Modern Anti-windup Synthesis: Control Augmentation for Actuator Saturation

303 Pages·2011·15.39 MB·English

by  Luca Zaccarian

Modern Anti-windup Synthesis
PrincetonSeriesinAppliedMathematics
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ModernAnti-windupSynthesis:ControlAugmentationforActuatorSaturation,Luca
ZaccarianandAndrewR.Teel
Modern Anti-windup Synthesis
Control Augmentation for A
tuator Saturation
Lu
a Za

arian and Andrew R. Teel
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright c 2011byPrincetonUniversityPress
(cid:13)
PublishedbyPrincetonUniversityPress,41WilliamStreet,Princeton,NewJersey
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IntheUnitedKingdom: PrincetonUniversityPress,6OxfordStreet,Woodstock,
OxfordshireOX201TW
press.princeton.edu
Coverart: FioreB.Zaccarian,Letturasottoglialberi(readingunderthetrees),oil
oncanvas,1970. CourtesyofDr. PaoloZaccarian.
AllRightsReserved
LibraryofCongressCataloging-in-PublicationData
Zaccarian,Luca.
Modern anti-windup synthesis: control augmentation for actuator saturation /
LucaZaccarian,AndrewR.Teel.
p. cm. –(Princetonseriesinappliedmathematics)
Includesbibliographicalreferencesandindex.
ISBN978-0-691-14732-1(hardback)
1. Automaticcontrol–Mathematicalmodels. 2. Linearcontrolsystems. 3.
Actuators. I.Teel,AndrewR.,1965-II.Title.
TJ213.7.Z3262011
629.8’3–dc22 2010044248
BritishLibraryCataloging-in-PublicationDataisavailable
Thepublisherwouldliketoacknowledgetheauthorsofthisvolumeforproviding
thecamera-readycopyfromwhichthisbookwasprinted.
Printedonacid-freepaper. ∞
PrintedintheUnitedStatesofAmerica
10987654321
Contents
Prefa
e ix
Algorithms Summary xi
PART 1. PREPARATION 1
1. The Windup Phenomenon and Anti-windup Illustrated 3
1.1 Introduction
3
1.2 Illustrativeexamples
4
1.3 Summary
21
1.4 Notesandreferences
22
2. Anti-windup: De(cid:28)nitions, Obje
tives, and Ar
hite
tures 23
2.1 Preliminaries
23
2.2 Qualitativeobjectives
26
2.3 Anti-windupaugmentation
32
2.4 Quantitativeperformanceobjectives
40
2.5 Notesandreferences
47
3. AnalysisandSynthesisofFeedba
kSystems: Quadrati
Fun
tionsandLMIs 48
3.1 Introduction
48
3.2 Unconstrainedfeedbacksystems
50
3.3 Linearmatrixinequalities
51
3.4 Constrainedfeedbacksystems:globalanalysis
59
3.5 Constrainedfeedbacksystems:regionalanalysis
63
3.6 Analysisexamples
67
3.7 Regionalsynthesisforexternalstability
70
3.8 Notesandreferences
73
PART 2. DIRECT LINEAR ANTI-WINDUP AUGMENTATION 75
4. Stati
Linear Anti-windup Augmentation 77
4.1 Overview
77
4.2 Keystate-spacerepresentations
78
4.3 Algorithmsprovidingglobalguarantees
81
4.4 Algorithmsprovidingregionalguarantees
98
4.5 Notesandreferences
107
vi CONTENTS
5. Dynami
Linear Anti-windup Augmentation 109
5.1 Overview
109
5.2 Keystate-spacerepresentations
110
5.3 Factoringrank-deficientmatrices
113
5.4 Algorithmsprovidingglobalguarantees
114
5.5 Algorithmsprovidingregionalguarantees
141
5.6 Notesandreferences
152
PART 3. MODEL RECOVERY ANTI-WINDUP AUGMENTATION 155
6. The MRAW Framework 157
6.1 Introduction
157
6.2 Ablockdiagram/transferfunctiondescription
158
6.3 Astate-spacedescription(linearitynotneeded)
161
6.4 Robust,fragile,orboth?
164
6.5 Notesandreferences
167
7. Linear MRAW Synthesis 174
7.1 Introduction
174
7.2 Globalstability-basedalgorithms
176
7.3 Regionalstabilityandperformancealgorithms
195
7.4 Notesandreferences
199
8. Nonlinear MRAW Synthesis 200
8.1 Introduction
200
8.2 Switchingandschedulinglinearcontrollers
201
8.3 Modelpredictivecontrolforanti-windupdesign
208
8.4 Globaldesignsfornon-exponentiallyunstableplants
217
8.5 Designsforexponentiallyunstableplantsthatmaximizethebasinofattraction
222
8.6 Notesandreferences
225
9. The MRAW Stru
ture Applied to Other Problems 226
9.1 Rate-andmagnitude-saturatedplants
226
9.2 Anti-windupfordead-timeplants
232
9.3 Bumplesstransferinmulticontrollerschemes
235
9.4 Reliablecontrolviahardwareredundancy
240
9.5 Notesandreferences
243
10.Anti-windup for Euler-Lagrange Plants 245
10.1 FullyactuatedEuler-Lagrangeplants
245
10.2 Anti-windupconstructionandselectionofthestabilizerv
246
10.3 Simulationexamples
250
10.4 Notesandreferences
268
11.Annotated Bibliography 269
11.1 Overview
269
11.2 Problemdiscovery
269
11.3 Thefirstconstructivetechniques
270
CONTENTS vii
11.4 Callforsystematization
271
11.5 Modernanti-windupschemes
272
11.6 Additionalreferences
281
Index 285
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Prefa
e
When feedback control is synthesized for a linear plant, an often neglected but
importantfeatureofthefeedbackloopisasaturationnonlinearityattheplantinput.
A saturated input typically provides a better model of reality than a linear input
modeldoes. Indeed,allrealactuatorshavelimitedcapabilitiesandtheselimitscan
haveadramaticeffectontheperformanceofanotherwiselinearfeedbackloop. In
the face of input saturation, the control engineer must accept that the achievable
largesignalperformanceisinherentlylimited. Thenheorshemustdecidehowto
addressthisfactincontrolsynthesis.
When input saturation is expected to be a common occurrence in the plant’s
operation, it makes sensetolook for designmethodologies thataccountfor input
saturationdirectly.Therearemanyusefultechniquesinthiscategorythathavebeen
developed in the control literature, including the very popular “model predictive
control” framework. In the case where input saturation is expected to be a less
frequentoccurrenceandthesmallsignalperformancespecificationsaredifficultto
incorporateintoageneralframeworklikemodelpredictivecontrol,itmakessense
toconsiderthe“anti-windup”paradigm.
In the anti-windup synthesis problem, a controller for the saturation-free case
hasalreadybeensynthesizedbasedonsomeperformancecriterionandithasbeen
confirmedthatthiscontrollerdoesnotperformwellwheninputsaturationoccurs.
In this case, the task is to synthesize a controller augmentation that has no effect
whensaturationdoesnotoccurandthatotherwiseattemptstoprovidesatisfactory
performance for large signals. In this way, the small signal performance is not
compromisedforthesakeofguaranteeingacceptablelargesignalperformance.
This book is dedicated to the description of anti-windup synthesis algorithms.
Thefocusisonstate-spacemethodsandsynthesisalgorithmsthatrequiresolving
linearmatrixinequalities(LMIs). Manyefficientsoftwareprograms,suchasMAT-
LAB/Simulink,1arewidelyavailableforsolvingtheseLMIs.Inordertofollowthe
material in this book, the reader should be familiar with state-space descriptions
andbasicstabilitytheoryforlinear, continuous-timesystems. Inordertobecome
comfortablewiththealgorithmsandtheirbehaviors,thereadershouldbewillingto
investtheenergyrequiredtogaincompetencywithanLMIsolver. Severalexam-
plesareprovidedinthetextthatcanbeusedbythereadertotesttheunderstanding
1MATLAB(cid:13)R andSimulink(cid:13)R areregisteredtrademarksofTheMathWorksInc. andareusedwith
permission. TheMathWorksdoesnotwarranttheaccuracyofthetextorexercisesinthisbook. This
book’suseofMATLAB(cid:13)R andSimulink(cid:13)R doesnotconstituteanendorsementorsponsorshipbyThe
MathWorksofaparticularpedagogicalapproachorparticularuseoftheMATLAB(cid:13)R andSimulink(cid:13)R
software.