1 **An Overview and Brief History of Feedback Control 1**

A Perspective on Feedback Control 1

Chapter Overview 2

1.1 A Simple Feedback System 3

1.2 A First Analysis of Feedback 6

1.3 A Brief History 9

1.4 An Overview of the Book 14

Summary 16

Review Questions 16

Problems 17

2 **Dynamic Models 20**

A Perspective on Dynamic Models 20

Chapter Overview 21

2.1 Dynamics of Mechanical Systems 21

2.1.1 Translational Motion 21

2.1.2 Rotational Motion 27

2.1.3 Combined Rotation and Translation 36

2.1.4 Distributed Parameter Systems 38

2.1.5 Summary: Developing Equations of Motion

for Rigid Bodies 40

2.2 Models of Electric Circuits 41

2.3 Models of Electromechanical Systems 45

2.4 Heat and Fluid-Flow Models 50

2.4.1 Heat Flow 50

2.4.2 Incompressible Fluid Flow 54

2.5 Historical Perspective 60

Summary 62

Review Questions 63

Problems 64

3 **Dynamic Response 74**

A Perspective on System Response 74

Chapter Overview 75

3.1 Review of Laplace Transforms 75

3.1.1 Response by Convolution 75

3.1.2 Transfer Functions and Frequency Response 80

3.1.3 The *L*− Laplace Transform 87

3.1.4 Properties of Laplace Transforms 89

3.1.5 Inverse Laplace Transform by Partial-Fraction Expansion 91

3.1.6 The Final Value Theorem 93

3.1.7 Using Laplace Transforms to Solve Problems 94

3.1.8 Poles and Zeros 96

3.1.9 Linear System Analysis Using MATLAB 97

3.2 System Modeling Diagrams 102

3.2.1 The Block Diagram 102

3.2.2 Block Diagram Reduction Using MATLAB 107

3.3 Effect of Pole Locations 108

3.4 Time-Domain Specifications 116

3.4.1 Rise Time 116

3.4.2 Overshoot and Peak Time 117

3.4.3 Settling Time 118

3.5 Effects of Zeros and Additional Poles 120

3.6 Stability 130

3.6.1 Bounded Input—Bounded Output Stability 130

3.6.2 Stability of LTI Systems 131

3.6.3 Routh’s Stability Criterion 132

3.7 Obtaining Models from Experimental Data 140

3.7.1 Models from Transient-Response Data 142

3.7.2 Models from Other Data 146

3.8 Amplitude and Time Scaling 147

3.8.1 Amplitude Scaling 147

3.8.2 Time Scaling 148

3.9 Historical Perspective 149

Summary 150

Review Questions 151

Problems 152

4 **A First Analysis of Feedback 170**

A Perspective on the Analysis of Feedback 170

Chapter Overview 171

4.1 The Basic Equations of Control 171

4.1.1 Stability 173

4.1.2 Tracking 174

4.1.3 Regulation 174

4.1.4 Sensitivity 175

4.2 Control of Steady-State Error to Polynomial Inputs: SystemType 178

4.2.1 System Type for Tracking 179

4.2.2 System Type for Regulation and Disturbance Rejection 183

4.3 The Three-Term Controller: PID Control 186

4.3.1 Proportional Control (P) 187

4.3.2 Proportional Plus Integral Control (PI) 187

4.3.3 PID Control 188

4.3.4 Ziegler—Nichols Tuning of the PID Controller 192

4.4 Introduction to Digital Control 198

4.5 History of Control Theory and Practice 203

Summary 205

Review Questions 206

Problems 207

5 **The Root-Locus Design Method 220**

A Perspective on the Root-Locus Design Method 220

Chapter Overview 221

5.1 Root Locus of a Basic Feedback System 221

5.2 Guidelines for Determining a Root Locus 226

5.2.1 Rules for Plotting a Positive (180æ) Root Locus 228

5.2.2 Summary of the Rules for Determining a Root Locus 233

5.2.3 Selecting the Parameter Value 234

5.3 Selected Illustrative Root Loci 236

5.4 Design Using Dynamic Compensation 248

5.4.1 Design Using Lead Compensation 249

5.4.2 Design Using Lag Compensation 254

5.4.3 Design Using Notch Compensation 255

5.4.4 Analog and Digital Implementations 257

5.5 A Design Example Using the Root Locus 260

5.6 Extensions of the Root-Locus Method 266

5.6.1 Rules for Plotting a Negative (0æ) Root Locus 266

5.6.2 Consideration of Two Parameters 270

5.6.3 Time Delay 272

5.7 Historical Perspective 274

Summary 276

Review Questions 278

Problems 278

6 **The Frequency-Response Design Method 296**

A Perspective on the Frequency-Response 296

Design Method 296

Chapter Overview 297

6.1 Frequency Response 297

6.1.1 Bode Plot Techniques 304

6.1.2 Steady-State Errors 315

6.2 Neutral Stability 317

6.3 The Nyquist Stability Criterion 319

6.3.1 The Argument Principle 320

6.3.2 Application to Control Design 321

6.4 Stability Margins 334

6.5 Bode’s Gain—Phase Relationship 341

6.6 Closed-Loop Frequency Response 346

6.7 Compensation 347

6.7.1 PD Compensation 348

6.7.2 Lead Compensation 348

6.7.3 PI Compensation 360

6.7.4 Lag Compensation 360

6.7.5 PID Compensation 365

6.7.6 Design Considerations 371

6.7.7 Specifications in Terms of the Sensitivity Function 373

6.7.8 Limitations on Design in Terms of the Sensitivity

Function 377

6.8 Time Delay 381

6.9 Alternative Presentation of Data 382

6.9.1 Nichols Chart 382

6.10 Historical Perspective 386

Summary 386

Review Questions 388

Problems 389

7 **State-Space Design 413**

A Perspective on State-Space Design 413

Chapter Overview 413

7.1 Advantages of State-Space 414

7.2 System Description in State-Space 416

7.3 Block Diagrams and State-Space 421

7.3.1 Time and Amplitude Scaling in State-Space 424

7.4 Analysis of the State Equations 425

7.4.1 Block Diagrams and Canonical Forms 425

7.4.2 Dynamic Response from the State Equations 436

7.5 Control-Law Design for Full-State Feedback 442

7.5.1 Finding the Control Law 443

7.5.2 Introducing the Reference Input with Full-State

Feedback 451

7.6 Selection of Pole Locations for Good Design 455

7.6.1 Dominant Second-Order Poles 456

7.6.2 Symmetric Root Locus (SRL) 457

7.6.3 Comments on the Methods 466

7.7 Estimator Design 466

7.7.1 Full-Order Estimators 466

7.7.2 Reduced-Order Estimators 472

7.7.3 Estimator Pole Selection 476

7.8 Compensator Design: Combined Control Law and Estimator 478

7.9 Introduction of the Reference Input with the Estimator 491

7.9.1 A General Structure for the Reference Input 492

7.9.2 Selecting the Gain 501

7.10 Integral Control and Robust Tracking 502

7.10.1 Integral Control 503

7.10.2 Robust Tracking Control: The Error-Space Approach 505

7.10.3 The Extended Estimator 516

7.11 Loop Transfer Recovery (LTR) 519

7.12 Direct Design with Rational Transfer Functions 524

7.13 Design for Systems with Pure Time Delay 527

7.14 Historical Perspective 530

Summary 533

Review Questions 534

Problems 536

8 **Digital Control 558**

A Perspective on Digital Control 558

Chapter Overview 559

8.1 Digitization 559

8.2 Dynamic Analysis of Discrete Systems 561

8.2.1 *z*-Transform 561

8.2.2 *z*-Transform Inversion 562

8.2.3 Relationship between *s *and *z *565

8.2.4 Final Value Theorem 566

8.3 Design Using Discrete Equivalents 568

8.3.1 Matched Pole-Zero (MPZ) Method 571

8.3.2 Modified Matched Pole-Zero (MMPZ) Method 575

8.3.3 Comparison of Digital Approximation Methods 575

8.3.4 Applicability Limits of the Discrete Equivalent Design

Method 576

8.4 Hardware Characteristics 577

8.4.1 Analog-to-Digital (A/D) Converters 577

8.4.2 Digital-to-Analog (D/A) Converters 578

8.4.3 Anti-Alias Prefilters 578

8.4.4 The Computer 579

8.5 Sample-Rate Selection 580

8.5.1 Tracking Effectiveness 581

8.5.2 Disturbance Rejection 581

8.5.3 Effect of Anti-Alias Prefilter 582

8.5.4 Asynchronous Sampling 583

8.6 Discrete Design 583

8.6.1 Analysis Tools 583

8.6.2 Feedback Properties 585

8.6.3 Discrete Design Example 586

8.6.4 Discrete Analysis of Designs 588

8.7 Historical Perspective 590

Summary 591

Review Questions 592

Problems 593

9 **Nonlinear Systems 599**

Perspective on Nonlinear Systems 599

Chapter Overview 600

9.1 Introduction and Motivation: Why Study

Nonlinear Systems? 600

9.2 Analysis by Linearization 602

9.2.1 Linearization by Small-Signal Analysis 603

9.2.2 Linearization by Nonlinear Feedback 608

9.2.3 Linearization by Inverse Nonlinearity 608

9.3 Equivalent Gain Analysis Using the Root Locus 609

9.3.1 Integrator Antiwindup 615

9.4 Equivalent Gain Analysis Using Frequency

Response: Describing Functions 619

9.4.1 Stability Analysis Using Describing Functions 625

9.5 Analysis and Design Based on Stability 629

9.5.1 The Phase Plane 630

9.5.2 Lyapunov Stability Analysis 636

9.5.3 The Circle Criterion 642

Summary 649

Review Questions 650

Problems 650

10 **Control System Design: Principles and Case Studies 660**

A Perspective on Design Principles 660

Chapter Overview 661

10.1 An Outline of Control Systems Design 662

10.2 Design of a Satellite’s Attitude Control 667

10.3 Lateral and Longitudinal Control of a Boeing 747 684

10.3.1 Yaw Damper 689

10.3.2 Altitude-Hold Autopilot 696

10.4 Control of the Fuel—Air Ratio in an Automotive Engine 702

10.5 Control of the Read/Write Head Assembly

of a Hard Disk 709

10.6 Control ofRTP Systems in SemiconductorWafer Manufacturing 717

10.7 Chemotaxis or How *E. Coli *Swims Away

from Trouble 731

10.8 Historical Perspective 739

Summary 741

Review Questions 742

Problems 743

Appendix A **Laplace** ** Transforms 757**

A.1 The *L*− Laplace Transform 757

A.1.1 Properties of Laplace Transforms 757

A.1.2 Inverse LaplaceTransform by Partial-Fraction Expansion 766

A.1.3 The Initial Value Theorem 769

A.1.4 Final Value Theorem 770

Appendix B **Solutions to the End-of-Chapter Questions 772**

Appendix C **MATLAB** **® ** **Commands 788**

**Bibliography ** **793**

**Index ** **803**