A MULTIPLE OBJECTIVE

OPTIMIZATION APPROACH TO

QUALITY CONTROL

by

Christopher Michael Seaman

Rensselaer Polytechnic Institute

Electrical, Computer, and Systems Engineering

Troy, New York 12180-3590

April 1991

CIRSSE REPORT #89
CONTENTS

LIST OF FIGURES ................................ vi

LIST OF TABLES . .............. xii

DEDICATION ................................... xiii

ACKNOWLEDGEMENT ............................. xiv

ABSTRACT .................................... xv

1. Introduction ................................... 1

1.1 Motivation and Objectives ........................ 1

1.2 Contributions ............................... 2

1.3 Application to the Injection Molding Process .............. 3

1.4 Organization of the Thesis ........................ 5

2. The Injection Molding Process and Part Quality . .............. 9

2.1 Introduction ................................ 9

2.2 A Review of Injection Molding Research ................ 10

2.2.1 Research in Modeling and Control ................ ll

2.2.2 Research on the Effect of Microstructure on Mechanical Properties ................................ 14

2.2.3 Research on the Effect of Processing on Mechanical Properties 14

2.3 Preliminary Experiments ......................... 16

2.3.1 Sensitivity of Melt Temperature to Plastication Inputs .... 17

2.3.2 Part Quality Dependence upon Processing Conditions ..... 30

2.3.3 Sensitivity of Machine Dynamics to Operating Conditions 42

2.4 Summary . ............................... 48

3. Proposed Approach to the Quality Control Problem ............. 49

3.1 Introduction ................................ 49

3.2 Definition of the Quality Control Problem ............... 50

3.2.1 Performance Criteria ....................... 50

3.2.2 Inputs and Outputs ........................ 52

ii
.

.

3.2.3 The System Model ........................ 56

3.2.4 A Control Policy ......................... 58

3.3 A Review of the Multiple-Objective Optimization Problem ...... 59

3.3.1 Finding the Tradeoff Surface ................... 62

3.4 A Linear Programming Solution to Multiobjective Quality Control . . 64

3.5 Implementation of Automatic Quality Control ............. 66

3.5.1 A Proposed Feasible Directions Algorithm ........... 6T

3.5.2 Properties of this Algorithm ................... 71

3.5.3 Example .............................. 73

3.6 Summary . ................................ T4

Experimental Setup ............................... 76

4.1 The Injection Molding Machine ..................... T6

4.1.1 Actuators ............................. 76

4.1.2 Process Sensors .......................... 79

4.2 The Control Computer .......................... 80

Multiobjective Controller Tuning ....................... 84

5.1 Plastication Modeling and Identification ................ 87

5.1.1 Analytical Modeling of Plastication ............... 87

5.1.2 Identification of a Plastication Model .............. 91

5.2 Plastication Controller Tuning: Simulation Results .......... 95

5.2.1 Simulation Conditions ...................... 95

5.2.2 Discussion of the Simulation Tuning Results .......... 96

5.3 Plastication Controller Tuning: Experimental Results ......... 100

5.3.1 Discussion of the Experimental Tuning Results ......... 101

5.4 Injection Controller Tuning ....................... 110

5.4.1 Analytical Modeling of Injection ................. 110

5.4.2 Description of the Injection Controller Tuning Experiments . . Ill

5.4.3 Experimental Results ....................... 113

5.4.4 Discussion ............................. 121

5.5 Conclusions ................................ 123

.°.

111
6. Tuning the Process to Achieve Quality Objectives .............. 125

6.1 Spiral Mold Quality Tuning ....................... 125

6.1.1 Description of the Spiral Mold Quality Tuning Experiments . 125

6.1.2 Experimental Results ....................... 130

6.1.3 Discussion ............................. 135

6.2 ASTM Four Cavity Mold Quality Tuning ................ 136

6.2.1 Experimental Conditions ..................... 138

6.2.2 Discussion of the Tuning Results ................ 138

6.3 Conclusions ................................ 150

T. Conclusions ................................... 152

7.1 Contributions ............................... 154

T.2 Future Work ................................ 155

7.2.1 Improving Algorithm Efficiency ................. 155

7.2.2 Improving the Decision Maker's Interface ............ 15T

8. Literature Cited ................................. 159

A. The Injection Molding Process ......................... 16T

A.1 The Injection Molding Cycle ....................... 167

A.2 Plastication ................................ 169

A.2.1 Polymer Melting (Thermodynamics) .............. 169

A.2.2 Polymer Pumping (Fluid Dynamics) .............. 179

A.2.3 Overall Polymer Pumping Dynamics .............. 183

A.2.4 Overall Plastication Dynamics .................. 185

A.3 Injection .................................. 185

A.3.1 Injection Polymer Flow Dynamics ................ 187

A.3.2 Overall Injection Dynamic Model ................ 189

A.4 Holding .................................. 190

A.4.1 Holding Polymer Mass Balance ................. 191

A.4.2 Overall Holding Dynamic Model ................. 192

A.5 Cooling .................................. 193

A.6 Process Control and Phase Interaction ................. 194

A.7 Summary of Previous Research ..................... 196

A.8 Summary . ................................ 197

iv
B. Symbol Table and Nomenclature for Machine Dynamics ........... 198

B.1 Definition of the Quality Control Problem ............... 198

B.2 A Review of the Multiobjective Optimization Problem ........ 199

B.3 A Linear Programming Solution to Multiobjective Quality Control . . 199

B.4 Polymer Melting Due to Conduction Heating .............. 199

B.5 Polymer Melting Due to Viscous Dissipation .............. 200

B.6 Overall Melting Dynamics ........................ 200

B.7 Polymer Fluid Dynamics ......................... 201

B.8 Injection Dynamics ............................ 202

]3.9 Holding Dynamics ............................ 203

B.10 Cooling Dynamics ............................. 203

C. An Example of Approximation Programming ................. 204

C.1 Example .................................. 204
LIST OF FIGURES

Figure 1.1

Figure 1.2

Figure 1.3

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

Figure 2.10

Figure 2.11

Figure 2.12

Figure 2.13

Figure 2.14

Cut-away View of a Reciprocating Screw Injection Molding

Machine .............................. 6

The Injection Molding Phases .................. 6

Injection Molding Phases and Interactions ........... 7

Preparation of Specimens with Two Different Orientations 15

The relationship between melt temperature and melt pressure,

at three different barrel temperatures, for polystyrene ..... 19

Melt temperature error versus melt pressure for polystyrene. 19

Melt temperature versus recovery rate for polystyrene ..... 20

Recovery rate versus back pressure for polystyrene ....... 20

Melt temperature versus plastication time for polystyrene. . . 22

Melt temperature error versus plastication time for polystyrene. 22

Estimated versus actual temperature using a three parameter

model ................................ 23

Estimated versus actual temperature using a two parameter

model ................................ 23

Polypropylene melt temperature versus back pressure for four

different barrel temperatures at 200RPM. ............ 25

Polypropylene melt temperature versus back pressure for four

different barrel temperatures at 300RPM. ............ 25

Polypropylene melt temperature error versus back pressure for

four different barrel temperatures at 200RPM .......... 26

Polypropylene melt temperature error versus back pressure for

four different barrel temperatures at 300RPM .......... 26

Back pressure versus recovery rate: upper curve is 300RPM

data, lower curve is 200RPM data ................ 27

vi
Figure 2.15

Figure 2.16

Figure 2.17

Figure 2.18

Figure 2.19

Figure 2.20

Figure 2.21

Figure 2.22

Figure 2.23

Figure 2.24

Figure 2.25

Figure 2.26

Figure 2.27

Figure 2.28

Figure 2.29

Figure 2.30

Figure 2.31

Figure 2.32

Figure 2.33

Polypropylenemelt temperature versusplastication time for

four different barrel temperaturesat 200RPM. ......... 28

Polypropylenemelt temperature versusplastication time for

four different barrel temperaturesat 300RPM. ......... 28

Estimated temperatureversusactual temperaturefor polypropylene at 200RPM. ......................... 29

Estimatedtemperatureversusactual temperaturefor polypropyleneat 300RPM.......................... 29

Melt temperatureversusposition in the shot at four different

backpressures........................... 31

ASTM test mold usedfor part quality experiments....... 33

Velocity profiles usedin the quality experiments......... 35

Histogram of parallel dimensiondata for the initial experiment 37

Histogramof perpendiculardimensiondata for the initial experiment .............................. 37

Histogramof parallel dimensiondata for the experiment using

the optimal inputs......................... 39

Histogramof perpendiculardimensiondata for the experiment

using the optimal inputs...................... 39

Histogramof paralleldimensiondata for the experiment using

the adjustedoptimal inputs.................... 40

Histogramof perpendiculardimensiondata for theexperiment

using the adjustedoptimal inputs................. 40

Melt pressureresponseat a setpoint of 5000PSL ........ 43

Melt pressureresponseat a setpoint of 7000PSL ........ 44

Melt pressureresponseat a setpoint of 3000PSL ........ 44

Overshootfor 50 runs from a cold start............. 45

Steady-stateerror for 50 runs from a cold start......... 45

Plot showingevolution of the plastication step responseover

time................................. 47

vii
Figure 2.34 Plot showingevolution of the hydraulic oil temperature over

time................................. 47

Figure 3.1 Variability in the ManufacturingProcess ............ 53

Figure 3.2 Two dimensional exampleshowing how the quadratic constraint is approximatedby linear constraints........... 66

Figure 3.3 Completeconflict versuse-conflict ................ 70

Figure 3.4 Contour plot of objective functions showing algorithm iterations. 75

Figure 5.1 The algorithm used for performance monitoring and control. 86

Figure 5.2 PID controller block diagram ................... 86

Figure 5.3 Plastication dynamics block diagram ............... 90

Figure 5.4 Estimated poles and zeros with confidence intervals ....... 93

Figure 5.5 Process input produced by relay controller ............ 94

Figure 5.6 Comparison of process output and simulation output (simulation is dashed-line) ........................ 94

Figure 5.7 The tradeoff boundary (nondominated set) found using the

multiobjective algorithm ...................... 98

Figure 5.8 Efficient points: KI vs. Kv .................... 98

Figure 5.9 Efficient points: KD vs. Kp .................... 99

Figure 5.10 Efficient points: KD vs. K1 .................... 99

Figure 5.11 A comparison of the experimentally determined tradeoff boundary (x) with the "simulation" boundary (o) ........... 104

Figure

Figure

Figure

Figure

Figure

Figure

Figure

5.12 Comparison of efficient points: Kt vs. Kp ........... 104

5.13 Comparison of efficient points: Kp vs. I(D ........... 105

5.14 Comparison of efficient points: Kz vs. IfD ........... 105

5.15 Control effort using initial gains ................. 106

5.16 Process response using initial gains ................ 106

5.17 Control effort at tradeoff surface point A ............ 107

5.18 Process response at tradeoff surface point A ........... 107

°..

VIII
Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure 5.31

Figure 5.32

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

5.19 Control effort at tradeoff surface point B ............ 108

5.20 Process response at tradeoff surface point B ........... 108

5.21 Control effort at tradeoff surface point C ............ 109

5.22 Process response at tradeoff surface point C ........... 109

5.23 Injection dynamics block diagram ................. 113

5.24 Nondominated set of ITAE versus RMSE ............ 116

5.25 Nondominated set of ITAE versus maximum control deviation. 116

5.26 Nondominated set of maximum control deviation versus RMSE. 117

5.27 Efficient points: Kp versus/'_'_ .................. 117

5.28 Efficient points: Kp versus KD ................. 118

5.29 Efficient points: I(i versus I@ .................. 118

5.30 Melt pressure output at nondominated point A. The setpoint

is 2500 PSI ............................. 119

Servovalve input at nondominated point A ........... 119

Melt pressure output at nondominated point G. The setpoint

is 2500 PSI ............................. 120

5.33 Servovalve input at nondominated point G ........... 120

6.1 The spiral part ........................... 126

6.2 Plastication inputs for spiral mold optimization ......... 131

6.3 Injection inputs for spiral mold optimization ........... 131

6.4 Cycle time as the algorithm progressed .............. 132

6.5 Spiral length as the algorithm progressed ............. 132

6.6 Nondominated set after first line search ............. 133

6.7 Efficient points after first line search .............. 133

6.8 Comparison of first (o) and second (") nondominated sets... 134

6.9 Efficient points for first (o) and second (") nondominated sets. 134

6.10 The ASTM test part ........................ 137

ix
Figure 6.11

Figure 6.12

Figure 6.13

Figure 6.14

Figure 6.15

Figure 6.16

Figure 6.17

Figure 6.18

Figure 6.19

Figure 6.20

Figure 6.21

Figure 6.22

Figure 6.23

Figure 6.24

Figure A.1

Figure A.2

Figure A.3

Figure A.4

Figure A.5

Figure A.6

Nondominated set of cycle time versus flash, after the first

algorithm iteration ......................... 141

Nondominated set of underfill versus flash, after the first algorithm iteration .......................... 141

Nondominated set of cycle time versus underfill, after the first

algorithm iteration ......................... 142

Efficient plastication and injection setpoints for the nondominated set .............................. 142

Nondominated set of cycle time versus flash, after the first

algorithm iteration ......................... 143

Efficient plastication and injection setpoints for the nondominated set .............................. 143

Nondominated set of cycle time versus flash, after the second

algorithm iteration, under the new setpoint constraint ..... 146

Nondominated set of underfill versus flash, after the second

algorithm iteration, under the new setpoint constraint ..... 146

Nondominated set of cycle time versus underflll, after the second algorithm iteration, under the new setpoint constraint... 147

Efficient plastication and injection setpoints for the nondominated set, under the new setpoint constraint .......... 147

Final nondominated set of cycle time versus flash ........ 148

Final efficient plastication and injection setpoints ........ 148

Plastication inputs for ASTM four cavity mold optimization.. 149

Injection inputs for ASTM four cavity mold optimization. . . 149

Cross Section of One Screw Flight During Plastication .... 170

Two Parallel Plates Separated by a Newtonian Fluid ..... 173

The Melt Temperature Dynamic System ............ 178

Pumping Dynamics Block Diagram ............... 184

Plastication Dynamics ...................... 185

Injection Dynamics Block Diagram ............... 190
Figure A.7 Holding Dynamics ........................ 193

xi
LIST OF TABLES

Table 2.1

Table 2.2

Table 2.3

Table 2.4

Table 4.1

Table 4.2

Table 5.1

Table 5.2

Table 5.3

Table 5.4

Table 6.1

Table 6.2

Table 6.3

Quality Categories and Components .............. 11

Inputs used for the quality experiments ............. 41

Statistics of the dimensional data ................. 41

Flash output for the three experiments .............. 41

Arburg Allrounder Machine Specifications ........... 77

Solenoid Valve Activation Table ................. 77

Symbols used in the plastication dynamic equations ...... 89

Nominal Input Settings ...................... 101

Symbols used in the injection dynamic equations ........ 112

Nominal Input Settings ...................... 114

A Sequence of Randomized Runs ................. 128

Nominal Input Settings for the Spiral Mold ........... 129

Nominal Input Settings for the ASTM Mold .......... 139

xii
DEDICATION

This work is dedicatedto my wife Elaine, for her love and support, and to my son

Gregory, who may neverknow how muchhe'staught me.

oo°

Xlll
ACKNOWLEDGEMENT

I would like to express my gratitude to Dr. Marion Waltz and Mr. B.P. Wilkerson,

of Alcoa, for their support of this research.

I would like to thank Dr. Alan Desrochers for challenging me to do my best.

Also, I would like to thank Dr. George List for his valuable insight and inspiration.

Finally, I would like to thank Steve Murphy for his advice and friendship, Kurt

Redlitz for quality, custom computer work, Peter Gennes for his injection molding

expertise, and Kostas Kyriakopoulos, Deepak Sood, and the other denizens of RAL

and CIRSSE.

xiv
ABSTRACT

In this work, the use of product quality as the performance criteria for manufacturing

system control will be explored. The goal in manufacturing, for economic reasons, is

to optimize product quality. The problem is that since quality is a rather nebulous

product characteristic, there is seldom an analytic function that can be used as a

measure. Therefore standard control approaches, such as optimal control, cannot

readily be applied.

A second problem with optimizing product quality' is that it is typically measured along many dimensions: there are man3" aspects of quality which must be

optimized simultaneously. 'Very often these different aspects are incommensurate

and competing. The concept of optimality must now include accepting tradeoffs

among the different quality characteristics.

These problems are addressed using multiple objective optimization. In this

thesis, it is shown that the quality' control problem can be defined as a multiple

objective optimization problem. A controller structure is defined using this as the

basis. Then, an algorithm is presented which can be used by" an operator to interactively find the best operating point.

Essentially, the algorithm uses process data to provide the operator with two

pieces of information:

• If it is possible to simultaneously improve all quality criteria, then determine

what changes to the process input or controller parameters should be made to

do this;

• If it is not possible to improve all criteria, and the current operating point is

not a desirable one, select a criteria in which a tradeoff should be made, and

make input changes to improve all other criteria.

XV
The process is not operating at an optimal point in any sense if, no tradeoff has to

be made to move to a new operating point. This algorithm ensures that operating

points are optimal in some sense and provides the operator with information about

tradeoffs when seeking the best operating point.

This approach is new because it is an integration of quality control and automatic control. Quality control provides a way of rnonitorin 9 process quality but

does not specifically address the need to react to process changes so as to maintain or improve the level of quality. This is typically left up to a process expert.

Automatic control provides a way to methodically react to output deviations, decreasing the variability of the process, but it is assumed that there is an accurate

analytical model of the process. This can be a very limiting assumption, especially

when addressing complex process control problems where the underlying models are

nonlinear and time-varying.

In this work, the multiobjective algorithm was implemented in two different

injection molding scenarios: tuning of process controllers to meet specified performance objectives and tuning of process inputs to meet specified quality objectives.

Five case studies are presented.

xvi
1. Introduction

1.1 Motivation and Objectives

In manufacturing, the ultimate measureof any product is quality. Quality is

"the totality of featuresand characteristicsof a product ... that bearson its ability

to satisfy given needs"[1]. A product beingof "good" quality implies that there

exist measuresfor the characteristicsand that predefinedlevelsfor thesemeasures

are being met or exceededby the product. The goal in manufacturing, then, is to

control this quality.

Quality control is usualh" the responsibility of the operator. He makes subjective decisions based on observations and experience. One aspect of the problem

which makes the decision making process difl:icult is that quality is made up of

many characteristics that may be incommensurate and competing. The operator

will have to make tradeoffs amongst the criteria so that all of them are satisfied

simultaneously and the process is operating at the "best" point. One problem with

operator control is that the level of quality control which is achieved depends upon

the expertise of a given operator.

The goal is to control quality automatically. To do this, the process of making

tradeoffs must be dealt with. It is proposed that the quality control problem be

formulated as a multiple objective optimization problem. A motivation for this

approach is that any important quality characteristic can be represented by an

appropriate cost function. Optimization takes place with respect to the vectorvalued cost function, of which each quality characteristic is an element. In this way,

making tradeoffs among quality characteristics is an explicit part of the decision

process, rather than being hidden in some aggregate cost function. Furthermore,

the operator's role changes from one of monitoring and operating the process to one
of monitorin 9 and improvin 9 quality.

This approach goes beyond the typical methods of quality control, which consist mostly of charting measurements. In these methods, the charts are used to detect

process variations. Any decisions regarding control actions are left to the process

operator. His experience and intuition are the guide. Conversely, an automatic

controller, based on a process model, will predictably and methodically control the

process. An added advantage is that because this method is optimization-based,

improved quality performance may result over that obtained by a human operator

alone.

1.2 Contributions

In this thesis, a method for automatic control of quality is presented. This

method is based on building a quality-based model of the process. For this model,

output measurements are part quality characteristics, and inputs are process command signals and parameters of process controllers. This model is then used within

a multiobjective optimization algorithm, the quality controller. Essentially the quality controller iteratively tunes or adapts the process inputs and controllers on-line,

so that quality is continuously optimized.

This approach is new in several ways. First, automatic controllers are usually designed based on outputs which represent some physical measurement of the

process. For example, temperature is controlled to a setpoint, and the controller is

judged on how tightly this is done. A typical optimal control approach involves the

minimization of some function of the process measurements, the foremost example

being the minimization of a scalar linear quadratic function of the process inputs

and states (the LQR problem). Product quality is not explicitly part of the performance criteria when designing this controller. Good quality is achieved because a

process expert has given the control engineer a set of physical specifications which
the expert knows will yield good product. A good controller will make a process

less susceptible to variations, but this is only the first step towards good quality

product. The automatic quality control approach goesone step further by making the achievementof good quality the criteria by which controller performance is

measured.

The multiple-objective optimization approach is also unique. When controlling quality, this approach is necessary when one considers the nature of quality.

Ultimately, only one operating point is used, and therefore tradeoffs among quality

characteristics are made. One approach to optimizing quality would be to form an

aggregate cost measure and use this to determine optimal operating points. One

weakness of aggregation is that tradeoffs are made before optimization takes place.

It may also be necessary to choose individual quality measures to accommodate the

aggregate cost function (again, consider the requirements when designing using the

LQR formulation). By leaving the cost function as a vector, the system retains flexibility by allowing the operator to make tradeoff decisions based on current needs

and goals.

1.3 Application to the Injection Molding Process

The automatic quality controller will be implemented on the injection molding

process. Plastic injection molding is a cyclical process used to make complexly

shaped plastic parts. The goal of the injection molding machine operator is to

consistently produce parts which meet some predefined quality specifications. Part

quality can be measured in terms of suitable material properties such as strength.

surface finish, and shape. The key word above is consistently. The time-varying

nature of the process and the complex relationships between molding conditions

and part properties make this a difficult task.

Each cycle of the process has four phases, which occur in the following order:
4

. Plastication - the polymer is melted and deposited in front of an injecting

mechanism. This continues until the required amount of polymer has been

accumulated.

2. Injection - the molten polymer is forced into the mold under high pressure.

This continues until the mold is completely full.

. Holding - the molten polymer is held in the mold, under pressure, to ensure

complete mold filling. Cooling of the part begins at this point. Holding

continues until the gate has frozen, at which point no more polymer can enter

the mold cavity.

. Cooling- the polymer is allowed to cool in the mold. This continues until the

temperature of the formed part is low enough so that it will retain its shape

once ejected from the mold.

A cut-away view of a reciprocating screw injection molding machine is shown in

Figure [1.1]. In this type of machine, the screw is used to both melt the polymer and

pump it into the nozzle area. This melting and pumping is accomplished by rotation

of the screw. As the melting and pumping occurs, the screw retracts, creating the

volume occupied by the polymer melt. Injection and holding are accomplished by

forcing the screw forward, like a piston. (A check valve at the screw tip prevents

a flow of polymer back into the screw.) Once the polymer is in the mold, cooling

begins, and at a predetermined time, the mold opens and the solidified part is

ejected. There are other types of injection molding machines which accomplish these

four phases via different mechanisms, but the phases and sequencing are essentially

the same [2].

A sequential block diagram of the molding cycle is shmvn in Figure [1.2). Interactions between these phases occur when one phase establishes the initial conditions

for a subsequent phase [3]. It is the complexity of these interactions which motivates
the needfor control of the process.Theseinteractions areshown in Figure [1.3]and

are discussedin detail in Appendix A.

1.4 Organization of the Thesis

In Chapter 2, morebackgroundon the injection molding processis presented.

This discussionfocuseson processmodeling and the relationship betweeninputs,

processingconditions,and quality. Experimental data from the AdvancedManufacturing Laboratory's Arburg injection molding machine is presented. These results

motivate the need for control of quality, and not just of processing conditions.

In Chapter 3, the quality control problem will be defined rigorously. The quality control policy will be defined in terms of solving a multiple-objective optimization

problem. This chapter also includes a brief review of multiobjective optimization.

Finally, an algorithm for automatic quality control is presented.

As discussed previously, the multiobjective algorithm will be used for quality

control in an injection molding process. Details of the experimental equipment are

contained in Chapter 4.

Chapter 5 contains the discussion of the results of controller tuning on the

injection molding machine using the multiobjective optimization algorithm. Three

different case studies are presented: tuning of a plastication phase controller in

simulation, on-line tuning of this same controller, and the on-line tuning of an

injection phase controller.

Presentation of the quality control results is done in Chapter 6. Quality control

for two different molds was investigated. Three quality criteria are defined for a

spiral mold: spiral length, cycle time, and flashing. For an ASTM four cavity test

specimen mold, the following criteria are used: flashing, underfill, and cycle time.

For both studies, process setpoints are automatically adjusted to optimize quality.

Conclusions and suggestions for future research are presented in Chapter 7.
6

HOPPER

THERMOcOUPLE "'

@F_RR_L

Figure 1.1: Cut-away View of a Reciprocating Screw Injection Molding

Machine

'tlasticatio:

Cooling

S

Injection

\ )

Holding ]

Figure 1.2: The Injection Molding Phases
Rsw M_steriad Characteristics

I Temperature of Ezces_ Polymer

HydraulicBl Pressure --[ T_I • •

Nozzle Pressure ] Melt Temperature

.... remp....... ----'1 riasticat_o Sho,S,=e

Screw Speed |

Desired $ho_ Size /

Mold Temper_.ture

Nozzle P res_ure

Injection Speed

Bsrtel Temperature

Holding

Injection

_ydrauli¢ P tes, ure

Nozzle P resture

Holdin 8 Time

Barrel Temper&ture

Coollng Water Temper&ture_

Cooling Water Flow Rate

B&rtel Temper&t ute

Melt Temperature

Cuthion Size

Mold Temperature

t Melt Temperature

I Mold Temperature

Cooling F_.,,hedPart

] Extes_ Polymer

Figure 1.3: Injection Molding Phases and Interactions
8

Several appendices are included which will provide more extensive background

material. The modeling of the injection molding process and a derivation of models specific to the Arburg machine can be found in Appendix A. A guide to the

notation used in the equations throughout this thesis is found in Appendix B. In

Appendix C, a simple example is presented which highlights aspects of the mulitobjective optimization algorithm.
2. The Injection Molding Process and Part Quality

2.1 Introduction

Part quality is easyto label but difficult to define quantitatively. There are

two reasonsfor this: quality is specificto the function of the part, and not all quality

measureshavea quantitative basis[1]. Quality "measures"for an injection molded

part can fall into oneof the following categories[33]:

• Mechanical Properties;

• Dimensions;

• Appearance.

Each of these categoriesis made up of various componentsand each component

measuresa different aspectof the part. Thesecomponents come in two distinct

forms, variables or attributes [34]. A variable is a characteristic or property which

can be measured on a continuous scale. Part dimension is a variable, measured in

some appropriate unit of length. An attribute is a characteristic which the part does

or does not have. The presence of streaks in a colored part is an attribute: either

coloration is solid or it is not. Quality categories and typical components are listed

in Table 2.1. This is by no means an exhaustive list because with every new design

there will be some new measure of quality.

Kamal and Bata, in [35], stated

The thermo-mechanical history experienced by the polgmer during

flow and solidification results in the development of microstr'ucture (morphology, crystallinity, and orientation) in the manufactured article. The

ultimate properties of the article are closel 9 related to the microstructure.
10

Therefore, the control of the process and product quality must be based on

an understanding of the interactions between resin properties, equipment

design, operating conditions, thermo-mechanical history, microstructure,

and ultimate properties.

Polymer research invoh,es elucidating the following three relationships:

,, fp - the relationship between processing and microstructure;

•fm - the relationship between microstructure and part mechanical properties;

• fp- f,_ - the overall relationship between processing and mechanical properties.

Experiments with the goal of determining fm (or "f,_-research") would be considered fundamental research. The goal is to determine the relationship between the

microstructure and aggregate mechanical properties of the material, regardless of

the process which is used to produce the material. Experiments with the goal of determining fp. fm ("fp" fm research") is more oriented to manufacturing and control.

Here, the goal is to determine the relationship between processing and mechanical

properties, one application being to determine the limits of a particular processing

technique with respect to properties of the final product.

2.2 A Review of Injection Molding Research

In this section, the literature pertinent to the plastic injection molding process

is reviewed. The research described in this work draws from three fields: injection

molding, process control, and multiple objective optimization. Because of this diversity, there is no single literature review chapter. Literature pertinent to the

process modeling, control, and quality specifically for injection molding will be reviewed here. Otherwise, when appropriate, the literature on process control and

multiobjective optimization will be discussed.
11

Table 2.1: Quality Categories and Components

Categories Components

Mechanical Properties

Dimensions

Appearance

Tensile Strength

Flexural Strength

Izod Impact Strength

Ball-drop Impact Strength

Stiffness Modulus

Elastic Modulus

Absolute Size

Shrinkage

Part Weight

Part Density

Part Volume

Color

Color Mixing

Surface Smoothness

Clarity

2.2.1 Research in Modeling and Control

Dynamic studies of the process can be divided into two categories: those which

attempt to build a mechanistic model of the process and those which attempt to

build an empirical model of the process. A mechanistic model typically consists

of partial differential equations based on mass balance, momentum balance, and

energy balance equations. To these equations, the boundary conditions describing

a particular situation must be applied [4]. An empirical model of the process is

an attempt to form a rough approximation of the behavior of the process using

simple models. Empirical models are based on some convenient form such as a

linear static model or a transfer function [5, 6]. It must be emphasized that the best

mechanistic oz" empirical models will only provide a general clue to the input and

output relationships in the process, the actual situation is much more complex.

The foundation work on polymer processing was done by Spencer and Gilmore

in [7, 8]. Their goal was to determine the role of pressure, temperature, and time in
12

injection molding. The result wasan empirical equation of state for polymers.

The most comprehensivework on plastication was done by Donovan in [9, 10,

11]. The emphasis of this work was on melting due to conduction heating. The result

was a theoretical model which would predict the melt profile and temperature profile.

Donovan, Thomas, and Leverson then verified this using the cooling ezperiment.

Details of this can be found in [9]. Lipshitz, Lavie, and Tadmor [12] developed a

melting model which took viscous dissipation into affect. This was modeled as a

periodic step disturbance that occurred during screw rotation. Raimund used these

results as the foundation for his experimental work [13]. His experiments, and a

new analysis using his data, will be presented in Section 2.3.1. Kamal, Gomes, and

Patterson, in a series of papers [14, 15], studied the dynamics and control of melt

and barrel temperature. The focus of this work was on the implementation and

effect of feedback controllers. They developed empirical transfer functions of melt

temperature as a function of barrel temperature and melt pressure. The5" then used

these transfer functions to evaluate different control strategies.

The mechanistic modeling of injection has centered around the modeling of

the unsteady flow of a hot, non-Newtonian polymer melt into a cold cavity. These

dynamics are modeled as unsteady-state free surface flow coupled with transient

cooling and are described by the basic equations of change. Tadmor, Boyer, and

Gutfinger, in [16], modeled the flow of polymer into the melt cavity using the finite

element method. Experimental work by Kreuger and Tadmor [17] validated this

modeling method by studying the injection of polymer into a rectangular cavity with

various obstructions. One result was that it could be assumed that molten polymer

behaved as a Newtonian fluid, which simplified modeling and simulation. Later

work by Mavridis, Hyrmak, and Vlachopoulos [18], Kamal, et al. [19], and Gogos,

Huang, and Schmidt [20] included fountain flow in the modeling. Understanding of

the injection phase is important because shear and elongational stresses cause high
13

orientation of the polymer. Polymer which contacts the mold surface has frozenin orientation while the remainderof the polymer has time to relax while cooling.

These microstructure differenceswithin a part, due to the injection process, can

have a great impact on quality properties suchas strength and warpage.

Empirical modeling of the injection phasehas focusedon control of injection

velocity or pressure. In a sense,the filling of the mold can be considered "openloop" oncethe plastic leavesthe nozzle.Shankarin [21]andShankarand Paul in [22]

developeda lumped-parametermodelfor the injection processand evaluateda statespaceapproachto injection control. Costin, Okonski, and Ulicny in [23] examined

adaptive control of the injection process and compared it to the performance of a

PI controller for hydraulic pressure control. Haber and biamal [24] and Kamal et

al. [25] also studied the control of pressure during injection.

Ma presented one of the first analyses of the injection molding process as a

system [3]. The fundamental result is that no portion of the cycle exists in isolation. This was further emphasized by the experimental work of Whelan in [29].

Sanschagrin in [30], and the discussion of the importance of cycle-to-cycle control in

the survey of injection molding machine control done by Agrawal, Pandelidis. and

Pecht in [31].

All of the process control techniques have one thing in common: the process

is controlled but the effect of the control has not been related to the quality of the

product. The operator has been given new, more complicated knobs, but has nol been

told where or how to set them. Product quality must be used to provide feedback

on the dynamic performance of the process.
14

2.2.2 Research on the Effect of Microstructure on Mechanical Properties

Much research has been done in trying to determine the relationship between

microstructure and mechanical properties ("fro research"). One example is the

study of the influence of orientation on the mechanical properties of injection molded

polystyrene, by Hoare and Hull [26], Bayer et al. in [2T],and Lopez Cabarcos. et al.

in [28]. Other studies compare the affect of crystallinity, the ty'pe of polymer, the

length of the polymer chains, or an3: number of other characteristics. This research

was carried out to understand the correlations between processing and properties.

The goal was modeling, and not control.

A common thread through all of these ty'pes of studies is the approach. The

experimenters select a particular aspect of the microstructure to examine, find a

process which will produce this aspect preferentially, and allow some method to

vary the aspect over a defined range to determine the effect on the properties under

study. In the case of Hoare and Hull. two different processes were used to produce specimens with orientation. One set was made by using a well-characterized

polystyrene sheet and a second set was made bv injection molding thin plaques.

Molecular orientation was varied by cutting specimens out of the sheet or plaques in

different directions, as shown in Figure [2.1]. The processing to achieve the orientation was not examined, indeed the goal was to produce sheet or plaques as identical

as possible. The differences in microstructure in the specimens were induced bv

the method of sample preparation, the result being that the relationship between

microstructure and properties can be studied independently, of the process.

2.2.3 Research on the Effect of Processing on Mechanical Properties

Once the relationship between the microstructure and mechanical properties

is understood, it is possible to determine the right plastic for the job. Given the
15

\

/

/

Microstructure Orientation

Figure 2.1: Preparation of Specimens with Two Different Orientations

specifications for the part, a plastic is chosen. Now the product must be manufactured. An understanding of the relationship between processing and properties

("fp • fro-research") must be developed.

Whelan, in [29], showed that by varying the processing conditions, a wide range

of mechanical properties could be generated in the product. Similar conclusions

were reached by Lopez Cabarcos, et al. in [28]. The reason for this was that by

changing the processing conditions, different microstructures could be developed in

the molded part, thereby changing part properties. Whelan also showed that even

if machine settings were held strictly constant, there still remained variation in the

product due to factors not under direct control (e. 9. machine dynamic variations

and process disturbances).

In a later study [30], Sanschagrin tried to determine those inputs which were

most important to the production of consistent parts. He was able to show a substantial reduction in part variation through the implementation of some fairly simple

control schemes. More importantly, he showed that the relative importance of the
16

inputs changed when a different part was being made. Not only is it important to

control the process, but it is important to understand how to control the process for

any particular product.

Hsieh in [32] applied the Taguchi Method of quality control to injection molding. A new analysis based on this work is discussed extensively in Section 2.3.2. All

three of these works specifically address quality as a performance measure, but none

of them take advantage of feedback control.

2.3 Preliminary Experiments

As an illustration of the different types of investigations that can be carried out

on an injection molding machine and some of the problems with the process, three

different experimental programs will be discussed. These programs were carried out

on the AML Arburg injection molding machine, the target machine of the research

in this thesis.

The first set of experiments to be discussed was research conducted by Raimund

[13]. The purpose of these experiments was to develop an understanding of the relationship between the three different inputs which affect melt temperature and

the melt temperature that was actually generated. With respect to the research

described earlier this could be considered "fp-research". The second set of experiments was research conducted by Hsieh [32] to develop an understanding between

process input and quality. This would be "fp • f,.,-research'. Finally. experiments

were carried out by Redlitz [36] to develop an understanding of the problem of process control for the injection molding process. This would also be "fp-research".

These research programs will be described in the following sections.
17

2.3.1 Sensitivity of Melt Temperature to Plastication Inputs

In the research conducted by Raimund [13], the goal was to determine the

relative importance, to melt temperature, of conduction heating and shear heating

during plastication. Experiments were carried out in which the different plastication

inputs were manipulated and the resulting effect on melt temperature was recorded.

Conduction heating was directly affected by the setting of the barrel temperature.

Shear heating was directly affected by setting the screw speed and the melt pressure.

Screw speed set the initial shear rate on the polymer and melt pressure controlled

the time under which shearing took place bv lengthening the plastication time.

Raimund carried out the following type of experiment: a 5cm shot size was

generated under preset and constant barrel temperature, screw speed, and melt

(back) pressure conditions. Barrel temperature was set by selecting the desired

temperature on the barrel heater controllers. Screw speed was set using an adjustable hydraulic valve. Melt pressure was set by adjusting a hydraulic pressure

relief valve (in steady-state, melt pressure and hydraulic pressure are linearly related). During plastication, melt pressure and recovery rate data were collected

using the microprocessor controller [36, 37]. Finally, to collect temperature data,

the molten polymer was slowly injected, past the nozzle thermocouple, into the air

(no mold was used). A slow injection rate was used (approximately 0. l crn/s) so that

polymer temperature could be measured without inducing any shear heating errors

at the thermocouple tip. These experiments were carried out on two different polymers: polystyrene and polypropylene. In recording the data for these experiments,

the average of each process variable value over one cycle was used.

In the following section, a new analysis, based on the data collected by Raimund

will be presented and discussed.
18

2.3.1.1 Analysis of Polystyrene Data

Experiments on polystyrene were carried out using three different barrel temperatures and seven different melt pressures. Screw speed was not varied, possibly

because of difficulties in processing polystyrene. A constant value of 300RPM was

used for all experiments. In Figure [2.2], melt temperature is plotted with respect

to back pressure. Notice that melt temperature increases with both increasing back

pressure and increasing barrel temperature. To get a better feel for the shape of

the relationship, melt pressure versus the difference between barrel temperature and

actual melt temperature (or "melt temperature error") is shown in Figure [2.3]. In

this figure, the nonlinear relationship between pressure and temperature is much

more evident. Also, notice that as the barrel temperature gets higher, the error

decreases. This suggests that the dependence on shear heating decreases with increasing temperature. This is reasonable since the amount of shear heating that

takes place within the polymer is dependent upon the viscosity of the melt, and this

viscosity decreases as temperature increases.

Figure [2.4] shows melt temperature versus recovery rate. This figure is ver b'

similar to Figure [2.2]. The reason for this is that recovery rate is linearly dependent

on melt pressure and screw speed. This dependence is readily seen in Figure [2.5].

where recovery rate is shown versus melt pressure. Again, notice that this relationship is temperature dependent for this polymer.

In Figure [2.6], melt temperature is plotted versus plastication time. The

same data is used in Figure [2.7], except that melt temperature error is used. This

figure suggests a linear relationship between melt temperature and plastication time.

(Compare this with Figures [2.3] and [2.4].) This is further justified ,,'hen one

considers the approximate relationship between plastication time, shot size, and

recovery rate:

Shot Size

Plastication Time =

Recovery Rate
19

240 !

I

235 .................; .............................................: .........i-.._ ............... _ ......................................................

I i :

°

230 : : *

U

8 225 ...................................._ .............;....-._.................................................................................. _................ Legend

8.

E .?.2.0................................... _.................................... ;................................................................... ; ...............

,_=

_ 215 ................................... :........................................................................................... ;...............................

)

210 ................................ ................................ _................. ............... ;............... _................. :.............

",o # ! :

i I L _ I I L

2050 500 I003 1500 2000 2500 3000 3500 4000 4500

* -200C

÷ - 220C

x - 230C

Back Pressure(PSI)

Figure 2.2: The relationship between melt temperature and melt

pressure, at three different barrel temperatures, for

polystyrene.

14 ! ! ;

! : !

i •

12 .................................... :,....................................:,........................................................................ i.............

,,.,

: Legend

8 ................. !.................. !.................. i.................. !........... ;"'-:................................... :'""............. :................ *-200C * - 220C

= 6 .............. "-:............... ,..:................ L.._'.......... :........... ,.....i................................ 2............... i.......

! x :

zl : • :

2 ', , , -- , , --

0 500 lO00 1500 2000 2500 3000 3500 4000 4500

Back Pn:ssure(PSI)

Figure 2.3: Melt temperature error versus melt pressure for polystyrene.
2O

240 : ,

J _

i : lz

z

230 ........... "_......................................... :.................. ;.........................................................................................

_ '?,2_5 ......................................................................... :".................................... _ ......................._ ......................... Legend

_ * * - 200C

_. + - 220C

E 220 ....................................................................................................................................................................

N x - 230C

215 .........................................................................................................

210 ................................................................... ';,.................................................. ,................ ,..............

: .]°

i i t 1 t t i i i

205.5 0.6 0.7 0.8 0.9 I i.I 1.2 1.3 1.4

Recovery Raze (cm/s)

Figure 2.4: Melt temperature versus recovery rate for polystyrene.

la

e¢

e¢

1.4 ._ ! ! _ !

1.3 .................................... i..................................... :....................................................... :.................. _.................

_!.L :

!

1.2 ...................................................................................................................................................................

I

|.1 ............................................................ i*....................................................................................................

1 ...................................................................................... t ....................... ....................................................... Legend

i

0.9 ........................................................................................................... i ....................................... ':.................

0.8 .......................................................................................................... • ..................................... i.................

__o.7..........................................................................................................•............. ;.................

: X

: 4P

__0.6..........................................................................................................:....................................

0"50 500 1000L 1500, . 20_ 25_00 3000, 3500, 40_ z4500

* -200C

-,- 220C

x -230(?

Back Pmssur_ (PSI)

Figure 2.5: Recovery rate versus back pressure for polystyrene.
21

A linear regressionwasperformedon thedata shownin Figure [2.6]. The three

input variableswerebarreltemperature (Tb),plastication time (tp), and screwspeed

(,;). The output wasaveragemelt temperature (T,,,). The resulting model was:

Tm = 0.857Tb + 1.317tt, + 0.103,z (2.1)

This equation was used to estimate the melt temperatur% given the same input used

in the regression. Estimated versus actual temperature is shown in Figure [2.8]. The

fit seems to be quite good, but the following should be kept in mind:

1. There are only three different levels for the barrel temperature. Because of

this, it is difficult to judge how "linear" the response of melt temperature to

barrel temperature really is.

2. Onl)' one value of recover)," rate (a,,) was used in generating Equation (2.1).

Therefore, the model fits three coefficients to data wi_:h only two degrees of

freedom.

3. There is enough data that linearity with respect to plastication time seems to

be justified.

A second linear regression was run on the data, this time using only two

parameters: barrel temperature and plastica.tion time coefficients. The resulting

model was

Tm = 0.996Tb + 1.360tp (2.2)

Estimated versus actual temperature is shown in Figure [2.9]. As expected, the fit

is much worse (although not bad). If the plastication time coefficient is compared

between the two equations, it will be noticed that it is practically the same. In

comparison, the barrel temperature coefficient changes by about 15%. This supports

the linearitv assumption for plastication time and also underscores the need for more

barrel temperature and screw speed data.
22

240 _ J

235 .................... ............................ _................ i...s_ ........................................ _....................... i.....................

o

230 ..................................................................... i............................................... _............... _.............................

• ,_* :

'!r

_' 225 ..................... _ ....... "*......................................................................................'...................... Legend

•", ; * - 200C

_. + - 220C

220 .............................................................................................. i....................... _:............................................. x -230C

E'-

:_ 215 ..................... :"............................................... _....................... "............................................. i.....................

s i

210 ...................... ................................. "/................... : _ :

t

h _ • : :

L t t

l

2053 4 5 6 7 8 9 10

Plastication Time (s)

Figure 2.6: Melt temperature versus plastication time for polystyrene.

E

Figure 2.7:

polystyrene.

14 _ ! '.

o. !

_2....................... :..............................................:........... ::.......... ................................:!....................

t

i Legend

8 ......................................... _"............................................... _' ........................................................... * - 200(7.

." ÷ - 220C

x - 230C

e: • *

...."....._..! ....................... i.,, ......................................................

6 * ! :. •

4 ...................... _............................................... _ :..............................................

2 ',4 i 6' 7' i 9' 10

Plastication Time (s)

Melt temperature error versus plastication time for
23

245

240

235

230

_. 225

E

= 220

215

210

2O5

3

i_ 1_ • :

..................................................................... ............................................................... m_._ ....................

: !lJ i

• :8 :

: i

, i i / I i

4 5 6 7 8 9 10

Legend

x - Actual

o - Estimated

PLastication Time (s)

Figure 2.8: Estimated versus actual temperature using a three parameter model.

2

r.

tL

E

Figure 2.9:

245

240

235

230

225

220

215

210

205

200

! r r !

: o

........................................................................................... . ............ o ......... _..................... : .................. |-

o *" i z i i

............ _-_" ........... :.............. _............. :_ ............... i................................................. _....................

_t 1: m : . :

...................................................................... i....................... i....................... :....................................

:O

............ _"'""ib ........ tP ............................................................................................................ _.....................

............................................................ "l ............................... "................ _' ......................... ).................

4 5 8 9 10

Legend

x - Actual

o - Estimated

Plastication Time (s)

Estimated versus actual temperature using a two parameter model.
24

2.3.1.2 Analysis of Polypropylene Data

An identical set of experiments were run, this time using polypropylene. The

dependence of machine processing conditions on the type of polymer used could

be studied. In these experiments, it was possible to use two different screw speeds:

200RPM and 300RPM. For polypropylene, four barrel temperature levels were used:

180C, 190C, 200C, and 210C. The hydraulic pressure levels were kept the same as

in the previous experiments.

Melt temperature versus back pressure data for the two different screw speeds

are shown in Figures [2.10] and [2.11]. If these figures are compared with Figure [2.2]

it can be immediately seen that there is much less of a temperature dependence on

back pressure in polypropylene than in polystyrene. This is due mainly to the

difference in apparent viscosity of the two polymers.

In Figures [2.12] and [2.13], this same data is plotted with respect to melt

temperature error. If these figures are compared with Figure [2.3], it is immediately

seen that there is much less of a dependence on barrel temperature. Also, this data

is not nearly as "orderly" as that for polystyrene.

One hypothesis is that apparent viscosity of polypropylene is less dependent

upon the operating conditions investigated. This is supported by a plot of recovery

rate versus back pressure (Figure [2.14]). The linear relationship doesn't change

as a function of barrel temperature, only with screw speed. (Compare this with

Figure [2.5].) As is discussed in more detail in Appendix A, recovery rate, V_, is

assumed to be linearly related to screw speed, _, and melt pressure, Pro, in steadystate. Figure [2.14] justifies this assumption, over the range of temperatures tested.

In Figures [2.15] and [2.16], temperature is plotted versus plastication time.

Again, the phenomena of melt temperature being linearly related to plastication

time is evident.
25

215 ! , ,

210 .................................... :..................................... :............... : . .

o e ; : :............ ,o................. ;......e.......................

o !

205 ....................................................................... _...................................................... ;...................................

_, 200 .......... _..._................. _...................... _............ :.......... _..............................................................................

2 • : Lagend

+- 180C

* - 190C

E 195 ................. i........................................................................... ...................................................... _................

: x- 200

- : • o-210

I

, • • t

190 ......... ,_ ..i................. _ ......................... "................. ".......................................................................

$ * i

185 ......................................................................................... ....................................................................

# .t: _g

i I i I i _ i i

1800 500 1000 1500 2000 2500 3(300 3500 4000 4500

Back Pressu_ (PSI)

Figure 2.10: Polypropylene melt temperature versus back pressure for

four different barrel temperatures at 200RPM.

_=

E

215

: : ! i , : so

210 ................ # ................. _ ................ ;......._ ......... ;........... 0.......................................................... i.................

:o_........... ,_i............... _....................... t............. i........ ,.................... ,.................... !.................. i...............

200.................................... i........................................................................:.................. i.............

195 .................................... i.................. _..................................................................... i...................................

! :. ° * !

1901 : " :

............ _.................. _ !................. ._....................................................................... !....................................

.

• 4. : :

185 ....................................................... :.._.....................................................

¢ 4- .................

, , , _ , _ , ,

1800 500 I000 1500 2000 2500 3000 3500 4000 4500

Legend

+ - 180C

" - 190C

x - 200

o-210

Back P_ssun: ('PSI)

Figure 2.11: Polypropylene melt temperature versus back pressure for

four different barrel temperatures at 300RPM.
26

E

E

Figure 2.12:

7

6

5

4

3

2

1

0

-1

-2

"30 5()0

:g *

4

: :z

! = :.it i L :. i _ :

• _[ : : : o : :

o

: a o° .

O : : :

o : : : :

1000 1500 2000 2500 3000 3500 4000 4500

Back Pressure (PSI)

Legend

+ - 180C

* - 190C

x - 200

o -210

Polypropylene melt temperature error versus back pressure for four different barrel temperatures at 200RPM.

8

Figure 2.13:

lO, , :

6

z z! iz

: z i.

4 * **: _ s

: i : o

! : • : o

: .................................... !........................................................ !.......................................... i.......... ............ :-

g '_o o !

"20 5 1000 I 2 2500 3_,,_ 3500 4000 4500

Legend

+ - 180C

* - 190C

x - 200

o -210

Back Pressure (PSI)

Polypropylene melt temperature error versus back pressure for four different barrel temperatures at 300RPM.
!

,.¢

8

o

Figure 2.14:

27

1.6 : ! ,

_ '

1.4 ................................... i...................................................... :........................................................ i...............

1.2 .............................................. :.-# ................... :..................... :.............. :............

: : o _ ! :

....i-._) ...................... !.................. !.................. :............. :............ _: .................. :.......................

_ : : 0 f

t

06................................... :................. ................................... _............. ...... _,

o._o 5_o .... ooo ' ' ,ooo 15oo :ooo :5oo 3 s5oo 40o0 _5®

Legend

x - 200 RPM

o - 300 RPM

Back Pressure (PSI)

Back pressure versus recovery rate: upper curve is 300RPM

data, lower curve is 200RPM data.

As for the polystyrene data, a linear regression was run on the data shown in

Figures [2.15] and [2.16] to find a model relating melt temperature to plastication

time, barrel temperature, and screw speed. For this model, two levels of screw speed

data were available, so a full three parameter fit could be done. For polypropylene.

the model was

T,, = 0.9284Tb + 0.9136tp + 0.04373w (2.3)

Estimated versus actual temperature is shown in Figure [2.17] and [2.18]. The fit is

reasonably good although not as good as that for polystyrene. Again, the correct

trend is predicted by plastication time, but the prediction of absolute temperature

is not as good. Also, the model fits the data better at 200RPM than at 300RPM.

2.3.1.3 Different Processing for Different Materials

Now that the experimental data for processing two different polymers has been

presented, some of the differences in the effect of processing on these materials can

be discussed.
28

215

+ ........................ , 0..°. 0

21o...................... .io......... _............ ,...+...........;+:........ + " _ ............... +i ......................

;0

i

205 ................................................................................................................................................................

e 200 ..................... _ ........ Lx................................................................................................Legend

_* + - 180C

8. *- 190C

E 195 .................................................................................................................:...................... x - 200

•. o -210

190 .................... ,"_........ "......... _"......... _.................................................................................. i......................

: +*

185 =:.+.,. _ "- i ;

180 ' " ' ' ' '

4 5 6 7 8 9 I0 11

Plastication Time (s)

Figure 2.15: Polypropylene melt temperature versus plastication time

for four different barrel temperatures at 200RPM.

2

E

215

o0

o

210............... +.......... "........... _,._° ............. ,_.................. i.................... ;..................... .................... :,...................

0 : o o

i

: |

205 ............... _it: ........ _ ...... ;-"x .................... _,.............. _ ......................... _.........................................

/IL +

200 .................................................................................. i............................................................ :................. Legend

: + - 18{3(2

i

*- 190C

195 ............................................................ :.................... i....................................... ;_"'"_'"'_ ........ :"................... x - 200

" :. o-210 rl

190 !.............. ';'Z_'""_............ "_ ................. ......................................... i.................... _.................... ;..................

_85......................................... i':...................................... i.................... i.....................................;..................

I I t i

180 315 4 4.5 5 515 6 615 7

Plastication Time (s)

Figure 2.16: Polypropylene melt temperature versus plastication time

for four different barrel temperatures at 300RPM.
29

215

: o • •

........................................... o Q Z _ •

210 :I ° q? * " _ ................. _i ...........

205 ....................................................................... i....................... ;....................... ;............................................

: !

i : o io o

2 200...................... _......... I......... :"_............ "+t.................

_ " :........................................... ...................... Legend

)_ ! :; x -Acm_

E

195 ....................................................................... _....................... :................................................ i...................... o - Estimated

_ x°

.i 000

o zx •

Z,L

19o..................... b........ •.......... _-"........ _.,,,..i.+.::,+.,? ......................................

lz

ix 185 .......................................................... +.................................. ,....................... i................

oo

oo

180 ' _' , - , , t

4 5 6 7 8 9 lO l

Plastication Time (s)

Figure 2.17: Estimated temperature versus actual temperature for

polypropylene at 200RPM.

215

o oo

_o o o o _ o x 1

oi_ -+ . _ x • :

210 ............... .It,.+: ........ _...........:,._:_............. _ ................... i ................... _........................................... i.................

: !

G . i . i _ °°

oo! (3o : i) !

_= 2_ ................... _.................... ::......................................... i.................... _.................................................

E i . :

---+2 195 .................... ;>.................... ;+....................................... +_........................................... ® _: •°................ i

oo! • e. oe + z_ :

tgo............... i:+ ............. :................... _.................... !_..................................... :...................................

- i i• !

185 ! _x !t,, :

.............. i:_ ......... _o.......... _...................... !...................................... i...................................

1803 3 5 4 4 5 5 5 5 6 6.5 7

Legend

x - Actual

o - Estimated

Plastication Time (s)

Figure 2.18: Estimated temperature versus actual temperature for

polypropylene at 300RPM.
3O

First, consider barrel temperature. For polystyrene, the relationship between

barrel temperature and melt temperature is well-behaved. This is not the case for

polypropylene. For this polymer, there are some instances where the melt temperature is less than the barrel temperature and in other cases it is higher, but for

the most part, barrel and melt temperature are equal. This cannot be easily explained in terms of the processing conditions and must be related to some property

of polypropylene.

For both polymers, there is a strong linear relationship between melt temperature and plastication time. Since plastication time is the time under which

viscous heating of the polymer takes place, the importance of viscous heating to

melt temperature is emphasized.

Finally, all of the analysis carried out on this experimental data was done using statistics of the data. Plastication time was computed using average recovery

rate. The heater bands had a +5C error band on the setpoint so only average barrel

temperature was known. In Figure [2.19], the melt temperature profile versus linear position in the melt is shown (for polystyrene) for four different back pressures.

Notice that every one of the profiles has a hot spot, and this peak temperature spot

is at a different position in the melt. Since final molecular orientation is dependent upon the polymer temperature, this profile will result in a part with different

molecular orientation depending upon the location in the mold.

All of these observations only underscore the fact that the conditions of the

polymer must be measured and controlled, not just machine operating points.

2.3.2 Part Quality Dependence upon Processing Conditions

In the research conducted by Hsieh [32], the goal was to determine optimal

machine operating conditions for a particular part. Three different quality characteristics were defined for this part, and a series of experiments were run to determine
244

Position Dependence of Melt T_mp- Polystyrene at 230C. 300 RPNI

31

2

r242

240

238

_6

234

232

•.-......4_ iqi.................

°° _.,

• °

°°

. .

, .... •,..•

o•-"

3619 PSI

1677 PSI -"..............."'"......

•L

.°--- •

413 PSI

2300 0.5 i 1.5 2 2.5 3 3.5 4 4 5 5

Axial Melt Position (cm)

Figure 2.19: Melt temperature versus position in the shot at four different back pressures.

the sensitivities of these characteristics to the different machine inputs.

Hsieh used the Taguchi method [38] to determine the optimal input settings

using experimental data. This method is essentially a three step process. First, a

set of initial experiments are run to determine typical machine output under standard operating practice. This establishes a performance benchmark for the process.

Next, a set of inputs are chosen to be manipulated. The choice depends on the

experimenter's a priori knowledge as to which inputs will be important. An orthogonal array, appropriate for the number of inputs and levels to be manipulated,

is chosen and a series of experiments are run as prescribed by this orthogonal array. The use of this orthogonal array allows the maximum amount of information

to be gotten out of the minimum number of experimental runs. Analysis of this

data will yield the sensitivities of the outputs to the different inputs. When using

the Taguchi method, standard functions are established to determine optimums,

such as nominal is best or larger is best. These functions provide standard rules of

thumb for the process designer• For instance• nominal is best simply corresponds to
32

minimizing the squarederror about the desiredmean. When usedproperly, these

optimizing functions will determinea set of input conditions which reduceprocess

varianceas well as achievethe target quality output. Theseoptimizing functions

areconsideredseparatelyfor eachinput/output pair. It is up to the experimenterto

determine trade-offs betweencompetingoptimums, as was the case with this series

of experiments [32]. A thorough explanation of the Taguchi method can be found

in [38].

2.3.2.1 Operating Conditions for the Experiments

For these experiments, a test specimen mold was used (Figure [2.20]). This

mold has four cavities: two tensile test specimens, an impact strength test disc,

and a weld line test bar. Two quality variable outputs were defined for the part:

the dimension parallel to flow and the dimension perpendicular to flow on the large

tensile test specimen. These dimensions were measured by the experimenter to an

accuracy of 0.0001 inches. One quality attribute was defined: the amount of flashing

that occurs on the whole part. To make analysis easier, Hsieh designated four

categories of flash,

• Class I - No flash - part is acceptable;

• Class II - Minimal flash - part is acceptable;

• Class III - Medium flash - part is unacceptable;

• Class IV - Severe flash - part is unacceptable;

The definition of these classes gave the attribute some "variable-like" properties for

use in analyzing the data.

Seven inputs were manipulated with three different levels assigned to each

input. They were as follows:
33

Figure 2.20" ASTM test mold used for part quality experiments.
34

• Heater Base Temperature- this is the referencetemperature at which the

barrel was kept. Three levelswereused:210C, 240C,and 270C.

• Heater Temperature Profile - there arefour heaters which can be set independently, starting at the nozzle. Two different temperature profiles were

investigated:

1. Uniform - all heatersat the basetemperature (Tr,I)

2. Varied - the heaters are adjusted according to the following schedule:

(a) H, = Tr_/- 40C,

(b) H2 = T,,I + 10C,

H3 = T,o ,

(d) H, = T_,! - 10C,

• Injection Velocity Profile - this is the velocity trajectory that was used to

inject the shot. Three levels were used: 20% uniform, 20% ramped, and 40%

ramped. These profiles are shown in Figure [2.21].

• Injection Pressure - this is the maximum hydraulic pressure used during injection. The following levels were used: 900PS/, 1200PSI, and 1500PSL

• Injection Time - this is the amount of time allowed for injection of the part.

The levels used were ls, 2s, and 3s.

• Holding Pressure - this is the hydraulic pressure applied during holding. The

following levels were used: 50PSI, 120PSI, and 200PSI.

• Hold/Cool Time - the ratio of hold time to cool time was varied. Since cooling

takes place throughout the entire period, this amount was kept to 25s. The

following hold/cool ratios were investigated: 5/20s, 10/15s, and 15/10s.
0.5

0.45

0.4

= 0.35

o.31

II

0.25

0.2

.__

o.t5

0.1

0.05

0

0

Figure 2,21:

- - z

20 40 60 80 l O0 120

Legend

x - 20% Unif

'.*- 20% Ramp

o - 40% Ramp

Percent of Total Injection Time

Velocity profiles used in the quality experiments.

35

The following should be noted about the experiments that were run. The

injection molding machine was treated as a black box. No attempt was made to

choose inputs based on the expected effect on the polymer. This is evident by the fact

that injection time was chosen as an input to be manipulated. The end of injection

is usually signaled by the screw reaching a specific forward position, commonly

referred to as the cushion setting. The use of injection time as the terminating

signal in these experiments can have two possible consequences: either the injection

time will be too short, possibly causing an incomplete fill, or injection time is too

long and the plastic undergoes what is essentially a "high-pressure" holding phase

before actual holding commences, increasing the probability of flashing. One other

problem was that injection pressure and velocity were both used as control inputs.

These inputs are actually dynamically coupled and it is not possible to control both

at the same time. Finally, inputs were set but the actual machine response was

not measured. It is not known whether the machine actually achieved the desired

setpoints and profiles and therefore whether the input manipulated has any effect

upon the process.
36

2.3.2.2 Initial Machine Performance Experiments

Since the goal of these experiments is to determine optimal settings for machine

inputs, a set of experiments were run at commonly used machine settings to establish

a performance benchmark. Twenty-six experiments were run at the input settings

shown in the first column of Table [2.2].

In Figures [2.22] and [2.23], histograms of the dimensional data are shown. The

parts were grouped in 0.0002 inch increments for these charts. These histograms

roughly approximate the probability distribution of the random process affecting

these quality measurements. The mean and variance of the dimensional output are

given in Table [2.3 I. The number of parts in each of the classes of flash is given in

Table [2.4]

As can be seen from the histograms, the distribution is somewhat spread out.

Also the means are off of the target values of 5.0 inches for the parallel dimension

and 0.5 inches for the perpendicular dimension.

2.3.2.3 Machine Performance Using Optimal Inputs

A set of experiments was performed, based on an orthogonal array experimental design, and the resulting part data was analyzed, as prescribed by the Taguchi

method [38]. The method predicted that the process inputs shown in Table [2.2]

would optimize the outputs according to the nominal is better criteria for the dimensional outputs and the larger is better criteria for the flash output (larger is better

refers to the number of parts in Class I).

Twenty-nine experiments were then performed at the new settings and the

parts were measured. Histograms of the dimensional outputs are shown in Figures [2.24] and [2.25]. Both part quality distributions are much "tighter" than those

of the initial experiment and the product is definitely closer to the target. The

mean and standard deviation of the parts for this experimental run are given in
37

¢w

12 r :

!0 ................................................................................... i...................................................................................

6

4 ............... _..................................................................................................................................................

0

4.97 4.971 4.972 4.973 4.974 4.975 4.976 4.977 4.9"/8 4.979 4.911

Inches

Figure 2.22: Histogram of parallel dimension data for the initial experiment

o12

6 ...................................................................................................................................................................

i i ........... i........ : .................

0.5 0.5 0.501 0.501 0.502 0.502 0.503 0.503 0.504 0.504 0.505

Inches

Figure 2.23: Histogram of perpendicular dimension data for the initial

experiment
38

Table [2.3].

Also, accordingto Table [2A], all of the parts areacceptable.This is deceiving

in that someof the parts were actually incomplete. This quality characteristic is

important, but was not measured by any of the three quality characteristics actually

used, so it could not be taken into account in the choice of inputs.

2.3.2.4 Machine Performance Using Adjusted Optimal Inputs

A third set of experiments were run using the inputs shown in the last column

of Table [2.2]. These inputs were adjusted, based on recommendations from an

experienced operator, to compensate for the incomplete fill problem. For this run,

all parts were acceptable, although most parts fell into flash output Class II, rather

than Class I. Histograms of the part data from this run are shown in Figures [2.26]

and [2.2T], and the statistics are shown in Table [2.3]. From these figures, it can be

seen that machine performance is still quite good, as compared with Figures [2.22]

and [2.23].

2.3.2.5 Analyzing the Quality Control Approach

In these experiments, the relationship between part quality and operating

conditions of the injection molding conditions was investigated. Using a methodical

approach, it was quite easy to find new operating conditions for the injection molding

machine which significantly increased the quality of the product. In fact, if the

different sets of inputs are compared (Table [2.2]), large improvements in quality

can be achieved with relatively small adjustments in the operating conditions.

One problem with this approach is that it is somewhat open-loop. The process

variables are not measured, so the actual machine response to the chosen inputs is

not known. Also, since the process variables are not measured, if variations occur

in the output, it may not be possible to immediately determine which input should
39

12

10

4 !!!!!! ii!i!i .......... .......... -_

2 ...........

0 _J

4.97 4.971 4.972 4.973 4.974 4.975 4.976 4.977 4.978 4.979 4.98

Inches

Figure 2.24: Histogram of parallel dimension data for the experiment

using the optimal inputs.

t12 : _ !

6 !!!!!! i!!!!!! !!! !!i!! !!!!!i i!!!!!!!!!!i! !!!!!! •!!!!!!i! !!!!!!!!!!!

!!! !!!!!!

iiiiiiiiiii i!!i

0.5 0.5 0.501 0.501 0.502 0.502 0.503 0.503 0.504 0.504 0.505

Inches

Figure 2.25: Histogram of perpendicular dimension data for the experiment using the optimal inputs.
4O

tt,

4 i n.......

t i t

0

4.97 4.971 4.972 4,973 4.974 4.975 4.976 4.977 4.978 4.979 4.98

Inches

Figure 2.26: Histogram of parallel dimension data for the experiment

using the adjusted optimal inputs.

t_

12

I0

0

0.5 0.5 0.501 0.501 0.502 0.502 .503 0.503 0.504 0.504 0.505

Inches

Figure 2.27: Histogram of perpendicular dimension data for the experiment using the adjusted optimal inputs.
41

Table 2.2: Inputs used for the quality experiments.

Inputs Initial Optimal Adjusted

Heater Base Temperature 21sd

Heater Temperature Profile

Injection Velocity Profile

Injection Pressure

Injection Time

Holding Pressure

Hold/Cool Time

Uniform

20% Ramped

1100PSI

ls

1O0PSI

5/19s

210C

Uniform

20% Ramped

900PSI

is

50PSI

15/10s

220C

Uniform

20% Uniform

900PSI

ls

50PSI

15/10s

Table 2.3: Statistics of the dimensional data.

Initial

Dimension Mean St. Dev.

Parallel 4.9745 1.292.10 -J

Perpendicular 0.5032 7.137.10 -3

Optimal Adjusted

Mean St. Dev. Mean St. Dev.

4.9790 0.255-10 -a 4.9784 0.619.10 -a

0.5023 0.109.10 .3 0.5029 0.536.10 .3

Table 2.4: Flash output for the three experiments.

Flash Initial Optimal Adjusted

Class I 7 29 4

Class II 17 0 26

Class III 2 0 0

Class IV 0 0 0

Total 26 29 30
42

be changedto correct the problem. In the next section,a feedbackapproach to

machinecontrol will be discussed.

2.3.3 Sensitivity of Machine Dynamics to Operating Conditions

In research conducted by Redlitz [36, 37], a microprocessor controller was

implemented to study control of the plastic injection molding process. Redlitz implemented PID control of melt pressure with a hydraulic servovalve as the actuator.

The gains of the controller were tuned heuristically, and repeatability of the process

and sensitivity of the process dynamics to the operating point was investigated.

First, the process controller was tuned using a melt pressure setpoint of 5000PSI.

After several iterations, it was found that the best attainable process response was

that shown in Figure [2.28]. This run had the following statistics:

• Overshoot = 6.4%

• Undershoot = 7.1%

• Steady-state Error = 0.47%

Using the same controller gains, runs were made at two other setpoints: 7000PSI

and 3000PSL The melt pressure responses for these runs are shown in Figures [2.29]

and [2.30]. These figures show that the dynamic characteristics change drastically

with the operating point. It is important, therefore, to understand the range of

operating points for which a particular set of gains produces a good output, and

possibly to determine a gain schedule for different setpoints.

The controller was tuned on an injection molding machine that had already

been warmed up. A set of experiments were run to see how this warming affected

the machine dynamics. Fifty runs were made, using a setpoint of 5000PSI and the

previously determined controller gains. Two statistics were collected from each of
5500

5000

4500

4000

r,

" 3500

3OO0

2500

2000

0

i i i i _ i

0.5 1 1.5 _ 2.5 3 3.5 4 4.5

Time (s)

Figure 2.28: Melt pressure response at a setpoint of 5000PSL

43

the runs: overshoot and steady state error. The results of this experiment are shown

in Figures [2.31] and [2.32].

Both performance measures showed some stochastic behavior over the fifty

runs. The variance of steady-state error decreased somewhat, but not significantly

over the course of the experiments. The variance of the overshoot decreased significantly by the end of the experiments. This is an indication that the transient

dynamics of the system are affected by thermal transients in the machine.

Other experiments have shown that there is a marked change in performance

which is dependent upon the hydraulic oil temperature. Cooling of the injection

molding machine hydraulic oil was done using tap water and thermostat control.

The heat transfer is dependent upon the ambient temperature of the water, but

the temperature at which the thermostat opens is fxed. When ambient water

temperature is very low, approximately 18C, the thermostat control system enters

a limit cycle with an amplitude of 6C and a period of about 30 minutes. This had a

remarkable effect on the system dynamics as can be seen in Figures [2.33 and [2.34].

Figure [2.33] shows the evolution of the plastication step response over time. As the
44

7O00

6500

6OO0

55OO

_ 5ooo

_ 4500

4O00

35OO

300O

0

Figure 2.29:

j •

] i z

............. /! ....................... :................................................................................................ i......................

f

i

L _ L A i A

1 ._ 3 4 5 6 7

Time (s)

Melt pressure response at a setpoint of 7000PSI.

60OO

5500

5000

4500

:_ 3500

3130O

2500

2000 ' '

0

U

........... I........... il...................... :.....................................................................................................................

......... ,........... :._.................. :........................................................................._......................

0.5 I 1.5 2 2.5 3 3.5

Time (s)

Figure 2.30: Melt pressure response at a setpoint of 3000PSL
45

44

: , !

i

42 .......................................................... '..... "

,::J.......... ...................... :..:1........ A............. t....................................

0 5 10 15 20 25 30 35 40 45 50

Run Number

Figure 2.31: Overshoot for 50 runs from a cold start.

1.8

1.6

1.4

1.2

1

o.8

0.6

0.4

0.2

00 5' 10 15' 2_ 25' 3'o 35' 40' 45 50

Run Number

Figure 2.32: Steady-state error for 50 runs from a cold start.
46

runs progress,the systemchangesfrom onewith a very large,fast overshootto one

with a very slow,long overshoot.This changeis directly attributable to the change

in hydraulic oil viscosity causedby the limit cyclein the hydraulic oil temperature

controller.
47

Figure 2.33:

over time.

Plot showing evolution of the plastication step response

56

54

5o

42 , ,

0 10 20 30 40 50 60 70 80

Run Number

Figure 2.34: Plot showing evolution of the hydraulic oil temperature over

time.
48

2.4 Summary

In this chapter,different measuresof part quality and the factors that affect

this quality wereexplored. The ultimate propertiesof any part are dependentupon

both the material that makesup the part aswell asthe processingconditions under

which the part wasmade. Thesefactors must be controlled if high quality parts are

to be made consistently.

Three different aspectsof control of the injection molding machinewerediscussed,basedon researchperformed on the Arburg machine. Theseexperiments

highlighted variousaspectsof themachinecontrolproblem. The researchby Raimund

showedthe importance of measuringprocessvariablesand how processdynamics

are material dependent. The researchby Hsiehshowedthe importance of considering part quality in choosingthe operating point for the machine. The research

by Redlitz showedhow control performance depends upon the machine operating

point and that the dynamics are time-varying. One conclusion is that an automatic

quality control system must incorporate process measurements and quality measurements with a controller that can adapt to changing process conditions. A method

to provide this automatic quality control will be explored in Chapter 3.
3. Proposed Approach to the Quality Control Problem

3.1 Introduction

In this chapter, the quality control problem is posed as a multiple-objective

optimization problem. Multi-objective optimization provides a framework for understanding the relationships between the various performance criteria and allows

the operator to make decisions on how to trade off amongst the quality objectives

to achieve performance he defines as best. It is an inherently interactive process,

with the operator constantly making decisions.

In Section 3.2, the quality control problem is defined with respect to the components of a control system [39]. It is shown that a quality control policy can be

formulated as a multiobjective optimization problem.

In Section 3.3, the important concepts of multiple-objective optimization are

reviewed. When there are multiple objective functions, a tradeoff surface can be

defined. This surface corresponds to those points where performance in one objective

must be given up in order to achieve better performance in another. An5" point on

this surface represents a nondominated or Pareto optimal operating point. It can be

shown that this point is an extremum of a, probably unknown, scalar function of the

objective functions [40]. The most important result is that if the current operating

point is dominated, it is possible to determine how the inputs should be changed so

that the process moves towards the tradeoff surface. This feasible direction can be

found as the solution of a linear program, which is presented in Section 3.4.

In Section 3.5, an algorithm is presented which will automatically find operating points on the tradeoff surface given an initial feasible starting point. This algorithm incorporates the feasible direction finding linear program with a line search,

49
5O

so that the process will automatically iterate towards the tradeoff surface. An important feature of the algorithm is that the decision maker is a fundamental part

of the algorithm. Interaction of the decision maker with the algorithm allows the

operator to "steer" the system towards the operating point he defines as best. In

subsequent chapters, case studies are presented which demonstrate the operation of

this algorithm in the face of conflicting objectives.

3.2 Definition of the Quality Control Problem

In this work, a systems approach to quality control was taken.

system has the following four components:

• Performance Criteria

Any control

• Inputs and Outputs

• The System Model

• The Control Policy

The control problem is to define a control policy such that the inputs, acting through

the model, cause the outputs to meet the performance criteria. In the following sections, each of these components is defined with respect to the problem of controlling

quality.

3.2.1 Performance Criteria

The performance criteria for control of product quality can be summarized

quite simply: consistently maintain optimal product quality. Product quality is

measured in terms of meeting objectives with respect to many different part characteristics which must be satisfied simultaneously [1, 41]. Hsieh [32], for her experiments, defined three different quality characteristics: minimize the deviations
51

from the desired length in the "parallel to flow'" direction, minimize the deviations

from the desired length in the "perpendicular to flow" direction, and minimize the

number of parts with flashing. Two different performance objectives could be associated with the each of the dimension criteria: minimize the mean deviation from

the desired length and minimize the variation of the actual length.

The goal is to find and maintain operating conditions which produce the best

quality. When only one objective must be met, best corresponds to the minimization

of an objective function. Because of the multiple objective nature of the problem,

best cannot be defined as easily. Improving one objective function may only be done

at the expense of another; a tradeoff must be made. In multiobjective optimization,

those points at which a tradeoff decision must be made are known as nondominated

points. The locus of these points forms a tradeoff surface. It can be shown that any

point on this surface is optimal in some sense in that there exists a coordiaate-wise

increasing function for which this point is an optimum (see [40], p 148). An5" point

on the tradeoff surface will be preferable to a point off of the surface (a dominated

point). If the experiments described in Section 2.3.2 are considered, the operating

conditions used in the optimal and adjusted runs provide performance which dominates that of the initial run. (See Figures [2.22] to [2.27] in Section 2.3.2.) They

can be considered to lie on the tradeoff surface. Of these two sets. the experimenter

preferred the third because it was felt that a loss in meeting the dimensional criteria

was worth the gain in the flashing criteria. Here, the experimenter was optimizing

an implicit function, her utility function, in choosing the best operating conditions

from the nondominated set. More detailed discussion on the formulation of multiobjective optimization problems and approaches to the solution of these problems

can be found in Cohon [42], Zeleny [43], Steuer [40], and Sawaragi, et al. [44].

In this work, it will be assumed that quality is defined as a set of objective
52

functions which are to be minimized simultaneously. The solution to this multiple objective optimization problem exists and is representedas a tradeoff surface

correspondingto the achievablequality. The automatic quality controller has two

functions. First, mapenoughof this tradeoff surfacesothat the operator canchoose

a feasiblepoint at which the processcanoperate. This is the point that hasbeen

previously describedas best. It will be called the quality setpoint. Second, continually monitor the process to make sure that the quality setpoint is being maintained

and that it still lies on the tradeoff surface.

3.2.2 Inputs and Outputs

An open-loop manufacturing system is depicted in Figure [3.1]. For injection

molding manufacturing, the process is the injection molding machine dynamics.

The inputs would be any machine settings, such as desired injection velocity, melt

pressure, holding time, etc. (see Figure [1.3]). The outputs are process variables

such as screw velocities, pressures, and temperatures during the molding cycle. The

material relation represents the relationship between the processing and part quality.

In a sense, the process variables over the manufacturing cycle are the inputs and

part quality is the output.

One aspect of this "cascaded" system which must be taken into account in

controlIer design is that there are two time scales. Quality information is only

available once per cycle (and possibly delayed due to measurement complexity)

whereas process data is available throughout the cycle (at the speed of the computer

sampling rate). In modeling the system and designing the control, there must be

some way to integrate the information between these two subsystems. The method

of integration will be suggested through the definition of the inputs and outputs.
53

Operator

Variations

1

Operating

Policy

Disturbances &

Modeling Errors

1

Dynamics

Noise

Raw Material

Variations

Material I

Noise

Product

Output

Quality

Measurements

L

Process

Measurements

Figure 3.1: Variability in the Manufacturing Process

3.2.2.1 Inputs

For purposes of control, the inputs to the manufacturing system can be an)"

mechanism that can affect the part quality. These inputs can be qualitative or

quantitative [45]. Qualitative inputs are those inputs that can only take on a value

from a unique set. The particular batch of polymer used, or a particular mold

geometry are examples of these. Quantitative inputs are those that can take on ans

value over a continuous range. Since qualitative inputs usually can be changed only

when the process stops, they will not be considered for the purpose of automatic

quality control.

The set of quantitative inputs can be partitioned into two sets: those which are

command signals of the process, ui(t); and those which are controller parameters,

k,(t). In an abstract sense, there is no difference in these types of inputs. They can

be treated in the same manner when determining the quality control policy. The

partitioning is done simply to emphasize the different entry points of these inputs.

In general, all inputs will be continuous functions over the interval [0. t_]. Since

the injection molding process is cyclical, it will be convenient to define these inputs
54

with respectto the cyclenumber. Therefore,the mr, command signals and the mrp

parameters during the lth cycle are defined as

 d(t) c c[0,tc(l)] i= (3.1)

kl(t) E 35 C C[O,tc(1)] i = 1,2,-.-,ml. (3.2)

In this definition, N is the set of command signals, constrained by an5' actuator

limits and /C is the set of controller parameters, only constrained by the necessity

for closed-loop stability, to(l) is the length of the It_ cycle.

3.2.2.2 Outputs

There are three different types of measured outputs in the system; quality variables, quality attributes, and process variables. Of these, the quality variables and

attributes are more important, since this is what the performance criteria are based

on. The process variables are important because they are a direct measure of the

processing conditions, which ultimately determine part quality. Each of these measured outputs and their relationship to the manufacturing model will be discussed

in the following sections.

Quality Variables

Quality variables are measured on a continuous scale. Examples of these might

be product dimensions or weight. If there are pv different quality variables which

are measured, the quality variable space can be defined bv

qv(l) E Qv C Rpv (3.3)

qv(l) is the measurement of quality for the I th part (or cycle) and the set Qv is the

set of all achievable part qualities (good or bad). Note that each of these quality

variables can only be measured once per cycle.
oo

The performance criteria are specified in terms of the optimization of each of

these quality variables (i.e. , each element of the vector qv(l)). Also, a constraint

region can be defined which represents the minimal requirements for a part to be

gOOd.

_Yr

o v, c Qv (3.4)

is the quality variable target set.

When considered as an optimization problem, Qt_ represents the constraint set

for quality. Over this constraint set, the continuous functions representing quality.

qv(l), are optimized.

Quality Attributes

Quality attributes are measured on a discrete scale. This is a property which

a part either does or does not have. In this sense each quality attribute can take on

a discrete value. If PA quality attributes are measured,

qA,(l) E CA = {0,1} i= 1,2,-..,pa (3.5)

In this equation, qa,(1) is the quality attribute function for the i th attribute. This

function will only take on the value 0 or 1. The quality attribute also can only be

measured once per cycle.

For part I to be good, all elements of qA(l) must equal zero. Therefore, the

quality attribute target set is:

QA, = {0} (3.6)

If

qA,(1) e QAr Vi (3.7)

then the Ith part is said to satisfy quality with respect to the quality attributes.

When considered as an optimization problem, QAr represents the constraint

set for quality with respect to attributes. Since the quality attribute functions have
56

a discreterange,an optimum within this rangecannot bedefined, qa(l) are equality

constraints for the optimization problem.

Process Variables

The third type of measurable output of the manufacturing system is the process variable. This is an "intermediate" variable because it is the output of the processing equipment but is the input to the material relationship (or quality model).

These measurements, more than the machine inputs, describe what was done to the

material to produce the final part properties, and ultimately the quality.

Process variables can be measured continuously through the processing cycle.

Each output can be defined as a continuous function over the processing cycle, as

was the input. Therefore, the rnv process variables during the l th cycle are defined

as

yl(t) c y c i= (3.s)

In the above definition, y is the set of measurable outputs during the l th cycle. The

only constraints associated with the process variables will be those due to equipment

limitations.

These variables can be measured continuously (up to the sampling rate of

the available data acquisition system), so are availabie much more frequently than

quality measurements. This allows some process variations to be detected more

quickly than if only quality measurements are used.

3.2.3 The System Model

The system consists of the cascade of a process model and a material relation (or quality model). The process model is the mathematical description of

tile material processing, most often represented as dynamic equations (as shown in
57

Appendix A). The quality model is usedto describethe relationship betweenprocessingand part quality. Thereis no strict form for this relation, due mostly to the

situation-dependentdefinition of quality, which is the systemoutput. An important

consequenceof this is that the model largely determines the type and capabilities

of the control systemwhich canbe implemented.

3.2.3.1 The Process Model

For the purposeof integrating processinputs and outputs with the quality

control system,this processmust bemodeledwith respectto machinecycles. This

model can bedefinedas

Yl(t) = H[uZ(t),kt(t)] (3.9)

where

H :U x K: --, 3; (3.10)

In the above equation, H represents the system dynamics, including an)' automatic

controllers.

3.2.3.2 The Quality Models

There are two types of quality measurements, so two models will be defined.

For quality variables

qv(l) = Gy[yt(t)] (3.11)

where

Gv : Y _ _pv (3.12)

Gv is a vector function which maps the process variables into a real number, for

each of the quality variables.

The model for quality attributes is defined as

qA(l) : Ga[yt(t)] (3.13)
58

where

GA,:Y_ {0, i} i= 1,''',pA (3.14)

GA, maps the process variables into a discrete set depending upon whether the part

is good (0) or bad (1), with respect to each of the quality attributes.

It should be noted that both GA and Gv map a continuous function space

into the reals. Both of these functions perform feature detection on the output. The

feature detection will have to be determined ahead of time. Examples of this would

be to relate quality to the average value or peak value of a process variable.

8.2.4 A Control Policy

The control policy is driven by the performance criteria which are defined for

the system. For quality control, two performance criteria were defined:

• Find the tradeoff surface so that the operator can choose a quality setpoint.

• Ensure that the quality setpoint is on the tradeoff surface.

Given the objectives and constraints, it is possible to find a suitable control

policy bv solving the following multiobjective optimization problem:

subject to

min q_,; = Gv, [H(u(t),k(t))] i = 1,...,pv (3.15)

,4 0,k (O

Gv [H(u(t),k(t))] E Qvr

G,_[H(u(t),k(t))] E QAr

(3.16)

(3.17)

u(t) e lg k(t) E/C (3.18)

To solve this problem the pv different functions described by Equation (3.15) must

be minimized simultaneously. As was discussed before, this is the point where
59

the operator or processengineermust make tradeoffs in satisfying the different

performance criteria. Ultimately, the solution will be a set of command signals,

u'(t), and controller parameters, k'(t), which yield quality performance that lies on

the tradeoff surface.

The advantage of the multiobjective formulation of the problem is that a utility function (or preference function) is not an explicit part of the solution. As will be

seen when implementation is discussed, since experimental data is generated during

algorithm iterations, decisions made by the operator when evaluating different operating points can be based on actual performance, not on some a' priori aggregate

cost function. This also allows the operator to dynamically change the quality goal,

as manufacturing priorities change.

3.3 A Review of the Multiple-Objective Optimization Problem

In this section, a review of multiple-objective optimization will be provided.

This will include a general problem statement and pertinent definitions. The KuhnTucker Conditions for Nondominance will be presented. These conditions form the

basis of a feasible directions algorithm used to find the nondominated points of a

given multiobjective optimization problem.

The general form of the constrained, nonlinear, multiple-objective optimization

problem can be stated as follows:

minzi = fi(x) i = 1,...,p (3.19)

X

subject to

x E X = {x E _" [gj(z) <_ O,j = 1,...,rn} (3.20)

There are p objective functions which must be minimized simultaneously. The

minimization takes place over X which is a subset of _'_. This subset reflects the

m functional constraints, 9j(z), on the decision variables x E X. The p objectives

define a mapping from decision space into objective space, Z.
6O

In the case of a single objective function, the decision space is mapped to the

real line. Since the real line can be considered an ordered space, there will be only one

global minimum point in objective space. There may be many solutions, i.e. , many

points in decision space that achieve the same value in objective space, but there

will be only one global minimum value of the objective function. When there are

multiple objective functions, the objective space is multi-dimensional. Only a partial

ordering can be imparted. The point in decision space, x_', that minimizes fi(x) may

not minimize the other p- 1 objective functions. The concept of nondominance must

be introduced (Note: The following definitions are due to Steuer in [40].)

Definition: Let z" E Z. Then z" is nondominated (Pareto optimal)

iff there does not exist another z E Z such that 1 z _< z" and z 7_ z'.

Otherwise, z" is a dominated criterion vector.

All nondominated points in Z form the nondominated set A' (also known as the

tradeoff surface).

Nondominance refers to objective space. The points in decision space which

are mapped into the nondominated set are known as efficient points. This can be

defined as follows:

Definition: Let f : X ---* Z. A point z" E X is efficient if f(z') E ,'_".

Otherwise x" is inefficient.

In other words, an efficient point is a solution to the multiobjective optimization

problem which produces a nondominated vector of objective function values.

Even though there is typically no single point in decision space which simultaneously minimizes all of the objective functions, only one solution can be chosen.

The decision maker must weigh the various options (those points on the tradeoff

XLet u,v E _" and i = 1,...,n. Then u > v implies that u, > vi Vi; u _> v implies that

ui > vi ¥i.
61

surface) and choose the best one. In multiobjective optimization problems, it is

assumed that the decision maker has a utility function which maps the criterion

vectors to a scalar figure of merit but the mathematical form of this function is not

known. Define the utility function, U, as

U: 77.p _ 77. (3.21)

The following two theorems, presented without proof, show the relationship

between the utility function and the nondominated points [40].

Theorem: Let U be coordinate-wise increasing. 2 Then, if :" is optimal

for U it is nondominated.

Theorem: Let z" be nondominated. Then, there exists a coordinatewise increasing utility function, U, such that z" is optimal.

Therefore, if the objective function values are represented by _. and A" C _P

is the nondominated set, the best operating point can be defined as z" such that

z" = max U(z) (3.22)

zEN

where U is the operator's utility function. Any, not necessarily unique, decision

variable z', which is the inverse image of z', is a best operating input for the

process.

Typically, U is unknown and so multiobjective optimization algorithms rely

on interaction with the decision maker in order to proceed. Interactive algorithms

use two different general methods for finding the minimum of U: either the gradient

of U is determined locally such as with the STEM algorithm [46] or the GeoffrionDyer-Feinberg (GDF) algorithm [47]; or the nondominated set is iteratively reduced

2A coordinate-wise increasing utility function implies that all objective are in maximization

form and that monotonicity holds for each of them [40].
62

to exclude regions where the best point does not lie, such as in the tradeoff cut approach of Musselman and Talavage [48], the Zionts-Wallenius (Z-W) method [49],

or the interactive weighted sums algorithm of Steuer [40]. Convergence of these

types of algorithms can be shown under certain assumptions such as the constraint

set being closed, convex, and bounded, and the utility function being continuous,

differentiable, and concave. These assumptions may not hold globally and the operator must be relied upon to steer the algorithm clear of undesirable regions, such as

those which appear to be Pareto optimal but are not, as in the case of local optima

[5O].

3.3.1 Finding the Tradeoff Surface

The Kuhn-Tucker conditions for nondominance [51] are a generalization of

the necessary conditions for optimality of a scalar function. They can be stated as

follows.

The Kuhn-Tucker conditions for nondominance: Given the multiobjective optimization problem as stated in Equations (3.19) and (3.20),

with the fl and gj being continuous functions, if a solution z" = f(.7_'),

is nondominated, then there exist multipliers wi > O, i = 1,...,p and

u: _> 0, j = 1,..-,m such that, when

x" E X C 7U_, (3.23)

m

wiVf/(x') + _ ujVgj(x" ) = 0 (3.24)

i=1 j=l

usgj(x" ) = 0 j = 1,-.-,m (3.25)

These conditions are necessary for nondominance. They are also sufficient if

all of the objective functions are convex, if X is convex, and if w, > 0 for all i [42].
63

The above conditions are used to detect a nondominated point. What is needed

is an algorithm which iterates towards a nondominated point. This can be developed

using the Kuhn-Tucker conditions, Gordan's Theorem of the Alternative[52], and

duality. First, intuitive arguments will be given to justify this.

Consider the following. If x is a dominated point, then there is at least one

objective function that can be strictly decreased without increasing any of the other

objective functions (or violating any constraints). Therefore, a direction vector,

d E T/'_ exists such that for z + ed E X and for some e > 0,

+ ed) < (3.26)

This will hold for any d such that

Ufi(x). d < 0 (3.27)

When the above inner product is minimized with respect to any one objective function, d represents the steepest descent direction for that objective function. If a

d exists which satisfies (3.27) for all of the objective functions, this represents a

feasible direction to reduce all of the objective functions simultaneously.

This intuition can be verified rigorously using Gordan's Theorem of the Alternative. The theorem, the proof of which can be found in Mangasarian [52], is stated

as follows:

Gordan's Theorem of the Alternative: For a given matrix A E

_m×_, x E 7"4.'_, y E 7_", with y >_ 0, y 7_ 0 either

Az < 0 (3.28)

has a solution, x, or

ATy = 0 (3.29)

has a solution, y, but not both.
64

Gordan's Theorem can be related to the Kuhn-Tucker conditions in the following way. Let A be the Jacobian matrix of the objective functions and active

constraints. Let x be the direction vector, and let y be a vector with the first p-rows

being the wi's of (3.24) and the remaining rn - p rows are the u._'s of the active

constraints. If there is no solution for (3.24), (or (3.29)), then there is a solution for

(3.28) and z is the direction vector specified by (3.27).

3.4

Using the arguments in the previous section.

can be used to determine the descent direction:

subject to

A Linear Programming Solution to Multiobjective Quality Control

the following nonlinear program

max Po (3.30)

d

Po+ a,(_)v f,(:_) . d <_o = 1,--.,p (3.31)

-/?j(_')Vgj(_') - d < 0 j E [1,m] _ gi(5:) = 0

Po>O

n

Ed =l

k=l

where al and _?i are chosen to satisfy the following:

,_,(_)2vf,(_). vf,(_)= 1 i= 1,...,p

(3.32)

(3.33)

(3.34)

(3.35)

/3j(5:)2Vgj(k) • Vgj(_' ) = 1 j E [1,rn] _ gj(._)= 0 (3.36)

In this nonlinear program, the direction d which maximizes Po is the direction

which is a compromise descent direction among all of the objective functions. Po is

the inner product representing the projection of d on each of the objective function
65

gradients. Equation (3.32) ensuresthat no constraint of the original problem will

be violated. Equations (3.35) and (3.36) ensurethat the direction chosenis not

affected by the magnitudeof anyof the gradients. Equation (3.34)ensuresthat the

magnitude of d does not affect the result and that the solution is always bounded.

This optimization problem was approximated by a linear program when Equation (3.34) was replaced by the following equations:

--1 _dk _ 1 k= 1,..-,n (3.37)

n

Idol < (3,3s)

k=l

n n

(1 + E d_-)= E 2a_dk (3.39)

k=l k=l

These linear constraints represent a faceted approximation of the hypersphere constraint of Equation (3.34). This is illustrated by a two dimensional example of

this approximation, pictured in Figure [3.2]. Equation (3.39), represented by the

dashed line in the figure, is the Taylor Series expansion of Equation (3.34) around

the approximate direction r). This direction is determined by first solving the linear program without Equation (3.39) then resolving the larger program using the

previous result as d. This technique is called approximation programming and is

described in [53]. An example is presented in Appendix C which shows the necessity

of the approximation given by Equation (3.39). One point that should be emphasized is that the only constraint which is necessary is Equation (3.37), which ensures

that the solution is bounded and is a feasible direction.

The solution to this linear program is the direction which mazimizes the minimum gain in all objectives. This can be thought of as a "best compromise" direction. Implicitly. this formulation gives all objectives equal weight. It is possible

to conceive other formulations of this linear program, through the use of additional

constraints or weights such that certain objectives are minimized preferentially. It

is the authors' contention that this "steering" be done interactively by the decision
66

Figure 3.2:

d2

Eqn (3.34)

dl--1

1

_ Eqn (3.39)

dl

Two dimensional example showing how the quadratic constraint is approximated by linear constraints.

maker (DM). The DM will be better prepared to steer the algorithm while observing

the process, rather than choosing weights a'priori.

In the following section, this linear program will be incorporated into an algorithm which directs the system towards the tradeoff surface.

3.5 Implementation of Automatic Quality Control

Implementation of automatic quality control can be formulated as the iterative

solution of the multiobjective optimization problem described in the previous section. This tuning must be done in two phases. In the first phase, a feasible operating

condition is given and the system will iteratively track to a point on the tradeoff

surface. In the second phase, the best operating point, as defined by the decision

maker, will be chosen from among the available points on the tradeoff surface. The

best operating point can be defined as the one that minimizes or maximizes the

utility function.

Direct implementation of the direction finding linear program requires that

there be a functional description of both the objective functions and the constraints.

It is proposed that objective gradient information be obtained experimentally from
67

the system. A set of plannedperturbations of the operating point from somereference can be used, suchas in evolutionary operation [54]. These data can be used

to fit empirical modelsto the processin a regionabout the designcenter. This will

provide the necessarygradient information for the objective functions. Two different types of constraints will be considered. As shown in Equation (3.16), there may

be hard constraints on the objective functions which must be met by the system,

corresponding to the limits on the quality variables as defined by the quality variable

target set. The same experiments which provide estimates of the objective function

gradients will provide estimates of these constraint gradients. There may also be

other constraints not directly related to quality. One such constraint could be limits

on the controller gains which correspond to the stability of the closed-loop system

or of the ability of the actuator to output the required control signal. These types

of constraints do not appear in the statement of the original optimization problem

(Equations (3.15) to (3.18)). It may be argued that these types of constraints are

artificial since they do not correspond to product quality directly, but from a practical point of view, they exclude undesirable operating conditions, reducing the region

of decision space which must be searched. Any constraints of this type must have a

functional description.

3.5.1 A Proposed Feasible Directions Algorithm

The following algorithm is proposed as a feasible directions method for detecting the tradeoff surface.

Step 1: Choose an initial starting point, 5:. Form a set, A,r, of known nondominated operating conditions and associated efficient points. If this information does not exist, then A t. = ¢, the null set. Values for the following

three parameters must also be chosen: 8, the criterion which is used

to measure how well Equation (3.39) must be approximated (used in

68

Step 4), c, the stopping threshold which detects "nearness" to the tradeoff surface (used in Step 7), and 7?, the step size of the line search (used

in Step 9).

Step 2: Generate data around 5: using designed experiments. Use this data to

find the gradient for each objective function. Normalize each of the

estimated gradients, i.e. , satisfy Equations (3.35) and (3.36).

Step 3: Formulate and solve the linear program, specified by Equations (3.30)

to (3.33) and Equations (3.37) and (3.38). This yields the initial feasible

direction, d.

Step 4: If _=1 d_. < 1 + _5, then the constraint of Equation (3.34) is approximately satisfied; go to Step 7. Otherwise continue.

Step 5: If Ildll> Ildll,the approximating program is diverging. Set d = d and go

to Step 7. Otherwise continue.

Step 6: Set d = d. Solve the linear program, specified by Equations (3.30) to

(3.33) and Equations (3.37)to (3.39). Go to Step 4.

Step 7: Analyze the linear program solution. If Po > e, a stopping threshold, use

the feasible direction to find a new nondominated point; go to Step 8. If

Po <_ e, 5: is on the tradeoff surface; go to Step 11.

Step 8: Let 5:=5:.

Step 9: 5: = _ + Porld. Run an experiment with _" as the operating condition

and generate 0, the value of the performance objectives at 5:. 77 is a

step-size for the search algorithm chosen by the experimenter. Po, the

projection value returned by the linear program, is used to adjust the

step size so that smaller steps are taken as the algorithm gets nearer the

69

tradeoff surface, when objectives begin to conflict. (In this step, any line

search algorithm may be employed. This method was chosen because it

is relatively conservative.)

Step 10: Evaluate the performance of (k,q) against 3{. If 0 is nondominated,

update N" and keep searching along d (go to Step 9); otherwise continue.

Step 11: If 0 is dominated by an alternative in A/', stop the line search. Allow

the decision maker to evaluate the alternatives within A" and determine

a new operating point to begin evaluations from. Call this desired operating point k. The algorithm can be stopped at this point or a new

iteration can be started by returning to Step 2.

The algorithm has three phases. In the first phase, Steps 1 and 2, the system

is probed to generate a model of the response surface. This model provides the

objective function gradients which are used to determine the search direction. In the

second phase, Steps 3 to 7, a linear program is solved which determines the feasible

direction, d, which will move the system to an operating point on the tradeoff

surface. In the third phase, Steps 8 to 11, a line search is carried out until no

improvement in performance is obtained in the current feasible direction. Once

the feasible direction has been exhausted, the operator can be presented with the

current set of nondominated operating conditions. The operator will choose one,

and a new iteration can begin or the algorithm can be stopped if the operator is

satisfied with system performance.

There are three parameters, selected by the user, which control the performance of the algorithm. 3 is used as a threshold for determining when Equation (3.34) is approximately satisfied. For the case studies in this thesis, it was

found that 6 = 0.05 worked well. e was used as a "conflict threshold". Po is the

minimum value of the projection of the feasible direction vector onto each of the
70

d

Complete Conflict Conflict at e = 0.1

Figure 3.3: Complete conflict versus e-conflict.

gradients. The magnitude of Po is a function of the magnitude of the most conflicting gradients, the magnitude of d, and the cosine of the angle between them.

If two objectives are in complete conflict, i.e. , they are 180 ° apart, then Po = O.

Assuming the magnitude of all vectors is unity, Po _< 0.1 signifies that the angle

between the feasible direction and the most conflicting gradients is greater than 85%

or the two most conflicting gradients are greater than 170 ° apart. This is illustrated

in Figure [3.3]. In this thesis e = 0.1 was used as the threshold for detecting the

tradeoff surface. Finally, 77is selected to determine the step-size taken during the

line search. Selection of this parameter is completely dependent upon the particular

problem. It should be of the same order of magnitude as the perturbations used in

the experiments of Step 2.

There are two different methods by which the operator steers the algorithm,

both of which occur during Step 11. The simplest form is that the operator request

that certain regions of the tradeoff surface be explored by his choice ._ and a selection

of the step-size for input perturbations and the line search. The algorithm will

explore the local region around _:, with "local" being determined by the step-size

chosen.
71

The secondmethod of steering is wherethe operator selectively disablesobjective functions. If it is found that a region is encounteredwhere the marginal

value of a gain in one objective function is very low, this objective can be disabled

and all optimization will take placewith respectto the others. The objective will

continue to be monitored, and it's behavior will be reflected in the construction of

the nondominated set, but it will not affect the feasibledirection of search. This

is implemented directly via the ai weighting factors. If ai = O, the i t_ objective

function is removed from the linear program.

3.5.2 Properties of this Algorithm

This algorithm is a primal method, in that it directly attacks the original

problem, in this case through the application of the Kuhn-Tucker conditions. One

advantage of primal methods, that is particularly useful in this research, is that

each point generated in the search process is feasible. Therefore if the search is

terminated early, a feasible and probably nearly optimal operating point is still

obtained [55]. When the implementation is to be in real-time, feasibility is especially

important because an infeasible point may correspond to an operating condition that

is dangerous to the operator or the equipment. Operator safety is of paramount

importance.

One performance measure typically applied to optimization algorithms is that

of convergence. Insight into the convergence properties of the multiobjective optimization algorithm can be gained by considering the following theorem[55]:

Global Convergence Theorem: Let A be an algorithm on X. and

suppose that given Xo the sequence {xk}_'=o is generated satisfying

zk+l e A(zk)

Let a solution set F C X be given, and suppose,
72

(i) all points zk are contained in a compact set S C X

(ii) there is a continuous function f on X such that

(a) if x _ F, then f(y) < f(z)Vy E A(z)

(b) if z E r', then f(y) < f(z)Vy E A(z)

(iii) the mapping A is closed at points outside r'.

Then the limit of any convergent subsequence of {zk} is a solution.

Condition (iii), requiring closedness of the map, is the most important. The multiobjective algorithm can be considered as the sequential application of two maps, a

direction finding map and a line search map. Closedness of the algorithm requires

that both of these maps be closed. Closedness is defined as follows:

Definition: A point-to-set mapping A : X ---, Y is said to be closed at

x if the assumptions

(i) zk_z, zk_X

(ii) _tk _ Y, Yk E A(zk)

imply

(iii) y E A(z)

If A is a direction finding algorithm, then the sequence {yk) represents the direction

vector for each of the points in the sequence {zk}. For A to be closed, it is required

that the direction not change suddenly. This cannot be guaranteed by the linear

program of Equations (3.30) to (3.33) and (3.37) to (3.39). The direction will change

suddenly when a formerly inactive constraint becomes active. For this reason, this

algorithm is not globally convergent. (This problem is inherent to feasible direction

algorithms in general.)

When the constraints are analytic, this "direction change" problem can be

avoided by implementing penalty or barrier functions as part of the objective. This
73

way, gradient information from inactive constraints is included. This is not possible

for this problem due to the nature of the quality attribute constraints, as described

by Equation (3.17).

Convergence is really important only if an optimization algorithm is running

autonomously. For the multiobjective optimization algorithm, a decision maker is

an intrinsic part of the algorithm, (i.e. , Step 11). In this case, the decision maker

provides the "steering" that would be provided by the penalty or barrier function

in other feasible direction implementations. Furthermore, it is the decision maker

who decides when the algorithm is to stop iterating. When the algorithm "finds" a

point that the decision maker considers best, iterations are stopped and the process

is run using the associated operating conditions. The operator ultimately decides

when the optimization has converged.

3.5.3 Example

The following example is intended to show the use of the algorithm on a very

simple example. The objective functions are analytic, so there is no need to generate

gradients experimentally.

Consider the following optimization problem:

min f(x) = [ (xl -1)2 + (x2-1)2 I (3.40) =ere' (x, + 1) 2 + (x2 + 1) 2

The Jacobian of f(z), evaluated at 5: and then normalized such that Equation (3.35)

holds, is

] r" 1

022(zl + 1) a22(z2 + I) J [ Ja vf (z)

The analytic formulation of the objective functions allows Steps 1 and 2 of the

algorithm to be skipped. Then, the following tableau represents the linear program

which must be solved to determine the direction of the Pareto boundary. (Note: this
74

tableau representsthe initial LP to besolved.The constraint dueto Equation (3.39)

is not included.)

0

0

0

1

1

1

1

,/7

-1 0 0 0 0 0 0 0 0 0 0 0

1 f_(_) F_2(:_) -F_(:_) --r_2(_) 1 0 0 0 0 0 0

1 F2,(:_) P;2(:_) -P;,(:_) -F22(:_) 0 1 0 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

0 1 1 1 1

0 0 1 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1

The objective functions are two paraboloids, with global minima at (1, 1) and

(-1, -1), respectively. Due to symmetry, the Pareto optimal boundary is the line

connecting them. In terms of the geometry of the objective surface, this line forms

the set of all points for which the individual objective gradients point in opposing

directions.

The algorithm was run using three different initial points. A plot showing

the objective function contours and the iterations is given in Figure [3.4]. The zigzagging across the Pareto boundary is a result of the algorithm not being closed.

This behavior would be detected by the operator, in practice.

3.6 Summary

In this chapter, the automatic quality control problem has been defined formally. This led to the statement of the problem as a multiple objective function

optimization problem. The iterative solution to this problem forms the basis of

feedback control of part quality.
x

Figure 3.4:

ations.

. x_ ................... _

\

..r

...... •-C ...... X_ : .

Pareto Boundary '. " ':

C _

-... ....... .." ............ • .: . ._ .

, i i

-'1 (_ I 2 3 4

X1

75

Contour plot of objective functions showing algorithm iterA multiobjective algorithm for implementation of quality-based controller tuning was presented. This was a feasible directions algorithm based on the KuhnTucker conditions for nondominance.

Implementation of this algorithm on the injection molding machine is the focus

of the rest of this work. In the next chapter, details of the equipment on which the

implementation was carried out, are presented. In the two chapters following that,

the controller tuning and quality control case studies are presented.
4. Experimental Setup

Any control system will depend, to some degree, on the specific machine being

controlled and the equipment used to control this system. In this chapter, specific

details of the injection molding machine used, available sensors, actuators, and the

control computers will be discussed.

4.1 The Injection Molding Machine

The injection molding machine which will be used is an Arburg Allrounder,

Model #221-175-350. The specifications for this machine are given in Table [4.1].

This machine is equipped with actuators and sensors which can be used to monitor

and control the process. This equipment will be discussed in the following sections.

4.1.1 Actuators

The actuators available are used for machine sequencing, feedback control, or

as process settings. These actuators are the solenoid valves, the barrel heater bands.

the nozzle heater band, the hydraulic motor control valve, and the servovalve. The

capabilities of each of these actuators will be described below.

Solenoid Valves

Each of the machine phases is initiated by activating or deactivating-activating

solenoid controlled valves. These valves "route" hydraulic fluid through the hydraulic system. They are merely switches, and as such, cannot be used for feedback

control. Sequencing of the machine through the phases does depend upon the proper

switching of these valves. In Table [4.2], these valves are listed showing the phases

where they are active (ON) and inactive (OFF).

76
77

Table 4.1: Arburg Allrounder Machine Specifications

Shot Size

Clamp Force

Injection Force

Screw Speed

Screw Torque

Nozzle Pressure

Nozzle Retract

Effective Stroke Ma.x

Clamp Stroke

Open Daylight Max

Close Daylight

Platen Dimensions

Tie Rod Spacing

Tie Rod Diameter

Max Mold Weight

Max Movable Mold Weight

Mold Size

Mold Thickness

Hydraulic Ejector Force

Hydraulic Ejector Stroke

Barrel Heater (3)

Nozzle Heater

Total Connected Load

4.64 ounces

40 tons (toggle-type)

12.2 tons

Min 10 RPM

145 foot-pounds

6 tons

7.08 inches

5.7 inches

Min 2.4 inches

19.7 inches

Min 5.9 inches

13.5 x 9.8 inches

8.66 inches

1.8 inches

275 pounds

165 pounds

8.66 x 9.8 inches

Min 5.9 inches

2.7 tons

2.36 inches

1750 Watts

200 Watts

45 Amps

Max 320 RPM

Max 7.9 inches

Ma_x 11.8 inches

Max 11.8 inches

Table 4.2: Solenoid Valve Activation Table

Valve Plastication Injection Holding Cooling

System Pressure OFF ON ON X

Mold Close ON ON ON ON -- OFF

Mold Open OFF OFF OFF OFF -- ON

Holding Pressure OFF OFF ON OFF

Screw Forward OFF ON ON OFF

Screw Retract OFF OFF OFF ON (briefly)

Screw Rotate ON OFF OFF OFF

Carriage Forward ON once, to initiate the first cycle

Carriage Retract ON once. to end last cycle
78

Barrel and Nozzle Heater Bands

The heater bands are used for control of the barrel and nozzle temperature.

Each of these heater bands has its own closed-loop control system. Underneath each

heater, set in the barrel, is a thermocouple which provides temperature feedback to

the associated heater band. Each closed-loop controller provides the operator with

a knob to set the desired barrel temperature for that heater band. The controller

is an analog control loop, possibly a PID controller (observation of the controller in

action shows that there is some anticipatory response to the sensed temperature).

The control action is a simple ON/OFF. There is no readout of actual temperature;

a simple LED readout indicates when the temperature is within -4-5°C of the set

temperature. It is not possible to adjust temperature setpoints automatically, due

to the construction of these particular heater band controllers, although this is not

a limitation in general. If changes to the temperature setpoints is necessary, this

will be accomplished by notifying the operator.

Hydraulic Motor Control Valve

The hydraulic motor control valve is used to control screw rotation speed. This

is done by regulating the flow to the hydraulic motor which rotates the screw, using

a ball valve. In typical operation, this is set by the operator through a knob and

graduated dial. On the dial, 0 corresponds to the minimum screw speed and 5 corresponds to the maximum speed (see Table [4.1]). A stepper motor and gear system

has been implemented to allow computer control of screw speed. One limitation of

this actuator is that it can only be turned when the screw rotate solenoid valve is

off (the mechanical construction of this valve does not allow it to be turned while

under pressure). The result is that any feedback control scheme which incorporates

changing the screw speed can only make one control action per cycle, when the

machine is not rotating.
79

Servovalve

The servovalveis usedto control the flow of hydraulic fluid to the piston actuating the plasticating screw. This actuator controls any linear motion of the screw

as well as hydraulic pressureapplied by the screw during holding. This actuator

is the heart of the injection molding machinecontrol system. During plastication,

this valve controls the flow of hydraulic fluid out of the piston chamber_ thereby

controling the recovery rate of the screw and the pressure developed in the molten

polymer. During injection, this valve controls the flow of fluid into the piston chamber, thereby controlling the injection velocity and pressure. During holding, this

valve is primarily used to control the pressure on the polymer in the mold. Computer control of this actuator is accomplished through a D/A converter and amplifier

from the microprocessor.

4.1.2 Process Sensors

There are five sensors on the injection molding machine: hydraulic pressure,

melt pressure, screw position, melt temperature, and screw RPM. Each of these will

be discussed below.

Hydraulic Pressure

A hydraulic pressure sensor is located in the hydraulic piston chamber behind

the screw. This sensor can be used to record pressure on the screw during plastication, injection, and holding. This sensor has a range of 0-15000 PSI and the output

is connected to an A/D converter on the microprocessor.

Melt Pressure

An additional pressure sensor is located in the nozzle of the barrel. This

sensor measures the pressure on the molten polymer during plastication, injection,
8O

and holding. The sensorrangeis 0-3000 PSI and is also connectedto an A/D

converter on the microprocessor.

Screw Position

A linear potentiometer, attachedto the screw,is used to measurescrewposition. It's rangeis 0-15 cm and is alsoconnectedto an A/D converter.

Melt Temperature

A type-K thermocouplehasbeenmountedin the nozzlefor usein measuring

temperature of the molten polymer. This temperature will be different from that

measuredby the thermocouplesusedby the heater band controllers. This sensor

has beenusedin previousexperiments,but is not currently installed.

Screw RPM

ScrewRPM is sensedby a magneticpickup on the screw. This is then fed to

an analog gaugeto providethe operatorwith an indication of screwspeed(it is not

very accurate). Hardware hasbeendesignedto feed screwspeed, using a similar

magnetic pickup, to the control computer. This hardware has not been installed.

4.2 The Control Computer

Control of the plastic injection molding process is done using two computers:

an Intel 80286-based personal computer, used for process supervision, and a Motorola 68000-based microprocessor, used for real-time control and data acquisition

[36, 37]. Together, these computers comprise the plastic injection molding machine

quality evaluation and control system (also known as the Diatomic Spurtalyzer).

A typical cycle proceeds as follows:

O-2-1
81

1. Injection parameters are entered into the PC. They are then passed down to

the 68000 via a serial line. There are three types of parameters: termination

variables, process setpoints, and controller gains. These will be described later.

2. Upon receipt of the parameters, the 68000 begins the injection cycle. Process

data is stored and the process is controlled. For the most part, the PC is

dormant during this stage.

3. During the injection cycle, the 68000 can signal the PC to prompt the operator

for information. An example of this is information related to the operator's

inspection of the molded part. Any interaction will be specific to a particular

mold being used.

4. Upon completion of the injection cycle, the 68000 passes the logged process

data to the PC. All data is stored in files on the disc, and any data pertinent

to the optimization is retained in memory to be operated on by the algorithm.

During any operations on the PC, the 68000 is dormant.

Communication between the two processors takes place via a serial line. The

protocol is a pure master/slave relationship. When one processor is running, the

other halts until a message is received, then the roles are reversed. Three types of

variables are passed from the PC to the 68000 at the initiation of an injection cycle:

• Termination variables - these are used to end a particular phase. The following

five are used:

- Shotsize - the amount of polymer to be plasticated, in centimeters (cm)

of screw displacement.

- Pullback - the distance the screw is retracted after plastication to allow

decompression of the polymer, also in cm.
82

- Cushion- the amountof polymerthat is to remainin the barrel at the end

of injection. During holding, this polymer will beusedto fill any volume

due to shrinkage.This is alsomeasuredin cm of screwdisplacement.

- Holding Time - the lengthof time to maintain holding pressure,measured

in seconds.

- Cooling Time - thelengthof time after holding, that the part is to remain

in the mold beforeinjection, alsomeasuredin seconds.

- Maximum Plastication and' Injection Time - These are limits that are

used as a fail safein casea problem occursduring the cycle. They can

not be altered interactivelv.

• ProcessSetpoints- thesearethe setpointsthat the controllersmust maintain

during a particular phase.The following areused:

- Plastication Melt Pressure

- Injection Melt Pressure

- Holding Melt Pressure

• Controller Gains - the plastication, injection, and holding phaseseach are

controlled by a PID controller. During eachcycle, thesegains are passedto

the 68000.

During a set of experiments,anyoneof the aboveparameterscan bechanged

from one cycle to the next. The parameterswhich were changeddependedupon

the particular optimization beingdone. For example,during plastication controller

tuning, only the PID gainswerealtered, all other parameterswere held constant.

During quality experiments,only processsetpoints werevaried.

The machine also had other inputs which were not under computer control,

namely screwspeedand barrel temperature. Thesesettings are dependenton the
8a

polymer being used. In all of the experiments to be described, this polymer was

polypropylene. For these experiments, screw speed was held constant at 380 RPM.

The barrel temperature could be set independently in four zones along the barrel.

These temperatures were 220 C, 220 C, 220 C, and 205 C, from the nozzle to the

hopper respectively.

Programming on the 68000 was done in C. All real-time control code was

written on the PC, cross-compiled using Uniware software [56], then downloaded

to the 68000. Programming of the PC was done via Matlab [57]. All algorithmic

programming was done using custom Matlab functions, as well as those provided

within blatlab. Interaction with the injection molding machine, such as serial port

communication and control, was done using the Matlab-to-C interface. The user

operated the injection molding machine by calling a Matlab function called "inject".

Input parameters were those described above, and logged data was returned as

Matlab matrices, converted into the appropriate units.
5. Multiobjective Controller Tuning

The algorithm was applied to the tuning of two PID controllers, one used in the

plastication phase and one used in the injection phase. These case studies are

relevant for two reasons: injection molding requires the operator to control many

different parameters to produce a good part [29, 30]; and the process is cyclical,

so the operator (or algorithm) has the opportunity to evaluate performance and

update operating conditions between cycles. Use of the algorithm allows the loop

to be closed around process performance. This is shown in Figure [5.1].

The PID regulator is probably the most widely used controller in the process

industry. Different methods have been proposed for tuning these regulators, such

as Ziegler-Nichols [58]. Despite this, it is common experience to have the regulator

poorly tuned, yielding a less than adequate process response [59]. Also, these tuning

methods may optimize some aspect of the response which is not appropriate for

a particular application. It is not the author's intention to provide "yet another

controller tuning tool", but rather to explore the multi-objective nature of process

control.

An anti-reset windup implementation of this controller was chosen because the

control output is limited to +5V. Feedback to the integral term is used to "turn

off" the integrator when the actuator is in saturation. If this does not occur, when

the process has a constant error due to actuator limitations, the integrator "winds

up", resulting in poor performance once the system comes out of saturation.

84
85

The block diagram for this controller is shown in Figure [5.2]. The controller

= )o-i

.

- ¢,p(1 + Kz3)(" - (1 - - '+_"°) P_(z) (5.1)

i)

In this equation, the operator z is the z-transform variable.

Notice that the command input, R(z), does not pass through the differencing

term of the controller. This prevents spikes due to discontinuities in the command

input from passing through the system. Also, since the limiter is a nonlinear element,

it is not represented in the transfer function. It should be noticed that this controller

has three adjustable parameters, two zeros and a gain. One zero is dependent only

on the ratio of Kz to ]i'p and the other only on/i'D. The controller gain is dependent

on the product of/t'p and I(z3. These relationships will be important when analyzing

the results of the multiobjective controller tuning. Another important point is that if

control design is thought of in terms of choosing the zeros and gain of the controller,

there will be more than one set of PID gains that will achieve the same closedloop response. (It may be possible to use this "degree of freedom" to optimize an

additional objective at no "cost".)

Three different case studies are presented. In the first case study the plastication PID controller was tuned in simulation. An analytical model for plastication

has been derived, but this model contains many unknown and processing-dependent

parameters. Because of this, a model was identified from input/output data. Details

of the modeling and identification are presented in Section 5.1.

Using the identified model, the multiobjective algorithm was applied, in simulation, to the tuning of a PID regulator for control of the plastication phase. Two

objective functions were defined; one which was directed towards good trajectory

tracking and one which was directed towards avoiding controller saturation. Details

transfer function is
86

Machine

Inputs

Figure 5.1:

Process

Disturbances

L

Dynamics

___Performance

Evaluation t

Controller

Performance

Measure

Automatic ControLler

[ iMu,_-ou_ L

The algorithm used for performance monitoring and control.

R(._) +

+

_(z)

Figure 5.2: PID controller block diagram.
87

of this simulation, and algorithm results are presented in Section 5.2.

The ptastication controller was then tuned on-line. Essentially, the simulation

of the process was replaced by the Matlab function which controlled the injection

molding machine, as discussed in Chapter 4. With respect to the algorithm, there

was no difference in the two implementations, other than the fact that the real data

was noisy. Details of these results, and a comparison with those obtained from the

simulation, are presented in Section 5.3.

Finally, the injection controller was tuned on-line. In the case of injection,

the dynamics are very dependent upon the mold, so new gains must be scheduled

for every mold. Two different molds were used for the quality control case studies,

which are discussed in Chapter 6. For the spiral mold, a slight modification of

the plastication gains provided adequate control. For the ASTM four cavity mold,

new gains had to be found. Implementation details are no different than that for

plastication. Results of this study are presented in Section 5.4.

5.1 Plastication Modeling and Identification

During plastication, a prescribed amount of polymer is melted and deposited

in front of an injecting mechanism. Tight control of the processing conditions of the

polymer during plastication is very important since this has a large impact on final

part quality [29, 30, 35, 26, 28]. In this study, melt pressure is controlled as a singleinput single-output system. This affects both how quickly polymer is accumulated

as well as the melt temperature.

5.1.1 Analytical Modeling of Plastication

The plastication dynamics consist of the cascade of fluid dynamics associated

with the hydraulic portion of the injection system, a mass balance associated with
88

the motion of the screw, and fluid dynamics associated with the pumping and accumulation of the polymer. The following linear state equation can be used to describe

the system:

p --

Vm

_± _._

171 rn frt

o _

VH VH

Zp +

K_a____ 0

v_

0 0 u, (5.2)

In the above equation, the states are defined as Zpl = P,_, Zp2 = 1_, and Zp3 = PH.

The control inputs are defined as Up1 = _z and Uv2 = X,. (All of the symbols used

in this equation are defined in Table [5.1].) An output equation must be defined for

this state-space model. If melt pressure is the output, then the output equation is:

Y,- I 0 0 jZp (5.3)

A block diagram of this system is shown in Figure [5.3]. A detailed derivation of

this model is contained in Appendix A.

The dynamics of the servovalve are dependent on the internal construction

of the valve. On the Arburg machine, this is a three-port valve. An approximate

model of the valve dynamics is [61]

x+] -

KI/

s(r_s + 1) V[s] (5.4)

The input to the servovalve is a voltage, V, and the output is the position of the

spool, which regulates hydraulic flow out of the screw piston chamber. The integral

term of the transfer function is a result of the fact that there is no internal feedback

within the valve because of the solid spool [62]. The first order term, with time

constant r., is an approximation of the rest of the valve dynamics, including the

windings driving the pilot stage.

The problem with this model is that the parameters are difficult to determine,

and can be related non-linearly to system states. For example, both hydraulic oil
89

Table 5.1: Symbols used in the plastication dynamic equations.

AH

Am

b

Km

Kx,

I(pp

IQ

1'72

PM

Pm

u,

v_

Vm

T/;

V

X.

z_

Zm

/3H

hydraulic pressure effective area

melt pressure effective area

approximate viscous friction on screw

melt pressure empirical gain

servovalve spool position flow gain

servovalve hydraulic pressure flow gain

servovalve gain (voltage to spool position)

screw speed empirical gain

approximate mass of screw and polymer system

hydraulic pressure

polymer melt pressure

plastication input

hydraulic oil volume

melt volume

recovery rate (screw linear velocity)

servovalve input voltage

servovah'e spool position

plastication output

plastication state vector

melt bulk modulus

screw RPM (screw rotational velocity)

hydraulic oil bulk modulus

servovalve time constant

and polymer bulk moduli will be dependent on the overall machine temperature.

For this reason, it was decided to develop an empirical transfer function for the

system.
9O

Figure 5.3: Plastication dynamics block diagram.
91

5.1.2 Identification of a Plastication Model

Initial experiments showed that the open-loop system was marginally stable.

Input/output data had to be collected with the system under closed-loop control.

This situation can lead to problems with the identification. Specifically, the controller can induce correlation between the output and input, which can adversely

affect identification algorithms [63]. To avoid this, a relay with hysteresis was used

as the stabilizing feedback element. This element will stabilize a class of systems and

provides very good data for identification and because the control was nonlinear,

the correlation problem is less significant..3_rzen, in [64], discusses in detail the use

of relays as feedback elements. /_rzen presents a describing function analysis which

shows that the relay will induce limit cycle behavior in the closed loop system. Because of the relay, the input during this limit cycle is a square wave with a period

related to the system dynamics.

The nominal melt pressure reference was chosen to be 3000 psi. Other references were tested to determine the sensitivity of the model to this parameter. The

relay with hysteresis has two parameters which were changed routinely: relay amplitude and hysteresis width. These changes would generate limit cycles with different

periods and amplitudes, possibly exciting different dynamics in the system. These

parameter changes provided a large ensemble of input/output data sets which were

used to evaluate the accuracy of the identified model.

The sampling period used for these experiments was 12.5 ms. This choice was

dictated by the ability of the 68000 controller to compute the PID control law within

one sampling period.

Input and output data for the injection molding machine, using this feedback

are shown in Figures [5.5] and [5.6]. Model identification was done using the Identification Toolbox in Matlab [63]. The following ARMA model was chosen as the
92

representation.

A(q)Pm(k) = B,(q)V(k) + B_(q)_z(k) (5.5)

In this equation, q is the unit time delay operator and k is the sample index.

Various model orders were tried. The selection of the final model was done by

comparing output from the model, under the same relay feedback as the real process.

Also, models which displayed approximate pole-zero cancellation were deemed to

be of too high an order. Model identification was done using over 40 data sets. In

all cases, the seventh-order model, with respect to the voltage input, proved to be

the best. It was possible to identify only" one coefficient with respect to screw speed.

This input had to be a step, due to physical considerations. Because of this, it did

not provide enough excitation for more than one coefficient to be estimated. Also,

the poles of the empirical models were practically identical for the different data.

The zeros varied quite a bit. In almost all cases, the zeros were non-minimum phase,

and they typically appeared as complex-conjugate pairs.

After an analysis of process data, the following two-input, one-output ARMA

2.380P_(k - 1)- 1.908P_(k - 2) + 0.570Pm(k - 3)

0.796Pm(k - 4) - 1.914P,,,(k- 5)+ 1.392P_(k-6)

0.316P,_(k- 7)+ 0.371V(k- 1) - 0.974V(k - 2) (5.6)

+ 1.6 2v(k-a)- 1.616v(k-4)-0.026v(k-5)

- 0.285V(k - 6) + 0.275V(k - 7) + 0.00731w(k - 1)

In the above equation, k represents the sample index. The current output, P,.,,(k),

is a function of previous input and output values.

The poles and zeros of this transfer function are shown in Figure [5.4]. There

are no zeros associated with screw speed since only one coefficient is estimated for

this input. In this figure, confidence intervals representing one standard deviation

around each of the poles and zeros are shown. The confidence intervals for the

model was chosen,

Pm (a') =

+
Figure 5.4:

2

1.5

I

._ 0.5

K

,<

0

--_ -0.5

. .

-1

-1.5 ........... : ........................ ........ :. _-..-__ ................

2 L _ i i i i i

"-2 -1.5 -I -0.5 0 0.5 I 1.5 2

RealAxis

Estimated poles and zeros with confidence intervals.

93

poles are extremely tight, indicating a very good fit of the denominator polynomial.

Those for the zeros are not quite as good, especially for the non-minimum phase

zeros.
94

>=

>

o_

4

3

2

1

0

-1

-2

-3

......Ii..............

........... ..............

-4

0 1

i .........

i ...........

...........

...... i

.... q i

i i ..... I .. •

i t

3 4 5 6 7

Time(s)

Figure 5.5: Process input produced by relay controller.

8O0O

Iii...............

4000 ...............

2000

]000 "";........................................................... i..................................................................................

i i i ,

°o i 3 5 ;

Time(s)

Figure 5.6: Comparison of process output and simulation output (simulation is dashed-line).
95

5.2 Plastication Controller Tuning: Simulation Results

For this study, two objective functions were chosen: minimize the integral time

absolute error [65] and minimize control saturation. The following equations were

used in conjunction with a discrete-time simulation of the process:

• Minimize Integral Time Absolute Error (ITAE)

7"1

ITAE = _ kiR(k ) - Pm(k-)l (5.7)

k=l

• Minimize Maximum Control Deviation

]] _"11 16 = V(]_) 16 (5.S)

Equation (5.8) is the "16-norm" and approximates taking the maximum value of

the control voltage over the interval but is a continuous and convex function, which

is numerically better conditioned [50].

5.2.1 Simulation Conditions

The simulation scenario was that the closed-loop system consisting of the

transfer function, Equation [5.6], and the controller, Equation [5.1], was given a

step input of 3000 PSI for 4 seconds (320 samples). The resulting system response

was evaluated with respect to the performance objectives described above. A two

level factorial design with center point was used for experimental perturbations to

determine gradients. For experiments run using this design, each parameter can

take on two distinct values, its nominal value plus the desired perturbation, and its

nominal value minus the perturbation. An additional experiment was run with all

parameters unperturbed: the centerpoint. This yielded a total of nine input level

combinations. This particular experimental design allows the estimation of all firstorder interactions between inputs. Since only linear terms were to be estimated,
96

this design also provided enough degrees of freedom so that "goodness-of-fit" of the

linear model could be examined. This information would help determine whether a

particular gradient estimation is valid [45, 5].

A root-locus design was done to find an initial set of stabilizing gains to be

used as an initial feasible point for the multiobjectiv'e optimization algorithm. Approximately 100 iterations of the algorithm were performed to map the tradeoff

boundary and associated efficient points shown in Figures [5.71,[5.s],[5.9],and

[5.10]. Each iteration involved determining the gradients of the objective functions,

determining a feasible descent direction using the linear program, and performing a

line search in this direction. Each algorithm iteration will evaluate the system over

a minimum of ten input level combinations, so many nondominated points can be

generated in a single iteration. It is important to note that nothing prevented the

algorithm from generating a set of gains which produced an unstable closed-loop

system. It relied entirely on the choice of the performance criteria and the fact that

small perturbations of gains were used to determine parameter changes.

5.2.2 Discussion of the Simulation Tuning Results

Analyzing the results of the optimization yields some interesting insights. Even

though the tradeoff curve is continuous, there seems to be two disconnected efficient

regions in decision space. This bifurcation corresponds to the point where ITAE

is 500 and the control norm is 0.6. This disconnected behavior of the efficient set

occurs in multi-objective optimization problems where the objective functions are

functions of ratios of the decision variables [43]. In the two different regions, there

is a completely different, parametric relationship between the decision variables and

the tradeoff surface. This behavior may not be anticipated by an operator. In each

performance region, a parametric relationship seems to exist between performance

and controller gains. This could be used to give the operator one knob to select
97

desiredperformancerather than havehim selectthree gains.

The tradeoff curveshowsa lowerlimit of 200 for the ITAE objective function,

no matter how much control is used. This provides a good idea of the marginal value

of allowing more control action during any one cycle. At the same time, an upper

limit of the control norm is 1.1. Physically, this means that any desired, achievable

ITAE performance can be met without any actuator saturation.

In this example, all efficient gains yielded an asymptotically stable closed-loop

system.
98

0

l

0

0

La

Figure 5.7:

1.2" , _ ", •

...... ,<i ..........................................

0.6 ......................i....z.l ..................................................... _........................................................

: I11

Ix

: x i

0,5 .................... _.................. i ...............t-.-t_x............... _........................................................

I11 1

0.4 .................................. '-.....................i.................. : ......._ _._ j. .................. ..........

1 I I 1 l • 1 1 1

0'2300 400 600 800 l_ 1200 1400 1600

Integral Time Absolute Error

The tradeoff boundary (nondominated set) found using the

multiobjective algorithm.

xlO-_

-1

4","':

-1.5 ........................................................................................................................................................

-2 ................................... !................................... _.......................................................................................

-2 ._ ................................................................ .C... ..............................................................................

-3.5.................................. ..... N ........................................

_4 ........................................................................ .................................................. _...............................

"4"5-2 ' -1.4_ -1'.2 -I' -0.8' -0.6' -0.4' .2

Ki Gain xlO-S

Efficient points: KI vs. Kp.

_t

i

-1.8 -I'.6

Figure 5.8:
99

5_ ,,

xxx

1 •

50

t xiz

x

zz

45 ............................ I. f_

x

4o ................... _.................!...................... i....................................... :....} .........

: : x z

35

3O

.... x x_ ...................... ! .....................

x

z

x

z x

................................................. _t .........

: 1

it

5

2:4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1

Kp Gain xl0 -_

Figure 5.9: Efficient points: Kv vs. Kp.

55

: xx :

zxi

i Its!

45 .................................................................... i............................................. _:...............................

ix x

35 _ _ i

I

z_

30 ............ :..................................... '/ -

x

:l

i

I , i , i

_.2 .;.8 -1.6 -;., -_.2 -i _'.8 _.6 -0._

IG Gain

Figure 5.10: Efficient points: KD vs. Kl.
100

5.3 Plastication Controller Tuning: Experimental Results

The procedure for experimental tuning of the controller was identical to that

of the simulation. In fact, the only implementation difference was that the Matlab

function which simulated the system was replaced by the Matlab function which

accessed the 68000 microprocessor, as described in Chapter 4.

The machine operating conditions used for these experiments are given in

Table 5.2. Plastication was initiated with the nozzle empty and terminated when

the desired shotsize was achieved. During plastication, data was collected using a

sampling rate of 80 Hz, the same as was used for the identification experiments.

The barrel temperature could be set independently in four zones. The temperature

setpoints are given in Table 5.2 sequentially from the nozzle to the hopper. All of

these parameters were held constant for the experiments. Any one of them could

affect the empirically-determined transfer function. These parameter settings represent typical values. In the future, it may be useful to develop a family of transfer

functions for different values of these parameters.

A typical injection session proceeds as follows. The operator enters the desired

initial operating point. The system then runs a series of injection cycles, with no

operator interaction, where this operating point is perturbed according to some

experimental design. After this experimentation phase, the performance data is

processed to yield the gradient information. The linear program is then solved to

determine if the current operating point is nondominated. If not, the algorithm

proceeds with the line search, generating new PID gains and running the process.

Assuming no problems occur which require operator intervention, the line search

continues as long as non-dominated points (with respect to known performance)

are generated. Once a dominated point is generated, the operator is presented

with information about the known nondominated operating points and associated

efficient points. In this example, with only two objective functions, this information
1Ol

Table 5.2: Nominal Input Settings

Polymer

ScrewSpeed

Barrel Temperature

Shotsize

Plastication Setpoint

Polypropylene

380RPM

200 C, 200 C, 200 C, 185 C

5.0 cm

3OOOPSI

is presentedasa plot of ITAE versusthe control effort (suchasFigure [5.7]or [5.11]).

The operator then selectswhich of thesepoints is the desiredoperating point. The

associatedgainsarepasseddown to the microprocessorcontroller and the machine

continuescycling. Heis alsogiventhe opportunity to allowthe algorithm to improve

on this point. If he so chooses,the algorithm will begin the experimentation/line

searchprocessagain.

In the initial phasesof running a particular process,the algorithm is used

to map out the tradeoff surface. This experimentation phase is used to gain an

understanding of possiblemachineperformance,suitability of the chosenobjective

functions, suitableexperimentalperturbations of the inputs, a suitable step-sizefor

the algorithm, and other implementation-specificdetails. Oncethe tradeoff surface

and efficient points areestablished,this knowledgeprovidesdiagnostic information

about the process.For example,if the systemcanno longer maintain a previously

achievableoperating point, or if the efficient points change, there may be some

underlying physical causewhich should beinvestigated.

5.3.1 Discussion of the Experimental Tuning Results

Figures [5.11]showsboth the experimentallydeterminedtradeoff boundary (x)

and the appropriate regionfrom the simulationresults (o). The processperformance

usedto initialize the algorithm is alsoshown(*). Figures [5.12]through [5.14]show

the associatedefficientpoints and the initial PID gain values.
102

The experimental session was started by initializing the process with the gains

shown by the "_'" in Figures [5.12], [5.13], and [5.14]. A set of experiments were

run, which were based on a two-level factorial design plus the center point. A level

change consisted of a -1-5% gain change. The result was that 9 experiments were

run, which were then used to evaluate the gradient of the performance measure with

respect to the gains. This constituted Steps 1 and 2 of the algorithm described in

Section 3.5.1. N" was initialized as _, the null set. Once the initial search direction

was found (Steps 3 to 7), the algorithm proceeded with the line search in gain space

(Steps 8 through 9). Once the line search failed, the current set of nondominated

points was presented to the operator. A new point of exploration was chosen and

the algorithm was restarted at Step 2. After approximately 200 injection cycles, the

performance tradeoff boundary shown in Figure [5.11] was produced.

In Figures [5.15] through [5.22], the process input and response is shown for

the initial point, as well as for three points along the tradeoff surface, marked A,

B, and C. a From a practical point of view, process performance for each of these

points is adequate, so there is no reason not to use the gains that yield the minimum

ITAE value.

The following observations can be made about the experimental results:

• The simulation and experimental results are in good agreement. The tradeoff

curves and PID gains have the same basic shape and values.

• The experimental results show a lower achievable ITAE value. This is due

to the fact that nonlinearities in the real system, which are not modeled by

Equation (5.6), allow a much faster rise time.

• In general, the control norm has a higher value. This is a result of the noisiness

of the actual signals. This does not affect the ITAE criterion because the

3Note: No relationship between run number and performance should be inferred from any of

the tradeoff surface plots.
103

pressure signal has a much better signal-to-noise ratio.

• When compared to the simulation results, much less of the tradeoff surface

has been "mapped". This is due more to practical constraints; the low control

norm/high ITAE system gains yielded a response that could possibly damage

equipment, so these regions were not explored.
104

0

Z

0

_9

Figure 5.11:

2.5

1.5 ................ _ .................

A

x

i ................... _--_Qc ............. i....................................................................................

_* ooOO_o: o .

................................ i.................. i.................................................................... ;.............

° °:- °°° OOOooooo i i

! °!oo Oooo:o

!

L J i i _ i

O0 200 4(0 600 800 1000 1200 1400 1600

Integral Time Absolute Error

A comparison of the experimentally determined tradeoff

boundary (x) with the "simulation" boundary (o).

0.5

.s

xlO _

-2 ......................................................................................................... _ ...............................................

-3 ................................................................................................ _ ....... L ....................................

o

i

-7 .................................................................................................................... 2...............................................

"f_.5 -; -_.5 -_ -;.5 -i -;._ o

IG Gain xl0 "s

Figure 5.12: Comparison of efficient points: KI vs. Kp
105

55

.=

50 .................................................................................................................................

45 ........................................................................................................... .-.................... _......... _-

i oo: _o

40 .......... !.......... . ..................................................... _ "o

i • _h_ g

a X

30 ................................................................................. ,............... " ......)............................ o

: i ::

20 .................... ;................ i................... i......................................... : ...............................

i

i i ,i I k i

158 -7 -6 -5 -4 -3 -2 -I

Kp Gain xl0 _

Figure 5.13: Comparison of efficient points: 1(1, vs. /(D

I

30

55

45

40

35

25

20

'.%

Figure 5.14:

50 ............................................................................................ _.................................. _,_........_....................

oo o_ .

o :

: l:.

: o o

.................................................................... i......._ °° "....... _:"......

............................................................................ _, .................................. ,................... ;g....................

-; -2Y -2 -;.5

Ki Gain

Comparison of efficient points: K1

0

xl04

VS. KD
106

0

...................... i .....................................................................................................................

t

50

Figure 5.15:

I i I

100 150 200

Samples

Control effort using initial gains.

250

4000

3500

3000

2500

i 1500

I000

500

°o s_ _o _o _o 250

Samples

Figure 5.16: Process response using initial gains.
107

w

3

.I

.....i.........i......i_ • i_

1 ............... i.............................. i...................! i

"30 50 100 150 200 250

Figure 5.17:

Samples

Control effort at tradeoff surface point A

4° 3500 t.............................. _................................ i............................... i................................ i...........................

2000 .......

'7/.................... :.......................i....................j.................:.............

'°If ....................:..................i................:.............:...........

' ;o ; '

0 50 1 1 0 200 250

Samples

Figure 5.18: Process response at tradeoff surface point A.
108

>

:i.........i.....I I

0

: i

I I I !

"30 50 lO0 150 2O0 250

-I

Samples

Figure 5.19: Control effort at tradeoff surface point B.

Figure 5.20:

100' 150' 200 250

Samples

Process response at tradeoff surface point B.
109

"2 t

-3

0

Figure 5.21:

L i

50 100 150 200 250

Samples

Control effort at tradeoff surface point C.

250o3°°°II| .............................. ...... " i _ i

1500 ......

i..iii.i'.

0

o _; ,60 1;0 :60 _o

Samples

Figure 5.22: Process response at tradeoff surface point C.
II0

5.4 Injection Controller Tuning

During injection, the accumulated molten polymer is injected into the mold.

It is during this phase that the part is actually formed. Physical properties of the

part, such as flashing, underfill, and initial polymer chain alignment are dependent

on how the injection takes place. Injection develops the initial molecular structure

of the part. Any final physical properties of the part depend on this initial molecular

structure and the thermal history during holding and cooling.

The multiobjective optimization algorithm was used to tune injection PID

controller gains, as was done for the plastication phase. For this tuning, three

objective functions were chosen:

• Minimize Integral Time Absolute Error (ITAE)

T!

ITAE = _ k]R(k) - P,_(k)l (5.9)

k=I

• Minimize Maximum Control Deviation

IlVll_ = v(k) _ (5.1o)

• Root Mean Square Error (RMSE)

1

RMSE = (R(k)-Pro(k)) _ (S._)

The first and second of these were the same as was used for the plastication controller tuning. The third objective was added to ensure faster rise time, which is

deemphasized by the ITAE because of the time-based weighting.

5.4.1 Analytical Modeling of Injection

The injection dynamics have the same basic subsystems as plastication: the

polymer fluid dynamics subsystem; the screw force balance subsystem; and the
Iii

hydraulic subsystem. The injection phase dynamics are dependent upon the mold

design, as is discussed in more detail in Appendix A. From the formulas derived

there, the following state space model can be formulated:

Zi

RA._3,._ A,,tZ3m 0

V,. V.,

Am _2 _ A_ll

rfl Tn rfl

0 A_z2z

VH V_

Z, + U, (5.12)

In the above equation, the injection state variable, Zi, is defined using the same

physical variables as Zp. Again, if the desired output is melt pressure, then

}'i=[ 1 0 o]Zi (5.13)

A block diagram of this system is shown in Figure [5.23].

Notice that this system only has one input, namely servovalve spool position.

The servovalve transfer function is identical for this phase, as for the plastication,

namely Equation (5.4).

Because the dynamics are dependent upon the mold, different controller gains

must be used for every mold. In the quality experiments, the results of which will

be presented in the following chapter, two different molds were used; a spiral mold

and a four cavity test specimen mold. Because of this, two different sets of injection

controller gains had to be found. For the spiral mold, it was found that a slight

modification of the plastication PID gains resulted in gains that provided adequate

control for injection. For the four cavity mold, these gains yielded a completely

unsatisfactory response.

5.4.2 Description of the Injection Controller Tuning Experiments

All injection controller tuning experiments were run using the ASTM four cavity mold. This was the second mold used for the quality control experiments, which

are described in Chapter 6. Process input settings for these experiments are given
112

Table 5.3: Symbols used in the injection dynamic equations.

AH

A._

b

Igm

Kxi

Kpi

]IQ

:m

Pn

I'm

R

u.

y.

i

!y...

y_

V

X_

y_

i z_

i b.,"rn

I DH

i

] %

hydraulic pressure effective area

melt pressure effective area

approximate viscous friction on screw

melt pressure empirical gain

servovalve spool position flow gain

servovalve hydraulic pressure flow gain

servovalve gain (voltage to spool position)

screw speed empirical gain

approximate mass of screw and polymer system

hydraulic pressure

polymer melt pressure

effective mold flow resistance

plastication input

hydraulic oil volume

melt volume

recovery rate (screw linear velocity)

servovalve input voltage

servovalve spool position

plastication output

plastication state vector

melt bulk modulus

hydraulic oil bulk modulus

servovalve time constant
113

F---{ H

v. F

Figure 5.23: Injection dynamics block diagram.

in Table 5.4. These values were held constant during all of the experiments. Injection controller gains and holding controller gains were varied by the multiobjective

algorithm. During these experiments, holding gains were set equal to injection gains

because holding dynamics will be the same as those at the end of injection. Keeping

the gains the same reduced the deterministic disturbances applied to the machine

to those only caused by changing the setpoint. It was desirable to minimize these

disturbances because anything which caused a pressure spike during injection could

lead to flashing of the part. The goal was to make the transfer from injection to

holding as "bumpless" as possible.

5.4.3 Experimental Results

After approximately 100 runs, the nondominated sets shown in Figures [5.24],

[5.25], and [5.26] were found. The associated efficient points are shown in Figures [5.27], [5.28], and [5.29]. In these figures, the points are lettered A through F
114

Table 5.4: Nominal Input Settings

Polymer

ScrewSpeed

Barrel Temperature

Shotsize

Pullback

Cushion

Holding Time

Cooling Time

Plastication Setpoint

Injection Setpoint

Holding Setpoint

Plastication Kp

Plastication Kt

Plastication ffz)

Polypropylene

380 RPM

220 C, 220 C, 220 C, 205 C

3.25 cm

0.5 cm

0.75 cm

5.0 s

10.0 s

3OO0 PSI

2500 PSI

2000 PSI

-4.13.10 -4

-1.76.10 -5

29.3

for ease of comparison. This lettering does not indicate any of the decision maker's

preferences or any time relationship between the different runs. It should only be

used to associate nondominated points with the corresponding efficient points.

Since there were three objective functions, it was not possible to plot the tradeoff surface as a smooth curve, as was obtained for the plastication controller tuning.

In this case, presentation of the nondominated sets consists of pair-wise plots. Each

plot represents the projection of the nondominated set into the subspace defined by

the axes of the plot. The general shape of the tradeoff surfaces in Figures [5.24],

[5.25], and [5.26] lend some insights into the behavior of the system.

During algorithm iterations, the ITAE and RMSE objectives were not in conflict, as can be seen by the shape of the tradeoff surface in Figure [5.24]. If it is

possible to get an approximately deadbeat response with the given controller, these

two objectives would produce the same tuning results. Any difference would lie in

tradeoffs that had to be made of rise time over steady-state error. One big difference in these two objectives was their respective dynamic range. Over the same
115

responses, ITAE doubled whereas RMSE changed by about 13%. This difference

can be attributed to the time-dependent weighting. Over all of the gains found, the

transient response was about the same, the system immediately shot up to about

3000 PSI. After this, for half of the trajectories, lots of control energy was used to

try and force the system back to the setpoint quickly. For the other half, the system

was allowed to slowly settle to the setpoint, but because injection was so short, it

was never reached.

By examining the efficient points in Figures [5.27], [5.28], and [5.29], it can be

seen that controller saturation is directly tied to the magnitude of I(p. This seems

reasonable since the overall controller gain and the position of one of the controller

zeros is dependent on Kp (see Equation (5.1). As I(p increases, the bandwidth of

the open-loop system is increased, forcing the system to react much more quickly.

Gains defined by point G in these figures were used as controller gains for the

quality experiments. The only points at which the controller did not saturate were

E, F, and G. Point G was chosen because these gains gave the least saturation for

no meaningful gain in setpoint tracking. Furthermore, if one compares Figures [5.30]

with [5.32], the "G"-gains yield a response which is much flatter. Since inputs will

be tuned in the quality experiments, even though, the setpoint is not tracked, this

can be compensated for by adjusting the "applied" setpoint to be lower than the

"desired" setpoint. This will be taken care of automatically by the algorithm during

the quality tuning since the operator will be controlling product quality directly.

This is an important feature of this approach to quality control. Specific machine

idiosyncrasies are invisible to the operator.
116

G

9500 .........................................................................................................

9400 ...................................... _...................... _........................................

9300 .........................................................................................................

E

F

9200 ........................................................................................................

._ 9100 ..................................................................................................................................................

9000 ....................................................................................................................... _ ..............

8900 .............................................................................................................................................

B

C

8800 .....o................................ '...................... ! ............... ".......................................... _ ................

8700 ............................................................................................................................................

8600 ^ i i i J i

1O0 120 140 160 180 200 220 240

ITAE

Figure 5.24: Nondominated set of ITAE versus RMSE.

O

z

Figure 5.25:

ation.

5.5

5.4 8C D _ , .......................................................................................

_°3 ................................................................................................................................................................

5.2 ........................................................................ .........

5. I ............................................ ,....................... _....................... _"........................................... _..........

E

5 ................................................................................................................ ?.................................. I_.........

4.9

4.8 ................................................................................................................................................................

G

t i

4"7oo 28o 240

ITAE

Nondominated set of ITAE versus maximum control devi-
117

0

z

Figure 5.26:

RMSE.

5.2

5.11

5

4.9

4.g

8600 8700 8800 8900 9000 9100 9200 9300 940O

5.5

5.4......................... _ .c. B

5.3............ _............... i........... _...................................................................

...... i ........ i

........................

........................................................................................... !.........

F :

........................................................... i_.......il.................... .:.........

................................ i ...............................

!

.......................... i.................. !........................................... :.....

:G

i

9500 9600

RMSE

Nondominated set of maximum control deviation versus

-3.3....................................................................

-3.7

-3.4.........................................................................................................._.....................

;A

:,7 -3.5..........................................................................................................

)

E

......................... o ..................................................... : .................

i ",c i i i i i

-2.4 -2.3 -2.2 -2.1 -2 -1.9 -1.8 -1.7

Kp xl0 -3

Figure 5.27: Efficient points: Kp versus Kt

-3.8

-2.5
118

6.4

6.3

6.2

6.1

5.8

5.7

5

I

"-"2.5 -2.4 -2.3

6.6 _t

6.5 ........................................ _....................................................................................

................................. i ................................................................................................ :............ _n,

i F

B

.................. _, .................. ?........................................................... e ................... . ........... E ..........................

;,

D:.

i I I | _ I

-2.2 -2.1 -2 -1.9 -I.8

Kp

Figure 5.28: Efficient points: I(p versus I(D

-1.7

x10-3

6.6 c ^ i

6.5 .............................................................................................................. i...................................................

6.4 -..................... L.G: .............................................

6,3 .................................................................................................................................................................

B

6.2 .................................................................................................................................................................

6.[ ..................................................................................................................................................................

6 ............................................................................................................ i .....................................................

5.9 ............................._ .........................................................................._...........................;.........................

" .......................................................i........................ i......................

5.7 ................................................................................. '...................................................... ! .........................

D

I

5"63.8 -3'.7 -31.6 -3.5 -31.4 -3.3i -3.2

Kd xl0 6

Figure 5.29: Efficient points: Kz versus KD
119

35O0

3000

25O0

2OO0

r,

s.

_- 1500

5O0

Figure 5.30:

!

• i

J

i I i i i

00 20 40 60 80 100 120

Samples

Melt pressure output at nondominated point A. The setpoint is 2500 PSI.

O

3

2

1

0

-1

-2

-3

-4

-5

........... I ................................................................................................................................................

40 60 8 100 120

Samples

Figure 5.31: Servovalve input at nondominated point A.
120

35OO

3OOO

2500

2000

o

=- 1500

1000

500

Figure 5.32:

............... 17 .............................................................. ".....................................................

/

' ' ' '0 4_ '0 6_ ' 10 20 3 5 70 80

Samples

Melt pressure output at nondominated point G. The setpoint is 2500 PSI.

0

o -2

p,

-3

-4

-5

0

I

I

1

--.4 ....................

I

Figure 5.33:

1'0 20 3'0 40 5'0 60 7'0 80

Samples

Servovalve input at nondominated point G.
121

5.4.4 Discussion

In general, it was found that this system was not nearly as well behaved as the

plastication system. Some discussion of the behavior of the system will lend some

insight into the problem.

The actuator for this system is a three port servovalve. This valve has two

stages, a spool valve which regulates the hydraulic flow through the valve, and a

pilot stage which drives the spool. The pilot stage can be thought of as the hydraulic

equivalent to a "pre-amplifier". This pilot stage is actuated by an armature which

is voltage driven. Ideally, if the valve is balanced, an input of zero volts results in no

motion of the spool. This voltage is known as the null voltage. The valve used on the

Arburg was not balanced, and due to the valve's integrating action, the spool would

tend to drift to the completely closed or completely open position. Attempts were

made to identify the null voltage, but this proved to be fruitless. One assumption

was that the null voltage was the control output during plastication, when steadystate error was zero, but this did not prove to be correct. Apparently the null voltage

is dependent upon the spool position, which is not unreasonable 4

A second problem with the valve was stiction of the spool. In previous experiments, the valve was repeatedly given various open-loop inputs, to try and

characterize valve response, but no clear relationships could be determined.

A third problem was slew limit and position limits of the spool. No matter

how large an input was given to the valve, it could open only so fast and so far.

This also limited the ability of the valve to react to large command signals. These

effects were lessened by applying an empirically determined preset signal to the valve.

This was an attempt to position the spool near it's steady-state operating position,

hopefully making the valve behave "more linearly". This approach was successful

for the plastication phase. These problems were probably not as significant during

4A complete discussion of servovalve dynamics can be found in [62].
122

plastication becausethe demandsmadeon the servovalvewerenot asgreat.

The injection dynamics also contained a backlashnonlinearity with respect

to the melt pressuresensor. After plastication, the polymer is "decompressed"by

pulling the screwback slightly. This results in the polymer being pulled out of

contact with the melt pressuretransducer. During the initial stagesof injection

(typically the first eight samples),thereis no pressuresignal, causingthe controller

to maintain a "full on" signal. When the melt pressuredoes start registering, a

very fast rise time is detected which the controller must counter-act. Becauseof

the slow valve dynamics, due to the integrator, this reaction does not take place

quickly enough,resulting in a large overshootby the system. Before the controller

can recover,injection is over. (In Figures[5.30]and [5.32],there is a pressurerise

nearthe end of the cycle. This occurswhenthe mold is full. The injection dynamics

changesuddenly becausethere is no longera polymer flow. Ideally, switchoverfrom

injection to holding would occurpreciselyat this point.)

It wasalsofound that the injection screwitself wasrate limited. This problem

did not occurwhenusingthe spiral mold becausethe typical polymer flow path cross

section wasmuchsmaller. This causeda higher "flow resistance",which translated

into a higher melt pressurefor a givenvelocity. In the four cavity mold, the flow

resistancewas roughly one fourth that of the spiral mold becauseof the many

runners. At low injection pressures, the maximum screw velocity of 7 cm/sec was

easily achieved. Essentially, the system could not track setpoints above 3000 PSI.

The typical response, in this case, was a trajectory that ramped up to the setpoint

but might not have reached it by the time the mold was filled.

To put these problems in perspective, consider that an injection molding operator typically does not have the process data available to detect these control

problems. If process performance indicates that he should be increasing injection
123

pressure,hewill do this without knowing that there is "poor tracking" of the command. The only indication he may have is that abovea certain point, the process

output is insensitive to any changeshe makes. This information is built into the

operator's intuitive processmodel, which is derivedfrom experience.

Despitetheseproblems,a reasonableset of controller gains werefound. These

wereusedfor injection control during the quality experiments,that will bediscussed

in the next chapter.

5.5 Conclusions

In this chapter,resultson using the multiobjective optimization algorithm for

controller tuning werepresented.The algorithm wasusedto tune a PID controller in

three different situations; during a simulation of plastication, on-line for plastication,

and on-line for injection.

This particular implementationof controller tuning hasa very interestinganalogy to root-locus design. Oncea feasibledirection is determined by the algorithm,

the gainsare parameterizedalongthis direction and changedlinearly; this is the algorithm line searchphase.Performanceof the controller is evaluatedat eachstep of

this line searchwith the hopethat it is beingcontinually improved. The Evansroot

locus provides location of the closed-looppolesof the system as this gain is varied

linearly. The construction of an Evansroot locusrequiresa very specificparameterization of the controller, which does not occur with the line search in the algorithm,

but both show a continuous relationship between the closed-loop response and the

free parameter.

In general, the construction of the tradeoff surface, by the algorithm, for each

of the systems yielded information that was valuable in understanding the approach

to control. In the case of the plastication phase, the nondominated set revealed

that the PID controller provided quite good response. In fact, the efficient gains
124

from tradeoff surfacepoint A in Figure [5.11] provide practically deadbeatcontrol

to the input (this responseis shownin Figure [5.18]). Practically speaking,any of

the nondominated gainsfor this systemprovide very good control.

Conversely,achievablecontrol during injection is worse. Severalconclusions

can be drawn from this. One is that a different controller might possibly provide

adequate control. It is also possible that the actuator is insufficient for the system.

It should be noted that, even though control was lousy, if %etpoint-tracking" was

the major performance objective, a set of controller gains was found that produced a

machine response that had desirable response characteristics, namely being fairly flat

after the initial transient. Since the multiobjective optimization approach uses both

command inputs and control gains simultaneously, this could be taken advantage

of.

In all of the cases discussed, the multiobjective algorithm provided a very

methodical way of obtaining information about the relationship between system

response and controller gains.
6. Tuning the Process to Achieve Quality Objectives

The multiobjective optimization algorithm was applied to the tuning of process

inputs to achieve quality objectives. In this chapter, two different cases will be

discussed; a spiral mold and an ASTM four cavity test specimen mold. Each of

these molds present different control problems and have different quality objectives

associated with them.

6.1 Spiral Mold Quality Tuning

One of two molds used in a quality control case study was a standard spiral mold. The mold consists of a constant cross-section channel arranged in an

Archimedean spiral. The spiral is marked in one inch increments and has a total

length of 65 inches. The actual length of a part during any given injection cycle is a

function of the polymer viscosity, mold cavity temperature, the melt temperature,

and the injection pressure and velocity [66]. Figure [6.1] shows an end-on view of

the spiral part. From this vantage, the sprue is projecting out of the page.

6.1.1 Description of the Spiral Mold Quality Tuning Experiments

For this part, optimization was to take place with respect to the following

quality objectives:

• prevent flashing (quality attribute);

• minimize cycle time (quality variable);

• minimize the variance of the spiral about a nominal length of 35 inches (quality

variable).

125
126

Figure 6.1: The spiral part.

Flashing and spiral length were measured by operator inspection. Each time a part

was ejected, the presence or lack of flashing was noted and the length of the part

was measured. This information was then input to the algorithm and the cycle

was continued. Cycle time was measured as the elapsed time of plastication and

injection and was recorded automatically. All other phases of the cycle were of fixed

duration.

In running these experiments, it was decided that only plastication and injection setpoints would be varied. The philosophy behind this choice was that these

are inputs that are typically controlled by an operator. It would be possible to vary

controller gains as well, but it was felt that this would make the experimentation

unnecessarily complex with no added benefit. (One alternative would be to run

the algorithm sequentially, first finding the desirable operating points, then tuning

the controllers to optimize performance around these points. This would, to some

degree, require that the system response to setpoint changes be "orthogonal" to the

response to gain changes.)

The algorithm was initialized with a plastication setpoint of 4000 PSI and

an injection setpoint of 10000 PSI. All other process inputs were held constant,
127

at levels shown in Table 6.2. The algorithm was then used to determine those

setpoints that would minimize all performance criteria simultaneously. Since there

were only two inputs, a two level factorial design with center point was used for

experimental perturbations to determine gradients. For experiments run using this

design, each parameter can take on two distinct values, its nominal value plus the

desired perturbation, and its nominal value minus the perturbation. An additional

experiment was run with all parameters unperturbed: the centerpoint. This yielded

a total of five input level combinations. Each input level combination was replicated

four times so that deleterious effects due to process noise would be minimized.

Therefore, twenty runs were required when a gradient had to be determined. To

confound other unmeasured interactions, input levels for sequential experiments

were chosen randomly [45]. The initial perturbation value was to vary each setpoint

by 10%. Table 6.1 shows a possible randomized sequence of the 20 experiments.

During experimentation, a new random order was chosen every time.

Randomization of experiments is important. In this case, the gradient of the

performance objectives was being estimated. Essentially, this is the estimation of

a linear, static model. It is important to make sure that only effects due to the

desired inputs are measured in the output. A prime example of this type of problem

is shown in Figures [2.33] and [2.34]. The controller response was highly correlated

to hydraulic oil temperature which wasn't measured or controlled by the system.

Randomization of input changes help prevent this type of deterministic disturbance

from affecting the gradient estimation significantly.

Once a gradient was found, the line search was begun. During the line search,

each run was replicated four times. Since it was not possible to randomize these

experiments, data from these experiments were susceptible to the interaction that

was avoided when estimating gradients. The line search was continued until a dominated point, in objective space, was generated. This indicated that either the
128

Table 6.1: A Sequence of Randomized Runs.

Run Plastication

Setpoint

I 3600

2 4400

3 3600

4 4400

5 4400

6 4000

7 4400

8 4000

9 4O00

I0 3600

ii 4000

12 3600

13 3600

14 3600

15 4400

16 3600

17 3600

18 4400

19 4400

20 4400

Injection

Setpoint

II000

9000

9000

ii000

II000

10000

9000

10000

10000

9000

I0000

ii000

9000

II000

9000

9000

llO00

11000

9000

llO00
129

Table 6.2: Nominal Input Settings for the Spiral Mold

Polymer

Screw Speed

Barrel Temperature

Shotsize

Pullback

Cushion

Holding Time

Cooling Time

Holding Setpoint

Plastication/(p

Plastication Kt

Plastication Kz)

Injection K.

Injection KI

Injection KD

Holding/(p

Holding Ki

Holding KD

Polypropylene

380 RPM

220 C, 220 C, 220 C, 205 C

2.5 cm

0.25 cm

1.0 cm

5.0 s

10.0 s

3000 PSI

-4.13.10 -4

-1.76.10 -s

29.3

-4.13.10 -4

-1.76.10 -s

40.0

-4.13.10 -4

-1.76.10 -s

40.0
130

tradeoff surface had been reached or the current direction no longer pointed to this

surface.

6.1.2 Experimental Results

Process inputs generated by the algorithm are shown in Figures [6.2] and

[6.3]. Those runs which were used for gradient estimation, and those used for the

line search are easily distinguished. The resulting cycle time and spiral length are

shown in Figures [6.4] and [6.5]. The initial inputs yield both a long cycle time

and a spiral that is too short. The gradient identified by the algorithm directed

the system to decrease the plastication setpoint and increase the injection setpoint.

Both of these resulted in a faster cycle. At the same time, as the injection setpoint

was increased a longer spiral was produced. At input levels for Runs 40 to 44,

the spiral was too long. The algorithm was stopped because this operating point

was dominated by one that had already been determined. The operator was then

presented with the empirically determined nondominated set and efficient points

depicted in Figures [6.6] and [6.7].

The operator selected a new region on the tradeoff surface to explore, by

choosing a point from the experimentally generated non-dominated set. In this

case, it was point A shown in Figure [6.6]. Automatically, the corresponding efficient

operating points noted in Figure [6.7] were transmitted to the 68000 microprocessor.

The objective function gradients were again determined through a series of planned

perturbations from the new input settings. This series of experiments is depicted in

Runs 45 through 60 in Figures [6.2] and [6.3]. In this instance, only one step along

the feasible direction was made before the tradeoff surface was encountered. The

resulting nondominated set and efficient points are shown in Figures [6.8] and [6.9].
131

4200 = --.... =:

o oo:o

4100 ....................................

: i

:

-- 401)0 ...... a-.oo,. .............. : .................................................. ÷ .................................. 4................. : ............

! oo : o o0o +o

39130 ................................ :+*_i.......................................... +................................ :................. i........

'_ 0<3 0

3800 _.o.......... -+.+++oo............ _............................................................................ i.........

i ioooo .oooo

3700 i.......... oooo ..........

o_o

o o oo oo

:1 I i i ooo i i

36000 10 20 30 40 5; 6}) 70 80 90

Rtul

Figure 6.2: Plastication inputs for spiral mold optimization.

2=

1.1

1.05

1

0.95

0

xl04

1.35 : ,.o

! .3 ................................................................... _,_ .....................................................................................

i oooo

]._ ...................................................... _..................................................................... _...............................

ioooo o oo: o o :oo

1.2 ............................................................

o oo: o oo 9oo0

ooo0

: oo io o ooe

1.15 ................................. ::.................. :..................................................................... +...................................

oo o

................................ .i'+*++ ............................................ 7 ................. :................ -..................................

: o. :o ooo o:

e.. l_o-.......... o.<l.--.oo, o ...................................................... ;......................

:oooo .......... <...............................

-- .o+.oo:.o ............... : .......................................................................................................................

=: : 10: .... 20 3'0 40' 50' 6; 7'0 10 90

Rull

Figure 6.3: Injection inputs for spiral mold optimization.
132

3

._+

+-

(J

4.1

o

4 ............................ o ..................................................................

o

o i o

3.9 ................ +................................................................. ................................................................

o

3.8 ............ i.°............ .°..i ........

ooO o+ o ..............................................................................................................

o

o

3.7 ....................... ................................................................... o_ .....

o o

3.6 ......................................... o ............................................................. o............... _ .............................

o i i

3.5 ............................ o...i_% ............................................... i_ ............. i ........ +....i.......... o.._ .............

0 . 0 O_ ! !

0o 0 ° :. o o oo° o +o

• o _ o

3.4 ........................................... o"":_ ............. _................................ %_.......................................

i o :o o!

o io+ o ooo

3.3 • ,o ............................ : ............................................................................................ i........ o .o_.

: oo o : o

i :. !

I i i 0 1 0 I + I

3"20 I0 20 30 40 50 60 70 80 90

Run

Figure 6.4: Cycle time as the algorithm progressed.

d_

.2

"E.

38 _ i i

i o

37 .................................................................... _'-.:................. :- , .. -

o: :

o : o

i . • . °

36 ................................................................ + """io............................ +i.............. +..................... "...............

o o o 4 o _ o o :°o

• o: : : o

35 ................. :........................... _.0...,¢ ................ i ............. +"7................. _............. a_:_.................................

_ 9 io o

i i o i o o° oz o o

34 ............................................ P'-'-"; .................. i............... :................................................... _i...........

• ' o " o oi o i "

o o o o o _l

33 ...... +......o._..................... o.............. !o........................................................................................ _.................

o • i

G, "o :o • • i

32 "............. 4,............... !........................................................................................+.................

o o

• •

31 .................................................................................. o........ o...............................................................

o

30 • +

oo :

o

29 -.-,_........... :........ o........: .........................................................................................

2go l; 2; 3; 5; 40 ,'o

Rull

Figure 6.5: Spiral length as the algorithm progressed.
133

.ePareto Set

4 J,

3.5

3 .......................................... _ ........ i......................................................................... _ ............

2.5 ............................................................................................................................................

o

1.5

1 , ,

3.24 3.26 3.28 3.36 3.38 3.4

Figure 6.6:

L i i

3.3 3.32 3.34

Cycle Time

Nondominated set after first line search.

x I 0 4 Efficient Points

1.35

o

1.3 .............................. i...o ......................................................................................................

!

: o

1.15 ................................................................................................................................................

m

c

"_. 1.25

•=_ ! .2

I.I i

3600 3650

Figure 6.7:

i i i

3700 3750 3800 3850

Plastication Setpoint

Efficient points after first line search
134

4 i

Par_to Set

.=

>

Figure 6.8:

3.5

2.5

1,5 .... .................. _.................. r .............

o

i i i 3.25 3 3 3.35 3.4 3.45

Cycle Time

Comparison of first (o) and second (*) nondominated sets.

Figure 6.9:

sets.

xl04 Efficient Points

1.35

1.3 .................... e.................................... !...................... !...................

1.25 .......................................................................................................

1.2 .......................................................................................................

1.15 ...................................................................................................................................................................

I.I .................... _....................... !................_........................ _ ................

1"025600 3660 37'00 3450 38'00 3850

Plastication Sett_int

Efficient points for first (o) and second (*) nondominated
135

6.1.3 Discussion

The overall behavior of the algorithm seems to improve the performance of the

plastic injection molding process. Figures [6.4] and [6.5] show that there is a steady

decrease in cycle time and the target spiral length of 35 inches is being approached.

Also, the nondominated sets, shown in Figures [6.8] are beginning to take on the

characteristic "hyperbolic" shape.

Close inspection of the figures shows that some levels do not appear as if

they were replicated. In these instances, such as near Run 24, there was a control

computer failure that resulted in unusable data being collected. After Run 87, the

control computer failed completely.

Achieving the quality objectives of this mold was not pursued after these runs

because it was desirable to see how the algorithm operated on a quality attribute,

which this part did not contain. The problem is that, except for some uncontrolled

runs, it was not possible to cause flashing of this part. This is essentially due to

the mold design. In general, flashing occurs when the injection pressure exceeds

some threshold, after the mold is filled. When this happens, the mold is forced

open slightly, allowing the polymer to flow into this thin opening. In the case of the

spiral mold, it is "open-ended"; with a total length of 65 inches. It is impossible

to completely fill the mold. Any "over pressuring" that occurs merely results in a

lengthening of the spiral, rather than flash.

During experimentation, a second quality problem was observed; the occurrence of shrinkage voids. This also could not be removed, because of the mold

design. These voids occur when the plastic shrinks as it cools, allowing them to

form inside the body of the part. Formation of voids is avoided by maintaining

pressure on the polymer melt, during the holding phase, allowing more polymer to

flow into the mold as shrinking occurs. Since this mold is open-ended, maintaining

this pressure is nothing more than continuing injection except at a lower pressure.
136

6.2 ASTM Four Cavity Mold Quality Tuning

Quality experiments were run on the ASTM four cavity mold. The part from

this mold is pictured in Figure [6.10]. Quality objectives for this part were:

• Minimize cycle time (a quality variable).

• Prevent flashing (a quality attribute).

• Prevent underfill (a quality attribute).

The first two objectives were also used for" the spiral mold. The last objective,

underfill, represents the condition where an insufficient amount of polymer has been

injected into the mold such that the part does not fill completely. It is possibIe

for both flashing and underfill to occur simultaneously, possibly when the molten

polymer is too cold s'uch that the mold freezes off before injection is complete and the

remaining polymer flashes. Information about the quality attribute was provided via

operator inspection. After each part was ejected, the operator was asked to decide

whether the part had flashing or underfill. Two different approaches to inspection

were tried. One approach was that a threshold was defined and the operator was

allowed to answer either yes or no. For the second approach a fuzzy definition of these

attributes was used, where the attribute was from zero to one, and this information

was passed to the algorithm. Although this strictly violates the definition of a quality

attribute, as described in Section 3.2.2.2, the use of a fuzzy measure improved the

performance of the algorithm. The use of fuzzy measures for these constraints is

analagous to the use of penalty or barrier functions in optimization problems with

analytic constraints [55]. (It should be noted that, because of replications, even

non-fuzzy measures of flashing and underfill can result in a quality attribute value

that is between zero and one.)
137

Figure 6.10: The ASTM test part.
138

6.2.1 Experimental Conditions

An additional quality variability wasevaluatedfor feasibility, that of minimizing part weight variance. This would be another measureof uniformity of parts,

and could possibly be related to consistencyof part density. It turned out that for

thoseparts which showedno flashingor underfill, the weight variability wason the

order of a hundredth of a gram. This weight variation was consideredtoo small to

be meaningful and so this objective wasdropped.

For theseexperiments,controller gains were held constant, using the values

determined from the controllertuning experiments.As wasdonefor the spiral mold,

only the plastication andinjection setpointswerechanged.The initial setpointswere

4000 PSI for plastication and 2500 PSI for injection. Also, each experimental level

was replicated four times, and randomization was used in determining the gradients,

for the same reasons as discussed in Section 6.1. All of the machine inputs are given

in Table 6.3.

6.2.2 Discussion of the Tuning Results

In Figures [6.11], [6.12], and [6.13], the non-dominated set found from the initial point is shown. Interestingly enough, underfill and flash are mutually exclusive

for this part. Apparently there is no problem with the temperature of the polymer for these runs. The nondominated sets of Figures [6.11] and [6.13] have the

characteristic hyperbolic shape one expects of a tradeoff surface. This is because of

the "orthogonality" of flashing and underfill. It is especially easy to identify which

parts violate the flashing objective in Figure [6.13] because they all appear on the

zero-underfill axis, but with a faster cycle time than is possible for any of the parts

with underfill. (This is another indication of the relationship between mold filling

and injection pressure.)

At the end of thirty search iterations, the operator was presented with these
139

Table 6.3: Nominal Input Settings for the ASTM Mold

Polymer

ScrewSpeed

Barrel Temperature

Shotsize

Pullback

Cushion

Holding Time

Cooling Time

Holding Setpoint

Plastication Kp

Plastication Kz

Plastication K_

Injection Kp

Injection KI

Injection KD

Holding Kp

Holding Kz

Holding /(D

Polypropylene

380 RPM

220 C, 220 C, 220 C, 205 C

3.25 cm

0.5 cm

0.75 cm

5.0 s

10.0 s

Inj. Setpoint - 2000 PSI

--4.13. 10 -4

--1.76. 10 -s

29.3

-4.13- 10 .4

-1.76- l0 -s

40.0

-4.13.10 -4

--1.76.10 -s

40.0
140

non-dominated sets. The point denoted by A in Figure [6.11] was used to initiate the

algorithm for the second iteration. The corresponding efficient points are marked in

Figure [6.14]. This nondominated point was chosen because it exhibited no underfill

and the smallest amount of flashing. The approach was to try and satisfy the quality

attribute constraints before minimizing the quality variable.

An additional thirty search iterations yielded the nondominated set and associated efficient points shown in Figures [6.15] and [6.16]. After this iteration, there

was no underfill, whatsoever.

Evaluation of the data from these runs showed a curious phenomenon: as

plastication setpoint decreased, injection stroke increased. This increased stroke

was not correlated with an increased part weight or other physical characteristic.

It can only be surmised that the check valve in the plasticating screw, which is

supposed to prevent back flow of plastic, was not seating itself and therefore was

allowing leakage. This may also be a function of the low plastication pressure and

high injection pressure. In these experiments, a minimum setpoint constraint was set

to be 1000 PSI. This was increased to 2000 PSI and an additional set of experiments

were run.

An initial nondominated set was constructed from the data of the previous

runs. The algorithm was initialized at a suitable point and was allowed to run

through two iterations. The resulting nondominated sets and efficient points are

shown in Figures [6.17] to [6.20]. The algorithm was allowed one more iteration,

and the final nondominated set and efficient points are shown in Figures [6.21] to

[6.22]. Again, operating conditions were found such that underfill was removed

entirely.

Figures [6.23] and [6.24] show the input changes made during the last three

iterations. From the initial point, the plastication setpoint is decreased and the
141

Figure 6.11:

0.9............................................................................. i.......................... i....................

0.8 ........................................................................................ i......................... _......................

0.7 ........................................................................................... _........................... i ....

0.6 ............................... _.......................................................................................... r...........................

0.5.............................. i......................................................................!.............................

0.4 ............................. !..................................................................................................................

0.3 ..........................._ .............................................................................................................

0.2 ................................................................................. _.......................... !..................

0.1 ....................................... ,,.......................................................................

O: A

: o

t • I • s o

2.5 3.5 4 4.5

Cycle Time

Nondominated set of cycle time versus flash, after the first

algorithm iteration.

Figure 6.12:

I t

0.9 .........................................................................................................................

0.8 ....................................... ,................... _................. _..................................... :................... _ ..........

0.7 ..............................................................................................................................................................

0.6 ........................................................................................................................

0.5 ........................................................................................................................

0.4 ............................................................................................................................................................

0.3 .........................................................................................................................................................

0.2 ........................................ '.................... '.................... _.................... e................... _................... !

0. I ...........................................................................................................................................................

O0 0.1..... 0.2 0.3 0.4 015 0.6.... 0.7 0.8

Underftll

Nondominated set of underfill versus flash, after the first

algorithm iteration.
142

Figure 6.13:

0.8

0.5

0.4

0.3

0.2

O.l

o

0.7 ........................................................................................................

o

0.6 .........................................................................................................................................................

o

......................................................... ........................... j........................................................

........................ !........................ :............................................................................

!

. . i. . .i . , , ,

.5 3 3.5 4 4.5 5

Cycle Time

Nondominated set of cycle time versus underfill, after the

first algorithm iteration.

44OO

4200

4OOO

38OO

.__

_ 36170

46011 ̧ !

• :

o!

• 0

A

3400 .................................................................................. *"............................................................................

3200 ..................................................................................................................................................................

o

30(10 ...................................................................................................................................................................

o

2800 .................................................................................................................................................................

o o

2_00 !0_ 15'03 2000 2500 30_0 35'00 4000 4500

Plastication

Figure 6.14: Efficient plastication and injection setpoints for the nondominated set.
143

Figure 6.15:

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

................. i................. !................................... i................... !................ !................... !...........

................................... i ...............................................................................................................

o

, i i i i _ t Idl .8 2.85 2.9 2.95 3 3. 5 3.1 3.15 3.2

Cycle Time

Nondominated set of cycle time versus flash, after the first

algorithm iteration.

42130

4000

3800

.__

_ 36O0

Figure 6.16:

3400

3200

3000

2800

2600

4600

4400 ............................................................................................................................................................

O!

!

............................................................................... !-e. ......................................................................

........................ •"-....................................................... •........................................................... _.................

• o

o

• o

500 1000 1500 2000 25 3 3500 4000 4500

Flastication

Efficient plastication and injection setpoints for the nondominated set.
144

injection setpoint is increased to satisfy all objectives simultaneously. Once a dominated point was generated, a new efficient point was chosen, and iterations began

again. In this instance, and the following, one, both setpoints were decreased, but

with injection being decreased at a much slower rate. In the final iteration (beginning at Run 74), the injection setpoint is held constant and only the plastication

setpoint is changed. This "steady-state" injection setpoint represents a value where

neither underfill nor flashing will occur. The algorithm uses the remaining degree

of freedom, the plastication setpoint, to decrease the cycle time.

An additional observation involves the treatment of qualit 9 attributes. For the

ASTM mold trials, two different techniques were tried for rating these attributes:

• A crisp measure was defined for each of the attributes. In this case either the

attribute constraint was violated (given a value of "1") or it was not violated

("o").

• A fuzzy measure was defined for each attribute. In this case, the operator acted

as inspector and assigned a value to the severity of the attribute constraint

violation.

The algorithm still managed to find the tradeoff surface in either case. The difference was in how quickly this was found. The fuzzy measures gave better information

to the algorithm when it was necessary to determine a direction. Crisp measures

only worked when the perturbations, used to determine the gradients (algorithm

Step 2), "straddled" the boundary between violation and no violation. (In normal

constrained optimization, this could be likened to an inequality constraint being

suddenly activated.) In general, the trials where fuzzy measures were used involved

fewer iterations to "get the process out of trouble". One caveat: the fuzzy measures,

as implemented, took on a finite number of values between zero and one. If perturbations only caused the process output to fluctuate between two of these finite
145

values, the fuzzy measure performed no differently than the crisp measure.
146

0.03

0.0_

0.05 .* . , . !

0.045 .......................................................................................... Z.............................................

0.04 .................................................................................... _................................................

0,035 ............................................................................................................................................

................................... ! ................... ' ................ i.................... e................... _ .................... : ..............

........................ B........ t ...................................... i .............................................................................

0.o2 ................................. :_..........................................................................................................

o.o15 ................................. i..................z.......................................................................... _.............

o.oi .................................. i.................... i................ i................... ' ......

0.005 ........................................................... :................................................. : ..........

.4 3 5 ., 3.6_ 3.7i 3.8i C3.9I 4i D 4.1i £ 4.2

Cycle Time

Figure 6.17: Nondominated set of cycle time versus flash, after the second algorithm iteration, under the new setpoint constraint.

0.05_ , , ,,,

0.045 .....................................................................................................................................................

0.04 ..................................................................................................................................................

0.035 ...........................................................................................

0.03 ...........................................................................................

0.025 ...............................................................................................................................................................

0.02 ............................................................................................................................................................

0.015

0.011

0oo51

°o 0:1 01

!

, i I G I I D I C

0.3 04. 0.5 0.6 0.7 0.8

UndenC'dl

Figure 6.18: Nondominated set of underfill versus flash, after the second

algorithm iteration, under the new setpoint constraint.
147

Figure 6.19:

0.8 i

cl

0.7 .................................................................................... !.

0.6 ................................................. !...................

0.5 ..........

! D

E"

0.4 .................................................................................................................................. : ..............

0.3 ...................................................................................................................................

0.2 .........................................................................................................................................

0.1 ........................... i ........................... i ...........................................................

A L it i t i t I I I

.4 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2

Cycle Time

Nondominated set of cycle time versus underfill, after the

second algorithm iteration, under the new setpoint constraint.

3600

3500

3400

33O0

.__

'_ 3200

Figure 6.20:

.................................................................................................................................. ..................

3100

C

3000 ............................................................................................................. ..................................... E2900 .................................................................. i...................................................................................

: D

i

oo 28 ,_,;00 24 2 2800 3000 3200 3400 3600

Plastication

Efficient plastication and injection setpoints for the nondominated set, under the new setpoint constraint.
148

0.08

0.07

0.06

0.05

.__

_ 0.04

0.03

0.02

0.01

0

3.29

................................................ !................................................ . ......................................

z i

....................... 4.................... i..........................................................................................

i i i _ IA

3.3 3.31 3.32 3.33 3.34 3.35

Cycle T_ne

Figure 6.21: Final nondominated set of cycle time versus flash.

2950

2949 .............................................................................................................................................................

2948 .......................................................................................................................................................

B ^

2947 .........................................................................................................................................................

2946 .......................................................................................................................................................

.__

't_ 2945 ...............................................................................................................................................................

u

"__

2944 ...........................................................................................................................................................

2943 ...............................................................................................................................................................

2942 .......................................................................................................................................................

2941 ...................................................................................................................................................

2940

2080

Figure 6.22:

' ' ' '6O '8 '00 ' 2100 2120 2140 21 21 0 22 2220 2240

Plastication

Final efficient plastication and injection setpoints.
149

45O0

4OOO

_. 3500

N 3ooo

_ 2500

20OO

o o o-o o oo o:

............ i............... ; ........... _................: ........

.o i :

o

o_o :o ooo

oooo i o

° _i° i

:0 o .......... 0 oo0 oo ...................................

Z o ooo o

................................... i.ooo ......... o.o._.o_........

: o

o

o

o i ocx_o

oo: oo_oo

: o o

o

o o! ,

0

: i

i / I I i t

l0 20 30 40 50 60

Run

1500

0 70' 8 'o 90' I00

Figure 6.23: Plastication inputs for ASTM four cavity mold optimization.

3800

o

3600 .................................... a "o'"_ .......... aa ;a'a ........................................................................

3400 ................................. _ ...... !..o._o...................................................................................

°Oo_ i

3200 o :o +,, . o

oo: o o o!oo o

¢:

"_ • oo oo ooo

o o oo+o

3000 .. • '

............................. +.................................................................. :................ :o ............................. :............

O"J : o o:ooo : _oo

:o . . •

"- _ _ _ Z

,, 2800 ............. ;................ ;................................................................ _.............. ;....... ,,.......o...,.oo,. o ......

• _", oo Imooo o !

2600 ............................................................................................... :.............. :..................................

o, oi oi

2400 .............. _.............. i ...................................................................................................................

O0 00: 0000

22000 10 20 30 40 50 7 80 100

Run

Figure 6.24: Injection inputs for ASTM four cavity mold optimization.
150

6.3 Conclusions

In this chapter, two case studies were presented where the multiobjective optimization algorithm was used for quality control. For each of the molds, quality

objectives were defined and the algorithm was used to find inputs which were a

compromise between all of these objectives.

The main difference between these cases and those discussed in the previous chapter was the presence of attribute constraints. They served to define the

acceptable region of quality space, within which the quality variables would be optimized. These attributes presented the problem that they had to be integrated

into an algorithm which depended upon estimating objective gradients. Using the

experimental perturbations, it was possible to estimate a direction that was normal

to the constraint boundary and use this with the quality objective gradients to define a direction that was used to improve the process output. A further refinement

was implemented, where a fuzzy measure was used to evaluate these constraints and

generate the direction normal to the boundary.

These constraints can be thought of as defining a boundary separating two

regions. Far away from the boundary, the regions can be considered as taking a

single value; figuratively speaking this can be thought of as two planes of height "1"

outside the constraint boundary and of height "0" inside the boundary. In these

plane regions, there is no information which can be used to find the direction to the

other region. This information is present at the boundary. When crisp measures

of the attributes are used, the boundary can be pictured as a "cliff". When fuzzy

measures are used, this boundary becomes a slope, with the steepness dependent

upon the membership function. The advantage found with the fuzzy measures

is that they extend the range over which information about the boundary can be

detected. As was found in the second case, this resulted in some savings on iterations

towards the boundary. The fuzzy measure also provided a natural way to rate the
151

attributes. Experience showed that it tended to be harder to judge parts when the

their attribute was near the "crisp threshold" whereas the fuzzy measures were more

forgiving.

Behavior of the quality variables was very similar to the performance variables

in the previous chapter. In fact, gradient estimation proved to be less sensitive to

noisy data than was anticipated. This may also be a function of the fact that the

gradients were used to determine a feasible direction, but were not used explicitly

in the optimization. This additional step may provide some robustness to the algorithm. It must be remembered that this algorithm provides a ':slow, plodding"

approach towards optimization. There is no urgent need for speed, in fact, because

the algorithm is to be run on-line, it is important that it make no moves that surprise

the operator.
7. Conclusions

In this thesis the problem of quality control was considered from a control engineering perspective. The goal was to re-define the quality control problem such

that the techniques and tools of control theory could be used. A fundamental basis

of control theory is that there exists a system model which defines the functional

relationship between system inputs and system outputs. Performance objectives

are defined which reflect a desired relationship between these inputs and outputs.

The goal is to formulate a control policy which augments the system such that the

performance objectives are achieved.

In this context, the product quality was defined as the system output, and

the optimization of quality was the performance criteria for manufacturing system

control. The goal in manufacturing, for economic reasons, is to optimize product

quality. One problem is that, as quality is a rather nebulous product characteristic,

there is seldom an analytic function that can be used as a measure. Therefore,

standard control approaches, such as optimal control, cannot readily be applied.

A second problem with optimizing product quality is that it is typically measured along many dimensions: there are many aspects of quality which must be

optimized simultaneously. Very often these different aspects are incommensurate

and competing. The concept of optimality must now include accepting tradeoffs

among the different quality characteristics.

The solution to both of these problems was achieved by defining the quality

control problem as a multiobjective optimization problem. This solution had the

following advantages:

• Quality consists of many different factors. The multiobjective formulation

allowed these factors to be incorporated simultaneously.

152
153

• A fundamental premise of multiobjective optimization is that there is a decision maker in the "loop". The decision maker was relied upon to evaluate

those aspects of quality that could not be measured.

Automatic quality control was approached as the solution to a multiobjective optimization problem. The result was the development and implementation of automatic quality control using an algorithm which iteratively solved this multiobjective

optimization problem on-line.

Performance of the multiobjective optimization algorithm was investigated

using five case studies. Three case studies were centered around the tuning of PID

controller gains to achieve a desired process response. For the first case study, a

transfer function model of plastication was identified using input/output data. The

controller was tuned in simulation, using this transfer function and performance

criteria based on the process response and control action. In the second case study,

plastication controller tuning was implemented to run on-line. This verified the

simulation results and also demonstrated the ability of the algorithm to work in

real-time. In the third case study, the algorithm was used to tune an injection

PID controller. In the above cases, the algorithm was successful in generating a

tradeoff surface showing the limits of performance of this controller. Also, the

relationship between the nondominated points and efficient points showed how the

tradeoff surface is parameterized by the controller gains.

In the final two case studies, the algorithm was used for closed-loop quality

optimization for two different molded parts. For the spiral mold, quality criteria

were defined based on the length of the spiral, the length of the injection cycle,

and the presence of flashing. For the ASTM four cavity mold, quality criteria were

defined based on the injection cycle time, the presence of flashing and the presence

of underfill. In both cases, from an initial operating point, it was demonstrated that

the algorithm steadily improved the performance of the process with respect to the
154

stated criteria, and alsogeneratedtradeoff information to be usedby the operator

in selectingthe final operating point.

T.1 Contributions

The contributions of this work wereasfollows:

Quality control was formulated as a multiobjective optimization problem.

Quality is naturally measuredas many possibly incommensurateand competing objectives. This featurewasretained in the problem formulation.

The multiobjective optimization formulation allowedthe incorporation of both

quality attributes and variables. Attributes were used to definethe feasible

qualitv region and variables were used to optimize within this region.

• A feedback control policy for control of quality in manufacturing processes

was formulated as the solution to the multiobjective optimization problem.

• An algorithm was formulated which implemented automatic quality control.

The algorithm has the following features:

- An analytic formulation of the performance criteria (objective functions)

is not necessary.

- A' priori weighting of objectives is not necessary because of the multiobjective formulation.

- Feedback is provided which informs the decision maker (DM) if the operating point is on the tradeoff surface or how far away it is.

- If the operating point is not on the tradeoff surface, the algorithm generates input changes which will direct the system towards the tradeoff

surface.
- Determination of the tradeoff surface allows explicit tradeoff analysis by

the DM.

- The DM can interactively move along the tradeoff boundary, exploring

alternate nondominated points, until the best operating point is found.

- The evolutionary nature of the algorithm allows improvement to be made

and information to be gained without upsetting the process excessively.

The algorithm provides an interface between the process and the operator.

By the use of this algorithm, the operator's role has been changed from one

of monitorin 9 and operatin 9 the process to one of monitorin 9 and improvin 9

quality. This algorithm implements true quality feedback in that the operator's

goal is to select the desired quality.

The algorithm was implemented on the injection molding process and was used

successfully for tuning of inputs to meet quality objectives and for tuning of

controller gains to meet performance objectives on-line.

7.2 Future Work

This work can be extended in two general areas, improving the efficiency of

the algorithm, and improving the decision maker's interface. Specific research issues

are discussed in the following sections.

7.2.1 Improving Algorithm Efficiency

In the context of on-line quality optimization, requirements of the algorithm

are different than those typically applied in an optimization exercise. Efficiency

is important, because each iteration requires one injection cycle. More important

than efficiency, though, are safety and continuity. By safety, we mean that under

no circumstances should the algorithm make unpredictable operating point changes.
156

The algorithm relies only on data gathered from small perturbations about the nominal operating point. By continuity, we mean that new operating points are found

by moving smoothly from old ones, and consequently process output is improved

smoothly. This allows the operator to analyze trends in the process output to determine if the current operating conditions being explored are likely to yield improvements. Also, as operating conditions are changed automatically, any degradation of

the output quality is minimized. With this in mind, the following improvements to

the algorithm should be explored:

• The algorithm depends on estimating the objective function gradient when

determining the feasible direction. This estimate is sensitive to the noisiness

of the measurements, the presence of "second-order geographical features" in

the response surface, and the magnitude of the input perturbation levels. It

would be valuable to build in some mechanism that would provide feedback on

how good the estimate is. One possible approach would be to use an F-ratio

test as the measure of confidence for the gradient. This information could

also be used to adjust the perturbation levels to improve gradient estimation

accuracy.

• The same input perturbations are used to estimate the gradient for all of the

objectives. Perturbations that are good for one objective may not provide

good estimates for other objectives. Ways of dealing with the "insensitive"

objectives must be determined. One concern that arises is that if a particular

objective is insensitive, it may not be suitable for optimization. It may be

more reasonable to use that particular objective as a constraint.

• The experimental design used to estimate the gradient was a full factorial

design with center point. Given n inputs, 2 '_ + 1 experiments must be run.

As n gets large (more than four or five), this portion of the algorithm could
157

becometime consumingand expensive.Other types of experimental designs,

suchasorthogonal arraysor fractional factorial designs,shouldbe investigated

to seeif a savingscan be made in the number of experiments run, without

losing any of the accuracyin the gradient estimate.

• The speedwith which the algorithm convergesto the tradeoff surface is due, in

large part, to how efficiently the line search is conducted. Different techniques

of adaptive step-size control should be evaluated.

• As the algorithm iterates, the shape of the response surface with respect to

each of the objectives is built up locally. Methods of incorporating this information into a global picture of the process response should be explored.

One interesting question is, can this response surface be used as a "process

signature", providing a useful diagnostic tool?

7.2.2 Improving the Decision Maker's Interface

The operator interface performs two functions: it allows the operator to provide information to the process and it provides the operator with information about

the process.

Information from the process is presented in terms of the nondominated set

and efficient points. Once there are more than three objectives, it becomes very

difficult to present the tradeoff surface information in an intuitively meaningful

manner. Methods of presenting this information, such as graphical or symbolic,

should be investigated.

An important aspect of quality is its attribute nature. In the strictest sense,

this attribute represents a binary decision; the part is good or bad. Often there is

a gray area associated with this decision. A fuzzy interface was used for this work,

but this still depended on the operator's interpretation. This suggests two areas

of research: a determination of the effect of inconsistent decision making on the
158

algorithm and an investigation of ways in which the algorithm can automatically

calibrate itself to a decisionmaker.
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APPENDIX A

The Injection Molding Process

Plastic injection molding is a repetitive processused to make complexly shaped

plastic parts, with onepart being madeeachcycle. The goal of the injection molding machineoperator is to consistently produceparts which meet somepredefined

quality specifications. Part quality can be measuredin terms of suitable material

properties suchasstrength, surfacefinish, and shape. The key word aboveis consistently. The time-varying nature of the process and the complex relationships

between molding conditions and part properties make this a difficult task. In this

appendix, the dynamics of each of the phases will be presented. From this, an understanding of the important process parameters and the interaction between the

phases will be developed. This development is based on other studies of the dynamics of injection molding machines but will be specific to the AML Arburg machine

[21, 22].

A.1 The Injection Molding Cycle

Each cycle of the process has four phases, which occur in the following order:

1. Plastication - the polymer is melted and deposited in front of an injecting

mechanism. This continues until the required amount of polymer has been

accumulated.

2. Injection - the molten polymer is forced into the mold under high pressure.

This continues until the mold is completely full.

3. Holding - the molten polymer is held in the mold, under pressure, to insure

complete mold filling. Cooling of the part begins at this point. Holding

167
168

continues until the gate has frozen, at which point no more polymer can enter

the mold cavity.

4. Cooling - the polymer is allowed to cool in the mold. This continues until the

temperature of the formed part is low enough so that it will retain its shape

once ejected from the mold.

A cut-away view of a reciprocating screw injection molding machine is shown in

Figure [1.1]. In this type of machine, the screw is used to both melt the polymer and

pump it into the nozzle area. This melting and pumping is accomplished by rotation

of the screw. As the melting and pumping occurs, the screw retracts, creating the

volume occupied by the polymer melt. Injection and holding are accomplished by

forcing the screw forward, like a piston. (A check valve at the screw tip prevents

a flow of polymer back into the screw.) Once the polymer is in the mold, cooling

begins, and at a predetermined time, the mold opens and the solidified part is

ejected. There are other types of injection molding machines which accomplish these

four phases via different mechanisms, but the phases and sequencing are essentially

the same [2].

A sequential block diagram of the molding cycle is shown in Figure [1.2]. Interactions between these phases occur when one phase establishes the initial conditions

for a subsequent phase [3]. It is the complexity of these interactions which motivates

the need for control of the process. These interactions are shown in Figure [1.3] and

will be discussed in detail at the end of this appendix.

In this appendix, the underlying physical processes in each phase will be described in detail. The relationship between the inputs and outputs will be discussed, and where possible, these relationships will be quantified with a differential

equation model. All models will be specific to the Advanced Manufacturing Laboratory (AML) Arburg Allrounder injection molding machine, but the principles

under which they are derived will be general to the process. (Specifications for this
169

injection molding machine can be found in Chapter 4.)

A.2 Plastication

Plastication is the portion of the injection molding cycle in which the solid

polymer pellets are melted and deposited in the nozzle chamber (in front of the

retracting screw) prior to injection. The pellets are fed from the hopper into the

reciprocating screw. The screw rotates, simultaneously melting and pumping the

pellets forward. As the pumping progresses, a mass of molten polymer (the shot)

accumulates in the nozzle. Plastication continues until a predetermined shot size

has been accumulated. Plastication can begin any time after the gate has frozen

shut, which prevents the accumulating shot from "leaking" into the mold.

From the above scenario, it is obvious that there are two dynamic processes

occurring concurrently. There is a thermodynamic process because of the heat input

and associated melting. There is also a fluid dynamic process because of the pumping

which is used to accumulate the shot. Both dynamic processes interact. In the

following sections, the dynamics of each process and the mechanism of interaction

will be discussed.

A.2.1 Polymer Melting (Thermodynamics)

Polymer melting takes place due to heat input. There are two sources of heat

input:

• Conduction - energy from the barrel heater bands is conducted into the plastic;

• Viscous Dissipation - as the molten polymer is sheared, adjacent fluid laminae

rub, transforming kinetic energy into frictional heat.
170

Dirccdon of Barrel Motion

q

Melt

Pool

Barrel Wall

,_ 4 11

•".:.','.#.'.:.:.),;:. | :.:t,',:,;..,',)',')#,'.

...*._I.......I....° %.o__;°_.,..,......_ v. v° ......°° ....

Melt Film

Figure A.I: Cross Section of One Screw Flight During Plastication

During the injection molding cycle, conduction heating is always taking place because the heater bands are always on. On the other hand, viscous dissipation contributes to melt temperature only during screw rotation. The goal of modeling the

polymer melting is to determine the machine inputs which affect melt temperature

and to develop models which will allow the control of melt temperature.

Figure [A.1] shows a cut-away view of one screw flight during plastication [11].

The heater bands contact the barrel at this point, and maintain the barrel temperature, T_. This is the energy source for conduction heating. During plastication,

there is both an axial screw translation corresponding to the shot accumulation rate

and a rotation about the screw axis which corresponds to the screw rotational speed.

This motion provides the shearing of the plastic for frictional heat input. Melting

only takes place near the barrel. During plastication, this melt is swept into the melt

pool and the solid bed continuously rearranges itself. This phenomenon is described

more fully by Donovan, et al. , in [9].
171

The relative contribution of the conduction heat versus viscous dissipation

must be understood so that a control strategy for melt temperature can be developed. In the following sections, the temperature dynamics due to conduction and

viscous dissipation will be presented analytically. This will provide a functional

relationship between machine inputs and temperature which will be useful for experimental design and empirical modeling of the system. It must be emphasized

that this analytical modeling will only provide a general clue to the relationships,

the actual situation is much more complex.

Polymer Melting Due to Conduction Heating

Donovan, in [10], developed a theoretical melting model for the polymer. In

this paper, he was concerned with both the melt profile of the polymer in the

barrel (ratio of melted to unmelted polymer as a function of screw distance) as well

as the temperature of the molten polymer in the barrel. To develop an accurate

approximation of the conduction melting model, the problem was posed as a onedimensional heating problem with boundary conditions that depend on the melt

film thickness. The solid bed is assumed to be a semi-infinite mass. This is known

as Neumann's Problem [69]. Figure [A.1] shows the cross-section of one screw flight,

which was used as a control volume to develop the model.

In the melt film, the one-dimensional transient heat conduction equation is

=amo--V; 02 T_ O<y< (1.1)

T l is the melt film temperature profile, the melt film thickness is 6 and am is

the thermal diffusivity of the melt. y is the coordinate direction starting at the

barrel/melt film interface and increasing towards the solid bed.

In the solid bed, the conduction equation is

OO_ "- -_,0-7_, 02 T" " 6 < y < e_ (A.2)
172

Ts is the solid bed temperature profile and as is the thermal diffusivity of the solid

bed. The initial conditions and boundary conditions are

T_(_)= Tb; y = 0

T_(y) = Ts(y) = Tin; y =

Ts(y) _ T_; y ---,

(A.3)

Tb is the temperature of the barrel, Tm is the temperature at the melt film/solid

bed interface, and T, is a reference temperature, assuming the solid bed has infinite

depth. In practice, this would be the steady-state temperature that is achieved at

the center of the screw.

An energy balance was applied at the melting interface, and Donovan was able

to derive the following solutions to the transient temperature problem in the melt

film and the solid bed.

TI(Yl-Tm erf(_/2 _,/z77_t). 0 < y < _ (A.4) Tb-T,,., --'-- 1 - erf(8/2 _,/X_-7_0'

and

T,(y)-T,_ _ 1 -- erfc(y/2v'ET). _ < y < _ (A.5)

Tr-T,, -- erfc(6/: _,/ESt)'

Temperature predictions based on this analysis were found to correlate well

with the data near the start of melting but to underestimate it at the conduction

melting near the end of the screw. The authors attribute this to the neglection

of melting at the other three faces within the channel. (There may also be some

melting due to heat generation during viscous dissipation. There is no accounting

for an additional heat source in this model.) Equation (A.4) is the equation of

interest since this predicts the transient temperature behavior in the melt.

Polymer Melting Due to Viscous Dissipation

As a first step towards understanding the melting due to viscous dissipation,

a simple example from [70] will be presented. This example will then be compared

with the situation actually occurring during plastication.
173

Barrel

Velocity Temperat ur_

Profile Profile

0 T, y=5

Solid Bed

Figure A.2: Two Parallel Plates Separated by a Newtonian Fluid

Assume that the barrel and solid bed can be modeled by two parallel plates

separated by the thickness of the melt film, which stays constant. Also assume that

the molten polymer can be modeled as a Newtonian fluid with viscosity p. The

barrel is moving at a constant velocity 14, relative to the solid bed. This situation

is depicted in Figure [A.2]. This is analogous to the situation shown in Figure [A.1],

if the thickness of the melt film is small compared to the radius of the barrel and

solid bed.

This system has conduction heating as well as viscous dissipation, therefore

the energy equation is

d T (y) dYe(y)

0 = am + ru__ (a.6)

dy 2 dy

The first right hand side term in this equation is energy from heat conduction and the

second term is energy from viscous dissipation. V_,(y) is velocity in the x-direction

as a function of y and ru, is shearing stress in the x-direction due to a force in the

dYe(y)

rye, = # d-'-'_ (A.7)

The steady state velocity profile is

Y

V,:(y) = g>; (h.8)

y-direction. Since the fluid is Newtonian, the following relationship holds:
174

The boundary conditions are

y=O

y=,5

(A.9)

T (y)-Ts y 1 y y

Tb-T - 3 (A.10)

Br= (A.11)

Br is the Brinkman number. It is an important dimensionless quantity which is the

ratio of the heat generated via viscous dissipation (the numerator) to the ability of

the fluid to conduct this heat away (the denominator).

The steady-state temperature profile has two components. There is a conduction component which makes the temperature a linear function of the distance from

the top plate, as expected. The second term is the viscous dissipation component.

The temperature profile for this component is parabolic. If the Brinkman number

is large enough, there will be a maximum temperature, somewhere between the two

plates, which is greater than either Tb or T,, as shown in Figure [A.2].

An exact solution to the steady-state temperature profile of a screw flight

for a plasticating extruder has been carried out by Donovan in [71]. It will be

reviewed here briefly, as a comparison to the simplicity of Equation (A.10). Again,

the analysis is begun with the energy equation.

0 = am + ru=_ (A.12)

dy 2 dy

Now, the molten polymer is assumed to be non-Newtonian. The constitutive equation, assuming a power-law model, is

ru_=moe_(T'(u'-T') (dV_) " (A.13)

\dr]

where

resulting differential equation is

Equations (A.7) and (A.S) can be substituted into (A.6), and the solution for the
175

This is an empirical equation where too, a, and n must be determined through

experimentation.

If (A.13) is substituted into (A.12) and the resulting differential equation

solved, the following temperature distribution is found.

Tf(y) : Tm + na log [{cosh 2 (C1 - B1C2 2 y) Co 2}] (A.14)

The constants Co, C1, and C_ must be found using the following initial conditions.

T_(y) = Tb; y = o

E(y) = E; y= 0

V_(y)= 0; y =

is the average thickness of the melt film. The constant Bx is defined as

B1 = _/amo(V2+' )(_a-")/2n_m

(A.15)

(A.16)

Notice that this constant is a ratio similar to the Brinkman number defined by

Equation (A.11), but due to the complexity of the mathematics, a simple form

does not result. The point to be emphasized is that there is no longer an intuitive

relationship between the effect of conduction heating versus viscous heating. This

must be developed experimentally.

Overall Melting Dynamics

Two different mechanisms for heat generation have been investigated. Equations (A.4), (A.5), and (A.14) are useful for understanding the physical relationships

in the process, but they are too complex to be used for empirical modeling.

Fundamentally, the different mechanisms work on two different time scales.

The conduction transients are quite slow, because of the "thermal mass" associated

with the polymer and barrel. This dynamic response was investigated both theoretically and experimentally by Kamal, et aI. , in [14, 15]. Their results for the
176

melt temperature responseto the barrel heaters,using a lumped parameter model,

was a second-order,overdamped system with a time delay. This seems reasonable

since a thermal system can typically be modeled as a first-order system and that

the barrel and polymer melt would represent two cascaded first-order systems (i.e.

, the heater heats the barrel and the barrel heats the polymer). A transfer function

for this system would be

I(ce -sd

Tm (s) =

(_'bs + 1)(rms + 1) Ts(s)

(a.17)

In the above equation, T,,c is the component of the melt temperature due to conduction heating. The constants Kc, d, rb, and 7"m must be found via experimentation.

It should be noted that screw speed was not manipulated in any of the experiments

discussed by Kamal, et al. , therefore any effect due to changes in viscous dissipation

would not appear in their results.

Heat input from viscous dissipation only occurs during screw rotation. This

can be thought of as a periodic step disturbance to the melt temperature. A model

of this phenomenon was developed by Lipshitz, et al. , in [12]. This work was mainly

concerned with how this disturbance affected the solid bed profile, but analogous

conclusions can be drawn for the melt temperature profile.

During plastication, the melt film is swept into the melt pool, therefore the

melt pool will "asymptotically" approach the volume average temperature of the

melt film. The temperature of the melt pool is therefore governed by the speed of

screw rotation as well as the length of time of screw rotation.

Equation (A.10) is an approximation of the steady-state melt temperature

profile during plastication. This equation indicates that the melt temperature will

be a function of the shear velocity. Now, as the screw rotates, the melt film depth

stays approximately constant and the solid bed is continuously rearranged [71].

Therefore, the melt pool volume in the screw flight is increased by a constant stream

of polymer from the melt film, at a volume average temperature determined by the
177

viscousdissipation. This is analogousto the situation of a constant stream influx to

a well stirred tank. The lumped parameter temperature dynamicsof this situation

is representedquite well by a first-order system. Sinceconduction dynamics have

been accountedfor, the input to the mixing dynamic system is shear velocity. A

simple transfer function representingthis systemwould be

K_

T,_,(s) -- l/; (s) (A.18)

r_ds+ 1

T,_ is the component of temperature due to viscous dissipation. The shear velocity,

V_, is both a function of screw translational velocity, V_, known as recovery rate, and

screw rotational velocity, ,3, known as screw RPM. The following approximation to

the numerator of (A.18) can be made.

I(,t_ = K_V_ + A'_,_a., (A.19)

K_, and K._ are determined experimentally. This is essentially a linear approximation of the function between these velocities. Then, using this equation, (A.18)

becomes

K,,,v,(s) +

= (a.2o)

%_s + 1

Equation (A.20) can be used to determine the effect of the three primary

plastication variables, screw RPM, recovery rate, and recovery time (t,), on the

melt temperature. Recovery time and recovery rate are tightly coupled. Shot size,

X,, is typically measured in terms of the distance the screw must retract to develop

the required volume of melt. Therefore,

J

X, = t_V, (A.21)

In the above equation, 1_ is the average or steady-state recovery rate. The relationship between t_ and r_ will give some indication of the degree of control that can be

exercised over the melt temperature in the time available.
178

Barrel Melt

Temperature[ Barrel /Polymer Temperature ° Dynamics Dynamics

Polymer I

Shear Pate Mixing

" Dynamics

Figure A.3: The Melt Temperature Dynamic System

Recovery rate and screw RPM are loosely coupled through the screw pumping

dynamics, which will be discussed in the next sections. At this time it will just

be noted that it is important to determine the magnitudes of IQs and K_,. This

will indicate the degree of temperature control which can be attained using these

velocities. At the same time, this control is constrained by the dynamic coupling

between these velocities, as well as the range over which they can be varied.

In summary, the system, depicted in Figure [A.3], can be used to empirically

model the system temperature dynamics. In this model, the barrel and polymer

conduction dynamics are much slower than the viscous dissipation dynamics. With

respect to the conduction dynamics, the viscous dissipation dynamics would appear

as an exponential disturbance, with dynamics dependent upon the mixing, where

the duration of the disturbance depends upon the rotation time. With respect

to the viscous dissipation dynamics, the conduction dynamics establish the initial

temperature of the melt pool. The effect of the viscous dissipation is dependent on

the screw rotation time. This is directly related to the desired shot size and the

accumulation rate. In the following sections, the machine dynamics involved with

accumulating the shot will be derived.
179

A.2.2 Polymer Pumping (Fluid Dynamics)

In this section, the dynamics of pumping the polymer into the nozzle region

will be presented. Raimund, in [13], presented an analysis of the melt conveyance assuming the system was isothermal, non-Newtonian, and steady-state. A solution for

the volume flow rate was obtained. Since it was steady-state, it wasn't particularly

applicable for control system design. In the following section, a lumped-parameter

analysis of the system will be done, using simple mass and force balances. The

analysis will not be as rigorous as [13], but the basic dynamics and interactions of

the system will be captured. It should be noted at the outset that the coefficients of

the model will be dependent on the particular machine being studied, the polymer

being used, and the operating conditions. Ideally, these would be found through

designed experiments or on-line identification.

As the screw rotates, molten polymer is forced into the nozzle, generating a

pressure in the melt. The accumulation of the polymer is modeled by a simple

mass balance. The melt pressure in the nozzle forces the screw backwards against

the hydraulic oil in the piston chamber. These screw dynamics can be modeled by

a force balance between the "polymer-side" and the "hydraulic-side". Finally, the

hydraulic oil flow out of the piston is regulated by a servovalve. This can be modeled

by another mass balance plus the dynamics of the valve. In the following sections,

the analysis outlined above will be presented, yielding a simple melt-filling transfer

function. Finally, an overall dynamic model of the plastication stage, combining

melting and pumping, will be presented.

Polymer Flow Mass Balance

Polymer flow into the nozzle chamber is governed by the following mass balance

equation [62].

vm.

Q_ = A,,,V_ + "x-'Pro + _,,,o (A.22)
1_0

In the above equation, Qmi is the flow of molten polymer into the nozzle of the screw

and Qmo is flow out of the nozzle, which is zero during plastication. The right hand

side of (A.22) consists of two terms: the first is the change in volume in the nozzle,

which is proportional to screw recovery rate, V,, and the second is the change in

volume due to compression of the polymer, which is proportional to the derivative

of melt pressure, Pro. In this equation, Am is the effective cross-section of the melt,

Vm is the melt volume, and j3m is polymer melt bulk modulus.

Flow into the nozzle, Qmi, has two components: a drag flow component, Qma,

imparted to the polymer by the rotating action of the screw, and a pressure flow

component, Qmp, due to the pressure differential between the nozzle and the hopper.

Qm_ = Qm_ + Qmp (A.23)

The drag flow component is the flow imparted to the molten polymer by the rotating

action of the screw, driving it down the screw channel. The pressure flow component

is the flow of the polymer, back towards the hopper, imparted by the pressure

developed in the polymer in the nozzle. The exact solution to (A.23) involves nonNewtonian fluid flow analysis. The modeling goal, here, is to determine simple

relations, to be used for feedback control. Equation (A.23) can be approximated by

[13]

Qmi = K_ - KmPm (A.24)

In this equation, If, and Km are empirical gains related to screw speed and melt

pressure.

Combining this with (A.22) yields

V_ •

•"z-Pro = -AmVs- KmPm + K,_w (A.25)
181

Screw Force Balance

During plastication, the screw retracts, creating the volume in the nozzle in

which the molten polymer is deposited. The screw dynamics can be derived using

a force balance, shown below.

AHPH - A,_Pm = -mV, - bV, (A.26)

The inputs to the system are the hydraulic pressure, PH, and melt pressure, pro,

which apply a force through the system by acting through the hydraulic piston

surface, AH, and the nozzle piston surface, Am. m is the approximate mass of the

polymer and screw and b is the approximate viscous friction on the screw. The

resultant force is countered by an inertial term, and a viscous friction term. This

system can be linearized for a given operating point, and the resulting equation is

mI:':, = -bI,_ + AmPm - AHPH (A.27)

Hydraulic Fluid Mass Balance

The hydraulic pressure is supplied to the piston through the servovalve. These

dynamics can be found using a mass balance of the hydraulic fluid around the piston

chamber, similar to that done for polymer flow.

VH • A_V_ + QHo (A.28) Q.i = -

In this case, flow into the hydraulic piston chamber, QHi is zero. QHo is the flow of

hydraulic fluid out of the piston chamber. AH is the effective area of the hydraulic

piston, VH is the hydraulic oil volume, _g is the hydraulic oil bulk modulus, and

PH is the hydraulic pressure.

Fluid flow into or out of the piston chamber is a function of the servovalve spool

position. A reasonable approximation [62] can be obtained by using an equation for
182

turbulent flow through an orifice:

QHo = CeX_I_( PH - Pa) (A.29)

Flow through the orifice is a function of the orifice opening, X_, the pressure differential across the orifice, (PH --PR), the fluid mass density, p, and the valve discharge

coefficient, Ce. The pressure differential will always be positive because hydraulic

fluid will only be flowing out of the piston chamber during plastication. Equation (A.29) can be linearized around the valve operating point and the following

simplified relation results:

QHo = I(.xpX, + IippPH (A.30)

In this equation, Ifxp is the plastication valve position flow gain, and Kpp is the

plastication pressure flow gain. Strictly speaking, the pressure term in the above

equation should be a function of the pressure differential, not just PH, but since the

return pressure, Pa is typically zero, the approximation is valid.

Equation (A.30) can be substituted into (A.28) to yield the linear differential

equation:

-fi-_HPH = AHVs - Kp, P, - KxpX_ (A.31)

Modeling the Servovalve

The dynamics of the servovalve are dependent on the internal construction of

the valve. On the Arburg machine, the valve was a three-port valve. A reasonable

model of the valve dynamics is [61]

X,[s] -" s(r_s + 1) V[s] (A.32)

The input to the servovalve is a voltage, V, which drives the pilot stage. The valve

transfer function has two poles. The integral term is a result of the fact that there is

no internal feedback within the valve because of the solid spool [62]. The first order
183

term, with time constant r,,, is an approximation of the rest of the valve dynamics,

including the windings driving the pilot stage.

A.2.3 Overall Polymer Pumping Dynamics

The analysis above resulted in three first-order linear differential equations

which describe the machine dynamics during plastication. Equations (A.25), (A.27),

and (A.31) are essentially state equations. The system has three states: melt pressure, Pro, screw linear velocity, V_, and hydraulic pressure, PH. A block diagram

of this system is shown in Figure [A.4]. In the block diagram, the input to the

system is the servovalve opening, Xs. The screw rotational velocity, w, appears as a

disturbance input. The physical variables can be defined as state variables and the

following matrix equation results:

1,',_e,, A,,,e,_ 0

v,, v,,,

_± _ A_._

m rrt

o a__ __

VH Vn

Zp +

Vm

0

0

0

0 Up (A.33)

VH

In the above equation, the states are defined as Zpl = P,,,, Zp2 = V,, and Zp3 = PH.

The control inputs are defined as Up1 = w and Up2 = X,. An output equation must

be defined for this state-space model. If recovery rate is the output, then the output

equation is:

Yp= [0 1 0 I Z_, (A.34)

J

A transfer function relating recovery rate to servovalve spool position and

screw rotational speed, w, can be derived using (A.33) and (A.34), and is

nl_z[s] - n2X_[s] (A.35)

Vs[s] = das3 + d2s2 + dis + do
184

Figure A.4: Pumping Dynamics Block Diagram

The coefficients in the above equation are:

nl = A,,,I(., (S_v--_+ I(p.)

n_ = AHI(x, (s'_'_ + I(,n)

da = m p-n'y-'_

Vx V,,.,

d_ = m_v-_I(.,,, + mKpp-_ +-v. v,,,

dl = mKppK,_ + b_v-_xK., + bKpp-_--_ + "*,,, __. v+_as+ A 2H_9.._z.,

2 .

do = bKpp K,,, + A_ Kpp + A H I_ m

(A.36)

The servovalve dynamics can be included by substituting Equation (A.32)

into Equation (A.35). The inputs to the system are now screw speed and servovalve

voltage.
Barrel Dynamics

twd

V

Polymer

Dynamics

Dynamics Mixing I

-1 Valve & l V_ ,

q Dynamics Pumping X,

Figure A.5: Plastication Dynamics

A.2.4 Overall Plastication Dynamics

Earlier, it was stated that there are two interacting dynamic systems which

describe the plastication dynamics. After the individual dynamics have been described, it is now possible to understand these interactions. Figure [A.5] is a simplified block diagram showing the interaction between the thermodynamic system and

the fluid dynamic system. The goal of plastication is to generate the required shot

size, Xs, at the correct temperature, Tin, in the fastest possible time, tr (to maximize

productivity while maintaining part quality). As can be seen from Figure [A.5], the

interaction takes place through the recovery rate, Vs. It will be up to the quality

control system to determine the correct screw RPM, recovery rate profile, and barrel

temperature to meet the objectives.

A.3 Injection

Injection is the phase during which the molten polymer is injected into the

mold cavity. Initially, the molten polymer is in the nozzle, and the screw acts like

a piston, forcing the polymer through the nozzle into the mold. Injection continues

until the mold is completely filled, and is terminated based either on screw position
186

or time.

The microscopic modeling of injection has centered around the modeling of

the unsteady flow of a hot, non-Newtonian polymer melt into a cold cavity. These

dynamics are modeled as unsteady-state free surface flow coupled with transient

cooling and are described by the basic equations of change. Tadmor, Boyer, and

Gutfinger, in [16], modeled the flow of polymer into the melt cavity using the finite

element method. Experimental work by Kreuger and Tadmor [17] validated this

modeling method by studying the injection of polymer into a rectangular cavity

with various obstructions. One result was that it could be assumed that molten

polymer behaved as a Newtonian fluid, which simplified modeling and simulation.

Later work by Mavridis, Hyrmak, and Vlachopoulos [18], Kamal, et aI. [19], and

Gogos, Huang, and Schmidt [20] included fountain flow in the modeling. Understanding of the injection phase is important because shear and elongational stresses

cause high orientation of the polymer. Polymer which contacts the mold surface

has frozen-in orientation while the remainder of the polymer has time to relax while

cooling. These microstructure differences, within a part, due to the injection process

can have a great impact on quality properties such as strength and warpage. Empirical modeling of the injection phase has focused on control of injection velocity

or pressure. In a sense, the filling of the mold can be considered %pen-loop" once

the plastic leaves the nozzle. Shankar, and Shankar and Paul in [21, 29_] developed

a lumped-parameter model for the injection process and evaluated a state-space approach to injection control. Costin, Okonski, and Ulicny in [23] examined adaptive

control of the injection process and compared it to the performance of a PI controller

for hydraulic pressure control. Haber and Kamal [24] and Kamal et at. [25] also

studied the control of pressure during injection.

For implementation of screw velocity melt pressure control, injection dynamics

should include the following:
187

1. a mass balance model of molten polymer flow, relating melt pressure and

polymer flow into the mold;

2. a force balance model of the screw, relating polymer deposition rate to the

melt pressureand hydraulic pressure;

3. a mass balancemodel of hydraulic fluid flow, relating screw speed and hydraulic pressureto servovalvespool position;

The only differencebetweeninjection andplastication dynamicsis that now polymer

flow out of the nozzlethrough an orifice must be modeled instead of polymer flow

into the nozzle.

A.3.1 Injection Polymer Flow Dynamics

Polymer flow during injection can be describedas a fluid flow massbalance

(similar to plastication)[62].

v;..

Q,_, = A_Vs + .--_mPm + Qmo (A.37)

During injection, flow into the nozzle, Q_i, is zero. Flow out of the nozzle, Q,-,,o, is

described by a laminar flow equation [21].

Q_o = RA,,Pm (A.38)

In the above equation, A,_ is the nozzle cross-sectional area, and R is the effective

resistance to flow of the polymer through the nozzle. This equation is also a simplification of the flow behavior of the polymer. It should be noted that R will change

with time. There are two reasons for this: as the polymer enters different parts of

the mold, the area available for flow will change, and as the polymer freezes to the

sides of the mold, the area available for flow will decrease. Because of this resistance,

injection dynamics are also a function of the mold design.
188

Equations (A.37) and (A.38) can be combined to form the following state

equation:

vm.

"ff"£mPm= -RA,_Pm - A,_ V, (A.39)

Screw Force Balance

During injection, the screw dynamics are the same as those during plastication.

The equation will be re-stated for completeness.

m'_ = -bI_ + AmPm - AHPH (A.40)

Hydraulic Fluid Mass Balance

The hydraulic fluid dynamics are the same for injection as for plastication,

except that fluid is now flowing into the piston chamber, and the system is being

driven by the supply pressure. The mass balance equation is

I/H • AHV_ + QHo (A.41)

Q H i = "_u P H --

and the orifice flow equation is

QHi = CdXv_(Ps - PH) (A.42)

Notice that the only difference between Equations (A.29) and (A.42) is how the

pressure differential term is generated. During injection, the flow is driven by the

supply pressure, as opposed to plastication where pressure is developed due to screw

rotation. Again, the flow equation can be linearized to yield one in which hydraulic

flow is proportional to spool position and hydraulic pressure,

QHi -- KxiXv + KpiPH

I(xi and Ifpi are spool position and hydraulic pressure empirical gains.

Combining (A.41) and (A.43) yields the following state equation:

vu.

w'-PH = AHI_ + I(xiXv + I(piPu

PH

(A.43)

(A.44)
189

A.3.2 Overall Injection Dynamic Model

As for plastication, a state space model for injection can be formulated using

Equations (A.39), (A.40), and (A.44). The matrix equation is:

v_ v,.,,

_± _.de.

rtl m m

0 _

vH VH

Zi +

0

0 Ui (A.45)

In the above equation, the injection state variable, Z,, is defined using the same

physical variables as Zp. Notice that this system only has one input, namely servovalve spool position. Again, if the desired output is recovery rate, then

Y,=[0 1 0]Zi (A.46)

A transfer function can be derived from the above state and output equations.

The system input is servovalve spool position and the output is recovery rate:

AHKxi (s _'_ + RA,_)

Vs[$] = d383 + d2.s 2 -t- dis + do (A.47)

The denominator coefficients in the above equation are:

d 3 = m_V--/_-HB,'_

d2 = -m-_ Kp, + reRAn _ + _-¢-_vH

dl = -mRA,_Kp, - by__ Kp, + bRA,_-_ + .4_ _ H + A2n _y.. (A.48)

do = -bRA,_Kp, + A_Kp, + A:HRA,_

The servovalve dynamics can be included by substituting Equation (A.32) into

Equation (A.47). The input to the system is servovalve voltage. A block-diagram

for this system is shown in Figure [A.6].
190

5L{ L_.K

Figure A.6:

j

Injection Dynamics Block Diagram

Pm

t,;

A.4 Holding

Holding occurs after injection. At this point, the mold is full and cooling

begins. The gate is not frozen off completely, so polymer can still flow in or out. The

goal of holding is to ensure that no voids or porosity form due to shrinkage resulting

from the initial cooling and that part density is consistent. Holding continues until

the gate is frozen shut. Holding dynamics are identical to injection dynamics, except

for the polymer mass balance. This will be discussed in the following section.

The main phenomenon governing the holding phase is an equation of state

[7,s]:

(P_ +a)(u + _) = pTm (A.49)

This is a modified van der Waal's equation of state which relates melt pressure

and specific volume to melt temperature. In Equation (A.49), Pm and Tm are the

melt pressure and temperature respectively, u is the specific volume (the inverse of

density), and a,/3, and p are empirically determined constants.
191

Agrawal, et al. [31] discuss the use of this equation for control. One example

is the control of part density. If it is desired to maintain constant density, (A.49)

becomes

P= = xTm - y (A.50)

This relationship can be used to determine the input trajectory to a holding phase

pressure controller.

Since the mold is full at this point, the start of holding is effectively the start

of cooling. Physically, any pressure control on the polymer in the mold terminates

when the gate freezes shut, since there is no longer an}" way to maintain pressure on

the molded part. Therefore, there are three ways to terminate the holding phase:

pressure in the mold is monitored or melt temperature is monitored and some reference value is used as a termination threshold, or assumptions are made about

the heat dissipation of the mold and holding is terminated by time. In this work,

holding will be terminated by time. The length of this time will be dependent upon

the melt temperature at the beginning of holding, but a constant value was used

throughout.

A.4.1 Holding Polymer Mass Balance

Polymer flow during holding can be described by the following fluid flow mass

balance:

vm.

Q.,, = + Q.,o (A.51)

For holding, the control volume for the mass balance is now the mold. There is no

volume change term in this equation, as there was in plastication and injection. In

this case, there is no flow out of the mold, so Qmo is zero. There is a flow into the

mold, corresponding to the shrinkage of the cooling part. This flow is equal to the

polymer which flows out of the nozzle into the mold. This is proportional to the
192

velocity of the screw.

Q_i = -AmV, (A.52)

The negative sign in the above equation is a consequence of the fact that screw

motion towards the nozzle is considered to be in the negative direction so as to

conform with the previously derived dynamic equations. Now, substituting (A.52)

into (A.51) yields

vm.

_-_'Pm = -AmV_ (A.53)

A.4.2 Overall Holding Dynamic Model

The form of the state equations for hydraulic pressure and screw velocity are

identical to those of plastication. The only difference will be in the coefficients of

the linearization. For completeness, these state equations are:

mf/s = -bV_ + A,_P_ - AHPH (A.54)

and

vH.

_HPH = AHVs + KxhX,, + KphPH (A.55)

KXh and I(ph are spool position and hydraulic pressure empirical gains.

Again, defining the holding state vector, Zh, analogous to the plastication and

injection state vector, the following matrix equation can be formed:

Zh --

0 A_fl._ 0

v,.

Am _± _.4.u.

_t, tn

0 _

vn vn

Zh +

0

0

vn

Uh (A.56)

If melt pressure is the desired output, then

 -[10 01zh (A.57)
193

Figure A.7: Holding Dynamics

The melt pressure transfer function, for holding, is

K x h.k'_

Pm[s] = das3 + d2s2 + dis + do

The denominator coefficients in the above equation are:

d3 --_'_ -_

= "'7-g_ v.

d2 = bp--_-_'_a-- rrtl_ph v,_ vn

dl = A sn _'_ _ + Am_ _v_ - Kehb-_-g_

do = - A_ Kph

The servovalve dynamics can be included by using Equation (A.32), as in

plastication and injection. A block-diagram for holding is shown in Figure [A.7].

(A.58)

(A.59)

A.5 Cooling

During cooling, the part rests in the mold until it is cool enough to be ejected

and still retain its shape. The cooling dynamics are determined by the ability of
194

the mold to dissipatethe heat of the injected plastic. In general,this is a complex

transient heat conduction problem, akin to that describedfor the polymer during

plastication. In this case,the mold will typically havecooling waterrunning through

it, removingheat. For control purposesthe mold dynamicscanbeassumedto behave

as a first order system with a time delay, as in plastication [21]. The following

empirical model for cooling canbe used

/x" M

TraM[s]- Tw[s] (A.60)

rats + 1

Average melt temperature in the mold, TraM, is a function of the water temperature,

Tw, and the heat dissipation time constant, rM. KM is an empirical gain.

In the mold used for this study, there was a constant flow of cooling water

through the mold, and there was no instrumentation for collecting mold temperature. For all experiments, cooling was terminated by a time, which was determined

empirically. Again, the length of this time is dependent on the temperature of the

polymer in the mold but a constant value was used. This is typical of the methods

used in industry.

A.6 Process Control and Phase Interaction

In the previous section, the dynamics of each phase have been examined independently. The result was an approximate transfer function model for each phase.

In each case, the model parameters are dependent upon the specific machine, the

operating conditions, and the particular polymer used, so some experimentation and

identification would have to be done to get an accurate model. Once obtained, this

model would be suitable for use with traditional control system design techniques.

Control design for a specific phase would consist of determining desirable controller performance, making assumptions about the inputs and disturbances that

may be expected, etc. The result will be a controller which meets the design goals

within that phase, independent of the other phases. This is intra-phase control.
19.5

All phases of the injection molding cycle affect part quality. This is due to the

interaction among each of the phases of the injection molding cycle. This interaction

is a product of the fact that the output of a phase establishes the operating conditions

for subsequent phases and machine cycles.

The phase inputs and outputs, and interactions are shown in Figure [1.3]. Two

different types of inputs are defined, external and internal. External inputs are those

which the operator has direct control over. Typically, these are machine settings

which are used to control the process directly, or are inputs to controllers on the

process. Examples of external inputs are screw speed and the barrel temperature

setpoints. In Figure [1.3], these inputs are shown entering from the left side. Internal inputs are those inputs which cannot be controlled directly. These can be

inputs which are outputs from a previous phase or are disturbance inputs. Examples of internal inputs are polymer melt temperature and raw material variations.

In Figure [1.3], these inputs are shown entering from the top.

To better understand this interaction, the effect of melt temperature will be

traced through the cycle. During plastication, the melt is generated at a temperature which is a function of the barrel temperature, screw RPM, and the recovery

rate. The result is a melt with an axial temperature profile which varies with the

distance down the barrel from the nozzle. (This is discussed in more detail when

the experimental results of Raimund are considered, in Chapter 2.) Following plastication comes injection. The major physical effect of melt temperature is on the

viscosity of the polymer melt. Both the viscous friction term, b (Equation (A.40)),

and the effective mold resistance, R (Equation (A.39)), are functions of viscosity.

When holding commences, pressure is applied to the polymer in the mold until the

gate freezes off. During this time, the polymer cools and shrinks. The amount

of shrinkage that occurs, and therefore the amount of polymer which must be introduced to ensure complete mold filling (Equation (A.49)) is proportional to the
196

temperature drop which occurs through the holding cycle. Finally, the part cannot

be ejected from the mold until it has cooled below the point at which it would warp.

This time would depend upon the average melt temperature in the mold at the start

of cooling and the heat dissipation time constant (Equation (A.60)).

So far, the effect on subsequent cycles has not been discussed. Consider that

the total volume in the screw flights can be several times the volume of one shot.

Therefore, the melt temperature profile at the n *h cycle may be dependent upon the

temperature history of the k previous cycles. Sanschagrin analyzed this cyclical dependence with several different inputs and outputs in [30]. This interaction signifies

the need for inter-phase control and inter-cycle control [31].

Inter-phase and inter-cycle control is what an experienced operator does. He

starts with an initial set of control inputs;the operating policy. He will adjust these

inputs, as the process evolves, based on his monitoring of process outputs. The

size and type of adjustments to make are determined based on experience with

the process. The operator has developed an intuition about the interactions and

makes adjustments based on this knowledge. This control might be viewed as a

combination of automatic control of the process and operator control of quality.

A.7 Summary of Previous Research

In this appendix, the dynamics were described in detail. The most comprehensive work on plastication was done by Donovan, et al. in [9, 10, 11]. The emphasis

of this work was on melting due to conduction heating. A theoretical model of

conduction transients was developed, and then verified using cooling experiments.

Donovan applied earlier work in plastication extruders to understand melting dynamics due to viscous dissipation [71]. This work was extended by Lipshitz to the

injection molding process [12]. Raimund used these results as the foundation for

his experimental work [13]. Kamal, Patterson, and Gomes described the dynamics
197

and control of melt temperature in [14, 15]. This work was different from the previous work in that their concern was on the implementation and effect of feedback

controllers rather than a study of the process.

The remaining phases of the injection molding process were studied as a

lumped-parameter system. This analysis followed the development of Shankar in

[21, 22]. This was used to formulate the dynamic models of injection and holding,

as well as the melt accumulation dynamics during plastication.

Ma presented one of the first analyses of the injection molding process as a

system [3]. The fundamental result is that no portion of the cycle exists in isolation.

This was further emphasized by the experimental work of Sanschagrin in [30], and

the discussion of the importance of cycle-to-cycle control in the survey of injection

molding machine control done by Agrawal et al. in [31].

A.8 Summary

In this appendix, a detailed analysis of the plastic injection molding process

was carried out. This process consisted of several different dynamic processes that

occur sequentially. These are the phases of the injection molding process. Because

each of these phases establishes the operating conditions for subsequent phases, this

interaction must be understood before designing a controller for one phase and for

the whole process. This control is essential for maintaining product consistency.
APPENDIX B

Symbol Table and Nomenclature for Machine Dynamics

Below is a list of symbols used in formulas throughout the dynamics appendix.

It is arranged according to the sections in which the particular symbols appear.

A tradeoff has been made for clarity over duplicity. Also, due to the number of

equations, some symbols have been used to denote several different quantities. In

general, the meaning of the symbol should be evident from the context.

B.1 Definition of the Quality Control Problem

u(-) command input

k(.) controller parameter

/A set of feasible command signals

K set of feasible controller parameters

to(l) length of the I th cycle

qv quality variable

Qv quality variable space

QVr quality variable target set

qA quality attribute

QA quality attribute space

QAT quality attribute target set

y(.) process output

y set of measurable outputs

H(-) process model

Gv(.) quality variable model

GA(.) quality attribute model

198
199

B.2 A Review of the Multiobjective Optimization Problem

f(.)

Z

Z

X

X

9(.)

x

U

?2

W

d

vector objective function

objective function value

objective space

optimization variable

optimization feasible set

constriant function

nondominated set

utility function

Langrange constraint multiplier

Lagrange objective function multiplier

feasible direction

B.3

B.4

A Linear Programming Solution to Multiobjective Quality Control

Po feasible direction projection value

a objective gradient normalizing constant

f(.) objective function

d feasible direction

constraint gradient normalizing constant

j active constraint index

d approximate direction

N" nondominated set

feasible operating point

feasible direction magnitude threshold

e optimization stopping threshold

rt line search step size

5: operating condition generated by the line search

O process output during line search

Polymer Melting Due to Conduction Heating

erf()

erfc()

rl

r,

t

Y

O_rn

O_s

error function

error function complement

barrel temperature

melt film temperature profile

melt film/solid bed interface temperature

reference temperature

solid bed temperature profile

time

melt film coordinate direction

melt film thermal diffusivity

solid bed thermal diffusivity

melt film thickness
200

B.5

B.6

Polymer Melting Due to Viscous Dissipation

Br

rb

T:

E(y)

Y

_rn

#

Brinkman number

barrel temperature

melt film temperature profile

solid bed temperature (constant)

barrel velocity

velocity profile due to force in the x-direction

melt film coordinate direction

melt film thermal diffusivity

melt film thickness

average melt film thickness

Newtonian viscosity

shear stress in the x-dir due to force in y-dir

Overall Melting Dynamics

d

Kc

K_

gvs

/T( w

Tb

t_

X,

Y,

rb

Tm

a:

conduction heating time delay

conduction heating gain

viscous dissipation heating gain due to _';

viscous dissipation heating gain due to _.'

viscous dissipation heating gain

melt temperature due to conduction

melt temperature due to viscous dissipation

barrel temperature

recovery time

shear velocity

screw recovery rate

shot size (screw position)

average screw recovery rate

barrel heating time constant

melt film heating time constant

viscous dissipation heating time constant

screw rotational velocity
201

B.7 Polymer Fluid Dynamics

AH

Am

b

Cd

Km

It xp

[(Pp

K_

IC

PH

PR

Ps

QH_

QHo

Qmi

Qmo

Qmd

Q_p

V

X_

p

%

hydraulic pressure effective area

melt pressure effective area

approximate viscous friction on screw

orifice flow discharge coefficient

melt pressure empirical gain

servovalve spool position flow gain

servovalve hydraulic pressure flow gain

servovalve gain (voltage to spool position)

screw speed empirical gain

approximate mass of screw and polymer system

hydraulic pressure

polymer melt pressure

hydraulic return pressure

hydraulic supply pressure

hydraulic flow into piston chamber

hydraulic flow out of piston chamber

polymer melt flow into nozzle

polymer melt flow out of nozzle

polymer melt drag flow component

polymer melt pressure flow component

plastication input

hydraulic oil volume

melt volume

recovery rate (screw linear velocity)

servovalve input voltage

servovalve spool position

plastication output

plastication state vector

melt bulk modulus

screw RPM (screw rotational velocity)

hydraulic oil bulk modulus

fluid mass density

servovalve time constant
202

B.8 Injection Dynamics

AH

Am

A,_

b

Cd

Kxi

Kpi

K_

772

PH

pm

PR

Ps

Qm

QHo

Qmo

R

Ui

V.

E

V

X.

Z,

P

vv

hydraulic pressure effective area

melt pressure effective area

nozzle cross-sectional area

approximate viscous friction on screw

orifice flow discharge coefficient

servovalve spool position flow gain

servovalve hydraulic pressure flow gain

servovalve gain (voltage to spool position)

approximate mass of screw and polymer system

hydraulic pressure

polymer melt pressure

hydraulic return pressure

hydraulic supply pressure

hydraulic flow into piston chamber

hydraulic flow out of piston chamber

polymer melt flow into nozzle

polymer melt flow out of nozzle

effective resistance to flow in the mold

injection input

hydraulic oil volume

melt volume

recovery rate (screw linear velocity)

servovalve input voltage

servovalve spool position

injection output

injection state vector

melt bulk modulus

hydraulic oil bulk modulus

fluid mass density

servovalve time constant
203

B.9 Holding Dynamics

AH

Am

A,_

b

Cd

I(xh

I(ph

rrl

PH

P_

QHi

QHo

Uh

VH

V

X_

yh

Zh

a.

p

v.

hydraulic pressure effective area

melt pressure effective area

nozzle cross-sectional area

approximate viscous friction on screw

orifice flow discharge coefficient

servovalve spool position flow gain

servovalve hydraulic pressure flow gain

approximate mass of screw and polymer system

hydraulic pressure

polymer melt pressure

hydraulic flow into piston chamber

hydraulic flow out of piston chamber

holding input

hydraulic oil volume

melt volume

recovery rate (screw linear velocity)

servovalve input voltage

servovalve spool position

holding output

holding state vector

melt bulk modulus

hydraulic oil bulk modulus

fluid mass density

servovalve time constant

B.IO Cooling Dynamics

KM

T_,

Tw

TM

heat dissipation empirical gain

average melt temperature in the mold

cooling water temperature

heat dissipation time constant
APPENDIX C

An Example of Approximation Programming

In this thesis,a nonlinearprogram (NLP) wasdefined,the solution of which yielded

a feasible direction to the tradeoff surface. The solution to this NLP was obtained

by approximating the nonlinear constraints with linear constraints, forming a linear

program (LP) and iteratively solving this LP. This technique is known as approximation programming [53]. In this appendix, this process will be illustrated with a

simple example.

C.1 Example

The following nonlinear program is used to determine the descent direction:

subject to

maxPo (C.1)

d

Po + ai(5:)KTjq(:_), d <_ 0 i = 1,...,p (c.2)

-flj(k)Vgj(k).d <_ 0 j E [1,m] 9 gj(k) = 0 (C.3)

Po > o (c.4)

n

_ = 1 (c.5)

k=l

where ai and fli are chosen to satisfy the following:

(_,(:_)2Vfi(5:). Vfi(5:)= 1 i= 1,...,p (c.6)

and

flj(5:)2Vgj(5:). Vgj(k)- 1 j E [1,m] S 9j(5:) -- 0 (c.7)

204
205

The nonlinear constraint of Equation (C.5) is approximated by the following

three linear constraints:

-1 < d_ < 1 k = t,...,n (C.8)

Idol_< (C.9)

k--'-I

n

(I + E 3_)= E 2dkdk (C.10)

k=l k=l

These linear constraints represent a faceted approximation of the hypersphere constraint of Equation (C.5). One point that should be emphasized is that the only

constraint which is necessary is Equation (C.8), which ensures that the solution is

bounded and is a feasible direction.

Assume that the gradient vectors have been obtained for some operating point,

5:. After being normalized, they are:

Vf,(i',, _:2,&a)-'- [-0.8

and

0.56 -0.23 ] (C.11)

A physical interpretation of these gradients would be that to decrease fl one must

increase input 5:1, decrease input _:2, and increase input 5:a. To decrease f2 one

must decrease all three inputs. If the elements of fl and f2 are interpreted as

the sensitivity of the objective to each of the inputs, one would conclude that any

changes in 5:a would produce completely conflicting changes in the both objectives,

because the relative weights are equal; there is some conflict in the objectives with

respect to 5:1; and there is concurrence with respect to _2. The solution to the

original NLP is the following direction

d'= [-0.21 -0.98 0.02 ] (C.13)
/

/

/

206

This result suggests that 5:1 should be changed "a little bit", 5:2 should be changed

"a lot", and 5:3 should be "left alone". This supports the intuition discussed above.

If the approximating LP is solved, as is specified by the multiobjective algorithm, the problem is first solved without using Equation (C.10). This yielded the

following direction:

d'---[-0.31 -1.0 0.42 ] [[d'll =- 1.27 (C.14)

This direction is feasible, but there is a problem. The solution to a linear program

is always a vertex of the simplex formed by the constraints. In this case, because of

the poor approximation of the nonlinear constraint by the linear equations, there is

no vertex "along" the correct direction. For the second iteration, d was set to d 1, in

Equation (C.10), and the new LP was solved. This yielded the following direction:

d= = o 4] lld II=108

The solution is beginning to converge to d'. After the third iteration,

[-0.19 -1.0 0.05] I[d311=1.04 (C.16) d3 =

This solution is in good agreement with_'d ".

If one more iteration of the algorithm is run, the resulting direction is

[-o.al -0.980.44] IId'II=

d4=

Notice that this answer is almost exactly the same as d 1. This is another consequence

of the fact that the solution to an LP is a vertex of the simplex. As the LP result

begins to approach d', the approximating constraint of Equation (3.39) becomes

tangent to the hypersphere described by Equation (3.34) near d'. Vertices created

by this constraint are positioned away from d'. This problem of convergence is

inherent to approximation programming, as is noted by Avriel in [53]. Since any

direction generated by the multiobjective algorithm is feasible, if divergence of the

approximation program is detected, the last "best" feasible direction is used.