Control of Injection Timing in

Port Fuel-Injected Gasoline Engines

Dissertation submitted in partial fullfilment

of the requirements for the degrees of

B.Tech in Mechanical Engineering

and

M.Tech in Computer Aided Design and Automation

by

Pranav Rajendra Shah

(Roll 03D01010)

under the supervision of

Prof. Shashikanth Suryanarayanan

Department of Mechanical Engineering

Indian Institute of Technology Bombay

July 2008

Abstract

This thesis focuses on studying response of wall-wetting dynamics observed

in port-fuel injected gasoline engines to changes in fuel injection timing. In

particular, it investigates the manner in which real time changes in fuel injec-

tion timing influence the quantity of fuel that evaporates from the deposited

wall-film in the intake port. An advancement in injection timing is shown

to cause greater amount of fuel to enter the cylinder in the transient state

though with no change in the steady state. A mathematical model captur-

ing this observed phenomenon is derived and utilized to build a controller

to achieve tighter control of air-fuel ratio(AFR) in the transient state. Ex-

periments are also conducted to confirm the efficacy of the proposed control

action.

Contents

List of Symbols vi

List of Abbreviations viii

1 Introduction 1

1.1 Problem Addressed . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Contributions of the thesis . . . . . . . . . . . . . . . . . . . . 4

1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . 5

2 Preliminaries 6

2.1 Internal Combustion Engines . . . . . . . . . . . . . . . . . . 6

2.2 Control Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Control Systems for Internal Combustion Engines . . . . . . . 16

3 Experimental Setup and Operation 21

3.1 Sensing and Actuation Architecture . . . . . . . . . . . . . . . 21

3.2 Controlled operation of the engine . . . . . . . . . . . . . . . . 24

4 Wall-Wetting phenomenon 28

4.1 Aquino Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Droplet evaporation . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2.1 Motion of a droplet . . . . . . . . . . . . . . . . . . . . 32

i

4.2.2 Convective mass transfer . . . . . . . . . . . . . . . . . 33

4.2.3 Coupled differential equation analysis . . . . . . . . . . 36

4.3 Port-Wall Film Evaporation . . . . . . . . . . . . . . . . . . . 38

4.3.1 Estimation of air velocity . . . . . . . . . . . . . . . . . 38

4.3.2 Film evaporation . . . . . . . . . . . . . . . . . . . . . 42

4.4 Model sensitivity analysis . . . . . . . . . . . . . . . . . . . . 44

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Influence of injection timing 46

5.1 Experiments - Change in FIT . . . . . . . . . . . . . . . . . . 47

5.1.1 Measured quantities . . . . . . . . . . . . . . . . . . . 47

5.1.2 Experimental Observation . . . . . . . . . . . . . . . . 48

5.2 Phenomenological explanation . . . . . . . . . . . . . . . . . . 49

5.3 Modeling influence of injection timing . . . . . . . . . . . . . . 52

5.3.1 Analytical Formulation . . . . . . . . . . . . . . . . . . 52

5.3.2 Model Validation . . . . . . . . . . . . . . . . . . . . . 55

5.4 Controller development . . . . . . . . . . . . . . . . . . . . . . 57

5.4.1 State Space Model of the plant . . . . . . . . . . . . . 59

5.4.2 Feedback Controller . . . . . . . . . . . . . . . . . . . . 60

5.4.3 Results and Discussion . . . . . . . . . . . . . . . . . . 62

6 Conclusion 64

6.1 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . 64

6.2 Outline for future work . . . . . . . . . . . . . . . . . . . . . . 65

A LQR Controller Design 66

ii

List of Figures

1.1 Port-fuel injection engine architecture . . . . . . . . . . . . . . 2

1.2 Fuel injection advance diagram . . . . . . . . . . . . . . . . . 3

1.3 AFR variation for change in FIT . . . . . . . . . . . . . . . . 4

2.1 Four stroke engine operating cycle [5] . . . . . . . . . . . . . . 7

2.2 Time-trace of TVS Victor GLX following Indian Drive Cycle . 9

2.3 Conversion efficiency of three-way catalytic converter [17] . . . 10

2.4 Input-Output view of Engine . . . . . . . . . . . . . . . . . . 17

2.5 Engine feedback system architecture . . . . . . . . . . . . . . 19

3.1 Schematic of engine with sensors and actuators . . . . . . . . 22

3.2 Typical fuel injection pulse (Electronic) . . . . . . . . . . . . . 23

3.3 Schematic of engine with a computational element . . . . . . . 24

3.4 Engine fitted with transient dynamometer . . . . . . . . . . . 25

4.1 Wall-film deposition . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Demonstration of wall-wetting dynamics . . . . . . . . . . . . 31

4.3 Plot of droplet evaporation with time . . . . . . . . . . . . . . 38

4.4 Intracycle manifold pressure variation . . . . . . . . . . . . . . 39

4.5 Intake pipe air-flow . . . . . . . . . . . . . . . . . . . . . . . . 40

4.6 Model calibration based on observed manifold pressure . . . . 42

iii

5.1 Injection timing diagram . . . . . . . . . . . . . . . . . . . . . 46

5.2 Fuel flow dynamics study using AFR . . . . . . . . . . . . . . 47

5.3 AFR Variation with FIT . . . . . . . . . . . . . . . . . . . . . 49

5.4 Wall-film evaporation and FIT diagram (∆FIT = 0) . . . . . 50

5.5 Wall-film evaporation and FIT diagram(∆FIT = 200 ◦) . . . . 51

5.6 Fuel mass flow-rate signal . . . . . . . . . . . . . . . . . . . . 53

5.7 Unit step signal (Γ = 2sec) . . . . . . . . . . . . . . . . . . . . 54

5.8 Prediction of the analytical formulation for influence of injec-

tion timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.9 FIT model - fitting on observation . . . . . . . . . . . . . . . . 56

5.10 Comparison - AFR response with and without FIT change . . 57

5.11 MIMO control - feedback architecture . . . . . . . . . . . . . . 58

5.12 AFR response and actuation effort . . . . . . . . . . . . . . . 63

A.1 System Response - Only FID as input . . . . . . . . . . . . . . 67

A.2 System Response - FID and FIT as inputs . . . . . . . . . . . 69

iv

List of Tables

3.1 Experimental setup details . . . . . . . . . . . . . . . . . . . . 27

4.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . 37

v

List of Symbols

α Throttle position

γ Specific heat ratio of air

λ Air-fuel ratio

κ Injected-fuel impingement fraction

µ Dynamic viscosity of air

ρa Density of air

ρf Density of fuel

ω Engine crank-shaft speed

CAs Fuel vapor concentration at droplet surface

CD Droplet drag coefficient

DAB Diffusivity of gasoline

M Molecular weight of gasoline

R Universal gas constant

Sc Schmidt number

Sh Sherwood number

T Engine cycle time

Tatm Atmospheric air temperature

Ts Droplet surface temperature

Vman Volume of manifold

cd Throttle discharge coefficient

vi

hfg Specific enthalpy of gasoline

mcylinder Air mass flow rate into cylinder

mevp Mass rate of fuel-film evaporation

mf Mass of fuel-film

minj Mass rate of fuel injection

mman Mass of air in manifold

mpul Fuel injected per injection

mthrottle Air mass flow rate from throttle

pAs Vapor pressure at droplet surface

pman Average manifold pressure

sW Distance from injector to opposite wall

tinj Timing of fuel injection

tIV O Intake valve opening time

vii

List of Abbreviations

AFR Air-Fuel Ratio

EFI Electronic Fuel Injection

EOT Engine Oil Temperature

FID Fuel Injection Duration

FIT Fuel Injection Timing

MAP Manifold Air Pressure

MAT Manifold Air Temperature

PFI Port Fuel Injection

TDC Top Dead Center

viii

Acknowledgments

First and foremost, I would like to sincerely thank my advisor, Professor

Shashikanth Suryanarayanan, for his immense support and guidance right

through my undergraduate and graduate days at IIT. I greatly appreciate

his openness to whatever I wanted to express as well as the freedom that he

provided me to pursue research of my interest. A guide and a mentor like

him will always be hard to find.

I am also grateful to Pushkaraj Panse for all that I have learned from

him. His patience and mentoring were very vital for my understanding of

various aspects of the project. I am proud to have a person like him as a

very good friend of mine.

I thank all my friends at IIT - wingmates and mates from the mechanical

department - for all the fun that we had together and which has helped me

make my stay at IIT a memorable one.

Finally, I am deeply thankful to my parents and my family for always

being very supportive of whatever I wanted to pursue. Little could have

been achieved without their blessings and love.

ix

Chapter 1

Introduction

Gasoline engines today are fitted with three-way catalytic converters in the

exhaust pipe to curb tailpipe emissions. The efficiency of these catalytic con-

verters to curb hydrocarbon(HC), nitrogen oxide(NOx) and carbon monox-

ide(CO) emissions depends critically on the air-fuel ratio(AFR) of the mix-

ture undergoing combustion in the cylinder. In order to meet emission reg-

ulations set by governing bodies, most larger gasoline engines(greater than

500 cc) today are fitted with electronic fuel injection(EFI) systems for fuel

delivery rather than carburetors. This is because EFI systems help with

better metering of the injected fuel as well as better atomization of fuel into

droplets, both of which play an important role in achieving good performance

of the engine.

A fuel injection system delivers finely atomized droplets by allowing pres-

surized fuel to flow through a nozzle with fine orifices. The opening of the

nozzle is usually controlled by using an electronically actuated solenoid valve.

The time for which the nozzle is kept open decides the amount of fuel deliv-

ered per injection event. The work for this thesis is carried out on an engine

based on port-fuel injection(PFI) architecture(figure 1.1). In these engines

1

Figure 1.1: Port-fuel injection engine architecture

fuel is sprayed into the intake manifold which then forms a combustible mix-

ture with incoming air. Specifically, we study the influence of injection tim-

ing on engine performance in port-fuel injected, four stroke, single cylinder

gasoline engines.

1.1 Problem Addressed

The engine cycle can be looked upon as made up of a number of discrete

events synchronized with opening of the intake valve. The important deci-

sions to be made every cycle with fuel injected engines are the amount of

fuel to be injected and the timing of fuel injection with respect to top dead

center(TDC) of the intake stroke. We define fuel injection timing(FIT) as the

crank angle in degrees before the top dead center of intake stroke at which

injection happens(figure 1.2).

2

Figure 1.2: Fuel injection advance diagram

The usual methodology to achieve AFR control is to make appropriate

changes in amount of fuel quantity injected based on accurate air-flow esti-

mation schemes[2][7][8]. Injection timing is known to influence size of fuel

droplets entering the cylinder[12] and hence plays an important role in deter-

mining hydrocarbon(HC) emissions[20]. Injection timing is also important

from the point of view of drivability as it is believed to affect the torque

response by influencing air-fuel mixture formation in the intake-port[13].

Further, a commonly suggested heuristic while implementing AFR control

systems is to fix injection timing such that the end of fuel injection occurs

at the opening of the intake valve[7]. This heuristic helps minimize the error

between the sensed quantity of air flow at the time of injection and the ac-

tual air flow when intake valves open and thereby help achieve tighter air-fuel

ratio control.

3

Figure 1.3: AFR variation for change in FIT

In this thesis we investigate the influence of real-time changes in injection

timing on quantity of fuel entering the cylinder per cycle and thus on AFR

control. As far we know all studies previously made to study influence of

injection timing consider it as a fixed parameter. It is shown in this thesis

that changes in fuel injection timing have a significant influence on AFR

transients(figure 1.3). It is this conclusion that is exploited to achieve tighter

AFR control by controlling fuel injection timing in real-time.

1.2 Contributions of the thesis

Methodologies to control AFR by appropriately adjusting opening-closing

time of the injector is a well researched topic in the power-train control

research community. The work in this thesis will help put forth the view

that injection timing can act as an additional degree of freedom to achieve

4

AFR control. This claim is also intended to fuel further similar studies for

multi-cylinder and larger engines.

The more immediate contribution of this thesis is toward commercial en-

gine management system developers. With the goal of automobile manufac-

turers set to meet increasingly stricter emissions norms along with improved

fuel economy and drivability, the work in this thesis will help build better

AFR controllers for engines.

1.3 Outline of the thesis

The structure of the thesis is as follows. Chapter 2 briefly describes engine

and control preliminaries needed to understand the remainder of the thesis

while chapter 3 describes the experimental facilities used to carry out in-

vestigation for this thesis. Since injection timing is conjectured to play a

role through aperiodic excitation of the wall-wetting dynamics, chapter 4

describes a detailed model for this dynamics by separately considering the

phenomenon of droplet and wall-film evaporation. Chapter 5 presents the

investigation made to study influence of injection timing. This chapter also

describes a methodology to use it as a control input for AFR control while

chapter 6 gives a summary of the thesis, list important conclusions as well

as outlines direction for possible future work.

5

Chapter 2

Preliminaries

Internal combustion engines today are fitted with a number of sensors and

actuators for diagnosis and control. The need for these elements necessitates

out of the requirement to obtain improved performance from vehicular sys-

tems. This chapter focuses on few preliminaries related to engines and their

control which will aid understanding of rest of the thesis. Section 2.1 covers

basic material on internal combustion engines while section 2.2 covers mate-

rial on basic control theory. Section 3 details out the experimental facility

that was used for conducting experiments related to work in this thesis.

2.1 Internal Combustion Engines

The purpose of internal combustion engines is the production of mechanical

power through the release of chemical energy contained in the fuel. This

energy is released by burning fuel inside the engine. The majority of the

internal combustion engines are of reciprocating nature and operate on what

is known as the four-stroke cycle. In particular we would be looking at four-

stroke, port fuel-injected, spark ignited gasoline engines.

6

Figure 2.1: Four stroke engine operating cycle [5]

Engine Operating Cycle

Four-stroke internal combustion(IC) engines require each cylinder to undergo

four strokes for completion of the engine cycle as shown in figure (2.1). The

four strokes, which comprise the engine operating cycle, are described below:

• An intake stroke which starts with the piston at top center (TC) and

ends with the piston at the bottom center (BC) and which draws fresh

mixture into the cylinder. To increase the air-fuel mass inducted, the

inlet valve opens shortly before the stroke starts and closes after it ends.

• A compression stroke where both valves are closed and the mixture

inside the cylinder is compressed. Towards the end of the compression

stroke, combustion is initiated by sparking the in-cylinder mixture with

a spark plug.

• A power stroke which starts with the piston at the TC and ends at BC

7

as the high temperature, high pressure gases push the piston down and

force the crank to rotate. The crank in turn causes the road wheel to

rotate which makes the vehicle move.

• An exhaust stroke where the remaining burned gases in the cylinder are

driven out. As the piston approaches the TC the inlet valve opens and

just after the TC the exhaust valve closes and the cycle starts again.

Fuel Injection

Carburetors have been used extensively in commercial throttle operated gaso-

line vehicles for mixing of air and fuel in the intake manifold. A suction

created by flowing air in the intake manifold draws fuel from the fuel tank

into the intake manifold and forms air-fuel mixture. However, they do not

provide accurate fuel metering and good atomization of the incoming fuel.

Fuel injection using electronic fuel injectors is an increasingly common

alternative to carburetors. In electronic fuel injectors, pressurized fuel is

passed through very fine orifices which cause atomization of injected fuel.

Amount of fuel to be injected is decided by the opening and closing time of

the solenoid valve in these injectors (see[9]). This control over the amount

and timing of fuel injection helps with accurate metering of the fuel entering

the manifold as well as better air-fuel mixture preparation, both of which

have an influence on performance of the engine.

Engine Emissions

The exhaust gases of gasoline engines contain mixture of nitric oxide (NO)

and small amounts of nitrogen dioxide (NO2) and nitrous oxide (N2O) which

are collectively called NOx, un-burnt or partially burned hydrocarbons (HC)

8

Figure 2.2: Time-trace of TVS Victor GLX following Indian Drive Cycle

and carbon dioxide (CO), in addition to the predominant gases carbon diox-

ide (CO2) and water vapour(H2O).

Regulatory bodies have been setup around the world to set and implement

emission regulation laws in-order to curb current vehicular emissions. A

vehicle emission test consists of driving the vehicle through a velocity versus

time profile that the vehicle is required to go through within certain limits.

The Indian drive cycle(IDC) is as shown in the figure (2.2). The total mass

emission of each pollutant over the drive cycle is measured and in order for

the vehicle to be certified for usage by the regulatory authorities, these values

must be below the allowed limits of emissions.

Emission norms are getting stricter with time and hence mechanisms to

curb emissions are gaining increased importance.

9

Figure 2.3: Conversion efficiency of three-way catalytic converter [17]

Three-way Catalytic Converter

Engine out pollutants of spark-ignited gasoline engines greatly exceed levels

mandated by most regulatory authorities. These requirements can only be

satisfied if appropriate exhaust gas after-treatment systems are used. The

three-way catalytic (TWC) is one common after-treatment system used which

facilitates the conversion of three pollutant species, namely, NOx, CO and

HC present in the exhaust gas into less harmful water, carbon dioxide and

nitrogen. Most catalytic converters consist of noble metals platinum, palla-

dium and rhodium. It is these metals that dictate the conversion efficiency of

the converter. The conversion efficiency is high only if the air-to-fuel ratio by

mass (AFR) of the in-cylinder mixture is kept in a narrow band around the

stoichiometric value of 14.6 (figure 2.3) during operating conditions of the

engine (figure 2.3). Thus air-fuel ratio excursions are to be strongly avoided

10

and this requirement necessitates accurate metering and control of the in-

jected fuel. This is one of the vital reason for wide-spread replacement of

carburetors by electronic fuel injectors.

2.2 Control Preliminaries

This section gives brief introduction to the basic concepts and ideas in linear

control theory. For detailed discussion one may refer to [3][11].

Linear Time Invariant(LTI) Systems

Definition 2.2.1. A system L is a map from one set of signals u(t), called

as inputs to another set of signals y(t), called as outputs. Such maps are

often denoted as L : u → y

Definition 2.2.2. A map L : u → y is called linear if for every u1(t), u2(t)

and α, β ∈ <,

L(αu1(t) + βu2(t)) = αL(u1(t)) + βL(u2(t))

Definition 2.2.3. A system L is said to be time-invariant if for every τ > 0,

L(u(t)) = y(t) =⇒ L(u(t− τ)) = y(t− τ)

The class of linear, time-invariant systems are expressed mathematically

using linear, constant coefficient, ordinary differential equations of the form

yn(t) + an−1yn−1(t) + · · ·+ a0y(t) = bmum(t) + bm−1u

m−1(t) + · · ·+ b0u(t)

with u(t), y(t) as input, output signal respectively. All coefficients ai’s and

bi’s here are real. The above representation of LTI systems is in the time

11

domain. An equivalent representation of the system in the Laplace domain

can be expressed as

L : u → y :=Y (s)

U(s)=

bmsm + bm−1sm−1 + · · ·+ b0

sn + an−1sn−1 + · · ·+ a0

A convenient representation for linear, time-invariant systems with multiple

inputs(say,p) and outputs(say,q) in time domain is in the state-space form

expressed as

x(t) = Ax(t) + Bu(t) (2.1)

y(t) = Cx(t) + Du(t) (2.2)

where x(t) ∈ <n is the state vector, u(t) ∈ <p is the input vector to the

system, y(t) ∈ <q is the output vector of the system, A ∈ <n×n is the

state matrix, B ∈ <n×p is the input matrix, C ∈ <q×n is the output matrix

and D ∈ <q×p is the feedforward matrix. The state x(t) of the system is a

collection of variables that help predict future response of the system.

The state-space representation in the Laplace domain is of the form

X(s) = (sI − A)−1BU(s)

Y (s) = (C(sI − A)−1B + D)U(s)

where U(s), X(s) and Y(s) are Laplace transforms of u(t), x(t) and y(t)

respectively.

Linearization of Non-linear Systems

Non-linear systems of the form

x(t) = h(x(t), u(t))

12

y(t) = f(x(t), u(t))

may be approximated around an operating point under the assumption that

output deviation is small when the input perturbations are small. Suppose

that for some constant input vector uo, the state of the system is x0 i.e.

h(x0, u0) = 0

Now for a small deviation in the state from x0 to x0 + ∆x(t) due to a small

perturbation ∆u(t) in the input, we can write

∆x(t) = h(x0 + ∆x(t), u0 + ∆u(t))

≈ h(x0, u0) +∂h

∂x|x0,u0∆x(t) +

∂h

∂u|x0,u0∆u(t) · · · (Taylor series)

= A∆x(t) + B∆u(t)

where, A := ∂h∂x|x0,u0 and B := ∂h

∂u|x0,u0 are called Jacobians. The above is a

linear state-space approximation of the non-linear system. A similar linear

approximation can be obtained for y(t) = f(x(t), u(t))

Controllability and Observability of Systems

Definition 2.2.1. The system

x(t) = Ax(t) + Bu(t)

is said to be completely controllable if for x(0)=0 and any given state x1,

there exists a finite time t1 and a piecewise continuous input u(t), 0 ≤ t ≤ t1

such that x(t1) = x1.

Qualitatively, the above definition implies that if the system is completely

controllable then it is possible to drive the system from any initial value to

any specified final value in a finite time.

13

Theorem 2.2.1. The system

x(t) = Ax(t) + Bu(t)

is completely controllable if and only if the n× np matrix

M = [B, AB, · · · , An−1B]

has rank n.

The above matrix M is called the controllability matrix.

Definition 2.2.2. The system

x(t) = Ax(t) + Bu(t) (2.3)

y(t) = Cx(t) + Du(t) (2.4)

is completely observable if there exists t1 > 0 such that knowledge of u(t) and

y(t) for all t, 0 ≤ t ≤ t1 is sufficient to determine x(0).

Once x(0) is known, equation 2.1 can then be used along with known

input u(t) to determine x(t) for all t, 0 ≤ t ≤ t1. Control inputs are usually

decided upon the basis of outputs available. If the available outputs do not

convey complete information on the state vector, which along with inputs

govern the future response of the system, then it may not be possible to

achieve desired control performance. Thus, both complete controllability

along with observability is important to get good control performance.

Theorem 2.2.2. The system

x(t) = Ax(t) + Bu(t) (2.5)

y(t) = Cx(t) + Du(t) (2.6)

14

is completely observable if and only if the qn× n matrix

S =

C

CA

CA2

...

CAn−1

has rank n.

State Feedback and LQR control

The state variable x(t) along with the input signal u(t) decide the future

response of the system. Thus, in order to regulate the response of the system,

it is important to adjust the input signal in accordance with the value of the

state variable. A commonly used linear control strategy is of the form

u(t) = −Kx(t)

Equation 2.1 then turns into

x(t) = (A−BK)x(t) (2.7)

The eigenvalues of the matrix A − BK govern response of the system. In

order for the system to be stable, it is required that all eigenvalues of A−BK

have negative real parts i.e.

<(λ) < 0 where λ is an eigenvalue of A−BK

The value of the gain matrix K is chosen such that desired performance

specifications in terms of response of the system are met.

A commonly used methodology to specify the desired performance is in

the form of a cost function of the form

J =∫ ∞

t=0(xT Qx + uT Ru)dt (2.8)

15

which needs to be minimized by proper choice of the gain matrix K. The

first term in the above criteria represents the cost incurred when the system

response is away from the desired value (i.e. x 6= 0) while the second term

represents the cost incurred in using the actuator (higher is the actuation

effort higher is the cost). The key quantities in this criteria that a designer is

required to choose based on the performance specifications are the matrices

Q and R both of which have to be positive definite. They represent the

relative weightage given to the two cost criteria. Arriving at the right values

for the matrices Q and R based on the performance specifications is usually

an iterative process.

The gain matrix K that minimizes the integral 2.8 for the system 2.1 is

given by

K = −R−1BT P

where P is given by the Ricatti equation

AT P + PA− PBR−1BT P + Q = 0

2.3 Control Systems for Internal Combustion

Engines

This section looks at the important control problems of relevance in internal

combustion(IC) engines. In particular, control problems in port fuel-injected,

spark-ignited gasoline engines are discussed.

Input-Output view of IC engine

An engine can be viewed as a map from input variables like throttle angle,

fuel injection amount, fuel injection timing, spark timing and load torque to

16

Figure 2.4: Input-Output view of Engine

output variables like engine torque, engine speed and engine out emissions

(figure 2.4). The engine control problem is to decide on a strategy to control

the input variables of the engine in a manner such that desired performance

of the engine is obtained in terms of the output variables. However, not all

input variables may be available for control, e.g. like the throttle angle which

is used by the driver to express torque demand or the load torque which the

engine experiences due to external forces. All inputs which cannot be used

as control inputs as classified as disturbances.

The following sections discuss some important engine control problems.

Air-Fuel Ratio(AFR) Control

Port fuel-injected gasoline engines are fitted with three-way catalytic con-

verters to reduce emissions(section 2.1). In-order for these converters to do

17

well, the in-cylinder AFR needs to be kept in a narrow band around stoi-

chiometric value. Over the years various feed-forward and feedback schemes

have been proposed and devised for AFR control in gasoline engines.

• Feed-forward schemes depend on precise estimates of incoming air mass

flow rate. These estimates are made by means of engine maps developed

by performing a large number of controlled experiments under steady

state operating conditions. Based on an estimate of mass flow rate of

air from these maps, the controller issues command to the fuel injector

in real-time so that appropriate quantity of fuel is injected to keep

AFR at the desired value. However, these maps developed at steady

state operation cannot help overcome air-fuel ratio excursions during

engine transients. Also, performance of these feed-forward schemes

gets affected by external uncertainties like variations in atmospheric

conditions, fuel quality variations, engine aging, etc.

• Feedback schemes rely on measurement of air-fuel ratio made by an

oxygen sensor placed in the exhaust pipe. These measurements help the

controller decide on the correction to be made to the quantity of fuel

injected to keep the AFR at the desired value. Feedback systems are

much more robust in performance than feed-forward systems as they are

not as prone to uncertainties as the feed-forward systems are. However,

the sensor measurement process of air-fuel ratio is based on diffusion

and includes delay of the order of tens of milliseconds. In addition,

there exists a delay of the order of few hundreds of milliseconds due to

the time taken by the air-fuel mixture at the intake to reach the sensor

placed in the exhaust. These delays limit the maximum achievable

bandwidth (speed of response) of the feedback system.

18

Figure 2.5: Engine feedback system architecture

In practice, a combination of both schemes is used. The feed-forward sys-

tem helps achieve faster response while the feedback system helps achieve

robustness to disturbances and uncertainties.

In-order to obtain improved performance during transients, detailed dy-

namic control oriented modeling of the engine air-flow dynamics, fuel sup-

ply dynamics, air-fuel mixture formation dynamics, exhaust gas dynamics,

etc becomes necessary. Some of these issues have been covered in previous

projects on engine management design at IIT Bombay [6][14]. This project

looks at fuel supply dynamics and assesses the role of injection timing in it.

19

Ignition Control

For other inputs held constant, the brake torque of the engine varies with

the spark timing or spark advance. At each operating condition of the en-

gine, there exists a spark advance which maximizes the brake torque called

as Maximum brake torque (MBT) timing. With maximum brake torque a

higher fuel economy can be achieved. However, it may not always be possible

to operate the engine at MBT due to tendencies of knocking and also increase

in NOx production at MBT. Thus spark timing is controlled in a manner

such that maximum brake torque is obtained without engine knocking as

well as keeping NOx emissions under a desired value. Various feedback and

feed-forward control schemes exist for spark timing control. One such control

scheme has been implemented by [6].

20

Chapter 3

Experimental Setup and

Operation

An engine is a system whose performance depends on multiple variables. A

systematic study to understand behavior of the engine would require un-

derstanding the influence of a variable on engine performance with other

engine variables held fixed. Carrying out such controlled experiments re-

quires accurate monitoring and control of the engine through sophisticated

instrumentation. A state of the art facility to work on small engines (100-

300 cc) exists at the engine management systems(EMS) laboratory at IIT

Bombay, the details of which are given in this chapter.

3.1 Sensing and Actuation Architecture

The engine used to carry out investigations for this thesis is a single cylinder,

four-stroke, small-sized gasoline engine of the type which is commonly found

in motorcycles in India. The details of the engine are put in table 3.1a(page

27). This engine has been fitted with a number of sensors and actuators for

21

Figure 3.1: Schematic of engine fitted with sensors and actuators

its monitoring and control(figure 3.1), the details of which given below.

Actuators

The engine is fitted with an electronically controlled, solenoid actuated fuel

injector which is excited using a pulse train like one shown in figure 3.2. The

opening of the injector allows pressurized fuel to enter the intake manifold. A

higher voltage is applied across the injector for a time duration for which it is

desired to be open and we call this duration as fuel injection duration(FID).

An electric spark plug is used to initiate combustion of the air-fuel mixture

in the cylinder. A voltage greater than breakdown voltage of air is applied

across its terminals when combustion of the mixture is desired to be initiated.

22

Figure 3.2: Typical fuel injection pulse (Electronic)

Sensors

An universal exhaust gas oxygen sensor(UEGO) is fitted in the exhaust

tailpipe of the engine to monitor air-fuel ratio by mass(AFR) of the mix-

ture that had entered the cylinder for combustion in the just finished cycle.

The engine is also fitted with a manifold air pressure and temperature sensor

to monitor state of air moving into the cylinder while a thermocouple is used

to monitor engine oil temperature(EOT).

A sensor in the form of an inductive pickup is used to obtain position of

the crank shaft once every rotation. Detailed specifications for sensors and

actuators used are put in table 3.1b.

The important decisions to be made during engine operation are the

amount and timing of fuel injection and timing of the sparking event. In

addition to current engine state perceived using various sensors, an elec-

tronic hardware is usually employed to perform computations in order to

23

Figure 3.3: Schematic of engine with a computational element

arrive at these decisions. Architecture of such a system is shown in fig-

ure 3.3. dSpace R©DS1104 rapid prototyping card was used as the electronic

hardware device for all experiments. This has the provision to accept ana-

log/digital signals from various sensors as well as give actuator command

signals. Programming of the control algorithm is carried out in MATLAB-

SIMULINK R©and the developed program is then transferred onto the proto-

typing hardware for real-time operation.

3.2 Controlled operation of the engine

The mass of air inducted per cycle by the engine at steady state is a function

of engine speed(ω), throttle position(α) and manifold air temperature(MAT)

(see [7]). Thus, holding these variables constant helps ensure that the mean

mass flow rate of air into the engine is held at a steady value.

24

Figure 3.4: Engine fitted with transient dynamometer

(EMS Lab, IIT Bombay)

Engine speed(ω) is held at a steady value using a high bandwidth tran-

sient dynamometer coupled to the engine crank-shaft(figure 3.4). The tran-

sient dynamomter achieves this by appropriately applying load torque to

the engine crank-shaft. For engine operation at 3000 rpm, the transient

dynamometer helps hold the speed within a band of ±1.5%.

Manifold air-temperature(MAT) is held with an accuracy of±0.5 ◦C using

an external air-blower while a stepper motor attached to the engine throttle

helps hold the throttle at a fixed opening. It is found that such controlled

operation of the engine helps hold the mean mass flow rate of air with an

accuracy of ±2.0% about the nominal value.

With the mass flow rate of air held reasonably fixed, changes in AFR can

be attributed to changes in fuel mass inducted into the cylinder. AFR is thus

used as a measure of amount of fuel entering into the cylinder under such

25

controlled operation. Further, it is assumed that the mass of fuel delivered

through the fuel injector is proportional to the opening time of the injector

and hence for all experiments FID is taken as a measure of the amount of fuel

injected into the manifold. During AFR transients the mass of fuel injected

into the manifold is different from the mass of fuel that enters the cylinder

because of the wall-wetting phenomenon(chapter 4).

In this thesis change in injection timing is conjectured to influence re-

sponse of the wall-wetting phenomenon. We therefore describe a physics

based model for this phenomenon so that influence of injection timing can

be better understood. This is the subject of the next chapter.

26

No. of cylinders one

Cylinder Volume 125 cc

Max Power(at 7250 rpm) 7.16 kW

Max Torque(at 5000 rpm) 9.3 Nm

Compression Ratio 9.2 : 1

Intake Valve Opens (5% of maximum) 17 ◦ before intake TDC

Intake Valve Closes (5% of maximum) 210 ◦ after intake TDC

(a) Engine Specifications

Part Specifications

UEGO Response time = 25 ms

Pressure transducer Response time = 1 ms; Range-(0-3 bar abs)

Fuel Injector Used in Honda CBR 600RR motorcycle

(b) Sensor and Actuator Specifications

Motor Make Bosch Rexroth

Motor Type 3-Phase Synchronous Servo

Motor

Peak Power 39 kW

Peak Torque 231 Nm

Max Speed 2800 rpm

Speed Control rise time 5 ms

Torque Control rise time 0.6 ms

(c) Transient Dynamometer Specification

Table 3.1: Experimental setup details

27

Chapter 4

Wall-Wetting phenomenon

In port-fuel injected engines not all of the injected fuel enters the cylinder

for combustion but a significant fraction gets accumulated onto the intake

pipe walls and intake valves in the form of a liquid film (figure 4.1). This

accumulated fuel film evaporates over time and enters the cylinder for com-

bustion only in subsequent engine cycles. This phenomenon is described as

the wall-wetting phenomenon.

In this thesis fuel injection timing is shown to influence the quantity

of the wall-film that evaporates over an engine cycle. Thus, in order to

understand the influence of injection timing better it is important to gain

greater insight into the wall-wetting phenomenon. This is accomplished in

this chapter by considering a simplistic model for this phenomenon(section

4.1), parameters of which are then determined by considering the phenomena

of droplet(section 4.2) and wall-film evaporation(section 4.3).

28

4.1 Aquino Model

The accumulated wall-film mass experiences convective evaporation due to

flow of air in the intake pipe from the throttle side (figure 4.1). The evap-

Figure 4.1: Wall-film deposition

oration of this accumulated fuel film is considered to be proportional to the

area and hence to the mass of the wall-film itself(mf )[1]. Thus, it is modeled

as

mevp =mf

τ(4.1)

The parameter τ is the evaporation time constant which is expected to be

dependent on the engine operating conditions. Assuming κ as the fraction

of the injected fuel that hits the intake pipe walls, applying mass balance to

the wall-film gives,

mf = κminj − mevp

⇒ mf = κminj −mf

τ

29

⇒ τmf + mf = κτminj (4.2)

where minj is the mass rate of fuel injection. Equation 4.2 governs how the

fuel-film mass and hence the rate of evaporation of it varies with the rate of

fuel injection. It may be noted that at a steady rate of fuel injection into the

engine i.e. when minj = constant, the wall-film film mass settles to a steady

value. This implies that rate of evaporation of mass-film and therefore the

mass of fuel entering the cylinder also reaches a steady value.

Let ∆minj be deviation in the rate of fuel injection from a steady value

and let ∆mf be corresponding deviation in the mass of the wall-film. Then

from equation (4.2), we have

τ∆mf + ∆mf = κτ∆minj

Putting the above equation in Laplace domain gives,

∆mf (s)

∆minj(s)=

κτs

τs + 1

From equation (4.1) we have,

∆mevp(s)

∆minj(s)=

κ

τs + 1

Since any deviations in AFR at steady state operation of the engine can

be attributed to deviations in the mass of fuel entering the cylinder while

deviations in fuel mass injected are proportional to deviations in fuel injection

duration(FID)(section 3.2), we have

∆λ

∆FID=

−c1

τs + 1(4.3)

where c1 > 0 is a parameter which is experimentally determined. The nega-

tive sign in numerator indicates that an increase in FID causes a decrease in

AFR.

30

Figure 4.2: Wall-wetting dynamics

(ω = 3000 rpm, pman = 0.80 bar, MAT= 65 ◦C)

Figure (4.2) demonstrates the wall-wetting dynamics when a step change

in FID is given to the engine. The model fits the observation well for c1 = 10.0

ms−1 and τ = 0.3 s

The parameters κ and τ signify the impingement fraction and the evapo-

ration time constant of the deposited fuel film respectively. These parameters

vary based on engine operating variables and the conventional approach has

been to identify them experimentally at a few operating conditions of the en-

gine and interpolate for other operating conditions. However, this approach

is very time-consuming and would be not very efficient at meeting future

stricter emission standards. Moreover, a physics based understanding would

provide greater insight into the dynamics and its dependence on engine vari-

ables than one based on extensive experimentation. This chapter proposes

one such model for wall-wetting dynamics considering the phenomena of

31

droplet evaporation and wall-film evaporation.

Section 4.2 proposes a model for evaporation of a fuel droplet once it

is injected into the intake pipe while section 4.3 proposes an accumulated

wall-film evaporation model.

4.2 Droplet evaporation

With port-fuel injected engines, pressurized fuel is injected into the intake

pipe through fine orifices of the injector. This causes fuel to be injected in the

form of a fine spray of droplets. It is assumed that all droplets are spherical

in shape and having the same diameter. In addition, since the fuel injection

pressure is usually much higher (∼ 4-5 times) compared to the pressure in

the manifold, the speed of the injected droplet is assumed to be far greater

than the speed of air flowing in the intake pipe.

4.2.1 Motion of a droplet

A droplet injected at high speed into the intake pipe is expected to experience

high drag force during its travel in the surrounding air. A measure of this

drag force is given by coefficient of drag defined as [19]:

CD :=FD

12ρaAVD

2

The drag coefficient for a spherical object has been found to be[19]:

CD = 0.4 +24

ReD

+4√ReD

Applying Newton’s law to the droplet gives,

dVD

dt= − 1

2mD

ρaAVD2CD︸ ︷︷ ︸

f1(D,VD)

(4.4)

32

The initial value for the velocity of the droplet VDo can be obtained by

applying Bernoulli’s principle across the injector. Thus,

VDo =

√2

ρf

(pf − pman) (4.5)

Equation (4.4) is a nonlinear differential equation in velocity of the droplet

(VD) and also involving droplet diameter(D). Solution to this equation would

give variation of speed of droplet with time as it travels through intake pipe

after injection. The droplets would travel until either they hit the intake pipe

walls or the intake valves open, whichever occurs earlier. It is assumed that

when the intake valve opens, all airborne droplets would be drawn directly

into the combustion chamber.

However, the droplet diameter changes with time as in travels in air due

to fuel evaporation from the surface. This is discussed in the next section.

4.2.2 Convective mass transfer

During travel of fuel droplets in the intake pipe, in addition to the drag

experienced by them, vaporization of fuel also occurs from the surface. This

vaporization causes the droplet diameter to decrease continuously and the

vaporized fuel diffuses into the surrounding air contributing to the air-fuel

mixture for combustion.

The convective mass transfer rate of species A from a surface maintained

at species molar concentration of CAs and with a fluid of species molar con-

centration CA∞ flowing over it will depend not only on the differences in

the concentrations but also on the tendency of gasoline vapor to diffuse into

surrounding medium of air. This tendency of diffusion is captured by the

convective mass transfer coefficient hm. The molar rate of evaporation can

33

thus be expressed as:

N ′′A = hm(CAs − CA∞) (4.6)

where N ′′A is in kmol/s.m2. For a spherical droplet of diameter D, molar

mass transfer rate from the surface is given by:

N ′′A := −d(m/As)

dt= − ρf

6M

dD

dt(4.7)

The molecules of the fuel droplet at the surface, due to their intrinsic

kinetic energy, have a tendency to escape and form a layer of vaporized fuel

around the droplet. At the same time, the gaseous vapor has a tendency to

condense back onto the droplet. An equilibrium between liquid and gaseous

phase is reached at a particular partial pressure of the vaporized fuel based on

the temperature of the droplet. This partial pressure at a given temperature

Ts of the droplet is given by the Clausius-Clayperon equation as[18]:

pAs = pref exp

(hfgM

R

(1

Tref

− 1

Ts

))(4.8)

where hfg is the enthalpy of vaporisation of gasoline and Tref is the boiling

temperature of gasoline at reference pressure pref . It is assumed in this

analysis that the temperature of the droplet does not change during its travel

in the intake pipe. Thus, molar concentration of fuel vapor at the surface,

given as:

CAs =pAs

RTs

depends only on the temperature of the droplet, assuming ideal gas behavior

of gasoline vapor.

Several empirical relations exist expressing the convective mass transfer

coefficient from a spherical droplet as a function of physical properties of the

droplet and the surrounding medium. One widely used empirical relationship

34

given for freely falling droplets by Ranz and Marshal[16] is:

ShD :=hmD

DAB

= 2 + 0.6 Re1/2D Sc1/3 (4.9)

where Sc is the Schmidt number defined as Sc := νDAB

. The first term

represents natural convective mass transfer while the second term is for forced

transfer. DAB is a property of the binary mixture called as binary diffusion

coefficient. It is a measure of ease of diffusion of gasoline vapor into a gaseous

mixture of air and gasoline. Using kinetic theory of gases, it has been deduced

that DAB ∝ T 3/2p−1 where T and p are temperature and pressure of the

complete mixture respectively[18]. Physically, this can be justified as an

increase in temperature causes an increase in kinetic energy of the vapor

molecules and hence is expected to increase the tendency for diffusion while

an increase in pressure would increase the opposition to motion and hence

decrease diffusion.

Putting equations (4.7) and (4.9) in equation (4.6) gives:

− ρf

6M

dD

dt=(2 + 0.6Re

1/2D Sc1/3

) DAB

D(CAs − CA∞)

⇒ dD

dt=−6MDAB (CAs − CA∞)

ρf

2 + 0.6Re1/2D Sc1/3

D

The first factor in this nonlinear ordinary differential equation is a constant

for one operating condition of the engine while the second factor is a func-

tion of droplet diameter(D) and velocity of droplet(VD) through Reynolds

number. Substituting for the Reynolds number in terms of D and VD, the

equation can be written as:

dD

dt=−L

(2 + Λ

√VDD

)D︸ ︷︷ ︸

f2(D,VD)

where, (4.10)

35

L :=6MDAB (CAs − CA∞)

ρf

Λ := 0.6

√ρa

µSc1/3

L and Λ are constants at a given operating condition of the engine. The so-

lution to this equation, at a given operating condition, will give the variation

of droplet diameter during its motion in the intake pipe.

4.2.3 Coupled differential equation analysis

Equations (4.4) and (4.10) are coupled differential equations which are to

be solved simultaneously to get variation of droplet diameter and distance

traveled by the droplet as a function of time during its flight. The droplets are

assumed to remain airborne until either they hit the intake walls or intake

valve opens when all droplets are assumed to be drawn into the cylinder,

whichever occurs earlier. Thus equations are solved until time t = ξ where

ξ = min(tw, tivo − tinj). The time the droplet takes after injection to hit the

walls (tw) is obtained by solving the equation:

sw =∫ tw

0VD dt (4.11)

where sw is shortest distance of the wall across the injector.

A sample solution of these equations has been plotted for two different

initial droplet diameters. Data for the engine operating conditions was ob-

tained from measurements made on the engine. The data used for simulation

has been tabulated in table 4.1.

From figure 4.3a, it is observed that a droplet of initial diameter 150µm

takes around 4 ms to cover a distance of swall = 6.08 cm and reach the

wall on the opposite side. During this travel towards the opposite end the

droplet can lose at most around 14% of its total mass through evaporation.

Now, if injection of fuel happens less than 4 ms before intake valve opening

then most of the injected fuel would enter the cylinder under the assumption

36

ω 2500 rpm

pman 0.70 bar

MAT 56 ◦C

sw(manifold geometry) 6.08 cm

(a) Engine parameters

Tboil (1atm) 126 ◦C

hfg(octane) 300 kJ/kg [18]

M 112.2 kg/kmol

ρf 746.6 kg/m3

(b) Fuel properties

µair (T = 56 ◦C) 2× 10−5Ns/m2 [10]

Sc 1.5 [15]

(c) Air-Gasoline mixture properties

pinj 4 bar

Do(simulation) 50 & 150 µm

(d) Fuel injection properties

Table 4.1: Simulation Parameters

that when intake valve opens all airborne fuel droplets are sucked into the

cylinder. However, less than 14% of the fuel mass would be in vapor form

and other would be in liquid droplet form which are known to produce excess

hydrocarbons[20]. On the other hand, if fuel is injected more than 4 ms before

intake valve opening then only 14% of injected mass enters the cylinder, all

of which is in vapor form, and the remanining fuel forms a wall film in the

intake pipe.

Contrastingly, it is found that for initial droplet diameter of 50µm,

injected droplets never reach the opposite wall but always remain airborne

(figure 4.3b). Hence most of the fuel is expected to enter the cylinder on

intake valve opening. However, based on injection timing, there would be

varying proportion of fuel in droplets and vapor form entering the cylinder.

Similar computations can be done at each operating condition of the engine

pman, ω, MAT and for a given injection timing to predict the fraction of

37

(a) Initial Droplet Dia = 150µm (b) Initial Droplet Dia = 50µm

Figure 4.3: Droplet evaporation with time

injected fuel entering the cylinder. This would be the parameter κ in the

wall wetting dynamics model described at the beginning of the chapter.

4.3 Port-Wall Film Evaporation

The injected droplets which hit the intake pipe wall form a layer of fuel along

the length of the intake pipe. This fuel evaporates with time and mixes with

the incoming air to form the air-fuel mixture for combustion. The rate of

evaporation of the film will depend on the flow rate of air in the intake pipe

and a methodology to estimate it is proposed in section 4.3.1. Section 4.3.2

then proposes a model for predicting the rate of evaporation of wall film.

4.3.1 Estimation of air velocity

An engine cycle is composed of a number of discrete events. One impor-

tant discrete event that plays a role in the intake air dynamics is the open-

38

Figure 4.4: Intracycle manifold pressure variation

ing/closing event of the intake valve. This is reflected in the way in the way

manifold pressure varies over an engine cycle called as intra-cycle variation.

A sample plot of intra-cycle manifold pressure over an engine cycle is shown

in the figure 4.4. The manifold pressure rises during the period when intake

valve is closed due to filling of the manifold with air from the throttle side

and it falls when intake valve is open due to the air drawn into the cylin-

der causing the manifold to empty. This emptying-filling phenomena can be

modeled as follows:

Considering the manifold as a control volume, mass conservation equation

at any instant of time can be written as(figure 4.5):

dmman

dt= mthrottle − mcylinder (4.12)

The intake pipe of the engine is usually modeled as a converging-diverging

39

Figure 4.5: Intake pipe air-flow

nozzle with the throat occurring at the location of throttle plate[7][4]. It is

assumed that the flow in the intake pipe is isentropic and frictionless. Using

compressible fluid flow theory, the mass flow of air through the throttle can

then be expressed as[7]:

mthrottle = cd · Athrottle ·patm√RTatm

·Ψ(patm

pman

) (4.13)

Ψ

(patm

pman

)=

√

γ[

2γ+1

]γ+1/γ−1if pman ≤ pcr[

pman

patm

]1/γ·√

2γγ−1

·[1− pman

patm

] γ−1γ if pman ≥ pcr

(4.14)

(4.15)

and where

pcr =

[2

γ + 1

] γγ−1

patm

is the critical pressure where the flow reaches sonic conditions at the throttle

section. At patm = 1 bar, pcr turns out to be 0.528. The manifold pressure,

for most operating conditions, remains above this critical value and hence

it is assumed for this analysis that flow never chokes at the throttle. When

the intake valve is closed, no air flows into the cylinder and hence the second

40

term on the right of equation 4.12 can be dropped. Hence equation becomes:

dmman

dt= mthrottle (4.16)

Assuming ideal gas behavior of air we have:

dmman

dt=

VmanMa

RTman

dpman

dt(4.17)

Using equations 4.13, 4.16 & 4.17 we get:

dpman

dt= K ·

∫ IV O

IV C

[pman

patm

]1/γ

·

√√√√√[1− pman

patm

] γ−1γ

dt (4.18)

where the parameter K varies largely with the throttle position(Athrottle) as:

K :=cdAthrottlepatm√

RTatm

√2γ

γ − 1

MaVman

RTatm

patm (4.19)

The solution to this equation predicts the variation of manifold pressure

with time when the intake valve is closed. The value of the parameter K is

chosen such that the mean square error is minimized between the predicted

and actual value measured using a manifold air pressure sensor. A sample

calibration plot is as shown in figure 4.6.

The value of parameter K obtained by such a calibration is a measure of

throttle opening of the engine. This value can then be used in equation 4.13

to get:

mthrottle =cdAthrottlepatm√

RTatm

√2γ

γ − 1

[pman

patm

]1/γ

√√√√√[1− pman

patm

] γ−1γ

(4.20)

(4.21)

= K · MaVman

RTman

patm

[pman

patm

]1/γ

√√√√√[1− pman

patm

] γ−1γ

(4.22)

The average flow rate of air can then be obtained as:

mavg =K · MaVman

RTmanpatm

∫cycle

[pman

patm

]1/γ√[

1− pman

patm

] γ−1γ dt

tcycle

(4.23)

41

Figure 4.6: Model calibration based on observed manifold pressure

Once K is known, above equation can be integrated using measurements

made by a manifold pressure and temperature sensor to get the average air

mass flow rate. An estimate of velocity of air (Va) flowing in the intake

pipe can thus be obtained through real-time implementation of the above

methodology.

4.3.2 Film evaporation

The fuel film in the intake pipe is modeled as a liquid surface A exposed to

an air-stream flowing with a velocity Va over it. The analysis for wall film

evaporation is on similar lines as that for droplet evaporation. The rate of

evaporation from the film surface is expressed :

N ′′A = hm(CAs − CA∞)

42

⇒ mevp

MAwf

= hm(CAs)assuming CA∞ = 0

⇒ mevp = MAwfhmCA

⇒ mevp = Mmf

tfρf

hmCA (4.24)

The factor CA∞ is assumed 0 because incoming air is assumed to contain

hardly any gasoline vapors. The concentration (CA) is obtained using wall

film temperature(Tw) in a manner similar to that used for droplet evapo-

ration. The convective mass transfer coefficient (hm) is obtained using the

widely used empirical relation given below[10]:

ShF :=hmDpipe

DAB

= 1 + 0.023 Re0.83pipe Sc0.44

The estimate of velocity of air at a given operating point obtained using

methodology of Section 4.3.1 is used to obtain Reynolds number in the above

equation.

Comparing equations 4.24 and 4.1, we get:

τ =tfρf

MhmCA

(4.25)

The wall film thickness (tf ) and convective mass transfer coefficient (hm) are

different at different operating conditions of the engine. Hence the wall-film

dynamics is expected to change at different operating conditions. Further,

since wall-film evaporation is dependent of mass of the wall-film, the output

of the droplet evaporation model in terms of fraction κ is used as an input

to solve wall-film evaporation model.

To summarize, models for estimating fraction of injected fuel that evapo-

rates and rate of evaporation of wall-film have been proposed in sections 4.2

and 4.3. These models when put together can be used to estimate the mass

of fuel entering the cylinder per cycle. However, before these models can be

43

validated and used, it is important to determine how parameters of the model

are sensitive to operating conditions so that measurements can be made with

appropriate accuracy. This is the done in the next section.

4.4 Model sensitivity analysis

The equation governing evolution of the droplet in the evaporation model

proposed in section 4.2 is:

dD

dt=−L

(2 + Λ

√VDD

)D︸ ︷︷ ︸

f(D,VD)

where,

L :=6MDAB (CAs − CA∞)

ρf

The parameters DAB, CAs and Sc depend on temperature of the droplet and

thus we determine sensitivity of the model to droplet temperature estimates.

Now, as discussed previously, we have

DAB ∝ T 3/2p−1and

CAs =pAs

RTs

where,

pAs = pref exp

(hfgM

R

(1

Tboil

− 1

Ts

))Thus for a ∆TD change in temperature of the droplet ,assuming other

conditions remains the same, we have:

∆DAB

DAB

=3

2

∆TD

TD

and (4.26)

∆CA

CA

=hfgM

R

∆TD

T 2D

− ∆TD

TD

(4.27)

Thus ∆LL

can be written as:

∆L

L=

3

2

∆TD

TD

+hfgM

R

∆TD

T 2D

− ∆TD

TD

(4.28)

44

Using the above equation it is found that a 6% in change in TD at TD =

330 K gives a 76% change in L for parameter values stated in table 4.1. Hence

evolution of the droplet is very sensitive to the temperature of the droplet

and an accurate estimation of it is desired. This high sensitivity is due to

exponential dependence of CA on droplet temperature. This exponential

factor also makes the solution very sensitive to errors in values of enthalpy

(hfg) and molecular weight(M) of gasoline. Hence accurate composition of

gasoline used is also desired.

4.5 Summary

This chapter discusses a model to capture wall-wetting phenomenon observed

in port-fuel injected engines. The phenomenon has been modeled by first

considering evaporation of fuel droplets once fuel is injected. The output

result of this model is then used as an input for the wall-film evaporation

model and both these models together help determine the quantity of fuel

that enters the cylinder. The predictions of these models for droplet and

wall-film evaporation have been shown to be very sensitive to droplet and

wall-film temperature as well as to the physical properties of fuel and hence

accurate determination of these quantities is critical to validation. Further,

the model for wall-film evaporation depends on the thickness of the fuel

film deposited in the intake port and a measurement for this is necessary to

validate the model.

The influence of injection timing on engine behavior is shown to depend

on the parameters of the wall-wetting model in the next chapter and thus a

methodology to accurately estimate these parameters becomes important in

order to use injection timing as a control input.

45

Chapter 5

Influence of injection timing

The engine cycle is composed of a number of discrete events which occur

periodically. Fuel injection is one such discrete event which usually occurs

once every cycle. We define fuel injection timing(FIT) as the crank angle in

degrees before the top dead center of intake stroke at which injection happens

(figure 5.1). Advancement in injection timing is considered as an increase in

Figure 5.1: Injection timing diagram

46

injection timing as measured from TDC of intake stroke while retardation is

considered as a decrease in injection timing.

This chapter describes the experimental observation made on air-fuel ratio

when FIT is changed in real-time(section 5.1). A mathematical model is

derived to describe the observed phenomenon ans subsequently the model

is used for synthesis of a controller to achieve tighter control of AFR. The

efficacy of the control action has been verified experimentally.

5.1 Experiments - Change in FIT

5.1.1 Measured quantities

Figure 5.2: Fuel flow dynamics study using AFR

A dynamic relationship exists between the amount of fuel injected into

the manifold and the amount of fuel entering the cylinder because of the

wall-wetting phenomenon. In order to understand and validate any proposal

47

describing this dynamics it is important to have measures of both these

quantities.

An input-output view of the air and fuel flow dynamics is shown in figure

5.2. The mass of air inducted is a function of the throttle position(α), engine

speed(ω) and manifold air temperature(MAT)[7]. The throttle position is

held at a fixed position using a stepper motor while the engine speed is held

steady using the transient dynamometer. An external air blower helps hold

MAT with sufficient accuracy. All this helps ensure a steady flow rate of air

into the engine. AFR signal is then used as to monitor changes in fuel flow

into the cylinder.

Further, we assume that the amount of fuel injected into the manifold

is directly proportional to the opening time of the injector. Thus for all

experiments, fuel injection duration(FID) is taken as a measure of fuel that

is injected into the manifold every injection event.

5.1.2 Experimental Observation

Figure 5.3 shows the response of AFR to a step advance and a step re-

tardation in injection timing with the engine held at a steady state. This

experiment was conducted at an engine speed of 3000 rpm and MAT of 65 ◦C.

FID was fixed at a particular value to ensure that same amount of fuel was

injected every injection event.

As observed from figure 5.3, an advancement in injection timing from

100 to 300 degrees causes an initial sharp fall in AFR from 14.6 to 13.3

though the steady state AFR remains the same. It is thus concluded that an

advancement in injection timing by 200 degrees causes close to 9% change

in quantity of fuel that enters the cylinder initially. However, the steady

state value of amount of fuel entering the cylinder remains the same before

48

Figure 5.3: AFR variation with FIT

(ω = 3000 rpm, pm = 0.80 bar, MAT = 65 ◦C)

and after the change in timing is made. A similar conclusion is drawn for a

retardation in injection timing.

A phenomenological explanation for this observation is given in the next

section while a mathematical model capturing this phenomenon is proposed

and validated in sections 5.2 and 5.3. This observed phenomenon is then

exploited for tighter control of AFR in the transient state in section5.4.

5.2 Phenomenological explanation

The wall-wetting dynamics in port-fuel injected engines gets excited every

time fuel injection happens in its intake port. Every fuel injection event leads

to deposition of a fraction of the injected fuel onto the intake port walls which

49

Figure 5.4: Wall-film evaporation and FIT diagram (∆FIT = 0)

then evaporates and enters the cylinder for combustion in subsequent cycles.

A widely used model describing this evaporation process, as discussed in

section 4.1, is expressed mathematically as [1]:

mevp =mf

τ(5.1)

The wall-wetting dynamics thus turns into the following(section 4.2):

mf = kminj − mevp

⇒ τmf + mf = kτminj (5.2)

Periodic excitation(once every engine cycle) of this dynamics simulating

engine running conditions at 3000 rpm gives the following plot (figure 5.4).

This plot gives variation of wall-film mass(mf ) with time. From equation

5.1 it may be noted that the shaded area in the above figure is a measure of

the amount of fuel that enters the cylinder per cycle for combustion through

wall-film evaporation.

50

Figure 5.5 shows the response of wall-wetting dynamics to an advance-

ment in injection timing. It may be noted that a change in injection timing

causes a change in the time interval between a pair of injection pulses(see

the injection pulse below).

Figure 5.5: Wall-film evaporation and FIT diagram(∆FIT = 200 ◦)

As observed from this figure, an advancement in injection timing causes

an aperiodic deposition of fuel on the intake walls. This leads to an increased

amount of fuel entering the cylinder for few subsequent cycles through greater

evaporation. However, the steady state value of fuel amount entering the

cylinder remains unchanged. Further, it is observed that a change in FIT

causes a sharp initial change in amount of fuel that enters the cylinder. It

is this observed phenomenon that provides the motivation to use FIT as a

control input to achiever tighter transient AFR control.

We propose a mathematical model in the next section capturing this

phenomenon which is later used for systematic controller development to

51

control AFR using both FID and FIT.

5.3 Modeling influence of injection timing

5.3.1 Analytical Formulation

Combining the wall-film dynamics model (5.2) with the evaporation model

(5.1) we get,

τmevp + mevp = κminj

Expressing it as a transfer function gives,

Mevp(s)

Minj(s)=

κ

τs + 1(5.3)

where Mevp(s) and Minj(s) are the Laplace transforms of the signals mevp(t)

and minj(t) respectively. We now compare the mass of wall-film that evap-

orates when an advance in injection timing is made with the case when no

advance in timing is made. This will help us get an analytical formulation of

the excess mass of fuel that evaporates and enters the cylinder for combustion

for an injection timing advance.

A typical fuel mass flow-rate pulse train (minj signal) is shown in figure

(5.6). The area under each injection pulse numerically equals the mass of

fuel that is injected into the engine per cycle(mpul). Assuming each injection

pulse to be an impulse of area equal to mpul, the Laplace transform for the

injection pulse train with advance can be written as,

sMinj(s) = mpul + mpule−s(T−∆FIT ) + mpule

−s(2T−∆FIT ) + · · · (5.4)

From equation 5.3 we get,

Mevp(s) =κ

s(τs + 1)sMinj(s) = (

1

s− τ

τs + 1)κsMinj(s) (5.5)

52

Figure 5.6: Fuel mass flow-rate signal

Combining equations 5.4 and 5.3, we get

Mevp(s) = mpulκ

s(1 + e−s(T−∆FIT ) + e−s(2T−∆FIT ) + · · ·

−mpulκτ

τs + 1(1 + e−s(T−∆FIT ) + e−s(2T−∆FIT ) + · · ·)

The inverse Laplace transform to fetch the output signal in time domain

gives

mevp(t) = mpulκ[u(t− 0) + u(t− (T −∆FIT )) + · · ·]

−mpulκe−t/τ [1 + e−∆FIT/τ (eT/τ + · · ·)]

where u(t−Γ) is the unit step at t = Γ (figure 5.7). Thus, wall-film evaporated

after n-injection pulses is,

mevp(nT )|with advance = mpulκ(n + 1)

53

Figure 5.7: Unit step signal (Γ = 2sec)

−mpulκe−nT/τ [1 + e−∆FIT/τ (eT/τ + · · ·)]

= mpulκ(n + 1)

−mpulκ[e−nT/τ + e−∆FIT/τ 1− e−nT/τ

1− e−T/τ]

This gives the mass of wall-film that evaporates after n-injection pulses when

an injection timing advance of ∆FIT is given.

In order to obtain mass of wall-film that evaporates after n-injection

pulses without any advance in injection timing, we simply set ∆FIT = 0

in the above equation. Thus we get,

mevp(nT )|no advance = mpulκ(n + 1)

−mpulκ(e−nT/τ +1− e−nT/τ

1− e−T/τ) (5.6)

Thus the extra mass of wall-film that evaporates when an advance in injection

timing is given is expressed as:

∆mevp(nT ) = mevp(nT )|with advance −mEV P (nT )|no advance

= mpulκ(1− e−∆FIT/τ )1− e−nT/τ

1− e−T/τ(5.7)

54

Figure 5.8: Prediction of the analytical formulation

(∆FIT = 200 degrees, κ = 0.8, τ = 0.3, mpul = 0.3mg, T = 40 ms)

For sufficiently large n, the above equation can be turned into a continuous

time equation by putting nT ≈ t which gives

∆mevp(t) = mpulκ(1− e−∆FIT/τ )

(1− e−T/τ )(1− e−t/τ )

⇒ ∆mevp(t) = mpulκ

τ

(1− e−∆FIT/τ )

(1− e−T/τ )e−t/τ︸ ︷︷ ︸

evolves with time

(5.8)

The above equation gives the manner in which the excess wall-film mass

evaporates with time when a step change in injection timing is made. The

response of this equation for ∆FIT = 200 degrees is shown in figure 5.8.

5.3.2 Model Validation

As seen from figure 5.8, response of the analytical formulation to a change

in injection timing consists of an impulsive part(derivative action) and a

decay part similar to one observed for first order systems. This provides the

55

Figure 5.9: FIT model - fitting on observation

(ω = 3000 rpm, pm = 0. bar, MAT = 65 ◦C)

motivation to capture influence of injection timing using the following linear

model expressed in transfer function domain as:

∆λ(s)

∆FIT (s)=

−c2s

τs + 1(5.9)

The evaporation constant τ is determined experimentally by fitting the

wall-wetting dynamics model on actual response of AFR to changes in FID

(section 4.1). Fitting of the model on experimental observation made at

engine operating conditions of ω = 3000 rpm, pm = 0.80 bar, MAT = 65 ◦C

gives τ = 0.3.

Figure (5.9) shows how prediction of the model(equation 5.9) for influence

of injection timing compares with the observation. The parameter c2 was

found to be 2.05× 10−3 s/crank deg.

Looking at the above equation as well as the observed response of air-fuel

56

Figure 5.10: Comparison - AFR response with and without FIT change

(ω = 3000 rpm, pm = 0. bar, MAT = 65 ◦C)

ratio to changes in injection timing, it can be seen that though injection

timing cannot influence the steady state value of AFR, the rate of change of

∆FIT term on the input side can produce sharp changes in AFR when finite

changes in injection timing are made due to its derivative action. This opens

up the possibility to obtain tighter control in AFR by using injection timing

as a control input in conjunction with quantity of fuel delivered. This is the

focus of the next section.

5.4 Controller development

The speed of response of AFR to changes in fuel quantity injected is limited

by the wall-wetting dynamics. This speed of response can be significantly

57

Figure 5.11: MIMO control - feedback architecture

enhanced if appropriate changes in FIT are made in conjunction with FID.

The plots (figure 5.10) compare the experimentally observed performance of

system using both FID and FIT as inputs vis-a-vis one using only FID as an

input. It is clearly seen that faster AFR response is obtained through the

use of both inputs.

In order to achieve tight AFR control, an appropriate strategy has to be

decided upon to control the two available inputs. The approach we adopt

uses an AFR sensor in feedback to decide upon the inputs FID and FIT.

This particular system architecture is demonstrated in figure (5.11).

In-order to build a linear feedback controller, we express the two input

one output system in state space form as discussed in section 2.2.

58

5.4.1 State Space Model of the plant

The influence of fuel injection duration and injection timing on AFR are

captured through the following linear models (equations 4.3 and 5.9):

∆λ(s)

∆FID(s)=

−c1

τs + 1

&∆λ(s)

∆FIT (s)=

−c2s

τs + 1

On putting them together we obtain,

∆λ(s) =−c1

τs + 1∆FID(s) +

−c2s

τs + 1∆FIT (s) (5.10)

which on little manipulation gives,

∆λ(s) =−c2

τ∆FIT (s) +

1

τs + 1[−c1∆FID(s) +

c2

τ∆FIT (s)]︸ ︷︷ ︸

X(s)

(5.11)

Thus in the time domain,

∆λ(t) =−c2

τ∆FIT (t) + x(t) (5.12)

where x(t) is an intermediate variable given by

τ x + x = −c1∆FID(t) +−c2

τ∆FIT (t) (5.13)

Putting equations (5.13) and (5.12) in matrix form gives,

x =−1

τx +

[−c1τ

c2τ2

] ∆FID(t)

∆FIT (t)

(5.14)

∆λ(t) = x +[

0 −c2τ

] ∆FID(t)

∆FIT (t)

(5.15)

The above is the state space representation of the engine system. We

have a two-input, one-output system with one intermediate state. The state

59

x here represents the mass of the fuel film(mf ) on the walls of the intake

pipe. Comparing with the standard state-space form in section 2.2, we see

that in this case

A =−1

τ

B =[

−c1τ

c2τ2

]C = 1

D =[

0 −c2τ

]Thus the controllability and observability matrices are

M = [B] =[

−c1τ

c2τ2

]S = [C] = 1

Both these matrices have rank = 1 except for c1 = c2 = 0 and hence the

system is completely controllable and observable (section 2.2).

5.4.2 Feedback Controller

We employ state feedback to regulate output of the system around a desired

value. On substituting ∆FID(t)

∆FIT (t)

= −

k1

k2

x (5.16)

into equation 5.14, we get

x = (−1

τ+

c1k1

τ− c2k2

τ 2)︸ ︷︷ ︸

β

x (5.17)

In-order for the feedback system to be stable, we require that β < 0. Further,

β governs the speed of response of the system. A larger magnitude for β gives

faster convergence of x to 0 and hence faster response of the system.

We compare the following two scenarios:

60

• FIT is not used as control input i.e. k1 6= 0, k2 = 0

• Both FID and FIT are used as control inputs i.e. k1 6= 0, k2 6= 0

While comparing it is ensured that the AFR response for the two cases is

kept as close to each other as possible and then comparison is made between

the amount of input effort needed to achieve that response in either case.

Responses are kept similar by choosing the same value for β in both the

cases.

Figure 5.12a shows the response of the system using only FID as a control

input for k1 = −0.1 as well the actuator effort in terms of change in FID

(∆FID) required to achieve that response. In this case β turns out to be

−6.67.

With k1 and k2 both non-zero, it is seen that infinite combinations of

them can give β equal to −6.67. Thus, we impose additional constraint in

terms of the amount of control inputs that can be used in order to determine

the gain values. We express this constraint in the form of an LQR integral

to be minimized (section 2.2) with Q and R matrices as follows:

Q = 95.92

R =

3840 0

0 0.0058

The specific choice made for the matrices Q and R is explained in Appendix

A. The gain values obtained for these performance matrices by solving the

Ricatti equation are k1 = −0.0565, k2 = 51. Figure 5.12b shows the system

response along with the actuation effort for this case.

Since the state ’x’ of the system, which is the mass of the wall deposited in

the intake pipe, is a quantity not measurable usually, we employ an observer

to estimate the state which is then used for feedback. The value of the

61

observer gain is chosen such that estimated state converges to the actual by

an order of magnitude times faster than the performance desired out of the

feedback system.

5.4.3 Results and Discussion

It is observed from figures (5.12a) and (5.12b) that in order to achieve same

level of system performance in terms of AFR response, the maximum FID

actuation input required in the case where both inputs are utilized is close

to 0.04 ms compared with the case with only FID as the input where it is 0.1

ms. Thus, considerably lower actuation effort is required(lower by 60%) to

achieve the same level of system performace when both inputs are utilized.

Expressed in other terms, utilizing the same amount of actuator, a faster

response of the system can be obtained with the use of both FID and FIT

as inputs and thus tighter air-fuel ratio control is achievable.

62

(a) Only FID as input

(b) Both FID and FIT as input

Figure 5.12: AFR response and actuation effort

63

Chapter 6

Conclusion

This chapter provides a summary of work presented in this thesis as well as

outlines possible problems for future work.

6.1 Summary and Conclusion

The focus of this thesis has been on investigating influence that dynamic

changes in injection timing has on the quantity of fuel that evaporates from

wall-film and hence on mass of fuel entering the cylinder per cycle. The

investigation has been carried out by conducting experiments on a single-

cylinder, port-fuel injected small-sized gasoline engine. This study gains

importance with the underlying objective of achieving tighter air-fuel ratio

control to reduce engine-out emissions.

Through the work for this thesis, it has been found that changes in injec-

tion timing cause a sharp change in quantity of fuel that enters the cylinder

for a few cycles, though the steady state value remains the same before and

after the change(section 5.1). This occurs because a change in injection tim-

ing causes a change in mass of wall-film that evaporates over an engine cycle

64

for few subsequent cycles.

It is this observed phenomenon that opens up the possibility to use injec-

tion timing as a control input to influence mass of fuel entering the cylinder

and subsequently achiever tighter AFR control. This has been demonstrated

in section 5.4.

6.2 Outline for future work

The work for this thesis was carried out on a single cylinder, port-fuel in-

jected, small-sized(125 cc) gasoline engine. It would be interesting to note

if an observation similar to one made in this thesis regarding influence of

injection timing can be made in multi-cylinder engines, particularly on ones

based on multi-point port-fuel injection(MPFI) architecture.

Also while carrying out work for this thesis, it was always ensured that fuel

was injected into the intake port in the time interval when intake valve was

closed. An investigation can be carried out to compare influence of injection

timing with injection done when intake valve is open to that made when

intake valve is closed and understand the possible reasons for the difference,

if any.

Another possible extension for this work could be a study to determine the

correlation between the wall-wetting phenomenon and influence of injection

timing on AFR at different operating conditions of engine, particularly at

different wall-film temperatures. An experimental validation of the wall-

wetting dynamics model proposed in chapter 4 would be of substantial help

while carrying out this study.

65

Appendix A

LQR Controller Design

The choice of the gain values for a controller required to meet LQR criteria

is arrived at by solving the Ricatti equation for appropriate choice of the

matrices Q and R(section 2.2). The specific choice of matrices Q and R is

made on the basis of performance desired of the feedback system.

In this thesis, we compare the following two control strategies for AFR

performance:

• FIT is not used as control input i.e. k1 6= 0, k2 = 0

• Both FID and FIT are used as control inputs i.e. k1 6= 0, k2 6= 0

While comparing performances in the two cases, it is ensured that AFR re-

sponse for both cases is kept as close to each other as possible and comparison

is then made between the amount of input effort needed to achieve that re-

sponse. Responses are kept similar by choosing the same value for β in both

cases(section 5.4.2).

We set the following performance criteria to be achieved in both of the

above cases:

66

Figure A.1: System Response - Only FID as input

• AFR response for a unit initial displacement settles to within the 10%

band of its steady state value in a time period of less than 0.2 seconds

Mathematically, the specifications can be stated as

x(t) = 1.0 when t = 0,

x(t) < 0.1 for t > 0.2,

(A.1)

where x(t) is the state representing the mass of the wall-film. Figure 5.12a

shows system response when only FID is used as the control input. Specif-

ically, the gain values used are k1 = and k2 = 0.0 and the value of the

parameter β turns out to be -6.67.

For the case with both FID and FIT as control inputs, infinite combi-

nations of the gain values, k1 and k2, yield the same β value. Additional

constraints are thus imposed in the form of maximum limits on input signals

67

u1(t) and u2(t). We set the following limits:

|u1(t)| < 0.05 for all t > 0 and

|u2(t)| < 50 for all t > 0

Assuming a linear system, the free response of the state ’x(t)’ for a unit

initial deviation is of the form

x(t) = e−λt for all t > 0

where λ is chosen such that the set performance specifications are met. Here

we choose λ = 12.0

Thus, ∫ ∞

t=0x2(t)dt =

∫ ∞

t=0e−24tdt = 0.0417 (A.2)

Since for a state-feedback controller u(t) = −Kx(t) we assume

u1(t) = 0.05e−12t

& u2(t) = 50e−12t

The coefficients 0.05 and 50 help satisfy the performance requirements in

terms of bounds on input signals. Thus∫ ∞

t=0u2

1dt = 1.042 ∗ 10−4

&∫ ∞

t=0u2

2dt = 104.2

(A.3)

The LQR criteria can thus be expressed in the form:

J =∫ ∞

t=0

(q

x2(t)

0.0417︸ ︷︷ ︸xT Qx

+ (ru2

1

1.042 ∗ 10−4+ (1− r)

u22

104.2)︸ ︷︷ ︸

uT Ru

)dt where

q ≥ 0 and 0 ≤ r ≤ 1 (A.4)

68

Figure A.2: System Response - FID and FIT as inputs

Coefficient q decides which of the two criteria, accurate state regulation or

minimization of input effort, is given additional weightage while coefficient r

determines weightage for performance among the two inputs.

Trial and error to meet performance similar to that shown in figure A.1

leads to q = 4.0 and r = 0.5 and corresponding controller gain values ob-

tained by solving the Ricatti eqaution are k1 = −0.0565 and k2 = 51.76. The

system performance for these gain values is shown in figure A.2.

The Q and R matrices thus turn out to be:

Q =4

0.0417= 95.92 &

R =

r1.042∗10−4 0

0 1−r104.2

=

3840 0

0 0.0058

69

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