control systems technology

CHAPTER OUTLINE1.1 Purpose 1.2 Introduction 1.2.1 Control System Strategy 1.2.2 Examples of Control Systems 1.2.3 Analytical Issues 1.3 Analytical Descriptions 1.3.1 Block Diagram 1.3.2

CONTROL SYSTEMS TECHNOLOGY

Curtis D Johnson

University of Houston

Heidar A Malki

University of Houston

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10 9 8 7 6 5 4 3 2 1 ISBN 0-13-081530-6

This text was written to fill a very important educational niche in the broad spectrum of trol systems knowledge That niche lies between the hands-on electromechanical knowledge and skills needed by technicians and the highly abstract and theoretical knowledge required

con-by scholars who research and develop new control strategies This book focuses on the knowledge required by control systems practitioners to enable them to both understand and evaluate an existing control system and devise and design new control system applications The text presents classical and digital control systems with an emphasis on careful explanations of the concepts Many examples illustrate key topics and the operations re- quired to solve problems.

The text is an outgrowth of many years of teaching control systems to students in an engineering technology program It is written for a two-semester course, nominally sepa- rated into analog and digital control The difficulty with this approach is that much of dig- ital control is a spinoff of analog concepts Therefore, the analog material by itself is more extensive than the digital. In practice, we have found that some of the material on analog control must be delayed to the second course.

Although patterned after the course sequence expected for a particular educational program, this text can be adapted to other approaches For example, Chapter 2 (Mea- surement) can be omitted by those who prefer to cover sensors and measurement in other courses Likewise, if Laplace transforms are covered in an independent course, that sec- tion in Chapter 3 can be omitted or assigned as review It would be important to include,

however, the last section of Chapter 3, Analog Simulation.

The text emphasizes an understanding of control system concepts, but also requires the use of computers to implement practical solutions to problems There are a number of control and mathematical software packages which are of great value in the study of con- trol systems Throughout the text, the use of these packages to facilitate solving problems

is emphasized, and Mathcad or MATLAB is used to illustrate computer-based cal procedures An attempt has been made to emphasize the use of computers as a tool to implement the mathematical and graphical operations required to solve a problem.

mathemati-A Web page (www.uh.edu/~tech13v/ContSysTech) will be set up for this text as a means for communication between users and authors, and also for sharing ideas and tech- niques related to teaching control systems A solutions manual (ISBN: 0-13-090661-1) is available It contains examples of physical and simulation experiments that can be con- ducted to enhance learning.

Dr Malki would like to thank his parents, his wife Layla, and his son Armeen for their support and patience during the long task of writing this book Dr Johnson would like to thank his wife Helene and his mother-in-law Lois for their continuing kindness while he undertook this task.

1 INTRODUCTION TO CONTROL SYSTEMS 1

1.1 Purpose 2

1.2 Introduction 2

1.2.1 Control System Strategy 3

1.2.2 Examples of Control Systems 4

1.2.3 Analytical Issues 7

1.3 Analytical Descriptions 7

1.3.1 Block Diagram 8

1.3.2 Transfer Functions 10

1.3.3 Computer Applications Software 13

1.4 Analog and Digital Control 14

1.5 System Design Objectives 17

viii I CONTENTS

3.1 Purpose 52 3.2 Introduction 52 3.3 Definition of the Laplace Transform 54 3.3.1 Computer Applications 58

3.4 Properties of Laplace Transforms 59 3.5 Inverse Laplace Transform 65

3.5.1 Partial-fraction Expansion 66 3.5.2 Convolution 75

3.6 Analog Simulation 77

3.6.1 The Operational Amplifier 77 3.6.2 Simulation of Physical Systems 80 3.6.3 Simulation of Control Systems 85

Summary 87 Problems 89

4.1 Purpose 94 4.2 Transfer Functions 94

4.2.1 Block Transfer Functions 95 4.2.2 Transfer Function Properties 100 4.2.3 Graphing 106

4.3 Block Diagrams 113

4.3.1 Canonical Form 114 4.3.2 Block Diagram Reduction 115 4.3.3 Multiple Inputs 123

4.4 Mason's Gain Formula 126 4.5 Controller/Compensator Transfer Functiom 128 4.5.1 Proportional, Integral, and Derivative Contro~ 129 4.5.2 Lead and Lag Compensation 131

Summary 132 Problems 133

5.1 Purpose 140 5.2 Static Response 141

5.2.1 Steady-State Error 141 5.2.2 Disturbance Error 148

CONTENTS I ix

5.3 Dynamic Response of First- and Second-Order Plants 150

5.3.1 First-Order Plant 150

5.3.2 Second-Order Plant 153

5.4 Characteristics of Dynamic Response 158

5.5 Steady-State Error Versus Stability 165

6.2.1 Formal Definition of Stability 177

6.3 Routh-Hurwitz Stability Criterion 178

7.4 Bode Plot Applications 210

7.4.1 Gain and Phase Margins 212

x I CONTENTS

8.3 Root Locus Construction 228

8.3.1 Manual Construction 229 8.3.2 Computer Construction 240

8.4 Root Locus Applications 240

8.4.1 Gain and Phase Margin 240 8.4.2 Transient Response 242

Summary 249 Problems 249

9.1 Purpose 254 9.2 State-Space Definition 255 9.3 Solving State-Space Equations 259

9.3.1 Laplace Transform Solutions 259 9.3.2 Series Expansion Solution 264 9.3.3 Computer Simulation Solution 266

9.4 Simulation Diagrams and State-Space Equations 269

9.4.1 Simulation Diagram Definition 270 9.4.2 Generalized Rules of Simulation Diagram Construction 272

9.5 Transfer Function in State Space 275 9.6 Controllability and Observability 277

9.6.1 Controllability 277 9.6.2 Observability 279

Summary 281 Problems 282

10.1 Purpose 286 10.2 Definition of a Digital Control System 286

10.2.1 Digital Control System Hardware 287 10.2.2 Digital Control System Software 291 10.2.3 Simulations Using Computers 294

10.3 The Difference Equation 294

10.3.1 Finding the Difference Equation 295 10.3.2 Solution of the Difference Equation 298

Summary 302 Problems 303

11.3.2 Advance Theorem (Shift Left) 316

11.3.3 Delay Theorem (Shift Right) 317

11.3.4 Final Value Theorem 318 11.3.5 Initial Value Theorem 319

11.4 Inverse z-Transform 321

11.4.1 Partial-Fraction Expansion for Real Poles 321

11.4.2 Partial-Fraction Expansion for Complex Poles 325

11.4.3 Partial-Fraction Expansion for Repeated Poles 327

11.4.4 Direct Method: Long Division 329

11.4.5 Inverse z-Transforms by Software 330

11.5 Difference Equation Solution 330

Summary 334

Problems 335

12.1 Purpose 338

12.2 Discrete Transfer Function 338

12.3 Open-Loop Transfer Functions 347

12.4 Closed-Loop Transfer Functions 349

12.5 Static and Dynamic Response 355

13.3.1 Routh-Hurwitz Test with the Bilinear Transformation 374

13.3.2 Jury's Stability Test 376

xii ICONTENTS

13.4 Discrete System Root Locus 381

13.4.1 Root Locus Construction Rules 381

Summary 387Problems 388

14.1 Purpose 39214.2 State-Space Equations in the Discrete Domain 392

14.2.1 Discrete State Equations 392 14.2.2 Solution by Recursion 396 14.2.3 Solution byz-Transforms 398

14.3 Discrete State-Space Transfer Function 400

14.3.1 Generation of State-Vector Equations 401

14.4 Observability and Controllability 40314.5 Discrete Simulation Diagrams 406Summary 410

CONTROL SYSTEMS TECHNOLOGY

CHAPTER OUTLINE

1.1 Purpose

1.2 Introduction

1.2.1 Control System Strategy

1.2.2 Examples of Control Systems

1.2.3 Analytical Issues 1.3 Analytical Descriptions

1.3.1 Block Diagram

1.3.2 Transfer Functions

1.3.3 Computer Applications Sohware

1.4 Analog and Digital Control

1.5 System Design Objectives

1.5.1 Dynamic Response

1.5.2 Instability Summary

2 I CHAPTER ONE

This chapter presents an overview of control systems The basic strategy for trolling variables in a physical system is discussed and several examples of controlsystems are reviewed The analytical basis of control system analysis and design ispresented along with definitions of terms and expressions used throughout the book.The characteristics of both analog and digital control systems are presented Finally,the chapter explains why a study of control systems is always accompanied by astudy of the stability and instability of such systems

The modem world is described in terms of many physical variables Examples include

such things as the temperature of some quantity of material, the three-dimensional tion of a robot arm in a manufacturing operation, and the speed of a motor driving a

mo-conveyor belt The value of these variables is typically dynamic; i.e., the value tends

to change in time, either systematically or randomly These changes are caused by thedependence of the specific physical variable on many other variables For example,the temperature in a room may be a function of the outside temperature, air currents

in and out of the room, the number of people in the room, and any heat-generatingequipment operating in the toom Because of these dependencies, the room tempera-ture, if not controlled, will tend to fluctuate somewhat randomly in time

A control system is an automatic process by which the value of a physical

vari-able is forced to adopt specified values as a function of time "Automatic" means thatthis control is accomplished without human intervention Having a human maintainroom temperature at a constant value in time by manually adjusting a heater is an ex-ample of a human-aided control system In our definition, a control system is one thatmakes the required adjustments automatically, without human aid

It is common to separate control system descriptions into two broad gories-process control and servomechanisms depending upon how the value ofthe physical variable is expected to behave in time However, an analysis and under-standing of control system operation is fundamentally the same for both categories

cate-Process Control In many instances, the objective of a control system is to force

a physical variable to remain constant in time and equal to some desired value This

type of control is often called a process control system Process control is

en-countered, for example, in automated manufacturing operations, such as in thechemical and petrochemical industries where temperatures, flow rates, levels, and

so on are forced to maintain constant values Such control is often also called

reg-ulation and the desired value is called the setpoint.

INTROOUCTION TO CONTROL SYSTEMS I 3

Servomechanism Another type of control system objective is to force a cal variable to change in time, but in a precisely prescribed manner That is, the

physi-physical variable will be forced to follow or track some target value as it changes

in time The term servomechanism is frequently used to describe such control byreference to a historical approach to providing the control A common example ofthis kind of control system is in industrial robot arm motion, where the arm mustfollow a specific path in space as a function of time

EXAMPLE To which category of control system do the following examples belong, and why?

1.1

a A control system to steer a vehicle along a curving road

b A control system to maintain the liquid level in a chemical reaction vessel

1.2.1 Control System Strategy

The basic strategy by which control systems operate is quite simple and is the samefor both process control systems and servomechanisms This strategy is exactly thesame as that which a human would employ to affect the control manually Suppose

a system contains a physical variable requiring control that changes under the fluence of other external variables In general, control is accomplished via therepetitive application of the following four operations:

in-1 Determine the present value of the physical variable This is called

meas-urement.

2 Compare that value measured with the presently desired value of the

vari-able This is called error detection.

3 Determine a corrective action that will compensate for changes brought

about by the external influences? This is carried out by a controller or a

compensator.

4 Feed back is a corrective action to the system that contains the physical

variable under control such as to drive the error toward zero

The same four operations are applied for human-aided manual control, chanical automatic control, electricaVelectronic automatic control, or control pro-vided by a computer The difference lies only in how they are implemented

me-4 I CHAPTER ONE

1.2.2 Examples of Control Systems

This section presents three examples of control systems and explains how the fourbasic steps of the strategy are employed in their implementation

Engine Speed This is one of the original examples of modem automatic control

In this case, we want to control the rotational speed of a steam engine; that is, therate of revolution of the drive shaft We want the speed to remain constant and we

call that kind of control regulation. In about 1788, Matthew Boulton and JamesWatt came up with a method of doing this automatically by adjusting a valve thatcontrols steam to the piston The more the valve is open, the more steam is pro-duced and the faster the engine turns, and vice-versa if the valve is closed If a load

is placed on the shaft, the rotational rate will drop so that more steam is needed to

compensate Figure 1.1 shows the essential features of the regulator, or governor,

which was devised by Boulton and Watt

Notice the weights on the end of beams that rotate with the engine speed.Physics tells us that as the speed increases, the weights tend to move to the hori-zontal, as shown This resembles an umbrella, except instead of pushing the um-brella spines up from the shaft, the spinning weights pull it up If an umbrella isspun fast enough, it would also open up The angle of the weights is a measurement

of the engine speed

Let's see how this works Suppose the speed increases because of a reducedload on the shaft Then, as shown by the dashed lines in figure 1.1, the spinning

INTRODUCTION TO CONTROL SYSTEMS I 5

weights would rise, pulling up the hub, which slides on the vertical spinning shaft.Raising this hub pulls the valve linkages to the right as shown by the arrows, whichtends to close off the valve Thus, with less steam the engine will slow down its ro-tation and the spinning ball assembly will drop back down The error detector, con-troller and feedback element are combined in the linkages and the steam valve

As the speed increases and the balls move out, the valve closes a little, therebycutting down on the steam and slowing the engine Conversely, if a load is placed

on the engine and it starts to slow down, the balls move down toward the vertical.This opens the valve to let in more steam, compensating for the load so that thespeed remains constant

Water Heater Figure 1.2 illustrates an automatic water heater In this example,

temperature measurement is made by a sensor that provides a voltage proportional

to temperature The error detector and controller are embodied in a differential

am-plifier This amplifier outputs an error voltage, V H,proportional to the difference

between the desired temperature (expressed as a voltage), V n and the measured

temperature voltage, V To The controller function occurs when this difference is

multiplied by a gain, K, which determines how much voltage to apply to the heater

resistor, R H. The heater resistor is the feedback element that provides corrective tion Why is the diode used?

ac-If the temperature is quite low, V T will be much less than V rand the differential

amplifier applies a voltage, V H,to the heater resistor that is proportional to this

dif-ference The resistor heats up from V H2/RH heating and so the temperature rises As

the temperature rises, the difference between V T and V r reduces, so V His smaller andthe resistor produces less heat

An important feature of this type of control system is that it is the error thatproduces the heater signal Therefore, it follows that there must be error! The gaindetermines how much error will be necessary to produce the required heating

Crystal Puller The system shown in figure 1.3 is an example of a

servomech-anism type of control system In this case, a crystal is to be pulled from a melt of

the crystal material The rate of pull-out must follow a specific pattern, ramping upslowly to a certain speed that is held constant for a period of time, and then the pull-out rate is ramped back down to zero Thus, the reference value is changing in timeand the control system must track this while compensating for effects that maycause variations

INTRODUCTION TO CONTROL SYSTEMS I7

In this case, the sensor is a tachometer that measures motor speed as a voltage,

Vs The differential amplifier subtracts the motor speed voltage from the reference,

V p getting an error voltage, Ve = Vr - V S' The controller, or compensator, deduces

an appropriate drive voltage for the motor, Vrn A problem like this has many

dy-namic influences including such issues as motor start-up, speed reducer wind-up,system inertia, and varying loads as the crystal pulls Yet the control system must

force the motor speed to follow the input pattern of Vr as a function of time.

EXAMPLE Explain how the four-step control strategy applies to a human controlling the speed

1.2 of a vehicle What external variables influence the control system? Is this a

servo-mechanism or process control?

Solution:

Measurement is carried out by a combination of the electromechanical ter of the car and the visual observation of the human The human carries out er-ror detection in her cerebral cortex by comparing the measured value with the de-sired value A decision about corrective action is also carried out in the cerebralcortex Feedback control action is a combination of muscular signals to the feetand the mechanical action of the foot pedal on the gas feed to the engine Thereare many external influences, such as incline of the road, wind, road conditions,weather conditions, and road curvature It is a mixture of servomechanism andprocess control When the speed limit is constant, the target value is fixed and it

speedome-is like a process control system On the other hand, changing conditions, such asthe speed limit or safety considerations, may cause the target value to change intime like a servomechanism

1.2.3 Analytical Issues

This introduction has provided verbal and pictorial descriptions of control systemsand their operations The four strategy steps provide a descriptive mental image ofhow control systems operate However, to evaluate how well a control system isoperating' or to design a control system for a specific application, we need an ana-lytical description The next section sets the basis of how such analytical descrip-tions are constructed and describes the kinds of mathematical tools required to per-form evaluation and design

1.3 ANALYTICAL DESCRIPTIONS

In practical realizations of control systems, we need to be able to design the elements

of a control system for a particular application and we need to be able to evaluatehow well the control system is functioning For example, in the temperature control

8 ICHAPTER ONE

situation presented· in Section 1.2, nothing was said about how to detennine the

value of the gain, K Furthennore, we do not know how well the system can respond

to errors in temperature that may occur If the temperature suddenly dropped invalue due to external influences, how fast can the control system bring the temper-ature back to the desired value? These kinds of questions can be answered by de-vising an analytical basis for the study of control systems and their operation Thissection presents the fundamental ideas behind such analysis; later chapters study thetechniques in detail

1.3.1 Block Diagram

To conduct an analysis of a control system, it is first necessary to derive analyticaldescriptions of the strategy This is made easier by starting first with a pictorialblock diagram that demonstrates how the strategy is implemented We assumethere is a plant or system that is described by the values of many physical variables,all of which may vary in time, x(t), yet), z(t), u(t), wet), We assume that one ofthese variables has been identified to require control; let's say y(t). Then y(t) is

called the controlled variable The desired value ofyet)is called the reference, or

setpoint value and is described by the quantity, r(t).

In general, the value ofy(t) may depend explicitly on time and also upon some

or all of the other system variables,.

yet) =j[x(t), z(t), u(t), wet), tJ (1.1)

To provide corrective feedback as described in the basic control strategy,one of the system variables is selected as the controlling variable; let's say u(t).

Thus, if we need to make changes inyet), we do so by feeding back changes in

u(t) and letting the dependence of equation 1.1 cause changes in the controlledvariable

INTROOUCTION TO CONTROL SYSTEMS I 9

Figure 1.4 shows how the basic control strategy translates into a pictorialview There are three "blocks," one each for measurement, the controller or com-pensator, and the plant or system The fourth element is the error detector, whichfinds the present difference between the measurement value of the controlled vari-able and the reference value

The measurement block inputs the value of y(t) and outputs some value, b(t), which is proportional to y(t) The error detector subtracts this value from the de-

sired or reference value to form an error,

This error is provided as input to the controller or compensator After operation on

the error, this block outputs a new value of u(t) (control action) as necessary to drive

the error toward zero The operation of the controller is not simple or always ous For example, the controller may be required to provide a strong corrective feed-

obvi-back even when the error is zero! This would be the case, for example, if the error

at some time was zero but its rate of change (derivative) was not zero Thus, the troller must be designed to compensate for expected error as a function of time

con-The system forms a closed loop so that there is continuous correction to y(t)

as necessary Taken as a whole, the "input" of the control system is considered to

be the reference value and the "output" is the controlled variable

Closed Loop and Open Loop As noted, when a control system is implemented

to regulate the value of some variable, we describe the resulting system as a loop control system By contrast, in some cases a system may exhibit stabilization

closed-of a variable without the use closed-of a control system This is called an open-loop tem In some cases, such systems exhibit self-regulation It is important to realize

sys-that a system with self-regulation is not controlled because changes in the systemmay cause the variable of interest to adopt a new value It is stable, but the actualvalue to which it will stabilize is not controlled

An example of this is the level of liquid in a tank into which liquid is pumped

in at some rate, Qin, and flows out at some rate determined by the level, Qout =

[2gh]112.Clearly, the level, h, will adopt some constant value such that the flows in

and flow out are equal So, the level is self-regulated However, if the in-flowchanges, the level will also change, so the level is not controlled

Block diagrams such as that shown in figure 1.4 help in the analysis of trol systems in both evaluation and design Chapter 4 presents a more detailed study

con-of the block diagram analysis con-of control systems

EXAMPLE Construct a block diagram to represent the operation of the water heater control

1.3 system presented in figure 1.2 Identify the relationship of the physical system to

each block

sys-controller action is provided by multiplying the error by a gain, K (this is called

pro-portional control) The plant, or process, is the assemblage of water tank, in-flow,

out-flow, and heater The resistive heater is the feedback control element and theunspecified temperature sensor provides the measurement

1.3.2 Transfer Functions

In the block diagram shown in figure lA, each block has a single input and a

sin-gle output The block represents some operation that is performed on the input in

order to produce the output In a very general way, we define thetransfer function

of a block as the ratio of output to input For example, suppose a block simply

mul-tiplies the input, x(t), by some constant gain, K, to produce the output, y(t) Then

the equation for the block is,

and the transfer function for the block is y(t)/x(t) = K.

Linear Operations One of the restrictions that must be placed upon the

con-trol system in order to devise an analytical approach is that the transfer functions

must represent linear operations This is because our mathematical analysis isonly valid for linear operations; that is, we don't know how to solve nonlinearproblems Fortunately, many if not most, physical control systems can be repre-sented, at least to a good approximation, by linear operations

A linear operation is one that obeys a specific mathematical definition Let's

represent a mathematical operation performed on some variable, x, by the symbol,

14 I CHAPTER ONE

Now powerful computer applications software is available that can readilyobtain solutions to even very complex control systems Throughout this book, in-formation about how these software packages can be applied is presented How-ever, the approximate techniques are also explained because they enhance an un-derstanding of the nature of control systems In many cases, these approximationtechniques present a better view of system behavior when parameters are varied.Two application software packages will be used in the book: Mathcad andMATLAB In general, either software can be used in the applications presented inthe book Use one or both of these packages as you work through the problems andexamples in this book Both software packages can perform real and complex num-ber calculations, find roots of higher-order polynomials, perform graphing, findLaplace transforms and their inverses, and perform symbolic mathematical opera-tions including integration and differentiation and, in general, nearly any type ofmathematical operation

1.4 ANALOG AND DIGITAL CONTROL

In the early days of control systems, purely mechanical means were employed toperform the operations required, such as the steam engine speed regulator operationspresented previously Subsequently, methods were developed to employ pneumatictechniques that used air pressure in pipes to process and transmit signals required toperform control actions Then electrical and electronic methods were adopted to per-form the control action In recent years, the computer has become a valuable tool incontrol system implementation In the simple view, any control system that does not

use a computer is referred to as an analog control system; those that do use puters for control are called digital control systems.

com-Analog Control System In general, the physical variables of control systems'vary smoothly and continuously in time Thus, the temperature of a reaction vesselmay exhibit rapid variation, but it will not change in a discontinuous fashion intime The rotational rate of a motor may vary, but it will do so continuously in time

If these vari'ables are provided as input to some block of a control system, one mayexpect the output to vary smoothly and continuously in time as well Such varia-tion is the nature of an analog control system

Suppose x(t) is the input to a control system element and the output is an

ana-log function, yet) If all possible values of y versus a range of variation of x are

plot-ted, a smooth and continuous static graph like that shown in figure 1.7 will result

If xU) is changing in time and the time variation of yet) versus time is ted, a graph like that in figure 1.8 will result Note that the values of yet) in fig-

plot-ure 1.8 arenot found by translating values ofx(t) from figure 1.7 This is because

there will be some dynamic response difference between yet) and x(t) determined

FIGURE 1.8

Dynamic analog variation of y(t) as x(t) varies in time.

by the differential equation relating the two variables These are examples of log variations

ana-The first part of this book concentrates on design and analysis of analog trol systems The knowledge gained in this study is easily transferred to a study ofdigital control systems

con-Digital Control System When a computer is used as part of a control system,

it is not possible to maintain the analog relationships, for two reasons First, thecomputer obtains values of a variable through an analog-to-digital converter(ADC) The output of an ADC is discrete in nature, so that a static graph of the

output of an ADC, n, versus input of a physical variable, x, might look like that shown in figure 1.9 Note that there is not a smooth and continuous relationship

between the output and the input There are incremental ranges of change of x for

which the ADC output does not change at all

Second, the computer can only obtain sample values of the physical variable

at specific times because of the intervening time required by the ADC to obtain thesamples and the time required by the computer to process the data and provide afeedback Therefore, even though the variable xU) changes smoothly and continu-

ously in time, variation of the output of an ADC, net), in time consists of discrete

samples, as shown in figure 1.10 Note that the variation is not smooth and

contin-uous in time, as is the physical variable

So in digital-based (computer) control systems, knowledge of the physicalvariable is both discontinuous in value and discontinuous in time

INTRODUCTION TO CONTROL SYSTEMS I 17

Despite the lack of smooth and continuous knowledge about the variables in acomputer-based control system, the use of computers in control is growing because

of the many other advantages they offer The last part of this book is devoted to astudy of modifications of the evaluation and design methods when computers areused in control systems

1.5 SYSTEM DESIGN OBJECTIVES

The simple objective of a control system is to control the value of some physicalvariable In the design and evaluation of control systems, such an objective must

be put on a more quantitative basis The following paragraphs define the more cific objectives of designing a control system and evaluating its performance

In general, the dynamic response is described by specification of two quantities:

the steady-state error and the transient response Steady-state error refers to a

continuous deviation between the desired value of the controlled variable and itsactual value By selection of parameters in the control system, we can specify thevalue of such steady-state error

Transie~t response is defined as excursions that the controlled variablemakes from the desired value when some transient disturbance occurs Transientresponse also defines how the controlled variable behaves in time when a change

is made willfully, such as a change in setpoint In the design of control systems, wecan select parameters to assure that the transient response takes on some specifiedform and magnitude Figure 1.11 shows two common forms of transient response:

overdamped and underdamped, or cyclic For these curves it is assumed that some

transient disturbance occurs at tIand the controlled variable exhibits a temporarydeviation from the desired value, as shown You will1eam how such responses arecharacteriz_ed and how system parameters affect such responses

1.5.2 Instability

INTRODUCTION TO CONTROL SYSTEMS I 19

Notice in the previous argument that if the original error source had peared by the time the feedback signal was presented back at the input, the feed-

disap-back would still show up as -8yo, meaning that the error is caused to persist even

when the source is gone In fact, if the overall gain of the feedback were more thanunity, the original error would grow in time even though the source is gone

Thus, instability is connected to both the phase lag (delay) of the control loop

and the gain of the loop in feeding back a correction Later chapters will be devoted

to trying to find the conditions for which a system can become unstable and whatcan be done to prevent those conditions from occurring On the basis of the pre-ceding argument, we can make the following general observation about controlsystem stability:

If there is any condition for which the feedback, apart from the error detector, has a net phase shift of - 180 0 and a gain greater than unity, the control system will

ba-a Measure the controlled variable.

b Find the error between the measured value and the desired value.

c Decide on corrective action to compensate for any error.

d Feed back a correction to the system to change the controlled variable.

3 A block diagram is used to pictorially display the operational units of a controlsystem Each block represents some part of the control system, such as meas-urement, compensation, error detection, and the plant itself

4 The various blocks of a control system have a transfer function that defines howthe block output varies as a function of the input These transfer functions aremost often represented by differential equations

5 Block transfer functions may be linear or nonlinear Analysis and design of controlsystems as presented in this book require the transfer functions to be linear Lin-earization refers to methods used to force a block transfer function to be linear

6 An analog control system has smooth and continuous relationship between theinputs and outputs of all system blocks Digital control systems most often usecomputers and represent conditions wherein the output of a block may vary in

a discontinuous fashion as the input changes

20 ICHAPTER ONE

7 The objectives of control system design are to make the steady-state error assmall as possible and to make the response to a transient disturbance follow aspecified pattern, either over-damped or under-damped

8 One of the biggest problems in control system design is that a control systemcan cause a controlled variable to become unstable This means that the value

of the variable, instead of being controlled, begins to increase without limit Thisinstability is linked to the phase shift and gain of a control loop as signals arefed back to the error detector

PROBLEMS

1.1 Which of the following control examples are more like process control andwhich are more likely servomechanisms? Explain

a A common home refrigerator

b A solar panel tracking sun movement across the sky

c An automobile's cruise control

d The flow rate of oil in a pipeline1.2 Identify the four elements of control: measurement, error detection, compen-sation, and feedback for the following control systems:

a A common home refrigerator

b A home air cOnditioner

c The steering of an automobile

d An automobile's cruise control1.3 Study the steam engine governor shown in figure 1.1 What element of thesystem would you adjust to change the speed range of the engine, and why?Suppose the system is regulating the rotational rate from 15 to 25 revolutions-per-minute (rpm) What would you change so the range was, say, 25 to 50rpm?

1.4 Why is the diode used in the water heater shown in figure 1.2?

1.5 Consider the water heater example of a proportional control system as sented in figure 1.2 The system should maintain the tank temperature at 80°Cand has the following characteristics: (1) the sensor has a transfer function of

pre-5 mVJOC, (2) the amplifier gain is pre-50, (3) the heater resistor is 2.0n,(4) thetank loses temperature at O.2°C per minute, (5) the heater resistor causes thetank temperature to increase 0.05°C per minute for every watt of dissipatedpower, and (6) the diode forward voltage drop is 0.7 volts Assuming the start-ing tank temperature is 20°C, answer the following questions:

a What voltage should be used for the amplifier reference?

b What is the initial tank heating rate in °c per minute?

c What is the tank heating rate for temperatures of 40°, 60°, and 80°C?

23

24 I CHAPTER TWO

As stated in chapter 1, the objective of a control system is to control the value ofsome dynamic variable in a system To do this, the control system designer employs

a control strategy that starts with a comparison of the present value of the variable

to be controlled with the desired value The resulting error is processed in ways tated by the control system designer to assure a stable system with the best possi-ble control There are many ways to process the error to provide good control inany given system

dic-The act of measurement determines the present value of the controlled

vari-able The control system designer does not have many choices on how to make ameasurement Only a certain number of sensors are available to measure specificvariables Yet the static and dynamic characteristics of measurement can have aprofound effect on control This chapter describes the basic characteristics of meas-urement and its analytical description

Whatever strategy is employed, it is essential to have good knowledge of thevalue of the controlled variable This is provided by the measurement system, asnoted in the control system block diagram of figure 1.4 This chapter presents theessential dynamic and static characteristics of the measurement system

The system block diagram in Figure 1.4 shows where the measurement operationsappear in a control system The measurement operation is composed of two ele-

ments, as shown in figure 2.1: the sensor and signal conditioning The variable to

be measured, yet), is provided as an input to the sensor The sensor output, x(t), is modified by signal conditioning to provide a signal, bet), for use in the rest of the control system In general, there will be differences between yet) and bet) in mag- nitude and time The time difference is a lag between the present actual value of

yet) and its measured indication, bet) In addition, bet) is usually in an entirely

dif-ferent form, such as a voltage to indicate position, and has a difdif-ferent scale factor;that is, its magnitude is not the same as the actual variable magnitude

MEASUREMENT I 25

2.2.1 Sensor

The sensor converts information about the measured variable into another formmore suitable for the control system Historically, the sensor was also called a

transducer The output in modem sensors is usually some electrical quantity such

as resistance or voltage For example, a thermocouple outputs a voltage that varies

with temperature An analytical description of a sensor includes its static response, when the input is not changing in time, and its dynamic response, when the input

changes in time

The expression transfer function, defined in chapter I as the relationship

be-tween an object's input and output, is used to describe the sensor static and dynamictransfer functions If the input is not changing in time, the transfer function definesthe static response If the input is changing in time, the transfer function embodiesboth the dynamic and static responses

Static Response If the measured variable is not changing in time, the sensorwill have some static output; that is, the output will not be changing in time either

The static relationship between input and output must be single valued-for everyinput value of the measured variable, YI, the sensor output will have one uniqueoutput, XI' We frequently draw a graph of sensor output versus sensor input to vi-sualize the static response

Figure 2.2a shows an example of a linear static response graph The teristic of such a response is that the derivative of sensor output to input is a con-stant Regardless of the value of Y, a change will always produce the samechange in output Lll

charac-MEASUREMENT I 27

ranges of 0 kPa to 40 kPa and 10 kPa to 30 kPa Linearity can be compared by ing the deviation of actual data from that predicted by a straight line drawn throughthe endpoints

find-Solution:

Figure 2.4 shows a straight line drawn through the 0 kPa and 40 kPa endpoints Wecan find a linear equation for this line by finding the slope and intercept of the equa-tion, VLl = ml p + VI fitted to these endpoints

o = ml(O) + VI> so that VI = 00.632 = ml(40), so that ml = 0.01575The error at the midpoint is determined from the actual sensor output, which is

V20 = 0.1[20]1/2 = 0.447 V, compared to the linear prediction of VLl =

0.01575(20) = 0.315 V, so the error is about 29% This is much too great for mostcontrol system applications

Suppose now the measurement range only needs to be from 10 kPa to 30 kPa ure 2.5 shows a straight line drawn between these endpoints We find the linear ap-proximation by the slope and intercept of VLZ= m2P + V2.Fitting to the endpoints,

Fig-0.316 = 10m2 + V2

0.548 = 30m2 + V2

subtracting the first from the second gives 0.232 = 20m, orm = 0.0116 Using this

in the first equation gives V2 = 0.316 - 10(0.0116) '* 0.20 Thus, the linear proximation is VLZ = 0.0116p +0.20

ap-FIGURE 2.4

Large span linear approximation

of example 2.1.