FEEDBACK CONTROL SYSTEMS: A Fast-Track Guide for Scientists and Engineers FEEDBACK CONTROL SYSTEMS: A F ast-Track Guide for Scientists and Engineers by ALEX ABRAMOVICI, Ph.D. JAKE CHAPSKY, P.E. Jet Propulsion Laboratory, California Institute of Technology SPRINGER SCIENCE+BUSINESS MEDIA, LLC Library of Congress Cataloging-in-Publication Data Abramovici, Alex. Feedback control systems: a fast-track guide for scientists and engineers / by Alex Abramovici, JaIce Chapsky. p.em. ISBN 978-1-4613-6952-3 ISBN 978-1-4615-4345-9 (eBook) DOI 10.1007/978-1-4615-4345-9 1. Feedback control systems. 1. Chapsky, Jake. ll. Title. TJ216 .A2? 2000 629.8'3-dc21 00-057785 Copyright © 2000 Springer Science+Business Media New York Originally published by Kluwer Academic Publishers, New York in 2000 Softcover reprint of the hardcover 1s t edition 2000 All rights reserved. No part of this publieation may be reprodueed, stored in a retrieval system or transmitted in any form or by any means, mechanical, photo copying, recording, or otherwise, without the prior written permission of the publisher, Springer Science+Business Media, LLC. Printed on acid-free paper. To our parents. Contents Summary of Notations x List of Examples x List of Figures xi Preface xv Acknowledgments xxiii Part I INTRODUCTION TO FEEDBACK CONTROL SYSTEMS 1. CONTROL SYSTEM DIAGRAMS 3 2. SIGNALS IN THE FEEDBACK LOOP 9 2.1 Relationships between Signals in a Loop 9 2.2 Simplified FCS Diagrams 13 2.3 Actuator Output Range 15 3. STABILITY 17 3.1 Differential Equations, Laplace Transforms and Stability 17 3.2 The Nyquist Stability Criterion and Bode Diagrams 19 3.3 System Stability: Intuitive Approach 26 4. EXAMPLES 35 4.1 Camera for Aircraft Tracking 35 4.1.1 Camera: Range and Lock Acquisition 35 4.1.2 Electrical Motors as Actuators 36 4.2 Nd:YAG Laser Frequency Stabilization 39 4.2.1 Free Running Laser Frequency Noise 40 4.2.2 System Concept 41 4.2.3 Tracking Requirement 46 vii viii FEEDBACK CONTROL SYSTEMS 4.2.4 Environmental Parameters 47 4.2.5 In-Band, Out-of-Band Frequency Ranges 47 Part II DESIGN AND IMPLEMENTATION 5. DESIGN PRINCIPLES 51 5.1 Design Approach 51 5.1.1 Assumptions 52 5.1.2 Error Budget 53 5.1.3 Design Sequence 54 5.2 Input Data for FCS Design 57 6. DESIGN AND TROUBLESHOOTING 59 6.1 Sensor Specification 60 6.1.1 Sensor Range 60 6.1.2 Sensor Error 62 6.1.3 Sensor Transfer Function 64 6.1.4 Sensor Nonlinearity 66 6.2 Actuator Specification 67 6.3 Shaping the Open-Loop Response 68 6.4 Compensator Specification 72 6.4.1 Compensator Frequency Response Specification 72 6.4.2 Compensator Hardware Specification 72 6.5 Lock Acquisition 75 6.6 System Integration: Making It All Work 78 6.6.1 Achieving Closed-Loop Operation 79 6.6.2 Measuring the Open-Loop Frequency Response 89 6.6.3 Measuring the Free-Running Variable 91 6.6.4 Evaluating Tracking Performance 91 6.7 Lock Acquisition Efficiency 95 6.8 Refining the system 96 7. MULTIPLE SIGNAL PATHS 99 7.1 The Need for Multiple Paths: Examples 101 7.1.1 Parallel Actuators for Laser Frequency Noise Suppression 101 7.1.2 Parallel Low Frequency Gain Boost 105 7.1.3 Two Actuators with Nested Loops for Keeping the Optical Path Constant 108 7.2 Parallel and Nested Loops: Equivalence and Stability 110 7.2.1 Equivalence 110 7.2.2 Stability Criterion 113 7.3 The PID Compensator 114 7.4 Choosing a Multiple-Path Configuration 116 Contents ix 8. DIGITAL COMPENSATORS 119 8.1 When Should One Use Digital Compensators? 121 8.2 Aspects of Digital Compensator Design 123 8.2.1 AID Converter and Anti-Aliasing Filter 124 8.2.2 Range Matching Amplifier (RMA) 127 8.2.3 Digital Filter Block 127 8.2.4 Digital-to-Analog Converter 132 8.2.5 Smoothening Filter 132 8.3 Step-by-Step Specification Recipe 133 Part III SPECIAL TOPICS 9. ACTIVE NULL MEASUREMENTS 137 10. TWO SENSORS FOR ONE VARIABLE 141 10.1 Stability of Control Systems with Two Sensors 143 10.2 Noise Considerations 144 l1.FLEXIBLE ELEMENTS AND STABILITY 147 11.1 Effect of Structure Flexibility on Loop Response 147 11.2 Resonance-Induced Instability and Ways to Prevent It 150 Appendices 153 A Poles and Zeros 153 B Stability of Operational Amplifiers 159 B.1 OpAmp as Feedback Amplifier 160 B.2 Unity Gain Stability and Compensation 164 B.3 Driving Capacitive Loads 165 B.4 Input Capacitance and Photodiode Preamplifiers 167 C Quantization Error 171 C.1 Assumptions and Notations 171 C.2 Quantization Error Suppression 173 C.2.1 Signal/Error Ratio with Dithering and Filtering 173 C.2.2 Signal Recovery 174 C.3 Example 176 Index 179 x FEEDBACK CONTROL SYSTEMS Summary of Notations Variables Subscripts x(t): variable, X(s): its Laplace transform fr: free-running X(J): frequency domain representation of x(t) CI: closed-loop e(t),E(s),E(J): electronic signals (Voltages) 0: output A( s): transfer function, A(jw): frequency response d: disturbance s = 17 + jw: Laplace variable r: reference 17: damping, real part of s G: compensator w: 27rJ c: correction f: frequency or Fourier frequency s: sensor N(J): amplitude spectral density of noise List of Examples Example Page Tracking camera 35 Laser frequency noise suppression 39 Optical path stabilization 108 Two-sensor motion control 142 Control-structure interaction 147 Amplifier with capacitive load 165 Photodiode preamplifier 167 List of Figures 1.1 General schematic representation of a feedback con- trol system. 5 1.2 Algebraic meaning of blocks in feedback control system diagrams. 6 1.3 Example of equivalent blocks. 7 2.1 Schematic representation of a feedback control sys- tem, showing the signals and the noise contributions. 11 2.2 Simplified diagrams of reference-following (track- ing) and disturbance suppression feedback systems. 14 3.1 Nyquist diagram. 20 3.2 The concepts of phase and gain margins. 20 3.3 Example of Bode plot. 21 3.4 Connection between phase margin and time do- main behavior of a closed-loop system. 23 3.5 Example of asymptotic Bode diagram. 25 3.6 Diagram of feedback system used for disturbance suppression. 26 3.7 Forcing function used to examine system stability. 27 3.8 Bode plot of an open-loop transfer function with 0.10 phase deficit. 28 3.9 System response to a 4s pulse (0.25 Hz bandwidth). 29 3.10 System response to a 0.5 s pulse (2 Hz bandwidth). 30 3.11 Bode diagram of system with 73 with 0.10 phase deficit. 31 3.12 Input signal used to illustrate the effect of a large phase deficit at the unity gain frequency. 32 3.13 Instability build-up for a system with large phase deficit. 33 4.1 Example of a tracking system. 37 Xl xu FEEDBACK CONTROL SYSTEMS 4.2 Azimuth-elevation pointing for CCD camera. 38 4.3 Effect of current/voltage drive for a motor. 39 4.4 Example of upper limit for the free-running fre- quency noise of a 1 W monolithic Nd:YAG laser. 40 rv 4.5 Concept of laser frequency noise suppression system. 42 4.6 Essential features of the Pound-Drever frequency noise sensor. 43 4.7 Sample Bode plot of output characteristic from a Pound-Drever-Hall frequency fluctuation sensing arrangement. 45 4.8 Sample Bode plots for frequency-tuning devices in a typical monolithic Nd:YAG laser. 46 5.1 Schematic representation of the FCS design process. 55 6.1 Multiple-stage sensor configuration. 64 6.2 Example of lower bound of open-loop gain for laser frequency noise suppression. 69 6.3 Example of "optimum" open loop gain. 70 6.4 Concept of induced lock acquisition arrangement. 77 6.5 Use of a network analyzer for measuring the fre- quency response of an amplifier stage. 80 6.6 Modified version of Fig. 2.1, relevant to trouble- shooting and testing FCS performance. 82 6.7 Build-up of oscillation as a result of closed loop instability, seen at the sensor output. 84 6.8 Example of system that goes unstable when the overall gain is either too high or too low. 85 6.9 Sensor output for insufficient open-loop gain. 87 6.10 Lag-lead circuit used to increase open-loop gain at low frequencies. 88 6.11 Time record of actuator driver output ec(t) for a situation where tracking fails as a result of insuffi- cient actuator range. 89 6.12 Use of an external sensor for testing tracking performance. 94 7.1 Diagram of Fig. 2.1, modified for the discussion of multiple control paths. 100 7.2 Modified version of the laser frequency noise sup- pression arrangement shown in Fig. 4.5. 101 7.3 Example of upper limit to the spectrum of fre- quency fluctuations in a free-running laser. 103 7.4 Closed-loop system with two parallel signal paths. 104