Analysis and Control of Linear Systems - Chapter 0 pptx

Analysis and Control of Linear Systems




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Analysis and Control
of Linear Systems











Edited by
Philippe de Larminat

First published in France in 2002 by Hermès Science/Lavoisier entitled “Analyse des
systèmes linéaires” and “Commande des systèmes linéaires”
First published in Great Britain and the United States in 2007 by ISTE Ltd
Apart from any fair dealing for the purposes of research or private study, or criticism or
review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may
only be reproduced, stored or transmitted, in any form or by any means, with the prior
permission in writing of the publishers, or in the case of reprographic reproduction in
accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction
outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd ISTE USA
6 Fitzroy Square
4308 Patrice Road
London W1T 5DX
Newport Beach, CA 92663
UK USA
www.iste.co.uk


© ISTE Ltd, 2007

© LAVOISIER, 2002
The rights of Philippe de Larminat to be identified as the author of this work have been
asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

[Analyse des systèmes linéaires/Commande des systèmes linéaires. eng] Analysis and control
of linear systems analysis and control of linear systems/edited by Philippe de Larminat.
p. cm.
ISBN-13: 978-1-905209-35-4
ISBN-10: 1-905209-35-5
1. Linear control systems. 2. Automatic control. I. Larminat, Philippe de.
TJ220.A5313 2006
629.8'32 dc22
2006033665

British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 10: 1-905209-35-5
ISBN 13: 978-1-905209-35-4
Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire.
Table of Contents
Preface xv

Part 1. System Analysis 1

Chapter 1. Transfer Functions and Spectral Models 3
Dominique BEAUVOIS and Yves TANGUY
1.1. System representation 3
1.2. Signal models 4

1.2.1. Unit-step function or Heaviside step function U(t) 4
1.2.2. Impulse 4
1.2.3. Sine-wave signal 7
1.3. Characteristics of continuous systems 7
1.4. Modeling of linear time-invariant systems 8
1.4.1. Temporal model, convolution, impulse response and
unit-step response 8
1.4.2. Causality 9
1.4.3. Unit-step response 10
1.4.4. Stability 10
1.4.5. Transfer function 12
1.4.6. Causality, stability and transfer function 16
1.4.7. Frequency response and harmonic analysis 17
1.5. Main models 21
1.5.1. Integrator 21
1.5.2. First order system 23
1.5.3. Second order system 27
1.6. A few reminders on Fourier and Laplace transforms 33
1.6.1. Fourier transform 33
1.6.2. Laplace transform 34
1.6.3. Properties 38

vi Analysis and Control of Linear Systems
1.6.4. Laplace transforms of ordinary causal signals 40
1.6.5. Ordinary Fourier transforms 41
1.7. Bibliography 42

Chapter 2. State Space Representation 43
Patrick BOUCHER and Patrick TURELLE
2.1. Reminders on the systems 44

2.1.1. Internal representation of determinist systems: the concept of state 44
2.1.2. Equations of state and equations of measurement for
continuous systems 46
2.1.3. Case of linear systems 47
2.1.4. Case of continuous and invariant linear systems 48
2.2. Resolving the equation of state 48
2.2.1. Free state 48
2.2.2. Forced state 49
2.2.3. Particular case of linear and invariant systems 50
2.2.4. Calculation method of the transition matrix
0
A(t -t )
e
51
2.2.5. Application to the modeling of linear discrete systems 55
2.3. Scalar representation of linear and invariant systems 57
2.3.1. State passage → transfer 57
2.3.2. Change of basis in the state space 60
2.3.3. Transfer passage → state 60
2.3.4. Scalar representation of invariant and linear discrete systems 65
2.4. Controllability of systems 66
2.4.1. General definitions 66
2.4.2. Controllability of linear and invariant systems 66
2.4.3. Canonic representation of partially controllable systems 69
2.4.4. Scalar representation of partially controllable systems 73
2.5. Observability of systems 74
2.5.1. General definitions 74
2.5.2. Observability of linear and invariant systems 74
2.5.3. Case of partially observable systems 77
2.5.4. Case of partially controllable and partially observable systems. . . 78

2.6. Bibliography 79

Chapter 3. Discrete-Time Systems 81
Philippe CHEVREL
3.1. Introduction 81
3.2. Discrete signals: analysis and manipulation 83
3.2.1. Representation of a discrete signal 83
3.2.2. Delay and lead operators 84
3.2.3. z-transform 85

Table of Contents vii
3.3. Discrete systems (DLTI) 88
3.3.1. External representation 88
3.3.2. Internal representation 89
3.3.3. Representation in terms of operator 91
3.3.4. Transfer function and frequency response 96
3.3.5. Time response of basic systems 98
3.4. Discretization of continuous-time systems 99
3.4.1. Discretization of analog signals 100
3.4.2. Transfer function of the discretized system 101
3.4.3. State representation of the discretized system 102
3.4.4. Frequency responses of the continuous and discrete system 103
3.4.5. The problem of sub-sampling 104
3.4.6. The problem of over-sampling 105
3.5. Conclusion 107
3.6. Bibliography 107

Chapter 4. Structural Properties of Linear Systems 109
Michel MALABRE
4.1. Introduction: basic tools for a structural analysis of systems 109

4.1.1. Vector spaces, linear applications 110
4.1.2. Invariant sub-spaces 111
4.1.3. Polynomials, polynomial matrices 113
4.1.4. Smith form, companion form, Jordan form 114
4.1.5. Notes and references 115
4.2. Beams, canonical forms and invariants 115
4.2.1. Matrix pencils and geometry 117
4.2.2. Kronecker’s canonical form 118
4.2.3. Controllable, observable canonical form (Brunovsky) 121
4.2.4. Morse’s canonical form 125
4.2.5. Notes and references 128
4.3. Invariant structures under transformation groups 128
4.3.1. Controllability indices 128
4.3.2. Observability indices 129
4.3.3. Infinite zeros 129
4.3.4. Invariants, transmission finite zeros 131
4.3.5. Notes and references 132
4.4. An introduction to a structural approach of the control 132
4.4.1. Disturbance rejection and decoupling: existence of solutions 133
4.4.2. Disturbance rejection and decoupling: existence
of stable solutions 135
4.4.3. Disturbance rejection and decoupling: flexibility in
the location of poles/fixed poles 135
4.4.4. Notes and references 136

viii Analysis and Control of Linear Systems
4.5. Conclusion 137
4.5.1. Optimal attenuation of disturbance 137
4.6. Bibliography 137

Chapter 5. Signals: Deterministic and Statistical Models 141
Eric LE CARPENTIER
5.1. Introduction 141
5.2. Signals and spectral analysis 141
5.3. Generator processes and ARMA modeling 150
5.4. Modeling of LTI systems and ARMAX modeling 153
5.4.1. ARX modeling 153
5.4.2. ARMAX modeling 154
5.4.3. Output error model 154
5.4.4. Representation of the ARMAX model within the state space 155
5.4.5. Predictor filter associated with the ARMAX model 155
5.5. From the Markovian system to the ARMAX model 156
5.6. Bibliography 157

Chapter 6. Kalman’s Formalism for State Stabilization
and Estimation 159
Gilles DUC
6.1. The academic problem of stabilization through state feedback 159
6.2. Stabilization by pole placement 161
6.2.1. Results 161
6.2.2. Example 163
6.3. Reconstruction of state and observers 164
6.3.1. General principles 164
6.3.2. Continuous-time observer 165
6.3.3. Discrete-time observer 166
6.3.4. Calculation of the observer by pole placement 167
6.3.5. Behavior of the observer outside the ideal case 168
6.3.6. Example 169
6.4. Stabilization through quadratic optimization 171
6.4.1. General results for continuous-time 171

6.4.2. General results for discrete-time 173
6.4.3. Interpretation of the results 174
6.4.4. Example 175
6.5. Resolution of the state reconstruction problem by duality of
the quadratic optimization 177
6.5.1. Calculation of a continuous-time observer 177
6.5.2. Calculation of a discrete-time observer 178
6.5.3. Interpretation in a stochastic context 179
6.5.4. Example 181

Table of Contents ix
6.6. Control through state feedback and observers 183
6.6.1. Implementation of the control 183
6.6.2. Dynamics of the looped system 184
6.6.3. Interest and limitations of this result 185
6.6.4. Interpretation in the form of equivalent corrector 186
6.6.5. Example 187
6.7. A few words on the resolution of Riccati’s equations 189
6.8. Conclusion 192
6.9. Bibliography 192

Chapter 7. Process Modeling 195
Alain BARRAUD, Suzanne LESECQ and Sylviane GENTIL
7.1. Introduction 195
7.2. Modeling 198
7.3. Graphic identification approached 204
7.3.1. Pseudo-periodic unit-step response 205
7.3.2. Aperiodic unit-step response 207
7.3.3. Partial conclusion 213
7.4. Identification through criterion optimization 214

7.4.1. Algorithms 214
7.4.2. Models 215
7.4.3. Methods 215
7.4.4. Optimization criteria 216
7.4.5. The problem of precision 217
7.4.6. How to optimize 218
7.4.7. Partial conclusion 219
7.4.8. Practical application 220
7.5. Conclusion around an example 222
7.5.1. Simulated procedure 222
7.5.2. In search of a model 223
7.6. Bibliography 226

Chapter 8. Simulation and Implementation of Continuous Time Loops . . 227
Alain BARRAUD and Sylviane GENTIL
8.1. Introduction 227
8.1.1. About linear equations 228
8.1.2. About non-linear equations 228
8.2. Standard linear equations 228
8.2.1. Definition of the problem 228
8.2.2. Solving principle 229
8.2.3. Practical implementation 229
8.3. Specific linear equations 231
8.3.1. Definition of the problem 231

x Analysis and Control of Linear Systems
8.3.2. Solving principle 232
8.3.3. Practical implementation 233
8.4. Stability, stiffness and integration horizon 234
8.5. Non-linear differential systems 235

8.5.1. Preliminary aspects 235
8.5.2. Characterization of an algorithm 236
8.5.3. Explicit algorithms 239
8.5.4. Multi-interval implicit algorithms 240
8.5.5. Solver for stiff systems 242
8.5.6. Partial conclusion 243
8.6. Discretization of control laws 244
8.6.1. Introduction 244
8.6.2. Discretization 244
8.6.3. Application to PID regulators 247
8.7. Bibliography 250

Part 2. System Control
251

Chapter 9. Analysis by Classic Scalar Approach 253
Houria SIGUERDIDJANE and Martial DEMERLÉ
9.1. Configuration of feedback loops 253
9.1.1. Open loop – closed loops 253
9.1.2. Closed loop harmonic analysis 255
9.2. Stability 258
9.2.1. Nyquist criterion 259
9.2.2. Routh’s algebraic criterion 265
9.2.3. Stability margins 267
9.3. Precision 270
9.3.1. Permanent error 272
9.3.2. Transitional error 277
9.4. Parametric sensitivity 278
9.4.1. Open loop sensitivity 278
9.4.2. Closed loop sensitivity 280

9.5. Bibliography 282

Chapter 10. Synthesis of Closed Loop Control Systems 283
Houria SIGUERDIDJANE and Martial DEMERLÉ
10.1. Role of correctors: precision-stability dilemma 283
10.1.1. Analysis of systems’ behavior 284
10.1.2. Serial correction 288
10.1.3. Parallel correction 289
10.1.4. Correction by anticipation 290
10.1.5. Conclusions 292

Table of Contents xi
10.2. Serial correction 293
10.2.1. Correction by phase lead 293
10.2.2. Correction by phase delay 300
10.3. Correction by combined actions 303
10.3.1. Transfer function 303
10.4. Proportional derivative (PD) correction 306
10.4.1. Transfer function 306
10.5. Proportional integral (PI) correction 307
10.5.1. Transfer function 307
10.6. Proportional integral proportional (PID) correction 310
10.6.1. Transfer function 310
10.6.2. Experimental adjustment method 313
10.7. Parallel correction 315
10.7.1. General principle 315
10.7.2. Simple tachymetric correction (C(p) = λp) 317
10.7.3. Filtered tachymetric correction 320
10.7.4. Correction of delay systems: Smith predictor 323
10.8. Bibliography 325

Chapter 11. Robust Single-Variable Control through Pole Placement . . . 327
Gérard THOMAS
11.1. Introduction 327
11.1.1. Guiding principles and notations 327
11.1.2. Reminders on polynomial algebra 329
11.2. The obvious objectives of the correction 332
11.2.1. Internal stability 332
11.2.2. Stationary behavior 333
11.2.3. General formulation 335
11.3. Resolution 336
11.3.1 Resolution of a particular case 337
11.3.2. General case 342
11.4. Implementation 344
11.4.1. First possibility 345
11.4.2. Minimal representation 345
11.4.3. Management of saturations 349
11.5. Methodology 354
11.5.1. Intuitive approach 354
11.5.2. Reduction of the noise on the control by choice of degrees 356
11.5.3. Choice of the dynamics of A
m
and A
o
357
11.5.4. Examples 363
11.6. Conclusion 370
11.7. Bibliography 370

xii Analysis and Control of Linear Systems
Chapter 12. Predictive Control 373
Patrick BOUCHER and Didier DUMUR
12.1. General principles of predictive control 373
12.1.1. Anticipative aspect 373
12.1.2. Explicit prediction of future behavior 374
12.1.3. Optimization by minimization of a quadratic criterion 374
12.1.4. Principle of the sliding horizon 374
12.2. Generalized predictive control (GPC) 376
12.2.1. Formulation of the control law 376
12.2.2. Automatic synthesis of adjustment parameters 380
12.2.3. Extension of the basic version 382
12.3. Functional predictive control (FPC) 389
12.3.1. Definition of numerical model 390
12.3.2. Choice of a reference trajectory 390
12.3.3. Object-model difference 391
12.3.4. Structure of the future control 391
12.3.5. Structure of the optimal predictor 392
12.3.6. Definition of quadratic criterion, concept of match points 393
12.3.7. Adjustment parameters 394
12.4. Conclusion 394
12.5. Bibliography 396

Chapter 13. Methodology of the State Approach Control 399
Philippe CHEVREL
13.1. Introduction 400
13.2. H
2
control 402
13.2.1. Standards 403

13.2.2. H
2
optimization 407
13.2.3. H
2
– LQG 412
13.2.4. H
2
– LTR 414
13.2.5. Generalization of the H
2
standard problem 416
13.2.6. Generalized H
2
problem and robust RPIS 421
13.2.7. Discretization of the H
2
problem 423
13.3. Data of a feedback control problem 426
13.3.1. Model of the process 427
13.3.2. Modeling the environment of the system to adjust 428
13.3.3. Additional data 430
13.4. Standard H
2
optimization problem 431
13.4.1. Construction of the conceptual control model 431
13.4.2. Definition of the H
2
optimization problem 433
13.4.3. The interest in standardization 435

13.4.4. Management of the control compromises 436
13.5. Conclusion 437

Table of Contents xiii
13.6. Appendices 438
13.6.1. Resolution of the Lyapunov equations 438
13.6.2. Duality principle 439
13.6.3. Another useful interpretation of grammians 440
13.6.4. Zeros of a multivariable system 440
13.6.5. Standardization of a system 441
13.7. Bibliography 441

Chapter 14. Multi-variable Modal Control 445
Yann LE GORREC and Jean-Francois MAGNI
14.1. Introduction 445
14.2. The eigenstructure 446
14.2.1. Notations 446
14.2.2. Relations among signals, modes and eigenstructures 448
14.3. Modal analysis 453
14.3.1. Introduction 453
14.3.2. Modal simulation 453
14.3.3. Controllability 455
14.4. Traditional methods for eigenstructure placement 457
14.4.1. Modal specifications 458
14.4.2. Choice of eigenvectors of the closed loop 460
14.4.3. State feedback and output elementary static feedback 465
14.5. Eigenstructure placement as observer 468
14.5.1. Elementary observers 468
14.5.2. Observer synthesis 469
14.5.3. Synthesis of output dynamic feedback in the form of observer . . 471

14.6. Conclusion 476
14.7. Bibliography 477

Chapter 15. Robust H
∞
/LMI Control 479
Gilles DUC
15.1. The H
∞
approach 480
15.1.1. The H
∞
standard problem 480
15.1.2. Example 482
15.1.3. Resolution methods 488
15.1.4. Resolution of
∞
H standard problem through
the Riccati equations 488
15.1.5. Resolution of the
∞
H standard problem by LMI 490
15.1.6. Restricted synthesis on the corrector order 493
15.2. The µ-analysis 493
15.2.1. Analysis diagram and structured single value 495
15.2.2. Main results of robustness 495
15.2.3. Example 496
xiv Analysis and Control of Linear Systems
15.2.4. Evaluation of structured single value 500
15.3. The µ-synthesis 502

15.3.1. A H
∞
robust synthesis 502
15.3.2. Approach by D-K iterations 504
15.3.3. Example 506
15.4. Synthesis of a corrector depending on varying parameters 507
15.4.1. Problem considered and L
2
gain 507
15.4.2. Polytopic approach 509
15.4.3. A more general approach 512
15.4.4. Example 515
15.5. Conclusion 518
15.6. Bibliography 518

Chapter 16. Linear Time-Variant Systems 521
Michel GUGLIELMI
16.1. Ring of non-commutative polynomials 522
16.1.1. Division and the right highest divisor (RHD) 523
16.1.2. Right least common multiple (RLCM) 523
16.1.3. Explicit formulation of RLCM 524
16.1.4. Factoring, roots, relations with the coefficients 524
16.2. Body of rational fractions 525
16.3. Transfer function 526
16.3.1. Properties of transfer functions 527
16.3.2. Normal modes 528
16.3.3. Stability 528
16.4. Algebra of non-stationary linear systems 529
16.4.1. Serial systems 529
16.4.2. Parallel systems 530

16.5. Applications 531
16.5.1. Modeling 531
16.5.2. Pole placement 532
16.6. Conclusion 534
16.7. Bibliography 535

List of Authors 537

Index 541
Preface
This book is about the theory of continuous-state automated systems whose
inputs, outputs and internal variables (temperature, speed, tension, etc.) can vary in a
continuous manner. This is contrary to discrete-state systems whose internal
variables are often a combination of binary sizes (open/closed, present/absent, etc.).

The word “linear” requires some explanation. The automatic power control of
continuous-state systems often happens through actions in relation to the gaps we
are trying to control. Thus, it is possible to regulate cruise control by acting on the
acceleration control proportionally to the gap observed in relation to a speed
instruction. The word “proportional” precisely summons up a linear control law.

Some processes are actually almost never governed by laws of linear physics.
The speed of a vehicle, even when constant, is certainly not proportional to the
position of the accelerator pedal. However, if we consider closed loop control laws,
the return will correct mistakes when they are related either to external disturbances
or to gaps between the conception model and the actual product. This means that
modeling using a linear model is generally sufficient to obtain efficient control laws.
Limits to the automated systems performances generally come from the restricted
power of motors, precision of captors and variability of the behavior of the
processes, more than from their possible non-linearity.

It is necessary to know the basics of linear automated systems before learning
about the theory of non-linear systems. That is why linear systems are a fundamental
theory, and the problems linked to closed-loop control are a big part of it.

Input-output and the state representations, although closely linked, are explained
in separate chapters (1 and 2). Discrete-time systems are, for more clarity, explained
in Chapter 3. Chapter 4 explains the structural properties of linear systems. Chapter
xvi Analysis and Control of Linear Systems
5 looks into deterministic and statistical models of signals. Chapter 6 introduces us
to two fundamental theoretical tools: state stabilization and estimation. These two
notions are also covered in control-related chapters. Chapter 7 defines the elements
of modeling and identification. All modern control theories rely on the availability
of mathematical models of processes to control them.

Modeling is therefore upstream of the control engineer. However, pedagogically
it is located downstream because the basic systems theory is needed before it can be
developed. This same theory also constitutes the beginning of Chapter 8, which is
about simulation techniques. These techniques form the basis of the control laws
created by engineers.

Chapter 9 provides an analysis of the classic invariable techniques while Chapter
10 summarizes them. Based on the transfer function concept, Chapter 11 addresses
pole placement control and Chapter 12 internal control. The three following chapters
cover modern automation based on state representation. They highlight the
necessary methodological aspects. H
2
optimization control is explained in Chapter
13, modal control in Chapter 14 and H
∞

control in Chapter 15. Chapter 16 covers
linear time-variant systems.