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Doctoral Thesis

Optimal operation of CVT-based powertrains

Author(s): Pfiffner, Rolf Andreas Josef

Publication Date: 2001

Permanent Link: https://doi.org/10.3929/ethz-a-004142150

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Diss. ETH No. 14136

Optimal Operation of

CVT-Based Powertrains

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGY

ZÜRICH

for the degree of

Doctor of Technical Sciences

presented by

Rolf Andreas Josef Pfiffner

Dipl. El.-Ing. ETH

born 18. January 1969

citizen of Mels and Mels-Weisstannen SG

accepted on the recommendation of

Prof. Dr. L. Guzzella, examiner

Prof. Dr. H. P. Geering, co-examiner

2001

contact: [email protected]

X-J -L H r/V lui O

page size: DIN A5

documentclass: book

standard character size: 10 pt

packages: ainsinath, caption2, graphicx, harvard, here, latexsymRolf Pfiffner

2001

LITTERA SCRIPTA MANET

ROSS KING, Ex-Libris

To Tamara, with love

ROGFZNTLXYDAWEUT

V

Preface

This thesis is based on my research performed at the Engine SystemsLaboratory of the Swiss Federal Institute of Technology (ETH) in

Zürich between 1996 and 2001.

It was part of the PALOS (PArt-Load Optimized System) projectwhich was financially supported by the Swiss Federal Office of Energy(BFE), Grant No. 15752.

I would like to thank my supervisor, Professor Dr. Lino Guzzella,

for the organization of the funding and for providing guidance duringthe course of this work as well as Professor Dr. Hans Peter Geering for

accepting to be my co-examiner. My sincere thanks go to Dr. Chris

Onder for his sustained interest in my work and for the numerous

technical discussions with him.

I am also grateful to the whole staff of the Engine Systems Lab

for the great time we had together. I will never forget the fellowshipand the amiable atmosphere in the office we had during this time.

I want to extend my special thanks to Elena Gortona and Patrik

Soltic for their natural helpfulness during our project. Further I am

indebted to Brigitte Rohrbach for her many helpful comments duringthe proofreading of the text.

My overwhelming gratitude is kept finally for my wife Tamara.

Her daily support and encouragement prevented me from abandoningthe project during difficult times. Nevertheless, it was a fascinatingand challenging project that taught me a lot.

Zürich, March 2001

Inhaltsverzeichnis

Abstract

Zusammenfassung

Abbreviations, Acronyms, Symbols, Subscripts

Xll

xvi

1 Introduction

1.1 Motivation for Using CVTs

1.2 Classical vs. Optimal Control Approaches .

1.3 Contribution

1.4 Structure of this Thesis

1

1

3

4

5

2 Classical Control Approaches2.1 "Quasi Static" Operation

2.2 Transient Operation

2.2.1 Strategy No. 1: Speed Envelope , . .

2.2.2 Strategy No. 2: Single Track ....

2.2.3 Strategy No. 3: Off the Beaten Track

2.3 Summary of Chapter 2

3 Powertrain Modeling

3.1 Engines3.1.1 Downsized Supercharged Engine . .

3.1.2 Conventional Engine

3.2 CVT

3.2.1 Detailed Model

3.2.2 Simplified Model

7

8

9

10

12

15

17

19

20

20

23

25

26

40

viii

3.3 Powertrain 45

3.4 Combined System 48

3.5 Summary of Chapter 3 48

4 "Quasi Static" Optimization 51

4.1 Problem Definition 51

4.2 Without CVT Losses 53

4.3 Considering the CVT Losses 56


5 Optimal Transient Operation 61

5.1 Problem Definition 62

5.2 A Direct Collocation Method 64

5.3 Optimal Solution 68

5.3.1 Downsized Supercharged Engine 69

5.3.2 Conventional Engine 73

5.4 Common Features 75

5.5 An Applicable Siiboptimal Strategy 77

5.6 Efficiency Improvements Estimation 79


6 Conclusions and Outlook 83

IX

Abstract

This thesis analyzes the fuel-optimal operation of powertrains that

are equipped with a Continuously Variable Transmission (CVT). Alt¬

hough the use of CVTs is commonly considered as a promising ap¬

proach for the improvement of the fuel economy of passenger cars, the

operation mode that achieves minimum fuel consumption is known

for stationary vehicle conditions only. For transient vehicle conditi¬

ons, however, three approaches have been proposed so far. All of them

rely on heuristic arguments, and the reasoning for fuel-optimality is

based on "quasi static" considerations only. A truly dynamic solution

is not yet available.

In order to fill this gap, a detailed model of the complete power-

train is developed and a meaningful performance index (cost function)is defined. The fuel-optimal control problem for transient vehicle ope¬

ration is brought into an appropriate mathematical framework, i.e.,it is formulated as a nonlinear optimal control problem which is then

solved for various predefined vehicle trajectories using state-of-the-art

numerical optimization software.

These investigations are carried out for a representative of the

class of lightweight passenger cars. Within these investigations, two

different engines types are considered: one is a conventional SI engineand the other is a downsized supercharged (DSC) SI engine. In both

cases, the optimal solutions are shown to be superior to standard

CVT control algorithms, yielding larger gains in fuel economy for

DSC engines.The optimal control problem has been solved for various different

transient vehicle trajectories. Common features of these solutions are

identified. They have been used to derive a causal suboptimal control

strategy that realizes almost the same benefits in fuel economy, but

requires a substantially reduced computational effort. Hence, it is

realizable on-line.

On the basis of a comparison between the suboptimal control stra¬

tegy and the classical control strategies, some adjustments for the

X

well-known classical single track modified control strategy are propo¬

sed. The resulting improvements in fuel consumption are similar to

those of the suboptimal control strategy.

XI

Zusammenfassung

Das Thema der vorliegenden Arbeit ist der verbrauchsoptimale Be¬

trieb von Fahrzeugantrieben, die mit einem stufenlosen Getriebe aus¬

gestattet sind. Obwohl die Anwendung von stufenlosen Getrieben als

ein vielversprechender Ansatz zur Verbrauchsverbesserung von Per¬

sonenkraftwagen angesehen wird, ist die verbrauchsoptimale Strate¬

gie nur für stationäre Fahrzustände bekannt. Für transiente Fahr¬

vorgänge wurden bis anhin drei Steuerstrategien angewendet, die auf

heuristischen Überlegungen aufgebaut sind. Die Grundlage dieser

Strategien besteht in der Annahme von „quasistationären" Fahrzeug¬bedingungen. Die dynamische Lösung war bisher nicht bekannt.

Zur Lösung des dynamischen Problems wurde ein Modell des ge¬

samten Fahrzeugantriebes entwickelt sowie ein sinnvoller Güteindex

(Kostenfunktion) definiert. Durch diese Massnahmen konnte das dy¬namische Problem in eine geeignete mathematische Darstellung über¬

führt werden, d.h. es wurde als nichtlineares dynamisches Optimie¬rungsproblem formuliert. Die Lösung dieses Problems wurde unter

Verwendung moderner Optimierungssoftware für verschiedene Fahr-

zeugtrajektorien berechnet.

Die Berechnungen wurden für einen typischen Vertreter der Klein¬

wagenklasse durchgeführt, wobei zum einen ein konventioneller Ver¬

brennungsmotor, zum anderen ein hubraumverkleinerter und aufgela¬dener Motor betrachtet wurde. Die optimale Lösung ergibt für beide

Motorkonzepte im Vergleich zu konventionellen Steuerstrategien eine

Verbrauchsminderung. Die Einsparung ist im Falle des hubraumver-

kleinerten und aufgeladenen Motors grösser als für den konventionel¬

len Motor.

Anhand der optimalen Lösungen für verschiedene Fahrzeugtrajek-torien wurden Charakteristika der optimalen Lösung identifiziert, wel¬

che die Herleitung einer suboptimalen Steuerstrategie erlaubten. Die

suboptimale Steuerstrategie ist im Gegensatz zur Berechnung der op¬timalen Lösung in Echtzeit realisierbar, wobei die Verbrauchseinspa¬rung nur marginal abnimmt.

Xll

Der Vergleich der suboptimalen Steuerstrategie mit den klassischen

Steuerstrategien ermöglichte eine Weiterentwicklung der bekannten

single track modified Steuerstrategie, Mit diesem modifizierten Algo¬rithmus können ähnliche Verbrauchseinsparungen wie bei der subop¬timalen Steuerstrategie erreicht werden.

Xlll

Abbreviations, Acronyms

CVT Continuously Variable Transmission

DBW Drive-by-WireDIRCOL Direct Collocation Method

DSC Downsized SuperchargedECU Engine Control Unit

EHV Electrohydraulic Valve

ETH Eidgenössische Technische Hochschule

(Swiss Federal Institute of Technology)FEM Finite-Element Method

FTP Federal Test Procedure

FTP75 U.S. city cycleFWV Four-Way Valve

IC Internal Combustion

IEEE The Institute of Electrical and Electronics Engineers, Inc.

MTL Maximum Torque Line

NLP Nonlinearly Constrained Optimization Problem

ODE Ordinary Differential Equation

PWSC Pressure-Wave Supercharger

SI Spark Ignition

SQP Sequential Quadratic Programming

TCU Transmission Control Unit

W0T Wide 0pen Throttle

Symbols

Symbol Description

a throttle positionr stationary peak efficiency curve

5 angle difference

A difference

( pressure ratio

Unit

%

rad

XIV

r)

ß

ê

K

A

A

P

S

r

Lu

a

A

bsfc

cd

c

d

F

9

H

i

J

k

Ipm

*

m

"îfuel*

mfuel

M

n

ne

P

PUP2

PS

p

efficiency —

multipliers of state constraints

moment of inertia kgm2angle rad

constant coefficient

adjoint variables

max. gear ratio driving resistance curve _

Lagrangian multipliers

density kg/m3optimal operating line __

time constant s

torsion angle rad

angular velocity rad/s"quasi static" peak efficiency curve _

area m2 (mm2)brake-specific fuel consumption kg/Ws (g/kWh)drag coefficient —

rolling resistance coefficient %

constant coefficient

length m (mm)force N

acceleration due to gravity m/s2Hamiltonian

geometrical gear ratio —

performance index

gain —

load-pedal position %

mass kg

mass flow rate kg/sfuel mass kg

fuel mass flow rate kg/storque in the context of the CVT Nm

rotational speed Hz (rpm)engine speed Hz (rpm)pressure Pa (bar, kPa)pressures in the variators Pa (bar, kPa)control pressure of the four-way valve Pa (bar, kPa)power W (kW)

XV

q generalized coordinate m, rad

Q generalized force N, Nm

Quv lower fuel heating value J/kg (MJ/kgr gear ratio (output/input speed) —

radius m (mm)rf final drive gear ratio _

r gear ratio change rate s-1

s length, distance m (mm)t time s

T torque Nm

kinetic energy J

Te brake torque Nm

u system input

Us control voltage V

V vehicle speed m/s (km/h)v volume m3 (1)*

v volume flow rate m3/s (1/s)X system state

Subscripts

a input to the CVT

ace acceleration

act actual

ao after orifice

app approximatedbo before orifice

c chain

car car, vehicle

d desired, demanded

ds drive shaft

e engine

f final

F torque sensor

hm hydraulic and mechanical

is input shaft

XVI

I losses

L leakagem mechanical

max maximum

min minimum

ml mechanical losses

os output shaft

P pump

rot rotational

R friction, resistance

ss steady-stateb translatoryth theoretical

V vehicle

ID wheel

X cuttingZ output of the CVT

Chapter 1

Introduction

Traditionally an automobile powertrain or driveline consists of an in¬

ternal combustion (IC) engine, a clutch, a transmission, a differential,and axles that join the differential with the wheels. Using a continu¬

ously variable transmission (CVT) instead of a conventional gear box

is considered by many as a promising approach to the improvementof the fuel economy of passenger cars. An overview of the differ¬

ent CVT designs can be found in (Beachley and Frank, 1980) and

(Kluger and Fussner, 1997). CVTs have now reached a technologi¬cal level that permits a large-scale introduction of these devices even

in the class of full-size passenger car. Maximum torques of approxi¬

mately 300 Nm can be handled with push belts (cf. (Gesenhaus, 2000)or (Goppelt, 2000)). Other structures (e.g., toroidal drives (Dräger,Gold, Kammler and Rohs, 1998)), which promise to cover even larger

torque ranges, have been proposed as well.

.1 Motivation for Using CVIs

Efficiencies of CVTs are inherently lower than those of conventional

fixed-ratio transmissions, usually cog-wheel gear boxes. Nevertheless,the use of CVTs permits a more efficient system behavior by shiftingthe engine operating points for a certain demanded power towards

higher torques and lower speeds. This is best explained by an example.As shown in Figure 1.1, the engine of this example can produce 10 kW

i

2 CHAPTER 1. INTRODUCTION

of power with efficiencies anywhere between 77 = 0.1 and 0.33. Thus

using a CVT, the engine can be controlled to operate at maximal

efficiency for any demanded power (7? = 0.33 for 10 kW of power), i.e.

to operate on the so-called ''quasi static" peak efficiency curve fi.

The desired engine operating point is enforced by choosing appro¬

priate control inputs to the system, which are the throttle valve of the

engine and the CVT gear ratio change rate. Both control parametersare assumed to be accessible "by wire," which means that electric ac¬

tuators are present which permit the decoupling of the driver's inputfrom the actual system controls.

1 1 1 1 1 *~

1000 2000 3000 4000 5000 6000 n6 [rpm]

Figure 1.1: Map of a modern two-liter SI engine. Indicated are

iso-efficiency curves (thin black), iso-power curves (thingray), A (thick black), 0 (thick gray), and T (dashed).

In stationary conditions, however, the maximum gear ratio drivingresistance curve A (defined by the vehicle resistances for constant

speed and horizontal roads) imposes restrictions on the stationaryoperating points. Thus, for the example mentioned above, only an

efficiency of rj œ 0.27 can actually be realized, since in this case the

engine can only be operated below the A curve, cf. Figure 1.1.

For high stationary power values, O is below A, and hence the

engine can be operated at maximal efficiency. The resulting stationary

1.2. CLASSICAL VS. OPTIMAL CONTROL APPROACHES 3

operating line, where best efficiency is achieved, is the F line, the so-

called peak efficiency curve for stationary operation (Liu and Paden,

1997). Of course, the various curves just described are only valid if the

remainder of the driveline is considered to be lossless. As mentioned

above this is not true, especially for the CVT. Therefore the efficiency

map of the CVT also has to be taken into account for derivation of

these curves.

In summary, a CVT permits to operate the engine with a better

efficiency than it would be possible with a multi-step fixed-ratio trans¬

mission. Therefore, despite the relatively high transmission losses,the overall system efficiency can be increased. An additional advan¬

tage is the gear ratio shifting mechanism which produces a smooth

and comfortable behavior that cannot, or only with great difficulty,be achieved by conventional transmissions. Torque converters, which

also permit very smooth accelerations, have much higher efficiencylosses than CVTs. Continuously variable transmissions are therefore

a potentially very attractive alternative to existing devices.

1.2 Classical versus Optimal Control

Approaches

The fuel-optimal strategy for CVT-based powertrains in stationary ve¬

hicle conditions is the well-known operation of the engine on the peak

efficiency curve for stationary operation F described in Section 1.1.

For transient vehicle conditions, however, it is not obvious which tra¬

jectory in the engine map will produce the best possible fuel economy

for a given vehicle speed trajectory. In the literature several classical

transient control strategies are proposed which will be discussed in

detail in Chapter 2. They are all based only on heuristic arguments

as to how fuel-optimal control could be achieved. In fact, the most

promising idea is mainly based on the reasoning that operating the

system on the "quasi static'5 peak efficiency curve O has to be close

to the optimal control for transient vehicle operation, too.

Consequently, there is indeed a need to investigate what the fuel-

optimal control algorithm looks like in truly dynamic conditions and

how much the classical control approaches deviate from this optimalsolution.

4 CHAPTER 1. INTRODUCTION

1.3 Contribution

This optimal control problem for transient vehicle operation was for¬

mulated and solved for two different engine types: a downsized su¬

percharged (DSC) engine (Softie, 2000) and a conventional SI engineof approximately the same size. For this purpose, accurate models of

all various parts of the driveline had to be developed. The focus was

on the development of new models of the DSC engine and the CVT,whereas for the models of the conventional SI engine and of the pow-

ertrain without the CVT well-known approaches were used. Based on

work proposed by (Shafai. Simons, Neff and Geering, 1995), a more

detailed CVT model was derived which includes a measured mechan¬

ical efficiency map and a gear-type pump model for the hydraulic

supply. Besides these extensions, the equations of motion describingthe mechanical part were replaced by a new approach.

Using the new CVT model, an overall efficiency map 7]cvt of the

CVT was derived, which is a three-dimensional function of the in¬

put shaft speed ucvt, the input torque Tcvt, and the gear ratio

rCVT} see Section 3.2.2. The optimal control for "quasi static" condi¬

tions was then derived, once with and once without considering this

CVT efficiency map i)cvt for the driveline model. In the case where

the powertrain losses are neglected, the optimization is well-known,whereas it yields some new features in the other case. Actually, since

the CVT efficiency in this case also depends on the gear ratio, the

"quasi static" peak efficiency curve 0 is no longer a single line, but

becomes a particular curve for every vehicle speed v.

The optimal control problem for transient vehicle operation was

then solved for both engine types mentioned above using state-of-the-

art numerical optimization software. Compared to the best classical

control approaches, the expected fuel economy improvements are in

the range of 3% for the DSC engine,whereasfortheconventionalenginediscussedhereinthebestclassicalstrategyisshowntobeal¬mostoptimal.However,thefueleconomyimprovementscomparedtotoday'smostcommonlyimplementedstrategiesareshowntobeap¬proximately4%fortheDSCengineor6%fortheconventionalengine,respectively.Derivingcommonfeaturesoftheoptimalsolution,anewcontrolstrategythatisrealizableon-linewasproposed.Itachievesalmost

1.4. STRUCTURE OF THIS THESIS 5

the same fuel economy benefits but with a substantially lower control

complexity.

1.4 Structure of this Thesis

The objectives of this thesis outlined above were achieved by usingthe following approach.

Chapter 2 is concerned with the description of the classical control

approaches that can be found in the literature. Special focus is set on

the question of how these approaches improve fuel economy.

Chapter 3 covers the various models used to describe the wdiole

powertrain. First the modeling of the two engines analyzed in this

work is reported, followed by a very detailed model of a chain drive

CVT. This model is then used to derive a simpler model that can be

used for the optimization analysis.The next two chapters deal with the fuel-optimal control of the

powertrain introduced in Chapter 3. To be more specific, in Chapter 4

the steady-state optimal control problem is defined and analyzed for

two different cases: considering the CVT losses or neglecting them.

In Chapter 5 the optimal control for transient vehicle operation is

analyzed in the following way. Once the whole problem is defined, the

methods used to solve it are introduced before the optimal solutions

for both engine types and their common features are presented. In

addition, an applicable suboptimal strategy is derived and comparedwith the optimal solution. Finally, the chapter concludes with an

estimation of the improvements in fuel economy that can be gainedby applying the proposed strategy.

In Chapter 6 some conclusions are drawn and future research di¬

rections are proposed.

I

Chapter 2

Classical Control

Approaches

The literature concerning the control of CVTs in steady-state and

transient conditions is quite substantial. Unfortunately, the expres¬

sion steady-state operation is not uniformly used. In fact, the parts of

the system actually in steady-state conditions are not clearly defined.

Some authors use the expression for those cases where the whole pow-

ertrain including the vehicle is in steady-state condition, others take

it for the driver's input only. To prevent misunderstandings, the fol¬

lowing definitions are adopted throughout this thesis:

Steady-state operation: The whole powertrain including the ve¬

hicle is in steady-state condition. The vehicle in particular is

moving at a constant speed.

"Quasi static" operation: The driver is holding a constant accel¬

erator pedal position. The engine is thus assumed to operate in a

"quasi static" condition, i.e. the engine operation point changes

very slowly, if at all. The remainder of the powertrain is in tran¬

sient condition. The vehicle in particular is allowed to accelerate

or decelerate.

Operating the engine on the maximum gear ratio driving resistance

curve A is the best efficient operation mainly for steady-state oper¬

ation, because only for very high stationary power values a better

v

8 CHAPTER 2. CLASSICAL CONTROL APPROACHES

efficiency can be achieved as discussed in Chapter 1.1, cf. Figure 1.1.

For the "quasi static" operation, however, two classical approaches

depending on the vehicle's throttle equipment can be distinguished.

2.1 "Quasi Static" Operation

For "quasi static'' operation the main objective for choosing the gear

ratio of the CVT is to maximize the efficiency of the engine with

respect to the driver's demand. If the vehicle is equipped with a drive-

by-wire (DBW) system, the driver's demand set by the load pedal is

interpreted as a desired engine power (Chan, Yang, Volz, Breitweiser,

Jamzadeh, Frank and Omitsu, 1984; Stubbs, 1981). Since in this case,

the engine speed/torque combination can be chosen independently for

each demanded power level, the desired best efficiency operation leads

to the "quasi static" peak efficiency curve 0 as depicted in Figure 2.1.

Another strategy has to be followed if the vehicle is not equippedwith an electronic throttle. In that case the engine torque is deter¬

mined by the accelerator position and the engine speed, i.e. the engine

operation point cannot be chosen completely freely since it has to lie

on the corresponding iso-throttle position curve. Consequently one

degree of freedom is lost, and the engine efficiency can only be mini¬

mized by calculating the corresponding optimal engine speed value for

each throttle level. The resulting E line is called optimal operatingline (Vanvuchelen, Moons, Minten and De Moor, 1995), see Figure 2.1.

Notice that the optimal operating line E for values close to the maxi¬

mum torque sometimes has to be adjusted in such a manner that the

engine power is never reduced for increasing accelerator values.

All the classical approaches that can be found in the literature

are based on considerations of the efficiency map of the engine only.But obviously the efficiency of the other driveline elements, especiallythe one of the CVT, should also be included in the derivation of the

"quasi static" best efficiency curves (Stall, 1985). The reasons for this

omission and the consequences for A, O, and F when the losses of the

CVT are considered will be the topic of Chapter 4.

2.2. TRANSIENT OPERATION 9

Figure 2.1: The two classical "quasi static" operation lines O (thickgray) and S (thick black) for a modern 2 1 SI engine.Also indicated are iso-efficiency curves (thin black), iso-

power curves (thin gray), and iso-throttle position curves

(thin dashed).

2.2 Transient Operation

This section focuses on the basic control strategies once the vehicle

is launched and operates in forward driving conditions, i.e. after the

clutch or torque converter is fully engaged. Although a large num¬

ber of different control strategies exist, three main approaches can be

distinguished, namely speed envelope, single track, and off the beaten

track. A survey of these three classical control approaches to gear ra¬

tio control is presented in this section. It mainly reports, even if not

fully, the same topics and descriptions as outlined in (Pfiffner, 1999b).The nomenclature of each strategy follows the one proposed in (Liuand Paden, 1997).


2.2.1 Strategy No. 1: Speed Envelope

The speed envelope strategy originally proposed by (Deacon, Brace,Guebeli, Vaughan, Burrows and Dorey, 1994) proves to be the core

of the controller in most of today's CVT-equipped cars. Its name is

derived from the fact that the desired operating area of the system is

formed by two curves in the vehicle speed versus engine speed plane,see Figure 2.2.

ne [rpmj

6000

5000

4000

3000

2000

1000

30 60 90 120 150 180 vcar[km/h]

Figure 2.2: Limiting curves in a speed envelope strategy

The upper curve defines the engine speed trajectory when hard ac¬

celeration is requested, whereas the lower curve represents the desired

engine speed during overrun or cruising condition. Between these two

curves, the desired engine speed is determined by the actual vehicle

speed and the actual accelerator position. For any given vehicle speed,the desired engine speed is almost a linear function of the accelerator

position (small nonlinearities are only present due to the mechanical

characteristic of the accelerator pedal mechanism). The engine speedis then governed to this desired value by an underlying gear ratio

controller.

The improvements in the fuel economy of the vehicle are realized

simply by choosing a relatively low desired engine speed at cruising(overrun) conditions. Consequently the design of the lower curve of

,

hard acceleration s^ ,100%

/ accel erator

0%,-""overrun

__________---------~'"""~^

«i i i


the speed envelope strongly affects the fuel economy, whereas the de¬

sign of the upper curve determines the driveability of the vehicle. The

advantages of this strategy are its simplicity and the fact that it can

be realized without any DBW equipment since the engine torque is

only considered indirectly during the design of the speed envelope.

A more sophisticated implementation of the speed envelope strat¬

egy is used for controlling the Honda Multi Matic CVT, which is

installed in the 1996 Civic series (Funatsu, Koyama and Aoki, 1996).This extended speed envelope strategy considers various driving mode

selections such as sporty driving mode or fuel economy mode, which

can be chosen by the driver. For each mode the speed envelope is

specifically chosen such that the resulting driving behavior fulfills the

requirements of this mode. For example, if the driver changes from

fuel economy mode to the sporty mode, the speed envelope is shifted

towards higher engine speeds to improve the dynamic behavior of the

vehicle, see Figure 2.3.

ne l

''mm '

-

1 /

,

1 /1 /

1 /

1/

proved dynamic/ performance

j***"> krmax

/ --"' improved engine

/ --

~

brake performance

. '1 ,,, i „,,, i i .. i i ».

mm

Figure 2.3: Speed envelope shifting characteristics for the Honda

Multi Matic: sporty driving mode (thick black) and fuel

economy mode (thick gray)


2.2.2 Strategy No. 2: Single Track

A natural approach for transient ratio shift control is to maintain the

engine on the "quasi static" peak efficiency curve Q (Gui, Goeringand Buck, 1985; Yang and Frank, 1985), which corresponds to the as¬

sumption that any arbitrary speed/torque combination can be realized

instantaneously. In reality, however, due to the dynamics inherent to

the system, the engine cannot always be operated on this curve. Since

this approach interprets the accelerator as demanded power Pj, the

chronological locations of the desired operating points on Q during a

vehicle acceleration are directly related to the time trace of the ac¬

celerator position. This approach is called single track and is known

to perform well with respect to fuel consumption, but to cause some

problems with the driveability of the vehicle. Figure 2.4 shows the

system trajectory in the engine map for an accelerator tip-in.

•'

i ___—i 1 i i è ^

1000 2000 3000 4000 5000 6000 ne [rpm]

• initial steady-state operation a single track

O proposed final steady-state operation ± single track modified

Figure 2.4: System trajectories of the single track and single track

modified strategy resulting from an accelerator tip-in

In this strategy, the engine torque is brought to the "quasi static"

peak efficiency curve O as quickly as possible. Concurrently the gearratio is brought to the value which corresponds to the proposed final


steady-state operating point. As the driver will usually change the

accelerator position during the transient operation, the proposed final

steady-state operation point will not remain the same, as shown in

Figure 2.4, but will move on the peak efficiency curve.

Figure 2.5 shows a typical control structure needed to realize the

single track strategy (Stall, 1985). The desired engine speed u;ed is

TeactTed +"-

Tn(»eact)torque

controller

a„

drive train

engine &CVT

"*eact

&)ecVç r„<Mpd)

speed ratio

controller

"'eact.-

û,eact

Figure 2.5: Control structure scheme of the single track strategy

determined by the stationary peak efficiency curve F that correspondsto the actual power demanded by the driver Pd- A feedback controller

changes the gear ratio change rate r until the actual engine speed w6act

coincides with the desired value. Simultaneously the desired engine

torque Ted is derived such that the engine certainly operates at O,i.e. the desired engine torque is set to the value that belongs to the

actual engine speed cjeact on 0. Then the actual engine torque TP<ict is

brought to the desired value using the DBW equipment to correctlyset the throttle position a. The two functions lüt(.) and Tq(.) can be

derived from the F or 0 curve, respectively, and later be implementedas lookup tables. The reason for setting the engine speed and not the

engine torque to the proposed final value of the steady-state operationis that the bandwidth of the engine torque control loop is higher than

that of the gear ratio control loop. The possibilitytofulfilltheenginetorquedemandreasonablyfastenablesthesystemtooperatemostlyonthe"quasistatic"peakefficiencycurve0,theinverseapproachwouldnotbefeasible(Yang,GuoandFrank,1985).ThecarelectronicsmanufacturerBoschhasusedthesingletrackstrategyfortheeconomymodeoftheirproposedoverallcontrolstruc¬tureCartronic(Engelsdorf,SengerandBolz,1996).Thiscontrolstructureisphysicallybased,hierarchical,andfullyobject

oriented.


Moreover, it can be used for the software development of the enginecontrol unit (ECU) as well as of the transmission control unit (TCU).Within this flexible structure, it is an easy task to integrate different

modes or additional features. For example the desired behavior for

a kick-down maneuver, where the engine always should deliver full

engine power (Engelsdorf et al., 1996; Pellegrino, 1986), can easily be

attained.

The single track approach is straightforward to implement but it

has substantial drawbacks for the driveability of the vehicle. The

dynamics of the powertrain are artificially limited since the vehicle

acceleration is principally realized by changing the gear ratio of the

CVT rather than by increasing the engine torque. Moreover, as men¬

tioned above, the bandwidth of the gear ratio control loop is typicallymuch smaller than that of the engine torque control loop.

For these reasons a modified version of the single track strategy, the

so called single track modified strategy, has been proposed (Abromeitand Wilkinson, 1983; Kim, Song, Kim and Kim, 1996; Stubbs, 1981).In the most common case the actual engine torque is no longer con¬

trolled along the "quasi static" peak efficiency curve fl, but is brought

directly to the value of the proposed final steady-state operation pointon the peak efficiency curve, see Figure 2.4. Obviously, this modifica¬

tion increases the dynamics of the vehicle.

Figure 2.6 shows the block diagram of the modified control struc¬

ture. The two control structures differ in the control of the engine

Tf(Pd)

car(Pd)

Teact

.,ô<UtL torquecontroller

ffiev

weaet

speed ratio

controller

1

a^

drive train

engine & CVT

Teaet

f,

ffleac,

Figure 2.6: Modified control structure scheme

torque. For the single track modified strategy, the desired enginetorque Ted is immediately set to the corresponding value of the pro¬

posed final steady-stateoperatingpointonthepeakefficiency

curve.


As the gear ratio control loop shows a lower bandwidth than that of

the engine control loop, the engine torque reaches its desired value

much faster than the gear ratio. Thus the system leaves the desired

operation curve 0 towards the proposed steady-state operation torque

(Gui et al., 1985; Song, Kim, Kim and Kim, 1996).

Remark: The torque controller in Figures 2.5 and 2.6 is pictured as a

feedback controller, because generally a feedforward structure, using a

static map a = f(Te<i, weact) to compute the throttle position a, is too

inaccurate for this kind of application (Peng, 1987). A combination of

a feedforward and a feedback controller would obviously be the best

choice.

2.2.3 Strategy No. 3: Off the Beaten Track

Since the performance of the single track strategy is limited, one

approach is to use two different transient trajectories (Vahabzadehand Linzell, 1991). These two trajectories represent different driver-

selectable modes: the economy and the performance mode. Since both

trajectories are usually far off the "quasi static" peak efficiency curve

0, this strategy is known as the off the beuten track strategy.

During a transient maneuver, an increase in the demanded power

leads to a new proposed steady-state operating point on the peakefficiency curve F. In the economy mode, the throttle is opened as

quickly as possible to the level corresponding to the new steady-statepoint. Thus the engine operates along a constant throttle curve while

the gear ratio is continuously changed until the final steady-state op¬

erating point is reached, see Figure 2.7.

Once the driver has selected the performance mode, the engine is

forced to operate on the iso-power curve corresponding to the power

used at the final steady-state operating point. Therefore, the throttle

is opened as quickly as possible to achieve this power level. Obviously,there are times when this level exceeds the engine's maximum torque

capability. In this case the engine is operated at the maximum torqueline (MTL) until the demanded iso-power curve is reached.

Obviously a DBW throttle is required for the implementation of

the performance mode of this strategy, whereas such an equipment is


9 initial steady-state operation a economy mode strategyO proposed final steady-state operation a performance mode strategy

Figure 2.7: Resulting engine map trajectories of the two modes of

the off the beaten track strategy on an accelerator step.Also indicated are iso-efficiency curves (thin black), iso-

throttle position curves (thin dashed), and iso-powercurves (thick gray).

not necessary for the economy mode. In the Volvo 460 car for exam¬

ple, the economy mode strategy is implemented without an electronic

throttle (Vanvuchclcn et al., 1995).A typical control structure for an engine equipped with an elec¬

tronic throttle is depicted in Figure 2.8. This structure is capable of

handling both modes by changing the function a(.) only. Because for

the economy mode, this function becomes independent of the actual

engine speed we,u.t, whereas for the performance mode, the desired

throttle position o^ is a function of the demanded power Pd as well

as of the actual engine speed o;eact.

2.3. SUMMARY OF CHAPTER 2 17

throttle with

controllerdrive train

engine & CVT

aactêact

speed ratio

controller

Figure 2.8: Control structure scheme usable for both modes


In steady-state vehicle conditions, the fuel-optimal operation of a

CVT-based powertrain is the well-known operation of the engine on

the peak efficiency curve F. Unfortunately powertrain losses, espe¬

cially those of the CVT, are not taken into account for the derivation

of this curve. But since for reasonable vehicle speeds the T curve

coincides with the maximum gear ratio driving resistance curve A,these losses have almost no influence on the fuel-optimal steady-statecontrol.

Three different classical control approaches for transient vehicle

conditions can be found in the literature. All of them rely on heuristic

arguments, and considerations for fuel-optimality are based on "quasistatic" conditions only. Therefore using optimal control theory and

applying precise optimization techniques to solve the problem of the

fuel-optimal control of drivelines equipped with a CVT seems to be a

promising option.

Chapter 3

Powertrain Modeling

In order to formulate and solve an optimal control problem, the dy¬namic behavior of the system under consideration has to be quantified.

Traditionally this is done by ordinary differential equations (ODEs).This chapter therefore describes the modeling of the whole powertrainof a vehicle which is assumed to consist of an engine, a CVT, and a

final drive.

The drive train is modeled in its longitudinal behavior only and no

drive train elasticities are taken into account. Moreover, only non-zero

vehicle velocities are assumed, i.e., the vehicle launch process which

wTould need a clutch or torque converter is not analyzed.Sections 3.1 through 3.3 describe the models of all components

in detail. Special focus is set on the modeling of the CVT since this

component represents the most important part of the powertrain when

the overall efficiency is considered. Thus a detailed model of the CVT

has been developed which includes a measured mechanical efficiencymap and a gear-type pump model for the hydraulic supply of the

CVT. This detailed model was the starting point for the derivation of

a simpler model that could be used for the optimal control problem.This simplified model represents a good trade-off between complexityand accuracy.

The model parameters are chosen such that the model represents

approximately the lightweight vehicle described in (Guzzella and Mar¬

tin, 1998) which is powered by an IC engine of about 40 kW and which

is additionally equipped with a CVT.

1Q

20 CHAPTER 3. POWERTRAIN MODELING

3.1 Engines

Two different engine classes are considered: conventional port-injectedSI engines and downsized supercharged (DSC) port-injected SI en¬

gines. The first class represents state-of-the-art SI engines, whereas

the second class can be seen as a promising approach to improve the

fuel economy of IC engines (Soltic and Guzzella, 2000).Both engine types are assumed to be operated mostly with a stoi¬

chiometric air/fuel ratio. Mixture enrichment, however, is used close

to maximum torque conditions which affects the fuel economy behav¬

ior at full load. More information on the DSC approach can be found

in (Soltic, 2000).

3.1.1 Downsized Supercharged Engine

The engine under consideration is a small SI engine (360 ccm displace¬ment) supercharged by a pressure-wave supercharger (PWSC) (Gyar-mathy, 1983) yielding a maximum power of about 40 kW (Guzzellaand Martin, 1998). Figure 3.1 shows the corresponding engine fuel

consumption map which has rather unusual efficiency contours. The

consequences of this special characteristic will be discussed in Chap¬ters 5 and 6. The resulting peak efficiency curves Û and A defined in

Chapter 1 are also indicated.

The peak efficiency curves Q and A were introduced by choosingthe maximal engine efficiency ?/e for any given desired engine power

Pd- Since the brake-specificfuelconsumption(bsfc)isinverselypro¬portionaltotheengineefficiencyr\e(Heywood,1988),1bsfc=—,(3.1)f]c•Qïïvmaximizingtheengineefficiencyisequaltominimizingthebsfc,seeFigure3.1.ThefuelconsumptioniscalculatedbyintegrationofthismeasuredDSCenginemapovertime:mfuei=/mhieï(uje,Te)dt<(3.2)wherethefuelmassflowratern{uel(uje,Te)=bsfc(we,Te)•Pe=bsfc(we,Te)ueTe(3.3)

3.1. ENGINES 21

oI ' ' ' L ' l ' ' L

1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

engine speed n [rpm]

Figure 3.1: DSC engine map, fuel consumption in g/kWh. Also in¬

dicated are the corresponding A, O, and F curves intro¬

duced in Chapter 1.

is a function of the measured brake-specific fuel consumption map.

The system model structure used for the load-pedal to torque be¬

havior of the engine is pictured in Figure 3.2 where the following

definitions are used:

• Ip load-pedal position representing the system input «i, Ipwor

load-pedal position at wide open throttle (WOT), i.e., at the

limit of the naturally aspirated part (additional torque is pro¬

duced by boosting (Guzzella and Martin, 1998))

• Te engine torque and ne engine speed

• f'rc the mapping from the load-pedal position and from the en¬

gine speed to the engine torque

• Ti time constants and kt gains, as identified from measurements

with the engine mounted on a dynamometer (Soltic, 2000).


J-K

-y-i*-.j*

KkO-V--K1^)-^|

>_t^_*ri/^ J1/^ ]j

t.r^kty-1 J

Figure 3.2: Model of the DSC engine

The first low-pass element (r\) in Figure 3.2 represents the model

for the electronic throttle. The upper two first-order lags (r2l T3)describe the manifold dynamics for the lower torque range. More pre¬

cisely, the first-order lag with time constant t2 represents the manifold

dynamics of the naturally aspirated part without the PWSC, whereas

the other lag (V3) describes the manifold dynamics of the naturally

aspirated part with the additional receiver of the PWSC, i.e. T3 > t2.

The lower path (ta) finally represents the manifold dynamics whenever

the boosting process takes place. The last three low-pass elements rep¬

resent essentially the "empty-and-filling" phenomena in the manifolds

of the engine. The gasdynamicprocessinthePWSCitselfisveryfastcomparedtothedynamicsofthesereceivers(Amstutz,1991;WeberandGuzzella,2000)andcanthereforebeneglected.ThustheDSCenginemodelcanbewrittenasa4th-ordersystemofnonlinearODEs:X\x2a;3xa—{lp-xi)T\1/k2-fTe{s-ät(x1)\^wo\ne)^x2)T21h-fTX^l)\tVOr.nr-x3fTt(Sàt(xl)\}p?voT'ne)~~fn(lPWOT,ne)-XA(3.4)(3.5)(3.6)(3.7)

3.1. ENGINES 23

The sat(.)|;£ is the saturation function which imposes lower and upper

bounds on the argument:

( x if x < x

sat(:r)|£ = < .r if x < x < x (3.8)[x if x > x

The engine torque Te — ^2 + ^3 +£4 turns out to be the superpositionof the particular value of the three paths.

Remark: Notice that the engine speed nc is not an input of the

model, but it represents the system state of the powertrain model

which is described in Section 3.3.


The conventional engine chosen to be analyzed was a modern one-

liter SI engine of approximately the same peak power as the DSC

engine. The resulting brake-specific fuel consumption map is shown

in Figure 3.3.

Compared to the DSC engine, the best efficiency point is at almost

the same engine speed but at a substantially higher torque. Thus

the resulting "quasi static*' peak efficiency curve 0 is close to the

maximum torque line and the peak efficiency curve for stationaryoperation F coincides with the maximum gear ratio driving resistance

curve A. Again the fuel consumption is calculated by integration of

this measured bsfc map (Equations (3.2) and (3.3)) as described for

the DSC engine.The load-pedal to torque behavior of the conventional engine is

modeled as two first-order lags in series:

(3.9)

(3.10)

where x\ represents the actual load-pedal position and x2 is the en¬

gine torque. Again /^ is the measured mapping from the load-pedalposition and from the engine speed to the engine torque. Thus the

Xl = —(Ip-Xx)

à2 =— (/rf(ai>rcp


0 i I 1 1 1 L I I _l 1

1000 1500 2000 2500 3000 3500 4000 4500 5000 5500


Figure 3.3: Conventional engine map, fuel consumption in g/kWh.Also indicated are the corresponding A, O, and F curves

introduced in Chapter 1.

model structure of the conventional engine degenerates to the upper¬

most path of the DSC engine model depicted in Figure 3.2, which

describes exactly the behavior of the naturally aspirated path.This structure of the conventional model is also proposed in current

models of SI engines known in the literature (Butts, Sivashankar and

Sun, 1995; Shim, Park, Khargonekar and Ribbens, 1995), in case the

various nonlinear static functions are combined into the measured /ye

map.

Another simplification is made by neglecting the induction to power

stroke delay (Powell and Cook, 1987) for the conventional enginemodel as well as for the model of the DSC engine. This is done

for three reasons. First, optimal control theory is not yet extended

to systems with pure delays. Although a first approach has been pro¬

posed in (Göllmann, 1996) for constant delays, there is still no theorythat can handle state-dependent delays such as the induction to power

stroke delay. Second, an approximation of this pure delay, as for ex¬

ample in (Pfiffner and Guzzella, 1999) or (Girijashankar, Srikantiah

and Pai, 1976), results in an unnecessarily high system order. Third,

3.2. CVT 25

the induction to power stroke delay is rather small compared to the

other time constants of the powertrain and therefore it is admissible

to neglect it.

3.2 CVT

In the literature many publications can be found that treat the control

of CVTs, but relatively few describe the modeling of such a device.

A complete model, including a model of the hydraulic and the me¬

chanical part as well, is described in (Shafai et al., 1995). Other

publications propose models of the different parts of the whole CVT

or assume too many simplifications.In order to derive a simple model that can be used later for the

derivation of the fuel-optimal control algorithm, it is necessary to have

an accurate, detailed model of this device. Therefore the same type of

CVT as modeled in (Shafai et al., 1995; Schmid, 1993) was used. To

be precise, the chain drive CVT RHVF 147 manufactured by P.I.V.

Reimers with a spreading of 5.5, a maximal power of 55 kW, and a

maximal input torque of about 180 Nm was chosen. The mechanical

parts of such a device are shown in Figure 3.4.

Figure 3.4: Mechanical parts of a P.I.V. chain drive CVT: the two

pulleys connected by the chain


3.2.1 Detailed Model

The core of the CVT considered consists of a pair of steel V-shapedpulleys (primary and secondary pulleys) connected by a chain. This

chain is used to transmit torque or power, respectively, between the

pulleys, see Figure 3.6. Each pulley consists of a fixed and an axi-

ally slidable sheave. Each of the two moving sheaves is connected to

a corresponding hydraulic cylinder and a piston. The primary pul¬

ley (indicated with I) is regarded as the drive (input) end and the

secondary pulley (II) as the driven (output) end of the CVT. The

CVTs manufactured by P.I.V. consist of a double torque sensor sys¬

tem that prevents a slipping of the chain at the critical secondary pul¬ley by producing always a slightly greater pressure p2 than it would

be necessary to fulfill the clamping force requirement of the pulleys

(Sauer, 1994; Rattunde, Schönnenbeck and Wagner, 1991).A torque at the input of a torque sensor causes an axial sliding of

its piston shaft which is provoked by a rolling of some spheres in the

torque sensor, see Figure 3.6. This displacement of the piston shaft

(caused by a force Fa corresponding to the input torque Ma) changesthe drain orifice as depicted in Figure 3.5.

" drain

Figure 3.5: Clarification of the torque sensor operation

Depending on the drain orifice area and the fluid flow Vf into the

torque sensor, a pressure pp results which is a measure for the torque

3.2. CVT

torque sensor

slidable

^Hsheave

Figure 3.6: Scheme of the CVT considered


Ma transmitted by the CVT. The two torque sensors are hydraulicallyconnected in series, cf. Figure 3.6. Hence, for two identical torque

sensors, the sensor which measures the higher torque determines the

pressure in the cylinders and thus the clamping force of the pulleys(Dittrich and Simon, 1988).

The control input for changing the gear ratio of the CVT is the

actuator input voltage Us of the electrohydraulic valve (EHV). This

EHV can be used to control the pressure p$ of the four-way valve

(FWV) and thus to displace the valve spool. As shown in Figure 3.6,the two control lines of the FWV are connected to the cylinders, and

both drain lines are connected to the primary torque sensor. Conse¬

quently, displacing the valve spool changes the pressures pi and p2

in the cylinders causing the two slidable sheaves to also change their

positions. Thus the gear ratio changes accordingly.The following subsections describe the modeling of the CVT de¬

picted in Figure 3.6. For this purpose, the CVT is divided in two sub¬

systems, one mechanical and one hydraulic, as in (Shafai et al., 1995).The interface between these two subsystems is characterized by the

two torque sensors which are described in the mechanical part of the

system. Since the whole modeling is based on the work described in

(Shafai et al., 1995), only a short summary of the models therein is

given. Focus is set on the extensions and the modifications of those

models.

Mechanical Part

The simplified scheme of the mechanical subsystem which is used for

the modeling is shown in Figure 3.7, where the torque sensors are

represented by the torques Mi and Mj.

Figure 3.7: Scheme of the mechanical part

3.2. CVT 29

For small torques transmitted by the CVT or for high fluid flows

through the torque sensors, the input and the output shaft of the

torque sensor are rigidly connected by squeezing the spheres from

both sides. In this case the torque sensors represent rigid connections.

Thus, for the primary torque sensor, the inertias 0a and 9\ can be

combined into one inertia which is the sum of these two inertias. Of

course, this procedure is also applicable for the torque sensor II bycombining 9Z and #2-

By increasing the torque or reducing the flow, a relative motion

between the input and the output shaft of the torque sensor is possi¬ble. Thus the torque sensor represents an elastic connection, and the

torques Mi and Mi are determined by the corresponding pressures

in the torque sensors as well as by the friction torque Mr as follows

(Shafai et al, 1995):

Mi = sgn(Aî) cpPFi + sgn(Aî) MR (3.11)

M2 = sgn(A(/?2) cpPF2 + sgn(Awn) Mr, (3.12)

where c.p is a characteristic quantity of the torque sensor that relates

the rotation of the shaft to the translatory motion of the piston ring,see Figure 3.5.

Therefore four different system structures of the mechanical model

of Figure 3.7 have to be distinguished, sec Table 3.1.

Table 3.1: The four possible system structures

DescriptionSystem order

Torque sensor I Torque sensor II

rigid

rigidelastic

elastic

rigidelastic

rigidelastic

1

2

2

3

A change from an elastic to a rigid connection (or vice versa) of

each torque sensor causes a change in the structure of the mechanical

scheme. The simulation software used in (Shafai et al., 1995) was not

able to deal with such structural changes. Hence, a stiff spring-damper


element was used instead of a rigid connection. However, nowadaysmost of the simulation software on the market are capable of han¬

dling structural changes. Therefore the model has been implemented

considering these structure changes.The additional state equation for the input shaft of the CVT is

ÖJa = ^- (Ma - Mi) (3.13)

if the torque sensor I becomes elastic. In addition, the state equationfor the output shaft of the CVT is

ùs = 2-{M2-Mz) (3.14)

if the torque sensor II changes from a rigid to an elastic connection.

The conditions for a torque sensor to represent an elastic or a rigidconnection between its input and output shaft are defined regardingthe relevant cutting torque

Mis eos + mos eisMx =

j— , (3.15)

where M[s and Mos are the torques acting on the input or the output

shaft, respectively. The inertias 8-1H and 0OS represent the correspond¬ing input and output shaft inertias up to the first non-rigid connection.

Thus the torque sensor is considered to be elastic for

\Mx\>cF-pF (3.16)

and to be rigid for

(|A/*| < cF'PF) A(\A(p\ < A<po). (3.17)

The value Acp thus represents the torsion over the torque sensor

whereas A</?o represents a small angle to detect zero distortion.

Aside from the implementation of the changes of the system struc¬

ture just described being different, another differenceispresentintheequationofmotionfortheinnerrigidpartofthemodel,i.e.forthechainandallitsrigidlyconnectedparts.Obviouslythispartisdiffer¬entforeachofthefoursystemstructuresofTable3.1.However,itcan

3.2. CVT 31

be represented by a common scheme for all these different structures,

namely by the two constant inertias 9\ and 62 which represent the pul¬leys and are connected by the chain with mass mc. The input torque

Mi as well as the output torque il/2, on the other hand, have to be

interpreted differently for each of these various system structures. For

example, if the primary torque sensor is assumed to be rigid and the

secondary torque sensor represents an elastic connection, the inputtorque Mi represents the input torque of the CVT Ma and not the

output shaft torque of the torque sensor Mi. The corresponding inputinertia 61 becomes the inertia of the whole primary pulley 6a + #1, see

Figure 3.7. In contrast, the output torque Mi remains M2 and the

inertia of the secondary shaft #2 remains 9u.

Figure 3.8 shows a scheme of this inner rigid part of the system,where the chain is already split into rotating and translational partsin order to graphically assist the derivation of the equation of motion.

The gray disks represent the sheaves of the pulleys and their borders

are the circular contact curves of the chain and of the pulleys with

radii 7*1 and ri. Both pulleys are assumed to consist of a constant

inertia 6q or 82, respectively, and the chain mass mc is assumed to be

uniformly distributed over the whole length of the chain. In addition,it is assumed that no slip occurs between the belt and the pulleys.The external torques Mi and M2 acting on this inner rigid part of the

system and the generalized velocities are also shown.

(i) ©

Figure 3.8: Split model used for the derivation of the equation of

motion of the inner rigid part

The length of the whole chain is assumed to be constant and equal


to di + 2df + d2 = 2(da + 7rf), where da is the distance between the

two axes of the pulleys and f represents the radius when both contact

curves are the same size (f = rj — r2), i.e. when the gear ratio i of

the power transmission element of a CVT defined as

(3.18)

is equal to 1. The length of the different parts of the chain can be

written as follows (Micklem, Longmore and Burrows, 1994):

n_

u*

r2 wi

dt = y/dl-fa-ri)* (3.19)

representing the length of the two straight lines, and

di = (tt — 2<5)r-1 (3.20)

and

d2 = {tt + 26)7-2 (3.21)

the length of the two arcs. Thereby the angle S represents half the re¬

duction/enlargement of the contact angle from w. It can be expressed

by the following equation (Pfiffner, 1999a):

S = arcsin ["

,

- j (3.22)

In order to derive the equations of motion, the model is split into

the three different parts depicted in Figure 3.8. One consists of the

inertia 6\ representing the primary part and the corresponding part of

the chain belonging to the arc of the length d\. Another part consists

of the two straight lines of the chain, whose total length is 2dt. The

last part is similar to the first one but for the secondary side. Therefore

the system pictured in Figure 3.8 has three degrees of freedom and the

generalized coordinatescanbechosenasqi=$i,q2=$2,andQ3=s.Thesystemisnonholonomicsincethevelocitiesofthedifferentpartsofthechainatthepartinglines(Y)and{2)obviouslyhavetocoincide.ThekineticLagrangeequationfornonholonomicsystemsisdis¬cussedin(Hiller,1983)whereitisformulatedasfollows:_qi_\^\ivavl=0,i—1,...,m(3.23)

3.2. CVT 33

The nonholonomic constraints are thus given as

TO.

â„;ft+ a„ = 0, v = l,...,k, (3.24)i=l

where m represents the number of generalized coordinates, k the num¬

ber of nonholonomic constraints, and \iv the Lagrangian multipliers.For the model shown in Figure 3.8, m is equal to 3 and k equal

to 2. Thus the kinetic energy T of the whole system can be expressedas

(3.25)

(3.26)

(3.27)

T =: ±M? + ±M| + ^rritàwhere

01 = 0X +dl 2

try} <r*

2{da + TIT )1

0-2 = 02 +d,9

9<YY) t*

*~*

2(da + Kf)mc1'2

represent the inertias of the rotating parts and

TTlf =dt

(da + TTf)(3.28)

is the mass of the translational part of the chain.

The two nonholonomic constraints at the parting lines (Y) and (2)are

rî - £ = 0 (3.29)

r2#2 - s = 0. (3.30)

Thus for the nonholonomic constraints (3.29) and (3.30) the values

avi and av of equation (3.24) become;

«Li = ri «12 = 0 «i3 = -1 «1=0(SSI)

«21=0 «22=^2 «23 = -1 «2=0 '

The three generalized forces Qi of the system can be expressed as

functions of the external torques actingonthesystem:Qi=MQ2=-M2Q3=0(3.32)


Inserting (3.25), (3.31), and (3.32) in (3.23) leads to three equa¬

tions. All three are dependent on the two Lagrangian multipliers fi\

and fi2. Elimination of these two dependent variables results in the

following equation of motion for this inner rigid part of the CVT:

Ùjl^ë1 = M\ A/l"d, (3.33)Aired

where

-*-red= $1 + i2B2 +

,

mcri,

(dt + (I b)ri + V% + S)r2) (3.34)(da + 7rr)

v z Â

= êl + i292-±mcr2 (3.35)

and

Mlred = iM2 + (i02 + mcrx

(da + irr)

(dt + I (tt - 2£)r! + | (tt + 2<5) r2) ^- (3.36)<%

— (tt + 2(5) r2di-r Mi¬di

Remark: For the derivation of Equations (3.33)-(3.36), the followingrelation was used as well (Pfiffner, 1999a):

dr* " ' "" l

;3.37)

Comparing the above approach with the simpler approach de¬

scribed in (Shafai et al., 1995), herein referred as the "old" approach,

yields the following differences. In (Shafai et al, 1995) the chain is

not split into three different parts, but the whole mass of the chain

is considered to be constant betweenthepartinglines(Y)and(2)(cf.Figure3.8).Thusthemassmtandtheinertias6\and62becomeconstant.Thatapproachledtothesameequationofmotionas(3.33)aboveand,surprisingly,thereducedinertiaremainsunchanged,i.e.OirPd=0i+2202+morî.(3-38)

3.2. CVT 35

However, in (Shafai et al., 1995) the reduced torque was found to be

Aflrpd = iM2 + (i02 + mcrl~^\ ^l (3.39)

The reduced inertias 9[red of both approaches are the same, whereas

the reduced torques M\xed are different. In order to compare the re¬

duced torques of the two approaches a special symbol is introduced

for the deviating parts. Therefore Mired is written as follows:

diAfired = iM2 + (i02 + A) — uu (3.40)

where A becomes

Aoid = mcr i-^rr- (3.41)O'l

for the old approach and

3 <V _ MV, 4-^7r4- 9/HïO -

di

Arncri

(da + irr)(dt + § (tt

-

25)n+ f (tt +

2£)r2)^~ - (n + 2£)r^

(3.42)for the approach considered herein.

dSince the kinematic relation 4p- can be expressed in a closed form

(3.43)ßr r2 f tt — 2 arcsin [ r?d ri

°^7T (ri + r2) + 2 arcsin ( £2tj:i- ) (r\ — r2)rfa

(cf. (Pfiffner and Guzzella, 2001)), it is possible to calculate the dif¬

ference of the two As analytically:

A a —

87TY)rfr2 .

ôld ~

"new—

"71 ; ZT} } j C T~niT7 \T (o.44j(4 + 7rr)(7r(r1 + r2) + 2ö(r1

-

r2))

Expression (3.44) was used to modify the equation of motion for the

inner rigid part of the model derived in(Shafaietal.,1995).MoreinformationaboutthemodelingoftheinnerrigidpartoftheCVTincludingaquantitativecomparisonofthenewandoldapproachcanbefoundin(PfiffnerandGuzzella,2001).


Remark: Due to the unbalanced force distribution in the chain, the

contact curve between the pulleys and the chain is not a circular curve

but a spiral (Wang, 1991). In (Sauer, 1996) this effect was analyzed.The change in the radii turns out to be less than 1% so that the as¬

sumption of a circular contact curve is justified. Q

The last modification that is made on the model introduced by

(Shafai et al., 1995) concerns the efficiency of the mechanical part

of the CVT. The mechanical losses can be accurately modeled and

obtained by the finite-element method (FEM) as described in (Srnikand Pfeiffer, 1999) or (Sauer, 1996). Unfortunately simpler models

have failed to be reasonably accurate. Therefore another approachis used herein. Based on measurements (Wittmer, 1996) from a very

similar CVT which was used in the ETH Hybrid III project, a three-

dimensional map of the mechanical power losses of the CVT is built.

These power losses turn out to be a function of the input speed coaj

the input torque Ma, and the gear ratio i of the CVT. They can be

transformed into a friction force JF"mi in the chain which affects the

torque transmitted:

Fm\ = Fmi(ua, Ma,i,uji). (3.45)

The equation of motion (3.33) of the inner rigid part of the CVT has

therefore to be adjusted by the following term;

xred

In summary, three changes were made in the mechanical part of

the model described in (Shafai et al., 1995):

• the structural changes of the system have been considered

• the equation of motion of the inner, rigid part of the CVT has

been adjusted such that the mass of the chain is more preciselytaken into account

• mechanical losses of the CVT arc introduced by a friction force

in the chain which is obtained from a measured map.

3.2. CVT 37

Hydraulic Part

In order to obtain a causal model and to avoid solving algebraic loops

during the simulations, the fluid is assumed to be compressible for the

calculation of the relevant pressures of the hydraulic system. There¬

fore in (Shafai et al., 1995) five volumes were chosen where the fluid

was assumed to be compressible, cf. Figure 3.6:

* the two cylinders with the corresponding pressures pi and p2

* the two torque sensors with the pressures ppi or pp2, respec¬

tively

* the pipe that connects the pump and the FWV with supply

pressure pq.

The relevant pressure p in each volume is given by its equation of

motion

P = comp • (vin - Vout - v) , (3.47)

where comp is the compressibility factor for that pressure, Vin and

yout are the volumetric fluid flows in and out of this volume, and V

is the change of the considered volume V caused by a motion. For

the two cylinders, the latter is the motion of the piston surface of the

slidable sheave and for the two torque sensors, it is the motion of the

piston caused by a change in the corresponding torque as indicated in

Figure 3.5.

The input and output flows Fin and Vout usually consist of partswhich are piped through an orifice. These parts are functions of the

pressures before and after the corresponding orifice and of the valve

characteristic. The flow through each of the orifices is assumed to be

turbulent but incompressible and can be written as

*_

,,!„ Vf,1-1

V- A • SgIl(j?bo - Pao)V\Pbo -pZ~\, (3.48)

where A is the valve characteristic, pbothepressurebefore,andpao*aftertheorifice,whenlookinginthedirectionofapositiveflowV-ThevalvecharacteristicAofeachorificeisidentifiedasafunctionofthecontrolpressurepsbysteady-statemeasurementsoftheFWV.However,thevalvecharacteristicofthetorquesensorsisadaptedby


comparing the measured pressures with simulation results and is ob¬

viously a function of the torsion Acp over this torque sensor.

Another part of the output flows is the leakage flow of the cor-

responding volume Vl which is assumed to be proportional to the

pressure in the volume considered.

Vl=Cl-p (3.49)

An exception is made for the leakage of the volume that connects the

pump with the FWV which is assumed to depend on the control unit

pressure ps-

The kinetics of the chain (in the direction the slidablc sheave can

move) and of the slidable sheaves themselves are based on steady-statemeasurements. In steady-state conditions, the spool of the FWV is

displaced from the middle position such that the ratio of the pressures

Pi and p2 have a certain value depending on the gear ratio i and the

input torque of the transmission Ma (Pietz, 1993):

< = <(i,M0) =£!

P'2(3.50)

steady-state

The equation of motion for the displacement of the primary slidable

sheave is derived under the assumption that an acceleration of this

sheave is possible if the ratio of the pressures p\ and p2 differs from

the steady-state value (:

mss s'i = Ass (p1 - (p2) - Css ai (3.51)

where mss is the representative mass of the sheave and the chain, and

Css is the friction coefficient.

Since the gear ratio i is directly related to the position of the

slidable sheave s\ through the geometry of the sheaves i = i(si),the gear ratio and hence the position of the slidable

sheaveIIs2isdeterminedaswell.Finally,theelectrohydraulicvalve(EHV)ismodeledasafirst-orderlagwithanonlinearsteady-stateinput-outputcharacteristic.Formoreinformationincludingalltheequationsandcharacteristicsusedreferto(Barandun,1999;Schmid,

1993).

3.2. CVT 39

Up to now the modeling of the hydraulic part of the system coin¬

cides very much with the one proposed in (Shafai et al., 1995). Unfor¬

tunately the pump for the supply is not modeled in that work but the

volumetric flow of the pump is assumed to be constant. However, it is

exactly this device that very much affects the maximum possible gear

ratio change as well as the overall efficiency of the whole transmission.

Therefore, in this research, a gear-type pump is chosen to supply the

hydraulic of the CVT. It is directly connected to the input shaft of

the CVT. This is the most common configuration (Heidemeyer and

Scholz, 1987) because of its simplicity and since it permits to changethe gear ratio even if the vehicle is at a standstill. However, this

pump is designed to supply the necessary volumetric flow already at

idle conditions. Thus at normal and high engine speeds, the pump

supplies too much oil, most of wdiich is absorbed by an outlet valve.

Obviously this significantly reduces the overall efficiency of the CVT

(Sauer, 1994).More recent designs tend to overcome this problem by using mod¬

ern types of pumps where the volumetric fiow can be controlled inde¬

pendently of the engine speed. One example is a variable displacementvane-type pump with flow control (Sauer, 1994), another the internal-

gear pump used in the Audi Multitronic (Gesenhaus, 2000), or yetanother a variable displacement ball pump (Kluger and Long, 1999).Nevertheless the configuration with the gear-type pump was chosen

and investigated herein, because it is used in most actual systems.

A 5.5 cm3/rev gear-type pump from Bosch was chosen (Bosch Au¬

tomation) which is also used by the manufacturer of the CVT for

studies on the test bench. It can supply the necessary volumetric flow

already at idle speed.

The theoretical volume flow Vth of a gear-type pump depends on

the geometry of the pump and is proportional to the input shaft speed(Pohlenz, 1975). In this configuration it is equal to the input shaft

speed of the CVT uja and can be written as

7th=A'i-wa, (3.52)

where K\ is a specific value for each pump.

The volumetric flow losses of the pump Vi are described in (Karassik,


Krutzsch, Fraser and Messina, 1976) and (Pohlenz, 1984):

Vi= K2 Ap + K3 oja + A'4, (3.53)

where K% and K4 are coefficients that depend on the viscosity of the

fluid, and Ap represents the pressure difference over the pump. Thus

the actual volumetric flow supplied by the pump can be written as

Vp = Fth - Vi = Ki Ap + K2 • cüa + K3. (3-54)

To calculate the hydraulic and mechanical power losses Piim the fol¬

lowing approach proposed in (Pohlenz, 1984) is used:

i\m = iv5 Ap + K'e cj2a, (3.55)

which yields the mechanical power acting on the input shaft of the

pump

Pm =V> -Ap + Phm. (3.56)

Notice that the coefficient Kq also depends on the viscosity of the

fluid. However, all these coefficients are determined from technical

data published in (Bosch Automation, ).Since the pump is directly connected to the input shaft of the CVT,

a new and artificial input torque of the overall CVT system termed

Ma is introduced. It is the sum of the actual torque at the input shaft

of the CVT Ma and the torque that is required to drive the pump

MP:

Ma = Ma + Mp = Ma + Pm/üJa. (3.57)

In this section, a very detailed model of the chain drive CVT

RHVF 147 was derived. The model was based on the work proposedin (Shafai et al., 1995) and extended with measured mechanical losses

and a pump model. The equation of motion of the inner rigid partand the structural changes were also modified.

3.2.2 Simplified Model

The detailed model just described is the starting point for the deriva¬

tion of a simpler model that can be used for the optimal control prob¬lem. A good trade-off between complexity and accuracy is achieved

3.2. CVT 41

using an input- and state-constrained integrator:

-irrcvT = "2, (3.58)at

where 112 is the system input corresponding to the gear ratio changerate. It can be interpreted as a nonlinear mapping of the control

voltage Us of the EHV.

Notice that the gear ratio of the CVT for the simplified model is

denoted by rcvT and not by i as in Subsection 3.2.1. This is due to

the different definition of the gear ratio for the two models, because

the definition of the gear ratio denoted by i, see Equation (3.18), is

based on the relation between the two radii of the contact curve of

the chain and the pulleys r\ and r2, whereas the simplified model has

to be defined as

rcvT'Ûte/ue, (3.59)because the simplified model does not explicitly contain these radii.

Since the powertrain is assumed to have the structure shown in Figure3.10, the drive shaft speed u^s represents the output shaft speed of

the CVT, and the engine speed uje is obviously the input shaft speedof the CVT.

The state constraints of the integrator (3.58) are given by

rcvrmia < rcvT < rcvTmàx (3.60)

and reflect the fact that the gear ratio of the CVT can only take valuesbetweenaminimumrnvrandamaximumrc\rrHowever,thely^lmm<_/\tmax'state-dependentinputconstraintsê•fr2(rcvT)<"2<^•/n(rcvr),(3.61)wherefriandfr2arespecialkinematicrelationsoftheCVT,de¬scribethelimitedvelocitytochangethegearratiooftheCVT.Thespecialapproach(3.61)isverydifferentfromtheoneproposedin(Naunheimer,1995),wherenodependencyontheenginespeedujeisconsidered.ThederivationoftheseinputconstraintsisthetopicofthefollowingparagraphandisbasedonconsiderationsofthedetailedmodeldepictedinFigure3.6.Thetimederivativeofthetworadiir\andr2ofthecontactcurveofthechainandthepulleyscanbewrittenasdndin=(3.62)


and

01 at

where both partial derivatives represent geometrical static functions

of the gear ratio. Thus the maximum possible gear ratio change is

proportional to the maximum possible change of the radii r\ or 7"2.

Moreover, Figure 3.6 clearly shows that r\ is also proportional to

the position of the slidable sheave I s\ and that, analogously, r2 is

proportional to the position of the slidable sheave II s2. Therefore

their time derivatives are also proportional:

s2 ~r2.

When the gear ratio is demanded to be increased with maximum

possible velocity, the relevant cylinder is the one of the primary side

because in this case the control pressure p$ becomes minimal and all

the volumetric flow of the fluid supplied by the pump Vp enters this

cylinder. Assuming that the maximum change of the position of the

slidable sheave of this cylinder simax is proportional to this volumetric

flow into the cylinder, the following relation can be stated:

rimax ~ 5lmax ~ Vp - (3.64)

In the opposite case, when the gear ratio must be decreased with

full speed, the relevant cylinder changes to the one of the secondaryside. In fact, the control pressure ps rises to its maximum value,

and the whole volumetric flow supplied by the pump Vp flows into

the cylinder of the secondary side. Hence, considering the same as¬

sumptions as those elaborated in the previous case, the relation of the*

maximum change of the radius r2 and Vp is

r-> ~ s-> ~I/p . (3.65)

The fluid flow supplied by the pump Vp can be approximated bya linear function of the engine speed

Vp& K2-Lüe, (3.66)

3.2. CVT 43

because the influence of the pressure difference Ap and the constant

term A3 in Equation (3.54) turns out to be only marginal for the

pump used.

Therefore the following approach can be used for the limit of the

gear ratio change when the gear ratio is demanded to be increased

(î) -' ($)"'-—(£)" <»"

or when it is decreased

(îL-M$r—(&)" <«>

This yields the functions fri or /r2, respectively, as follows:

/-, = «• (~)'

(3-69)

and

where k is a constant coefficient and both partial derivatives represent

static maps of the gear ratio which specifically depend on the geometryof the mechanical part of the CVT.

This approach was verified with simulations of the detailed model.

The maximum relative difference resulted to be less than 1%.

The CVT efficiency "map" t\cvt models the losses within the CVT

system, including all auxiliaries, as a static function of the three vari¬

ables input torque Ma, input speed cja, and gear ratio vcvt- Since

the input shaft of the CVT is directly connected to the output shaft

of the engine (cf. Figure 3.10), the input shaft speed of the CVT be¬

comes equal to the engine speed ua = toC) and the input torque of the

CVT coincides with the engine torque Ma = Te. Therefore the CVT

efficiency tjcvt can be written as

Vcvt = ï]cvt (^V> Te,7'cvt)•

(3.71)


Notice that the power losses described by rjcvT afreet only the torquetransmitted and not the shaft speeds:

7}CVT =

Rout

TJ- fout k-Ms

T,out

Jrni 11 • U)e 1

c

T* s~*\/~ ~p , (3.72)

The overall efficiency map was obtained from simulations of the de¬

tailed model. A downscaled version of this map is shown in Figure 3.9.

The CVT was downscaled such that the power requirements of both

engines discussed in Section 3.1 were fulfilled.

1000

2 35

100

gear ratio [-] ° 426 0input torque [Nm]

Figure 3.9: Overall efficiency map of the CVT

As mentioned above, the efficiency of current CVTs is not goodenough that it could be neglected. The overall efficiency is strongly af¬

fected by the input torque and the input speed, whereas the gear ratio

shows a smaller dependency on the CVT efficiency. Roughly stated,the efficiency decreases with increasing input speed and decreasinginput torque and reaches maximal values for a gear ratio equal to 1.

The dependency and the order of magnitude of the overall ef¬

ficiency just described coincides with measurements of chain drive

CVTs published in the literature as for example (Dittrich, 1986),

3.3. POWERTRAIN 45

(Heidemeyer and Scholz, 1987), (Höhn, 1988), (Renius, 1996), (Sauer,

1994), or (Vahabzadeh, Macey and Dittrich, 1990).

Remark: The efficiency of the CVT, i]cvt, is developed under

steady-state conditions and is therefore assumed to be independent

of the gear ratio change rate of the CVT tcvt- Simulations of the

detailed model described in Subsection 3.2.1 as well as efficiency mea¬

surements under dynamic conditions (Bonthron, 1985) show that this

assumption is valid.

3.3 Powertrain

A sketch of the whole powertrain structure including the engine and

the CVT is shown in Figure 3.10. In Sections 3.1 and 3.2 the models

of most of the components have been described in detail. In this

section the equation of motion for the longitudinal behavior of the

whole powertrain including the vehicle will be derived and analyzed.

Engine Vehicle

Inertia Inertia

Figure 3.10: Drive train structure

In addition to the CVT and the engine, the final drive and the

vehicle itself have also to be taken into account, cf. Figure 3.10. The

final drive is modeled as a gear ratio only, rj — ujw/ujds, producing an

overall gear ratio r = tqvt '

r/ of lne whole powertrain. Its inertia is

considered in the inertia of the rotating powertrain elements 9Vrot, see

Table 3.2.

The vehicle dynamic is represented as an inertia 0V and a wheel

resistance force Fr. The vehicle inertia consists of the vehicle mass


niv and the inertia of the rotating elements 0Vtof including the wheels,the final drive, and the secondary inertia of the CVT:

Ov - #tvot +mv -rl,. (3.73)

The resistance force Fr is a function of the vehicle speed v

and is modeled as

FR = cr • m„ g+ - • cd A- pair v2.

The vehicle and drive train parameters used are listed in Table 3.2.

They were chosen to match the prototype vehicle described in (Guzzel-la and Martin, 1998). Note that the engine inertia 9e represents the

engine inertia including the primary inertia of the CVT.

Table 3.2: Main drive train parameters

Parameter Symbol Value Unit

curb mass mv 800 kginertia of the rotating 0*'rot 2.0 kgm2powertrain elements

engine inertia Oe 0.05 kgm2frontal area A 2.0 m2

air drag coefficient Cd 0.25 _

rolling resistance coeff. Cr 0.8 %

wheel radius rw 0.25 m

final drive gear ratio rf 0.1582 _

In order to derive the equation of motion for the powertrain de¬

picted in Figure 3.10, the dynamic torque balance at the drive shaft

has to be formulated:

(Te - deÙje)-ricvT(ue,Te, rCvT)/rcvT = rrrw-FR±r2f-9v-Lüds (3.75)

Since the drive shaft speed is a function of the engine speed and the

gear ratio of the CVT

(3.74)

^ds = 'ê • t'CVT, (3.76)

3.3. POWERTRAIN 47

the time derivative of the drive shaft speed can be written as

^ds = Ùe • rCVT + ê ' ÏCVT = ê ' rCVT + We • W2, (3.77)

where u2 is the system input of the CVT, see Equation (3.58).Substituting (3.76) and (3.77) into (3.75) yields the following equationof motion for the engine speed:

.

_

Te-7icvT(ue,Te,rCvT) - r-rw-FR - r-rf-u2-0v-ue0e i]cvr (we, Tc, rcvT ) + Vv- r~

This equation of motion was also proposed in (Jones, Kuriger, Hughes,Holt, Ironside and Langley, 1986), but therein the losses of the CVT

considered by T]cvrr were neglected.A particular characteristic of the drive train shown in Figure 3.10

can be recognized if the equation of motion (3.78) is not formulated

for the engine speed, but for the drive shaft speed:

rcvT ( Tc ~r-rw>FR + u2-zP^ wds)V

______________

rCVT Jùds = -^^^^^^^ (3.79)

where for simplicity the CVT losses are neglected, i.e. r/cvr is set

equal to 1. The right-hand side of Equation (3.79) shows that the

drive shaft or the vehicle, respectively, can be accelerated by either

increasing the engine torque Te or by increasing the gear ratio of the

CVT rcvT-, he. by a positive value of the control input u2 = fcvT-

However, in order to produce more engine power Pe = Te ue, the

engine speed lo as well as the engine torque Te should be increased.

But increasing the engine speed requires a negative control input u2

causing the third term of Equation (3.79) to be negative. Therefore

to guarantee a positive vehicle acceleration, the gear ratio change rate

u2 has to be carefully limited as a function of the actual engine torque

(Yang et ah, 1985). The condition for a positive resulting drive shaft

acceleration {lüc\s > 0) can be easily derived from Equation (3.79)

u2 = rcvT > :^x- ——

-rcvT, (3-80)

which has to be fulfilled at each time instant. Therefore, an increase

of the engine power Pc may result in a deceleration of the vehicle if


condition (3.80) is violated, even though the power demand is positive.This reaction of the system is very similar to the non-minimum phasebehavior well-known from certain linear systems.

For parallel hybrid vehicles equipped with a CVT, the electric mo¬

tor can be used to compensate this negative torque term just described

(Shafai and Geering, 1996).

3.4 Combined System

The complete model of the powertrain equipped with the DSC enginediscussed in Subsection 3.1.1 becomes a 6th-ordcr system of nonlinear

ODEs. Because four state variables are associated with the engine

model, see Equations (3.4)-(3.7), one state variable is necessary to

describe the simplified model of the CVT (Equation (3.58)), and fi¬

nally one is needed to reflect the drive train dynamic itself (Equation

(3.78)).For the model of the powertrain with a conventional SI engine in¬

troduced in Subsection 3.1.2. the order is reduced by two, because the

conventional SI engine is described by two ODEs only, i.e. Equations

(3.9) and (3.10).No matter which engine type is used, the complete system has two

inputs, namely

• the load pedal signal u\ and

• the CVT gear ratio change rate ii2-

The "quasi static" optimization of Chapter 4 as well as the optimalcontrol problem formulated and solved in Chapter 5 are based on the

equations of motion just described. In fact, only a steady-state version

of these ODEs is used for the "'quasi static" optimization.


A model of the whole powertrain of a vehicle consisting of an engine, a

CVT, a final drive, and the vehicle itself has been derived. Two differ¬

ent engine types were modeled: the DSC SI engine which represents

a promising approach for improving fuel economy and a state-of-the-

art conventional SI engine of about the same peak power. For each


of the two engine types, the load-pedal to torque behavior has been

described by nonlinear ODEs, and a measured fuel consumption mapwras presented.

A CVT model of the chain drive CVT RHVF 147 manufactured byP.I.V. Reimers was derived. Based on the model proposed in (Shafaiet al., 1995), a more detailed description was derived which includes a

measured mechanical efficiency map and a gear-type pump model for

the hydraulic supply. In addition to these extensions, the mechanical

equations of motion were replaced by a new approach and the struc¬

ture changes of the mechanical part of the system were introduced.

This detailed model of the CVT was the starting point for the deriva¬

tion of a simpler model which is a good trade-off between complexityand accuracy so that it can be used for the optimal control problem.

The dynamic of a CVT-equipped powertrain including the vehicle

was derived and analyzed. The so-called non-minimum phase behavior

of such a powertrain was also discussed.

Finally, all the models of the various components of the whole

powertrain were combined so that their equations of motion can be

used for the optimization discussed in the remainder of this thesis.

Chapter 4

"Quasi Static"

Optimization

The fuel-optimal operation for "quasi static" conditions of a vehicle

equipped with a CVT and an electronic throttle is the operation of

the system on the corresponding peak efficiency curve 0 as discussed

in Section 2.1. Up to now, the correct problem definition for this

"quasi static" fuel-optimal operation as well as the resulting static

optimization problem has not been introduced. This is the topic of

the present chapter.

Moreover, for the derivation of this "quasi static" peak efficiencycurve O so far only the efficiency map of the engine has been considered

and the efficiency of the remaining powertrain, especially that of the

CVT r]cvT, has been neglected. The derivation of the fuel-optimal

operation including the CVT losses has therefore also to be analyzedand solved. This is done for the system introduced in Chapter 3,where the values of the DSC engine have been chosen for the numerical

analysis.

4.1 Problem Definition

The fuel-optimal operation problem of a powrertrain equipped with a

CVT for "quasi static" conditions can be defined as follows:

The vehicle is assumed to move with a certain actual velocity vact and

52 CHAPTER 4. "QUASI STATIC" OPTIMIZATION

the driver demands a desired wheel power PWd by holding a constant

accelerator pedal position. On these conditions, find the optimal en¬

gine speed uj", the optimal engine torque Te°, and the optimal gear

ratio of the CVT rCVT such that fuel mass flow riifue\ related to the

wheel power Pw is minimized, i.e.

.

r, , mfuei {ue,Te) .

mm J {ue,Ie)rcvT) =

77^—-^— r- (4.1)lw {ue,le,rcvT)

This choice of the performance index J can be interpreted as the

specific fuel consumption of the whole system. It indicates how ef¬

ficiently the whole powertrain is using the fuel supplied to producework on the wheel. If the powertrain is assumed to be lossless, i.e. the

wheel power is equal to the power supplied by the engine (Pw = Pe),this choice coincides with the engine's brake-specific fuel consumptionbsfc, cf. Equation (3.3). However, considering or neglecting the CVT

losses, the performance index J can be written as

J (ue,Te,rcvT) = — 7—~7F T> (4-2)VcvT Kê, L

o rcvr)

where in the latter case the efficiency of the CVT is equal to 1 and the

performance index J becomes independent of the gear ratio rcvr-

Obviously the three control parameters coe, Te, and rcvr of the

static optimization problem have to fulfill certain conditions. In fact,all of them obviously have to be between a minimum and a maximum

bound:

uje < ue < ue (4.3)f min — t

— cmax V /

TemJoje)<Tc<Te_(uje) (4.4)

r'CVTmin < r'cVT < rCVTmax- (4-5)

In addition to these inequality constraints (4.3)-(4.5), two equalityconstraints have to be met. First, the engine speed ue and the gear

ratio of the CVT rcvr have to be chosen such that the resultingvehicle speed v is equal to the one presumed, i.e. ?'act

v - vact = r-f rw uc • rcvr ~ vact = 0, (4.6)

4.2. WITHOUT CVT LOSSES 53

and second, the resulting power at the wheel Pw has to be the same

as the value demanded by the driver PWd

Pw - PWd -

ue ' Te t]cvt (we, Te, rcvr) - PWd = 0. (4.7)

The solution of the above stated static optimization problem is

generally defined as Q(PWd1 t?act) and represents three two-dimensional

static maps uj°(PWd1vact), T°(PWd, uact), and r^vr(PWd,v3LCt)- Since

the optimal gear ratio r^,VT can be easily expressed by the optimal

engine speed uj° and the actual vehicle speed vact

^{p-'•^) = 77^fc^• (4'8)

only the other two maps are shown in the numerical solutions of this

chapter.

4.2 Without CVT Losses

This section shows the well-known solution of the fuel-optimal opera¬

tion for "quasi static" conditions when the efficiency of the SI engineonly is taken into account. However, no publication could be found

that has formulated the problem in such a mathematical framework

as it is done in Section 4.1. Therefore the problem definition is brieflyrepeated,

butwiththenecessaryadjustmentsforthecaseinwhichtheCVTislossless,i.e.rjcvr=!•Findthevaluesofwe,Tc,andrcvrthatminimizetheperformanceindexJ(uje,Te)=bsfc(we,Te),(4.9)subjecttotheinequalityconstraints(4.3)-(4.5),theequalitycon¬straint(4.6)andPe-Plrd=ê-Te^PWd=0.(4.10)NoticethattheperformanceindexJdefinedin(4.9)isindependentofthegearratiooftheCVTrcvr-Furthermore,thegearratiooccursonlyinoneofthetwoequalityconstraints,namelyinEquation(4.6).Butforacertainvalueofa;P,thisequalityconstraintcanalwaysbefulfilledbychoosingthecorrectvalueforthegearratio,atleast

as


long as this correct value is within in the feasible range defined by

(4.5). Therefore, the optimal solution usually is independent of the

equality condition of the vehicle speed (4.5) and becomes a function

of the desired wheel power PWd only.This special characteristic is illustrated in Figures 4.1 and 4.2,

where the optimal engine speed map cü°(PWd,vaCi) and the optimal

engine torque map T°(PWd, uact) for the powertrain powered by the

DSC engine are shown.

desired wheel power [kW]u 0

vehicle speed [km/h]

Figure 4.1: Optimal engine speed map u°(PWd, väc\) of the DSC en¬

gine in case the CVT losses are neglected

For small values of the vehicle speed ?;act and large values of the desired

wheel power PWd, the optimal value of the gear ratio r^VT becomes

equal to the maximum gear ratio rcvrmAX >the upper limit of the in¬

equality constraint (4.5). Consequently, the optimal engine speed u°

becomes dependent on the engine speed ?;act because of the equalityconstraint (4.6). The same dependency occurs for large values of the

vehicle speed ?;act and for small values of the desired wheel power PWd.

4.2. WITHOUT CVT LOSSES 55

desired wheel power fkW]° 0

vehicle speed [km/h]

Figure 4.2: Optimal engine torque map T°(PWd}vact) of the DSC

engine in case the CVT losses are neglected

But in this case, the optimal gear ratio r^VT becomes equal to the

other extreme value of (4.5), i.e. the minimum gear ratio rcvTmin- Be¬

tween these two extreme areas of operation, the fuel-optimal solution

is independent of the vehicle speed vact and degenerates to a function

of the desired wheel power PWd only. Thus for the area of operation,where the solution is independent of the vehicle speed uact, the tra¬

jectory of the optimal engine speed u° and the optimal engine torque

Te° in dependence on the desired wheel power PWd can be drawn on

the engine map which leads to the well-known "quasi static" peakefficiency curve O pictured in Figure 3.1.

In the literature this dependence of the optimal engine values on

vehicle speed is usually omitted and only the O curve is used. But

since the classical structures described in Chapter 2 bring the gear

ratio and not the engine speed to the corresponding value on 0, the

physical limitations of the gear ratio of the CVT (4.5) automaticallyforce the system to operate with the optimal values that correspondto uj°e{PWd1vACt) and T°(PUit, ract).


The numerical solution of the constrained and nonlinear static op¬

timization problem defined at the beginning of this section and pic¬tured in Figures 4.1 and 4.2 has been obtained using standard Se¬

quential Quadratic Programming (SQP) methods. To be precise, the

Matlab Optimization Toolbox (Branch and Grace, 1996) and the

NPSOL package (Gill, Murray, Saunders and Wright, 1998) of FOR¬

TRAN 77 subroutines were used. The results of both methods coincide

to five decimal places. For the interpolation of the fuel consumption

engine map bsfc(we, TP), the spline interpolation proposed in (Press,Teukolsky, Vetterling and Flannery, 1992) has been applied for both

methods.

4.3 Considering the CVT Losses

The more comprehensive constrained and nonlinear static optimiza¬tion problem stated in Section 4.1 (Equations (4.2)-(4.7)) has to be

solved if the CVT losses are not neglected. As mentioned above, the

efficiency of the CVT i]cvt hi that case is not assumed to be constant

and equal to 1, as in the case before, but is a function of the three

control parameters we, Te, and tcvt, see Figure 3.9. Thus the perfor¬mance index J is a proper function of the three control parameters, as

defined in Equation (4.2), and the optimal solution will never become

independent of the vehicle speed vact- This fact is clearly visible in

Figures 4.3 and 4.4, where the maps of the optimal engine parametersWeftd,«act)andT°{PWd)v.ACt)aredepicted.Thesurfacebetweenthetwoextremeareasofoperation,wheretheoptimalgearratioisattheupperorlowerboundofthefeasiblerangedefinedby(4.5),isalwaysafunctionofthedesiredwheelpowerPWdandthevehiclespeedi>act,cf.Figures4.1and4.3orFigures4.2and4.4.Therefore,drawingthetrajectoryoftheoptimalenginespeeduj°andtheoptimalenginetorqueTe°independenceonthedesiredwheelpowerPWdandtheactualvehiclespeedvactontheenginemapnolongeryieldsthesingleandwell-known"quasistatic"peakefficiencycurveO,showninFigure3.1,butanentireregion.Itcanberegardedastheunionofthevarious"quasistatic"peakefficiencycurveswhichariseforeverypossibleconstantvalueofthevehiclespeedt>act.Thisspecialcharacteristicofthefuel-optimalsolutionobviously

4.3. CONSIDERING THE CVT LOSSES 57

max gear ratio

200

desired wheel power [kW vehicle speed [km/h]

Figure 4.3: Optimal engine speed map u>°(PWcnvaci) of the DSC en¬

gine in case the CVT losses are taken into account

affects all the classical control structures described in Chapter 2 that

use the "quasi static" peak efficiency curve Q as a lookup table (singletrack strategy and off the beaten track strategy). An additional inputfor these lookup tables has to be introduced, so that the actual vehicle

speed vact can also be taken into account for the derivation of the

desired controller values.

The numerical solution pictured in Figure 4.3 and 4.4 has been

obtained using the same methods as discussed in Section 4.2. More¬

over, a spline interpolation has been used for the numerical evaluation

of the CVT efficiency map r}cvT(ê,TP,rcvT) using the algorithmsproposed in (Press et al, 1992).

The maximum gear ratio driving resistance curve A as well as the

peak efficiency curve for stationary operation V remain single curves

on the engine map even when the CVT losses are not neglected. This

is obvious for the A curve, where the gear ratio is fixed to its max¬

imum value and thus the CVT losses simply cause a shift towards

higher engine torques, whereas for the T curve this is more difficult to


desired wheel power [kW]° 0

veh.cle speed [km'h]

Figure 4.4: Optimal engine torque map T°(PWd,vact) of the DSC

engine in case the CVT losses are taken into account

understand. However, the fuel-optimal operation for stationary con¬

dition F as well as the stationary wheel power Pw(vact) are only a

function of the actual vehicle speed vact. Thus the fuel-optimal op¬

eration for stationary condition F can be seen as a special subset of

the optimal solution for "quasi static" conditions, namely when the

desired wheel power coincides with the stationary value for the actual

vehicle speed

r(üAct) = fi(P«,(yact)âct)- (4-11)

In addition, the function PtlJ(vac\) has a one-to-one correspondenceand consequently the peak efficiency curve for stationary conditions F

is a single line on the engine map, no matter whether the efficiency of

the CVT is considered or neglected. Moreover, this single line is partof the region of the optimal solution for "quasi static" conditions on

the engine map.



The fuel-optimal operation problem for "quasi static" conditions has

been generally formulated as a constrained and nonlinear optimization

problem. It was numerically solved for the powertrain powered by the

DSC engine, once neglecting and once considering the CVT losses.

Both cases were analyzed and the difference between their solutions

has been demonstrated.

In the case in which the CVT losses are not taken into account,the optimal solution is mostly independent of the actual vehicle speedand yields the well-known peak efficiency curve for "quasi static" con¬

ditions 0. However, the solution obtained demonstrates that specialattention has to be paid to the fact that the possible gear ratio range

is bounded.

The optimal solution for the case in which the CVT losses are

taken into account always shows a dependency on the actual vehicle

speed and can therefore not be drawn as a single line on the engine

map, but it covers an entire region. This affects the classical control

structures such that they have to be modified for this case. However,under realistic conditions, where CVT losses are certainly present, the

fuel-optimal operation for "quasi static" conditions has to be adjustedby including the actual vehicle speed.

Finally, the consequences of the CVT losses for the maximum gear

ratio driving resistance curve A as well as for the peak efficiency curve

for stationary operation T have been discussed. Both of them remain

single lines on the engine map, even in the presence of CVT losses.

\

Chapter 5

Optimal Transient

Operation

Although Optimal Control Theory in the present formulation was al¬

ready known in the fifties of the 20th century by means of the formu¬

lation of the Maximum Principle of Pontryagin (Pontryagin, Boltyan-skii, Gamkrelidze and Mislichenko, 1964) and Bellman's concept of

Dynamic Programming (Bellman, 1957), it has since been applied on

relatively few technical applications. This is due to the high com¬

putation requirement to solve optimal control problems numerically.This has become less of a problem in the last few years, since the

performance of state-of-the-art processors is now reaching a level that

allows to solve technical problems within a reasonable period of time.

The work presented in this chapter deals with such a technical

application of the optimal control theory, namely for the CVT-based

powertrain introduced in Chapter 3. In fact, the problem considered

herein is the minimization of the specific fuel consumption for such a

powertrain during transient system operations. Only two other publi¬cations were found in the literature that apply optimal control theoryto such a type of system (Wu. Shen and Wang, 1997) and (Kleimaierand Schroder, 2000), both analyzing a completely different problemfrom the one at hand.

However, before Optimal Control Theory can be applied, the prob¬lem has to be formulated and brought into a proper mathematical

description.

äj

62 CHAPTER 5. OPTIMAL TRANSIENT OPERATION

5.1 Problem Definition

The optimal control problem considered in this thesis is the mini¬

mization of the specific fuel consumption of the vehicle modeled in

Chapter 3 during transient system operations of the vehicle. The con¬

trol functions, load pedal u\ — Ip and gear ratio change «2 = ïcvt,are to be found such that a prescribed vehicle speed change within a

given time range tf is achieved with a minimum of fuel consumed per

distance traveled. In addition, some vehicle speed constraints have to

be met during this maneuver.

Although the optimal solution was investigated for several different

vehicle trajectories, the solution of one particular trajectory only is

herein presented, viz. the solution for the specific acceleration part of

the FTP75 driving cycle (CFR, 1984) depicted in Figure 5.1.

80 r

70-

60

50:-'

40

upper limit of v„

-

«-

*

prescribed v ,,.-•'''

lower limit of v

5

time [s]

10

Figure 5.1: The selected part of the FTP75 driving cycle

The vehicle starts at a speed of 49.1 km/h and is set to accelerate

within 10 s to a final speed of 75.3 km/h. The FTP75 driving cyclerequires the vehicle speed to remain between certain upper and lower

speed limits, as indicated in Figure 5.1. The trajectory has to be

finished in exactly 10 s, i.e., the optimal control problem has a fixed

5.1. PROBLEM DEFINITION 63

final time of tf — 10 s.

Explicitly, in Mayer form (Föllinger, 1985) the problem is given by

. , m{uci(tf) , .

mm J = ;—-—-, 5.1

s(tf)l ;

where rnfue\(if) is the mass of fuel used and s(tf) is the distance trav¬

eled. The optimization is subject to the equations of motion (3.58),(3-78),

which represent the engine dynamics from load-pedal position to brake

torque (i.e. Equations (3.4)-(3.7) for the DSC engine or Equations

(3.9) and (3.10) for the conventional engine) and the two additional

ODEs which are needed to compute the performance index J

mfuel = mfuel (ue,Te) (5-3)

s = v. (5.4)

The special choice of the performance index J corresponds to the

regular approach for fuel consumption measurements during drivingcycles where the fuel consumed is not related to the prescribed dis¬

tance, but rather to the one actually traveled.

The control constraints are given by the nonlinear state-dependentinput constraints of the CVT (3.61) and the permissible load pedal

range

Ipe [0,1]. (5.5)

In addition to the linear and constant state inequalities (3.60) of

the gear ratio of the CVT. the two nonlinear time-dependent state

inequalities of the allowed vehicle speed range of Figure 5.1 have to

be considered as well:

«lower(0 < V(t) < Supper© (5-6)

The system is assumed to operate in steady-state conditions on

the peak efficiency curve for stationary operation T with a vehicle

speed v(0)—49.1km/hbeforeandv(tj)=75.3km/hafterthetransient.Thesystemstatesthereforehavetofulfillthe

following


boundary conditions:

rcvrify — rcvT0 fcvritf) = fcvTf

^fuei(O) = 0 mfuei(tf) = free

s(0) = 0 *(*/) = free

Since the vehicle speed is not a system state but a combination of the

two system states cuc and revt

v(t) = rf rw rCvr(t) we(£) (5.8)

it does not appear explicitly in the boundary conditions (5.7).

5.2 A Direct Collocation Method

Numerical solutions of optimal control problems, such as the one just

stated, can be obtained in two different ways: by so-called direct or

indirect methods (von Stryk and Bulirsch, 1992).The latter method takes full advantage of the Optimal Control

Theory, especially of the Maximum Principle, and converts the orig¬inal problem into a two-point boundary value problem that can be

solved either by the well-known shooting method (Bryson, 1999) or bythe relaxation method (Press et al., 1992).

However, direct methods are not based on Optimal Control Theory.They try to find the optimal solution by applying other algorithms.Direct methods are usually very effective for solving nonlinearandhigh-orderoptimalcontrolproblems.ThussuchamethodwaschosentoobtainthenumericalsolutionoftheproblemdefinedinSection5.1.ItiscalledDIRCOL(DIRectCOLlocationmethod)andisimple¬mentedinanumericaloptimizationpackagewiththesamename(vonStryk,1999).ThispackageisacollectionofFORTRAN77subrou¬tines.Theunderlyingmethodisbrieflyintroducedintheremainderofthissection.Themainideaofthismethodcanbedescribedasfollows:Bydiscretizingthecontinuoustimeinterval0<t<tfintoanJV-dimensionaltimeinterval0=ti<t2<...<tN=tf(5.9)

5.2. A DIRECT COLLOCATION METHOD 65

and by an appropriate interpolation of the control and state vari¬

ables between these time steps, the constrained infinite dimensional

control problem is transformed into a finite dimensional nonlinearlyconstrained static optimization problem (NLP) whose dimension de¬

pends on the discretization grid. This NLP is then solved either bythe dense Sequential Quadratic Programming (SQP) method NPSOL

(Gill, Murray, Saunders and Wright, 1998) or by the sparse SQPmethod SNOPT (Gill, Murray and Saunders, 1998).

Consider the following generally defined infinite dimensional opti¬mal control problem. The aim is to find the control input u(t) and

the final time tf that minimize the Mayer-type performance index

J(u(t),tf) = $(x(tf),tf) (5.10)

subject to a system of n nonlinear differential equations

Xi(t) = fi(x(t), u(t).t), i = 1,..., n, 0<t<tf, (5.11)

the nr boundary conditions

ri(x(0), x(tf), tf) = 0, i = l,...,nr, nr < 2n, (5.12)

the rih equality constraints

hl(x(t),u(t),t) = Q< i = l,...,nh, 0<t<tf} (5.13)

and the ng inequality constraints

gi(x(t),u(t),t)>Q, i = l,...,ng, 0<t<tf, (5.14)whereu(t)=(ui(t)....,'Ui(t))Trepresentsthe/-dimensionalcon¬trolvectorandthen-dimensionalstatevectorisdenotedbyx(t)=(x$t),...,xn(t))T.Thevariousfunctions<£,/,r,h,andgareas¬sumedtobecontinuouslydifferentiable.Thecontrolvectoruisas¬sumedtobeboundedandtfmaybefixedorfree.Noticethatotherproblemformulations,e.g.BolzaandLagrange-typeperformancein¬dex,integral-typeconstraint,ormin-maxperformanceindexcanbetransformedintotheformulationjustdescribed(vonStryk,

1999).


As a result of the discretization (5.9), the parameter vector Y of

the NLP consists of the values of the control and state variables at

the grid points tj, j = 1,..., N, and the final time tf

Y = (u(h),..., u(tN), x(h),..., x(tN), tN) e RNV+^+1. (5.15)

The control vector u(t) is approximated as a piecewise linear interpo¬

lating function between u(tj) and u(tj+$ for t E [tj,tj+\)

ut-t3

appM = u(tj) + (u(tj+1) - ufy)). (5.16)tj4-l - tj

The state vector x(t) is chosen as a continuously differentiablc vector

function which represents a cubic polynomial approximation between

x{tj) and x(tj+i) with xa,pp(s) := f(x(s),u(s),s) at s = tj, £j+i:

3/t — t-\k

fc=0^ J ^

4 = x(tj) (5.18)

c? = A,-/; (5-19)

4 - -3^) - 2A,-/,- + 3s(*i+1) - Ajfj+1 (5.20)

4= 2^) + Ajfj-2x(tj+i) + Ajfj+1 (5.21)

The following notation has been used in Equations (5.17)-(5.21)

fj := f(x(tj),u(tj),tj) and Aj := tJ+i — tj. (5.22)

The approximating function of the state vector has to satisfy the dif¬

ferential equations (5.11) of the original problem at the grid points tj,j = 1,..,, N, as well as at the centers of the discretization intervals

tc,j "•= (tj + tjjri)/2 for j = 1,..., N — 1. Since the chosen approxi¬mation (5.17)-(5.21) of x(t) already fulfills these constraints at tj, the

only remaining constraints of the NLP are

• the collocation constraints at the centers of the discretization

intervals tcj

J \xapp\tc,j)j Uapp\tc,j j) t'c,j) ~ xa,pp{tCij) — U, [u.Zoj

5.2. A DIRECT COLLOCATION METHOD 67

• the equality constraints at the grid points tj

HXzppitj), Wappfe), tj) = 0, (5.24)

• the inequality constraints at the grid points tj

9(x&pp(tj), wappfe), tj)>0, (5.25)

• and the initial and end point constraints at ti and tjy

r(xaPP(h),xHJ)p(tN),tN) = 0. (5.26)

Hence the original continuous problem of finite dimension is trans¬

formed in the following large-scale but finite dimensional NLP:

Find Y such that the performance index

$(Y), $ : S'V(/^nKl -> R (5.27)

is minimized, subject to the equality and inequality constraints (5.23)-(5.26).

This NLP can be solved using state-of-the-art SQP software as for

example the dense SQP method NPSOL (Gill, Murray, Saunders and

Wright, 1998) or the sparse SQP method SNOPT (Gill, Murray and

Saunders, 1998).

Moreover, it can be shown that the first-order necessary conditions

of optimality of the discretized problem reflect the necessary condi¬

tions of the original continuous problem (von Stryk, 1993). Hence,relatively accurate estimates of the adjoint variables X(t) and of the

multipliers of state constraints r)(t) of the original continuous prob¬lem can be obtained from the solution of the NLP. Therefore, the

computation of the Hamiltonian H and the inspectionofthefirst-orderoptimalityconditions(BrysonandHo,1975)canbeperformedeasily.Theinspectionofthefirst-ordernecessaryconditionsofopti¬malityisalsointegratedinthesoftwarepackagecalledDIRCOL(vonStryk,1999)whichmakesthistoolveryvaluableandpowerfulforsolvingoptimalcontrolproblemsnumerically.


5.3 Optimal Solution

The solution of the optimal control problem defined in Section 5.1 was

obtained using the DIRCOL software package just described. Unfor¬

tunately, DIRCOL requires feasible and relatively "good" initial esti¬

mates of the system states x(t) and of the control inputs u(t) at the

grid points to solve the problem properly. But at the beginning, this

information is usually not available. Therefore a continuation tech¬

nique (known as homotopy) was applied: The original problem was

simplified in such a way that the resulting problem could be solved.

Then the solution of the simplified problem was used as input (initialestimate) for a less simplified problem, and so on, until the originalproblem was reached.

Examples for such simplifications, which may be used in combina¬

tions also, are:

• Choosing a completely different performance index J as for ex¬

ample the minimum-time optimal control problem, because its

solution can often be obtained by physical considerations.

• Relaxing some of the inequality constraints which may be too

restrictive. The less restrictive problem may be easier to solve.

• Expand the prescribed initial or final values of the state variables

x to a whole feasible area.

Once the solution of the simpler problem is obtained, it has to be used

for the initial estimates of the original problem. Usually this step is

too large and DIRCOL fails to find the correct solution even with

these initial estimates. In this case, a new problem has to be solved

first which is a combination of the already solved simplified problemand the original one. The size of this step has to be carefully chosen

such that the numerical algorithm does not fail.

This technique was also applied to obtain the solution of the opti¬mal control problem considered herein. The initial problem that was

solved was the minimum-time optimal control problem for the lossless

powertrain. This minimum-time optimal control problem then was

slowly transformed to the original one described in Section 5.1 and

hence the numerical solution could be obtained.

5.3. OPTIMAL SOLUTION 69

5.3.1 Downsized Supercharged Engine

For the DSC engine, the optimal solution of the lossless powertrain

(r/cvT = 1) as well as the solution when the CVT losses are taken

into account are presented.

Without CVT Losses

The solution of the optimal control problem defined in Section 5.1,when the powertrain is assumed to be lossless, can be seen as an in¬

termediate stage of the continuation technique applied to solve the

problem with the losses taken into account. However, in order to

demonstrate the influence of the CVT losses on the optimal control

problem, it is presented anyway. The optimal engine operation tra¬

jectory is pictured in Figure 5.2.

70 r— i i—

i t~~~~ t r—

0I J _J, 1 1_ 1 L_ ! I

, 1 1

1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000


Figure 5.2: Engine map trajectory of the optimal solution

As described earlier, the system starts at the point on the peakefficiency curve for stationary operation F which corresponds to the

vehicle speed v(0) = 49.1 km/h. Then the engine speed is quicklyraised to approximately 3500 rpm by changing the gear ratio of the

70 CHAPTER, 5. OPTIMAL TRANSIENT OPERATION

CVT as fast as possible, before the engine gets back on the best effi¬

ciency curve for stationary operation V at 1 s. Actually, it gets on the

"quasi static" best efficiency curve Q which coincides with T for enginespeed values greater than 2800 rpm. After the system has reached $7,it stays on this '"quasi static'' best efficiency curve O for a relatively

long time (8.6 s) before it leaves it again to return to the final sta¬

tionary operating point on V. The resulting time traces of engine and

vehicle speed are depicted in Figure 5.3.

4000

3000

CD

to

5o>

w

2000 S

1000

01 23456789 10

time [s]

Figure 5.3: Vehicle and engine speed traces of the optimal solution:

vehicle speed (solid black) and engine speed (solid gray)

It can be shown that the system is on a singular arc (Äthans and

Falb, 1966) if the engine operates on the O curve, because during this

time period (t G [1.0 s, 9.6 s]) the Hamiltonian H ceases to be an ex¬

plicit function of the gear ratio change rate u<i. Since the Hamiltonian

H of the optimal control problem considered herein is linear in the

control variable 112, optimal control in the regular case would be of the

bang-bang type (Johnson and Gibson, 1963). Obviously, this is not

the case while the system operates on the O curve, because during this

period the gear ratio of the CVT changes only slightly, see Figure 5.3.

The optimal control input trajectories ui{t) and m2(£) have been

inspected as to whether or not they coincide with feasible candidates

.__r x —^—T_._„__ .

T r^. -j- r


derived from an extended version of the Maximum Principle of Pon-

tryagin (Chen and Desrochers, 1989; Leitmann, 1981). This extended

version of the Maximum Principle permits to find possible candidates

for the control inputs even in the presence of active state constraints.

The optimal control inputs obtained with DIRCOL indeed coincide

with some of the possible candidates.

The behavior of the optimal solution described above does not

correspond to any of the classical control strategies known in the lit¬

erature and described in Chapter 2. Therefore controlling the engineon the "quasi static" peak efficiency curve Q during transient vehicle

operation, as proposed in the single track strategy, cannot be fuel-

optimal, although so far this has been assumed to be true. However,an estimation of how much can be gained compared to the classical

strategies will be the topic of Section 5.6.

Considering the CVT Losses

The solution of the optimal control problem for the powertrain in¬

cluding the CVT losses and powered by the DSC engine is shown in

Figures 5.4 and 5.5.




As described earlier, the system starts on the peak efficiency curve

for stationary operation F derived including the CVT efficiency map(solid gray), to be compared with the resulting F curve for the lossless

powertrain depicted in Figure 5.2. Then the engine speed is quicklyincreased to approximately 3500 rpm by changing the gear ratio as fast

as possible, before the engine gets back close to F at 2.1 s. Actually, it

reaches the "quasi static" optimal operation area fl(PWd, v&ct) defined

in Chapter 4. Since the optimal operation for "quasi static" conditions

is no longer a single curve, but a single line for any constant vehicle

speed vact, it is difficult to indicate it on the engine map because the

engine speed is obviously quite far from being constant. Therefore it

is not drawn. During a relatively long time (7.1 s), the system stayson this optimal area for "quasi static" conditions fl(PWd,vSLCt) before

it leaves it again to return to the stationary peak efficiency curve F.

The resulting time traces of the engine and vehicle speed are depictedin Figure 5.5.

80 —

70

1

>

"O

§60Q,to

o

xz

>

50

40

0123456789 10

time [s]



Notice that the system really reaches the Cl(PWd, vHCt) area alreadyat t = 1.4 s, but due to the special form of this area, it is far from co-


inciding with the peak efficiency curve F. The optimal "quasi static"

operation area ù(PW(l, vaci) prevents the gear ratio from taking values

smaller than 1 if it is possible to satisfy the power requirement by

increasing the torque. That is exactly what happens at t = 1.4 s, be¬

cause it is more efficient to increase the engine torque than to increase

the engine speed by decreasing the gear ratio to more than 1 to fulfill

the power requirement.

Remark: Notice that the system is again on a singular arc if the

engine operates at the Q(PWd, uact) area as it did in the lossless case.

Thus the optimal solution above is on a singular arc between t = 1.4 s

and 9.2 s.

In principle the optimal solution of the powertrain without the

CVT losses and the optimal solution when the CVT losses are taken

into account behave very similarly. Thus the behavior of the optimalsolution considering the CVT losses also does not correspond to any

classical control strategy as described in Chapter 2. Both have in com¬

mon that they are on a regular arc for a short period only, at the begin¬ning and at the end of the prescribed vehicle trajectory. However, the

system spends most of the time on a singular arc which correspondson the optimal "quasi static" operation analyzed in Chapter 4. This

behavior is plausible, because if the dynamics of the whole powertrainare assumed to be infinitely fast, then the system would always be

in a "quasi static" condition, which would correspond to the classical

considerations for fuel-optimal operation, see Chapter 2.


At first glance, the optimal solution for the conventional engine shownin Figures 5.6 and 5.7 looks very different from that of the DSC en¬

gine. In fact, there is only a minor difference in the structure of the

optimal solution. The main difference is only due to the different

characteristics of the fuel consumption map of the two approaches.Again, the system starts on the peak efficiency curve for stationaryoperation I\ Then the engine speed is increased as fast as possibleto approximately 3000 rpm where it stays for almost i.2 s before it

is decreased again to about 2Ü0Ü rpm. It stays on this level untilthe


end of the trajectory. At t = 9.7 s, the engine torque decreases to the

corresponding level of the stationary peak efficiency curve.

1000 1500 2000 2500 3000 3500 4000 4500 5000 5500



The system reaches the corresponding Q(PW(l, fact) area at t = 1.2 s

and stays there for about 1.2 s before it leaves this area again to re¬

turn at t = 3.1 s, but at a smaller engine power. Thus as for the DSC

engine, the optimal solution tends to operate in this "quasi static"

optimal operation area Q(PWd1 vact). Unlike the solution of the DSC

engine, however, the conventional engine solution leaves it twice. This

is due to the small separation between the ^(Pu^, t>act) area and the

maximum torque curve. Thus for the conventional engine, it is onlyat the beginning of the prescribed vehicle trajectory that the power

required to accelerate the vehicle is higher than the power that can be

achieved by just increasing the torque. Afterwards, the engine power

requirements can be fulfilled almost without changing the gear ratio

of the CVT.

Nevertheless, the system is on singular arcs whenever it operatesat the Çî(PWd, yact) area, viz. also in the case of the conventional SI

engine.

5.4. COMMON FEATURES 75

403 5

time [s]

6

3000

2500

2000

1500

101000



5.4 Common Features

The computation requirements to solve the optimal control problemon-line are too high even with modern state-of-the-art processors (sev¬eral minutes with a Pentium© III, 450 MHz). Therefore, optimal tra¬

jectories for all possible acceleration maneuvers have to be computedoff-line as described in Section 5.3 and saved as lookup tables in the

controller. Another possibility is to try to derive a causal control

strategy that is realizable on-line and can be applied for any transient

vehicle operation.

In order to find such a control strategy that approximates the opti¬

mal solution for any transient system operation, the optimal solutions

for various prescribed vehicle trajectories were investigated and solved.

Common features of all these optimal solutions were then analyzed to

understand the rules for optimality which then allow the derivation

of a suboptimal strategy. The common features of all these optimalsolutions of the two different engine types as well as for the various

different vehicle trajectories can be summarized as follows:


• bring the system to the "quasi static" optimal operation area

Q(PWd,viiCt) as fast as possible

• operate it on this area as long as possible

• during the periods when the system moves towards or away from

this area the gear ratio has to be changed with maximum pos¬

sible speed, i.e. bang-bang type control

• the optimal engine power trajectory Pe{t) is almost constant

over time, with a small overshooting at the beginning.

The last feature is due to the facts that the system starts at steady-state operation and that the dynamics of the system do not allow an

instantaneous wheel power change. Thus the actual wheel power Pw is

delayed compared to the required PW(i. For that reason, the dynamicsof the system are "inverted" at the beginning by choosing an engine

power Pe greater than the one that would be best for the rest of the

maneuver.

This overshooting of the engine power is also the reason for the

difference between the optimal solutions of the DSC and the con¬

ventional engine as described in Section 5.3, because the overshoot¬

ing for the DSC engine has almost no influence on the engine map

trajectory, whereas it substantially affects the behavior of the con¬

ventional engine. This is mainly due to the different characteristics

of the efficiency maps, because with increasing wheel power PWd the

corresponding engine torque and speed of the Ct(PW(i,va,ct) area also

increase for the DSC engine. For the conventional engine, on the other

hand, the engine torque is first increased while the engine speed is held

constant, followed by an increase in speed while holding the torquealmost constant.

These common features are not fulfilled by any classical control

strategy known in the literature, cf. Chapter 2. Thus it is indeed

necessary to derive a novel control strategy that approximates the

behavior of the optimal solution and preserves the fuel-optimality to

the greatest extent possible.

5.5. AN APPLICABLE SUBOPTIMAL STRATEGY 77

5.5 An Applicable Suboptimal Strategy

Fortunately, it is possible to derive such a suboptimal strategy which

shows only a small degradation in fuel economy as promised by the

optimal solution. This suboptimal strategy takes advantage of the

common features of the optimal solution outlined in Section 5.4. The

accelerator (the input of the driver) is mainly assumed to be pro¬

portional to a desired vehicle acceleration acar(t). This suboptimal

strategy can be described as follows:

• Filter the accelerator signal such that it "inverts" the power

dynamics of the system.

• Calculate the total desired wheel power PWd(t) that will accel¬

erate the vehicle with this filtered accelerator signal äcar(£), i.e.

PWd(t) = Pioad(i) + 5car(*) • mcar v(t). (5.28)

» Derive the engine speed and engine torque of the "quasi static''

optimal operation area i1(PWd(t),v(t)) that corresponds to this

desired wheel power and to the actual vehicle speed v(t).

• Bring the engine speed and torque to these desired values as fast

as possible.

• When the accelerator signal is decreased, apply the same al¬

gorithm as above, but hold whichever control input is faster

(engine speed or torque) at the current value as long as possiblebefore changing it, again as fast as possible, to the new desired

value.

Obviously this suboptimal strategy needs an accelerator pedal or a

sophisticated driver pedal interpretation. Figure 5.8 shows the result¬

ing trajectory in the DSC engine map for an accelerator pedal tip-in.

Although it does not necessarily look like the optimal trajectory,the degradation in fuel economy is only marginal, see Section 5.6. It

seems that the exact trajectory towards and awayfromthe"quasistatic"optimaloperationareaQ(PW(l(t),v(i))islessimportantthanthechoiceofaclevertrajectoryatthisarea.However,itseemsthattheapproach(5.28)reflectsthisquitewell.


70 , , T r

g I I I I 1, , I I -L- ! L_ I

1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000


Figure 5.8: Comparison of the optimal solution for the DSC engine

(dashed) and its approximation (solid) in the engine map

Notice that the accelerator command is set back to zero at t — 9.2 s

to fulfill the boundary conditions at the end of the trajectory. Since

the engine torque dynamics have a higher bandwidth than that of the

gear ratio change, the driver command for the engine torque is delayedand held a little bit longer at the most recent value.

Surprisingly, this suboptimal strategy strongly resembles the well-

known single track modified strategy described in Subsection 2.2.2,but only if the pedal signal is interpreted as described above and the

gear ratio is always changed by its maximum rate. In addition, the

CVT efficiency has to be taken into account for the derivation of the

correct "quasi static" optimal operation area Q(PWj(t),v(t)). If these

modifications are not performed, the fuel consumption remains on the

same order as that of the single track strategy, cf. Section 5.6.

5.6. EFFICIENCY IMPROVEMENTS ESTIMATION 79

5.6 Efficiency Improvements Estimation

The previous sections of this chapter have shown that it is indeed

possible to derive the optimal solution as well as a realizable subop¬timal control strategy. The improvements in fuel economy can now

be compared. The most commonly used strategy (speed envelope)and the one presently known as the most efficient were thus imple¬mented and simulated for typical driver inputs. Each of these driver

inputs was obtained by means of measurements in a conventional car

equipped with a CVT while driving the corresponding vehicle trajec¬

tory, as for example the one shown in Figure 5.1 above. The systemas well as the classical control strategies investigated were simulated

using the Matlab/Simulink environment (Frediani, 1999; Maag and

Schoedler, 2000). The resulting gains in fuel consumption are listed

in Table 5.1. The most commonly used strategy serves as reference

for the comparison.

Table 5.1: Relative fuel consumption for different strategies

Type of engine

StrategyConventional

Downsized

supercharged

Speed envelope base base

Single track (modified) -5.8% -1.0%

Optimal solution -6.0% -4.0%

Suboptimal solution -5.9% -3.5%

For both engine types, there is a remarkable gain in efficiency whenthe optimal solution instead of the speed envelope strategy is applied.Therefore changing the algorithms of today's CVT controllers to the

herein proposed strategy will improve the fuel consumption in the

order of magnitude of about 4 to 6% during transient vehicle oper¬

ations. In the case of the DSC engine the single track is quite far

away from the optimal solution (3%). Thus the suboptimal strategycan gain approximately 2.5% in efficiency compared to the best clas¬

sical approach. This is quite a substantial improvement, especially if


one considers that this strategy has long been accepted as the most

efficient. Moreover, this substantial improvement can be realized bychanging only controller algorithms, i.e. no additional hardware equip¬ment is necessary.

In the case of the conventional engine, the numbers are different.

Although the absolute gain compared to the speed envelope strategyis greater than for the DSC engine, almost nothing can be gainedcompared to the single track strategy. One reason for this is certainlythe special characteristics of the engine efficiency, i.e. the fact that

the "quasi static" optimal operation area Q(PWcn vact) is quite close to

the maximum engine torque curve. Thus optimal control is achieved

almost exclusively by changing torque while maintaining the gear ratio

constant, which is exactly what occurs with the single track strategy.

Generally it can be stated that the farther away the Q(PWd, vact)area is from the maximum torque curve of the engine, the more im¬

portant optimal control becomes for CVT-based powertrains. Hence,since new engine designs aim at moving the maximum efficiency pointever closer to the part-load area of the SI engine, optimal control of

CVT-equipped powertrains will become more important.


Optimal control theory has been successfully applied on the fuel-

optimal operation of CVT-based powertrains. After the problem was

defined in Section 5.1 and stated in a proper mathematical framework,it was solved using a state-of-the-art numerical optimization software

called DIRCOL. A brief introduction of this package as well as its

underlying method was given in Section 5.2. The optimal control

problem for particular vehicle trajectories and for two different enginetypes was solved with DIRCOL. The resulting optimal solution for a

part of the FTP75 driving cycle was presented in Section 5.3. Vari¬

ous inspections for optimalitv as for example the first-order conditions

were performed and approved.Common features of the optimal trajectories computed for differ¬

ent prescribed vehicle trajectories were listed in Section 5.4. Based

on these common features, a novelsuboptimalcontrolstrategythatisrealizableon-lineandcanbeusedforanytransientvehicleoperationwasderivedinSection5.5.Thissuboptimalstrategyiseasytoimpie-


ment and turns out to realize almost the same fuel economy benefits

as the optimal solution, but with substantially lower control complex¬

ity. The relation of this new strategy to the well-known classical singletrack modified strategy were explained.

In Section 5.6, the suboptimal control strategy obtained as well

as the optimal solution were compared to the classical control ap¬

proaches regarding the fuel consumption during a prescribed vehicle

trajectory. The expected fuel economy improvements are in the range

of 3% for the downsized and supercharged engine, whereas the well-

known single track strategy is shown to be almost optimal for the

conventional engine analyzed in this thesis. However, the fuel econ¬

omy improvements compared to today's most commonly implemented

strategy (speed envelope) are shown to be approximately 4% (for the

DSC engine) or 6% (for the conventional engine), respectively.Since these improvements in fuel consumption are obtained by

modifying the controller software only, no new hardware componentsare needed, thus costs should not be an issue.

Chapter 6

Conclusions and

Outlook

A brief introduction of classical CVT control strategies for transient

vehicle operation has shown that there is indeed a need to investi¬

gate the fuel-optimal control algorithm and how much the classical

strategies deviate from the optimal control solution. Furthermore,the classical control strategies are based on considerations of the effi¬

ciency map of the engine only, which represents a simplification that

is certainly not valid for powertrains equipped with a CVT, since the

losses of current CVTs are still too high to be neglected.

Therefore, an appropriate system model of the whole powertrain

including the efficiency maps of the engine as well as of the CVT is

developed. The overall efficiency map of the CVT is derived from a

detailed model of the chain-drive CVT RHVF 147 manufactured byP.I.V. Reimers. It describes the behavior of the mechanical part, the

hydraulic part, as well as that of the gear-type pump for the hydraulicsupply.

Since the efficiency of the CVT is not only a function of the inputspeed and torque of the CVT, but also depends on the gear ratio,the classical fuel-optimal considerations for "quasi static" conditions

are no longer valid. The fuel-optimal operation problem is therefore

brought into an appropriate mathematical framework and solved for

the lossless powertrain as well as for the case where the CVT losses

0 9

84 CHAPTER 6. CONCLUSIONS AND OUTLOOK

are taken into account. It is shown that the classical "quasi static"

peak efficiency curve 0 is no longer a single line, but rather a whole

area on the engine map (it can be represented by a single line for a

constant vehicle speed, only).Finally, the optimal control problem for transient vehicle opera¬

tions has been formulated and solved using state-of-the-art numerical

optimization software. By inspection of the optimal solution for var¬

ious preselected vehicle trajectories, common features of the optimalsolutions were identified. Based on these common features, a sub-

optimal strategy is proposed that is applicable on-line and realizes

almost the same fuel economy benefits as the optimal solution, but

with substantially lower control complexity. This suboptimal strategyis strongly related to the classical single track modified strategy, at

least for the cases in which the driver pedal interpretation is adjustedin a particular way and a bang-bang type control of the gear ratio is

implemented.

Compared to the best classical control approaches, the expectedfuel economy improvements are in the range of 3% for a downsized su¬

percharged engine concept, whereas the well-known single track strat¬

egy is shown to be almost optimal for the conventional engine dis¬

cussed herein. However, the fuel economy improvements compared to

today's most commonly implemented strategy (speed envelope) are

shown to be approximately 4% (for a DSC engine) or 6% (for a con¬

ventional engine), respectively.The optimal control algorithm can be used on speed trajectories

that are known a priori, e.g. in regulated driving schedules. For normal

driving conditions, where the simpler suboptimal but causal controller

has to be used, it is essential to have a sophisticated driver interpreta¬tion. Notice that the improvements in fuel consumption are obtained

by modifying the controller software only. Since no new hardware

components are needed, costs should not be an issue.

The optimization problem considered and solved in this thesis is

focused on the minimization of the fuel consumption only. Driveabil-

ity issues of such a fuel-optimal controlled vehicle were completelyneglected. Nevertheless, the algorithm proposed herein can be seen as

the best solution for the economy mode of future CVT control units,where the dynamics of the vehicle are of minor importance. More

85

especially the fuel-optimal vehicle trajectories presented demonstrate,at first glance, no unacceptable behavior concerning driveability.

Further work in this area could aim at finding an accurate and rep¬

resentee quantification of driveability to define a performance index

for the optimization problem that considers both fuel consumptionand driveability. The resulting optimal control problem could then be

solved using the same methods and procedures as proposed herein.

Finally, since the computation performance of modern state-of-the-

art processors is still increasing by a factor of 2 to 3 per year, it will

be possible to solve optimal control problems in real-time in the near

future.

I

I

Bibliography

Abromcit, G. and Wilkinson, A. C. N. (1983). An Electronic Control

Concept for a Continuously Variable Transmission, Proceedingsof the International Symposium on Automotive Technology and

Automation (ISATA), Vol. 1, Croydon, U.K., pp. 31-45.

Amstutz, A. (1991). Geregelte Abgasrückführung /Air Senkungder Stickoxid- und Partikelemissionen beim Dieselmotor mit

Comprex-Aufladung, Dissertation no. 9421, Swiss Federal Insti¬

tute of Technology, Zürich, Switzerland.

Äthans, M. and Falb, P. L. (1966). Optimal Control: An Introduction

to the Theory and Its Applications, McGraw-Hill Book Company,New York, NY.

Barandun, G. A. (1999). Entwicklung eines dynamischen CVT-

Modells, Term paper, Laboratory of Engine Systems, Swiss Fed¬

eral Institute of Technology, Zürich, Switzerland.

Beachley, N. H. and Frank, A. A. (1980). Principles and Definitions for

Continuously Variable Transmissions, with Emphasis on Auto¬

motive Applications, Proceedings of the Century 2 International

Power Transmissions & Gearing Conference, American Societyof Mechanical Engineers (ASME) 80-C2/DET-95, San Francisco,CA.

Bellman, R. (1957). Dynamic Programming, Princeton UniversityPress, Princeton, NJ.

Bonthron, A. (1985). CVT-Efficiency Measured under Dynamic Run¬

ning Conditions, SAE Technical Paper 850569, Detroit, MI.

Ü2L

88 BIBLIOGRAPHY

Bosch Automation. Zahnradpumpen, Datenblätter, Robert Bosch

GmbH, Automationstechnik, Mobilhydraulik, Stuttgart, Ger¬

many.

Branch, M. A. and Grace, A. (1996). Matlab Optimization Toolbox

User's Guide. Version 1.5, The MathWorks, Inc., Natick, MA.

Bryson, A. E. (1999). Dynamic Optimization, Addison Wesley Long¬

man, Inc., Menlo Park, CA.

Bryson, A. E. and Ho, Y.-C. (1975). Applied Optimal Control, Revised

printing, Hemisphere Publishing Corporation, Washington, D.C.

Butts, K., Sivashankar, N. and Sun, J. (1995). Feedforward and Feed¬

back Design for Engine Idle Speed Control Using £i Optimization,

Proceedings of the American Control Conference, Vol. 4, Seattle,

WA, pp. 2587-2590.

CFR (1984). Code of Federal Regulations, Title 40, Parts 81 to 99,

The Office of the Federal Register, Washington, D.C, pp. 756-

767.

Chan, C, Yang, D., Volz, T., Breitweiser, D., Jamzadeh, F. S., Frank,A. and Omitsu, T. (1984). System Design and Control Consider¬

ations of Automotive Continuously Variable Transmissions, SAE

Technical Paper 840048, Detroit, MI.

Chen, Y. and Desrochers, A. A. (1989). Structure of Minimum-Time

Control Law for Robotic Manipulators with Constrained Paths,

Proceedings of the IEEE International Conference on Robotics

and Automation, Vol. 2, Scottsdale, AZ, pp. 971-976.

Deacon, M., Brace, C. J., Guebeli, M., Vaughan, N. D., Burrows, C. R.

and Dorey, R. E. (1994). A Modular Approach to the ComputerSimulation of a Passenger Car Powertrain Incorporating a Diesel

Engine and Continuously Variable Transmission, Proceedings ofthe International Conference on Control '94, Vol. 1, Coventry,

U.K., pp. 320-325.

Dittrich, O. (1986). Der stufenlose Kettenwandler im Kraftfahrzeug,Getriebe im Fahrzeugbau, VDI-Berichte 579, VDI-Verlag GmbH,

Düsseldorf, Germany, pp. 151-168.

BIBLIOGRAPHY 89

Dittrich, O. and Simon, H. (1988). The Continuously Variable Chain

Transmission In Motor-Cars, SAE Technical Paper 885064, De¬

troit, MI.

Dräger, C, Gold, P. W., Kammler, M. and Rohs, U. (1998). Das

Kegelringgetriebe - Ein stufenloses Reibradgetriebe auf dem

Prüfstand, Automobütechnische Zeitschrift (ATZ) 100(9): 640-

646.

Engelsdorf, K., Senger, K.-H. and Bolz, M.-P. (1996). Electronic CVT

Control for Power Train Optimization, Yokohama Proceedings

of the International Conference on Continuously Variable Power

Transmissions CVT '96, Yokohama, Japan, pp. 71-76. (Conf.Pub. No. 111).

Föllinger, O. (1985). Optimierung dynamischer Systeme, Methoden

der Regelungstechnik, Oldenbourg Verlag GmbH, München, Ger¬

many.

Frediani, L. (1999). Quantitativer Vergleich von Steuerstrategien für

CVTs, Term paper, Laboratory of Engine Systems, Swiss Federal

Institute of Technology, Zürich, Switzerland.

Funatsu, K., Koyama, H. and Aoki, T. (1996). Electronic Control

System of Honda for CVT, Yokohama Proceedings of the Inter¬

national Conference on Continuously Variable Power Transmis¬

sions CVT '96, Yokohama, Japan, pp. 43-51. (Conf. Pub. No.

107).

Gesenhaus, R. (2000). Audi Multitronic: The New Generation of Au¬

tomatic Transmission, Proceedings of the 1st Ricardo Interna¬

tional Conference, Vehicle Systems Integration - the Way Ahead,

Brighton, U.K.

Gill, P. E., Murray, W. and Saunders, M. A. (1998). User's Guide for

SNOPT 5.3: A Fortran Package for Large-Scale Nonlinear Pro¬

gramming, Technical report, Department of Operations Research,Stanford University, Stanford, CA.

Gill, P. E., Murray, W., Saunders. M. A. and Wright, M. H. (1998).User's Guide for NPSOL 5.0: A Fortran Package for Nonlinear

90 BIBLIOGRAPHY

Programming, Technical Report SOL 86-1, Department of Oper¬ations Research, Stanford University, Stanford, CA.

Girijashankar, P. V., Srikantiah, G. and Pai, M. A. (1976). Dynamic

Simulation of a Boiling Water Nuclear Reactor, Annals of Nu¬

clear Energy 3(2-3): 133-445.

Göllmann, L. (1996). Numerische Behandlung optimaler Steuer¬

prozesse mit konstanten Verzögerungen unter gemischten Steuer-

Zustandsbeschränkungen, Dissertation, Westfälische Wilhelms-

Universität Münster, Germany.

Goppelt, G. (2000). Stufenloses Automatikgetriebe Multitronic von

Audi, Automobiltechnische Zeitschrift (ATZ) 102(2): 110-111.

Gui, X. Q,, Goering, C. E. and Buck, N. L. (1985). Simulation of a

Fuel-Efficient Augmented Engine, Transactions of the American

Society of Agricultural Engineers (ASAE) 32(6): 1875-1881.

Guzzella, L. and Martin, R. (1998). The SAVE Engine Concept, Mo-

tortcchnische Zettschrift (MTZ) 59(10): 644-650.

Gyarmathy, G. (1983). How Does the Comprex Pressure-Wave Su¬

percharger Work?, SAE Technical Paper 830234, Detroit, MI.

Heidemeyer, P. and Scholz, R. (1987). Kettenwandlcrgetriebe der

VW-Forschung, 1. Aachener Kolloquium für Fahrzeug- und Mo¬

torentechnik, Aachen, Germany, pp. 487-522.

Heywood, J. B. (1988). Internal Combustion Engine Fundamentals,McGraw-Hill Book Company, New York, NY.

Hiller, M. (1983). Mechanische Systeme - Eine Einführung in die ana¬

lytische Mechanik und Systemdynamik, Springer-Verlag, Berlin,

Germany.

Höhn, B.-R. (1988). The Potential And Advantages Of ContinuouslyVariable Transmissions For Automotive Applications, SAE Tech¬

nical Paper 885063, Detroit, MI.

Johnson, C. D. and Gibson, J. E. (1963). Singular Solutions in Prob¬

lems of Optimal Control, IEEE Transactions on Automatic Con¬

trol 8: 4-14.

BIBLIOGRAPHY 91

Jones, R. P., Kuriger, I. F., Hughes, M. T. G., Holt, M. J., Ironside,J. M. and Langley, P. A. (1986). Modelling and simulation of

an automotive powertrain incorporating a Perbury continuoslyvariable transmission, Proceedings of the International Confer¬ence on Integrated Engine Transmission Systems, Institution of

Mechanical Engineers (IMechE), Bath, U.K., pp. 4P 49.

Karassik, I. J., Krutzsch, W. C, Fraser, W. H. and Messina, J. P.

(eds) (1976). Pump Handbook, McGraw-Hill Book Company,New York, NY.

Kim, H., Song, H., Kim, T. and Kim, J. (1996). Metal Belt CVT and

Engine Optimal Operation by PWM Electro-Hydraulic Control,Yokohama Proceedings of the International Conference on Con¬

tinuously Variable Power Transmissions CVT '96, Yokohama,Japan, pp. 157-160. (Conf. Pub. No. 304).

Kleimaier, A. and Schröder, D. (2000). Optimized Design and Con¬

trol of a Hybrid Vehicle with CVT, Proceedings of 1st IFAC-

Conference on Mechatronic Systems, Vol. 1, Darmstadt, Ger¬

many, pp. 197-202.

Kluger, M. A. and Fussner, D. R. (1997). An Overview of Current

CVT Mechanisms, Forces and Efficiencies, SAE Technical Paper970688, Detroit, MI.

Kluger, M. A. and Long, D. M. (1999). An Overview of Current Auto¬

matic, Manual and Continuously Variable Transmission Efficien¬

cies and Their Projected Future Improvements, SAE Technical

Paper 1999-01-1259, Detroit, MI.

Leitmann, G. (1981). The Calculus of Variations and Optimal Con¬

trol: An Introduction, Plenum Press, New York, NY.

Liu, S. and Paden, B. (1997). A Survey of Today's CVT Controls,Proceedings of the 36t/l Conference on Decision & Control, San

Diego, CA, pp. 4738-4743.

Maag, J. and Schoedler, B. (2000). Optimierung und Vergleich klas¬

sischer Steuerstrategien für CVTs, Term paper, Laboratory of

Engine Systems, Swiss Federal Institute of Technology, Zürich,Switzerland.

92 BIBLIOGRAPHY

Micklem, J. D., Longmore, D. K. and Burrows, C. R. (1994). Mod¬

elling of the steel pushing V-belt continuously variable transmis¬

sion, Journal of Mechanical Engineering Science 208(C1): 13-27.

Naunheimer, H. (1995). Beitrag zur Entwicklung von Stufenlosgetrie-ben mittels Fahrsimulation, Dissertation, Universität Stuttgart,Germany.

Pellegrino, E. (1986). Powertrain Optimal Control, Automotive Micro¬

electronics: Proceedings of the Advanced Course on InformationTechnology for Fuel Economy, Emission Control and Safety in

Motor Vehicles, Capri, Italy, pp. 497-523.

Peng, L. (1987). Optimal Control System Design in Vehicles

with CVT, Modelling, Simulation & Control, B, AMSE Press

7(4): 19-26.

Pfiffner, R. (1999a). Kinematic and Kinetic Properties of Chain Drive

CVTs, Technical Report LMS 1999P2, Laboratory of Engine Sys¬tems, Swiss Federal Institute of Technology, Zürich, Switzerland.

http://www.lms.ethz.ch/Int/Berichte/1999.ll.22.98.pdf.

Pfiffner, R. (1999b). Steuerstrategien von CVT-Getrieben: Übersichtund Ausblick, UMSCHLINGUNGSGETRIEBE: Systemelementeder modernen Antriebstechnik VDI-Berichte 1467, VDI-VerlagGmbH, Düsseldorf, Germany, pp. 313-331.

Pfiffner, R. and Guzzella, L. (1999). Feedback Linearization

Idle-Speed Control: Design and Experiments, Kybernetika

35(4): 441-458.

Pfiffner, R. and Guzzella, L. (2001). A Control-Oriented CVT Model

with Non-Zero Belt Mass, Journal of Dynamic Systems, Measure¬

ment, and Control, American Society of Mechanical Engineers(ASME). Submitted for publication.

Pietz, A. (1993). Betriebsverhalten von CVT-Riementrieben für den

Antrieb von Automobilen, Dissertation, Universität Hannover,Germany.

Pohlenz, W. (ed.) (1975). Grundlagen für Pumpen, Pumpen für

Flüssigkeiten und Gase, VEB Verlag Technik Berlin, Berlin, Ger¬

many.

BIBLIOGRAPHY 93

Pohlenz, W. (ed.) (1984). Pumpen für Flüssigkeiten, Pumpen für

Flüssigkeiten und Gase, VEB Verlag Technik Berlin, Berlin, Ger¬

many.

Pontryagin, L, S., Boltyanskii. V. G., Gamkrelidze, R. V. and

Mishchenko, E. F. (1964). The Mathematical Theory of Opti¬mal Processes, International Series of Monographs in Pure and

Applied Mathematics, Pergamon Press, Oxford, U.K.

Powell, B. K. and Cook, J. A. (1987). Nonlinear low frequency phe-nomenological engine modeling and analysis, Proceedings of the

American Control Conference, Vol. 1, Minneapolis, MN, pp. 332-

340.

Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P.

(1992). Numerical Recipes in FORTRAN: The Art of ScientificComputing, second edn, Cambridge University Press, Cambridge,U.K.

Rattunde, M., Schönnenbeck, G. and Wagner, P. (1991). Bauelemente

stufenloser Kettenwandler und deren Einfluß auf den Wirkungs¬grad, Getriebe in Fahrzeugen, heute und morgen, VDI-Berichte

878, VDI-Verlag GmbH, Düsseldorf, Germany, pp. 259-275.

Renius, K. T. (1996). Traktoren 1995/96, Automobiltechnische

Zeitschrift (ATZ) 98(9): 468-473.

Sauer, G. (1994). Untersuchungen an stufenlosen Umschlingungs-getrieben für Kraftfahrzeuge, Innovationen im Antriebsstrangvon Kraftfahrzeugen, VDI-Berichte 1170, VDI-Verlag GmbH,Düsseldorf, Germany, pp. 173-186.

Sauer, G. (1996). Grundlagen und Betriebsverhalten eines Zugketten-Umschlingungsgetriebes, Fortschritt-Berichte VDI, Reihe 12, Nr.

293, VDI-Verlag GmbH, Düsseldorf, Germany.

Schmid, A. (1993). Modellbildung der Hydraulik und der Mechanik

des Kettenwandlers, Technical Report Mot 2-93, Measurement

and Control Laboratory, Swiss Federal Institute of Technology,Zürich, Switzerland.

94 BIBLIOGRAPHY

Shafai, E. and Geering, H. P. (1996). Control Issues in a Fuel-

Optimal Hybrid Car, Proceedings of the IFAC 13th Triennial

World Congress, Session 8b-07 4, San Francisco, CA, pp. 231-

236.

Shafai, E., Simons, M., Neff, U. and Geering, H. P. ( 1995). Model of a

Continuously Variable Transmission, Proceedings of the 1st IFAC

Workshop on Advances in Automotive Control, Monte-Vcrità,

Switzerland, pp. 99-107.

Shim, D., Park, J., Khargonekar, P. P. and Ribbons, W. B. (1995).Engine Idle Speed Control, Proceedings of the American Control

Conference, Vol. 4, Seattle, WA, pp. 2587-2590.

Soltic, P. (2000). Part-Load Optimized SI Engine Systems, Disser¬

tation no. 13942, Swiss Federal Institute of Technology, Zürich,Switzerland.

Soltic, P. and Guzzella, L. (2000). Optimum SI Engine Based Pow-

ertrain Systems for Lightweight Passenger Cars, SAE Technical

Paper 2000-01-0827, Detroit, MI.

Song, H., Kim, T., Kim, J. and Kim, H. (1996). Shifting Charac¬

teristics of PWM Electro-Hydraulic Controlled Metal Belt CVT

for the Engine Optimal Operation, Proceedings of the 7th Inter¬

national Power Transmission and Gearing Conference, Vol. 88,American Society of Mechanical Engineers (ASME), San Diego,

CA, pp. 81-88.

Srnik, J. and Pfeiffer, F. (1999). Dynamics of CVT chain drives,International Journal of Vehicle Design 22(1/2): 54-72.

Stall, E. (1985). Enhuicklung einer elektronischen Regelung für

kraftstoffsparendes Motor-Getriebe-Management im Forschungs-Pkw Uni-Car, Schriftenreihe Automobiltechnik, Forschungs- gesellschaftKraftfahrwesenmbHAachen,Germany.Stubbs,P.W.R.(1981).TheDevelopmentofaPerburyTractionTransmissionforMotorCarApplications,JournalofMechanicalDesign103:29-41.

BIBLIOGRAPHY 95

Vahabzadeh, H. and Linzell, S. M. (1991). Modeling, Simulation,and Control Implementation for a Split-Torque, Geared Neutral,

Infinitely Variable Transmission, SAE Technical Paper 910409,

Detroit, MI.

Vahabzadeh, H., Macey, J. P. and Dittrich, O. (1990). A Split-

Torque, Geared-Neutral Infinitely Variable Transmission Mech¬

anism, Proceedings of the 18f/l FISITA Congress - The Promise

of New Technology in the Automotive Industry, Society of Auto¬

motive Engineers (SAE) 905089, Torino, Italy, pp. 687-695.

Vanvuchelen, P., Moons, C, Minten, W. and De Moor, B. (1995).Electronic Control of Continuously Variable Transmissions, Pro¬

ceedings of the 2nd International Conference on Road Vehicle Au¬

tomation, Bolton, U.K., pp. 100-109.

von Stryk, O. (1993). Numerical Solution of Optimal Control Prob¬

lems by Direct Collocation, in R. Bulirsch, A. Miele, J. Stoer and

K.-H. Well (eds), Optimal Control - - Calculus of Variations, Op¬timal Control Theory and Numerical Methods, number 111 in In¬

ternational Series of Numerical Mathematics, Birkliäuser, Basel,

Switzerland, pp. 129-143.

von Stryk, O. (1999). User's Guide for DIRCOL. A Direct Colloca¬

tion Method for the Numerical Solution of Optimal Control Prob¬

lems, Version 2.0, Technische Universität München, Germany,Lehrstuhl M2 Höhere Mathematik und Numerische Mathematik.

von Stryk, O. and Bulirsch, R. (1992). Direct and Indirect Meth¬

ods for Trajectory Optimization, Annals of Operations Research

37: 357-373.

Wang, G. (1991). Untersuchungen an stufenlos einstellbaren Zug-imd Schubgliedergetrieben, Dissertation, Technische Universität

Clausthal, Germany.

Weber, F. and Guzzella, L. (2000). Control Oriented Modeling of

a Pressure Wave Supercharger, SAE Technical Paper 2000-01-

0567, Detroit, MI.

96 BIBLIOGRAPHY

Wittmer, C. G. (1996). Entwurf und Optimierung der Fahrstrate¬

gien für ein Personenwagen-Antriebskonzept mit Schwungrad¬komponente und stufenlosem Zweibereichsgetriebe, Dissertation

no. 11672, Swiss Federal Institute of Technology, Zürich, Switzer¬

land.

Wu, G., Shen, S. and Wang, H. (1997). The Optimal Control of the

Vehicular Powertrain with a Continuously Variable Transmission,

Proceedings of International Truck & Bus Meeting & Exposition,

Society of Automotive Engineers (SAE) 973281, Cleveland, OH,

pp. 47-51.

Yang, D. and Frank, A. A. (1985). An Optimisation Technique for

the Design of a Continuously Variable Transmission Control Sys¬tem for Automobiles, International Journal of Vehicle Design6(1): 41-54.

Yang, D., Guo, Z. and Frank, A. A. (1985). Control and Responseof Continuously Variable Transmission (CVT) Vehicles, Proceed¬

ings of the 1985 American Control Conference, Vol. 3, Boston,MA, pp. 1438-1444.

Curriculum Vitae

Personal Data

Name:

Date of birth:

Place of birth:

Parents:

Rolf Andreas Josef Pfiffner

18. January 1969

Liestal BL, Switzerland

Andreas Pfiffner and Esther Biirgi-Leuenbcrger

Education

1976-1981

1981-1985

1985-1989

1989

1989-1990

1990-1996

1996

1996-2001

Primary school in Reinach BL, Switzerland

Secondary school in Reinach BL, Switzerland

Gymnasium in Münchenstein BL, Switzerland

Matura certificate, type C

Military service

Studies in electrical engineering, Swiss Federal insti¬

tute of Technology (ETH) Zürich, Switzerland

Major in automatic control

Diploma as Dipl. El.-Ing. ETH

Doctoral student and research assistant at the Insti¬

tute for Energy Technology, Swiss Federal institute of

Technology (ETH) Zürich, Switzerland

Professional Experience

2000 For five months I worked on the Automatic Trans¬

mission Control System project initiated by the Ford

Dunton Engineering Centre, Essex, England. Within

this project a control strategy for a new transmission

was developed.

fl7

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