control algorithms, tuning, and adaptive gain tuning in

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1981

Control Algorithms, Tuning, and Adaptive GainTuning in Process Control.Alberto Arner RoviraLouisiana State University and Agricultural & Mechanical College

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R ovira, A lberto A rner

CONTROL ALGORITHMS, TUNING, AND ADAPTIVE GAIN TUNING IN PROCESS CONTROL

The Louisiana State University and Agricultural and Mechanical Col PhJO. 1981

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CONTROL ALGORITHMS, TUNING, AND ADAPTIVE GAIN TUNING IN PROCESS CONTROL

A Dissertation

Submitted to the Graduate Faculty of the Louisiana State University and

Agricultural and Mechnical College in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in

The Department of Chemical Engineering

byAlberto Amer Rovira

B.S., Louisiana State University, 1964 M.S., Louisiana State University, 1966

August 1981


ACKNOWLEDGEMENT

This work was conducted under the direction of Dr. Paul W.

Murrill, Chancellor and Professor of Chemical Engineering at

Louisiana State University. The author wishes to express his appreci

ation to Chancellor Murrill for his guidance, encouragement and

patience.

Sincere thanks are also extended to the faculty and staff of The

Department of Chemical Engineering in particular to Dr. Cecil L.

Smith and Dr. Armando B. Corripio for their generous assistance.

The author is grateful for the use of the computer facilities

provided by Project THEMIS Contract No. F44620-68-C-0021 administered

for the U.S. Department of Defense by the U.S. Air Force Office of

Scientific Research.

The author also wishes to express his thanks to Miss Helen

Chisholm and Mrs. Jan Easley for the typing and correcting of this

manuscript.

XI


TABLE OF CONTENTS

111

Page

ACKNOWLEDGEMENTS ii

LIST OF FIGURES viLIST OF TABLES XABSTRACT xiCHAPTER

I INTRODUCTION 1II STUDY OF CONTROL ALGORITHMS: LINEAR AND NONLINEAR

Introduction 4

Control Loop 4

Algorithms Considered 6

Performance Criteria 8Sensitivity 8Parameter Optimization 9

Algorithms Which Show Steady State Error or Offset 9

Algorithms Which Show No Steady State Offset 12Summary 34

References 36

III TUNING CONTROLLERS FOR SET POINT CHANGES

Introduction 37

Control Loop 37


Performance Criteria 40

Tuning Relationships 40

Application To Direct Digital Control 42

Comparison Of Set Point vs Disturbance Tuning 49

References 59

IV EFFECT OF APPLYING THE PROPORTIONAL ACTION TO THEFEEDBACK VARIABLE IN A PI ALGORITHM

Introduction 60

Control Loop 60

Control Algorithms 60

Process 62

Performance Criterion 63

Results and Discussion 63

Summary 71

References 72

V AUTOMATIC TUNING TECHNIQUE

Introduction 73

Outline of the Method 73

Process 74

Performance Criteria 77

Characterization of the Process 77

Tuning Relationships 78

Results 78

Conclusions 90

References 100

I V


VI ADAPTIVE GAIN TUNING USING PARAMETERS SENSITIVITYCOEFFICIENTS

Introduction 101

Theory of Inverse Sensitivity 102

Identification of a Second Order System 106

Application to Adaptive Gain Tuning 106

Reactor Control System 110

Results 112

References 117

VII CONCLUSIONS AND RECOMMENDATIONS 118

APPENDIX

A COMPUTER PROGRAMS 120

VITA 152

V


LIST OF FIGURES

Figure Page

II-1 Direct Digital Control Loop 5

II-2 Optimum IAE-1 vs Sampling Time (T/t). 8^/% = .5 10

II-3 Optimum IAE-2 vs Sampling Time (T/t). 8q/X = .5 11

II-4 Manipulated Variable vs Error For The Algorithmm = K |e j e. 13

11-5 Optimum lAE vs Sampling Time (T/t) For A Unit StepChange In Set Point. 8g/t =0.1 15

11-6 Optimum lAE vs Sampling Time (T/t) For A Unit StepChange in Set Point. 8g/t =0.5 16

11-7 Optimum lAE Responses To A Step Change In Set PointFor A Sançling Time (T/t) of 0.1. Process G^/t =0.5 18

11-8 Optimum lAE Responses To A Step Change In Set PointFor A Sampling Time (T/t) of 0.5. Process G^/t = 0.5 19

11-9 Optimum lAE Responses To A Step Change In Set PointFor A Sampling Time (T/t) of 1.0. Process G^/t =0.5 20

11-10 Optimum lAE Response To A Step Change In Set PointFor A Sangling Time (T/t) of 2.0. Process Gg/t = 0.5 21

11-11 Optimum lAE vs Sampling Time (T/t) For A Unit StepDisturbance. 8g/t =0.1 23

11-12 Optimum lAE vs Sangling Time (T/t) For A Unit StepDisturbance. 8g/t =0.5 24

11-13 Optimum lAE Response To A Unit Step Disturbance ForA Sampling Time (T/t) of 0.1. Process 6g/t =0.5 26

11-14 Optimum lAE Responses To A Unit Step Disturbance ForA Sampling Time (T/t) of 0.5. Process 8g/t =0.5 27

11-15 Optimum lAE Responses To A Unit Step Disturbance ForA Sampling Time (T/t) of 1.0. Process 8g/t = 0.5. 28

V I


Figure Page

II-16 Optimum lAE Responses To A Unit Step Disturbance ForA Sampling Time (T/t) of 2.0. Process 8^/t = 0.5. 29

III-l Block Diagram Of Basic Control Loop 39

III-2 Process Reaction Curve 41

III-3 Optimum Settings For Proportional Plus Integral (PI)Controller. lAE Criterion. 43

III-4 Optimum Settings For Proportional Plus Integral (PI)Controller. ITAE Criterion. 44

III-5 Optimum Settings For Proportional Plus Integral PlusDerivative (PID) Controller. lAE Criterion. 45

III-6 Optimum Settings For Proportional Plus Integral PlusDerivative (PID) Controller. ITAE Criterion. 46

III-7a Block Diagram Of Typical Digital Control Loop With ASampler And A Zero Order Hold 48

III-7b Block Diagram Of Equivalent Continuous Control Loop 48

III-8 Difference Between Set Point And Disturbance TuningParameters For A PI Controller. lAE Criterion 50

III-9a Response To A Set Point Change. Disturbance vs SetPoint Tuning. PI Algorithm. lAE Criterion 51

III-9b Response To A Disturbance Change. Disturbance vsSet Point Tuning. PI Controller. lAE Criteron. 52

Ill-lOa Response To A Set Point Change. Disturbance vsSet Point Tuning. PID Controller. ITAE Criterion 53

Ill-lOb Response To A Disturbance Change. Disturbance vsSet Point Tuning. PID Controller. ITAE Criterion 54

Ill-lla Response To A Set Point Change. Disturbance vsSet Point Tuning. PI Controller. lAE Criterion 55

Ill-llb Response To A Disturbance Change. Disturbance vsSet Point Tuning. PI Controller. lAE Criterion 56

vxx


Figura Page

III-12a Response To A Set Point Change. Disturbance vsSet Point Tuning. PID Controller. ITAE Criterion 57

III-12b Response To A Disturbance Change. Disturbance vsSet Point Tuning. PID Controller. ITAE Criterion 58

IV-1 Direct Digital Control Loop 61

IV-2a Optimum Parameters For PI and PCI Algorithms.lAE Criterion. Sampling Time of 0.1 64

IV-2b Optimum Parameters For PI and PCI AlgorithmslAE Criterion. Sang)ling Time of 0.1 65

IV-3 Response To A Step Disturbance. PI AlgorithmTuned For Set Point And For Disturbances 67

IV-4 Response To A Step Disturbance. PCI AlgorithmTuned For Set Point And For Disturbances 68

IV-5 Response To A Setp Change In Set Point. PIAlgorithm Tuned For Set Point And For Disturbances 69

IV-6 Response To A Step Change In Set Point. PCIAlgorithm Tuned For Set Point And For Disturbances 70

V-1 Stirred Tank Chemical Reactor 75

V-2a Upward Series Of Steps In Level Of Production 81

V-2b Adjusted Response. First Order Lag Plus Dead TimeTuning 82

V-2c Unadjusted Response. First Order Lag Plus Dead TimeTuning 83

V-2d Adjusted Response. Second Order Lag Pius Dead TimeTuning 84

V-2e Unadjusted Response. Second Order Lag Plus DeadTime Tuning 85

V-2f Adjusted Response. Optimum Parameters For A 10%Step Change In Feed Flow Rate 86

vxii


Figure Page

V-2g Unadjusted Response. Optimum Parameters For A 10%Step Change In Feed Flow Rate 87

V-3a Downward Series Of Steps In Level Of Production 92

V-3b Adjusted Response. First Order Lag Plus Dead TimeTuning 93

V-3c Unadjusted Response. First Order Lag Plus DeadTime Tuning 94

V-3d Adjusted Response. Second Order Lag Plut DeadTime Tuning 95

V-3e Unadjusted Response. Second Order Lag Plus DeadTime Tuning 96

V-3f Adjusted Response. Optimum Parameters For A 10%Step Change In Feed Flow Rate 97

V-3q Unadjusted Response. Optimum Paraemters For A 10%Step Change In Feed Flow Rate 98

VI-1 Tomovic's Method For Calculating Ax In The Method OfInverse Sensitivity 103

VI-2 Adaptive Gain Tuning Technique 109

VI-3 Response To Step Changes In Temperature 113

VI-4 Manipulated Variable Response 115

VI-5 Response To Step Changes In Temperature 116

I x


LIST OF TABLES

Table Page

II-l Controller Parameter Sensitivity For Set Point Changes 17

11-2 Controller Parameter Sensitivity For Disturbances 25

ll-3a Optimum lAE Parameter Settings For PI Controller 30

ll-3b Optimum lAE Parameter Settings For P^l Controller 31

ll-3c Optimum lAE Parameter Settings For PID Controller 32

II-3d Optimum lAE Parameter Settings For Hosier's Controller 33

III-1 Tuning Equations For Set Point Changes 47

V-1 Model Parameters 79

V-2 Control Parameter Settings 80

V-3 ITAE For Upward Series Of Steps 88

V-4 ITAE For Downward Series Of Steps 99

VI-1 Parameter Identification Of A Second Order System 107

Vl-2 Model Parameters From Reactor Response To A StepChange In Flow Rate At Different Temperature Levels Using Meyer's Approximation 111

X


ABSTRACT

Controller timing relationships based on optimizing the response of

a first order lag plus dead time process are developed for proportional-

plus -integral (PI) and proportional-plus-integral-plus-derivative (PID)

control algorithms. Minimum error integrals were used as criteria of

performance for the feedback control loop. The relationships presented

are shown to provide excellent response characteristics in particular

for processes where little or no overshoot is desired.

The effect of applying the proportional action to the feedback

variable in a PI control algorithm is analyzed and found to provide a

more consistent response to set point changes and load changes than the

conventional PI algorithm, tuned for either set point or load changes.

A technique of adaptive gain tuning for a PI controller is deveJ-

oped. The method is based on the use of sensitivity coefficient

analysis to the identification of model parameters. The model chosen is

a second order lag, and the technique is applied to a stirred tank

chemical reactor temperature control system. The response of the pro

posed adaptive gain tuning technique proved to be superior to the

response of the unadapted algorithm. Digital computers were used to

simulate the control systems studied.

XI


CHAPTER I

INTRODUCTION

The chemical industry has always been one of the most competi

tive and profitable sectors of our economy. In recent years the effect

of high energy costs and shortages, foreign con5>etition, and environ

mental and safety regulations have contributed to a significant decrease

in profit potential. As a result of these pressures, improved design,

operation and regulation of processes has become of utmost importance

to chemical engineers.

The proliferation of process control computers and more

recently the introduction of microprocessors have enhanced the capability

of process and control engineers to study, develope and implement new and

more sophisticated control techniques to achieve improved performance in

the operation of process plants.

Recently, one of the most active research in the field of

process control has been in the area of feedback control systems tuning,

including related areas like adaptive, direct digital and nonlinear con

trol. It is the purpose of this work to present and evaluate several

techniques in the area of control algorithm tuning and adaptive gain

tuning that may be applied to improve the performance of control loops

in process plants. Chapter II is a comparative study of selected control

algorithms with major emphasis in the sensitivity to parameter settings

and their response performance to both set point and load (disturbance)

changes. Most controller tuning relationships are based on optimizing


the responses to load changes. In Chapter II significant dif

ferences were found to exist between optimal tuning parameters for

set point vs. load changes. The advantages and disadvantages of both

types of tuning approaches are discussed in Chapter III. Tuning

relationships for proportional plus integral (PI) and proportional plus

integral plus derivative (PID) algorithms based on optimal set point

change response are presented for a first order lag plus dead time model.

A technique used to avoid fast rising and overshooting res

ponses to set point changes is to apply the proportional action of a PI

or PID controller to the controlled variable as opposed to the error

signal which is a more conventional approach. Chapter IV presents a

comparison of both methods and discusses the response characteristics to

be considered in selecting one of them. Tuning relationships to be used

are also discussed.

Most chemical processes are of a nonlinear nature. As levels

of operation and/or process parameters change, controller parameters

have to be adjusted or retuned to maintain acceptable behavior of the

process. Chapters V and VI present techniques to compensate for these

nonlinearities. The first is a simple automatic tuning technique in which

a nonlinear process is characterized by a linear model at different

operating conditions. Tuning relationships for the model are then used

to adjust controller settings as variations in process conditions occur.

Chapter VI demonstrates an adaptive gain tuning technique for

automatically adjusting the gain of a PI or PID controller. It is based

on using parametric sensitivity theory to the identification of model

parameters. The method was applied to a stirred tank reactor control


system and its performance compared to the unadapted behavior of the

system.

The results presented in this work were obtained by numerical

simulation in a digital computer. The main programs used are included

in the Appendix.


CHAPTER II

STUDY OF CONTROL ALGORITHMS: LINEAR AND NONLINEAR

Introduction

Many different types of algorithms and tuning techniques

have been proposed in the literature (1-8) for conventional analog

control and for direct digital control of processes. The ones most

commonly used are the proportional, proportional plus integral

and proportional plus integral plus derivative. These algorithms

are very effective, linear, and easily implemented in analog hard

ware; which accounts for their popularity. With the introduction

of the digital computer as a direct process controller a variety

of control functions may easily be implemented, which previously

had not been tried, in the hope of improving the control per

formance of the system. It is the purpose of this chapter to

investigate and compare different algorithms which have been

selected on the basis of their attractiveness and potential.

Tiie responses were analyzed for disturbance and set point changes.

The effect of sampling time as well as the sensitivity to tuning

parameters were also considered.

Control Loop

The algorithms were tested as part of a direct digital

control loop as the one shown in Figure II-l. In this simplified

unity feedback loop the controlled or feedback variable is fed into


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the computer at regular sampling intervals. The computer compares

this feedback variable with the set point input producing an error

signal. A control algorithm will calculate the new value of the

manipulated variable. This position of the manipulated variable

is held constant between sampling periods by a zero order hold. The

transfer function of the zero order hold is written as:

1HCs) = (II-l)

where: H(s) = zero order hold transfer function

T = sampling time

s = Laplace transform variable

The process model used is the familiar first order lag

plus dead time vdiich may be written as:

where: G(s) = process transfer function

K = process gain

0 = dead timeoT = time constant

Algorithms Considered

The algorithms studied may be divided into two groups :

a) Algorithms which show steady state offset.

b) Algorithms lAich show no steady state offset.

The main difference between these two groups is that the


algorithms which do not exhibit an offset have an integral

element which eliminates the steady state error.

The algorithms which show steady state offset

considered are;

= Kê^ proportional (II-3)

= Kg I proportional squared (II-4)

m^ = Kê^ + K g I p r o p o r t i o n a l plusproportional squared (XI-5)

= ^l®n ^2™n-l two parameter (II-6)

where: e^ = error at sampling interval n

m^ = manipulated variable at sampling interval n

= controller constant

In the second group which does not show steady state

offset the following algorithms were considered:

"n " PI (II-7)

( + PÎ (II-S)"n “ *=c

m = K n c

“n = Vn - 2Vl + - 3 Vl + %_2“n = ^c (=8° % 2 ®n (11-12)


where; T = sampling time

T^ = integral constant

Tjj = derivative constant

Performance Criteria

Several authors (9,10,11) have recommended the use of error

integrals as figures of merit or performance criteria in comparing

control system responses. In this work the integral of the absolute

value of the error (lAE) was used. Two different forms of this in

tegral were used:

a) lAE - 1 = I c - c(“) I dt (11-14)0

b) lAE - 2 = f |c - c(«>) jdt (11-15)0

where c = controlled variable

0Q = dead time of system

t = time

The lAE - 2 was used only in the study of the algorithms that

show steady state offset.

Sensitivity

Controller parameters are very seldom tuned optimally. Lack

of accuracy and approximations used in modeling the process may affect

the values obtained from tuning relationships. Therefore, a quantity


xdiich will measure the sensitivity of the response to controller

tuning is desired. For purposes of this study we shall define

sensitivity to controller parameters as:

.5(IAE^0^05^+ IAZ^_o,o5^)- lAE^ (H-16)lAEa

This is the average deviation of the criterion function

(lAE) when the parameter is 5% above and below its optimum value (a) .

Parameter Optimization

The parameters of the different algorithms were optimized

for a range of sampling times using a numerical Pattern Search ver

sion programmed by Moore (12). In order to simplify the character

ization of the process, the model parameters were nondimensionalized.

Therefore the first order lag plus dead time model may be characterized

by the ratio of dead time to time constant (G^/t) and the sampling

time becomes the ratio of sampling time to time constant (T/x).

Algorithms Which Show Steady State Error Or Offset

In this class the following algorithms were tested.

m = K e P (II-3)n c n

m = K le le p2 (II-4)n c‘ n' n

" n = + Kgm^_i 2P (H-5)

”n PP^ (II-6)

For a process with 8^/x = .5, Figure II-2 shows the optimum

values of lAE - 1 for the response to a unit disturbance of the algor

ithms listed above as a function of sampling time. Figure II-3 shows


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2an equivalent plot for lAE - 2. The proportional squared (P ) proved

to be the poorest and will not be discussed any further. The two

parameter algorithm (2P) performed better than any other, especially

at low sampling times; one drawback of this algorithm is that it is

very sensitive to tuning. It may be noticed that for sampling times

(T/t) greater than 1.6 the difference between the performance of the

algorithms became very small. The proportional plus proportional

squared algorithm performed better than the proportional algorithm;

this algorithm may be useful for nonlinear processes.

As part of the study on algorithms which show steady state

offset, the algorithm

m^ = K|e|*e (11-17)

was optimized for a unit step disturbance and a sampling time of

0.1 in order to have an indication of which would be the most desirable

type of gain in a controller. It was found that K = 1.1682 and

a = -.7405 and the algorithm showed a slight improvement over pro

portional control. Figure II-4a shows a plot of the manipulated vari

able as a function of the error. A proportional controller produces

a straight line function.

Algorithms Which Show No Steady State Offset

Analysis of the responses of the algorithms which show no

steady state offset described in equations II-7 to 11-13 showed that

for the first order lag plus dead time process studied, only four of

these algorithms show adequate behavior. These are:

m = K n c ( + ÎT ( " - 7 )


13

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1.5

1.0

1.5.5 1Error (e)

Figure II-4. Manipulated Variable vs Error for the Algorithm

m = k | e I %.

Constants: K = 1.1682

a = -.7405


14

”n = ( I f E % ")

= K-e^ - K,e„ . + (1 - K_)m. , + K-m <, Hosier (11-11) n JL n z n—X o n—x j n—z

The last algorithm is equivalent to the one proposed by

Hosier, et al (8), which in z-transform notation may be written as:

K - K_z"lD(z) = ---------- ^ (11-18)

(1 - z~^)(1 + Kgz"^)

Following is a comparative discussion of these algorithms,

which will be referred to as: P^I, PID, and Hosier's. The PI

algorithm was the basis of comparison since this is the most widely

used algorithm. Responses to both distrubance and set point changes

were compared.

1. Set point changes - The minimum lAE of the responses to set point

changes are shown for the four algorithms under consideration in

Figures II-5 and II-6, for 9^/% of .1 and .5 respectively, as

function of the sampling time (T/t). Figures II-7, II-8, XI-9 and

11-10 show the optimum (lAE) responses of each algorithm for a process

with of .5, and values of sampling time of .1, .5, 1., and 2.

Table II-l shows the sensitivities to controller parameters of each algorithm. The following observations based on these results may be made:

a) For sampling times greater than 2., the four algorithms performed similarly, meaning that the third parameter in PID and Hosier's algorithm may be omitted; thus, becoming a PI algorithm.

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TABLE II-l

Controller Parameter Sensitivity for

Set Point Changes

17

Sampling Time KjKl 1/1%iKg D ‘ 3

0.2 PI

P^I

PID

Mosler

00.69

16.40

01.95

44.52

00.963

19.82

02.84

34.54

0.25

1.74

0.5 PI

P^I

PID

Mosler

00.94

12.44

01.83

45.54

01.17

14.97

01.53

30.08

0.08

8.78

2.0 PI

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Mosler

01.51

10.26

02.0203.30

02.07

10.660.246

0.72

0.09

0.29


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LUtno

aoB.oaG.00 10.00z .oo1.00 TIME

Figure II-7. Optimum lAE Responses to a Step Change in Set

Point for a Sampling Time (T/t) of 0.1.

Process 6 /t = 0.5. o


19

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MoslerPID PIo

üJ^-tn

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0.00B.OOG.002.000-00 TIME

Figure XI-8. Optimum lAE Responses to a Step Change in Set

Point for a Sampling Time (T/x) of 0.5.

Process 6 /x = 0.5. o


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PIDPI

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2.000.00 S. 00 B.OO 10.00TIMEFigure II-9. Optimum lAE Responses to a Step Change in Set

Point for a Sampling Time (T/x) of 1.0.



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Point for a Sampling Time (T/t) of 2.0.

Process 6 /t = .5. o


22

2b) The optimum responses of P I showed little or no overshoot, but being a nonlinear algorithm, it was found to be sensitive to parameter variations.

c) Hosier's algorithm was best at sampling times in the vicinity of the dead time, but this was also an area of very high sensitivity to tuning.

2d) For PI, PID and P I there is a plateau in the lAE plots (Figures II-5 and II-6) after a sanqjling time of 1.. This indicates that sampling less often will not cause deterioration of the response, and is due to the fact that in one sampling time, the response has almost reached steady state.2e) P I with only two parameters performs as well as PID

with three. The disadvantage of P^I being its nonlinearity.

2. Disturbances - The optimum values of the lAE of the responses to

a unit step disturbance are shown in Figures 11-11 and 11-12 for

6^/t of .1 and .5 respectively. Table II-2 shows the sensitivity

to controller parameters and Figures 11-13, 11-14, 11-15, and 11-16

show optimal responses of each algorithm for sampling times of .1,

.5, 1., and 2.. The following observations, very similar to the set

point case, may be made;

a) For sampling times greater than 1.5 the lAE curves bunch together, indicating that nothing is gained by using a third parameter as in PID and Hosier's algorithms.

2b) The optimal responses of P I showed little or no undershoot.

c) Hosier's algorithm performed best at sampling times in the vicinity of the dead time, but was highly sensitive to tuning in this area.

A comparison of optimum lAE controller settings for dis

turbances and set point changes is shown in Tables II-3a,b,c and d,2for PI, P I, PID, and Hosier's algorithms respectively. They are

for a dead time (0 /t) of .5. It should be remembered that the


23

0.0 1.0Sampling Time (T/x)

Figure 11-11. Optimum lAE vs Sampling Time (T/x) for A Unit Step Disturbance. G^/x = .1


24

oen

CM

wCM

M

OO

HH

%

oag

îCO

0)og•s34JCO•HQCLm4JCO

kgHHOJB

W)c

ICOg§

ICLoCM I—II

Ou360•H&L

LAoIIH

en CM

3VI


25

TABLE II-2

Controller Parameter Sensitivity for Disturbances

Sampling Time kJ k l/ïjKj V K 3

0.2 PI 00.75 00.63

PÎ 07.38 11.24

PID 01.67 00.98 0.35

Mosler 23.53 17.44 1.38

0.5 PI 00.78 00.61

PÎ 05.28 07.24

PID 01.13 00.82 0.14

Mosler 06.31 03.26 7.24

2.0 PI 00.48 00.74

PÎ 00.17 00.52

. PID 00.55 00.87 0.09

Mosler 00.92 00.38 0.29


26

ru

PIMosler

UJ^-cn PID

Oû_COo!i lO

a

D-OD 2.00 G.00 B.OO 10.00TIMEFigure 11-13. Optimum lAE Response to a Unit Step Disturbance

for a Sampling Time (T/t) of 0.1.Process 6 /t = 0.5.O


27

ru

PIPIDMosler

2.00‘o.oo 5.00 6.00 10.00TIMEFigure 11-14. Optimum lAE Responses to a Unit Step Disturbance

for a Sampling Time (T/x) of 0.5.



28

ooeu

PI

PID■Mosler

0.00 2.00 y.00 6.00 .00 10.00TIMEFigure 11-15. Optimum lAE Response to a Unit Step Disturbance for

a Sampling Time (T/t) of 1.0.



29

eu

ÜJ^-cnoCOolljO

PI, PID, Mosler

‘d .oo 2.00 B.OO 10.00TIMEFigure 11-16. Optimum lAE Responses to a Unit Step Disturbance

for a Sampling Time (T/t) of 2.0.

Process 0 /t = 0.5.O


30

TABLE II-3a

Optimum lAE Parameter Settings for PI Controller.

Sampling Time Set Point Changes Disturbances

T K 1/T^ K 1/T^

.1 1.169 .873 1.652 .932

.5 .771 .976 1.005 .988

1.0 .709 .934 .908 .921

2.0 .297 1.587 .324 1.661

3.0 .095 3.564 .097 3.673

4.0 .073 3.435 .071 3.596


31

TABLE II-3b

2Optimum lAE Parameter Settings for P I Controller


T K 1/T\ K 1/T^

.1 1.308 .800 2.810 .527

.5 .777 .961 1.401 .682

1.0 .745 .906 1.663 .567

2.0 .252 1.801 .328 1.591

3.0 .095 3.516 .100 3.497

4.0 .071 3.558 .034 7.587


32

TABLE II-Sc

Optimum lAE Parameter Settings for PID Controller

1 ( % - e - 1)" n = + T [ 2: e .T + ?D î------- >


T K 1/T^ K 1/T^ ?D

.1 1.584 .719 .188 2.015 1.453 .251

.5 .8582 .929 .204 .885 1.553 .345

1.0 .800 .908 .425 .631 1.772 .726

2.0 .267 1.802 .252 .254 2.190 .283

3.0 .092 3.698 .109 .099 3.746 .125

4.0 .034 7.633 .042 .083 3.076 .103


33

TABLE II-3d

Optimum lAE Parameter Settings for Mosler’s Controller

“n = h .% - ^2%-l + ^ - ^3^“n-l + % _ 2


T %1 %2 %3 %1 %2 %3

.1 2.621 -2.403 1.014 4.291 -3.909 1.143

.5 2.62 -1.589 1.031 5.162 -2.530 1.642

1.0 1.682 - .619 .401 2.421 - .738 .479

2.0 1.233 - .167 .108 1.403 - .171 .110

3.0 1.110 - .041 .050 1.165 - .046 .042

4.0 1.060 - .021 .050 1.078 - .015 .015


34

sampling time and controller parameters have been nondimensionalized

(i.e., process gain and time constant equal to 1.)

The differences between set point and disturbance tuning

are considerable for every algorithm, PI showing the smallest dif

ferences. The differences in tuning parameters are larger at fast

sampling rates, decreasing as sampling becomes less frequent. The

case of the PI and PID algorithms will be discussed in greater detail

in Chapter III.

Simimary

Of the algorithms which show steady state offset only the 2ones with two parameters (PP and 2P) performed better than a propor

tional algorithm. It is questionable if the improvement in perform

ance is large enough to justify going to a two parameter algorithm.

Of the algorithms which show no offset it was found that2PI, PID, and Hosier’s performed as well or better than a PI algo-

2rithm. P I has the main disadvantage of being nonlinear and there

fore its tuning depends on the size of the disturbances. PID may be

used at fast sampling rates with success. Hosier's algorithm should

perform very well at sampling times somewhat larger than the dead

time of the system in order to avoid the highly sensitive region in

the vicinity of the dead time.

In the cases studied it was found that except for Hosier’s

algorithm at a sampling time equal to the dead time, the performance

of the algorithms did not improve greatly over PI. This is due to

the fact that in a linear first order lag plus dead time model the

response is unaffected by disturbances a length of time equal to the

dead time, therefore the disturbance will go undetected for that time


35

plus a fraction of the sampling time (depending on when the distur

bance entered the system). It is then very difficult to make the

process return to steady state in a more efficient manner than PI

does, using a conventional type of feedback controller. It may also

be concluded that a given nonlinear algorithm may be used to improve

the response of a given nonlinear process, but it is doubtful that

much can be done with linear models.


36

REFERENCES

1. Ziegler, J.G. and Nichols, N.B., "Optimum Settings for Automatic Controllers", Trans. ASME, 64, 1942.

2. Smith, C.L. and Murrill, P.W., "A More Precise Method for Tuning Controllers", Instrument Society of America Journal, May 1966.

3. Bates, H.T. and J.C. Webb, "Nonlinear Controller for Underdamped Systems", Chemical Engineering Progress, Vol. 57, p. 49 - 53, September, 1961.

4. Jones C.A. , et al., "Design of Optimum Dynamic Control Systems for Nonlinear Processes", Industrial and Engineering Chemistry Fundamentals, Vol. 2, No. 2, pp. 81-89, May 1963.

5. Lopez, A.M., J.A. Miller, P.W. Murrill and C.L. Smith, "Controller Tuning Relationships Based on Integral Criteria", Instrumentation Technology, Vol. 14, No. 11 (1967) p. 57.

6. Lopez, A.M., P.W. Murrill, and C.L. Smith, "Optimal Tuning of Proportional Digital Controllers”, Instruments and Control Systems, October, 1968.

7. Lopez, A.M., P.W. Murrill, and C.L. Smith, "Tuning PI and PID Digital Controllers", Instruments and Control Systems, Feb. 1969.

8. Mosler, H.A., et al., "Process Control by Digital Compensation",AIChE Journal, Vol. 13, No. 4., pp. 768 - 778, July, 1967.

9. Graham, D. and R.C. Lathrop, "The Synthesis of Optimum Transient Response: Criteria and Standard Forms", Transactions AIEE, Vol. 72,pt. II, pp. 273 - 288, Nov., 1953.

10. Schultz, W.C. and V.C. Rideout, "Central Systems Performance Measures: Past, Present, and Future", IRE Transactions on Automatic Control,pp. 22 - 35, February, 1961.

11. Lopez, A.M., Ph.D. Dissertation, Louisiana State University, 1967.

12. Moore, C.F., Ph.D. Dissertation, Louisiana State University, 1967.


CHAPTER III

TUNING CONTROLLERS FOR SET POINT CHANGES

Introduction

Techniques for tuning controllers based on the open loop

response of a system or process reaction curve were developed by

Ziegler and Nichols (1) and later refined by Cohen and Coon (2) and

Smith and Murrill (3). These methods were based on the use of the

one quarter decay ratio of the response as the performance criterion.

Lopez, et al (4) developed open loop tuning relationships based on

the minimization of error integral criteria. Miller, et al (5) com

pared the above mentioned tuning techniques and concluded that

techniques based on the use of error integrals as performance criteria

are superior to others.

In his work Lopez developed tuning relationshiops which

minimized the error integrals of the response to disturbance changes.

Optimum settings for disturbance changes are not optimum for changes

in set point; and in certain applications set point changes are

rather common. It is the purpose of this chapter to develop con

troller tuning relationships based on the minimization of error

integrals for set point changes. It will also be shown how these

relationships may be used to tune both conventional and direct

digital control loops.

Control Loop

This work will be concerned with single input-single output

37


38

control loops such as the one shown in the block diagram in Figure III-l.

The process block includes the dynamics of sensing elements, valves,

etc., in addition to the actual process. The controller block repre

sents the control algorithm.

The control algorithms which will be tuned are the propor

tional plus integral (PI) and the proportional plus integral plus

derivative (PID). In both cases the controllers are assumed to be

"ideal", i.e. no interaction between modes and no lags associated with

them. An ideal PID controller may be represented in transfer function

form by the following equation:

where: M(s) = manipulated variable

E(s) = error

K« = controller gain

T^ = reset time

Tp = derivative time


Open loop tuning methods are based on the characterization of the pro

cess reaction curve, the response of the process to a step change in

the controller output or manipulated variable. The process may be

represented by a first order lag plus dead time:

Output _ Ke80SInput TS + 1


39

CLoo

0 uAJ1COCOCQ

BCOu60CO*i4QUOfQ

auD60f-lb

0) o03 Pk


40

where: K = process gain

T = time constant

0Q = dead time

We may calculate the model parameters (K, T, from the

process reaction curve by the method shown in Figure III-2.


Error integral criteria have been recommended (5,6,8) for

the analysis and comparison of controller performance. In this work

two error integrals will be used:

1. The integral of the absolute value of the error, lAE.

r " lAE = J |e(t)|di

2. The integral of the time and the absolute value of the

error, ITAE.

ITAE = t|e(t)|dt

Tuning Relationships

As stated earlier in this work the tuning relationships will

be based on optimizing the response to a step change in set point as

opposed to a step change in disturbance. This problem is one of find

ing the minimum value of the integral criteria with respect to the

controller parameters. In other words the integral criteria 0 is a

function of K^, T , Tq for a given process and the optimal controller

parameters are those that minimize 0. This optimization of the


41

o

a.

CSJmvo o o

CMvO

n

5§ •H JJ . Ücd(ScoCQQ)aou04

CMIMMM<DWD00•HPK4

42

controller parameters was accomplished by using a numerical Pattern

Search version programmed by Moore (7). The calculations were per

formed in a digital computer.

To generalize the results the parameters have been non-

dimensionalized and the controller settings will be expressed as

functions of the ratio of dead time to time constant (Sq/t ). Figures

III-3 and III-4 show the tuning relationships for PI controllers and

Figures III-5 and III-6 for PID controllers. Empirical equations have

been fitted to these curves and the resulting equations are shown in

Table III-l.

Application to Direct Digital Control

Consider the block diagram of a typical direct digital con

trol loop shown in Figure III-7a. Notice that the sampler and the

zero order hold are the main difference from the conventional analog

loop considered up to now. Moore (7) showed that the sampler and

hold could be approximated by a dead time equal to one half the

sampling time of the sampler. The effect of this approximation is

shown in the equivalent diagram of Figure III-7b.

Based on this approximation the tuning technique developed

for continuous controllers in this article may be extended to the

more complex sampled data case. This is accomplished by considering

an ’’effective dead time” for direct digital control loops. This

"effective dead time" is the sum of the process model dead time plus

one half the sampling time of the loop, i.e.

6' = Go + i Ts


43

10KK

orKK

■=v

0. .2 .4 .6 .8 1

Figure III-3. Optimum Settings for Proportional Plus Integral (PI) Controller. lAE Criterion.


44

10

KK irec

or

0. .2 .4 .6 .8 1.8o/T

Figure III-4. Optimum Settings for Proportional plus Integral (PI) Controller. ITAE Criterion.


45

10

4

KKc, 3

TT,-

or

IOTt1

KK

lOT.

.5

.69^/t

Figure III-5. Optimum Settings for Proportional Plus Integral Plus Derivative (PID) Controller. lAE Criterion.


46

10

KK

lOT,

or

lOT-

0 .2 4 6 8 1

Figure III-6. Optimum Settings for Proportional Plus Integral Plus Derivative (PID) Controller. ITAE Criterion.


TABLE III-l

Tuning Equations for Set Point Changes

Controller Algorithm: M(S) = Kc ( 1 + + Tg^ E(s)

47

Tuning Equations: = a(0o/T)‘

= c + d(6o/T)

T = e(8o/T)f

Controller Criterion a b c d e f

PI lAE .758 -.861 1.02 -.323

PI ITAE .586 -.916 1.03 -.165

PID lAE 1.086 -.869 .74 -.13 .348 .914

PID ITAE .965 -.85 .796 -.1465 .308 .929


48

SamplerSetPoint

Zero Order Hold

1-e-Ts

ProcessController

Figure III-7a. Block Diagram of Typical Digital Control Loop with a Sampler and a Zero Order Hold.

- — sProcessController

Figure III-7b. Block Diagram of Equivalent Continuous Control Loop. The Dead Time Approximates the Sampler and the Zero Order Hold.


49

Using this effective dead time we may find the control param

eters in the same figures and equations used for the continuous case

(Figures III-3 to III-6, Table III-l).

Comparison of Set Point vs. Disturbance Tuning

The difference between set point and disturbance tuning

parameters for a PI controller based on the lAE criterion is shown in

Figure III-8. Values of controller gain (K ) are larger for disturbance

tuning, and values of reset time (T ) are smaller (larger 1/Ti). The

difference is the largest for small 6q/t , decreasing as 6q/t increases.The responses to a set point change for a first order lag

plus dead time with Sq/t = .5 are shown in Figure III-9a for a PI

controller. Figure III-9b shows the responses for a disturbance change.

As expected, set point tuning is best for set point changes, and dis

turbance tuning is best for disturbance changes. Neither method may be

judged better than the other since they are based on different criteria.

Processes where seldom set point changes occur should be tuned for dis

turbances. Processes where frequent set point changes occur, as is

often the case in batch processes or where supervisory control is used,

should be tuned accordingly. In other words a compromise should be

reached on what is best for each particular case in question. Figures

Ill-lOa and Ill-lOb show the same conclusions for a PID controller.

Figures III-ll and III-12 are equivalent to III-9 and III-IO,

but for Sq/t = .2. In this case the difference in tuning methods is

appreciably larger since Ôq/t is smaller.


50

or

Ti

10.k k d i s t

5. KK SET POINT4.

3. DIST

2.

1.

SET POINT.5

ns

0. 2 4 6 8 1

Figure III-8. Difference Between Set Point and Disturbance Tuning Parameters for a PI Controller. lAE Criterion.


51

ru

Dist. Tunedoo

Set Point TunedO

20.008.00S.000.00 2.00 4.00TIME

Figure III-9a. Response to a Set Point Change. Disturbance vs. Set Point Tuning. PI Controller. TAB Criterion.


52

eu

T =O

tn

Set Point Tuned

Dist. Tuned

O

10.002.00 6.00 6.00TIME

Figure III-9b. Response to a Disturbance Change. Disturbancevs. Set Point Tuning. PI Controller. lAE Criterion.


53

eu

Dist. Tunedoo

enzo Set Point Tuned

o

2.00 S . 00 8.00 10.000.00 H . 00TIME

Figure IlI-lOa. Response to a Set Point Change. Disturbance vs. Set Point Tuning. PID Controller.• ITAE Criterion.


54

ru

T =Oa

cn z ■ o

Set Point TunedCOoÜJO

Dist. Tuned

2.00 8.00 10.004 . 0 0 6.000.00 TIME

Figure IlI-lOb. Response to a Disturbance Change. Disturbance vs.Set Point Tuning. PID Controller. ITAE Criterion.


55

m

Disturbance Tuned

lUrw"enzoû-COou lO

Set Point Tuned

o

2.00 6.00 8.000.00 3 0 . 0 0TIME

Figure Ill-lla. Response to a Set Point Change. Disturbance vs.Set Point Tuning. PI Controller. lAE Criterion.


56

oaeu“

oaÎTÏzoÛ.COotu aOC

ao

0.00

e^T = .2

Set Point Tuned

Disturbance Tuned

2‘.0Q H . 00 6 . 0 0TIME 8.00 10.00

Figure Ill-llb. Response to a Disturbance Change. Disturbance vs.Set Point Tuning. PI Controller. lAE Criterion.


57

<u

Disturbance Tunedoo

V)zo

Set Point Tuned

COoli l O

o

2.00 B.OC0.00 8.00 10.00TIME

Figure III-12a. Response to a Set Point Change. Disturbance vs.Set Point Tuning. PID Controller. ITAE Criterion.

58

Oeseu'

oo

«nzon_cnotxjo

oo

6 /t - .2

Set Point Tuned

Disturbance Tuned

,00 2.00 «i-.OO 6.00TIME 8.00 10.00

Figure III-12b. Response to a Disturbance Change. . Disturbance vs.Set Point Tuning. PID Controller. ITAE Criterion.


59

REFERENCES

1. Ziegler, J. G., and N. B. Nichols, "Optimum Settings for Automatic Controllers," Trans. ASME. 64. pp. 759-765 (1942).

2. Cohen, G. H., and G. A. Coon, "Theoretical Considerations of Retarded Control," Taylor Technical Data Sheet No. TDS-IOAIOO,Taylor Instrument Companies, Rochester, Nev York.

3. Smith, C. L., and P. W. Murrill, "A More Precise Method for Tuning Controllers," ISA Journal. May 1966.

4. Lopez, A. M., et al, "Tuning Controllers with Error-Integral Criteria," Instrumentation Technology. Vol. 14, pp. 57-62, Dec. 1967.

5. Miller, J. A., et al, "A Comparison of Controller Tuning Techniques," Control Engineering. Vol. 14. pp. 72-75, Dec. 1967.

6. Graham, D., and R. C. Lathrop, "The Synthesis of Optimum Transient Response: Criteria and Standard Forms," Transactions IEEE. Vol. 72.pt. II, pp. 273-288, Nov. 1953.

7. Moore, C. F., Ph.D. Dissertation, Louisiana State University, 1969.

8. Lopez, A. M., Ph.D. Dissertation, Louisiana State University, 1967.


CHAPTER IV

EFFECT OF APPLYING THE PROPORTIONAL ACTION TO THE FEEDBACK VARIABLE IN A PI ALGORITHM

Introduction

The difference existing between the controller tuning par

ameters obtained by optimizing the response to disturbances and the

parameters obtained by optimizing the response to set point changes

has been shown in Chapter III. In a proportional plus integral (PI)

algorithm, the proportional action may be applied to the error signal

or to the feedback variable (1,2). Both variations respond in the same

manner to disturbances, but they respond differently to changes in

set point. In this chapter the responses of both algorithms to the

different types of inputs will be studied.

Control Loop

A typical single input-single output direct digital control

feedback loop is shown in Figure IV-1. The loop differs from a con

tinuous control system by the introduction of the samplers and zero

order hold. The computer, represented by the dotted block, comprises

the comparator and the control algorithm.

Control Algorithms

An ideal continuous proportional plus integral control al

gorithm may be written as

60


61

COü

COXJ

4J4JCOCO

§ ë Sw oNOS:

COCO

ouAJc8

tapÜouP

O300•H


62

where: m = manipulated variable

e = error signal

t = time

Kg = controller gain

Ti = reset or integral time

■When used in direct digital control, the discrete form of

this algorithm is:

“n = Kc [ en + ^lAt]i=0

where: At = sampling time

n = sampling interval

A variation of the common PI algorithm described above is to

apply the proportional action to the feedback or controlled variable (c)

instead of to the error signal (e). This algorithm which will be

called proportional on the controlled variable plus integral (PCI) may

be expressed as:

CO

“n = ^c [-Cn + 2 ®nn=l

where: c = controlled or feedback variable

It may be noted that both algorithms are equivalent when

no set point changes occur. It is when set point changes take place

that these two algorithms behave differently.

Process

A first order lag plus dead time model will be used to test

the algorithms; i.e..


63

G(s) = K =•©oS

TS + 1

where: G(s) = process transfer function

K = process gain

T = time constant

0Q = dead time


For convenience the process gain and time constant will be

considered equal to one. This is equivalent to non-dimensionalization

of the equation. The process may then be characterized by the ratio

of dead time to time constant (©q/t ).

Performance Criterion

The integral of the absolute value of the error (lAE) will be

used as the criterion of performance. It may be expressed as:

lAE = J lejdto

Results and Discussion

The parameters which minimize the integral of the absolute

error (lAE) of the responses to disturbance and set point changes were

found for both PI and PCI algorithms. These optimal parameters are

plotted in Figure IV-2a and IV 2b for a range of ©q/t from 0.1 to 1.0,

and a sampling time of 0.1. Since disturbance tuning is the same for

both algorithms only one curve is shown on each graph corresponding to

disturbances, while two curves are needed for set point tuning: one

for each algorithm. It may be noticed that the set point tuning curves


64

KK

5.

4.Disturbance Tuned |~

3. Set Point Tuned PCI

Set Point Tuned PI2.

1.

0.2 .4 . 6 1.0

Figure IV-2a - Optimum Parameters for PI and PCI Algorithms, based on

the Response to Disturbance and Set Point Changes. lAE Criterion.

Sampling Time of 0.1.


65

-U.4-

1.4 ,i_L.

a1.2

1.0■TT

Set Point Tuned PCI

Disturbance Tuned

Set Point Tuned PI

u.

0.2 .4 .8 1.0

Figure IV-2b - Optimum Parameters for PI and PCI Algorithms, based

on the Response to Disturbance and Set Point Changes. lAE Criterion.

Sampling Time of 0.1.


66

for the PCI algorithms are much closer to the disturbance tuning curve

than those for the PI algorithm. Therefore, if both a PI and a PCI al

gorithms are tuned for disturbances, the PCI will be closer to optimal

set point tuning than the PI algorithm. It may also be pointed out

that as 6o/t increases, the parameters are less dependent on the type

of tuning for both algorithms, and the differences between disturbance

and set point tuning become smaller.

Responses to a step change in set point and disturbance for a

process with a Qq/t of 0.2 are shown in Figures IV-3 to IV-6. Figures

IV-3 and IV-5 show the responses of a PI algorithm to disturbance and

set point changes respectively for both types of tuning. Figures IV-4

and IV-6 show equivalent responses for a PCI algorithm.

Figure IV-4 shows that there is almost no difference between

disturbance and set point tuning for a PCI algorithm when the system is

subject to disturbances, while Figure IV-3 shows that PI responds much

slower to disturbances when set point tuned than when tuned for dis

turbances. Figure IV-5 shows the large overshoot inherent in PI

algorithms tuned for disturbances when a set point change occurs. This

overshoot is not present in the response of a PCI algorithm as shown in

Figure IV-6, but a slower rising response is noticed.

Tuning relationships have been developed for PI controllers

based on minimizing error integrals by Lopez, et al (3,4) for dis

turbances and in Chapter III of this work for set point changes.

Lopez's tuning relationships also apply for a PCI controller since they

are based on the response to disturbances. From the closeness of the

parameters discussed above, it seems that the development of set point

tuning relationships for PCI algorithms is not justified, and the


67

oooj'

oabJ^-en

aO

ÜJC3

Set Point Tuning - lAE = .425

Dist. Tuning - lAE = .201

'O.QO 2 .0Q *i‘.OD b '.OOTIME 8.00 10.00

Figure IV-3 - Response to a step disturbance.PI Algorithm tuned for set point and for disturbances.


68

aoeu

cnzoa .COoÜ J OCCn-

Dist. Tuning - lAE = .201

Set Point Tuning - lAE = .212oo

G.00 8.002.00 10.00TIME

Figure IV-4 - Response to a step disturbance.PCI Algorithm tuned for set point and for disturbances.


69

eu

Disturbance Tuning - lAE = .48

OÙ_COoUjo

Set Point Tuning - lAE = .265

G.DO 8.00 10.00TIME

Figure IV-5 - Response to a step change in set point. PI Algorithm tuned for set point and for disturbances.


70

CM

Set Point Tuning - lAE = .457oa

W.J-z

COor , l ODisturbance Tuning - lAE = .595

G.OO0.00 2.00 8.00 20.00TIME

Figure IV-6 - Response to a step change in set point. PCI Algorithm tuned for set point and for disturbances.


71

tuning relationships developed by Lopez for disturbances are good

enough for any types of input when a PCI controller is used.

Summary

The following points should be made before concluding this

discussion:

a) PCI seems to be a more consistent algorithm, tuning being less dependent on the type of input considered than a PI controller.

b) The PCI algorithm may be implemented in continuous control systems.

c) The principle of applying the proportional action to the feedback or controlled variable may also be applied to proportional-plus-integral-plus-derivative (PID) algorithms with similar results.

d) The PCI algorithm is not recommended for use in the slave controller of a cascade system, because in this case it is desired that the proportional action respond immediately to the set point change.


72

REFERENCES

1. Bernard, J.W. and Cashen, J.F., "Direct Digital Control", Instruments and Control Systems, Sept. 1965.

2. Woodley, G.V., "Building Block Approach to DDC Algorithm Implementation", 22nd Annual International ISE Instrumentation - Automation Conference, Chicago, Illinois, Sept. 11-14, 1967.

3. Lopez, A.M. , et al, "Tuning Controllers with Error-Integral Criteria", Instrumentation Technology, Vol. 14, pp. 57-62, Nov. 1967.

4. Lopez, A.M., et al, "Tunign PI and PID Digital Controllers", Instruments and Control Systems, Feb. 1969.


CHAPTER V

AUTOMATIC TUNING TECHNIQUE

Introduction

The tuning of process controllers in the chemical industries

is usually done manually in the field or in the control room by instru

ment engineers and technicians. In many processes where levels of

production are changed fairly often, the controller parameters have to

be readjusted or retuned after each one of these changes due to the

nonlinear nature of most processes. A method is proposed herein, where

control parameters are tuned automatically when changes in level of

production occur. The method is easily implemented when direct digital

control is being used. A small variation may also be implemented in

continuous conventional analog control.

The method proposed is simulated as part of a chemical reactor

temperature control system, different alternatives are also evaluated.

Outline of the Method

In the proposed automatic parameter adjusting technique out

lined here, a nonlinear process is characterized by a linear model at

different levels of production. With sufficient points, the parameters

of the linear model may be expressed as functions of the level of pro

duction. This may be accomplished by fitting the data points with an

empirical equation, or by using tables and linear interpolation between

points. The model used may be one obtained from experimental plant

73


74

tests, as the ones used in this work. Tuning relationships for the

model being used must be available.

With the information described above stored in the control

computer, and sampling the variable which defines the level of pro

duction; i.e., feed flow rate, as soon as a change in level of pro

duction is noticed, the model parameters may be calculated for this

new level and from the tuning relationships the new control param

eters obtained and adjusted. It will be noted that the controller

parameters may be stored directly as functions of the level of pro

duction, therefore saving storage and one step in the calculation

procedure; on the other hand, tuning relationships stored may be used

for several loops.

An explanation of the chemical reactor model in which the

proposed method is simulated, the control strategy, and analysis of

the performance of the method follows.

Process

The process for evaluating the proposed control parameter

adjustment scheme is the continuous stirred tank chemical reactor

system shown in Figure V-1. Feed at a flow rate W, temperature Tj,

and concentration enters the reactor of volume V . Two moles of

component A react to produce one mole of B according to the equation:

A — Y

The reaction is second order, irreversible, and exothermic.

The rate of reaction being given by

rA = kC^


75

Wo 0000000 T,C,

(p

A — D-------~(tRC)------- ^

Parameters :

w = 50 Ib/min Pt = 55 Ib/ft^

^i = 198°F Pw = 62.4 Ib/ft^

^Ao " .5975 Ib-mole/ft^ = .9 BTU/lb “F

Ttt- = 80°F C — 1 BTU/lb °FWi pw

\ = 13.38 ft^ m = -6000 BTU/lb-mole2 -a/TA = 100 ft k — V

= 8.64 ft^ = 8.33 X 10® ft®/lb-

U = 25 BTU/hr °F ft^ a = 14000°R

Figure V-1. Stirred Tank Chemical Reactor


76

The product stream leaves at a temperature T and concentra

tion C . To remove the heat evolved by the reaction, cooling water is

circulated through a tube bundle, water entering at temperature T j

and leaving at T^q.

The basic assumptions made in the development of the mathe

matical model are:

1. The contents of the tank are perfectly mixed, so that the concentration and temperature of the reacting mass and exit stream maybe considered equal.

2. The contents of the tube bundle are recirculated fast enough sothat the temperature of the cooling water may be considered uniformand equal to the exit water temperature.

3. The physical properties of the different streams, and the tank level are considered to be constant.

4. Heat losses to the surroundings and heat generated by the impeller are negligible.

5. There is no change in volume due to the reaction.

6. The valve and temperature sensor dynamics may be represented by first order lags with time constants of and Tg minutes respectively.

The equations which describe the system are:

Material Balance on A:

Heat Balance on reactor:

Heat Balance on tube bundle:

_ M , UA'"Pwdt“ VsPw VbP^C- ■ ^wo)


77

Valve dynamics:

Temperature sensor dynamics:

TR(s) = T r r V i T(s)

The controller acting on the cooling water flow rate is an "ideal" pro

portional plus integral controller which may be represented as:

MV(s) = Kc (1 + =1-) E(s)i-iS


The performance criterion used to evaluate and compare the

control parameter adjustment scheme is the integral of the time weighted

absolute error, namely:

ITAE =o

t e dt

The ITAE criterion was selected since it produces less oscil

lations in the response and in general lower values of gain and l/T^,

when compared to the integral of the absolute value of the error.

Characterization of the Process

Two different models were used to characterize the reactor

system considered. Both of these methods are based on the response to

a step change in the manipulated variable, or process reaction curve.


78

A process reaction curve was obtained at each of the three levels of

production considered, namely 50%, 75% and 100%. The two models obtained

were the popular first order lag plus dead time model, and the second

order lag plus dead time model approximation proposed by Meyer (2). The

values of the model parameters are shown in Table V-1. The variation of

these parameters from level to level points out the nonlinearity of the

reactor system. These parameters may be fitted with an equation or used

in table form, interpolating for intermediate points.

Tuning Relationships

Another reason which supports the choice of the models discus

sed above is that tuning relationships are readily available for both of

them. Lopez, et al (1) and Lopez (3) developed tuning relationships

based on the optimization of error integrals.

For the first order plus dead time model, empirical equations

for the gain and the reset time based on minimizing the integral of

time weighted absolute error (ITAE) are (1):

il T

For the second order plus dead time model equations are not

available and graphs (3) have to be used.

Results

Table V-2 shows the values of the control parameters for a

PI controller at three different levels of production (50%, 75% and

100%) as obtained by:


79

TABLE V-1

MODEL PARAMETERS

First order lag plus dead time model

G(s) K e"®°® s + 1

Feed Flow Rate (lb/hr)50

K

.782@o

3.43

T

15.41

75 .3185 2.7 12.55

100 .1305 2.25 9.25

Second order lag plus dead time model

e-®o®G(s) - -rj-

s + 2gwn= +

Feed Flow Rate (lb/hr) K ®o C Wn

50 .782 1.1 1.18 .118

75 .3185 1.0 1.1 .154

100 .1305 0.85 0.97 .197


80

c.

TABLE V-2

CONTROL PARAMETER SETTINGS

Based on First Order Lag Plus Dead Time Model and Tuning Relationships

Feed Flow Rate (lb/hr) %c

1Ti

50 4.73 .0875

75 11.3 .1065

100 25.3 .1375

Based on Second Order Lag Plus Dead Time Model and Tuning Relationships

Feed Flow Rate (lb/hr)

Kc 1

50 10.25 .063

75 17.3 .082

100 39.8 .089

Based on Minimizing ITAE for a 10% Upward Step Change in ] Flow Rate at Each different Level

Feed Flow Rate (lb/hr) Kc 1

?i50 10.9 .047

75 18.24 .07

100 40.8 .089


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00

88

TABLE V-3

ITAE FOR UPWARD SERIES OF STEPS

a. Using first order lag plus dead time tuning

50-75% 75-100%

Adjusted 2110 1109

Unadjusted 1124 894

b. Using second order lag plus dead time tuning

50-75 % 75-100 %

Adjusted 706 338

Unadjusted 594 1162

c. Using settings from optimization of the ITAE of the response to a 10% step in feed flow rate

50-75 % 75-100 %

Adjusted 606 352

Unadjusted 933 1869


89

a) Using the first order lag plus dead time model and corresponding ITAE tuning relationships.

b) Using the second order lag plus dead time model and corresponding ITAE tuning relationships.

c) By minimizing the ITAE of the response of the system to a10% upward step change in feed flow rate.

From this table it can be seen that the second order lag plus

dead time model yields parameter values much closer to the parameters

obtained by optimizing the response of the system to a 10% step change

in feed flow rate, than those obtained by using the first order lag plus

dead time model.

A sequence of two step changes in feed flow rate were applied

to the reactor system to compare the use of the parameter adjustment

technique with the unadjusted case for each one of the three sets of

parameters under consideration.

The sequences of steps taken are shown in Figures V-2a and

V-3a. The first sequence is from 50% to 75% and then from 75% to 100%

level of production. A time of 120 minutes elapses between steps so

that steady state may be reached. The second sequence is in the re

verse direction; i.e., from 100% to 75% and then from 75% to 50% level

of production.

Figures V-2b, V-2d and V-2f show the responses of the auto

matically adjusted control loop for the upward series of steps, while

Figures V-2c, V-2e and V-2g show the unadjusted case. Table V-3 shows

the values of the integral of time weighted absolute error (ITAE) for

each one of the steps. The use of the adjustment technique proved to

be very good when the second order model or the optimization parameters

were used. When used with the first order model there is a deteriora

tion of control for the first step but an improvement in the second


90

step. This poor performance is attributed to the fact that the first

order lag plus dead time model does not approximate the dynamics of the

reactor system with sufficient accuracy.

For the downward series of steps, the responses for the ad

justed and unadjusted cases are shown in Figures V-3b to V-3g, and

values of the ITAE criterion are shown in Table V-4 for each step.

The responses of the unadjusted system turned out to be unstable, show

ing that some form of tuning adjustment is necessary, either manually

by operators or automatically as proposed in this article. The re

sponses of the automatically adjusted system turned out to be very good,

and again the second order lag plus dead time tuning proved to be better

than the first order lag plus dead time tuning.

Conclusions

From the results of this study, two basic conclusions may be

drawn. The first is the superiority in this case of the second order

lag plus dead time model over the first order lag plus dead time model,

and the closeness of the predicted parameters from the second order

model to those obtained by the optimization of the response. The first

order model has been very popular in use by the control engineers, and

the second order model has been generally rejected since it introduces

one extra parameter. From this work it appears that it should perhaps

be considered since increase in accuracy of results may pay for the

little extra work in its use.

The second conclusion is that it is necessary in many cases

to retune control parameters when production changes occur, and a

simple adjusting technique, like the one discussed, may be developed.


91

This technique may be easily implemented in direct digital control com

puters and also in continuous analog control installations.


92

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00

99

TABLE V-4

ITAE FOR DOWNWARD SERIES OF STEPS

a. Using first order lag plus dead time tuning

100-75 % 75-50 %Adjusted 2394 3505

Unadjusted unstable unstable

b. Using second order lag plus dead time tuning

100-75 % 75-50 %Adjusted 803 1488


c. Using settings from optimization of the ITAE of the response to a 10% step in feed flow rate

100-75 % 75-50 %Adjusted 701 1212



100

REFERENCES

1. Lopez, A.M., et al, "Controller Tuning Relationships Based onIntegral Criteria", Instrumentation Technology, Vol. 14, No. 11 (1967) p. 57.

2. Meyer, J.R., at al, "Simplifying Process Response Approximation",Instruments and Control Systems, Vol. 40, December 1967.

3. Lopez, A.M., Ph.D. Dissertation, Louisiana State University, 1967.


CHAPTER VI

ADAPTIVE GAIN TUNING USING PARAMETER SENSITIVITY COEFFICIENTS

Introduction

The nature of most chemical processes is that their response

characteristics vary both with time and operating conditions. The

inherent linearities of most conventional controllers require that

controller parameter changes or retuning be performed to compensate

for the process nonlinearities. In Chapter V a simple automatic

tuning technique was presented in which the model parameters and con

troller settings were identified a priori from the responses of the

process to step changes at different levels of operation. In this

chapter a real time adaptive gain tuning technique is presented.

The method is based on Tomovic’s (1) application of sensi

tivity coefficients to the identification of model parameters, also

referred to as the problem of inverse sensitivity.

The adaptive technique will be applied to a chemical reactor temperature control system, and the model reference will be a second order system.

The purpose of this work is to investigate the use of the identification technique and its applicability to adaptive control.

101


102

Theory of Inverse Sensitivity

Given a dynamic systemF(x, X, X, t, q^) = 0 (VI-1)

where; x = independent variable

t = time

= parameters ; i = 1, ... m the sensitivity or influence coefficients are defined as:

u = (VI-2)Sq.

where; u = sensitivity coefficient of ^i parameter q^X = system output q^ = model parameter

If we designate the variations of the system caused by parameter changes as:

Ax = x(t, q^^ + Aq ) - x(t, q ^) (VI-3)

then we can say that:

Ax = f(Aq^) i = 1, 2, 3, ... m (VI-4)

or formally:

Aq^ = f"^(Ax) (VI-5)

This is a statement of the problem of inverse sensitivity which

relates the system parameter variations to the deviations of the system

output from the ideal system or model. In his work, Tomovic (1) pro

poses a method to solve this problem of inverse sensitivity. It is

shown pictorially in Figure VI-1, and is based in comparing the output

of the dynamic system with the output fo the model to produce the

function Ax in equation VI-5.


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The Method of Inverse Sensitivity.

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104

Since there is no one-to-one solution to equation VI-5,

the solution will be accomplished by reducing the problem to one of

optimization. Tomovic proposes optimizing the expression:

I = Opt (Ax + AXq) (VI-6)

where Ax is the difference between system and model as shown in Figure

VI-1^ and AXq will be the linear approximation of the system:m m

\ = Z 9 ^ • = Z io*1=1 ^lo i=l

where u^^ are by definition the sensitivity coefficients of the system

parameters. By substituting VI-7 into VI-6 and using the minimum of

the integral of the square of the error over an interval of time T

as the optimization criteria, we get:

T 2I . = f {Ax(t) + (u.Aq. + ... u Aq^)} dt (VI-8)min 1 1 m nn

Differentiating with respect to each parameter, for a three parameter

system. Equation VI-8 reduces to a system of algebraic equations

Oil Aq^ + 0^2 ^ 2 + ^13 ^ 3 = ^i ^ = 1,2,3 (VI-9)

Twhere: b^ = - Ax(t)u^(t)dt (VI-10)

and IC.. = C.. = I u.(t) • u.(t)dt for i = 1,2,3 CVI-11)ij 1 3

The system of algebraic equations in VI-9 can be solved for the

parameter variations Aq^ which may then be used to update the model

parameters. The linear approximation used in equation VI-7 will

place restrictions in the region of convergence. In cases in which

the starting points of the model parameters are outside this region


105

of convergence, a steepest descent method proposed by Marquardt (2)

will be used. In this method a constant factor X is added to the

diagonal elements of the matrix described by Equations VI-9. We

may express this in matrix notation as:

(Ç + XI) = B (VI-12)

As the value of X increases a gradient method is obtained.

For X equal to zero, the system of equations reduces to the original

set of equations (VI-9).

Finally, in order to calculate the coefficients of Equation

VI-9 we must know the sensitivity coefficients of the model parameters.

In this work a second order system will be used as the mathematical

model:

ÿ + Ay + By = K • f(t) (VI-13)

Taking partial derivatives with respect to each parameter and inverting

the order of differentiation we arrive at the set of differential

equations:

Ü, + AÙ, + Bu. = 0 A A Aüg + AUg + BUg = -y

u^ + Au^ + Bu^ = f(t) (VI-14)

with initial conditions, u_ = uu = 0. A numerical solution of these

equations simultaneously with Equation VI—13 will provide the para

meter sensitivity coefficients required to calculate the coefficients

of Equation VI-9.


106

Identification of a Second Order System

In order to verify the applicability of the identification technique, similar second order systems were used for both process and model. The process equation used was:

X + Ax + Bx = Ky (VI-15)with parameters A = .16

B = .01 K = .01

To test convergence, different starting points were chosen for

the model parameters and a step change in the input (y) was used for

excitation. Iterative computations of new model parameters were per

formed at time intervals of 4 minutes. The results are shown in Table

VI-1.

For most cases convergence occurred before the response reached

steady state usually within 5 iterations. In one case, where the start

ing points were quite far from the actual process parameters, convergence

was not achieved except when using Marquardt's gradient method. Trial

and error produced a value of X equal to 0.5 as being adequate for con

vergence. These results validate the applicability of the use of sensi

tivity coefficients to continuous identification of process parameters.

Application to Adaptive Gain Tuning

Having proven the effectiveness of the method of inverse sensitivity to the identification of model parameters, a technique for applying this concept to adaptive gain tuning of a control system was developed.


107

TABLE VI-1

Parameter Identification of a Second Order System

Process: x + Ax + Bx = Ky

where K = .01

A = cl5

B = .01

Starting Values ÿ of Intervals

K A B For Convergence

.1 .16 .01 1

.009 .13 .013 3

.001 .13 .013 10 (1)

.01 .16 .002 2

.01 .16 .1 3

.01 .03 .01 4

.01 .25 .01 3

.006 .13 .018 4

(1) Marquardt's Method Required for Convergence. X - .5


108

This technique is also based in using a second order system as

the model reference in the identification algorithm but allowing only the

gain of the model (K) to vary. In this fashion the identification is

reduced to one parameter. The value of this parameter is then used to

adjust the gain or proportional action of a proportional plus integral

(PI) control algorithm that controls the manipulated variable of the

process. The reasons or assumptions for developing this technique

follow:

- reducing the identification of model parameters to the

process gain simplifies the computations to be performed

in real time by a computer.

- the greatest return or benefit in the dynamic response of a

system is obtained by maintaining a constant loop gain.

- in tuning a PI controller for a second order system, the

proportional action is influenced more by the system gain

than by other system parameters.

- the system gain varies more than the other parameters for

most systems. As will be shown later, this is the case for

the reactor control system to which the technique will be

applied.

The proposed adaptive gain tuning method is illustrated in Figure VI-2.

Inverse sensitivity is used to identify the model gain (K) in real time

from the difference between process and model outputs (Ax). The model

is updated with the new value and a new controller gain (K ) is calcu

lated from it.


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110

Reactor Control System

The process selected for testing the proposed adaptive gain

tuning technique was the simulated stirred tank chemical reactor temper

ature control system described in Chapter V. The reader is referred

to Chapter V for a detailed discussion of the process and the equations

that describe it.

In order to select initial parameters for the identification of

the model, the method developed by Meyer (3) for approximating process

response with second order systems was used. Step changes in cooling

water flow rate were applied to the reactor and from the open loop res

ponse, Meyer's graphical method was used to calculate the second order

model parameters. Table VI-2 shows the parameters calculated at three

temperature levels. In this table it can be noticed that the process

gain (K) exhibits variations (non-linearities) much larger than the

other two parameters (Ç and W^), lending support to the assumption made

previously.

Averages of these parameters were used as starting points in

the identification of the process. As a result of several test runs

it was found that values of = 0.08 and Ç = 1.0 could be used satis

factorily over the temperature range of operation.

The temperature controller algorithm used in the reactor

system is the velocity form of the standard PI algorithm:

m — m - = K (e - e . + * e * AT) (VI-16)n n-1 c n n-1 T^ n


Ill

TABLE VI-2

Model Parameters From Reactor Response to a

Step Change in Flow Rate at Different Temperature

Levels Using Meyer's Approximation

C(s) KModel Equation: -- ~ 9----------- 9s + 2ÇW + W n n

Temperature K Ç(°F)

178 .23 .90 .10

198 .97 1.01 .09

218 1.65 1.25 .08


112

where = manipulated variable

e^ = error signal

= controller gain

= integral time

The controller gain was adjusted from the model parameters using an

equation of the form

^1K = -# (VI-17)c K

where = constant

K = model gain

The integral parameter (1/T^) was kept constant since the contributing

parameters to the process time constants (| and were not allowed to

vary. Values of and 1/T^ were estimated from the controller settings

used for the reactor system in Chapter V and from optimization of the

response to a 5®F step change in set point.

Results

The adaptive gain tuning technique was tested by applying a

series of step changes in the temperature set point of the reactor

system. Figure VI-3 shows the responses to step changes from 175°F

to 188“F and from 188°F to 198°F. For comparison purposes, both the

adapted and the unadapted cases were plotted. The adaptive technique

showed improvement over the unadapted case as shown by the integral of

the absolute value of the error (lAE) values: 142 vs 201 for the first


113

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114

step, and 114 vs 163 for the second step. Figure VI-4 shows a plot of

the manipulated variable (cooling water rate) for the steps. It" can be

seen that the adapted case controlled the water flow rate in a smoother

fashion. Note that after the initial effect of the set point change, the

adaptive case did not overcool the reactor; this was due to the lower

controller gain calculated because of the increased process gain at

higher temperatures.

Figure VI-5 shows the responses for larger step changes: from

175°F to 198°F and from 198°F to 218°F. The results are similar to

the ones discussed for the first series of steps.


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REFERENCES

1. Tomovic, Raj ko. Sensitivity Analysis of Dynamic Systess, McGraw-Hill, New York, 1963.

2. Marquardt, D.W. , "An Algorithm for Least-Squares Estimation of Nonlinear Parameters", J.S.I.A.M., Vol. 2, June 1963, pp. 431-441.

3. Meyer, J.R. et al, "Simplifying Process Response Approximation", Instruments and Control Systems, Vol. 40, December 1967.

4. Mollenkamp, R.E., et al, "Using Models to Tune Industrial Controllers", Instruments and Control Systems, Vol. 46, September 1973.

5. Meissinger, H.F., "The Use of Parameter Influence Coefficients in the Computer Analysis of Dynamic Systems", Proceedings of the Western Joint Computer Conference, May 1960.


CHAPTER VII

CONCLUSIONS AND RECOMMENDATIONS

This work was intended to present and evaluate control

algorithms, tuning techniques, and their use in industrial control

applications. Chapters II, III and BT covered the performance

of control algorithms and tuning techniques. Chapters V and VI

discussed how to automatically retune controllers when used in

nonlinear processes. Chapter VI addressing a new adaptive gain tuning

technique using inverse sensitivity theory. As a result of this

work the following general conclusions are presented:

- Sensitivity of control algorithm response to tuning para

meters is a very important consideration in algorithm selection.2Although some algorithms like P I and Hosier showed an improvement

over conventional PI and PID algorithms when optimally tuned, they

also showed a sensitivity to parameter settings that may not be

desirable in practical applications.

- Tuning parameters developed from optimizing responses to

step changes in set point can differ significantly from those de

veloped by optimizing response to disturbance changes. The set

point tuning relationships presented in Chapter III are more con

servative than disturbance tuning relationships, and are recommended

for use in processes where frequent set point changes occur, or

when overshoot in the response is not desirable.

118


119

- The Proportional on the Control Variable plus Integral

(PCI) algorithm was found to be less dependent on the type of

tuning than the PI algorithm, and produced more consistent res

ponses for both types of excitation, showing little or no over

shoot to set point changes.

- The adaptive gain tuning method proposed in this work

using inverse sensitivity theory proved to be an effective techr-

nique for automatically adjusting the controller gain in non

linear processes.

Based on this work, it is recommended that additional

research be conducted to expand the use of inverse sensitivity

to identify and adapt all three parameters of the second order

model, and also include adjustments to the reset and rate para

meters of the controller.

Finally, recognizing that computer simulation is one step

removed from actual process plant operation, it is felt that the

true test of the techniques proposed in this work will come when

they are implemented in a live process control enviroment.


APPENDIX A

Computer Programs

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VITA

Alberto Amer Rovira was b o m in Santiago de Cuba, Cuba on

January 30, 1943. He obtained his elementary and secondary education

at the Colegio de Dolores in Santiago de Cuba graduating in 1959.

He attended the Universidad de Oriente in Cuba for one year and

subsequently transfered to Louisiana State University where he re

ceived his Bachelor of Science Degree in Chemical Engineering in May

1964. He was awarded a Master of Science Degree in Chemical

Engineering in May 1966.

In October 1979, he married the former Jo Eva Peak of Walker,

Louisiana.

From 1969 to the present he has been employed by the

International Business Machines Corporation and is currently a

Systems Engineering Manager in Houston, Texas.

152


EXAMINATION AND THESIS REPORT

Candidate: Alberto Amer Rovira

Major Field: Chemical Engineering

Title of Thesis: Control Algorithms, Timing, and Adaptive Gain Tuningin Process Control

Approved:

Major Professor and Chairman

^ Dean of the Graduate^chool

EXAMINING COMMITTEE:

Date of Examination:

December 12, 1980


control algorithms, tuning, and adaptive gain tuning in

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