Feedback Control of Dynamic Systems

THIRD EDITION Feedback Control of Dynamic Systems Gene F. Franklin Stanford University J. David Powell Stanford University Abbas Emami-Naeini Integrated Systems, Inc. TT Addison-Wesley Publishing Company Reading, Massachusetts Menlo Park, California New York Don Mills, Ontario Wokingham, England Amsterdam Bonn Sydney Singapore Tokyo Madrid San Juan Milan Paris

Contents 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 2 1.2 A First Analysis of Feedback 4 1.3 A Brief History 7 1.4 An Overview of the Book 13 Summary 15 Problems 15 2 Dynamic Models 19 A Perspective on Dynamic Models 19 Chapter Overview 20 2.1 Dynamics of Mechanical Systems 20 2.2 Differential Equations in State-variable Form 35 2.3 Models of Electric Circuits 39 2.4 Models of Electromechanical Systems 43 2.5 Heat and Fluid-flow Models 51 2.5.1 Heat Flow 51 2.5.2 Incompressible Fluid Flow 55

XIV Contents 2.6 Linearization and Scaling 61 2.6.1 Small-signal Linearization 62 2.6.2 Linearization by Feedback 65 2.6.3 Amplitude Scaling 66 2.6.4 Time Scaling 67 Summary 68 Problems 70 J Dynamic Response 85 A Perspective on System Response 85 Chapter Overview 86 3.1 The Laplace Transform 86 3.1.1 Response by Convolution 86 3.1.2 Transfer Functions 89 3.1.3 The jsp-laplace Transform 92 3.1.4 Properties of Laplace Transforms 95 3.1.5 Partial-fraction Expansion 98 3.1.6 Laplace Transform Theorems 102 3.1.7 Using Laplace Transforms to Solve Problems 106 3.1.8 Laplace Transforms Using CACSD Software 108 3.2 System Modeling Diagrams 111 3.2.1 The Block Diagram 111 3.2.2 Mason's Rule and the Signal-flow Graph 114 3.3 Response versus Pole Locations 118 3.4 Time-domain Specifications 126 3.5 Effects of Zeros and Additional Poles 131 3.6 Numerical Simulation 138 3.6.1 Solution of Nonlinear Differential Equations 138 3.6.2 Solution of Linear Differential Equations 140 3.7 Obtaining Models from Experimental Data 144 3.7.1 Models from Transient-response Data 145 3.7.2 Models from Other Data 150 Summary 150 Problems 151

Contents XV т" Basic Properties of Feedback 167 A Perspective on Properties of Feedback 167 Chapter Overview 167 4.1 A Case Study of Speed Control 168 4.1.1 Disturbance Rejection 170 4.1.2 Sensitivity: Effects of Gain Changes 173 4.1.3 Dynamic Tracking 176 4.2 The Classical Three-term Controller 179 4.2.1 Proportional Feedback Control 179 4.2.2 Proportional-Integral (PI) Feedback Control 181 4.2.3 Derivative Feedback Control 183 4.2.4 Proportional-Integral-Derivative (PID) Control 185 4.2.5 Time-response Sensitivity to Parameters 187 4.2.6 Ziegler-Nichols Tuning of PID Regulators 191 4.2.7 Integrator Antiwindup 196 4.3 Steady-state Tracking and System Type 200 4.3.1 A Special Case of System Type: Unity Feedback 203 4.3.2 System Type with Respect to Disturbance Inputs 206 4.3.3 Truxal's Formula 211 4.4 Stability 212 4.4.1 Bounded Input-Bounded Output Stability 213 4.4.2 Stability of Constant Systems 214 4.4.3 Routh's Stability Criterion 215 Summary 223 Problems 224 5 The Root-locus Design Method 243 A Perspective on the Root-locus Design Method 243 Chapter Overview 244 5.1 Root Locus of a Basic Feedback System 244 5.2 Guidelines for Sketching a Root Locus 249 5.3 Selected Illustrative Root Loci 260 5.4 Other Root-locus Usage 281 5.4.1 Loci versus Other Parameters 282 5.4.2 Zero-degree Loci for Negative Parameters 285

XVI Contents 5.5 Selecting Gain from the Root Locus 289 5.6 Dynamic Compensation 292 5.6.1 Lead Compensation 293 5.6.2 Lag Compensation 298 5.7 Extensions of the Root-locus Method 300 5.7.1 Time Delay 300 5.7.2 Nonlinear Systems 304 5.8 Computer-aided Determination of the Root Locus 310 Summary 318 Problems 320 О The Frequency-response Design Method 337 A Perspective on the Frequency-response Design Method 337 Chapter Overview 338 6.1 Frequency Response 338 6.1.1 Bode Plot Techniques 345 6.1.2 Steady-state Errors 357 6.2 Stability 359 6.3 The Nyquist Stability Criterion 361 6.4 Stability Margins 375 6.5 Bodes Gain-phase Relationship 383 6.6 Closed-loop Frequency Response 388 6.7 Compensation 389 6.7.1 PD Compensation 389 6.7.2 Lead Compensation 390 6.7.3 PI Compensation 400 6.7.4 Lag Compensation 401 6.7.5 PID Compensation 407 6.8 Alternate Presentations of Data 412 6.8.1 Nichols Chart 412 6.8.2 Inverse Nyquist 417 6.9 Sensitivity 418 6.9.1 Sensitivity Functions 420 6.9.2 Stability Robustness 427 6.10 Time Delay 432

Contents XV11 6.11 Obtaining a Pole-zero Model from Frequency-response Data 435 Summary 438 Problems 440 State-space Design 469 A Perspective on State-space Design 469 Chapter Overview 470 7.1 Advantages of State Space 470 7.2 Analysis of the State Equations 472 7.2.1 Block Diagrams and Canonical Forms 472 7.2.2 Dynamic Response from the State Equations 485 7.3 Control-law Design for Full State Feedback 493 7.3.1 Finding the Control Law 494 7.3.2 Introducing the Reference Input with Full State Feedback 500 7.4 Selection of Pole Locations for Good Design 505 7.4.1 Dominant Second-order Poles 505 7.4.2 Prototype Design 507 7.4.3 Symmetric Root Locus 510 7.4.4 Comments on the Methods 514 7.5 Estimator Design 515 7.5.1 Full-order Estimators 515 7.5.2 Reduced-order Estimators 521 7.5.3 Estimator Pole Selection 525 7.6 Compensator Design: Combined Control Law and Estimator 527 7.7 Introduction of the Reference Input 541 7.7.1 A General Structure for the Reference Input 542 7.7.2 Calculation of the System Zeros 545 7.7.3 Selecting the Gain 550 7.8 Integral Control and Robust Tracking 551 7.8.1 Integral Control 552 7.8.2 Robust Tracking Control: the Error-space Approach 554 7.8.3 Disturbance Rejection by Disturbance Estimation 560 7.9 Direct Design with Rational Transfer Functions 564 7.10 Design for Systems with Pure Time Delay 568

XVÜi Contents 7.11 Lyapunov Stability 573 Summary 578 Problems 581 8 Digital Control 601 A Perspective on Digital Control 601 Chapter Overview 601 8.1 Digitization 602 8.1.1 Eulers Method 604 8.1.2 Digitization Using CACSD Software 607 8.2 Dynamic Analysis of Discrete Systems 609 8.2.1 г-transform 609 8.2.2 z-transform Inversion 610 8.2.3 Relationship between s and z 613 8.2.4 Final Value Theorem 616 8.3 Design by Emulation 617 8.3.1 Digitization Procedures 617 8.3.2 Design Example 623 8.3.3 Applicability Limits of the Emulation Design Method 625 8.4 Discrete Design 626 8.4.1 Analysis Tools 626 8.4.2 Feedback Properties 628 8.4.3 Discrete Design Example 629 8.4.4 Discrete Analysis of Designs 632 8.5 State-space Design Methods 634 8.6 Hardware Characteristics 641 8.6.1 Analog-to-Digital (A/D) Converters 641 8.6.2 Digital-to-Analog (D/A) Converters 642 8.6.3 Analog PrefiIters 642 8.6.4 The Computer 643 8.7 Word-size Effects 644 8.7.1 Random Effects 644 8.7.2 Systematic Effects 646

Contents XIX Sample-rate Selection 647 8.8. f Tracking Effectiveness 647 8.8.2 Disturbance Rejection 648 8.8.3 Control Systems Design Methodology Summary 650 Problems 652 649 У Control Systems Design: Principles and Case Studies 663 A Perspective on Design Principles 663 Chapter Overview 663 9.1 An Outline of Control Systems Design 664 9.2 Design of a Satellite's Attitude Control 669 9.3 Lateral and Longitudinal Control of a Boeing 747 684 9.3.1 Yaw Damper 685 9.3.2 Altitude-hold Autopilot 693 9.4 Control of the Fuel-air Ratio in an Automotive Engine 700 9.5 Control of a Digital Tape Transport 707 Summary 719 Problems 721 Appendix A Laplace Transforms 733 Appendix В A Review of Complex Variables 736 B. 1 Definition of a Complex Number 736 B.2 Algebraic Manipulations 737 B.3 Graphical Evaluation of Magnitude and Phase B.4 Differentiation and Integration 740 B.5 Euler's Relations 741 B.6 Analytic Functions 741 B.7 Cauchy's Theorem 742 B.8 Singularities and Residues 742 B.9 Residue Theorem 743 B. 10 The Argument Principle 743 739

XX Contents Appendix С Summary of Matrix Theory 746 C.l Matrix Definitions 746 С 2 Elementary Operations on Matrices 746 C.3 Trace 747 С4 Transpose 747 C.5 Determinant and Matrix Inverse 748 C.6 Properties of the Determinant 749 С 7 Inverse of Block Triangular Matrices 750 C.8 Special Matrices 750 C.9 Rank 751 C. 10 Characteristic Polynomial 751 С11 Cayley-Hamilton Theorem 751 C.12 Eigenvalues and Eigenvectors 752 С13 Similarity Transformations 752 С14 Matrix Exponential 753 С15 Fundamental Subspaces 754 C.16 Singular-value Decomposition 754 C.17 Positive Definite Matrices 755 Appendix D Controllability and Observability 756 Appendix E Ackermann's Formula for Pole Placement 762 Appendix F CACSD Cross-references 766 References 767 Index 773