physics-based modeling and control of residual-affected hcci

PHYSICS-BASED MODELING AND CONTROL OF

RESIDUAL-AFFECTED HCCI ENGINES USING

VARIABLE VALVE ACTUATION

a dissertation

submitted to the department of mechanical engineering

and the committee on graduate studies

of stanford university

in partial fulfillment of the requirements

for the degree of

doctor of philosophy

Gregory Matthew Shaver

September 2005

c© Copyright by Gregory Matthew Shaver 2006

All Rights Reserved

ii

I certify that I have read this dissertation and that, in

my opinion, it is fully adequate in scope and quality as a

dissertation for the degree of Doctor of Philosophy.

J. Christian Gerdes(Principal Adviser)




Christopher F. Edwards




Sanjay Lall

Approved for the University Committee on Graduate

Studies.

iii

iv

Abstract

A wonderful opportunity exists to capitalize on recent improvements in actuator and

sensing technologies in the pursuit of cleaner and more efficient automobiles. While

a considerable amount of attention has been given to hybrid and fuel cell approaches,

significant improvements can be made in the area of advanced combustion strategies.

One such strategy, Homogeneous Charge Compression Ignition (HCCI), combines

benefits of both diesel and spark ignition (SI) methodologies to produce a strategy

that is cleaner than either approach. Residual-affected HCCI uses variable valve

actuation (VVA) to reinduct or trap hot combustion gases, enabling dilute, stable

autoignition. As a result, residual-affected HCCI has an efficiency exceeding SI and

matching diesel. While these characteristics of HCCI can address increasing environ-

mental regulatory demands, there are some fundamental challenges. To practically

implement residual-affected HCCI, closed-loop control must be used for two reasons:

there is no direct combustion trigger and cycle-to-cycle dynamics exist through the

residual gas temperature. Although HCCI is a complex physical process, this the-

sis shows that the aspects most relevant for control - in cylinder pressure evolution,

combustion timing, work output and cycle-to-cycle dynamics - can be captured in

relatively simple and intuitive physics-based simulation and control models. From

the physics-based control model, a variety of control strategies are outlined and im-

plemented in experiment to successfully track desired work output, in-cylinder peak

pressure and combustion timing.

Specifically, the simulation model captures the general behavior of residual-affected

HCCI, including the dependence of combustion timing, work output and in-cylinder

pressure evolution on the inducted gas composition, effective compression ratio and

v

cycle-to-cycle coupling. While the simulation model is accurate, and provides valu-

able insight, it is cumbersome to use in the direct synthesis of control strategies.

For this reason a simpler physics-based model was formulated. This control model

still captures the dynamics of the residual-affected HCCI process while allowing the

synthesis of control strategies. The first control strategy outlined is developed from

the peak pressure dynamics in the control model and allows the modulation of work

output via closed-loop control of peak in-cylinder pressure at constant combustion

timing. Peak pressure tracking is rapid (step change tracking within about 5 engine

cycles), and cyclic dispersion is reduced. The second control approach allows the

simultaneous control of combustion timing and peak pressure, or work output, using

a decoupled control approach. Specifically, combustion timing and peak pressure (or

work output) are controlled on different time scales through modulation of effective

compression ratio and inducted gas composition, respectively. The combustion tim-

ing controller is designed to be notably slower (tracking within 30 cycles) than the

control of peak pressure (tracking within 5 cycles), so the effect of combustion timing

variation on cycle-to-cycle peak pressure dynamics can be neglected. The final and

most capable strategy allows the simultaneous, coordinated control of combustion

timing and peak pressure via an H2 controller synthesized from the complete control

model, using effective compression ratio and inducted gas composition as inputs. The

coordinated approach exhibits step change tracking within 4-5 engine cycles for both

combustion timing and work output (or peak pressure). Furthermore, by directly

coordinating the modulation of the two inputs, a reduction in controller effort is re-

alized. The successful implementation of these control approaches demonstrate the

utility of physics-based modeling and control, and represent a positive step toward

the practical implementation of clean and efficient HCCI engines.

vi

Acknowledgements

The last five years at Stanford have been truly wonderful. I recall being asked a

couple of years ago what I would do if I won the lottery. I thought about it for a

few moments and replied that as far as where I was, and what I was doing, I would

not change a thing. This realization made me very aware of how lucky I was to be

surrounded by such great people in an environment where I could learn and grow as

a person. Above all it has been the people that I have met along the way that has

been the most special part of both my life and time in graduate school.

First and foremost, I want to thank my parents, for all of the wonderful things

they have done for me. Although my mother, Lee, passed away when I was young,

I vividly remember her beauty and intelligence. She gave my brother, Jeff, and I all

she could and I will never forget her. Likewise, my dad, Paul, continues to be my

biggest supporter and role model. He has always been there for me, in good times

and bad.

I also owe a great deal to a large number of other people that have been involved

in my life. In the years following my mother’s passing, two women, Mary Fox and

Linda Dixon, played significant roles in my childhood, and I want to thank them.

I have also had many outstanding mentors, role models and friends along the way:

my little brother, Jeff, a dedicated teacher, great person, and fabulous bro; Professor

Chris Gerdes, a great thesis advisor and dear friend, Chris never ceases to amaze me

with his passion for life, science and learning; Chris Evans, who gave me a chance to

work at AlliedSignal Aerospace (now Honeywell) under his guidance, an experience

that coupled with my experiences at Purdue lead to my decision to go to graduate

school; Professor Klod Kokini who gave me a chance to see what research was like as

vii

an undergraduate at Purdue; Professor Matt Franchek, who provided guidance about

graduate school, and today continues to be a wonderful friend and role model; Chris

Carlson and Eric Rossetter, a couple of years ahead of me in the graduate program

these guys became not only great friends but (I can not believe I am going to say

this), role models!, proving to me that you can work hard and have fun at the same

time; Shannon Miller and Sara Mark, great friends and avid 80s music fans; Professor

Chris Edwards, who taught me how to think like an engines guy, and who has the

most incredible repertoire of terrible puns I have ever heard; Matt Roelle, another

character who’s hard work and incite has played a significant role in the work outlined

in this thesis; Josh Switkes a decent tip-10 player who is nice enough to let me win

most of the time....; Andy Schober and Joe Matteo, two good guys who prove that you

can be incredibly smart, hairy and strong at the same time; Chuck Booten, a great

friend who never ceases to surprise with the stories he tells; my pledge brothers (Ken

Fischer, Chad Goze, Joe Martin, Jon Helman, Rahul Oltikar, Dan McKechnie, Wyatt

Meek, Greg Goodrich, Joe Toniolo and Marty Daiga) at the Phi Chapter of Theta Tau

Fraternity, with whom I share some great memories; the Dynamic Design Lab, a great

group of people; Brucek Khailany, a good friend who plays off the ”cool dork” better

than anyone; the Hank’s family (my goddaughter Shardane, Camden, Ed, Elizabeth

and Pat), who have always treated me like family; and special acknowledgements to

Heather and Hayden Harding, two very special people in my life.

viii

ix

Contents

iv

Abstract v

Acknowledgements vii

ix

1 Introduction 1

1.1 Context - Environmental Challenges and New Technologies . . . . . . 1

1.1.1 Homogeneous Charge Compression Ignition . . . . . . . . . . 2

1.1.2 Challenges with Making HCCI Practical . . . . . . . . . . . . 4

1.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 HCCI Research . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 HCCI Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 HCCI Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Simulation Model 14

2.1 Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Volume Rate Equation . . . . . . . . . . . . . . . . . . . . . . 16

2.1.2 Valve Flow Equations . . . . . . . . . . . . . . . . . . . . . . 16

2.1.3 Species Concentration Rate Equations . . . . . . . . . . . . . 18

x

2.1.4 Temperature Rate Equations . . . . . . . . . . . . . . . . . . 20

2.1.5 Exhaust Manifold Modeling . . . . . . . . . . . . . . . . . . . 22

2.2 Combustion Chemistry Modeling . . . . . . . . . . . . . . . . . . . . 25

2.2.1 Temperature Threshold Approach . . . . . . . . . . . . . . . . 26

2.2.2 Integrated Global Arrhenius Rate Threshold . . . . . . . . . . 30

2.2.3 Knock Integral Technique . . . . . . . . . . . . . . . . . . . . 34

2.3 Transients and Mode Transitions . . . . . . . . . . . . . . . . . . . . 35

2.3.1 Validation in Transient Operation . . . . . . . . . . . . . . . . 36

2.3.2 Validation during an SI-to-HCCI Mode Transition . . . . . . . 37

2.4 Extension to More Complex Fuels . . . . . . . . . . . . . . . . . . . . 39

2.5 Physical Insight from Simulation Modeling . . . . . . . . . . . . . . . 40

2.5.1 Modulation of Inducted Gas Composition . . . . . . . . . . . 41

2.5.2 Self-stabilizing Nature of Residual-Affected HCCI . . . . . . . 41

2.5.3 Cycle-to-cycle Coupling in Residual-Affected HCCI . . . . . . 42

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Control Modeling 45

3.1 Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1.1 Instantaneous Mixing of Species . . . . . . . . . . . . . . . . . 48

3.1.2 Isentropic Compression to Pre-Combustion State . . . . . . . 50

3.1.3 Constant Volume Combustion . . . . . . . . . . . . . . . . . . 50

3.1.4 Isentropic Expansion and Exhaust . . . . . . . . . . . . . . . . 52

3.2 Peak Pressure Equation . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Combustion Timing Modeling Approach . . . . . . . . . . . . . . . . 53

3.4 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.1 Model Validation in Steady State . . . . . . . . . . . . . . . . 57

3.4.2 Model Validation During Transients . . . . . . . . . . . . . . . 60

4 Control of Peak Pressure 62

4.1 Control Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.1.1 Linearization of Pressure Relation . . . . . . . . . . . . . . . . 64

4.2 LQR Controller Synthesis . . . . . . . . . . . . . . . . . . . . . . . . 66

xi

4.3 Valve Timing Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.4 Implementation on Simulation Model . . . . . . . . . . . . . . . . . . 68

4.5 Implementation on Research Engine . . . . . . . . . . . . . . . . . . . 69

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5 Stability Analysis 74

5.1 Closed-loop HCCI Dynamics . . . . . . . . . . . . . . . . . . . . . . . 74

5.2 Estimating the Domain of Attraction using the Sum of Squares De-

composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 75

5.2.2 The Positivstellensatz . . . . . . . . . . . . . . . . . . . . . . 77

5.2.3 Sum of Square Programs . . . . . . . . . . . . . . . . . . . . . 79

5.3 Domain of Attraction for the HCCI System . . . . . . . . . . . . . . 81

5.3.1 Quadratic Lyapunov Function . . . . . . . . . . . . . . . . . . 82

5.3.2 Quartic Lyapunov Function . . . . . . . . . . . . . . . . . . . 83

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6 Decoupled Control of HCCI 86

6.1 Control Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.2 Controller Development . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.2.1 Combustion Timing Control . . . . . . . . . . . . . . . . . . . 89

6.2.2 Peak Pressure Control . . . . . . . . . . . . . . . . . . . . . . 90

6.2.3 H2 Control Formulation . . . . . . . . . . . . . . . . . . . . . 90

6.3 Experimental Implementation . . . . . . . . . . . . . . . . . . . . . . 93

6.4 Extension to Work Output Control . . . . . . . . . . . . . . . . . . . 95

6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7 Coordinated Control of HCCI 99

7.1 Control Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.1.1 Linearization of Full Control Model . . . . . . . . . . . . . . . 100

7.2 H2 Controller Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.3 Experimental Implementation . . . . . . . . . . . . . . . . . . . . . . 104

xii

7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

8 Conclusions and Future Work 112

8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

8.2 Future Research Efforts . . . . . . . . . . . . . . . . . . . . . . . . . . 114

8.3 The Future of HCCI . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

A Alternative Exhaust Manifold Model 116

A.1 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Bibliography 121

xiii

List of Tables

2.1 Engine Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Propane Fuel Simulation Parameters . . . . . . . . . . . . . . . . . . 29

2.3 Temperature Threshold Approach: Comparison of experiment and

simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4 Integrated Arrhenius Rate Threshold Approach: Comparison of exper-

iment and simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5 Knock Integral Threshold Approach: Comparison of experiment and

simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1 Experimentally Monitored Values . . . . . . . . . . . . . . . . . . . . 58

3.2 Estimated Experimental Values . . . . . . . . . . . . . . . . . . . . . 58

3.3 Static Validation of Control Model . . . . . . . . . . . . . . . . . . . 59

6.1 Engine Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

xiv

List of Figures

1.1 Key Processes in Residual-Affected HCCI . . . . . . . . . . . . . . . . 2

2.1 Valve Mass Flows: left - induction flows with intake and exhaust valves

open, right - exhaust flow . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Valve profiles used for residual-affected HCCI using variable valve ac-

tuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Schematic of exhaust manifold control mass: (a) residual mass from

previous exhaust cycle, θ = EV O; (b) increase in mass due to cylinder

exhaust, EV O < θ < 720; (c) maximum amount of exhaust manifold

mass, θ = 720; (d) decrease in mass due to reinduction, 0 < θ < EV C;

(e) post-reinduction mass, θ = EV C; (f) decrease in mass to residual

value, EV C < θ < EV O . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Single Cylinder Research Engine Outfitted with VVA . . . . . . . . . 27

2.5 Operating manifold [11] . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.6 Temperature Threshold Approach: Simulated HCCI combustion dur-

ing steady state: dashed - simulation, solid - experiment; left - IVO/EVC

= 25/165, middle - IVO/EVC = 45/185, right - IVO/EVC = 65/205 30

2.7 Integrated Arrhenius Rate Threshold Approach: Simulated HCCI com-

bustion during steady state: dashed - simulation, solid - experiment;

left - IVO/EVC = 25/165, middle - IVO/EVC = 45/185, right -

IVO/EVC = 65/205 . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

xv

2.8 Knock Integral Threshold Approach: Simulated HCCI combustion

during steady state: dashed - simulation, solid - experiment; left -

IVO/EVC = 25/165, middle - IVO/EVC = 45/185, right - IVO/EVC

= 65/205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.9 Simulated HCCI combustion over a valve timing change: top - experi-

ment, bottom - simulation . . . . . . . . . . . . . . . . . . . . . . . . 36

2.10 Simulated SI combustion during steady state: solid - simulation, dashed

- experiment; work carried out with fellow student Matthew Roelle . 38

2.11 Simulated HCCI combustion over a valve timing change: left - experi-

ment, right - simulation; work carried out with Matthew Roelle . . . 39

2.12 Integrated Arrhenius Rate Threshold Approach for Gasoline: Simu-

lated HCCI combustion during steady state: dashed - simulation, solid

- experiment; left - equivalence ratio = 1, middle - 0.91, right - 0.83,

steady-state experiment plots are cycle averaged; work carried out with

Nikhil Ravi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1 General view of partitioned HCCI cycle . . . . . . . . . . . . . . . . . 46

3.2 Block Diagram of Control Model . . . . . . . . . . . . . . . . . . . . 47

3.3 Representation of control model . . . . . . . . . . . . . . . . . . . . . 54

3.4 Dynamic Validation of Control Model . . . . . . . . . . . . . . . . . . 61

4.1 Operating manifold [11] . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 Effect of valve timings on inducted gas composition . . . . . . . . . . 68

4.3 Block diagram of controller implementation . . . . . . . . . . . . . . 68

4.4 Simulation of tracking controller on 10-state model . . . . . . . . . . 69

4.5 Experimental results of closed loop control on research engine, dashed

line shows simulation result . . . . . . . . . . . . . . . . . . . . . . . 70

4.6 Experimental results of closed loop control on research engine . . . . 72

5.1 Quadratic level set Vquad(x) = 0.21 with vector plot (only direction

shown): shaded region is typical operation region . . . . . . . . . . . 83

xvi

5.2 Quartic level set, Vquart(x) = 0.0175, with vector plot (only direction

shown): shaded region is typical operation region . . . . . . . . . . . 84

6.1 Control Strategy for Simultaneous Decoupled Control of Peak Pressure

and Combustion Timing . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2 General control configuration considered for the synthesis of the in-

cylinder peak pressure controller . . . . . . . . . . . . . . . . . . . . . 91

6.3 The frequency dependent weights used for synthesis of the peak pres-

sure control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.4 Effect of valve timing on inducted gas composition . . . . . . . . . . . 94

6.5 Comparison of system response with combustion timing control only

(a) and both combustion timing and peak pressure control simultane-

ously (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.6 Simultaneous control of both peak pressure and combustion timing . 96

6.7 Direct control of work output, (a) - step response, (b) - step followed

by a sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7.1 Control Strategy for Simultaneous Coordinated Control of Peak Pres-

sure and Combustion Timing . . . . . . . . . . . . . . . . . . . . . . 100

7.2 The frequency dependent weights used for synthesis of the H2 controller103

7.3 Experimental control result showing negative step change in peak pres-

sure with constant combustion timing . . . . . . . . . . . . . . . . . . 107

7.4 Experimental control results showing positive step change in peak pres-

sure with constant combustion timing . . . . . . . . . . . . . . . . . . 108

7.5 Experimental control result showing simultaneous changes in combus-

tion timing and peak pressure . . . . . . . . . . . . . . . . . . . . . . 109

7.6 Experimental control result showing simultaneous changes in combus-

tion timing and peak pressure . . . . . . . . . . . . . . . . . . . . . . 110

7.7 Zoomed view of Figure 7.5 . . . . . . . . . . . . . . . . . . . . . . . . 111

xvii

A.1 Comparison of two heat transfer models for usage in the control model

dynamics, simple model - Equation A.2, physically-motivated - Equa-

tion A.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

xviii

Chapter 1

Introduction

1.1 Context - Environmental Challenges and New

Technologies

Human civilization is facing significant socioeconomic and environmental challenges

in the next 100 years. Despite major improvements in emission-reduction technology,

the production and release of smog generating chemicals, such as oxides of nitrogen

(NOx), are still major issues as the number of people and their energy needs continue

to increase. Additionally, as fuel costs reach new levels, the economic repercussions

of ‘business as usual’ could be devastating. There are 200 million vehicles on the road

in the United States alone, resulting in 600 billion gallons of fuel being consumed

each year [21]. With annual growth rates of light-duty sales and miles driven at 0.8%

and 0.5% [21], there is little evidence that emissions reduction will occur through a

decrease in the amount of vehicle use. There is a substantial opportunity to reduce

the amount of carbon and smog-generating chemicals released into the atmosphere

by concentrating on cleaner and more efficient transportation strategies.

There are several options for realizing more efficient and less polluting trans-

portation powerplants, including continued advancement in internal combustion (IC)

strategies, hybrid powertrains and fuel cells. Advanced IC engines and hybrid power-

trains are more practical today than fuel cells and will likely continue to be so for the

1

CHAPTER 1. INTRODUCTION 2

forseeable future. Among advanced IC engine strategies the most promising is Homo-

geneous Charge Compression Ignition (HCCI), the compression-induced autoignition

of a uniform fuel and air mixture.

1.1.1 Homogeneous Charge Compression Ignition

HCCI is an approach for increasing efficiency and reducing NOx emissions in internal

combustion engines. Improvements in efficiency of up to 15 to 20% compared to a

conventional Spark Ignited (SI) engine are possible [58], making HCCI efficiencies

comparable to diesel engines. Unlike diesel combustion, however, the lack of fuel

rich regions in HCCI results in little or no particulate emissions, a common issue

with diesel strategies. Furthermore, combustion of a homogeneous reactant mixture

during HCCI leads to a reduction in the peak combustion temperature, lowering NOx

levels compared to conventional SI and diesel strategies.

One effective strategy for achieving HCCI is through the reinduction [11, 32] or

trapping [32] of residual exhaust gas via variable valve actuation (VVA). In this

thesis, the methodology of using residual gas is called residual-affected HCCI. The

key processes in residual-affected HCCI are depicted in Figure 1.1.

Figure 1.1: Key Processes in Residual-Affected HCCI

Residual-affected HCCI via exhaust reinduction is achieved by using the flexible

valve system to hold the intake and exhaust valves open during a portion of the intake

stroke. This leads to the induction of both reactant (fuel and air) and residual (previ-

ously exhausted combustion products) gases from the intake and exhaust manifolds,

respectively. Residual-affected HCCI can also be achieved by retaining some exhaust


gas in the cylinder by closing the exhaust valve early during the exhaust stroke. Again,

this is made possible with a flexible valve system. The specific amounts of reactant

and residual are varied through modulation of the intake and exhaust valves. Follow-

ing the induction process, the compression of the reactant/residual mixture results

in increases of both the in-cylinder mixture concentrations and temperature. If the

reactant concentration and temperature reach sufficient levels, a uniform autoignition

process occurs. A key characteristic of the autoignition process is that there is no

direct trigger for its initiation. In the conventional SI and diesel strategies the onset

of the combustion event is triggered with the application of a spark or fuel injection,

respectively. In HCCI the combustion process has no direct combustion initiator. If

autoignition occurs, the conversion of reactants to combustion products during the

combustion event elevate the in-cylinder gas pressure and temperature. During the

expansion stroke this elevated pressure is used to effectively push the piston, resulting

in the extraction of useful work. The expansion stroke then expels the hot combus-

tion products into the exhaust manifold. Unlike conventional strategies, a portion of

the exhausted gas is then reinducted or trapped for use during the subsequent engine

cycle. It is this reinduction/trapping process that couples engine cycles through the

exhaust gas temperature.

Even though other methods exist for achieving HCCI, including intake air pre-

heating or pre-compression [53, 35, 43, 12] or some combination of methods [5, 24],

the focus of this thesis is residual-affected HCCI. There are at least three reasons for

focusing efforts on residual-affected HCCI:

1. Using flexible valve actuation to achieve HCCI is more feasible than pre-heating.

2. Residual-affected HCCI can be achieved with lower peak in-cylinder pressures

than pre-compression strategies, reducing the strength requirements of the en-

gine.

3. No throttling is required to modulate the work output in residual-affected HCCI

achieved via exhaust reinduction. This increases the efficiency of the HCCI

process.


1.1.2 Challenges with Making HCCI Practical

Regardless of the method chosen, HCCI combustion exhibits some fundamental chal-

lenges with regards to combustion timing and dilution limits. Additionally, for

residual-affected approaches, cycle-to-cycle coupling through the exhaust gas tem-

perature plays a critical role.

Combustion timing challenges

As noted previouly, HCCI has no specific initiator of combustion. Ensuring that

combustion occurs with acceptable timing, or at all, is more complicated than in the

case of either SI or diesel combustion. Combustion timing in HCCI is dominated by

chemical kinetics, which depends on the in-cylinder concentrations of reactants and

products, their temperature and the amount of compression.

Cycle-to-cycle coupling challenges

When HCCI is achieved via trapped or reinducted residual gases, subsequent engine

cycles are coupled through the residual temperature. Since the inducted reactant gas

is heated by the residual, the residual temperature from an engine cycle directly affects

the chemical kinetic-dominated combustion event on the subsequent cycle. The cyclic

coupling plays a fundamental role in steady state operation and during operating point

changes. If care is not taken during transient and mode transitions, combustion timing

can become unstable, leading to misfire, an unacceptable condition.

Dilution limit challenges

For any practical HCCI strategy, the reactants are diluted with either residual gas or

air. This dilution leads to the presence of upper and lower load limits. Furthermore,

the dilution decreases the amount of work that can be extracted for a given engine

geometry. For these reasons, practical HCCI will be accompanied with either con-

ventional SI or diesel strategies in a multi-mode engine. At very low and high load

conditions the engine will run in the conventional mode. At low to moderate load

conditions the engine will run in HCCI mode. A key issue is how to transition from


the conventional mode to HCCI. For residual-affected strategies, the cyclic coupling

exists during transitions into the HCCI mode. In fact, due to the higher exhaust tem-

peratures associated with the conventional modes, the dynamics of a mode transition

into HCCI are even more dramatic than intra-HCCI operating point changes.

To control HCCI through VVA it is essential to understand how the valves in-

fluence the inducted gas composition, combustion timing and cycle-to-cycle coupling

during steady state, transients and mode transitions.

1.2 Related Work

The application of HCCI to internal combustion engines has been studied since 1979

when it was concurrently applied to two-stroke engines by Onishi et al. [44] and

Noguchi et al. [39] through use of hot residual gas retained in the cylinder. Since

then a large number of studies of HCCI from a experimental and modeling point of

view have been completed. More recently, control of the process has become another

focus of the research community.

1.2.1 HCCI Research

In the very earliest experimental work of [44] several of the key characteristics of

residual-affected HCCI were identified, including the importance of achieving neces-

sary levels of hot residual to achieve autoignition, uniform mixing between residual

and reactant, and repeatable presence of residual gases on a cycle-to-cycle basis. In

both [44] and [39] the improvements in efficiency and emissions reduction were well

identified. The work of Najt et al. [37] outlines the first use of HCCI in a 4-stroke

engine. Furthermore this work outlines a physical explanation of the process, utiliz-

ing a simple single-zone model of the process. Building on the work of [37], Thring

et al. [52] examined the use of a gasoline fuel in an HCCI engine. In these early

works it was pointed out that HCCI exhibits some fundamental challenges, including

a part-load limitation and lack of a direct combustion initiator. Since these early


studies, significant efforts have been dedicated to HCCI research.

1.2.2 HCCI Modeling

To understand HCCI, the dependence of the combustion process on the gas exchange,

the composition of inducted gases, the level of mixedness and heat transfer must be

well understood. The development and validation of models facilitates this under-

standing. A wide variety of modeling approaches have been considered, including

from least to most complex:

1. zero-dimensional thermo-kinetic

2. quasi-dimensional thermo-kinetic

3. segregated, sequential fluid mechanics - thermo-kinetic multizone approaches

4. multidimensional fluid mechanics with coupled kinetics

Each approach presents its own set of advantages and disadvantages. The implemen-

tation of one approach over another depends on the aim of the user.

Zero-dimensional thermo-kinetic models

Models of this type utilize a single-zone approach to modeling the in-cylinder gases.

There are a large number of efforts in this area [49, 13, 1, 30, 57, 56, 34, 14, 19,

16, 22, 41]. The first law of thermodynamics is applied to a homogeneous mixture

of in-cylinder gases. The effects of the fluid mechanics are not directly considered

except, in some cases, when deriving relevant heat transfer coefficients. The com-

position of the gases are either determined with standard valve flow relations or

through approximation from steady state experiment. In this approach, the largest

computational cost is associated with the chemical kinetics model, which can vary

from a very simple temperature threshold to detailed kinetics utilizing hundreds of

rate equations. The benefits of the zero-dimensional approach follow from the ho-

mogeneity assumption of the in-cylinder gases, resulting in a dramatic reduction in

complexity and computational costs relative to more complex modeling approaches.


Zero-dimensional thermo-kinetic models can capture main HCCI engine outputs, in-

cluding work output, combustion timing and peak in-cylinder pressures. However,

due to the homogeneity assumption, near wall/piston quenching effects are difficult

to capture, leading to inaccuracies predicting emissions and completeness of com-

bustion. Additionally these approaches can not capture the mixing process during

induction.

Quasi-dimensional thermo-kinetic models

This modeling approach [38, 15, 17, 28] builds on the zero-dimensional modeling ap-

proach by considering multiple zones in the cylinder. Using this methodology the ef-

fects of temperature stratification and near wall/crevice quenching can be considered.

This leads to improvements in emissions prediction compared to the zero-dimensional

approaches, albeit at the expense of added modeling complexity and computation.

Segregated, sequential fluid mechanical - thermo-kinetic multi-zone ap-

proaches

These approaches [2, 6] attempt to more tightly couple the mixing process prior to

combustion and the chemical kinetics of the autoigntion process. The distribution

of reactant and diluting gas is modeled with a fluid mechanics solver. Prior to the

combustion event the gases are sequestered into a number of zones. The combustion

process is then carried out using a multi-zone combustion approach like that used

in the quasi-dimensional models. This approach allows the mixing process during

induction to be modeled, so that the effects of inhomogeneity on the autoigntion

process can be explicitly captured.

Multidimensional fluid mechanics with coupled kinetics

In this approach [36, 31, 3, 27, 23], by far the most complex and computationally

intensive, an attempt is made to completely couple the fluid mechanics and chemical

kinetics in three dimensions. In this case the fluid mechanics and chemical kinetics

solvers are run in parallel so that the effect of the combustion process on the fluid


motion, and vice versa, can be explicitly captured. This approach allows more accu-

rate representation of composition and temperature inhomogeneities, in some cases

leading to more accurate predictions of NOx and soot formation.

During the last 10 years substantial progress has been made in HCCI modeling.

With a variety of approaches a large number of important engine characteristics have

been captured, including combustion timing, peak in-cylinder pressure, work output,

maximum rate of pressure rise, exhaust gas temperature, emissions and extent of

combustion. To date, however, the dynamic cycle-to-cycle coupling via exhaust gas

temperature that exists with residual-affected strategies has not been considered in

a modeling strategy. For residual-affected HCCI this coupling plays a fundamental

role in steady state operation, during operating point changes, and across SI-to-HCCI

mode transitions. The dynamic nature of the cycle-to-cycle coupling also has critical

implications for controlling the process, since the control inputs depend not only on

the desired engine behavior for the current engine cycle, but also on what occurred

during the previous cycle. For this reason, the modeling of the cyclic coupling is a

central focus in this thesis.

1.2.3 HCCI Control

In a number of studies, closed-loop control has been utilized to fix combustion tim-

ing. Several approaches have been demonstrated [5, 20, 43, 9]. Agrell et al. [5] used

valve timings to effectively alter the compression ratio and control combustion tim-

ing. Haraldsson et al. [20] modulated the fuel amount to vary the work output while

altering the mixture ratio of two fuels to control combustion timing, a timing control

strategy also adopted by Bengtsson et al. [9]. Olsson et al. [43] took a similar ap-

proach but used compression ratio instead of fuel mixture to shift combustion timing.

These studies indicate the usefulness of effective compression ratio and inducted gas

composition as control inputs for HCCI.

While all of these authors either used tuned controllers or synthesized a strategy

from a black-box model, the work outlined in this thesis demonstrates that HCCI


controllers can also be synthesized using physics-based approaches. A physics-based

control approach allows for a fundamental understanding of how control inputs affect

the dynamics of the HCCI process. Furthermore, the approach is easily generaliz-

able to other HCCI engines since model parameters are directly based on physical

quantities, such as the cylinder geometry and fuel used.

1.3 Thesis Contributions

In the thesis, a 10-state simulation model predicts the effects of the VVA system on

the HCCI combustion process. A single-zone model of the in-cylinder gases captures

the compression, combustion initiation, energy release and expansion processes. An

integrated Arrhenius rate describes the dependence of combustion timing on reactant

concentration, temperature and amount of compression. The in-cylinder dynamics

are coupled with a single-zone model of the exhaust manifold gases to predict the

cycle-to-cycle coupling through the exhaust gas temperature. The resulting model

agrees with experimental values of inlet reactant flow rate, combustion timing, in-

cylinder pressure evolution, work output and exhaust gas temperature. The dynamics

of operating point change and mode-transition dynamics are also captured.

To synthesize control strategies, a low-order physics-based control model is for-

mulated by discretizing the distinct processes that occur during HCCI, including

induction, compression, combustion and exhaust. From the control model several ap-

proaches for closed-loop control of peak in-cylinder pressure, combustion timing and

work output are outlined. The candidate approaches are first tested on the simulation

model and then implemented on a single-cylinder research engine.

The thesis contributions include:

• Residual-affected HCCI simulation model that:

– Captures the dependence of the system outputs (inlet reactant flow rates,

exhaust gas temperature, maximum rates of pressure rise, combustion tim-

ing, peak in-cylinder pressure and work output) on the VVA-controllable

system inputs (valve opening and closing times)


– Captures the cycle-to-cycle coupling through exhaust gas temperature

– Captures ignition via kinetics with a simple, intuitive model that captures

the effects of reactant concentration, temperature and amount of compres-

sion

– Shows that the dynamics of mode transitions into HCCI are a simple ex-

tension of transient HCCI dynamics

– Exhibits run times at around 15 seconds per engine cycle (more amenable

to use as a control testbed than more complex modeling approaches)

• Physics-based control model of residual-affected HCCI that:

– Has two inputs (inducted gas composition and effective compression ratio),

two-states (combustion timing and peak pressure) and one output (work

output)

– Captures the dependence of the system outputs (combustion timing, peak

in-cylinder pressure and work output) on the VVA-controllable system

inputs (inducted gas composition & amount of compression)

– Captures the cycle-to-cycle coupling through exhaust gas temperature

– Captures ignition via kinetics with a simple, intuitive model that captures

the effects of reactant concentration, temperature and amount of compres-

sion

– Is simple enough to be directly used in the synthesis of control strategies

– Has been validated against the simulation model

• Several control approaches synthesized from the control model and implemented

in experiment:

– Closed-loop control of peak pressure (or work output) with constant com-

bustion timing

– Decoupled closed-loop control of peak pressure (or work output) and com-

bustion timing


– Simultaneous coordinated control of peak pressure (or work output) and

combustion timing

1.4 Thesis Outline

The material in this thesis is organized into eight chapters.

Chapter 2 formulates a simulation model of the residual-affected HCCI process

and shows that although HCCI represents a complex physical process, the aspects

most relevant for control - pressure evolution, combustion timing, work output, max-

imum rate of pressure rise and exhaust gas temperature - can be captured with rela-

tively simple models. By combining a simple ignition model, the integrated Arrhenius

rate, with single-zone first law analysis of the in-cylinder and exhaust manifold gases,

the steady state, transient and mode transition behaviors of residual-affected HCCI

are captured. The simulation model development also provides valuable insights for

the formulation of control strategies - inducted gas composition can be varied via

modulation of the valves, residual-affected HCCI exhibits a self-stabilizing behavior

due to the competing influences of mixture temperature and reactant concentration,

and cyclic coupling is inherent to the process and must be included. Additionally, the

simulation model provides an excellent virtual testbed for analyzing feedback control

strategies.

Chapter 3 Using insights gained from the simulation modeling, a nonlinear

control-oriented model with inducted gas composition and effective compression ra-

tio as inputs and peak pressure and combustion timing as outputs is developed by

discretizing the HCCI process into six distinct stages: induction, compression, com-

bustion, expansion, exhaust and residence in the exhaust manifold. This control

model is the launching point for the development of the control strategies outlined in

following chapters, resulting in the first generalizable, validated and experimentally

implemented controls-oriented modeling approach of residual-affected HCCI engines.

Chapter 4 Based on the insights developed in the preceding chapters, the first

control approach, outlined in this chapter, relies on the ability to vary the inducted

gas composition with the VVA system and the existence of an operating manifold with


nearly constant combustion timing. The peak pressure portion of the control model,

outlined in Chapter 3, is used to synthesize a strategy capable of cycle-to-cycle control

of peak pressure through modulation of the inducted gas composition. Specifically

a linear control law is synthesized from a linearized version of the nonlinear peak

pressure dynamics. The self-stabilizing nature of the process is used to maintain

nearly constant combustion timing without direct control of the timing.

Chapter 5 This chapter examines the stability of the linear controller utilized in

Chapter 4 in closed-loop with the full nonlinear peak pressure dynamics. Specifically,

a Lyapunov-based analysis utilizing sum of squares decomposition and a theorem

from real algebraic geometry, the Positivstellensatz, is used to estimate the domain of

attraction for the nonlinear system with the linear control law. The resulting region

of attraction proves stability of the system over the desired portion of the state space.

Physically, this means that the linear control stabilizes the nonlinear system over the

desired operating range of HCCI.

Chapter 6 examines a natural next step, the simultaneous control of peak pres-

sure and combustion timing. The approach outlined in this chapter approximately

decouples the cycle-to-cycle dynamics of combustion timing and peak in-cylinder

pressure by controlling them on separate time scales with different control inputs -

inducted composition and effective compression ratio, respectively. A physics-based

H2 framework is used to determine a linear control law. Timing controller gains are

selected via pole placement to achieve a response time that is slightly slower than the

pressure controller.

Chapter 7 outlines the most complete control approach, allowing the simul-

taneous, coordinated control of combustion timing and peak pressure on the same

time scale through modulation of inducted gas composition and effective compres-

sion ratio. The controller used is directly synthesized from a linearized version of

the complete control model developed in Chapter 3. The approach represents the

most capable approach to control of residual-affected HCCI presented in the thesis.

Tracking responses for combustion timing and peak pressure occur within 4-5 engine

cycles. Additionally, a reduction in control effort is realized due to the coordinated

modulation of the control inputs.


Chapter 8 summarizes the results of the thesis and discusses the direction of

continuing work.

.

Chapter 2

Simulation Model

The aim of this chapter is the development of an accurate, simple and intuitive model

of the residual-affected HCCI process. The goal is to be able to capture the response

of the system outputs most relevant to the control problem:

• inlet flow rates

• combustion timing

• cycle-to-cycle coupling

• in-cylinder pressure evolution

• work output

• exhaust gas temperature

to system inputs:

• the VVA-modulated intake and exhaust valve motions.

The model is validated against experimental results during steady-state conditions,

transients and SI-to-HCCI mode transitions.

Specifically, the chapter outlines an overall model for residual-affected HCCI uti-

lizing a premixed reactant at fixed equivalence ratio. The model is based on a first-law

thermodynamic analysis of the cylinder and exhaust manifold and compressible flow

14

CHAPTER 2. SIMULATION MODEL 15

expressions for gas exchange. Since a major goal of this work is to capture the HCCI

combustion process in a simple and intuitive way, this chapter compares three simple

submodels for combustion initiation: a simple temperature threshold, an integrated

Arrhenius rate expression, and the knock integral. The Arrhenius expression, which

relates the in-cylinder temperature, reactant concentrations and the amount compres-

sion to the start of combustion, matches experimental data for propane and gasoline

combustion quite closely. In comparison, the other methods, which do not reflect

concentration changes, exhibit considerable deviation in predictions of combustion

phasing. Subsequent validation during transients and SI-to-HCCI mode transitions

shows that the combined model is capable of reproducing the cycle-to-cycle dynamics

of VVA-induced HCCI and serving as a basis for controller design and validation.

Furthermore, the process of developing the simulation model makes clear several key

insights for developing control strategies. These include the ability to vary the in-

ducted gas composition and amount of compression via VVA on a cycle-to-cycle basis,

the self-stabilizing nature of residual-affected HCCI due to the competing influences

of reactant concentrations and mixture temperature, and the fundamental role that

cycle-to-cycle coupling plays during steady state, transients and mode transitions.

2.1 Modeling Approach

The modeling is based on an open system first law analysis, with steady state com-

pressible flow relations used to model the mass flow through the intake and exhaust

valves. The model includes ten states: the crank angle, θ; the cylinder volume, V;

the temperature, T; the concentrations of the fuel used, [fuel], oxygen, [O2], Nitrogen,

[N2], carbon dioxide, [CO2], water, [H2O]; the mass in the exhaust manifold, me; and

the internal energy of the product gases in the exhaust, ue.


2.1.1 Volume Rate Equation

The in-cylinder volume and its derivative are given by the slider-crank equations:

V = Vc +πB2

cyl

4

(Lcyl + acyl − acylcosθ −

√L2

cyl − a2cylsin

2θ)

(2.1)

V =π

4B2

cylacylθsinθ

1 + acyl

cosθ√(L2

cyl − a2cylsin

2θ)

(2.2)

where:

ω = θ (2.3)

is the rotational speed of the crankshaft, acyl is half of the stroke length, Lcyl is the

connecting rod length, Bcyl is the bore diameter and Vc is the clearance volume at

top dead center.

2.1.2 Valve Flow Equations

The mass flow through the valves consists of flow from intake manifold to cylinder,

mic, from cylinder to exhaust manifold, mce, and from exhaust manifold to cylinder,

mec, as shown in Figure 2.1. Figure 2.2 also shows the general shape of the valve lift

profiles used for the experimental validation of the simulation model.

Equations for the mass flow rates are developed using a compressible, steady state,

quasi-one-dimensional, isentropic flow analysis for a restriction, where real gas flow

effects are included by means of a discharge coefficient, CD. The relations for the

mass flows are:

m =CDARpo√

RTo

(pT

po

)1/γ[

2γ

γ − 1

[1−

(pT

po

)(γ−1)/γ]]1/2

(2.4)

for subsonic flow (pT /po > [2/(γ + 1)]γ/(γ−1)), and:

m =CDARpo√

RTo

√γ

[2

γ + 1

](γ+1)/2(γ−1)

(2.5)


INTAKE

VALVE

EXHAUST

VALVE

mic

.m

ec

EXHAUST

VALVE

INTAKE

VALVE

. mce

.

Figure 2.1: Valve Mass Flows: left - induction flows with intake and exhaust valvesopen, right - exhaust flow

for choked flow (pT /po≤[2/(γ + 1)]γ/(γ−1)), where AR is the effective open area for

the valve, po is the upstream stagnation pressure, To is the downstream stagnation

temperature and pT is the downstream stagnation pressure.

For the mass flow of the reactant gas into the cylinder through the intake valve,

mic, po is the intake manifold pressure, assumed to be atmospheric, and pT is the

cylinder pressure, p. For the mass flow of burnt products out of the cylinder through

the exhaust valve, mce, po is the cylinder pressure, p, and pT is the exhaust manifold

pressure, assumed to be atmospheric. For the reinducted exhaust from the previous

cycle through the exhaust valve, mec, po is the exhaust manifold pressure, and pT is

the cylinder pressure, p. Note that manifold pressure dynamics due to valve timing

transients [18] are not included, but could be implemented if desired. Additionally,

note that it is assumed that there is no flow from cylinder to intake manifold. This is

a reasonable assumption for the experimental system studied in this paper. However,

allowing flow from cylinder to intake manifold would be simple to include in the model

if necessary.


0 100 200 300 400 500 600 700

Va

lve

Lift

[cm

]

Crankshaft ˚

Intake Valve

Exhaust Valve

IVO EVC

IVC=θ1 EVO=θ4

Figure 2.2: Valve profiles used for residual-affected HCCI using variable valve actua-tion

2.1.3 Species Concentration Rate Equations

The rate of change of concentration for species i, [Xi], is related to number of moles

of species i in the cylinder, Ni, by:

˙[Xi] =d

dt

(Ni

V

)=

Ni

V− V Ni

V 2= wi − V Ni

V 2(2.6)

where wi, the rate of change of moles of species i per unit volume has been defined

as:

wi =Ni

V(2.7)

It has two contributions: the rate of change of moles of species i per unit volume

due to the combustion reactions, wrxn,i, and due to flow through the valves under the

control of the VVA system, wvalves,i, such that:

wi = wrxn,i + wvalves,i (2.8)


The combustion reaction rate, wrxn,i, is determined through the use of a combustion

chemistry mechanism. The three combustion chemistry approaches considered are

outlined in Section 2.2.

Given the mass flow rates (mic,i, mec,i and mce,i) from the analysis in Section 2.1.2,

the rate of change of moles of species i per unit volume due to flow through the valves,

wvalves,i, can be found using the species mass fractions:

wvalves,i = wic,i + wec,i − wce,i (2.9)

where:

wic,i =Yi,imic

V MWi

(2.10)

wec,i =Ye,imec

V MWi

(2.11)

wce,i =Yc,imce

V MWi

(2.12)

Here Yi,i, Ye,i and Yc,i are the mass fractions of species i in the inlet manifold, exhaust

manifold and cylinder, respectively. It is assumed that a lean or stoichiometric reac-

tant mixture with an equivalence ratio of φ is present in the intake manifold. Further,

it is assumed that only the major combustion products of CO2, H2O, N2 and O2 (for

the lean case) are reinducted into the cylinder through the exhaust. Therefore Yi,i

and Ye,i are constant. Note that other intake and exhaust manifold compositions can

be considered, but in any case the manifold mass fractions are constant during an

engine cycle. However, the mass fraction of species i in the cylinder, Yc,i, is constantly

changing, and can be related to the concentration states as:

Yc,i =[Xi]MWi∑[Xi]MWi

(2.13)


2.1.4 Temperature Rate Equations

In order to derive a differential equation for the temperature of the gas inside the

cylinder, the first law of thermodynamics for an open system and the ideal gas law

are combined as outlined below. The first law of thermodynamics for the cylinder is:

d(mcuc)

dt= Qc − Wc + michi + meche − mcehc (2.14)

where mc is the mass of species in the cylinder, uc is the in-cylinder internal energy,

Qc is the heat transfer rate into the cylinder, Wc = pV is the work output rate, hi

is the enthalpy of species in the intake manifold, he is the enthalpy of species in the

exhaust manifold, and hc is the enthalpy of the species in the cylinder. The convective

heat transfer rate is modeled as:

Qc = −hcAs(T − Twall) (2.15)

where As is the in-cylinder surface area and Twall is the average cylinder wall tem-

perature. The convection coefficient, hc, is modeled using the Woshni heat transfer

correlation [50], so that:

hc = 194.7p0.8(C1Vp)0.8B−0.2

cyl T−0.55 (2.16)

where Vp is the mean piston velocity, and C1 = 2.28. The wall temperature is approx-

imated as 400K, a common assumption for evaporativly cooled engines [50]. Now,

given that the enthalpy is related to the internal energy as:

hc = uc + pV/mc (2.17)

Equations 2.14 and 2.17 can be combined to yield:

d(mchc)

dt= Qc + pV + michi + meche − mcehc (2.18)


Expanding the enthalpy to show the contributions of the species in the cylinder yields:

mchc = Hc =∑

Nihc,i (2.19)

where Ni is the number of moles of species i in the cylinder, Hc is the total enthalpy

of species in the cylinder, and hc,i is the enthalpy of species i in the cylinder on a

molar basis. Noting that the rate of change of enthalpy per unit mole of species i can

be represented as˙hc,i = cp,i(T )T , where cp,i(T ) is the constant pressure specific heat

of species i per mole at temperature T , Equations 2.19 and 2.18 can be combined to

give:d(mchc)

dt= V

(∑˙[Xi]hc,i + T

∑[Xi]cp,i(T )

)+ V

∑[Xi]hc,i (2.20)

In-cylinder pressure and its derivative can be related to the concentrations and

temperature through the ideal gas law as:

p =∑

[Xi]RT (2.21)

p =p∑ ˙[Xi]∑

[Xi]+

pT

T(2.22)

The in-cylinder mass and its derivative may be related to the species concentrations,

molecular weights and volume as:

mc = V∑

[Xi]MWi (2.23)

mc = V∑

[Xi]MWi + V∑

˙[Xi]MWi (2.24)

Equating the right sides of Equations 2.18 and 2.20, substituting Equations 2.22,

2.23, and 2.24, and rearranging yields a differential equation for temperature:

T =Q− V

∑ ˙[Xi]hc,i − V∑

[Xi]hc,i + RTV∑ ˙[Xi] +

∑mh

V (∑

[Xi]cp,i(T )−R∑

[Xi])(2.25)


where:

∑mh = michi + meche − mcehc (2.26)

This completes the thermodynamic model of the engine cylinder.

2.1.5 Exhaust Manifold Modeling

The exhaust manifold model attempts to capture the thermodynamic properties of

the reinducted exhaust gas by following the evolution of a variable amount of mass

in the manifold. The relevant mass at any time includes the exhaust from the most

recent cycle and a small residual, mres, from the previous exhaust cycle. Retain-

ing this amount of mass from the previous cycle allows the internal energy to exist

continuously and provides a means of modeling various amounts of mixing between

cycles.

Figure 2.3 shows the progression of mass in the exhaust manifold model. When the

exhaust valve opens, flow through the valve increases the mass in the manifold until

the piston reaches top dead center and reinduction begins. Reinduction similarly

decreases the mass in the manifold until the point where the exhaust valve closes.

After this point, the boundary defining the control volume smoothly resets to the

exhaust manifold residual mass (as reflected by Equation 2.29), reflecting the fact

that combustion products tend to flow away from the valve and exert less influence

on the next reinduction. Mathematically, this model can be described by:

EV O < θ < 720 : me = mce (2.27)

0 < θ < EV C : me = −mec (2.28)

EV C < θ < EV O : me = −me,EV C −me,res

EV O − EV Cω (2.29)


(a)

(b)

(c)

(d)

(e)

(f)

mce

.

mec

.

me,max

me,res

me,EVC-me,res

EVO-EVCω

me,EVC

Figure 2.3: Schematic of exhaust manifold control mass: (a) residual mass fromprevious exhaust cycle, θ = EV O; (b) increase in mass due to cylinder exhaust,EV O < θ < 720; (c) maximum amount of exhaust manifold mass, θ = 720; (d)decrease in mass due to reinduction, 0 < θ < EV C; (e) post-reinduction mass,θ = EV C; (f) decrease in mass to residual value, EV C < θ < EV O

In order to obtain a governing expression for the exhaust manifold internal en-

ergy, ue, the first law of thermodynamics is applied in conjunction with the ideal gas

assumption and a simple convective heat transfer model. The first law of thermody-

namics for product gases in the exhaust manifold is:

d (meue)

dt= Qe − We + mcehc − meche (2.30)

where ue is the internal energy of product gases in the manifold, Qe is the manifold

heat transfer rate and

We = patmVe (2.31)

is the boundary work for the control mass. The exhaust volume is related to the mass


through the ideal gas assumption:

Ve =meRTe

MWe patm

(2.32)

and MWe is the molecular weight of the major products of combustion. The convec-

tive heat transfer model for the manifold is:

Qe = −heAe (Te − Tambient) (2.33)

where he is the convection coefficient of exhaust over the area Ae. This effective area

is taken to be the cylindrical surface area of the exhaust pipe used in the experiment.

With a diameter, De, the heat transfer area can be related to the exhaust volume, as

Ae = 4Ve/De. The temperature of the exhaust is a function of the internal energy,

given that pressure is assumed to be constant at one atmosphere.

Te = f (ue| patm) (2.34)

The enthalpy of the exhaust can be expressed as a function of the internal energy and

temperature:

he = ue + RTe (2.35)

Combining Equations 2.30, 2.32, 2.33, 2.34 and 2.35, a governing equation for the

internal energy of the gases in the exhaust manifold can be expressed as:

ue =1

meγ

[mce (hc − he) + heAe (Tambient − Te)

](2.36)

This completes the modeling of the exhaust manifold dynamics.


2.2 Combustion Chemistry Modeling

The final step necessary to complete the HCCI model is to define the species of

interest and specify the reaction rate terms, wrxn,i, in Equation 2.8. While the model

to this point has been independent of the particular fuel chosen, the fuel species

and reaction rates are clearly specific to the fuel. Although the development in the

following section uses the example of propane to enable a straightforward comparison

with experimental data on an existing test stand, the general modeling approach can

handle a variety of fuels. Changes necessary to model other fuels and experimental

evidence that more complex fuels can be described with similar models are discussed

in Section 2.4.

From a control standpoint, the combustion model must accurately capture the

pressure evolution in the cylinder and, most importantly, the timing of combustion.

The following sections present three different approaches to modeling the onset of

combustion in order to establish the level of modeling necessary to reproduce exper-

imentally observed behavior: a temperature threshold, an integrated Arrhenius rate

model and a knock integral. In each of these approaches, stoichiometric (φ = 1)

and lean (φ < 1) reactions of propane and air are considered. Since HCCI is a lean

strategy, rich mixtures are not considered. With the assumption of major products,

the global reaction for combustion in each modeling approach is:

φC3H8 + 5O2 + 18.8N2 → 3φCO2 + 4φH2O + 5(1− φ)O2 + 18.8N2 (2.37)

For each of the three approaches, the simulation can be compared to the results

from a single-cylinder engine testbed, Figure 2.4, with VVA (characteristics given

in Table 2.1). In [11] the valving strategy shown in Figure 2.2 was used to study

HCCI via exhaust reinduction on the engine. Figure 2.5 shows the experimental

load, emissions and efficiency characteristics of the engine for a variety of intake valve


parameter symbol value unitsequivalence ratio φ 0.93 —

engine speed ω 1800 rpmstroke acyl 9.2 cm

connecting rod length Lcyl 25.4 cmbore diameter Bcyl 9.7 cm

compression ratio 13 —valve diameter, intake Ar,i 4.8 cm

valve diameter, exhaust Ar,e 4.3 cmexhaust manifold diameter De 5 cmvalve rise/fall durations 90 CAD

intake valve closing IVC 210 CADexhaust valve opening EVO 480 CAD

Table 2.1: Engine Parameters

opening (IVO) and exhaust valve closing (EVC) times. A key observation in [11]

was the existence of an operating trajectory, shown with solid points in Figure 2.5,

which exhibits the upper and lower ranges of efficiency and emissions, respectively.

Simulation results are compared to the experiment along this operating trajectory

to validate the modeling approach. The parameters used in each of the modeling

approaches are given in Table 2.2. Experimental pressure-crank angle diagrams were

averaged to eliminate the cycle-to-cycle dispersion inherent in IC engine combustion

processes. At this point no process noise is added into the model to simulate the cyclic

dispersion, however, nothing in the modeling strategy precludes the introduction of

such a noise model.

2.2.1 Temperature Threshold Approach

The simplest approach to modeling HCCI combustion is to assume the combustion

reactions start once the in-cylinder temperature reaches a critical value. From this

point forward, the rate of reaction of the propane is approximated with a Wiebe

function [50], such that:


Figure 2.4: Single Cylinder Research Engine Outfitted with VVA

T ≥ Tth : wC3H8 =[C3H8]i Vi θ a (m + 1)

(θ−θi

∆θ

)m

V ∆θ exp[a

(θ−θi

∆θ

)m+1] (2.38)

T < Tth : wC3H8 = 0 (2.39)

where θi, Vi and [C3H8]i are the crank angle, volume and propane concentration,

respectively, at the point where combustion begins (i.e. where T = Tth). The duration

of combustion is ∆θ. The parameters a and m shape the Wiebe function. Note that


Figure 2.5: Operating manifold [11]

other functions could be used to model the reaction rate of propane as a function of

crank angle. The Wiebe approach is one of the most popular.

By inspection of Equation 2.37 the reaction rates of the other species follow di-

rectly:

wO2 = 5wC3H8 (2.40)

wN2 = 0 (2.41)

wCO2 = −3wC3H8 (2.42)

wH2O = −4wC3H8 (2.43)

Equations 2.2, 3.29, 2.25-2.29, 2.36 and 2.38-2.43 therefore represent the complete

set of nonlinear differential equations for the model when the temperature threshold

approach is used.

The constants used in Equation 2.38 are selected in order to correlate the simu-

lation with experiment at a valving condition of IVO/EVC = 25/165. Comparison


parameter symbol value units

A 8.6e11 (gmol/m3)−0.75

sEa 15098 K

Arrhenius rate parameters

ak 0.1 —(source: Turns)

bk 1.65 —a 1 —m 4.2 —

Wiebe function parameters

∆θ 12 CADtemperature threshold Tth 1000 K

knock integral threshold∫RRth 190

(gmolm3

)−0.75

int. Arrhenius rate threshold∫RRth 7.2e-6 gmol/cm3

coeff. of discharge, intake Cd,i 0.68 —coeff. of discharge, exhaust Cd,e 0.49 —

cyl. wall temperature Twall 400 Kcyl. wall heat transfer coeff. hc modeled W/m2K

ex. port heat transfer coeff. he 72 W/m2K

ex. manifold residual me,res 0.24 g

Table 2.2: Propane Fuel Simulation Parameters

of simulation (without changing any constants) and experiment at the IVO/EVC =

45/185 and 65/205 conditions (as shown in Figure 2.6 and Table 2.3) shows that

a single temperature threshold fails to capture combustion phasing at different op-

erating conditions. This is due to the fact that the initiation of the combustion

reaction depends not only on the temperature, but also on the concentration of reac-

tant (i.e. fuel and oxygen) species present in the cylinder. This dependence on both

temperature and reactant concentrations is especially important in the case of VVA

induced HCCI, where reinducted or trapped exhaust species from the previous cycle

both dilute and increase the temperature of the reactant species. While the decrease

in reactant concentration delays the onset of combustion, the increase in reactant

temperature advances the onset of combustion.


0

10

20

30

40

50

60

70

-80 -40 0 40 80-80 -40 0 40 80 -80 -40 0 40 80Crankshaft (o ATC)

Pre

ssu

re (

ba

r)

Figure 2.6: Temperature Threshold Approach: Simulated HCCI combustion duringsteady state: dashed - simulation, solid - experiment; left - IVO/EVC = 25/165,middle - IVO/EVC = 45/185, right - IVO/EVC = 65/205

2.2.2 Integrated Global Arrhenius Rate Threshold

The conclusion that both temperature and reactant concentration must be considered

in a model of combustion initiation motivates a new combustion trigger. In reality,

the combustion process consists of numerous reactions through which the reactants

are transformed to products. As a simplification, the model here assumes that the

combustion initiation point can be modeled with a single global reaction rate. Mathe-

matically, this involves integrating a single Arrhenius reaction rate expression similar

to those used for each individual reaction in models with more detailed chemistry,

e.g. [40]. This integrated reaction rate,∫

RR, takes the form:

∫RR =

∫ θ

IV O

AT nexp(−Ea/(RT ))[C3H8]ak [O2]

bk/ωdθ (2.44)

Once this integrated Arrhenius rate crosses a pre-set threshold, the rate of reaction

of propane proceeds according to the same Wiebe function used for the temperature


metric operating cond.,IVO/EVC25/165 45/185 65/205

exp 59.5 52.5 44.4peak pressure

sim. 61.2 60.6 54.9[bar]

% error 2.81 14.32 21.15

combustion phasing exp 12.2 11.4 11

(angle of pk. press.) sim. 13 7.4 4.6[CAD] % error 6.35 -42.55 -82.05

Table 2.3: Temperature Threshold Approach: Comparison of experiment and simu-lation

threshold approach, such that:

∫RR ≥

∫RRth : wC3H8 =

[C3H8]i Vi θ a(m + 1)(

θ−θi

∆θ

)m

V ∆θ exp[a

(θ−θi

∆θ

)m+1] (2.45)

∫RR <

∫RRth : wC3H8 = 0 (2.46)

The values A, Ea/R, ak, bk and n are empirical parameters determined from

propane combustion kinetics experiments [54] and thus are independent of the par-

ticular engine. The threshold value of the integrated reaction rate,∫

RRth, is set

empirically according to the value which most closely correlates with experiment at

the IVO/EVC = 25/165 operating condition. It is not changed at other operating

points. The replacement of the temperature threshold with the integrated Arrhenius

rate expression is thus the only change in the model.

Figure 2.7 shows that the integrated rate threshold approach predicts the com-

bustion phasing and peak pressure quite well. Instead of predicting that the phasing

shifts progressively earlier as the valve timings change, this model reproduces the more

consistent phasing of the experiment. The results have a clear physical interpretation.


metric operating cond.,IVO/EVC25/165 35/175 45/185 55/195 65/205

Average exp 234 219 198 178 156Inlet Airflow sim. 262 227 196 168 143[liters/min] % error 12.0 3.5 -1.0 -5.6 -8.3

exp 59.5 57.7 52.5 49.6 44.4peak pressuresim. 61.3 59.5 56.0 51.3 45.4

[bar] % error 3.0 3.1 6.7 3.4 2.3

combustion timing exp 12.2 11.6 11.4 11.6 11(angle of pk. press.) sim. 13.6 12.1 11.3 11.5 12.7

[CAD] % error 11.5 4.3 -0.9 -0.9 14.5exp 5.8 5.35 5.05 4.5 3.8IMEPsim. 6.33 5.47 4.69 3.97 3.31

[bar] % error 9.1 1.5 -7.1 -11.8 -12.9exp 5.2 5.1 4.6 3.6 3.5max. rate of risesim. 5.7 5.25 4.6 3.8 3.0

[bar/CAD] % error 9.6 2.9 0.0 5.6 -14.3

Average exp 697 666 655 644 638Exhaust Gas Temp. sim. 739 705 675 646 620

[K] % error 6.0 5.9 3.1 0.2 -2.8

Table 2.4: Integrated Arrhenius Rate Threshold Approach: Comparison of experi-ment and simulation


)

280 320 360 400 440

Crankshaft Position [Degrees] 280 320 360 400 440 280 320 360 400 440

10

20

30

40

Pre

ssure

[bar]

0

50

60

Figure 2.7: Integrated Arrhenius Rate Threshold Approach: Simulated HCCI com-bustion during steady state: dashed - simulation, solid - experiment; left - IVO/EVC= 25/165, middle - IVO/EVC = 45/185, right - IVO/EVC = 65/205

As the IVO/EVC timings shift progressively later, more products are reinducted from

the exhaust and fewer reactants are inducted through the intake. This raises the ini-

tial temperature, leading the temperature threshold model to predict earlier phasing.

However, the increase in products reduces the reactant concentration, increasing igni-

tion delay. The integrated Arrhenius rate correctly captures both of these competing

effects, predicting a relatively constant phasing.

Table 2.4 gives a direct comparison between experiment and simulation for average

inlet flow rate, peak pressure, combustion timing, IMEP, maximum rate of pressure

rise and average exhaust gas temperature for five different operating points. The

average inlet air flow rate, ¯V , is calculated by evaluating the pressure drop across

a laminar flow element located on the inlet section. Additionally, the exhaust tem-

perature is measured using a thermocouple. Although the model used is extremely

simple, the agreement with experiment demonstrated in Table 2.4 is comparable to

values achieved by more sophisticated models such as [40] and [7]. In fairness, more


complicated models provide a much deeper picture of the HCCI process, enabling

factors such as emissions and the effects of inhomogeneity to be considered. From a

control standpoint, however, nothing is lost by using the simple combustion model to

predict pressure evolution and phasing. Thus, this approach appears to present an

ideal combination of simplicity and accuracy.

2.2.3 Knock Integral Technique

The integrated Arrhenius rate threshold model bears a considerable resemblance to

integral threshold techniques used to predict engine knock (often called “Livengood-

Wu” integrals after [33]). In addition to this original use, knock integrals have been

used more recently as a simple model for HCCI ignition [4, 42]. The knock integral

basically has the same form as Equation 3.23 but with values of ak and bk set to 0.

Since this removes the dependence on reactant concentration, using the knock integral

as the trigger for HCCI combustion would appear to present the same problems as

using the temperature threshold.

As demonstrated by the results in Figure 2.8 and Table 2.5, this is indeed the case.

While a single value of the knock integral can be chosen for any particular operating

condition, no one value accurately predicts the combustion phasing when the amount

of inducted reactants is varied. This result agrees with previous remarks that the

threshold value of the knock integral must be changed to handle different operating

conditions of HCCI [4]. In contrast, the threshold level for the integrated Arrhenius

rate expression need only be fit for a single operating point. Once set, the predicted

impact of reactant concentrations, temperature and valve timing on combustion phas-

ing matches that described in previous experimental work [11]. Given the need for a

control model to predict transient dynamics over a range of conditions, the integrated

Arrhenius rate represents an intuitive, low-order approach to combustion modeling.


0

10

20

30

40

50

60

70

-80 -40 0 40 80-80 -40 0 40 80 -80 -40 0 40 80Crankshaft (o ATC)

Pre

ssu

re (

ba

r)

Figure 2.8: Knock Integral Threshold Approach: Simulated HCCI combustion duringsteady state: dashed - simulation, solid - experiment; left - IVO/EVC = 25/165,middle - IVO/EVC = 45/185, right - IVO/EVC = 65/205

2.3 Transients and Mode Transitions

The process of achieving HCCI by reinducting or trapping exhaust gas fundamentally

couples the behavior during a given engine cycle to the results of the previous cycle.

This coupling must be represented correctly in a model of the HCCI process centered

metric operating cond., IVO/EVC25/165 45/185 65/205

exp 59.5 52.5 44.4peak pressuresim. 60.7 59.4 55

[bar] % error 2.00 12.33 21.33

combustion phasing exp 12.2 11.4 11(angle of pk. press.) sim. 13 8.2 4.6

[CAD] % error 7.87 -32.65 -82.05

Table 2.5: Knock Integral Threshold Approach: Comparison of experiment and sim-ulation


on control since it dictates the dynamics through which valve timings influence com-

bustion. The modeling strategy outlined here captures these dynamics through the

exhaust manifold model and the integrated Arrhenius rate model for combustion ini-

tiation. A benefit of this modeling approach is the capability to capture the behavior

during changes in HCCI operating point and during mode transitions.

2.3.1 Validation in Transient Operation

0

10

20

30

40

50

60

0

10

20

30

40

50

60

Pre

ssu

re [

ba

r]

Cycle 1

Cycle 2

Cycle 6

Cycle 1

Cycle 2 Cycle 6

EVC 180 IVO 70EVC 185

IVO 50

Valve Profiles 1 2-6

Crankshaft Position [Degrees]

280 320 360 400 440420380340300 340

Figure 2.9: Simulated HCCI combustion over a valve timing change: top - experiment,bottom - simulation

Figure 2.9 shows simulation and experimental results of the in-cylinder pressure

during a step change in the valve timing. Cycle 1 corresponds to the steady state

solution at a valve timing of IVO/EVC= 50/180. A step change to a valve timing


of IVO/EVC= 70/185 is then made. Unlike the previous results, the pressure traces

are not averaged. Consequently, a rippling in the experimental pressure curves can

be seen on the expansion side due to the fact that the auto-ignition process is not

completely homogeneous. The simulation matches the experimental results within

the range of cyclic dispersion normally present on any particular steady-state engine

cycle.

The simulation also correctly predicts the advancing phenomenon present in the

experimental data. Following the step change in valve timings, the combustion event

occurs first earlier and then progressively later as the system converges toward a

steady state solution for the new valve timing. This effect is due to the cycle-to-cycle

coupling through the exhaust gas temperature. The exhaust temperature for the

steady state operating condition at the 70/185 timing is lower than that corresponding

to the 50/180 timing. The elevated temperature on the engine cycle following the

step change produces an acceleration of the combustion kinetics, leading to earlier

phased combustion. This effect is correctly predicted by the basic modeling approach

outlined in this paper.

2.3.2 Validation during an SI-to-HCCI Mode Transition

The dynamics of SI-to-HCCI mode transitions can also be captured using this mod-

eling approach. As it does during steady-state and transient HCCI, the exhaust gas

temperature plays a fundamental role during SI-to-HCCI mode transitions. In fact,

the effect is often more pronounced since the SI exhaust temperature is significantly

higher than steady state HCCI exhaust temperatures. The simulation model cap-

tures these dynamics with the inclusion of the exhaust manifold model. During SI

cycles the Wiebe function is activated following the application of the spark. Figure

2.10 shows that steady SI is captured. Figure 2.11 shows simulation and experimen-

tal results of the in-cylinder pressure during a SI-to-HCCI mode transition. Cycle


1 corresponds to an SI operating point, while subsequent cycles are in HCCI mode.

The mode transition simulation, though not exact, shows definite agreement with the

general trend from experiment. The rippling on the expansion side of the experimen-

tal pressure curves is due to the fact that the auto-ignition process is not completely

homogeneous, leading to modest pressure waves at higher loads.

Following the mode transition, the combustion event occurs first earlier and then

progressively later as the system converges toward a steady state solution for the new

valve timing and combustion mode. This effect is due to the cycle-to-cycle coupling

through the exhaust gas temperature. The exhaust temperature for the steady state

HCCI operating condition, 625K, is notably lower than the SI exhaust temperature,

810K. The elevated exhaust gas temperature from the SI engine cycle accelerates the

combustion kinetics on the subsequent HCCI cycle, leading to earlier combustion.

This effect is correctly predicted by the basic modeling approach outlined in this

paper. Both the transient and mode transition examples show that appropriate

10

20Pre

ssu

re [

ba

r]

0

10

20

0

250 300 350 400 450Crankshaft Position [Degrees]

250 300 350 400 450

100/25 100/20

85/3090/25

Figure 2.10: Simulated SI combustion during steady state: solid - simulation, dashed- experiment; work carried out with fellow student Matthew Roelle

valve timings depend not only on desired values of combustion timing and load, but


270 340 360 380 4500

10

20

30

40

50

60

70

270 340 360 380 450

Pre

ssu

re [

ba

r]

-1 SI

0 HCCI

12

3

45,6

0 HCCI

1 23

4,5,6

-1 SI

Figure 2.11: Simulated HCCI combustion over a valve timing change: left - experi-ment, right - simulation; work carried out with Matthew Roelle

also on the behavior of the previous cycle. Care must be taken to avoid misfire and

erratic combustion timing. The model presented here provides a basis for developing

and validating controllers capable of modulating HCCI combustion on a cycle-by-cycle

basis. The model offers a simple but very useful tool for the study or prototyping of

HCCI controllers.

2.4 Extension to More Complex Fuels

The model is also applicable to more complex fuels, such as gasoline. The change

required is fairly minor, since differences in fuels only involve the chemical reaction

equation and some parameters in the combustion model. Specifically, the reaction

in Equation 2.37 must be replaced by the valid reaction for the fuel being modeled.

Subsequently, the reaction rates of O2, CO2 and H2O relative to this global reaction

rate can be determined and Equations 2.40-2.46 modified accordingly.


For validation, data from a gasoline HCCI engine (for details see [29, 45]) is com-

pared with model results. The parameters in Equations 2.40-2.46 are modified accord-

ing to the single step kinetics model in Westbrook and Dryer [55]. Figure 2.12 shows

the comparison between experiment and simulation. In the figure, the equivalence

ratio is varied from 0.83 to 1. Within the range examined, the model satisfactorily

matches the experimental pressure evolution, peak pressure and combustion timing.

10

20

30

40

Pre

ssure

[bar]

0280 320 360 400 440

Crankshaft Position [Degrees] 280 320 360 400 440 280 320 360 400 440

Figure 2.12: Integrated Arrhenius Rate Threshold Approach for Gasoline: SimulatedHCCI combustion during steady state: dashed - simulation, solid - experiment; left- equivalence ratio = 1, middle - 0.91, right - 0.83, steady-state experiment plots arecycle averaged; work carried out with Nikhil Ravi

2.5 Physical Insight from Simulation Modeling

A number of valuable insights result from the simulation modeling and validation:

1. Modulation of IVO and EVC allow variation of the inducted gas composition

on a cycle-to-cycle basis.

2. A simple, intuitive model of combustion timing, accounting for reactant con-

centration, mixture temperature and amount of compression, captures the self-

stabilizing behavior often seen with residual-affected HCCI for both propane


and gasoline. This behavior explains the existence of an IVO/EVC operating

trajectory on the experimental engine that exhibits nearly constant combustion

timing.

3. Cycle-to-cycle coupling through the exhaust gas temperature exists, and plays

a fundamental role during steady-state, transients and mode transitions.

2.5.1 Modulation of Inducted Gas Composition

The intake and exhaust valve profiles used to achieve residual-affected HCCI are

shown in Figure 2.2. The reinduction is achieved by having both the intake and

exhaust valves open during portions of the induction stroke. For later intake valve

opening (IVO), the amount of time the intake valve is open during induction de-

creases, causing a decrease in the amount of fuel and air inducted. Likewise, for later

exhaust valve closing (EVC), the amount of time the exhaust valve is open during

induction increases, causing an increase in the amount of reinducted products. So

through modulation of IVO and EVC, the relative amounts of reactants and rein-

ducted exhaust can be varied. One strategy is to move IVO and EVC in tandem. As

IVO/EVC is increased, the amounts of residual and reactant increase and decrease,

respectively, as shown in Table 2.4.

2.5.2 Self-stabilizing Nature of Residual-Affected HCCI

A second key insight from the modeling work is that the combustion timing can be

predicted with a simple and intuitive model - the integrated Arrhenius rate thresh-

old. The dependence of combustion timing on the in-cylinder temperature, reactant

concentrations and final valve closure is evident from Equation 3.23. This relation

captures the fact that the combustion timing will occur earlier for any of the follow-

ing: increased reactant concentrations ([C3H8], [O2]), increased mixture temperature


or increased compression. This dependence on both temperature and reactant con-

centrations is especially important for residual-affected HCCI, where reinducted or

trapped exhaust species from the previous cycle both dilute and increase the tem-

perature of the reactant species. While the dilution of reactants delays the onset of

combustion, the increase in reactant temperature advances the onset of combustion.

Thus, as residual mass fraction increases (i.e. as dilution increases) the reactant con-

centration decreases while the initial mixture temperature increases. This leads to

very little change in combustion timing for varying amounts of dilution, giving the

system a self-stabilizing characteristic. In fact, as noted previously for the engine

studied, there exists a tandem-shift IVO/EVC strategy that spans the load range

and exhibits nearly constant combustion timing.

2.5.3 Cycle-to-cycle Coupling in Residual-Affected HCCI

The last key characteristic of residual-affected HCCI is the exhaust gas coupling

between subsequent engine cycles. The modeling approach captures this coupling

during steady state, transient and mode transition conditions. It is shown that the

coupling plays a fundamental role in all cases, and plays a dominant one during rapid

transients and mode transitions. For this reason, the control strategy depends not

only on the desired behavior of the current engine cycle, but also on the behavior of

the previous engine cycle. Furthermore, control input modulation on one cycle will

affect subsequent cycles. This coupling must be accounted for in the development of

control strategies with cycle-to-cycle capability.

These insights provide valuable clues about how to approach the control problem.

The first step, outlined in the following chapter, is the development of a model more

appropriate for the synthesis of control strategies.


2.6 Conclusion

Although residual-affected HCCI is a complex physical process, the system char-

acteristics most important for control - in-cylinder pressure evolution, combustion

timing, work output and cyclic-to-cycle coupling - can be modeled in a simple and

intuitive manner. The onset of combustion can be predicted by integrating a single

global reaction in the form of an Arrhenius rate expression and determining when

this value crosses a threshold level. Unlike simpler temperature thresholds or knock

integrals, this method reflects the importance of both temperature and reactant con-

centration on the start of combustion. The combination is particularly important

for residual-affected HCCI since the process of trapping or reinducting exhaust gas

links concentration and temperature directly. When this ignition is coupled to simple

thermodynamic models of the cylinder and exhaust manifold, the combined system

predicts both steady-state and transient behavior of HCCI combustion. The ability to

predict the transient behavior and inherent cycle-to-cycle coupling that occurs with

exhaust gas reinduction or trapping makes the model a useful tool for control system

design and validation.

Other more detailed models of HCCI combustion have been developed to capture

the combustion process and kinetics, including multi-zone models [40, 7] and multi-

dimensional CFD models [26] using detailed chemistry. While this level of detail

is necessary for accurately predicting the overall process of HCCI combustion, in

particular the emissions, these models are often far too detailed for controller design

or validation. As the model in this chapter illustrates, simple models can accurately

capture the properties most relevant to control with comparable levels of fidelity.

With run times of about 15 seconds/cycle, the simulation model presents a useful

virtual testbed for control strategies. However, the 10-state model is still slightly

too complex for the synthesis of control strategies. For this reason, the next chapter


outlines a control model of residual-affected HCCI formulated through a reduction of

the simulation model and application of some additional assumptions.

Chapter 3

Control Modeling

The simulation model developed in Chapter 2 captures the aspects most relevant

for control - cyclic coupling, in-cylinder pressure evolution, work output and ignition

via kinetics - with a level of fidelity matching more complex multi-zone CFD models

with detailed chemical kinetic mechanisms. While very helpful in gaining intuition

and providing an accurate virtual testbed for control, a slightly simpler model of the

process is desirable for synthesis of control strategies.

This chapter outlines a physics-based control-oriented system model for peak pres-

sure and combustion timing to address this need. Since the inducted gas composition

can be controlled through valve timing modulation, it is chosen as one of the inputs

to the model. Since final valve closure (IVC in this study) alters the effective com-

pression ratio, IVC is the second control input. By discretizing the various processes

which occur during a HCCI combustion engine cycle and linking them together, a

mathematical relation for the peak pressure dynamics is formulated. A simplified

version of the integrated Arrhenius rate threshold approach outlined in Chapter 2

produces an expression for the combustion timing dynamics. Together the dynamic

equations for peak pressure and combustion timing result in a nonlinear two input,

two output control model of residual-affected HCCI. This control model provides a

45

CHAPTER 3. CONTROL MODELING 46

natural launching point for developing control strategies.

3.1 Modeling Approach

The framework for developing the control model is to partition the engine cycle into

five stages, as shown in Figure 3.1:

4900

intakeexhaust

VBDC

VTDC

V1

V23

V4

IVO EVC EVO=θ4IVC=θ1

crank angle [deg.]

cylin

der

volu

me

valv

e op

enin

g

stage 1:

adiabatic, constant

pressure induction

stage 2:

isentropic

compression

stage 3:

constant volume

combustion

approximation

stage 4:

isentropic

expansion

stage 5:

isentropic

exhaust

θ23

in-c

ylin

de

r p

ressu

re

0 540 720180

actual

control model

Figure 3.1: General view of partitioned HCCI cycle

1. mixing of reactant and reinducted product gases during a constant pressure,

adiabatic induction process

2. isentropic compression to the point where combustion initiates


3. constant volume combustion to major products with heat transfer

4. isentropic expansion to the cylinder volume at exhaust valve opening

5. isentropic expansion to atmospheric pressure through the exhaust valve

The reinducted product temperature is directly related to the exhaust temperature

from the previous cycle. The general model structure is shown in Figure 3.2.

peak pressure

P

composition

a = Np/Nr

amount of compression

q1, V1 = V(q1)

VVA-modulated Control Inputs

Model Outputs

combustion timing

q23, V23 = V(q23)

Residual-affected HCCI

Figure 3.2: Block Diagram of Control Model

The first model input is the inducted gas composition. The inducted gas compo-

sition is formulated as the ratio of the moles of reinducted product Np to the moles

of inducted reactant charge Nr:

α ≡ Np/Nr (3.1)

The second model input is the final valve closure, which dictates the volume, V1 =

V (θ1), at the start of compression and therefore the effective compression ratio. Model

outputs are the peak pressure, P , and the volume at the constant volume combustion

event, V23 = V (θ23), which acts as a proxy for combustion timing. By linking the

thermodynamic states of the system together, a dynamic model of peak pressure,

P , and phasing, θ23, for residual-affected HCCI is formulated. Note that at points

between stages, the cylinder volume (see Figure 3.1) is either known or is a model

output (as is the case for V23 = V (θ23)). These volumes effectively split up the dif-

ferent processes (induction, compression, combustion, expansion and exhaust). The


modeling techniques are applied to propane-fueled HCCI. The model will also apply

to other fuels by making appropriate changes to the fuel-specific model constants.

3.1.1 Instantaneous Mixing of Species

The mixing of the reactant and reinducted product species during the induction

process for lean or stoichiometric propane HCCI can be represented as:

α (3φCO2 + 4φH2O + 18.8N2 + 5(1− φ)O2)

+ (φC3H8 + 5O2 + 18.8N2) →φC3H8 + 5((α(1− φ) + 1)O2 + 18.8(1 + α)N2+

3αφCO2 + 4αφH2O(3.2)

where φ is the equivalence ratio, defined as the ratio of moles of fuel to the amount

required for the complete combustion of both fuel and oxygen, such that the reactant

mixture is φC3H8 + 5O2 + 18.8N2. The first law of thermodynamics applied to an

assumed adiabatic, constant pressure induction process is:

mprodh1,prod + mrcth1,rct =d(mh)

dt(3.3)

The reactant mass flow rate through the intake and reinducted product mass flow rate

through the exhaust are mrct and mprod, with corresponding enthalpies in the intake

and exhaust manifolds of h1,rct and h1,prod. When this equation is integrated from

the beginning to the end of the induction process with the assumption that manifold

conditions do not vary during induction, the resulting expression for the kth engine

cycle is:∑

stage1prods.

Ni,khi(T1prod,k) +∑

stage1react.

Ni,khi(T1rct,k)=∑

stage1

Ni,khi(T1,k) (3.4)


where Ni,k is the number of moles of species i, hi is the molar enthalpy of species i,

T1prod,k is the reinducted product temperature, T1rct,k is the inducted reactant tem-

perature and T1,k is the temperature of the reactants and products after full mixing.

Assuming that the molar enthalpy of species i can be approximated using a specific

heat, cp,i that is constant with temperature, then:

hi(T ) = ∆f hi + cp,i(T − Tref ) (3.5)

where ∆f hi is the molar heat of formation of species i, and Tref is the reference

temperature corresponding to the heat of formation. Equation 3.4, applied to Equa-

tion 3.2, yields after rearrangement the following in-cylinder mixture temperature at

θ1,k = IV C1,k:

T1,k =c1Tinlet + c2αkT1prod,k

c1 + c2αk

(3.6)

where

c1=φcp,C3H8 + 5cp,O2 + 18.8cp,N2 (3.7)

c2=3φcp,CO2 +4φcp,H2O +18.8cp,N2 +5(1− φ)cp,O2 (3.8)

are the specific heats of the inducted reactant and reinducted exhaust gas, respec-

tively. The reinducted product species are assumed to have a temperature, T1prod,k,

that is directly related to the temperature of the exhausted products from the last

cycle, T5,k−1, as:

T1prod,k = χT5,k−1 (3.9)

This simple relation is meant to represent heat transfer. A more physically motivated

exhaust manifold heat transfer model with similar form and calculated T1prod,k values

is developed in the Appendix. Equation 3.9, however, matches experimental obser-

vations reasonably well while keeping the relation as simple as possible. Substituting


Equation 3.9 into Equation 3.6, leads to:

T1,k =c1Tinlet + c2χαkT5,k−1

c1 + c2αk

(3.10)

3.1.2 Isentropic Compression to Pre-Combustion State

With the assumption that the compression stage occurs isentropically, the thermo-

dynamic state of the system prior to and following the stage may be related with the

following isentropic relations for an ideal gas:

T2,k =

(V1,k

V23,k

)γ−1

T1,k P2(k) =

(V1,k

V23,k

)γ

Patm (3.11)

where γ is the specific heat ratio.

3.1.3 Constant Volume Combustion

In order to model HCCI combustion in a very simple way, it is assumed that the

combustion reaction, from reactants to products, occurs instantaneously uniformly

throughout the combustion chamber. The instantaneous combustion assumption is

justified by the fact that HCCI combustion is typically very fast. It is also assumed

that all in-cylinder wall/piston heat transfer occurs during the combustion event.

The location of the combustion event, θ23, is modeled in Section 3.3 from a simplified

version of the integrated Arrhenius rate threshold model. It is further assumed that

only major products result from the combustion event, such that the combustion

reaction can be written as:

φC3H8 + 5(αk(1− φ) + 1)O2 + 18.8(1 + αk)N2

+3φαkCO2 + 4αkφH2O →(1 + αk)(3φCO2 + 4φH2O + 18.8N2 + 5(1− φ)O2) (3.12)


For a constant volume combustion process, the total internal energy before and after

combustion can be related as:

U2,k = U3,k + Qk = U3,k + LHVC3H8NC3H8,kε (3.13)

where the total amount of wall/piston heat transfer, Qk, has been modeled as a

certain percentage, ε, of the chemical energy available from the combustion reaction,

LHVC3H8NC3H8,k. Here LHVC3H8 is referred to as the lower heating value for propane,

and is defined as:

LHVC3H8 = 3∆f hCO2 + 4∆f hH2O −∆f hC3H8 (3.14)

Equation 3.13 can then be expanded to:

∑2 Ni,khi(T2,k)−RuT2,k

∑2 Ni,k =∑

3 Ni,khi(T3,k)−RuT3,k

∑3 Ni,k + LHVC3H8NC3H8,kε

(3.15)

Applying the constant specific heat assumption to the expanded form of the post-

combustion internal energy expression, Equation 3.15, gives:

T3,k =c3 + (c1 + c2αk)T2,k

c2(1 + αk)(3.16)

where:

c3 = (1− ε)φLHVC3H8 (3.17)

The number of moles in the cylinder following combustion, N3, can be related to N2

by inspection of Equation 3.12:

N3,k =

(1 + αk

f + αk

)N2,k (3.18)


where f = 24.8/25.8 is the reactant to product species molar ratio. The in-cylinder

pressure following the constant volume combustion stage, P3,k, can be related to the

temperature at that point, T3,k, by invoking the ideal gas assumption at states 2 and

3, and combining Equation 3.18 with Equation 3.11 to arrive at:

P3,k =1 + αk

f + αk

(V1,k

V23,k

)γc1 + c2αk

c2(1 + αk)T3,k − c3

PatmT3,k (3.19)

3.1.4 Isentropic Expansion and Exhaust

The fourth stage of HCCI is approximated as isentropic volumetric expansion follow-

ing the constant volume combustion stage. The exhaust stage is also assumed to be

isentropic, with the additional assumption that the pressure in the exhaust manifold

is atmospheric. This results in the relations:

T4,k =

(V23,k

V4

)γ−1

T3,k, P4,k =

(V23,k

V4

)γ

P3,k (3.20)

T5,k =

(Patm

P4,k

)γ−1γ

T4,k (3.21)

3.2 Peak Pressure Equation

By linking the distinct processes which occur during HCCI combustion - combining

Equations 3.10, 3.11, 3.16, and 3.19-3.21 with the approximation that (1 + αk)/(f +

αk) ≈ 1 and cv ≡ mean(c1, c2) ≈ c1 ≈ c2 - a dynamic model of the peak in-cylinder

pressure can be formulated:

Pk =

((1− ε)LHVC3H8+ cv

V γ−11,k

V γ−123,k

Tin

)(1 + αk−1)

(Pk−1− V γ

1,k−1

V γ23,k−1

)V1,k

V23,k+χαk(1− ε)LHVC3H8

V γ1,k

V γ23,k

P1/γk−1

cvTin(1 + αk−1)(Pk−1 − V γ

1,k−1

V γ23,k−1

)+ χαk(1− ε)LHVC3H8P

1/γk−1

= f1(states; inputs)

= f1(Pk−1, θ23,k, θ23,k−1;αk, αk−1, θ1,k, θ1,k−1) (3.22)


Here Pk is normalized by Patm, and is therefore unitless. The presence of cycle-to-

cycle dynamics is evident by inspection of Equation 3.22, as the current peak pressure

Pk depends on the previous cycle peak pressure Pk−1 and combustion timing θ23. This

is a very powerful expression as it relates a desired model output, the peak pressure,

to the model inputs, the molar ratio of the reinducted products and reactants, α, and

the IVC timing θ1,k (via V1,k). Additionally note the dependence on the combustion

phasing (represented by the combustion volume, V23,k). What is now required is a

physics-based expression for the combustion phasing.

3.3 Combustion Timing Modeling Approach

Chapter 2 shows that an integrated Arrhenius model of combustion is a simple and

accurate way to mathematically describe HCCI combustion phasing. For propane

fuel this integrated reaction rate model takes the form:

Kth =

∫ θth

IV C

Aexp(Ea/(RuT ))[C3H8]a[O2]

bdθ/ω (3.23)

where ω is the engine speed. The values A, Ea/Ru, a, b and n are empirical parameters

determined from combustion kinetics experiments. Once a pre-defined threshold,

denoted as Kth, for this integral is exceeded, the combustion process is initiated and

assumed to proceed as a function of crank angle using a Wiebe function. The crank

angle at peak in-cylinder pressure, θ23, can then be related to threshold crossing point,

θth, as: θth = θ23 − ∆θ, as shown in Figure 3.3, where ∆θ is assumed constant as

a consequence of approximating the combustion event as a function of crank angle.

Applying the threshold approach to Equation 3.23, then yields:

Kth =

∫ θ23−∆θ

IV C

Aexp(Ea/(RT ))[C3H8]a[O2]

bdθ/ω (3.24)


crank angle [deg.]

in-c

ylin

der

pre

ssure

540180

actual

control model

θth

θ23

∆θ

Figure 3.3: Representation of control model

Equation 3.24 captures the dependence of combustion phasing on the in-cylinder

temperature, reactant concentrations and the start of compression (i.e IVC). The

integration in Equation 3.24 can be simplified by approximating the integrand at the

end of the compression stroke (i.e. at top dead center (TDC)) and beginning the start

of integration at this point. This is a justifiable approximation since the integrand

takes on its largest value at this point. Then Equation 3.24 becomes:

Kth ≈∫ θ23−∆θ

θTDC

Aexp(Ea/(RTTDC))[C3H8]aTDC [O2]

bTDCdθ/ω (3.25)

= Aexp(Ea/(RTTDC))[C3H8]aTDC [O2]

bTDC/ωθc (3.26)

where:

θc ≡ θ23 −∆θ − θTDC (3.27)

Note that the value of the integrated Arrhenius rate threshold in Equation 3.26,

Kth, will not be the same as the threshold in Equation 3.24 due to the constant


integrand and lower integration limit approximations. The value of Kth is set to

reflect experimental combustion timing results using Equation 3.26. The in-cylinder

temperature at TDC is:

TTDC = T (T1,k, V1,k) =

(V1,k

VTDC

)γ−1

T1,k (3.28)

Before combustion and after the point in time when both the intake and exhaust valves

are shut (i.e. at IVC) the reactant concentrations can be derived from Equation 3.2

at TDC as:

[C3H8]TDC,k = φNr,k/VTDC [O2]TDC,k = 5(α(1− φ) + 1)Nr,k/VTDC (3.29)

Furthermore, the total number of moles of all species is:

Ntotal,k = (2φ + 23.8)Np,k + (23.8 + φ)Nr,k (3.30)

Then, by invoking the ideal gas assumption at IVC:

Nr,k =PatmV1,k

RT1,k

1

(2φ + 23.8)αk + (23.8 + φ)(3.31)

Substitution of Equations 3.10, 3.28-3.31 into Equation 3.26, applied to the kth engine

cycle, gives:

θc =C1

((1+αk)T1,k

V1,k

)a+b

(exp

[−Ea

RT1,k

(VTDC

V1,k

)γ−1]) = f(T1,k, V1,k, αk) (3.32)

where:

C1 =

(Kthω

Aφa5b

)(25VTDCR

Patm

)a+b

(3.33)


is a positive constant. Equation 3.32 is essentially a simplified version of the simula-

tion ignition model 3.23, the integrated Arrhenius rate threshold approach. Like that

model, Equation 3.32 captures the advance in combustion timing θ23 = θc + ∆θ due

to:

1. an increase in the inducted gas mixture temperature T1,k

2. an increase in the in-cylinder volume at final valve closure V1,k (i.e increased

amount of compression)

3. a decrease in ratio of reinducted product to inducted reactant αk (i.e increase

in reactant concentration)

By combining Equations 3.6, 3.21, 3.16 and 3.11, the following expression for the

pre-compression mixture temperature T1,k can be found:

T1,k =cvTi(1 + αk−1)

(Pk−1 −

(V1,k−1

V23,k−1

)γ)+ χαkP

1/γk−1(1− ε)LHVC3H8

cv

(Pk−1 −

(V1,k−1

V23,k−1

)γ)(1 + αk−1)(1 + αk)

(3.34)

Combining Equations 3.34, 3.32 and 3.27 yields:

θ23,k =

C1

cvTi(1+αk−1)

�Pk−1−

�V1,k−1V23,k−1

�γ�+χαkP

1/γk−1c3

cvV1,k

�Pk−1−

�V1,k−1V23,k−1

�γ�(1+αk−1)

a+b

exp

−Ea

R

cv

�Pk−1−

�V1,k−1V23,k−1

�γ�(1+αk−1)(1+αk)

cvTi(1+αk−1)

�Pk−1−

�V1,k−1V23,k−1

�γ�+χαkP

1/γk−1c3

(VTDC

V1,k

)γ−1

+ ∆θ + θTDC

(3.35)


which has the following functional form:

θ23,k = f2(system states; system inputs) (3.36)

= f2(Pk−1, θ23,k−1; αk, V1,k, αk−1, V1,k−1) (3.37)

Equation 3.35 captures the dependence of combustion timing on the system inputs

(inducted gas composition and effective compression ratio) and the system states (the

peak pressure and combustion timing on the previous cycle).

Together, Equations 3.22 and 3.35 complete the physics-based control model of

residual-affected HCCI. Although these mathematical expressions are complex non-

linear functions, they are nevertheless well-behaved and amenable for controller de-

velopment.

3.4 Model Validation

With the number of assumptions made in the control modeling approach, a compar-

ison with results from experiment and the simulation work in Chapter 2 is necessary

to gain confidence in the resulting model. The control model is first validated in

steady state and then during dynamic operation.

3.4.1 Model Validation in Steady State

A series of experiments at five different operating conditions on a single-cylinder

research engine is described in Chapter 2. Variations in operating condition were

made by adjusting the exhaust valve closing (EVC) and intake valve opening (IVO)

positions, effectively changing the ratio of reinducted products and reactants. Figure

3.1 shows the general valve profile used on the research engine. The experimental

values are given in Table 3.1.


case IVO/EVC ¯V [m3

s] Tex[K] Pmax [atm] θ23 [CAD]

1 25/165 0.00390 697 59.5 12.22 35/175 0.00365 666 57.7 11.63 45/185 0.00330 655 52.5 11.44 55/195 0.00298 644 49.6 11.65 65/205 0.00260 638 44.5 11.0

Table 3.1: Experimentally Monitored Values

case IVO/EVC α1 25/165 0.682 35/175 0.803 45/185 0.954 55/195 1.125 65/205 1.37

Table 3.2: Estimated Experimental Values

Experimental estimates of α are calculated given values of ¯V , Tex and V1 from

experiment. The total volume flow of reactant mixture through the intake during an

engine cycle, Vinlet, is related to the average inlet air flow rate and the cycle time,

tcycle, by:

Vinlet = ¯V tcycle (3.38)

With the assumption that the reactant charge is inducted under atmospheric

conditions and behaves as an ideal gas, the total number of moles of reactant species

inducted is:

Nr =PatmVi

RuTatm

(3.39)

Using the ideal gas law for the mixture of products and reactants in the cylinder at

state 1:

Nr

(α

f+ 1

)= Np + Nr = Ntotal =

PatmV1

RuT1

(3.40)


case IVO/EVC Pmax,model [atm] % error θ23 % error1 25/165 59.7 0.33 10.97 -10.02 35/175 57.3 -0.69 11.70 0.93 45/185 54.5 3.81 11.49 0.754 55/195 50.9 2.62 11.51 -0.85 65/205 48.4 8.76 11.34 3.1

Table 3.3: Static Validation of Control Model

Utilizing the argument presented in Section 3.1.1 of constant pressure, adiabatic mix-

ing of the reactants and reinducted products, Equation 3.10, can be used:

T1(k) =cp,r(Tinlet)Tinlet + cp,p(Tex)χα(k)Tex

cp,r(T1) + cp,p(T1)α(k)(3.41)

Here the specific heats are allowed to vary with temperature to provide the most

accurate expression of the first law. Equations 3.38-3.41 can be solved simultaneously

for experimental estimates of α given values of ¯V , Tex and V1 from experiment. The

results of these calculations are in Table 3.2. Equation 3.22 is then used to find the

values of peak pressure predicted by the control model. Model-predicted values of

combustion timing are found using Equation 3.35. These values are presented in Table

3.3. The control model is calibrated here by choosing heat transfer coefficients χ and

ε (from Equations 3.9 and 3.13) that result in correlation between experimental and

model calculated values of peak pressure and combustion timing. The values selected

are χ = 0.94 and ε = 0.12. All other parameters in the control model equations

are either taken from, or directly calculated using, the physical specifications of the

system, as shown in Table 2.1.

By inspection of Tables 3.1 and 3.3 it can be seen that the experimental and model

predicted values of peak pressure and combustion timing show good correlation in

steady state, within 10% of experimental values.


3.4.2 Model Validation During Transients

This section outlines the dynamic validation of the control model. Cycle-to-cycle

values of α, calculated from the simulation (itself validated in transient operation in

Chapter 2) during a series a valve timing changes, are used as inputs to the control

model. Comparisons of combustion timing and peak pressure validate the control

model in dynamic operation.

Figure 3.4 shows the series of step changes in valve timing used, and the resulting

simulation outputs: α, average exhaust gas temperature, peak pressure and combus-

tion timing. The combustion timing and peak pressure values calculated with the

control model using the cycle-to-cycle α values are plotted on top of the simulation.

As shown in Figure 3.4, the control model captures the general characteristics of

the combustion timing and peak pressure transients during the 25/165 to 65/205 to

45/165 step changes. As first discussed in Section 2.3.1 of Chapter 2, the advance

in combustion timing during the 25/165 to 65/205 is due to the higher steady state

exhaust temperature at the 25/165 operating condition. Likewise, the delay in com-

bustion timing during the 65/205 to 45/185 step changes is due to the lower exhaust

gas temperature at the 65/205 operating condition. The control model captures this

behavior through the inclusion of reactant concentration and mixture temperature

effects.

While the level of accuracy does not match that of the simulation model developed

in Chapter 2, the simplicity and predictive capabilities of this low-order modeling ap-

proach make it a good candidate for model-based controller synthesis. In the following

chapters several control strategies synthesized from the control model developed in

this chapter will be studied in detail. The control model also allows system stability

to be studied with a number of available theoretical tools. One such example is given

in Chapter 5, in which convex optimization and Lyapunov stability theory are used

to prove stability of a closed-loop control strategy.


0.5

1.0

1.5

600

700

800

360

370

380

40

50

60

25/165

405/185

65/205

Va

lve

Tim

ing

, IV

O/E

VC

α f

rom

sim

ula

tion

Co

mb

ust

ion

Tim

ing

10 15 20 25 30 35 40

Engine Cycle

Pe

ak

Pre

ssu

re [

atm

]

0

Control Model ResultSimulation Model Result

E

xha

ust

Ga

s T

em

pe

ratu

re [

K]

Figure 3.4: Dynamic Validation of Control Model

Chapter 4

Control of Peak Pressure

Through the process of developing the simulation and control models in Chapter 2 and

Chapter 3, a number of valuable insights were developed. The first control approach,

outlined in this chapter, relies on the self-stabilizing nature of HCCI combustion

timing and the ability to vary the inducted gas composition with the VVA system to

regulate the peak pressure at nearly constant combustion timing. The control strategy

is synthesized from a linearized version of the nonlinear control model formulated in

Chapter 3. Implemented on the 10-state model, this control law is able to successfully

track peak in-cylinder pressure. Experimental results show good tracking of peak in-

cylinder pressures. As desired, combustion timing is nearly constant, with small

deviations (< 3 degrees) attributed to unmodeled wall temperature dynamics.

4.1 Control Development

As outlined in Chapter 2, Caton et al. [11] observed an operating trajectory spanning

the load range while also exhibiting low NOx emissions and high efficiency. This

trajectory is shown with solid points in Figure 4.1. Additional characteristics of

the operating trajectory are nearly constant combustion timing and a monotonic

62

CHAPTER 4. CONTROL OF PEAK PRESSURE 63

relationship between peak pressure and IMEP, as shown in Table 2.4.

Figure 4.1: Operating manifold [11]

One strategy, then, for HCCI control is to modulate work output by dynamically

moving along this operating manifold, using peak pressure as a proxy for load. To

develop a control strategy to do this, the cycle-to-cycle dynamics from inducted gas

composition to peak in-cylinder pressure must be well understood. To address this,

the HCCI control model from Chapter 3 is used. Since smooth transients in operat-

ing conditions lead to only modest changes in combustion timing, the approach used

in this chapter will neglect the combustion timing dynamics portion of the control

model. This approach simplifies the control law development and experimental im-

plementation. Approaches for adding in direct control of combustion timing will be

explored in subsequent chapters.

With the assumption of constant combustion timing

V23,k = V23,k−1 = V23 (4.1)


and no modulation of the effective compression ratio

V1,k = V1,k−1 = V1 (4.2)

the peak pressure dynamics of the control model (Equation 3.22), simplify to the

following form :

Pk =

((1− ε)LHVC3H8 + cv

V γ−11

V γ−123

Tin

)(1 + αk−1)

(Pk−1 − V γ

1V γ23

)V1V23

+ χαk(1− ε)LHVC3H8

V γ1

V γ23

P1/γk−1


1V γ

23


1/γk−1


= f1(Pk−1; αk, αk−1) (4.3)

where Pk is the peak in-cylinder pressure at cycle k, αk ≡ Np/Nr is the VVA-

controllable molar ratio of inducted residual to reactant at cycle k. From Equation

4.3, a variety of control strategies can be explored. A simple approach is to linearize

the nonlinear model about one operating point and then to synthesize a simple linear

control law.

4.1.1 Linearization of Pressure Relation

The peak pressure model, Equation 4.3, can be linearized about an operating point

(α, P ). Straightforward linear expansions for α and P give:

αk = α + αk (4.4)

Pk = P + Pk (4.5)

(4.6)


and since the peak pressure, P , is always positive, the Taylor expansion of P1/γk−1

approximates the term as:

P1/γk−1 ≈ P 1/γ + Pk−1

P (1−γ)/γ

γ(4.7)

Applying these to Equation 4.3, and neglecting second order terms of fluctuations

(i.e. αkPk, αkαk, Pk˜Pk−1, Pk ˜αk−1αk, etc) leads to:

c1βk = c2βk−1 + c3αk + c4 ˜αk−1 (4.8)

where βk ≡ (Pk − P )/P is the normalized difference between desired and actual

pressure, and:

c1 =cvTin

(1 + α)−1

(P− V γ

1

V γ23

)+ χ(1− ε)LHVC3H8αP 1/γ (4.9)

c2 =−cvTin

(1 + α)−1

(P− V γ

1

V γ23

)

− LHVC3H8

(1− ε)−1γ

(χα

P (γ−1)

(P− V γ

1

V γ23

)+

V1

V23

(1 + α)

)(4.10)

c3 =−cvTinP − χ(1− ε)LHVC3H8P(1−γ)/γ

(P− V γ

1

V γ23

)(4.11)

c4 =cv

T−1in

V γ1

V γ23

+

(LHVC3H8

(1− ε)−1

V1

V23

+cv

T−1in

V γ1

V γ23

)1

P

(P− V γ

1

V γ23

)(4.12)

Equation 4.8 can also be written as a low-order discrete linear transfer function:

β(z)

α(z)=

c3 + c4z−1

c1 − c2z−1(4.13)


or in state space form:

xk+1 = Axk + Buk (4.14)

yk = Cxk + Duk (4.15)

where:

A =

[0 0c4c1

c2c1

], B =

[1c3c1

], C = [0 1] , D = 0 (4.16)

xk =

[αk−1

βk−1

]un = αk yn = βk (4.17)

Equations 4.9-4.17 yield a linearized version of the control model peak pressure dy-

namics.

4.2 LQR Controller Synthesis

From the low-order linearized model of HCCI combustion, a variety of different con-

trollers can be synthesized to track the desired in-cylinder peak pressure. For illus-

trative purposes, a state feedback control law can be found with the form:

un = −Kxn (4.18)

which minimizes the cost function:

J =∑

x′nQxn + u′nRun + 2x′nNun (4.19)

This feedback control law is the standard linear quadratic regulator (LQR) with a

controller output of α and full state feedback consisting of the previous cycle’s α and


β, two easily attained values, so that Equation 4.18 can be written as:

αk = K1βk−1 + K2αk−1 (4.20)

With Q, R and N selected as:

Q =

[0 0

0 q

]R = 1 N =

[0

0

](4.21)

the cost function becomes:

J =∑ (

qβ2k−1 + α2

k

)(4.22)

With this formulation, weights can be placed directly on both the output and input

of the system, β and α, respectively. In this way, the tradeoff between control effort

and tracking performance is explicitly accounted for in the control design process.

4.3 Valve Timing Map

The control strategy presented uses the previous cycle’s peak pressure, the desired

peak pressure for the current cycle, and the inducted mixture composition, α, on the

previous cycle to determine the desired α on the current cycle according to Equation

4.20. Before this can be implemented experimentally, however, a map from desired α

to required valve timing (IVO/EVC) is necessary since it is in fact the valve timings

that are the input to the experimental system. As presented in Chapter 2 a simple set

of IVO/EVC valve timings, shown in Figure 4.2, exist that span the load range and

exhibit low emissions of NOx species for the engine used. The required valve timings

are obtained from the desired α and the loop can be closed around the peak pressure in

the cylinder. Figure 4.3 shows how the LQR controller is used in conjunction with the

valve timing map in closed loop. Note that the any imprecision of the pre-determined


75/215 65/205 55/195 45/185 35/1750.60

0.75

0.90

1.05

1.20

1.35

1.50

IVO /EVC

ind

uc

ted

co

mp

osi

tio

n, α

Figure 4.2: Effect of valve timings on inducted gas composition

LQR

controller

Engine

Cylinder

α' to

IVO/EVC

map

IVO

EVC

α'β' Pmax+

-1

P

let: P=Pmax,desired

P

1

z

Figure 4.3: Block diagram of controller implementation

α to IVO/EVC relationship will be corrected due to feedback.

4.4 Implementation on Simulation Model

To test the LQR controller strategy it was implemented in closed loop with the

10-state simulation model introduced in Chapter 2. The linearization of the control

model was completed about the IVO/EVC=45/185 operating point. Figure 4.4 shows

the closed-loop control simulation results for combustion timing, peak pressure, valve

timings, and IMEP as the desired peak pressure goes through a series of step changes.

As expected, the combustion phasing is very consistent. Also note the correlation

between IMEP and peak pressure, validating the claim that peak pressure can be used

as a proxy for work output when the phasing is constant. The closed loop response to


53

54

55

56

57

58

59

60

61

62

pe

ak

in-c

ylin

de

r p

ress

ure

[atm

] desired pressure

5

5.2

5.4

5.6

5.8

6

6.2

6.4

6.6

IME

P [

atm

]

actual pressure IMEP

0 15 30 45 60 75 90 105 120 135

362

364

366

368

370

372

374

376

378

time [seconds]

engine cycles1 2 3 4 5 6 7 8 90

Co

mb

ust

ion

Tim

ing

Figure 4.4: Simulation of tracking controller on 10-state model

the step changes in desired peak pressure are rapid, and accurate after about 6 cycles.

The specific response in desired peak pressure shown here is essentially arbitrary and

controllers with more specific design objectives could certainly be synthesized from

the control model.

4.5 Implementation on Research Engine

Having achieved success with the closed-loop simulation, the control strategy was

experimentally implemented on the single cylinder research engine. Figure 4.5 shows

the closed-loop controlled system response to a substantial step change in desired peak

pressure. Despite the large change, peak pressure tracking is rapid (within about 5

engine cycles), and the cyclic dispersion is reduced. Furthermore, no change in the


360

370

380

40

50

60

40/180

70/210

3

4

5

Pe

ak

Pre

ssu

re[a

tm]

IVO

/EV

C [

CA

D]

IME

P[a

tm]

Ign

itio

n T

imin

g[C

AD

]

0 50 100 150 200 250 300 350 400 450

Engine Cycle

CONTROL OFF CONTROL ON

mean: 69.1/209.1 std dev: 1.61

mean: 4.75 std dev: 0.11






Figure 4.5: Experimental results of closed loop control on research engine, dashedline shows simulation result

calculated control gains was required during either the simulation or experimental

implementations, illustrating the utility of a physics-based approach to control design.

Additionally, note that there is a subtle change in the combustion phasing during

the transition, followed by a slow drift to the pre-transition phasing. Although the

magnitude of these fluctuations in phasing are quite small (< 3 degrees on the mean),

there is a corresponding effect on the IMEP. The phasing changes are most likely

due to exhaust manifold gas and wall temperature changes. In general, the steady

state exhaust gas temperatures are higher for higher load conditions (when more


reactant gas is inducted and burned). As noted, for the engine being studied, the

combustion phasing is fairly constant in steady state. When a transition is made

from a high load to low load condition, the re-inducted exhaust gas temperature is

elevated in comparison to the steady-state value. This leads to slightly earlier phasing,

which follows from physical intuition and is captured by the integrated Arrhenius

rate, Equation 3.23. This temperature effect explains the subtle change during the

transition. Note that the simulation model result in Figure 4.5 captures this deflection

in phasing during the transition. The slow drift of the combustion phasing back to

the pre-control phasing is most likely due to the exhaust manifold wall temperature

cooling following the change to a lower load condition. As the wall temperature

decreases, the heat transfer from exhaust manifold gas to the wall increases, leading

to lower inducted gas temperatures and later phasing. The simulation model does

not capture the dynamic wall temperature effect since the wall temperature has been

assumed constant. In general, however, the correlation between the experimental and

simulation control results show that the 10-state simulation model predicts not only

the open-loop behavior, but also the closed-loop behavior of the system.

Figure 4.6 shows the open and closed-loop operation of the engine near the limits.

The closed-loop system response shows strong evidence that in addition to accurate

mean tracking, the controller substantially reduces cyclic dispersion. Prior to the

controller being turned on, the system is operating at the edge of the operating

range (note the significance of the pressure and phasing dispersion, and the presence

of misfires). Once the controller is activated the 65atm peak pressure set-point is

tracked rapidly, making an unacceptable open-loop operating point more acceptable

with the application of closed-loop control. Note that even though the peak pressure

and IMEP dispersion have been reduced, the phasing dispersion is still fairly large.

Despite this, the peak pressure controller still performs as desired. This is a clear

example of an operating condition that would not be acceptable without control.


360

370

380

20

40

60

25/165

35/175

4

6

2

IVO

/EV

C [

CA

D]

0 100 200 300 400 500Engine cycle

CONTROL OFF

CONTROL ON CONTROL OFF

mean: 365.29 std dev: 5.54

mean: 367.54 std dev: 4.96




mean: 34.1/174.1 std dev: 1.17

mean: 361.83 std dev: 4.05




misfires

IME

P[a

tm]

Ign

itio

n T

imin

g[C

AD

]P

ea

k P

ress

ure

[atm

]

Figure 4.6: Experimental results of closed loop control on research engine

4.6 Conclusion

While cycle-to-cycle dynamics and chemical kinetics make VVA-induced, residual-

affected HCCI a complex process, a simplified version of the control model developed

in Chapter 3 can be used to synthesize a control strategy for peak pressure control.

The simplified control model is linearized about an operating condition and used to

synthesize an LQR controller. Paired with a map from inducted gas composition

to required IVO/EVC timing, this represents a complete approach for modulating

engine load while minimizing emissions and maximizing efficiency. For both the


closed-loop simulation and experiment, the peak in-cylinder pressure is tracked well.

Furthermore, control experiments show that cycle-to-cycle variations are reduced from

the uncontrolled cases, introducing more stability and decreasing the likelihood of

misfire. Tracking and dispersion reduction reflect a successful peak pressure LQR

control implementation. This control approach represents the simplest approach for

residual-affected HCCI control considered in the thesis. In Chapter 6, the ability

to directly control combustion timing and work output will be added to the peak

pressure control, at the expense of a marginal increase in control design complexity.

Chapter 5

Stability Analysis

To address the issues of cyclic coupling and lack of a combustion trigger, a low-order

nonlinear model of the HCCI combustion process was developed in Chapter 3 and

used to synthesize a simple, yet effective peak pressure control strategy in Chapter 4.

In this chapter, a Lyapunov-based stability analysis utilizing sum of squares decom-

position and a theorem from real algebraic geometry, the Positivstellensatz, estimates

the domain of attraction for the nonlinear system in closed-loop with the linear con-

troller. The resulting region of attraction proves stability of the system over all of the

desired portion of the state space. Physically, this means that the linear controller

stabilizes the nonlinear system over the entirety of the desired operating range of

HCCI.

5.1 Closed-loop HCCI Dynamics

By linking the distinct processes which occur during HCCI combustion, a nonlinear

relation between the molar ratio of reinducted product to inducted reactant αk and the

peak in-cylinder pressure, Pk was developed in Chapter 3, as shown in Equation 3.22.

From the nonlinear model of HCCI peak pressure dynamics, a variety of different

74

CHAPTER 5. STABILITY ANALYSIS 75

controllers can be synthesized to track the desired in-cylinder peak pressure. For

instance, an LQR control law can be synthesized from a linearization of the system

about an operating point (α, P ), as shown in Chapter 4.

5.2 Estimating the Domain of Attraction using the

Sum of Squares Decomposition

A key question is whether it can be proven that the simplified nonlinear peak dynamics

from Chapter 4

Pk =

((1− ε)LHVC3H8 + cv

V γ−11

V γ−123

Tin

)(1 + αk−1)

(Pk−1 − V γ

1V γ23

)V1V23

+ χαk(1− ε)LHVC3H8

V γ1

V γ23

P1/γk−1


1V γ

23


1/γk−1


= f1(Pk−1; αk, αk−1) (5.1)

are stable in closed-loop with the LQR control law (Equation 4.20) synthesized from

the linearization of the nonlinear model at a single operating point. In the next two

sections it is shown that the closed-loop dynamics described by Equations 3.22 and

4.20 are amenable to stability analysis using sum of squares (SOS) techniques. In

particular, stability in the sense of Lyapunov will be shown for a candidate Lyapunov

function. Additionally, the domain of attraction is estimated.

5.2.1 Problem Formulation

In this section, a methodology, outlined in [25, 46], for finding the domain of attraction

for systems of a specific form is presented. In Section 5.3 it will be shown how this

approach is applicable to the nonlinear closed-loop HCCI dynamics, proving stability

of the control strategy in Chapter 4.


Consider the discrete nonlinear system:

xk = f(xk−1, uk−1) (5.2)

with the following equality constraints:

ei(xk−1, uk−1) = 0, for i = 1, ..., Ne (5.3)

where x ∈ Rn is the state of the system, and u ∈ Rm is a collection of auxiliary

variables (such as non-polynomial functions of states, uncertain parameters, etc.).

The ei’s are polynomial functions in (x, u) and f(x, u) is a vector of polynomial or

rational functions in (x, u) with f(0) = 0.

If a polynomial function of the states V (xk−1) with V (0) = 0, Ψ > 0 and ∆V (x) ≡V (xk)− V (xk−1), such that:

{x ∈ Rn|V ≤ Ψ}\{0} ⊆ {x ∈ Rn|∆V < 0} (5.4)

{x ∈ Rn|V ≤ Ψ}\{0} ⊆ {x ∈ Rn|V > 0} (5.5)

can be found, then the system is asymptotically stable about the fixed point x = 0

and has a region of attraction which includes D = {x ∈ Rn|V (x) ≤ Ψ}. To see this,

just note that for the above to hold, Vk(x)− Vk−1(x) must be negative in the set D.

This proves that D is a region of attraction since a smooth positive function of the

states continually decreases within the set D.

We can pose the problem of showing feasibility of the candidate Lyapunov function

and finding the largest value of Ψ (thus maximizing the stability region defined by

D) as an optimization problem of the form:


maxV ∈Rn,V (0)=0

Ψ

s.t.

{x ∈ Rn|V ≤ Ψ, x 6= 0, w∆V ≥ 0} = φ (5.6)

{x ∈ Rn|V ≤ Ψ, x 6= 0, V ≤ 0} = φ (5.7)

where w(x, u) is a positive polynomial. This is just a set emptiness form of the

constraints given in Equations 5.4 and 5.5.

5.2.2 The Positivstellensatz

By utilizing a theorem from real algebraic geometry called the Positivstellensatz, the

above optimization problem can be re-cast into a form that can be solved via convex

optimization.

Theorem 3: The Positivstellensatz Given sets of polynomials f1, ..., fr, g1, ..., gt,

and h1, ..., hu in Rn+m, the following are equivalent:

1.) The set, (x, u) ∈ Rn+m for which:

f1(x, u) ≥ 0, ..., fr(x, u) ≥ 0 (5.8)

g1(x, u) 6= 0, ..., gt(x, u) 6= 0 (5.9)

h1(x, u) = 0, ..., hu(x, u) = 0 (5.10)

is empty.

2.) There exist k1, ..., ku ∈ Z+, si ∈ Σn , rk ∈ Rn+m such that


f +g2 + h = 0 (5.11)

where: (5.12)

f = s0 +r∑

i1=1

si1fi1 + · · ·+r∑

i1=1

· · ·r∑

ir=1

si1···irfi1 · · · fir (5.13)

g =(gk11 · · · gku

u

), h =

u∑

k=1

hkrk (5.14)

Proof: See [10] Theorem 4.2.2 2

The set of multivariate polynomials in (x, u) ∈ Rn+m is denoted above as Rn+m,

while Σn+m represents the set of sum of squares polynomials. A multivariate poly-

nomial s(x, u) is sum of squares (SOS) if there exist polynomials p1(x, u), ..., pq(x, u)

∈ Rn+m such that:

s(x, u) =

q∑i=1

p2i (x, u) (5.15)

Note that there is no upper bound given on the required polynomial degrees of the

s’s and r’s and the value of the k’s that might be needed to satisfy Equation 5.11 if

Equations 5.8-5.10 hold, however, there are finite degrees that will.

By recognizing the correspondence between Equations 5.3, 5.6-5.7 and Equations

5.8 and 5.10, the domain of attraction problem can be re-cast as:

maxsi∈Σn,V ∈Rn,V (0)=0

Ψ

s.t.

s1+(Ψ−V )s2−V s3−V (Ψ−V )s4+Ne∑

k=1

rkek+L21 = 0 (5.16)

s5+(Ψ−V )s6+w∆V s7+w∆V (Ψ− V )s8+Ne∑

k=1

rkek+L22 = 0 (5.17)


where the Li(x)s take the form (xk1,i

1 · · · xkn,in ). For the case where f(xk−1, uk−1) is a

rational vector field, such as n(x,u)d(x,u)

, the multiplier w(x, u) > 0 should be chosen so that

w(x, u)∆V (x) is a polynomial. Obviously w(x, u) can be chosen as the denominator

of ∆V (x) as long as it is always positive in the region of interest. For more detail see

[46, 25].

5.2.3 Sum of Square Programs

The above optimization can be cast as a sum of squares program (SOSP). Sum of

squares programs are formulated as follows:

Find the coefficients of:

polynomials ri(x), for i = 1, 2, ..., N1 (5.18)

sum of squares sj(x), for j = 1, 2, ..., N2 (5.19)

with a pre-defined structure, such that:

ak(x)+

N1∑i=1

ri(x)bi,k(x)+

N2∑i=1

sj(x)cj,k(x)=0 (5.20)

for k = 1, 2, ..., N3, where: ak(x), bi,k and cj,k are constant coefficient polynomials.

Using this formulation, the domain of attraction problem as given at the end of

Section 5.2.2, for a given value of Ψ, can be written as a sum of squares program:

Find the coefficients of:

polynomials ri(x), for i = 1, 2, ..., Ne (5.21)

sum of squares sj(x), for j = 1, 2, ..., 8 (5.22)


with a pre-defined structure, such that:

s1+(Ψ−V )s2−V s3−V (Ψ−V )s4+Ne∑

k=1

rkek+L21 = 0 (5.23)

s5+(Ψ−V )s6+w∆V s7+w∆V (Ψ− V )s8+Ne∑

k=1

rkek+L22 = 0 (5.24)

The candidate polynomial Lyapunov function V (x) is chosen as part of the design

process. As long as the chosen degrees of the s and r polynomials are large enough,

the SOS program will show feasibility of Equations 5.23 and 5.17 if Equations 5.4 and

5.5 hold, for a given value of Ψ. A linesearch of Ψ can be made to find the largest

Ψ, Ψmax, that satisfies Equations 5.23 and 5.17. This yields an the estimate of the

domain of attraction:

{x ∈ Rn|V (x) < Ψmax} (5.25)

for a given V (x). Other polynomial Lyapunov functions can be chosen to search for

larger domains of attraction.

Sum of squares programs can be solved using convex optimization, in particular

semi-definite programming (SDP) [47]. Feasibility of a candidate Lyapunov function

and an estimate for the domain of attraction can therefore be obtained efficiently

from SDP.

In Section 5.2 an approach for showing stability and finding a region of attraction

for discrete dynamics represented by rational vector fields is outlined. In the following

section it is shown that the closed-loop HCCI dynamics have this form and are stable

according to this technique.


5.3 Domain of Attraction for the HCCI System

Rewriting the states of the system as:

x1,k−1 = βk−1 (5.26)

x2,k−1 = αk−1 (5.27)

and an auxiliary variable

u1,k−1 = (P1γ

k−1 − P1γ )/P

1γ (5.28)

The nonlinear closed-loop HCCI dynamics depicted by Equations 3.22 and 4.20

can be rewritten as:

x1,k =c11(1 + α + x2,k−1)(P (1 + x1,k−1)− V γ

1

V γ23

) + c12(α + x2,k)P1/γ(1 + u1,k−1)

P(c13(1 + α + x2,k−1)(P (1 + x1,k−1)− V γ

1

V γ23

) + c14(α + x2,k)P 1/γ(1 + u1,k−1))− 1

x2,k =K1x1,k−1 + K2x2,k−1

e1 (xk−1, uk−1) = (1 + u1,k−1)y − (1 + x1,k−1)

z = 0

where c11, c12, c13 and c14 are constants:

c11 = (1− ε)LHVC3H8 + cvTin

(V1

V23

)γ−1

(5.29)

c12 = χ(1− ε)LHVC3H8

(V1

V23

)γ

(5.30)

c13 = cvTin (5.31)

c14 = χ(1− ε)LHVC3H8 (5.32)


Here y and z are the smallest integers such that y/z = γ (with γ=1.4, y = 7 and

z = 5). Note that the system dynamics are represented by rational vector fields and

are therefore amenable to the stability analysis approach outlined in Section 5.2. The

multiplier w(x, u) is set equal to the denominator of ∆V (x), so that w(x, u)∆V (x) is

a polynomial. For the region of interest in the system state space, the denominator

of ∆V (x) is always positive, so that w(x, u) is always positive, as required.

A domain of attraction using sum of squares decomposition can be found, as

outlined in Section 5.2. In order to solve the problem, the sum of squares Toolbox for

Matlab (SOSTOOLS, [48]) is used. This software package automates the conversion

from the sum of squares program to SDP, calls the SDP solver (SeDuMi, [51]), and

converts the SDP solution back to the form of the original sum of squares program.

5.3.1 Quadratic Lyapunov Function

Using a candidate Lyapunov function

Vquad(x) = x21 + 0.135x2

2

a linesearch of Ψmax for the SOSP given in Section 5.2.3 yields a Ψmax of 0.21. Fig-

ure 5.1 shows the phase plot for the system with the level curve corresponding to

Vquad(x) = 0.21. This corresponds to a region of attraction guaranteed through use

of the technique outlined in Section 5.2. The shaded area shows the typical operating

range of the HCCI engine. This method guarantees stability over the vast majority of

that region. Physically this means that regulation about the desired operating point

is guaranteed, even when the system is perturbed away from the local equilibrium

region.


1. 5 1 0. 5 0 0.5 1 1.50. 8

0. 6

0. 4

0. 2

0

0.2

0.4

0.6

0.8

normalized peak pressure, x2

rea

tan

t/p

rod

uc

t m

ola

r ra

tio

, x1

Figure 5.1: Quadratic level set Vquad(x) = 0.21 with vector plot (only directionshown): shaded region is typical operation region

5.3.2 Quartic Lyapunov Function

In order to show stability over the entire operating range, a more complex quartic

candidate Lyapunov function is used:

Vquart(x) = x41 + 0.005x4

2

The SOSP given in Section 5.2.3 yields a Ψmax of 0.0175 in this case. As shown in

Figure 5.2 the resulting region of attraction, Vquart(x) < 0.0175, guarantees stability

over the entire operating regime.


1. 5 1 0. 5 0 0.5 1 1.5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

normalized peak pressure, β

rea

tan

t/p

rod

uc

t m

ola

r ra

tio

, α

Figure 5.2: Quartic level set, Vquart(x) = 0.0175, with vector plot (only directionshown): shaded region is typical operation region

While the stability and region of attraction appear to be easily estimated visu-

ally by inspection of the phase plot, there are several issues with relying on a visual

inspection. First, relying on a visual inspection is prone to error, while the method-

ology outlined in this chapter provides a mathematical guarantee. Furthermore, once

the system dynamics are described by more than two states, phase plot visualization

become nearly impossible. In future work, the stability of the full dynamics (includ-

ing the combustion timing and effective compression ratio dynamics) will be assessed

using the technique outlined in this chapter.


5.4 Conclusion

A guarantee of closed-loop stability of the LQR controller (synthesized from the lin-

earized peak pressure dynamics in the control model) in closed loop with the nonlinear

peak pressure dynamics, is shown by invoking convex optimization, the Positivstellen-

satz and sum of squares programs. This proves that the linear controller will stabilize

the nonlinear dynamics over the operating range, a result that is not possible by just

considering the linearized dynamics.

Chapter 6

Decoupled Control of HCCI

In Chapter 4, cycle-to-cycle peak pressure control using a physics-based approach is

demonstrated. Due to the self-stabilizing nature of the residual-affected HCCI, the

combustion timing remains fairly constant for the engine studied. This chapter builds

on that approach with the development of a simple, experimentally implementable

approach to dual peak pressure and combustion timing control.

The simplified strategy outlined in this chapter is to approximately decouple the

control of peak pressure and combustion timing by controlling them on separate time

scales with different control inputs (inducted composition and effective compression

ratio, respectively). By designing the combustion timing controller to be notably

slower than the cycle-to-cycle control of peak pressure, the effect of cycle-to-cycle

combustion timing variation on peak pressure can be neglected. Combustion timing

and intake valve closing (IVC) become slowly varying parameters in the peak pressure

dynamics. This simplifies the peak pressure control problem, allowing the use of an

approach very similar to the pressure control strategy in Chapter 4. Since combustion

timing in residual-affected HCCI is more dependent on IVC than inducted gas com-

position, IVC is used to modulate combustion timing. The combustion timing control

gains are intentionally selected to achieve transient responses that are slightly slower

86

CHAPTER 6. DECOUPLED CONTROL OF HCCI 87

than the cycle-to-cycle pressure control. While modulation of IVC does influence

in-cylinder pressure, the cycle-to-cycle pressure controller compensates for it.

Experimental results show accurate tracking of both combustion timing and peak

in-cylinder pressure. Controlled combustion timing and peak pressure response times

are on the order of 30 (2 seconds) and 5 engine cycles (0.3 seconds), respectively,

validating the claim that these two system outputs can be controlled simultaneously

on different time scales. A strategy for extending the peak pressure control strategy to

the direct, model-based control of work output is also presented. Experimental control

results show the capability of explicit cycle-to-cycle regulation of work output.

6.1 Control Approach

In Chapter 3 the combustion event is modeled as a constant-volume process at an

in-cylinder volume of V23, producing the major products of combustion at an ele-

vated temperature and pressure. The integrated Arrhenius rate threshold model of

combustion is used as a simple and accurate way to mathematically describe HCCI

combustion timing. The model takes the form:

Kthresh =

∫ θcomb

IV C

exp(−Ea/(RuT ))[fuel]a[O2]bdθ/ω (6.1)

where ω is the engine speed and θcomb corresponds to the combustion timing, such

that V23 = V (θcomb). The values A, Ea/Ru, a, b and n are empirical parameters

determined from combustion kinetics experiments for the particular fuel. Kthresh is set

at one experimental operating condition. Note the dependence of combustion timing

on in-cylinder temperature, reactant concentration and the start of compression.

A very interesting aspect of residual-affected HCCI, captured by Equation 6.1,

is the self-stabilizing nature of the process due to the competing affects of reactant


concentration and mixture temperature. For increasing amounts of hot reinducted

exhaust, the reactant concentration drops while the mixture temperature increases.

This trend causes little to no change in combustion timing as inducted composition is

varied. On the other hand, effective compression ratio via IVC (dictating final valve

closure) has a direct and substantial affect on combustion timing. As the amount of

compression is increased, the integrand increases, causing Equation 6.1 to be satisfied

for lower values of the upper limit, the combustion timing. In other words, increasing

effective compression ratio leads to earlier combustion timing. For this reason IVC is

used as a direct control input for combustion timing.

6.2 Controller Development

A variety of closed-loop controllers can be synthesized to track desired values of peak

pressure and combustion timing. A fairly simple approach is to try to approximately

decouple the regulation of peak pressure and combustion timing by controlling them

on separate time scales with different control inputs: inducted composition and ef-

fective compression ratio (ECR), respectively (Figure 6.1). The approach outlined

here is to control the peak pressure on a cycle-to-cycle basis. This is accomplished

in this chapter by using exhaust valve closing (EVC) to modulate the inducted gas

composition. As EVC is delayed, the exhaust valve is open for an increased amount

of time during induction, leading to an increase in the molar ratio of reinducted prod-

ucts to reactants. Combustion timing control is achieved on an intentionally slower

time scale through use of IVC to vary the effective compression ratio. This is justified

by the observation in Section 6.1 that combustion timing is directly affected by the

amount of compression.


αdes

Closed-loop

controlled

VVA system

IVCdes

EVCdes

Single

Cylinder

EngineV

1

actual

αactual

P3

des

AOPdes

Pmeasured

AOPmeasured

valve

timing

map

IVCdes

cycle-to-cycle

peak pressure

controller

"slow"

combustion

timing

controller

IVC

dynamically de-coupled

peak pressure and

combustion timing

control

1

z

1

z

Figure 6.1: Control Strategy for Simultaneous Decoupled Control of Peak Pressureand Combustion Timing

6.2.1 Combustion Timing Control

Previous results [5] have shown that IVC timing can effectively control combustion

timing. In that study, a simple proportional-integral (PI) control scheme controlled

combustion timing with responses on the order of 10 cycles. Due to the successful

implementation of this approach elsewhere, a slow PI combustion timing controller

has been adopted here as well, with the form:

utc,k = utc,k−1 + Kp(etc,k − etc,k−1) + KIetc,k (6.2)

where the intake valve closing time on cycle k is:

IV Ck = ¯IV C + ∆IV C = ¯IV C + utc,k (6.3)


and

etc,k = AOPmeasuredk − AOP desired

k (6.4)

is the error in the combustion timing (angle of peak pressure (AOP)) on cycle k. In

order to choose controller gains, the IVC to combustion timing dynamics around a

desired operating point are simplified to a static gain by dividing the steady state

change in combustion timing by the IVC change required to achieve it. The PI com-

bustion timing control gains are selected via pole placement to achieve a response time

of about 2 seconds. Since the relationship between final valve closure and effective

compression ratio are approximately parabolic about the bottom dead center piston

position [8], care must be take in the linearization of the combustion timing/IVC rela-

tionship. In this chapter, the IVC values used occur well after the bottom dead center

piston position, resulting in a monotonic relationship between final valve closure (via

IVC) utilized and combustion timing, making a linearization appropriate.

6.2.2 Peak Pressure Control

In order to address the need for mean tracking and a reduction in cycle-to-cycle

variation of peak in-cylinder pressure while bounding the “energy” of the control

input, a local H2 controller is synthesized from a linearized version of the peak pressure

model. This approach allows limitations of the control inputs to be handled, by

bounding the amount and speed at which they are actuated. The following sections

outline the linearization and control synthesis approaches.

6.2.3 H2 Control Formulation

The peak pressure model, Equation 3.22, can be linearized about an operating point

(α, P , V23, V1). At this point, the combustion timing (V23) and IVC (V1) are considered

slowly varying parameters in the peak pressure model since they are modulated at


slower time scales. Thus the linearization for the peak pressure dynamics outlined in

Chapter 4, Equations 4.14-4.17, is appropriate for use here.

From Equations 4.14-4.17 an H2 control strategy is developed. The general feed-

back control problem formulation is shown in Figure 6.2, where the weights on state

noise, measurement noise, and performance noise are depicted in detail. The standard

H2 optimal control problem is to find a stabilizing feedback controller which mini-

mizes the H2 norm from the system “noise” inputs, w, to “performance” outputs,

z.

zu

Apc

zx

zr

xr

z

x

xr

w

u

Bpc

Cpc Dpc

H2 controller

R1/2

W1/2

z-1 A

Wu(z)

C

V1/2

Q1/2 Wx(z)

Wr(z)

Generalized Plant

xk

wn

wd

++

++++ +

-

B

xk+1

S1/2

+

Figure 6.2: General control configuration considered for the synthesis of the in-cylinder peak pressure controller

The noise weights, W and V, depict the variances of the state (i.e. “process”)

and output (i.e. measurement) noise. The system control input, states and tracking

error are each weighted with a constant (R, Q and S, respectively) and frequency

dependent transfer function (Wu(z), Wx(z) and Wr(z), respectively). In order to

stress the desire for tracking, the transfer function for tracking error, Wr(z), is a low

pass filter. This introduces the same sort of effect that the integrator portion of a


PID controller achieves. In order to reduce cyclic dispersion, the portion of the state

weighting transfer function which corresponds with the normalized peak pressure,

Wx(z), is a high pass filter. This weights the higher frequency components of the

peak pressure, yielding a control law which attenuates cycle-to-cycle variation. In a

similar manner, the control input transfer function weighting, Wu(z), is a high pass

filter, emphasizing the fact that a rapidly changing control input is not desirable.

Overall, the formulation shown in Figure 6.2 allows a tradeoff to be made between

mean tracking, cycle-to-cycle variation reduction and control effort. Figure 6.3 shows

the performance weights (RWu(z), QWx(z) and SWr(z)) used in this study.

-150

-100

-50

0

50

Mag

nit

ude

(dB

)

0.01

on tracking error

on control input

on normalized pressure

Frequency (1/ engine cycle)10.1

Figure 6.3: The frequency dependent weights used for synthesis of the peak pressurecontrol

From the solution of the H2 synthesis problem, the controller gains (Apc,Bpc,Cpc)

are found. The general form of the H2 peak pressure controller is:

xpc,k+1 = Apcxpc,k + Bpc

αk

βk

βdesired,k

(6.5)

αk = Cpcxpc,k (6.6)


parameter symbol value unitsequivalence ratio φ 0.93 —

engine speed ω 1800 rpmstroke acyl 9.2 cm

connecting rod length Lcyl 25.4 cmbore diameter Bcyl 9.7 cm

compression ratio 15.5 —valve diameter, intake Ar,i 4.8 cm

valve diameter, exhaust Ar,e 4.3 cmexhaust manifold diameter De 5 cmvalve rise/fall durations 90 CAD

intake valve closing IVC 210 CADexhaust valve opening EVO 480 CAD

Table 6.1: Engine Parameters

As shown in Equations 6.5 and 6.6, the inducted gas composition, α = α + α

is the output of the peak pressure controller. The inputs to the controller are the

inducted gas composition from the previous cycle, the normalized peak pressure from

the previous cycle βk and the desired normalized peak pressure on the current cycle.

Note that the combustion timing (i.e. V23) and IVC (i.e. V1) are slowly varying pa-

rameters in the peak pressure dynamics, Equation 3.22. In this study, these dynamics

have been linearized about a single operating point (α, P , θ23, θ1)=(0.68, 59.7, 366, 225).

It is likely that a parameter-varying control strategy directly accounting for variation

in V23 and V1 would improve the control response, albeit at the expense of additional

complexity.

6.3 Experimental Implementation

The controller has been experimentally implemented on the single cylinder research

engine with general engine characteristics given in Table 6.1. Note that the com-

pression ratio used is different than that used for work presented in the Chapters


2-4. This difference is due to changes made to the research engine piston geometry

to achieve higher compression ratios during the evolution of the thesis research. As

a consequence, the operating region (i.e. IVO, EVC and IVC) is different than the

configuration used in the previous chapters. The valve timing to inducted gas com-

position map is also different, as shown in Figure 6.4. In this case only EVC is varied

to modulate inducted gas composition. Despite these differences in compression ratio

and valve map, the modeling and control approach is still applicable, a characteristic

and benefit of using a physics-based strategy. Experimental results are shown in Fig-

165175185195

induct

ed g

as c

om

posi

tion, α

=N

p/N

r

exhaust valve closing, EVC

1.3

1.1

0.9

0.7

Figure 6.4: Effect of valve timing on inducted gas composition

ures 6.5 and 6.6. Figure 6.5 shows the benefit of simultaneous control. If only control

of combustion timing is considered (plot on left), then there is no regulation of peak

in-cylinder pressure. In order to hold a desired peak pressure while modulating the

desired timing, simultaneous control of both timing and peak pressure (plot on right)

must be used. Figure 6.6 shows the decoupled controller performance over a range

of desired pressures and combustion timings. Peak pressure and combustion tracking

responses occur within 5 and 25 engine cycles, respectively. Using this scheme rapid

modulation of work output is possible while maintaining combustion timing within a

desired region.


369

375

40

60

150

160

0 20 40 60

220

230

240

time [sec.]

IVC

EV

Cp

k.

pre

ss.

[

atm

]A

OP

0 20 40 60time [sec.]

Figure 6.5: Comparison of system response with combustion timing control only (a)and both combustion timing and peak pressure control simultaneously (b)

6.4 Extension to Work Output Control

In the previous sections, the HCCI model motivates a decoupled pressure and com-

bustion timing control strategy. In practice however, it may be more useful to directly

regulate work output and combustion timing. This section outlines a simple model

of work output for the HCCI process and shows how work output control is a simple

extension of peak pressure control in the case where ECR and combustion timing are

slowly varying parameters.

In a piston engine, the work output is due to boundary work, W =∫

PdV , which

can be split into contributions before combustion (BC) and after combustion (AC)

(maintaining the assumption of a constant volume combustion process):

W =

∫PdV =

∫ Vcomb

VIV C

PBCdV +

∫ VEV O

Vcomb

PACdV (6.7)

Assuming isentropic compression and expansion processes, the pressure before and


369

375A

OP

40

60

80

pk

pre

ss.

160

170

EV

C

0 20 40 60 80

220

240

IVC

time [seconds]

pressure control ON

timing control ON

Figure 6.6: Simultaneous control of both peak pressure and combustion timing

after the combustion event can be modeled as:

PBC(θ) =PatmV γ

IV C

V (θ)γPAC(θ) =

PpkVγcomb

V (θ)γ(6.8)

Substituting these expressions into Equation 6.7, and evaluating yields the following

model of work output:

W =PatmV γ

IV C

(V 1−γ

comb − V 1−γIV C

)+ PpkV

γcomb

(V 1−γ

EV O − V 1−γcomb

)

1− γ(6.9)

As expected, the work output depends on the system states (peak in-cylinder pressure

and combustion timing) and control input, IVC. Again, combustion timing and ECR

(via IVC) are considered slowly varying parameters in Equation 6.9 since the timing

control via IVC modulation is achieved at an intentionally slower time scale than

the work output control. Under the assumptions of isentropic compression, constant


volume combustion and isentropic expansion, Equation 6.9 shows that work output

is strongly dependent on peak in-cylinder pressure in a linear fashion. With this

linear dependence on peak pressure, the work output control problem is simply an

extension of peak pressure control. Figure 6.7 shows successful experimental results

of direct work output control utilizing the same H2 control framework outlined in

Section 6.2.2. Desired step responses are achieved within about 5 engine cycles. In

addition, both positive and negative load transients (via a sine function) are tracked.

In each case the combustion timing remains nearly fixed, with a modest amount of

additional cyclic dispersion.

6.5 Conclusion

This chapter augments the peak pressure control results from the previous chapter

by adding the capability to directly control work output and combustion timing. The

scheme approximately decouples the cycle-to-cycle dynamics of combustion timing

and peak in-cylinder pressure (or work output) by controlling them on separate time

scales with different control inputs - inducted composition and ECR, respectively.

The cycle-to-cycle control is formulated from a physics-based H2 framework. Timing

controller gains are selected via pole placement to achieve a response time that is

slightly slower than the pressure controller. Although in-cylinder pressure depends on

the combustion timing and ECR, the effects of their slow variation on peak pressure

can be compensated by the cycle-to-cycle peak pressure controller. Experimental

results show that this simple framework effectively controls peak pressure or work

output on cycle-to-cycle basis while desired combustion timing is achieved over a

slightly slower time scale.


engine cycle0 100 200 300

(b)

(a)

AO

P E

VC

W

ork

Ou

tpu

t

(IM

EP

) [a

tm]

360

380

370

4

3

170

190

AO

P E

VC

W

ork

Ou

tpu

t

(IM

EP

) [a

tm]

360

380

370

4

3

170

190

control ON

control ON

Figure 6.7: Direct control of work output, (a) - step response, (b) - step followed bya sine

Chapter 7

Coordinated Control of HCCI

In Chapter 6 a decoupled approach to peak pressure and combustion timing control

was outlined and validated. An obvious next step is the control of both combustion

timing and peak pressure in a simultaneous, coordinated approach. This is possible

through the direct synthesis of control strategies from the complete control model,

Equations 3.22 and 3.35. In this chapter of the thesis an H2 controller is synthesized

from a linearized version of the full control model. Once implemented, the controller

allows the coordinated control of both combustion timing and peak pressure on com-

parable time scales.

7.1 Control Development

The approach used in this chapter is to synthesize a controller directly from the

complete control model, where inducted gas composition and effective compression

ratio act as control inputs, while the combustion timing and peak in-cylinder pressure

are the outputs being regulated (Figure 7.1). Figures 6.1 and 7.1 show the key

difference between the coordinated and decoupled control strategies. The approach

outlined in this chapter results in one controller, while the decoupled approach from

99

CHAPTER 7. COORDINATED CONTROL OF HCCI 100

Chapter 6 results in two distinct controllers.

αdes

Closed-loop

controlled

VVA system

IVCdes

EVCdes

Single

Cylinder

EngineV

1

actual

αactual

ControllerP

des

AOPdes

Pmeasured

AOPmeasured

valve

timing

mapIVC

des

Single, coordinated

peak pressure and

combustion timing

controller

1

z

1

z

Figure 7.1: Control Strategy for Simultaneous Coordinated Control of Peak Pressureand Combustion Timing

7.1.1 Linearization of Full Control Model

The control model dynamics, Equations 3.22 and 3.35, can be linearized about an

operating point (α, θ1, P , θ23), and put in state space form:


βk

ν23,k

αk

ν1,k

=

�c101+c106

c201c200

�

c100

�c107+c106

c207c200

�

c100

�c103+c106

c203c200

�

c100

�c105+c106

c205c200

�

c100

c201c200

c205c200

c203c200

c205c200

0 0 0 0

0 0 0 0

βk−1

ν23,k−1

αk−1

ν1,k−1

+

�c102+c106

c202c200

�

c100

�c104+c106

c204c200

�

c100

c202c200

c204c200

1 0

0 1

αk

ν1,k

where the constants are functions of system parameters defined in previous chapters:

c100 =cvTin(1 + α) + χ(1− ε)LHVC3H8αP 1/γ (7.1)

c101 =−(

cvTin(1 + α) +χ(1− ε)LHVC3H8αP

1−γγ

γ

) (P− V γ

1

V γ23

)(7.2)

+(1− ε)LHVC3H8

V1

V23

(1 + α)

c102 =−χ(1− ε)LHVC3H8P1/γ

(P− V γ

1

V γ23

)P−1 (7.3)

c103 =

(P− V γ

1

V γ23

)(−cvTin

(P− V γ

1

V γ23

)+ (1− ε)LHVC3H8

V1

V23

)P−1 (7.4)

c104 =

(P− V γ

1

V γ23

)(1 + α)

(cvTin

V γ1

V γ23

γ + (1− ε)LHVC3H8

V1

V23

)P−1 (7.5)

+V γ

1

V γ23

χ(1− ε)LHVC3H8αP1−γ

γ

c105 =(1 + α)V γ

1

V γ23

γ

(−cvTin

(P− V γ

1

V γ23

)+ (1− ε)LHVC3H8

V1

V23

)P−1 (7.6)


c106 =−c104 (7.7)

c107 =−c105 (7.8)

c200 =π2B2

cylacylsin(θ23)Aφa5bexp(− Ea

RT1

V γ−1TDC

V γ−11

)(PatmV1)

a+b

4V23180Kthω(VTDCR25T1(1 + α))a+b(7.9)

c201 =

(a + b− Ea

RT1

V γ−1TDC

V γ−11

) ((1− γ)P − V γ

1

V γ23

)χ(1− ε)LHVC3H8αP 1/γ

γT1

(P− V γ

1

V γ23

)2

(1 + α)2cv

(7.10)

c202 =a + b

1 + α+

a + b− EaRT1

V γ−1TDC

V γ−11

T1(1 + α)

χ(1− ε)LHVC3H8P

1/γ

cv(1 + α)2(P− V γ

1

V γ23

) − Tatm

1 + α

(7.11)

c203 =−a + b− Ea

RT1

(VTDC

V1

)γ−1

T1(1 + α)

χ(1− ε)LHVC3H8αP 1/γ

cv(1 + α)2(P− V γ

1

V γ23

) (7.12)

The above equations yield a linearized version of the full control model peak

pressure and combustion timing dynamics. Model states include the normalized peak

pressure β, the normalized cylinder volume at final valve closure ν23 ≡ (V23−V23)/V23,

the normalized volume at combustion ν23 ≡ (V23 − V23)/V23, and the inducted gas

composition α ≡ Np/Nr. The operating point about which the control model is

linearized is (α, P , θ23, θ1)=(0.68, 59.7, 366, 210).

7.2 H2 Controller Synthesis

From the linearized control model equations, a number of control strategies could be

used. For illustrative purposes an H2 control strategy will be implemented. This

strategy is comparable to the peak pressure portion of the control strategy used in

Chapter 6, but with two inputs, inducted gas composition and effective compression

ratio, and two outputs, combustion timing and peak pressure. In order to synthesize a

controller, a set of frequency dependent weights for the control inputs, system outputs


and tracking errors are designated, as shown in Figure 7.2.

-300

-200

-100

0

100

Mag

nit

ud

e (d

B)

on normalized pressure &

combustion timing

on inducted gas composition

on effective compression ratio

on pressure tracking

on combustion timing tracking

Frequency [1/engine cycle]

1.00.10.01

Figure 7.2: The frequency dependent weights used for synthesis of the H2 controller

From the solution of the H2 synthesis problem, the controller gains (Ac,Bc,Cc) are

found. The general form of the H2 peak pressure controller is:

xc,k+1 = Acxc,k + Bc

αk

θ1,k

βk

θ23,k

βdesired,k

θ23,desired,k

(7.13)

[αk

θ1,k

]= Ccxc,k (7.14)

As shown in Equations 7.13 and 7.14, the inducted gas composition, α = α + α

and the normalized final valve closure volume are the outputs of the controller. The

inputs to the controller are the inducted gas composition and normalized final valve


closure cylinder volume from the previous cycle, the normalized peak pressure and

combustion timing from the previous cycle and the desired normalized peak pressure

and combustion timing on the current cycle.

7.3 Experimental Implementation

The simultaneous, coordinated controller developed in this chapter was implemented

on the same single-cylinder testbed utilized in the previous chapters. During these

tests the engine was in the same configuration as during the decoupled control test-

ing outlined in Chapter 6. For this reason, the same valve timing to inducted gas

composition map was used. Figures 7.3 - 7.6 show results of the experimental im-

plementation. Figures 7.3 and 7.4 exhibit the capability of the control approach to

vary peak pressure while holding combustion timing constant. Figures 7.5 and 7.6

show the ability to simultaneously change both peak pressure and combustion timing.

Figure 7.3 shows characteristic statistical values for the controller performance. Mean

tracking performance for combustion timing and peak pressure are quite good, with

mean errors not exceeding +/-0.4 degrees and +/-0.1 atm, respectively. In general,

the standard deviation values are elevated compared to what would be possible at

equivalent compression ratios on a modern engine with optimized gas exchange. En-

hanced gas exchange with the purpose of homogenizing the reactant/residual mixture

during induction allows more consistent cycle-to-cycle behavior, reducing the cyclic

dispersion. Nevertheless, reductions in the standard deviation for both combustion

timing and peak pressure are seen once the controller is activated around the 90th

engine cycle. The controller therefore provides good mean tracking, with a modest

reduction in cyclic dispersion for these results.

Another benefit of the simultaneous coordination of both control inputs is a re-

duction in the control effort required to elicit the desired response. Instead of using a


peak pressure controller that must compensate for the effects of a combustion timing

controller, and vice versa, the coordinated approach optimizes the use of both control

inputs to regulate both outputs. This is made clear by direct comparisons of Figures

7.3 - 7.6 and Figures 6.5 and 6.6 from Chapter 6. For comparable changes in the

peak pressure and combustion timing, the modulation of inducted gas composition

and IVC are reduced for the coordinated approach.

Figure 7.7 is an exploded view of Figure 7.5, and shows the controller’s capability

to dictate step changes in both combustion timing and peak pressure within about

4-5 engine cycles. This performance is a significant improvement over both the peak

pressure and decoupled control strategies developed in Chapter 4 and Chapter 6,

respectively. Furthermore, the fastest HCCI control responses achieved in other work

are around 4 engine cycles for combusting timing control [8]. The results outlined

here allow responses for both combustion timing and peak pressure (or work output)

to occur within 4-5 engine cycles.

The control approach in this chapter was made possible by synthesizing a con-

troller from the full 2-input, 2-output control model dynamics presented in Chapter 3.

Furthermore, the control model development relies heavily on the intuition and model

validation opportunity proved by the simulation modeling work presented in Chap-

ter 2. The results in this chapter therefore symbolize the culmination of all the work

outlined in the thesis. In addition, these results demonstrate the power of using

physics-based modeling and control, and represent another step toward the practical

implementation of HCCI engines.

7.4 Conclusions

The control-oriented model developed in Chapter 3 leads to a feedback controller that

uses the inducted gas composition and the the effective compression ratio as inputs.


The approach represents a more complete and capable approach to control of residual-

affected HCCI, albeit at the expense of some of the transparency of the approaches

presented in Chapter 4 and Chapter 6. Experimental results shows that the control

strategy is quite effective. Specifically, the results show that the simultaneous control

of both combustion timing and peak pressure on an experimental system is possible.


365

370

375

380

40

60

80

0.6

0.8

160

170

180

220

240

θco

mb

pk p

ress

.

[a

tm]

EV

CIV

Cα

=N

p/N

r

0 150 300 450

0 10 20 30time [seconds]

engine cycles

des.: 368mean: 367.6

st dev: 1.5

des.: 368mean: 368.3

st dev: 1.7

des.: 65mean: 65.1

st dev: 4.4

des.: 60mean: 60

st dev: 4.5

mean: 0.78

st dev: 0.05

mean: 0.88

st dev: 0.05

mean: 170.9

st dev: 2.8

mean: 176.9

st dev: 2.6

mean: 218.8

st dev: 1.8

mean: 223.7

st dev: 1.6

mean: 367.3

st dev: 1.9

mean: 69.3

st dev: 6.5

Figure 7.3: Experimental control result showing negative step change in peak pressurewith constant combustion timing


365

370

375

380

40

60

80

0.6

0.8

160

170

180

220

240

θco

mb

pk

pre

ss.

[a

tm]

EV

CIV

Cα

=N

p/N

r

0 150 300 450

0 10 20 30time [seconds]

engine cycles

Figure 7.4: Experimental control results showing positive step change in peak pressurewith constant combustion timing


750

50

θco

mb

pk p

ress

.

[a

tm]

EV

CIV

Cα

=N

p/N

r

0 150 300 450 600

0 10 20 30 40time [seconds]

engine cycles

365

370

375

380

40

60

80

0.6

0.8

160

170

180

220

240

Figure 7.5: Experimental control result showing simultaneous changes in combustiontiming and peak pressure


365

370

375

380

40

60

80

0.6

0.8

160

170

180

220

240

0 150 300 450 600

0 10 20 30 40

time [seconds]

engine cycles

θco

mb

pk p

ress

.

[a

tm]

EV

CIV

Cα

=N

p/N

r

Figure 7.6: Experimental control result showing simultaneous changes in combustiontiming and peak pressure


365

370

375

380

40

60

80

0.6

0.8

1

160

170

180

220

240

θco

mb

pk pr

ess.

[atm

]EV

CIV

Cα

engine cycles675 725700

Figure 7.7: Zoomed view of Figure 7.5

Chapter 8

Conclusions and Future Work

8.1 Conclusions

Residual-affected HCCI is a promising strategy for increasing efficiency and reducing

NOx emissions in internal combustion engines. In addition to these benefits, there

also exist two significant challenges: the lack of a direct combustion initiator and

cycle-to-cycle coupling through the residual gas temperature.

Although residual-affected HCCI is a complex process, Chapter 2 shows that an

HCCI engine outfitted with VVA can be modeled in a fairly simple and straightfor-

ward way. The emphasis here is on capturing the combustion timing and work output

of the engine with simple and intuitive models, so that a sound understanding about

how variable valve actuation affects these system outputs can be realized. A single-

zone HCCI combustion model, including exhaust manifold dynamics was presented to

achieve this. Given the accuracy, intuition, and simplicity of the integrated Arrhenius

rate approach, it has been selected as the method of choice for relating combustion

phasing to parameters controlled through the use of the VVA system.

While the simulation model presented in Chapter 2 accurately captures the steady-

state, transient and SI-to-HCCI mode transition dynamics, it is cumbersome to use in

112

CHAPTER 8. CONCLUSIONS AND FUTURE WORK 113

the direct synthesis of control strategies. For this reason, a nonlinear low-order control

model that correlates well with both the experiment results and the simulation model

is outlined in Chapter 3. The low-order model was developed by splitting several key

processes that occur during HCCI combustion into discrete steps. These include

constant pressure, adiabatic mixing of inducted reactants and reinducted products

from the previous cycle, isentropic compression up to the point where combustion

initiates, constant volume combustion, isentropic volumetric expansion and isentropic

exhaust.

The resulting model can be linearized about an operating condition and used to

synthesize controllers, such as the LQR controller developed in Chapter 4. Paired with

a map from desired inducted gas composition to required IVO/EVC timing, this rep-

resents a complete approach for varying work output at nearly constant combustion

phasing. Closed-loop simulations with the more complex 10-state HCCI simulation

model from Chapter 2 and experimental results show that despite the large number

of simplifications, the control strategy is quite effective at tracking the desired peak

pressure at nearly constant combustion timing, allowing modulation of work output.

Chapter 5 outlines a strategy for proving the stability of the LQR controller,

synthesized from a linearized version of the peak pressure dynamics, in closed-loop

with the nonlinear peak pressure dynamics. This result proves that the LQR controller

will stabilize the nonlinear dynamics over the entire operating range, a result that is

not possible by only considering the linearized dynamics.

In Chapter 6 the approach outlined in Chapter 4 was extended to include feed-

back control of combustion timing. This is accomplished by controlling the peak

pressure (or work output) and combustion timing on different time scales with differ-

ent actuators, inducted gas composition and effective compression ratio, respectively.

Since combustion timing in residual-affected HCCI is more dependent on amount of

compression (via IVC in this study) than inducted gas composition, IVC is used to


modulate combustion timing. The combustion timing controller is designed to be

notably slower than the cycle-to-cycle control of peak pressure, so that the effect of

cycle-to-cycle combustion timing variation on peak pressure can be neglected. Com-

bustion timing and amount of compression become slowly varying parameters in the

peak pressure dynamics. This simplifies the peak pressure control problem, allowing

the use of an approach very similar to the pressure control strategy in Chapter 4.

Chapter 7 outlines a strategy for the simultaneous, coordinated control of combus-

tion timing and peak pressure on the same time scale through modulation of inducted

gas composition and effective compression ratio. The controller is directly synthesized

from a linearized version of the control model developed in Chapter 3. This strategy

represents the most capable control approach presented in the thesis since it allows

the coordinated regulation of combustion timing and peak pressure (or work output)

within 4-5 engine cycles.

8.2 Future Research Efforts

Closed-loop stability of the peak pressure controller (outlined in Chapter 4) with the

complete nonlinear peak pressure dynamics was shown in Chapter 5 through use of

concepts from convex optimization, sum of squares analysis and real algebraic geom-

etry. In future studies, nonlinear stability analysis of the decoupled and coordinated

control strategies will be carried out using the same framework as that outlined in

Chapter 5.

In addition, while the modeling strategies outlined in Chapter 2 and Chapter 3

show good correlation with deterministic behavior of residual-affected HCCI, stochas-

tic cycle-to-cycle variations of the combustion process (or cyclic dispersion) have not

been explicitly included. One approach would be to develop a noise model to char-

acterize the dispersion seen on the experiment. This will enable the implementation


of even more capable control techniques.

Having shown that the reduced-order modeling approach outlined in Chapter 3

is applicable for controller design, next steps involve extending it to variable engine

speed and multi-cylinder operation. Additionally, although the LQR and H2 control

schemes show good results, other control strategies are available, and may further

enhance system performance for this application.

Furthermore, the current state of the art in direct injection technology allows the

use of injected fuel mass as a control input on a cycle-to-cycle basis. The modeling

and control approaches outlined in the thesis can be extended to include in-cylinder

injection of fuel. The relative merits of using inducted gas composition, effective

compression ratio or injected fuel mass could then be assessed in a physically based

analysis.

8.3 The Future of HCCI

At the time of the writing of this thesis, several automobile manufacturers have near-

term plans for the on-road application of HCCI. It is the authors opinion that people

will have the option to drive a car with an HCCI engine within the next 10 years.

This is possible because of the hard work of many dedicated individuals, companies

and government agencies, in addition to the convergence of key developments in

actuation, sensing and in-vehicle computer technology. It is the author’s hope that

the work outlined in this thesis has played some small role in this large effort to bring

HCCI to our highways.

Appendix A

Alternative Exhaust Manifold

Model

This appendix presents an alternative model of exhaust manifold heat transfer to

that outlined in Section 3.1.1. In the control model the reinducted product species

are assumed to have a temperature, T1prod,k, that is directly related to the temperature

of the exhausted products from the last cycle, T5,k−1. In the simple model presented

in Section 3.1.1, the relationship is approximated as:

T1prod,k = χT5,k−1 (A.1)

and is meant to represent heat transfer from the hot product gas to the exhaust

manifold prior to the reinduction of a portion of the gas during the subsequent engine

cycle. In Chapter 3 a value of χ = 0.94 was used to calibrate the model, resulting in

Equation A.1 becoming:

T1prod,k = 0.94T5,k−1 (A.2)

In this appendix a more physically motivated approach is taken, although still

116

APPENDIX A. ALTERNATIVE EXHAUST MANIFOLD MODEL 117

resulting in an equation very similar to Equations A.1 and A.2.

A.1 Model Development

As outlined in Section 2.1.5 a governing equation for the internal energy of the gases

in the exhaust manifold can be expressed as:

ue =1

meγ

[mce (hc − he) + heAe (Tambient − Te)

](A.3)

With the exhaust valve closed, and with the assumption that flow leaving the exhaust

manifold is negligible, Equation A.3 reduces to:

ue =1

meγ

[heAe (Tambient − Te)

](A.4)

With the constant specific heats assumption implemented in the control modeling,

we have:

ue = cvT (A.5)

Applying Equation A.5 to Equation A.4 yields:

T =−heAe

cpme

(T − Tambient) (A.6)

which has the solution at time, tf :

T (tf ) = Tambient + (T (t = t0)− Tambient) exp(−heAe

cpme

tf ) (A.7)

Within the control model framework the initial temperature is:

T (t = t0) = T5,k−1 (A.8)


The time, tf represents the length of time the hot exhaust gases are in the manifold

prior to reinduction. T1prod,k is therefore given by:

T (tf ) = T1prod,k (A.9)

Application of this result with Equations A.7-A.9, gives:

T1prod,k = k1T5,k−1 + k2 (A.10)

where:

k1 = exp

(−heAe

cpme

tf

)(A.11)

k2 =

(1− exp

(−heAe

cpme

tf

))Tambient (A.12)

are assumed constant during the period of time that subsequently reinducted com-

bustion products are in the exhaust manifold. The exhaust volume is related to the

mass through the ideal gas assumption:

Ve =meRTe

MWe patm

(A.13)

With a diameter, De, the heat transfer area can be related to the exhaust volume, as

Ae = 4Ve/De. Combing this equation with Equation A.13 yields:

heAe

cpme

tf =he

cp

RuTe

DeMWePatm

tf (A.14)


Plugging Equation A.14 into Equations A.15 and A.16 gives:

k1 = exp

(−he

cp

RuTe

DeMWePatm

)tf (A.15)

k2 =

(1− exp

(−he

cp

RuTe

DeMWePatm

tf

))Tambient (A.16)

The approximated exhaust manifold residence time tf for the portion of hot combus-

tion gas that is reinducted can be calculated from the exhaust valve opening (EVO)

and intake valve opening (IVO) times as:

tf =(720− EV O) + IV O

720∗ tcycle (A.17)

where tcycle is the time duration of one engine cycle. At 1800RPM, tcycle = 1/15seconds.

Utilizing values of exhaust manifold diameter De = 0.05cm and heat transfer coeffi-

cient he = 72W/(m2 K) from Table 2.2, average exhaust gas temperature Te = 655K

and IV O = 45 from case 3 in Table 2.4, and EV O = 480 and Tambient = 400K

yields k1 and k2 values of 0.98 and 4.10, respectively. Substituting these values into

Equation A.10 give:

T1prod,k = 0.98T5,k−1 + 4.10K (A.18)

Equation A.10 is a more physically motivated model of exhaust manifold heat transfer

than Equation A.1, however, note the strong similarity, both in their forms (differing

by only a bias) and the T1prod,k values calculated, as shown in Figure A.1. Although

Equation A.1 ultimately works quite satisfactorily, and exhibits the same basic be-

havior as Equation A.10, Equation A.10 can be substituted into the control modeling

formulation if desired.


300 400 500 600 700250

350

450

550

650

750

Calc

ula

ted T

1,p

rod,k

T5,k-1

Simple model

Physically-motivated

model

Figure A.1: Comparison of two heat transfer models for usage in the control modeldynamics, simple model - Equation A.2, physically-motivated - Equation A.18

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Documents

physics-based modeling and control of residual-affected hcci