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MODELING OF TURBOCHARGED SPARK IGNITED ENGINE AND MODELPREDICTIVE CONTROL OF HYBRID TURBOCHARGER
By
KANG RONG
A THESIS PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCE
UNIVERSITY OF FLORIDA
2014
c⃝ 2014 Kang Rong
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Special thanks to everyone that helped!
3
ACKNOWLEDGMENTS
I am very grateful to my advisor Dr. Carl Crane for letting join the CIMAR group
and providing me with such a precious opportunity to work on this excellent project.
Special thanks to Olugbenga Moses Anubi and Darsan Patel for their wonderful
instruction, selfless help and great support to me on this research. This work could
not be completed without your help.
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TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1 Introduction of Conventional Engine Charging Method . . . . . . . . . . . 121.2 Introduction of Hybrid Turbocharger . . . . . . . . . . . . . . . . . . . . . 13
1.2.1 How Hybrid Turbocharger Works . . . . . . . . . . . . . . . . . . . 131.2.2 Why Use Hybrid Turbocharger . . . . . . . . . . . . . . . . . . . . . 14
1.3 Problem Formulation and Thesis Outline . . . . . . . . . . . . . . . . . . . 15
2 BATTERY MODELING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Battery Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Battery Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 Mathematical Equations of Battery Discharging and Charging . . . 182.2.1.1 State of Charge . . . . . . . . . . . . . . . . . . . . . . . 182.2.1.2 Discharging Mode . . . . . . . . . . . . . . . . . . . . . . 182.2.1.3 Charging Mode . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Simulink Battery Modeling and Validation . . . . . . . . . . . . . . . . . . 192.3.1 Simulink Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.2 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 TURBOCHARGED SI ENGINE MODELING . . . . . . . . . . . . . . . . . . . . 22
3.1 Model Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1.1 Model Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1.2 Model States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.3 Model Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Compressor Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.1 Pressure Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.2 Temperature Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.3 Mass Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.4 Efficiency Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Intercooler Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3.1 Pressure Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.2 Mass Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
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3.3.3 Temperature Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.4 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Throttle Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4.1 Mass Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4.2 Throttle Pressure Model . . . . . . . . . . . . . . . . . . . . . . . . 313.4.3 Temperature Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5 Intake Manifold Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.6 Combustion Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.7 Exhaust Manifold Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.7.1 Mass Flow Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 343.7.2 Pressure Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.7.3 Temperature Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.8 Turbine Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.8.1 Mass Flow Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 353.8.2 Pressure Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.8.3 Temperature Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 373.8.4 Efficiency Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.9 Exhaust System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.10 Wastegate Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.11 Turbocharger Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.12 Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 HYBRID TURBOCHARGER . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.1 Hybrid Turbocharger Simulink Model . . . . . . . . . . . . . . . . . . . . . 444.1.1 DC Motor Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.1.2 Hybrid Turbocharger Modeling . . . . . . . . . . . . . . . . . . . . . 44
4.2 Advantages of Hybrid Turbocharger . . . . . . . . . . . . . . . . . . . . . 464.2.1 Comparison with Conventional Turbocharger . . . . . . . . . . . . 464.2.2 Comparison with Naturally Aspirated Engine . . . . . . . . . . . . . 47
4.3 Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5 INTRODUCTION TO MODEL PREDICTIVE CONTROL . . . . . . . . . . . . . 49
5.1 Why Use MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2 MPC Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3 MPC Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.4 Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6 MODEL PREDICTIVE CONTROL OF HYBRID TURBOCHARGER . . . . . . . 54
6.1 Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.2 Model Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.3 MPC Implementation In Matlab . . . . . . . . . . . . . . . . . . . . . . . . 596.4 MPC With Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.4.1 Overview of MPC with Constraints . . . . . . . . . . . . . . . . . . 626.4.2 Add Constraints to the System . . . . . . . . . . . . . . . . . . . . 63
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6.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.6 Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
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LIST OF TABLES
Table page
2-1 Battery Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3-1 Model Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3-2 Model States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3-3 Model Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
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LIST OF FIGURES
Figure page
1-1 Turbocharged SI engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1-2 Scheme of hybrid turbocharger workflow . . . . . . . . . . . . . . . . . . . . . . 15
2-1 Battery model validation in discharge mode . . . . . . . . . . . . . . . . . . . . 20
2-2 Battery model validation in charge mode . . . . . . . . . . . . . . . . . . . . . . 21
3-1 Validation of the compressor temperature model . . . . . . . . . . . . . . . . . 25
3-2 Validation of the compressor mass flow model . . . . . . . . . . . . . . . . . . 26
3-3 Validation of the intercooler temperature model . . . . . . . . . . . . . . . . . . 28
3-4 Validation plot for Q-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3-5 Validation of throttle mass flow model . . . . . . . . . . . . . . . . . . . . . . . 30
3-6 Validation of volumetric efficiency model . . . . . . . . . . . . . . . . . . . . . . 32
3-7 Validation of torque model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3-8 Validation of exhaust manifold temperature model . . . . . . . . . . . . . . . . 36
3-9 Validation of turbine mass flow model . . . . . . . . . . . . . . . . . . . . . . . 37
3-10 Validation of turbine temperature model . . . . . . . . . . . . . . . . . . . . . . 38
3-11 Validation of turbine efficiency model . . . . . . . . . . . . . . . . . . . . . . . . 39
3-12 Validation of exhaust system mass flow model . . . . . . . . . . . . . . . . . . 39
3-13 Turbocharged SI engine model . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4-1 Hybrid turbocharged SI engine model . . . . . . . . . . . . . . . . . . . . . . . 45
4-2 Turbo lag elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4-3 Engine downsizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6-1 Validation of linearized mass flow model . . . . . . . . . . . . . . . . . . . . . . 57
6-2 Validation of linearized engine torque model . . . . . . . . . . . . . . . . . . . . 58
6-3 Fuel consumption before and after optimization . . . . . . . . . . . . . . . . . . 64
6-4 Engine toruque tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6-5 Required battery voltage input . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
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6-6 Required wastegate opening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
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Abstract of Thesis Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science
MODELING OF TURBOCHARGED SPARK IGNITED ENGINE AND MODELPREDICTIVE CONTROL OF HYBRID TURBOCHARGER
By
Kang Rong
May 2014
Chair: Carl CraneMajor: Mechanical and Aerospace Engineering
The idea of a hybrid turbocharger is demonstrated in this thesis. First a battery
model and a turbocharged spark ignited(SI) engine is modeled using Simulink. The
hybrid turbocharger is obtained by replacing the turbo shaft with a battery, which is
connected to the turbine through a generator and to the compressor through a motor.
The main idea of the hybrid turbocharger is that the compressor is driven by the battery
and the battery is charged by the generator, which is driven by the turbine.
Comparisons of the performance of the hybrid turbocharger to the conventional
turbocharger and the naturally aspirated engine has been made in a few aspects. The
comparison to the naturally aspirated engine shows that the hybrid turbocharger plays a
significant role in engine downsizing. In comparison to the conventional turbocharger, it
shows that the hybrid turbocharger eliminates the turbo lag.
The last step is applying model predictive control (MPC) to the hybrid turbocharger
model to minimize the fuel consumption. This is achieved by controlling two inputs of the
system. First is controlling the battery output voltage in order to change the compressor
speed and influence the air mass flow into the engine. Second is the control of the
open angle of the waste gate in order to improve the turbine efficiency and decrease
emissions.
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CHAPTER 1INTRODUCTION
1.1 Introduction of Conventional Engine Charging Method
The most common type of engine is the naturally aspirated engine. In a naturally
aspirated engine, air for combustion (diesel cycle in a diesel engine, or specific types of
Otto cycle in gasoline engines namely gasoline direct injection), or an air/fuel mixture
(traditional Otto cycle petrol engines) is drawn into the engine’s cylinders by atmospheric
pressure acting against a partial vacuum that occurs as the piston travels downwards
toward bottom dead center during the intake stroke. Most automobile petrol engines, as
well as many small engines used for non-automotive purposes, are naturally aspirated.
A supercharged engine is an engine that uses an air compressor as the supercharger
to increase the pressure or density of air supplied to an internal combustion engine.
This gives each cycle of the engine more oxygen, letting it burn more fuel and do
more work, thus increasing power. Power for the supercharger can be provided
mechanically by means of a belt, gear, shaft, or chain connected to the engine’s
crankshaft. Superchargers (and turbochargers) have been widely applied to racing
and production cars, although the supercharger’s technological complexity and cost
have largely limited it to expensive, high-performance cars.
When power is provided by a turbine powered by exhaust gas, a supercharger is
known as a turbosupercharger typically referred to simply as a turbocharger or just
turbo. A large amount of work has already been done on the design and control of
the turbocharged engine, as described in [5], [8], [9],[12]. The working principal of the
turbocharger is utilizing the high pressure and temperature of the exhaust gas to drive
the turbine, which is connected to the compressor through a shaft, in order to drive
the compressor to increase the air flow rate into the engine. A turbocharged engine is
more powerful and efficient than a naturally aspirated engine because the turbine forces
12
more air, and proportionately more fuel, into the combustion chamber than atmospheric
pressure alone.
Figure 1-1 shows the workflow of how a conventional turbocharger works.
Figure 1-1. Turbocharged SI engine
1.2 Introduction of Hybrid Turbocharger
1.2.1 How Hybrid Turbocharger Works
A hybrid turbocharger is an electric turbocharger consisting of an ultra high speed
turbine-generator and an ultra high speed electric air compressor. The turbine and
compressor are high-speed aeromachines, as in a conventional turbocharger. The
electrical motors run at speeds in excess of 120,000 rpm and when used as generators,
generate electricity at up to 98.5% electrical efficiency. High electrical efficiency is
paramount, because there is no mechanical link between the turbine and compressor. In
other words, hybrid turbocharger refers to a series hybrid setup, in which the compressor
speed and power are independent from the turbine speed and power. This design
13
flexibility leads to further improvements in turbine and compressor efficiency, beyond a
conventional turbocharger.
Discharging Mode. When the driver depresses the throttle, the HTT initially acts
like an electric supercharger. The compressor motor is powered from the energy storage
medium, which in this thesis is the battery, allowing it to accelerate to full operating
speed in approximately 500 ms. During this transient stage, the engine control unit
(ECU) on a standard turbocharged engine uses a combination of sensors such as
lambda sensors and air mass flow sensors to regulate the fuel flow rate. In an HTT
equipped engine the ECU can deliver the precise fuel flow rate for complete combustion
more accurately. This is achieved by directly controlling the air flow rate and boost
pressure via control of the compressor speed.
Charging Mode. When the state of charge(SOC) [2] of the battery drops to some
certain level, the generator starts to charge the battery until the SOC returns to a
fixed level. The discharging and charging mode will repeat again and again during
the working process to keep the battery in a good working mode, so as to keep the
compressor speed at a high level to supply sufficient air to the engine.
Figure 1-2 shows the main idea of the hybrid turbocharger and how it works.
1.2.2 Why Use Hybrid Turbocharger
Even though turbo charging technology has already been fully developed and the
turbocharger shows excellent performance in engine downsizing and increasing engine
power, it still has inevitable shortcomings. The most obvious one is the turbo lag, which
means it takes a very long time for the vehicle to reach the required speed after the
driver depresses the gas pedal.
The idea of the hybrid turbocharger solves this problem. Since the compressor is no
longer connected directly to the turbine, the turbo shaft inertia is not important any more.
The compressor is driven by the battery, and is able to reach full operating speed in less
than 0.5s. This rate of acceleration eliminates the turbo lag significantly.
14
Figure 1-2. Scheme of hybrid turbocharger workflow
On the other hand, the hybrid turbocharger keeps the advantage in engine
downsizing with respect to the naturally aspirated engine. The engine with the hybrid
turbocharger installed can have a smaller size than ordinary ones to provide even more
power due to the increased air flow rate and higher combustion efficiency.
1.3 Problem Formulation and Thesis Outline
Two problems have been solved in this thesis.
The first one is the modeling of the SI engine with the hybrid turbocharger installed.
Comparison with the naturally aspirated engine and with the turbocharged SI engine
will be made to show that the hybrid turbocharger does play an important role in engine
downsizing and eliminating the turbo lag.
The second one is to apply Model Predictive Control (MPC) to the system to
minimize the fuel consumption and in the meantime prevent the generated engine
15
torque deviating too much from the required torque. This is achieved by controlling the
voltage output of the battery and the open angle of the wastegate at the same time.
The outline of this thesis is as following:
1. Modeling the battery using Simulink.
2. Develop the conventional turbocharged engine model and validate each componentof the engine.
3. Form the hybrid turbocharger model by replacing the turbo shaft with the battery.
4. Run simulations to demonstrate the performance of the hybrid turbocharger inengine downsizing and eliminating turbo lag.
5. Apply model predictive control to the system to minimize the fuel consumption.
16
CHAPTER 2BATTERY MODELING
2.1 Battery Description
There are a lot of proposed battery models that have been made previously, as
described in [2], [3], [4]. However, they do not provide a good parameter estimation
result for this thesis either in the battery discharging mode or charging mode.
The battery used in this thesis to charge the compressor is a sealed 50-module
nickel metal hydride (NiMH) battery pack providing 60.5 volts and 6.5 Ah capacity. In
this chapter, only one module of the battery pack is modeled and validated. The specific
value of the battery module is as following (E0 is battery constant voltage(V), Q is
battery capacity(Ah),R is internal resistance())[1]:
Table 2-1. Battery ParameterParameter Value (Unit)
E0 1.2101 (V)Q 6.5 (Ah)R 0.002 ()
2.2 Battery Modeling
The battery model is achieved by making the Simulink model according to the
mathematical equations of the battery charging and discharging mode. The model is
validated using the manufacturer’s data. The following assumptions have been made [1]:
• The internal resistance is assumed constant during the charge and dischargecycles and does not vary with the amplitude of the current.
• The model’s parameters are deduced from the discharge characteristics andassumed to be the same for charging.
• The capacity of the battery does not change with the amplitude of the current (noPeukert effect).
• The temperature does not affect the model’s behavior.• The self-discharge of the battery is not represented.• The battery has no memory effect.
17
2.2.1 Mathematical Equations of Battery Discharging and Charging
2.2.1.1 State of Charge
The state of charge (SOC) is a very important parameter of the battery. It
represents how much longer the battery can discharge. The mathematical equation
for SOC is given as:
SOC = SOCinitial −∫idt
Q(2–1)
where
SOCinitial=the initial state of charge of the battery
i = the current in the battery
Q = battery capacity
2.2.1.2 Discharging Mode
The proposed discharge model can be represented accurately by the voltage
dynamics when the current varies and takes into account the open circuit voltage as a
function of SOC. The battery voltage obtained is given by:
Vbatt = E0 − KQ
Q − it· it − R · i + Aexp(−B · it)− K
Q
Q − it· i∗ (2–2)
where
Vbatt=battery voltage(V)
E0=battery constant voltage(V)
K=polarisation constant(V/(Ah))
Q=battery capacity(Ah)
it=∫
idt=actual battery charge(Ah)
A=exponential zone amplitude(V)
B=exponential zone time constant inverse(Ah)−1
R=internal resistance()
i=battery current(A)
18
i∗=filtered current(A)
The exponential zone of equation (2-1) can be obtained by the following equation:
_exp(t) = B · |i(t)| · (−exp(t) + A(t)) (2–3)
where
exp(t)=exponential zone voltage(V)
i(t)=battery current(A)
u(t)=charge or discharge mode
2.2.1.3 Charging Mode
For a NiMH battery, after the battery has reached the full charge voltage, the voltage
decreases slowly, depending on the current amplitude. This behavior is represented
by modifying the charge polarisation resistance. When the battery is fully charged,
the voltage starts to drop. This phenomenon can be represented by decreasing the
polarisation resistance when the battery is overcharged by using the absolute value of
the charge (it):
Pol .Resistance = KQ
|it| − 0.1Q(2–4)
Thus, the mathematical equation for the charging mode is:
Vbatt = E0 − R · i − KQ
|it| − 0.1Q· i∗ − K
Q
Q − it· it + exp(t) (2–5)
Now, the Simulink battery model is ready to be made according to the mathematical
equations for battery discharging and charging mode.
2.3 Simulink Battery Modeling and Validation
2.3.1 Simulink Model
The basic modeling of the battery in Simulink is based on equation (2-1) and (2-4).
However, the detailed modeling is more complicated.
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Parameter estimation is performed in Simulink to obtain all the unknown parameters
given the input and output of the battery according to the manufacturer’s data. The next
step is to validate the battery model.
2.3.2 Model Validation
Figure 2-1 and 2-2 shows the results of validation of the battery model in discharging
and charging modes respectively. It could be obviously found that the model is validated
very well according to the validation results.
Figure 2-1. battery model validation in discharge mode
2.4 Chapter Conclusion
The battery is a very important component of the hybrid turbocharger system. In
this chapter, a NiMH battery pack is formed based on mathematical equations, modeled
20
Figure 2-2. battery model validation in charge mode
and validated in Simulink. According to the model validation, it can be found that the
estimated parameters of the battery provide reasonable results. Now the battery model
is ready to be used in the engine model which will be made in the following chapters.
21
CHAPTER 3TURBOCHARGED SI ENGINE MODELING
3.1 Model Overview
In this chapter, a mean value engine model (MVEM) with turbocharger installed is
presented.The model is completed by firstly creating subsystems of each component of
the engine and then connecting each subsystems to form the final engine model. The
mass flow through the engine is central in the modeling, and thus the modeling is based
on the air flow path. The air enters the engine through the air filter to be cleaned. Then
the clean air enters the compressor where the pressure and temperature increase. The
air needs to be cooled down before entering the engine cylinder to avoid knock, and this
process is done via a heat exchanger called the intercooler. The amount of air into the
engine cylinder is controller by the throttle in order to control the engine output power.
Then the air is mixed with fuel in the intake manifold. The mixture enters the cylinder,
where combustion takes place. The pressure and temperature increases significantly
after combustion. The hot gas, which gets out of the engine via the exhaust manifold,
is the power to drive the turbine. The turbine then drives the compressor to spin at very
high speed through the turbo shaft. A wastegate is used to regulate the air flow into the
turbine. Finally the wasted gas leaves the engine through the exhaust system. All the
components will be modeled in the following sections and finally the whole engine model
will be validated according to experimental data.
One simplification has been made here. The air filter does not have significant
influence on neither the pressure nor the temperature of the air. Since this model is only
for simulation, the air filter will not be modeled in the following sections.
3.1.1 Model Input
The input into the model is shown in Table 3-1:
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Table 3-1. Model InputName Description Unit
N Engine speed rpmα Throttle angle deg
uwg Wastegate opening -pamb Ambient pressure PaTamb Ambient temperature K
Table 3-2. Model StatesState Description Unit
pc pressure after compressor PaTc temperature after compressor Kpic pressure after intercooler PaTic temperature after intercooler Kpi intake manifold pressure PaTi intake manifold temperature Kpe exhaust manifold pressure PaTe exhaust manifold temperature Kpt pressure after turbine PaTt temperature after turbine Kωtc turbocharger speed rad/s
3.1.2 Model States
After subtracting the air filter from the engine model, the system contains 11 states,
including the pressure and temperature after each component and the turbocharger
speed. All the states are listed in Table 3-2:
3.1.3 Model Constants
Table 3-3 shows all the constants of the engine model that will mentioned in the
following chapters [11]:
In the following chapters, each component of the hybrid turbocharged engine will be
modeled in the order of the air flow path.
23
Table 3-3. Model ConstantsName Description Value(unit)
R Gas constant 287(J/(kg · K))γ Heat capacity ratio 1.4cp Heat capacity at constant pressure 1003.4(J/K )
Qhv Heating value of fuel 44×106(J/kg)pamb Ambient pressure 101.7(kPa)Tamb Ambient temperature 296(K)Vc Compressor volume 0.005(m3)Vic Intercooler volume 0.005(m3)Vim Intake manifold volume 0.002(m3)Vem Exhaust manifold volume 0.002(m3)Ves Exhaust system volume 0.01(m3)J Turbo shaft inertia 0.15(kg/m2)
3.2 Compressor Modeling
3.2.1 Pressure Model
The dynamic equation for the compressor can be derived from the first law of
thermodynamics, and is given as:
∂pc∂t
=RTc
Vc
( _mc − _mic) (3–1)
where
Vc=compressor volume,
_mic=mass flow rate through the intercooler,kg/s
_mc=mass flow rate through the compressor,kg/s
3.2.2 Temperature Model
If the expansion of gases through the compressor was isentropic, i.e. ηc=1,the
temperature after the compressor could be modeled as:
Tc = k1Tamb(pc
pamb
)(γ−1)/γ (3–2)
Despite its simplicity, due to the high efficiency of the compressor which makes
the isentropic approximation more appropriate, this model works well according to the
experimental data, which can be shown from Figure 3-1:
24
Figure 3-1. Validation of the compressor temperature model
3.2.3 Mass Flow Model
The compressor mass flow depends mainly on the compressor speed and the
pressure ratio. Some basic requirements are that the mass flow must be zero when the
turbo shaft speed is zero, and when there is no pressure difference before and after the
compressor. One possible model is presented as:
_mc = k1(1−pamb
pc) + k2ωtc
√1− pamb
pc+ k3ωtc
4
√1− pamb
pc+ k4ωtc (3–3)
where k1 to k4 are unknown parameters to be determined.
However, this model is difficult to tune since it produces imaginary numbers for
some circumstances. Therefore, another mathematical equation is used to model the
compressor mass flow:
N̂T =ωtc − 8 · 104
2 · 104
p̂r = �c − b4N̂T − b5N̂2T
_mc = b1p̂r + b2p̂2r + b3p̂
2r
(3–4)
where
�c =pcpamb
25
In this model, b1 to b5 are determined by using the lsqcurve�t function in Matlab.
The validation shows a good result of the parameter estimation for the compressor mass
flow model, shown in Figure 3-2:
Figure 3-2. Validation of the compressor mass flow model. The group of pointsrepresent turbo shaft speeds 80000 RPM,10000 RPM, 12000 RPM and14000 RPM respectively from left to right.
3.2.4 Efficiency Model
The efficiency is defined by the ratio of the isentropic and the actual specific input
work. The mathematical equation for the efficiency model is given as:
ηc =( pcpamb
)γ−1
γ − 1Tc
Tamb− 1
(3–5)
The efficiency model is difficult to be estimated, however, equation (3-5) stills yields a
reasonable result of the compressor efficiency.
3.3 Intercooler Modeling
Due to the first law of thermodynamics
ρ =P
RT
26
Thus when the pressure of air increases when being compressed by the compressor,
the air temperature also rises. The high temperature of the intake air into the engine will
cause knock in the cylinder. Therefore, the air needs to be cooled and this is the main
reason why the intercooler is modeled.
3.3.1 Pressure Model
The intercooler can be treated as a static flow restriction.The dynamic equation for
the intercooler can be derived from the first law of thermodynamics, and is given as:
∂pic∂t
=RTic
Vic
( _mic − _mth) (3–6)
where
Vic=intercooler volume
_mic=mass flow rate through the intercooler,kg/s
_mth=mass flow rate through the throttle,kg/s
3.3.2 Mass Flow Model
The relationship between the pressure drop in the intercooler and the mass flow
rate has been found to fit the following equation:
pc − pic = kTc_m2
ic (3–7)
Then the mass flow through the intercooler can be modeled as:
_mic =
√pc − pic
kTc
(3–8)
where k is the unknown parameter to be estimated in Matlab using lsqcurve�t function.
The validation shows that this model fits well with the experimental data.
3.3.3 Temperature Model
The ability of the intercooler to lower the temperature of the compressed air
depends on the intercooler efficiency ϵ. For perfect gas the heat capacity is a function of
27
the temperature only, and thus the intercooler efficiency can be expressed as:
ϵ =Tc − Tic
Tc − Tcool
(3–9)
Thus the intercooler temperature is ready to be expressed as:
Tic = Tc − ϵ(Tc − Tcool) (3–10)
where Tcool in this thesis is equal to the atmospheric pressure Tamb.
3.3.4 Model Validation
Figure 4-1 shows the result of the parameter estimation of the intercooler temperature
model. It could be seen that the temperature model fits the experimental data well.
Figure 3-3. Validation of the intercooler temperature model
3.4 Throttle Modeling
3.4.1 Mass Flow Model
In gasoline engines, a throttle is used to control the air mass flow into the cylinders.
Thus it is important to model the throttle mass flow rate precisely. The mass flow
through the throttle can be modeled like the flow of an ideal gas through a venturi. A
28
standard model for this type of flow is
_mth =pic√RTic
A · C ·�(pr) (3–11)
where
A=throttle opening area
C=discharge coefficient
C depends on the shape of the flow area. (pr) is a function of the pressure ratio
given as:
(pr) =
√
2γγ−1
(p2
γr − p
γ−1
γr ), if pr > ( 2
γ+1)
γγ−1√
2γγ−1
(( 2γ+1
)2
γ−1 − ( 2γ+1
)γ+1γ−1 ), otherwise
(3–12)
where pr is the pressure ratio
pr =pth
pic
Since both the opening area A and the discharge coefficient C depend on the
throttle plate opening angle α, it is reasonable to lump A and C together to form another
equation Qth(α) to expressing the opening of the throttle. There are many validated
model for Qth(α) according to previous researches. The model used in this thesis is
given as
Qth(α) = Q1(1− cos(a0α+ a1)) +Q0 (3–13)
where Q1, Q0, a1 and a0 are unknown parameters that will be determined by using the
lsqcurve�t function in Matlab. Figure 3-4 shows the validation of the parameters of the
function Qth(α):
Now the mathematical equation of throttle mass flow rate model can be expressed
as a function of α, pic ,Tth and pim, which is given as:
_mth(α, pic ,Tic , pim) =pic√RTic
Qth(α)(pr) (3–14)
29
Figure 3-4. Validation plot for Q-function
The throttle mass flow is a central quantity in the engine. It has a significant
influence on the combustion process, and therefore determines the output power of
the engine. Thus the accuracy of the mass flow model is important. Figure 3-5 shows
the validation of the throttle mass flow model.
Figure 3-5. Validation of throttle mass flow model. This shows that the the model fits theexperimental data well
30
3.4.2 Throttle Pressure Model
Similarly to the pressure model of the intercooler and the compressor, the throttle
pressure model can be derived from the first law of thermodynamics and is given as:
∂pth∂t
=RTic
Vim
( _mth − _mac) (3–15)
where
Vim=intake manifold volume
_mac=the mass flow rate into the cylinder
3.4.3 Temperature Model
The temperature change in the throttle is neglected, which means the temperature
after the throttle is the same as the one after the intercooler.
Tth = Tic (3–16)
3.5 Intake Manifold Modeling
The intake manifold is where the air and fuel are mixed and is the path where the
mixture enters the cylinders. The pressure and temperature are just considered to be
the ones that are after the throttle. So in this section, only the mass flow into the cylinder
is modeled. One of the parameters that governs the mass flow into the cylinder is the
volumetric efficiency ηvol . Many mathematical equations have been used by previous
researchers for modeling ηvol . In this thesis, ηvol is modeled as a function of the intake
manifold pressure pth and the engine speed N, which is given as [14]:
ηvol(N, pth) = a0 + a1N + a2N2 + a3pth (3–17)
Figure 3-6 shows the validation of the volumetric efficiency.
Now the mass flow into the cylinder is ready to be modeled as:
_mac =ηvol(N, pth)VdNpth
120RTth
(3–18)
31
Figure 3-6. Validation of volumetric efficiency modell
where N is the engine speed in [RPM] and Vd is the displacement volume of the engine
in [m3].
3.6 Combustion Modeling
During the combustion precess, the mixture of air and fuel is burnt to generate
torque and power. The amount of air into the cylinder affects the extension of the
combustion, so that it will influence the output power of the engine. In order to inject a
correct amount of fuel into the engine, it is important to know the theoretical proportion
of air and fuel, which is called the stoichiometric air to fuel ratio
A
F s=
mac
mfc
(3–19)
In this thesis, this ratio is set to be 14.7. An important parameter is the ratio
between the true air to fuel ratio (A/F) and (A/F )s
λ =(A/F )
(A/F )s(3–20)
When there is excess air in the combustion(λ > 1), the mixture is referred to as lean
and when there is excess fuel in the combustion(λ < 1),the mixture is called rich. An
32
engine that runs under lean conditions will emit large amount of NOx and if the mixture
is rich there will inevitably be unburned hydro carbons and CO in the exhaust gases.
Thus, it is essential to keep λ close to one in order to maintain good catalyst function,
which will yields the fuel mass flow rate as:
_mfc =1
14.7_mac (3–21)
The torque generated by the engine depends on the work produced and consumed
in the engine, which is given as [18]:
Me =Wig −Wp −Wf
2πnr(3–22)
where nr is the number of engine revolutions per cycle. In this thesis, the model is a
2-stroke engine, so nr=2. Wig is the indicated gross work produced by the engine, Wp is
the pumping work consumed and Wf is the friction work consumed. The mathematical
expressions for these three terms are as follows:
Wig = Vd · _mfuelQhv
60
N
2
Vd
ηe
Wp = Vd · (pem − pim)
Wf = Vd · [0.97 + 0.15(N
1000) + 0.05(
N
1000)2]
(3–23)
where ηe is the combustion efficiency. There are also many validated mathematical
expressions for ηe . In this thesis, the equation is given as following:
ηin = 0.588(1− 0.392N−0.36)
ηip = 0.9301 + 0.2154pth − 0.1657pth2
ηe = Cηinηip
(3–24)
where pth is the intake manifold pressure in bar and C is an unknown parameter to be
estimated using the lsqcurve�t function in Matlab.
33
The engine torque model is validated according to the experimental data, which is
shown in Figure 3-7.
Figure 3-7. Validation of engine torque model
Then the output engine power can be easily expressed as:
P = 2πMN (3–25)
where M is the engine torque in N _m, N is the engine speed in rpm.
3.7 Exhaust Manifold Modeling
The mixture burnt in the engine cylinder generates very high pressure and
temperature into the exhaust manifold, which is used by the turbine to drive the
compressor. Thus the pressure and temperature out of the exhaust manifold is modeled
here.
3.7.1 Mass Flow Modeling
Firstly, the mass flow through the exhaust manifold is modeled because it will be
used to model the pressure. The exhaust manifold mass flow is just the sum of the air
34
mass and fuel mass flow through the engine, which is given as:
_mem = _mac + _mfuel (3–26)
where _mac and _mfuel are expressed in section 3.6
3.7.2 Pressure Modeling
The dynamics of the exhaust manifold can be described by the familiar tank
model approach. The pressure pem is built up by exhaust manifold mass flow _mem and
temperature Te , the turbine mass flow _mt and the wastegate mass flow _mwg which will
be modeled in the following section.
∂pem∂t
=RTem
Vem
( _mem − _mt − _mwg) (3–27)
3.7.3 Temperature Modeling
The exhaust temperature increases with increasing engine speed, load, and spark
retard. Different combinations of these properties have been tested as a model for Tem.
Correlation analysis was used to find a combination of powers of N and M that works
fine. Thus the exhaust temperature is given as:
Tem = k1M2 + k2
4√N + k3MN (3–28)
where ki are the parameters to be determined by using lsqcurve�t in Matlab. The
exhaust manifold temperature model is validated as shown in Figure 3-7
3.8 Turbine Modeling
3.8.1 Mass Flow Modeling
Unlike the compressor, the mass flow rate through the turbine does not depend on
the turbo shaft speed. According to a large amount of previous researches, it can be
accurately modeled by using only the pressure ratio, pem/pt . The basic requirement for
the modeling is that when there is no pressure difference the mass flow rate needs to
be zero. It turns out that the following mathematical equation for the turbine mass flow
35
Figure 3-8. Validation of exhaust manifold temperature model
makes a good fit with the experimental data provided by the manufacturer.
_mt = k1(pem
pt) + k2
√pem
pt− 1 (3–29)
Equation (3-29) is linear in parameters k1 and k2 so that the parameters can be
adjusted to measured data by using standard least square methods. Actually, the
parameters are estimated by using the lsqcurve�t function in Matlab. Figure 3-9 shows a
good validation result.
3.8.2 Pressure Modeling
Similar to other components, the pressure modeling of the turbine can also be
derived from the first law of thermodynamics, which is given as:
∂pt∂t
=RTt
Ves
( _mwg + _mt − _mes) (3–30)
where _mes is the exhaust system mass flow rate in (kg/s).
36
Figure 3-9. Validation of turbine mass flow model
3.8.3 Temperature Modeling
As what was done to the compressor, it is possible to model the turbine temperature
in the same way given as:
Tt = k1Tem(pt
pem)(γ−1)/γ (3–31)
However, this model does not hold as expected. One possible explanation is that
the great heat transfer form the turbine to the surroundings makes the model fail to
capture Tt . Thus, another model is introduced here, which shows a good fit with the
experimental data.
Tt = k1(Tem − Tamb)(pt
pem)(1−γ)/γ + k2(Tem − Tamb)
2 + k3 (3–32)
where ki are parameters to be determined by using lsqcurve�t function in Matlab.Figure
3-10 shows the validation result of this turbine temperature model.
3.8.4 Efficiency Model
The turbine efficiency model can be calculated by the equation given as:
ηt =1− Tt
Tem
1− ( ptpem
)γ−1
γ
(3–33)
37
Figure 3-10. Validation of turbine temperature model
The turbine efficiency is important since it determines the power delivered to the
compressor. It is not clear what the efficiency is when there is no mass flow through
the turbine. The relationship between the turbo shaft speed and the turbine efficiency
is complicated and very difficult to measure. As an approximation, a model which is
independent of turbine speed is used in this thesis given as:
ηt = k1
√pem
pt− 1 + k2
4
√pem
pt− 1 + k3 (3–34)
where ki are the unknown parameters to be estimated using the lsqcurve�t function in
Matlab. Figure 3-11 shows the validation of the turbine efficiency model.
3.9 Exhaust System Modeling
The pressure drop from the turbine through the exhaust system to the surrounding
air is significant, therefore it is necessary to model this pressure loss. The exhaust
system can be regarded as a tube with a sudden restriction. The mathematical equation
for the pressure drop in this tube is given as:
pt − pamb = k1 _mes + k2 _m2es (3–35)
38
Figure 3-11. Validation of turbine efficiency model
Figure 3-12. Validation of exhaust system mass flow model
In order to form the exhaust system mass flow rate to be used for calculating turbine
pressure, equation (3-35) is redefined as following:
_mes = −k1k2Tt
2+
√(
k1k2Tt
2)2 +
pt − pamb
k2(3–36)
This exhaust system mass flow model is validated in Figure 3-12
39
3.10 Wastegate Modeling
The turbocharger will overspeed very easily at high loads, which causes excessive
boost pressure or even turbocharger bearing failure. If the turbo shaft speed is too high,
the compressor will consequently spin at a very high speed, resulting in high intake
manifold pressure, which will cause knock in the engine cylinder. To prevent the turbine
from over speeding, it is necessary to deviate some amount of exhaust gases away
from flowing into the turbine directly. This is achieved by using a valve called wastegate.
This will keep the driving torque and therefore the turbine speed at a lower level when
the wastergate is open. The wastgate can be modeled in a similar way as the throttle
modeling. Equation (3-12) and (3-13) will be used. The only difference is that the Q
function will be replaced with a function of the opening area of the wastegate. The
mathematical equation for wastegate modeling is given as:
Awg = CdAwgmaxuwg
Awgmax=
πD2
4
_mwg =pem√RTem
Awg(pr)
(3–37)
where
Cd=wastegate flow coefficient, 0.9
uwg=the opening of the wastegate, uwg ∈ [0, 1]
D=the diameter of the wastegate tube
pr=the pressure ratio pt/pem
3.11 Turbocharger Dynamics
According to Newton’s Second Law for rotating systems, the turbine and compressor
are connected by the mathematical equation given as:
Tqt − Tqc = Jtc _ωtc (3–38)
40
where Tqt stands for the driving torque of the turbine and Tqc is the braking torque
acting on the compressor. Jtc denotes the inertia of the turbo shaft and ωtc is the
rotational speed of the turbo shaft.
The theoretical maximum torque delivered by the turbine depends on the exhaust
manifold temperature Tem and the ratio pem/pt . If the process would be reversible, which
means there would not be any frictional losses in the turbine, the work would be called
isentropic. However, since the turbine gets red hot at high loads, this is not an isentropic
case. Thus the turbine efficiency ηt is introduced here to calculate the true portion of
power delivered by the turbine. The mathematical equation for the turbine power is given
as
Pt = ηt _mtcpTem[1− (pem
pt)(1−γ/γ)] (3–39)
The mathematical equation for compressor power can be modeled in a similar way.
Since the compressor consumes energy, the net amount of produced power is negative.
Moreover, the compressor is not ideal, so the efficiency ηc is also introduced here.
Pc = _mccpTamb
1
ηc[(
pc
pamb
)(γ−1)/γ − 1] (3–40)
The torque and the power are connected through equationP = Tq · ω, thus the
equations for the turbine and compressor torques are given as:
Tqt =ηt _mtcpTem[1− (pem
pt)(1−γ/γ)]
ωtc
Tqc =_mccpTamb
1ηc[( pc
pamb)(γ−1)/γ − 1]
ωtc
(3–41)
Substituting equation (3-41) into equation (3-38) yields a differential equation
of ω|tc . The turbo shaft rotating speed can be calculated by solving this differential
equation for ωtc .
3.12 Chapter Conclusion
This chapter covers the modeling and the validation of all the components in the
turbocharged SI engine. Validation results show that each component works well
41
separately, which will provide a reasonable turbocharged engine system. The next step
is to combine the battery model with the engine model to form the hybrid turbocharger
and make comparisons with the conventional turbocharger and the natural aspirated
engine to demonstrate the advantages of the hybrid turbocharger. Figure 3-13 shows
the top level of the Simulink model of the turbocharged SI engine.
42
Figure 3-13. Turbocharged SI engine model43
CHAPTER 4HYBRID TURBOCHARGER
4.1 Hybrid Turbocharger Simulink Model
4.1.1 DC Motor Modeling
The first task is to model the dc motor which is used to drive the compressor. The
requirements for the motor are:
• Reach the full operating speed within a very short time.
• The full operating speed should be over 10000 rad/s
The mathematical equation for the dc motor is given as:
_ω =1
J(kt I − bω)
_I =1
L(−RI + V − keω)
(4–1)
where
J=moment of inertia of the rotor, 0.01J/m2
b=motor viscous friction constant, 0.01(Nms)
R=electric resistance, 1
L=electric inductance, 0.1H
ke=electromotive force constant,0.001V /(rad/s)
kt=motor torque constant, 2N ·m/A
4.1.2 Hybrid Turbocharger Modeling
The battery model and turbocharged SI engine model have already been completed
in previous chapters. Therefore, it is ready to make the engine with the hybrid turbocharger.
This is done by replacing the turbo shaft with the battery and power electronics. Power
electronics consist of the motor and the generator. The battery is used to start the
motor to drive the compressor. The generator, which is driven by the turbine, is used for
charging the battery when the state of charge (SOC) of the battery drops to a certain
level.
44
Figure 4-1. Hybrid turbocharged SI engine model45
Figure 4-2. Hybrid Turbocharger working process showing the charging and dischagringmode
In this thesis, the motor used is modeled as a DC motor which can reach the full
operating speed (12000 rad/s) in about 0.8s. The generator in this thesis is just picked
from the SimPowerSystem DC Machine Library which is powered by the turbine torque.
The top level Simulink model of the hybrid turbocharger is shown in Figure 4-1. The
working process is to use the battery to drive the motor in order to drive the compressor.
When the SOC of the battery drops to 40%, the switch is turned on to start the generator
to charge the battery until the SOC reaches 80%. This repeats during the whole
working process, which is shown in Figure 4-2. The input to the system are atmospheric
pressure, temperature and the throttle angle. The important outputs are the engine
torque and power.
4.2 Advantages of Hybrid Turbocharger
4.2.1 Comparison with Conventional Turbocharger
The most significant advantage of the hybrid turbocharger against conventional
ones is that it eliminates the turbo lag. Turbo lag means the time it takes the engine to
generate required engine torque, or in other words, the vehicle reaches the required
speed after the driver depresses the gas pedal. The turbo lag is resulted from the
46
Figure 4-3. Turbo lag elimination
inertia of the turbo shaft, which is the main reason that causes the turbo shaft to reach
the required speed in a few seconds. The hybrid turbocharger solves this problem
well because the compressor is driven by the motor directly, which is able to reach full
operating speed in less then 1s and independent of the turbine. Since there is no shaft
between the compressor and the turbine, turbo inertia is not a problem any more.
This comparison is completed by using a step throttle angle input to simulate the
case that the driver depresses the gas pedal to make the throttle angle increase from
25◦ to 35◦, and check the time it takes the two engine to reach the required speed, as
shown in Figure 4-3.
It is easy to found the difference between the time it takes the engines with the two
types of turbocharger installed to reach the required torque. The result demonstrates
well that the hybrid turbocharger eliminates the turbo lag significantly.
4.2.2 Comparison with Naturally Aspirated Engine
The second advantage of the hybrid turbocharger is compared to the naturally
aspirated engine, since they are able to eliminate the turbo lag. Then why should we
use the hybrid turbocharger but not just the naturally aspirated engine? The answer is
47
Figure 4-4. Engine downsizing with hybrid turbocharger Installed. The two engines arein the same condition(same throttle angle and same load). The solid lineshows a 1L engine with hybrid turbocharger installed. The dashed lineshows a 2L naturally aspirated engine.
that the hybrid turbocharger plays an important role in engine downsizing, which means
the engine with the hybrid turbochager installed is able to generate equal or even more
power than the naturally aspirated engine of a larger size. This is shown by Figure 4-4.
It can be seen in Figure 4-4 that with the hybrid turbocharger installed, the 1L
engine has a even larger output power and higher engine speed than the 2L naturally
aspirated engine, which means the engine downsizes about 50%.
4.3 Chapter Conclusion
In this chapter, comparisons between the hybrid turbocharger with the conventional
turbocharger and the naturally aspirated engine have been made. It can be found that
the hybrid turbocharger has a great significance in eliminating the turbo lag and engine
downsizing. In the next chapter, the controller will be made to control the system.
48
CHAPTER 5INTRODUCTION TO MODEL PREDICTIVE CONTROL
5.1 Why Use MPC
Even though PID-control is normally used in industry, the control method used in
this thesis is model predictive control(MPC), an advanced method of process control.
The reasons are as follows [19], [21].
• MPC allows the current timeslot to be optimized, while keeping future timeslots inaccount.
• MPC has the ability to anticipate future events and can take control actionsaccordingly. PID and LQR controllers do not have this predictive ability.
• MPC can handle safety constraints.
• More than one input and output (MIMO-systems) can be handled using MPC.
5.2 MPC Overview
Model Predictive Control is an advanced process control technique widely adopted
in industry as an effective method to deal with large multivariable constrained control
problems. MPC uses a model of the system to predict its future behavior, and then
optimizes a quadratic performance based on the prediction. The main idea is to choose
the control input by solving an on line optimal control problem repeatedly, aiming at
minimizing a performance criterion over a future horizon. This future horizon is called
the prediction horizon Np, which means the number of samples one looks ahead.
Another important term is the control horizon Nc , meaning the number of samples that
the optimal input is calculated for. Np and Nc are not necessarily the same. If the Nc is
shorter than Np, the complexity of the problem is reduced. In this thesis, Np is picked to
equal to Nc .
The procedure of the MPC is as following: Assume the system is running during the
period of time T. Discretize the time period T into N pieces of equal length, which is the
sampling time Ts = T/N. Then perform the discretization of the continuous system.
Assume starting at time k which is given as the initial condition, predict the states from
49
k+1 to k + Np. The optimal input u is calculated at time k by solving the optimal control
problem. Select the first element of u as uk and substitute it into the dynamic equation
to calculate xk+1, which is regarded as the initial condition of the next prediction horizon.
Since the input is optimized at each time step, finally the best U will be obtained.
5.3 MPC Derivation
Considering the dynamic equation:
_x = Ax + Bu
y = Cx +Du
(5–1)
The first step is to dicretize the continuous equation. This can be completed by
using the c2d function in Matlab. The discretized equation is given as:
xk+1 = Amxk + Bmuk
yk = Cmxk +Dmuk
(5–2)
Substitute xk+1 in to the state space equation to obtain xk+2. Repeat this process Np
times to obtain all the predicted states from xk+1 to xk+Np. This is given as follows:
xk+1 = Amxk + Bmuk
xk+2 = A2mxk + AmBmuk + Bmuk+1
xk+3 = A3mxk + A2
mBmuk + AmBmuk+1 + Bmuk+2
...
xk+Np= ANp
m xk + ANp−1m Bmuk+1 + · · ·+ AmBmuk+Np−2 + Bmuk+Np−1
(5–3)
Put all these equations into matrices to rewrite the state space equation as [20]
xk+1
xk+2
...
xk+Np
=
Am
A2m
...
ANp
m
xk +
Bm 0 ... 0
AmBm Bm ... 0
...... ...
...
ANp−1m Bm A
Np−2m Bm ... Bm
uk
uk+1
...
uk+Np−1
(5–4)
50
Define
X =
xk+1
xk+2
...
xk+Np
,U =
uk
uk+1
...
uk+Np−1
(5–5)
and let
Ex =
Am
A2m
...
ANp
m
(5–6)
Fx =
Bm 0 ... 0
AmBm Bm ... 0
...... ...
...
ANp−1m Bm A
Np−2m Bm ... Bm
(5–7)
Now the state space equation is rewritten as
X = Exxk + FxU (5–8)
In a similar way, it is easy to form the expression for the output state space
equation. It can be done by calculating xk+1 to xk+Npand substituting into the state
space equation to form the output vector, which is given as:
yk
yk+1
...
yk+Np−1
=
CAm
CA2m
...
CANp
m
xk +
CBm 0 ... 0
CAmBm CBm ... 0
...... ...
...
CANp−1m Bm CA
Np−2m Bm ... CBm
uk
uk+1
...
uk+Np−1
(5–9)
Therefore equation (5-9) can be rewritten as
Y = Eyxk + FyU (5–10)
51
Now it is ready to solve the optimal problem. The cost function is designed
depending on what to minimize. If it is required to minimize the error between the
actual output and the desired output , then the cost function looks like:
J =1
2(
Np∑s=0
||yk+s − rk+s ||2Q +
Np−1∑s=0
||uk ||2P) (5–11)
where rs is the discretization of the desired trajectory r(t) in the continuous domain. If
we define
ek+s = yk+s − rk+s (5–12)
then the error matrix can be obtained as
�e =
yk − rk
yk+1 − rk+1
...
yk+Np−1 − rk+Np−1
= Y − R (5–13)
Now equation (5-11) can be rewritten as:
J =1
2[Eyxk + FyU − R]TQ[Eyxk + FyU − R] +
1
2UTPU (5–14)
where Q and P are weighting functions.
The optimal problem can be solved by taking the first derivative of J with respect to
U and making it equal to zero:∂J
∂U= 0 (5–15)
This yields to an equation containing U and xk . Therefore, the optimized input U could
be expressed by the initial state xk as
U = (FTy QF + P)−1(FT
y QR − RTy QEyxk) (5–16)
52
Since xk is the initial state which is given, the optimized input vector U at time k is
obtained by plugging in the value of xk . Then take the first element of U, which is uk out
of the vector:
uk = [1 0 ... 0]U (5–17)
Substitute the value of uk back into equation (5-2) and together with the value of
xk , it is easy to calculate the value of xk+1. Then use xk+1 as the new initial condition
to repeat the process above to obtain uk+1. After repeating the process for N times
(N is defined previously), the best input U which is optimized at each time step will be
obtained.
5.4 Chapter Conclusion
In this chapter, the benefits, introduction, and derivation of MPC is discussed in
detail. Now it is ready to apply this control method to the hybrid turbocharger system to
achieve the desired goal.
53
CHAPTER 6MODEL PREDICTIVE CONTROL OF HYBRID TURBOCHARGER
6.1 Model Reduction
In this chapter, the MPC will be applied to the hybrid turbocharger system. As
mentioned at the beginning of chapter 3, the turbocharged SI engine model contains
11 states, which is too many for MPC. Therefore, model simplification is necessary.
According to previous researches, some states do not have significant influence on the
system performance so it is reasonable to have them truncated. [17] has proposed a
reduced engine model with 5 states. Therefore, the simplest proposed model for the
hybrid turbocharger consists of 6 states together with the states of the motor, which is
given as follows:
_ω =1
J(kt I − bω)
_I =1
L(−RI + V − keω)
_pc =RTc
Vc
( _mc(ω, pc)− _mic(pic , pc))
_pic =RTic
Vic
( _mic(pic , pc)− _mth(pic , pim))
_pim =RTic
Vim
( _mth(pic , pim)− _mac(pim))
_pem =RTem
Vem
( _mem(pim)− _mt(pem, pt)− _mwg(pt , pem))
(6–1)
6.2 Model Linearizion
According to the reduced model, all the mass flow rate functions are nonlinear in the
states. Since the control method used in this thesis is just linear MPC, it is necessary
to linearize the model. Before the linearizion, some parameters need to be set to
constants.Tc = 330K , Tic = 302K , Tem = 1264K
pt = 149kPa, α = 30◦, ug = 0.7
(6–2)
54
Linearizion is completed around the equilibrium point, which is obtained by
equalizing all the differential equations to zero.
0 =1
J(kt I − bω)
0 =1
L(−RI + V − keω)
0 =RTc
Vc
( _mc(ω, pc)− _mic(pic , pc))
0 =RTic
Vic
( _mic(pic , pc)− _mth(pic , pim))
0 =RTic
Vim
( _mth(pic , pim)− _mac(pim))
0 =RTem
Vem
( _mem(pim)− _mt(pem, pt)− _mwg(pt , pem))
(6–3)
Substitute all the constants into the equations and solve for the solution. The
equilibrium point is obtained as:
ω
I
pc
pic
pim
pem
=
9628.2
48.8
124.1
113.7
73.4
125.7
(6–4)
The linearized model is given as:
_ω = −ω + 200I
_I = −0.01ω − 10I + 10V
_pc = 5.278pic − 7.238pc − 1.185ω + 224.9
_pic = 11.866pc − 12.6pic − 6.134α+ 294.4
_pim = 7.321pic − 0.7274pim + 6.14α− 178.4
_pem = 2.766pim − 3.864pem − 3.453uwg + 255.9
(6–5)
55
In equation(6-5), the battery voltage V and the wastegate opening uwg are the inputs
to the system u. The throttle angle and all the constants in the equation are considered
to be the measured disturbance umd . Then it is now possible to rewrite equation (6-5) in
state space form as
_ω
_I
_pc
_pic
_pim
_pem
=
−1 200 0 0 0 0
−0.01 −10 0 0 0 0
−1.185 0 −7.238 −5.278 0 0
0 0 11.866 −12.6 0 0
0 0 0 0.732 −0.727 0
0 0 0 0 2.766 −3.864
ω
I
pc
pic
pim
pem
+
0 0
1 0
0 0
0 0
0 0
0 −3.453
V
uwg
+
0 0
0 0
224.9 0
294.4 −6.134
−178.4 6.14
355.9 0
1
α
(6–6)
The output of the system is the fuel mass flow _mfuel which is given as equation
(3-18) and (3-21), and the engine torque Me , which is given as equation (3-22) and
(3-23). These two equations are also nonlinear in states, so they need to be linearized
too.
A expression for the air mass flow into the cylinder linear in the state pim is given as
[21]
_mac = (a0pim + a1)VdN
60RTimnr(6–7)
where a0 and a1 are parameters to be estimated by using the lsqcurve�t equation in
Matlab. The validation of the linearized model is shown in Figure 6-1.
56
Figure 6-1. Validation of linearized mass flow into the cylinder
Here assume that the engine speed is constant N=3000 rpm and the intake
manifold temperature is constant Tim=302K. Then the air mass flow into the cylinder can
be expressed linearly in the state pim, so is the fuel mass flow rate mfuel , given as
_mfuel = k1pim + k2 (6–8)
The other output of the system is the engine torque, given as
Wig = Vd · _mfuelQhv
60
N
2
Vd
ηe
Wp = Vd · (pem − pim)
Wf = Vd · [0.97 + 0.15(N
1000) + 0.05(
N
1000)2]
Me =Wig −Wp −Wf
2πnr
(6–9)
For the equation to be linear in states pim and pem, the same assumption has to
be made, that is, N=3000. Moreover, set the combustion efficiency ηe to be a constant
0.3469. Thus the engine torque is able to be expressed linearly in states pim and pem,
which is given as:
Me = k3pim + k4pem + k5 (6–10)
57
Figure 3-2 shows the validation of the linear engine torque model.
Figure 6-2. Validation of linearized engine torque model
In equations (6-8) and (6-10), k1 to k5 are all constants that are easily calculated.
Combining equations (6-8) and (6-10), it is possible to form the output matrices of
the system, given as
_mfuel
Me
=
0 0 0 0 3.92× 10−5 0
0 0 0 0 2.1 −0.16
ω
I
pc
pic
pim
pem
+
−3.13× 10−4 0
−15.2 0
1
α
(6–11)
Now according to equations (6-6) and (6-11), it is ready to form the state space
equation for the linearized system as
_x = A0x + B0u + B0mdumd
y = C0x +D0mdumd
(6–12)
58
where
x =
ω
I
pc
pic
pim
pem
, y =
_mfuel
Me
, u =
V
uwg
, umd =
1
α
(6–13)
Now everything is prepared, thus it is ready to apply MPC to the system according
to the procedure described in Chapter 5.
6.3 MPC Implementation In Matlab
The objective is to minimize the fuel consumption and the deviation from the
requested engine torque. In this thesis, the required engine torque Mreq=200Nm. There
fore, the cost function can be expressed as:
J = (
Np∑i=1
[ _mTfuelQf _mfuel + (Me −Mreq)
TQM(Me −Mreq)] +
Np−1∑i=0
(uTQuu) (6–14)
where Qf , QM and Qu are weighting functions in the form of
Qi =
qi I
qi I
. . .
qi I
(6–15)
where the subscript i stands for f, M and u respectively.
59
According to the procedure described in chapter 5, the first step is the discretization
of the continuous system. This can be completed in Matlab by doing
sysC = ss(A0,B1,C0,D1)
sysD = c2d(sysC ,T )
[A,B,C ,D] = ssdata(sysD)
(6–16)
whereT = 0.6s is the sampling time.
Here, the matrices A and C are the same as the ones of the original system. The B1
matrix is the combination of B0 and B0md , and the same goes with the D matrix, given as
B1 =
[B0 B0md
], D1 =
[D0 D0md
](6–17)
where matrix D0 is just 2 × 2 zero matrix. Now the discretization can be expressed
asxk+1 = ADxk + BDuk + BDmduDwg
yk = CDxk +DDmduDwg
(6–18)
where
AD=A
BD=B1(:, 1 : 2)
BDmd=B1(:, 3 : 4)
CD=C
DD=D1(:, 3 : 4)
Following the procedure described in chapter 5, the prediction state space equation
can be expressed as:
X = Exxk + FxU + GxUmd
Y = Eyxk + FyU + GyUmd
(6–19)
60
where Ex , Fx and Eyare of the same form of the ones described equations (5-6), (5-7)
and (5-9) in Chapter 5. The measured disturbance term Gx and Gy are given as
Gx =
BDmd 0 ... 0
ADBDmd Bmd ... 0
...... ...
...
ANp−1D BDmd A
Np−2D BDmd ... BDmd
(6–20)
Gy =
CDBDmd DDdm 0 ... 0
CDADBDmd CDBmd Ddm ... 0
...... ...
...
CDANp−1D BDmd CDA
Np−2D BDmd ... CDBDmd Ddm
(6–21)
Since there are two outputs of the system _mfuel and Me which are both in the cost
function, there needs to be two matrices that will separate the two outputs, given as
C1 =
1
0
. . .
1
0
, C2 =
0
1
. . .
0
1
(6–22)
Thus the cost function can be expressed now as
J = (C1Eyxk + C1FyU + C1GyUmd)TQf (C1Eyxk + C1FyU + C1GyUmd)
+(C1Eyxk + C2FyU + C2GyUmd − R)TQM(C1Eyxk + C2FyU + C2GyUmd − R)
+uTQuu (6–23)
61
where R is the reference matrix:
R =
200
200
...
200
(6–24)
Take the first derivative of J with respect to U and solve the differential equation:
∂J
∂U= 0 (6–25)
Then U can be expressed in the initial state xk . Since the system has two inputs, the
optimized uk should be the first column of the 2byNp matrix U, given as:
uk =
1 0 0 ... 0
0 1 0 ... 0
U (6–26)
Repeating the process described in chapter 5, it is easy to obtain the optimized input U
6.4 MPC With Constraints
In reality, there should be constraints on the input of the system, which means the
battery voltage input and wastegate opening should be both in a reasonable range. In
this thesis, the input constraints are
0 ≤ uv ≤ 60
0 ≤ uwg ≤ 1
(6–27)
Now it is ready to add constraints to the system.
6.4.1 Overview of MPC with Constraints
With the same cost function, now the goal is to solve the optimal problem subject to
some constraints, described as
min1
2UTHU + UTF
s.j LU ≤ b
(6–28)
62
where matrices H, F, L and b are formulated by C1, C2, Ey , Fy , Gy , Umd , xk , Qfuel , Qtorque
and Qu given before.The next step is to form the matrices H, F, L and b and apply the
constraints to the system.
6.4.2 Add Constraints to the System
It is necessary to rewrite the equation (6-27) to form the matrices L and b in the
following way:
uv ≤ 60
−uv ≤ 0
uwg ≤ 1
−uwg ≤ 0
(6–29)
Rewrite (6-29) into matrix format as
1 0
−1 0
0 1
0 −1
uv
uwg
=
60
0
1
0
(6–30)
Define
l =
1 0
−1 0
0 1
0 −1
, m =
60
0
1
0
(6–31)
Now it is ready to form the L and b matrices as
L =
l
l
. . .
l
, b =
m
m
. . .
m
(6–32)
63
The next step is to rewrite the cost function J to form the matrices H and F. Based
on the original cost function, it is easy to rewrite it in the following form
J =1
2UTHU + UTF (6–33)
where
H = FTy C
T1 QfuelC1Fy + FT
y CT2 QtorqueC2Fy +Qu
F = FTy C
T1 QfuelC1(Eyxk + GyUmd) + FT
y CT2 QtorqueC2(Eyxk + GyUmd)− FT
y CT2 QtorqueR
Now using the Hildreth Quadratic Programming procedure described in [19], it is
ready to run the simulation of the system with constraints.
6.5 Simulation
After implementing the MPC controller, it is ready to run the simulation. Set the
control horizon and predict horizon to be the same Np = Nc = 90 and run the simulation
for 90s. Figure 6-3 and Figure 6-4 shows the simulation results.
Figure 6-3. Fuel consumption after optimization
The results show that the designed MPC controller achieves the goal of minimizing
fuel consumption and in the meantime preventing the engine torque from deviating too
much from the desired torque.
Figure 6-5 and 6-6 shows the optimized inputs of the system. It can be concluded
that in the case the throttle angle at 30◦ and engine speed at 3000 rpm, it requires
64
Figure 6-4. Tracking of the desired engine torque
54.76V battery voltage and 26% wastegate opening to obtain the minimum fuel
consumption and deviation from the desired engine torque.
Figure 6-5. The required battery voltage to minimize the fuel consumption and deviationfrom desired engine torque
6.6 Chapter Conclusion
In this chapter, the hybrid turbocharger model designed in Chapter 4 is reduced
and linearized so that the linear MPC controller could be designed and applied to it. The
simulation results show that the designed controller works well on the linearized system
65
Figure 6-6. The required wastegate opening to minimize the fuel consumption anddeviation from desired engine torque
by achieving the goal of minimizing the fuel consumption and and the deviation of the
real engine torque output.
66
CHAPTER 7CONCLUSION AND FUTURE WORK
In this thesis, the SI engine with a hybrid turbocharger installed is designed by
building the battery model and each component of the turbocharged SI engine. The
comparison has been made between the hybrid turbocharger with the conventional
turbocharger and the naturally aspirated engine respectively, demonstrating the
two main advantages of the hybrid turbocharger in eliminating the turbo lag and
engine downsizing. Then the linear MPC controller is designed and applied to the
simplified hybrid turbocharger system after model linearization to realize the objective of
minimizing the fuel consumption and deviation from the required engine torque.
Future works that needs to be done focus on two aspects: the first one is to design
the nonlinear MPC controller which can be applied directly to the nonlinear hybrid
turbocharger model without model linearization. This will provide a more accurate result
and can be used in experiment with real hardware. The second one, as mentioned in the
first aspect, is to perform the real-time MPC by designing the nonlinear MPC controller
and applying it to the real engine model, which has a very practical and important
significance.
67
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69
BIOGRAPHICAL SKETCH
Kang Rong received his Bachelor of Science degree in Automotive Engineering
in 2008 in Shandong University, Jinan, China. He is now pursuing his Masters
of Science degree in the Department of Mechanical and Aerospace Engineering
in University of Florida. His research interests are: Turobocharged SI engine
modeling, Hybrid turbocharger design and Application of nonlinear control and
Model Predictive Control.
70
