Corripio, armando b tuning of industrial control systems ISA (2001)

Tuning of
Industrial
Control Systems
Second Edition

by Armando B. Corripio, Ph.D., P.E.
Louisiana State University


Notice
The information presented in this publication is for the general education of the reader. Because
neither the author nor the publisher have any control over the use of the information by the reader,
both the author and the publisher disclaim any and all liability of any kind arising out of such use.
The reader is expected to exercise sound professional judgment in using any of the information
presented in a particular application.
Additionally, neither the author nor the publisher have investigated or considered the affect of any
patents on the ability of the reader to use any of the information in a particular application. The
reader is responsible for reviewing any possible patents that may affect any particular use of the
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make any representation regarding the availability of any referenced commercial product at any
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times, even if in conflict with the information in this publication.

Copyright © 2001 ISA—The Instrumentation, Systems, and Automation Society.
All rights reserved.
Printed in the United States of America.
No part of this publication may be reproduced, stored in retrieval system, or
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recording or otherwise, without the prior written permission of the publisher.
ISA
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P.O. Box 12277
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Library of Congress Cataloging-in-Publication Data
Corripio, Armando B.
Tuning of industrial control systems / Armando B. Corripio.-- 2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN 1-55617-713-5
1. Process control--Automation. 2. Feedback control systems. I. Title.
TS156.8. C678 2000
670.42’75--dc21
00-010127


TABLE OF CONTENTS
Unit 1: Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-1.
Course Coverage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-2.
Purpose. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-3.
Audience and Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-4.
Study Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-5.
Organization and Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1-6.
Course Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-7.
Course Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
3
4
4
4
4
5
6

Unit 2: Feedback Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2-1.
The Feedback Control Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2-2.
Proportional, Integral, and Derivative Modes . . . . . . . . . . . . . . . . 13
2-3.
Typical Industrial Feedback Controllers. . . . . . . . . . . . . . . . . . . . . 19
2-4.
Stability of the Feedback Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2-5.
Determining the Ultimate Gain and Period . . . . . . . . . . . . . . . . . . 24
2-6.
Tuning for Quarter-decay Response . . . . . . . . . . . . . . . . . . . . . . . . 25
2-7.
Need for Alternatives to Ultimate Gain Tuning . . . . . . . . . . . . . . 31
2-8.

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Unit 3: Open-Loop Characterization of Process Dynamics . . . . . . . . . . . . . . . . .
3-1.
Open-Loop Testing: Why and How. . . . . . . . . . . . . . . . . . . . . . . . .
3-2.
Process Parameters from Step Test . . . . . . . . . . . . . . . . . . . . . . . . .
3-3.
Estimating Time Constant and Dead Time. . . . . . . . . . . . . . . . . . .
3-4.
Physical Significance of the Time Constant . . . . . . . . . . . . . . . . . .
3-5.
Physical Significance of the Dead Time. . . . . . . . . . . . . . . . . . . . . .
3-6.
Effect of Process Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-7.
Testing Batch Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-8.
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35
37
39
41
45
49
52
55
56

Unit 4: How to Tune Feedback Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-1.
Tuning for Quarter-decay Ratio Response . . . . . . . . . . . . . . . . . . .
4-2.
A Simple Method for Tuning Feedback Controllers . . . . . . . . . . .
4-3.
Comparative Examples of Controller Tuning . . . . . . . . . . . . . . . .
4-4.
Practical Controller Tuning Tips . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-5.
Reset Windup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-6.
Processes with Inverse Response . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-7.
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59
61
64
65
74
77
78
81

Unit 5: Mode Selection and Tuning Common Feedback Loops . . . . . . . . . . . . .
5-1.
Deciding on the Control Objective. . . . . . . . . . . . . . . . . . . . . . . . . .
5-2.
Flow Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-3.

Level and Pressure Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-4.
Temperature Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-5.
Analyzer Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-6.
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83
85
86
88
94
96
97

Unit 6: Computer Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6-1.
The PID Control Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6-2.
Tuning Computer Feedback Controllers . . . . . . . . . . . . . . . . . . . 108
6-3.
Selecting the Controller Processing Frequency . . . . . . . . . . . . . . 115
6-4.
Compensating for Dead Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6-5.
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
vii

viii

Table of Contents

Unit 7: Tuning Cascade Control Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-1.
When to Apply Cascade Control . . . . . . . . . . . . . . . . . . . . . . . . . .
7-2.
Selecting Controller Modes for Cascade Control. . . . . . . . . . . . .
7-3.
Tuning Cascade Control Systems. . . . . . . . . . . . . . . . . . . . . . . . . .
7-4.
Reset Windup in Cascade Control Systems . . . . . . . . . . . . . . . . .
7-5.
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125
127
130
131
139
142

Unit 8: Feedforward and Ratio Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8-1.
Why Feedforward Control? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8-2.
The Design of Linear Feedforward Controllers . . . . . . . . . . . . . .
8-3.
Tuning Linear Feedforward Controllers . . . . . . . . . . . . . . . . . . . .

8-4.
Nonlinear Feedforward Compensation . . . . . . . . . . . . . . . . . . . .
8-5.
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143
145
150
152
157
164

Unit 9: Multivariable Control Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
9-1.
What Is Loop Interaction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
9-2.
Pairing Controlled and Manipulated Variables. . . . . . . . . . . . . . 173
9-3.
Design and Tuning of Decouplers . . . . . . . . . . . . . . . . . . . . . . . . . 183
9-4.
Tuning Multivariable Control Systems . . . . . . . . . . . . . . . . . . . . . 188
9-5.
Model Reference Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
9-6.
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Unit 10: Adaptive and Self-tuning Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10-1. When Is Adaptive Control Needed? . . . . . . . . . . . . . . . . . . . . . . .
10-2. Adaptive Control by Preset Compensation . . . . . . . . . . . . . . . . .
10-3. Adaptive Control by Pattern Recognition . . . . . . . . . . . . . . . . . .
10-4. Adaptive Control by Discrete Parameter Estimation . . . . . . . . .

10-5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

197
199
202
209
212
220

Appendix A: Suggested Reading and Study Materials. . . . . . . . . . . . . . . . . . . . 223
Appendix B: Solutions to All Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251


Unit 1:
Introduction and
Overview


UNIT 1
Introduction and Overview
Welcome to Tuning of Industrial Control Systems. The first unit of this selfstudy program provides the information you will need to take the course.
Learning Objectives — When you have completed this unit, you should be
able to:

A. Understand the general organization of the course.
B.

Know the course objectives.

C. Know how to proceed through the course.

1-1.

Course Coverage
This book focuses on the fundamental techniques for tuning industrial
control systems. It covers the following topics:
A. The common techniques for representing and measuring the dynamic
characteristics of the controlled process.
B.

The selection and tuning of the various modes of feedback control,
including those of computer- and microprocessor-based controllers.

C. The selection and tuning of advanced control techniques, such as
cascade, feedforward, multivariable, and adaptive control.
When you finish this course you will understand how the methods for
tuning industrial control systems relate to the dynamic characteristics of
the controlled process. By approaching the subject in this way you will
gain insight into the tuning procedures rather than simply memorizing a
series of recipes.
Because microprocessor- and computer-based controllers are now widely
used in industry, this book will extend the techniques originally
developed for analog instruments to digital controllers. We will examine
tuning techniques that have been specifically developed for digital
controllers as well as those for adaptive and auto-tuning controllers.
No attempt is made in this book to provide an exhaustive presentation of
tuning techniques. In fact, we have specifically omitted techniques based
on frequency response, root locus, and state space analysis because they
are more applicable to electrical and aerospace systems than to industrial

3


4

Unit 1: Introduction and Overview

processes. Such techniques are unsuitable for tuning industrial control
systems because of the nonlinear nature of industrial systems and the
presence of transportation lag (dead time or time delay).

1-2.

Purpose
The purpose of this book is to present, in easily understood terms, the
principles and practice of industrial controller tuning. Although this
course cannot replace actual field experience, it is designed to give you the
insights into the tuning problem to speed up your learning process during
field training.

1-3.

Audience and Prerequisites
The material covered will be useful to engineers, first-line supervisors,
and senior technicians who are concerned with the design, installation,
and operation of process control systems. The course will also be helpful
to students in technical schools, colleges, or universities who wish to gain
some insight into the practical aspects of automatic controller tuning.
There are no specific prerequisites for taking this course. However, you
will find it helpful to have some familiarity with the basic concepts of

automatic process control, whether acquired through practical experience
or academic study. In terms of mathematical skills, you do not need to be
intimately familiar with some of the mathematics used in the text in order
to understand the fundamentals of tuning. This book has been designed to
minimize the barrier that mathematics usually presents to students’
understanding of automatic control concepts.

1-4.

Study Materials
This textbook is the only study material required in this course. It is an
independent, stand-alone textbook that is uniquely and specifically
designed for self-style.
Appendix A contains a list of suggested readings to provide you with
additional reference and study materials.

1-5.

Organization and Sequence
This book is organized into ten separate units. The next three units (Units
2-4) are designed to teach you the fundamental concepts of tuning,
namely, the modes of feedback control, the characterization and
measurement of process dynamic response, the selection of controller


Unit 1: Introduction and Overview

5

performance, and the adjustment of the tuning parameters. Unit 5 tells

you how to select controller modes and tuning parameters for some
typical control loops. An entire unit, Unit 6, is devoted to the specific
problem of tuning computer- and microprocessor-based controllers. The
last four units, Units 7 through 10, demonstrate how to tune the more
advanced industrial control strategies, namely, cascade, feedforward,
multivariable, and adaptive control systems.
As mentioned, the method of instruction used is self-study: you select the
pace at which you learn best. You may browse through or completely skip
some units if you feel you are intimately familiar with their subject matter
and devote more time to other units that contain material new to you.
Each unit is designed in a consistent format with a set of specific learning
objectives stated at the very beginning of the unit. Note these learning
objectives carefully; the material in the unit will teach to these objectives.
Each unit also contains examples to illustrate specific concepts and
exercises to test your understanding of these concepts. The solutions for
all of these exercises are contained in Appendix B, so you can check your
own solutions against them.
You are encouraged to make notes in this textbook. Ample white space has
been provided on every page for this specific purpose.

1-6.

Course Objectives
When you have completed this entire book, you should:
• Know how to characterize the dynamic response of an industrial
process.
• Know how to measure the dynamic parameters of a process.
• Know how to select performance criteria and tune feedback controllers.
• Know how to pick the right controller modes and tuning parameters to match the objectives of the control system.
• Understand the effect of sampling frequency on the performance of

computer-based controllers.
• Know when to apply and how to tune cascade, feedforward, ratio,
and multivariable control systems.
• Know how to apply adaptive and auto-tuning control techniques to
compensate for process nonlinearities.


6

Unit 1: Introduction and Overview

Besides these overall course objectives, each individual unit contains its
own set of learning objectives, which will help you direct your study.

1-7.

Course Length
The basic premise of self-study is that students learn best when they
proceed at their own pace. As a result, the amount of time individual
students require for completion will vary substantially. Most students will
complete this course in thirty to forty hours, but your actual time will
depend on your experience and personal aptitude.


Unit 2:
Feedback Controllers


UNIT 2
Feedback Controllers

This unit introduces the basic modes of feedback control, the important
concept of control loop stability, and the ultimate gain or closed-loop
method for tuning controllers.
Learning Objectives — When you have completed this unit, you should be
able to:

A. Understand the concept of feedback control.
B.

Describe the three basic controller modes.

C. Define stability, ultimate loop gain, and ultimate period.
D. Tune simple feedback control by the ultimate gain or closed-loop
method.

2-1.

The Feedback Control Loop
The earliest known industrial application of automatic control was the
flywheel governor. This was a simple feedback controller, introduced by
James Watt (1736-1819) in 1775, for controlling the speed of the steam
engine in the presence of varying loads. The concept had been used earlier
to control the speed of windmills. To better understand the concept of
feedback control, consider, as an example, the steam heater sketched in
Figure 2-1.

Steam
FS
Process
F

Fluid

C

Ti
Steam
Trap

Condensate
Figure 2-1. Example of a Controlled Process: A Steam Heater

9


10

Unit 2: Feedback Controllers

The process fluid flows inside the tubes of the heater and is heated by
steam condensing on the outside of the tubes. The objective is to control
the outlet temperature, C, of the process fluid in the presence of variations
in process fluid flow (throughput or load), F, and in its inlet temperature,
Ti. This is accomplished by manipulating or adjusting the steam rate to the
heater, Fs, and with it the rate at which heat is transferred into the process
fluid, thus affecting its outlet temperature.
In the example in Figure 2-1, the outlet temperature is the controlled,
measured, or output variable; the steam flow is the manipulated variable; and
the process fluid flow and inlet temperature are the disturbances. These
terms refer to the variables in a control system. They will be used
throughout this book.

Now that we have defined the important variables of the control system,
the next step is to decide how to accomplish the objective of controlling
the temperature. In Figure 2-1, the approach is to set up a feedback control
loop, which is the most common industrial control technique—in fact, it is
the “bread and butter” of industrial automatic control. The following
procedure illustrates the concept of feedback control:
Measure the controlled variable, compare it with its desired value,
and adjust the manipulated variable based on the difference
between the two.
The desired value of the controlled variable is the set point, and the
difference between the controlled variable and the set point is the error.
Figure 2-2 shows the three pieces of instrumentation that are required to
implement the feedback control scheme:
1.

A control valve for manipulating the steam flow.

2.

A feedback controller, TC, for comparing the controlled variable
with the set point and calculating the signal to the control valve.

3.

A sensor/transmitter, TT, for measuring the controlled variable
and transmitting its value to the controller.

The controller and the sensor/transmitter are typically electronic or
pneumatic. In the former case, the signals are electric currents in the range
of 4-20 mA (milliamperes), while in the latter they are air pressure signals

in the range of 3-15 psig (pounds per square inch gauge). The control valve
is usually pneumatically operated, which means that the electric current
signal from the controller must be converted to an air pressure signal. This
is done by a current-to-pressure transducer.


Unit 2: Feedback Controllers

11

Steam

Setpoint

r

FS
m

TC
b

F

TT

Ti

C

Process
Fluid
Steam
Trap

Condensate
Figure 2-2. Feedback Control Loop for Heater Outlet Temperature

Modern control systems also use digital controllers. There are three basic
types of digital controllers: distributed control systems (DCS), computer
controllers, and programmable logic controllers (PLC). Some of the more
modern installations use the “fieldbus” concept, in which the signals are
transmitted digitally, that is, in the form of zeros and ones.
This book is in accordance with standard ANSI/ISA S5.1-1984 (R1992),
Instrumentation Symbols and Identification. Further, the degree of detail
is per Section 6.12, Example 2, “Typical Symbolism for Conceptual
Diagrams,” that is, diagrams that convey the basic control concepts
without regard to the specific implementation hardware. The diagram in
Figure 2-2 is an example of a conceptual diagram.
Figure 2-2 shows that the feedback control scheme creates a loop around
which signals travel. A change in outlet temperature, C, causes a
proportional change in the signal to the controller, b, and therefore an
error, e. The controller acts on this error by changing the signal to the
control valve, m, causing a change in steam flow to the heater, Fs. This
causes a change in the outlet temperature, C, which then starts a new cycle
of changes around the loop.
The control loop and its various components are easier to recognize when
they are represented as a block diagram, as shown in Figure 2-3. Block
diagrams were introduced by James Watt, who recognized that the
complex workings of the linkages and levers in the flywheel governor are

12

Unit 2: Feedback Controllers

Heater

Heater

Controller

Valve

Heater

Sensor

Figure 2-3. Block Diagram of Feedback Control Loop

easier to explain and understand if they are considered as signal
processing blocks and comparators. The basic elements of a block diagram
are arrows, blocks (rectangles), and comparators (circles). The arrows
represent the instrument signals and process variables, for example,
transmitter and controller output signals, steam flow, outlet temperature,
and so on. The blocks (rectangles) represent the processing of the signals
by the instruments as well as the lags, delays, and magnitude changes of
the variables caused by the process and other pieces of equipment. For
example, the blocks in Figure 2-3 represent the control valve, the sensor/
transmitter, the controller, and the heater. Finally, the comparators (circles)

represent the addition and/or subtraction of signals, for example, the
calculation of the error signal by the controller.
The signs in the diagram in Figure 2-3 represent the action of the various
input signals on the output signal. That is, a positive sign means that an
increase in input causes an increase in output or direct action, while a
negative sign means that an increase in input causes a decrease in output
or reverse action. For example, the negative sign by the process flow into
the heater means that an increase in flow results in a decrease in outlet
temperature. By following the signals around the loop you will notice that
there is a net reverse action in the loop. This property is known as negative
feedback and, as we will show shortly, it is required if the loop is to be
stable.
The most important component of a feedback control loop is the feedback
controller. It will be the subject of the next two sections.


Unit 2: Feedback Controllers

2-2.

13

Proportional, Integral, and Derivative Modes
The previous section showed that the purpose of the feedback controller is
twofold. First, it computes the error as the difference between the
controlled variable and the set point, and, second, it computes the signal
to the control valve based on the error. This section presents the three basic
modes the controller uses to perform the second of these two functions.
The next section (2-3) discusses how these modes are combined to form
the feedback controllers most commonly used in industry.

The three basic modes of feedback control are proportional, integral or reset,
and derivative or rate. Each of these modes introduces an adjustable or
tuning parameter into the operation of the feedback controller. The
controller can consist of a single mode, a combination of two modes, or all
three.
Proportional Mode

The purpose of the proportional mode is to cause an instantaneous
response in the controller output to changes in the error. The formula for
the proportional mode is the following:
Kce

(2-1)

where Kc is the controller gain and e is the error. The significance of the
controller gain is that as it increases so does the change in the controller
output caused by a given error. This is illustrated in Figure 2-4, where the
response in the controller output that is due to the proportional mode is
shown for an instantaneous or step change in error, at various values of
the gain.
Another way of looking at the gain is that as it increases the change in
error that causes a full-scale change in the controller output signal
decreases. The gain is therefore sometimes expressed as the proportional
band (PB) or the change in the transmitter signal (expressed as a
percentage of its range) that is required to cause a 100 percent change in
controller output. The relationship between the controller gain and its
proportional band is then given by the following formula:
PB = 100/Kc

(2-2)

Some instrument manufacturers calibrate the controller gain as
proportional band, while others calibrate it as the gain. It is very important
to realize that increasing the gain reduces the proportional band and vice
versa.


14

Unit 2: Feedback Controllers

Figure 2-4. Response of Proportional Controller to Constant Error

Offset

The proportional mode cannot by itself eliminate the error at steady state
in the presence of disturbances and changes in set point. The
unavoidability of this permanent error or offset can best be understood by
imagining that the steam heater control loop of Figure 2-2 has a controller
that has proportional mode only. The formula for such a controller is as
follows:
m = m0 + Kce

(2-3)

where m is the controller output signal and m0 is its bias or base value.
This base value is usually adjusted at calibration time to be about 50
percent of the controller output range so as to give the controller room to
move in each direction. However, assume that the bias on the temperature
controller of the steam heater has been adjusted so as to produce zero error

at the normal operating conditions, that is, to position the steam control
valve so that the steam flow is that flow required to produce the desired
outlet temperature at the normal process flow and inlet temperature. In
this manner the initial error of the controller is zero and the controller
output is equal to the bias term.
Figure 2-5 shows the response of the outlet temperature and of the
controller output to a step change in process flow for the case of no control
and for the case of two different values of the proportional gain. For the
case of no control, the steam rate remains the same, which causes the
temperature to drop because there is more fluid to heat with the same
amount of heat. The proportional controller can reduce this error by
opening the steam valve, as shown in Figure 2-5. However, it cannot


Unit 2: Feedback Controllers

15

Figure 2-5. Response of Heater Temperature to Step Change in Process Flow Using a
Proportional Controller

eliminate it completely because, as Eq. 2-3 shows, zero error results in the
original steam valve position, which is not enough steam rate to bring the
temperature back up to its desired value. Although an increased controller
gain results in a smaller steady-state error or offset, it also causes, as
shown in Figure 2-5, oscillations in the response. These oscillations are
caused by the time delays on the signals as they travel around the loop
and by overcorrection on the part of the controller as the gain is increased.
To eliminate the offset a control mode other than proportional is required,
namely, the integral mode.

Integral Mode

The purpose of the integral or reset mode is to eliminate the offset or
steady-state error. It does this by integrating or accumulating the error
over time. The formula for the integral mode is the following:
Kc
------ e dt
TI

∫

(2-4)

where TI is the integral or reset time, and t is time. The calculus operation
of integration is somewhat difficult to visualize, and perhaps it is best
understood by using a physical analogy. Consider the tank shown in
Figure 2-6. Assume that the liquid level in the tank represents the output
of the integral action, while the difference between the inlet and outlet
flow rates represents the error e. When the inlet flow rate is higher than
the outlet flow rate, the error is positive, and the level rises with time at a
rate that is proportional to the error. Conversely, if the outlet flow rate is
higher than the inlet, the level drops at a rate proportional to the negative


16

Unit 2: Feedback Controllers

error. Finally, the only way for the level to remain stationary is for the inlet
and outlet flows to be equal, in which case the error is zero. The integral

mode of the feedback controller acts exactly in this manner, thus fulfilling
its purpose of forcing the error to zero at steady state.
The integral time TI is the tuning parameter of the integral mode. In the
analogous tank in Figure 2-6, the cross-sectional area of the tank represents
the integral time. The smaller the integral time (area), the faster the
controller output (level) will change for a given error (difference in flows).
As the proportional gain is part of the integral mode, integral time means
the time it takes for the integral mode to match the instantaneous change
caused by the proportional mode on a step change in error. This concept is
illustrated in Figure 2-7.

Figure 2-6. Tank Analog of Integral Controller

Figure 2-7. Response of PI Controller to a Constant Error


Unit 2: Feedback Controllers

17

Some instrument manufacturers calibrate the integral mode parameter as
the reset rate, which is simply the reciprocal of the integral time. Again, it is
important to realize that increasing the integral time results in a decrease
in the reset rate and vice versa.
Although the integral mode is effective in eliminating offset, it is slower
than the proportional mode in that it must act over a period of time. A
faster mode than the proportional is the derivative mode, which we
discuss next.
Derivative Mode

The derivative or rate mode responds to the rate of change of the error
over time. This speeds up the controller action, compensating for some of
the delays in the feedback loop. The formula for the derivative action is as
follows:
de
K c TD -----dt

(2-5)

where TD is the derivative or rate time. The derivative time is the time it
takes the proportional mode to match the instantaneous action of the
derivative mode on an error that changes linearly with time (a ramp). This
is illustrated in Figure 2-8. Notice that the derivative mode acts only when
the error is changing with time.

Figure 2-8. Response of PD Controller to an Error Ramp


18

Unit 2: Feedback Controllers

On-Off Control

The three basic modes of feedback control presented in this section are all
proportional to the error in their action. That is, a doubling in the
magnitude of the error causes a doubling in the magnitude of the change
in controller output. By contrast, on-off control operates by switching the
controller output from one end of its range to the other based only on the
sign of the error, not on its magnitude. On-off controllers are not generally

used in process control, and when they are it is very simple to tune them.
Their only adjustment is the magnitude of a dead band around the set
point.
The next section, 2-3, discusses the procedures for combining the three
basic control modes to produce industrial process controllers. However,
before doing this we need to simplify the notation for the integral and
derivative modes; a simple look at Eqs. 2-4 and 2-5 makes it clear why. A
simpler notation is achieved by introducing the Heaviside operator “s.”
Oliver Heaviside (1850-1925) was a British physicist who baffled
mathematicians by noting, without proof, that the differentiation operator
d/dt could be treated as an algebraic quantity, a quantity we will represent
by the symbol “s” here. Heaviside’s concept makes it easy to simplify our
notation as follows:
• se will denote the rate of change of the error
• e/s will denote the integral of the error
Integration is the reciprocal operation because the rate of change of the
output is proportional to the input. This allows us to write the formulas
for the integral and derivative modes as follows:
K
Integral mode: -------c- e
TI s

(2-6)

Derivative mode: K c T D s e

(2-7)

These expressions are easier to manipulate than Eqs. 2-4 and 2-5. For those
readers who are not comfortable with the mathematics, be assured that we

will use these expressions only to simplify the presentation of the material.
Nevertheless, it is important to associate the s operator with rate of change
and its reciprocal with integration. It is also important to realize that since
s is associated with rate of change, it takes on a value of zero (that is, it
disappears) at steady state, when variables do not change with time.


Unit 2: Feedback Controllers

2-3.

19

Typical Industrial Feedback Controllers
Most industrial feedback controllers, about 75 percent, are
proportional-integral (PI) or two-mode controllers, and most of the rest are
proportional-integral-derivative (PID) or three-mode controllers. As Unit
6 will show, there are a few applications for which single-mode
controllers, either proportional or integral, are indicated, but not many. It
is also rather easy to tune a single-mode controller, as only one tuning
parameter needs to be adjusted. In this section, we will look at PI and PID
controllers in terms of how the modes are combined and implemented.
The formula for the PI controller is produced by simply adding the
proportional and integral modes:
K
m = K c e + -------c- e = K c [ 1 + ( 1 ⁄ T I s ) ] e
Ts

(2-8)

I

Eq. 2-8 shows that the PI controller has two adjustable parameters, the
gain Kc and the integral or reset time TI. Figure 2-9 presents a block
diagram representation of the PI controller.
The simplest formula for the PID or three-mode controller is the addition
of the proportional, integral, and derivative modes, as follows:
K
m = K c e + -------c- e + K c TD s e = K c [ 1 + ( 1 ⁄ T I s ) + TD s ] e
Ts

(2-9)

I

This equation shows that the PID controller has three adjustable or tuning
parameters, the gain Kc, the integral or reset time TI, and the derivative or

1
Tls
r

m

e

KC

b
Figure 2-9. Block Diagram of PI Controller

20

Unit 2: Feedback Controllers

rate time TD. The block diagram implementation of Eq. 2-9 is sketched in
Figure 2-10. The figure also shows an alternative form that is more
commonly used because it avoids taking the rate of change of the set point
input to the controller. This prevents derivative kick, an undesirable pulse of
short duration on the controller output that would take place when the
process operator changes the set point.
The formula of Eq. 2-9 is commonly used in computer-based controllers,
as Unit 6 will show. This form is sometimes called the “parallel” PID
controller because, as Figure 2-10 shows, the three modes are in parallel.
All analog and most microprocessor (distributed) controllers use a
“series” PID controller, which is given by the following formula:
m = Kc ′ [ 1 + ( 1 ⁄ TI ′ s ) ] [ ( 1 + TD ′ s ) ⁄ ( 1 + α TD ′ s ) ]

(2-10)

The last term in brackets in Eq. 2-10 is a derivative unit and is attached to
the standard PI controller of Figure 2-9 to create the PID controller, as
shown in Figure 2-11. It contains a filter (lag) to prevent the derivative
mode from amplifying noise. The derivative unit is installed on the
controlled variable input to the controller to avoid the derivative kick, just
as in Figure 2-10. The value of the filter parameter α in Eq. 2-10 is not
adjustable; it is built into the design of the controller. It is usually of the
order of

1
T ls
r

e

KO

m

TDs

b

1
T ls
r

e

b

KO

m

TDs

Figure 2-10. Block Diagram of Parallel PID Controller with Derivative on the Error Signal, and
with Derivative on the Measurement

Unit 2: Feedback Controllers

21

0.05 to 0.1. The noise filter can and should be added to the derivative term
of the parallel version of the PID controller. Its effect on the response of the
controller is usually negligible because the lag time constant, αTD, is small
relative to the response time of the loop.
The three formulas in Eq. 2-11 convert the parameters of the series PID
controller to those of the parallel version:
Kc = Kc'Fsp TI = TI'Fsp TD = TD'/Fsp

(2-11)

where
Fsp = 1 + (TD'/TI')
The formulas for converting the parallel PID parameters to the series are
as follows:
Kc' = KcFps

TI' = TIFps TD' = TD/Fps

(2-12)

where
Fps = 0.5 + [0.25 - (TD/TI)]0.5
Because of this difference between the parameters of the series and
parallel versions of the PID controller, this will be indicated explicitly

whether the tuning parameters are for one version or the other. It follows
that in tuning a controller you must determine whether it is the series or
parallel form by using the manuals for the specific controllers. Notice that
there is no difference when the derivative time is zero (PI controller).

Figure 2-11. Block Diagram of Series PID Controller with Derivative on the Measurement


22

Unit 2: Feedback Controllers

All industrial feedback controllers, whether they are electronic,
pneumatic, or computer-based, have the following features:
Features intended for the plant operator—
• Controlled variable display
• Set point display
• Controller output signal display
• Set point adjustment
• Manual output adjustment
• Remote/local set point switch (cascade systems only)
• Auto/manual switch
Features intended for the instrument or control engineer—
• Proportional gain, integral time, and derivative time adjustments
• Direct/reverse action switch
The operator features are on the front of panel-mounted controllers or in
the “menu” of the computer control video display screens. The
instrument/control engineer features are on the side of panel-mounted
controllers; in computer control systems they are in separate computer
video screens that can be accessed only by a key or separate password.

Now that we have described the most common forms of feedback
controllers, we will turn in the next section to the concept of loop stability,
that is, the interaction between the controller and the process.

2-4.

Stability of the Feedback Loop
One of the characteristics of feedback control loops is that they may
become unstable. The loop is said to be unstable when a small change in
disturbance or set point causes the system to deviate widely from its
normal operating point. The two possible causes of instability are that the
controller has the incorrect action or it is tuned two tightly, that is, the gain
is too high, the integral time is too small, the derivative time is too high, or
a combination of these. Another possible cause is that the process is
inherently unstable, but this is rare.
When the controller has the incorrect action, you can recognize instability
by the controller output “running away” to either its upper or its lower


Unit 2: Feedback Controllers

23

limit. For example, suppose the temperature controller on the steam heater
of Figure 2-2 was set so that an increasing temperature increases its
output. In this case, a small increase in temperature would result in an
opening of the steam valve, which in turn would increase the temperature
further, and the cycle would continue until the controller output reached
its maximum with the steam valve fully opened. On the other hand, a
small decrease in temperature would result in a closing of the steam valve,

which would further reduce the temperature, and the cycle would
continue until the controller output is at its minimum point with the steam
valve fully closed. Thus, for the temperature control loop of Figure 2-2 to
be stable, the controller action must be “increasing measurement decreases
output.” This is known as reverse action.
When the controller is tuned too tightly, you can recognize instability by
observing that the signals in the loop oscillate and the amplitude of the
oscillations increases with time, as in Figure 2-12. The reason for this type
of instability is that the tightly tuned controller overcorrects for the error
and, because of the delays and lags around the loop, the overcorrections
are not detected by the controller until some time later. This causes a larger
error in the opposite direction and further overcorrection. If this is allowed
to continue the controller output will end up oscillating between its upper
and lower limits.
As pointed out earlier, the oscillatory type of instability is caused by the
controller having too high a gain, too fast an integral time, too high a
derivative time, or a combination of these. This is a good point to
introduce the simplest method for characterizing the process in order to
tune the controller: determining the ultimate gain and period of oscillation
of the loop.

Figure 2-12. Response of Unstable Feedback Control Loop