Feedback Control of Dynamic Systems

Fifth Edition

Gene F. Franklin Stanford University

J. David Powell Stanford University

Abbas Emami-Naeini SC Solutions, Inc.

PEARSON

Prentice Hall

Upper Saddle River, New Jersey 07458

Contents

Preface xii

1 An Overview and Brief History of Feedback Control xviii A Perspective on Feedback Control xviii Chapter Overview 1

1.1 A Simple Feedback System 2 1.2 A First Analysis of Feedback 5 1.3 A Brief History 9 1.4 An Overview of the Book 15

Summary 16 End-of-Chapter Questions 17 Problems 18

2 Dynamic Models 20 A Perspective on Dynamic Models 20 Chapter Overview 21

2.1 Dynamics of Mechanical Systems 22 2.2 Models of Electric Circuits 34 2.3 Models of Electromechanical Systems 39

• 2.4 Heat and Fluid-Flow Models 44 A 2.5 Complex Mechanical Systems 55

Summary 61 End-of-Chapter Questions 61 Problems 62

3 Dynamic Response 72 A Perspective on System Response 72 Chapter Overview 73

3.1 Review of Laplace Transforms 74 3.2 System Modeling Diagrams 102 3.3 Effect of Pole Locations 107

3.4 Time-Domain Specifications 115 3.5 Effects of Zeros and Additional Poles 121 3.6 Amplitude and Time Scaling 127 3.7 Stability 130

A 3.8 Obtaining Models from Experimental Data 139 A 3.9 Mason's Rule and the Signal-Flow Graph 141

Summary 145 End-of-Chapter Questions 147 Problems 148

4 Basic Properties of Feedback 166

A Perspective on the Properties of Feedback 166 Chapter Overview 167

4.1 The Basic Equations of Control 168 4.2 Control of Steady-State Error: System Type 176 4.3 Control of Dynamic Error: PID Control 186

A 4.4 Extensions to the Basic Feedback Concepts 191 Summary 210 End-of-Chapter Questions 211 Problems 212

5 The Root-Locus Design Method 230

A Perspective on the Root-Locus Design Method 230 Chapter Overview 231

5.1 Root Locus of a Basic Feedback System 232 5.2 Guidelines for Sketching a Root Locus 237 5.3 Selected Illustrative Root Loci 249 5.4 Selecting the Parameter Value 263 5.5 Design Using Dynamic Compensation 266 5.6 A Design Example Using the Root Locus 278 5.7 Extensions of the Root-Locus Method 284

Summary 292 End-of-Chapter Questions 294 Problems 295

6 The Frequency-Response Design Method 314 A Perspective on the Frequency-Response Design Method 314 Chapter Overview 315

6.1 Frequency Response 316 6.2 Neutral Stability 338 6.3 The Nyquist Stability Criterion 340 6.4 Stability Margins 353

6.5 6.6 6.7 6.8 6.9

6.10

Bode's Gain-Phase Relationship 361 Closed-Loop Frequency Response 365 Compensation 366 Alternative Presentations of Data 392 Specifications in Terms of the Sensitivity Function 398 TimeDelay 407 Summary 409 End-of-Chapter Questions 412 Problems 413

7 State-Space Design 438 A Perspective on State-Space Design 438 Chapter Overview 439 Advantages of State Space 440 System Description in State Space 441 Block Diagrams and State Space 448 Analysis of the State Equations 451 Control-Law Design for Full-State Feedback 471 Selection of Pole Locations for Good Design 485 Estimator Design 497 Compensator Design: Combined Control Law and Estimator 511 Introduction of the Reference Input with the Estimator 524 Integral Control and Robust Tracking 536 Loop Transfer Recovery (LTR) 554 Direct Design with Rational Transfer Functions 559 Design for Systems with Pure Time Delay 563 Summary 568 End-of-Chapter Questions 569 Problems 572

8 Digital Control 594 A Perspective on Digital Control 594 Chapter Overview 595

8.1 Digitization 596 8.2 Dynamic Analysis of Discrete Systems 598 8.3 Design Using Discrete Equivalents 606 8.4 Hardware Characteristics 615 8.5 Sample-Rate Selection 619

A 8.6 Discrete Design 622 • 8.7 State-Space Design Methods 629

Summary 637 End-of-Chapter Questions 639 Problems 639

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10

A 7.11 A 7.12 A 7.13

Vlll Contents

9 Nonlinear Systems 652 Perspective on Nonlinear Systems 652 Chapter Overview 653

9.1 Introduction and Motivation: Why Study Nonlinear Systems? 654 9.2 Analysis by Linearization 656 9.3 Equivalent Gain Analysis Using the Root Locus 663 9.4 Equivalent Gain Analysis Using Frequency Response: Describing

Functions 673 • 9.5 Analysis and Design Based on Stability 684

Summary 706 End-of-Chapter Questions 706 Problems 707

1 0 Control System Design: Principles and Case Studies 716 A Perspective on Design Principles 716 Chapter Overview 717

10.1 An Outline of Control Systems Design 718 10.2 Design of a Satellite's Attitüde Control 723 10.3 Lateral and Longitudinal Control of a Boeing 747 742 10.4 Control of the Fuel-Air Ratio in an Automotive Engine 761 10.5 Control of the Read/Write Head Assembly of a Hard Disk 769 10.6 Control of Rapid Thermal Processing (RTP) Systems in Semiconductor

Wafer Manufacturing 777 Summary 791 End-of-Chapter Questions 793 Problems 793

Appendix A Laplace Transforms 807

A.l The =C_ Laplace Transform 807 A.2 Final Value Theorem 821

Appendix B A Review of Complex Variables 823

B.l Definition of a Complex Number 823 B.2 Algebraic Manipulations 825 B.3 Graphical Evaluation of Magnitude and Phase 827 B.4 Differentiation and Integration 828 B.5 Euler's Relations 828 B.6 Analytic Functions 829 B.7 Cauchy's Theorem 829 B.8 Singularities and Residues 830 B.9 Residue Theorem 830 B.10 The Argument Principle 831 B.ll Bilinear Transformation 833

Appendix C Summary of Matrix Theory 835

C.l Matrix Deflnitions 835 C.2 Elementary Operations on Matrices 835 C.3 Trace 836 C.4 Transpose 836 C.5 Determinant and Matrix Inverse 837 C.6 Properties of the Determinant 838 C.7 Inverse of Block Triangulär Matrices 839 C.8 Special Matrices 839 C.9 Rank 840 CIO Characteristic Polynomial 840 C.ll Cayley-Hamilton Theorem 840 C.l2 Eigenvalues and Eigenvectors 840 C.13 Similarity Transformations 841 C.14 Matrix Exponential 842 C.15 Fundamental Subspaces 843 C.l6 Singular-Value Decomposition 843 C.17 Positive Defmite Matrices 844 C.l8 Matrix Identity 844

Appendix D Controllability and Observability 845

D.l Controllability 845 D.2 Observability 851

Appendix E Ackermann's Formula for Pole Placement 853

Appendix F MATLAB Commands 857

Appendix G Solutions to the End-of-Chapter Questions 859

References 875

Index 885