John wiley sons principles and practice of automatic process control smith corripio1997(782s)

SELECTED TABLES AND FIGURESTYPICAL RESPONSES TRANSFORMS TUNING FORMULAS INSTRUMENTATION ISA standard instrumentation symbols and labels Control valve inherent characteristics Control val

SELECTED TABLES AND FIGURES

TYPICAL RESPONSES

TRANSFORMS

TUNING FORMULAS

INSTRUMENTATION

ISA standard instrumentation symbols and

labels

Control valve inherent characteristics

Control valve installed characteristics

Flow sensors and their characteristics

Temperature sensors and their

characteristics

Classification of filled-system thermometers

Thermocouple voltage versus temperature

Valve capacity (Cv) coefficients

699-706 211 217 724-725 736-737 739 740 754-755

BLOCK DIAGRAMS

Principles and Practice of Automatic Process Control

Second Edition

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.

New York l Chichester l Weinheim l Brisbane l Singapore l Toronto

This work is dedicated with all our love to The Lord our God, for all his daily blessings made this book possible

The Smiths:

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 subjectshave been added, notably the z-transform analysis and discrete controllers, and severalother 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 processcontrol 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 theprinciples are still very much the same, the “tools” to implement the controls strategieshave certainly advanced The use of computer-based instrumentation and control sys-tems is the norm

Chapters 1 and 2 present the definitions of terms and mathematical tools used inprocess control In this edition Chapter 2 stresses the determination of the quantitativecharacteristics of the dynamic response, settling time, frequency of oscillation, anddamping 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 fractionexpansion Typical responses of first-, second-, and higher-order systems are now pre-sented in Chapter 2

The derivation of process dynamic models from basic principles is the subject ofChapters 3 and 4 As compared to the first edition, the discussion of process modellinghas been expanded The discussion, meaning, and significance of process nonlinearitieshas been expanded as well Several numerical examples are presented to aid in theunderstanding of this important process characteristic Chapter 4 concludes with a pre-sentation of integrating, inverse-response, and open-loop unstable processes

Chapter 5 presents the design and characteristics of the basic components of a controlsystem: sensors and transmitters, control valves, and feedback controllers The presen-tation of control valves and feedback controllers has been expanded Chapter 5 should

be studied together with Appendix C where practical operating principles of somecommon 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 andperiod 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 termination of the characteristics of the response of the closed loop, not with the exactanalytical response of the loop Chapter 7 keeps the same tried-and-true tuning methodsfrom 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 frequencyresponse techniques These techniques are principally used to study the stability ofcontrol systems

de-V

vi Preface

The additional control techniques that supplement and enhance feedback control havebeen distributed among Chapters 10 through 13 to facilitate the selection of their cov-erage 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 helpunderstanding of this important and common control technique

Chapter 11 presents different computing algorithms sometimes used to implementcontrol schemes A method to scale these algorithms, when necessary, is presented Thechapter also presents the techniques of override, or constraint, control, and selectivecontrol Examples are used to explain the meaning and justification of them

Chapter 12 presents and discusses in detail the techniques of ratio and feedforwardcontrol Industrial examples are also presented A significant number of new problemshave been added

Multivariable control and loop interaction are the subjects of Chapter 13 The culation and interpretation of the relative gain matrix (RGM) and the design of de-couplers, are kept from the first edition Several examples have been added, and thematerial has been reorganized to keep all the dynamic topics in one section

cal-Finally Chapters 14 and 15 present the tools for the design and analysis of data (computer) control systems Chapter 14 presents the z-transform and its use toanalyze sampled-data control systems, while Chapter 15 presents the design of basicalgorithms for computer control and the tuning of sampled-data feedback controllers.The chapter includes sections on the design and tuning of dead-time compensationalgorithms and model-reference control algorithms Two examples of Dynamic MatrixControl (DMC) are also included

sampled-As in the first edition, Appendix A presents some symbols, labels, and other notationscommonly used in instrumentation and control diagrams We have adopted throughoutthe book the ISA symbols for conceptual diagrams which eliminate the need to differ-entiate between pneumatic, electronic, or computer implementation of the various con-trol 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 providethe student/reader an opportunity to design control systems from scratch

During this edition we have been very fortunate to have received the help and couragement of several wonderful individuals The encouragement of our students,especially Daniel Palomares, Denise Farmer, Carl Thomas, Gene Daniel, Samuel Pee-bles, 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

en-in the en-initial stages His comments were very helpful and resulted en-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 icas who volunteered to review the manuscript and offered many useful suggestionsfrom his industrial background

Amer-In the preface to the first edition we said that “To serve as agents in the training anddevelopment of young minds is certainly a most rewarding profession.” This is still ourconviction and we feel blessed to be able to do so It is with this desire that we havewritten this edition

A Process Control System 1

Important Terms and the Objective of Automatic Process Control

Regulatory and Servo Control 4

Transmission Signals, Control Systems, and Other Terms 5

Chapter 2 Mathematical Tools for Control Systems Analysis

2-1 The Laplace Transform 11

2- 1.1 Definition of the Laplace Transform 12

2-1.2 Properties of the Laplace Transform 14

2-2 Solution of Differential Equations Using the Laplace Transform 21

2-2.1 Laplace Transform Solution Procedure 21

2-2.2 Inversion by Partial Fractions Expansion 23

2-2.3 Handling Time Delays 27

2-3 Characterization of Process Response 30

2-4.4 Response with Time Delay 45

2-4.5 Response of a Lead-Lag Unit 46

2-5 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 Linearization 59

2-6.1 Linearization of Functions of One Variable 60

2-6.2 Linearization of Functions of Two or More Variables 62

2-6.3 Linearization of Differential Equations 65

2-7 Review of Complex-Number Algebra 68

2-7.1 Complex Numbers 68

2-7.2 Operations with Complex Numbers 70

11

vii

Processes and the Importance of Process Characteristics

Thermal Process Example 82

3-6.2 Chemical Reactor Example 111

Effects of Process Nonlinearities 114

4- 1.1 Noninteracting Level Process 135

4- 1.2 Thermal Tanks in Series 142

4-2 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

4-3 Response of Higher-Order Systems 164

4-4 Other Types of Process Responses 167

4-4.1 Integrating Processes: Level Process 168

4-4.2 Open-Loop Unstable Process: Chemical Reactor 172

4-4.3 Inverse Response Processes: Chemical Reactor 179

4-5 S u m m a r y 1 8 1

4-6 Overview of Chapters 3 and 4 182

P r o b l e m s 1 8 3

Chapter 5 Basic Components of Control Systems

5-1 Sensors and Transmitters 197

5-2 Control Valves 200

5-2.1 The Control Valve Actuator 200

5-2.2 Control Valve Capacity and Sizing 202

5-2.3 Control Valve Characteristics 210

5-2.4 Control Valve Gain and Transfer Function 216

5-2.5 Control Valve Summary 222

5-3 Feedback Controllers 222

5-3.1 Actions of Controllers 223

1 3 5

1 9 7

5-3.2 Types of Feedback Controllers 225

5-3.3 Modifications to the PID Controller and Additional Comments

5-3.4 Reset Windup and Its Prevention 241

5-3.5 Feedback Controller Summary 244

S u m m a r y 2 4 4

P r o b l e m s 2 4 5

238

Chapter 6 Design of Single-Loop Feedback Control Systems 252

6-1 The Feedback Control Loop 252

6- 1.1 Closed-Loop Transfer Function 255

6-1.2 Characteristic Equation of the Loop 263

6-1.3 Steady-State Closed-Loop Gains 270

6-2 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 283

6-2.4 Effect of Dead Time 285

Quarter Decay Ratio Response by Ultimate Gain 304

Open-Loop Process Characterization 308

7-2.1 Process Step Testing 310

7-2.2 Tuning for Quarter Decay Ratio Response 319

7-2.3 Tuning for Minimum Error Integral Criteria 321

7-2.4 Tuning Sampled-Data Controllers 329

7-2.5 Summary of Controller Tuning 330

Tuning Controllers for Integrating Processes 331

7-3.1 Model of Liquid Level Control System 331

7-3.2 Proportional Level Controller 334

7-3.3 Averaging Level Control 336

7-3.4 Summary 337

Synthesis of Feedback Controllers 337

7-4.1 Development of the Controller Synthesis Formula 337

7-4.2 Specification of the Closed-Loop Response 338

7-4.3 Controller Modes and Tuning Parameters 339

7-4.4 Summary of Controller Synthesis Results 344

7-4.5 Tuning Rules by Internal Model Control (IMC) 350

Tips for Feedback Controller Tuning 351

7-5.1 Estimating the Integral and Derivative Times 352

7-5.2 Adjusting the Proportional Gain 354

Summary 354

Problems 355

Analysis of Feedback Control Systems by Root Locus

Rules for Plotting Root Locus Diagrams 375

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 431

10-3 Implementation and Tuning of Controllers 445

10-3.1 Two-Level Cascade Systems 446

10-3.2 Three-Level Cascade Systems 449

10-4 Other Process Examples 450

1 1 - 1.1 Scaling Computing Algorithms 464

1 l-l.2 Physical Significance of Signals 469

11-2 Override, or Constraint, Control 470

Chapter 13 Multivariable Process Control 5 4 5

13-1 Loop Interaction 545

13-2 Pairing Controlled and Manipulated Variables 550

13-2.1 Calculating the Relative Gains for a 2 X 2 System 554

13-2.2 Calculating the Relative Gains for an n X n System 561

13-3 Decoupling of Interacting Loops 564

13-3.1 Decoupler Design from Block Diagrams 565

13-3.2 Decoupler Design for n X IZ Systems 573

13-3.3 Decoupler Design from Basic Principles 577

13-4 Multivariable Control vs Optimization 579

13-5 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

13-6 S u m m a r y 5 9 2

P r o b l e m s 5 9 2

12-2.1 The Feedforward Concept 494

12-2.2 Block Diagram Design of Linear Feedforward Controllers 496

12-2.3 Lead/Lag Term 505

12-2.4 Back to the Previous Example 507

12-2.5 Design of Nonlinear Feedforward Controllers from Basic Process

P r i n c i p l e s 5 1 112-2.6 Some Closing Comments and Outline of Feedforward Controller

D e s i g n 5 1 512-2.7 Three Other Examples 518

14-2.1 Definition of the z-Transform 601

14-2.2 Relationship to the Laplace Transform 605

14-2.3 Properties of the z-Transform 609

14-2.4 Calculation of the Inverse z-Transform 613

Pulse Transfer Functions 616

14-3.1 Development of the Pulse Transfer Function 616

14-3.2 Steady-State Gain of a Pulse Transfer Function 620

14-3.3 Pulse Transfer Functions of Continuous Systems 621

14-3.4 Transfer Functions of Discrete Blocks 625

14-3.5 Simulation of Continuous Systems with Discrete Blocks 627

Sampled-Data Feedback Control Systems 629

14-4.1 Closed-Loop Transfer Function 630

14-4.2 Stability of Sampled-Data Control Systems 632

Modified z-Transform 638

14-5.1 Definition and Properties of the Modified z-Transform 639

xii Contents

14-6

14-5.2 Inverse of the Modified z-Transform 642

14-5.3 Transfer Functions for Systems with Transportation Lag

S u m m a r y 6 4 5

P r o b l e m s 6 4 5

643

Chapter 15 Design of Computer Control Systems

15-1 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

15-2 Tuning of Feedback Control Algorithms 662

15-2.1 Development of the Tuning Formulas 662

15-2.2 Selection of the Sample Time 672

15-3 Feedback Algorithms with Dead-Time Compensation 674

15-3.1 The Dahlin Algorithm 674

15-3.2 The Smith Predictor 677

15-3.3 Algorithm Design by Internal Model Control 680

15-3.4 Selection of the Adjustable Parameter 685

15-4 Automatic Controller Tuning 687

15-5 Model-Reference Control 688

15-6 S u m m a r y 6 9 5

P r o b l e m s 6 9 6

Appendix A Instrumentation Symbols and Labels

Case 1: Ammonium Nitrate Prilling Plant Control System 707

Case 2: Natural Gas Dehydration Control System 709

Case 3: Sodium Hypochlorite Bleach Preparation Control System 710

Case 4: Control Systems in the Sugar Refining Process 711

Case 5: CO, Removal from Synthesis Gas 712

Case 6: Sulfuric Acid Process 716

Case 7: Fatty Acid Process 717

Appendix C Sensors, Transmitters, and Control Valves

c-9

C-l0

C-l1

Control Valve Actuators 750

C-g.1 Pneumatically Operated Diaphragm Actuators 750

Chapter 1

Introduction

The purpose of this chapter is to present the need for automatic process control and tomotivate you, the reader, to study it Automatic process control is concerned withmaintaining process variables, temperatures, pressures, flows, compositions, and thelike 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, thenthe important process variables-those related to safety, product quality, and produc-tion rates-will not achieve design conditions

This chapter also introduces two control systems, takes a look at some of their ponents, and defines some terms used in the field of process control Finally, the back-ground needed for the study of process control is discussed

com-In writing this book, we have been constantly aware that to be successful, the engineermust be able to apply the principles learned Consequently, the book covers the prin-ciples that underlie the successful practice of automatic process control The book isfull of actual cases drawn from our years of industrial experience as full-time practi-tioners or part-time consultants We sincerely hope that you get excited about studyingautomatic process control It is a very dynamic, challenging, and rewarding area ofprocess engineering

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 ofthis unit is to heat the process fluid from some inlet temperature T,(t) up to a certaindesired outlet temperature T(t) The energy gained by the process fluid is provided bythe latent heat of condensation of the steam

In this process there are many variables that can change, causing the outlet ature to deviate from its desired value If this happens, then some action must be taken

temper-to correct the deviation The objective is temper-to maintain the outlet process temperature atits desired value

One way to accomplish this objective is by measuring the temperature T(t), ing it to the desired value, and, on the basis of this comparison, deciding what to do tocorrect 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

compar-1

Condensate return

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 thesteam flow to the exchanger All of this can be done manually by the operator, and theprocedure is fairly straightforward However, there are several problems with such

manual control First, the job requires that the operator look at the temperature

fre-quently to take corrective action whenever it deviates from the desired value Second,different operators make different decisions about how to move the steam valve, andthis results in a less than perfectly consistent operation Third, because in most processplants 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 havesystems 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

imple-mented A possible control system for our heat exchanger is shown in Fig 1-1.2

(Ap-Steam

return

Figure l-l.2 Heat exchanger control system

1-2 Important Terms and the Objective of Automatic Process Control 3pendix 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, orthe 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, thecontroller decides what to do to maintain the temperature at the desired value On thebasis 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 control.

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 sensorcan 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 controlsystem, and it is imperative that they be in a loop That is, on the basis of the mea-surement 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 Thedecision 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

tem-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 controlledvariable 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

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, possibledisturbances include the inlet process temperature, T,(t), the process flow, f(t), the en-ergy content of the steam, ambient conditions, process fluid composition, and fouling

It is important to understand that disturbances are always occurring in processes Steadystate is not the rule, and transient conditions are very common It is because of thesedisturbances 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 notdeciding how to maintain the controlled variable at set point It is up to the operator tomanipulate 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 andtaking 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 ation

consider-2 Maintain product quality (composition, purity, color, and the like) on a continuousbasis 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 onhuman labor

1-3 REGULATORY AND SERVO CONTROL

In some processes, the controlled variable deviates from set point because of

distur-bances 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 batchreactor where the temperature must follow a desired profile), and therefore the con-

trolled variable must follow the set point Systems designed for this purpose exert servo

control.

Regulatory control is much more common than servo control in the process

indus-1-4 Transmission Signals, Control Systems, and Other Terms 5tries 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

TERMS

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 forpneumatic 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 ofAmerica (ISA) when a control concept is shown without concern for specific hardware.The reader is encouraged to review Appendix A, where different symbols and labelsare 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 vices-transmitters, controllers, final control elements, and the like-to communicate

de-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 controlledvariable 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 controlleruses 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); seeFig 1-4.1 The input signal may be 4 to 20 mA and the output 3 to 15 psig An analog-to-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-to-current (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 matic or electrical Most controllers, however, are computer-based, or digital By com-

pneu-puter-based we don’t necessarily mean a main-frame computer but anything startingfrom a microprocessor In fact, most controllers are microprocessor-based Chapter 5presents different types of controllers and defines some terms related to controllers andcontrol systems

1-5 CONTROL STRATEGIES

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 loopshown in Fig l-l.2 is presented to foster this understanding

If the inlet process temperature increases, thus creating a disturbance, its effect mustpropagate through the heat exchanger before the outlet temperature increases Once thistemperature 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 occurredand that it must compensate for the disturbance by manipulating the steam valve Thecontroller signals the valve to close and thus to decrease the steam flow Fig 1-5.1shows 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 theincrease in inlet temperature, but it then decreases even below set point and continues

to oscillate around set point until the temperature finally stabilizes This oscillatoryresponse is typical of feedback control and shows that it is essentially a trial-and-erroroperation That is, when the controller “notices” that the outlet temperature has in-creased above the set point, it signals the valve to close, but the closure is more thanrequired Therefore, the outlet temperature decreases below the set point Noticing this,

Ti(t)

L

Fraction of valve opening

Figure 1-5.1 Response of a heat exchanger to a disturbance: feedback control

1-5 Control Strategies 7the 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

compen-sates for all disturbances Any disturbance affects the controlled variable, and once thisvariable deviates from set point, the controller changes its output in such a way as toreturn 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 controlledvariable at set point and in so doing compensates for all disturbances The feedbackcontroller 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

distur-bance only after the controlled variable has deviated from set point That is, the turbance must propagate through the entire process before the feedback control schemecan initiate action to compensate for it

dis-The job of the engineer is to design a control scheme that will maintain the controlledvariable at its set point Once this is done, the engineer must adjust, or tune, the con-troller 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 Oncethese characteristics are known, the control system can be designed and the controllertuned Process characteristics are explained in Chapters 3 and 4, Chapter 5 presents themeaning 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 Itssimplicity accounts for its popularity In some processes, however, feedback controlmay not provide the required control performance For these processes, other types ofcontrol strategies may have to be designed Chapters 10, 11, 12, 13, and 15 presentadditional control strategies that have proved profitable One such strategy is feedfor-ward control The objective of feedforward control is to measure disturbances andcompensate for them before the controlled variable deviates from set point When feed-forward 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

mea-sured, and then a decision must be made about how to manipulate the steam valve tocompensate for them Fig 1-5.2 shows this control strategy The feedforward controllermakes the decision about how to manipulate the steam valve to maintain the controlledvariable 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 Thefeedforward control system shown in Fig 1-5.2 compensates for only two of them Ifany of the others enter the process, this strategy will not compensate for it, and theresult will be a permanent deviation of the controlled variable from set point To avoidthis 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”

stream T

c;]

Y

Condensate return

Figure l-S.2 Heat exchanger feedforward control system

bances, while feedback control compensates for all other disturbances Chapter 12 sents the development of the feedforward controller Actual industrial cases are used todiscuss this important strategy in detail

pre-It is important to note that the three basic operations, M, D, A, are still present inthis more “advanced” control strategy Measurement is performed by the sensors andtransmitters 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

Conde’nsate return

Figure 1-5.3 Heat exchanger feedforward control with feedback compensation

Problems 9hardware, computing power, and the effort involved in designing, implementing, andmaintaining them Therefore, the expense must be justified before they can be imple-mented 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 strategymay be justifiable It is important, however, to recognize that these advanced strategiesstill require some feedback compensation.

1-6 BACKGROUND NEEDED FOR PROCESS CONTROL

To be successful in the practice of automatic process control, the engineer must firstunderstand the principles of process engineering Therefore, this book assumes that thereader 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 processesbehave 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 differentialequations Laplace transforms are used heavily in process control This greatly simpli-fies the solution of differential equations and the dynamic analysis of processes andtheir control systems Chapter 2 of this book is devoted to the development and use ofthe Laplace transforms, along with a review of complex-number algebra Chapters 3and 4 offer an introduction to the modeling of some processes

1-7 SUMMARY

In this chapter, we discussed the need for automatic process control Industrial

pro-cesses 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 iables that must be controlled

var-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 decisionmade on the basis of the measurement of those process variables Finally, A refers tothe 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 signals-pneumatic, electrical, and digital-were introduced, along with the purpose of trans-ducers

Two control strategies were presented: feedback and feedforward control The vantages and disadvantages of both strategies were briefly discussed Chapters 6 and 7present the design and analysis of feedback control loops

ad-PROBLEMS

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

(A) functions, and classify the action function as “On/Off’ or “Regulating.” Alsodraw a process and instrumentation diagram (P&ID), using the standard ISA sym-bols given in Appendix A, and determine whether the control is feedback or feed-forward.

(a) House air conditioning/heating(b) Cooking oven

(c) Toaster(d) Automatic sprinkler system for fires(e) Automobile cruise speed control(f) Refrigerator

1-2 Instrumentation Diagram: Automatic Shower Temperature Control Sketch theprocess and instrumentation diagram for an automatic control system to controlthe temperature of the water from a common shower-that is, a system that willautomatically do what you do when you adjust the temperature of the water whenyou take a shower Use the standard ISA instrumentation symbols given in Ap-pendix A Identify the measurement (M), decision (D), and action (A) devices ofyour control system

Chapter 2

M,athematical Tools for

Control Systems Analysis

This chapter presents two mathematical tools that are particularly useful for analyzingprocess dynamics and designing automatic control systems: Laplace transforms andlinearization Combined, these two techniques allow us to gain insight into the dynamicresponses of a wide variety of processes and instruments In contrast, the technique ofcomputer simulation provides us with a more accurate and detailed analysis of thedynamic 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 thedynamic 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

trunsjierfinction, from what is characteristic of the input forcing functions Because

the differential equations that represent most processes are nonlinear, linearization isrequired 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 todesign control systems Also presented are the responses of some common processtransfer functions to some common input functions These responses are related to theparameters of the process transfer functions so that the important characteristics of theresponses can be inferred directly from the transfer functions without having to re-invert them each time Because a familiarity with complex numbers is required to workwith 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 isessential for understanding the fundamentals of process dynamics and control systemsdesign

2-1 THE LAPLACE TRANSFORM

This section reviews the definition of the Laplace transform and its properties

11

2-1.1 Definition of the Laplace Transform

In the analysis of process dynamics, the process variables and control signals are tions of time, t The Laplace transform of a function of time, f(t), is defined by theformula

where

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 Laplacetransform variable, F(s) The limits of integration show that the Laplace transformcontains information on the function f(t) for positive time only This is perfectly ac-ceptable, 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 followingexample uses the definition of the Laplace transform to develop the transforms of a fewcommon forcing functions

The four signals shown in Fig 2-1.1 are commonly applied as inputs to processes andinstruments to study their dynamic responses We now use the definition of the Laplacetransform to derive their transforms

(a) UNIT STEP FUNCTION

This is a sudden change of unit magnitude as sketched in Fig 2-l.la Its algebraicrepresentation is

u(t) = -I0 t-co

Substituting into Eq 2-1.1 yields

z[u(t)] =

-I u(t)e-sf dt = - 1 e-"'

m

2-1 The Laplace Transform 13

The pulse sketched in Fig 2-1.1 b is represented by

This function, also known as the Dirac delta function and represented by t?(t), is

sketched in Fig 2-1.1~ It is an ideal pulse with zero duration and unit area All ofits 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 Laplacetransform is

Y[8(t)] = S(t)emsf dt = 1

Note that the result of the integration, 1, is the area of the impulse The same resultcan 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

e-(s-iw)t

=- - - + e- (s+ioJ)t m

1[

= w

s2 + cl?

=-The preceding example illustrates some algebraic manipulations required to derivethe Laplace transform of various functions using its definition Table 2- 1.1 contains ashort list of the Laplace transforms of some common functions

2-1.2 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 realdifferentiation and integration theorems are essential for transforming differential equa-tions 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

n!

p+l

1S+U1(s + a>*

n.I

(s + a)“+1ws* + w2

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

Real Differentiation Theorem

This theorem, which establishes a relationship between the Laplace transform of afunction and that of its derivatives, is most important in transforming differential equa-tions into algebraic equations It states that

2 df(O

[ 1 - = SF(S) -f(O)dt Proof From the definition of the Laplace transform, Eq 2-1.1,

-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

(2-1.6)

This means that for the case of zero initial conditions at steady state, the Laplacetransform of the derivative of a function is obtained by simply substituting variable sfor the “dldt” operator, and F(s) forf(t).

Real Integration Theorem

This theorem establishes the relationship between the Laplace transform of a functionand that of its integral It states that

Real Translation Theorem

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

Because the Laplace transform does not contain information about the original tion for negative time, the delayed function must be zero for all times less than the timedelay (see Fig 2- 1.2) This condition is satisfied if the process variables are expressed

func-as deviations from initial steady-state conditions

Proof From the definition of the Laplace transform, Eq 2- 1.1,

W(t - 4Jl =

t=o t=to t

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

= eesfOF(s) q.e.d

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 fromits 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 pre-sented

2-1 The Laplace Transform 19

Complex Differentiation Theorem

This theorem is useful for evaluating the transforms of functions that involve powers

of the independent variable, t It states that

Complex Translation Theorem

This theorem is useful for evaluating transforms of functions that involve exponentialfunctions of time It states that

Z[eatf(t)] = F(s - a) (2-1.11)

Initial Value Theorem

This theorem enables us to calculate the initial value of a function from its transform

It would provide another check of the validity of derived transforms were it not for thefact that in process dynamic analysis, the initial conditions of the variables are usuallyzero The theorem states that

Then apply the real differentiation theorem, Eq 2-1.6.

9s2Y(s) + 6sY(s) + Y(s) = 2X(s)Finally, solve for Y(s)

2Y(s) =9s2 + 6s + 1X(s)

The preceding example shows how the Laplace transform converts the original ferential equation into an algebraic equation that can then be rearranged to solve forthe dependent variable Y(s) Herein lies the great usefulness of the Laplace transform,because algebraic equations are a lot easier to manipulate than differential equations

dif-Obtain the Laplace transform of the following function:

c(t) = u(t - 3)[1 - e-(r-3)‘4]

Note: The term u(t - 3) in this expression shows that the function is zero for t < 3.

We recall, from Example 2-l.l(a), that u(t - 3) is a change from zero to one at t =

3, which means that the expression in brackets is multiplied by zero until t = 3 and is

multiplied by unity after that Thus the presence of the unit step function does not alter

the rest of the function for t 2 3.

2-2 Solution of Differential Equations Using the Laplace Transform 21Next apply the real translation theorem, Eq 2-1.8.

C(s) = Lqf(t - 3)] = C3”F(S)

C(s) =

e-3s

s(4s + 1)

We can check the validity of this answer by using the final value theorem, Eq 2-1.9

lim c(t) = lim u(t - 3) [l - e-(‘-3)‘4] = If-m t-+m

2-2.1 Laplace Transform Solution Procedure

The procedure for solving a differential equation by Laplace transforms consists ofthree steps:

1 Transform the differential equation into an algebraic equation in the Laplace form variable s

trans-2 Solve for the transform of the output (or dependent) variable

3 Invert the transform to obtain the response of the output variable with time, t.Consider the following second-order differential equation:

d2Y(d + a dY(t)

The problem of solving this equation can be stated as follows: Given the constantcoefficients a,, a,, a2, and b, the initial conditions y(O) and dy/dt I l= , and the function x(t), find the function y(t) that satisfies the differential equation.

We call the function x(t) the “forcing function” or input variable, and we call y(t)

the “output” or dependent variable In process control systems, a differential equationsuch as Eq 2-2.1 usually represents how a particular process or instrument relates itsoutput signal, y(t), to its input signal, x(t) Our approach is that of British inventor James

Watt (1736- 1819), who considered process variables as signals and processes as signalprocessors.

The first step is to take the Laplace transform of Eq 2-2.1 We do this by applyingthe linearity property of Laplace transforms, Eq 2-1.3, which allows us to take theLaplace transform of each term separately:

u,%[~] + @[fy] + a,~ely(t)] = b%w (2-2.2)

Assuming for the moment that the initial conditions are not zero, the indicated Laplacetransforms are obtained by using the real differentiation theorem, Eq 2-1.5

2 [-Id2Y(4dt2 = s2Y(s) - sy(0) - 5 f 0 =

f 0

The second step is to manipulate this algebraic equation to solve for the transform

of the output variable, Y(s)

MS) + @2s + ~,>Y(O) + a2 2

Y(s) =

u*s* + u,s + a,

This equation shows the effect of the input variable, X(s), and of the initial conditions

on the output variable Our objective is to study how the output variable responds tothe input variabIe, so the presence of the initial conditions complicates our analysis Toavoid this unnecessary complication, we assume that the initial conditions are at steady

state, dy/dt I t=O = 0, and define the output variable as the deviation from its initial value,

thus forcing y(O) = 0 We will show in the next section how this can be done withoutloss of generality With zero initial conditions, the equation is reduced to

The form of Eq 2-2.4 allows us to break the transform of the output variable into

the product of two terms: the term in brackets, known as the trunsfirfunction, and the

transform of the input variable, X(s) The transfer function and its parameters

charac-2-2 Solution of Differential Equations Using the Laplace Transform 23terize the process or device and determine how the output variable responds to the inputvariable The concept of transfer function is described in more detail in Chap-ter 3.

The third and final step is to invert the transform of the output to obtain the timefunction y(t), which is the response of the output Inversion is the opposite operation

to taking the Laplace transform Before we can invert, we must select a specific inputfunction for x(t) A common function, because of its simplicity, is the unit step function,u(t), which was introduced in Example 2- 1.1 From that example, or from Table 2- 1.1,

we learn that for x(t) = u(t), X(S) = l/s We substitute into Eq 2-2.4 and invert toobtain

2-2.2 Inversion by Partial Fractions Expansion

The mathematical technique of partial fractions expansion was introduced by the Britishphysicist Oliver Heaviside (1850- 1925) as part of his revolutionary “operational cal-culus.” The first step in expanding the transform, Eq 2-2.5, into a sum of fractions is

to factor its denominator, as follows:

(a22 + a,s + ~2,)s = u2(s - r,)(s - rJs (2-2.6)

where r, and r2 are the roots of the quadratic term-that is, the values of s that satisfy

2a2

(2-2.7)

For higher-degree polynomials, the reader is referred to any numerical methods text for

a root-finding procedure Most electronic calculators are now able to find the roots of

third- and higher-degree polynomials Computer programs such as Mathcad’ andMATLAB provide functions for finding the roots of polynomials of any degree.Once the denominator is factored into first-degree terms, the transform is expandedinto partial fractions as follows:

A2 Y(s) = * + - +A3

provided that the roots, r, , r2, and r3 = 0, are not equal to each other For this case ofunrepeated roots, the constant coefficients are found by the formula

A, = lim (s - rJY(s) s+q

We can now carry out the inversion of Eq 2-2.8 by matching each term to entries inTable 2-1 l; in this case the first two terms match the exponential function with a =

- r,, and the third term matches the unit step function The resulting inverse tion is

func-y(t) = A,erlr + A2erzf + A+(t)

Repeated Roots

For the case of repeated roots, say rl = r2, the expansion is carried out as follows:

y(s) = (s - r,)2

A, = !i; i $ [(s - rl)2Y(s)]

Again, we carry out the inversion of Eq 2-2.10 by matching terms in Table 2-1.1 Thefirst term matches the sixth term in the table with a = - rl, to give the inverse

y(t) = A,terlf + Azerlr + A+(t) (2-2.11)

’ Mathcad User’s Guide, by MathSoft, Inc., 201 Broadway, Cambridge, MA, 02139, 1992.

ZMATLAB User’s Guide, The MathWorks, Inc., 24 Prime Park Way, Natick, MA, 01760, 1992.

2-2 Solution of Differential Equations Using the Laplace Transform 25

In general, if root Y, is repeated m times, the expansion is carried out as follows:

The coefficients are calculated by

The following example is designed to illustrate numerically the partial fractionsexpansion procedure and the entire inversion process Three cases are considered: un-repeated real roots, repeated roots, and complex conjugate roots

Given the quadratic differential equation considered in the preceding discussion, Eq.2-2.1, with zero steady-state initial conditions, we will obtain the unit step response ofthe output variable y(t) for three different sets of parameters.

Let a2 = 9, a, = 10, a, = 1, and b = 2, in Eq (2-2.1) Then the unit step

re-sponse is, from Eq 2-2.5,

The coefficients are calculated using Eq 2-2.9.

Invert by matching entries in Table 2-1.1 to obtain the step response

y(t) = - 2.25edg + 0.2.W’ + 2u(t)

(b) REPEATED ROOTS

Let a, = 6, and let the other parameters be as before The roots, from the quadratic

formula, are Y, = r2 = - 1/3, and the Laplace transform of the output response is

1 ss+-3The coefficients are, from Eq 2-2.13,

2-2 Solution of Differential Equations Using the Laplace Transform 27

(c) PAIR OF COMPLEX CONJUGATE ROOTS

Let a, = 3, and let the other parameters be as before The roots, from the quadratic

formula, are r,,* = - 0.167 k i0.289, where i = a is the unit of the imaginary

numbers The transform of the output is then

2Y(s) =

and A, = 2, as before The inverse response is again obtained by matching entries

in Table 2- 1.1 Note that the fact that the numbers are complex does not affect thispart of the procedure

y(t) = (- 1 + j0.577)e(-0.167+i0.289)r + (- 1 - j0.577)e(-O.l67-iO.289)t + ‘&4t)

It is evident from the preceding example that calculating the coefficients of the partialfractions expansion can be difficult, especially when the factors of the transform arecomplex numbers As we shall see in the next section, the roots of the denominator ofthe transfer function contain most of the significant information about the response.Consequently, in analyzing the response of process control systems, it is seldom nec-essary to calculate the coefficients of the partial fractions expansion This is indeedfortunate

2-2.3 Handling Time Delays

The technique of partial fractions expansion is restricted to use with Laplace transformsthat can be expressed as the ratio of two polynomials When the response contains timedelays, by the real translation theorem, Eq 2-1.8, an exponential function of s appears

in the transform Because the exponential is a transcendental function, we must priately modify the inversion procedure

appro-If the denominator of the transform contains exponential functions of s, it cannot befactored because the exponential function introduces an infinite number of factors Onthe other hand, we can handle exponential terms in the numerator of the transform, as

we shall now see