)
shown in (3.32):
𝜔
𝑚̇ 3 𝑓 ( )
𝜔
𝑚
̇ +𝑇
𝑚̇𝑐𝑝 (𝑇𝑜𝑢𝑡 − 𝑇𝑖𝑛 ) = 𝑚̇ 𝑓 ( ) ⇒ 𝑇̂𝑜𝑢𝑡 =
𝑖𝑛
𝑚̇
𝑚̇𝑐𝑝

(3. 32)

3

1.2

1

0.8

0.6

0.4

0.2

0
30

Measured compressor mass flow rate/ Max measured mass flow rate
Measured TC speed/ Max measured TC speed
Measured temperature after compressor/ Max measured temperature
35

40

45

50

55

60

65

Time [s]

Figure 25. Normalized measurements for mass flow rate, TC speed and Compressor downstream
temperature for a load step

65

The corrected temperature, downstream of the compressor, is compared against the measured
signal in Figure 26. The same filter was then applied to the measured temperature signal for the
full cycle and the corrected temperature was used to evaluate the measured value of the
compressor power using (3.26). The corrected measured temperature signals are shown in Figure
26. The predicted compressor power was established from the predicted 𝐶𝑝𝑜𝑤𝑒𝑟 and known mass
flow rate. The predicted and calculated values are compared in Figure 26. The filter is seen to
converge at 160secs, after which, it is clear that the predicted compressor power is capable of
reproducing the actual compressor reasonably well. With this confidence, it is reasonable to
claim that using the proposed method the compressor power can be predicted reasonably well
based on measurements of compressor mass flow rate and TC speed.
800
Measured temperature
Model predicted temperature by (32)
Reconstructed temperature with phase lead-lag compensation

Temperture [K]

700
600
Filter converged
500
400
300
200

0

100

200

300

400

500
Time [s]

600

700

800

900

1000

150

Power [kW]

Measured power
Model predicted power by Compressor-1 model (22)
Reconstructed power by temperature with phase lead-lag compensation
100

50

0

0

100

200

300

400

500
Time [s]

600

700

800

900

1000

Figure 26. Model validation against transient engine test data for a FTP 75 cycle

66

3.6 Conclusion
A compressor power model, based on the Euler turbomachinery equations with realistic
assumptions, was developed. Two new performance coefficients, the power and speed
coefficients were proposed as an alternative to multiple performance maps. The proposed
correlation between

𝐶𝑝𝑜𝑤𝑒𝑟 and

𝐶𝑠𝑝𝑒𝑒𝑑

is especially useful in defining the compressor power

necessary for achieving a desired compressor mass flow rate. This compressor power demand
can then be translated into a VGT (Variable Geometry Turbo) vane position or an assist demand
for assisted boosting systems. This relationship can also be easily used to compare compressor
design variants with respect to performance and range. The model is validated against data sets
from standard turbocharger flow bench tests, steady-state engine dynamometer tests as well as
transient engine simulations and test. Validation results indicate that the proposed model
provides accurate compressor power prediction over a broad range of compressor operating
conditions and provides for an easy and reliable extrapolation for operating conditions outside
the standard mapping domain. Further, the proposed model reduces the dimensionality of the
parameter space typically necessary for such applications. The reduced order, reduced
complexity model is especially useful for the control applications. Future work will focus on
improving prediction accuracy in the face of measurement noise as previously discussed. Modelbased control design based upon proposed model will be investigated.

67

CHAPTER 4: MODELLING OF HYDRAULIC ASSISTED AND REGENERATIVE
TURBOCHARGED D ENGINE
4.1 Abstract
In this chapter, a systematic modelling approach for engine air-path system and hydraulic
assisted and regenerative turbocharger system are presented. New developed turbocharger submodels are integrated with engine air-path and EGR system. Further new modelling approaches
for high speed hydraulic turbine and hydraulic centrifugal turbo-pump are proposed. System
level model integration and plant behavior investigation are carried out for engine air-path
system and hydraulic assisted turbocharger. The results show proposed reduced order engine
modelling has high fidelity. It could be used for model-based analysis and model-based
controller design.
4.2 Regenerative Hydraulic Assisted Turbocharger With VGT-EGR Overview
Regenerative and assisted turbocharger system are used to assist turbocharger during engine
acceleration and to recover exhaust energy during engine during engine deceleration. A most
common studied actuation system for assisted and regenerative turbocharger are electric based
and hydraulic based. Different actuation systems are reported in [53][54][63][65]. In this chapter,
we introduce the modelling hydraulic assisted and regenerative turbocharger system. More
details of system introduction can be found in [60]. A high speed hydraulic turbine and high
speed hydraulic turbo-pump are added inside turbocharger center housing as shown in Figure 27.

68

Figure 27. Diesel engine air-path system with hydraulic actuation system
Turbocharger turbine, compressor, hydraulic turbine and hydraulic turbo-pump share the
same shaft. With adding two hydraulic actuators on TC shaft, this provides extra control inputs
for the turbocharged air-path system. For traditional EGR-VGT system (Only HP EGR system),
VGT vane control and EGR valve control are coupled for its sharing the same source to drive the
exhaust flow and EGR flow. VGT vane position is normally used to track the target boost
pressure when the engine needs higher boost command. For a transient tip-in, VGT is used to
close further to build up exhaust pressure. To have demanded boost pressure, EGR valve is
intended to close further to help to build up exhaust pressure, such that EGR flow capacity is
reduced. To have the higher EGR mass flow rate, EGR valve open action is intended to drop
exhaust pressure, which will lead to lower exhaust pressure and turbine flow. Then turbine power
would drop. In such way, engine transient response would be compromised with lower
compressor mass flow rate. This is a classic control problem for VGT-EGR system, which has
been studied for almost three decades in control community. VGT-EGR control needs to be
coordinated based on different engine operating condition. .

69

When variable geometry turbocharger is interacting with high-pressure EGR loop, additional
assisted and loaded power will serve as useful inputs to decouple this dynamic system. For
instance, boost pressure might not need aggressive VGT and EGR closing with extra assisted
during tip-in. Since the turbocharger can be driven by assisted power, VGT control can be used
for partially boost control or EGR control. In such way, both EGR demanded and target boost
pressure will be met. During transient deceleration without regenerative power, VGT is used to
open to drop the pressure ratio across turbine wheel, reducing extracted turbine power. With this
VGT open action, EGR valve cannot have the right high-pressure ratio to drive EGR. With
additional loading power on TC shaft, TC shaft speed can be managed independently with VGT
action. Hence, with external control inputs, EGR mass flow rate, and boost pressure can be well
regulated.
Regenerative hydraulic assisted turbocharger system modelling consists of traditional engine
air-path modelling and assisted and regenerative turbocharger modelling. The challenges are to
identify a high fidelity mean value model for model-based analysis and controller design.

4.2.1 Engine air-path modelling overview
In this section, the subcomponents of engine block are discussed. To use mathematical
equations to represent a complex physic dynamic process, subsystem introduction for the diesel
engine system are presented. As shown in Figure 27, the diesel engine air-path system normally
consists of air intake system and exhaust system. For turbocharged diesel engine, the turbine is
driven by high temperature and high-pressure exhaust gas, which also drives the compressor to
pump air into the intake manifold. To have turbine accommodate wide flow range, vane nozzle is
used to adjust turbine inlet flow rate. Another approach is to size turbine smaller, bypassing
turbocharger with waste-gate at higher mass flowrate avoiding turbocharger over speed. This

70

approach is widely used in gasoline engine. Part of exhaust gas is guided though high pressure
EGR valve into intake manifold, then back into cylinder. Two coolers are normally deployed in
the intake manifold system: turbocharger inter cooler and EGR cooler, which cools high
temperature compressor mass flow and EGR mass flow, respectively.

Figure 28. Diesel engine control volume lay out [52]
Typically mean value diesel engine model with VGT and EGR can be represented in
equation as below (4.1) [52]. This is a seven states dynamics equation, which represents
thermodynamics process of diesel engine air-path system. Further crankshaft rotational dynamics
state and engine torque production can be added to make a comprehensive engine model, which
can be coupled with transmission or driveline model. All the variables in the equation can be
defined explicitly as a function of system states or other parameters. Experimental validation
data can be used to calibrate these models.

71

𝑑𝑚1
= 𝑊𝑐1 + 𝑊21 − 𝑊1𝑒 − 𝑊12
𝑑𝑡
𝑑𝑚2
= 𝑊12 + 𝑊𝑒2 − 𝑊21 − 𝑊2𝑡
𝑑𝑡
𝑊21 (𝐹2 − 𝐹1 ) − 𝐹1 𝑊𝑐1
𝐹1̇ =
𝑚1
𝑊𝑒2 (𝐹𝑒2 − 𝐹2 ) − 𝑊12 (𝐹1 − 𝐹2 )
𝐹2̇ =
𝑚2
𝑑𝑇1 𝑊21 (ℎ21 − 𝑢1 ) − 𝑊𝑐1 (ℎ𝑐1 − 𝑢1 ) − (𝑊1𝑒 − 𝑊12 )𝑅1 𝑇1 − 𝑚1 𝜒𝐹1 𝐹1̇
𝑄̇1
=
−
𝑑𝑡
𝑐𝑣1 𝑚1
𝑐𝑣1 𝑚1
𝑑𝑇2 𝑊𝑒2 (ℎ21 − 𝑢1 ) − 𝑊12 (ℎ12 − 𝑢2 ) − (𝑊21 − 𝑊2𝑡 )𝑅2 𝑇2 − 𝑚2 𝜒𝐹2 𝐹2̇
𝑄̇2
=
−
𝑑𝑡
𝑐𝑣2 𝑚2
𝑐𝑣2 𝑚2
𝑑𝑃𝑐
1
(−𝑃𝑐 + 𝜂𝑡𝑚 𝑃𝑡 )
=
𝑑𝑡
𝜏𝑡𝑐

(4.1)

But this high order system is difficult to do the controller design. Most of controller design
framework for diesel engine air-path system used a simplified three states [5] [7] with the
assumptions: 1. No thermodynamic dependence, all thermodynamic properties with respect to air;
2. Temperature dynamics are neglected.
Further simplification such as: air fraction states for the intake manifold and exhaust
manifold are not measured directly, these parameters need estimation or can be removed by the
assumption that emission control target can be met with designed target value. Then the seven
dynamic states can be reduced into 3rd system as follows:
𝑃̇ 3 =
𝑃̇ 2 =

𝑅𝑇3

(𝑚̇ 𝑜𝑢𝑡 − 𝑚̇ 𝑒𝑔𝑟 − 𝑚̇ 𝑡 )

𝑉3
𝑅 𝑇2
𝑉2

(𝑚̇ 𝑐 − 𝑚̇ 𝑖𝑛 + 𝑚̇ 𝑒𝑔𝑟 )

(4.2)

𝐽𝜔𝜔̇ = 𝑊̇𝑇 − 𝑊̇𝐶 − 𝑊̇𝐿𝑜𝑠𝑠

Here we redefined boost pressure as 𝑃2 and exhaust pressure as 𝑃3 . The three states diesel
engine air-path system includes intake manifold pressure, exhaust manifold pressure and TC
shaft speed. Based on previous researchers [5][7], three states model can well represent engine
air-path dynamics and be used for controller design. Hence, in this investigation, we start with

72

three states model for modelling the diesel engine air-path system coupled with hydraulic
assisted system.
4.2.2 Regenerative hydraulic assisted turbocharged engine modelling
Regenerative hydraulic assisted turbocharged engine modelling has three major subsystems,
which are engine system, turbocharger and hydraulic system as shown in Figure 29. Simplified
three state diesel engine air-path model structures are adopted from previous literatures. The
model has eight states as indicated in Eq(1). The eight states are the engine intake and exhaust
manifold pressures (P2 , P3), the turbocharger speed (ω), the hydraulic accumulator pressure (Pacc ),
the pre-turbine hydraulic pressure (Pt ), and the pump discharge pressure(Pp), and the hydraulic
accumulator piston position (x) and piston speed (v). The 8 states are highlighted with green in
Figure 29.
The control inputs of the system are the VGT vane position ( uvgt ), the EGR valve
position (uegr ) the hydraulic turbine inlet valve position ( uturbine ) and the hydraulic pump
discharge valve position(upump ). The system modelling interaction and causality for each sub
components are summarized in Figure 29. Each solid block represents a modelling subcomponent. Each line delivers parameter from one block to another block. For instance, ‘VGT
Turbine’ block needs temperature and pressure from exhaust pressure block ‘P3’ and
turbocharger speed from ‘Inertia+Fric’ block to calculate turbine mass flow rate and turbine
power. Figure 29 can be represented as the dynamic equations as in (4.3).
𝑇2
(𝑚̇𝑐 − 𝑚̇𝑖𝑛 + 𝑚̇𝑒𝑔𝑟 )
𝑉2
𝑇3
𝑃3̇ = (𝑚̇𝑜𝑢𝑡 − 𝑚̇𝑒𝑔𝑟 − 𝑚̇ 𝑇 )
𝑉3
𝐽𝜔𝜔̇ = 𝑊̇𝑇 − 𝑊̇𝐶 − 𝑊̇𝐿𝑜𝑠𝑠 + 𝑊̇𝑡𝑢𝑟𝑏𝑖𝑛𝑒 − 𝑊̇𝑝𝑢𝑚𝑝
𝛽
𝑃𝑃̇ = (𝑚̇𝑝𝑢𝑚𝑝 − 𝑚̇𝑣𝑎𝑙𝑣𝑒_𝑝 )
𝑉𝑝
𝛽
𝑃𝑡̇ = (𝑚̇𝑣𝑎𝑙𝑣𝑒_𝑡 − 𝑚̇𝑡𝑢𝑟𝑏𝑖𝑛𝑒 )
𝑉𝑡
𝑃2̇ =

73

(4.3)

𝛽
(𝑚̇
− 𝑚̇𝑡𝑢𝑟𝑏𝑖𝑛𝑒 − 𝑣𝐴)
𝑉0 + 𝑥𝐴 𝑝𝑢𝑚𝑝
𝑥̇ = 𝑣
(𝑃𝑎𝑐𝑐 − 𝑃𝑟𝑒𝑡𝑢𝑟𝑛 )𝐴 − 𝐹0 − 𝐹𝑓 (𝑣) − 𝑐𝑣 − 𝑘𝑥
𝑣̇ =
𝑥𝜌𝐴
̇ =
𝑃𝑎𝑐𝑐

Figure 29. System modelling architecture and calculating loop.

Five control volumes are identified and they are engine intake manifold, engine exhaust
manifold, pipe volume (Vt ) between the hydraulic turbine and hydraulic turbine inlet valve, pipe
volume ( Vp ) between the hydraulic pump and hydraulic pump valve, and high pressure
accumulator displacement (V0 + xA), where V0 is the initial displacement, A is the piston area
surface. The TC shaft dynamics model couples the hydraulic loop to the engine air-path loop.
The shaft dynamics equation represents the power balance between the four components on the
turbocharger shaft: turbine, compressor, hydraulic turbine and hydraulic pump. The power
balance is adjusted for shaft friction. The hydraulic loop interacts with air-path loop directly
through turbocharger shaft by transferring assist power from hydraulic turbine and extracting
regeneration power by the hydraulic pump. The hydraulic turbine and pump are controlled by
linear valves.

74

Figure 30. System inputs and outs for control perspective

In summary, the system can be represented as in Figure 30. Four inputs and two outputs
(compressor mass flow rate (or boost pressure) and EGR mass flow rate), which is a multi-input
and multi-output (MIMO) system for the model-based analysis and controller design.
Modelling assumptions:
1. Engine intake and exhaust manifolds are modeled as thermodynamics control volumes
with mass and flows in and out of these volumes.
2. For intake and exhaust manifolds volumes contains an ideal gas mixture of air and
combustion gas. The ideal gas is assumed to be uniform across the volume.
3. No inverse flow through EGR valve.
4. The temperature in intake and exhaust manifold are assumed to be uniform across the
controlled volume. ( T3  T3 , T2  T2 ).
5. No heat loss in air-path system is considered in this study.
6. The pressure of low pressure hydraulic accumulator is assumed as constant.
7. Parasitic loss of hydraulic turbine and hydraulic pump are not considered during
inactivate mode.

75

8. Heat loss and pipe loss in the hydraulic system is not considered in this study.
9. Constant hydraulic fluid bulk modulus.
10. Constant hydraulic fluid density.
11. Pressure of low pressure hydraulic tank is treated as constant.
As shown in the modelling governing equation (4.3), most of the modelling in this study
involves mass flow rate modelling. The mass flow rate models can be overviewed in Table 11.
The mass flow rate for each subcomponent depends on the pressure difference between
subcomponents upstream and downstream. For engine air-path and hydraulic fluid path, control
actuators directly impact the mass flow rate. Further, mass flow rate serves the outputs of control
results, such as demanded fresh air amount, EGR mass flow rate or engine charged air mass flow
rate.
Table 11. Mass flow rate governing equation
Engine intake mass flow
Engine exhaust mass flow

Governing equation
𝑚̇𝑖𝑛 = 𝑓(𝑁𝑒𝑛𝑔𝑖𝑛𝑒 , 𝑃2 )
𝑚̇𝑜𝑢𝑡 = 𝑚̇𝑓𝑢𝑒𝑙 + 𝑚̇𝑖𝑛

EGR mass flow

𝑚̇𝑒𝑔𝑟 = 𝑓 (𝑢𝑒𝑔𝑟 , 3)

Turbine mass flow

𝑚̇𝑡𝑢𝑟𝑏𝑖𝑛𝑒 = 𝑓 (𝑢𝑣𝑔𝑡 , 3)

Compressor mass flow

𝑚̇𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑜𝑟 = 𝑓 (𝜔 , 2)

Hydraulic pump mass flow
Hydraulic turbine mass flow
Hydraulic control valve flow

𝑚̇ℎ𝑦𝑑_𝑝𝑢𝑚𝑝 = 𝑓(𝜔, 𝑃𝐴𝑐𝑐 )
𝑚̇ℎ𝑦𝑑_𝑡𝑢𝑟𝑏𝑖𝑛𝑒 = 𝑓(𝜔, 𝑃𝐴𝑐𝑐 )
𝑚̇ℎ𝑦𝑑_𝑣𝑎𝑙𝑒 = 𝑓(𝑢𝑣𝑎𝑙𝑣𝑒 , ∆𝑃)

𝑃

𝑃2

𝑃

𝑃2
𝑃

𝑃1

4.3 Engine Modelling and Validation
For simplified three states engine air-path modelling as shown in (4.2). Intake pressure,
exhaust pressure and TC shaft speed are the three dynamic states. An expanded version for
engine air-path system can be viewed as in (4.4). As shown in (4.4), all the parameters mainly
depend on states and control inputs, with the assumption that turbine outlet pressure and
compressor inlet pressure are set as constant. (𝑃1 ≈ 𝑃4 ≈ 𝑃𝑎𝑖𝑟 ).

76

𝑅𝑓𝑇2 (𝜔,

𝑃2
)
𝑃1

𝑃2
𝑃3
) − 𝑓𝑚𝑖𝑛 (𝑁𝑒𝑛𝑔𝑖𝑛𝑒 , 𝑃2 ) + 𝑓ℎ𝑝𝑒𝑔𝑟 (𝑢𝑒𝑔𝑟 , ))
𝑉2
𝑃1
𝑃2
𝑅𝑓𝑇3 (𝑃2 , 𝑚̇𝑓𝑢𝑒𝑙 )
𝑃3
𝑃3
𝑃3̇ =
(𝑓𝑚𝑜𝑢𝑡 (𝑁𝑒𝑛𝑔𝑖𝑛𝑒 , 𝑃2 ) + 𝑚̇𝑓𝑢𝑒𝑙 − 𝑓𝑚𝑇 (𝑢𝑣𝑔𝑡 , ) − 𝑓ℎ𝑝𝑒𝑔𝑟 (𝑢𝑒𝑔𝑟 , ))
𝑉3
𝑃4
𝑃2
𝑃2̇ =

(𝑓𝑚𝑐 (𝜔,

𝐽𝜔𝜔̇ = 𝑓𝑊̇𝑇 ( 𝑢𝑣𝑔𝑡 ,

(4.4)

𝑃3
𝑃2
, 𝜔, 𝑇3 ) − 𝑓𝑊̇𝑐 ( , 𝜔) − 𝑓𝑊̇𝐿𝑜𝑠𝑠 ( 𝜔)
𝑃4
𝑃1

Each of the nonlinear functions is investigated and validated in this chapter. Developed
engine sub-models are identified from a medium duty diesel engine steady state mapping data.
The 195 engine steady state operation points are used for model identification. Then engine airpath model is further validated through transient engine test.
4.3.1 Engine intake and exhaust mass flow rate
It is common to model the engine breathing process in four stroke engines with the speed
density equation. The mass flow rate of intake charge ṁin in equation (4.2) can be modelled as:
𝑚̇ 𝑖𝑛 =

𝜂𝑣𝑜𝑙 𝑃2 𝑁𝑒 𝑉𝑑
120𝑅𝑇2

=

𝜂𝑣𝑜𝑙 𝜌2 𝑁𝑒 𝑉𝑑
120

(4.5)

Where, 𝜂𝑣𝑜𝑙 : volumetric efficiency of the engine
𝑃2

𝑅𝑇2

: density of the gas mixture in the intake manifold.

𝑉𝑑 : engine displacement volumes.

The volumetric efficiency is a function of engine speed and boost pressure. From [8], the
volumetric efficiency can be taken as:
𝜂𝑣𝑜𝑙 = (𝑐𝑜𝑒1 ∙ √𝑃2 + 𝑐𝑜𝑒2 ∙ √𝑁𝑒 + 𝑐𝑜𝑒3 )

(4.6)

With knowing engine intake mass flow rate, density of gas mixture in the intake manifold,
engine displacement volume, the volumetric efficiency can be easily identified.
shown in Figure 31.

77

Results are

The engine exhaust mass flow rate 𝑚̇𝑜𝑢𝑡 can be approximated as the sum of charged air
ṁin and fuel flow rate ṁfuel . Obviously, this is based on the assumption of negligible and zero
residual gas fractions in the cylinder. Thus:
𝑚̇𝑜𝑢𝑡 = 𝑚̇𝑖𝑛 + 𝑚̇𝑓𝑢𝑒𝑙

(4.7)

Fuel mass flow rate is based on the commanded fuel injection rate into each cylinder. For
eight cylinder engine, the fuel mass flow rate is as in (4.8). Unit for fuel command is injected
fuel quantity, which has the unit of mg/stroke.

Model predicted intake mass flow rate [kg/s] Engine intake mass flow rate[kg/s]

𝑚̇𝑓𝑢𝑒𝑙 =

8 × 10−6
𝑢𝑓𝑢𝑒𝑙 𝑁𝑒
120

(4.8)

1

0.5

0
400
300
200
P2 [kPa]

100

0

1000

2000

4000

3000

Engine speed [rpm]

0.5
0.4

y=x

0.3
0.2
0.1
0

0

0.1

0.2
0.3
0.4
Engine intake mass flow rate [kg/s]

0.5

0.5

Error [%]

Figure 31. Model identification results for volumetric efficiency
0

-0.5

-1

0

78

0.1
0.2
0.3
0.4
Measured intake mass flow rate [kg/s]

0.5

4.3.2 Exhaust manifold temperature
The engine exhaust manifold temperature model is based on two sub-models, which are
cylinder-out temperature and heat loss model for exhaust pipes. The cylinder out temperature is
modelled as in (4.9) [55]. This approach is based upon ideal-gas Seliger cycle [56].
𝑚̇𝑓𝑢𝑒𝑙 𝐿𝐻𝑉
𝑃𝑒𝑚 1−1/𝛾𝑎𝑖𝑟 1−𝛾𝑎𝑖𝑟 1/𝛾𝑎𝑖𝑟 −1
1 − 𝑥𝑐𝑣 𝑥𝑐𝑣
𝛾 −1
𝑇𝑒𝑜𝑢𝑡 = (
)
∙ 𝑟𝑐
∙ 𝑥𝑝
∙(
∙ (1 − 𝑥𝑟 ) ∙ (
+
) + 𝑇2 𝑟𝑐 𝑎𝑖𝑟 )
𝑃𝑖𝑚
𝑚̇𝑎𝑖𝑟 + 𝑚̇𝑓𝑢𝑒𝑙
𝐶𝑝𝑎
𝐶𝑣𝑎

(4.9)

This temperature doesn’t take account for temperature loss through exhaust pipe. With the
same approach in [55], exhaust manifold temperature T3 can be modelled as a function of the
exhaust mass flow rate and the engine exhaust temperature.
𝑇3 = 𝑇𝑎𝑖𝑟 + (𝑇3 − 𝑇𝑎𝑖𝑟 )exp

−

ℎ𝑡𝑜𝑡 𝜋𝑑𝑝𝑖𝑝𝑒 𝑙𝑝𝑖𝑝𝑒 𝑛𝑝𝑖𝑝𝑒
𝑚̇𝑎𝑖𝑟 +𝑚̇𝑓𝑢𝑒𝑙 𝑐𝑝𝑒

(4.10)

Where Tair is ambient temperature, htot the total heat transfer coefficient of exhaust manifold.
The dpipe , lpipe , and , npipe are pipe diameter, pipe length and number of pipes, respectively. The
s identified results can be viewed in Figure 32

79

Exhaust temperature [c]

1000
800
600
400
200
0
400
150

300

100

200

Model predicted Exhaust temperature [c]

Engine intake pressure [kPa]

50
100

0
Fuel injection [mg/hub]

900
y=x
800
700
600
500
400
300
200
100
100

200

300
400
500
600
700
Measured Exhaust temperature [c]

800

900

Figure 32. Model identification results for exhaust temperature
0.15
0.1

4.3.4 Intake manifold temperature
0.05
Error [%]

0
-0.05 engine includes an intercooler after the compressor, and EGR
All turbocharged diesel
-0.1
-0.15

charger cooler. The temperature in the intake manifold is the combined temperature from both
-0.2
-0.25

intake charger cooler and EGR cooler. These components can be treated as standard heat
-0.3
100

200

300
400
500
600
700
Measured Exhaust temperature [c]

800

900

exchangers. Normally, a heat exchanger effectiveness 𝜖 is used to modelling the heat transfer
effect through coolers. Then the downstream of cooler temperature can be taken as:
𝑇𝑑𝑜𝑤𝑛𝑠𝑡𝑟𝑒𝑎𝑚 = 𝑇𝑢𝑝𝑠𝑡𝑟𝑒𝑎𝑚 (1 − 𝜖) + 𝑇𝑐𝑜𝑜𝑙𝑎𝑛𝑡 𝜖

(4.11)

Assume temperature mixture based on mass flow mixture, then intake manifold temperature
can be modelled as:
𝑇2′ =

𝑚̇𝑐𝑜𝑚 𝑇2∗ + 𝑚̇𝑒𝑔𝑟 𝑇3∗
𝑚̇𝑐𝑜𝑚 + 𝑚̇𝑒𝑔𝑟

Where 𝑇2∗ = 𝑇2 (1 − 𝜖) + 𝑇𝑐𝑜𝑜𝑙𝑎𝑛𝑡 𝜖

80

(4.12)

𝑇3∗ = 𝑇3 (1 − 𝜖) + 𝑇𝑐𝑜𝑜𝑙𝑎𝑛𝑡 𝜖

There are two ways to model temperature after compressor. One can be based on compressor
power equation, which is discussed in Chapter 3. Another way is that the temperature after the
compressor can be modeled as high order polynomial equation as below.

Intake manifold temperature [c]

2
𝑇2′ = 𝑓(𝜔, 𝑚̇𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑜𝑟 ) = 𝑐𝑜𝑒1 + 𝑐𝑜𝑒2 𝑚̇𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑜𝑟 + 𝑐𝑜𝑒3 𝜔 + 𝑐𝑜𝑒4 𝑚̇𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑜𝑟
2
3
2
+𝑐𝑜𝑒5 𝑚̇𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑜𝑟 𝜔 + 𝑐𝑜𝑒6 𝜔 + 𝑐𝑜𝑒7 𝑚̇𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑜𝑟 + 𝑐𝑜𝑒8 𝑚̇𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑜𝑟 𝜔 + 𝑐𝑜𝑒9 𝑚̇𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑜𝑟 𝜔2

(4.13)

200

100

0
15
10
5

Model predicted Intake temperature [c]

4

x 10
TC speed [RPM]

0.2

0

0

0.3

0.4

0.5

0.1
Compressor mass flow rate [kg/s]

200
150
y=x
100
50
0

0

50

100
150
Measured Intake temperature [c]

200

Figure 33. 0.01
Model identification results for temperature after compressor
Error [%]

4.3.3 EGR mass flow
rate modelling
0.005
0

EGR mass flow rate is controlled through EGR valve position. The EGR flow rate can be
-0.005

modeled by the conventional orifice flow equation, which has major two inputs: valve
-0.01
20

40

60

80
100
120
140
160
P3 [c]
Measured Intake temperature

180

200

position uegr , upstream and downstream pressure ratio P . Orifice flow equation for compressor
2

81

mass flow rate is usually modeled as a one dimensional, steady, compressible flow. Two
different flows can be distinguished by using pressure ratio as below.
𝑚̇𝑒𝑔𝑟 = 𝐴𝑒𝑓𝑓 𝑃3 √

2
ℜ𝑇3

𝜓

𝑘

2

𝑘+1
𝑘

𝑃3
2 𝑘−1
𝑘
𝑃3 −𝑘
𝑃3 −
𝐼𝑓
>(
)
,𝜓 = √
[( ) − ( )
𝑃2
𝑘+1
𝑘 − 1 𝑃2
𝑃2
𝑘

(4.14)

]

𝑘+1

𝑃3
2 𝑘−1
2 𝑘−1
𝐼𝑓
>(
)
, 𝜓 = √2𝑘 (
)
𝑃2
𝑘+1
𝑘+1

The EGR valve flow effective area Aeff is affected by engine RPM, pressure ratio and load as
show in Figure 35 . This can be explained with engine pulsation flow effect on EGR mass flow
rate. In this modelling work, since this is not our primary interest, so a subsonic model with
Aeff = f(uegr ) is adopted for this study. The identified results are shown in Figure 34.
35
test data
Model

30

EGR effective area

25

20

15

10

5

0

0

10

20

30
40
EGR valve position

50

Figure 34. Identified EGR effective area

82

60

70

Model predicted HP EGR mass flow rate [kg/s] Model predicted HP EGR mass flow rate [kg/s]

200
100
0
0
50
100
150

20

0

40

80

60

valve opening percentage [%]

EGR delta P [kPa]

200
y=x

150
100
50
0

0

50
100
150
Measured HP EGR mass flow rate [kg/s]

200

Figure 35. Model identification results for EGR mass flow rate
4.4

Variable Geometry Turbocharger modelling and validation
The two major components are connected through turbine shaft for the turbocharger.

Turbocharger speed dynamic equation has four terms for turbocharger itself, the exhaust gas
driven turbine power, compressor power, friction power and kinetic energy term. Based on
Newton's second law, power balance equation (without assisted and regenerative power) is:
𝐽𝜔𝜔̇ = 𝑊̇𝑇 − 𝑊̇𝐶 − 𝑊̇𝐿𝑜𝑠𝑠

(4.15)

Energy balanced equation would serve as the link between assisted and regenerative devices.
With external add-on power, turbine power will be balanced at higher speed region. For
modelling of turbocharger power, most of the approaches are based turbine and compressor

83

efficiency map provided by manufactures. For integrity, some results from previous chapters are
adopted here. But for details, please refer to previous chapters. Friction model and identification
for turbocharger are presented in Chapter 2. New modelling approach for compressor mass flow
rate modelling is discussed in details here.

4.4.1 VGT turbine model
The turbine utilizes the exhaust gas residual enthalpy to drive the compressor. Energy is
extracted from engine exhaust, through expansion across turbine blades. Turbine power is used
to overcome the compressor load and friction loss as well as acceleration inertia. Hence accurate
turbine power is needed for modelling turbocharger rotational speed dynamics. The overall
modelling structure of turbine block is as shown in Figure 36. The turbine model consists of submodels for turbine mass flow rate and turbine power. The turbine modelling depends on inputs
from the engine block, which are exhaust manifold pressure, exhaust manifold temperature.

Figure 36. Turbine modelling layout
4.4.1.1 Turbine power
In this section, turbine operation depends on turbine inlet condition, turbine power based on
inlet condition [23] are utilized here:
𝑊̇ 𝑇 =

1
60

𝑁𝑇𝐶 (𝑚̇ 𝑒𝑥 )

2

1 ℜ𝑇∗
𝐵 𝑃

84

∗

tan 𝛼1

(4.16)

Temperature T ∗ and𝑃∗ is temperature and pressure between the turbine blade and nozzle. In
this study, these two parameters are approximated as temperature and pressure in the exhaust
manifold. Model identification can be referred to Chapter 2.
4.4.1.2 Turbine mass flow rate
The mass flow rate through the turbine can be modeled using the similar flow equations in
EGR section. The effective valve area is experimentally determined to be a polynomial function
of VGT position. Two other variables which impact the flow rate are critical pressure and
pressure ratio when flow rate is zero. From [1], the mass flow rate through turbine can be
modelled as in (4.17). Turbine mass flow rate depends on pressure ratio across turbine as well as
effective nozzle areas, the identified results can be found at Figure 37.
𝑚̇𝑡𝑢𝑟𝑏𝑖𝑛𝑒 = 𝑓 (𝑢𝑣𝑔𝑡 ,
𝑚̇𝑡𝑢𝑟𝑏𝑖𝑛𝑒 = 𝐴𝑒𝑓𝑓 𝑝3 √
𝜓=

𝑘

2

𝑝 𝑘
√
[( 4)
𝑘−1 𝑝3

𝑝4

−( )
𝑝3

𝑘+1
𝑘

𝑃3
)
𝑃4
2

ℜ𝑇3
1

]=

𝜓

𝑝 𝑘 𝑐
( 4) √ 𝑝 [1
𝑝3
ℜ

𝐴𝑒𝑓𝑓 = 𝑓 (𝛼(𝑢𝑣𝑔𝑡 ))

85

𝑝4

−( )
𝑝3

𝑘−1
𝑘

(4.17)
]

Model predicted turbine mass flow rate [kg/s]

0.5
0.4
0.3
0.2
0.1
0
1

0.5

0

2

1

Model predicted turbine mass flow rate [kg/s]

VGT

3

5

4

Turbine PR

0.4

y=x

0.35
0.3
0.25
0.2
0.15
0.1
0.05
0

0

0.05

0.1

0.15
0.2
0.25
0.3
0.35
Measured turbine mass flow rate [kg/s]

0.4

0.45

Figure
37. Turbine mass flow rate model identification
0.1
0
Error [%]

4.4.2 Compressor modelling
-0.1
Inputs for compressor-0.2
modelling are intake manifold pressure and TC shaft speed. The output
-0.3

of compressor model is compressor power and compressor mass flow rate. In this section, we
-0.4

0

0.05

0.1

0.15
0.2
0.25
0.3
0.35
0.4
Measured turbine mass flow rate [kg/s]

0.45

0.5

investigate different approaches for compressor mass flow rate modelling in details.

Figure 38. Compressor model layout
4.4.2.1 Compressor power
As shown in [84], compressor power can be expressed as a function of TC speed and mass
flow rate. More detailed model development for compressor power can be found in chapter 3.

86

𝜔 2
𝜔
𝑊̇𝑐 = 𝑚̇𝑐3 𝜀1 ( ) − 𝜀2 ( ) + 𝜀
𝑚̇𝑐
𝑚̇𝑐

(4.18)

4.4.2.2 Compressor mass flow rate
The compressor receives power from the turbine, driven by engine exhaust gas. High
pressure is built by compression process from compressor inlet port to outlet port. High pressure
in volute behaves as the resistance force to slow down compressor speed. Other resistance forces
are compressor incidence losses (impeller wheel and diffuser) as well as aerodynamic friction
losses (impeller wheel and diffuser). The resistance forces reduce compressor efficiency as well
as mass flow rate for the compression process.

Figure 39. Automotive compressor layout
In this section, we redefine the P3 , P4 as volute pressure and intake manifold pressure in this
study. From compressor dynamic mass flow rate equation (4.19) [57], compressor mass flow rate
dynamics depends on volute pressure and intake manifold pressure as well as the geometry of
volute and intake pipe.
𝐴

𝑚̈ 𝑐 = (𝑃3 − 𝑃4 ) 𝐿

(4.19)

Volute pressure 𝑃3 can be calculated by pressure rise from the compressor. Based on forward
flow, pressure rise can be modelled as:

87

𝛾

(4.20)

𝜂Δℎ 𝛾−1
𝑃3 = (1 +
)
𝑃1
𝑇1 𝐶𝑝

From [84], increased enthalpy by impeller can be approximated with:
𝜂Δℎ = (𝑚̇𝑐 2 𝑓 (

𝜔
𝑚̇𝑐

) − 𝑚̇𝑐 2 𝑘𝑓 ) /𝜓

(4.21)

Combining (4.19), (4.20) and (4.21), compressor mass flow rate dynamic equation is:
𝛾

𝑚̈𝑐 = ((1 +

𝜔
(𝑚̇𝑐 2 𝑓(𝑚̇ )−𝑚̇𝑐 2 𝑘𝑓 )/𝜓 𝛾−1
𝑐

)

𝑇1 𝐶𝑝

𝑃1 − 𝑃4 )

𝐴

(4.22)

𝐿

With constant inlet conditions, the dynamic mass flow rate can be expressed as a function of
mass flow rate, turbocharger speed and intake manifold pressure:
(4.23)

𝑚̈𝑐 = 𝑓(𝑚̇ 𝑐 , 𝜔, 𝑃4 )

This means the compressor mass flow rate dynamic system can take TC shaft speed and
intake manifold pressure as inputs. Compressor mass flow rate is the dynamic state and output of
this dynamic system. Hence, compressor mass flow rate can be expressed as the integration of
derivative of mass flow rate with given initial condition and given time step.
𝑡

𝑡

(4.24)

𝑚̇ 𝑐 = ∫ 𝑚̈ 𝑐 = ∫ 𝑓(𝑚̇ 𝑐 , 𝜔, 𝑃4 ) + 𝑚̇ 𝑐_0
0

0

In model (4.22), there are three parameters to be identified: 𝑘𝑓 , 𝜓, 𝐴𝐿.
found in chapter 3.

𝐴
𝐿

𝑘𝑓 , 𝜓 identification

can be

is the geometry parameter from volute to the location (from 3 to 4 in Fig.1).

a. Hammer-Winner model for compressor mass flow rate
Based on (4.20), compressor mass flow rate can be taken as output of a transfer functions
with TC speed and intake manifold pressure as inputs. Alternative approach is to use ‘Black box’
identification method to investigate the transfer function between inputs and outputs. Proposed
‘Black box’ dynamic system takes TC speed and intake manifold pressure as input as shown in
(4.22) and Figure 40. It includes input nonlinearities, linear systems and output nonlinearities.

88

This is so called Hammer-Wiener (H-W) model. This system can also be expressed as state space
representation as in (4.23).
𝑚̇𝑐 = 𝑓(𝑇𝐹1 (g1 (𝜔)) + 𝑇𝐹2 (g 2 (𝑃4 )))

(4.25)

Figure 40. Hammer-Wiener Model

𝑚
̂̇ = 𝐴𝑚
̂ + 𝐵𝑔(𝑢) 𝑚̇𝑐 = 𝑓(𝑚
̂)

𝜔

(4.26)

𝑔 (𝜔)

Where, 𝑢 = [𝑃 ], 𝑔(𝑢) = [𝑔 1(𝑃 )], 𝑔1 , 𝑔2 and f are nonlinearity for inputs and output. 𝑚
̂ is defined
4

2

4

as the Intermediate variable. In this case, the coefficients need to be identified are A, B matrix
and 𝑔1 , 𝑔2 and f nonlinearity function. The system output value depends on input and output
nonlinearities as well as linear systems.
b. Static nonlinear model compressor mass flow rate
Base on the power model from Chapter 3, the compressor mass flow rate can be obtained
through turbomachinery flow coefficient equation. The speed coefficient is defined in Chapter 3.
The flow coefficient is defined as in (4.27). The identified results can be shown in Figure 41.
𝐶𝑓𝑙𝑜𝑤 =

𝑚̇𝑝2
𝛾−1
𝛾
(𝑃𝑟

− 1)

89

(4.26)

1.4
Test data
Model

1.2

Flow coefficient

1

0.8

0.6

0.4

0.2

0

0

2

4

6
Speed coefficient

8

10

12
x 10

5

Figure 41. Compressor mass flow rate static nonlinear model

c. Compressor mass flow rate model validation
For physics based mass flow rate model, 𝑘𝑓 , 𝜓 are identified through steady state flow bench
test data [84]. Geometry parameter

𝐴
𝐿

is tuned online in dynamic simulation to match with

vehicle test results in this study. Geometry parameter

𝐴
𝐿

could be further identified through

For H-W compressor mass flow rate model identified through vehicle

experimental data.

transient test data. Identified linear transfer function is as below:
𝑧 −1 − 0.9993 𝑧 −2
1 − 1.437 𝑧 −1 + 0.7298 𝑧 −2 − 0.2922 𝑧 −3
𝑧 −1 − 0.9974 𝑧 −2
𝑇𝐹2 =
1 − 1.119 𝑧 −1 + 0.06416 𝑧 −2 + 0.05707 𝑧 −3
𝑇𝐹1 =

(4.27)

𝑔 (𝜔)

̂ can be obtained as:
Let 𝑣 = 𝑔(𝑢) = [𝑔 1(𝑃 )], Hence, intermediate 𝑚
2

4

𝑚
̂ = [𝑇𝐹1
𝑚
̂=

𝑣1
𝑇𝐹2 ] [ ]
𝑣2

(4.28)

𝑧−1 − 0.9993 𝑧−2

𝑣1
1 − 1.437 𝑧−1 + 0.7298 𝑧−2 − 0.2922 𝑧−3
−1
−2
𝑧 − 0.9974 𝑧
+
𝑣2
−1
1 − 1.119 𝑧 + 0.06416 𝑧−2 + 0.05707 𝑧−3

(4.29)

Form above equation, it can be concluded that, current state of intermediate variable m
̂ is
based on previous two steps of inputs v and previous five steps of m
̂.

90

𝑚
̂ 𝑘 = 𝑓(𝑣1(𝑘−1), 𝑣1(𝑘−2), 𝑣2(𝑘−1), 𝑣2(𝑘−2) , 𝑚
̂ 𝑘−1 , 𝑚
̂ 𝑘−2 , 𝑚
̂ 𝑘−3 , 𝑚
̂ 𝑘−4 , 𝑚
̂ 𝑘−5 , 𝑚
̂ 𝑘−6 )

(4.30)

This is an interesting finding. Nonlinear physics based model can be converted to a
‘sandwich’ model; system dynamic is purely based on linear model, and input and out
nonlinearity. Identified results can be shown in Figure 42 and Figure 43. It is clearly to show that
compressor system has strong nonlinearity. Identified linear system has six states in this study. It
varies with TC speed, intake manifold pressure, as well as mass flow rate dynamic system itself.
The poles and zeros of linear systems show that this open loop system is not stable for some
modes. It also shows non-minimum phase behaviors due to positive zeros for both input channels.
This non-minimum phase could be potentially eliminated by the control technique of poles and
zeros placement.

u

-0.8

x 10

u

y

Linear Block

NL

5

2.4
g

-1

Linear Block

NL

NL

x 10

y

NL

5

g

1

2

2.3

-1.2

2.2

-1.4

2.1

-1.6
g2

g1

2
-1.8

1.9
-2

1.8

-2.2

1.7

-2.4

1.6

-2.6
-2.8

0

2

4

6
8
TC speed

10

12

u

NL

a.

1.5

14
x 10

4

Nonlinearity for TC speed input

0

Linear Block

500

1000
Pressure

y

1500

2000

NL

b. Nonlinearity for pressure input
f

0.45
0.4
0.35

f

0.3
0.25
0.2
0.15
0.1
0.05
-2.4875

-2.4875

-2.4875

-2.4875
m hat

-2.4875

-2.4875

-2.4875
x 10

4

c. Nonlinearity for mass flow rate output

Figure 42. Nonlinearity for H-W compressor mass flow rate model

91

u

To Mass flow rate

0.6

1
Zeros
Poles
unit circle

0.8
0.6

Zeros
Poles
unit circle

0.4

From Pressure

From TC speed

0.4
0.2
0
-0.2

0.2
0
-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1
-1

NL

To Mass flow rate

1
0.8

y

Linear Block

NL

0

1

-1
-1

0

1

Figure 43. Zeros and poles for linear system in H-W model
H-W compressor mass flow rate model has good accuracy compared to vehicle transient test
data as shown in Figure 42 and Figure 45. The only small error happens at extreme light load and
very fast transient operations. For model accuracy, proposed model shows 5% accuracy within
90.75% of the operating points. The error function is defined as in (9). This compressor model
could be used as a subcomponent model for compression system (compressible flow),
turbocharged internal combustion engine as well as a gas turbine engine, theoretically.
Application for incompressible flow needs further investigation. For most turbocharged internal
combustion engine air-path modelling methods, intake manifold pressure and TC speed are
modelled as dynamic states. Hence, in those engines modelling approaches, proposed
compressor mass flow rate sub-model can directly use intake manifold pressure and TC speed as
model inputs to compute compressor mass flow rate.
𝑒𝑟𝑟𝑜𝑟 =

|𝜔𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑(𝑖) − 𝜔𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑(𝑖) |
|𝜔𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑(𝑖) |

(4.31)

Where, i is each sample data. By comparing with physics based model and H-W model as
shown in Figure 44. Identification results for H-W compressor mass flow rate model. Both of
these two models are simulated with same inputs, which are TC speed and intake manifold

92

pressure. Initial conditions for these two dynamic models are set as zero. The results for both
models agree with test data, but physics based dynamic model has high error in some region.
These could be due to unmodelled dynamics and simplification. For instance, for idling
operation (480s-600s) in Figure 45, the high predicted compressor mass flow rate from physicsbased model might be due to not considering the heat transfer effect from turbine side.
One potential application for these two models is to have more accurate compressor mass
flow rate estimation by combining two models through sensor fusion techniques, such as using
extended Kalman filter. By combining two models, the model output would have better accuracy
as well as representing the physics of compression process. This approach can also be used as a

Mass flow rate [kg/s]

virtual sensor for compressor mass flow rate.
0.5

0.3
0.2
0.1
0

Delta mass flow rate [kg/s]

Measured mass flow rate
Model predicted mass flow rate

0.4

0

100

200

300

400
Time [s]

500

600

700

800

0

100

200

300

400
Time [s]

500

600

700

800

0

100

200

300

400
Time [s]

500

600

700

800

0.02
0.01
0
-0.01
-0.02
-0.03

Absolute error [%]

20
15
10
5
0

Figure 44. Identification results for H-W compressor mass flow rate model

93

Figure 45. Comparison between physics based mass flow rate model and H-W compressor mass
flow rate
4.5 Hydraulic System Modelling
4.5.1 System overview
The RHAT system consists of two linear solenoid valves, a centrifugal hydraulic turbine, a
centrifugal hydraulic turbo pump and a spring-loaded hydraulic accumulator as shown in Figure
46.. In this study, both hydraulic pump and turbine are radial flow, centrifugal turbomachines.
Two solenoid valves are used to control the mass flow rates of the hydraulic turbine and
hydraulic pump. The Dynamic states are pressures between turbine and valve, the pressure
between pump and pump valve, pressure in the high-pressure hydraulic tank, the piston position
and velocity. For simplicity the turbine downstream pressure and pump upstream pressure are
treated as a constant 𝑃0 (pressure in the low-pressure accumulator).
𝛽
𝑃̇ 𝑝 = (𝑚̇ 𝑝 − 𝑚̇ 𝑣𝑎𝑙𝑣𝑒_𝑝 )
𝑉
𝑝

𝛽

𝑃̇ 𝑇 = (𝑚̇ 𝑇 − 𝑚̇ 𝑣𝑎𝑙𝑣𝑒_𝑡 )
𝑉
𝑡

𝑃̇ 𝐴𝑐𝑐 =
𝑣̇ =

1
𝑚

𝛽

𝑉0

(𝑚̇ 𝑣𝑎𝑙𝑣𝑒𝑝 − 𝑚̇ 𝑣𝑎𝑙𝑣𝑒𝑡 − 𝑣𝐴)
+𝑥𝐴
𝑥̇ = 𝑣

((𝑃𝐴𝑐𝑐 − 𝑃𝑟𝑒𝑡𝑢𝑟𝑛 )𝐴 − 𝐹0 − 𝐹𝑓 (𝑣) − 𝑐𝑣 − 𝑘𝑥)

94

(4.32)

Figure 46. Layout of hydraulic actuation system
The governing equations of the hydraulic system, mass flow rates of turbine, pump, and
valve are provided in (4.33):

𝛽
(𝑓
(𝑃 − 𝑃𝑡 , 𝑢𝑣𝑎𝑙𝑣𝑒_𝑡 ) − 𝑓𝑚̇𝑇 (𝑃𝑡 , 𝜔))
𝑉𝑡 𝑚̇𝑣𝑎𝑙𝑣𝑒_𝑡 𝐴𝑐𝑐
𝛽
𝑃𝑝̇ = (𝑓𝑚̇𝑝 (𝑃𝑝 , 𝜔) − 𝑓𝑚̇𝑣𝑎𝑙𝑣𝑒_𝑝 (𝑃𝑝 − 𝑃𝐴𝑐𝑐 , 𝑢𝑣𝑎𝑙𝑣𝑒_𝑝 ))
𝑉𝑝
𝑃𝑡̇ =

̇ =
𝑃𝐴𝑐𝑐

𝛽
(𝑓
(𝑃 − 𝑃𝐴𝑐𝑐 , 𝑢𝑣𝑎𝑙𝑣𝑒_𝑝 ) − 𝑓𝑚̇𝑣𝑎𝑙𝑣𝑒 (𝑃𝐴𝑐𝑐 − 𝑃𝑡 , 𝑢𝑣𝑎𝑙𝑣𝑒_𝑡 ) − 𝑣𝐴)
𝑡
𝑉0 + 𝑥𝐴 𝑚̇𝑣𝑎𝑙𝑣𝑒_𝑝 𝑝
𝑥̇ = 𝑣
1
𝑣̇ = ((𝑃𝐴𝑐𝑐 − 𝑃𝑟𝑒𝑡𝑢𝑟𝑛 )𝐴 − 𝐹0 − 𝐹𝑓 (𝑣) − 𝑐𝑣 − 𝑘𝑥)
𝑚

(4.33)

The system dynamics depends on the mass flow rates through the subcomponents. For
instance, pressure before hydraulic turbine 𝑃𝑡 is based on the mass flow rate change in the small
volume 𝑉𝑡 between the hydraulic turbine and valve. The flow rate range is based on hydraulic
turbine and valve mass flow rates. Furthermore, the turbine mass flow rate is based on the
hydraulic turbine speed and the pressure difference across the turbine. For the control valve,
mass flow rate is based on the pressure difference across control valve (or pressure difference
between the hydraulic accumulator pressure 𝑃𝐴𝑐𝑐 and pressure after the valve 𝑃𝑡 ). The dynamics
for hydraulic accumulator pressure is mainly dependent on the valve mass flow rate, piston
position 𝑥, velocity,𝑣, and acceleration 𝑣̇ . Energy stored in the pre-loaded spring pressurizes the

95

hydraulic fluid used to drive the hydraulic turbine. The hydraulic turbine output power is
dependent on turbine mass flow rate 𝑚̇ 𝑇 and speed 𝜔 . Thus, with the known control input
(turbine valve position), hydraulic turbine power can be calculated based on the flow rate and
current TC shaft speed.
During the TC shaft deceleration process, the hydraulic turbine is deactivated by closing the
turbine valve. The pump valve is opened to apply load onto the TC shaft. The pump drives the
low-pressure fluid through the pump valve into the high-pressure accumulator and builds up
pressure before and after the pump valve, causing fluid flow into the accumulator. The pump
valve flow rate is based on the pressure difference across the valve (or the pressure difference
between pre-valve pressure, 𝑃𝑝 and the accumulator pressure,𝑃𝐴𝑐𝑐 ). The valve mass flow rate is
regulated by the valve position, which is the control mechanism for the pump flow. Thus, the
pump power can be regulated using the pump valve position. TC shaft kinetic energy can be
stored in the form of the accumulator spring potential energy.
4.5.2

Hydraulic centrifugal pump modelling

4.5.2.1 Hydraulic centrifugal pump power model
Most of the early modeling approaches for hydraulic component power are based on
efficiency map as shown in Figure 47. These modeling approaches are based on the utilization of
the efficiency map by extrapolation and interpolation. Given the nonlinearity of these maps, it is
typically difficult to obtain an analytical model describing these efficiency maps. However, since
the hydraulic pump power is the target parameter for the turbocharger shaft speed dynamic
equation, it is meaningful to model the hydraulic turbine power directly.

96

Hydraulic pump efficiency
0.7
50
0.65

Hydraulic pump power [kW]

45
40

0.6

35

0.55

30
0.5
25
0.45

20
15

0.4

10

0.35

5
0.3
3

4

5

6
7
TC speed [RPM]

8

9

10
x 10

4

Figure 47. Hydraulic pump efficiency (y-axis is pump power)

Based on the Euler turbomachinery equation in [84], the pump performance can be modeled
using the defined speed coefficient ṁ

ω
pump

and power coefficient

Ẇpump
(ṁpump )

3

; as discussed in [84].

The power coefficient can be a quadratic function of speed coefficient (or linear function in the
logarithmic scale). Based on the manufacture-provided performance map, these two variables
are plotted in Figure 48.
𝑊̇𝑝𝑢𝑚𝑝

3

(𝑚̇𝑝𝑢𝑚𝑝 )

= 𝑎(

𝜔
𝑚̇𝑝𝑢𝑚𝑝

2

) +𝑏(

𝜔

𝑚̇𝑝𝑢𝑚𝑝

3
𝑊̇𝑝𝑢𝑚𝑝 = 𝑚̇𝑝𝑢𝑚𝑝
∗ 10𝑝 (

𝜔
𝑚̇𝑝𝑢𝑚𝑝

97

)+𝑐

(4.34)
𝑞

)

3

x 10

6

Test data
Model
2.5

Power coeff

2

1.5

1

0.5

0

0

0.5

1

1.5

2
2.5
Speed coeff

3

3.5

4

4.5
x 10

6

Figure 48. Identified pump model and test data

Coefficients in (4.34) can be determined by using the same approach in the previous
compressor modeling section. The results in Figure 48 validate that turbo pump's operation
characteristics agree with proposed centrifugal compression machine power model. With
knowing pump speed and pump mass flow rate, pump power can be computed through proposed
model.
4.5.2.1 Centrifugal pump flow rate model
Centrifugal pump flow rate modelling is based on TC shaft and pressure head of pump.

𝑚̇𝑝 = 𝑓(𝜔, Δ𝑃)

(4.35)

Based on turbomachinery fluid dynamics for the centrifugal pump, two dimensionless
parameters are used to characterize pump operation, the dimensionless flow coefficients: flow
coefficient 𝐶𝑞 and head coefficient 𝐶ℎ [27], as in (4.36) and (4.37).
𝑚̇
𝑉𝑥 𝜋(𝐷2 − 𝑑 2 )/4
𝐶𝑞 = 𝛼
𝜋𝑁𝐷
𝑈
60

98

(4.36)

𝑉𝑥
𝑔𝐻
𝛼
𝑈
𝜋𝑁𝐷 2
(
)
60

𝐶ℎ =

(4.37)

These two parameters are related to the mass flow rate, angular speed, and the pressure
difference across the pump :
𝐶𝑞 ∝

𝑚̇
𝜔

(4.38)

𝐶ℎ ∝

∆𝑃
𝑚̇𝑝2

(4.39)

Alternately two new flow coefficients can be defined for the centrifugal pump. A modified
flow coefficient Cflow and a modified head coefficient, Chead , are defined as shown in (4.40) and
(4.41) respectively. Since the modified mass flow rate model depends only on two states (TC
shaft speed and hydraulic accumulator pressure), this significantly reduces the model complexity.
𝐶𝑓𝑙𝑜𝑤 =

𝑚̇
𝜔

(4.40)

𝐶ℎ𝑒𝑎𝑑 =

𝑚̇𝑝2
∆𝑃

(4.41)

Based on flow coefficient and head coefficient, mass flow rate for centrifugal mass flow rate
can be derived as in (4.42) and (4.43). The model calibration results are as shown in Figure 49.
The model calibration results are shown in Figure 49 and it shows that this method looks
promising for modeling the turbo pump mass flow rate.
𝑚̇𝑝2 = ∆𝑃10𝑟 (
𝑚̇ 𝑝 =

𝑚̇𝑝 𝑢
𝜔

2/𝑢

)
1

(4.42)
𝑢

√∆𝑃10𝑟 ( )
𝜔

99

(4.43)

6

x 10

-6

Test
Model

5

Head coeff

4

3

2

1

0

-1

0

0.2

0.4

0.6

0.8
Flow coeff

1

1.2

1.4

1.6
x 10

-7

Figure 49. Identified mass flow rate through modified flow coefficient and modified head
coefficient

4.5.3 Hydraulic turbine modelling
4.5.3.1 Turbine power model
Hydraulic turbine is driven by high pressure fluid in high pressure accumulator. Hydraulic valve
is used to control the mass flow rate through hydraulic turbine. A typical high speed hydraulic
turbine efficiency is as shown in Figure 50.

Hydraulic turbine efficiency
0.7
0.65

20
Hydraulic turbine power [kW]

0.6
0.55
15
0.5
0.45
10

0.4
0.35

5

0.3
0.25

0

0.2
3

4

5
6
TC speed [RPM]

7

8
x 10

4

Figure 50. Hydraulic turbine efficiency (y-axis is turbine power)

100

Similar to the pump model, the hydraulic turbine is characterized using a map. The hydraulic
turbine power model is investigated in this section. Turbine performance characteristics can be
expressed in terms of coefficient of performance in [58].
Cp =

Ẇ
3
0.5ρAt V∞

(4.44)

The coefficient of performance Cp is calibrated using the experiment data as a function of
hydrofoil Reynolds number and tip speed ratio. Tip speed ratio is defined as the ratio of the rotor
tangential velocity to flow velocity:
Îť=

ωR
V∞

(4.45)

Reynolds number is due to the sensitivity of hydrofoil lift and drag characteristics on blade
Reynolds number [51]. This dependency is relatively small for a fixed turbine size. Thus, the
coefficient of performance can be assumed as invariant with respect to Reynolds number. With
fixed density assumption, the coefficient of performance can be taken as a function of turbine
power and mass flow rate, which is analogous to power coefficient for compressor and pump in
the previous discussion for the hydraulic pump and the centrifugal compressor. Further, tip speed
ratio is also a function of shaft speed and mass flow rate, which is speed coefficient defined
previously.
Cp ∝

Ẇ
V3∞

λ∝

∝

Ẇ

(ṁ )3

ω
ω
∝
V∞ ṁ

(4.46)
(4.47)

With this simplification, the hydraulic turbine performance can be represented by the speed and
power coefficients. Based on the data provided by manufacturers, the relationships for these two

101

coefficients are shown in Figure 51. In this case, a third order polynomial is used to approximate
the turbine performance.
Ẇp
ω 3
ω 2
ω
=
a
(
)
+
b
(
) + c( )+ d
(ṁT )3
ṁT
ṁT
ṁT
18

x 10

(4.48)

4

Test data
Model

16
14

Power coeff

12
10
8
6
4
2
0

0

50

100

150
200
Speed coeff

250

300

350

Figure 51. Identified hydraulic turbine model and test data
4.5.3.2 Turbine mass flow rate model
Investigating turbine flow using the same approach as a centrifugal turbo pump, the results are
shown in Figure 52. Using flow and head coefficients as discussed in hydraulic pump model
section, the mass flow rate of the turbine can be modeled in a similar way.
𝐶𝑓𝑙𝑜𝑤 =

𝑚̇ 𝑇
𝜔

(4.49)

𝐶ℎ𝑒𝑎𝑑 =

𝑚̇2𝑇
∆𝑃

(4.50)

102

1.4

x 10

-7

1.2
Test
Model

Head coeff

1

0.8

0.6

0.4

0.2

0

0

0.5

1
Flow coeff

1.5
x 10

-7

Figure 52. Identified turbine mass flow rate model

4.5.3 Valve model
Valve flow is modelled based on flow factor 𝐾𝑣 , which can be obtained using experimental
data. Flow factor can be modelled as a nonlinear function of valve position (control current input)
as shown in Figure 53. The mass flow rate 𝑚̇𝑣𝑎𝑙𝑣𝑒 through valve can be calculated by flow factor
𝐾𝑣

and pressure difference across valve ∆𝑃 [59]. In this modelling, valve loss is not modeled

explicitly. However, the loss for the valve is already included in the valve performance map.
𝑚̇𝑣𝑎𝑙𝑣𝑒 = 𝐾𝑣 √∆𝑃

(4.51)

𝐾𝑣 = 𝑓(𝜃𝑣𝑎𝑙𝑣𝑒 )

(4.52)

In this study, the turbine and pump valves use the same actuator. The mass flow rate for both
hydraulic turbine and pump valve can be calculated using the following equations:
𝑚̇𝑣𝑎𝑙𝑣𝑒_𝑃 = 𝑓 (𝜃𝑣𝑎𝑙𝑣𝑒𝑝 , 𝑃𝐴𝑐𝑐 , 𝑃2 )
𝑚̇𝑣𝑎𝑙𝑣𝑒_𝑇 = 𝑓(𝜃𝑣𝑎𝑙𝑣𝑒_𝑡 , 𝑃𝐴𝑐𝑐 , 𝑃3 )

103

(4.53)
(4.54)

1
Test data
Model

0.9

Normalized flow factor

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

0.4

0.5

0.6
0.7
Normalized current

0.8

0.9

1

Figure 53. Control valve flow factor

4.5.4 Modelling for piston accumulator
For the spring actuated accumulators the dynamics are governed by the spring constant and
the combination of spring and piston masses.

The natural frequency of a spring loaded

accumulator is given by:
𝜔𝑛 = √

𝑘𝑠

(4.55)

𝑚𝑝

A dynamic model for a spring loaded accumulator consists of a first order differential
equation of accumulator pressure and a second order differential equation of position mass,
damper, and spring dynamics as shown in (4.56). This model is further intergraded with other
subcomponents.
𝛽
(𝑚̇
− 𝑚̇𝑡𝑢𝑟𝑏𝑖𝑛𝑒 − 𝑣𝐴)
𝑉0 + 𝑥𝐴 𝑝𝑢𝑚𝑝
𝑥̇ = 𝑣
(𝑃𝑎𝑐𝑐 − 𝑃𝑟𝑒𝑡𝑢𝑟𝑛 )𝐴 − 𝐹0 − 𝐹𝑓 (𝑣) − 𝑐𝑣 − 𝑘𝑥
𝑣̇ =
𝑥𝜌𝐴
̇ =
𝑃𝑎𝑐𝑐

104

(4.56)

4.6 Model Validation and Plant Investigation
4.6.1 Model validation
A mean value diesel engine model coupled with regenerative assisted hydraulic assisted
turbocharger was built in Simulink environment based on interconnected sub-models as shown in
Figure 54. The governing equations and identification results have been presented in previous
sections. The Simulink model includes engine air-path and hydraulic systems model. Since only
transient test data for engine system without hydraulic system are available, only model
validation results for engine air-path modelling are presented here.

Figure 54. Engine system modeling in Simulink
First, a load step simulation is carried out. The integrated diesel engine air-path model is
simulated with the inputs of engine speed, fuel injection, VGT vane position and EGR valve
position with a transient load step in open loop sense. The simulation results are shown in Figure
55. Engine speed is kept at 2000 RPM. The initial increase of turbine mass flow rate is because

105

of VGT open action. Total turbine power is the sum of compressor power, friction power, and
shaft power. Shaft power imbalance leads to TC shaft deceleration or acceleration action. During
turbocharger acceleration, positive shaft power drives TC shaft to higher speed. Comparing with
test data for three states (intake pressure, exhaust pressure, and TC speed), proposed model well
represent system dynamics with reasonable agreement, some error happens for light load range.
Further, an FTP cycle dynamometer test inputs (engine speed, fuel injection, VGT position, EGR
position) are used to drive developed engine air-path model in simulation. The results are shown
in Figure 57. Relative error distribution for intake pressure and exhaust pressure are presented in
Figure 58. The relative errors for both states concentrate between Âą10% bound. This provides

0.4

Turbine mass flow rate
Compressor mass flow rate

0.2
0

5

10

15

20
Time [s]

25

Power [kW]

20

30

35

40

Engine Speed (RPM)
Turbo Shaft Speed (RPM)

0

shaft power
Assist(-) Regen(+) power

0
-20

Power (kW)

0

5

10

15

20
Time [s]

25

30

35

40

40

0

0

5

10

15

20
Time [s]

25

30

35

0

5

10

15

20
Time [s]

25

30

35

40

2000

Temp [k]

5

P
2
0

12
10
8
6
4
2

3

P2
0

x 10

5

10

15

20
Time [s]

25

30

35

40

4

TC speed

0

5

10

15

20
Time [s]

25

30

35

40

Exhaust temp
Intake temp

1000
0

5

10

15

20
Time [s]

25

30

35

Engine speed

1999

40

VGT control
EGR control

50

0

x 10

2000

100

0

4

2001

Turbine power
Compressor power
Friction power

20

Compressor mass flow rate [kg/s] P3 dot [pa]

Control signal

Pressure [Pa]

Mass flow rate [kg/s]

confidence for future model-based analysis and controller design.

40

1

0

x 10

5

10

15

25

30

35

40

7

0
-1
-6

-5

-4

-3

-2

-1
P2 dot [pa]

0

1

2

3

4
x 10

5

0.2
0.1
0
0.05

0.1

0.15
Turbine mass flow rate [kg/s]

Figure 55. Open loop simulation with test inputs.

106

20
Time [s]

0.2

0.25

10

x 10

4

TC speed

x 10

3

P

5

3

3
Model
Engine dyno Test data

2.8

x 10

P

5

2

Model
Engine dyno Test data

2.8

9
2.6

2.4

2.4

Pressure [bar]

7

2.2

2.2

6

5

Pressure [bar]

8

2

2

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

4

0

10

20
Time[s]

30

1

40

0

10

20
Time[s]

30

40

1

0

10

20
Time[s]

Figure 56. Model validation results for load step test
x 10

5

1.4

Intake pressure [pa]

3

Model predicted
Measured

1.3
1.2
1.1
1
0.9
700

x 10

Exhaust pressure [pa]

TC speed [RPM]

2.6
Model
Engine dyno Test data

720

740

760
Time [s]

780

800

820

740

760
Time [s]

780

800

820

5

Model predicted
Measured

2
1.8
1.6
1.4
1.2
1
700

720

Figure 57. Model validation results for FTP 75 driving cycle

107

30

40

4

Histogram of intake pressure modeling error over FTP 75
Histogram of intake pressure modeling error over FTP 75

x 10
4
6
x 10
6
5

=4.7
=4.7
=1.8
=1.8

5
4
4
3
3
2
2
1
1
0
-25
0
-25

8

87

x 10

x 10

-20

-15

-10

-5

-20

-15

-10

-5

4

0
Error
0 [%]
Error [%]

5

10

15

20

5

10

15

20

25

25

Histogram of exhaust pressure modeling error over FTP 75

4

=5.0
=-2.8
=5.0
=-2.8

Histogram of exhaust pressure modeling error over FTP 75

76
65
54
43
32
21
10
-25
0
-25

-20

-15

-10

-5

-20

-15

-10

-5

0
Error [%]
0
Error [%]

5

10

15

20

5

10

15

20

25

25

Figure 58. Modelling error for FTP 75 driving cycle
4.6.2 Plant behavior investigation
4.6.2.1 Engine air-path response with three actuators
To investigate plant dynamic behavior with three different actuator inputs, engine is
simulated at 1200 RPM, 35 mg/cc fuel injection. Three different actuators change with different
step variations. If one actuator has step action as shown in Table 12, the other two actuators are
kept as the same as base value as shown in Table 12. The open loop simulation results are shown
in Figure 59 Figure 60, and Figure 61.
Table 12. Load step simulation inputs
Base actuator position
Base VGT=50% opening
Base EGR=15% opening
Base RHAT=0kW

Step change
5%
5%
0.5kW

108

Note
100% =full close
100% =full open
Negative=assist

The response of mass flow rate, pressure and TC speed are expected with respect to different
actuators. With VGT vane close action in Figure 59, exhaust pressure dynamics is faster than
intake pressure, which is due to small exhaust volume and fast change for turbine mass flow rate.
Because EGR mass flow rate is dominated by pressure ratio across EGR valve with fixed valve
position. With faster response of exhaust pressure, EGR mass flow rate increases, and then
converges to steady state. The inverse response of compressor mass flow rate due to step change
of VGT position shows the non-minimum phase behavior as in Figure 59. When the VGT vane
open, an initially increased flow across the turbine results in increased compressor power 𝑊̇𝑐 ,
hence increased compressed air flow rate out of the compressor 𝑚̇𝑐 . This results in the emptying
of the exhaust manifold at a much faster rate than the filling rate of the intake manifold.
Eventually the exhaust manifold pressure 𝑃3 will drop resulting in a reduction of the compressor
power 𝑊̇𝑐 and the compressor flow rate ṁc .
Similar behavior is found when the engine intake mass flow rate and intake manifold
pressure response to step change of EGR valve position as in Figure 60. With a positive step
change to the EGR valve, there is an initial flow, over the EGR valve, from the exhaust manifold
(at pressure 𝑃3 ) into the intake manifold. This results in an initial increase in the intake manifold
pressure 𝑃2 , leading to an initial increase in the engine intake flow rate. However, flow of exhaust
gas over the EGR valve implies a reduced flow across the turbine resulting in reduced turbine
power and an eventual decrease in 𝑃2 and 𝑚̇𝑖𝑛 to the actual steady state values. The effect of the
reduced boost on the overall intake manifold dynamics is delayed owing to the turbocharger
dynamics. The asymmetry responses show highly nonlinearity of the engine air-path plant. This
brings challenges for VGT-EGR system control. But, intake pressure, compressor mass flow rate
well behaved response to step changes for VGT and EGR action.

109

With externally assisted power and regenerative power, there is no such non-minimum phase
behavior as shown in Figure 61. With assisted power on TC shaft, compressor mass flow rate
increases due to higher compressor power, hence, intake pressure increases. In the exhaust
manifold side, increased exhaust pressure is due to increased engine exhaust mass flow rate,
which is driven by intake pressure. However, the increased turbine power is not directly from
VGT vane closing action, which results in the emptying of the exhaust manifold with similar
filling rate of the intake manifold during transient operation. For steady state, since the VGT is
kept at the same opening, the exhaust flow is not restricted by vane closing, which leads to
exhaust manifold pressure is relatively lower than VGT closing case. Hence, pressure ratio
across EGR valve decreases with assisted power. EGR mass flow rate decreases to its steady
state flow. But this quit depends on how much assisted power on TC shaft. With high assisted
power and wide open VGT position, pressure drops across EGR valve might not be able to drive
demanded EGR mass flow rate anymore. In some extreme case, intake manifold pressure might
be higher than exhaust manifold pressure. Then EGR valve will lose control authority of EGR
mass flow rate. With regenerative power on TC shaft, exhaust energy extracted by turbine is
used to drive both compressor and hydraulic pump. TC shaft speed drops because of reduced
turbine power with reduced turbine mass flow rate. In extreme case, if constant TC shaft loading
power exceeds the total turbine available power ( 𝑊̇𝑇 − 𝑊̇𝐶 − 𝑊̇𝐿𝑜𝑠𝑠 ) , then TC shaft speed
intended to converge to zero, leading turbocharger to stall. With loading on TC shaft, turbine
flow is restricted by slowing down turbine speed, leading to lower exhaust manifold pressure.
Meanwhile, decreased compressor power leads to lower compressor mass flow rate. Hence,
lower intake manifold pressure is observed. In this case, pressure ratio across EGR valve
increases, which leads a higher EGR mass flow rate.

110

Table 13. Operating points for model linearization
Engine speed
[RPM]

Fuel injection
[mg/stroke]

VGT position
[%]

EGR valve position
[%]

External power
on TC shaft
[kW]

800

25

55

6

0

1200

35

50

12

0

1500

45

45

10

0

2000

50

40

8

0

Eigenvalue
-150.2801
-14.5672
-0.3581
-93.0635
-9.5879
-0.8150
-95.6492
-12.4370
-1.7939
-86.3667
-5.6772
-5.6772

These behaviors further are confirmed by analyzing the linearized model. Three states model
are linearized symbolically through using Matlab. Four equilibrium points were analyzed
through linearization as shown in Table 13. Equilibrium points were simulation results from
nonlinear plant simulation with steady state inputs. Since assisted hydraulic turbine or
regenerative hydraulic pump only provides assisted or regenerative power for transient operation,
base hydraulic power was set as zero. Results show real stable eigenvalue for all the equilibrium
points as in Table 13.
Based on bode plot in Figure 62, at high load operating condition, VGT has more control
authority over boost pressure relative to light load. It can be expected that EGR valve will have
less control authority for boost pressure compared to light load. However, with assisted power on
TC shaft, this shape the current VGT-EGR system control scenario; from Figure 62, it shows the
benefit of assisted turbocharger for boost pressure and TC speed control for its higher DC gain
and higher bandwidth with input unit of kW. It also shows higher control authority for EGR
mass flow rate than VGT. Hence with assisted power, boost pressure might not depend on VGT
action; VGT vane position might be used to control EGR mass flow rate. It also shows that VGT

111

actuator might be redundant for both EGR control and boost pressure control, since EGR valve
dominates EGR mas flow rate regulation. Assisted power dominates boost pressure regulation.
In this case, boost control and EGR mass flow rate control can be decoupled with assisted and

VGT position
Pressure [pa]

1.4

1.6

MFR [kg/s]

60

Pressure [pa]

regenerative power on TC shaft.

VGT position
50
40

30
x 10

MFR [kg/s]

40

45
Time

50

55

P2

1.3
1.2

30
x 10

35

40

45
Time

50

55

60

5

P3

1.4
1.2

60

5

30

35

40

45
Time

50

55

60

0.09
0.08
Compressor mass flow rate
30

35

40

45
Time

50

55

60

45
Time

50

55

60

0.015
0.01
EGR mass flow rate
0.005

30

35

40

0.1
Engine intake mass flow rate
0.095
0.09

Speed [rpm]

MFR [kg/s]

35

5

30
x 10

35

40

45
Time

50

55

4

TC speed

4.5
4

60

30

35

40

45
Time

50

55

60

Figure 59. Engine air-path response respect to VGT position with step change

112

EGR position
Pressure [pa]

0

30
x 10

35

40

45
Time

1.25

1.5

50

55

60

50

55

60

5

P2

1.3

Pressure [pa]
MFR [kg/s]
MFR [kg/s]

EGR position
20

1.35

30
x 10

35

40

45
Time

5

P3
1

30

35

40

45
Time

50

55

60

0.1
Compressor mass flow rate
0.05

30

35

40

45
Time

50

55

60

0.02
EGR mass flow rate
0.01
0

30

35

40

45
Time

50

55

60

0.1
Engine intake mass flow rate
0.095
0.09

Speed [rpm]

MFR [kg/s]

40

5

30
x 10

35

40

45
Time

50

55

60

40

45
Time

50

55

60

4

TC speed
4.5
4

30

35

Figure 60. Engine air-path response with respect to EGR valve with step change

113

RHAT power [kW]

0.5
0
RHAT power

1.4

30
x 10

35

40

45
Time

50

55

60

5

P2

1.2
1

1.6

30
x 10

35

40

45
Time

50

55

60

5

P3

1.4
1.2

30

35

40

45
Time

50

55

60

55

60

0.1
Compressor mass flow rate
0.05

30

35

40

45
Time

50

0.02
0.01
0

EGR mass flow rate
30

35

40

45
Time

50

55

60

0.12
Engine intake mass flow rate
0.1
0.08

Speed [rpm]

MFR [kg/s]

MFR [kg/s]

MFR [kg/s]

Pressure [pa]

Pressure [pa]

-0.5

5

30
x 10

35

40

45
Time

50

55

60

4

TC speed
4
3

30

35

40

45
Time

50

55

60

Figure 61. Engine air-path response with respect to assisted and regenerative power with step
change

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Figure 62. Frequency analysis for different engine operating points

To: Pim

To: Pim

-360
-1
10

-180

0

-50
180

0

50

-360
100

-180

0

-80
180

-60

-40

-20

0

20

-180
40

-90

0

90

-100
180

0

1

10

2

10

-1

10

0

10

1
Frequency (rad/s)
10

2

10

-1

10

0

10

1500 RPM
2000 RPM

10

From: EGR

800 RPM
1200 RPM

From: VGT

-50

50

Bode Diagram

0

To: EGR mass
To: EGR mass
Magnitude (dB) ; Phase (deg)

To: TC speed

To: TC speed

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From: RHAT

1

10

2

10

4.7 Conclusion
In this chapter, a diesel engine air-path with regenerative hydraulic assisted turbocharger
system is developed. The model (simulator) is developed by integrating engine system, variable
geometry turbocharger system and hydraulic system. The exhaust and intake manifolds are
modeled as volumes with ideal gas having constant specific heats. The EGR valve is modeled
with the valve flow equations for flow through orifices. The effective area is determined
experimentally. Volumetric efficiency, temperatures rise are modeled as static nonlinearities.
Physics based compressor and turbine model are integrated with engine air-path system. For the
turbine modelling, it uses VGT as direct input for turbine power and flow regulation.
Compressor power uses speed and compressor mass flow rate. The hydraulic system is
connected with turbocharger system through shaft speed dynamic equation. New sub-models for
hydraulic turbine and centrifugal hydraulic pump are developed.
Engine air-path model is validated through engine transient test data. It shows proposed
modelling approach has high fidelity with only three states. It could be used for model-based
analysis and controller design. The complexity of the interactions between the VGT, EGR and
hydraulic power has been illustrated through simulation of the nonlinear model. The model

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CHAPTER 5: SYSTEM ANALYSIS FOR HYDRAULIC ASSISTED TURBOCHAED
DIESEL ENGINE THROUGH 1-D SIMULATION
5.1 Abstract
Engine downsizing and down-speeding are essential to meet future fuel economy mandates.
Further pushing the envelope for even better fuel economy improvement without compromising
vehicle drivability via a turbocharged engine will run into a major constraint, i.e. transient
response of the turbocharger. When high torque is demanded during acceleration, it is not
available until a few seconds later with conventional turbocharger technologies, which is called
"turbo lag." The key to the market acceptance of the downsized turbocharged engine to reduce
petroleum consumption is to bring the torque to demand level without a noticeable delay. A
regenerative hydraulic assisted turbocharger (RHAT) system is proposed in this chapter. In this
new system, a hydraulic turbine is used to spin the turbocharger shaft via high-pressure fluid out
of a tank; a turbo pump is used to absorb excessive power from the turbocharger shaft while
pressurizing the fluid and pumping back into the tank. A driveline pump is also used to recover
vehicle kinetic energy during vehicle deceleration mode and pump the fluid into the highpressure tank. Both the hydraulic turbine and the turbo pump are packaged inside the
turbocharger center housing. The RHAT concept itself will fundamentally change the operation
format of a turbocharged engine, with reduced turbo lag, engine pumping loss as well as
improved surge margin. Compared to traditional electric assisted and regenerative turbocharger,
RHAT has a much higher assist and regenerative capability, except that it is more durable and
cost effective. The abundance hydraulic energy that is recovered during vehicle deceleration can
be used to assist the turbine so the variable geometry turbine (VGT) can operate at most efficient
open positions rather than at small and inefficient positions to meet the compressor power

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demand. With two extra actuators on turbocharger shaft, variable geometry turbocharger (VGT)
could be potentially replaced with fixed geometry turbocharger for higher efficiency, lower cost,
and better durability. A 1D production vehicle with medium duty turbocharged diesel engine
model was used in the investigation. The hydraulic turbine and turbo pump maps and driveline
pump were predicted out of 1-dimensional hydraulic model analysis from suppliers. A baseline
controller was coupled with the 1D model and upgraded to control the engagement and
disengagement of RHAT and energy management in the hydraulic energy storage tank. The
preliminary 1-D simulation demonstrates that the proposed RHAT turbocharger system can
significantly improve engine transient response. The 1D vehicle level simulation shows that 3-5%
fuel economy improvement for FTP 75 driving cycle, depending on different sub-component
sizing. The study also identifies technical challenges for optimal design and operation of RHAT,
as well as additional fuel economy improvement opportunities that are enabled by the RHAT.
5.2 Introduction
5.2.1 Assisted turbocharger
Downsizing the engine for better fuel economy run into the turbo lag issue, among others.
When high torque is demanded during acceleration, it is not readily available until a few seconds
later for a typical turbocharged engine, which is called "turbo lag." Drivers would consider this
turbo lag "lack of power." Conventional turbocharged engines have no effective synchronization
between the engine and the turbocharger. That means that during the engine tip-in operation, the
turbo lags behind the engine. This turbo lag results in sluggish air supply that is a root cause of
transient smoke emission and lack of torque response. On the other hand, during the engine tipout, i.e. sudden slowdown in engine speed and air requirement, the turbocharger, due to high

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inertia, may continue to spin and pump the air more than required, driving the compressor into
surge (typically high-pressure ratio and low mass flow) area.
For a turbocharged engine, diesel or gasoline, the instantaneous engine power or torque lug
curve is different from the peak power or torque curve when the engine reaches steady state.
Depending on the turbocharger system and the EGR purging time, a turbocharged engine may
take 1 to 20 seconds to reach steady state torque or power. Figure 1 shows the instantaneous
power of a light-duty diesel engine, measured out of load step response tests on an engine
dynamometer. With the sluggish transient response of this engine, the vehicle transmission has to
downshift to give the customer “crisp” response perception, thus pushes the engine to operate at
a higher speed where the engine efficiency is much lower than the medium engine speed. Should
the turbo have assisted power to accelerate to high boost/speed without turbo lag, the
transmission downshift may be avoided, which will translate to 1-2% fuel economy
improvement, besides the transient response improvement.

Figure 63. Instantaneous engine power lug curve based on a light duty diesel load step test

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Figure 64. Instantaneous Engine EGR Rate, Soot and NOx Emissions during d light duty diesel
FTP transient test
To address the impact of turbo lag on vehicle transient performance of turbocharged engines,
typical measures taken by vehicle manufacturers include:
a. Transmission downshifting: shifting the transmission to a lower gear. Thus the engine
operates at a higher speed to gain more power since the turbo lag prevents the engine from
gaining torque instantly. The operation of the engine at a higher speed, thus higher friction and
pumping loss, will result in fuel economy penalty (Figure 63);
b. Reduction of exhaust gas recirculation (EGR) during the tip-in operation replaces the
exhaust gas with fresh air for better torque response. The momentary reduction of EGR will
certainly lead to NOx emission spikes as shown in Figure 64, thus future NOx emission control
challenges;
c. Reduction of smoke limited air-fuel ratio (AFR) allows the engine to momentarily run
rich (thus higher soot emission) for better torque response. The smoke spike during this transient
operation (Figure 64) may not be visible on diesel engine due to the wide application of diesel

120

particulate filters (DPF). However, accelerated soot loading on the DPF will result in DPF
durability and fuel economy penalty concerns due to frequent DPF regenerations;
d. Increase in “air reserve” before the engine tip-in operation to allow more fuel to be
injected into the cylinder and more exhaust energy to spin the turbine for more boost, more air,
thus more torque at the moment of tip-in. This high “air reserve” at light load to ensure "crisp"
transient response causes high pumping loss on diesel engines.
e. To address the tip-out surge issue, a high surge margin is needed for automotive
compressors. The design requirement for wide operation range of the compressor also
compromises the peak efficiency.
To ensure the right amount of air at any moment for the turbocharged engine during steady
state or transient state, certain types of synchronization (like the mechanical connection between
the engine and a supercharger) between the engine and the turbocharger, i.e. an assisted
turbocharger, is needed. Since an assisted turbocharger will need substantial power during
transient operation, in order to avoid FE penalty on the assisted turbocharger system, any assisted
boost system has to be energy regenerative, i.e. it should be able to recover energy, e.g. during
engine deceleration, engine or vehicle braking. With such a regenerative and responsive
turbocharger system, even a downsized engine can work with a stiffer torque converter, more
aggressive transmission shifting schedule and gear ratio, thus enabling substantial FE savings.
5.2.2

Hydraulic assisted and regenerative turbocharger

Different to other exhaust waste heat recovery technologies, such as turbocompound, which
compete with reciprocal internal combustion engines for combustion energy, the regenerative
assisted turbocharger proposed in this chapter, recovers waste energy, mainly during engine
deceleration, throttling, or exhaust braking modes and vehicle braking events.

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The hydraulic-assisted turbocharger has been studied since the early-1980's [64]. There have
been publications and patents on using high–pressure, oil-driven hydraulic turbines on the
turbocharger shaft to accelerate the turbocharger during acceleration [60]-[63]. A hydraulic
turbine was added between the conventional compressor and the turbine wheels to assist the
transient acceleration of the turbocharger, whenenergizedby high-pressure fluid. The hydraulic
wheel was very compact and could be integrated into the turbocharger center housing. However,
all the previous arts require a stand-alone hydraulic pump to build up and maintain high pressure
in the hydraulic accumulator.
One of the main reasons that the hydraulic-assisted turbo technology has not been widely
accepted was that the stand-alone mechanical-driven hydraulic pump to pressurize the fluid had
very high fuel economy penalty, which was not acceptable.
Two energy recovery devices are proposed for this hydraulic assisted turbocharger [59],
including a turbo pump on turbocharger shaft and a driveline driven pump that is only engaged
during vehicle deceleration. The primary energy recovery would be fromduring vehicle
deceleration; The secondary energy recovery would be from turbo pump on the turbo shaft. The
total system benefits come from the total amount of energy that can be recovered.
Every turbo acceleration will be followed by a deceleration. The exhaust gas energy should
and could be recovered during engine deceleration, throttling, or exhaust braking mode rather than
firing mode, as far as an energy storage device is available. Firing mode energy recovery without
fuel penalty is only limited to very high engine load [66]. A hydraulic turbo pump on the same
compressor/turbine wheel can recover part of the exhaust energy except it is much more compact,
durable, and cost effective than the electric motor/electric energy storage system [59], [63]. The
pressurized fluid out of the turbo pump can then be saved in the highpressure hydraulic/pneumatic

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accumulator to be used later as a driving force on the hydraulic turbine that shares the same turbo
shaft. The high-speed turbo pump, like the hydraulic turbine, is also a matured technology in the
aerospace industry [67]. NASA in 1974 published a report indicating the efficiency of a hydraulic
turbo pump to be around 73% for rocket applications [67]. Other studies (including research at
Ford Motor Company a few years ago on a hydraulic assisted turbocharger), also indicated around
72% efficiency on large-size hydraulic turbines [59],[68],[69].
A combination of the hydraulic turbo pump and the hydraulic turbine on TC shaft can provide
extra actuator for turbo speed control, i.e. to spin up the turbocharger for more air flow during
engine acceleration or slow down the turbocharger to recover exhaust energy and to reduce air
supply to the engine, avoiding surge during deceleration, as well as manipulating the turbo speed
in steady state conditions to high efficiency areas of compressor and turbine to maximize
turbocharger, thus engine system efficiency.
In order to harvest vehicle brake energy during vehicle tip-out, another hydraulic energy
regeneration device (a driveline pump) is added on the driveline, which recovers vehicle brake
energy during vehicle brake events. This idea is similar to the regeneration mode of the hydraulic
hybrid vehicle However, for a hydraulic hybrid vehicle, a hydraulic tank with large volume is
needed to store hydraulic energy, which is used to launch the whole vehicle. The tank size could
be 112 liters with 420 bar designed pressure for a 15,000kg vehicle [73]. But in this study, a
compact size tank would be enough to drive a hydraulic turbine for turbocharger assist due to the
small inertia of a turbocharger. With the vehicle brake energy recovery, the assist capability of a
hydraulic turbine would be further enhanced.
The regenerative hydraulic assisted turbo (RHAT) concept itself may fundamentally change
the operation format of a turbocharged engine, i.e. to provide a means to have a "synchronized"

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transient operation between the engine and turbocharger, thus addressing turbo lag and tip-out
surge issues for turbocharged engines. The focus of this study was: using a hydraulic driveline
pump and turbo pump (integrated in a turbocharger) to recover part of vehicle braking energy
and engine exhaust energy to drive a hydraulic turbine (also inside the turbocharger) during
engine acceleration. The energy from the hydraulic driveline pump and turbo pump is saved in a
pressurized hydraulic/pneumatic storage. The details of this concept were included in a patent
[59].
The main advantage of RHAT, compared to the electrically assisted turbocharger system, is
power density and compactness of hydraulic wheels. Although the regenerative electric assisted
turbocharger (REAT) is very attractive, it has the following concerns for light and medium duty
automotive diesel applications:
1. As shown in Table 14, most of current electric assisted motor lies in the range of 1.5-10 kW.
The desired power requirement to assist the acceleration of large turbochargers may be more
than 10kW, depending on turbocharger size. This high assisted power is a challenge for
vehicles with 12 volts electric infrastructure
Table 14. A Survey for current electric motor/ generator design space
Reference
Ibaraki [75]
Pfister, P. D [76]
Takata [77]

Designed speed
120,000 RPM
200, 000 RPM
220,000 RPM

Supply voltage
72V
65V
72V

150,000 RPM

Max power
3 kW
2 kW
2 kW
3.5kW(motor)
5.4kW(generator)
1.5kW

Terdich [78]

120,000 RPM

Noguchi [79]
David [80]

75,000 RPM

10kW

200V

Winward [82]

180,000 RPM

5kW

-

650V
12V

2. The power capacity needed to recover exhaust energy during deceleration and
braking/motoring should be in the range of 15-40 kW, since the turbo would be operating at

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high speed with high kinetic energy when the excessive exhaust energy is available to be
harvested. It is challenging that an electric motor of this power can be packaged and integrated
into an automotive turbocharger center housing without a dramatic increase in inertia and
mechanical stress when operating at high speed. That means the current production-ready
2kW electric motor supported by 12 volts infrastructure will not effectively recover exhaust
energy when it is abundant.
3. For the 2 kW REATs that are available today, they typically have substantial increase in
rotational inertia over conventional turbochargers [65] , i.e. during acceleration, more energy
will be used to rotate the motor itself.
4. A permanent magnetic electric motor is more efficient and compact than an induction motor,
but the performance may deteriorate over time under high temperature; the material strength
of the permanent magnet may limit its application to low operation speed, which is
incompatible to smaller turbochargers.
The design target of the RHAT system will be 50% “round trip” efficiency, which may be
slightly lower than a state-of-the-art regenerative electric-assisted turbocharger. But the
advantages like lower inertia, cost, smaller packaging space, and potential better durability make
the RHAT an attractive alternative option, especially for medium and heavy-duty turbocharged
diesel applications.

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Figure 65. System layout of diesel engine with hydraulic assisted and regenerative turbocharger

Figure 66. Hydraulic assisted turbocharger layout
Figure 65 illustrates the proposed Regenerative Hydraulic Assisted Turbocharger (RHAT)
system. Whenever the vehicle or engine has ‘free’ energy (e.g. during vehicle or engine
deceleration, exhaust braking, or steady state when the intake throttle is used for intake oxygen
control, or when a wastegate is used, etc.), the driveline pump will be engaged to recover vehicle
kinetic energy, while the hydraulic turbo pump is powered by gas turbine. The power collected
by both driveline pump and the turbopump on TC shaft will pressurize the fluid and at the same
time the turbo slows down to "synchronize" with the decelerating engine to avoid "tip-out" surge.
The pressurized fluid out of the two pumps will pass a check valve and be saved in a highpressure hydraulic accumulator. During the engine acceleration, the high-pressure fluid from the
accumulator will be discharged into and drive the hydraulic turbine to accelerate the turbocharger.

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When the TC turbine wheel gets external hydraulic energy, VGT can open up for better efficiency.
Thus the enthalpy drop across the turbine will be reduced, thus engine pumping loss is reduced
and engine net power output is increased. With higher enthalpy at TC turbine outlet, the
temperature for after-treatment would be higher with hydraulic assist. Hence, after-treatment
conversion efficiency would be improved due to increased temperature. With external energy
input on the shaft, the turbine works in lower expansion ratios (i.e. at high turbine speed ratio,
U/C as shown in Figure 67. Also, the turbine does not count on the small nozzle to collect
sufficient exhaust energy to drive the compressor. Therefore, the turbine operates with lower
expansion ratio, wider open nozzle, and higher speed, thus operating more efficiently (see Figure
67). Both the high U/C and relatively large nozzle may potentially make the turbine 30-50% more
efficient than the turbocharger without external energy input. That means the RHAT converts
exhaust energy into mechanical energy more efficiently, further improving the engine transient
response with improved fuel economy, i.e. the improvement in transient response is partly from
external hydraulic energy and partly from improved turbine and compressor efficiency. During
very aggressive tip-out, the hydraulic loading on the turbo shaft will slow down the turbo to avoid
trip-out surge and recover the aerodynamic and kinetic energy of the turbocharger.

Figure 67. With assisted turbocharger, turbocharger can be regulated to its higher efficiency
operation range.

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During very aggressive tip-out, the hydraulic loading on the turbo shaft will slow down the
turbo to avoid trip-out surge and to recover the aerodynamic and kinetic energy of the
turbocharger, as shown in Figure 68.

Figure 68. The Hydraulic turbo pump can recover exhaust energy and avoid tip-out surge.
5.2.3 System causality
In order to mitigate the tradeoffs between fuel benefit, emission reduction and performance
improvement, a system approach is investigated. System causality with hydraulic assisted
turbocharger is shown in Figure 69. For a given turbocharger, engine and vehicle, there would be
an optimal design solution to achieve different design target with given design constraints.

128

Figure 69. System causality for regenerative hydraulic assisted turbocharger

For performance improvement consideration, the hydraulic system should be designed to
provide high assisted power as well fast response through a hydraulic turbine. To provide
sufficient energy for the assist, tank volume should be designed as large as possible with high
pressure. However, the largest tank volume size is limited by physical packaging size as well as
the driveline pump and turbo pump energy recovery capability.
To meet stringent emission constraint, the air-fuel ratio (AFR) would be optimized with
assisted turbocharger. For instance, the combustion air-fuel ratio set point can be dropped to a
lower level with assisted turbocharger. An engine without assisted turbocharger needs higher
AFR ratio setpoint to accommodate potential driver’s aggressive tip-in, by avoiding smoke limit.
However, higher AFR set point will result in higher target boost pressure. In this case, higher
exhaust pressure is needed to drive turbocharger with vane nozzle closing for VGT turbocharger.
This will result in high pumping loss. With assisted turbocharger, air can be supplied to engine
cylinder with a faster response by applying energy directly to turbocharger shaft.

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With assisted turbocharger, variable geometry turbocharger can be replaced with fixed
geometry turbocharger with waste-gate. But the system interaction depends on turbine sizing and
hydraulic components sizing. With properly components sizing, the hydraulic assist can be used
for engine light load tip-in. During engine medium and high load operations, the turbocharger is
controlled through waste-gate and hydraulic components. This structure could also be applied to a
turbocharged gasoline engine. For diesel engine application, this might eliminate nozzle vane for
the turbocharger, which leads to potential cost reduction.
Driveline energy recovery capability depends on driveline pump size and vehicle weight.
Hydraulic turbine and pump sizing on TC shaft depends on engine sizing and turbocharger sizing
with considering the design target. Optimal designs of the hydraulic turbine and turbo pump that
can deliver high efficiencies over wide operation range, regarding the pressure and turbo speed.
An energy storage management strategy should regulate the tank energy so that hydraulic
pump and hydraulic turbine can work in high-efficiency areas at a wide range of flow and speed.
The optimal control strategy is needed to best utilize the RHAT and operate the turbocharger in
high-efficiency areas to achieve maximum engine system efficiency over entire customer driving
cycles within the constraint of emission.
Governing the hydraulic pump and turbine wheels provides a means to "synchronize" the
turbocharger with engine operation conditions to ensure the compressor and turbine working in a
narrower but more efficient area, i.e. the hydraulically governed turbocharger will allow the
compressor and turbine to be designed for higher efficiency, since they don't have to trade the
efficiency for operation range. The hydraulic-governed turbocharger may control the airflow
independently from engine operation conditions, thus eliminating "turbo lag" and the necessity of
intake throttle and wastegate for turbocharged engines.

130

In this study, 1-D simulation approach is used to investigate the benefits and design trade-offs
for regenerative hydraulic assisted turbocharger. The chapter is organized into three major parts:
1. Engine steady state operation with assisted and regenerative capability
2. Transient response improvement with hydraulic assisted turbocharger
3. RHAT design trade-off investigation through driving cycle study.
5.3 Vehicle Level Integrated Simulation in GT-Power / Simulink
5.3.1 Simulation platform and control algorithm
Vehicle level simulation is utilized to investigate hydraulic assisted turbocharger fuel benefit
and design trade-offs. The 1-D model as shown in Figure 70 includes a vehicle model, driver,
engine, torque converter, transmission, driveline, regenerative hydraulic-assisted turbocharger.
The control includes a production version engine control, 6 speed transmission control, torque
converter control, and designed RHAT control.

Figure 70. Simulation platform and control structure

131

Figure 71. Modeling layout in GT-suites

132

TABLE 15. VEHICLE INFORMATION
Vehicle Weight
Engine

10000 Lbs.
Heavy duty Diesel
Variable geometry
turbocharger
6. speed

Turbocharger
Transmission

In simulation, a production control algorithm for regular VGT turbocharger diesel engine
control was modified with the following changes in the air handling and control strategies:


Due to the energy input at the turbo shaft, the expansion ratio across the turbine may be too
low to pump the conventional high-pressure exhaust gas recirculation (HP-EGR). Thus the
low-pressure EGR (LP-EGR) is used, when necessary, in the analysis. In the LP-EGR, having
EGR flowing through the compressor and turbine, instead of taking a short cut and bypassing
the compressor and turbine, will have the turbine and compressor operate at more efficient
areas at low engine speed and load conditions



RHAT control is designed to track the target boost pressure with a fixed gain PI controller



Driveline pump is controlled to recover vehicle brake energy within the constraint of max
hydraulic tank pressure.
The base transmission shifting strategy, boost pressure set-point, fuel injection set-point and

EGR fraction set-point are kept the same for all cases in this study. The engine air handling
control is to track target engine boost pressure and EGR rate as shown in Figure 72.

Figure 72. Engine air-path controller overview

133

The vehicle simulation is to track the target vehicle speed. A driver model is used to control
gas and brake pedal positions by feedback action. Target engine brake power is based on
calibrated map with inputs of engine speed and gas pedal position. Base fuel injection is based
on current engine speed and demanded engine torque. Fuel injection is controlled by both
feedback and feedforward loops to achieve target torque output. VGT vane postion is feedback
controlled to track the required boost pressure, which is from pre-calibrated boost map. The
transmission control is based on shift schedule map with inputs of engine speed and gas pedal
position. Both HP-EGR and LP-EGR are used to achieve target EGR mass flow rate. Both VGT
and EGR valve controllers are map based with gain scheduled PI controller. Detailed engine and
turbocharger model validation can be found in [25]. The vehicle system model (including engine
and turbocharger) validation is shown in in Figure 73.

Dyno test
GT simulation

80
60
40
20
0

Transmission gear number

Vehicle speed [km/h]

vehicle model validation
100

200

400

600
800
Time [s]

1000

1200

6
Dyno test
GT simulation

5
4
3
2
1

200

400

600
800
Time [s]

1000

1200

Figure 73. Vehicle model validation through FTP_75 driving cycle

134

5.3.2

Hydraulic components

Preliminary meanline analyses of the hydraulic turbine and turbopump were conducted by a
supplier. The regular engine oil at 100 deg C was assumed in the meanline analyses. From friction
loss perspective, engine oil may not be the optimal choice, due to high viscosity. However, the
fluids between the high-pressure RHAT and low-pressure lubricant in the turbo center housing
can't be sealed 100%; the engine oil is the best choice at this point. The preliminary predicted
hydraulic turbine map (Figure 74) shows that when the turbine power reaches above 10 kW,
substantial operation area of the hydraulic turbine can have an efficiency above 70%, even though
the high-efficiency operation area is not as large as a typical electric motor. When managed
carefully between flow rate and pressure ratio, hydraulic turbine efficiency can be above 60%
(Figure 74). is the preliminary meanline analysis of the performance of a turbo pump, indicating
that the turbo pump can achieve 70% or better efficiency as far as the pressure in the hydraulic
energy storage is managed to match the oil flow rate or the turbo speed.
The aforementioned analytical maps of a hydraulic turbine, turbo pumps and driveline pump
out of meanline analyses (Figure 75, Figure 74, Figure 76) were integrated with GT-Power
vehicle model for the engine system transient response investigation as well as fuel economy
assessment over the FTP 75 driving cycle. The oil temperature was kept at 100 deg C throughout
the cycle.
Other assumptions of the RHAT system include:
a. Hydraulic power 25 kW on RHAT_Turbine for assisted mode and 25 kW on RHAT_Pump
in energy recovery mode; a 25 kW driveline pump to recovery vehicle kinetic energy during
vehicle brake.

135

b. Hydraulic fluid pressures: 100-150 bar in Accum-HP tank, and 10-20 bar in Accum-LP
tank; The pressure of low-pressure tank is set to avoid cavitation in the RHAT_Pump.
c. Hydraulic storage space varies for design trade-off investigation.
d. Hydraulic valve actuator response time<50 ms.
e. With assisted power on the shaft, the TC turbine will operate in lower expansion ratio and
higher speed region, compared to operation without assisted power. Thus assisted and
regenerative turbocharger needs further data extrapolation out of current turbine flow bench
tested map. A further experimental test should be used for wide range turbine operation
investigation. In this study, map extrapolations are based on 1-D simulation software [82].
A comparable study between hydraulic components and electric components for assisted
turbocharger is illustrated in Figure 77. Hydraulic components (turbine and pump) have much
higher power density than an electric motor or generator. It clearly shows that hydraulic turbine
has higher torque than electric motor below 70K rpm, where the assist is needed for TC shaft for
engine light load tip-ins. For the low-speed region, the assisted torque from the hydraulic turbine
is three times of that from the electric generator. Hydraulic pump has much higher torque above
50K rpm, where high regeneration load is needed for energy recovery from TC shaft. These
results clearly show the power density of regenerative hydraulic assisted turbocharger is greater
than that of regenerative electric assisted turbocharger.

136

Figure 74. Hydraulic turbine efficiency

Figure 75. Hydraulic pump efficiency

Figure 76. Driveline pump efficiency

137

1
Hydraulic pump torque
Hydraulic turbine torque
Electric motor/generator torque

0.9
0.8

Normalized torque

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

2

4

6

8
Speed [rpm]

10

12

14
x 10

4

Figure 77. Torque comparison between electric motor and hydraulic components

5.4 Simulation results and discussion
5.4.1 Engine alone steady state investigation for feedforward calibration
For a given engine speed and fixed fuel injection amount, with assisted power, exhaust
pressure intends to decrease. Thus pumping mean effective pressure (PMEP) and engine brake
specific fuel consumption (BSFC) will decrease. However, this improvement can only be
maximized with optimal VGT vane position. This means VGT needs to be coordinately
controlled with assisted power.
With loading power on TC shaft on the other hand, e.g. during engine throttling mode, TC
speed intends to decrease. Lower TC speed will reduce compressor power. Thus compressor
mass flow rate and intake manifold pressure decrease. Turbine mass flow rate decreases with
lower TC speed. Hence, exhaust pressure intends to increase. Higher exhaust pressure and lower
intake manifold pressure will lead to high pumping loss and lower engine efficiency. For
different engine operations, thermal energy available to the TC turbine is dependent on engine
operation point as well as VGT position. Thus regenerative power on TC shaft will be defined by
engine load as well as VGT vane position.

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Based on turbocharger power balance equation , [83]:
𝐽𝜔̇ 𝜔 = (𝑚̇𝑡 )2 𝜔𝐷

𝑅𝑇3
𝑃3

tan(𝑓1 (𝑉𝐺𝑇)) − (𝑚̇𝑐 )3 𝑓(

𝑚̇𝑐
𝜔

) + 𝑃𝑟ℎ𝑎𝑡

(5.1)

Where, 𝑓1 (𝑢𝑉𝐺𝑇 ) is vane angle; 𝑢𝑉𝐺𝑇 is vane position control input. For steady state with
assisted power, Jω̇ω = 0, power balance equation becomes:
(𝑚̇𝑡 )2 𝜔𝐷

𝑅𝑇3
𝜔
+
tan(𝑓1 (𝑉𝐺𝑇)) + P𝑟ℎ𝑎𝑡
= (𝑚̇𝑐 )3 𝑓( )
𝑃3
𝑚̇𝑐

(5.2)

+
Max assisted power P𝑟ℎ𝑎𝑡_𝑚𝑎𝑥
can be defined by:

𝜔
𝑅𝑇3
+
P𝑟ℎ𝑎𝑡_𝑚𝑎𝑥
= (𝑚̇𝑐 )3 𝑓 ( ) − (𝑚̇𝑡 )2 𝜔𝐷
tan(𝑓1 (𝑉𝐺𝑇))
𝑚̇𝑐
𝑃3

(5.3)

Max hydraulic assisted power is defined by demanded boost requirement and VGT position,
and is limited by compressor surge. For steady state TC shaft energy regeneration, power balance
equation is:
(𝑚̇𝑡 )2 𝜔𝐷

𝑅𝑇3
𝜔
−
tan(𝑓1 (𝑉𝐺𝑇)) = (𝑚̇𝑐 )3 𝑓( ) + P𝑟ℎ𝑎𝑡
𝑃3
𝑚̇𝑐

(5.4)

−
Max regeneration P𝑟ℎ𝑎𝑡_𝑚𝑎𝑥
power is defined by
−
P𝑟ℎ𝑎𝑡_𝑚𝑎𝑥
= (𝑚̇𝑡 )2 𝜔𝐷

𝑅𝑇3
𝜔
tan(𝑓1 (𝑉𝐺𝑇)) − (𝑚̇𝑐 )3 𝑓( )
𝑃3
𝑚̇𝑐

(5.5)

Max regeneration power from TC shaft depends on total energy availability to turbine as well
as VGT position. Turbine mass flow rate can be represented as a function of VGT position and
turbine pressure ratio Prt as [84]:
𝑚̇𝑡 = 𝑓2 (𝑉𝐺𝑇) ∗ 𝑓(𝑃𝑟𝑡 )

Comparison of flow and power function, VGT position:

139

(5.6)

2

0

0

20

40
60
VGT closing [%]

80

100

VGT imapct on turbine power
VGT impact on pumping loss based on turbine flow

1
0.8

0.6

0.4

0.2

0

0

20

40
60
VGT closing [%]

80

100

Figure 78. VGT impact on turbine power and mass flow rate

Hence, for a given turbine power demand without RHAT, there is an optimal VGT position
for best system efficiency. With assisted power on TC shaft, compressor mass flow rate increases,
and turbine mass flow rate increases. In order not to restrict turbine flow, VGT position should
open with regenerative load on TC shaft. With opening of VGT position, turbine efficiency
changes.
The aim of this study is to show that how assisted and regen power on TC shaft affects
engine performance through steady state simulation. The engine is simulated at 2000 RPM with
60 mg/stroke fuel injection rate. Both EGR valves are fully closed for this case, VGT position
varies from 0.15 to 1 (0.15 is fully close and 1 is fully open), and TC shaft loading power varies
from -7kW to 7kW (positive is regenerative and negative is assisted power). It shows that engine
BSFC changes with different VGT position and different assisted and regenerative power on TC
shaft, as illustrated in Figure 79. For a typical system without assisted power or regenerative
power, best turbine efficiency is located at VGT position=0.6. In order to have optimal
(minimum) BSFC with assisted power on TC shaft at light engine speed and load, VGT should
be open wider for maximum turbine efficiency. For example, best VGT position for 7kW assisted

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power is VGT position=0.8. On the contrary, with loaded regenerative power on TC shaft at high
engine speed and load, VGT should be closed down to achieve best turbine efficiency. In order to
achieve optimal BSFC, both VGT position and assisted or regenerative power on TC shaft
should be controlled coordinately as shown in Figure 79. If the flow is restricted too much such
as VGT being closed to 0.4, this will lead to high fuel consumption.
The loading capacity on TC shaft depends on VGT position for a given engine operation. For
this case in Figure 79, wider open VGT position decreases energy availability of turbine, leading
to lower TC shaft loading capacity. For example, for VGT position=0.8, max TC shaft steady
state loading power is 2kW. Power higher than 2kW will lead to turbocharger shaft stall, which
means steady state turbine power cannot balance with compressor power.

Figure 79. Engine BSFC under assist and regeneration
For assisted mode and regeneration mode, turbine power and compressor power change with
different level of input external shaft power. VGT fixed at 0.6 is investigated in this case. For
baseline case, the compressor power is the same as turbine power without external power, as
shown in Figure 80. During assisted mode, lower extracted turbine power leads to lower exhaust

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pressure, and temperature. Compressor power is the sum of assisted hydraulic turbine power and
turbine power. During regen mode, both compressor and TC turbine power drop because of
lower TC speed and lower mass flow rate. The loss of engine power is mainly due to increased
pumping loss thus lower engine efficiency.
Power distribution between turbine and compressor

Compressor power

Kw

Turbine power

2 kW regen

1 kW regen

baseline

1 kW assist

2 kW assist

5 kW assist

Figure 80. Compressor and turbine power distribution under assist and regeneration

Engine air-fuel ratio (AFR) increases with higher assisted power as shown in Figure 81.
reducing smoke emission during transient tip-in operation. This also indicates that more fuel can
be injected into the cylinder to maintain the same AFR to achieve higher engine brake power
with higher level of assisted power. This would be beneficial for engine performance for both
steady state and transient operations. For steady state case as shown in Figure 82, the baseline
case without assist and experimental case with 19kW TC shaft hydraulic assisted power have the
same AFR. Engine max power can be significantly improved at light load with 19kW assisted
power. Low-speed engine torque increases almost 4 times with the assisted power, compared to
the case without assisted power. The results are based on the assumption that energy in the
hydraulic tank is available to drive hydraulic turbine on TC shaft. For transient improvement,

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engine operation during the transient process would avoid smoke limit with the external assisted
power. This is discussed in next section.

AFR

Air-Fuel Ratio

2 kW regen

1 kW regen

baseline

1 kW assist

2 kW assist

5 kW assist

Figure 81. Air fuel ratio under assist and regeneration
As mentioned previously, TC shaft loading capacity also changes with engine operation
point. As shown in Figure 83, a case at multiple engine load points was studied with different
assisted and regenerative power on TC shaft as well as varying VGT position. It is concluded
that, for light engine load, it might not be possible to load any power on TC shaft to recover
energy during steady state condition, such as engine speed=500 RPM. At engine speed= 1500
RPM, the TC shaft valid loading region expands with higher fuel injection level, which is due to
higher engine exhaust energy. For medium engine speed from 1500 RPM to 2500 RPM, loading
power on TC shaft will lead to high fuel consumption by increasing engine pumping loss and
reducing engine efficiency. For high engine speed operations (3500 RPM), loading capacity on
TC shaft is much wider than lower engine speed operations (<3500 RPM). Further, TC shaft
energy can be recovered with minimum fuel penalty with high engine load region, which agrees
with the finding in the literature [66], [70]. It can also be concluded that engine fuel consumption
can be reduced with power regeneration at certain conditions, depending on engine load
condition. Steady state optimization table for each engine operating points can serve as

143

feedforward calibration table for RHAT-VGT control. Note that, the investigation, in this case,
does not include the impact of EGR (No EGR mass flow rate).
The hydraulic assisted and regenerative turbocharger introduces extra control inputs into
engine system. It increases optimization dimension. With assisted power on TC shaft, engine
BSFC will be improved. The proposed high power hydraulic turbine on TC shaft can
significantly improve engine light load torque response. TC shaft loading capacity depends on
engine speed and load condition. Coordinated control for VGT and RHAT should be
implemented for optimal engine BSFC.

Engine brake power

1
0.8
0.6
0.4
Without assist
With 19 kW assist on TC shaft

0.2
0
500

1000

1500
2000
2500
Engine speed [rpm]

3000

3500

Engine brake torque

1
0.8
0.6
0.4
0.2
0
500

Without assist
With 19 kW assist on TC shaft
1000

1500
2000
2500
Engine speed [rpm]

3000

3500

Figure 82. Max engine power with 19 kW assist with same AFR

144

Figure 83. Loading power on TC shaft for engine BSFC impact (+: regeneration,-: assist)

145

5.4.2

Transient response improvement investigation

With assisted power from the hydraulic turbine, the transient response of engine would be
significantly improved. In this study, transient behaviors under the different level of assisted
power are investigated through simulation. Engine speed is kept the same for all cases while the
torque increases. The control target is to track a same aggressive engine torque demand. Fuel
injection is feedback controlled to meet the target torque, with smoke limit constraint. Air-fuel
ratio (AFR) must be higher than 16 for avoiding rich burn during transient tip-in, to prevent
excessive soot emission. VGT is used to meet the target boost pressure. HP and LP EGR valve
controls are used to track the target EGR mass flow rate. Hydraulic turbine is feedforward
controlled with different hydraulic valve position. Total 8 different hydraulic valve opening
levels were studied, corresponding to different assisted power from 0kW to 14kW. The 0kW
assisted power with fully closed valve position is the baseline case, which has only VGT to
control the boost. With assisted power, VGT control also follows the same boost pressure
tracking control algorithm.
5.4.2.1 Engine transient performance improvement
As shown in Figure 84, all the simulation cases start with 25kW engine power with the same
engine speed; engine transient response is improved with different level of assisted power over
the duration of 0.7 seconds. With 14 kW assisted power, engine power increased 60kW at 11.2s,
compared to the baseline case. Without assisted power, base VGT case cannot achieve aggressive
target torque set-point, due to insufficient air introduced into the cylinder.

146

Figure 84. Engine brake power under different assist

The engine produces more power with the higher level of assisted power since more air is
introduced into the cylinder and more fuel can be injected into the cylinder with respect to the
smoke limit. With insufficient fresh air in the engine cylinder with traditional VGT, engine fuel
injection is limited by air-fuel ratio limit to mitigate soot emission. Thus, less fuel results in less
TC turbine extracted power to drive the compressor This is one of the major reasons for so-called
‘turbo lag'. With assisted power on TC shaft, the power to drive compressor is not totally
dependent on TC turbine power. Hence, with hydraulic assisted power on the shaft, engine
transient response can be significantly improved with more air compressed by the turbocharger.
As show in Figure 85, a higher air-fuel ratio is achieved with assisted power. Note that smoke
limit is used for rich fuel injection. The engine without assisted power has 50% of time operating
below AFR=17 during this transient tip-in event. With increased assisted power, the operating
time when AFR below 17 decreases. With 5.5 kW assisted power, AFR is higher than AFR=17
over the whole simulation duration. With less time operating below the smoke limit, less soot

147

emission is expected with assisted power. Note that, the fluctuation for AFR is due to different
sampling rate for GT simulation and Simulink.
Air Fuel Ratio
24
0kW
2kW
4kW
5.5kW
7kW
9kW
12kW
14kW

23
22

AFR

21
20
19
18
17
16
10.4

10.5

10.6

10.7

10.8
10.9
Time (s)

11

11.1

11.2

11.3

Figure 85. Air fuel ratio with different assist
As shown in Figure 85, the total time duration for engine acceleration from 28kW to 90kW is
0.3 second with 14 kW assisted power, compared to 0.55 second without assisted power. The
acceleration time is reduced by 45% from baseline case. Although engine fuel injection rate
increased with different level of assisted power, total fuel consumption for 14 kW is reduced by
42.7% (from 2.69g to 1.54g) achieving the same target engine power(90 kW) compared to the
baseline case with no assisted power.
5.4.2.2 Turbocharger transient performance improvement
Turbocharger response can be significantly improved with different level assisted power as
shown in Figure 86. With higher assisted power, turbocharger speed increases faster. The
acceleration time from 20K rpm to 40K rpm can be reduced by 75% compared to non-assisted
case.

With higher TC shaft speed, both compressor and turbine operating efficiency are

148

improved. Associated with higher TC shaft speed, compressor and turbine achieves higher
efficiency as shown in Figure 87, Figure 88.
8

x 10

4

Turbocharger Speed
0kW
2kW
4kW
5.5kW
7kW
9kW
12kW
14kW

7

6

RMP

5

4

3

2

1
10.4

10.5

10.6

10.7

10.8
10.9
Time (s)

11

11.1

11.2

11.3

Figure 86. Turbocharger speed transient profile with different hydraulic assisted power
Turbine Efficiency
0.65
0kW
2kW
4kW
5.5kW
7kW
9kW
12kW
14kW

0.6
0.55

Efficiency

0.5
0.45
0.4
0.35
0.3
0.25
10.4

10.5

10.6

10.7

10.8
10.9
Time (s)

11

11.1

11.2

11.3

Figure 87. TC turbine efficiency transient profile with different hydraulic assisted power.

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Figure 88. Compressor efficiency during 10.5 s to 11.2 s with different assisted power

5.4.2.3 Hydraulic components transient performance
Figure 89 shows the hydraulic turbine power and hydraulic tank energy profile for different
level of assisted power. Since different hydraulic power is based on different hydraulic valve
position, pressure difference across turbine as well as TC shaft speed. Although hydraulic valve
position is fixed for each case, the hydraulic power changes according to tank pressure and TC
shaft speed with respect to designed turbine efficiency. During these transient operations,
hydraulic turbine could be controlled through hydraulic valve position to achieve higher
operating efficiency.
In summary, hydraulic assisted turbocharger can significantly achieve faster engine response
than non-assisted turbocharger. It can increase AFR, TC speed, TC turbine efficiency, TC

150

compressor efficiency and engine operating efficiency during transient engine tip-in. All these
benefit quite depend on how energy can be recovered through driveline pump and turbo pump
without fuel penalty. In order to evaluate the fuel benefit level for proposed hydraulic assisted
turbocharger, driving cycle simulation is investigated in next section.
Hydraulic turbine power
14
12

Power [kW]

10
8
6
4

0kW
2kW
4kW
5.5kW
7kW
9kW
12kW
14kW

2
0
-2
10.4

10.5

10.6

10.7

10.8
10.9
Time (s)

11

11.1

11.2

11.3

11.2

11.3

High pressure hydraulic tank pressure
1.005

Normalized tank pressure

1
0.995
0.99
0.985
0.98
0.975
0.97
10.4

0kW
2kW
4kW
5.5kW
7kW
9kW
12kW
14kW
10.5

10.6

10.7

10.8
10.9
Time (s)

11

11.1

Figure 89. Hydraulic turbine output power transient profile
5.4.3 Design trade-offs for fuel benefit through driving cycle simulation
In the previous section, the benefit of hydraulic assist operation is discussed. However, in
order to achieve sustainable power assisted operation, sufficient hydraulic energy needs to be
recovered from both vehicle and exhaust gas. Total recovered energy depends on driveline pump
sizing as well as hydraulic tank sizing for a given vehicle and an engine for a fixed driving cycle.
Hydraulic tank size would be limited by physical packaging space for design considerations.

151

In this study, vehicle simulation for FTP 75 driving cycle was investigated to understand the
design trade-offs for fuel benefit and hydraulic components sizing and turbine sizing with vehicle
level simulation. In traditional VGT control, in order to achieve higher turbine power to drive
compressor at engine light load, VGT vane position is closed to a small opening to increase preturbine pressure. In this case, pumping loss increases due to higher engine exhaust pressure,
which leads to fuel economy penalty. With assisted power on TC shaft, VGT position can avoid
operating at small opening, which increases turbocharger efficiency and reduces engine pumping
loss. But the capability of assisted power depends on energy availability in hydraulic tank.
Baseline in this study is still the traditional VGT for boost control with both high-pressure
exhaust gas recirculation (HP-EGR) and lower pressure exhaust gas recirculation (LP-EGR). For
RHAT-VGT turbocharger, VGT position is fixed at 0.5, 0.65, 0.75 and 1 position. Position 1 is
fully open. RHAT is used to control boost pressure to track target pressure. There are two
advantages for simulating with fixed VGT positions: first, fuel saving can be easily evaluated
without sophisticated control system dealing with interaction with VGT and RHAT control.
Second, the result would be used to exam feasibility of fixed geometry turbocharger (FGT) with
hydraulic assisted and regenerative turbocharger, compared to a VGT turbocharger. Increased
inertia by hydraulic turbocharger is considered in this simulation.
Based on the distribution of VGT position on FTP driving cycle for traditional VGT control,
vane positions 0.5-0.6 and 0.2-0.3 are the most frequent opening positions during the driving
cycle. Vane position 0.2-0.3 is used to build up turbine back pressure to drive compressor at very
light engine load. This results in high pumping loss and low turbine efficiency. Vane position 0.50.6 is the most efficient turbine operation range from design prospect. When the vane position is
fixed at large openings, extra assisted power and regenerative power is needed to meet engine

152

boost target and torque demand. Note that, in this simulation study, performance map for FGT
turbine inherits from VGT turbocharger with fixed vane position for comparison. But in a
practical case, FGT turbine efficiency would be higher without losses across turbine vane
nozzles.

2500

2000

1500

1000

500

0
0.1

0.2

0.3

0.4

0.5
0.6
VGT position

0.7

0.8

0.9

1

Figure 90. VGT position distribution for traditional VGT control over FTP 75 cycle.

The simulation results are shown in Figure 91 and Table 16. All the simulation cases meet
the driver’s demand by achieving the same target vehicle speed. In order to achieve the same
vehicle speed, the engine needs to be provided with sufficient fresh air for combustion process to
produce the right amount of torque. Thus, different level of energy is needed for a hydraulic
turbine with different fixed geometry turbine. The results show that system fuel benefit mainly
depends on tank size for a given VGT size. The tank could be sized for largest energy drop
during the driving cycle which may be dictated by the highway portion of the FTP cycle if
supplemental hydraulic energy input to the TC is required. Thus, small VGT opening position
results in small tank size since small VGT open position requires less supplemental hydraulic
energy. Note that only VGT=0.5 and VGT=0.65 can achieve tank energy balance with current

153

tank size and driveline pump size. In these two cases, RHAT application results in 0.8% and
4.0% fuel saving, respectively. In other two cases, the RHAT can't recover enough energy to
balance tank pressure or the required supplemental hydraulic energy with current driveline pump
size due to reduced turbine energy extraction with large VGT open positions.
Vehicle Speed

Vehicle speed [km/h]

100

target vehicle speed
VGT without RHAT
fixed VGT=0.5+RHAT+Driveline pump
fixed VGT=0.65+RHAT+Driveline pump
fixed VGT=0.75+RHAT+Driveline pump
fixed VGT=1+RHAT+Driveline pump

80
60
40
20
0

200

400

600

800
1000
Time [s]
MPG

1200

1400

1600

1800

20

MPG

15
VGT without RHAT
fixed VGT=0.5+RHAT+Driveline pump
fixed VGT=0.65+RHAT+Driveline pump
fixed VGT=0.75+RHAT+Driveline pump
fixed VGT=1+RHAT+Driveline pump

10
5
0

200

400

600

800
1000
Time [s]

1200

1400

1600

1800

Tank energy balance

Normalized energy

1
0.8
0.6
fixed
fixed
fixed
fixed

0.4
0.2
0

200

400

600

VGT=0.5+RHAT+Driveline pump
VGT=0.65+RHAT+Driveline pump
VGT=0.75+RHAT+Driveline pump
VGT=1+RHAT+Driveline pump

800
1000
Time [s]

1200

1400

1600

1800

Figure 91. Fuel benefit for driving cycle with different FGT turbine

Table 16. Vehicle fuel benefits and tank energy
Turbine size
(VGT position)
Fuel Saving
Tank energy
end of cycle
Max tank
energy drop
State of charge

0.5

0.65

0.75

1

0.8%

4.0%

5.2%

6.6%

100%

100%

75%

30%

20%

38%

56%

83%

balanced

balanced

unbalanced

unbalanced

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Pumping loss reduction is shown in Figure 92. Vane position is closed aggressively for high
boost pressure demand during vehicle tip-in without assisted power, leading to high pumping
loss. With different fixed vane position and assisted power, pumping losses are reduced while
meeting the same boost target. This is one of the main reasons for fuel consumption reduction
over the driving cycle when assisted power is used. But the level of fuel savings depends on how
much hydraulic energy that can be recovered during the whole driving cycle. Energy usage and
recovery for the hydraulic turbine, TC shaft pump, and driveline pump are shown in Figure 95. It
indicates that in order to achieve higher fuel benefit, more hydraulic energy is needed. With
wider turbine vane open position, there are less energy recovery opportunities for the hydraulic
pump on TC shaft. Total recovered energy from the hydraulic turbopump on TC shaft alone may
not be enough to drive turbocharger to meet the same boost demand. Energy recovery mainly
depends on the driveline pump. It is concluded that, energy recovery from hydraulic driveline
pump itself would limit total system fuel benefit. Simulation results show that fuel saving rate is
less than 0.5% without driveline energy recovery on the FTP cycle. Note that only turbine size
0.5 and turbine size 0.65 have balanced tank pressure over the cycle in this study.
Note that since dual EGR loop is utilized in this system to meet demanded EGR mass flow
rate. Results show that total EGR mass fraction in the intake manifold is equivalent for all the
VGT and FGT with RHAT cases. But, with hydraulic assisted power, more LP-EGR is needed to
compensate insufficient HP-EGR, because the pre-turbine pressure is decreased with assisted
power.
Engine system with RHAT needs to be optimally designed such that: system performance
and fuel economy meets the design target within constraints of hydraulic tank packaging space
and cost. Hence, with given hydraulic turbine and hydraulic pumps (turbo-pump and driveline

155

pump) as well as engine and vehicle size, the design matrices for RHAT system would be TC
turbine size, hydraulic tank size as shown in Figure 96. Tank size is identified through largest
tank energy drop during the driving cycle for four different cases. It clearly shows that, in order
to have higher fuel benefit, larger hydraulic tank needed to be utilized to store hydraulic fluids.

Engine pumping power

pumping Power [kW]

15
VGT no RHAT
fixed VGT=0.5+RHAT
fixed VGT=0.65+RHAT
fixed VGT=0.75+RHAT
fixed VGT=1+RHAT

10

5

0
345

350

355

360
Time [s]

365

370

375

VGT position

VGT position

1
0.8
0.6
0.4
0.2

Hydraulic turbine power [kW]

0
345

350

355

360
365
Time [s]
Hydraulic turbine power [kW]

370

375

350

355

370

375

5
4
3
2
1
0
345

360
Time [s]

365

Figure 92. Pumping loss reduction with hydraulic assisted power

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With improved transient response, the engine operating trajectory can be moved to more efficient
operating region for both transient tip-in, and steady state. The results for FTP driving cycle can
be found as shown in Figure 93 and Figure 94. The results agree with the discussion above.

Figure 93. Improved engine efficiency during transient

157

Figure 94. Engine brake efficiency [%] distribution

158

1
Hydraulic turbine energy used
Hydraulic TC pump recovered energy
Driveline pump recovered energy

0.9
0.8

Normalized energy

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

VGT=0.5+RHAT VGT=0.65+RHAT VGT=0.75+RHAT VGT=1+RHAT

Figure 95. Energy usage and recovery for hydraulic components
Design matrices

Fuel saving [%]

8

6

4

2
0
1
0.8

1
0.9

0.6

0.8
0.7

0.4
Tank size

0.2

0.6
0.5

FGT (VGT position)

Figure 96. Design space for hydraulic components with normalized tank size

In summary, hydraulic assisted turbocharger with driveline energy recovery is a feasible
technology to improve engine transient response as well as improve vehicle fuel efficiency.

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5.5 Conclusion
Combining a turbo pump and driveline pump with the hydraulic assisted turbocharger to
recover the exhaust energy and vehicle brake energy to offset the fuel economy penalty as a
result of hydraulic power input to the turbo shaft for turbocharger transient response
improvement, is a novel idea.
The preliminary one-dimensional GT-Power analysis indicated that it is possible to gain 3-5%
fuel economy improvement with the RHAT system, compared with the base turbocharged diesel
engine, over the FTP 75 transient cycle. This FE improvement does not include the other FE
benefits that may be enabled by RHAT technology, such as engine downsizing, transmission
optimization, etc.
The uncertainties in the numerical study include: dynamic responses and flow losses on the
hydraulic actuation system and energy storage tank, especially during the warm-up period. These
uncertainties will be addressed as the component level technologies are studied and refined.
Like many other innovative technologies, there are some key technical challenges that have
yet to be tackled before this RHAT system can see wide application in turbocharged engines, e.g.
 Development of fast-response hydraulic valves and actuation systems with high-flow
capacity and low-flow losses;
 The reduction of parasitic or windage loss when the hydraulic wheels are not engaged;
 Management of energy storage tank pressure so the turbo pump can work in highefficiency areas at different flow rate or turbo speed;

160

 Optimal designs of the hydraulic turbine and turbo pump that can deliver high efficiencies
over wide operation range, in terms of pressure and turbo speed, without adding excessive
thrust loading and excitation on the turbo rotor system.
 Optimal control strategy to best utilize the RHAT and operate the turbocharger in highefficiency areas to achieve maximum engine system efficiency over entire customer driving
cycles;

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CHAPTER 6: LINEAR QUADRATIC CONTROLLER DESIGN FOR VGT-EGR
DIESEL ENGINE AIR-PATH
6.1 Abstract
This chapter is focused on control design for a diesel engine air-path system equipped with
variable geometry turbocharger (VGT) and exhaust gas recirculation (EGR). The controller
design is further extended to the assisted and regenerative turbocharger system with VGT and
EGR.
Diesel engines are of great challenges due to stringent emission and fuel economy
requirements. Compared with the conventional turbocharger system, regenerative assisted
system provides additional degree of freedoms for the turbocharger speed control. Hence, it
significantly improves control capability for exhaust-gas-recirculation (EGR) and boost pressure.
This paper focuses on control design for the diesel engine air-path system equipped with an EGR
subsystem and a variable geometry turbocharger (VGT) coupled with a regenerative hydraulic
assisted turbocharger (RHAT). The challenges lie in the inherent coupling among EGR,
turbocharger performance, and high nonlinearity of engine air-path system. A linear quadratic
(LQ) controller design approach is proposed in this paper for regulating the EGR mass flow rate
and boost pressure simultaneously and the resulting closed-loop system performance can be
tuned by properly selecting the LQ weighting matrices. Multiple LQ controllers with integral
action are designed based on the linearized system models over a gridded engine operational map
and the final gain-scheduling controller for a given engine operational condition is obtained by
interpreting the neighboring LQ controllers. The gain-scheduling LQ controllers for both
traditional VGT-EGR and VGT-EGR-RHAT systems are validated against the in-house baseline
controller, consisting of two single-input and single-output controllers, using the nonlinear plant.

162

The simulation results show that the designed multi-input and multi-output LQ gain-scheduling
controller is able to manage the performance trade-offs between EGR mass flow and boost
pressure tracking. With the additional assisted and regenerative power on turbocharger shaft for
the RHAT system, engine transient boost pressure performance can be significantly improved
without compromising the EGR tracking performance, compared with the baseline control.

6.2 Introduction
Control system for diesel engine must meet driver’s performance demand, meanwhile satisfying
emission constraint. Two major emission pollutions for diesel engine are visible smoke or particle
matter and NOx (nitric oxide and nitrogen dioxide). Visible smoke can be avoided by keeping the
air-fuel ratio high for lean combustion. For reduction of NOx , commonly method is to use exhaust
gas recirculation and 𝑁𝑂𝑥 aftreatment system. Inside engine cylinder, recirculated exhaust gas
acts as inert gas, it can increases the specific heat capacity and decreases the oxygen concentration
of charged air. Hence it can reduces burn rate, lower peak flame temperature and reduce the
formulation of NOx . In order to meet the driver’s demand and emission requirement, the air
handling system for diesel engine must supply diesel engine with right amount of fresh air and
desired EGR fraction in intake charge for a given engine operating condition.

163

Figure 97. Diesel engine with VGT and EGR system
To fulfill these purposes, modern diesel engines are normally equipped with Variable
Geometry Turbocharger (VGT) and high-pressure Exhaust Gas Recirculation (EGR) as shown in
Figure 97. The VGT turbocharger is driven by exhaust gas turbine. The gas turbine is used to
drive the compressor to compress fresh air into engine intake manifold. Since, charged air
quantity can be increased by the compressor, a larger amount of fuel can be burnt to provide
higher engine torque, compared to the non-turbocharged engine. With variable geometry turbine,
changing turbine vane nozzle position can accommodate both engine low flow and high flow
conditions for better turbocharger system efficiency and transient response. The high-pressure
EGR system connects exhaust gas manifold with the intake manifold, driving exhaust gas into
the intake manifold. EGR valve position is adjusted to achieve desired EGR mass flow rate.
However, the nonlinear multivariable nature of the diesel engine VGT-EGR system and dual
objectives make the close loop control design problem arduous. Traditional control strategy treat
these two control actions as two single input and single output system by using only one actuator
at one time. Reported in [53] is an overview of the control strategies used in heavy-duty Diesel

164

engines with EGR-VGT. During tip-ins, the control system closes the EGR valve and uses a
Proportional Integral Derivative (PID) controller feedback on boost pressure to rapidly increase
air supply to the intake using the VGT. In steady state operation, the controller fully opens the
EGR valve regulates the EGR flow rate by open loop vane control of the VGT. This strategy is
satisfactory for the vehicle with prolonged steady state operation as in commercial vehicles. For
passenger car applications transient operation is more frequent and often more severe. For
instance, the natural feedback established by the VGT can be compromised by the presence of
high-pressure EGR valve action. The inverse response type behavior for the compressor mass
flow by the step response of change of EGR increases the control design challenges. Therefore to
fully exploit the potential of these devices, the control design needs to consider the problem in
the context of multivariable control, with expectations of a better performance.
A good control solution for production application must provide a robust controller that does
not use up many resources of the ECU and is simple to implement and calibrate. Most of the
papers discussed the stability and robustness of controller design for diesel engine air flow
regulation. A Control Lyapunov Function (CLF) based controller design is introduced in [7] This
method is constructed for a simplified model by using input-output linearization. Robustness is
achieved by using the domination redesign. In [5], a multivariable controller is designed based
on input and output linearization with sliding model control. It provides a systematic method to
regulate the EGR mass flow rate and intake manifold pressure with choosing different sliding
surface. However, few papers discussed a systematic approach for close loop controller design
with respect to engine performance, emission and fuel economy during transient operation,
specifically, not only considering stability and robustness but also taking engine transient

165

performance into account during control design process. The engine transient performance here
means engine response, emission, and fuel economy during transient operation.
The linear quadratic regulator (LQR) is a well-known design technique that provides
practical steady feedback gains which are also known for its robustness which has been
interpreted as gain margin (1/2, +∞) and 60𝑜 phase margin. LQR techniques has been widely
investigated in other nonlinear system [85][86][87][88]. For diesel engine VGT-EGR air-path
control, one of the challenges is to handle the trade-offs between engine performance, emission
and fuel economy. In LQR design routine, weighting matrix can be directly used to design the
controller for different purposes. It eventually provides a multi-input and multi-output (MIMO)
controller to coordinate VGT vane position and EGR valve action. However, LQR-controller
designs are only applicable for linear systems. Gain scheduling is a natural approach to extend
the linear control design to a nonlinear system by using a family of linear controllers, which
individually provides satisfactory control for a different local operating point of the system.
In this chapter, we show the development of gain scheduling control design for VGT-EGR
system. Controller design is based on the high fidelity reduced order nonlinear diesel engine
model developed in Chapter 4. Linearized models are obtained for the nonlinear system along
with engine operating trajectory. A linear quadratic technique with integral action is used to
design the local linear controller to regulate the boost pressure and EGR mass flow rate tracking
error to zero and keep steady state error to zero. Controller design for different target scheme is
proposed with weighting selection. Different performance indexes are used to evaluate the
controller design. This provides flexibility for controller design with respect to different engine
performance and emission trade-offs. Proposed MIMO linear controllers are scheduled by engine
speed and fuel injection to be implemented with the nonlinear plant. By comparing with baseline

166

controller, the proposed controller shows its advantage of the multivariable controller over SISO
controllers. It also shows a systematic method for engine performance and emission trade-offs
mitigation during controller design. Further controller design is extended to assisted and
regenerative turbocharger system with VGT and EGR system.
6.3 Controller design
6.3.1 Control objective and problem formulation
For diesel engine VGT-EGR air-path system, the control objective is to regulate the
demanded fresh air and oxygen concentration in the intake manifold to the desired levels as
determined from an optimized engine static calibration. These static maps are generated based on
a trade-off between maximal fuel economy and minimal 𝑁𝑂𝑥 generation, without violating the
constraints on soot formation. Demanded fresh air and oxygen concentration can be transferred
to air fuel ratio and EGR mass flow fraction. Hence, while the set-point for AFR determines the
engine response and prevents smoke, the EGR flow fraction seeks to minimize in-cylinder 𝑁𝑂𝑥
generation. If the fueling rate is known (from drivers pedal position), then the set-point for AFR
pressure can be transformed into a set-point for compressor flow rate 𝑚̇𝐶𝑑 . Similarly, the set-point
𝑑
for EGR flow fraction can be expressed regarding the desired quantities 𝑚̇𝑒𝑔𝑟
and𝑚̇𝐶𝑑 . Further

set-points can be reformulated to intake pressure, exhaust pressure and TC speed and oxygen
concentration intake manifold as shown in Figure 98. These are common states to be regulated to
the desired valve in previous control design.

167

Figure 98. Set-point for diesel engine air-path control

In order to compare with in-house baseline controller, the control objective of this study is to
regulate the boost pressure (𝑃2𝑑 )) and EGR mass flow rate(𝑚̇𝑒𝑔𝑟 ) to their desired values. The
reference trajectories are generated by calibration tables, which are not discussed in this study.
With driver's pedal position input, desired engine torque is used to define desired fuel injection
amount. Target boost pressure and target EGR mass flow rate are based on current engine speed
and fuel injection. Meanwhile, optimal steady state set-point for VGT vane position (𝑢𝑣𝑔𝑡 ) and
EGR valve position ( 𝑢𝑒𝑔𝑟 ) are obtained based on steady state emission and performance
requirement. Note that, for each pair of

(𝑁𝑒 , 𝑚̇𝑓𝑢𝑒𝑙 , 𝑢𝑣𝑔𝑡 , 𝑢𝑒𝑔𝑟 ) , there exists a unique

equilibrium of diesel engine plant. Hence, with a given engine speed and fuel injection for steady
state, engine plant will operate at a given operating point. For this reason, engine speed and fuel
injection would be a natural candidate for gain scheduling.

Figure 99. Tracking reference generation in baseline controller

168

In this study, the close loop control design is formulated as a regulation problem. Its target is
to regulate the boost pressure tracking error and EGR mass flow rate tracking error to zero,
meanwhile keeping the steady state errors to zero. For a regulation problem, the control design is
based on error system. Linearization of the nonlinear plant is investigated first.
6.3.2 Linearization of Nonlinear System
Consider a nonlinear differential equation:
𝑥̇ (𝑡) = 𝑓(𝑥(𝑡), 𝑢(𝑡))

(6. 1)

where 𝑓 is a function mapping Rn × Rm Rn , a point x̅ ϵ Rn is called an equilibrium point, if
there is a corresponding 𝑢̅ 𝜖 𝑅 𝑚 such that:
𝑧𝑒𝑟𝑜(𝑛) = 𝑓(𝑥̅ (𝑡), 𝑢̅(𝑡))

(6. 2)

System (6.1) starts from initial conditions x(t 0 ) = x̅, with input u(t) ≡ u̅ for all t ≥ t 0 . The
resulting solution x(t) satisfies:
x(t) = x̅ For all t ≥ t 0

(6. 3)

Suppose(𝑥,
̅ 𝑢̅) is an equilibrium point and input for the nonlinear system (6.1). Considering
states and control inputs deviate from the equilibrium point, the deviation variables can be
defined as:
𝛿𝑥(𝑡) = 𝑥(𝑡) − 𝑥̅

(6. 4)

𝛿𝑢(𝑡) = 𝑢(𝑡) − 𝑢̅

(6. 5)

Variables x(t) and u(t) are governed by the system (6.1). Substitute equations (6.4) and (6.5)
into dynamic system (6.1):

169

𝛿𝑥̇ (𝑡) = 𝑓(𝑥̅ + 𝛿𝑥(𝑡), 𝑢̅ + 𝛿𝑢(𝑡)) − 𝑓(𝑥̅ , 𝑢̅)

(6. 6)

Applying Taylor expansion of the right hand side and neglect all higher order (>2) terms:
𝛿𝑥̇ (𝑡) + 𝑓(𝑥̅ , 𝑢̅) ≅ 𝑓(𝑥̅ , 𝑢̅) +

𝜕𝑓
𝜕𝑓
|𝑥=𝑥̅ 𝛿𝑥(𝑡) + | 𝑥=𝑥̅ 𝛿𝑢(𝑡)
𝜕𝑥
𝜕𝑢
̅
𝑢=𝑢

̅
𝑢=𝑢

(6. 7)

leads to
𝛿𝑥̇ (𝑡) ≅

𝜕𝑓
𝜕𝑓
| 𝑥=𝑥̅ 𝛿𝑥(𝑡) + |𝑥=𝑥̅ 𝛿𝑢(𝑡)
𝜕𝑥
𝜕𝑢
̅
𝑢=𝑢

(6. 8)

̅
𝑢=𝑢

The differential equation governs system dynamics with the infinitely small deviations of
δx(t) and 𝛿𝑢(𝑡).

Note that the resulting dynamic system is linear time-invariant since derivative of

δx(t) are the linear combination of 𝛿𝑥(𝑡) and δu(t). For the state-space realization, matrices A
and B can be expressed as
𝐴=

𝜕𝑓

|
𝜕𝑥 𝑥=𝑥̅
̅
𝑢=𝑢

∈ 𝑅𝑛×𝑛 , 𝐵 =

𝜕𝑓
|
𝜕𝑢 𝑥=𝑥̅
̅
𝑢=𝑢

∈ 𝑅𝑛×𝑛

(6. 9)

They are constant matrices under the given equilibrium condition. Thus linear approximation for
the nonlinear system (6.1), around equilibrium point (x,
̅ u̅), can be expressed as:
𝛿𝑥̇ (𝑡) = 𝐴𝛿𝑥(𝑡) + 𝐵𝛿𝑢(𝑡)

(6. 10)

This is called Jacobian linearization. For small values of 𝛿𝑥(𝑡) and 𝛿𝑢(𝑡), the linearized system
approximately represents the dynamic relationship between 𝛿𝑥(𝑡) and 𝛿𝑢(𝑡).
6.3.3 Model linearization for diesel engine air-path system
Consider a three states diesel engine air-path model with exhaust pressure ( 𝑃3 ), boost
pressure (𝑃2 ) and TC shaft speed (𝜔) as states. Control inputs are VGT vane position (𝑢𝑣𝑔𝑡 ) and
EGR valve position (𝑢𝑒𝑔𝑟 ); and control outputs are boost pressure (𝑃2 ) and EGR mass flow rate

170

(𝑚̇𝑒𝑔𝑟 ). The control target is to minimize the tracking errors of both boost pressure and EGR
mass flow rate.
𝑃3̇ =

𝑅𝑇3
(𝑚̇𝑜𝑢𝑡 − 𝑚̇𝑒𝑔𝑟 − 𝑚̇𝑡 )
𝑉3

𝑃2̇ =

𝑅𝑇2
(𝑚̇𝑐 − 𝑚̇𝑖𝑛 + 𝑚̇𝑒𝑔𝑟 )
𝑉2

(6. 11)

𝐽𝜔𝜔̇ = 𝑊̇𝑇 − 𝑊̇𝐶 − 𝑊̇𝐿𝑜𝑠𝑠

The nonlinear plant (6.11) can be expanded as shown in (6.12) and is highly nonlinear. Plant
dynamics varies under different engine operational conditions ( 𝑁𝑒 , 𝑚̇𝑓𝑢𝑒𝑙 , 𝑢𝑣𝑔𝑡 , 𝑢𝑒𝑔𝑟 ). More
details about the nonlinear plan can be found in Chapter 4.
𝑅𝑓𝑇3 (𝑃2 , 𝑚̇𝑓𝑢𝑒𝑙 )
𝑃3
𝑃3
(𝑓𝑚̇𝑖𝑛 (𝑁𝑒 , 𝑃2 ) + 𝑚̇𝑓𝑢𝑒𝑙 − 𝑓𝑚̇𝑡 (𝑢𝑣𝑔𝑡 , ) − 𝑓𝑚̇𝑒𝑔𝑟 (𝑢𝑒𝑔𝑟 , ))
𝑉3
𝑃4
𝑃2
𝑃2
𝑅𝑓𝑇2 (𝜔, )
𝑃2
𝑃3
𝑃1
𝑃2̇ =
(𝑓𝑚̇𝑐 (𝜔, ) − 𝑓𝑚̇𝑖𝑛 (𝑁𝑒𝑛𝑔𝑖𝑛𝑒 , 𝑃2 ) + 𝑓𝑚̇𝑒𝑔𝑟 (𝑢𝑒𝑔𝑟 , ))
𝑉2
𝑃1
𝑃2

𝑃3̇ =

𝐽𝜔𝜔̇ = 𝑓𝑊̇𝑇 ( 𝑢𝑣𝑔𝑡 ,

(6. 12)

𝑃3
𝑃2
, 𝜔, 𝑇3 ) − 𝑓𝑊̇𝑐 ( , 𝜔) − 𝑓𝑊̇𝐿𝑜𝑠𝑠 (𝜔)
𝑃4
𝑃1

A linearized model about an equilibrium point of the nonlinear diesel engine air-path model
(6.12) was developed using analytical methods. The linearized equations are shown below
𝛿𝑥̇ 𝑝 (𝑡) = 𝐴𝑃 (𝑡)𝛿𝑥𝑝 (𝑡) + 𝐵𝑃 (𝑡)𝛿𝑢(𝑡)
𝑦𝑝 (𝑡) = 𝐶𝑃 (𝑡)𝛿𝑥𝑝 (𝑡) + 𝐷𝑃 (𝑡)𝛿𝑢(𝑡)

(6. 13)

𝑧(𝑡) = 𝐹𝛿𝑥𝑝 (𝑡) + 𝑊(𝑡)𝛿𝑢(𝑡)

where, 𝑥(𝑡) ∈ 𝑅 3 is the states of the plant (𝑃3 , 𝑃2 , 𝜔), 𝑢(𝑡) ∈ 𝑅 2 is the control inputs (𝑢𝑣𝑔𝑡 , 𝑢𝑒𝑔𝑟 );
yp ∈ R2 is the output available ( 𝑚̇𝑒𝑔𝑟 , 𝑃2 ) for feedback control; and z(t) is the performance
output to be regulated to the desired reference value. In this study, 𝑦𝑝 = 𝑧(𝑡), assuming both
outputs are measurable. The states, inputs, and outputs in (6.13) are all deviations of the

171

corresponding trajectories of the nonlinear system (6.12) from the equilibrium operation
condition. For a typical diesel engine VGT-EGR air-path system, the linearized model in in this
general form below in (6.14). More detailed plant linearization can be found in Appendix.
𝐴11
𝛿𝑃3̇
[𝛿𝑃2̇ ] = [𝐴21
𝐴31
𝛿𝜔̇
𝛿𝑦(𝑡) = [

𝐴12
𝐴22
𝐴32

𝛿𝑚̇𝑒𝑔𝑟
𝐶
] = [ 11
0
𝛿𝑃2

𝐵11
0 𝛿𝑃3
𝐴13 ] [𝛿𝑃2 ] + [ 0
𝐵31
𝐴33 𝛿𝜔
𝐶12
𝐶22

𝐵12 𝛿𝑢
𝑣𝑔𝑡
𝐵22 ] [
]
𝛿𝑢𝑒𝑔𝑟
0

𝛿𝑃3
0
0
] [𝛿𝑃2 ] + [
0
0
𝛿𝜔

(6. 14)
𝐷21 𝛿𝑢𝑣𝑔𝑡
][
]
0 𝛿𝑢𝑒𝑔𝑟

Linear controllers are designed based on the linearized models at different equilibrium points,
and the associated performances are evaluated through simulation studies based on the full-scale
nonlinear model. The desired outcome of controller design is for the system to track the target
EGR mass flow rate and boost pressure.
6.3.4 Augmented with actuator dynamics
For model accuracy, simple actuator dynamics are added to the plant model. The EGR valve
actuator dynamics is modelled as:
𝛿𝑢̃̇𝑒𝑔𝑟 (𝑡) = 𝐴𝑑 (𝑡)𝛿𝑢̃(𝑡) + 𝐵𝑑 𝛿𝑢𝑒𝑔𝑟 (𝑡)

(6. 15)

In this case, only EGR actuator dynamics is modelled. The parameters of the actuator model
are chosen to approximate values for an actual EGR valve. In this case, EGR actuator response
time is about 0.03s. The model (6.15) is as follows:
𝛿𝑢̃̇𝑒𝑔𝑟 (𝑡) = −

1
1
𝛿𝑢̃ (𝑡) +
𝛿𝑢 (𝑡)
0.03 𝑒𝑔𝑟
0.03 𝑒𝑔𝑟

(6. 16)

After augmenting the actuator dynamics, the new system is:
𝛿𝑥̇1 (𝑡) = 𝐴1 (𝑡)𝛿𝑥1 (𝑡) + 𝐵1 𝛿𝑢1 (𝑡)
𝛿𝑦1 (𝑡) = 𝐶1 (𝑡)𝛿𝑥1 (𝑡)

172

(6.17)

where
𝑥1 (𝑡) = [

𝛿𝑢𝑣𝑔𝑡
𝐴𝑝
𝛿𝑥𝑝 (𝑡)
] ; 𝛿𝑢1 (𝑡) = [
]; 𝐴1 = [
𝛿𝑢𝑒𝑔𝑟
0
𝛿𝑢̃(𝑡)

𝐵𝑃𝑒𝑔𝑟
𝐵
]; 𝐵1 = [ 𝑃𝑣𝑔𝑡
𝐴𝑑
0

0
] ; 𝐶1 = [𝐶𝑃
𝐵𝑑

𝐷𝑃𝑒𝑔𝑟 ].

Note that matrix D of (6.14) is only associated with the EGR control input that is the direct input
for the EGR mass flow rate. With the actuator augmented actuator dynamics, matrix D in (6.17)
is zero.
6.3.5 Integral action
It is desirable to include integral action into the state feedback control to eliminate the steady
state errors. With model uncertainties presented in system, controller design must be able to
compensate those uncertainties to eliminate steady state errors. Note that the standard LQR
results in a proportional state feedback controller without integral action. This implies that at
steady state the tracking errors of LQR are not zero. By augmenting the system with its integral
errors it is possible to add integral action to the LQR with integral gains selected automatically.
The advantage of adding the integral action is that it eliminates the steady state tracking errors.
Define a new system model with integral action that has the time derivative of output and states
as states:
𝑑 𝛿𝑥̇1 (𝑡)
𝐴 (𝑡)
(
)=[ 1
𝐶1 (𝑡)
𝑑𝑡 𝛿𝑦1 (𝑡)

0 𝛿𝑥̇1 (𝑡)
𝐵 (𝑡)
][
] + [ 1 ] 𝛿𝑣(𝑡)
0 𝛿𝑦1 (𝑡)
0

(6. 18)

The new augmented system is:
𝑑
(𝑀(𝑡)) = 𝐴̃(𝑡)𝑀(𝑡) + 𝐵̃ (𝑡)𝑈(𝑡)
𝑑𝑡

where
𝑈(𝑡) = 𝛿𝑣(𝑡) =

𝛿𝑥̇ (𝑡)
𝐴 (𝑡)
, 𝑀(𝑡) = [ 1 ], 𝐴̃(𝑡) = [ 1
𝛿𝑦1 (𝑡)
𝐶1 (𝑡)

𝑑(𝛿𝑢1 (𝑡))
𝑑𝑡

(6.19)
0 ̃
𝐵 (𝑡)
], 𝐵 (𝑡) = [ 1 ]
0
0

Based on equations (6.17), (6.18) and (6.19), the final augmented linear system have the
following system matrices:

173

𝐴11
𝐴21
𝐴̃ = 𝐴31
0
𝐶11
[ 0

𝐴12
𝐴22
𝐴32
0
𝐶12
𝐶22

0
𝐴13
𝐴33
0
0
0

𝐵12
𝐵22
0
𝐴𝑑
𝐷21
0

0
0
0
0
0
0

0
𝐵11
0
0
0 , 𝐵̃ = 𝐵31
0
0
0
0
[ 0
0]

0
0
0
𝐵𝑑 .
0
0]

(6.20)

For instance, the linearized plant matrices at engine speed of 𝑁𝑒 =1200 rpm with fuel injection
quantity of 𝑚̇𝑓𝑢𝑒𝑙 =35mg/cc are
−73.94
2.48
0.04
𝐴̃ =
0
2.8𝑒 − 07
[
0

77.13
−78.94
0.023
0
−2.6𝑒 − 07
−1

0
−1.61𝑒5
1.15𝑒3 2.03𝑒4
−5.1246
0
0
−33.33
0
−0.0024
0
0

0
0
0
0
0
0

0
0
0
,
0
0
0]

1.129𝑒5
0
−20
𝐵̃ =
0
0
[ 0

0
0
0
0.0003
0
0 ]

(6. 21)

6.3.6 Linear quadratic regulator with integral action (LQI)
The target for this control design is to find a linear optimal controller that minimizes the
tracking errors of both boost pressure and EGR mass flow rate for the given LQI cost function
below
∞

0
𝐽 = ∫ (𝑀(𝑡)𝑇 𝑄̃ 𝑀(𝑡) + 𝑈(𝑡)𝑇 𝑅𝑈(𝑡)) 𝑑𝑡, 𝑄̃ = [
0
0

0
6
]∈𝑅
𝑄

(6.22)

To minimize the tracking errors, the integral of 𝑀(𝑡)𝑇 𝑄̃𝑀(𝑡) and 𝑈(𝑡)𝑇 𝑅𝑈(𝑡) should be
nonnegative and small. As a result, matrix 𝑄 must be positive semi-definite and matrix R must ne
positive definite to make sure all control channels are finite. In this case, 𝑄 is 2x2 and 𝑅 is 2x2.
Note that 𝑄 is the weighting matrix for tracking errors of both boost pressure and EGR mass
flow rate; and R is the weighting matrix for the derivative control inputs, VGT vane and EGR
valve positions. Now the problem is formulated as a standard LQR problem. The observability
and controllability conditions required for the Riccati equation to have stationary positive
definite solution must be checked for the matrix pairs

174

1

(𝐴̃ (𝑡), 𝑄2 [0𝑛 𝐼𝑚 ]) and(𝐴̃ (𝑡), 𝐵̃ (𝑡))

(6.23)

Assume observability and controllability conditions are satisfied, then feedback law for 𝛿𝑣(𝑡) =
𝑈(𝑡)

is given by:
𝛿𝑣(𝑡) = 𝑈(𝑡) = −𝑅 −1 𝐵̃ (𝑡)𝑃̃ 𝑀(𝑡) = −𝐾𝑥 𝛿𝑥̇1 (𝑡) − 𝐾𝑦 𝛿𝑦1 (𝑡)

(6.24)

where, 𝐾𝑥 , 𝐾𝑦 and 𝑃̃ are given by:
⌊𝐾𝑥 𝐾𝑦 ⌋ = [𝑅 −1 𝐵̃ (𝑡)𝑃̃𝑥𝑥 𝑅 −1 𝐵̃ (𝑡)𝑃̃𝑥𝑦 ]
𝑃̃

𝑃̃𝑥𝑦
]. 𝑃̃𝑥𝑦 = 𝑃̃𝑦𝑥 . 𝑃̃
𝑃̃𝑦𝑦

and 𝑃̃ = [ ̃𝑥𝑥
𝑃𝑦𝑥

is obtained by solving matrix algebraic Riccati equation:

0
𝐴̃(𝑡)𝑇 𝑃̃ + 𝑃̃ 𝐴̃(𝑡) − 𝑃̃ 𝐵̃ (𝑡)𝑅 −1 𝐵̃ (𝑡)𝑇 𝑃̃ + [
0

Thus:
[
𝑃̃𝑥𝑥
−[
𝑃̃𝑦𝑥

𝐴(𝑡)
𝐶(𝑡)

𝑇 ̃
𝑃𝑥𝑥
0
] [
0
𝑃̃𝑦𝑥

(6.25)

𝑃̃𝑥𝑦
𝑃̃𝑥𝑥
]+[
𝑃̃𝑦𝑦
𝑃̃𝑦𝑥

0
]=0
𝑄

𝑃̃𝑥𝑦
𝐴(𝑡)
][
̃
𝐶(𝑡)
𝑃𝑦𝑦

𝑇
𝑃̃𝑥𝑦
𝐵 (𝑡)
𝐵 (𝑡) 𝑃̃𝑥𝑥
] [ 1 ] 𝑅 −1 [ 1 ] [
0
0
𝑃̃𝑦𝑦
𝑃̃𝑦𝑥

(6.26)

0
]
0

𝑃̃𝑥𝑦
0
]+[
0
̃
𝑃𝑦𝑦

(6.27)
0
]=0
𝑄

Expand each term for algebraic Riccati equation:
2𝐴𝑃̃𝑥𝑥 + 2𝐶𝑃̃𝑥𝑦 − 𝑃̃𝑥𝑥 𝐵𝑅 −1 𝐵𝑃̃𝑥𝑥 = 0
𝐴𝑃̃𝑥𝑦 + 𝐶𝑃̃𝑦𝑦 − 𝑃̃𝑥𝑥 𝐵𝑅 −1 𝐵𝑃̃𝑥𝑦 = 0

(6.28)

𝑄 − 𝑃̃𝑥𝑦 𝐵𝑅−1 𝐵𝑃̃𝑥𝑦 = 0

Thus, we can obtain control law by integrating from initial time 0 to t. This control is a type
of PI controller with state error and integration of output error.
𝑡

𝑡

𝑡

𝛿𝑢(𝑡) = ∫ (𝛿𝑣(𝑡)) 𝑑𝑡 = ∫ (−𝐾𝑥 𝛿𝑥̇1 (𝑡) − 𝐾𝑦 𝛿𝑦1 (𝑡)) 𝑑𝑡 = −𝐾𝑥 𝛿𝑥1 (𝑡) − 𝐾𝑦 ∫ (𝛿𝑦1 (𝑡)) 𝑑𝜏
0

where,

0

0

⌊𝐾𝑥 𝐾𝑦 ⌋ = [𝑅 −1 𝐵̃ (𝑡)𝑃̃𝑥𝑥 𝑅 −1 𝐵̃ (𝑡)𝑃̃𝑥𝑦 ]

175

(6.29)

To test the effectiveness of each controller developed in this study, their performance is
evaluated through simulation studies using the nonlinear model (6.11). In order to implement the
linear controller design for the nonlinear plant, the equilibrium states value must be subtracted
from the states of the nonlinear model and the control inputs are fed by the outputs of the linear
controller with equilibrium controls as in (6.28).
𝛿𝑥1 (𝑡) = 𝑥1 (𝑡) − 𝑥̅1

(6.30)

𝑢1 (𝑡) = 𝑢̅1 + 𝛿𝑢1 (𝑡)

Then the final control inputs for nonlinear plant are:
𝑡

𝑢1 (𝑡) = 𝑢̅1 − 𝐾𝑥 (𝑥1 (𝑡) − 𝑥̅1 ) − 𝐾𝑦 ∫ (𝛿𝑦1 (𝑡)) 𝑑𝜏
0

(6.31)

𝑇
𝑑
where 𝑢̅1 = [𝑢̅𝑣𝑔𝑡 𝑢̅𝑒𝑔𝑟 ]𝑇 , 𝑥̅1 = [𝑃2𝑑 𝑃3𝑑 𝜔𝑑 𝑢𝑒𝑔𝑟
] . Specifically, the steady state control gain has

the following structure
𝐾𝑥
𝐾𝑥 = [ 11
𝐾𝑥21

𝐾𝑥12
𝐾𝑥22

𝐾𝑦
𝐾𝑦 = [ 11
𝐾𝑦21

𝐾𝑥13 𝐾𝑥14
],
𝐾𝑥23 𝐾𝑥24
𝐾𝑦12
]
𝐾𝑦22

(6.32)

where 𝐾𝑥 is the proportional gain; and 𝐾𝑦 is the integral gain. First row of 𝐾𝑥 and 𝐾𝑦 are control
gains for VGT vane position (𝛿𝑢𝑣𝑔𝑡 ). The second row of 𝐾𝑥 and 𝐾𝑦 are control gains for EGR
valve position ( 𝛿𝑢𝑒𝑔𝑟 ). The control actions for both VGT vane and EGR valve positions are
coupled through state deviation and output error. In this study, the overall control architecture is
shown in Figure 100, where 𝑥1 (𝑡) is nonlinear plant states value; and 𝑥̅1 are equilibrium values
𝑑
for each state that are desired value along the regulating trajectory ( 𝑃2𝑑 , 𝑃3𝑑 , 𝜔𝑑 , 𝑢𝑒𝑔𝑟
). Boost

tracking error (𝛿𝑃2 ) and EGR mass flow rate tracking error (𝛿𝑚̇𝑒𝑔𝑟 ) are used to drive the integral
control. The feedforward controls (𝑢̅𝑣𝑔𝑡 𝑎𝑛𝑑 𝑢̅𝑒𝑔𝑟 ) are generated from feedforward calibration map

176

based on engine operating points (𝑁𝑒 , 𝑚̇fuel ) that are used to drive nonlinear plant close to the
steady state operating point.

Figure 100. Proposed linear quadratic regulator for Engine EGR-VGT air-path system
6.3.7 Observability and controllability analysis
Consider system described in (6.11), (6.12) and (6.13) with matrix Q as below:
0
0
0
0
]=
𝑄
0
0
[0

0
𝑄̃ = [
0

0
0
0
0
0
0

0
0
0
0
0
0

0
0
0
0
0
0

0
0
0
0
𝑄11
0

0
0
0
0
0
𝑄22 ]

(6.33)

For the standard LQR controller design, the solution relies on solving the Riccati equation. In
this case, we are dealing with infinite LQR design, in order to have a finite solution for the given
̃ 〉 needs to be detectable. Consider
cost function (6.22), 〈Ã, B̃〉 needs to be stabilizable; and 〈Ã, √Q

the following observability Gramian
1

1

1

1

1

1

𝑇

𝑂 = [𝑄̃ 2 𝑄̃ 2 𝐴̃(𝑡) 𝑄̃ 2 𝐴̃(𝑡)2 𝑄̃ 2 𝐴̃(𝑡)3 𝑄̃ 2 𝐴̃(𝑡)4 𝑄̃ 2 𝐴̃(𝑡)5 ] 𝑂𝜖𝑅36×6

and controllability Gramian:

177

(6.34)

𝐶 = [𝐵̃ (𝑡)

𝐴̃(𝑡)𝐵̃ (𝑡) 𝐴̃(𝑡)2 𝐵̃ (𝑡) 𝐴̃(𝑡)3 𝐵̃ (𝑡) 𝐴̃(𝑡)4 𝐵̃ (𝑡) 𝐴̃(𝑡)5 𝐵̃ (𝑡)] 𝐶𝜖𝑅6×36

(6.35)

As shown in Table 17, the condition for controllability Gramian losing rank is when 𝑃2 =
𝑃3 𝑜𝑟 𝑃3 = 𝑃4 . For instance, considering 𝐵12 , 𝐵22 𝑎𝑛𝑑 𝐷21 , they are block parameters for EGR
control inputs for exhaust pressure, boost pressure and EGR mass flow rate. When 𝑃2 = 𝑃3 ,
𝐵12 = 𝐵22 = 𝐷21 = 0 , EGR valve lose control authority over EGR mass flow rate. And
controllability Gramian rank (C)=4. Further, the system is not fully controllable under this
condition. This can be explained by physical meaning such that when HP EGR valve upstream
pressure is equal to or smaller than EGR valve downstream pressure; EGR valve position would
not have any effect for EGR control. For instance, 𝐵12 term is for EGR valve control input for
EGR mass flow rate as shown as in (6.36). When, 𝑃2 = 𝑃3 , 𝐵12 = 0. When, 𝑃2 > 𝑃3 , inverse
flow is not allowed, EGR valve needs to be fully closed. The same happens for the condition of
P3 < P4. There would be turbine mass flow, hence system’s controllability Gramian losses rank.
3
2874

𝐵12 = −

5
𝑃2 7

1
2 4
𝑃2 7

∗ 𝑇2 ∗ 𝑃3 ∗ 2 ∗ ( ) 𝐶𝑝 ∗ (1 − ( ) )
𝑃3
𝑃3

(6.36)

𝑇3 𝑉𝑖𝑚

Based on physical limitation, such that TC shaft cannot be stalled (𝑃2 ≤ 𝑃1 ) or over
speed(𝑃3 ≤ 𝑃4 ). Further, control target is to have exhaust pressure higher than intake manifold
pressure( 𝑃2 < 𝑃3 ). This will avoid losing controllability as well as keeping EGR flow capability.
In this case, operating space 𝜓 = {(𝑃1 , 𝑃2 , 𝑃3 , 𝑃4 ), 𝑃3 > 𝑃4 , 𝑃2 > 𝑃1 , 𝑃3 > 𝑃2 )} will be
guaranteed such that system will be fully observable and controllable. One thing needs to be
noticed that, control inputs are bounded. VGT and EGR positions are bounded by their physical
hardware actuator position. For instance, VGT position is [0-100] and EGR valve position is [0-

178

100]. However, if control design is targeted for engine performance, the EGR valve might need
to be fully closed for aggressive boost demand during transient tip-in and tip-out.
Table 17. Controllability and observability analysis for VGT-EGR system
Condition 1

Condition 2

𝑃1 ≥ 𝑃2

𝑃2 ≥ 𝑃3

Block value =0

None

𝐵12 𝐵22 𝐵𝑑 𝐴𝑑 𝐷21

𝐵31 𝐵21

Nominal
condition
𝑃1 < 𝑃2
𝑃2 < 𝑃3
𝑃3 > 𝑃4
None

Singularity

𝐴11 𝐴13
𝐴31 𝐴33

𝐴12 𝐴21 𝐴22

𝐴22

None

Physical meaning

Leaking in the intake
manifold

EGR valve closed to
prevent inverse flow
No EGR mass flow
rate

No turbine
mass flow
rate

None

3(full)

3(full)

3(full)

3(full)

3(full)

3(full)

3(full)

3(full)

6(full)

5
(loss rank)

6(full)

6(full)

4
(loss rank)

3
(loss rank)

6(full)

Observability Gramian
Rank with 3 states
(𝐴𝑝 (𝑡), 𝐶𝑝 (𝑡) )
Controllability Gramian
Rank
with 3 states
(𝐴𝑝 (𝑡), 𝐵𝑝 (𝑡) )
Observability Gramian
Rank
with 6 states(including
actuator dynamics with
integral action)

Condition 3
𝑃3 ≤ 𝑃4

1

(𝐴̃(𝑡), 𝑄̃ 2 )
Controllability Gramian
Rank
with 6 states (including
actuator dynamics with
integral action)
(𝐴̃(𝑡), 𝐵(𝑡))

6(full)

6.3.8 Plant scaling
For the linearized plant, the output and input matrices are badly conditioned, which could
lead to numeric issues when solving the Riccati equation. As shown in Table 18, the parameters
for the linearized model has very different numerical range. Especially for the boost pressure and
EGR mass flow rate. This results in a badly conditioned C matrix since the tracking targets are

179

boost pressure and EGR mass flow rate. In order to compensate this issue, input and output
scaling are carried out. The scaling factor for control input is 1000, scaling factor EGR mass
flow rate is 3600*1000. After that, matrix C has proper scale for both boost pressure and EGR
mass flow rate as shown in Table 19.
Table 18. Parameters numeric range

States

Control inputs
Plant outputs

Parameter

Unit

Boost pressure
Exhaust pressure
TC speed
EGR valve dynamic
VGT position
EGR valve position
Boost pressure
EGR mass flow rate

Pa
Pa
Rpm
Percentage
Percentage
Percentage
Pa
kg/s

Numeric range for normal
operation in the model
1e5-3e5
1e5-3e5
2e4-12e4
0-100
0-100
0-100
1e5-3e5
1e-3-1e-1

Table 19. C matrix reformulation
From

To
Boost pressure
EGR mass flow rate

Boost pressure

Exhaust pressure

Before
scaling

After
scaling

Before
scaling

After
scaling

1
−2.6𝑒 − 07

1
-0.9406

0
2.8𝑒 − 07

0
1.0359

TC
speed

EGR
dynamic
state

0
0

0
0

6.4 Controller Design Validation
6.4.1 Baseline controller for control development
Gain-scheduling control of Proportional Integral Derivative (PID) is commonly used in
industry to control a nonlinear system due to its advantages of simplicity and easy calibration.
The current in-house baseline controller has two independent control loops for tracking both
boost pressure and EGR mass flow rate as shown in Figure 101. EGR valve position is used to
track target EGR mass flow rate. Note that EGR valve opening is directly related to EGR mass
flow rate and pressure changes (exhaust pressure and boost). VGT vane position is feedback

180

controlled for boost pressure tracking. Each PID controller is well calibrated for given engine
operating point. The controller gains are scheduled based on the engine speed and fuel injection
quantity. The major drawback of this control structure is that the VGT and EGR actions are not
controlled coordinately. This leads to non-optimal control of VGT and EGR systems. In certain
extreme cases, the one controller performance (e.g., EGR) needs to be compromised for the other
one (e.g., VGT). Because the boost pressure and EGR mass flow rate are strongly coupled since
they share the same exhaust flow used to drive the turbine and EGR flow.
In order to understand the behavior of current baseline controller, a simulation study is
carried out by integrating the baseline controller with the developed 1-D nonlinear diesel engine
air-path model in Simulink environment. Although the baseline controller is not well tuned for
this developed nonlinear engine model, the simulation results provide the benchmark
performance for the LQI controller.

Figure 101. Baseline controller for control development

Different engine operational conditions are studied and shown in Figure 102. The results
show that the baseline controller can track the target boost pressure and EGR mass flow rate for
the developed nonlinear engine model. However, during the aggressive tip-in and tip-out

181

operations, the EGR tracking is not satisfactory. One of the reasons behind this is that the VGT
and EGR control is not coordinated for both targets. This is the typical disadvantage of using
multiple single input single outputs (SISO) controllers for a multi-input and multi-output (MIMO)
system. The detailed tip-in and tip-out investigation results can be found in Figure 103 and
Figure 104.
For tip-in results shown in Figure 103, VGT vane position is closed to build up exhaust
pressure and extract more engine exhaust energy to turbocharger to have enough compressor
power so that fresh air can be charged into intake manifold. This action leads to the increased
boost pressure. Meanwhile, exhaust manifold pressure (𝑃3 ) increases rapidly because of flow
restriction caused by the reducing effective turbine area. Because dynamics of exhaust pressure
is much faster than that of boost pressure due to small exhaust manifold volume compared to that
of intake manifold. This leads to the increased pressure ratio (𝑃3 /𝑃2 ) ; and high-pressure ratio
(𝑃3 /𝑃2 ) results in closing EGR valve to reduce the EGR mass flow rate. The maximal pressure
difference across EGR valve is determined by the VGT vane position. When VGT vane closes,
the EGR flow rate increases; and meanwhile EGR valve opens to decrease EGR mass flow rate.
The combined effect makes the overshoot of EGR mass flow rate overshot due to different
dynamics of EGR and VGT systems. This results in unsatisfactory tracking results for EGR
mass flow rate during tip-in.
For the tip-out case shown in Figure 104, the VGT vane position is fully opened to reduce
boost pressure to target value. During this process, exhaust pressure drops much faster than boost
pressure, which leads to negative pressure difference across the EGR valve and the EGR valve is
fully closed to prevent the reversed EGR flow from intake to exhaust manifold. When boost
pressure reduced down to below the exhaust pressure, EGR valve opens to track the target EGR

182

mass flow rate, leading to reduced exhaust pressure again. Hence the fluctuations in both exhaust
pressure and EGR flow occur around 102 second. Fast exhaust pressure reduction leads to the
EGR valve losing during aggressive transient tip-out.
The current two SISO controllers cannot not well track EGR mass flow rate during transient
tip-in and tip-out operations, which provides the improvement opportunities for the proposed
coordinated control design. The main task for VGT and EGR control is to control the intake and
exhaust pressures coordinately to achieve the desired boost pressure and EGR mass flow rate.
Good boost pressure tracking leads to good engine transient response and combustion efficiency.
Good EGR mass flow rate tracking leads to low NOx emissions during transient operations. To
improve tracking performance, an MIMO controller needs to be designed for VGT and EGR.
1500
1450
Target boost pressure
Baseline controller

Pressure [hPa]

1400
1350
1300
1250
1200
1150
1100

10

20

30

40

50

60
Time [s]

70

80

90

100

110

100
Target EGR mass flow rate
Baseline controller

90
80

MFR [kg/s]

70
60
50
40
30
20
10
0

10

20

30

40

50

60
Time [s]

70

80

90

Figure 102. Simulation results with baseline controller

183

100

110

1000

pressure [hpa]

Pressure [hPa]

1400
1300
Target boost pressure
Baseline controller

1200
1100
89

90

91

92

93

P3-P2
500

0
89

94

90

91

Time [s]

MFR [kg/s]

80

EGR flow parameter=f(P /P )
3

2

200

90

91

92

93

0
89

94

90

EGR effectative area

Time [s]
80

VGT position

94

400

40
20
89

VGT position
70
60
50
89

90

91

92

93

94

EGR mass flow rate [kg/h]

Time [s]
20

EGR positon

93

600

Target EGR mass flow rate
Baseline controller

60

92
Time [s]

15
10
EGR valve position
5
89

90

91

92

93

94

2

x 10

91

92

93

94

Time [s]

-4

EGR effectative area=f(uegr)
1

0
89

90

91

92

93

94

Time [s]
100
EGR mass flow rate
50

0
89

90

91

Time [s]

92

93

94

Time [s]

Figure 103. Tip-in investigation for baseline controller
pressure [hpa]

100
Target boost pressure
Baseline controller

1300
1200
1100
98

100

102

104
Time [s]

106

108

3

2

100

102

104
Time [s]

106

108

110

120

40
100

20

3

100

102

104
Time [s]

106

108

110

80
60
VGT position

40
20
100

102

104
Time [s]

106

108

110

EGR mass flow rate [kg/h]

0
98

30
EGR valve position
20
10
0
98

EGR flow parameter=f(P /P )

80
98

EGR effectative area

MFR [kg/s]

P -P

-200

140
Target EGR mass flow rate
Baseline controller

60

0
98

VGT position

0
-100

-300
98

110

80

EGR positon

Pressure [hPa]

1400

100

102

104
Time [s]

106

108

110

2

x 10

100

102

104
Time [s]

100

102

104
Time [s]

-4

2

108

110

1

0
98

EGR effectative area=f(uegr)
106

108

110

100
EGR mass flow rate
50

0
98

100

102

104
Time [s]

Figure 104. Tip-out investigation for baseline controller

184

106

106

108

110

6.4.2 LQI controller design for different design target
For the proposed LQI design, matrices 𝑄 and 𝑅 are design parameters used to penalize the
outputs and the control inputs. In this case, weighting coefficients 𝑄11 and 𝑄22 take account for
the tracking errors of boost pressure and EGR mass flow rate. Meanwhile, R11 and R 22 are
weighting parameters for VGT vane and EGR valve positions. When 𝑅 ≫ 𝑄, the cost function
is dominated by the control effort 𝑈, and the controller minimizes the control action. When
𝑅 ≪ 𝑄 , the cost function is dominated by the output tracking errors, and there is almost no
penalties for using large control efforts. It is challenging to tune the weighting matrices to
achieve acceptable responses for all the performance outputs and keep the control inputs within
their actuation limits.
In order to tune matrices Q and R for different controller design target, three different
evaluation indexes are defined as Performance index, Emission Index, and Fuel Economy index
as below.
𝑡

𝑖
𝑖
𝑃𝑒𝑟𝑓𝑜𝑟𝑚𝑎𝑐𝑒𝑖𝑛𝑑𝑒𝑥 (𝑄11
, 𝑄22
)=1−

𝑡

𝑛
(∫𝑡 2|𝑃2𝑑 − 𝑃2 |) − 𝑚𝑖𝑛𝑖=1
((∫𝑡 2|𝑃2𝑑 − 𝑃2 |) )
1

𝑛
𝑚𝑎𝑥𝑖=1

1

𝑖

𝑡
((∫𝑡 2|𝑃2𝑑
1

− 𝑃2 |) ) −
𝑖

𝑡

𝑖
𝑖
𝐸𝑚𝑖𝑠𝑠𝑖𝑜𝑛𝑖𝑛𝑑𝑒𝑥 (𝑄11
, 𝑄22
)

=1−

𝑖

𝑡
((∫𝑡 2|𝑃2𝑑
1

(6.37)

− 𝑃2 |) )
𝑖

𝑡

𝑛
𝑑
𝑑
(∫𝑡 2|𝑚̇𝑒𝑔𝑟
− 𝑚̇𝑒𝑔𝑟 |) − 𝑚𝑖𝑛𝑖=1
[(∫𝑡 2|𝑚̇𝑒𝑔𝑟
− 𝑚̇𝑒𝑔𝑟 |) ]
1

𝑛
𝑚𝑎𝑥𝑖=1

1

𝑖

𝑡
𝑑
[(∫𝑡 2|𝑚̇𝑒𝑔𝑟
1

− 𝑚̇𝑒𝑔𝑟 |) ] −
𝑖

𝑡

𝐹𝑢𝑒𝑙

𝑛
𝑚𝑖𝑛𝑖=1

𝑖
𝑖
𝑒𝑐𝑜𝑛𝑜𝑚𝑦𝑖𝑛𝑑𝑒𝑥 (𝑄11
, 𝑄22
)

=

𝑛
𝑚𝑖𝑛𝑖=1

𝑖

𝑡
𝑑
[(∫𝑡 2|𝑚̇𝑒𝑔𝑟
1

− 𝑚̇𝑒𝑔𝑟 |) ]
𝑖

𝑡

𝑛
(∫𝑡 2|𝜏𝑒𝑛𝑔𝑖𝑛𝑒 |) − 𝑚𝑖𝑛𝑖=1
[(∫𝑡 2|𝜏𝑒𝑛𝑔𝑖𝑛𝑒 |) ]
1

𝑛
𝑚𝑎𝑥𝑖=1

(6.38)

1

𝑖

𝑡
[(∫𝑡 2|𝜏𝑒𝑛𝑔𝑖𝑛𝑒 |) ]
1
𝑖

−

𝑛
𝑚𝑖𝑛𝑖=1

𝑖
𝑡2
[(∫𝑡 |𝜏𝑒𝑛𝑔𝑖𝑛𝑒 |) ]
1
𝑖

(6.39)

𝑓𝑜𝑟 𝑖 = 1 … 𝑛

The Performance index is defined as normalized boost pressure tracking error. Boost
pressure tracking error is normalized to its maximum and minimum values of testing points for

185

different 𝑄11 and 𝑄22 combinations. The same normalized method is also applied to the
Emission index that is used to evaluate the EGR mass flow rate tracking error. Note that, the
index values vary from 0-1. For Performance and Emission indices, the highest value represents
the best tracking results (minimum tracking error). For fuel economy evaluation, the Fuel
Economy index is defined as the accumulated output engine torque. Since fuel injection quantity
and engine speed are given for the gain-scheduling controller and engine model, engine output
torque would be a good parameter to evaluate engine combustion efficiency and pumping loss.
For the Fuel Economy index, the highest value is for the best fuel efficiency. The higher index,
the better control performance.
For the control input weighting matrix R, the ratio between VGT vane and EGR valve
positions is chosen to be 10:1. The reason is that, slow VGT action leads slow exhaust pressure
dynamics; and selecting the weighting matrix in such a way gives more control authority for
EGR mass flow rate tracking.
10
𝑅=[
0

0
]
1

(6.40)

Weighting matrix for tracking boost pressure and EGR mass flow rate are tuned based on a
sweep study. This sweep study is used to achieve different controller performance as defined
previously in (6.37), (6.38), and (6.39). The range for the normalized block value of matrix Q is
shown in Table 20. The simulation studies under different Q matrix are based on a small step
perturbation around a base engine speed and fuel injection quantity. In this study, the small step
change (100rpm and 2mg/cc) is simulated around a light load engine operational condition with
20mg/cc fuel injection quantity at 800 rpm. In the simulation study, the target engine boost
pressure and EGR mass flow rate for each case are shown in Figure 105. The target tracking

186

values are generated using engine fuel injection quantity and engine speed as discussed
previously. Each designed linear controller is simulated based on the nonlinear plant.

23

950
Engine speed

Engine speed [RPM]

Fuel injection [kg/cc]

Engine fuel injection
22

21
20

19

50

55

60
Time [s]

65

70

900

850
800

750

1190

60
Time [s]

65

70

MFR [kg/s]

20

1170
1160

19
18

1150
Target boost pressure
1140

55

21

1180

Pressure [hPa]

50

50

55

60
Time [s]

65

Target EGR mass flow rate
70

17

50

55

60
Time [s]

65

70

Figure 105. A load step test profile for engine operating at 800 RPM
Table 20. Normalized range for Q matrix
Min value
Max value

𝑄11
1
10

𝑄22
3
10

With the defined performance indices in (6.37), (6.38) and (6.39) for controller design,
simulation results for different weighting matrix Q are shown in Figure 106. The results clearly
show the controller design trade-offs for different design targets. Note that 𝑄11 and 𝑄22 are
penalty parameters for both tracking errors of boost pressure and EGR flow rate, respectively.
With higher 𝑄11 , the designed controller emphasizes more on the EGR flow tracking error,
resulting a tight tracking error bound. The same works for 𝑄22 . Larger 𝑄22 leads to smaller boost
pressure tracking error than EGR mass flow rate. The different selection of 𝑄11 and 𝑄22 leads to
different tracking error performances for the boost pressure and EGR flow rate.

187

Performance index

Emission index
1

9

0.5
5

0.3

3
2
1

3

4

5

6

7

8

9

10

Q

22

(a)

0.7

7

0.6

6

6
0.5
5

0.4
4

0.8

0.6

Q11

Q11

5

0.9

8

0.7

7

1

9

0.8

0.6
6
0.5

10

0.9

8

0.7

7

1

9

0.8

8

Fuel economy index

10

0.9

Q11

10

0.4
4

0.2

3

0.1

2

0

1

0.3
0.2

3

4

5

6

7

8

9

10

0.4
4

0.3

3

0.1

2

0

1

0.2
0.1
3

4

5

6

7

8

9

10

0

Q

Q

22

22

(b)

(c)

Figure 106. Weighting selection for different controller design index for 800 rpm engine speed /
20 mg/cc fuel injection
Large weighting coefficients for both boost pressure and EGR mass flow rate errors leads to
reduced boost pressure tracking error as shown in Figure 106 (a). Increasing weighting on boost
pressure error (𝑄22 ) leads to improved boost pressure performance index. Since, smaller EGR
tracking error helps to increase exhaust pressure by improving controllability of EGR flow rate,
and hence, increasing weighting 𝑄11 on EGR mass flow rate error will also reduce the boost
pressure tracking bound. However, only large weighting Q11 leads to improved EGR tracking
(high Emission index) results as shown in Figure 106 (b). Sine improved EGR flow rate and
boost pressure tracking results lead to increased exhaust pressure; and engine output torque will
decrease with higher pumping loss. Reduced engine output torque leads to worse Fuel Economy
index as shown in Figure 106 (c).This also shows that the best boost pressure tracking, the EGR
tracking and fuel economy cannot be achieved at the same time. From the fuel efficiency
prospective, the best fuel efficiency controller design region is also opposite to the best
performance design region.
Figure 107 shows the trade-offs for the VGT-EGR system; and it also shows that the design
target for the performance, emission reduction, and fuel economy can be coordinated through
weighting selection. It needs to compromise other two targets to improve the other. This also

188

shows that with proper selection of weighting parameters, the controller can be designed for the
different targets. As shown in Figure 107, with coordinated weighting for 𝑄11 and 𝑄22 , it is
possible to achieve similar engine performance target (boost pressure tracking) with better
emission target (EGR mass flow tracking) without degrading fuel economy. Same for the other
two targets.
Trade off between performance, emission, fuel efficiency
10
Performance index
Emission index
Fuel efficiency index

0.8
0.6

9

0.2

8
7

0.6
0.6
0.8

Q11

6

0.6

5

0.4
0.8

4
3

0.4

0.4

0.6
0.4

0.4

0.2
0.2

0.2

0.8
0.2

2
1

0.6

0.6
0.2
0.6

3

4

5

6

0.4

0.8

7

8

9

10

Q

22

Figure 107. Trade-off between performance, emission and fuel efficiency for controller design
for 800 rpm engine speed / 20 mg/cc fuel injection

5
0

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure 108. Trade-off between performance, emission and fuel efficiency for controller design
for 800 rpm engine speed / 20 mg/cc fuel injection

189

Since in this study, we focus on boost pressure and EGR mass flow rate tracking, the
controller designed for the best performance ( 𝑄11 =10 and 𝑄22 =10) and for best emission ( 𝑄11
=10 and 𝑄22 =3) are simulated in the nonlinear engine plant. For simplicity, we call these two
controllers performance and emission controllers, respectively. The simulation results are
compared with the baseline controller shown in Figure 109, and it shows that the designed
MIMO controllers can achieve different target performances. For instance, for the performance
controller, it has faster boost pressure response and smallest boost tracking error under the
transient tip-in operation. Since more weights are put for the boost pressure tracking error, the
designed controller penalizes more on boost tracking error than EGR mass flow tracking error.
As shown in Figure 109, EGR valve opens for the performance controller to build up boost
pressure through EGR flow during the tip-in operation. Although the EGR tracking suffers for
the performance controller during transient tip-in, EGR mass flow rate can be well regulated
during tip-out, compared to the baseline controller. For the emission controller, the EGR flow
rate has the smallest tracking error with a larger weighting on EGR tracking error used for
controller design; and boost pressure response lags behind, compared with the other two
controllers. During the tip-out, only the designed MIMO controllers can handle the aggressive
exhaust pressure drop. The control actions for both MIMO controllers are to regulate both boost
pressure and EGR mass flow rate at the same time. For the baseline controller, as mentioned
previously, it cannot track EGR mass flow rate properly without coordinated control action of
both VGT and EGR control during transient tip-out operation. As shown in Figure 110, pressure
difference across EGR valve for the baseline controller drops down to zero, leading to no
controllability for EGR mass flow rate through the EGR valve.

190

1190
Target boost pressure
Designed controller for performance
Designed controller for emission
Baseline controller

Pressure [hPa]

1180
1170
1160
1150
1140
48

50

52

54

56

58
60
Time [s]

62

64

66

68

70

40
Target EGR mass flow rate
Designed controller for performance
Designed controller for emission
Baseline controller

MFR [kg/s]

30

20

10

0
48

50

52

54

56

58
60
Time [s]

62

64

66

68

70

70
Designed controller for performance
Designed controller for emission
Baseline controller

VGT position

65
60
55
50
45
48

50

52

54

56

58
60
Time [s]

62

64

66

68

70

12
Designed controller for performance
Designed controller for emission
Baseline controller

EGR positon

10

8

6

4
48

50

52

54

56

58
60
Time [s]

62

64

66

68

70

Figure 109. Comparison controller design with baseline controller.
20000
Designed controller for performance
Designed controller for emission
Production controller

Pressure hpa [pa]

15000

10000

5000

0

-5000
48

50

52

54

56

58

60

62

64

66

68

70

Time [s]

Figure 110. Pressure difference across EGR valve for three different controllers

191

6.4.3 Gain-scheduling for linear controllers
In order to implement the linear controller for the nonlinear engine plant, gain-scheduling
control approach is used in this study. With the same LQI control and weighting matrix selection
approach, two sets of controllers are designed for four different engine operating points. One set
is targeted for performance to minimize boost pressure tracking error; and the other is targeted
for emissions to minimize EGR mass flow tracking error. Meanwhile, both set of controllers take
account for the other tracking purpose. Designed controllers are shown in Table 21.
Table 21. Controller designs for different engine operation point

Controller
gain

Controller
design for
emission

Controller
design for
performance

Engine
Speed
[rpm]

Fuel
Injection
[mg/stroke]

-

-

800

20

[

1200

20

[

800

35

1.83 ∗ 10−5
[
3.30 ∗ 10−4

1200

35

800

20

1200

Proportional gain 𝐾𝑥
𝐾𝑥
[ 11
𝐾𝑥21

𝐾𝑥12
𝐾𝑥22

𝐾𝑥13 𝐾𝑥14
]
𝐾𝑥23 𝐾𝑥24

Integral gain 𝐾𝑦
𝐾𝑦
[ 11
𝐾𝑦21

2.19 ∗ 10−5
4.34 ∗ 10−4

5.25 ∗ 10−5
−2.43 ∗ 10−4

4.66 ∗ 10−3 6.6 ∗ 10−2 ]
−7.5 ∗ 10−3 1.60

[ 5.06 ∗ 10 4
11.06 ∗ 10

1.08 ∗ 10−5
2.96 ∗ 10−4

1.3881 ∗ 10−5
−1.09 ∗ 10−4

4.3 ∗ 10−3 8.07 ∗ 10−2 ]
−2.9 ∗ 10−3 3.13

[ 3.03 ∗ 10 4
11.06 ∗ 10

1.32 ∗ 10−5
[
2 ∗ 10−4

1.48 ∗ 10−5
−1.7154 ∗ 10−4

5.4 ∗ 10−3 6.4 ∗ 10−2 ]
−5.6 ∗ 10−3 1.65

𝐾𝑦12
]
𝐾𝑦22

3

3.49 ∗ 10−4 ]
−1.60 ∗ 10−4

3

3.52 ∗ 10−4 ]
−9.58 ∗ 10−5

3

[4.74 ∗ 104
1.10 ∗ 10

1.76 ∗ 10−5
−1 ∗ 10−4

4.61 ∗ 10−3 9.3 ∗ 10−2 ]
−3 ∗ 10−4 2.71

[ 4.13 ∗ 10 4
11.10 ∗ 10

3.44 ∗ 10−5
[
4.29 ∗ 10−4

7.9 ∗ 10−5
−2.7 ∗ 10−4

2.53 ∗ 10−2 6.54 ∗ 10−2 ]
−1.71 ∗ 10−2 1.60

[ 5.07 ∗ 10 4
11.10 ∗ 10

20

2.28 ∗ 10−5
[
2.92 ∗ 10−4

8.82 ∗ 10−5
−1.31 ∗ 10−4

2.19 ∗ 10−2 6.54 ∗ 10−2 ]
−0.8 ∗ 10−2 3.13

[ 3.11 ∗ 10 4
11.10 ∗ 10

800

35

3.37 ∗ 10−5
[
3.28 ∗ 10−4

9.58 ∗ 10−5
−2.08 ∗ 10−4

2.28 ∗ 10−2 6.32 ∗ 10−2 ]
−1.35 ∗ 10−2 1.65

[ 4.93 ∗ 10 4
11.10 ∗ 10

1200

35

2.61 ∗ 10−5
[
2.7 ∗ 10−4

1.08 ∗ 10−4
−1.4 ∗ 10−4

2.24 ∗ 10−2 9.25 ∗ 10−2 ]
−1.05 ∗ 10−2 2.71

[ 4.21 ∗ 10 4
11.10 ∗ 10

4

3

3

3

3

3.50 ∗ 10−4 ]
−1.49 ∗ 10−4
3.51 ∗ 10−4 ]
−1.30 ∗ 10−4
2.5 ∗ 10−3 ]
−1.1 ∗ 10−3
2.5 ∗ 10−3 ]
−6.95 ∗ 10−4
2.5 ∗ 10−3 ]
−1.1 ∗ 10−3
2.51 ∗ 10−3 ]
−9.41 ∗ 10−4

A bilinear interpolation is used to schedule controller gain inside the operating envelop. The
gain scheduling is based on engine speed and fuel injection quantity as shown in Figure 111.

192

Figure 111. Gain scheduling for local linear controllers

With designed controller gains for the four boundary points in this study, each controller
gain inside the operational range can be obtained as shown below:
𝑘 ∗ (𝑥 ∗ , 𝑦 ∗ ) ≈

1
𝑘
[𝑥 − 𝑥 ∗ 𝑥 ∗ − 𝑥1 ] [ 1
𝑘3
(𝑥2 − 𝑥1 )(𝑦2 − 𝑦1 ) 2

𝑘2 𝑦2 − 𝑦 ∗
][
]
𝑘4 𝑦 ∗ − 𝑦1

(6.41)

where, 𝑥 is the engine speed (𝑁𝑒 ; 𝑦 is the fuel injection (𝑚̇𝑓𝑢𝑒𝑙 ), and 𝑘 is the controller gain.
The final control law for the nonlinear plant is:
𝑡

𝑢1 (𝑡) = 𝑢̅1 − 𝐾𝑥(𝑁𝑒 ,𝑚̇𝑓𝑢𝑒𝑙) (𝑥1 (𝑡) − 𝑥̅1 ) − 𝐾𝑦(𝑁𝑒,𝑚̇𝑓𝑢𝑒𝑙) ∫ (𝛿𝑦1 (𝑡)) 𝑑𝜏

(6.42)

0

The gain-scheduling for each controller can be found in Figure 113, Figure 114, Figure 115,
Figure 116. The gain 𝐾𝑥12 is the proportional gain for the boost pressure error used for the VGT
control. As shown in Figure 113 and Figure 115, the control gain 𝐾𝑥12 (from boost pressure to
VGT control) for performance controller is higher than the emission controller. The gain 𝐾𝑥12
increases as engine load increases in terms of engine speed and fuel injection. Meanwhile, the
gain 𝐾𝑥22 is negative, which is boost pressure error proportional gain for EGR valve control.

193

Negative 𝐾𝑥22 gain means in order to have higher boost pressure, EGR value needs to be closed
to build up exhaust pressure. Since large weighting for the boost pressure tracking are used
during controller design to improve performance, the magnitude of 𝐾𝑥22 for boost pressure
tracking is larger than 𝐾𝑥22 for emission controller. On the exhaust pressure side, the gain 𝐾𝑥11
is the proportional gain for VGT to regulate exhaust pressure error. For the performance
controller, 𝐾𝑥11 , 𝐾𝑥12 and 𝐾𝑥13 are much larger than that of emission controller to achieve
better boost pressure tracking. For the emission controller, the magnitude of 𝐾𝑥22 is smaller
than that of performance controller, leading to less effort from EGR valve to boost pressure error.
All the calculated gains distinguish the two different controller designs for different targets.
Meanwhile, VGT and EGR actions are coordinated.

Figure 112. Controller based on gain scheduling

194

35

1.8
1.6

30

1.4
1.2
1000
1200
Engine speed [rpm]
Kx
13

35

x 10

0.8

4.6
4.4

0.08
30
0.075
0.07

11

35

Fuel injection [mg/cc]

Fuel injection [mg/cc]

0.065
1000
1200
Engine speed [rpm]
Ky
12

5000
4500
4000
3500

25
800

0.09
0.085

25
800

1000
1200
Engine speed [rpm]
Ky

30

0.6
1000
1200
Engine speed [rpm]
Kx
14

4.8

35

1

35

5

25
800

-5

1.2
30

-3

5.2

30

x 10
1.6
1.4

25
800

Fuel injection [mg/cc]

Fuel injection [mg/cc]

Kx12

-5

2

25
800

Fuel injection [mg/cc]

x 10

Fuel injection [mg/cc]

Kx11
35

-4

3.52
3.515
30

3.51
3.505

25
800

1000
1200
Engine speed [rpm]

x 10

3.5
1000
1200
Engine speed [rpm]

Figure 113. Gain-scheduling for VGT control (target for emission)

3

23

x 10
4

-2
-4
-6
1000
1200
Engine speed [rpm]
Ky
21

35

-2.4
1000
1200
Engine speed [rpm]
Kx
3

2.5
30
2

5

1000
1200
Engine speed [rpm]
Ky
22

x 10
1.113

35

1.111
1.11
1.109
1.108
25
800

-2.2

25
800

1.112

30

-2

24

0

25
800

-1.8

35

2

30

-4

-1.6
30

-3

Fuel injection [mg/cc]

1000
1200
Engine speed [rpm]
Kx

x 10

-1.4

25
800

Fuel injection [mg/cc]

Fuel injection [mg/cc]

35

3.5

30

35

Fuel injection [mg/cc]

Kx22

-4

4

25
800

Fuel injection [mg/cc]

x 10

Fuel injection [mg/cc]

Kx21
35

1.107

-4

-1.2
30
-1.4

25
800

1000
1200
Engine speed [rpm]

x 10
-1

1000
1200
Engine speed [rpm]

-1.6

Figure 114. Gain-scheduling for EGR controller (target for emission)

195

Fuel injection [mg/cc]

x 10
3.4

Kx12

-5

35

Fuel injection [mg/cc]

Kx11
35

3.2
3
30

2.8
2.6
2.4

25
800

10
9.5
30

13

Fuel injection [mg/cc]

30
0.023

0.022

5000

4000
3500

25
800

0.08
30

0.075
0.07
0.065
1000
1200
Engine speed [rpm]
Ky12

35

4500
30

0.09
0.085

25
800

1000
1200
Engine speed [rpm]
Ky11

35

Fuel injection [mg/cc]

35

0.024

25
800

8
1000
1200
Engine speed [rpm]
Kx
14

0.025

Fuel injection [mg/cc]

Fuel injection [mg/cc]

35

9
8.5

25
800

1000
1200
Engine speed [rpm]
Kx

-5

x 10
10.5

2.485
30
2.48

25
800

1000
1200
Engine speed [rpm]

-3

x 10
2.49

2.475
1000
1200
Engine speed [rpm]

Figure 115. Gain-scheduling for VGT (target for performance)

3

0.01
30

0
-0.01

25
800

1000
1200
Engine speed [rpm]
Ky21

35

-2.6
1000
1200
Engine speed [rpm]
Kx24
3

2.5
30
2

1000
1200
Engine speed [rpm]
Ky22

35

1.111
1.11
1.109
1.108
25
800

-2.4

5

1.112

30

-2
-2.2

25
800
x 10
1.113

1.107

x 10
-7

-4

-8
-9

30

-10
25
800

1000
1200
Engine speed [rpm]

-4

-1.8
30

35

Fuel injection [mg/cc]

0.02

x 10

-1.6

25
800

1000
1200
Engine speed [rpm]
Kx23

Fuel injection [mg/cc]

Fuel injection [mg/cc]

35

3.5

30

35

Fuel injection [mg/cc]

Kx22

-4

4

25
800

Fuel injection [mg/cc]

x 10

Fuel injection [mg/cc]

Kx21
35

-11
1000
1200
Engine speed [rpm]

Figure 116. Gain-scheduling for EGR (target for performance)

196

A transient profile shown in Error! Reference source not found. within the envelop of light
load operations is simulated for the three sets of controller: performance MIMO controller,
emission MIMO controller and baseline controller (two SISO controllers).

Both MIMO

controllers are gain-scheduled based on engine speed and fuel injection quantity. The three
different control algorithms, evaluated with the same set-point strategy, show comparable
performance, emissions and fuel economy.

35
Gain scheduling route

Fuel injection [mg/cc]

Tip -in
30

25
Tip -out

20
800

850

900

950
1000
1050
Engine speed [rpm]

1100

1150

1200

Figure 117. Gain scheduling route for controller validation.

The simulation results are shown in Figure 118 and Figure 119. The simulation results of the
gain-scheduling controller are shown in Figure 120.

Performance controller shows more

aggressive boost pressure tracking during tip-in and tip-out operations, compared to both
emission and baseline controllers, and also has good EGR rate tracking performance. For the

197

emission controller, it has best EGR tracking performance among the three controllers. However,
the transient response of boost pressure is compromised by reduced aggressiveness of the VGT
vane control. As shown in Figure 120, the proportional gain of the performance controller for
VGT vane position is higher than that of the emission controller. Also EGR control gain from
TC speed channel 𝐾𝑥23 is higher for the performance controller than that of the emission
controller. This leads to higher VGT and higher EGR control gains for the performance
controller. Both proportional and integral gains from boost pressure channel (𝐾𝑥22 and 𝐾𝑦22 ) are
negative for EGR valve control, which means the EGR valve is intended to be closed for
reducing boost pressure error. However, the EGR valve position is still driven by associated
tracking error and the VGT and EGR control is coordinated. The highest proportional gain for
both EGR and VGT control is from TC speed, which dominates the closed-loop response. This
perhaps could provide new thoughts for future gain-scheduling control design based on only TC
speed. As engine load gets increased, VGT gain decreases and EGR control gain increases.
Compared to the baseline controller, the two MIMO controllers is able to track both boost
pressure and EGR mass flow.

198

50
1450
45
40

1350

MFR [kg/h]

Pressure [hPa]

1400

1300
1250

1150
48

50

52

54

56

58
Time [s]

60

62

30
25

Target boost pressure
Designed controller for performance
Designed controller for emission
Baseline controller

1200

35

Target EGR mass flow rate
Designed controller for performance
Designed controller for emission
Baseline controller

20

64

66

15
48

68

75

50

54

56

58
Time [s]

60

62

64

66

68

62

64

66

68

13
Designed controller for performance
Designed controller for emission
Baseline controller

70

Designed controller for performance
Designed controller for emission
Baseline controller

12
11

EGR positon

65

VGT position

52

60
55

10
9
8
7

50
6
45
40
48

5
50

52

54

56

58
Time [s]

60

62

64

66

4
48

68

50

52

54

56

58
Time [s]

60

Figure 118. Gain scheduling controller validation with baseline controller
As shown in Figure 119, different controller leads to the different pressure difference across
EGR valve. Due to the aggressive VGT action of the performance controller, pressure difference
between P3 and P2 is larger than the two other controllers, which results in different pumping
losses. As shown in Figure 121, by comparing the boost pressure tracking, EGR mass flow rate
tracking, and output torque, the performance controller has better boost tracking and EGR
tracking than baseline controller. But it has lower output torque due to increased pumping loss.
Emission controller has the smallest EGR tracking error with the best engine output torque. But
the boost pressure tracking for the emission controller is the worst among the three cases. This
agrees with the controller design trade-off shown in Figure 107 that the three design targets
cannot be optimized at the same time.

199

2

x 10

5

Designed VGT-EGR control for performance
Designed VGT-EGR control emission
Baseline

Pressure difference across EGR valve [pa]

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
48

50

52

54

56

58
Time [s]

60

62

64

66

68

Figure 119. Pressure difference across EGR valve
The simulation results confirm the proposed gain-scheduling MIMO controller for the VGTEGR system, and also demonstrate that the EGR-VGT control can be designed for different
targets by selecting different weighting parameters. Comparing with the baseline controller, the
designed performance controller has better tracking results with respect to both boost pressure
and EGR mass flow rate. Furthermore, the proposed controller design method introduces
flexibility for multi-targets closed-loop control design for the VGT-EGR system.

200

-5

1.2

Kx12

0.03

0.12

0.025

0.11

0.8

0.02

0.6

0.09

0.4

0.01

0.08

0

0.2

0.005

0.07

-1

0

0

55

60
Time [s]

65

6000

3

5000

2.5

60
Time [s]

65

-3

6

1.5
1

55

60
Time [s]

0

65

0.08

x 10

55

60
Time [s]

0.06

65

-4

2

50

55

60
Time [s]

5

1.12

x 10

55

60
Time [s]

65

55

60
Time [s]

65

55

60
Time [s]

65

-4

0

3

1

65

50

1

4

-1
-2

2

0.5
50

50

5

Kx21

Ky12

3000

-3
50

x 10

55

60
Time [s]

-4

65

5

0

50

x 10

-3

-0.2

0.06

4

Kx24

0.02

3

-0.4

Ky22

1.115

0.04

-0.6

1.11

-0.8

1.105

-1.2

2

0

-1
55

60
Time [s]

65

1

50

55

60
Time [s]

65

50

55

60
Time [s]

65

50

Figure 120. Gain-scheduling for VGT-EGR
Accumulated boost tracking error

50

6

x 10

4

5
4
3
2
1
0

Accumulated EGR mass flow rate tracking error

-0.02

x 10

55

2

4000

1000

50

Kx22

50

2000

Kx23

0.1

0.015

Ky21

Kx11
Ky11

1

Designed controller for performance
Designed controller for emission

-4

1

3
2

x 10

Kx14

x 10

Kx13

4

Performance controller

Emission controller

Baseline controller

Performance controller

Emission controller

Baseline controller

2000

1500

1000

500

0

Figure 121. Benchmarking with baseline controller

201

6.4.4 Extended controller design for assisted and regenerative turbocharger
The controller design scheme for the VGT-EGR system can be easily extended to the
regenerative hydraulic assisted turbocharger system shown in Figure 122. In this study, The
hydraulic power is only treated as control inputs for the diesel engine air-path system. The two
actuators are treated as one control input 𝑢𝑟ℎ𝑎𝑡 , where positive input of 𝑢𝑟ℎ𝑎𝑡 is for hydraulic
pump power and negative power is for hydraulic turbine power. The hydraulic valve is assumed
to have fast response time with neglecting actuator dynamics. Hydraulic energy in the hydraulic
tank is not considered during the control design. The hydraulic system is only activated during
transient operations to reduce the boost pressure and EGR mass flow rate errors. By analyzing
both nonlinear and linearized plants for the hydraulic assisted and regenerative turbocharger, the
difference between EGR-VGT and EGR-VGT-RHAT appears in the control input matrix 𝐵̃ .
Since hydraulic power is a direct input to the TC shaft in the speed dynamic equation (6.34), 𝐵̃
matrix can be reformatted into 𝐵̃ ∗ by adding column 𝐵32 to take account of the assisted and
regenerative power as shown in (6.35).

202

Figure 122. Regenerative hydraulic assisted turbocharged diesel engine
𝐽𝜔𝜔̇ = 𝑊̇𝑇 − 𝑊̇𝐶 − 𝑊̇𝐿𝑜𝑠𝑠 + 𝑢𝑟ℎ𝑎𝑡
𝐵11
0
𝐵
∗
𝐵̃ = 31
0
0
[ 0

(6.43)

0 0
0 0
0 𝐵32
𝐵𝑑 0
0 0
0 0 ]

(6.44)

Table 22. Control input value range and physical interpretation

EGR valve (𝑢𝑒𝑔𝑟 )
VGT vane position (𝑢𝑣𝑔𝑡 )
RHAT power (𝑢𝑟ℎ𝑎𝑡 )

Minimum value

Maximum value

Note

0%
0%
-25kW

100%
100%
25kW

Valve control for EGR valve position
VGT vane position
Hydraulic system power range

As long as the operational space 𝜓 = {(𝑃1 , 𝑃2 , 𝑃3 , 𝑃4 ), 𝑃3 > 𝑃4 , 𝑃2 > 𝑃1 , 𝑃3 > 𝑃2 )} is
guaranteed for engine air path system, pairs (<𝐴̃, 𝐵̃ ∗> and <𝐴̃, √𝑄̃ > ) are fully controllable and
observable for the VGT-EGR-RHAT system. The same gain-scheduling LQI approaches for the
VGT-EGR system is used to design coordinated controller for the VGT-EGR-RHAT system
shown in Figure 123. Note that there is no set-point for RHAT input and the RHAT control

203

is 𝑢𝑟ℎ𝑎𝑡 = 𝛿𝑢𝑟ℎ𝑎𝑡 . The final controller for the VGT-EGR-RHAT system is shown in (6.45). The
RHAT control gain appears in the third row of control gain matrix.
𝑡

𝑢(𝑡) = 𝑢̅1 − 𝐾𝑥(𝑁𝑒,𝑚̇𝑓𝑢𝑒𝑙 ) (𝑥1 (𝑡) − 𝑥̅1 ) − 𝐾𝑦(𝑁𝑒,𝑚̇𝑓𝑢𝑒𝑙) ∫ (𝛿𝑦1 (𝑡)) 𝑑𝜏

(6.45)

0

𝐾𝑥11
where 𝐾𝑥 = [𝐾𝑥21
𝐾𝑥31

𝐾𝑥12
𝐾𝑥22
𝐾𝑥32

𝐾𝑦11
𝐾𝑥13 𝐾𝑥14
𝐾𝑥23 𝐾𝑥24 ] and 𝐾𝑦 = [𝐾𝑦21
𝐾𝑦31
𝐾𝑥33 𝐾𝑥34

𝐾𝑦12
𝐾𝑦22 ]
𝐾𝑦32

Figure 123. Controller design for VGT-EGR-RHAT system
With the assisted and regenerative power on TC shaft, extra energy is used to drive the
compressor to achieve faster boost response. Based on previous analysis, assisted power on the
TC shaft increases the bandwidth for boost pressure control, comparing with the traditional VGT,
and the VGT vane position does not need to be closed tightly to build up high exhaust pressure
for turbine power extraction. VGT can be used for both boost pressure control and EGR mass
flow regulation. With three actuators, the VGT-EGR-RHAT system would have much better
control for diesel air-path system. First control design trade-offs are investigated. Performance
index, Emission index, and Fuel Economy index for the VGT-EGR system are extended to the

204

VGT-EGR-RHAT system. For the RHAT system, since assisted and regenerative power uses the
external energy; additional RHAT energy index is defined for assisted and regenerative power in
(6.46). The energy index is used to evaluate the RHAT control action. The higher energy index,
the less hydraulic actuation energy is used. Frequent hydraulic actuation leads to low energy cost
index.
𝑡

𝑅𝐻𝐴𝑇

𝑖
𝑖
𝑒𝑛𝑒𝑟𝑔𝑦𝑖𝑛𝑑𝑒𝑥 (𝑄11
, 𝑄22
)

=1−

𝑡

𝑛
(∫𝑡 2|𝑢𝑟ℎ𝑎𝑡 |) − 𝑚𝑖𝑛𝑖=1
[(∫𝑡 2|𝑢𝑟ℎ𝑎𝑡 |) ]
1

1

𝑖

𝑡
𝑛
𝑚𝑎𝑥𝑖=1
[(∫𝑡 2|𝑢𝑟ℎ𝑎𝑡 |) ]
1
𝑖

−

𝑛
𝑚𝑖𝑛𝑖=1

𝑖
𝑡2
[(∫𝑡 |𝑢𝑟ℎ𝑎𝑡 |) ]
1
𝑖

𝑓𝑜𝑟 𝑖 = 1 … 𝑛

Figure 124. Normalized indexes with normalized Q matrix for controller design

205

(6.46)

The different indexes results for different combinations of weighting matrix are obtained
from the same load step simulations for the same engine operated at 800RPM shown in Figure
105. As shown in Figure 124, compared with the VGT-EGR case, Performance index for the
VGT-EGR-RHAT system is mainly dependent on the boost pressure weighting 𝑄22 . This is
because the external power can be used to control boost pressure instead of using VGT. The
EGR weighting 𝑄11 has less influence on boost pressure tracking, compared to the VGT-EGR
case. With larger 𝑄11, EGR tracking error reduction is expected. Higher exhaust pressure leads to
high pumping losses and decreases the engine fuel efficiency. For the Fuel eEonomy index, since
the boost control can be independent on the exhaust pressure with the RHAT system,
improvement on the boost pressure tracking has less effect on EGR mass tracking. Hence, with
the RHAT system, high energy usage (low RHAT energy index) leads to high Fuel Economy
index as shown in Figure 124. Based on the analysis, fuel economy improvement is expected
with extra assisted and regenerative power on TC shaft, assuming the hydraulic energy is mainly
from the driving shaft during braking.
In this study, we pay more attention to the hydraulic actuation energy; hence two sets of
controllers for the VGT-EGR-RHAT system are designed within the same operational range as
the previous VGT-EGR case. The two sets of controllers are based on high and low energy
indices. High energy index controller uses relative low RHAT energy for both hydraulic turbine
and pump with weighting matrix (𝑄11 = 2 , 𝑄22 = 2). Low energy index controller uses relative
high RHAT energy with (𝑄11 = 10 , 𝑄22 = 10). The same gain-scheduling approaches for the
VGT-EGR system are used for the nonlinear engine plant with the same operating range as the
VGT-EGR case. The simulation results, compared with the corresponding VGT-EGR cases, are
shown in Figure 125, Figure 126, and Figure 127.

206

For the low energy index controller (high assisted and regenerative power), RHAT provides
aggressive assistant during the initial tip-in operations as shown in Figure 126. Turbine mass
flow rate increases with the increased TC speed, leading to reduced exhaust pressure. This leads
to the aggressive VGT closing action to build up exhaust pressure; meanwhile, EGR valve closes
to keep EGR mass flow rate error small. Next, exhaust manifold pressure increases due to the
increased exhaust mass flow. Since less energy is needed from the turbo side for the low energy
index case, then VGT vane opens to reduce the exhaust pressure. With the increased boost and
exhaust pressures, both VGT vane and EGR valve open, leading to a constant pressure drops
between exhaust and intake pressures shown in Figure 127. This small pressure different across
EGR valve keeps EGR mass flow through EGR valve with reduced pumping loss. Hence, the
low energy index controller has the best fuel efficiency due to reduced pumping loss. With less
assisted power (high energy index), VGT action is still trying to build up exhaust pressure to
drive the compressor for improving boost pressure tracking performance. With the increased
exhaust pressure, EGR valve is closing to reduce the effect of higher exhaust pressure to keep the
right amount of EGR mass flow rate during the tip-in operations. The coordinated control for the
VGT-EGR-RHAT system improves engine tracking performances for both boost pressure and
EGR mass flow rate.
The benefit of hydraulic assisted and regenerative turbocharger is obvious as shown in Figure
127 and Figure 129. The pressure difference between the exhaust and intake manifolds reduces
with the RHAT system, leading to reduced pumping loss. Meanwhile, boost pressure and EGR
mass flow rate tracking performance can be well maintained through the interaction among VGT
vane and EGR valve positions, and RHAT turbine and pump power. As shown in Figure 129 and
Table 23, RHAT system can reduce boost pressure tracking error, EGR mass flow rate tracking

207

error, and at the same time improve engine output torque, compared to baseline VGT-EGR
system. By comparing with traditional VGT-EGR case, the performance, emissions, and fuel
economy can be achieved at the same time with external assisted and regenerative power on TC
shaft. However, the cost for these benefits is hydraulic energy.
In summary, the proposed control design approach is extended to the VGT-EGR-RHAT
system successfully with promising results.
50

1450

45

1400

40

MFR [kg/h]

1300
1250
1200
Target boost pressure
VGT-EGR performance
VGT-EGR emission
Baseline
VGT-EGR-RHAT with high energy usage
VGT-EGR-RHAT with low energy usage

1150
1100
1050
1000
48

50

52

54

56

58
Time [s]

60

62

35
30

Target EGR mass flow rate
VGT-EGR performance
VGT-EGR emission
Baseline
VGT-EGR-RHAT with high energy usage
VGT-EGR-RHAT with low energy usage

25
20

64

66

15
48

68

75

50

52

54

56

58
Time [s]

60

62

64

66

68

62

64

66

68

14
VGT-EGR performance
VGT-EGR emission
Baseline
VGT-EGR-RHAT with high energy usage
VGT-EGR-RHAT with low energy usage

65

VGT-EGR performance
VGT-EGR emission
Baseline
VGT-EGR-RHAT with high energy usage
VGT-EGR-RHAT with low energy usage

13
12
11

EGR positon

70

60
55

10
9
8
7

50

6
45
40
48

5
50

52

54

56

58
Time [s]

60

62

64

66

4
48

68

50

52

54

56

58
Time [s]

60

Figure 125. Simulation results for different controller design
1.5
VGT-EGR-RHAT with high energy usage
VGT-EGR-RHAT with low energy usage
1

Hydraulic actuation power [kW]

VGT position

Pressure [hPa]

1350

0.5

0

-0.5

-1

-1.5
48

50

52

54

56

58
Time [s]

60

62

64

66

68

Figure 126. Hydraulic actuation inputs for VGT-EGR-RHAT

208

Pressure difference across EGR valve [pa]

2

x 10

5

VGT-EGR performance
VGT-EGR emission
Baseline
VGT-EGR-RHAT with high energy usage
VGT-EGR-RHAT with low energy usage

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
48

50

52

54

56

58
Time [s]

60

62

64

66

68

Figure 127. Pressure difference across EGR valve

209

13
Target EGR mass flow rate fraction
VGT-EGR performance
VGT-EGR emission
Baseline
VGT-EGR-RHAT with low energy index
VGT-EGR-RHAT with high energy index

12.5
12

EGR percentage [%]

11.5
11
10.5
10
9.5
9
8.5
8
48

50

52

54

56

58
Time [s]

60

62

64

66

68

62

64

66

68

62

64

66

68

50
45

MFR [kg/h]

40
Target EGR mass flow rate
VGT-EGR performance
VGT-EGR emission
Baseline
VGT-EGR-RHAT with low energy index
VGT-EGR-RHAT with high energy index

35

30
25

20

15
48

50

52

54

56

58
Time [s]

60

14
VGT-EGR performance
VGT-EGR emission
Baseline
VGT-EGR-RHAT with low energy index
VGT-EGR-RHAT with high energy index

13
12

EGR positon

11
10
9
8
7
6
5
4
48

50

52

54

56

58
Time [s]

60

Figure 128. EGR mass fraction, EGR mass flow rate and EGR valve position for different
control design

210

Accumulated EGR mass flow rate tracking error

Accumulated boost tracking error

600
500
400
300
200
100

Performance

Emission

Baseline

RHAT low

RHAT high

18
16
14
12
10
8
6
4
2
0

4500

5

4400

4.5

Energy of hydraulic actuation [kJ]

Accumulated engine torque

0

20

4300
4200
4100
4000
3900
3800
3700
3600
3500

Performance

Emission

Baseline

RHAT low

RHAT high

Performance

Emission

Baseline

RHAT low

RHAT high

4
3.5
3
2.5
2
1.5
1
0.5

Performance

Emission

Baseline

RHAT low

0

RHAT high

Figure 129. Benchmarking with baseline controller

Figure 130. Comparison of different control designs

211

Table 23. Benchmarking with baseline controller

Baseline controller
VGT-EGR controller designed for performance
VGT-EGR controller designed for emission
VGT-EGR-RHAT controller with low energy
index
VGT-EGR-RHAT controller with high energy
index

Boost
pressure
tracking
improvement
0
14.6%
-92.9%

EGR mass
flow rate
tracking
improvement
0
44.1%
92.9%

Accumulated
engine
power
improvement
0
-0.48%
0.89%

Accumulated
hydraulic
actuation
energy usage
0
0
0

93.5%

29.4%

5.68%

4.6kJ

49.3%

31.3%

2.8%

2.8kJ

6.5 Conclusion
In this chapter, a systematic control design approach for a diesel engine air-path subsystem
with an assisted and regenerative turbocharger is proposed. Linear quadratic integral (LQI)
controllers are designed based on the linearized models over the gridded engine operational
conditions for tracking both boost pressure and EGR flow rate, and gain-scheduling controller
for a given operational condition is obtained by interpreting the LQI controllers around the
operational conditions. The gain-scheduled control strategy is then used for evaluation using the
nonlinear air-path engine system. Comparing to the baseline VGT-EGR controller, this approach
provides a way to design the VGT and EGR controller with trade-off between boost pressure and
EGR flow rate trade-offs by tuning the LQI weighting. Comparing with the dual-loop baseline
single-input and single-output controller, the designed multi-input and multi-output controllers
show improved tracking performance for both boost pressure and EGR mass flow rate, and nice
trade-off characteristics between engine performance and emissions through weighting selection.
With the added regenerative hydraulic assisted turbine system, the VGT-EGR-RHAT controller
further improves the transient engine performance without compromising EGR tracking
performance due to additional power available on the turbocharger shaft.

212

CHAPTER 7. CONCLUSIONS AND FUTURE WORK
7.1 Conclusions
A physics-based turbine power model of a variable geometry turbocharger (VGT) is
proposed along with its thrust friction model. The turbine power model is derived based on the
Euler turbine equations with the VGT vane position as the control input. A generalized method
for identifying mechanical friction is also proposed. The proposed model has better accuracy
than the traditional map-based model. The proposed the turbine power and its mechanical
efficiency models are suitable for the model-based VGT control due to the analytic nature of the
proposed models as a function of the VGT vane angle.

A compressor power model, based on the Euler turbomachinery equations with realistic
assumptions, was developed. Two new performance coefficients, the power and speed
coefficients (𝐶𝑝𝑜𝑤𝑒𝑟 and 𝐶𝑠𝑝𝑒𝑒𝑑 ), were proposed as an alternative to multiple performance maps.
The proposed correlation between 𝐶𝑝𝑜𝑤𝑒𝑟 and 𝐶𝑠𝑝𝑒𝑒𝑑 is especially useful to determine the
compressor power necessary for a given compressor mass flow rate. This compressor power
demand can then be translated into a VGT (Variable Geometry Turbo) vane position or an assist
demand for an assisted boost systems. This relationship can also be easily used to compare
compressor design variants with respect to performance and range. The reduced-order and
reduced-complexity model is especially useful for the control applications. Developed
compressor model is further extended to centrifugal hydraulic pump and hydraulic turbine
system and show promising results.

213

A systematic approach for diesel engine air-path system modelling with regenerative
hydraulic assisted turbocharger was developed. The model (simulator) is developed by
integrating engine system, variable geometry turbocharger system, and hydraulic system. Engine
air-path model is validated through engine transient test data. It shows that the proposed
modelling approach has adequate accuracy with only three states for the engine air-path system.
It could be used for model-based analysis and control design. The interactions between the VGT,
EGR, and hydraulic actuators has been illustrated through validation simulations with the
nonlinear engine model with good agreement.
Utilizing the combined efforts from the turbo and driveline pumps, used to recover both
exhaust and vehicle braking energies, to offset the fuel economy penalty as a result of hydraulic
power applied to the turbo shaft for improving turbocharger transient response is a novel idea.
The 1-D GT-Power analysis indicates that it is possible to gain 3-5% fuel economy improvement
with the RHAT system, compared with the baseline turbocharged diesel engine, over the FTP 75
transient cycle. This FE improvement does not include the other FE benefits that may be enabled
by the RHAT technology such as engine downsizing, transmission optimization, etc.
A systematic approach for diesel engine air-path controller design is proposed. Gainscheduling control is designed based on linear quadratic regulator with integral action. Gainscheduling controller is implemented for the nonlinear VGT-EGR engine system. Comparing
with the traditional VGT- EGR controller design, this approach provides a novel method to
design the controller for different design targets by selecting different weighting matrices. By
benchmarking with the baseline controller, the designed gain-scheduling multi-input and multioutput (MIMO) controller shows improved tracking results for both boost pressure and EGR
mass flow rate, and it also provides design flexibility for the trade-offs between engine

214

performance and emissions. Further, this control design method is extended to the engine with an
assisted and regenerative turbocharger. The simulation results show improved performance and
fuel economy with reduced emissions at the same time with the additional hydraulic energy for
the VGT-EGR-RHAT system. Note that the additional hydraulic energy can be recovered from
driveline during vehicle braking.
7.2 Future work
1. Model turbocharger heat transfer effect to improve turbocharger system model accuracy.
2. Study the effect of operating the turbocharger outside its traditional region due to the
utilization of assist or regeneration power. When a diesel engine is equipped with both low
pressure EGR and high pressure EGR, the gas turbine could work as a compressor to have
large amount of low pressure EGR mass; and compressor might work as a turbine when
compressor upstream pressure is higher than the downstream one. Both lead to distinct
physical characteristics, compared to traditional turbocharger. Both experimental and
modelling approaches are needed to study the two new operation modes.
3. Validate the designed controller experimentally. The prototype hardware needs to be
developed to assess this state of art technology.

215

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