Automatic control systems

Ninth Edition

V ■■

Automatic Control System

T h a V lM D H K T C N .1

K NV .1400in0

Farid Golnaraghi • Benjamin c. Ki

WIL

Laplace Transform Table

Laplace Transfom F(s) Time Function ^(0

1 Unit-impulse function ổ(r)

Unit-step function u,(r)

Unit-ramp function I

i"(n = positive integer)

1

i + a

f'e “ '(/1 = positive integer)

1

{s + a){s + ^)

s {i + a)

1

í ( ĩ + «)^

1

+ù}ị

s^ + ùiị

-{ ạ e - a e

Laplace Transform Table (cont.)

Laplace Transform F(s) Time Function yĩí)

1 - cos o>„r

co„\Ja- + cứị sin(ti>ni + 0)

v v h ereớ = ta n “ ‘ (oin/of)

(5 + a ) ( i2 + o ^ )

a, I

\/a ^ + 0)1

sin(ù;„í - 6

where Ỡ = tan“ '(a ;„/ữ )

-J= ^= = e ự \ — t (< < 1 )

y / T ^

M-

________ 2_______

5(52 + 2fcu„j + tư*)

I - - j à = = ị e s in (a .,x /l - { 2 , + e j

where Ớ = c o s '' f ( f < l )

iw *

+ 2Ỉ0J„S + lứị

where ớ = C O S " ( ? < 1 )

(C<1)

o>;

s^{s~ + 2^w„s + <dI)

i - ^ - \ --------. . e s in iw „ \/l - f- r + ỡ)

u > n \/\- ỉ' ^ '

whereớ = c o s " '( 2 r - 1) ( c < 1)

I EDITION

Automatic Contro

Systems

FARID GOLNARAGHI

Siiium Fraser U nivenitij

BENJAMIN c. KUO

Uiùccrsiti/ o f Illiium at Urhnmi-Cìitimpaiịíiì

WILEY

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Printed ill tlif Unitftl Stiitfs of Ami't

11) ‘J s 7 fi Õ 4

To my wife, Mitra, and to Sophia and Carmen, the joys o f my life.

—M. Farid Golnaraghi

Preface (Readme)

This is the ninth edition o f the text but the first w ith Farid G olnaraghi as the lead author.

For this edition, we increased the num ber o f examples, added M A T L A B ® ' toolboxes, and

enhanced the M A T L A B G U I software. A C S Y S . We added more com puter-aided tools fo r

students and teachers. The prepublication m anuscript was reviewed by many professors,

and most o f the relevant suggestions have been adopted. In this edition. Chapters 1 through

4 are organized to contain all background m aterial, w hile Chapters 5 through 10 contain

m aterial d irectly related to the subject o f control.

In this edition, the fo llo w in g m aterials have been moved in to appendices on this book's

Web site at w w w .w iley.com /college/golnaraghi.

Appendix A : Elem entary M a trix Theory and A lgebra

Appendix B; D ifference Equations

Appendix C: Laplace Transform Table

Appendix D: r-Transform Table

Appendix E; Propenies and Consữuction o f the R oot L o ci

Appendix F; General N yquisi C riterion

Appendix G; A C S Y S 2008: D escription o f the Software

Appendix H: Discrete-Data C onưol Systems

In addition, the Web site contains the M A T L A B files fo r A C S Y S . w hich are software

tools fo r solving control-system problem s, and Pow erPoint files fo r the illu stra tio n s in the

texl.

The fo llo w in g paragraphs are aimed at three groups: professors w ho have adopted the

book or who we hope w ill select it as their text; practicing engineers looking fo r answers to

solve their day-to-day design problem s: and. finally, students who are going to live w ith the

book because it has been assigned fo r the conirol-system s course they are taking.

T o the Professor: The materia] assembled in this book is an outgrow th o f senior-level

control-system courses taughi by the authors at their universities throughout theứ teaching

careers. The first eight editions have been adopted by hundreds o f universities in the U nited

States and around the w orld and have been translated in to at least six languages. P ractically

all the design topics presented in the eighth edition have been retained.

This text contains not only conventional M A T L A B toolboxes, where students can

learn M A T L A B and utilize their program m ing skills, bui also a graphical M A Tl-.^B -based

software. A C SYS. The A C S Y S software added to this edition is very different from the

software accompanying any other control book. Here, through extensive use o f M A T L .^B

G U I program m ing, we have created software that is easy to use. As a result, students w ill

need to focus only on learning control problem s, not program m ing! We also have added

tw o new applications. S IM Lab and V iitu a l Lab. through w hich students w ork on realistic

problems and conduct speed and position control labs in a software environm ent. In

SIM Lab. students have access to the system parameters and can alter them (as m any

sim ulation). In V irtual Lab. we have introduced a b)ack-box approach in w hich die students

' MATLAB '* is a registered ưademark of The MaihWorks. Inc.

have no access to the plant parameters and have to use some sort o f system identification

technique to find them. Through V irtu a l Lab we have essentially provided students w ith a

realistic online lab w ith a ll the problem s they w ould encounter in a real speed- or position￾control lab— fo r example, a m p lifie r saturation, noise, and nonlinearity. We welcom e your

ideas fo r the future editions o f this book.

Finally, a sample section-by-section fo r a one-semester course is given in the

Instructor’s Manual, w hich is available fro m the publisher to qualified instructors. The

M anual also contains detailed solutions to a ll the problem s in the book.

T o P ra c tic in g E ngineers: This book was w ritten w ith the readers in m ind and is very

suitable fo r self-study. O ur objective was to treat subjects clearly and thoroughly. The book

does not use the th eorem -proof-Q .E .D . style and is w ith o u t heavy mathematics. The

authors have consulted extensively fo r w ide sectors o f the industry fo r m any years and have

participated in solving numerous control-system s problem s, from aerospace systems to

industrial controls, autom otive controls, and control o f com puter peripherals. A lthough it is

d iffic u lt to adopt all the details and realism o f practical problem s in a textbook at this level,

some examples and problem s reflect sim p lifie d versions o f rea l-life systems.

T o Students: You have had it now that you have signed up fo r this course and your

professor has assigned this book! You had no say about the choice, though you can form

and express your opinion on the book after reading it. W orse yet. one o f the reasons that

your professor made the selection is because he o r she intends to make you w o rk hard. B ut

please d o n 't misunderstand us: w hat we really mean is that, though this is an easy book to

study (in our opinion), it is a no-nonsense book. It doesn’ t have cartoons o r nice-looking

photographs to amuse you. From here on. it is all business and hard w ork. You should have

had the prerequisites on subjects found in a typical linear-systems course, such as how to

solve linear ordinary differential equations. Laplace transform and applications, and tim e￾response and frequency-dom ain analysis o f linear systems. In this book you w ill not find

too much new mathematics to w hich you have not been exposed before. W hat is interesting

and challenging is that you are going to leam how to apply some o f the mathematics that

you have acquired during the past tw o or three years o f study in college. In case you need to

review some o f the mathem atical foundations, you can find them in the appendices on this

book’s Web site. The W eb site also contains lots o f other goodies, including the A C S Y S

software, w hich is G U I software that uses M A T LA B -based programs fo r solving linear

control systems problems. You w ill also find the S im u lin k ” ^-based S IM L a b and V irtual

Lab, w hich w ill help you to gain understanding o f real-w orld control systems.

T his book has numerous illustrative examples. Some o f these are deliberately simple

fo r the purpose o f illusư ating new ideas and subject matter. Some examples are more

elaborate, in order to bring the practical w o rld closer to you. Furthermore, the objective o f

this book is to present a com plex subject in a clear and thorough way. One o f the im portant

learning strategies fo r you as a student is not to rely strictly on the textbook assigned. W hen

studying a certain subject, go to the lib ra ry and check out a few sim ilar texts to see how

other authors treat the same subject. You may gain new perspectives on the subject and

discover that one author may treat the m aterial w ith more care and thoroughness than the

others. D o not be disưacted by w ritten-dow n coverage w ith oversim plified examples. The

m inute you step into the real w orld, you w ill face the design o f control systems w ilh

nonlinearities and/or tim e-varying elements as w ell as orders that can boggle your m ind. It

^ SImulink’' is a registered trademark of The Math Works. Inc.

may be discouraging to te ll you now that strictly linear and first-order systems do not exist

in the real w orld.

Some advanced engineering students in college do not believe that die m aterial they

leam in the classroom is ever going to be applied dữectly in indusưy. Some o f our students

come back from field and interview trips totally surprised 10 find that the m aterial ihev

learned in courses on control systems is actually being used in indusưy loday. They are

surprised to find that this book is also a popular reference fo r practicing engineers.

U nfortunately, these fact-finding, eye-opening, and self-m otivating uip s usually occur near

the end o f their college days, w hich is often too late fo r students to get m otivated.

There are many learning aids available to you: the M A TLA B -based A C S Y S software

w ill assist you in solving a ll kinds o f control-system s problems. The S IM Lab and VirruaJ

Lab software can be used fo r sim ulation o f virtu a l experim ental systems. These are all

found on the Web site. In addition, the R eview Questions and Summaries at ứie end o f each

chapter should be useful to you. A lso on the Web site, you w ill find the eưaia and other

supplemental material.

We hope that you w ill enjoy this book. It w ill represent another m ajor textbook

acquisition (investm ent) in your college career. O ur advice to you is not to sell it back to the

bookstore at the end o f the semester. I f you do so but fin d out later in your professional

career that you need to refer to a control systems book, you w ill have to buy ii again at a

higher price.

S pecial A ckn o w le d g m e n ts: The authors wish to thank the reviewers fo r ih e ir invaluable

comments and suggestions. The prepublication reviews have had a great im pact on the

revision project.

The authors thank Dr. Earl Foster. Dr. Vahe Caliskan. Sim on Fraser students and

research associates M ichael Ages, Johannes M in o r, L inda Franak. Arash Jamalian.

Jennifer Leone. Neda Pamian. Sean MacPherson. A m in Kamalzadeh. and Nathan

(W uyang) Zheng fo r their help. Farid Golnaraghi also wishes to thank Professor Benjam in

K uo fo r sharing the pleasure o f w ritin g this w onderful book, and fo r his teachings, patience,

and suppon throughout this experience.

M . F. Golnaraghi.

Vancouver. British Columbia,

Canada

B. c . Kuo.

Champaign. Illinois. U S A .

Contents

:e iv

lAPTER 1

luction 1

Introduction 1

1-1-1 Basic Components o f a Control

System 2

2-3

1-1-2 Examples o f Control-System

AppLcations 2

1-1-3 Open-Loop Control Systems

(Nonfeedback Systems) 5

1 -M Closed'Lxwp Control Systems

(Feedback Control Systems) 7

What Is Feedback, and w h a t Are Its Effects? 8 2-4

1-2-1 Effect o f Feedback on Overall Gain 8

1-2-2 Effect o f Feedback on Stability 9

1-2-3 Effect o f Feedback on External

Disturbance or Noise 10

Types o f Feedback Control Systems 11

1-3-1 Linear versus Nonlinear Control 2-5

Systems 11

1-3-2 Time-Invariant versus Time-Varying

Systems 12 2-6

Summary 14

AFTER 2

ỉm atical Foundation 16

Complex-Variable Concept 16

2-1-1 Complex Numbers 16 2-7

2-1-2 Complex Variables 18

2-1-3 Functions o f a Complex Variable 19

2-1-4 Analytic Function 20

2-1-5 Singularities and Poles o f a

Function 20

2-1-6 Zeros o f a Function 20

2-1-7 Polar Rfpresentation 22

Frequency •Domain Plots 26

2-2-1 Computer-Aided Constniction o f the 2-8

Frequency-Domain Plots 26 2-9

2-2-2 Polar Plots 27

2-2-3 Bode Plot (Comer Plot or As)inptotic

Plot) 32 2-10

2-2-4 Real (Constant K 34

2-2-5 Poles and Zeros at the Origin,

M * ' 34

2-11

2-2-6 Simple Zero, 1 +ju>T 37 2-12

2-2-7 Simple Pole. 1 /(1 + > 7 -) 39 2-13

2-2-8

2-2-9

2-2-10

2-2-11

2-2-12

2-3-2

2-3-3

2-3-4

2-3-5

Quadratic Poles and Zeros 39

Pure Time Delay, 42

Magnitude-Phase Plot 44

Cain- and Phase-Crossover Points 46

Minimum-Phase and Nonminimum￾Phase Functions 47

on to Differential Equations 49

Linear Ordinary DifTerential

Equations 49

Nonlinear Differential Equations 49

First-Order Differentia]

Equations; State EquaHons 50

Definition o f State Variables 50

The Output Equation 51

2-4-1 Definition o f the Laplace

Transform 52

2-4-2 Inverse Laplace Transformation 54

2-4-3 Important Theorems o f the Laplace

Transform 54

Inverse Laplace Transform by

Partial-Fraction Expansion 57

2*5'1 Partial-FracHon Expansion 57

Application o f the Laplace Transform

to the Solution o f Linear Ordinary

Differential Equations 62

2-6-1 First-Order Prototype System 63

2-6-2 Second-Order Prototype

System 64

Impulse Response and Transfer Functions

o f Linear Systems 67

2-7-1 Impulse Response 67

2-7-2 Transfer Function (Single-Input,

Single-Output Systems) 70

2-7-3 Proper Transfer Functions 71

2-7-4 Characteristic Equation 71

2-7-5 Transfer Funcdon (Multivariable

Systems) 71

Stability’ o f Linear Control Systems 72

Boiinclecl-Inptit, Bounded-Output

(BIBO ) Stability— ContìniiodS-Data

Systems 73

Roots an«l Stabilitv' 7*1

Zero-Input and Asvmpỉ

Continnoiis-Data Systei

2-13-1 Routh’s Tabulation 79

2-13-2 Special Cases when Routh's

Tabulation Terminates

Prematurelv 80

1 M ATLAB Tools and Case Studies 84

2-14-1 Description and Use o f Transfer

Function Tool 84

2-14-2 M ATLAB Tools for Stability 85

5 S u iiiin a n ' 9 0

:hapter 3

ck Diagrams and Signal-Flow Graphs 104

Block Diagrams 104

3-1-1 T\pical Elements o f Block Diagrams

in Control Systems 106

3-1-2 Relation between Mathematical

Equations and Block Diagrams 109

3-1-3 Block Diagram Reduction 113

3-1-4 Block Diagram o f M ulti-Input

Systems— Special Case; Svstems with

a Disturbance 115

3-1-5 Block Diagrams and Transfer

Functions o f Multivariable

Systems 117

Signal-Flow Graphs (SFGs) 119

3-2-1 Basic Elements o f an SFG 119

3-2-2 Summan' of the Basic Properties of

3-2-3 Definitions o f SFG Terms 120

3-2-4 SFG .-Ugebra 123

3-2-5 SFG o f a Feedback Control

System 124

3-2-6 Relation between Block Diaịírams

and SFGs 124

3-2-7 Gain Formula for SFG 124

3-2-S Application o f the Gain Formula

beh\ een Output Nodes and

Noninpiit Nodes 127

3-2-9 Application ol tlip Gain Formula tn

Block Diagrams 12S

3-2-10 Simplified Cain Formula 129

MATLAB Tools and Case Studies 129

Suinman' 133

30retical Foundation and Background

Ite ria l: M odeling of Dynamic Systems 147

IiUrodiiL'tidii to Níodeliiiíí o f Medianical

Systems 148

4-1-1 Transiatinnul M otio n ]4S

4-1-2 Rotatidiial M titiiin 157

4-1-3 (Jo m i'i'sio ii h ftw e e ii T ran-iliitio iia l ii

R o[;ifi(inal Moticms Ifil

4-1-4 C i-ar T rains lf>2

4-1-5 Backlash and Dead Zone (Nonlinear

Characteristics) 164

4-2 Introduction to Modeling o f Simple Electrical

Systems 165

4-2-1 Modeling o f Passive Electrical

Elements 165

4-2-2 Modeling o f Electrical Netv%-orks 165

4-3 Modeling o f Active Electrical Elements:

Operational Amplifiers 172

4-3-1 The Ideal Op-Amp 173

4-3-2 Sums and Differences 173

4-3-3 First-Order Op-Amp

Configurations 174

4-4 Introduction to Modeling o f Thermal S\-stems 177

4-4-1 Elementary Heat Transfer

Properties 177

4-5 Introduction to Modeling o f Fluid Systems 180

4-5-1 Elementary Fluid and Gas Svstem

Properties’ 180

4-6 Sensors and Encoders in Control Systems 189

4-6-1 Potentiometer 189

4-6-2 Tachometers 194

4-6-3 Incremental Encoder 195

4-7 DC Motors in Control Systems 198

4-7-1 Basic Operational Principles o f DC

Motors 199

4-7-2 Basic Classifications o f PM DC

Motors 199

4-7-3 Mathematical Modeling o f PM DC

Motors 201

4-8 Systems u itli Transportation Lags

(Time Delays) 205

4-8-1 Approximation o f the Tiine-Delav

Function by Rational

Functions 206

4-9 Linearization o f Nonlinear Systems 206

4-9-1 Linearization Using Tavlor Series:

Classical Representation 207

4-9-2 Linearization Using the State space

Approach 207

4-10 Aiuilopes 213

4-11 Case Studies 216

4-12 M ATLAB Tools 222

4 -1 3 S iiiiim iiiy 22 3

Time-Domain Analysis of Control Systems 253

5-1 Time Ri'spnnse o f Qintimious-Data Systems:

Introduction 253

5-2 T\pical Test Signals for tlu' Time Response of

Control Systems 254

5-3 Tlie Unit-Stcp Response and Time-Doiniiin

Specifa'iitions 25fi

5-4 Steaclv-Stiite Error 258

Contents ix

5-4-1 Steady-State Error o f Linear

Contìnuous-Data Control Systems 258

5-4-2 Steady-State Error Caused by

Nonbnear System Elements 272

Time Response o f a Prototype First-Order

System 274

Transient Response o f a Prototype

Second-Order System 275

5-6-1 Damping Ratio and Damping

Factor 277

5-6-2 Natural Undamped Frequenc)’ 278

5-6-3 Maximum Overshoot 280

5-6-4 Delav Tim e and Rise Time 283

5-6-5 Settling Time 285

Speed and Posítíon Control o f a DC M otor 2B9

5-7-1 Speed Response and the Effects of

Inductance and Disturbance-Open

Loop Response 289

5-7-2 Speed Control o f DC Motors;

Closed-Loop Response 291

5-7-3 Position Control 292

Time-Domain Analysis o f a Position-Control

System 293

5-8-1 Unit-Step Transient Response 294

5-8-2 The Steady-State Response 298

5-8-3 Tim e Response to a Unit-Ramp

Input 298

5-8-4 Time Response o f a Tliircl-Order

System 3(H)

Basic Control Systems and Effects of

Adding Poles and Zeros to Trdnsfer

Functtons 304

5-9-1 Addition o f a Pole to the

Forward-Path Transfer Function;

Unity-Feedback S\-steins 305

5-9-2 Addition o f a Pole to the

Closed-Loop Transfer Function 307

5-9-3 Addition o f a Zero to tlie

Closetl-Loop T ransfer P u n ư io n 308

5-9-4 Addition o f a Zero to file

Forward-Path TransfiT Function:

Unity-Feedback Systems 309

Dominant Poles and Zeros o f Transfer

Functions 311

5-10-1 Suinmar)' o f Effec-ts o f Poles and

Zeros 313

5-10-2 Tlie Relative Dainpinji Bati(j 3 i:ỉ

5-10-3 The Proper Way of Ni-iiUxling tlic

Itisiiĩiiiíìcaiit Poles \\iỉti Consideration

o f the Steatlv-Statf lU-spoiise 31-3

Basic Control Sysh-nis Utiiizinji Addition OÍ' Poll's

and Zeros 314

M A T L A B T cm)1s 319

SiimmarN 320

p- CHAPTER 6

The Control Lab 337

6-1 Introduction 337

6-2 Description o f the N'irtua] Experimental

System 338

6-2-1 M otor 339

6-2-2 Position Sensor or speed Sensor 339

6-2-3 Power Am plifier .340

6-2*4 Interface 340

6-3 Description o f SIM Lab and Virtual Lab

Software 340

6-4 Simulation and Virtual Experiments 345

6-4-1 Open-Loop Speed 345

6-4-2 Open-Loop Sine Input 347

6-4-3 Speed Control 350

6-4-4 Position Control 352

6-5 Design Project 1— Robotic Arm 354

6-6 Design Project 2— Quarter-Car Model 357

6-6-1 Introduction to the Quarter-Car

Model 357

6-6-2 Closed-Loop Acceleration

Control .3.59

6-6-3 Description o f Quarter Car

Modeling Tool 360

6-6-4 Passive Suspension 364

6-6-5 Closed-Loop Relative Position

Control 365

6-6-6 Closed-Loop Acceleration

Control 366

6-7 Siunman 367

v CHAPTER 7

Root Locus Analysis 372

7-1 Introduction 372

7-2 Basic Properties o f till- Root

Loci (RL) 373

Properties o f the Root I^ c i 377

7-3-1 ^ = 0 and K = ±oc Points 377

7-3-2 Number o f Branches on the R<K)t

Loci 378

7-3-3 Sviiiiiietr)- of the RL 37S

7-3-4 Angles o f Asvmptotes o f the RL;

Behavior o f the RL at |f| = oc 378

7-3-5 Intersect o f tlie AsMiiptoics

(Centroid) 379

7-3-6 Boot u > d on tl.e Rciil Axis 380

7-3-7 Angles o f Dcparturi' and Angies of

Arrival o f file B I. 380

7-3-8 Intersection o t'tlie HI. with the

Itiia0iiiir\' /V\is 380

7-3-y Breakiiwav Points iSiul(ili> Pointsl

(.11 the RÍ, 3S0

7-3-10 T lif Root SensilivitN 3S2

4 Design Aspects o f the Root Loci 385

7-4-1 Effects o f Adding Poles and Zeros

to G{s)H[s) 385

5 Root Contours (RC); Multiple-Parameter

Variation 393

6 M ATLAB Tools and Case Studies 400

7 Summar>’ 400

equency-Domain Analysis 409

1 Introdiiction 409

8-1-1 Frequenc%' Response of

Closed-Loop Systems 410

8-1-2 Frequencv-Doinain Specifications 412

2 M ,. Ur. and Bandwidth o f the Protohpe

Second-Order S\-stem 413

8-2-1 Resonant Peak and Resonant

Frequenc) 413

8-2-2 Bandwidth 416

3 Effects o f Adding a Zero to the Forward-Path

Transfer Function 418

4 EfTects o f Adding a Pole to the Fonvard-Patli

Transfer Function 424

5 Nyquist StabiliU' Criterion: Fundamentals 426

8-5-1 Stabilih’ Problem 427

8-5-2 Definition o f Encircled and

Enclosed 428

8-5-3 Number o f Encirclements and

Enclosures 429

8-5-4 Principles o f the Argument 429

8-5-5 Nvquisf Path 433

8-5-6 Nyquist Criterion and the Lis) or

the G(s)H{s) Plot 434

6 Nyquisl Criterion for Systems with

Mininium-Phase Transfer Functions 435

8-6-1 Application o f the Nyquist Criterion

to Miniinum-Pliase Tranfer

Functions That Are Not Strictly

Proper 436

Ĩ Relation between the Root Loci and the

Nvquist Plot 437

8 Illustrative Examples: Nvqiiist Criterion

for Minimum-Phase Transfer

Functions 440

9 Effects o f Adding Poles and Zeros

to L(s) on the Sliape o f the Nvquist

Plot 444

10 Relative Stabi!it\’: Gain Margin and Pliase

Margin 449

S-10-] G ain M argin (G M ) 451

S-10-2 PhiLst- M argin (P M ) 453

] ] s ta b ilih ' Analysis u itli tlie Bode Plot 455

s - 1 ] - 1 ‘ Bock- P lo ts (,f S y stem s \n tli P u re

T in it- Delays 458

8-12 Relative Stability Related to the slope o f the

Magnitude Curve o f the Bode Plot 459

8-12-1 Conditionally Stable Sv-stem 459

8-13 Stability Analysis w ith the Magnihide-Phase

Plot 462

8-14 Constant-M Loci in the Magnitude-Phase Plane:

The Nichols chart 463

8-15 Nichols Chart Applied to Nonunit\-Feedbacfc

Systems 469

S-16 Sensitivity Studies in the Frequenc\ Domain 470

8-17 M ATLAB Tools and Case Shidies ' 472

8-18 Summan- 472

Design of Control Systems 487

9-1 Introduction 487

9-1-1 Design Spetifications 487

9-1-2 Controller Configurations 489

9-]-3 Fundamenta] Principles o f Design 491

9-2 Design with the PD Controller 492

9-2-1 Time-Domain Interpretation o f PD

Control 494

9-2-2 Frequenc\’-Domain Interpretation o f

PD Control 496

9-2-3 Summary-of Effects o f P D C onfrol 497

9-3 Design with tlie PI ControUer 511

9-3-1 Time-Domain Interpretation and

Design o f PI Control 513

9-3-2 Frequenc\'-Domain Interpretation and

Design t)f PI Control 514

9-4 Design \nth the P ID ControUer 528

9-5 Design with Phase-Lead Controller 532

9-5-1 Time-Doinain Interpretation and

Design o f Phase-Lead Control >34

9-5-2 Frequencv-Doinajn Interpretation and

Design o f Phase-Lead Control 53Õ

9-5-3 Effects o f Phase-Lead

Compensatíon 5.54

9-5-4 Limitatíons o f Single-Stage Phase-Lead

Control 555

9-5-5 Multistage Phase-Lead Controller 555

9-5-6 Sensiti\it\’ Considerations 559

9-6 Design with Phase-Lag Controller -561

9-fi-l Tinie-Domain Interpretation and

Design o f Phase-Lag Control .561

9-6-2 FrequeiiC'v-Doinain Interpretation

and Design o f Phasc'-Lag Control 563

9-6-3 Eíĩetts and Limitations o f Phase-Lag

Control 574

9-7 Design u ith Lfad-Lag Controller 574

9-8 P<)lf-Zero-Cance!Iatinn Design: Notch Filter 576

9-S-l Second-Order Active Filter 579

9-8-2 Frequencv-Doinain Interpretation and

Design 5H0

Forward and Feedforward ControUers 588

Design o f Robust Control Systems 590

M inor-Loop Feedback Control 601

9-11-1 Rate-Feedback or

Tachometer-Feedback Control 601

9-11-2 M inor-Loop Feedback Control with

Actíve Filter 603

A Hydraulic Control System 605

9-12-1 Modeling Linear Actuator 605

9-12-2 Four-Way Electro-Hydraulic

Valve 606

9-12-3 Modeling tlie Hydraulic System 612

9-12-4 Applications 613

Controller Design 617

9-13-1 p Control 617

9-13-2 PD Control 621

9-13-3 PI Control 626

9-13-4 P ID Control 628

M ATLAB Tools and Case Studies 631

Plotting Tutorial 647

Summary 649

APTER V

Variable Analysis 673

Introduction 673

Block Diagrams, Transfer Functions, and State

Diagrams 673

10-2-1 Transfer Functions (Multivariable

Systems) 673

10-2-2 Block Diagrams and Transfer Functions

o f Multivariable Systems 674

10-2-3 State Diagram 676

10-2-4 From Differentia] Equations to State

Diagrams 678

10-2-5 From State Diagrams to Transfer

Funcdon 679

10-2-6 From State Diagrams to State and

Output Eijuations 680

Vecfor-Malrix Representation o f State

Equations 682

State-Transition Matrix 684

10-4-1 Significance o f the State-Transition

Matrix 685

10-4-2 Properties o f the State-Transition

Matrix 685

State-Transitíoii Equation 687

10-5-1 State-Transition Equatíon Determined

from the State Diagram 689

Relationship between State Equations and

High-Order Differentia] Equations 691

Relationship between State Equations and

Transfer Functions 693

Characteristic Efjuations, Eigenvalues,

and Eigenvectors 695

10-8-1 Characteristic Equation from a

Differential Equation 695

10-8-2 Characteristic Equation from a Transfer

Function 696

10-8-3 Characteristic Equation from State

Equations 696

10-8-4 Eigenvalues 697

10-8-5 Eigenvectors 697

10-8-6 Generalized Eigenvectors 698

Similarity Transformation 699

10-9-1 Invariance Properties o f tlie Similarity

Transformations 700

10-9-2 ControUabilitv- Canonical Form (CCF)

701

10-9-3 Obsewabilit>'CanonicalFonn(OCF) 703

10-9-4 Diagonal Canonical Form (D C F) 704

10-9-5 Jordan Canonical Form (JCF) 706

Decompositions o f Transfer Func-tions 707

10-10-1 Direct Decomposition 707

10-10-2 Cascade Decomposition 712

10-10-3 Parallel Decomposition 713

Conirollability o f Control Systems 714

10-11-1 General Concept o f Controllability

716

10-11-2 Definition o f State ControllabUity 716

10-11-3 Alternate Tests on Controllability 717

Observability o f Linear Systems 719

10-12-1 Definition o f ObservabOity 719

10-12-2 Alternate Tests on observabilitv 720

Relationship among Controllability.

Observability, and Transfer Functions 721

Invariant Theorems on Conirollabilitv and

Observabilitv 723

Case Study; Magiietic-Ball Suspension

System 725

State-Feedback Control 728

Pole-Placement Design Through State

Feedback 730

State Feedback with Integra] C;oiitrol 735

M ATLAB Tools and Case Studies 741

10-19-1 Description and Use o f the State-Space

Analysis Tool 741

10-19-2 Description and Use o f tfsym for

State-Space Applications 748

Summary 751

Appendices cun be found on tliis book's companion Web site:

www.uilev-c'oin/collegt'/golnaraghi.

Eletnentarv M atrix Theory and Algebra A-1

A -l Flenientan’ Matrix Theon A -I

A-1-1 Definition o f a Matrix A-2

10-12

10-13

10-14

10-15

10-16

10-17

10-18

10-19