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

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1 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 Feedback System Fundamentals A Brief History 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 Dynamics of Mechanical Systems Translational Motion Rotational Motion Combined Rotation and Translation Complex Mechanical Systems (W)** Distributed Parameter Systems Summary: Developing Equations of Motion for Rigid Bodies Models of Electric Circuits Models of Electromechanical Systems Loudspeakers Motors Gears Heat and Fluid-Flow Models Heat Flow Incompressible Fluid Flow Historical Perspective 68 Summary 71 Review Questions 71 Problems 72 Preface xiii **Sections with (W) indicates that additional material is located on the web at v A00_FRAN6598_07_SE_FM 2014/3/17 15:15 page v #5

2 vi Contents 3 Dynamic Response 84 A Perspective on System Response 84 Chapter Overview Review of Laplace Transforms Response by Convolution Transfer Functions and Frequency Response The L Laplace Transform Properties of Laplace Transforms Inverse Laplace Transform by Partial-Fraction Expansion The Final Value Theorem Using Laplace Transforms to Solve Differential Equations Poles and Zeros Linear System Analysis Using Matlab R System Modeling Diagrams The Block Diagram Block-Diagram Reduction Using Matlab Mason s Rule and the Signal Flow Graph (W) Effect of Pole Locations Time-Domain Specifications Rise Time Overshoot and Peak Time Settling Time Effects of Zeros and Additional Poles Stability Bounded Input Bounded Output Stability Stability of LTI Systems Routh s Stability Criterion Obtaining Models from Experimental Data: System Identification (W) Amplitude and Time Scaling (W) Historical Perspective 156 Summary 157 Review Questions 159 Problems A First Analysis of Feedback 180 A Perspective on the Analysis of Feedback 180 Chapter Overview The Basic Equations of Control Stability Tracking Regulation Sensitivity 186 A00_FRAN6598_07_SE_FM 2014/3/17 15:15 page vi #6

3 Contents vii 4.2 Control of Steady-State Error to Polynomial Inputs: System Type System Type for Tracking System Type for Regulation and Disturbance Rejection The Three-Term Controller: PID Control Proportional Control (P) Integral Control (I) Derivative Control (D) Proportional Plus Integral Control (PI) PID Control Ziegler Nichols Tuning of the PID Controller Feedforward Control by Plant Model Inversion Introduction to Digital Control (W) Sensitivity of Time Response to Parameter Change (W) Historical Perspective 215 Summary 217 Review Questions 218 Problems The Root-Locus Design Method 234 A Perspective on the Root-Locus Design Method 234 Chapter Overview Root Locus of a Basic Feedback System Guidelines for Determining a Root Locus Rules for Determining a Positive (180 ) Root Locus Summary of the Rules for Determining a Root Locus Selecting the Parameter Value Selected Illustrative Root Loci Design Using Dynamic Compensation Design Using Lead Compensation Design Using Lag Compensation Design Using Notch Compensation Analog and Digital Implementations (W) A Design Example Using the Root Locus Extensions of the Root-Locus Method Rules for Plotting a Negative (0 ) Root Locus Consideration of Two Parameters Time Delay (W) Historical Perspective 287 A00_FRAN6598_07_SE_FM 2014/3/17 15:15 page vii #7

4 viii Contents Summary 289 Review Questions 290 Problems The Frequency-Response Design Method 308 A Perspective on the Frequency-Response Design Method 308 Chapter Overview Frequency Response Bode Plot Techniques Steady-State Errors Neutral Stability The Nyquist Stability Criterion The Argument Principle Application of The Argument Principle to Control Design Stability Margins Bode s Gain Phase Relationship Closed-Loop Frequency Response Compensation PD Compensation Lead Compensation (W) PI Compensation Lag Compensation PID Compensation Design Considerations Specifications in Terms of the Sensitivity Function Limitations on Design in Terms of the Sensitivity Function Time Delay Time Delay via the Nyquist Diagram (W) Alternative Presentation of Data Nichols Chart The Inverse Nyquist Diagram (W) Historical Perspective 404 Summary 405 Review Questions 408 Problems State-Space Design 433 A Perspective on State-Space Design 433 Chapter Overview Advantages of State-Space System Description in State-Space Block Diagrams and State-Space 442 A00_FRAN6598_07_SE_FM 2014/3/17 15:15 page viii #8

5 Contents ix 7.4 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 Symmetric Root Locus (SRL) Comments on the Methods Estimator Design Full-Order Estimators Reduced-Order Estimators Estimator Pole Selection Compensator Design: Combined Control Law and Estimator (W) Introduction of the Reference Input with the Estimator (W) General Structure for the Reference Input Selecting the Gain Integral Control and Robust Tracking Integral Control Robust Tracking Control: The Error-Space Approach Model-Following Design The Extended Estimator Loop Transfer Recovery Direct Design with Rational Transfer Functions Design for Systems with Pure Time Delay Solution of State Equations (W) Historical Perspective 559 Summary 562 Review Questions 565 Problems Digital Control 590 A Perspective on Digital Control 590 Chapter Overview Digitization Dynamic Analysis of Discrete Systems z-transform z-transform Inversion 595 A00_FRAN6598_07_SE_FM 2014/3/17 15:15 page ix #9

6 x Contents Relationship Between s and z Final Value Theorem Design Using Discrete Equivalents Tustin s Method Zero-Order Hold (ZOH) Method Matched Pole Zero (MPZ) Method Modified Matched Pole Zero (MMPZ) Method Comparison of Digital Approximation Methods Applicability Limits of the Discrete Equivalent Design Method Hardware Characteristics Analog-to-Digital (A/D) Converters Digital-to-Analog Converters Anti-Alias Prefilters The Computer Sample-Rate Selection Tracking Effectiveness Disturbance Rejection Effect of Anti-Alias Prefilter Asynchronous Sampling Discrete Design Analysis Tools Feedback Properties Discrete Design Example Discrete Analysis of Designs Discrete State-Space Design Methods (W) Historical Perspective 628 Summary 629 Review Questions 631 Problems Nonlinear Systems 637 A Perspective on Nonlinear Systems 637 Chapter Overview Introduction and Motivation: Why Study Nonlinear Systems? Analysis by Linearization Linearization by Small-Signal Analysis Linearization by Nonlinear Feedback Linearization by Inverse Nonlinearity Equivalent Gain Analysis Using the Root Locus Integrator Antiwindup 655 A00_FRAN6598_07_SE_FM 2014/3/17 15:15 page x #10

7 Contents xi 9.4 Equivalent Gain Analysis Using Frequency Response: Describing Functions Stability Analysis Using Describing Functions Analysis and Design Based on Stability The Phase Plane Lyapunov Stability Analysis The Circle Criterion Historical Perspective 690 Summary 691 Review Questions 691 Problems Control System Design: Principles and Case Studies 703 A Perspective on Design Principles 703 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 the Read/Write Head Assembly of a Hard Disk Control of RTP Systems in Semiconductor Wafer Manufacturing Chemotaxis or How E. Coli Swims Away from Trouble 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 A00_FRAN6598_07_SE_FM 2014/3/17 15:15 page xi #11

8 xii Contents Appendix B Solutions to the Review Questions 819 Appendix C Matlab Commands 835 Bibliography 840 Index 848 List of Appendices on the web at 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 W : Routh Special Cases Appendix W3.7: System Identification Appendix W3.8: Amplitude and Time Scaling Appendix W : The Filtered Case Appendix W : 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 A00_FRAN6598_07_SE_FM 2014/3/17 15:15 page xii #12

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