**Preface xiii**

**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 Feedback System Fundamentals 10

1.4 A Brief History 11

1.5 An Overview of the Book 17

Summary 19

Review Questions 19

Problems 20

**2 Dynamic Models 23**

A Perspective on Dynamic Models 23

Chapter Overview 24

2.1 Dynamics of Mechanical Systems 24

2.1.1 Translational Motion 24

2.1.2 Rotational Motion 31

2.1.3 Combined Rotation and Translation 39

2.1.4 Complex Mechanical Systems (W)** 42

2.1.5 Distributed Parameter Systems 42

2.1.6 Summary: Developing Equations of Motion

for Rigid Bodies 44

2.2 Models of Electric Circuits 45

2.3 Models of Electromechanical Systems 50

2.3.1 Loudspeakers 50

2.3.2 Motors 52

2.3.3 Gears 56

2.4 Heat and Fluid-Flow Models 57

2.4.1 Heat Flow 58

2.4.2 Incompressible Fluid Flow 61

2.5 Historical Perspective 68

Summary 71

Review Questions 71

Problems 72

**3 Dynamic Response 84**

A Perspective on System Response 84

Chapter Overview 85

3.1 Review of Laplace Transforms 85

3.1.1 Response by Convolution 86

3.1.2 Transfer Functions and Frequency Response 91

3.1.3 The *L*− Laplace Transform 101

3.1.4 Properties of Laplace Transforms 103

3.1.5 Inverse Laplace Transform by Partial-Fraction Expansion 105

3.1.6 The Final Value Theorem 107

3.1.7 Using Laplace Transforms to Solve Differential Equations 109

3.1.8 Poles and Zeros 111

3.1.9 Linear System Analysis Using Matlab_ 112

3.2 System Modeling Diagrams 118

3.2.1 The Block Diagram 118

3.2.2 Block-Diagram Reduction Using Matlab 122

3.2.3 Mason’s Rule and the Signal Flow Graph (W) 123

3.3 Effect of Pole Locations 123

3.4 Time-Domain Specifications 131

3.4.1 Rise Time 132

3.4.2 Overshoot and Peak Time 132

3.4.3 Settling Time 134

3.5 Effects of Zeros and Additional Poles 137

3.6 Stability 146

3.6.1 Bounded Input—Bounded Output Stability 147

3.6.2 Stability of LTI Systems 148

3.6.3 Routh’s Stability Criterion 149

3.7 Obtaining Models from Experimental Data: System Identification (W) 156

3.8 Amplitude and Time Scaling (W) 156

3.9 Historical Perspective 156

Summary 157

Review Questions 159

Problems 159

**4 A First Analysis of Feedback 180**

A Perspective on the Analysis of Feedback 180

Chapter Overview 181

4.1 The Basic Equations of Control 182

4.1.1 Stability 183

4.1.2 Tracking 184

4.1.3 Regulation 185

4.1.4 Sensitivity 186

4.2 Control of Steady-State Error to Polynomial Inputs: System Type 188

4.2.1 System Type for Tracking 189

4.2.2 System Type for Regulation and Disturbance Rejection 194

4.3 The Three-Term Controller: PID Control 196

4.3.1 Proportional Control (P) 196

4.3.2 Integral Control (I) 198

4.3.3 Derivative Control (D) 201

4.3.4 Proportional Plus Integral Control (PI) 201

4.3.5 PID Control 202

4.3.6 Ziegler—Nichols Tuning of the PID Controller 206

4.4 Feedforward Control by Plant Model Inversion 212

4.5 Introduction to Digital Control (W) 214

4.6 Sensitivity of Time Response to Parameter Change (W) 215

4.7 Historical Perspective 217

Summary 217

Review Questions 218

Problems 218

5 **The Root-Locus Design Method**

A Perspective on the Root-Locus Design Method 234

Chapter Overview 235

5.1 Root Locus of a Basic Feedback System 235

5.2 Guidelines for Determining a Root Locus 240

5.2.1 Rules for Determining a Positive (180¿Root Locus 242

5.2.2 Summary of the Rules for Determining a Root Locus 248

5.2.3 Selecting the Parameter Value 249

5.3 Selected Illustrative Root Loci 251

5.4 Design Using Dynamic Compensation 264

5.4.1 Design Using Lead Compensation 266

5.4.2 Design Using Lag Compensation 270

5.4.3 Design Using Notch Compensation 272

5.4.4 Analog and Digital Implementations (W) 274

5.5 A Design Example Using the Root Locus 275

5.6 Extensions of the Root-Locus Method 281

5.6.1 Rules for Plotting a Negative (0¿Root Locus 281

5.6.2 Consideration of Two Parameters 284

5.6.3 Time Delay (W) 286

5.7 Historical Perspective 287

Summary 289

Review Questions 290

Problems 291

**6 The Frequency-Response Design Method**

A Perspective on the Frequency-Response Design Method 308

Chapter Overview 309

6.1 Frequency Response 309

6.1.1 Bode Plot Techniques 317

6.1.2 Steady-State Errors 330

6.2 Neutral Stability 331

6.3 The Nyquist Stability Criterion 333

6.3.1 The Argument Principle 334

6.3.2 Application of The Argument Principle to Control Design 335

6.4 Stability Margins 348

6.5 Bode’s Gain—Phase Relationship 357

6.6 Closed-Loop Frequency Response 361

6.7 Compensation 363

6.7.1 PD Compensation 363

6.7.2 Lead Compensation (W) 364

6.7.3 PI Compensation 374

6.7.4 Lag Compensation 375

6.7.5 PID Compensation 381

6.7.6 Design Considerations 387

6.7.7 Specifications in Terms of the Sensitivity Function 389

6.7.8 Limitations on Design in Terms of the Sensitivity Function 394

6.8 Time Delay 398

6.8.1 Time Delay via the Nyquist Diagram (W) 400

6.9 Alternative Presentation of Data 400

6.9.1 Nichols Chart 400

6.9.2 The Inverse Nyquist Diagram (W) 404

6.10 Historical Perspective 404

Summary 405

Review Questions 408

Problems 408

**7 State-Space Design 433**

A Perspective on State-Space Design 433

Chapter Overview 434

7.1 Advantages of State-Space 434

7.2 System Description in State-Space 436

7.3 Block Diagrams and State-Space 442

7.4 Analysis of the State Equations 444

7.4.1 Block Diagrams and Canonical Forms 445

7.4.2 Dynamic Response from the State Equations 457

7.5 Control-Law Design for Full-State Feedback 463

7.5.1 Finding the Control Law 464

7.5.2 Introducing the Reference Input with Full-State Feedback 473

7.6 Selection of Pole Locations for Good Design 477

7.6.1 Dominant Second-Order Poles 477

7.6.2 Symmetric Root Locus (SRL) 479

7.6.3 Comments on the Methods 488

7.7 Estimator Design 489

7.7.1 Full-Order Estimators 489

7.7.2 Reduced-Order Estimators 495

7.7.3 Estimator Pole Selection 499

7.8 Compensator Design: Combined Control Law and Estimator (W) 501

7.9 Introduction of the Reference Input with the Estimator (W) 514

7.9.1 General Structure for the Reference Input 515

7.9.2 Selecting the Gain 524

7.10 Integral Control and Robust Tracking 525

7.10.1 Integral Control 526

7.10.2 Robust Tracking Control: The Error-Space Approach 528

7.10.3 Model-Following Design 539

7.10.4 The Extended Estimator 543

7.11 Loop Transfer Recovery 547

7.12 Direct Design with Rational Transfer Functions 552

7.13 Design for Systems with Pure Time Delay 556

7.14 Solution of State Equations (W) 559

7.15 Historical Perspective 559

Summary 562

Review Questions 565

Problems 566

**8 Digital Control 590**

A Perspective on Digital Control 590

Chapter Overview 591

8.1 Digitization 591

8.2 Dynamic Analysis of Discrete Systems 594

8.2.1 *z*-Transform 594

8.2.2 *z*-Transform Inversion 595

8.2.3 Relationship Between *s *and *z *597

8.2.4 Final Value Theorem 599

8.3 Design Using Discrete Equivalents 601

8.3.1 Tustin’s Method 602

8.3.2 Zero-Order Hold (ZOH) Method 605

8.3.3 Matched Pole—Zero (MPZ) Method 607

8.3.4 Modified Matched Pole—Zero (MMPZ) Method 611

8.3.5 Comparison of Digital Approximation Methods 612

8.3.6 Applicability Limits of the Discrete Equivalent Design Method 613

8.4 Hardware Characteristics 613

8.4.1 Analog-to-Digital (A/D) Converters 614

8.4.2 Digital-to-Analog Converters 614

8.4.3 Anti-Alias Prefilters 615

8.4.4 The Computer 616

8.5 Sample-Rate Selection 617

8.5.1 Tracking Effectiveness 618

8.5.2 Disturbance Rejection 618

8.5.3 Effect of Anti-Alias Prefilter 619

8.5.4 Asynchronous Sampling 620

8.6 Discrete Design 620

8.6.1 Analysis Tools 621

8.6.2 Feedback Properties 622

8.6.3 Discrete Design Example 623

8.6.4 Discrete Analysis of Designs 626

8.7 Discrete State-Space Design Methods (W) 628

8.8 Historical Perspective 628

Summary 629

Review Questions 631

Problems 631

** **

**9 Nonlinear Systems 637**

A Perspective on Nonlinear Systems 637

Chapter Overview 638

9.1 Introduction and Motivation: Why Study Nonlinear Systems? 639

9.2 Analysis by Linearization 641

9.2.1 Linearization by Small-Signal Analysis 641

9.2.2 Linearization by Nonlinear Feedback 646

9.2.3 Linearization by Inverse Nonlinearity 647

9.3 Equivalent Gain Analysis Using the Root Locus 648

9.3.1 Integrator Antiwindup 655

9.4 Equivalent Gain Analysis Using Frequency Response: Describing Functions 658

9.4.1 Stability Analysis Using Describing Functions 665

9.5 Analysis and Design Based on Stability 670

9.5.1 The Phase Plane 670

9.5.2 Lyapunov Stability Analysis 677

9.5.3 The Circle Criterion 683

9.6 Historical Perspective 690

Summary 691

Review Questions 691

Problems 692

** **

**10 Control System Design: Principles and Case Studies 703**

A Perspective on Design Principles 703

Chapter Overview 704

10.1 An Outline of Control Systems Design 705

10.2 Design of a Satellite’s Attitude Control 711

10.3 Lateral and Longitudinal Control of a Boeing 747 729

10.3.1 Yaw Damper 733

10.3.2 Altitude-Hold Autopilot 741

10.4 Control of the Fuel—Air Ratio in an Automotive Engine 747

10.5 Control of the Read/Write Head Assembly of a Hard Disk 755

10.6 Control of RTP Systems in SemiconductorWafer Manufacturing 763

10.7 Chemotaxis or How *E. Coli *Swims Away from Trouble 777

10.8 Historical Perspective 786

Summary 788

Review Questions 790

Problems 790

**Appendix A Laplace Transforms 804**

A.1 The *L*− Laplace Transform 804

A.1.1 Properties of Laplace Transforms 805

A.1.2 Inverse Laplace Transform by Partial-Fraction Expansion 813

A.1.3 The Initial Value Theorem 816

A.1.4 Final Value Theorem 817

**Appendix B Solutions to the Review Questions 819**

**Appendix C Matlab Commands 835**

**Bibliography 840**

**Index 848**

List of Appendices on the web at www.fpe7e.com

** **

**Appendix WA: A Review of Complex Variables**

**Appendix WB: Summary of Matrix Theory**

**Appendix WC: Controllability and Observability**

**Appendix WD: Ackermann’s Formula for Pole Placement**

**Appendix W2.1.4: Complex Mechanical Systems**

**Appendix W3.2.3: Mason’s Rule and Signal Flow Graph**

**Appendix W3.6.3.1: Routh Special Cases**

**Appendix W3.7: System Identification**

**Appendix W3.8: Amplitude and Time Scaling**

**Appendix W4.1.4.1: The Filtered Case**

**Appendix W4.2.2.1: Truxal’s Formula for the Error Constants**

**Appendix W4.5: Introduction to Digital Control**

**Appendix W4.6: Sensitivity of Time Response to Parameter Change**

**Appendix W5.4.4: Analog and Digital Implementations**

**Appendix W5.6.3: Root Locus with Time Delay**

**Appendix W6.7.2: Digital Implementation of Example 6.15**

**Appendix W6.8.1: Time Delay via the Nyquist Diagram**

**Appendix W6.9.2: The Inverse Nyquist Diagram**

**Appendix W7.8: Digital Implementation of Example 7.31**

**Appendix W7.9: Digital Implementation of Example 7.33**

**Appendix W7.14: Solution of State Equations**

**Appendix W8.7: Discrete State-Space Design Methods**

** **

****Sections with (W) indicates that additional material is located on the web at www.FPE7e.com **