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LSU Historical Dissertations and Theses Graduate School
1981
Control Algorithms, Tuning, and Adaptive GainTuning in Process Control.Alberto Arner RoviraLouisiana State University and Agricultural & Mechanical College
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Recommended CitationRovira, Alberto Arner, "Control Algorithms, Tuning, and Adaptive Gain Tuning in Process Control." (1981). LSU HistoricalDissertations and Theses. 3656.https://digitalcommons.lsu.edu/gradschool_disstheses/3656
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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;
= K^e^ proportional (II-3)
= Kg I proportional squared (II-4)
m^ = K^e^ + 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)
( + P^I (II-S)"n “ *=c
m = K n c
“n = Vn - 2Vl + - 3 Vl + %_2“n = ^c (=8° % 2 ®n (11-12)
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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
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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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10
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12
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 )
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13
5mrH•H
'O04JCÜiH1 I
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
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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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15
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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
p2l
PID
Mosler
01.51
10.26
02.0203.30
02.07
10.660.246
0.72
0.09
0.29
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18
oatu
.PID Moslera
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
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19
ni
MoslerPID PIo
üJ^-tn
o
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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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20
ruMosler
PIDPI
Lüen
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.
Process 0 /x = 0.5. o
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21
aaru
PI, PID, Mosler
OÛ.CODll lOÛC«-
o
10.006.00 B.OO0.00 2.00 TIMEFigure 11-10. Optimum lAE Responses to a Step Change in Set
Point for a Sampling Time (T/t) of 2.0.
Process 6 /t = .5. o
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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 non- linearity.
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
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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
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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
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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
P^I 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
P^I 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
P^I 00.17 00.52
. PID 00.55 00.87 0.09
Mosler 00.92 00.38 0.29
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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
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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.
Process 0 /x = 0.5. o
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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.
Process 6 /x = 0.5. o
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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
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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
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31
TABLE II-3b
2Optimum lAE Parameter Settings for P I Controller
Sampling Time Set Point Changes Disturbances
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
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32
TABLE II-Sc
Optimum lAE Parameter Settings for PID Controller
1 ( % - e - 1)" n = + T [ 2: e .T + ?D î------- >
Sampling Time Set Point Changes Disturbances
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
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33
TABLE II-3d
Optimum lAE Parameter Settings for Mosler’s Controller
“n = h .% - ^2%-l + ^ - ^3^“n-l + % _ 2
Sampling Time Set Point Changes Disturbances
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
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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
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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.
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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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
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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
s = Laplace transform variable
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
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39
CLoo
0 uAJ1COCOCQ
BCOu60CO*i4QUOfQ
auD60f-lb
0) o03 Pk
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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.
Performance Criteria
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
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41
o
a.
CSJmvo o o
CMvO
n
<u>5§ •H JJ . Ücd(ScoCQQ)aou04
CMIMMM<DWD00•HPK4
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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
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43
10KK
orKK
■=v
0. .2 .4 .6 .8 1
Figure III-3. Optimum Settings for Proportional Plus Integral (PI) Controller. lAE Criterion.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
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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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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.
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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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
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61
COü
COXJ
4J4JCOCO
§ ë Sw oNOS:
COCO
ouAJc8
tapÜouP
O300•H
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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..
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63
G(s) = K =•©oS
TS + 1
where: G(s) = process transfer function
K = process gain
T = time constant
0Q = dead time
s = Laplace transform variable
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
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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.
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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.
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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
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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.
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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.
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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.
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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.
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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.
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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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
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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^
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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
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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)
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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
Performance Criteria
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.
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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:
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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
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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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81
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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
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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
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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.
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91
This technique may be easily implemented in direct digital control com
puters and also in continuous analog control installations.
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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
Unadjusted unstable unstable
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
Unadjusted unstable unstable
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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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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
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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.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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The Method of Inverse Sensitivity.
ow
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
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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.
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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.
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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
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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
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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
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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
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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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115
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117
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.
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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
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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.
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APPENDIX A
Computer Programs
120
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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
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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
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