Preface v
CHAPTER 1
Introduction 1
1-1 Introduction 1
1-1-1 Basic Components of a Control System 2
1-1-2 Examples of Control-System
Applications 2
1-1-3 Open-Loop Control Systems (Nonfeed-
back Systems) 6
1-1-4 Closed-loop Control Systems (Feedback
Control Systems) 7
1-2 What is Feedback and What are its Effects? 8
1-2-1 Effect of Feedback on Overall Gain 8
1-2-2 Effect of Feedback on Stability 9
1-2-3 Effect of, Feedback on External
Disturbance or Noise 10
1-3 Types of Feedback Control Systems 11
1-3-1 Linear versus Nonlinear Control
Systems 11
1-3-2 Time-Invariant versus Time-Varying
Systems 12
1-4 Summary 15
CHAPTER :2
Mathematical Foundation 16
2-1 Introduction 16
2-2 Laplace Transform 17
2-2-1 Definition of the Laplace Transform 17
2-2-2 Inverse Laplace Transformation 18
2-2-3 Important Theorems of the Laplace
Transform 19
2-3 Inverse Laplace Transform by Partial-Fraction
Expansion 21
2-3-1 Partial-Fraction Expansion 22
2-4 Application of the Laplace Transform to the Solution
of Linear Ordinary Differential Equations 25
2-5 Impulse Response and Transfer Functions of Linear
Systems 27
2-5-1 Impulse Response 27
2-5-2 Transfer Function (Single-Input, Single-
Output Systems) 27
2-5-3 Transfer Function (Multivariable
Systems) 29
2-6 MATLAB Tools and Case Studies 30
2-6-1 Description and Use of Transfer Function
Tool 30
2-7 Summary 41
CHAPTER 3
Block Diagrams and Signal-Flow Graphs 44
3-1 Block Diagrams 44
3-1-1 Block Diagrams of Control Systems 45
3-1-2 Block Diagrams and Transfer Functions of
Multivariable Systems 46
3-2 Signal-Flow Graphs (SFGs) 48
3-2-1 Basic Elements of an SFG 49
3-2-2 Summary of the Basic Properties of
SFG 50
3-2-3 Definitions of SFG Terms 51
3-2-4 SFG Algebra 53
3-2-5 SFG of a Feedback Control System 54
3-2-6 Gain Formula for SFG 54
3-2-7 Application of the Gain Formula between
Output Nodes and Noninput Nodes 56
3-2-8 Application of the Gain Formula to Block
Diagrams 57
3-3 State Diagram 58
3-3-1 From Differential Equations to State
Diagram 59
3-3-2 From State Diagram to Transfer
Function 61
3-3-3 From State Diagram to State and Output
Equations 61
3-4 MATLAB tbols and Case Studies 63
3-5 Summary 65
CHAPTER 4
Modeling of Physical Systems 77
4-1 Introduction 77
4-2 Modeling of Electrical Networks 77
4-3 Modeling of Mechanical Systems Elements 80
4-3-1 Translational Motion 80
4-3-2 Rotational Motion 83
4-3-3 Conversion Between Translational and
Rotational Motions 85
4-3-4 Gear Trains 86
4-3-5 Backlash and Dead Zone (Nonlinear
Characteristics) 88
4-4 Equations of Mechanical Systems 89
4-5 Sensors and Encoders in Control Systems 94
4-5-1 Potentiometer 94
4-5-2 Tachometers 99
4-5-3 Incremental Encoder 100
4-6 DC Motors in Control Systems 103
4-6-1 Basic Operational Principles of DC
Motors 104
4-6-2 Basic Classifications of PM DC
Motors 104
4-6-3 Mathematical Modeling of PM DC
Motors 107
4-7 Linearization of Nonlinear Systems 110
4-8 Systems with Transportation Lags (Time Delays) 114
4-8-1 Approximation of the Time-Delay Function
by Rational Functions 115
4-9 A Sun-Seeker System 116
4-9-1 Coordinate System 117
4-9-2 Error Discriminator 117
4-9-3 Op-Amp 118
4-9-4 Servoamplifier 118
4-9-5 Tachometer 118
4-9-6 DC Motor 118
4-10 MATLAB Tools and Case Studies 120
4-11 Summary 120
CHAPTER 5
State Variable Analysis 138
5-1 Introduction 138
5-2 Vector-Matrix Representation of State
Equations 138
5-3 State-Transition Matrix 140
5-3-1 Significance of the State-Transition
Matrix 141
5-3-2 Properties of the State-Transition
Matrix 142
5-4 State-Transition Equation 143
5-4-1 State-Transition Equation Determined from
the State Diagram 145
5-5 Relationship between State Equations and High-
Order Differential Equations, 147
5-6 Relationship between State Equations and Transfer
Functions 149
5-7 Characteristic Equations, Eigenvalues, and
Eigenvectors 151
5-7-1 Eigenvalues 152
5-7-2 Eigenvectors 153
5-8 Similarity Transformation 155
5-8-1 Invariance Properties of the Similarity
Transformations 156
5-8-2 Controllability Canonical Form (CCF) 156
5-8-3 Observability Canonical Form (OCF) 158
5-8-4 Diagonal Canonical Form (DCF) 159
5-8-5 Jordan Canonical Form (JCF) 160
5-9 Decompositions of Transfer Functions 161
5-9-1 Direct Decomposition 162
5-9-2 Cascade Decomposition 166
5-9-3 Parallel Decomposition 167
5-10 Controllability of Control Systems 169
5-10-1 General Concept of Controllability 170
5-10-2 Definition of State Controllability 171
5-10-3 Alternate Tests on Controllability 171
5-11 Observability of Linear Systems 173
5-11-1 Definition of Observability 173
5-11-2 Alternate Tests on Observability 174
5-12 Relationship Among Controllability, Observability,
and Transfer Functions 175
5-13 Invariant Theorems on Controllability and
Observability 177
5-14 A Final Illustrative Example: Magnetic-Ball
Suspension System 178
5-15 MATLAB Tools and Case Studies 181
5-15-1 Description and Use of the State-Space
Analysis Tool 182
5-15-2 Description and Use of tfsym for State-
Space Applications 189
5-15-3 Another Example 189
5-16 Summary 195
CHAPTER 6
Stability of Linear Control Systems 211
6-1 Introduction 211
6-2 Bounded-Input, Bounded-Output (BIBO) Stability--
Continuous-Data Systems 212
6-2-1 Relationship between Characteristic
Equation Roots and Stability 212
6-3 Zero-Input and Asymptotic Stability of Continuous-
Data Systems 213
6-4 Methods of Determining Stability 215
6-5 Routh-Hurwitz Criterion 216
6-5-1 Routh's Tabulation (1) 217
6-5-2 Special Cases When Routh's Tabulation
Terminates Prematurely 219
6-6 MATLAB Tools and Case Studies 222
6-7 Summary 226
CHAPTER 7
Time-Domain Analysis of Control
Systems 233
7-1 Time Response of Continuous-Data Systems:
Introduction 233
7-2 Typical Test Signals for the Time Response of
Control Systems 234
7-3 The Unit-Step Response and Time-Domain
Specifications 236
7-4 Steady-State Error 237
7-4-1 Steady-State Error of Linear Continuous-
Data Control Systems 237
7-4-2 Steady-State Error Caused by Nonlinear
System Elements 249
7-5 Time Response of a First-Order System 251
7-5-1 Speed Control of a DC Motor 251
7-6 Transient Response of a Prototype Second-Order
System 253
7-6-1 Damping Ratio and Damping Factor 253
7-6-2 Natural Undamped Frequency 255
7-6-3 Maximum Overshoot 257
7-6-4 Delay Time and Rise Time 259
7-6-5 Settling Time 261
7-7 Time,Domain Analysis of a Position-Control
System 265
7-7-1 Unit-Step Transient Response 268
7-7-2 The Steady-State Response 271
7-7-3 Time Response to a Unit-Ramp
Input 271
7-7-4 Time Response of a Third-Order
System 273
7-8 Effects of Adding Poles and Zeros to Transfer
Functions 276
7-8-1 Addition of a Pole to the Forward-Path
Transfer Function: Unity-Feedback
Systems 276
7-8-2 Addition of a Pole to the Closed-Loop
Transfer Function 277
7-8-3 Addition of a Zero to the Closed-Loop
Transfer Function 279
7-8-4 Addition of a Zero to the Forward-Path
Transfer Function: Unity-Feedback
Systems 280
7-9 Dominant Poles of Transfer Functions 281
7-9-1 The Relative Damping Ratio 282
7-9-2 The Proper Way of Neglecting the
Insignificant Poles with Consideration
of the Steady-State Response 282
7-10 The Approximation of High-Order Systems by Low-
Order System the Formal Approach 283
7-10-1 Approximation Criterion 284
7-11 MATLAB Tools and Case Studies 293
7-12 Summary 307
CHAPTEF 8
Root-Locus Technique 318
8-1 Introduction 318
8-2 Basic Properties of the Root Loci (RL) 319
8-3 Properties of the Root Loci 323
8-3-1 K = 0 and K = _+c, Points 323
8-3-2 Number of Branches on the Root
Loci 324
8-3-3 Symmetry of the RL 324
8-3-4 Angles of Asymptotes of the RL: Behavior
of theRLatls]= ~ 324
8-3-5 Intersect of the Asymptotes (Centroid) 325
8-3-6 Root Loci on the Real Axis 325
8-3-7 Angles of Departure and Angles of Arrival
of the RL 325
8-3-8 Intersection of the RL with the Imaginary
Axis 326
8-3-9 Breakaway Points (Saddle Points) on the
RL 326
8-3-10 The Root Sensitivity [17, 18, 19] 326
8-4 Design Aspects of the Root Loci 330
8-4-1 Effects of Adding Poles and Zeros to
G(s)H(s). 330
8-5 Root Contours (RC): Multiple-Parameter
Variation 336
8-6 Root Locus with the MATLAB Toolbox 342
8-7 Summary 345
CHAPTER 9
Frequency-Domain Analysis 352
9-1 Introduction 352
9-1-1 Frequency Response of Closed-Loop
Systems 353
9-1-2 Frequency-Domain Specifications 355
9-2 Mr, mr, and Bandwidth of the Prototype Second-
Order System 356
9-2-1 Resonant Peak and Resonant
Frequency 356
9-2-2 Bandwidth 358
9-3 Effects of Adding a Zero to the Forward-Path
Transfer Function 360
9-4 Effects of Adding a Pole to the Forward-Path
Transfer Function 364
9-5 Nyquist Stability Criterion: Fundamentals 365
9-5-1 Stability Problem 366
9-5-2 Definition of Encircled and
Enclosed 366
9-5-3 Number of Encirclements and
Enclosures 367
9-5-4 Principle of the Argument 368
9-5-5 Nyquist Path 372
9-5-6 Nyquist Criterion and the L(s) or the
G(s)H(s) plot 373
9-6 Nyquist Criterion for Systems with Minimum-Phase
Transfer Functions 374
9-6-1 Application of the Nyquist Cri~terion to
Minimum-Phase Transfer Functions that
Are Not Strictly Proper 375
9-7 Relation Between the Root Loci and the Nyquist
Plot 376
9-8 Illustrative Examples: Nyquist Criterion.for
Minimum-Phase Transfer Functions 378
9-9 Effects of Addition of Poles and Zeros to L(s) on the
Shape of the Nyquist Plot 382
9-10 Relative Stability: Gain Margin and Phase
Margin 386
9-10-1 Gain Margin (GM) 388
9-10-2 Phase Margin (PM) 389
9-11 Stability Analysis with the Bode Plot 392
9-11-1 Bode Plots of Systems with Pure Time
Delays 394
9-12 Relative Stability Related to the Slope of the
Magnitude Curve of the Bode Plot 396
9-12-1 Conditionally Stable System 396
9-13 Stability Analysis with the Magnitude-Phase
Plot 399
9-14 Constant-M Loci in the Magnitude-Phase Plane: The
Nichols Chart 400
9-15 Nichols Chart Applied to Nonunity-Feedback
Systems 406
9-16 Sensitivity Studies in the Frequency Domain 407
9-17 MATLAB Tools and Case Studies 409
9-18 Summary 421
CHAPTER t0
Design of Control Systems 433
10-1 Introduction 433
10-1-1 Design Specifications 433
10-1-2 Controller Configurations 435
10-1-3 Fundamental Principles of Design 437
10-2 Design with the PD Controller 438
10-2-1 Time-Domain Interpretation of PD
Control 440
10-2-2 Frequency-Domain Interpretation of PD
Control 442
10-2-3 Summary of Effects of PD Control 442
10-3 Design with the PI Controller 454
10-3-1 Time-Domain Interpretation and Design of
PI Control 456
10-3-2 Frequency-Domain Interpretation and
Design of PI Control 456
10-4 Design with the PID Controller 468
10-5 Design with Phase-Lead Controller 471
10-5-1 Time-Domain Interpretation and Design of
Phase-Lead Control 473
10-5-2 Frequency-Domain Interpretation and
Design of Phase-Lead Control 474
10-5-3 Effects of Phase-Lead
Compensation 489
10-5-4 Limitations of Single-Stage Phase-Lead
Control 489
10-5-5 Multistage Phase-Lead Controller 489
10-5-6 Sensitivity Considerations 493
10-6 Design with Phase-Lag Controller 494
10-6-1 Time-Domain Interpretation and Design of
Phase-Lag Control 494
10-6-2 Frequency-Domain Interpretation and
Design of Phase-Lag Control 496
10-6-3 Effects and Limitations of Phase-Lag
Control 506
10-7 Design with Lead-Lag Controller 507
10-8 Pole-Zero Cancellation Design: Notch Filter 508
10-8-1 Second-Order Active Filter 511
10-8-2 Frequency-Domain Interpretation and
Design 512
10-9 Forward and Feedforward Controllers 520
10-10 Design of Robust Control Systems 521
10-11 Minor-Loop Feedback Control 530
10-11-1 Rate-Feedback or Tachometer-Feedback
Control 531
10-11-2 Minor-Loop Feedback Control with Active
Filter 532
10-12 State-Feedback Control 534
10-13 Pole-Placement Design through State
Feedback 535
10-14 State Feedback with Integral Control 540
10-15 MATLAB Tools and Case Studies 545
10-16 Summary 558
CHAPTE 10
The Virtual Lab 578
11-1 Introduction 578
11-2 Important Aspects in the Response of a DC
Motor 579
11-2-1 Speed Response and the Effects of
Inductance and Disturbance-Open Loop
Response 579
11-2-2 Speed Control of DC Motors: Closed-Loop
Response 581
11-2-3 Position Control 582
11-3 Description of the Virtual Experimental
System 583
11-3-1 Motor 584
11-3-2 Position Sensor or Speed Sensor 584
11-3-3 Power Amplifier 584
11-3-4 Interface 584
11-4 Description of SIMLab and Virtual Lab
Software 585
11-5 Simulation and Virtual Experiments 589
11-5-1 Open-Loop Speed 589
11-5-2 Open-Loop Sine Input 591
11-5-3 Speed Control 593
11-5-4 Position Control 596
11-6 Design Project 598
11-7 Summary 603
iNDEX 606
Complex Variable Theory CD-ROM
APPENDX B
Differential and Difference Equations CD-ROM
Elementary Matrix Theory and Algebra CD-ROM
APPENDlX D
Laplace Transform Table CD-ROM
Operational Amplifiers CD-ROM
APPENDIX
Properties and Construction of the Root
Loci CD-ROM
APPENDIX, G
Frequency-Domain Plots CD-ROM
APPENDIX H
General Nyquist Criterion CD-ROM
APPENDIX.
Discrete-Data Control Systems CD-ROM
APPENDIX J
z-Transform Table CD-ROM
APPENDIX K
ACSYS 2002: Description of the Software CD-ROM
ANSWERS TO SELECTED PROBLEm, S CD-ROM
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