Control system desing guide 3rd ed george ellis

Chapter 10, Intro-duction to Observers in Control Systems, is a general presentation of observers.Section II, Modeling, has three chapters.. Chapter 1Introduction to Controls Control the

Control System Design Guide

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Library of Congress Cataloging-in-Publication DataEllis, George (George H.)

Control system design guide: a practical guide/George Ellis.Ð3rd ed

p cm

ISBN 0-12-237461-4 (hardcover : alk paper)

1 Automatic control 2 System design I Title

TJ213.E5625 2003

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British LibraryISBN: 0-12-237461-4

For all information on all Academic Press publicationsvisit our website at www.academicpress.com

Printed in the United States of America

To my loving wife, LeeAnn, and to Gretchen and Brandon, who both make us proud.

Preface xxi

vii

2.3.2 Transfer Functions of Power Conversion 16

4.2 Sources of Delay in Digital Systems 58

4.3 Experiment 4A: Understanding Delay in Digital Control 61

4.4.2 Aggressive Assumptions for Position-Based

5.6.2.2 Experiment 5B: Inverse Trapezoidal

5.9 Quantization 91

6.2.1.1 How to Tune a Proportional Controller 100

6.4.1.4 Popular Terminology for PID Control 1176.4.1.5 Analog Alternative to PID: Lead-Lag 117

8.2.1 Experiment 8B: Power Converter Compensation 1568.2.2 Increasing the Bandwidth vs Feed-Forward

8.3.2 Experiment 8D: Power Converter Compensation

9.1.1.1 Using Low-Pass Filters to Reduce Noise

9.1.1.2 Using Low-Pass Filters to Reduce Aliasing 1739.1.1.3 Using Notch Filters for Noise and Resonance 174

9.2.1.3 A Simple Model for a Closed Loop System 178

9.3.3 Switched Capacitor Filters 184

10.1.2.2 Simulating the Plant and Sensor in Real

10.2 Experiments 10A±10C: Enhancing Stability with an Observer 196

10.4.3.4 Experiment 10E: Determining the Gain

10.5.1 Step 1: Temporarily Con®gure the Observer for

10.5.3 Step 3: Restore the Observer to the Normal

11.3.2.1 The Differential Equation Solver 22611.3.2.2 Advanced Differential Equation Solvers 228

11.3.4 Frequency Information from Time-Domain

14.4 Position Resolution, Velocity Estimation, and Noise 283

14.5 Alternatives for Increasing Resolution 28714.5.1 The 1/T Interpolation, or Clock Pulse Counting

14.7.3 Relationship Between Bandwidth and Ripple 296

14.7.7 Relationship Between Ripple in the Actual and

16.2 Tuned Resonance vs Inertial-Reduction Instability 345

16.3.1 Increase Motor Inertia/Load Inertia Ratio 352

17.1.4 Tuning P/PI Loops with Velocity

17.1.5 Acceleration Feed-Forward in P/PI Loops 37117.1.6 Tuning P/PI Loops with Acc/Vel

17.3.1.1 Selective Zeroing of the PID Integral

17.4.1 Positioning, Velocity, and Current Drive

17.5.1 Bode Plots for Systems Using Velocity Drives 38517.5.2 Bode Plots for Systems Using Current Drives 386

18.2.1 Eliminate Phase Lag from Simple Differences 391

18.2.1.2 Experiment 18A: Removal of Phase Lag

18.2.1.3 Experiment 18B: Tuning the Observer 396

18.2.2.1 Experiment 18C: Verifying the Reduction

18.2.2.2 Experiment 18D: Tuning the Observer

18.3 Acceleration Feedback 406

18.3.2 Experiment 18E: Using Observed Acceleration

C Programming Language Version of the Runge±Kutta Algorithm 426

The basics of control systems were developed in the ®rst half of the 20th century Our

predecessors aimed a cannon or warmed a bath using many of the same concepts we

use Of course, time and technology have generated many re®nements Digital

pro-cessors have changed the way we implement a control law, but in many cases they

haven't changed the law itself Proportional integral differential (PID) control works

about the same today as it did four or ®ve decades ago

Control systems are broadly used and are thus well integrated into our educational

system Courses are offered at most engineering universities, and a few disciplines even

require students to undergo modest training in the subject Given the longevity of the

principles and the number of trained engineers engaged in their use, one might expect

most of the trade's practitioners to be comfortable with the basics Unfortunately, that

does not seem to be the case

Over the past several years, I've had the opportunity to teach a total of about 1500

engineers through a daylong seminar entitled ``How to Improve Servo Systems.'' These

are motivated people, willing to spend time listening to someone who might provide

insight into the problems they face Most are degreed engineers who work in industry;

roughly half have taken at least one controls course A few minutes into the seminar,

I usually ask, ``How many of you regularly apply principles of controls you learned at

school?'' Normally, fewer than one in ten raises a hand It's clear there is a gap between

what is taught and what is used

So why the gap? It might be because the subject of controls is so often taught with

an undue emphasis on mathematics Intuition is abandoned as students learn how to

calculate and plot one effect after another, often only vaguely understanding the

signi®cance of the exercise I was one of those students years ago I enjoyed controls

and did well in all my controls classes, but I graduated unable to design or even tune a

simple PI control system

It doesn't have to be that way You can develop a feel for controls! This book

endeavors to help you do just that Principles are presented along with practical

methods of analysis Dozens of models are used to help you practice the material,

xxi

for practice is the most reliable way to gain ¯uency A goal of every chapter is to fosterintuition.

What's New in This Edition?

This third edition of Control System Design Guide includes several improvements overthe previous edition First, ModelQ, the modeling environment from the secondedition, has been rewritten to create Visual ModelQ; the preprogrammed models havebeen replaced with a fully graphical modeling environment You should ®nd it easier

to follow what is being modeled Second, two chapters have been added, both cerning observers: Chapter 10 is a general presentation of observers; Chapter 18focuses on observers in motion-control systems I hope these presentations will conveythe power of these remarkable software mechanisms as well as the ease with which theycan be implemented Also, a question set has been added to the end of almost everychapter, with answers provided in Appendix G

con-Organization of the BookThe book is organized into three sections Section I, Applied Principles of Controls,consists of ten chapters Chapter 1, Introduction to Controls, discusses the role ofcontrols and controls engineers in industry Chapter 2, The Frequency Domain, reviewsthe s-domain, the basis of control systems Chapter 3, Tuning a Control System, givesyou an opportunity to practice tuning; for many, this is the most dif®cult part ofcommissioning control systems

Chapter 4, Delay in Digital Controllers, culls out the fundamental difference in theapplication of digital and analog controllers, the contribution of instability fromsample delay Chapter 5, The z-Domain, discusses z-transforms, the technique thatextends the s-domain to digital control Chapter 6, Six Types of Controllers, coverspractical issues in the selection and use of six variations of PID control Chapter 7,Disturbance Response, provides a detailed discussion of how control systems react toinputs other than the command Chapter 8, Feed-Forward, presents techniques thatcan substantially improve command response Chapter 9, Filters in Control Systems,discusses the use of ®lters in both analog and digital controllers Chapter 10, Intro-duction to Observers in Control Systems, is a general presentation of observers.Section II, Modeling, has three chapters Chapter 11, Introduction to Modeling,provides overviews of time- and frequency-domain modeling methods Chapter 12,Nonlinear Behavior and Time Variation, addresses how to deal with nonlinear opera-tion when using linear control techniques Unfortunately, this subject is missing frommost texts on controls, although signi®cant nonlinear effects are common in industrialapplications Chapter 13, Seven Steps to Developing a Model, gives a step-by-stepprocedure for developing models

Section III, Motion Control, concentrates entirely on motion control using electricservomotors Chapter 14, Encoders and Resolvers, discusses the most common feed-

back sensors used with electric servomotors Chapter 15, Basics of the Electric

Servo-motor and Drive, reviews the operation of these Servo-motors Chapter 16, Compliance and

Resonance, is dedicated to one of the most widely felt problems in motion control,

instability resulting from mechanical resonance Chapter 17, Position-Control Loops,

addresses the control of position, since the great majority of applications control

position rather than velocity or torque Chapter 18, Using the Luenberger Observer

in Motion Control, focuses on observers in motion-control systems

Reader Feedback

I have endeavored to right the errors of the second edition; for those errata that slip

through into this edition, corrections will be posted at qxdesign.com Please feel free to

contact me at george.ellis@DanaherMotion.com or gellis@qxdesign.com

AcknowledgmentsWriting a book is a large task and requires support from numerous people, and those

people deserve thanks First, I thank LeeAnn, my devoted wife of more than 25 years

She has been an un¯agging fan, a counselor, and a demanding editor She taught me

much of what I have managed to learn about how to express a thought in writing

Thanks also to my mother, who, when facts should have dissuaded her, was sure

I would grow into someone of whom she would be proud And thanks to my father,

for his unending insistence that I obtain a college education, a privilege that was

denied to him, an intelligent man born into a family of modest means

I am grateful for the education provided by Virginia Tech Go Hokies The basics ofelectrical engineering imparted to me over my years at school allowed me to grasp the

concepts I apply regularly today I am grateful to Mr Emory Pace, a tough professor

who led me through numerous calculus courses and who, in doing so, gave me the

con®dence on which I relied throughout my college career and beyond I am especially

grateful to Dr Charles Nunnally; having arrived at university from a successful career

in industry, he provided my earliest exposure to the practical application of the

material I strove to learn I found him then, as now, an admirable combination of

analytical skill and practical application

I also thank Dr Robert Lorenz of the University of Wisconsin at Madison, the manmost in¯uential in my education on controls since I left college His instruction has been

well founded, enlightening, and thoroughly practical Several of his university courses are

available in video format and are recommended for those who would like to extend their

knowledge of controls I took the video version of ME 746 and found it quite useful;

much of the material of Chapter 7, Disturbance Response, is derived from that class

Thanks to the people of Danaher (manufacturer of Kollmorgen products), mylong-time employer, for their continuing support in the writing of this book Mygratitude to each of you is sincere.

Section I

Applied Principles of Controls

Important Safety Guidelines for Readers 3Chapter 1 Introduction to Controls 5Chapter 2 The Frequency Domain 11Chapter 3 Tuning a Control System 31Chapter 4 Delay in Digital Controllers 57Chapter 5 The z-Domain 69

Chapter 6 Six Types of Controllers 97Chapter 7 Disturbance Response 127Chapter 8 Feed-Forward 151Chapter 9 Filters in Control Systems 171Chapter 10 Introduction to Observers in Control

Systems 191

. Large amounts of heat

. High-voltage potentials

. Movement of objects or mechanisms that can cause harm

. Flow of harmful chemicals

. Flames

. Explosions or implosions

Unsafe operation makes it more likely for accidents to occur Accidents can cause

personal injury to you, your coworkers, and other people Accidents can also damage

or destroy equipment By operating control systems safely, you make it less likely that

an accident will occur Always operate control systems safely!

You can enhance the safety of control system operation by taking the following

steps:

1 Allow only people trained in safety-related work practices and lock-out/tag-out

procedures to install, commission, or perform maintenance on control systems

2 Always follow manufacturer-recommended procedures

3 Always follow national, state, local, and professional safety code regulations

4 Always follow the safety guidelines instituted at the plant where the equipment

will be operated

5 Always use appropriate safety equipment, for example, protective eyewear,

hearing protection, safety shoes, and other protective clothing

6 Never attempt to override safety devices such as limit switches, emergency stop

switches, light curtains, or physical barriers

7 Always keep clear from machines or processes in operation

8 Provide reliable protection, such as mechanical stops and emergency off

switches, so that unanticipated behavior from the controller cannot harm you

or anyone else and cannot damage equipment

Remember that any change of system parameters (for example, tuning gains),

components, wiring, or any other function of the control system may cause unexpected

results, such as system instability or uncontrolled system excitation

Remember that controllers and other control system components are subject to

failure For example, a microprocessor in a controller may experience catastrophic

failure at any time Leads to or within feedback devices may open or short together at

any time Failure of a controller may cause unexpected results, such as system

instability or uncontrolled system excitation

3

This book presents observers, the use of which within control systems poses certainrisks, including that the observer may become unstable or otherwise fail to observesignals to an accuracy necessary for proper system operation Ensure that observersbehave properly in all operating conditions.

If you have any questions concerning the safe operation of equipment, contact theequipment manufacturer, plant safety personnel, or local governmental officials, such

as the Occupational Health and Safety Administration

Always operate control systems safely!

Chapter 1

Introduction to Controls

Control theory is used for analysis and design of feedback systems, such as thosethat regulate temperature, fluid flow, motion, force, voltage, pressure, tension, and

current Skillfully used, control theory can guide engineers in every phase of the

product and process design cycle It can help engineers predict performance, anticipate

problems, and provide solutions

Colleges teach controls with little emphasis on day-to-day problems The academiccommunity focuses on mathematical derivations and on the development of advanced

control schemes; it often neglects the methods that are commonly applied in industry

Students can complete engineering programs that include courses on controls and still

remain untutored on how to design, model, build, tune, and troubleshoot a basic

control system The unfortunate result is that many working engineers lay aside

analysis when they practice their profession, relying instead on company history and

trial-and-error methods

This book avoids the material and organization of most control theory textbooks

For example, design guidelines are presented throughout; these guidelines are a

combination of industry-accepted practices and warnings against common pitfalls

Nontraditional subjects, such as filters and modeling, are presented here because they

are essential to understanding and implementing control systems in the workplace

The focus of each chapter is to teach how to use controls to improve a working

machine or process

The wide availability of personal computers and workstations is an importantadvance for control system designers Many of the classical control methods, such as

the root locus method, are graphical rather than analytical Their creators sought to

avoid what was then the overwhelming number of computations required for

analy-tical methods Fortunately, these calculations no longer present a barrier Virtually

every personal computer can execute the calculations required by analytical methods

With this in mind, the principles and methods presented herein are essentially

analy-tical, and the arithmetic is meant to be carried out by a computer

5

1.1 Visual ModelQ Simulation EnvironmentMost engineers understand the foundations of control theory Concepts such astransfer functions, block diagrams, the s-domain, and Bode plots are familiar to most

of us But how should working engineers apply these concepts? As in most disciplines,they must develop intuition, and this requires fluency in the basics In order to befluent, you must practice

When studying control system techniques, finding equipment to practice on is oftendifficult As a result, designers often rely on computer simulations To this end, theauthor developed, as a companion to this book, Visual ModelQ, a stand-alone,graphical, PC-based simulation environment The environment provides time-domainand frequency-domain analysis of analog and digital control systems Visual ModelQ

is an enhancement of the original ModelQ, in that Visual ModelQ allows readers toview and build models graphically Dozens of Visual ModelQ models were developedfor this book These models are used extensively in the chapters that follow Readerscan run these experiments to verify results and then modify parameters and otherconditions to experiment with the concepts of control systems

Visual ModelQ is written to teach control theory It makes convenient thoseactivities that are necessary for studying controls Control law gains are easy tochange Plots of frequency-domain response (Bode plots) are run with the press of abutton The models in Visual ModelQ run continuously, just as real-time controllers

do The measurement equipment runs independently, so you can change parametersand see the effects immediately

1.1.1 Installation of Visual ModelQVisual ModelQ is available at www.qxdesign.com The unregistered version is avail-able free of charge Although the unregistered version lacks several features, it canexecute all the models used in this book Readers may elect to register their copies ofVisual ModelQ at any time; see www.qxdesign.com for details

Visual ModelQ runs on PCs using Windows 95, Windows 98, Windows 2000,Windows NT, and Windows XP Download and run the executable file setup.exe forVisual ModelQ V6.0 or later Visual ModelQ installs with both a User’s Manual and aReference Manual After installation, read the User’s Manual Note that you canaccess the Reference Manual via Internet Explorer by pressing the F1 key Finally,check the Web site from time to time for updated software

1.1.2 ErrataCheck www.qxdesign.com for errata It is the author’s intention to regularly updatethe Web page as corrections become known

1.2 The Control SystemThe general control system, as shown in Figure 1-1, can be divided into the controller

and the machine The controller can be divided into the control laws and the power

converter The machine may be a temperature bath, a motor, or, as in the case of

a power supply, an inductor/capacitor circuit The machine can also be divided into

two parts: the plant and the feedback device(s) The plant receives two types of signals:

a controller output from the power converter and one or more disturbances Simply

put, the goal of the control system is to drive the plant in response to the command

while overcoming disturbances

1.2.1 The ControllerThe controller incorporates both control laws and power conversion Control laws,

such as proportional-integral-differential (PID) control, are familiar to control

engi-neers The process of tuning — setting gains to attain desired performance — amounts

to adjusting the parameters of the control laws Most controllers let designers adjust

gains; the most flexible controllers allow the designer to modify the control laws

themselves When tuning, most control engineers focus on attaining a quick, stable

command response However, in some applications, rejecting disturbances is more

important than responding to commands All control systems should demonstrate

robust performance because even nearly identical machines and processes vary

some-what from one to the other, and they change over time Robust operation means

control laws must be designed with enough margin to accommodate reasonable

changes in the plant and power converter

Virtually all controllers have power converters The control laws produce tion, but power must be applied to control the plant The power converter can be

informa-driven by any available power source, including electric, pneumatic, hydraulic, or

chemical power

Control

Power Converter

Feedback Sensor

+

Figure 1-1 The general control system.

1.2.2 The MachineThe machine is made of two parts: the plant and the feedback The plant is the element

or elements that produce the system response Plants are generally passive, and theyusually dissipate power Examples of plants include a heating element and a motorcoupled to its load

Control systems need feedback because the plant is rarely predictable enough to becontrolled open loop — that is, without feedback This is because most plants integratethe power converter output to produce the system response Voltage is applied toinductors to produce current; torque is applied to inertia to produce velocity; pressure

is applied to produce fluid flow In all these cases, the control system cannot controlthe output variable directly but must provide power to the machine as physics allowsand then monitor the feedback to ensure that the plant is on track

1.3 The Controls EngineerThe focal task of many controls engineers is system integration and commissioning.The most familiar part of this process is tuning the control loops This process can beintimidating Often dozens of parameters must be fine-tuned to ensure that the systemlives up to the specification Sometimes that specification is entirely formal, but moreoften it is a combination of formal requirements and know-how gained with years ofexperience Usually only the most senior engineers in a company are capable ofjudging when a system is performing well enough

For some control systems, each installation may require days or weeks to becorrectly commissioned In a complex machine such as a rolling mill, that processcan take months Each piece of the machine must be carefully tuned at the site So evenafter the design of the machine is complete, the expertise of a controls engineer isrequired each time a unit is installed

Although most controls engineers focus on installation, their job should begin whenthe machine is designed Many companies fail to take advantage of their controlsexpertise early in a project; this is shortsighted A controls engineer may suggest animproved feedback device or enhancements to a machine that will help overcome astubborn problem Ideally, the project manager will solicit this input early, becausechanges of this nature are often difficult to make later

The controls engineer should also contribute to the selection of the controller.There are many controls-oriented factors that should be taken into account Doesthe controller implement familiar control laws? For digital controllers, is the processorfast enough for the needs of the application? Is the unit appropriate for the supportteam and for the customer base? The selection and specification of a controller ofteninvolve input from many people, but some questions can be answered best by a skilledcontrols engineer

What is the role for control theory in the daily tasks of controls engineers? At itsroot, control theory provides understanding and, with that, intuition Should the

company purchase the controller that runs four times faster even though it costs 25%

more? Should they invest in machine changes, and what will be the expected

improve-ment from those efforts? How much will the better feedback device help a noise

problem? Understanding controls doesn’t guarantee that the engineer will have the

correct answer But a firm grasp on the practical side of controls will provide correct

answers more often and thus position the controls engineer to provide leadership in

process and product development and support

Chapter 2

The Frequency Domain

The frequency domain provides intuition on practical subjects traversing the ®eld of

control theory How can responsiveness be quanti®ed? How stable is a system and how

stable should it be? How do tuning gains work? How well does a system reject

disturbances? The frequency domain is the beginning of control theory

The competitor to the frequency domain is the time domain The time domain is,above all, convenient It is easy to comprehend and easy to measure The measure-

ments of an oscilloscope are immediately understood The time domain is often the

best way to communicate with customers and colleagues Thus, the controls engineer

should be ¯uent in both time and frequency They are two corroborating witnesses,

furnishing together a clearer understanding than either can alone

This chapter will present the frequency domain, beginning with its foundation,the Laplace transform Transfer functions are presented with examples of common

control elements and plants Bode plots, the favored graphic display in this text,

are presented next The chapter will discuss two important measures of control

system performance, stability and response, and then conclude with a Visual ModelQ

experiment

2.1 The Laplace TransformThe Laplace transform underpins classic control theory [32,33,85] It is almost uni-

versally used An engineer who describes a ``two-pole ®lter'' relies on the Laplace

transform; the two ``poles'' are functions of s, the Laplace operator The Laplace

transform is de®ned in Equation 2.1

F…s† ˆ

Z 1

11

The function f(t) is a function of time, s is the Laplace operator, and F(s) is thetransformed function The terms F(s) and f(t), commonly known as a transform pair,represent the same function in the two domains For example, if f (t) ˆ sin (ot), thenF(s) ˆ o/(o2‡ s2) You can use the Laplace transform to move between the time andfrequency domains.

The Laplace transform can be intimidating The execution of Equation 2.1 iscomplex in all but the simplest cases Fortunately, the controls engineer need investlittle time in such exercises The most important bene®t of the Laplace transform isthat it provides s, the Laplace operator, and through that the frequency-domaintransfer function

2.2 Transfer FunctionsFrequency-domain transfer functions describe the relationship between two signals as

a function of s For example, consider an integrator as a function of time (Figure 2-1).The integrator has an s-domain transfer function of 1/s (see Table 2-1) So it can besaid that

Vo…s† ˆ Vi…s†

Similarly, a derivative has the transfer function s; differentiating a time-domainsignal is the same as multiplying a frequency-domain signal by s Herein lies theusefulness of the Laplace transform Complex time-domain operations such as differ-entiation and integration can be handled with algebra Dealing with transfer functions

in the time domain (that is, without bene®t of the Laplace transform) requiresconvolution, a mathematical process that is so complicated that it can be used only

on the simplest systems

The constant j isp1 The o term translates to a sinusoid in the time domain; stranslates to an exponential (est) term Our primary concern will be with steady-state

sinusoidal signals, in which case s ˆ 0 The frequency in hertz, f, is de®ned as

f  o/2p So for most of this book, Equation 2.3 will simplify to Equation 2.4:

sSTEADY STATEˆ j2pf …2:4†

The practical side of Equation 2.4 is that the response of an s-domain transferfunction to a steady-state sine wave can be evaluated by setting s ˆ j2pf

2.2.1.1 DC GainOften it is important to evaluate the DC response of a transfer function, in other words, to

determine the output of the transfer function subjected to a DC input To ®nd the DC

response, set s to zero For example, we discussed before that the transfer function for

differentiation is Vo(s) ˆ Vi(s)  s What happens when a DC signal is differentiated?

Intuitively, it produces zero, and that is con®rmed by setting s to zero in this simple equation

2.2.2 Linearity, Time Invariance, and Transfer Functions

A frequency-domain transfer function is limited to describing elements that are linear

and time invariant These are severe restrictions and, in fact, virtually no real-world

system fully meets them The criteria that follow de®ne these attributes, the ®rst two

being for linearity and the third for time invariance

TABLE 2-1 TRANSFER FUNCTIONS OF CONTROLLER ELEMENTS

Simple ®lters

Double-pole low-pass ®lter oN2/(s2‡ 2oNs ‡ oN2)

Bilinear-quadratic (bi-quad) ®lter (oD2/oN2)(s2‡ 2NoNs ‡ oN2)/(s2‡ 2DoDs ‡ oD2)Compensators

1 Homogeneity Assume that an input to a system r(t) generates an output c(t).For an element to be homogeneous, an input k  r(t) would have to generate anoutput k  c(t), for any value of k An example of nonhomogeneous behavior issaturation, where twice as much input delivers less than twice as much output.

2 Superposition Assume that an element subjected to an input r1(t) will generatethe output c1(t) Further, assume that the same element subjected to input r2(t)will generate an output c2(t) Superposition requires that if the element is sub-jected to the input r1(t) ‡ r2(t), it will produce the output c1(t) ‡ c2(t) [32,80]

3 Time invariance Assume that an element has an input r(t) that generates an put c(t) Time invariance requires that r(t t) will generate c(t t) for all t > 0

out-So we face a dilemma: Transfer functions, the basis of classic control theory, requirelinear, time-invariant (LTI) systems, but no real-world system is completely LTI This

is a complex problem that is dealt with in many ways, some of which are detailed inChapter 12 However, for most control systems, the solution is simple: design compo-nents close enough to being LTI that the non-LTI behavior can be ignored or avoided

In practice, most control systems are designed to minimize non-LTI behavior This

is one reason why components used in control systems are often more expensive thantheir noncontrol counterparts For most of this text, the assumption will be that thesystem is LTI or close enough to it to use transfer functions Readers who are troubled

by this approximation should consider that this technique is commonly applied byengineers in all disciplines For example, Ohm's law, v ˆ iR, is an approximation thatignores many effects, including electrical radiation, capacitive coupling, and leadinductance Of course, all those effects are important from time to time, but few wouldargue the utility of Ohm's law

2.3 Examples of Transfer Functions

In this section, we will discuss the transfer functions of common elements in controlsystems The discussion is divided along the lines of Figure 2-2

Control

Power Converter

Feedback Sensor

+

Figure 2-2 Elements in the control system.

2.3.1 Transfer Functions of Controller ElementsThe controller elements divide into the control laws and power conversion Examples

of operations used in control laws are shown in Table 2-1 Note that Appendix A

shows the implementation of many of these laws using operational ampli®er (op-amp)

are added to reduce noise The use of ®lters in control systems is detailed in Chapter 9

Table 2-1 lists the s-domain representation for a few common examples

2.3.1.3 CompensatorsCompensators are specialized ®lters A compensator is a ®lter that is designed to

provide a speci®c gain and phase shift, usually at one frequency The effects on gain

and phase either above or below that frequency are secondary Table 2-1 shows a lag

compensator, a proportional-integral (PI) compensator, a

proportional-integral-differential (PID) compensator, and a lead compensator The principles of

compensa-tors are described in Chapters 3, 4, and 6

2.3.1.4 DelaysDelays add time lag without changing amplitude For example, a conveyor belt with a

scale can cause a delay Material is loaded at one point and weighed a short distance

later; that distance causes a delay that is inversely proportional to the belt speed

A system that controls the amount of material to be loaded onto the conveyor must

take the time delay into account Since microprocessors have inherent delays for

calculation time, the delay function is especially important to understanding digital

controls, as will be discussed in Chapters 4 and 5

A delay of T seconds is de®ned in the time domain as

2.3 EXAMPLES OF TRANSFER FUNCTIONS 315