Nonlinear and Adaptive Control with Applications

301 Pages·2008·5.388 MB·English

by  Alessandro Astolfi PhD, Dimitrios Karagiannis PhD, Romeo Ortega PhD (auth.)

Communications and Control Engineering
Series Editors
E.D. Sontag(cid:3) (cid:120)(cid:3) M. Thoma(cid:3) (cid:120)(cid:3) A. Isidori(cid:3) (cid:120)(cid:3) J.H. van Schuppen

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Alessandro Astolfi • Dimitrios Karagiannis
Romeo Ortega

Nonlinear and
Adaptive Control
with Applications

123
Alessandro Astolfi, PhD Dimitrios Karagiannis, PhD
Department of Electrical and Electronic Department of Electrical and Electronic
Engineering Engineering
Imperial College London Imperial College London
London SW7 2AZ London SW7 2AZ
UK UK

and
Romeo Ortega, PhD
Dipartimento di Informatica, Sistemi Centre National de la Recherche Scientifique
e Produzione Laboratoire des Signaux et Systèmes
Università di Roma “Tor Vergata” Supélec
00133 Roma 91192 Gif-sur-Yvette
Italy France

ISBN 978-1-84800-065-0 e-ISBN 978-1-84800-066-7
DOI 10.1007/978-1-84800-066-7
Communications and Control Engineering Series ISSN 0178-5354
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Library of Congress Control Number: 2007941070
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To Elisabetta and the kids (A.A.)
To Lara (D.K.)
To the memory of my father (R.O.)
Preface
Inthelastfew yearswehavewitnessedtheappearanceofaseriesofchalleng-
ing control engineering problems. Two common features of these new control
problems are that the interesting range of operationof the system is not nec-
essarily close to an equilibrium, hence nonlinear effects have to be explicitly
taken into account for a successful controller design, and that, even though
physicalmodellingallowstoaccuratelyidentifycertainwell-definednonlinear
effects, the controllerhasto cope witha highlevelofuncertainty,mainly due
to lack of knowledge on the system parameters and the inability to measure
the whole system state.
Thissituationjustifies the needforthe developmentoftoolsforcontroller
design for uncertain nonlinear systems, which is the main topic of this book.
Numerous theoretical control design methodologies for nonlinear sys-
tems have emerged over the last two decades. When viewed from a con-
ceptual standpoint, they can be broadly classified into analytically-oriented
and computationally-oriented. (The qualifiers analytical and computational
are used to distinguish between symbolical analysis and numerical computa-
tions.)
Theformerapproach,whichistheoneadoptedinthisbook,proceedsfrom
ananalyticalmodelofthe system,andthecontrollerdesignis theoutcomeof
asystematicprocessthatguaranteessomespecificbehaviour.Sincestabilityis
asine qua non condition,researchfollowingthisapproachusuallyrunsunder
the heading robust stabilisation, and it includes Lyapunov-based methods,
gain-assignmentmethods, and classical robust and adaptive tools.
Computationally-orientedtechniques,on the other hand, do not necessar-
ily require an analytical model, and they may be developed on the basis of a
numericalmodelofthesystemtobecontrolled—obtained,forinstance,bycol-
lecting large amounts of data to approximate its behaviour. Neural networks
based control, fuzzy control and intelligent control are the more conspicuous
representatives of this school. Recently, a second class of computationally-
orientedtechniques,thatreliesonanalyticalmodelsofthesystem,hasgained
some popularity. In an attempt to mimic the developments of linear systems
viii Preface
theory, piecewise linear (or linear parameter-varying) models are proposed
to capture nonlinear effects. Typically some optimal control objective is for-
mulated and the task of the controller design is to prove that, for the given
numerical values of the system model, the optimisation is feasible, e.g., it
can be translated into linear matrix inequalities, and a control signal can be
numerically computed. The optimal control approach suffers from two draw-
backs. First, the solutions are fragile with respect to plant uncertainty, e.g.,
lack of full state measurement and parametric uncertainty, which is the pre-
vailingconcerninmany,ifnotall,practicalapplications.Second,computation
of the optimal controllaw is feasible only for low-dimensionalsystems, which
puts a serious questionmark on the applicability ofthe method for nonlinear
systems. In addition there is not alwaysa clear reason,besides mathematical
convenience,to expressthe desiredbehaviourofadynamicalsysteminterms
of a scalar criterion to be optimised.
In summary, computationally-oriented approaches, while leveraging off a
swiftly growingcomputertechnology,providesolutionstosome specific prob-
lems,butdo notaimatexplainingwhy, how and when these solutionsindeed
work. At a more philosophical level, it is the authors’ opinion that controller
designshould not be reduced to the generationof numericalcode that imple-
mentsacontrollaw,withoutanyattempttotrytounderstandtheunderlying
mechanism that makes it work. This information is encoded in the nonlinear
system dynamics and revealed through a full-fledged nonlinear analysis.
We consider nonlinear control systems subject to various types of uncer-
tainty,includinglackofknowledgeontheparameters,partialmeasurementof
the systemstatesand uncertaintyonthe systemorderandstructure.To deal
with all these situations we follow a common thread encrypted in the words
immersion and invariance (I&I).
In the I&I approach we propose to capture the desired behaviour of the
system to be controlled introducing a target dynamical system. The control
problem is then reduced to the design of a control law which guaranteesthat
the controlled system asymptotically behaves like the target system. More
precisely,theI&Imethodologyreliesonfindingamanifoldinstate-spacethat
canberenderedinvariantandattractive,withinternaldynamicsacopyofthe
desired closed-loop dynamics, and on designing a control law that robustly
steers the state of the system sufficiently close to this manifold.
I&I should be contrasted with the optimal control approach where the
objective is captured by a scalar performance index to optimise. In addition,
because of its two-step approach, it is conceptually different from existing
(robust) stabilisationmethodologiesthat rely on the use of controlLyapunov
functions. However, it resembles the procedure used in sliding-mode control,
where a given manifold—the sliding surface—is rendered attractive by a dis-
continuous controllaw. The key difference is that, while in sliding-mode con-
trol the manifold must be reached by the trajectories, in the proposed ap-
proachthemanifoldneednotbereached.(Thisfeatureisessentialinadaptive
control and in output feedback design.)
Preface ix
Thebookisorganisedasfollows.Afterabriefintroductionwherethemain
ideasofI&Iareillustratedbymeansofexamples(Chapter1),inChapters2
and 3 we introduce the I&I framework and show how it can be used to solve
stabilisation and adaptive control problems.
InChapter 4themethodisappliedtononlinearsystemswithparametric
uncertainties,where it is assumedthat the full state is availablefor feedback.
InChapter 5weshowthatI&Iprovidesanaturalframeworkforobserver
design for general nonlinear systems.
In Chapter 6 the problem of output feedback stabilisation is solved for
classes of nonlinear systems, which include systems with unstable zero dy-
namics and the well-known output feedback form. Furthermore, the method
isextendedtoallowunstructureduncertaintiestoenterthesystemequations.
In Chapter 7 the I&I approach is used to design a class of nonlinear
proportional-integral controllers,where the gains of the controller are nonlin-
earfunctionsthatarechosentoguaranteestabilityforsystemswithunknown
parameters and uncertain disturbances.
Chapters 8, 9 and 10 are devoted to applications from electrical, me-
chanical and electromechanical systems, including power converters, power
systems, electric machines and autonomous aircraft.
AppendixAprovidesthebasicdefinitionsandrecallsbrieflyresultsused
throughout the book. In particular, characterisations of Lyapunov stability,
input-to-state stability and a nonlinear versionof the small-gaintheorem are
given along with some useful lemmas.
Acknowledgements
Thisbookistheresultofextensiveresearchcollaborationsduringthelastfive
years. Some of the results of these collaborations have been reported in the
papers[20,19,14,15,157,162,21,99,105,16,92,186,95,106,100,103,101,
93, 94, 102, 96, 39, 17, 97, 40, 104, 98, 18]. We are gratefulto our co-authors,
Nikita Barabanov,DanieleCarnevale,GerardoEscobar,Micka¨elHilairet,Liu
Hsu, Zhong-Ping Jiang, Eduardo Mendes, Mariana Netto, Hugo Rodr´ıguez,
and Aleksandar Stankovi´c, for several stimulating discussions and for their
hospitality while visiting their institutions. We also thank the research staff
of the Laboratoire de G´enie E´lectrique de Paris, for rendering available their
experimental facilities.
Someofthetopicsofthisbookhavebeentaughtbytheauthorsinaseries
of one-week graduate control courses offered in Paris for the last four years.
These have been organisedby the EuropeanCommission’s Marie Curie Con-
trol Training Site (CTS) and by the European Embedded Control Institute
(EECI) in the framework of the European Network of Excellence HYCON.
We would like to thank Franc¸oise Lamnabhi-Lagarriguefor giving us the op-
purtunity to teach during these schools.
Workshops on the topics presented in this book were organised at the
IEEEConference onDecisionandControl,Las Vegas,USA, 2002,andatthe
x Preface
XII Latin-American Congress on Automatic Control, Bahia, Brazil, 2006. A
mini-tutorial was given at the European Control Conference, Kos, Greece,
2007. We have delivered lectures on selected topics of the book in the DISC
SummerSchool,Eindhoven,TheNetherlands,2007,andinnumerousresearch
seminars.
Finally, a large part of this work would not have been possible without
the financial support of several institutions. The first author would like to
thank the Engineeringand PhysicalSciences ResearchCouncil(EPSRC)and
the Leverhulme Trust. The second author’s work was supported first by the
European Commission’s Training and Mobility of Researchers (TMR) pro-
grammethroughtheNonlinearandAdaptiveControl(NACO2)network,then
byBAESystems andthe EPSRCviathe FLAVIIRIntegratedProgrammein
Aeronautical Engineering, and finally by EPSRC via the Control and Power
Portfolio Partnership. The third author would like to thank the European
Network of Excellence HYCON for supporting part of his work.
Rome, London, Paris Alessandro Astolfi
April 2007 Dimitrios Karagiannis
Romeo Ortega
Contents
Notation....................................................... xv
1 Introduction............................................... 1
1.1 An I&I Perspective of Stabilisation ........................ 2
1.2 Discussion and Literature Review.......................... 4
1.3 Applications of I&I ...................................... 6
1.3.1 Robustification of Control Laws ..................... 6
1.3.2 Underactuated Mechanical Systems.................. 9
1.3.3 Systems in Special Forms........................... 9
1.3.4 Adaptive Control.................................. 11
1.3.5 Observer Design................................... 12
1.3.6 Nonlinear PI Control .............................. 13
2 I&I Stabilisation........................................... 15
2.1 Main Stabilisation Result................................. 15
2.2 Systems with Special Structures........................... 19
2.3 Physical Systems ........................................ 24
3 I&I Adaptive Control: Tools and Examples ................ 33
3.1 Introduction ............................................ 33
3.2 An Introductory Example ................................ 35
3.3 Revisiting Classical Adaptive Control ...................... 38
3.3.1 Direct Cancellation with Matching .................. 38
3.3.2 Direct Cancellation with Extended Matching.......... 39
3.3.3 Direct Domination................................. 40
3.3.4 Indirect Approach ................................. 40
3.3.5 Discussion........................................ 40
3.4 Linearly ParameterisedPlant ............................. 42
3.5 Linearly ParameterisedControl ........................... 47
3.6 Example: Visual Servoing................................. 49