Principles and practice of automatic process control, 2° ed. carl a. smith & armando b. corripio
PRACTICE OF A
ARMAND~ B. Co
SELECTED TABLES AND FIGURES
Common input signals
Stable and unstable responses
First-order step response
First-order ramp response
First-order sinusoidal response
Lead-lag step response
Lead-lag ramp response
Second-order step response
z-transforms and modified z-transforms
On-line quarter decay ratio
Open-loop quarter decay ratio
Minimum error integral for disturbance
Minimum error integral for set point
Controller synthesis (IMC) rules
Computer PID control algorithms
Dead time compensation algorithms
ISA standard instrumentation symbols and
Control valve inherent characteristics
Control valve installed characteristics
Flow sensors and their characteristics
Temperature sensors and their
Classification of filled-system thermometers
Thermocouple voltage versus temperature
Valve capacity (Cv) coefficients
Unity feedback loop
Temperature control loop
Flow control loop
Pressure control loop
Level control loop
Multivariable (2 X 2) control loop
Decoupled multivariable (2 X 2) system
Sampled data control loop
Internal Model Control (IMC)
Dynamic Matrix Control (DMC)
Principles and Practice of
Automatic Process Control
Carlos A. Smith, Ph.D., P.E.
University of South Florida
Armando B. Corripio, Ph.D., P.E.
Louisiana State University
John Wiley & Sons, Inc.
This work is dedicated with all our love to The Lord our God,
for all his daily blessings made this book possible
Cristina, Carlos A. Jr., Tim, Cristina M., and Sophia C. Livingston,
and Mrs. Rene M. Smith,
my four grandsons:
Nicholas, Robert, Garrett and David
and to our dearest homeland, Cuba
This edition is a major revision and expansion to the first edition. Several new subjects
have been added, notably the z-transform analysis and discrete controllers, and several
other subjects have been reorganized and expanded. The objective of the book, however,
remains the same as in the first edition, “to present the practice of automatic process
control along with the fundamental principles of control theory.” A significant number
of applications resulting from our practice as part-time consultants have also been added
to this edition.
Twelve years have passed since the first edition was published, and even though the
principles are still very much the same, the “tools” to implement the controls strategies
have certainly advanced. The use of computer-based instrumentation and control systems is the norm.
Chapters 1 and 2 present the definitions of terms and mathematical tools used in
process control. In this edition Chapter 2 stresses the determination of the quantitative
characteristics of the dynamic response, settling time, frequency of oscillation, and
damping ratio, and de-emphasizes the exact determination of the analytical response.
In this way the students can analyze the response of a dynamic system without having
to carry out the time-consuming evaluation of the coefficients in the partial fraction
expansion. Typical responses of first-, second-, and higher-order systems are now presented in Chapter 2.
The derivation of process dynamic models from basic principles is the subject of
Chapters 3 and 4. As compared to the first edition, the discussion of process modelling
has been expanded. The discussion, meaning, and significance of process nonlinearities
has been expanded as well. Several numerical examples are presented to aid in the
understanding of this important process characteristic. Chapter 4 concludes with a presentation of integrating, inverse-response, and open-loop unstable processes.
Chapter 5 presents the design and characteristics of the basic components of a control
system: sensors and transmitters, control valves, and feedback controllers. The presentation of control valves and feedback controllers has been expanded. Chapter 5 should
be studied together with Appendix C where practical operating principles of some
common sensors, transmitters, and control valves are presented.
The design and tuning of feedback controllers are the subjects of Chapters 6 and 7.
Chapter 6 presents the analysis of the stability of feedback control loops. In this edition
we stress the direct substitution method for determining both the ultimate gain and
period of the loop. Routh’s test is deemphasized, but still presented in a separate section.
In keeping with the spirit of Chapter 2, the examples and problems deal with the determination of the characteristics of the response of the closed loop, not with the exact
analytical response of the loop. Chapter 7 keeps the same tried-and-true tuning methods
from the first edition. A new section on tuning controllers for integrating processes,
and a discussion of the Internal Model Control (IMC) tuning rules, have been added.
Chapter 8 presents the root locus technique, and Chapter 9 presents the frequency
response techniques. These techniques are principally used to study the stability of
The additional control techniques that supplement and enhance feedback control have
been distributed among Chapters 10 through 13 to facilitate the selection of their coverage in university courses. Cascade control is presented first, in Chapter 10, because
it is so commonly a part of the other schemes. Several examples are presented to help
understanding of this important and common control technique.
Chapter 11 presents different computing algorithms sometimes used to implement
control schemes. A method to scale these algorithms, when necessary, is presented. The
chapter also presents the techniques of override, or constraint, control, and selective
control. Examples are used to explain the meaning and justification of them.
Chapter 12 presents and discusses in detail the techniques of ratio and feedforward
control. Industrial examples are also presented. A significant number of new problems
have been added.
Multivariable control and loop interaction are the subjects of Chapter 13. The calculation and interpretation of the relative gain matrix (RGM) and the design of decouplers, are kept from the first edition. Several examples have been added, and the
material has been reorganized to keep all the dynamic topics in one section.
Finally Chapters 14 and 15 present the tools for the design and analysis of sampleddata (computer) control systems. Chapter 14 presents the z-transform and its use to
analyze sampled-data control systems, while Chapter 15 presents the design of basic
algorithms for computer control and the tuning of sampled-data feedback controllers.
The chapter includes sections on the design and tuning of dead-time compensation
algorithms and model-reference control algorithms. Two examples of Dynamic Matrix
Control (DMC) are also included.
As in the first edition, Appendix A presents some symbols, labels, and other notations
commonly used in instrumentation and control diagrams. We have adopted throughout
the book the ISA symbols for conceptual diagrams which eliminate the need to differentiate between pneumatic, electronic, or computer implementation of the various control schemes. In keeping with this spirit, we express all instrument signals in percent
of range rather than in mA or psig. Appendix B presents several processes to provide
the student/reader an opportunity to design control systems from scratch.
During this edition we have been very fortunate to have received the help and encouragement of several wonderful individuals. The encouragement of our students,
especially Daniel Palomares, Denise Farmer, Carl Thomas, Gene Daniel, Samuel Peebles, Dan Logue, and Steve Hunter, will never be forgotten. Thanks are also due to Dr.
Russell Rhinehart of Texas Tech University who read several chapters when they were
in the initial stages. His comments were very helpful and resulted in a better book.
Professors Ray Wagonner, of Missouri Rolla, and G. David Shilling, of Rhode Island,
gave us invaluable suggestions on how to improve the first edition. To both of them
we are grateful. We are also grateful to Michael R. Benning of Exxon Chemical Americas who volunteered to review the manuscript and offered many useful suggestions
from his industrial background.
In the preface to the first edition we said that “To serve as agents in the training and
development of young minds is certainly a most rewarding profession.” This is still our
conviction and we feel blessed to be able to do so. It is with this desire that we have
written this edition.
Tampa, Florida, 1997
Baton Rouge, Louisiana, 1997
Chapter 1 Introduction
A Process Control System
Important Terms and the Objective of Automatic Process Control
Regulatory and Servo Control
Transmission Signals, Control Systems, and Other Terms
Control Strategies 6
1-5.1 Feedback Control 6
1-5.2 Feedforward Control 7
Background Needed for Process Control
Chapter 2 Mathematical Tools for Control Systems Analysis
The Laplace Transform 11
2- 1.1 Definition of the Laplace Transform 12
2-1.2 Properties of the Laplace Transform 14
Solution of Differential Equations Using the Laplace Transform
2-2.1 Laplace Transform Solution Procedure 21
2-2.2 Inversion by Partial Fractions Expansion 23
2-2.3 Handling Time Delays 27
Characterization of Process Response 30
2-3.1 Deviation Variables 3 1
2-3.2 Output Response 32
2-3.3 Stability 39
Response of First-Order Systems 39
2-4.1 Step Response 41
2-4.2 Ramp Response 43
2-4.3 Sinusoidal Response 43
2-4.4 Response with Time Delay 45
2-4.5 Response of a Lead-Lag Unit 46
Response of Second-Order Systems 48
2-5.1 Overdamped Responses 50
2-5.2 Underdamped Responses 53
2-5.3 Higher-Order Responses 57
2-6.1 Linearization of Functions of One Variable
2-6.2 Linearization of Functions of Two or More Variables
2-6.3 Linearization of Differential Equations 65
Review of Complex-Number Algebra 68
2-7.1 Complex Numbers 68
2-7.2 Operations with Complex Numbers 70
Chapter 3 First-Order Dynamic Systems
Processes and the Importance of Process Characteristics
Thermal Process Example
Dead Time 92
Transfer Functions and Block Diagrams
3-4.1 Transfer Functions 95
3-4.2 Block Diagrams 96
Gas Process Example
3-6.1 Introductory Remarks
3-6.2 Chemical Reactor Example
Effects of Process Nonlinearities
Chapter 4 Higher-Order Dynamic Systems
Noninteracting Systems 135
4- 1.1 Noninteracting Level Process 135
4- 1.2 Thermal Tanks in Series 142
Interacting Systems 145
4-2.1 Interacting Level Process 145
4-2.2 Thermal Tanks with Recycle 151
4-2.3 Nonisothermal Chemical Reactor 154
Response of Higher-Order Systems 164
Other Types of Process Responses 167
4-4.1 Integrating Processes: Level Process 168
4-4.2 Open-Loop Unstable Process: Chemical Reactor
4-4.3 Inverse Response Processes: Chemical Reactor
Overview of Chapters 3 and 4 182
Basic Components of Control Systems
Sensors and Transmitters 197
Control Valves 200
5-2.1 The Control Valve Actuator 200
5-2.2 Control Valve Capacity and Sizing
5-2.3 Control Valve Characteristics 210
5-2.4 Control Valve Gain and Transfer Function
5-2.5 Control Valve Summary 222
Feedback Controllers 222
5-3.1 Actions of Controllers 223
5-3.2 Types of Feedback Controllers
5-3.3 Modifications to the PID Controller and Additional Comments
5-3.4 Reset Windup and Its Prevention
5-3.5 Feedback Controller Summary
Chapter 6 Design of Single-Loop Feedback Control Systems
The Feedback Control Loop 252
6- 1.1 Closed-Loop Transfer Function
6-1.2 Characteristic Equation of the Loop
6-1.3 Steady-State Closed-Loop Gains
Stability of the Control Loop 274
6-2.1 Criterion of Stability 274
6-2.2 Direct Substitution Method 275
6-2.3 Effect of Loop Parameters on the Ultimate Gain and Period
6-2.4 Effect of Dead Time 285
6-2.5 Routh’s Test 287
Tuning of Feedback Controllers
Quarter Decay Ratio Response by Ultimate Gain
Open-Loop Process Characterization
7-2.1 Process Step Testing
7-2.2 Tuning for Quarter Decay Ratio Response
7-2.3 Tuning for Minimum Error Integral Criteria
7-2.4 Tuning Sampled-Data Controllers
7-2.5 Summary of Controller Tuning
Tuning Controllers for Integrating Processes
7-3.1 Model of Liquid Level Control System
7-3.2 Proportional Level Controller
7-3.3 Averaging Level Control
Synthesis of Feedback Controllers
7-4.1 Development of the Controller Synthesis Formula
7-4.2 Specification of the Closed-Loop Response
7-4.3 Controller Modes and Tuning Parameters
7-4.4 Summary of Controller Synthesis Results
7-4.5 Tuning Rules by Internal Model Control (IMC)
Tips for Feedback Controller Tuning
7-5.1 Estimating the Integral and Derivative Times
7-5.2 Adjusting the Proportional Gain
Chapter 8 Root Locus
Analysis of Feedback Control Systems by Root Locus
Rules for Plotting Root Locus Diagrams
Chapter 9 Frequency Response Techniques
Frequency Response 389
9- 1.1 Experimental Determination of Frequency Response
9-1.2 Bode Plots 398
Frequency Response Stability Criterion 407
Polar Plots 419
Nichols Plots 427
Pulse Testing 427
9-5.1 Performing the Pulse Test 428
9-5.2 Derivation of the Working Equation 429
9-5.3 Numerical Evaluation of the Fourier Transform Integral
Chapter 10 Cascade Control
A Process Example 439
Stability Considerations 442
Implementation and Tuning of Controllers
10-3.1 Two-Level Cascade Systems
10-3.2 Three-Level Cascade Systems
Other Process Examples 450
Further Comments 452
Override and Selective Control
Computing Algorithms 460
1 1 - 1.1 Scaling Computing Algorithms
1 l-l.2 Physical Significance of Signals
Override, or Constraint, Control 470
Selective Control 475
Chapter 12 Ratio and Feedforward Control
The Feedforward Concept 494
Block Diagram Design of Linear Feedforward Controllers
Lead/Lag Term 505
Back to the Previous Example 507
Design of Nonlinear Feedforward Controllers from Basic Process
12-2.6 Some Closing Comments and Outline of Feedforward Controller
12-2.7 Three Other Examples 518
Chapter 13 Multivariable Process Control
Loop Interaction 545
Pairing Controlled and Manipulated Variables 550
13-2.1 Calculating the Relative Gains for a 2 X 2 System
13-2.2 Calculating the Relative Gains for an n X n System
Decoupling of Interacting Loops 564
13-3.1 Decoupler Design from Block Diagrams 565
13-3.2 Decoupler Design for n X IZ Systems
13-3.3 Decoupler Design from Basic Principles 577
Multivariable Control vs. Optimization 579
Dynamic Analysis of Multivariable Systems 580
13-5.1 Signal Flow Graphs (SFG) 580
13-5.2 Dynamic Analysis of a 2 X 2 System 585
13-5.3 Controller Tuning for Interacting Systems 590
Mathematical Tools for Computer Control Systems
Computer Process Control
14-2.1 Definition of the z-Transform
14-2.2 Relationship to the Laplace Transform
14-2.3 Properties of the z-Transform
14-2.4 Calculation of the Inverse z-Transform
Pulse Transfer Functions
14-3.1 Development of the Pulse Transfer Function
14-3.2 Steady-State Gain of a Pulse Transfer Function
14-3.3 Pulse Transfer Functions of Continuous Systems
14-3.4 Transfer Functions of Discrete Blocks
14-3.5 Simulation of Continuous Systems with Discrete Blocks
Sampled-Data Feedback Control Systems
14-4.1 Closed-Loop Transfer Function
14-4.2 Stability of Sampled-Data Control Systems
14-5.1 Definition and Properties of the Modified z-Transform
14-5.2 Inverse of the Modified z-Transform
14-5.3 Transfer Functions for Systems with Transportation Lag
Design of Computer Control Systems
Development of Control Algorithms 650
15- 1.1 Exponential Filter 651
15- 1.2 Lead-Lag Algorithm 653
15- 1.3 Feedback (PID) Control Algorithms 655
Tuning of Feedback Control Algorithms 662
15-2.1 Development of the Tuning Formulas 662
15-2.2 Selection of the Sample Time 672
Feedback Algorithms with Dead-Time Compensation
15-3.1 The Dahlin Algorithm 674
15-3.2 The Smith Predictor 677
15-3.3 Algorithm Design by Internal Model Control
15-3.4 Selection of the Adjustable Parameter 685
Automatic Controller Tuning 687
Model-Reference Control 688
Appendix A Instrumentation Symbols and Labels
Appendix B Case Studies
Ammonium Nitrate Prilling Plant Control System
Natural Gas Dehydration Control System
Sodium Hypochlorite Bleach Preparation Control System
Control Systems in the Sugar Refining Process
CO, Removal from Synthesis Gas
Sulfuric Acid Process
Fatty Acid Process
Appendix C Sensors, Transmitters, and Control Valves
C-6.1 Pneumatic Transmitter
C-6.2 Electronic Transmitter
Types of Control Valves
C-7.1 Reciprocating Stem
C-7.2 Rotating Stem
Control Valve Actuators
C-g.1 Pneumatically Operated Diaphragm Actuators
C-8.2 Piston Actuators
C-8.3 Electrohydraulic and Electromechanical Actuators
C-8.4 Manual-Handwheel Actuators
Control Valve Accessories
C-9.3 Limit Switches
Control Valves-Additional Considerations
C- 10.1 Viscosity Corrections
C-lo.2 Flashing and Cavitation
The purpose of this chapter is to present the need for automatic process control and to
motivate you, the reader, to study it. Automatic process control is concerned with
maintaining process variables, temperatures, pressures, flows, compositions, and the
like at some desired operating value. As we shall see, processes are dynamic in nature.
Changes are always occurring, and if appropriate actions are not taken in response, then
the important process variables-those related to safety, product quality, and production rates-will not achieve design conditions.
This chapter also introduces two control systems, takes a look at some of their components, and defines some terms used in the field of process control. Finally, the background needed for the study of process control is discussed.
In writing this book, we have been constantly aware that to be successful, the engineer
must be able to apply the principles learned. Consequently, the book covers the principles that underlie the successful practice of automatic process control. The book is
full of actual cases drawn from our years of industrial experience as full-time practitioners or part-time consultants. We sincerely hope that you get excited about studying
automatic process control. It is a very dynamic, challenging, and rewarding area of
l-l A PROCESS CONTROL SYSTEM
To illustrate process control, let us consider a heat exchanger in which a process stream
is heated by condensing steam; the process is sketched in Fig. 1-1.1. The purpose of
this unit is to heat the process fluid from some inlet temperature T,(t) up to a certain
desired outlet temperature T(t). The energy gained by the process fluid is provided by
the latent heat of condensation of the steam.
In this process there are many variables that can change, causing the outlet temperature to deviate from its desired value. If this happens, then some action must be taken
to correct the deviation. The objective is to maintain the outlet process temperature at
its desired value.
One way to accomplish this objective is by measuring the temperature T(t), comparing it to the desired value, and, on the basis of this comparison, deciding what to do to
correct any deviation. The steam valve can be manipulated to correct the deviation.
That is, if the temperature is above its desired value, then the steam valve can be
2 Chapter 1 Introduction
Figure 1-1.1 Heat exchanger.
throttled back to cut the steam flow (energy) to the heat exchanger. If the temperature
is below the desired value, then the steam valve can be opened more to increase the
steam flow to the exchanger. All of this can be done manually by the operator, and the
procedure is fairly straightforward. However, there are several problems with such
manual control. First, the job requires that the operator look at the temperature frequently to take corrective action whenever it deviates from the desired value. Second,
different operators make different decisions about how to move the steam valve, and
this results in a less than perfectly consistent operation. Third, because in most process
plants there are hundreds of variables that must be maintained at some desired value,
manual correction requires a large number of operators. As a result of these problems,
we would like to accomplish this control automatically. That is, we would like to have
systems that control the variables without requiring intervention from the operator. This
is what is meant by automatic process control.
To achieve automatic process control, a control system must be designed and implemented. A possible control system for our heat exchanger is shown in Fig. 1-1.2. (Ap-
Figure l-l.2 Heat exchanger control system.
Important Terms and the Objective of Automatic Process Control
pendix A presents the symbols and identifications for different devices.) The first thing
to do is measure the outlet temperature of the process stream. This is done by a sensor
(thermocouple, resistance temperature device, filled system thermometer, thermistor, or
the like). Usually this sensor is physically connected to a transmitter, which takes the
output from the sensor and converts it to a signal strong enough to be transmitted to a
controller. The controller then receives the signal, which is related to the temperature,
and compares it with the desired value. Depending on the result of this comparison, the
controller decides what to do to maintain the temperature at the desired value. On the
basis of this decision, the controller sends a signal to the final control element, which
in turn manipulates the steam flow. This type of control strategy is known as feedback
Thus the three basic components of all control systems are
1. Sensor/transmitter Also often called the primary and secondary elements.
2. Controller The “brain” of the control system.
3. Final control element Often a control valve but not always. Other common final
control elements are variable-speed pumps, conveyors, and electric motors.
These components perform the three basic operations that must be present in every
control system. These operations are
1. Measurement(M) Measuring the variable to be controlled is usually done by the
combination of sensor and transmitter. In some systems, the signal from the sensor
can be fed directly to the controller, so there is no need for the transmitter.
2. Decision (0) On the basis of the measurement, the controller decides what to do
to maintain the variable at its desired value.
3. Action (A) As a result of the controller’s decision, the system must then take an
action. This is usually accomplished by the final control element.
These three operations, M, D, and A, are always present in every type of control
system, and it is imperative that they be in a loop. That is, on the basis of the measurement a decision is made, and on the basis of this decision an action is taken. The
action taken must come back and affect the measurement; otherwise, it is a major Jaw
in the design, and control will not be achieved. When the action taken does not affect
the measurement, an open-loop condition exists and control will not be achieved. The
decision making in some systems is rather simple, whereas in others it is more complex;
we will look at many systems in this book.
1-2 IMPORTANT TERMS AND THE OBJECTIVE OF AUTOMATIC
At this time it is necessary to define some terms used in the field of automatic process
control. The controlled variable is the variable that must be maintained, or controlled,
at some desired value. In our example of the heat exchanger, the process outlet temperature, T(t), is the controlled variable. Sometimes the term process variable is also
used to refer to the controlled variable. The set point (SP) is the desired value of the
controlled variable. Thus the job of a control system is to maintain the controlled
variable at its set point. The manipulated variable is the variable used to maintain the
controlled variable at its set point. In the example, the steam valve position is the
4 Chapter 1 Introduction
manipulated variable. Finally, any variable that causes the controlled variable to deviate
from the set point is known as a disturbance or upset. In most processes there are a
number of different disturbances. In the heat exchanger shown in Fig. 1-1.2, possible
disturbances include the inlet process temperature, T,(t), the process flow, f(t), the energy content of the steam, ambient conditions, process fluid composition, and fouling.
It is important to understand that disturbances are always occurring in processes. Steady
state is not the rule, and transient conditions are very common. It is because of these
disturbances that automatic process control is needed. If there were no disturbances,
then design operating conditions would prevail and there would be no need to “monitor”
the process continuously.
The following additional terms are also important. Manual control is the condition
in which the controller is disconnected from the process. That is, the controller is not
deciding how to maintain the controlled variable at set point. It is up to the operator to
manipulate the signal to the final control element to maintain the controlled variable at
set point. Closed-loop control is the condition in which the controller is connected to
the process, comparing the set point to the controlled variable and determining and
taking corrective action.
Now that we have defined these terms, we can express the objective of an automatic
process control system meaningfully: The objective of an automatic process control
system is to adjust the manipulated variable to maintain the controlled variable at its
set point in spite of disturbances.
Control is important for many reasons. Those that follow are not the only ones, but
we feel they are the most important. They are based on our industrial experience, and
we would like to pass them on. Control is important to
1. Prevent injury to plant personnel, protect the environment by preventing emissions
and minimizing waste, and prevent damage to the process equipment. SAFETY
must always be in everyone’s mind; it is the single most important consideration.
2. Maintain product quality (composition, purity, color, and the like) on a continuous
basis and with minimum cost.
3. Maintain plant production rate at minimum cost.
Thus process plants are automated to provide a safe environment and at the same
&me maintain desired product quality, high plant throughput, and reduced demand on
1-3 REGULATORY AND SERVO CONTROL
In some processes, the controlled variable deviates from set point because of disturbances. Systems designed to compensate for these disturbances exert regulatory
control. In some other instances, the most important disturbance is the set point itself.
That is, the set point may be changed as a function of time (typical of this is a batch
reactor where the temperature must follow a desired profile), and therefore the controlled variable must follow the set point. Systems designed for this purpose exert servo
Regulatory control is much more common than servo control in the process indus-
Transmission Signals, Control Systems, and Other Terms
tries. However, the same basic approach is used in designing both. Thus the principles
in this book apply to both cases.
1-4 TRANSMISSION SIGNALS, CONTROL SYSTEMS, AND OTHER
Three principal types of signals are used in the process industries. The pneumatic signal,
or air pressure, normally ranges between 3 and 15 psig. The usual representation for
pneumatic signals in process and instrumentation diagrams (P&IDS) is v.
The electrical signal normally ranges between 4 and 20 mA. Less often, a range of 10
to 50 mA, 1 to 5 V, or 0 to 10 V is used. The usual representation for this signal in
P&IDS is a series of dashed lines such as - - - - -. The third type of signal is the digital,
or discrete, signal (zeros and ones). In this book we will show such signals as N
(see Fig. l-1.2), which is the representation proposed by the Instrument Society of
America (ISA) when a control concept is shown without concern for specific hardware.
The reader is encouraged to review Appendix A, where different symbols and labels
are presented. Most times we will refer to signals as percentages instead of using psig
or mA. That is, 0%- 100% is equivalent to 3 to 15 psig or 4 to 20 mA.
It will help in understanding control systems to realize that signals are used by devices-transmitters, controllers, final control elements, and the like-to communicate.
That is, signals are used to convey information. The signal from the transmitter to the
controller is used by the transmitter to inform the controller of the value of the controlled
variable. This signal is not the measurement in engineering units but rather is a mA,
psig, volt, or any other signal that is proportional to the measurement. The relationship
to the measurement depends on the calibration of the sensor/transmitter. The controller
uses its output signal to tell the final control element what to do: how much to open if
it is a valve, how fast to run if it is a variable-speed pump, and so on.
It is often necessary to change one type of signal into another. This is done by a
transducer, or converter. For example, there may be a need to change from an electrical
signal in milliamperes (mA) to a pneumatic signal in pounds per square inch, gauge
(psig). This is done by the use of a current (I) to pneumatic (P) transducer (I/P); see
Fig. 1-4.1. The input signal may be 4 to 20 mA and the output 3 to 15 psig. An analogto-digital converter (A to D) changes from a mA, or a volt signal to a digital signal.
There are many other types of transducers: digital-to-analog (D to A), pneumatic-tocurrent (P/I), voltage-to-pneumatic (E/P), pneumatic-to-voltage (P/E), and so on.
The term analog refers to a controller, or any other instrument, that is either pneumatic or electrical. Most controllers, however, are computer-based, or digital. By computer-based we don’t necessarily mean a main-frame computer but anything starting
from a microprocessor. In fact, most controllers are microprocessor-based. Chapter 5
presents different types of controllers and defines some terms related to controllers and
Figure 1-4.1 I/P transducer.
6 Chapter 1 Introduction
1-5.1 Feedback Control
The control scheme shown in Fig. l-l.2 is referred to as feedback control and is also
called afeedback control loop. One must understand the working principles of feedback
control to recognize its advantages and disadvantages; the heat exchanger control loop
shown in Fig. l-l.2 is presented to foster this understanding.
If the inlet process temperature increases, thus creating a disturbance, its effect must
propagate through the heat exchanger before the outlet temperature increases. Once this
temperature changes, the signal from the transmitter to the controller also changes. It
is then that the controller becomes aware that a deviation from set point has occurred
and that it must compensate for the disturbance by manipulating the steam valve. The
controller signals the valve to close and thus to decrease the steam flow. Fig. 1-5.1
shows graphically the effect of the disturbance and the action of the controller.
It is instructive to note that the outlet temperature first increases, because of the
increase in inlet temperature, but it then decreases even below set point and continues
to oscillate around set point until the temperature finally stabilizes. This oscillatory
response is typical of feedback control and shows that it is essentially a trial-and-error
operation. That is, when the controller “notices” that the outlet temperature has increased above the set point, it signals the valve to close, but the closure is more than
required. Therefore, the outlet temperature decreases below the set point. Noticing this,
Fraction of valve opening
Figure 1-5.1 Response of a heat exchanger to a disturbance: feedback control.
1-5 Control Strategies 7
the controller signals the valve to open again somewhat to bring the temperature back
up. This trial-and-error operation continues until the temperature reaches and remains
at set point.
The advantage of feedback control is that it is a very simple technique that compensates for all disturbances. Any disturbance affects the controlled variable, and once this
variable deviates from set point, the controller changes its output in such a way as to
return the temperature to set point. The feedback control loop does not know, nor does
it care, which disturbance enters the process. It tries only to maintain the controlled
variable at set point and in so doing compensates for all disturbances. The feedback
controller works with minimum knowledge of the process. In fact, the only information
it needs is in which direction to move. How much to move is usually adjusted by trial
and error. The disadvantage of feedback control is that it can compensate for a disturbance only after the controlled variable has deviated from set point. That is, the disturbance must propagate through the entire process before the feedback control scheme
can initiate action to compensate for it.
The job of the engineer is to design a control scheme that will maintain the controlled
variable at its set point. Once this is done, the engineer must adjust, or tune, the controller so that it minimizes the amount of trial and error required. Most controllers have
up to three terms (also known as parameters) used to tune them. To do a creditable job,
the engineer must first know the characteristics of the process to be controlled. Once
these characteristics are known, the control system can be designed and the controller
tuned. Process characteristics are explained in Chapters 3 and 4, Chapter 5 presents the
meaning of the three terms in the controllers, and Chapter 7 explains how to tune them.
14.2 Feedforward Control
Feedback control is the most common control strategy in the process industries. Its
simplicity accounts for its popularity. In some processes, however, feedback control
may not provide the required control performance. For these processes, other types of
control strategies may have to be designed. Chapters 10, 11, 12, 13, and 15 present
additional control strategies that have proved profitable. One such strategy is feedforward control. The objective of feedforward control is to measure disturbances and
compensate for them before the controlled variable deviates from set point. When feedforward control is applied correctly, deviation of the controlled variable is minimized.
A concrete example of feedforward control is the heat exchanger shown in Fig.
1-1.2. Suppose that “major” disturbances are the inlet temperature, T,(t), and the process
flow,f(t). To implement feedforward control, these two disturbances must first be measured, and then a decision must be made about how to manipulate the steam valve to
compensate for them. Fig. 1-5.2 shows this control strategy. The feedforward controller
makes the decision about how to manipulate the steam valve to maintain the controlled
variable at set point, depending on the inlet temperature and process flow.
In Section 1-2 we learned that there are a number of different disturbances. The
feedforward control system shown in Fig. 1-5.2 compensates for only two of them. If
any of the others enter the process, this strategy will not compensate for it, and the
result will be a permanent deviation of the controlled variable from set point. To avoid
this deviation, some feedback compensation must be added to feedforward control; this
is shown in Fig. 1-5.3. Feedforward control now compensates for the “major” distur-
8 Chapter 1 Introduction
Figure l-S.2 Heat exchanger feedforward control system.
bances, while feedback control compensates for all other disturbances. Chapter 12 presents the development of the feedforward controller. Actual industrial cases are used to
discuss this important strategy in detail.
It is important to note that the three basic operations, M, D, A, are still present in
this more “advanced” control strategy. Measurement is performed by the sensors and
transmitters. Decision is made by both the feedforward and the feedback controllers.
Action is taken by the steam valve.
The advanced control strategies are usually more costly than feedback control in
Figure 1-5.3 Heat exchanger feedforward control with feedback compensation.
hardware, computing power, and the effort involved in designing, implementing, and
maintaining them. Therefore, the expense must be justified before they can be implemented. The best procedure is first to design and implement a simple control strategy,
keeping in mind that if it does not prove satisfactory, then a more advanced strategy
may be justifiable. It is important, however, to recognize that these advanced strategies
still require some feedback compensation.
1-6 BACKGROUND NEEDED FOR PROCESS CONTROL
To be successful in the practice of automatic process control, the engineer must first
understand the principles of process engineering. Therefore, this book assumes that the
reader is familiar with the basic principles of thermodynamics, fluid flow, heat transfer,
separation processes, reaction processes, and the like.
For the study of process control, it is also fundamental to understand how processes
behave dynamically. Thus it is necessary to develop the set of equations that describes
different processes. This is called modeling. To do this requires knowledge of the basic
principles mentioned in the previous paragraph and of mathematics through differential
equations. Laplace transforms are used heavily in process control. This greatly simplifies the solution of differential equations and the dynamic analysis of processes and
their control systems. Chapter 2 of this book is devoted to the development and use of
the Laplace transforms, along with a review of complex-number algebra. Chapters 3
and 4 offer an introduction to the modeling of some processes.
In this chapter, we discussed the need for automatic process control. Industrial processes are not static but rather very dynamic; they are continuously changing as a
result of many types of disturbances. It is principally because of this dynamic nature
that control systems are needed to continuously and automatically watch over the variables that must be controlled.
The working principles of a control system can be summarized with the three letters
M, D, and A. M refers to the measurement of process variables. D refers to the decision
made on the basis of the measurement of those process variables. Finally, A refers to
the action taken on the basis of that decision.
The fundamental components of a process control system were also presented: sensor/
transmitter, controller, and final control element. The most common types of signalspneumatic, electrical, and digital-were introduced, along with the purpose of transducers.
Two control strategies were presented: feedback and feedforward control. The advantages and disadvantages of both strategies were briefly discussed. Chapters 6 and 7
present the design and analysis of feedback control loops.
l-l. For the following automatic control systems commonly encountered in daily life,
identify the devices that perform the measurement (M), decision (D), and action
10 Chapter 1 Introduction
(A) functions, and classify the action function as “On/Off’ or “Regulating.” Also
draw a process and instrumentation diagram (P&ID), using the standard ISA symbols given in Appendix A, and determine whether the control is feedback or feedforward.
(a) House air conditioning/heating
(b) Cooking oven
(d) Automatic sprinkler system for fires
(e) Automobile cruise speed control
1-2. Instrumentation Diagram: Automatic Shower Temperature Control. Sketch the
process and instrumentation diagram for an automatic control system to control
the temperature of the water from a common shower-that is, a system that will
automatically do what you do when you adjust the temperature of the water when
you take a shower. Use the standard ISA instrumentation symbols given in Appendix A. Identify the measurement (M), decision (D), and action (A) devices of
your control system.
M,athematical Tools for
Control Systems Analysis
This chapter presents two mathematical tools that are particularly useful for analyzing
process dynamics and designing automatic control systems: Laplace transforms and
linearization. Combined, these two techniques allow us to gain insight into the dynamic
responses of a wide variety of processes and instruments. In contrast, the technique of
computer simulation provides us with a more accurate and detailed analysis of the
dynamic behavior of specific systems but seldom allows us to generalize our findings
to other processes.
Laplace transforms are used to convert the differential equations that represent the
dynamic behavior of process output variables into algebraic equations. It is then possible
to isolate in the resulting algebraic equations what is characteristic of the process, the
from what is characteristic of the input forcing functions. Because
the differential equations that represent most processes are nonlinear, linearization is
required to approximate nonlinear differential equations with linear ones that can then
be treated by the method of Laplace transforms.
The material in this chapter is not just a simple review of Laplace transforms but is
a presentation of the tool in the way it is used to analyze process dynamics and to
design control systems. Also presented are the responses of some common process
transfer functions to some common input functions. These responses are related to the
parameters of the process transfer functions so that the important characteristics of the
responses can be inferred directly from the transfer functions without having to reinvert them each time. Because a familiarity with complex numbers is required to work
with Laplace transforms, we have included a brief review of complex-number algebra
as a separate section. We firmly believe that a knowledge of Laplace transforms is
essential for understanding the fundamentals of process dynamics and control systems
2-1 THE LAPLACE TRANSFORM
This section reviews the definition of the Laplace transform and its properties.
Chapter 2 Mathematical Tools for Control Systems Analysis
Definition of the Laplace Transform
In the analysis of process dynamics, the process variables and control signals are functions of time, t. The Laplace transform of a function of time, f(t), is defined by the
F(s) = W(Ql =
F(s) = the Laplace transform off(t)
s = the Laplace transform variable, time-’
The Laplace transform changes the function of time, f(t), into a function in the Laplace
transform variable, F(s). The limits of integration show that the Laplace transform
contains information on the function f(t) for positive time only. This is perfectly acceptable, because in process control, as in life, nothing can be done about the past
(negative time); control action can affect the process only in the future. The following
example uses the definition of the Laplace transform to develop the transforms of a few
common forcing functions.
The four signals shown in Fig. 2-1.1 are commonly applied as inputs to processes and
instruments to study their dynamic responses. We now use the definition of the Laplace
transform to derive their transforms.
(a) UNIT STEP FUNCTION
This is a sudden change of unit magnitude as sketched in Fig. 2-l.la. Its algebraic
Substituting into Eq. 2-1.1 yields
2-1 The Laplace Transform 13
Figure 2-1.1 Common input signals for the study of control system response. (a) Unit step
function, u(t). (b) Pulse. (c) Unit impulse function, s(t). (d) Sine wave, sin cot (w = 27~/T).
(b) A PULSE OF MAGNITUDE HAND DURATION T
The pulse sketched in Fig. 2-1.1 b is represented by
t < 0, t 2 T
Substituting into Eq. 2-1.1 yields
= -s e-sr
= He-“’ dt
T = - y (e-sT - 1)
= s (l - e-sT)
(c) A UNIT IMPULSE FUNCTION
This function, also known as the Dirac delta function and represented by t?(t), is
Chapter 2 Mathematical Tools for Control Systems Analysis
sketched in Fig. 2-1.1~. It is an ideal pulse with zero duration and unit area. All of
its area is concentrated at time zero. Because the function is zero at all times except
at zero, and because the term e-“’ in Eq. 2- 1.1 is equal to unity at t = 0, the Laplace
S(t)emsf dt = 1
Note that the result of the integration, 1, is the area of the impulse. The same result
can be obtained by substituting H = l/T in the result of part (b), so that HT = 1,
and then taking limits as T goes to zero.
(d) A SINE WAVE OF UNITY AMPLITUDE AND FREQUENCY o
The sine wave is sketched in Fig. 2-1. Id and is represented in exponential form by
where i = ,/? is the unit of imaginary numbers. Substituting into Eq. 2- 1.1 yields
s - iw
=- 2i [
s - iw
o - 1
O - l- + -
=-2i s2 + 69
s + iw
=s2 + cl?
The preceding example illustrates some algebraic manipulations required to derive
the Laplace transform of various functions using its definition. Table 2- 1.1 contains a
short list of the Laplace transforms of some common functions.
Properties of the Laplace Transform
This section presents the properties of Laplace transforms in order of their usefulness
in analyzing process dynamics and designing control systems. Linearity and the real
differentiation and integration theorems are essential for transforming differential equations into algebraic equations. The final value theorem is useful for predicting the final
2-1 The Laplace Transform 15
Table 2-1.1 Laplace Transforms of Common
e-a’ sin ot
e-a’ cos wt
F(s) = Km1
(s + a>*
(s + a)“+1
s* + w2
s* + 6.2
(s + s+a+ wz
(s + a>* + a?
steady-state value of a time function from its Laplace transform, and the real translation
theorem is useful for dealing with functions delayed in time. Other properties are useful
for deriving the transforms of complex functions from the transforms of simpler functions such as those listed in Table 2-1.1.
It is very important to realize that the Laplace transform is a linear operation. This
means that if a is a constant, then
a$(01 = 4m1 = am
The distributive property of addition also follows from the linearity property:
.Z[uf(t) + bg(t)] = uF(s) + bG(s)
where a and b are constants. You can easily derive both formulas by application of
Eq. 2- 1.1, the definition of the Laplace transform.
Chapter 2 Mathematical Tools for Control Systems Analysis
This theorem, which establishes a relationship between the Laplace transform of a
function and that of its derivatives, is most important in transforming differential equations into algebraic equations. It states that
[ - = SF(S) -f(O)
Proof From the definition of the Laplace transform, Eq. 2-1.1,
Integrate by parts.
& = dfo dt
du = - semS’ dt
y = [f(t)emSf];
v = f(t)
The extension to higher derivatives is straightforward.
= s[sF(s) - f(O)1 - 5
= s2F(s) - s,(O) - $
= s”F(s) - s”-‘f(O) - . . . - dt”-’
2-1 The Laplace Transform 17
In process control, it is normally assumed that the initial conditions are at steady state
(time derivatives are zero) and that the variables are deviations from initial conditions
(initial value is zero). For this very important case, the preceding expression reduces to
This means that for the case of zero initial conditions at steady state, the Laplace
transform of the derivative of a function is obtained by simply substituting variable s
for the “dldt” operator, and F(s) forf(t).
This theorem establishes the relationship between the Laplace transform of a function
and that of its integral. It states that
The proof of this theorem is carried out by integrating the definition of the Laplace
transform by parts. This proof is similar to that of the real differentiation theorem and
is left as an exercise. The Laplace transform of the nth integral of a function is the
transform of the function divided by P.
This theorem deals with the translation of a function in the time axis, as shown in Fig.
2-1.2. The translated function is the original function delayed in time. As we shall see
in Chapter 3, time delays are caused by transportation lag, a phenomenon also known
as dead time. The theorem states that
1 T[f(t - to)] = e-T(s) 1
Because the Laplace transform does not contain information about the original function for negative time, the delayed function must be zero for all times less than the time
delay (see Fig. 2- 1.2). This condition is satisfied if the process variables are expressed
as deviations from initial steady-state conditions.
Proof. From the definition of the Laplace transform, Eq. 2- 1.1,
W(t - 4Jl =
Chapter 2 Mathematical Tools for Control Systems Analysis
Figure 2-1.2 Function delayed in time is zero for all times less
than the time delay to
Let r = t - to (or t = to + T) and substitute.
Note that in this proof, we made use of the fact thatf(r) = 0 for r < 0 (t < to).
Final Value Theorem
This theorem allows us to figure out the final, or steady-state, value of a function from
its transform. It is also useful in checking the validity of derived transforms. If the limit
of f(t) as t - w exists, then it can be found from its Laplace transform as follows:
The proof of this theorem adds little to our understanding of it.
The last three properties of the Laplace transform, to be presented next without proof,
are not used as often in the analysis of process dynamics as are the ones already presented.