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Michigan State

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This is to certify that the
thesis entitled

COMBUSTION MODELING AND AlR-TO-FUEL RATIO AND DUAL-
FUEL RATIO CONTROL OF AN INTERNAL COMBUSTION
ENGINE

presented by
Stephen Daniel Pace

has been accepted towards fulfillment
of the requirements for the

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COMBUSTION MODELING AND AIR-TO-FUEL RATIO AND DUAL-FUEL RATIO
CONTROL OF AN INTERNAL COMBUSTION ENGINE

By

Stephen Daniel Pace

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTERS OF SCIENCE

Electrical Engineering

2008

ABSTRACT

COMBUSTION MODELING AND AIR-TO-FUEL RATIO AND DUAL-FUEL RATIO
CONTROL OF AN INTERNAL COMBUSTION ENGINE

By
Stephen Daniel Pace

Internal combustion (IC) engines are optimized to meet exhaust emission
requirements with the best fuel economy. Closed loop combustion control is a key
technology that is used to optimize the engine combustion process to achieve this goal. In
order to conduct research in the area of closed loop combustion control, a control
oriented cycle-to-cycle engine model, containing engine combustion information for each
individual engine cycle as a function of engine crank angle, is a necessity. In this research,
the in-cylinder pressure of an IC engine is modeled using the mass fraction burned
(MFB), which is represented by the Wiebe function. Using MFB, along with other engine
control inputs, an engine in-cylinder pressure signal can be constructed as a function of
crank angle over an engine operational map.

Air-to-fuel (A/F) ratio is the mass ratio of air to fuel trapped inside a cylinder
before combustion begins, and it affects engine emissions, fuel economy, and other
performances. In this research, a multi-input-multi-output sliding mode control scheme
is used to simultaneously control the mass flow rate of both port fuel injection (PFI) and
direct injection (DI) systems to regulate the A/F ratio and fuel ratio of PFI to DI to
desired levels. The control target is to maintain the A/F ratio at stoichiometry and the
fuel ratio to a desired value between zero and one. The performance of the sliding mode
controller is compared with that of a baseline proportional, integral, and derivative (PID)

controller.

Copyright by
Stephen Daniel Pace

2008

T0 Daddy, Mommy, Stacey, and my entire family
For All Their
Love and Support

iv

Acknowledgements

I would like to thank my advisor Dr. Guoming (George) Zhu for supporting me
through my graduate work. His knowledge in the area of automotive engine modeling
and control was invaluable in my quest to complete successful research. Also Dr. Zhu’s
guidance challenged me to achieve and allowed me to flourish as a graduate student. I
would also like to thank Dr. Hassan Khalil for his advice and direction while
investigating the sliding mode controller design.

I would also like to thank Dr. Percy Pierre and Dr. Barbara O’Kelly for their
guidance and support in helping me attend and be successful in graduate school.

Finally, I would like to thank Dr. Uchechukwu Wejinya, Anthony Plummer, and

my closest friends for supporting me through this process.

TABLE OF CONTENTS

LIST OF TABLES ................................................................................. viii
LIST OF FIGURES ................................................................................. xi
CHAPTER 1: INTRODUCTION ........................................................................................ l
1.1 Objective .............................................................................................................. 1
1.2 Engine Operating Cycle ....................................................................................... 2
1.3 Air-to-Fuel Ratio ................................................................................................. 4
CHAPTER 2: ENGINE MODELING AND SIMULATION ............................................. 6
2.1 Background .......................................................................................................... 6
2.1.1 Combustion Process .................................................................................... 6
2.1.2 Abnormal Combustion ................................................................................. 6
2.1.3 Cylinder pressure ......................................................................................... 7

2.2 Cylinder Pressure Model ..................................................................................... 8
2.2.1 Wiebe Function ............................................................................................ 8
2.2.2 Wiebe Function Calibration ...................................................................... 10
2.2.3 Motoring Pressure ..................................................................................... 13
2.2.4 Total Cylinder Pressure ............................................................................. 15
2.2.5 Cylinder Pressure Model Simulation ......................................................... 18
CHAPTER 3: AIR-TO-FUEL AND FUEL RATIO MODELING ................................... 23
3.1 Air-to-Fuel Ratio Modeling ............................................................................... 23
3.1.1 Air Flow ..................................................................................................... 23
3.1.2 Fuel Flow ................................................................................................... 24
3.1.3 Air-to-Fuel Ratio Model Operating Speed ................................................ 25
3.1.4 Physical System Dynamics ........................................................................ 26
3.1.5 Air-to-Fuel Ratio Sensor ............................................................................ 26
3.1.6 Stoichiometric Air-to-Fuel Rat1027
3.1.7 Air-to-Fuel Ratio Calculation .................................................................... 28

3.2 Fuel Ratio Modeling .......................................................................................... 29
3.2.1 Fuel Ratio Uses .......................................................................................... 29
3.2.2 Fuel Ratio Calculation .............................................................................. 30

3.3 Air-to-Fuel Ratio and Fuel Ratio Model Simulation ......................................... 31
CHAPTER 4: BASELINE CONTROL DESIGN ............................................................. 35
4.1 Previous Controller Designs .............................................................................. 35
4.2 Baseline Controller Design ................................................................................ 36
4.3 Simulation Results ............................................................................................. 39
4.3.1 Simulation 1 ............................................................................................... 40
4.3.2 Simulation 2 ............................................................................................... 41
CHAPTER 5: SLIDING MODE CONTROL DESIGN .................................................... 44
5.1 Sliding Mode Control Theory ............................................................................ 44
5.2 Sliding Mode Control Model ............................................................................. 44
5.3 Sliding Mode Controller Design ........................................................................ 49
5.4 Sliding Mode Controller Simulations ................................................................ 52

vi

5.4.] Simulation 1 ............................................................................................... 52

5.4.2 Simulation 2 ............................................................................................... 57
5.4.3 Simulation 3 ............................................................................................... 60
5.4.4 Simulation 4 ............................................................................................... 62

5.5 Observer for Sliding Mode Controller ............................................................... 66

5. 5. 1 Observer Design ........................................................................................ 67
5.5.2 Observer Simulation .................................................................................. 68
CHAPTER 6: CONCLUSIONS ........................................................................................ 70
6.1 Cylinder Pressure Model Conclusions ............................................................... 70
6.2 Air-to-Fuel Ratio and Fuel Ratio Control Conclusions ..................................... 71
BIBLIOGRAPHY .............................................................................................................. 74

vii

Table 2-1

Table 2-2:

Table 2—3:

Table 2-4:

Table 2-5:

Table 2-6:

Table 2-7:

Table 6-1:

Table 6-2:

LIST OF TABLES

: Data from five DI engine tests ......................................................................... l I
Data from five DI engine tests with parameters a, m, and A0 ......................... 11
Values for Wiebe parameter a for extended engine operating conditions ....... 12
Values for Wiebe parameter m for extended engine operating conditions ...... 12

Values for Wiebe parameter A6 for extended engine operating conditions ..... 13

Values for W, for extended engine operating conditions ................................. 18
Values of the variance for the extended engine operating conditions .............. 18
Comparison of controllers for equivalence ratio .............................................. 72
Comparison of controllers for fuel ratio ........................................................... 73

viii

Figure l-l:
Figure 2-1:
Figure 2—2:
Figure 2-3:
Figure 2-4:
Figure 2-5:
Figure 2-6:
Figure 2-7:
Figure 2-8:
Figure 3-1:
Figure 3-2:
Figure 3-3:
Figure 4- 1:
Figure 4-2:
Figure 4-3:
Figure 4-4:
Figure 4-5:
Figure 4-6:
Figure 4-7:
Figure 5-1:
Figure 5-2:

Figure 5-3:

LIST OF FIGURES

The four stroke engine cycle [1] ....................................................................... 3
A typical MFB vs. crank angle curve [I] .......................................................... 9
MFB vs. crank angle (Wiebe function: 60 = 340, A6 = 50, a = 5, m = 2) ....... 10
Motoring pressure raw data ............................................................................ 14
Two-way filter applied to motoring pressure ................................................. 15
Volume in the cylinder at each crank angle .................................................... l6
Cylinder pressure plot (2500RPM, 100% load, stoich. A/F ratio) ................. 19
Cylinder pressure plot (1500RPM, 100% load, stoich. A/F ratio) ................. 20
Cylinder pressure plot (1500RPM, 50% load, stoich. A/F ratio) ................... 21
Equivalence ratio and fuel ratio model in Simulink ....................................... 32
Subsystem of equivalence ratio and fuel Ratio model in Simulink ................ 33
Open loop response of equivalence ratio and fuel ratio model ....................... 34
Cascaded PID and PI controllers .................................................................... 37
PID controller for the equivalence ratio error.......... ....................................... 38
PI controller for the fuel ratio error ................................................................ 39
Closed loop response of simulation 1 ............................................................. 40
PFI and DI control inputs for simulation 1 ..................................................... 41
Closed loop response of simulation 2 ............................................................. 42
PFI and DI control inputs for simulation 2 ..................................................... 43
Equivalence ratio and fuel ratio model divided into 3 sections ...................... 45
Oxygen sensor dynamics of G1 ...................................................................... 45
Air flow dynamics of G2 ................................................................................ 46

ix

Figure 5-4: Fuel flow dynamics of G3 .............................................................................. 46

Figure 5-5: Open loop response of equivalence ratio and fuel ratio transformed model ..49

Figure 5-6: States of the system for simulation 1 .............................................................. 53
Figure 5-7: States for the system to achieve desired equivalence and fuel ratios ............. 56
Figure 5-8: Closed loop response of simulation 1 ............................................................. 57
Figure 5-9: Closed loop response of simulation 2 ............................................................. 59
Figure 5-10: PF] and DI control inputs for simulation 2 ................................................... 60
Figure 5-11: Closed loop response of simulation 3 ........................................................... 61
Figure 5-12: PFI and DI control inputs for simulation 3 ................................................... 62
Figure 5-13: Maximum and minimum eigenvalues of X(e) .............................................. 63
Figure 5-14: Closed loop response for simulation 4 .......................................................... 65
Figure 5-15: PFI and DI control inputs for simulation 4 ................................................... 66
Figure 5-16: Estimated states generated by state estimator ............................................... 69
Figure 6-1: Integration of mean value engine model and cylinder pressure model .......... 71

Chapter 1: Introduction

The internal combustion (IC) engine is designed to produce power from the
energy that is contained in its fuel. More specifically, its fuel contains chemical energy
and together with air, this mixture is burned to output mechanical power. There are
various types of fuels that can be used in IC engines which include petroleum, bio-fuels,
and hydrogen. The output power produced by an IC engine results from the fuel, that it

uses, and also its mechanical parts.

1.1 Objective

The first objective of this work is to develop an engine combustion model for an
IC engine. Modeling of an entire IC engine is a very complicated process, which is due to
the many parameters and characteristics that an engine can have. One purpose of
modeling is to save development costs of real engines and minimizing the risks of
damaging an engine when validating controller designs. Nevertheless, developing a
small model, for specific controller design purposes, can be done and then validated on a
larger, more complicated model. The smaller model that is of interest in this work is a
cylinder pressure model.

Once an accurate cylinder pressure model is achieved, the controller can be
designed and validated based upon this model. Although this is not part of this research,
designing a controller based on a specific model is the idea of the second objective.
There are many types of controllers developed for IC engines, which include spark-

timing control, idle speed control, and engine knock control. In this work, a multi-input

 

and multi-output air-to-fuel ratio and fuel ratio controller will be designed for an IC

engine, which is the second objective.

1.2 Engine Operating Cycle

In an internal combustion engine, a piston moves up and down in a cylinder and
power is transferred through a connecting rod to a crank shaft. The continual motion of
the piston and rotation of the crank shaft as air and fuel enter and exit the cylinder
through the intake and exhaust valves is known as an engine cycle.

The first and most significant engine among all internal combustion engines is the
Otto engine, which was developed by Nicolaus A. Otto in 1876 [1]. In his engine, Otto
created a unique engine cycle that consisted of four piston strokes. These strokes are:

1. Intake stroke
2. Compression stroke
3. Expansion stroke

4. Exhaust stroke

 

Intake Compression Expansion Exhaust

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Figure [—12 The four stroke engine cycle [1]

During the intake stroke, the piston begins at top-dead-center (TDC) and ends at bottom-
dead-center (BDC). An air and gasoline mixture enters the cylinder through the intake
valve and in some cases this valve opens slightly before the intake stroke begins to allow
more air-fuel mixture into the cylinder.

During the compression stroke, the intake and exhaust valves are closed and the
mixture is compressed to a very small fraction of its initial volume. The compressed
mixture is then ignited by a spark causing the pressure to rise very rapidly.

During the expansion stroke, the piston begins at TDC. Due to the high pressure
and temperature gases in the cylinder, the piston is now pushed down, causing the crank
to rotate. As the piston approaches BDC the exhaust valve opens.

During the exhaust stroke, the burned gases exit the cylinder due to the high
cylinder pressure and low exhaust pressure and also due to the piston moving up towards

TDC. The cycle starts again after the exhaust valve closes.

A complete engine cycle is divided into 720 crank angle degrees, where the crank
angle is between the piston connecting rod at TDC and the connecting rod away from
TDC. This means that the piston will move up and down in the cylinder two times during
one complete engine cycle. Since there are two revolutions in one engine cycle, time
duration (in seconds) of one engine cycle can be found given the rotations-per-minute
(RPM). For example, at 1500 RPM, an engine cycle lasts 80 milliseconds (ms) and at
3000 RPM an engine cycle lasts 40 ms.

Although, the Otto cycle was created many years ago, it remains a commonly
used engine design. As previously mentioned, the modeling of the entire process of the
internal combustion engine is a very complicated one, which involves modeling of
thermal dynamics. This research intent is to develop a simple cylinder pressure model

that can be used in real-time simulation for controller design and validation purposes.

1.3 Air-to-Fuel Ratio

Air—to-fuel (A/F) ratio is the mass ratio of air and fuel trapped inside of the
cylinder of an engine before combustion starts. When all of the fuel in the cylinder is
combined with all the oxygen in the combustion chamber (cylinder), the mixture of air
and fuel is a stoichiometric mixture. For gasoline, stoichiometry is achieved when the
A/F ratio is 14.6.

In internal combustion engines, the air-to-fuel ratio is measured by a device
known as an oxygen sensor, or sometimes called a lambda sensor. The sensor is located
in the exhaust manifold and its main purpose is to determine how far away from
stoichiometry the air-fuel mixture is. This unique location of the oxygen sensor is

important in reducing the response time from the fuel injector to the sensor, which is a

very important time delay that is taken into consideration of A/F ratio feedback control

systems. The control of the A/F ratio in an engine will be discussed in a later chapter.

Chapter 2:Engine Modeling and Simulation

This chapter provides the background on the design of a cylinder pressure mode]
that can be used for controller design purposes. It also provides simulation results of the

model and an analysis of the results.

2.1 Background

Background information about concerning the modeling of the combustion process and

the cylinder pressure of internal combustion engines will be discussed in this section.

2.1.1 Combustion Process

In developing a valid engine model of spark-ignition engines, the concept of the
combustion process must be understood. The combustion process is relatively simple and
it begins with fuel and air being mixed together in the intake manifold and cylinder. This
air-fuel mixture is trapped inside cylinder after the intake valve(s) is closed and then gets
compressed. Thereafter, the compressed mixture is combusted, usually close to the end of
the compression stroke, due to an electric discharge from the spark plug. The flame that
is produced near the spark electrode travels through the unburned air-fuel mixture and
extinguishes when it hits the combustion chamber walls. This combustion process varies
from engine cycle-to-cycle and also varies from cylinder-to-cylinder. The actual
combustion of the air-fuel mixture begins before the end of the compression stroke,

extends through combustion stroke, and ends after the peak cylinder pressure occurs [1].

2.1.2 Abnormal Combustion

The previous explanation of the combustion process can be described as the

normal combustion phenomenon. A very important abnormal combustion event is

known as knock and its name arises from the audible noise that resonates from the pre-
ignition of the air-fuel mixture. When the air-fuel mixture is compressed it causes the
pressure and temperature to increase inside the cylinder as previously discussed. Unlike
normal combustion, the cylinder pressure and temperature can rise so rapidly that it can
spontaneously ignite the air-fuel mixture causing high frequency cylinder pressure
oscillations. These oscillations cause the metal cylinders to produce sharp noises called

knock [1].

2.1.3 Cylinder pressure

The pressure in the cylinder is a very important physical parameter that can be
analyzed from the combustion process. The pressure in the cylinder is at a certain point
(in the absence of combustion) because the air-fuel mixture within the cylinder is
compressed. Immediately after the flame develops, the cylinder pressure steadily rises (in
the presence of combustion), reaches a maximum point after TDC, and finally decreases
during the expansion stroke when the cylinder volume increases.

The time at which the electrical discharge from the spark plug occurs is very
important to the combustion event and must be designed to occur at the peak cylinder
pressure which occurs very close to top dead center. This is done so that the maximum
power or torque can be obtained. As a result, this optimum timing is called Minimal
advance for the Best Torque or MBT timing. The spark timing can sometimes be
advanced or retarded due to various operating conditions, which include engine speed
and load, and this will result in reduced output torque or power. The optimal spark
timing (or MBT timing) can also be determined using cylinder pressure signals and mass

fraction burned (MFB) derived from the cylinder pressure. In recent years, two important

criteria have been found using in-cylinder pressure signals: peak cylinder pressure occurs
around 15 degrees after TDC and 50% mass fraction burned occurs at 8 to 10 degrees
after TDC [2]. The velocity and acceleration of combustion can be obtained by taking the
first and second derivatives of the MFB signal, which can be parameterized by a so called
Wiebe function [1]. Using the peak cylinder pressure location, 50% MFB location, and
maximum acceleration of MFB as a closed loop control criterion, the MBT spark timing

can be optimized in real-time.

2.2 Cylinder Pressure Model

Since cylinder pressure is very important to the combustion event and the engine
cycle in spark ignition engines, the development of a model that produces the cylinder

pressure for each crank angle degree is necessary.

2.2.1 Wiebe Function

As previously stated, peak cylinder pressure is important in determining the
optimal spark timing that occurs during combustion. The optimum timing for MBT is
found relative to the peak cylinder pressure. If the timing is advanced or retarded from
this peak pressure, the engine will produce lower output power and torque.

The combustion process can be viewed as both a chemical and physical process
described by the MFB in the cylinder. MFB signifies how much and how fast chemical
energy is released during the combustion cycle and can be parameterized by the Wiebe
function. Thus, the Wiebe function is used to mathematically represent the MFB vs.

crank angle curve and has been known to model the engine combustion process very well

[1]. A typical MFB vs. crank angle has a smooth curve that is “s-shaped.” Figure 2-1

shows a typical MFB vs. crank angle curve.

 

 

 

 

 

 

100 .— --
'0 —
Q) .
E z
:3
m —
c:
o
3::
8 —
E
Q ~—
2
0 - -_--— 1r —w v :7
Crank Angle (degrees)

 

 

 

Figure 2—1: A typical MFB vs. crank angle curve [1]

The Wiebe function is given by the formula,

m+1
xb (6) = l — exp|:— a( 6;:0 j J (l)

where xb is the mass fraction burned, 60 is the start of the combustion, A0 is the

 

combustion duration (x1, = 0 to xb = 1), and a and m are calibration parameters [3].

Modifying the values of a and m can significantly change the shape of the s—curve. The

60 is commonly known as spark timing or ignition timing, which is the time (or crank

angle) where the air-fuel mixture is ignited. Figure 2-2 shows a MFB curve that was

generated by the Wiebe function.

 

MFB using Wiebe function

Mass Fraction Bumed

 

 

 

l l __
340 350 360 370 380 390 400
Crank Angle

 

 

 

Figure 2-2: MFB vs. crank angle (Wiebe function: 60 = 340, A6 = 50, a = 5, m = 2)

2.2.2 Wiebe Function Calibration

Although the Wiebe function can be used to represent MFB, it must be calibrated
at various engine operational conditions to provide an accurate MFB representation. To
achieve this, a direct injection gasoline engine was tested at various combinations of
engine speeds, loads, and air-to-fuel ratios. The data from these tests allowed us to
calculate the actual MFB for the various operating conditions. The speed of the engine
was measured in rotation per minute (RPM), the load, which is a percent measurement of
how hard an engine is working, ranged from zero to one, and finally the it (Lambda),
which is calculated by dividing air—to-fuel ratio by stoichiometric (14.6). The five tests

that were conducted are shown in Table 2-1.

10

 

 

 

 

 

 

TEST RPM LOAD A (Lambda) 60
l 1500 0.25 1 330
2 1500 0.5 1 330
3 1500 1 l 330
4 2500 l 0.92 340
5 3000 1 0.92 340

 

 

 

 

 

 

 

Table 2-1: Data from five DI engine tests
The actual MFB was then plotted vs. crank angle to obtain “s-curves” that could be
represented using the Wiebe function. The 60 for each engine test was known and could
be used in the Wiebe function equation. The other parameters a, m, and A6 were
unknown and had to be found. To determine these values, the actual MFB plot at a
certain engine speed, load, and air-to-fuel ratio was compared to a Wiebe function plot
for that operating condition. The correct values for a, m, and A6 were found by

matching the actual MFB curve with the Wiebe function plot, which was a trial and error

process done using Matlab. These values for the five engine tests are shown in Table 2-2.

 

 

 

 

 

 

TEST RPM LOAD A. (Lambda) 60 a A m A6
1 1500 0.25 l 330 0.5 2.3 40
2 1500 0.5 l 330 0.8 3 29
3 1500 1 1 330 0. 15 3.1 30
4 2500 l 0.92 340 0.4 2.4 27
5 3000 1 0.92 340 0.3 2.7 27

 

 

 

 

 

 

 

 

 

 

Table 2-2: Data from five DI engine tests with parameters a, m, and A9
The five engine tests were extending to more operating conditions to obtain more MFB
curves via the Wiebe functions. These operating conditions are under typical real-world
engine conditions, and within these conditions engine control systems are designed to
achieve their desired outcome. The operating conditions that the tests were extended to

W218

ll

0 RPM: 500-5500
0 Load: 0-1
0 ll. (Lambda): 8-16
The map of the values for a, m, and A6 for the operating conditions above are shown in

Table 2-3, Table 2-4, and Table 2-5, respectively.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

LOAD

RPM 0 0.25 0.5 0.75 1 1. (Lambda)
500 0.1 0.1 0.1 0.1 0.1 1
1000 0.3 0.3 0.5 0.3 0.1 1
1500 0.5 0.5 0.35 0.15 0.15 1
2000 0.3 0.3 0.5 0.3 0.25 0.95
2500 0.3 0.3 0.3 0.3 0.4 0.92
3000 0.3 0.3 0.3 0.3 0.3 0.93
3500 0.3 0.3 0.3 0.3 0.3 0.92
4000 0.3 0.3 0.3 0.3 0.3 0.91
4500 0.3 0.3 0.3 0.3 0.3 0.9
5000 0.3 0.3 0.3 0.3 0.3 0.9
5500 0.3 0.3 0.3 0.3 0.3 0.9

 

 

 

 

 

 

 

 

Table 2-3: Values for Wiebe parameter a for extended engine operating conditions

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

LOAD

RPM 0 0.25 0.5 0.75 1 1. (Lambda)
500 2 2.5 2.5 2.5 3.3 1
1000 2.5 2.5 3 3 3 1
1500 2.3 2.3 2.6 3 3.1 1
2000 2.5 2.5 3 3 2.7 0.95
2500 2.5 2.5 2.6 2.6 2.4 0.92
3000 2.5 2.5 2.5 2.5 2.7 0.93
3500 2.5 2.5 2.5 2.5 2.7 0.92
4000 2.5 2.5 2.5 2.5 2.5 0.91
4500 2.5 2.5 2.5 2.5 2.5 0.9
5000 2.5 2.5 2.5 2.5 2.5 0.9
5500 3 2.5 2.5 2.5 3 0.9

 

 

 

 

 

12

Table 2-4: Values for Wiebe parameter m for extended engine operating conditions

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

LOAD

RPM 0 0.25 0.5 0.75 1 1. (Lambda)

500 38 33 33 33 33 1
1000 41.5 36.5 31.5 31.5 31.5 1
1500 45 40 37 30 30 1
2000 38.5 33.5 26.5 26.5 26.5 0.95
2500 32 27 24 24 27 0.92
3000 25.5 22.5 22.5 22.5 27 0.93
3500 30 27 27 27 27 0.92
4000 30 27 27 27 27 0.91
4500 30 27 27 27 27 0.9
5000 30 27 27 27 27 0.9
5500 30 27 27 27 35 0.9

 

 

 

 

 

 

 

 

 

Table 2—5: Values for Wiebe parameter A6 for extended engine operating conditions
Once all the parameters for the Wiebe function were determined for the engine

operating conditions, the MFB was known for these engine operating conditions. Thus,

by giving the engine speed, load, and air-to-fuel ratio, the MFB (xb) can be found via the

Wiebe function.

2.2.3 Motoring Pressure
Motoring pressure, Pm, is the pressure in a cylinder when there is no combustion.

Naturally a cylinder contains air inside of it even in the absence of combustion. As the

piston compresses and expands this air, the pressure rises and falls. This pressure is

known as motoring pressure, and it occurs before the start of combustion at 60. More

specifically, it is present when the exhaust valve open at the end of an engine cycle

(where xb = l) to 60, when combustion begins (where xb = 0). Thus, if 60 is given and

the crank angle where xb equals 1 can be determined, then the time that motoring

pressure is present can ultimately be found. The raw data of the motoring pressure of a

13

cylinder was found by using a cylinder pressure sensor that calculated the pressure at
each crank angle. Figure 2-3 shows the motoring pressure raw data (in Pascal) vs. crank

angle.

 

 

 

Pressure (Pa)

 

 

 

I

O l I L I I I
O 1 00 200 300 400 500 600 700

 

Crank Angle (degrees)

 

 

 

Figure 2-3: Motoring pressure raw data
A two-way filter was applied to the raw data off-line to filter out any noise that was
present. The advantage of the two-way filter is that it introduces no phase change, thus
keeping the data accurate. The plot of the motoring pressure (in bar) with the filter

applied to it versus crank angle is shown in Figure 2-4.

14

 

 

 

Pressure (bar)

 

 

 

0 l J 1 l 1 1 1
0 100 200 300 400 500 600 700
Crank Angle (degrees)

 

 

 

Figure 2-4: Two-way filter applied to motoring pressure

2.2.4 Total Cylinder Pressure

A cylinder pressure model that calculates the total cylinder pressure over 720
crank angle degrees was created based upon the previous two ideas, mass fraction burned

(using the Wiebe function) and the motoring pressure of a cylinder. Therefore,
Pcy, (a) = Pm (a) + PM, (a) (2)

It is known that the net pressure change between two crank angles is [4]:

1.3
_ __ we) _ we)
AP(6)—{P(6+l) P(6)[V(6+1)] H—Vr ] (3)

15

where P is the pressure, V is the volume in the cylinder, and V18 is the cylinder volume at

the ignition point [4]. Solving the previous equation for P(6), the net pressure at each
crank angle is:

Pne,(6) = P,,e,(6— 1) + AP(6— 1)

1.3
= net (9 -1)[Yfl:1_l.] + pr _1{_Yll§_] (4)
we) V(6— 1)

The volume V( 6) of the cylinder at each crank angle is plotted in Figure 2-5.

 

x 10'4

8 -a———T—‘_V_U _. T I '_ I I I TT—T—TW

 

 

Cylinder Volume (m3)

 

 

O 100 200 300 400 500 600 700
Crank Angle (degrees)

 

 

 

Figure 2-5: Volume in the cylinder at each crank angle

Since MFB is proportional to the net pressure AP,,e,, the net pressure at each crank angle

can be rewritten as

 

 

1.3
_ _ V(9—1) _ Vlg 5
Rename 1)[ vw) ] +Wbeb(9 1)[V(6_1)] H

where Wb is a scaling coefficient and Axb is
Axb = xb (6) - X), (6 — I) (6)

Since the combustion process varies from cycle to cycle, the cylinder pressure during one
engine cycle will be slightly different from the previous one, causing the combustion

process to contain stochastic properties. A random property is assigned to coefficient

Wb, which is a function of engine speed, engine load, and A/F ratio, to model the cycle to

cycle variations. Therefore, the pressure can be calculated by the following equation,

l.3
V(6—1) ‘ V18
= _ . _ _ (7)
10(0) Pm(6)+Pnet(6 1>[—V(9) j W!) ”7‘6 ulna—1)]

The W), can be viewed as a scaling factor of xb, and its value will increase as the engine

speed, load, or A/F ratio increases. Similar to the Wiebe parameters a, m, and A6, Wb

had to be calibrated for the engine operating conditions that was previously described.

The map of W), for the engine operating conditions is shown in Table 2-6.

 

 

 

 

 

 

 

 

 

 

 

 

 

LOAD

RPM 0 0.25 0.5 0.7 5 1 2. (Lambda)

500 5.00 11.75 18.50 25.25 32.00 1
1000 7.10 12.57 19.32 26.07 34.03 1
1500 9.20 10.98 20.13 28.10 36.06 1
2000 11.30 13.88 22.08 30.18 38.15 0.95
2500 13.40 16.78 24.98 33.17 40.23 0.92
3000 15.50 19.69 27.88 36.08 44.27 0.93
3500 17.60 22.59 30.78 38.98 48.20 0.92
4000 19.70 25.49 33.68 42.65 52.14 0.91
4500 21.80 28.39 37.10 46.58 56.07 0.9
5000 23.90 31.55 41.03 50.52 60.00 0.9

 

 

 

 

 

 

 

 

 

I 5500 I 26.00 I 36.67 L4733 l 58.00] 64.00 L 0.9 I

 

Table 2-6: Values for W), for extended engine operating conditions

The W}, in the cylinder pressure model was changed at the beginning of each engine cycle

to account for the cycle to cycle variations. For one engine cycle, it used one value, but

for the next engine cycle it was changed to a different value. The variance of W), was

chosen between 3% and 30% of its mean value. Its mean value at a specific engine
speed, load, and A/F ratio was the values that are shown in Table 2-6. Its variance was a

function of engine speed, load, and A/F ratio, and was calibrated. The values of the

variance for the extended engine operating conditions are shown in Table 2-7.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

LOAD

RPM 0 0.25 0.5 0.75 1 2. (Lambda)
500 0.100 0.090 0.080 0.070 0.060 1
1000 0.105 0.087 0.077 0.067 0.057 1
1500 0.110 0.084 0.074 0.064 0.054 1
2000 0.1 15 0.087 0.071 0.061 0.051 0.95
2500 0.120 0.092 0.068 0.058 0.048 0.92

V 3000 0.125 0.097 0.069 0.055 0.045 0.93
3500 0.130 0.102 0.074 0.052 0.042 0.92
4000 0.135 0.107 0.079 0.051 0.039 0.91
4500 0.140 0.1 12 0.084 0.056 0.036 0.9
5000 0.145 0.117 0.089 0.061 0.033 0.9
5500 0.150 0.117 0.083 0.050 0.030 0.9

 

 

Table 2-7: Values of the variance for the extended engine operating conditions

2.2.5 Cylinder Pressure Model Simulation

After finding the equation for the cylinder pressure that is given above, the
implementation of the total cylinder pressure model in Matlab Simulink was begun. The

goal of the model was to calculate the cylinder pressure at each crank angle.

18

The cylinder pressure model in Matlab had the ability to output the cylinder

pressure at any crank angle, engine speed, engine load, A/F ratio, crank angle, and spark

timing (60) are used as the inputs. Figure 2-6 shows the plot of the cylinder pressure

signal generated by the model when the engine is simulated at 2500 RPM, 100% engine

load, and stoichiometric A/F ratio.

 

 

60 I—— T ‘F .T ‘ I U I ‘—

A
O
r
J

 

Pressure (bar)
00
O

N
O
I

 

 

 

 

 

 

 

 

 

0 0.05 ' 0.1 0.15 0.2 0.25 0.3
time

 

 

 

Figure 2—6: Cylinder pressure plot (2500RPM, 100% load, stoich. A/F ratio)
Similarly, Figure 2—7 shows the plot of the cylinder pressure signal generated by
the model when the engine was simulated at 1500 RPM, 100% engine load, and

stoichiometric A/F ratio.

19

 

 

60 I Tl I I' I

50~

b
O
r

I

 

Pressure (bar)
0.)
O
I
L

N
O
l

 

 

 

10~ _.

 

 

 

 

 

O I J I l __._._-

0 0.05 0.1 0.15 0.2 0.25 0.3
time

 

 

 

Figure 2-7: Cylinder pressure plot (1500RPM, 100% load, stoich. A/F ratio)

Notice that Figure 2-7 shows the engine simulated at a lower RPM than Figure
2-6, which resulted in lower peak cylinder pressure. The same can be said if the engine
were simulated at a lower engine load, the peak cylinder pressure would in fact be lower.
Figure 2—8 shows the plot of the cylinder pressure signal generated by the model when

the engine was simulated at 1500 RPM, 50% engine load, and stoichiometric A/F ratio.

20

 

 

 

30 "————I_— l I ' _ —— T I

25» ~

20~ ~

 

15»

Pressure (bar)

10» -

 

 

 

 

 

 

 

- I ____ . l______

0 0.05 0.1 0.15 0.2 ”0.25 0.3
time

 

 

 

 

Figure 2-8: Cylinder pressure plot (1500RPM, 50% load, stoich. A/F ratio)

Figure 2-8 shows the plot of the cylinder pressure for three full engine cycles and
a portion of a four cycle. The first peak or the first cycle can be viewed as an
initialization cycle and is similar to a cycle that contains no combustion. Thus, the first
cycle is essentially the motoring pressure. Also from Figure 2-8, it can be seen that the
peak cylinder pressure varies during each cycle and this is due to the cycle to cycle

variations of the combustion process. More specifically, the peak value of the cylinder

pressure that is calculated varies over each engine cycle due to the variance of Wb, which
causes the combustions variations to be modeled very well.
Since an engine cycle consists of 720 crank angle degrees, the cylinder pressure

model gives the cylinder pressure signal for each engine cycle operating under the engine

conditions that were previously discussed. Consequently, this model can be integrated

2l

with a mean valve engine model to produce cylinder pressure signals using the engine
speed, load, air-to-fuel ratio, and ignition timing from the mean valve engine model.
Based on the cylinder pressure signal, engine combustion control strategies can be
developed to control the peak cylinder pressure location, the intake and exhaust valve
opening and closing timings, ignition timing, etc. Therefore, this cylinder pressure model,
integrated with the mean valve engine model, provides a tool for developing closed loop

combustion control strategies.

22

Chapter 3: Air-to-Fuel and Fuel Ratio Modeling

In this chapter, an air-to-fuel ratio and fuel ratio model is introduced. The purpose
of this model is to use simple dynamics to model an engine’s A/F ratio for control design
and evaluation. In addition to the calculation of the A/F ratio in this model, the ratio of a

dual-fuel system is also developed.

3.1 Air-to-Fuel Ratio Modeling

Air fuel ratio is the mass ratio of air and fuel trapped inside the cylinder before
combustion starts. Mathematically it is the mass of the air divided by the mass of the fuel

as shown in the equation below.

A/ F ratio = M (8)

m fuel
If the ratio is too high or too low, it can be adjusted by adding or reducing the amount of
fuel per engine cycle that is injected into the cylinder. The engine A/F ratio that is
modeled in this research can be adjusted by both a port fuel injector and a direct fuel

injector.

3.1.1 Air Flow

The air mass used for calculating the A/F ratio is physically the air mass trapped
inside the cylinder of an engine. This air mass enters the intake manifold through the
throttle and can be measured using a mass-air-flow sensor. The throttle is a valve that
directly regulates the amount of air entering the cylinder, and it can have a direct effect
on the A/F ratio. In automotive vehicles, the throttle valve opening amount is determined

by acceleration pedal or controlled by an engine controller.

23

Since the air mass flow is adjusted for regulating engine output torque (or power),
it can not be adjusted for controlling engine A/F ratio and the fuel mass flow is used to

control A/F ratio. In this model, the air mass flow rate in Air is modeled as a nominal

value plus a disturbance input as shown in the equation below

mm, =w0+Aw whereabzl (9)
The disturbance A0) is due to the change in the air flow when the engine operational
condition changes.

The air mass flow at certain engine operating conditions could be constant, while
at others, the air mass flow could abruptly increase or decrease. One example of this is
when the throttle valve remains open at a constant angle and then is quickly changed to a
larger angle, allowing more air to enter the cylinder. When simulating the abrupt change
of air mass flow described in the previous example, this change can be modeled as a step
input

Similarly, the air mass flow changes over engine cycles due to manifold fiow
dynamics. In this case, the air mass flow variations are modeled as a constant signal plus

a random signal. The random signal is bounded by a constant value.

3.1.2 Fuel Flow

The fuel mass used to calculate the A/F ratio is the fuel flow rate, mfm), that

enters the cylinder. In the engine that is being studied in this research there are two fuel
systems providing fuel for the cylinder. The first fuel system is known as a port fuel
injection (PFI) system and is very common in gasoline engines. The liquid fuel that is

sprayed by a PFI injector does not directly enter the cylinder at the time that it is initially

24

sprayed. Instead it is injecting into the intake port and will enter the cylinder when the
intake valve opens. Thus, there is a very small amount of time between when that fuel is
injected and when it actually enters the cylinder. Also, the liquid fuel that is sprayed by
into the intake port only partially enters the cylinder after the intake valve opens. Some
of the fuel is stored as small fuel puddles on the intake port walls (which can evaporate)
and some at the back face of the intake valve. The fuel flow dynamics of a port fuel
injector is known as wall-wetting dynamics [5]. The A/F ratio wall-wetting dynamics
can be model by the following transfer function

m _a'-s+l
PFI ,B-s+l (10)

 

where the values for a and 6 that were used in the model were 0.5 and 0.8, respectively.

The second fuel system that sprays fuel into the cylinder is known as a direct
injection (DI) system. Obviously there is no wall-wetting dynamics involved when a D1
is used because fuel is directly injected into the cylinder.

The amount of fuel per engine cycle that is injected by both injectors is done so
by applying electric pulses to the injector solenoid that is located at one end of the
injector. The opposite end of the injector is the spray tip where fuel is released. When the
solenoid is energized, the plunger is lifted slightly to allow pressurized fuel to be sprayed.
The electric pulses that are applied to the solenoid are proportional to the amount of fuel

to be sprayed.

3.1.3 Air-to-Fuel Ratio Model Operating Speed

To begin developing a system that models the A/F ratio, the first assumption that

is made is that the engine will be running at a constant speed. For a given engine speed,

25

the amount of time for an engine cycle to complete can be found. The engine speed
chosen for this model was 1500 RPM, thus an engine cycle will last 80 milliseconds
which is shown below.

minute 605econds 2rounds

 

t .. = =.08seconds
W16 1500rounds minute cycle (11)

Knowing the amount of time it takes to complete one engine cycle is important in

modeling the physical time delays of the engine.

3.1.4 Physical System Dynamics

The biggest challenge when modeling an engine that will be used for controller
design purposes, is determining how to model the delays of the engine. From previous
work, it has been concluded that there are three characteristic delays that must be
accounted for during transient operation of an engine. These delays are the time-delay of
the control system, a transport delay in the intake manifold, and a physical delay of the
fuel flow [6]. There are other delays that occur in an engine, but the previous three will
be the ones of interest in this work.

The first delay that will be accounted for in this A/F ratio model is the delay of
the fuel flow. The delay of the fuel flow results from the finite rate of evaporation of the
fuel film on the intake manifold and port walls. This occurs when a port fuel injector is

used.

3.1.5 Air-to-Fuel Ratio Sensor

The A/F ratio is dependent on how far away from stoichiometry the air-fuel
mixture is. Stoichiometry is achieved when all the available fuel molecules are combined

with the available oxygen molecules. Specifically, for gasoline, when the mass ratio is

26

14.6 to l, the mixture is said to be stoichiometric. In IC engines, the A/F ratio is denoted
by the variable )1. and a stoichiometric mixture has a )1 value equal to 1. This A is found by
dividing the current A/F ratio by the stoichiometric A/F ratio, as shown in equation (12).

_ A/F ratio
A/F ratiostoich

 

(12)

If ,1 is less than I, then the mixture is said to be rich, and a mixture is said to be lean if 1. is
greater than one.

As previously stated, the air-to-fuel ratio is measured by a device known as an
oxygen sensor, or sometimes called a lambda sensor. The sensor is located in the exhaust
manifold and its main purpose is to determine how far away from stoichiometry the air-
fuel mixture is. The time it takes for the sensor to determine that the mixture is above or
below the value of 1, can be viewed as a time delay and must be modeled as such.

Typically this delay is about 40 milliseconds when an engine is running at 1500RPM.

3.1.6 Stoichiometric Air-to-Fuel Ratio

Engines are designed to maintain a stoichiometric A/F ratio in order to maximize
the efficiency and life of the three—way catalyst found in the oxygen sensor. Air-to-fuel
ratios that are richer than stoichiometry allow for greater peak engine power, leading to
poor fuel economy. Similarly, an A/F ratio leaner than stoichiometry will result in near
optimum fuel mileage and will produce the least amount of carbon dioxide (C02)
emissions, but can also produce sharp rises in nitrogen oxides (NOx) since the three-way
catalyst becomes less efficient under lean operation conditions. Also, if the A/F ratio
becomes too lean, the air-fuel mixture may sometimes fail to ignite, causing misfire,

leading to high hydrocarbon (HC) emissions. Although it may be possible to run

27

smoothly at a lean A/F ratio, manufacturers must focus more on engine emissions due to

federal and state environmental regulations.

3.1.7 Air-to-Fuel Ratio Calculation

Since air-to-fuel ratio is the ratio of the mass of air to the mass of fuel, then the
mathematical calculation of this ratio is found by dividing the mass of the air by the mass

of the fuel. The total fuel in the cylinder, mm, , is the fuel flow from the port fuel
injector and the direct injector as shown in the equation below.

mFuel = then + "'11)! (13)

Engines are design to achieve a target stoichiometric A/F ratio, thus the target ratio,

Atarget, should be equal to one. The equation below gives the expressing for itarget.

mAir
’11 t Z _———. 14
arge 14.6 ' mpue, ( )

The inverse of A/F ratio 1/1 is known as the equivalence ratio (D , and is given as

14.6-mm?)

¢target : (15)

mAir
The remainder of this research will use ¢target instead of Atarget because it will allow for

easier calculations. The air flow as discussed in equation (9) was
mm, = (1)0 +Aa) wherewo :1

Therefore, marge, becomes

. l
¢target = 146 ' szre/[Wj (I6)

28

. . . l
Usmg the Taylor ser1es expansmn about 600 = l of =

mm, ((00 + Aw

,I = —1-——-1-§Aa) (17)
mair 0'1) (lb

Finally, the target equivalence ratio is given in the equation below.

 

) results in

 

. l l
¢target : 14:6 ' mFtiel[a_b"—%'2_Aw] (18)

The above target equivalence ratio calculation will be combined with a fuel ratio

calculation that will be used for controller design purposes.

3.2 Fuel Ratio Modeling

The sum of the fuel that is injected into the cylinder by the port fuel injector and

the direct fuel injector is the total fuel, thuel. The amount of fuel injected by one

injector divided by the sum of the two is the fuel ratio of PFI to DI.

3.2.1 Fuel Ratio Uses

The fuel ratio can be used to determine which fuel system should have a larger
impact on how much fuel is injected into the cylinder. Since a direct fuel injector has
immediate injection of its fuel with significant charge cooling effect, it can have a
quicker response to the desired amount of fuel that is needed by an engine. Although a
port fuel injector may have a slower response due to its wall-wetting dynamics, the fuel
ratio will impact the combustion characteristics of an engine.

Fuel ratio also can be used to regulate or control two fuel types. For example, an

engine may have the ability to run on gasoline and ethanol. The gasoline could be

29

injected by a port fuel injector, while the ethanol could be injected by a direct injector.
Although, implementation of this may require to separate fuel lines and separate fuel
tanks, the ratio of gasoline to ethanol, or two other types of fuels, may be of interest to

future engine control designers.

3.2.2 Fuel Ratio Calculation

The calculation of the fuel ratio is found by dividing the fuel in the cylinder from
one injector by the total fuel. For PFI, the fuel ratio calculation is shown in the equation

below.

FuelRatioz . PF]. 2 .PF’ (19)
mPFI + m0] mFuel

 

Solving the target equivalence ratio for m Fuel gives

mair

 

 

m , = ——
Furl 14-6'¢targct (20)
Substituting equation (20) for raw, into equation (19)
Fuel Ratio: . mPF’ = PF] ¢lar°el (21)
mair l14-6'¢target mair
From the Taylor expansion of . and using charge. = l , the fuel ratio becomes
mair
. . l 1
Fuel Ratio 21461on, ————2—Aa) (22)
‘00 wt)

The above fuel ratio calculation will be combined with previous A/F ratio calculation and

will be used for controller design purposes.

30

3.3 Air-to-Fuel Ratio and Fuel Ratio Model Simulation

Knowing the equations for the calculation of the air-to-fuel ratio (using the
equivalence ratio) and the fuel ratio, the development of the model was begun. The
model was developed in Matlab Simulink and it included the modeling of the wall
wetting dynamics of the port fuel injector, the air flow dynamics, and three appropriately
modeled time delays. These time delays were the time delay of the fuel injected by the
port fuel injector, the time delay of the fuel injected by the direct injector, and the oxygen
senor calculation delay. A delay was also included for the air flow dynamics.

The air-to-fuel ratio and fuel ratio model was assumed to be operating at
1500RPM thus the amount of time per engine cycle would be 80 milliseconds. Knowing
this, the time delays in the model were:

1. Time delay of PFI response: 50 milliseconds

2. Time delay of D1 response: 50 milliseconds

3. Time delay of oxygen senor calculation: 40 milliseconds

4. Time delay of air flow: 200 milliseconds

These time delays were linearized as first order transfer functions as shown below

1. Transfer function of PFI delay: ——1—
0 05s + l

2. Transfer function of DI delay:
055 + l

3. Transfer function of oxygen senor calculation delay : —L—
0.04s + l

 

4. Transfer function of air flow delay:
0.23 + 1

3|

The time delays were cascaded together with the calculation of the equivalence ratio and

fuel ratio in Simulink as shown below.

 

 

 

 

.9
E .9
C .—
9 8
.2 LL
3
CT
1 DJ
#3
s
L c;
\
<9
3 \
2 "-.--,_\
X

 

 

 

 

l”

 

 

 

 

 

 

 

 

 

I 14.6

 

0.055+1

 

 

 

 

 

 

 

 

j

l l/ ‘\\
' . — l N i a
b 9L L/

Figure 3—1: Equivalence ratio and fuel ratio model in Simulink

 

A subsystem of the model shown in Figure 3-1 is shown below.

32

 

> PFI
Equivalence Ratio >

> Air Fuel Ratlo >

 

 

 

 

“Eq. Ratioland Fuel Ratio Model

Figure 3-2: Subsystem of equivalence ratio and fuel Ratio model in Simulink
From Figure 3-2 it is clear that the system model has two outputs, the equivalence ratio
and the fuel ratio. Likewise, the model will has 3 inputs, the PFI input, the DI input and
air mass input. As previously discussed, the air mass is a disturbance input, and the PFI
and DI fuel mass are the control inputs. Figure 3-3 shows the open loop response of the
equivalence ratio and fuel ratio signals when the PFI and DI inputs are a constant value of

0.04 and the air input is a constant value of 0.1.

33

 

 

 

 

 

 

 

 

l! a _ H H M 40'
-_ _ . w _
.2 _ _ 2 . o
—_ _ . _ _

-_ _ _ _ .mw
I--+.ri:tfi ....... n ma 19
-_ _ _ . A I
- . _ _ . 0.1%

— _ H . .EF
_ _ _ .
Il11IAI411111iII|111l-_. 1111111 _ 111111.._..— 18
. _ . _ -

. . _ _ —

__ _ . _ u
_ . _ _

Flill-IL 11111111111111 L IIIIII L1 11L 17
_ A . _

_ . . a
_ _ . a
_ _ . A

I111 1121111111611111111 111161111112 111111
1_ , _ _ _ L6
_ _ . _ _
“_ _ _ _ _
—_ _ _ _ _
1.. _ _ . _ 5
I111—11_1111111_11111I1_ 1111111 _ 11111111 _ 111111 1
_ _ _ _ _
u. _ _ . _
_ _ _ _ _
u. _ _
1d
.
—
-
.

 

 

 

 

 

036m. 6:". new ozmm 605.6233

time

 

 

Figure 3-3: Open loop response of equivalence ratio and fuel ratio model

The open loop time response of the system shows that given the constant inputs for PFI,

DI, and air, the equivalence and fuel ratios will become constants at steady state.

34

Chapter 4: Baseline Control Design

This chapter will discuss the baseline control design that was developed to control
the air-to-fuel ratio of IC engines. It will begin by discussing several controller designs
that have already been developed and designs that are currently being used in production.
The chapter will conclude by discussing the design of a baseline proportional integral

derivative (PID) controller.

4.1 Previous Controller Designs

There have been several engine controller designs over the past 40 years in which
the goal is to improve the efficiency and exhaust emissions of the automotive engine. A
key development in the evolution was the introduction of a closed loop fuel injection
control algorithm by Rivard in the 1973 [7]. This strategy was followed by an innovative
linear quadratic control method in 1980 by Cassidy [8] and an optimal control and
Kalman filtering design by Powers [9]. Although the theoretical design of these
controllers was valid, at that time it was not realistic to implement such complex designs.
Therefore, the production of these designs did not exist and engine designers did adopt
the methods.

Due to the increased production of the microprocessor in the 1990’s, it became
practical to use these microprocessors in developing more complex control and
estimation algorithms that could potentially be used in production automotive engines.
Specific applications of A/F ratio control based on observer measurements in the intake
manifold was developed by Benninger in 1991 [10]. Another approach was to base the
observer on measurements of exhaust gases measured by the oxygen sensor and on the

throttle position, which was researched by Onder [l I]. These observer ideas used linear

35

observer theory. Hedrick also used the measurements of the oxygen sensor to develop a
nonlinear, sliding mode approach to control the A/F ratio [12].

All of the previous control strategies were applied to engines that used only port
fuel injections, where fuel was injected in the intake manifold. The development of these
control strategies for direct injection was not practical because the production of direct
injection automobiles did not begin until the mid 1990’s. Mitsubishi began to investigate
combustion control technologies for direct injection engines in 1996 [13]. Furthermore,
engines that used both port fuel and direct systems appeared a couple years ago, leading
to the interest of developing the corresponding control strategies.

Current production A/F ratio controllers use closed loop feedback and feed
forward control to achieve the desired stoichiometric mixture. These controllers use
measurements from the oxygen sensor to control the desired amount of fuel that should

be injected over the next engine cycle and have been able to control the A/F very well.

4.2 Baseline Controller Design

Two controller designs for the contr01 of the A/F ratio and the fuel ratio were
developed in this research. The first design was a baseline controller, in which the
control design target was two-fold:

1. Control the air-to-fuel ratio at a desired value of unity

2. Control the fuel ratio at a desired value between zero and unity
The design of a baseline controller to control the A/F ratio and the fuel ratio was very
straight forward. Since there were two outputs from the A/F ratio and fuel ratio model,

this means that there would be two inputs into the baseline controller. Similarly, the two

36

outputs of the controller result from the two control inputs of the port fuel injector signal
and direct injector signal.

In a typical PID controller, the controller corrects the error between the desired
output value and the measured value. Since the equivalence ratio and fuel ratio are the
two measured signals, two controllers were cascaded together to control the PFI and DI
inputs. The first was a PID controller that corrected the error between the desired
equivalence ratio and the measured equivalence ratio; while the second was only a
proportional integral (PI) controller that corrected the fuel ratio error. Figure 4-1 shows

the two cascaded controllers.

 

 

 

 

 

 

 

 

PFI > PF|_input PFI >
> Eq. Ratio Error ____ _> Dl input
9' -——j 01 >
1 > Fuel Ratio Error
59- Ratio Fuel Ratio U ‘—
PID Controller Pl Controller

Figure 4-l: Cascaded PID and PI controllers
There are three parameters or control gains of a PID controller that must be
determined. These gains are KP, K, and Kd, which correspond to the proportional,
integral, and derivative terms. Within the equivalence ratio PID controller, there where
six gains, three for the PFI PID terms and three for the DI PID terms. These parameters
were found by tuning the response of closed loop system to be stable and relatively fast.

The values of the six gains were:

KP] = sz = 0.000005

K11=K12=-3

37

Kd1 = Kdg = 0.0207

Figure 4-2 shows the PID controller for the equivalence ratio error signal using the
previous gains. The equivalence ratio error signal is decoupled into one signal for the

PFI output and one for the DI output.

~—e —>15e- 00? "M

 

 

 

 

 

 

 

 

 

  

 

 

 

Kd1 Derivative _
— 1
Q 1 y ——-——-— 0.3 I———> g 2 —--~——w* —————— {I >
Eq. Ratio K't R——- PFI
Error 1 Integrator_ 1
IL A ———>{ 0.0207 -— ——~
Proportional_ 1
Kp1
r—'———' _-_ -- ..
,%,_2___ I5e-006 -———} du/dt — -
Kd2nJ Deng/e;
1 F” “ 1
”I 0.3 -—-—> g — x3
__ .- _ ., DI
K12 Integrator _2
H 0.0207
_} Proportional _2
Kp2

Figure 4-2: PID controller for the equivalence ratio error
Similar to the PID controller gains for the equivalence ratio, the PI controller of
the fuel ratio has only two control gains. These controls gains correspond to the

proportional and integral terms and were:

Figure 4-3 shows the PI controller for the fuel ratio error using Kp = 0.5 and Ki = 5.

Within this PI controller, the fuel ratio error signal was multiplied by the PFI signal,

38

while the DI signal was fed through. Thus, the fuel ratio is controlled only by using fuel
ratio error and the PFI signal. The DI signal will be adjusted to be one minus the product
of the fuel ratio signal and the PFI signal, since the total ratio can only be between zero

and one. This adjustment is done within the A/F ratio and fuel ratio model.

-.—_—é

1
5 F‘i E +4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fuel Ratio -- _
Error K1 1 Integrator
l--- l
T 0'5 J Proportional
Kp ‘—> ,
[ _1_ JP —— ,, —»———> X fl
PFl_input Product PFI
Q2 -—--~ - ——-—+<:>
Dl_input DI

Figure 4-3: PI controller for the fuel ratio error

4.3 Simulation Results

The baseline controller was implemented and simulated with the A/F ratio and fuel
ratio model in Matlab/Simulink. Several simulations that modeled real world A/F ratio
and fuel ratio situations were conducted. For all the simulations, a target equivalence
ratio was taken to be unity and a target fuel ratio was 0.6. The selection of unity for the
target equivalence ratio was based on the idea of stoichiometry, since engines are design
to run at such, while the selection of the target fuel ratio was arbitrary. The main
difference between each simulation was how the closed loop system responded to various
air flow inputs. Since the air flow is not a controlled input, it can be view as a
disturbance or uncertainty, Aw. Although air flow can only be modeled to be between

zero and unity, the effectiveness of the controller will be validated by its response to

39

various air input disturbances. These simulations using several air input disturbance will

now be discussed and analyzed.

4.3.1 Simulation 1

The first simulation begins using a constant value for the air input and at a later
time sums a step input to this initial constant value. This simulates a situation where the
throttle (air) initially remains open at a constant angle and then quickly (almost
instantaneously) opens or closes to a larger or smaller angle. The constant value was
chosen to be 0.1 and the step input value is chosen to be 0.15. The time at which the step

input would be added to the constant value was chosen to be at 6 seconds.

 

 

 

 

 

 

 

 

 

 

 

 

 

T I ' — ‘17— I— _ II ”I"- I T
1 1 I l I I 1 _____ '
1.2-.__-L.-__ 12___11 _____ l _____ L__--J_-___J_I Eq'Ratlo _
I I I I I I I FuelRatio
l I 1 I | I | 1 I
| | I I I 1 | I I
l 1 l 1 I 1
o 1 _----I------: — —- — —' - I I --|-----'- I I .1
._ 1 \ w""— l 1---1 1
a I 1‘-”‘1 1 I I 1 l l
m I I l 1 l 1 1 l I
_ l | 1 l 1 l | 1 |
g l I | l l | I l I
u- 0.84——-—I—--—I—— ——I ------ I ————— I----I—--4 ————— I ----- I—n—a
“o I l 1 l l I l l
C 1 l | 1 I I I 1 I
(U l l \l 1 1 I | 1 1
O I I 1 I I | I 1 |
'5: 1 I \ I 1 I
m 1 1___7 II“. 1 g
(I 0'6 1 W I 1 I
a) I 1\ 1 l 1 1 1 1
Q I 1\ 1 1 1 1 1
C 1 \ 1 1 I 1 l
2 l \ l I 1 l 1
g 0.4~—--«-.— ——————————— .'— ————— : ----------------
8- : '
LU 1

Controller responds
to step input

1'
l
| l
l
I I
|
l

 

 

7 7.5 ‘8 8:5 '9 9.5 “10
time (seconds)

 

 

 

Figure 4-4: Closed loop response of simulation 1

40

The closed loop response of the system in simulation 1 shows a fluctuation at 6 seconds,
resulting from the step input that was applied at that time. The controller quickly

adjusted the control inputs to maintain the desired equivalence ratio and fuel ratio.

 

 

 

 

 

 

 

 

 

 

 

0.07 I I I T l I ‘l ‘F l __
: : : I I I : : """ PF'
I r : : : r : : DI
0.06 ..... 1.---4 ..... 1-- ._I ______ I ...... -
: : ’;,,__; _____ L : : I I
1 1 I ----—-——--i ----- b—---d -----
I | ”I l l | | | |
"’ I I / I I I I I I I
20.05 HHHH T_-__'l""—_-I """" I' "T-_"‘l _____ l ————— I _____ r"— "‘
5 I " I I I I I I I
‘e "T"""I 't : I I : : :
E 1 1 1 I I I 1 I I
a 0041—-——:~—————:——- —: ----- ————— : ------ : ————— :— —————
5 : I | | I I l
'0 I I I I I | |
c | l I I I l l
(u 1 1 1 I I I
E 0-03 ”*I—“HI-H-"I ----- I ----- -----
D. 1 l I 1 I 1
0'02 Control inputs responds _ _ I _____ L _ _ _
tostepinput ; ;
0.01 I I 1 : 1 I
7 7.5 8 8.5 9 9.5 10
time(seconds)

 

 

 

Figure 4-5: PFI and DI control inputs for simulation 1
The control inputs, PFI and DI fueling, responses in simulation 1 show the control efforts

of PFI and DI were increased to achieve and maintain the desired ratios.

4.3.2 Simulation 2

The second simulation begins using a uniform random number for the air input
and adds a step input of 0.15 at 6 seconds. The random number had a minimum value of
-O.lS, a maximum value of 0.15, and the value changed to a number between these two

extremes every time it was sampled.

4|

 

 

 

 

 

I. I fl l T I l l i 1
1_2;,___-.' ______ : _____ L____I _____ ----- EqRatio I_-
I I I I l
| I I I f I I I I
I 1 1 1 i 1 I I I
1 a "9 f S ,5 1’ 1 I a I 1-
1 VV .3! ’ ' ‘ I}; p; s; “I 2 ‘\‘:5’\' 4‘WV" ' “fi-fi‘y‘V‘f ‘29:.”

 

Controller responds
to step input

Equivalence Ratio and Fuel Ratio

0.2 —————

I T _____ 1 ————— FT-" T
|

 

 

_ I I I I I
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
time (seconds)

 

 

 

 

Figure 4-6: Closed loop response of simulation 2
The closed loop response of the system in simulation 2 was very similar to simulation 1
although there was fluctuating random air flow disturbance throughout the entire
response. The controller achieved and maintained the desired ratios but due to the

random input disturbances could not maintain a smooth response.

42

 

 

 

 

 

 

 

0.07
0.06
.9
3. 0.05
E
"Q
E
8 0.04
5
'O
C
(U
E 0.03
:1
0'02 } I 1 , Control inputs responds I _ :
: i ; j to step input 1 1
0.01 4 I 1 1 I 1_ 1_..__ -1 ___ _I_-_
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
time (seconds)

 

 

 

Figure 4-7: PFI and DI control inputs for simulation 2
The control inputs in simulation 2 were similar to those of simulation 1 except for the
variations in its signal. The control efforts of PFI and DI achieved the desired ratio but
its variations caused fluctuations in the desired NP and fuel ratios. These fluctuations

corresponded to the changing random number that was modeled as the air input.

43

Chapter 5: Sliding Mode Control Design

The previous baseline PID control design was the first controller that was
designed for the A/F ratio and fuel ratio model. The second design was a sliding mode

nonlinear controller and will be discussed.

5.1 Sliding Mode Control Theory

Before the design of the sliding mode controller, a thorough understanding of the
sliding mode control theory must be done. Sliding mode control theory has one key
feature that is of intereSt as it applies to the air flow disturbance and that is its robustness
to matched uncertainties [14].

In this theory there is a sliding surface or manifold s that must be chosen. During
one phase of the system, known as the reaching phase, the states of the system can be
forced toward the sliding surface and maintained there. This is achieved by a sign
function switching control. During the sliding phase of the system, the states remain on
the sliding surface through the use of stabilization [l4].

Designing a sliding mode controller requires states of the system to be known and
transformed into a particular from. Therefore, the states of the equivalence ratio and fuel
ratio model must be brought to this form. This form, the states of the equivalence ratio

and fuel ratio model, and the validation of this form will be discussed.

5.2 Sliding Mode Control Model

The A/F ratio and fuel ratio model discussed in Chapter 3 was a simplified model

of a portion of an IC engine. It modeled the fuel flow dynamics, air flow dynamics, and

44

calculated both the A/F ratio and fuel ratio. The fuel flow and air flow dynamics along

with the calculated ratios led to a nonlinear model.

In order to begin the design of a sliding mode controller, the A/F ratio and fuel

ratio model had to be transformed into a particular form. This was done by first

separating the model into three sections: fuel flow dynamics, air flow dynamics, and

oxygen sensor dynamics.

Figure 3-1 showed the model in its original form in

Matlab/Simulink, while Figure 5-1 shows the same model divided into the three sections.

 

 

 

 

 

 

 

T.— _ _-
< _1 Iblni Out1
PFI
(mm Out2
DI
Fuel Flow Dynamics
(G3)
I 3 j’lm Out1

Air"

 

 

 

Air Flow Dynamics

(G2)

 

 

+1n1 Out1—fl)

 

 

 

Equiwlence Ratio

 

Oxygen Sensor

 

 

 

Iv H

 

Dynamics (G 1)

@

 

 

 

 

 

Fuel Ratio

Figure 5-1: Equivalence ratio and fuel ratio model divided into 3 sections

In Figure 5-1, the oxygen sensor dynamics are denoted as (31, the air flow dynamics are

denoted as G2, and the fuel flow dynamics are denoted as G3. The contents of 61, G2,

and G3 are shown in Figure 5—2, Figure 5-3, and Figure 5-4, respectively.

\ \\\‘
KL)— 14.6 /;‘_>%-->
|n1 //.
GaIn

 

 

 

» --->C:>

Cut 1

Figure 5-2: Oxygen sensor dynamics of GI

45

I\‘\.‘
I \\

\\

1

_ I‘ _— _-
C -(1/ (onA2\ C _i ___ Iw—‘flj
In 1 I/// 2:124 Out 1

Gain LIE

Figure 5—3: Air flow dynamics of G2
alpha-1’3: T T— 1__ \
In1 I’f’ta 9“ I I 231's” + (it—1)

1
< 2 ~
|n2 aid

Out 2

 

 

 

 

 

 

Figure 5—4: Fuel flow dynamics of G3
The state space equations of G1, G2, and G3 were found to obtain the states of each of

the three dynamics and are shown in equations (23), (24), and (25).

X] = Alx] + Blul

23
==Cix1 ( )
5c =A x +B a)
2 2 2 2 (24)
3’2 = C2ch
X3 = A3X3 + B3143
T T T (25)
)3 = C3x3 +D3u3, C3 = [C31 C32]
Combining equations (23), (24), and (25) resulted in
A] Bl O O
X = A2 x+ O C2C3lx3x2 + 32 (0+ 0 M3 (26)
A3 0 0 B3

The output equations for the equivalence ratio and fuel ratio using equation (26) resulted
in
I’EquIo = C 1X1 = C2C31~W3 (27)

(28)
.I’FueIRano = 14-6 ' C2X2 (C32x3 +-625u1)

46

The values for the matrices were:
AI = -25;B1=14.6;Cl= 25; D] =0;

A2 = -5;BZ = -1;C2 =5;D2 =0;

- 20 0 0.4688 0.625 0
20 20 0 0 0
A3: 0 -20 0 ;Bq= 0 1;Cq= ;D‘;= ;
‘ ‘ ‘ 0 0 0.46875 ‘ 0.625 0
O 0 -l.25 l 0

It can be seen that the state vector x3 has three elements denoted as

x31
x3 = x32
x33

States x1 and x3 were scalars. The X, state equation contains nonlinearities which results
from the product of the two states x2 and x3. The remaining .122 and 5:3 state equations are

linear equations that contain the air input disturbance and the fueling inputs, respectively.
Now that the states of the equivalence ratio and fuel ratio model are known, it was

transformed into the form [15]
'7 = fam, z)+ 6(7). z) (29)

2= fbm, Z) +007, z)" (30)

This form, usually referred to as the regular from [14], is special because it allows

fa(77,z) to contain the nonlinear portion of the state equation I7 and isolates the
disturbance 607,2). Similarly, the fi,(l],z) portion could be nonlinear as well and it is

isolated from the input portion G(I],z)u of 2.

47

A change of variables [77] = T - x transformed the original system into the regular
2

form, where x: [x1 x2 X31 x32 x33]T and the transformation matrix Twas

— —

1 O 0 O 0
0 l O 0 0
T: 0 O l 0 -0.625 (31)
0 O O O 1
_O 0 0 l 0 J

 

 

Using the transformation matrix T, which lets 77=[x1 x2 x31—0.6ZSX33]T and

Z = [x33 x32]T , then original system (represented in the regular form) becomes

 

-25 0 0 n1 0 1460 0

f7 : O -5 0 772 + 0 21+ 0 772(773+0.625Z1+Z2)+ -I 6 32
0 0 —20 773 —11.25 0 0 ( I

L ‘C J W—J

fam’ZI 507,2)

where 6260-1
, -125 0 z. 1 o u,
0 -20 z2 0 1 uz (33)
\ J \__V_.J

 

ffirrm) C
To ensure that this transformed model was exactly that of the original model, the
open loop response was compared to that of the original. Figure 5-5 shows the open loop
response of the transformed equivalence ratio and fuel ratio signals when the PFI and DI
inputs are a constant value of 0.04 and the air input is a constant value of 0.1. These were
the same values for the PFI, DI, and air inputs that were used in the open loop response of

the original model.

48

 

 

___—..___....__._..-————4

.0
co

.0
ca

______..4

.o
4:.

Avg—__._ _____

———~—-—<

Equivalence Ratio and Fuel Ratio

----- Eq. Ratio
Fuel Ratio

0.2

 

 

 

 

 

 

 

i l

8 9 10

 

 

 

 

 

Figure 5-5: Open loop response of equivalence ratio and fuel ratio transformed model
The error between the open loop response of the model transformed into the regular form

and the original model was extremely small. For the equivalence ratio, the steady state

-l3 .
error was smaller than 10 and for the fuel who the steady state error was smaller than

-5 . . . .
10 . Thus, the transformed model was valid and essentially an exact representation of

the original system.

5.3 Sliding Mode Controller Design

In sliding mode control theory, the control input 14 is designed to bring the sliding
manifold s to zero in finite time and maintain it at zero. To design the sliding manifold

s=z—¢(77)=0 a continuously differentiable (0(7)) must be found to stabilize

49

’7=fa(77,z)+5(77,z) such that its origin is asymptotically stable and ¢(0)=0.

Letting z = ¢(77) (shown below), (2)07) can be used to stabilize 17.

It should be noted that

[21 ] = [691(0)]
Z2 ¢2(77)
Also, it should be noted that (31(0) and ¢2(77) replaced z] and 22 in the f], state equation.

To find ¢(7]) , the partial derivative of 7) (equation (32)) with respect ton was calculated

as shown below.

 

 

a“ "-25 0 0 0 1460
_71 = O _5 O + 0 [30(0) a¢l(77) a¢l(77):|+ O
977 8m 8772 3773

L 0 0 -20 —11.25

_ -T

625772 a¢l_a_(__177)+712§__(32’7(177)
8 a
77~~23+625¢1(77)+¢2(77)+62577 —§",,——(’”+ +772 33,——‘2—"’
a and
I 772 + .625772 317;") + 772 “Sign—:1

 

Choosing a (1)07) that will cancel out the last portion of the f7l state equation was

achieved by letting (9(7)) be

1.6
¢<n> =[ '7' I (34)
'771 ‘773

Once the function ¢(I]) was found, it stabilized the system about the origin. Furthermore,
the system was asymptotically stable. Interestingly, the chosen (2)07) decoupled the
nonlinear portion of fam, z.) from the linear part. Also, (2(7)) does not contain the state

’72 and removes it from having any affect on the system. This was significant because the

50

state :72 contained the input disturbance, (5:01-1. The design of the input vector u

began by using 3 = z — ¢(n) and then finding .1.

s = z-¢(77)
.1: z—a—¢77 = fb(77,z)+G(77,z)u-§£Ifa(77,Z)+5(77,Z)I (35)
an 877

The u is selected to be

u = ueq +G_l(7],z)v (36)

The um, is chosen to cancel the known terms on the right-hand side of equation (35).

_ a
“661:0 R7793 [‘fb(’79z)+5%fa(n’z)] (37)
Thus,
3 = v -a—¢5(77, z) (38)
an
and,
a 16 0 0 O
-—¢-§( ’2' :l: . ]-1 =0
87] -1 0 -l 0

Finally the 3' equation became

5:2—5377=V (39)

The selected v forces 3 toward zero and by using V2592 as a Lyapunov function
candidate fors" = v — A07, 2) , then
V=s~$=s-v.<_|s|-|v| (40)

Selecting v as follows

5]

v = —,Bsgn(s), where ,6 2 l + b, for some b > 0
leads
VS—fl-Isl (4|)
The above inequality guarantees asymptotic stability and also ensures that all states that
are not on the manifold s = 0 will converge to the manifold in finite time and remains on
it.
For simulation purposes the sign function was replaced with a saturation function

to have a smooth landing to the manifold. This also makes implementation easier.

5.4 Sliding Mode Controller Simulations

The sliding mode controller of the A/F ratio and fuel ratio was implemented and
simulated in Matlab/Simulink. Simulations that model real world A/F ratio and fuel ratio
situations were conducted, similar to simulations of the baseline controller. The main
difference between each simulation was how the closed loop system responded to various
air flow inputs. For comparison purposes to the baseline controller simulations, the same

target equivalence ratio was taken to be unity and the same target fuel ratio was 0.6.

5.4.1 Simulation 1

The first simulation uses a constant value for the air input, which models a
situation that the throttle (air) opens at a constant angle. The constant value was chosen

to be 0.15 and the states of the closed loop system are shown in Figure 5-6.

52

 

 

0.18 nW-rm-—mm~- — -I----wT-m—v-1

 

0.16 ——————————— ' —————————— I ———————— ' ----- 4
0.144---- —\ —————————— : ------- 4 ———————— i ————————
_. _____________ ‘l‘\. _______ I _______ I _____ _________
0.12 , \\ . 1 :
1 statex2 1 1
O1 ------------ ~~-1 -------- 1 ---------
w I 1 :
,9 I I .
cc 1 _________ I __________________
.5 0.08 ~~~~~~~~~~ I I i:
0.06 ——————— I ------------------ I ———————— I ———————— I ——————

  

004 states X1, X3, X4, X5 J ________ i _________

 

__-_.___2___-_________‘__._-_,__.__.__________..__.___._‘

 

 

time (seconds)

 

 

 

Figure 5—6: States of the system for simulation 1

The initial conditions of the states of the transformed equivalence ratio and fuel

ratio model were chosen to be 0.01 as can be seen in Figure 5-6. The states x1, x3], X32,

and x33 are forced to the sliding manifold, s =0, in a finite time of less than I second.

Theses states remained at zero for all future time, which showed that the sliding mode

controller theory was valid.

Meanwhile the state x2 does reach the sliding manifold due to the constant

disturbance input but it reaches steady state in finite time. From the state space model,

the state x2 is affected by the input air disturbance. The sliding mode controller isolated

that state and decoupled it from the remaining states, causing the rest of the states to

53

reach the sliding surface of the system. Therefore, the input air disturbance affect on the
equivalence ratio and fuel ratio was minimized.
To achieve the desired ratios of unity for the equivalence ratio and 0.6 for the fuel

ratio, consider

771—7710

All=(l7"70)= ’72‘7720 30=A77+770 (42)
[73—7730
2.1-Z]

Az=(z—z0)= 0 Dz=Az+z0 (43)
ZZZ-'ZZO

where A17 and Az go to zero, leading 17 and 2 going to 770 and Z0 as time goes to infinity
due to the sliding mode controller regulating the states Am, A113 ,Az], and Azz to the

sliding manifold s = O. For a fixed input disturbance, the selection of 770 and Z0 begins

by determining the PFI and DI inputs values that will make the open loop response of the

system equal to unity for the equivalence ratio and 0.6 for the fuel ratio. The open loop

response will then give the steady state values for the states 17], 173, z], and Q. Using

these values for 770 and 2:0 can bring the equivalence ratio and fuel ratio to their desired

values. Thus, by using (42) and (43), (32) and (33) become

-25 -l460(7730+.6252]0+220) 44607720 A77] 4460-0625020 44607720

A17: 0 -5 0 A172 + 0 0 [A2]
0 0 -20 A773 -1 1.25 0 2
1460 0 14607720 (’730 +0-625810 + 320 l + 25’710
+ 0 A172(A173 +0.625m1 +A12)+ -1 5+ 5’720
0 O 20]]30 +1 1.25310
(44)

54

. - 1.25 O Az l 0 u 1.25 0 21
AZ = l + l + 0 (45)
0 - 20 Azz 0 l 142 0 20 220
To investigate stability of the system with these new target states, ¢(77) from (34) is used

and by substitution into (44) and (45), the system becomes

— 25 — 14600730 +5252l0 + zgo) 0 An, 0
A77: 0 —5 0 A772 + -1 6
—11.25-l.6 o -20 A773 0

14607720 (7730 + 0.625210 + 2'20 ) + 257710

+ 57720
207730 +11.25z.0

(46)

._-1.25 0 A2, 1 0 u, 1.25 o zlo
Az-lo liar IlliHo filial

It can be seen that the nonlinear term of the 7) equation of (44) is now eliminated, leaving

a linear term with constant matrices plus the forcing 6 term, while the 2 equation

remains linear. For stability of the entire system, it can be seen that the characteristic

equation of the linear matrix A~ will be

det(AI — A") = (A + 25)(/i + 5)(/i + 20)

and has ei gen values in the left half plane, where

- 25 - 14600730 +.625zlO + zzo) 0
A” = 0 —5 0
-ll.25-1.6 0 -20

Thus, the same (0(1)) can be used to stabilize the system with the new target states and the

previous sliding mode theory will remain valid for this system. The original x states for

55

the desired equivalence ratio and fuel ratio are shown in Figure 5-7 and these ratios are

shown in Figure 5—8.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.05 Ti [r I i ll _. - _._,
0.04 ' 1
0.03 x1 r
8 x2
:6. 0.02 x3-
(0
x4
0.01 _ ___xgw
o ---------------------------------- g —————————
-0.01 i
5.5 6

 

time (seconds)

 

 

 

Figure 5-7: States for the system to achieve desired equivalence and fuel ratios

56

the desired equivalence ratio and fuel ratio are shown in Figure 5-7 and these ratios are

shown in Figure 5-8.

 

 

0.05 It i r

-____-__1
___..__.__‘

0.04

 

 

,J.-/
l
l

0.03

 

0.02

states

 

 

0.01

 

 

 

 

 

-0.01
3

 

time (seconds)

 

 

 

Figure 5-7: States for the system to achieve desired equivalence and fuel ratios

56

 

 

 

 

 

_-__l,_____._,

——————————————————————————————————

Equivalence Ratio and Fuel Ratio

i
i
l
i
i
I

 

 

 

 

 

 

 

 

 

i ______ _l _______ L __ _____________ .1

i I ----- Eq. Ratio

3 Fuel Ratio

1 __ l l l

6 7 8 9 10
time (seconds)

 

 

 

Figure 5-8: Closed loop response of simulation 1
The simulation began at 3rd second and the equivalence ratio’s closed loop
response to the 0.] air input settles relatively fast (about 1.5 seconds). The fuel ratio’s
closed loop response to the same constant air input settles similar to the equivalence ratio.

To achieve an acceptable closed loop response, the ,8 value in the simulation was tuned to

[:3]

be

E
ii

5.4.2 Simulation 2

The ability to use the target states no and 20 to bring the equivalence ratio to
unity and fuel ratio to 0.6 can be extended to any desired ratio under any input

disturbance. This requires that 770 and 20 be determined as a function of the desired

57

equivalence ratio, fuel ratio, and input disturbance. To begin the calculation of 770 and

Z0 , consider the output equations (27) and (28) transformed using (31), resulting in

yEqRatio = 25771 = 5 ' 20 ‘ 772073 + O.625Z1 + 22)

yFuelRatio 214.6 ' 5 ' 772 (0.4687521 + 0.625111)

Next, consider the system of (32) and (33) initially at zero, leading to

l

771:2—514.6-5-20'772(773+0-625Z1+32)

 

l
772 = —g5
— —l—(l —— (a)
5
77 _ l 1.25 Z
3 20 l
u] =1.25Z]
L12 = 2022
Using (49) and (53),
yFuelRatio

 

Zl

Finally from (50), 22 can be expressed as follows

 

25
22 = ”1 -773 —0.625zl.
14.6'5-20-772
Note from (48), m can also be expressed as
: yEqRatio

771 25

2 14.6 - 5 - 772(0.46875+ 0.625-1.25) '

(48)

(49)

(50)

(51)

(52)

(53)

(54)

(55)

(56)

(57)

Now that the expressions for )7 and 2: as a function of the desired output ratios and input

disturbance have been found, they can be used as 170 and Z0. This will cause to the

58

states of the closed loop system to go to I70 and 2:0 as time goes to infinity, thus

achieving any desired equivalence ratio and fuel ratio under any input disturbance.

The second simulation begins using a constant value for the air input and at a later
time sums a step input to this initial constant value. This simulates a situation where the
throttle (air) initially remains open at a constant angle and then quickly (almost
instantaneously) opens or closes to a larger or smaller angle, much like simulation 2 of
the baseline controller. The constant value was chosen to be 0.1 and the step input value
was chosen to be 0.15. The time at which the step input would be added to the constant

value was chosen to be at 6 seconds.

 

 

T l— l i—fi I
I i , ——-Eq. Ratio M

 

. . ----- Fuel Ratio

 

 

 

 

____-_-

 

 

Controller responds
to step input

Equivalence Ratio and Fuel Ratio

____,__._...___.____

 

 

 

time (seconds)

 

 

 

Figure 5-9: Closed loop response of simulation 2
Figure 5-9 shows the closed loop response of the step input applied at 6 seconds. The

controller adjusts the control inputs to maintain the desired equivalence and fuel ratios.

59

 

 

 

0.09 T

i l l _“T— "”— T ‘ T l ' _2‘
; : I I i i 2 : PFl'
0.08 ————— : ....... l ----- ————— 1 OIL

 

 

 

 

 

 

PFI and DI Control Inputs

 

0.02 i_i‘\\\

0.01 _ __ -l. _. -- T' _ -5 _:_-. Control inputs responds . ____________ _
' ; to step input

 

 

l _L.

0 _ . 1
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
time (seconds)

 

 

 

 

Figure 5-10: PFI and DI control inputs for simulation 2
The responses of the control inputs, PFI and DI fueling, are shown in Figure 5-10. The
PFI and DI fueling inputs were increased to achieve and maintain the desired ratios,

which show the controller’s quick response to the step input is applied at 6 seconds.

5.4.3 Simulation 3

The third simulation uses a constant input disturbance of 0.1 with 5 percent noise
added to it, which is similar to simulation 2 of the baseline controller. It also decreases
the desired fuel ratio from 0.6 to 0.4 at 6 seconds and then increases it back to 0.6 at 8

seconds.

60

 

 

 

 

 

 

 

 

 

 

 

 

)‘ ""I I I I “ “T—fil—m' T 2:?!
1.2: 7 _ w _I _____ l - . _ l._ _ _ ; ‘_:‘ _____ lI _____ :_ 7 q ' Eq. Rat' I _
. . , I 7i ----- FuelRano:
_9 1 - '. L1 : 1 _' : _1 _
*6 I I ' i I I I i L
m I I l l l I | l I
F) I I I ' |
3 | I l I I
LL08 77' ————————————————————————— eeeeeeeeee 1 ~—-a
.o I | I I I
C
‘5 1 1 , I l . I I I
.9 I I l I I l “‘ l l I
a 6 --.-~q‘-h-- ... _ - _ ’r < _ - - 1|. ,,,,,, i _ _ _ . I _ _T‘rt'tfiw-n!s-——-—.—an—JI
m 0 I ‘1 I I I It I I I
8 I g I I I : I I I
0:) I I I I I | \ I I I
.5 I I I I I | I I
.2 o4~- i ..... I 5-- :-~~“--~-LI ----- . -----------
CDT I k/"flflI I I : I I
m I i (‘\+\\\ ' l 1 i i .
I : ; \T\\71I‘2‘ I l I I
0-2 [ ‘ ' j “““ } """ f """ j * 5 ‘ j’ ’ 5“ Controller responds
; ; ; ; ; to change in desired
; ; ; ; ; ‘ fuel ratio .
O H_ L _ 4' l l L l _l_ L__ _l__.___.
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
time (seconds)

 

 

 

Figure 5—l 1: Closed loop response of simulation 3

The closed loop response of simulation 3 shows the sliding mode controller’s
ability to achieve the desired ratios in spite of the 5 percent noise that was used on the
input disturbance. Compared to the baseline controller, the sliding mode controller
proves to be more robust in its response to the noisy disturbance. The instantaneous
change in the desired fuel ratio required an immediate change in the control input, which
was achieved as shown in Figure 5-1 1. Although instantaneously large fueling may not
be possible in a real world situation, rapid small amounts of fuel will ultimately make the

desired ratios achievable.

6l

 

 

 

 

 

 

 

 

 

 

 

 

 

0.09 I I T l l T — — T "T _ ' '____'T'——_‘

I I I I : : I :——_—-j -----

, Control inputs respond ; 1 ; PFI L

0'08?” tostepinput “““ (”'“T'H‘j ““““ : DI

\\ I ------
I \x I I I I l l
g I \. :\\: . :I ' ' :

0.06““———r—--X4 ————— I ————— I\ ~\—\~—--I ~44”: ————— I —————— I ——————
g I it I i “\1 " ‘\ : : :
6 I I I l\ ‘I I I

E 0-05 “""7r\"\lI—A““‘\I‘~~I ““““ I ““““

o —--—$---- ' A _ *‘ - I -*-'-§d-___‘.
0 I 1 I I . 2 I I

a 0.04 ————— -___I _____ I _____ . ..... IMI ______________ I ______ I
U l l l l I l I
g I I I I I I I

E 0.03 ”“7 """""""" I“;;:—:ia"“‘+‘WIIt/“J”r I '“I'” '
O. I I’ I I I I I I
I [If I I I I I I

0-02 ----- I””I"”/“I ----- ; ----- ------ : ----- : ------ .
. 1U : : : : : . :

0.01~—-——~'r ————— I —————— : ——————— : ————— I-———I ————— : ————— : ------
___; _I_.__I _;_ I I I I I

5 55 6 65 7 7.5 8 8.5 9 9.5 10

time(seconds)

 

 

 

Figure 5-12: PFI and DI control inputs for simulation 3

5.4.4 Simulation 4

To improve the performance of the sliding mode controller, the stabilizing function ¢(7])

can be generalized as follows

1.6m; +£A773
z =¢(An) =[ I ] (58)
-A771'A773
where 8 is a positive constant. Substitution into (32) yields,
- 25 O 0 A77] 1460 - 0.625A172
A7] = O - 5 0 A772 + 0 A7738 . (59)
-ll.25-1.6 O -20-ll.25£ A773 0
To determine stability, (59) can be viewed as

A17=AA77+g(A77,8), (60)

where,

62

-25 O O dAnz
A(8) 2 0 -5 0 ,g(AI],£‘) = 0 A1738, and d = 1460-0.625-
-ll.25-l.6 O -20-ll.25£ 0

It can been seen that A(8) is Hurwitz ifs > 0 and also

dATIZ dAUZ

2
0 A7738 Sllell2 0 "A773“2 SdllEllzllA'Illz
0 2 0 2

Let Q = I and define X(8) > O as the solution of the Lyapunov equation

AT(e)X (e) + X<e>A<e> = -Q
Solve for the maximum and minimum eigenvalues of X(e) and plot these valves as

function of a , see Figure 5-13.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.02 ”___—T ——_‘T —'t_ "—"t l 1 fl» ‘::“r>::
"“ l
----- A - 0fX(£)'
0.015 ; __.. "1— J1
\
0.01 f“ ~
\\
0.005 ‘x a
\~.._- - l!
O _ L __l__. ____L__ l_ i——_ ;____L___.__: T ___—J
0 1o 20 30 4o 50 60 7o 80 90 100
g > 0
O 2 '—— ‘_V—'—7fi‘_'uT_'_ if fi'ifiT_ l w :Y—* :fi _—‘—7
__x of mg]
0.15- E ma"
0.1
0.05—
0 *_l __ L__.i_. ___—L— L___ _¥ _ fl
0 1o 20 30 4o 50 60 7o 80 90 100
g > 0

 

 

 

Figure 5-13: Maximum and minimum eigenvalues of X(a)

Define the Lyapunov function V(A27) = AIITXAI] with the following properties

63

2min (X<e>>||An||§ s vmm s 1.1m mall/ml;

3V
___/1A”: -AI7TQA77 S ”1min (Q)"A77"§,

”3‘:— V,,=|| um xll <2nxu2uAnn =22m<x>umn2

The derivative of V(A17) satisfies

r
A772

V(A77)=AI]T(AT(£)X(£)+X(£)A(£))AI]+2&IA773 0 X(8)A77
0

S "1min (Q)"A’7": + 21mm (X (5))8‘IIA’73I ' "A77":

Therefore the origin is exponentially stable if

1

V(A )<0 or £< .
'7 21mm<X<e>>d|Anal

 

(61)

The stability condition will hold if we assume that |A773| < yfor all time, where y is a

positive constant, and can be restated as below:

1

 

 

 

 

E < ~ — . (62)
22m (X(e))d|773 -— 0301
When '03 — 1730' S y, then (62) becomes
< 1 63

22mm (X (8))(1 - 7 ( )

From Figure 5-l3 it can be seen that 2.max(X(8)) = 0.1 for all 8 > 0, thus

1 l l
< < = (64)
2~0.l-d-}/ 2-0.l-]460-0.625~y 1825-}!

Equation (64) shows that if y is small, say y < 0.01, from (64) 8 can be selected to be 0.5

with guaranteed stability. Choosing [X as follows

64

1.6
flzimi’

the response of the sliding mode controller was improved. Simulation 4 uses a constant
input disturbance of 0.1 with 5 percent noise added to it. The disturbance is increased to
0.25 at 6 seconds, the desired fuel ratio is decreased from 0.6 to 0.4 at 9 seconds, and

finally, the equivalence ratio is increased to value of 1.1 at 13 seconds.

 

 

 

 

 

 

 

 

 

 

 

 

Equivalence Ratio and Fuel Ratio

Eq. Ratio (sliding mode) i

 

 

 

 

 

0.2 L» — - — - l - - . ~C - - — - « - - Fuel Ratio (sliding mode) f
: i l : ----- Eq. Ratio (PID) i

. : 1 ; j ----- Fuel Ratio (PID) J

0 __ _ l __- L__l__._._L. I I 7*: _Ljff:

5 6 7 8 9 1O 11 12 13 14 15
time (seconds)

 

 

 

Figure 5-14: Closed loop response for simulation 4
It can be seen from Figure 5-14 that the response of the sliding mode controller is better
than the baseline PID controller. Specifically, the fuel ratio change at 9 seconds and the
equivalence ratio change at 13 seconds show a faster response than the baseline
controller. Furthermore, Figure 5-15 shows that the closed loop response of the control

inputs of the sliding mode controller are much better than the baseline controller.

65

 

0.09

 

0.08

0.07

 

0.05

 

0.04

 

 

0.03 -

PFI and DI Control Inputs

 

PFI (sliding mode)
DI (sliding mode)

0.02

 

 

 

 

 

 

 

001 : 1 i i .......... PF|(P|D)
' T T i I T l ---------- DI(PID)
0 i 1 a 1 1 1 1 1 :
5 6 7 8 9 1o 11 12 13 14 15

time (seconds)

 

 

 

Figure 5-15: PFI and DI control inputs for simulation 4

5.5 Observer for Sliding Mode Controller

Many times in control theory, the states of a system contain useful information that
is used in the controller design. This is the case in the sliding mode control scheme that
was introduced earlier in this chapter. The transformed states, I] and z, of the equivalence
ratio and fuel ratio model were fed back and used in design of the sliding manifold.

Often times it is impossible or impractical to obtain the measurements of all the
states and some of the states may not be available for measurement. This may be due to
economic reasons, as some sensors may be too expensive, or due to physical reason in
which the surrounding environment may be too noisy for measurement. For this reason,
the states must be estimated from the available measurements, which typically are the

outputs and inputs of the system [16].

66

5.5.1 Observer Design

Since our system is nonlinear, a nonlinear estimator is needed. For linear systems
of the form

5c=Ax+Bu
y=Cx+Du

where the matrices A, B, C, and D are known, a linear state estimator can be constructed

as

=A5E+Bu+L(y—y)

f (65)
5? = C55+ Du

The matrix L is a matrix that is designed to make A+LC stable. Equation (65) can be

rewritten as
5c: (A—LC)5c+[B—LD L][”] (66)
y

This estimator is known as the Luenberger observer [17]. The error between the actual
state x(t) and the estimated state fr(t) is given as
é(t) = in) — 2(1) = [Ax + Bu]— [142+ Bu + LC(x — 2)] (67)
Since the eigenvalues of A+LC are in the left half plane then,
e‘(t) : 5r(r) — fc(t), é(t) —> O as t —> 0

Now, consider the plan (26), it can be reorganized into two system equations below:

. 5c, Al Bl 0
xs= x, = A} xs+ 0 C2C31X3X2+ B3 “3 (68)

132 = A2X2 + 8260 (69)

67

For a fixed operational condition, we can estimate state x3 independently since input (0
can be measured physically be a mass air flow sensor. Assuming that x; is known, we can

rewrite equation (68) as follows:

° A BC C x 0
- l l l 2 31 2
x = = Ax +8 2 x + u 70
S [x3] 5 “3 [0 A3 :| S [B3] 3 ( )

The output matrix can also be rewritten as

C + D I O + O 0 (7 I)
= x u = x U
y S 3 0 14.6C2C32X2 S 0.625 O 3

Using the steady state value of x2 = 0.17 and the linear system estimation equation (65), a

linear state estimator can be designed with the following system parameter matrices:

" .4 0 ‘ ’- 25 248.2 248.2 0 ‘ ’ 0 0‘
.0014 0.137 0 —20 0 .4688 .625 0
L: , A: 9 B: 9
0 0 0 0 —20 0 0 1
_ 0 5.4794 1 0 0 0 —1.25_ _ 1 0.

 

 

 

 

 

 

25 0 0 0 0 0
C = , and D = .
0 0 0 5.8172 0.625 0

5.5.2 Observer Simulation

The observer was designed to estimate the states of the equivalence ratio and fuel
ratio model. It used the outputs (the equivalence ratio and fuel ratio) and the inputs (PFI

and DI fueling) to generate the states.

68

 

 

 

states

 

 

 

 

time (seconds)

 

 

 

Figure 5-16: Estimated states generated by state estimator
Figure 5-16 shows the estimated states, 2], £3, £4 ,and 25, generated by the estimator.
This simulation shows that the estimator was able to estimate the states and the sliding
mode controller was able to bring those states to the sliding manifold. Theses states
remained at zero for all future time, which again showed that the sliding mode controller

theory was valid. Note that the air flow used for the simulation is 0.15, the actual

disturbance input 0) is 0.85 (see Figure 5-3), leading the steady state valve for x2 to be

0.17.

69

Chapter 6: Conclusions

6.1 Cylinder Pressure Model Conclusions

After understanding the combustion process of a spark ignition engine and
developing a cylinder pressure model, this model is combined with a mean valve engine
model, developed by a colleague, to have a mean valve engine model with cycle-to-cycle
cylinder pressure output. This engine model can be calibrated using engine mapping data
such as the engine torque, pressure in intake and exhaust manifolds, and the air-to-fuel
ratio.

The mean-value engine model can be used to validate control strategies for
different types of controllers that are model-based. The equations that are used to
calculate the outputs of the model are approximated over an engine cycle. A significant
advantage of the mean-value engine model is its low computational throughput which
makes it possible for real—time simulation. In order to reduce engine emissions and
improving engine fuel economy, closed loop combustion control, which requires cycle-
to-cycle combustion measurement such as cylinder pressure, is a necessity.

The addition of a cylinder pressure signal to a mean value engine model will
allow for developing closed loop combustion control strategies (or other strategies that
involve cylinder pressure) to be validated. This is because the cylinder pressure model
can produce a cylinder pressure signal for a complete engine operational map and a mean

value engine model can produce real world engine parameters and conditions.

70

 

Calibrations

 

 

 

 

  
   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1
RPM p
RPM BMEP
’ P_intake
Throttle .
Spark Mean-value My
Valve Engine Model T_exhaust
’( ——>
Fuel Lamda .
> Crank Cylinder P cylinder
EGR Pressure -
\ CAM Model
L J

 

 

 

\

Prototype
Controller

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1111111

 

 

 

 

 

 

Figure 6-1: Integration of mean value engine model and cylinder pressure model
Figure 6-1' shows the integration 0f the mean value engine model with the
cylinder pressure model. The cylinder pressure model uses signals generated by the
mean value engine model, and also sends its own signals to a controller. Based on a
cylinder pressure signal, a controller can be designed improve MBT timing which is an

area of study for future research.

6.2 Air-to-Fuel Ratio and Fuel Ratio Control Conclusions

The A/F ratio and fuel ratio control designs were able to achieve the desired

performances. Specifically, the A/F ratio was regulated and maintained at its desired

71

level with various input disturbances. Likewise, the fuel ratio was regulated at a selected
level of 0.6.

The baseline controller is a multi-input and multi-output (MIMO) PID controller.
This controller, which had two inputs, equivalence ratio and fuel ratio, and two outputs,
PFI and DI fueling, is useful for A/F ratio and fuel ratio control. This is primarily due to
the fact that PID controllers are widely used in automotive applications and can be easily
implemented.

Likewise, the sliding mode controller design was also an excellent MIMO
controller. The ability to use sliding mode control theory on a MIMO case was
significant. Extending this theory to achieve and maintain any target output (desired
equivalence ratio and fuel ratio) is the most important contribution of this research.
Compared to the baseline controller, the sliding mode controller had a significantly lower
steady state error in response to the random input disturbance. The percent overshoot for
the sliding mode controller was also noticeably better and in response to the step input,
had a better settling time than the baseline controller.

For comparison purposes, Table 6-1 and Table 6-2 show the percent overshoot of
the disturbance step input, settling time after the fuel ratio change, and steady state error

under 5% noise of both controllers for the equivalence ratio and fuel ratio, respectively.

 

 

 

 

 

 

 

 

 

 

Disturbance Input % Overshoot Settling Time
PID Sliding Mode PID Sliding Mode
step 4.42% 2.44% 0.543 sec 0.782 sec
Steady State Error
PID I Sliding Mode
constant plus 5% noise .001 1 I .0002

 

 

 

Table 6-1: Comparison of controllers for equivalence ratio

72

 

Disturbance Input % Overshoot Settling Time
PID Sliding Mode PID ISliding Mode

 

 

 

 

 

 

 

step 2.16% 2.64% 1.95 sec | 0.745 sec
Steady State Error
PID Sliding Mode
constant plus 5% noise .024 .0005

 

 

 

 

 

Table 6-2: Comparison of controllers for fuel ratio
The linear observer that was designed for the sliding mode controller, by
linearizing the plan at a given operational condition, estimated the states well. It proved

to be effective for the potential implementation.

73

[ll

[2]

[3]

l4]

l5]

[6]

[7]

[8]

l9]

['0]

[ll]

[12]

[13]

[I4]

[151

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75

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