Sampling in Digital Signal Processing and Control

569 Pages·1996·7.951 MB·English

by  Arie Feuer, Graham C. Goodwin (auth.)

Systems & Control: Foundations & Applications
Founding Editor
Christopher I. Byrnes, Washington University
Arie Feuer
Graham C. Goodwin
Sampling in
Digital Signal Processing
and Control
1996
Birkhauser
Boston • Basel • Berlin
Arie Feuer Graham C. Goodwin
Dept. of Electrical Engineering Faculty of Engineering
Technion-Israel Institute of The University of Newcastle
Technology Callaghan NSW 2308
Haifa 32000 Australia
Israel
Library of Congress Cataloging-in-Publication Data
Feuer, Arie, 1943-
Sampling, in digital signal processing and control / Arie Feuer,
Graham C. Goodwin.
p. cm. -- (Systems & control)
Includes bibliographical references.
ISBN-13: 978-1-4612-7546-6 e-ISBN-13: 978-1-4612-2460-0
DOl: 10.1007/978-1-4612-2460-0
1. Signal processing--Digital techniques. 2. Signal processing-
-Statistical methods. 3. Sampling (Statistics) 4. Control theory.
I. Goodwin, Graham C. (Graham Clifford), 1945- II. Title.
III. Series.
TK5102.9.F48 1996 96-24284
621.382'23--dc20 CIP
Printed on acid-free paper
© 1996 Birkhauser Boston Birkhiiuser
Softcover reprint of the hardcover 1st edition 1996
Copyright is not claimed for works of U.S. Government employees.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording,
or otherwise, without prior permission of the copyright owner.
Permission to photocopy for internal or personal use of specific clients is granted by
Birkhauser Boston for libraries and other users registered with the Copyright Clearance
Center (CCC), provided that the base fee of$6.00 per copy, plus $0.20 per page is paid directly
to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be
addressed directly to Birkhauser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139,
U.S.A.
ISBN -13 : 97 8-1-4612-7546-6
Typeset by the Authors in LATEX.
987 6 5 432 1
Contents
Preface ................................ xiii
Notation ............................... xix
Chapter 1 Fourier Analysis
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Basic Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Fourier Series ................................... 3
1.2.2 The Continuous-Time Fourier Transform . . . . . . . . . . . . . . 4
1.2.3 The Discrete-Time Fourier Transform (DTFT) . . . . . . . . . . 5
1.2.4 The Discrete Fourier Transform (DFT) . . . . . . . . . . . . . . . . 7
1.3 Properties of Continuous-Time Fourier Transforms. . . . . . . . . . . . 8
1.4 Properties of Discrete-Time Fourier Transforms .............. 18
1.5 The A -Impulse Stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.6 Inter-relating the Various Transforms . . . . . . . . . . . . . . . . . . . . . . . 26
1.6.1 Fourier Series Revisited ........................... 26
1.6.2 The Discrete-Time Fourier Transform Revisited ........ 29
1.6.3 The Discrete Fourier Transform Revisited ............. 33
1.6.4 Summary....................................... 36
1. 7 Special Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.7.1 Sampling of Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.7.2 Irregular Sampling in Periodic Patterns ............... 51
vi Contents
1.8 Further Reading and Discussion ........................... 59
1.9 Problems ............................................. 61
Chapter 2 Sampling and Reconstruction
2.1 Introduction........................................... 71
2.2 Sampled Data Sequences - A Representation of
Continuous Signals ..................................... 71
2.3 Continuous Signal Reconstruction from a Sampled
Data Sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.4 Shannon's Reconstruction Theorem ........................ 75
2.5 Practical Methods of Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 78
2.5.1 Zero-Order-Hold (ZOH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.5.2 First-Order-Hold (FOH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.6 Signal Reconstruction from Periodic Samples ................ 82
2.7 Further Reading and Discussion ........................... 103
2.8 Problems ............................................. 103
Chapter 3 Analysis of Discrete-Time Systems
3.1 Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 109
3.2 Shift Operator Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 109
3.3 z-Transforms.......................................... 112
3.4 The Delta Operator ..................................... 114
3.5 Difference Equations in Delta Operator Form. . . . . . . . . . . . . . . .. 119
3.6 Discrete Delta Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 122
3.7 Use of Discrete Delta Transforms to Solve Difference Equations 126
3.8 The Discrete Transfer Function. . . . . . . . . . . . . . . . . . . . . . . . . . .. 130
Contents vii
3.9 Summary of Delta Transform Properties .................... 130
3.10 Stability of Discrete Systems ............................. 133
3.11 Discrete Frequency Response. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 135
3.12 Frequency Domain Stability Criteria for Discrete-Time Systems. 138
3.13 Digital Filter Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 139
3.14 Further Reading and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . .. 142
3.15 Problems............................................. 146
Chapter 4 Discrete-Time Models of Continuous
Deterministic Systems
4.1 Introduction........................................... 151
4.2 State-Space Development. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 152
4.3 Transform Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 156
4.4 Continuous-Time and Discrete-Time Poles and Zeros . . . . . . . . .. 164
4.4.1 Poles .......................................... 164
4.4.2 Zeros .......................................... 168
4.5 Numerical Issues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 172
4.6 Frequency Domain Development . . . . . . . . . . . . . . . . . . . . . . . . .. 175
4.7 Further Reading and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . .. 182
4.8 Problems............................................. 182
Chapter 5 Optimal Linear Estimation with Finite
Impulse Response Filters
5.1 Introduction........................................... 185
5.2 Problem Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 186
5.3 Sampled Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 187
viii Contents
5.4 The Discrete Lattice Filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 189
5.5 Continuous-Time Lattice Structure. . . . . . . . . . . . . . . . . . . . . . . .. 201
5.6 Relationships between the Discrete and Continuous Lattice Filters 203
5.7 Further Reading and Discussion ........................... 206
5.8 Problems ............................................. 206
Chapter 6 Optimal Linear Estimation with
State-Space Filters
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 209
6.2 Signal Model .......................................... 209
6.3 The Sampling Process ................................... 211
6.4 Discrete Stochastic Model. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . .. 213
6.5 The Discrete Kalman Filter .... . . . . . . . . . . . . . . . . . . . . . . . . . .. 217
6.5.1 Model Simplification .............................. 217
6.5.2 The Optimal Filter ................................ 218
6.5.3 Relationship to Finite Impulse Response Filters ......... 226
6.6 Continuous-Time State Estimation ......................... 229
6.6.1 Continuous-Time Data .. . . . . . . . . . . . . . . . . . . . . . . . . . .. 229
6.6.2 Sampled Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 230
6.6.3 Relationships to Shannon Reconstruction Theorem. . . . . .. 232
6.7 Further Reading and Discussion ........................... 235
6.8 Problems ............................................. 235
Chapter 7 Periodic and Multirate Filtering
7.1 Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 239
7.2 Models for Periodic Linear Systems ........................ 240
7.3 The Raising Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 242
Contents ix
7.4 Frequency Domain Analysis of Periodic Filters. . . . . . . . . . . . . .. 251
7.5 Models for Sampled Periodic Stochastic Systems . . . . . . . . . . . .. 256
7.6 Periodic Optimal Filtering ............................... 259
7.7 Further Reading and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .. 268
7.8 Problems............................................. 270
Chapter 8 Discrete-Time Control
8.1 Introduction........................................... 273
8.2 Closed-Loop Stability and Pole Assignment ................. 280
8.3 Some Special Discrete-Time Control Laws ........... . . . . . .. 285
8.3.1 Deadbeat Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 285
8.3.2 Model Reference Control .......................... 285
8.3.3 Minimal Prototype Control .. . . . . . . . . . . . . . . . . . . . . . .. 287
8.4 Sensitivity and Complementary Sensitivity Functions . . . . . . . . .. 291
8.5 All Stabilizing Control Laws ............................. 294
8.5.1 Open Loop Stable Plants . . . . . . . . . . . . . . . . . . . . . . . . . .. 295
8.5.2 Open-Loop Unstable Plants. . . . . . . . . . . . . . . . . . . . . . . .. 297
8.6 State Estimate Feedback. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 308
8.7 Rapprochement Between State Estimate Feedback and
All Stabilizing Controllers ............................... 314
8.7.1 Implicit Disturbance Modelling ..................... 314
8.7.2 Explicit Disturbance Modelling ..................... 318
8.8 Linear Quadratic Optimal Regulator ....................... 323
8.9 Duality Relationships ................................... 326
8.9.1 Filtering as a Control Problem ...................... 327
8.9.2 Associations..................................... 329
8.10 Further Reading and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . .. 329
8.11 Problems ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 331
x Contents
Chapter 9 Sampled Data Control
9.1 Introduction................ . . . . . . . . . . . . . . . . . . . . . . . . . .. 343
9.2 Mixing Continuous and Discrete Transfer Functions ........... 344
9.3 Sensitivity Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 349
9.4 Modified Discrete Transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 353
9.5 Examples............................................. 355
9.5.1 Servo System with Minimal Prototype Controller. . . . . . .. 355
9.5.2 Servo System with Dead-Beat Control. . . . . . . . . . . . . . . .. 360
9.5.3 Resonant System #1 ............................... 363
9.5.4 Resonant System #2 with Anti-aliasing Filter ........... 366
9.6 Observations and Comments from the Examples .............. 370
9.7 The Class of All Stabilizing Sampled-Data Controllers ......... 371
9.8 Linear Quadratic Design of Sampled-Data Controllers. . . . . . . . .. 374
9.9 Duality Relationships for Hybrid Optimal Controller. . . . . . . . . .. 381
9.9.1 Review of Hybrid Optimal Controller ................. 381
9.9.2 Review of Hybrid Optimal Filter . . . . . . . . . . . . . . . . . . . .. 383
9.9.3 Duality ......................................... 384
9.10 Further Reading and Discussion ........................... 390
9.11 Problems ............................................. 392
Chapter 10 Generalized Sample-Hold Functions
10.1 Introduction ........................................... 397
10.2 Generalized Sample-Hold Function: A Time Domain Perspective 398
10.3 Other Applications of Generalized Sample-Hold Functions. . . . .. 414
10.4 Frequency Domain Analysis ofGSHF ...................... 417
10.5 Sensitivity Considerations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 427
10.6 Further Reading and Discussion ........................... 432