MODELING AND CONTROL OF AN ENGINE FUEL INJECTION SYSTEM
TAN CHEE WEI
A project report submitted in partial fulfilment of the
requirements for the award of the degree of
Master of Engineering (Electrical – Mechatronic and Automatic Control)
Faculty of Electrical Engineering
University Teknologi Malaysia
NOVEMBER 2009
iii To my beloved father, mother and brothers.
iv ACKNOWLEDGEMENT
I would like to express my sincere appreciation and thankfulness to my final year
project supervisor, Dr. Hazlina Binti Selamat, regarding his guidance, support and
willingness to help throughout my final year project progress. She has provided me with
her valuable advice and suggestion so that I can follow the right track in performing all
necessary tasks and complete the project as well. Besides, she also acts as language
supervisor to check on my documentation. I believe that without her assistance, my
project will not be able to operate smoothly and complete on time.
I am also indebted to librarians for their assistance in supplying the relevant
literatures. My sincere appreciation also extends to my friends who have provided
assistance at various occasions. Their views and tips are useful indeed. Finally, I would
like to thank my parents and brothers for their encouragement and support who had
helped me go through all the difficulties that I faced throughout my project.
v ABSTRACT
Control of automotive exhaust emission has become an important research area
to meet the more stringent automotive emission regulations. Beside the modification on
internal combustion engine, control engineering is seen as another approach to improve
and meet these requirements. This project focuses on the design and development of a
control system to reduce the harmful waste of automotive exhaust emission. The control
system aims to regulate the amount of fuel injected into the combustion chamber such
that the air to fuel ratio (AFR) is maintained within the allowable range. The control
process in this project is demonstrated based on an analytical engine model that clearly
describe engine’s air and fuel dynamic with no loss of engine system performance. Since
the dynamics of the internal combustion engine and fuel injection systems are highly
nonlinear, a linear model is obtained in this project, based on a system identification
approach to allow methodical application of linear control theories. Two types of control
strategy are employed – the linear quadratic Gaussian (LQG) controller and the fuzzy
logic controller (FLC). The LQG controller, designed based on the linear model of the
engine system, results in good controlled output response but with large controller signal
variation. The FLC, however, provides better controlled output response by reducing
overshoot gain and transient effect occurred in LQG controller design.
vi ABSTRAK
Kajian dalam mengurangkan lepasan toksik dari ekzos semakin penting pada
masa kini demi memenuhi peraturan yang semakin ketat. Pada hari ini, melibatkan
sistem kawalan dalam enjin telah menjadi salah satu jalan penting dalam mengurangkan
lepasan toksik selain menjalankan modifikasi pada enjin. Projek ini akan fokus pada
penghasilan dan penciptaan system kawalan yang mampu mengurangkan pelepasan gas
toksik dari enjin ekzos ke udara. Sistem kawalan yang dicipta bertujuan untuk mengawal
jumlah kuantiti petrol yang dibenarkan untuk menyembuh ke dalam chamber enjin dan
menetapkan AFR pada jumlah yang dibenarkan. Dalam projek ini, penghasilan sistem
kawalan akan bergantung pada simulasi enjin model. Projek ini telah memilih enjin
model berasaskan cara analisasi yang mampu menterjemaahkan petrol dan udara proses
dalam enjin dengan kejituan yang tinggi. Akan tetapi, ciptatan sistem kawalan dalam
simulasi gagal diterima disebabkan oleh enjin proses yang tidak linear. Oleh itu, teknik
berasaskan sistem identification dipakai demi menghasilkan enjin model yang linear.
Dua jenis sistem kawalan akan dibincang dalam projek ini iaitu Linear Quadratic
Gaussian (LQG) dan Fuzzy Logic Controller (FLC). Sistem kawalan LQG dihasil
berasaskan enjin model yang linear manakala FLC dihasil berasaskan model enjin yang
tidak linear. Keseluruhnya, LQG mampu memberi bacaan AFR yang bagus. Akan tetapi,
ia menyebabkan signal kawalan yang berulang alik. Sistem kawalan FLC pula, mampu
member bacaan AFR yang lebih bagus daripada LQG. Kelemahan sistem kawalan LQG
telah dibaiki sepenuhnya dalam implikasi sistem kawalan FLC.
vii TABLE OF CONTENTS
CHAPTER
1
2
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENTS
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
x
LIST OF FIGURES
xi
LIST OF ABBREVIATIONS
xiv
LIST OF SYMBOLS
xvi
LIST OF APPENDICES
xviii
INTRODUCTION
1
1.1
1.2
1.3
1.4
1
2
6
7
Control System Overview
Background of the Study
Objectives of the Research
Organization of the Report
LITERATURE REVIEW
9
2.1
Introduction
9
2.2
Engine Model
10
viii 2.2.1 Engine Parts Description
2.2.1.1 Inlet Manifold
11
2.2.1.2 Exhaust Manifold
12
2.2.1.3 The Intercooler, Compressor and Turbine
12
2.2.1.4 The Turbocharger
13
2.2.1.5 The Exhaust Gas Recirculation (EGR)
13
2.2.2
Review of Engine Modeling Method
13
2.2.3
Analytical Models
15
2.2.3.1 Filling and Emptying Model
15
2.2.3.2 CFD Approach
16
2.2.3.3 Mean Value Model
17
Empirical Models
17
2.2.4.1 Neural Network
18
2.2.4.2 Polynomial Method
20
2.2.4.3 Interpolation from Steady State Maps
21
2.3
Review of Control System Applied to Engine Model
21
2.4
Conclusion
24
2.2.4
3
11
METHODOLOGY
26
3.1
Introduction
26
3.2
Engine Mathematic Model
27
3.2.1
The Air Dynamic
28
3.2.2
The Fuel Dynamic
32
3.2.3
The Rotation Torque Dynamic
35
3.3
Engine System Identification Theory
38
3.3.1
Experiment Design
40
3.3.2
Data Preprocessing
40
3.3.3
Model Estimation
41
3.3.3.1 State Space Model Using a Subspace
3.3.4
Method
42
Model Validation
45
ix 3.4
Linear Quadratic Gaussian (LQG) Controller
45
3.5
Fuzzy Logic Controller (FLC)
49
3.5.1
Fuzzification
50
3.5.2
Rule Base
52
3.5.3
Defuzzification
54
Conclusion
57
3.6
4
5
RESULT AND DISCUSSION
58
4.1
Introduction
58
4.2
Engine Model Using System Identification Technique
62
4.2.1
Import Data, Select Range and Data Preprocessing 63
4.2.2
Estimat Model Structure
67
4.2.3
Validation Estimated Model Performance
68
4.3
Linear Quadratic Gaussian (LQG) Controller
69
4.4
Fuzzy Logic Controller (FLC)
78
CONCLUSION AND FUTUREWORK
85
5.1
Conclusion
85
5.2
Futurework
86
REFERENCES
87
Appendices A-D
92-97
x LIST OF TABLES
TABLE NO.
1.1
TITLE
Fuzzy Rules
PAGE
53
xi LIST OF FIGURES
FIGURE NO.
TITLE
1.1
Historical view of emission legislation for vehicle
1.2
Percentage of pollutant conversion due to engine
PAGE
4
air fuel ratio
6
2.1
Conventional engine model types
10
2.2
Schematic representation of the diesel engine
11
2.3(a)
Analytical engine model types
14
2.3(b)
Empirical engine model types
14
2.4
Typical multi layer perception neural network structure
18
3.1
Diesel engine model implement in MATLAB-SIMULINK 28
3.2
Schematic of the air system
29
3.3
Schematic of fuel injection system
32
3.4
Piston engine model
35
3.5
Cycle of system identification function
39
3.6
State space structure model
42
3.7
LQG controller structure model
46
3.8
Fuzzy logic controller block diagram
49
3.9
Inputs membership function of error (a)
and change in error(b) contain in fuzzification process
3.10
51
Fuzzy output membership function with
participation of 5 fuzzy set ZO, ML, MM, MH.
54
xii 3.11
Graphical construction of the control system
in a fuzzy controller
55
3.12
defuzzification process
56
3.13
Fuzzy logic controller structure model
57
4.1
Variation of engine air throttle
59
4.2
Engine’s Air fuel ratio
59
4.3
Effect of air-fuel ratio on power, fuel consumption,
and emission
4.4
Engine output torque due to variation of input
air throttle angle value.
4.5
60
61
Engine’s acceleration reading due to variation
of input air throttle angle value.
61
4.6
System identification toolbox in MATLAB software
62
4.7
Engine model with assigned random signal into
engine’s input signals of beta and Alfa.
4.8
63
(a) and (b) shows output and input response from
engine model due to assigning of random signal
as model input and work for system identification purpose. 64
4.9
Estimate and validate data for randomness
input Beta, u1, Alfa, u2 and output AFR, y1.
4.10
66
System identification toolbox with linear parametric
model window
66
4.11
Actual and estimated plant output response
69
4.12
Output response and controller gain performance
under large and small weighting gain
4.13
Air fuel ratio response with LQG compensator (blue)
and without LQG compensator (green)
4.14
73
75
Air fuel ratio response with modified LQG
compensator (blue)and without LQG compensator (green) 76
xiii 4.15
Air fuel ratio response with modified LQG
compensator (blue)and without LQG compensator (green)
at time 350s to 550s
76
4.16
LQG controller output response
77
4.17
LQG controller output response display at
time 350 s to 550s
4.18
77
AFR response from engine model with fuzzy logic
controller (green) and without fuzzy logic controller (blue) 78
4.19
AFR response from engine model with fuzzy logic
controller (green) and without fuzzy logic
controller (blue) crop from time in between 350s to 550s
4.20
Square error value from engine AFR without fuzzy logic
controller (blue) and with fuzzy logic controller (green)
4.21
80
Square error value from engine AFR without LQG
controller (blue) and with LQG controller (green)
4.22
79
80
Close view of Square error response from engine
AFR without controller and with controller of
(a)LQG and (b) FLC
81
4.23
FLC controller output response
82
4.24
FLC controller output response display at
time 350 s to 550s
82
4.25
Effective fueling time constant
84
4.26
Engine rotational torque
84
xiv LIST OF ABBREVIATIONS
AFR
-
Air Fuel Ratio
FLC
-
Fuzzy Logic Control
CO
-
carbon monoxide
HC
-
Hydrocarbons
NOx
-
Nitrogen Oxides
CFD
-
Computational Fluid Dynamic
PI
-
Proportional Integral
LQG
-
Linear Quadratic Gaussian
LQR
-
Linear Quadratic Regulator
ze
-
estimated model
zv
-
validated model
A
-
an n-by-n system matrix
B
-
an n-by-m input matrix
C
-
an r-by-n output matrix
D
-
an r-by-m transmission matrix
Co
-
controllability
Ob
-
observability
H,Q, R
-
weighting matrix
Rk,Qk
-
noise covariance data
v
-
measurement noise
w
-
process noise
e
-
system error
xv ECU
-
Electronic control units
K
-
LQG controller gain
xvi LIST OF SYMBOLS
-
mass rate of air in the intake manifold
-
mass of air in the intake manifold
-
mass rate air entering the intake manifold
-
mass rate of air leaving the intake manifold and entering the combustion
MAX -
the maximum flow rate corresponding to full open throttle
TC
-
Normalized throttle characteristic
PRI
-
Normalized pressure influence function
α
-
the throttle angle
-
intake manifold pressure
-
atmosphere pressure
-
constant value
-
gas constant
-
gas temperature
-
intake manifold volume
-
engine angular velocity
-
volumetric efficiency
-
fuel rate entering the combustion chamber
-
command fuel rate
-
effective fueling time constant
-
desired air fuel ratio
∆
-
intake to torque production delay
∆
-
compression to torque production delay
xvii AFI
-
normalized air fuel ratio influence function
CI
-
normalized compression influence function
-
the maximum torque production capacity of an engine given that
AFI=CI=1
A/F
-
actual air fuel ratio of the mixture in the combustion chamber
CA
-
tuning parameter of cylinder advance at the Top Dead Center
MTB -
minimum tuning such that best torque acquire
-
effective inertia of the engine
-
engine indicated torque
-
engine friction torque
-
accessories torque
-
Cost funtion
-
Ricatti gain
-
expected states
xviii LIST OF APPENDICES
APPENDIX
A1
TITLE
Engine’s air flow dynamic represent
in MATLAB-SIMULINK
A2
95
LQG compensator with engine model
in MATLAB-SIMULINK
D
94
The Fuzzy Logic controller and engine
model in MATLAB-SIMULINK.
C
93
Engine’s rotational torque dynamic
represent in MATLAB SIMULINK.
B
92
Engine’s fuel injection dynamic
represent in MATLAB-SIMULINK
A3
PAGE
96
Performance enhancement to LQG
compensator with extra derivative block
97
CHAPTER 1
INTRODUCTION
1.1
Control System Overview
Control is defined as maintaining desired conditions in a physical system by
adjusting selected variable in the system (Stewart, 1995). There exist several reasons
why control system is necessary to implement in human life. The major reason of
control system application is to maintain desired output even when external disturbance
is occurred. For example control of temperature in a room, water level in a tank, power
supply of control room and etc while the second reason for control is to respond to
change in the desired value. For example, if the fluid level in a tank is increased,
percentage opening of control valve will be decreased in order to maintain desired value
of fluid level (Stewart, 1995).
In general, there are two types of control system structure- open loop control and
close loop control. For systems in which the output has no effect on the control action
they are called open loop control systems. In this case, output of open loop control
system is neither measured nor fedback for comparison with the input. On the other
hand, a closed loop control system or commonly called feedback control is capable in
2 feeding in an actuating error signal, which is the difference between the input signal and
the feedback signal( from output) to a controller so as to reduce the error and bring
output of the system to desired value (Lukáš, 2008). As a result, the controller design
become an important part yet critical in control system since it determines whether
performance of a system is good or poor.
1.2
Background of the Study
In the past decades, development of earth moving vehicle’s engine was mainly
focused on fuel efficiency and performance increment such as torque, horse power and
revolution of vehicle without worry on emission legislation. However in today situation,
emission legislation is no longer an easy challenge for vehicle manufacturer to pass
through when the numbers of vehicle all around the world has reached 50 millions in
2007 and expected to increase by 5% every year and reach approximate 60 million at
year 2010(Chang, 2007). The development of automotive market would bring many
negative effects that require serious consideration by automotive industrial. For example,
today, large quantity of earth moving vehicles has turned internal combustion engine
exhaust emission one of the main contributors to environment pollution with harmful
gases such as
•
carbon monoxide(CO)
•
Hydrocarbons(HC)
•
Nitrogen Oxides(NOx)
•
Particulate emission.
Carbon monoxide is a very toxic, colorless and odorless gas, which is generated
in the exhaust gas, as the result of incomplete combustion of fuel. As engines operate at
enclose spaces such as car park or tunnel, it can accumulate very quickly and reach
3 concentration which could harm humans health by causing headaches, lethargy or
dizziness. As well as carbon monoxide, hydrocarbons are also produced due to the
incomplete combustion of fuel. Generally, it causes bad impact to environment by
influencing earth ozone reactivity with contribution of smoke and has characteristic of
nuisance smell. Nitrogen oxides on the other hand are generated from nitrogen and
oxygen from air intake manifold of engine when air flow through the engine cylinder
under high pressure and temperature. Nitrogen oxides is a reactive gas and very toxic to
human. Emission of nitrogen oxides will also deteriorate ozone reactivity and cause
smog formation, which is a serious environment concern in today situation.
Therefore, due to global warming effect and environment protection, a lot of
attention has been focused on automotive industry and it started to become a hot topic in
climate discussion. These has force cars manufacturer and their supplier to develop new
engine control strategies within short time period instead of using traditional technology
to meet strict and stricter emission legislation from government(Ericson, 2007).
There are different control methods available for reducing pollutant components,
such as control of engine speed, engine torque, fuel injection timing, AFR and so on.
Among all, control of AFR is related to fuel efficiency, emission reduction and
drivability improvement, furthermore maintaining AFR at stoichiometric level can
obtain best balance between power output and fuel consumption (Muske, 2008).
Control of AFR also guarantee reduction of pollutant emission to atmosphere since
variation of AFR greater than 1% below 14.7 can result in significant increase of CO
and HC emission. An increase of more than 1% will produce more NOx up to 50%
(Kenneth, 2006).
Figure 1.1 shows historical view of worldwide emission legislation. It shows
that the allowable nitrogen oxide emission was reduce from 7 g/kWh in the year 1996 to
less than 1 g/kWh in the year 2010. Emission legislation Euro III at year 2000 shows
limits on allowable vehicle NOx emission, which reduce to less than 5 g /kWh, and this,
has been achieved through application of higher injection pressure to result in low
4 particulate emission and retarded injection. However, emission legislation Euro IV and
Euro V are no longer achievable by using the technology applied in Euro III. Therefore,
car manufacturers have introduced new technologies such as cooled Emission Gas
Recirculation (EGR) and Selective Catalytic Reduction to reduce NOx emission in order
to meet legislation requirement. Today, the technology of Selective Catalytic Reduction
is still applied in most vehicles due to its simple, practical and cost effective benefits.
Figure 1.1: Historical view of emission legislation for vehicle
(Ericson, 2007)
In general, Selective Catalytic Reduction can be divided into two types-:
Oxidation catalyst system and 3-way catalyst system. In this case, oxidation catalyst
system is effective in reducing two major exhaust pollutants of carbon monoxide and
hydrocarbons, through oxidation to carbon dioxide and water vapor (Tetsuji, 2004) as
shown in Equation (1) and Equation (2).
1
2
(1)
(2)
5 However, this method is not longer used for emission control due to its low
performance on reducing NOx components and meet stricter emission registration.
Therefore, a newer catalyst technology, which is known as 3-way catalyst, was
introduced (Tetsuji, 2004). In 3-way catalyst, three major pollutants, carbon monoxides,
hydrocarbons and nitrogen oxides are simultaneously convert to carbon dioxide, oxygen
and nitrogen. Equation 3 shows chemical conversion of pollutant within 3-way catalyst
into environment friendly components.
(3)
The fundamental reaction in 3-way catalyst is between CO, HC and NOx.
Therefore in order to achieve high percentage of conversions from all three environment
pollutants- HC, CO and NOx into environmental friendly components, their
concentration must be in stoischiometric ratio (Ali, 2008). This means that total amount
of HC and CO should match the amount of NOx present in the system, in such a way
exact equations of chemical reaction can be occurred in catalyst.
However, there is no way both of the components can meet stoichiometric ratio
all the time, since concentrations of pollutants in the exhaust gas are highly depend on
the fuel mixture composition. For example, at lean fuel mixtures the exhaust gases
contain little carbon monoxide and hydrocarbons but high concentrations of NOx. On
the other hand, at rich fuel mixtures the exhaust gas contains high concentrations of
carbon monoxide and hydrocarbons but low concentration of NOx. Therefore, amount
of engine’s fuel injection should be controlled in such a way so that engine’s air fuel
ratio (AFR) is at the stoischiometric value of 14.7 and achieve full conversion of
pollutant components as shown in Figure 1.2.
6 lean rich Figure 1.2 Percentage of pollutant conversion due to engine air fuel ratio
(Ali Ghaffari, 2008)
As a result, a compatible and suitable controller is required to be applied into
engine’s system such that engine’s AFR can be maintained at stoischiometric range, thus
resulting in high conversion efficiency of pollutant components.
1.3
Objectives of the Research
Based on the issue that variation of AFR deviating away from stoichimetric ratio
can result in high concentration of pollutant from exhaust emission as discussed in the
previous section (section 1.2), the project objective is therefore to maintain the engine’s
Air Fuel Ratio at stoischiometric level. This objective can be achieved through the
following efforts:
7 i)
To identified suitable mathematical engine model for AFR controller
design purpose.
ii)
To design and develop FLC and LQG control system for AFR control
purpose in MATLAB-SIMULINK
iii)
1.4
To ascertain the performance of the developed control system
Organization of the Report
The scope of work in this project concentrates on the engine and control system
modeling follows by ascertain of control system performance using MATLABSIMULINK. This report will be build up by five chapters, which are introduction in
chapter one, methodology in chapter two, literature review in chapter three, result and
discussion in chapter four, last but not least conclusion and future work in chapter five.
Following are important content and description of each chapter.
Chapter two will concentrate on literature review of engine and controller
modeling method. In this project, mean value method been applied for engine modeling.
Besides that, there are several available engine modeling method existed. For example,
CFD method, Filling and Emptying method, Polynomial method and so on. Advantages
and disadvantages of each method also application method of each method will be
explained in this chapter. For controller modeling, several types of controller shall be
reviewed. Performance and advantages of each controller will be discussed for decide
and decision making purpose on suitable controller.
Chapter three is descript the methodology of engine plant and control system
modeling. The simulated engine model is modeled by three blocks: Fuel dynamic, Air
dynamic and rotation torque dynamic. Each block is correlating between each other.
8 Sets of model equation and formula which contribute to each block will be explain and
descript in this chapter. Two types of controller will be discussed in this report:-fuzzy
logic controller (FLC) and linear quadratic Gaussian (LQG) controller. In chapter three,
the control algorithm and theory from each controller will be explained.
Chapter four will discuss FLC and LQG controller performance in controlling
engine model’s AFR. Simulation result from FLC and LQG controller will be compared
and investigated to determine suitable controller, which work well with engine plant.
Last but not least, chapter five is the project conclusion and future work description.
9 CHAPTER 2
LITERATURE REVIEW
2.1
Introduction
Design and modeling of engine and controller structure, which use for simulation
purpose been popular since past decade. In this case, modeling of engine model is
important for research and development purpose, such as development in design of
engine control system, without damaging real engine during test bed. Over the past
decade, a vast number of engine and controller models have been developed by
researchers in many fields. The developed engine and its control system model are
employing diverse technique between each other depend on the application requirement.
Therefore, design of any engine model needs to be done carefully and validated in order
to suit the particular characteristics of the engine for intended control application.
10 2.2
Engine Model
During modeling of engine parts, it is generally not possible to use another
research’s work verbatim, as the context will be different depend on someone’s work
objective. As such perhaps the most useful aspect is to review of previous work then
identify and evaluate most suitable structure of the models to be considered
(ANDERSSON, 2008).
In most of the developed engine models, they are tends to use a combination of
analytical and empirical methods to represent the engine components, but it may be
classified according to their majority content.
ENGINE MODEL
ANALYTICAL
EMPIRICAL
METHOD
METHOD
Figure 2.1: Conventional engine model types
According to Figure 2.1, show two of the most popular engine modeling method
of empirical and analytical method. In this case, analytically based models tend to focus
on degree by degree detail variation of engine variables in some considerable detail and
are hence generally has slow running speed during simulation. On the other hand,
empirically based models tend to take a wider view on engine components and
predicting the mean values or trends of the major engine variables with a subsequent
loss of resolution in the limit compare to analytically based method (.Lukáš ,2008).
However, empirically based model usually allows much shorter simulation run times
due to ignore of particular engine variable.
11 2.2.1 Engine Parts Description
An engine model is developed and make up by numbers of components such as
inlet and exhaust manifolds, the pumping action in engine, the turbocharger, intercooler
and EGR circuit.
Figure 2.2: Schematic representation of the diesel engine
(Stefanopoulou, 1999)
2.2.1.1 Inlet Manifold
Inlet manifold is the most important part of engine model. In this case, the
manifold is filled with air from the intercooler. And yet the gas flow out of the manifold
12 is dependent on the pumping action of the engine which in turn is affected by the
volumetric efficiency.
2.2.1.2 Exhaust Manifold
The exhaust manifold is function to transporting the incoming exhaust mass flow
come from engine cylinder. In this case, the outgoing mass flow rate from exhaust
manifold is coupled to the action of the exhaust turbine. The mass of gas in the exhaust
manifold gives the manifold gas pressure which then act on the turbine. Therefore, the
pressure and mass flow gives the operating point of the turbine (Stewart, 1995).
However, some developed engine models are neglecting the exhaust manifold modeling
since its time constant is so small that does not affect the accuracy of the simulation.
This is so because of the high exhaust temperature which causes the volume flow to be
high and directly reduce value of time constant yet exhaust manifold is still useful to
improve turbine model stability (Stewart, 1995)
2.2.1.3 The Intercooler, Compressor and Turbine
Intercooler is function to cool the compressed hot gas from the compressor. The
lower the inlet temperatures of the air entering the engine, the higher the density, which
mean more air mass will enter the engine. This has directly reduced the NOx emission
due to low temperature in engine cylinder. The model of intercooler is implementing
between the compressor and the inlet manifold within engine model. Therefore, the
intercooler is assumed to lower the temperature to a constant level without heat transfer
calculations on it. On the other hand, the influence of the turbine shaft speed on the
mass flow rate and pressure ratio is very small. Hence it is excluded from the engine
modeling, where the only pressure ratio is used to give the mass flow and efficiency.
13 2.2.1.4 The Turbocharger
Turbocharger is another important component of diesel engine. It is function to
increases the output power by allowing more fuel to be injected in to engine cylinder
through use of exhaust energy that supercharge the engine. In this case, the transient
responsiveness of the diesel engine can reach to a great extent due to ability of the
turbocharger to deliver sufficient air so that the air fuel ratio isn’t too low(RACHID,
1994).
2.2.1.5 The Exhaust Gas Recirculation (EGR)
EGR is function to lower NOx emissions by diluting the inlet air with inert gas
which lowers the combustion temperature. In this case, the EGR flow which replacing
part of the air that goes into the engine cylinder is driven by the pressure different
between the inlet and exhaust manifolds. Therefore, the EGR rate can be control
through the valve placing between inlet and exhaust manifolds (RACHID, 1994).
However, the EGR will reduce the air mixture into cylinder that can cause power
reduction to engine. So, the EGR flow is shut off to allow a faster response of the
turbocharger during transients process occurred.
2.2.2
Review of Engine Modeling Method
Mathematical model of each engine component as discussed in previous sub
chapter (2.2.1.1-2.2.1.5) will be combined to result a complete engine model with
14 reliable performance as well as real engine. However, there are different types of engine
models available that applying different mathematical function based on design needs
and requirement. Figure 2.3(a) and Figure 2.3(b) shows six popular engine models
applied in development and research work.
ANALYTICAL
MODELS
Mean value models
CFD approach
Filling and emptying model
Figure 2.3(a): Analytical engine model types
EMPIRICAL
MODELS
Interpolation from steady state maps
Neural network
Polynomial methods
Figure 2.3(b): Empirical engine model types
15 2.2.3
Analytical Models
The analytical models is describing engine model through detail mathematical
model and function with inclusion of physic laws. In this chapter, three most popular
analytical models of filling and emptying model, CFD approach and mean value model
will be reviewed.
2.2.3.1 Filling and Emptying Model
Filling and emptying engine modeling method is represented as a series of
control volumes linked by nodes or valves. In this case, it is applying the principles of
mass and energy conservation to predict engine’s performance.
For example, the
turbocharger performance is predicted by an empirical sub model, and engine’s emission
is predicted through complementary multi zone models that run alongside of single zone
model with inclusion of some input from the single zone model. On the other hand, this
model is also considered on engine’s rate of heat transfer. In this case, the rate of heat
release is determine through experimental data or simulated using a homogeneous heat
release function based on fuel burning rate. Therefore, this treatment achieves faster
simulation running time compare to those detailed simulations whilst retaining sufficient
complexity to make accurate predictions.
The filling and emptying model is useful as engine design tool since it predicts
the effects of varying engine parameters on the performance. Besides that, a high level
of analytical understanding of the engine operation is encapsulated within the code
which helps and assists people to appreciate the workings of the system. However, it
has a disadvantage of having long simulation time that might too long to be viable for
control system design or online predictions.
16 2.2.3.2 CFD Approach
The computational fluid dynamic (CFD) engine modeling is providing the most
detailed approach among all engines modeling approach. This technique is employed to
predict fluid flow properties within the engine. In this case, the engine combustion
chamber is split into many discrete volume, typically as 40000 element are used (Ericson,
2007). Differential equations governing the behavior of each element are then solved to
acquire desire result. The simulated result may be validated by comparing them to
measured results using laser Doppler or hot wire anemometry.
CFD approach is requiring extremely large computing overheads.
This is
directly caused the overall computation technique difficult to handle moreover time
consuming. However, the CFD approach provides a great understanding and knowledge
of the air flow pattern within the engine. In practical example, Somerville (1993) has
used this technique to investigate the air motion in a diesel engine to compare the
predicted result with experimental measurement.
Besides concentrate on calculating engine air flow process, Payri(1988) has
applying the same method to model the complete combustion process which include the
evolution of fuel from start of injection until end of combustion. In this case, the fuel is
split into many ‘packet’, one release at each degree of crank rotation in such a case to
predicting the evolution at each time step. Therefore, the resulted engine model can be
used to predict engine performance and emission through further correlations and
analytical steps. But, it is unlikely that all CFD based engine structure could make to
run at fast simulation time which makes it impractical if many tests are to be simulated.
17 2.2.3.3 Mean Value Model
In order to cope with problem of long simulation running time, Hendrick (1990)
has developed a new engine model of mean value engine model (Eko, 2001). Engine
model with mean value approach is function to speed up a simulation dramatically
which useful for test and simulation purpose. In this case, the mean value approach is
neglect the cycle variation of engine parameters and to use instead the mean value. Here,
the time scale is shorter compare to previous method such that mean value method just
adequate to describe accurately the changing mean value of the most rapidly changing
engine variable.
Hedrick (1990) has work out the mean value engine model with three differential
equations and a number of instantaneous expressions to represent the main engine sub
models, fuel dynamics, crankshaft dynamic and manifold air flow. On the other hand,
the model calibration constants are calculated from experimental data. In order to
improve the engine simulation running time, the experimental data is expressed as
relatively few constants to allow for quick running whilst retaining considerable
complexity and accuracy. Therefore, the improved simulation running time range has
make mean value model useful for control system design and possibly for a model based
controller.
2.2.4
Empirical Models
Empirical models are employing the experimental data for most of the predictive
engine process. There are many forms of empirical model available such as neural
18 network; polynomial and interpolation of steady state map depend on desire aims and
attributes.
2.2.4.1 Neural network
A neural network is useful for constructing nonlinear empirical models. In a
neural network, the engine model is constructed by large number of very simple units
which combine to represent any given relationship of input and outputs. Engine model
can be represented in various forms which typically design in term of multi layer
perception (MLP). Figure 2.4 shows the structure of typical network.
Figure 2.4: Typical multi layer perception neural network structure
A neural network consists of three groups of nodes or neurons: the input layer,
one or more hidden layers and an output layer. Each layer is fully interconnected to the
next via a series of connections, called synapses. The number of hidden layers and the
number of neurons in each is optional. Generally, accuracy will increase with the
complexity of the network until an optimum is reached. Thereafter accuracy will reduce.
The aim for this is to obtain the required accuracy with the most simple and therefore
quickest running network possible.
19 To make a network represent a real engine model, the weights and biases of
neural network must optimized by an iterative training process. Here, the training data
are presented many times and the network parameters incremented until the output
values converge to be the same as the desired output with an acceptably small error. In
this case, there are a number of algorithms available to perform the task of minimize
system error. The technique most commonly used for modeling applications is called
Stochastic Back Propagation. The code generates a rate of change of error and moves
across the error surface at a speed proportional to this slope. If the error gradient is steep;
large steps are taken. When the surface is flatter; smaller steps are more appropriate.
Stochastic back propagation differs in two important respects. The weights are changed
after each pattern is presented rather than after each epoch as for standard back
propagation. An epoch is the name given to one complete pass through all the training
data patterns. The other variation is that the patterns are presented in a random order to
prevent cyclic variations in the data affecting the learning process. Both of these changes
are designed to cope better with the scattered and noisy data often encountered in
modeling applications (Huang, 2006).
Once trained, the resulting structure can predict the outcome to scenarios it has
not previously encountered. Like any interpolation technique it is more accurate when
working within the boundaries of the training data, accuracy is also enhanced by
increasing the amount of training data presented. If highly non-linear behavior is being
modeled there must be enough data to define the relationship adequately. It is also
important that the inputs are sufficient to describe the behavior of the outputs. If this is
not the case there will be ambiguity in the data, leading to a failure to converge.
Furthermore, Inputs which do not have an influence on the output should be avoided as
the network may learn a coincidental relationship between two unrelated variables.
Once trained, the network is treated as a black box. Input values are presented
and the outputs are collected. A network can represent a huge volume of data in only a
few lines of code and runs very quickly once trained (Huang, 2006). For example,
O’Reilly (1994) has applied a neural network based air fuel ratio predictor for use in
20 engine control such that network models the complex behavior of the inlet manifold
using a time history of previous inputs to predict the future air fuel ratio. Shayler et al
(1995) has applied neural networks to the fuel consumption prediction of an engine with
varying operating temperatures.
2.2.4.2 Polynomial Method
First engine modeling uses of polynomial fit method was done by Stonach
(1988). This method is uses to predict engine torque from a turbocharger. In this case,
two pair of polynomials are used to predict the ratio of indicated torque to fuel rack
position and predict the ratio of net torque to fuel rack position. This model is useful for
control system design as it is compact and simulation of experimental data with
polynomial coefficient further reduce the simulation running time.
The new generation of polynomial engine modeling method is described by Jiang
(1992). This is a method combining analytical and empirical methods to model a diesel
engine. In this case, models of combustion and the turbocharger assembly are used to
predict an instantaneous equivalent ratio and engine speed. Then, the experimental data
of equivalent ratio and engine speed is used to predict the engine particle exhaust
emission through forth order polynomial curve fit. From the experimental data, different
polynomial coefficients are used for each of engine speeds spaced over the operating
range.
The polynomial method has advantage of result a compact model due to used of
polynomial to replace a look up table. However the model is not account for the effects
of varying engine temperatures or injection timing since both of these factors will also
affect the emission particle production. Therefore, it is not attempt to make an emission
prediction.
21 2.2.4.3 Interpolation from Steady State Maps
Interpolation from steady state maps is perhaps the simplest approach from
empirical engine modeling. In this case, experimental data of exhaust emission level at
a series of speeds and loads are captures from an engine running at steady state condition.
These data are then interpolated for speed and load to locate the instantaneous operating
point of the engine during a transient test.
The resulting map has a limitation of not account for transient accuracy. But the
emissions of a diesel engine are critically dependent on the turbocharger and intercooler
performance during transient and steady state condition. Therefore, extensions of the
mapping method are needed to include the effect of transient model.
One such engine modeling which included the steady state mapping and transient
mapping is developed by Watson (1989). In this case, the conventional engine map of
emission with data of torque and speed is extended to include speed and torque
derivative axes. Here it is requires the collection of transient data. Therefore, it shows
improvement in accuracy over the steady state method when used to predict transient
emissions. However, this technique may cause large error due to wide scattering of
experimental data points. As a result, a higher density of data would needs to improve
the model accuracy.
2.3
Review of Control System Applied to Engine Model
Now a day, development of automotive engine has indirectly brings to evolution
of control system, which work for control and improve efficiency of exhaust emission.
22 Initially, control effort is mainly concentrating on mechanical modification of engine
system. However, it is time consuming and cost large amount of money for investment
purpose. This approach is soon replaced by modern control with estimation algorithm as
the electronics transitioned to increasingly use in microprocessor in the 90s (Powell,
1998).
Over the past, there are numbers of researcher who have simulation or practically
implement engine microprocessor control and experimental evaluation for exhaust
emission control purpose. A team led by Hendricks, 1990 at the Technical University of
Denmark has become pioneer who introduced AFR control based on measurements in
the intake manifold. On the other hand, another approach which developed by ETH in
Zurich, Stanford University is to base the observer on measurements of oxygen quantity
from exhaust through oxygen sensor and on the throttle position. Furthermore, the
University of California at Berkeley has addressed the engine exhaust quality which use
for control purpose using a nonlinear, sliding mode approach (Powell, 1998).
Today, design and research of engine’s emission controllers are mostly aims for
AFR control, and exhaust quality control through sliding mode approach or through
oxygen sensor observer.
For example, Kenneth R. Muske and et al, (2008) from
Villanova University has introduced a classical way of PI controller which is developed
and applied for air fuel ratio control of engine. Here, an additional adaptive delay
compensator is introduced to overcome significant variation in the air fuel ratio dynamic
response as a function of engine operating condition. Whenever the engine is operated
lean or rich oxygen, PI compensator will result an output and accomplished AFR control
through adjusting the mass of fuel injected into the engine.
The engine control computer calculates a base fuel rate required to maintain
stoischiometric combustion using an estimate of the engine air flow. A multiplicative
correction to this base fuel rate, referred to as the base fuel multiplier, is then used as the
manipulated variable for air fuel ratio control. The actual air fuel ratio is measured by
an exhaust gas oxygen sensor placed in the exhaust pipe upstream of the catalytic
23 converter. However, PI compensator in this case face difficulty in achieving good AFR
control with rather long and time varying delay in the AFR output response due to
insensitive change in fueling.
On the other hand, Yildiz, Y. (2008) has introduced a conventional AFR control
system with inclusion of two nested controller:-adaptive controller. An adaptive control
system is divided into two loops. The outer loop controller is work to generates a
reference AFR for inner loop controller based; for instant, on the deviation of the
estimated three way catalyst, stored oxygen state that is only operate effectively when
the stored oxygen level was regulated at a range to accommodate further release or
storage during transient conditions is available. The inner loop controller is assigned to
maintain the AFR upstream of the three way catalyst at reference air fuel ratio, by using
the measurements of the feed gas air to fuel ratio with a linear Universal Exhaust Gas
Oxygen sensor to appropriate engine fueling rate.
The design of the inner loop consists of a feed-forward component which is fast
but may not be always accurate and a feedback component that is slower but eliminates
steady state error. The feed-forward component consists of estimation of the air and fuel
path dynamics combined with appropriate compensations. These air and fuel dynamics
correspond, mainly, to the intake manifold lag that affects the air charge, and the wallwetting that determines the amount of fuel inducted into the cylinder for each fuel
injection event during transient operation.
Previous section shows design of PI controller and adaptive controller, that are
simulated based on a linearized engine model. Here, Yao Ju-Bian (2009) and his team
from Beijing University of Technology have work on a simulation of AFR control based
on neural network that is work well in nonlinear engine plant.
In this case, the
traditional PI controller is combined with modern artificial intelligent control of neural
network. Neural network is performed to estimate the AFR signal without occurrence of
transportation delay. This effort in turn enhances performance of classical PI controller
through control of the transient air fuel ratio by using the estimated signal from neural
24 network but not maladjustment AFR signal that cause by transportation delay from
feedback oxygen sensor.
J.K.Pieper (1999) from University of Calgary, Canada has performed a
simulation for controlling AFR in nonlinear engine model using sliding model control
method. Sliding mode control is a technique to achieve perfect robust performance with
magnitude limited uncertainly due to the uncertain and nonlinear natural of the dynamic
of engine, which is in the range space of the control derivative of the plant. Control
efficiency is achieved by using state feedback and theoretically infinity high gain
actuation. Generally, a sliding surface, S is needed to achieve sliding mode control effort.
Sliding surface, S is defined as the difference between the actual and the desired AFR:
S=
,
,
,
,
(4)
(5)
S=
For AFR control problem, this surface is used to design the sliding mode and
adaptive controller.
2.4
Conclusion
Mean value engine modeling method has been chosen to perform in this project
due to its capability to present in compact manner and fast running simulation time
without lost of actual engine performance. Detail development of engine sub models
will be described in chapter 3.
On the other hand as mention in section 2.3, several types of controller has been
designed for engine exhaust emission control, so to cope with air fuel ratio output
25 response from deviate away from stoischiometric ratio, which is cause by imperfect fuel
control during vehicle acceleration and deceleration, thus results in increased of
pollutant emissions. In this case, the designed controller may perform well during
steady state condition but maintaining AFR during transient condition can become
difficult to them. Therefore, two different types of nonlinear controller:-fuzzy logic
controller (FLC) and linear quadratic Gaussian (LQG) controller will be designed and
their performance in transient conditions is evaluated.
26 CHAPTER 3
METHODOLOGY
3.1
Introduction
Since vehicle’s exhaust emission could harm human life by bringing pollution
and become the main source supply to global warming in the earth. Therefore, an
immediate and hurry step needs to be taken, in order to cope with this problem. In this
case, a controller with high effective and high capability thus is needed. The designed
controller is playing an important role to observe and interpret the controller input signal,
which come from exhaust emission quality, and fire suitable controller output such to
reduce harmful components from engine exhaust emission.
In this case, simulation models of internal combustion engine and control system
are thus needed. But, why simulation is needed? This is the question one always asks
oneself before starting of design. Answer: To get a better understanding of how things
work and, when dealing with large complex machines such as engine dynamic, it is the
only possibility for understanding how their components interact, how to foresee and
how a machine is going to react to the desire setpoint when being run in a new way can
preferably be obtained with the use of simulation (Chen, 2008).
27 When a model could describe reality plant in a correct manner, it is possible to
determine how an actual plant is going to operate in different configurations. Here,
model can be used for optimization of the actual plant, where trying all the possible
configurations in practice is too expensive and time consuming. Moreover, simulation
models is useful for controller training purposes yet enable troubleshooting of control
concept in computer instead of using a real engine test bed, so that mistakes can be made
without causing damage to hardware.
3.2
Engine Mathematical Model
The internal combustion engine model in this project is developed in MATLABSIMULINK software. In this case, engine model is grouped by three sub models
corresponding to the intake manifold of air dynamic, fuel injection dynamic and rotation
torque dynamic equipped with engine’s crankshaft speed. The simulation engine model
is a model based on the generic mean value engine model developed by Hendricks, a
well-known and widely used benchmark of engine modeling and control purpose (Eko,
2001). The engine model has an input signal of percentage opening of air throttle valve
and an output signal of engine’s air fuel ratio which include with controller signal of
total fuel flow rate that results to three engine’s state variables.
Figure 3.1 shows the resulted mean value model of diesel engine based on engine
model developed by Hendricks.
In this case, the engine model is form by three
important sub models that functioned to generate engine air fuel ratio at the engine’s
output.
28 Figure 3.1: Diesel engine model implement in MATLAB-SIMULINK
According to Figure 3.1, engine’s air fuel ratio will become the system
controlled variable while input air throttle valve opening is the system disturbance that
tend to disturbing engine’s AFR by causing AFR signal deviate away from setpoint
value of 14.7. The basis for this model is data collected during stationary measurement
and particular implementation of physic law. Therefore, certain required data such as
torque, air flow, fuel flow and emissions is needed to compiling the engine model with
equation which able to react in similar manner as well as actual plant.
3.2.1
The Air Dynamic
The air flow dynamic in engine’s inlet manifold is the main part of mean value
engine modeling. In an internal combustion engine, air is induced into the cylinders.
The airflow is first passes through an air filter to get the qualified fresh air. Then it
29 flows into the compressor, during which the air pressure is increased to be higher than
the atmospheric pressure. The charge air then flow through an intercooler to decrease
the intake air temperature. Hence the air density is increased again prior to the cylinder.
Finally, the manifold is filled with air from the intercooler. The gas flow out of the
manifold into the cylinder through inlet valve is depending on the pumping action of the
engine which in turn is affected by the volumetric efficiency,
.
FRESH AIR Figure 3.2: Schematic of the air system
Air mass balance in the inlet manifold is described by Equation (6):
(6)
Where
=
mass rate of air in the intake manifold
=
mass of air in the intake manifold
= mass rate air entering the intake manifold
= mass rate of air leaving the intake manifold and entering the combustion
30 In this case, the mass flow rate of air entering the intake manifold will be
described by Equation (7), which is close related to the engine’s throttle body that
describing energy transformation process from throttle angle inputs to mass rates of air
entering the intake manifold,
·
.
·
(7)
Where MAX= the maximum flow rate corresponding to full open throttle
TC= Normalized throttle characteristic
PRI= Normalized pressure influence function
From the normalized throttle characteristic, it is represented by a function of
throttle angle α as shown in Equation (8) that determined by an experiment with a data
table for simulation purpose.
1
1
cos 1.14459 ·
1.06
79.46
79.46
(8)
Where α = the throttle angle
While PRI, the normalized pressure ratio influence is represented by Equation (9).
1
Where
exp 9
1
(9)
= intake manifold pressure
= atmosphere pressure
The pressure in the manifold,
· ·
is calculated by Equation (10) of ideal gas law,
(10)
31 ·
·
(11)
·
= constant value
Where
= gas constant
=gas temperature
= intake manifold volume
From mass balance at Equation 6, the mass flow rate that exiting the manifold,
thus entering the engine body of cylinder chamber is descript by following Equation (12):
·
·
·
(12)
= engine angular velocity
Where
=volumetric efficiency
In this case, mass of air flow rate leaving the manifold is dependent upon engine
characteristic such as volumetric efficiency
and engine angular velocity
volume
, displacement
, such that
, intake manifold
is represented by Equation (13)
as below:
(13)
·
The volumetric efficiency
from mass flow rate of air, which exiting the
manifold, is determined by a non-linear empirical relation as shown in Equation (14). It
is work to represent the effectiveness of engine’s induction process. In this case, it is
form up by a complex function of engine geometry and model engine parameter through
experimental data.
24.5 ·
0.352
3.1 · 10
·
0.167 ·
222 ·
8.1 · 10
·
(14)
32 3.2.2
The Fuel Dynamic
In an internal combustion engine, fuel is injected directly into the engine cylinder,
just before the combustion process is required to start. Load control which is one of the
engine’s model state variable is achieved by varying the amount of fuel injected at each
cycle. Here, the engine’s AFR can be maintained through control of quantity of injected
fuel.
In a large size engine such as vehicle engine, direct-injection systems are used.
In this case, the diesel fuel-injection system consists of an injection high pressure pump,
delivery pipes and fuel injector nozzles; the governor is used to control the injected fuel
pressure and a timing device. The injection high pressure pump generates the pressure
required for fuel injection. The fuel under pressure is forced through the high-pressure
fuel-injection tubing to the injection nozzle, which then injects it into the combustion
chamber.
Figure 3.3: Schematic of fuel injection system
33 Amount of fuel injected is determined by the injection pump cam design and the
position of the helical groove. As the pump plunger arrives at the bottom dead center
(BDC), the pump-barrel inlet ports are open. Through them, the fuel, which is under
supply-pump pressure, flows from the pumps fuel gallery into the high-pressure
chamber of the plunger and barrel assembly. Then during pre-stroke process, it will
cause the retraction stroke and make the fuel pressure increases even higher.
During the effective stroke, fuel is forced through the high-pressure line to the
nozzle. The effective stroke is terminated as soon as the plungers helix opens the spill
port. In this case, changing the plungers of effective stroke will vary the injected fuel
quantity. To do so, the control rack turns the pump plunger in the barrel so that helix,
which runs diagonally around the plunger circumference, can open the inlet port sooner
or later and in doing so change the end-of-delivery point and thus the injected fuel
quantity (RACHID, 1995).
The plunger speed, and therefore the duration of injection, depends upon the
plunger actuating cams lift relative to the angle of cam rotation. This is why a wide
variety of different cam contours are required for everyday operations.
Mathematical modeling of the cam contour and helix groove is up to specific
components used in real engine work. Besides, the fuel spay condition is difficult to
model either. Therefore, here the fuel injection system is assumed as a linear system
with the signal input from the governor, which is up to the load condition as shown in
Equation (15).
·
Where
(15)
= fuel rate entering the combustion chamber
= command fuel rate
= effective fueling time constant
34 Equation (15) describes the relationship between fueling commands and fuel
flow rate into the cylinder, which is characterized by a combination of lag and transport
delays due to the discrete nature of the intake process. For a sequential-fire port fuelinjection system, the fueling model is simplified as a first order equation in term of the
actual fuel rate entering the combustion chamber
fueling time constant
0.05
Where
. ·
·
. On the other hand, the effective
is modeled as show in Equation (16):
·
= desired air fuel ratio
MAX= maximum torque capacity
(16)
35 3.2.3
The Rotation Torque Dynamic
Figure 3.4: Piston engine model
Engine’s piston model show in Figure 3.4 functions to comprises the air flow
through the inlet valve and the combustion torque calculation. In this case, the effect of
AFR on combustion torque is included as torque efficiencies measurement. Besides that,
the friction torque is taken into account too for effective torque calculation.
The torque built up in the engine is a function of engine speed, fuel flow and air
flow. The combustion and torque production subsystem contain delays associated with
the four combustion processes as modeled in the engine’s indicated torque
(17).
Equation
36 ∆
·
∆
·
∆
·
∆
(17)
Where ∆ = intake to torque production delay
∆
= compression to torque production delay
AFI= normalized air fuel ratio influence function
CI= normalized compression influence function
= the maximum torque production capacity of an engine given that AFI=CI=1
From Equation (17), the normalized AFR influence function is described as:
cos 7.3834 ·
/
13.5
(18)
Where A/F = actual air fuel ratio of the mixture in the combustion chamber
(19)
While the normalized compression influence CI is described as:
cos
.
(20)
Where CA= tuning parameter of cylinder advance at the Top Dead Center
MTB= minimum tuning such that best torque acquire
Finally, the crankshaft rotation which follows the torque balance relationship about a
rigid shaft is described as:
·
(21)
37 Where
= effective inertia of the engine
= engine indicated torque
= engine friction torque
= accessories torque
In this case, the engine friction torque
is resulted from coulomb and viscous
friction torque such that friction torque results to:
0.1056 ·
15.10
Where 0.1056 denote viscous friction coefficient
15.10 denote coulomb friction coefficient
(22)
38 3.3
Engine System Identification Theory
System identification technique is defined as deriving a mathematical model of
certain system dynamic from measured data (Hespanha, 2007). In this project, system
identification technique is performed to determine linear engine model, which
represented by parametric state space equation. The state space dynamic is needed for
design and implementing high accuracy of LQG controller parameters, which
performing control algorithm using Kalman filter theory through estimation of states
from engine’s state space model.
A system identification technique requires application to the engine model with
some specific input signal since it relies on the analysis of input and output signal to
identify the relationship between them. Here, engine model is assumed as gray box
problem providing some basic characteristics of the model. State space model is then
estimated from experiment data of stochastic steady state information with use of natural
source of randomness value as input to engine model. There are four main steps to
determine dynamic model of engine system and Figure 3.5 shows cycle of system
identification steps.
39 Data
Experiment design and data collection Data
Data processing Model estimation Model Model validation Model
Model structure selection Correction model needed Model acceptance? Yes
Correction data needed Filtering required? Figure 3.5: Cycle of system identification function
40 3.3.1
Experiment Design
The design of a system identification experiment includes many important
choices. First of all, the engine system is subjecting to step, ramp, pulse or sinusoidal
input variables. This procedure can produces input output data from the system to be
modeled. Then, the designer has to decide what signals to be measure and when to
measure them such that maximum information regarding the system response is
contained in the input output data. When these have been defined the next issue is to
decide the sampling frequency. The rate is determined from the dynamic properties in
the input and output signals. To be able to identify this behavior, the sampling rate has
to be fast enough to get all the wanted dynamics, but not so fast as to generate
unnecessarily large amounts of data.
3.3.2
Data Preprocessing
When data is collected from experiment, immediate usages in identification
algorithms are often not possible. First the data has to be pre-processed in several ways
in order to eliminate low- and high-frequency disturbances, outliers, missing data, drifts
and offsets etc. Removal of offsets such as drifts and trends are especially important
when output error models are used as estimation output. If this is not considered,
difference in amplitude will dominate the fit criterion and the dynamic behavior will be
of less importance. For methods that use flexible noise models, removal of offsets is not
as crucial, since this approach, by design, means de-emphasis of drifts and trends. One
such method is the least-squares method,( Ljung, 1999).
41 The data measurement equipment is not faultless. Therefore, the data will most
likely include bad values due to obvious measurement error. Such data are called
outliers. These types of values may have negative effect on the estimate and it is
recommended to remove such data from the experiment. Undoubtedly, residual analysis
is good for identifying outliers and bad data ( Ljung 1999).
Furthermore, bad data might be included in measurements and other data might
be missing for any reason. Therefore, data set can be corrected by merging data sets
from experiment that has been repeated for a number of times and it is desired to design
only one model, based on the data from all experiments. Whatever the reason might be,
it is desired to exclude parts of bad data and concatenate other parts. As good as it might
sound; it is not possible to simply connect data segments together, since the joining
points would cause transient behaviour that might destroy the estimate. Therefore
merging data sets can be done with statistical methods, using covariance matrices (Ljung,
1999).
3.3.3
Model Estimation
There are a number of different model structures to choose between when
describing a system. First the user has to decide upon whether to use linear or nonlinear
models, black-box or physically parameterized state-space models etc. In this project,
the focus is to design linear models for MIMO systems. Not all model structures can
handle multivariable systems. State-space models using a subspace method is one of the
models useful for this purpose.
42 3.3.3.1 State Space Model Using a Subspace Method
A discrete state-space model is described in Equation (23) and Equation (24).
Measured inputs sampled at time k are denoted as u and outputs as y. The number of
inputs is nu and the number of outputs is ny. The vector x is the state vector and
contains numerical values of n states. w=Ke(t) and v=e(t) are immeasurable signals,
assumed to be white noise.
State transition equation:
(23)
Observations equation:
(24)
In Equation (23) and (24), A is an n-by-n matrix, which describes the dynamics
of the system. B is an n-by nu matrix and it describes the linear transformation by which
the inputs influence the next state. C is an ny-by-n matrix, which represents how the
internal state is transferred to the output y. D is an ny-by-nu matrix, which is the direct
feed through term. Complex behaviour in the measured outputs can be captured by
choosing n high enough in the model estimation.
Figure 3.6 show a graphical representation of a state-space model is made.
v w x
u B 1/z y C A D Figure 3.6: State space structure model
43 A subspace identification algorithm is performed to identify input-state-and
output model of engine system. In this case, if the states of the system are known and
input and output data are measured, it would be possible to solve state space model in
Equation (23) and (24) for the four matrices. The equation would be a linear regression
and the C and D matrices can be found by applying the least squares method. Hence, the
other unknown matrices in the equation can then be determined. The problem is thus to
find the states in Equation (23) and (24).
The states can be described as linear
combinations of the k-step-ahead predicted output. Once these predictors are found, the
problem is solved.
Finding of state can be achieved by using a subspace method. This method
determines the predictors by projections directly on the measured data sequences in a
satisfactory way, and subspace models have full freedom in the noise model. Therefore,
a lower order system can be used for subspace models. Subspace models are also very
easy to implement in control algorithms, since the system matrices are directly known.
The use of subspace algorithms to carry out state space based system
identification in a stationary framework has been widely explored since 80’s
(Segismundo et al, 2004). There are two main ideas exploited in subspace algorithms:
1) A sequence of (Kalman filter like) states can be estimated directly from
observations.
2) All system matrices can be estimated via least squares (provided that
observations and a sequence of states are known).
In this case, estimation of a sequence of states from observations is possible by
combining two facts:
a) Consider the set
observations
, made up by observation y(t) plus next f-1 as future
44 = [y(t)’, y(t+1)’, ..., y(t+f -1)’]’
1
and the set
1
(25)
of p as past observations
= [y(t-1)’, y(t-2)’, ..., y(t-p) ’]’
An expected value of
(26)
1 can be calculated by the linear
based on
projection theorem: The orthogonal projection (
observations (
) into p past observations (
1
/
1 ’) E
= E(
1
where the projection E(
) of f future
1 ) is
1
’) E
1
/
1
1
1
1
(27)
can be, for
stationary processes, consistently estimated directly from data.
b) Let ŷ(t|t-1) be the expected value of y(t)based on past values (up to time t-1).
Consider a system that evolves according to state space Equations (23) and (24). At
time t, the expected value (ŷ t|t
1 ) of
conditional on past observations is the
product of an (extended) observability matrix by the expected value (ž(t|t-1)) of the state
vector at time t conditional on past observations. This fact can be seen by recursive
substitution in the state space system equations:
ŷ(t|t-1)
=
C
ž (t|t-1)
(28)
ŷ(t+1|t-1)
=
CA
ž (t|t-1)
(29)
ž t|t-1
(30)
…
Ŷ(t+f-1|t-1)
=
C
which can be expressed as
ŷ t|t
where
1
=
ž (t|t-1)
is the (extended) observability matrix
(31)
= [C’ (C A)’ … (C
)’ ]’.
45 Combining the two previous facts of (a) and (b), subspace algorithms decompose
a matrix of estimated orthogonal projections (predictions) into the product of an
(estimated) observability matrix plus an (estimated) “sequence of states” matrix.
3.3.4
Model Validation
Last but not least, the resulted model output from system identification technique
needs to validate and compare from original system (Ljung, L., 1999). The purpose of
model validation is to verify the identified state space model such that it is represents the
engine model under consideration adequately.
This normally involves statistical
analysis of the residuals and predictive capabilities of the model (Hepanha, 1997).
3.4
Linear Quadratic Gaussian (LQG) Controller
Linear quadratic Gaussian (LQG) controller, which is a linear quadratic regulator
(LQR) combined with Kalman filter, has been widely used in active control of building
structures. A general LQG model is shows in Figure 3.7. It is designed in the time
domain in such a way to enhance performance from linear quadratic regulator. And it
has proven to be effective in reducing the dynamic response of structures
(Gawronski,1994).
46 Process
noise, v
Controller
signal, u
Kalman Filter
Model
output, y
Reference
setpoint, r
Optimum Gain
Matrix,-K
Integrator
Plant
Air throttle
angle,
Figure 3.7: LQG controller structure model
However, it requires an iterative procedure to obtain LQR weighting matrix, K
that used as a performance index, because there is no definite criterion for selecting a
weighting matrix. And the improvement in performance and robustness comes at the
prices of increased parameters tuning.
Linear quadratic Gaussian is one of the several optimal control strategies which
have been used for control purpose (Hespanha, 2007).
Generally, optimal control
system requires a performance measure with a maximization or minimization algorithm
that provides an optimal result in some sense, such as minimize the process time, system
error and so on.
Involvement of Kalman filter in linear quadratic regulator to result a linear
quadratic Gaussian compensator has made it different compare to state feedback
regulator such as fuzzy logic controller. In this case, presence of Kalman filter works to
estimate of all unmeasured state in the engine system. Initially, the linear quadratic
regulator should be work well with excellent stability margins. However, the presence
of process noise and the unavailability of an onboard sensor have made it fail to present
perfectly thus result to use of the LQG compensator.
47 Linear quadratic regulator is a static state feedback controller represented by a
constant gain matrix K. To close the loop with K in the feedback, Kalman filter
therefore is needed such that all states of the engine model are available for
measurement.
The control objective of the LQG is to minimize a criterion, which is a quadratic
function of the system states and control signals (Åström & Wittenmark, 1990), when
the system is subject to certain initial conditions. When designing an optimal controller,
the system is assumed to be linear or a linearised system model is used, and has a state
space equation such as given in Equations (23) and (24). The control law is chosen such
that it minimizes the cost function,
tf
(
)
J LQ = x (t f ) H x (t f ) + ∫ x (t )Q x (t ) + u T (t ) Ru (t ) dt
T
T
(32)
0
where H, Q and R are weighting matrices. H and Q are at least positive semidefinite and
R is positive definite. tf is the final time that the control is required. For a control
system that is designed to operate for a long time period, the following cost is used
(Burl, 1999).
∞
(
)
J LQ = ∫ x (t )Q x(t ) + u T (t ) Ru (t ) dt
T
(33)
0
The control law is given by Equations (34) and (35),
u(t ) = −K (t ) xˆ(t )
(34)
K = −R −1 B T Pr
(35)
where K is the feedback gain matrix, xˆ(t) is the state estimated using the Kalman filter
and Pr is obtained by solving the Ricatti equation given in Equation (36).
P&r = −Pr A − AT Pr − Q + Pr BR−1 BT Pr
(36)
The Ricatti equation has only final condition, Pr (t f ) = H , and the values of Pr
corresponding to the optimal trajectory can therefore be found by solving it backward in
48 time using any numerical integration method. In MATLAB®, Pr can be obtained using
the function, care(A,B,Q).
For the scalar case of Q and R, the cost function, JLQ, can be interpreted as the
weighted sum of the state and control. The choices of Q and R matrices allow the
respective weighting of the energies of different signals and through that, increases the
importance of keeping certain signals small in expense of the others. Generally speaking,
selecting Q large means that, to keep cost function JLQ, small, the state x(t) must be
smaller. On the other hand selecting R large means that the control input u(t) must be
smaller to keep JLQ small. Choices of these matrices follow no particular rules. They
depend on the designer’s understanding of the behaviour of the system to be controlled,
followed by some tuning by trial and error, until satisfactory performance is achieved.
However, as a guideline, Q can be chosen such that it results in the contribution of each
state being roughly equal (Burl, 1999). To solve the LQR problem described so far, the
system in Equation (33) must be completely controllable so that JLQ in Equation (33) is
finite
LQG controller then uses the estimate state in the feedback to result a control
signal of u=-K . In this case, an accuracy of the estimate state greatly depends upon the
accuracy of the linear state space model used for design.
Generally, there are four important parameters require to present a LQG
compensator that participate by Kalman filter and LQ-optimal gain. They are state
weighting matrix Q, input weighting matrix R, process noise and measurement noise
covariance matrices,
and . By using the developed state space engine model, four
main parameters of Q, R,
and
can be assigned. To design LQG regulators and set
point trackers, following steps have to be performed:
1.
Construct a Kalman filter (state estimator).
2.
Construct the LQ-optimal gain.
49 3.
Form the LQG design by connecting the LQ-optimal gain and the
Kalman filter. (MATLAB, 2009)
3.5
Fuzzy Logic Controller (FLC)
Fuzzy logic controller can be viewed as an artificial brain that applying a “soft
computing” technique, which aims to mimic the ability of human mind to learn and
make rational decision in an uncertain and imprecise environment (Jantzen, 1998). In
case of traditional engine AFR control system, conventional ECUs is work to determine
suitable control output value by loading saved data from 3-D maps. Therefore, by
replacing 3-D maps with fuzzy logic controller into engine’s control system has the
potential to decrease time and effort required in calibration of engine 3-D maps control
system.
In general, fuzzy logic controller involve of three process of fuzzification, design
of rule base and defuzzification (Jantzen, 1998).
The controller is between
preprocessing and post processing block as show in Figure 3.8. Here, fuzzy logic
controller design is accomplished by using fuzzy logic toolbox, which available in
MATLAB software.
Fuzzy logic controller
Rule base
preprocessing
fuzzification
Inference
engine
defuzification
Figure 3.8: Fuzzy logic controller block diagram
postprocessing
50 In this project, error values, which result from difference between desire AFR
and actual AFR, is assigned as fuzzy controller input. The control strategy is a static
mapping between input and control signal.
To enhance performance of FLC, an
additional input of derivative of error, which is an error measurement backwards in time,
is assigned. These are created in the preprocessor thus making the controller multi
dimension.
3.5.1
Fuzzification
Fuzzification works to converts each piece of input data to degrees of
membership by a lookup in one or several membership functions. The fuzzification
block thus matches the input data with the conditions of the rules to determine how well
the condition of each rule matches that particular input instance. There is a degree of
membership for each linguistic term that applies to input variables. Figure 3.9 shows
two input funzzification membership function of engine model. The input variable of ‘e’
has universe of discourse, range from -1 to 1 while input variable of ‘ce’ has universe of
discourse, range from -2 to 2.
51 (a)
(b)
Figure 3.9: Inputs membership function of error (a) and change in error(b) contain in
fuzzification process.
52 Engine’s fuzzy input variables are participated by error of FLC and change in
error of FLC. Each of them form a membership function and participate by five fuzzy
set with linguistic term of:
NH(negative high)
-large negative error value,
NL(negative low)
-small negative error value,
ZO(zero)
-zero error value,
PH(positive high)
- large positive error value,
PL(positive low)
-small positive error value.
During fuzzification process, crisp value of AFR error and change in error,
which participated in fuzzy input variable, are mapped into sets of membership function
of fuzzy sets. Every element in the universe of discourse is a member of a fuzzy set to
some grade, maybe even zero. The grade of membership for all its members describes a
fuzzy set. In fuzzy sets elements are assigned a grade of membership such that the
transition from membership crisp value to non-membership is gradual rather than abrupt.
The set of elements that have a non-zero membership is called the support of the fuzzy
set. The function that ties a number to each element x of the universe is called the
Membership function,µ(x)
3.5.2
Rule base
After fuzzification process, these membership function values are ready to
process in rule base through conditional “if-then” statements. Rule base is function as
human brain of thinking in such a way to process fuzzified input variables and fire
suitable controller signal, thus regulate engine’s air fuel ratio around a prescribed
setpoint. Table 1 shows total of 25 rules implemented for AFR control purpose.
53 Table1.1: Fuzzy rules
Error
Change in error
ce
e
NH
NL
ZO
PL
PH
NH
MH
MM
ML
ML
MM
NL
MH
MM
ML
ML
MM
ZO
MH
ML
ZO
ML
MH
PL
MM
ML
ML
MM
MH
PH
MM
ML
ML
MM
MH
There is no specific theorem available for designing a complete fuzzy rule.
Therefore, a full understanding of plant behavior is needed, in order to result a suitable
fuzzy rule base. Error value which is the difference of air fuel stoischiometric ratio and
actual air fuel ratio have to be regulated around zero value. So, whenever percentage of
input air throttle, which influencing the flow rate of air mass, is increasing, input fuel
flow rate, which is control by fuzzy output, need to increase too so that error is always
close to zero. For example, during controlling engine’s AFR, if error value is a large
negative value, NH and change in error is a large negative value, NH too; which result in
large and larger model output error, then fuzzy controller will fire a large control gain
value to bring error value back to zero state.
On the other hand, if error value is zero value, ZO and change in error is a small
positive value, PL; which descript a condition during error signal start to vary away from
zero state with small deviation. Therefore, fuzzy controller will result a small control
gain value, in order to bring back error value to zero state without causing overshoot to
output response, which is cause by overload of controller signal.
54 Figure 3.10 shows membership function of fuzzy output that participated by 5
fuzzy set of
ZO(zero)
-zero gain value
ML(medium low)
-small gain value
MM(medium medium)
-medium gain value
MH(medium high)
-medium large gain value
Figure 3.10: Fuzzy output membership function with participation of 5 fuzzy set ZO,
ML, MM, MH.
3.5.3: Defuzzification
During defuzzification process, 25 fuzzy controller output will be fired
respectively. Each rule from rule base will fire an output based on received error and
change in error signal value. Controller output gains from 25 rules are then summed and
defuzzified into a crisp analogue output value through an inference engine.
55 Figure 3.11 shows graphical construction of the algorithm in the core of
controller. In this case, each of the 25 rows refers to one rule. The rule reflects the
strategy that the control signal should be a combination of the reference error and the
change in error of fuzzy logic controller.
Figure 3.11: Graphical construction of the control system in a fuzzy controller
(generated in the Matlab Fuzzy Logic Toolbox)
Generally, there are several ways to define the result from a rule, but one of the
most common and simplest is the “min-max” inference method. The inference engine
looks up the membership values in the condition of the rule.
conclusion are accumulated, using the max operation.
Then all activated
56 For example by refer to Figure 3.11, only rule number 1, 2, 6 and 7 have fired an
output signal, where each of the output signals is result from min operation between
fuzzy set of “error” and fuzzy set of “change in error” as show in Figure 3.12. Then,
output signals will be accumulative using max operation and fire only a crisp value
through centroid defuzzification method.
Min operation Max operation Figure 3.12: Defuzzification process
The final graph of defuzzification process is shows in bottom right of Figure 3.12.
The resulting fuzzy set from final graph must be converted to a number that can be sent
to the process as a control signal. This process is so called defuzzification. The crisp
output value is the abscissa under the center of gravity of the fuzzy set,
∑
(37)
∑
, is a running point in a discrete universe, and μ
is its membership value in the
membership function (Yao Ju-Biao et al, 2009). Following the evaluation of rules, the
defuzzification process that transforms the fuzzy membership values into a crisp output
value can be the fuel pulse width or fuel injection valve opening. Figure 3.13 shows the
simulation model of engine system with fuzzy logic controller.
57 Reference
setpoint, r
u
Error ,e
y
Fuzzy logic
controller
Plant
Output ,y
∆
Figure 3.13: Fuzzy logic controller structure model
3.6
Conclusion
A simulation of engine model is work through applying all equation and formula
as stated in chapter 3. In this case, it is useful to represent and perform well as a real
engine for specific purpose such as control system design. FLC and LQG controller
therefore implement into engine model for AFR control purpose thus evaluation made
on the controller performance.
58 CHAPTER 4
RESULT AND DISCUSSION
4.1
Introduction
The simulated engine model is assumed to perform under steady state condition,
with percentage variation of engine’s air throttle angle as shown in Figure 4.1. Here,
percentage opening of input air throttle is proportional to automotive acceleration.
Therefore, variation of air throttle opening in Figure 4.1 is assumed as accelerate or
decelerate of automotive in real time through change of input air flow. With this,
engine’s AFR response and controller performance will be evaluated using MATLABSIMULINK software.
59 0.75
0.7
trottle angle percentage/100
Percentage of air throttle variation/100
0.8
0.65
0.6
0.55
0.5
0.45
0.4
0
100
200
300
400
500
sampling time,s
600
700
800
900
1000
Time (sec)
Figure 4.1: Variation of engine air throttle
Figure 4.2 shows output response of engine AFR due to variation of input air
throttle angle in Figure 4.1. In this case, AFR is no longer maintained at stoischiometric
value of 14.7 but vary away from it when engines try to accelerate or decelerate.
17
16.5
16
15
air fuel ratio
Air fuel ratio
15.5
14.5
14
13.5
13
12.5
12
0
100
200
300
400
500
sampling itme,s
600
700
Time(sec)
Figure 4.2: Engine’s Air fuel ratio
800
900
1000
60 Figure 4.3 shows uncontrolled condition of engine at lean and rich oxygen (air
throttle variation), which could bring bad effect to engine power reduction and fuel
consumption.
For example, variation of input air throttle during accelerating or
decelerating engine can cause transient effect to engine torque and acceleration response
as shown in Figure 4.4 and Figure 4.5. In this case, engine torque response is fail to
reach desire value immediately during engine is accelerating or decelerating but result to
a delay period before reaching target value. Furthermore, variation of AFR around
stoischiometric ratio can influence the exhaust emission control in engine model due to
its stoischiometric value of 14.7 can ensure maximum efficiency of conversion of
harmful components in catalyst converter.
In this case, variations of AFR greater than 1 percent below stoischiometric ratio
can result in significant increase of CO and HC emission and increase of more than 1
percent above stoischiometric ratio will produce more NOx up to 50 percent( Ali, 2008).
This is then degrades the control quality. To avoid this deficiency, two different type of
control strategy are introduced to improve the control quality and to reduce the workload
of calibration process thus survive this ratio in an acceptable region. The model of
control system was built with MATLAB-SIMULINK. Based on this, AFR control
strategy that works with fuzzy logic controller and linear quadratic Gaussian controller
was studied.
Figure 4.3: Effect of air-fuel ratio on power, fuel consumption, and emission
61 90
70
engine output torque
Engine output torque
80
60
50
40
30
20
0
100
200
300
400
500
600
sampling time,s
700
800
900
1000
Time(sec)
Figure 4.4: Engine output torque due to variation of input air throttle angle value.
300
engine output acceleration
Engine acceleration
200
100
0
-100
-200
-300
0
100
200
300
400
500
600
sampling time,s
700
800
900
1000
Time(sec)
Figure 4.5: Engine’s acceleration reading due to variation of input air throttle angle
value.
62 4.2
Engine Model Using System Identification Technique
Section 3.3 has described theory of system identification technique in
determining of linearize state space engine model, which use for LQG controller design
purpose. However, working steps and procedure for determining of engine state space
model seen to be complicated for implementation purpose.
Therefore, MATLAB
system identification toolbox is used to simplify the work effort. Figure 4.6 shows
available system identification window, which included the process of data
preprocessing, model estimation and model validation within a single window.
Figure 4.6: System identification toolbox in MATLAB software.
To design state space engine model, following steps have to be performed.
1) Import data from input and output of engine model.
2) Select estimate and validate data range
3) Preprocessing imported data
63 4) Estimate model structure
5) Validate estimated model performance
4.2.1
Import Data, Select Range and Data Preprocessing
First of all, a random signal was appointed as input of engine model as shown in
Figure 4.7. Then, the resulted model input and output data is loaded to MATLAB work
space for further process.
Random signal to beta Random signal to alfa Figure 4.7: Engine model with assigned random signal into engine’s input signals of
beta and Alfa.
64 Input noise responsenoresponse AFR
25
20
15
10
5
0
100
200
300
400
500
time,s
600
700
800
900
1000
700
800
900
1000
Time(sec) (a)
Output noise responsenoresponse Model output beta
30
25
20
15
10
5
0
0
100
200
300
400
500
time,s
600
Time(sec) (b)
Figure 4.8: (a) and (b) shows output and input response from engine model due to
assigning of random signal as model input and work for system identification purpose.
65 In workspace, properties of input output data values in time domain are
encapsulated into a single entity through creating of iddata data object with MATLAB
command of:
z=iddata(output, inputs, sampling time);
where z is a data object with model input and output properties.
“z” is then separated to create two data objects, “ze” and “zv”, which “ze”
contains data for model estimation purpose and “zv” contains data for model validation
purpose. In this case, 2000 samples from input and output engine model has been
captured. Therefore, first 1000 samples are used for system model estimation with
MATLAB command:
ze= z(1:1000);
Whilst the remaining samples are used for model validation purpose with MATLAB
command:
zv=z(1001:2000);
The working data object, “ze” and validated data object, “zv” are ready to import
into system identification toolbox for system identification process. Figure 4.9 shows
time domain data of engine model’s inputs: desired fuel flow rate, u1 and flow rate of air
mass, u2 and engine model’s output: AFR, y1 after preprocess of removing mean values
in system identification toolbox. Typically, aims of building a engine model is to
described the responses for deviations from a physical equilibrium. Therefore,with
steady-state data, it is reasonable to assume that the mean levels of the signals
correspond to such an equilibrium. Thus, models can be seek around zero without
modeling the absolute equilibrium levels in physical units yet having initial value of zero
at t=0.
66 Estimation data Validation data Figure 4.9: Estimate and validate data for randomness input Beta, u1, Alfa, u2 and
output AFR, y1.
67 4.2.2
Estimate Model Structure
After all, working data in system identification toolbox which load from data
object “ze” and validation data from data object “zv” are ready for estimation of state
space mathematic model. Figure 4.10 shows system identification window, which data
objects are place at left corner of toolbox window.
Data Object Working data Validation Data Figure 4.10: System identification toolbox with linear parametric model window
State space model, which categorized under linear parametric model been chosen
for estimating actual engine model. Following shows state space transfer matrix A, B, C
and D, which reflects physical characteristics of engine model, is resulted from
estimated state space using system identification toolbox.
Estimated state space:
(38)
(39)
68 0.18734 0.13306 0.10468
A= 0.08183 0.78614
0.54529
0.00054 0.10877 0.26882
0.00516
0.01720
B= 0.00073 0.09841
0.00011 0.13589
C= 158.16 8.4277
0.44246
D= 0
0
4.2.3
Validate Estimated Model Performance
Engine system is estimated as a third order state space equation with three state
variables. Here, the estimated engine model is required to validate its performance, such
that estimated engine model capable to represent actual engine model and work well for
controller design purpose.
Therefore, a random signal is injected into input of estimated state space model
and work for validation purpose. Figure 4.11 shows actual plant model output (black)
response and estimated model output (blue) response displayed at time 1450s to 1550s
by sharing identical input signal. In this case, engine’s estimated state space model is
validated enable to track actual plant response with best fit up to 90.69 percent through
use of MATLAB system identification toolbox. Therefore, a third order state space
equation from Equation (38) and (39) with transfer matrix A,B,C and D are validated to
represent linear engine plant model, with smallest loss function value of 0.0641 and
smallest final prediction error of 0.0643 compare to any other order of estimated engine
model.
69 Measured and simulated model output
Estimated model output 10
8
Actual model output Model output Model output
6
4
2
0
-2
-4
-6
1450 1460 1470 1480 1490 1500 1510 1520 1530 1540 1550
Time
Time(sec) Figure 4.11: Actual and estimated plant output response
4.3
Linear Quadratic Gaussian (LQG) Controller
To design LQG regulators and set point trackers, following steps have to be
performed as such descript in section 3.4
Design steps of LQG controller:
1. Construct a Kalman filter (state estimator).
2. Construct the LQ-optimal gain, K.
3. Form the LQG design by connecting the LQ-optimal gain and the Kalman filter.
(MATLAB, 2009)
70 First and foremost, linear state space model, which acquired from section 4.2,
needs to verify its observability and controllability such that estimated state space engine
model is valid for control purpose. In this case, controllability and observability matrix
from estimated state space engine model needs to be full rank.
To determine if a system is controllable, one can compute the controllability
matrix, which is defined as:
···
(40)
In this case, controllability can be easily defined through MATLAB command:
Co=ctrb(A,B);
where A and B are n by n system matrix and n by m input matrix from estimated engine
state space model. Result shows the system has full rank of 3. Thus it is proved to be
controllable. Next, one can compute the observability using observability matrix, which
is defined as:
Ob=
·
·
·
(41)
In this case, observability can be easily defined through MATLAB command:
Ob=obsv(A,C);
where C is a p by n output matrix from engine state space model. Result shows the
system has full rank of 3. Therefore, it is proved to be observable. Now, engine’s state
space model is validated for control purpose with full rank of observability and
controllability matrix
Next, Kalman filter strategy that providing model’s estimated state parameter is
to be determined. In MATLAB software, estimated state can be determined through
following command:
Kest=kalman(plant,Qn,Rn);
71 where plant=ss(A,B,C,D,0.1); as state space transfer matrices of engine model and
Qn=0;Rn=0.002; is noise covariance data from engine model. Here, value of noise
covariance data Qn and Rn are defined as:
Qn=
(42)
and Rn=
(43)
with
defined as process noise and
defined as system measurement noise.
Therefore, by having complete data of Qn, Rn and plant, estimated state matrix
can be determined using Kalman’s filter theorem and following shows transfer matrix of
estimated states.
Kalman ‘estimated states:
Ae=
0.1873
0.1331
0.08183 0.7861
0.00054 0.1088
Be=
0.005164
0
0.0007339 0
0.0001151 0
158.2 8.428
1
0
Ce=
0
1
0
0
0
0
De=
0
0
0.1047
0.5453
0.2688
0.4425
0
0
1
0
0
0
0
Following step is therefore to construct the LQ-optimal gain, K. By using
MATLAB command, LQ-optimal gain can be acquired by following command:
K=lqi(plant, Q, R);
72 In this case, value of weighting matrix Q and weighting matrix R should be
selected in such a case of Q to be positive semi-definite and R to be positive definite.
This means that scalar quantity
Qx from cost function J is always positive or zero at
each time t for all functions x(t), and scalar quantity u Ru from cost function J is
always positive at each time t for all values of u(t). These guarantee that cost function J
is well-defined.
Here, value of Q is assigned as diagonal matrix and acquired through MATLAB
command:
Q=blkdiag(0.1*eye(nx),eye(ny));
0.1 0
Q= 0 0.1
0 0
0 0
0
0
0.1
0
0
0
0
1
and weighting gain R=500. In this case, weighting gain R should be choose carefully,
where it may cause oscillation and unstable output response under small weighting gain
value such that controller gain is too small for regulating error signal; on the other hand,
it may cause agitated and oscillated output response but fast AFR converging effect
under large weighting gain result.
Figure 4.12 shows detail graphical response of
controller signal and output response at over large and small weighting gain. The
estimated states,
from Kalman filter and optimum gain, K are form up together as
LQG controller and its model structure is shown in Figure 3.10. Figure 4.12 shows
simulated response of engine’s AFR with apply of LQG controller. In this case, LQG
controller gain shows lagging effect in such a way fail to bring AFR back to steady state
of 14.7 within short period. However, it still showing control afford by trying to reduce
the overshoot and transient effect on engine’s AFR due to variation of air throttle, Alfa.
73 Weighting gain, R=250
16
16
15.5
15.5
15
15
AFR
AFR
controller output
Weighting gain, R=600
14.5
14.5
14
14
13.5
13.5
13
350
•
400
450
time,s
500
13
350
550
400
450
time,s
Time(sec) Time(sec) AFR response
AFR response
•
Agitated and oscilated response
500
550
Slow and transient response
16
16
15.5
14.5
14
controller output
Controller output controller output
Controller output 15.5
15
15
14.5
14
13.5
13
350
400
450
time,s
500
550
13.5
350
400
Time(sec) Controller output
•
450
time,s
500
550
Time(sec) Controller output
Large and oscilated gain
•
Small and slow gain
Figure 4.12: Output response and controller gain performance under large and small
weighting gain
The delayed control afford from LQG in contrast result to overshooting of AFR
response (Figure 4,13) when lagging controller gain has react slow and not giving
immediate feedback to variation of AFR on time. Therefore, LQG controller is work on
structure modification to improve its performance.
Figure 4.14 and 4.15 shows
simulation result of AFR response after modification on LQG compensator model with
extra derivative block. The derivative block is function to speed up controller gains,
74 which try to regulate AFR error on time and eliminate lagging effect as occur in
previous control model. Simulation result shows good and acceptable AFR response
from modified LQG controller signals that success to reduce AFR transient effect yet
result to shorter settling time. Furthermore, controller output response is showing small
variation gain.
In this case, controller signal with large variation and agitation condition is in
proportion to controller actuator action, which may cause defect to actuator hardware.
Therefore, smooth and small controller signal as performed in modified LQG controller
is needed in most of the plant controlling cases for prolong controller life time and
protection purpose. However, it is weak in overcome overshooting effect by causing
overshooting effect to AFR response whenever engine model is accelerating or
decelerating.
Through overall performance, LQG controller with extra derivative block shows
stable and acceptable performance at model output response also in response to
controller signal (Figure 4.16, 4.17) in such a way trying to regulate AFR as close as
possible to setpoint value.
75 16
15.5
AFR Air Fuel Ratio
15
14.5
14
13.5
13
0
100
200
300
400
500
600
sampling time,s
700
800
900
1000
Time(sec) Figure 4.13: Air fuel ratio response with LQG compensator (blue) and without LQG
compensator (green)
76 16
15.5
AFR Air Fuel Ratio
15
14.5
14
13.5
13
0
100
200
300
400
500
600
sampling time's
700
800
900
1000
Time(sec) Figure 4.14: Air fuel ratio response with modified LQG compensator (blue) and without
LQG compensator (green)
16
15.5
AFR Air Fuel Ratio
15
14.5
14
13.5
350
400
450
sampling time's
500
550
Time(sec) Figure 4.15: Air fuel ratio response with modified LQG compensator (blue)and without
LQG compensator (green)at time 350s to 550s
77 16
15.5
controller output
Controller output 15
14.5
14
13.5
13
0
100
200
300
400
500
time,s
600
700
800
900
1000
Time(sec) Figure 4.16: LQG controller output response
16
controller output
Controller output 15.5
15
14.5
14
13.5
350
400
450
time,s
500
550
Time(sec) Figure 4.17: LQG controller output response display at time 350 s to 550s
78 4.4
Fuzzy Logic Controller (FLC)
Fuzzy logic controller modeled in Figure 3.16 is simulated at time period 100s
under condition of engine air throttle variation as shown in Figure 4.1. In this case, 3-D
maps, which assigning suitable AFR through ECUs signal has been replaced by a fuzzy
logic controller. Here, simulation result will showing how well performance of fuzzy
logic controller in regulating AFR to desire ratio under condition of accelerate and
decelerate of engine model.
Figure 4.18 shows engine’s AFR response with apply of fuzzy controller. In this
case, AFR response of engine model is perform well under FLC controller, which trying
to return back to stoischiometric ratio of 14.7 after change of engine’s air throttle angle
that causing AFR vary away from stoischiometric ratio.
16
15.5
AFR Air Fuel Ratio
15
14.5
14
13.5
13
0
100
200
300
400
500
600
sampling time's
700
800
900
1000
Time(sec) Figure 4.18: AFR response from engine model with fuzzy logic controller (green) and
without fuzzy logic controller (blue)
79 15.5
Air Fuel Ratio
AFR 15
14.5
14
13.5
350
400
450
sampling time's
500
550
Time(sec) Figure 4.19: AFR response from engine model with fuzzy logic controller (green) and
without fuzzy logic controller (blue) crop from time in between 350s to 550s
Furthermore, Figure 4.18 and 4.20 shows clear view of controlled AFR response
(green) and square error performance (green). In this case, its overshoot effect is
completely reduced with percentage higher than 50% compare to uncontrolled AFR
response without causing large transient effect to engine’s AFR response (zoom in view
in figure 4.19).
Engine’s AFR square error response under FLC shows obvious
reduction compare to AFR error signal in actual engine plant without implementation of
FLC(blue), and the output response show large improvement compare to LQG control
afford (Figure 4.22). Here, square error response from LQG controller shows fast
converging effort in forcing plant error reduce to zero value in short time. However,
LQG controller, which showing fast converging effort, has result to large controller gain
(Figure 4.16). This effort on the other hand causes large overshooting and oscillating
effect at plant output response as show in Figure 4.14.
80 3
2.5
square error value
Square error 2
1.5
1
0.5
0
0
100
200
300
400
500
sampling time's
600
700
800
900
1000
Time(sec) Figure 4.20: Square error value from engine AFR without fuzzy logic controller (blue)
and with fuzzy logic controller (green)
3
2.5
square error
2
1.5
1
0.5
0
0
100
200
300
400
500
600
time(sec)
700
800
900
1000
Figure 4.21: Square error value from engine AFR without LQG controller (blue) and
with LQG controller (green)
81 Response without controller Response with controller (a)
Response without controller Response with controller (b)
Figure 4.22 : Close view of Square error response from engine AFR without controller
and with controller of (a)LQG and (b) FLC
Besides inspect on AFR response of engine model, controller gain response is no
doubt another option in determine of suitable controller, so that simulation model
practically can be implemented into actual engine system. Figure 4.23 and 4.24 shows
FLC controller gain effort in regulating error signal of engine’s AFR. Response shows
smooth and small controller signal variation.
In this case, performance of FLC
controller gain is improved compare to response of LQG controller gain as shows in
Figure 4.17. Here, agitated and oscillated controller gain, which occurs in LQG control
system, has been recover through implementation of FLC control system.
82 16
Controller output controller output
15.5
15
14.5
14
13.5
0
100
200
300
400
500
time,s
600
700
800
900
1000
Time(sec) Figure 4.23: FLC controller output response
16
controller output
Controller output 15.5
15
14.5
14
13.5
350
400
450
time,s
500
550
Time(sec) Figure 4.24: FLC controller output response display at time 350 s to 550s
83 Controller gain response from FLC is therefore igniting the fuelling actuator with
time constant as show in Figure 4.25. The filling time constant is showing proportional
response compare to flow rate of engine air throttle (Figure 4.1), which is actually make
sense, since increase of air flow to cylinder needs increase of fuel flow to cylinder either,
so that stoischiometric air to fuel ratio of 14.7 can be resulted.
In real engine test bed, an engine torque production is important such that any
changes in engine system will not affect or weaken total toque production. Figure 4.26
show simulated engine torque response with FLC as control system. In this case,
comparison between actual engine torque responses (Figure 4.6) and engine torque
response with FLC shows almost no different between each other. The result shows an
FLC controller is working well in regulating error of AFR while maintaining desire
torque response from engine system.
In general, the overall performance of engine’s AFR with FLC is performed
better than original engine system with conventional 3-D mappings. Besides earning
good performance from fuzzy logic controller, fuzzy methods in the application of
engine control also result to relatively small number of parameters needed to describe
the equivalent 3-D map. The time needed in tuning a FLC compared to the same
equivalent level of 3-D map look-up control can be significantly reduced.
84 effective fueling time constant
0.054
0.0535
0.053
0.0525
0.052
0.0515
0.051
0
100
200
300
400
500
time,s
600
700
800
900
1000
Time(sec) Figure 4.25: Effective fueling time constant
90
80
enigne rotation torque
70
torque Effective fuelling tiime constant 0.0545
60
50
40
30
20
0
100
200
300
400
500
time,s
600
700
800
Time(sec) Figure 4.26: Engine rotational torque
900
1000
85 CHAPTER 5
CONCLUSION AND FUTURE WORK
5.1
Conclusion
In this project, the nonlinear engine model was developed; also the linear engine
model has been built from nonlinear engine model through system identification
technique. Once the model was obtained, fuzzy logic controller and LQG controller are
design to regulate the AFR response, which try to run away from stoischiometric value
whenever engine is accelerating or decelerating. The complexity of a fuzzy logic system
with a fixed input output structure is determined by the number of membership functions
used for the fuzzification and defuzzification and by the number of inference levels. On
the other hand, a LQG controller shows its robustness, where Kalman filter is made
adaptive and integrated with the standard LQR to obtain a novel control scheme.
LQG control provides very good results in high speed control. It is shown that
the proposed strategy provides improved performance in terms of generating control
effort and following the desired trajectory. In addition, the proposed controller copes
well with external disturbances and modeling uncertainty. On the other hand, FLC
86 controller, which performing artificial intelligent technique shows improvement
compare to engine AFR response from LQG control system based on system robustness,
speed and overshooting effect.
In a nutshell, simulation result shows both control strategy of FLC and LQG,
which replacing 3D-maps in ECU, are perform well in regulating AFR thus reduce
quantity of pollutant release to atmosphere without loss of engine desire torque
performance. Each controller has their own benefits as well as weakness and the control
performance is highly depending on model behavior. With the contemporary software
tools for design and simulation, the above systematic approach can be very well guided
by design template files and graphical user interface. A total development environment
for rapid prototyping and experimentation allows for a seamless transition to the
experiment and concept proving. With the knowledge of the basic theory one can easily
develop advanced control concepts in a minimum of time.
5.2
Future Work
In this project, a mean value engine model been applied for controller design
purpose. However, it is tends to neglect some cycle variation of engine parameter.
Therefore in the future, project efforts will be emphasized on design of engine model
with detail components performance description and inclusion of transient, steady and
ideal engine state.
Robustness of designed FLC and LQG controller shall be tested in developed
engine model, such that they are capable to withstand controlled output in allowable
range. Finally, with the knowledge of controller design, one can develop control system
in hardware yet implant to automotive engine and to achieve the project objective.
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92 APPENDIX A1
Engine’s air flow dynamic represent in MATLAB-SIMULINK
93 APPENDIX A2
Engine’s fuel injection dynamic represent in MATLAB-SIMULINK
94 APPENDIX A3
Engine’s rotational torque dynamic represent in MATLAB SIMULINK.
95 APPENDIX B
The Fuzzy Logic controller and engine model in MATLAB-SIMULINK.
96 APPENDIX C
LQG compensator with engine model in MATLAB-SIMULINK
97 APPENDIX D
Performance enhancement to LQG compensator with extra derivative block