Feedback Control of Dynamic Systems

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THIRD EDITION Feedback Control of Dynamic Systems Gene F. Franklin Stanford University J. David Powell Stanford University Abbas Emami-Naeini Integrated Systems, Inc.

1 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.

2 Contents 1 An Overview and Brief History of Feedback Control 1 A Perspective on Feedback Control 1 Chapter Overview A Simple Feedback System A First Analysis of Feedback A Brief History An Overview of the Book 13 Summary 15 Problems 15 2 Dynamic Models 19 A Perspective on Dynamic Models 19 Chapter Overview Dynamics of Mechanical Systems Differential Equations in State-variable Form Models of Electric Circuits Models of Electromechanical Systems Heat and Fluid-flow Models Heat Flow Incompressible Fluid Flow 55

$1 The Laplace Transform 86 3.1.1 Response by Convolution 86 3.1.2 Transfer Functions 89 3.1.3 The jsp-laplace Transform 92 3.1.4 Properties of Laplace Transforms 95 3.1.5 Partial-fraction Expansion 98 3.$

3 XIV Contents 2.6 Linearization and Scaling Small-signal Linearization Linearization by Feedback Amplitude Scaling Time Scaling 67 Summary 68 Problems 70 J Dynamic Response 85 A Perspective on System Response 85 Chapter Overview The Laplace Transform Response by Convolution Transfer Functions The jsp-laplace Transform Properties of Laplace Transforms Partial-fraction Expansion Laplace Transform Theorems Using Laplace Transforms to Solve Problems Laplace Transforms Using CACSD Software System Modeling Diagrams The Block Diagram Mason's Rule and the Signal-flow Graph Response versus Pole Locations Time-domain Specifications Effects of Zeros and Additional Poles Numerical Simulation Solution of Nonlinear Differential Equations Solution of Linear Differential Equations Obtaining Models from Experimental Data Models from Transient-response Data 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.

4 Contents XV т" Basic Properties of Feedback 167 A Perspective on Properties of Feedback 167 Chapter Overview A Case Study of Speed Control Disturbance Rejection Sensitivity: Effects of Gain Changes Dynamic Tracking The Classical Three-term Controller Proportional Feedback Control Proportional-Integral (PI) Feedback Control Derivative Feedback Control Proportional-Integral-Derivative (PID) Control Time-response Sensitivity to Parameters Ziegler-Nichols Tuning of PID Regulators Integrator Antiwindup Steady-state Tracking and System Type A Special Case of System Type: Unity Feedback System Type with Respect to Disturbance Inputs Truxal's Formula Stability Bounded Input-Bounded Output Stability Stability of Constant Systems Routh's Stability Criterion 215 Summary 223 Problems The Root-locus Design Method 243 A Perspective on the Root-locus Design Method 243 Chapter Overview Root Locus of a Basic Feedback System Guidelines for Sketching a Root Locus Selected Illustrative Root Loci Other Root-locus Usage Loci versus Other Parameters 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.

5 XVI Contents 5.5 Selecting Gain from the Root Locus Dynamic Compensation Lead Compensation Lag Compensation Extensions of the Root-locus Method Time Delay Nonlinear Systems 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 Frequency Response Bode Plot Techniques Steady-state Errors Stability The Nyquist Stability Criterion Stability Margins Bodes Gain-phase Relationship Closed-loop Frequency Response Compensation PD Compensation Lead Compensation PI Compensation Lag Compensation PID Compensation Alternate Presentations of Data Nichols Chart Inverse Nyquist Sensitivity Sensitivity Functions Stability Robustness 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.

6 Contents XV 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 Advantages of State Space Analysis of the State Equations Block Diagrams and Canonical Forms Dynamic Response from the State Equations Control-law Design for Full State Feedback Finding the Control Law Introducing the Reference Input with Full State Feedback Selection of Pole Locations for Good Design Dominant Second-order Poles Prototype Design Symmetric Root Locus Comments on the Methods Estimator Design Full-order Estimators Reduced-order Estimators Estimator Pole Selection Compensator Design: Combined Control Law and Estimator Introduction of the Reference Input A General Structure for the Reference Input Calculation of the System Zeros Selecting the Gain Integral Control and Robust Tracking Integral Control Robust Tracking Control: the Error-space Approach Disturbance Rejection by Disturbance Estimation Direct Design with Rational Transfer Functions 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.

7 XVÜi Contents 7.11 Lyapunov Stability 573 Summary 578 Problems Digital Control 601 A Perspective on Digital Control 601 Chapter Overview Digitization Eulers Method Digitization Using CACSD Software Dynamic Analysis of Discrete Systems г-transform z-transform Inversion Relationship between s and z Final Value Theorem Design by Emulation Digitization Procedures Design Example Applicability Limits of the Emulation Design Method Discrete Design Analysis Tools Feedback Properties Discrete Design Example Discrete Analysis of Designs State-space Design Methods Hardware Characteristics Analog-to-Digital (A/D) Converters Digital-to-Analog (D/A) Converters Analog PrefiIters The Computer Word-size Effects Random Effects Systematic Effects 646

Contents XIX Sample-rate Selection 647 8.

8 Contents XIX Sample-rate Selection f Tracking Effectiveness Disturbance Rejection Control Systems Design Methodology Summary 650 Problems У Control Systems Design: Principles and Case Studies 663 A Perspective on Design Principles 663 Chapter Overview An Outline of Control Systems Design Design of a Satellite's Attitude Control Lateral and Longitudinal Control of a Boeing Yaw Damper Altitude-hold Autopilot Control of the Fuel-air Ratio in an Automotive Engine 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

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.

9 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

1 An Overview and Brief History of Feedback Control 1. 2 Dynamic Models 23. Contents. Preface. xiii

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 3 1.2 A First Analysis of Feedback 6 1.3 Feedback System