Title LMI conditions for robust consensus of uncertain nonlinear multi-agent systems Advisor(s)

150 Pages·2014·2.71 MB·English

LMI conditions for robust consensus of uncertain nonlinear
Title
multi-agent systems
Advisor(s) Chesi, G
Author(s) Han, Dongkun; 韓東昆
Citation
Issued Date 2014
URL http://hdl.handle.net/10722/206333
The author retains all proprietary rights, (such as patent rights)
Rights
and the right to use in future works.
Abstractofthesisentitled
“LMI Conditions for Robust Consensus of Uncertain
Nonlinear Multi-agent Systems”
Submittedby
Dongkun Han
forthedegreeofDoctorofPhilosophy
at TheUniversityofHong Kong
in August2014
Establishingconsensusisakeyprobleminmulti-agentsystems(MASs). Thisthesis
proposes a novel methodology based on convex optimization in the form of linear
matrix inequalities (LMIs) for establishing consensus in linear and nonlinear MAS
inthepresenceofmodeluncertainties,i.e., robustconsensus.
Firstly, this thesis investigates robust consensus for uncertain MAS with linear
dynamics. Specifically,itissupposedthatthesystemisdescribedbyaweightedad-
jacencymatrixwhoseentriesaregenericpolynomialfunctionsofanuncertainvec-
torconstrainedinasetdescribedbygenericpolynomialinequalities. Forcontinuous-
timedynamics,necessaryandsufficientconditionsareproposedtoensuretherobust
first-order consensus and the robust second-order consensus, in both cases of pos-
itive and non-positive weighted adjacency matrices. For discrete-time dynamics,
necessary and sufficient conditions are provided for robust consensus based on the
existence of a Lyapunov function polynomially dependent on the uncertainty. In
particular, an upper bound on the degree required for achieving necessity is pro-
vided. Furthermore,anecessaryandsufficientconditionisprovidedforrobustcon-
sensus with single integrator and nonnegative weighted adjacency matrices based
onthezerosofapolynomial. Lastly,itisshownhowtheseconditionscanbeinves-
tigatedthroughconvexoptimizationby exploitingLMIs.
Secondly,localandglobalconsensusareconsideredinMASwithintrinsicnon-
lineardynamicswithrespecttoboundedsolutions,likeequilibriumpoints,periodic
orbits, and chaotic orbits. For local consensus, a method is proposed based on the
transformation of the original system into an uncertain polytopic system and on
theuseofhomogeneouspolynomialLyapunovfunctions(HPLFs). Forglobalcon-
sensus, another method is proposed based on the search for a suitable polynomial
Lyapunovfunction(PLF).Inaddition,robustlocalconsensusinMASisconsidered
with time-varying parametric uncertainties constrained in a polytope. Also, by us-
ingHPLFs,anewcriteriaisproposedwheretheoriginalsystemissuitablyapprox-
imated by an uncertain polytopic system. Tractable conditions are hence provided
in terms of LMIs. Then, the polytopic consensus margin problem is proposed and
investigatedviageneralized eigenvalueproblems(GEVPs).
Lastly, this thesis investigates robust consensus problem of polynomial nonlin-
ear system affected by time-varying uncertainties on topology, i.e., structured un-
certain parameters constrained in a bounded-rate polytope. Via partial contraction
analysis, novel conditions, both for robust exponential consensus and for robust
asymptotical consensus, are proposed by using parameter-dependent contraction
matrices. In addition, for polynomial nonlinear system, this paper introduces a
new class of contraction matrix, i.e., homogeneous parameter-dependent polyno-
mial contraction matrix (HPD-PCM), by which tractable conditions of LMIs are
providedviaaffinespaceparametrizations. Furthermore,thevariantratemarginfor
robust asymptotical consensus is proposed and investigated via handling general-
ized eigenvalueproblems(GEVPs).
For each section, a set of representative numerical examples are presented to
demonstratetheeffectivenessoftheproposedresults.
Anabstract ofexactly473 words
LMI Conditions for Robust
Consensus of Uncertain Nonlinear
Multi-agent Systems
by
Dongkun Han
B.Eng., M.Eng.
A thesissubmittedinpartial fulfilmentoftherequirementsfor
theDegree ofDoctorofPhilosophy
at theUniversityofHongKong
Department ofElectrical and ElectronicEngineering
TheUniversityofHongKong
August2014
Declaration
I declare that the thesis and the research work thereof represent my own work,
except where due acknowledgement is made, and that it has not been previously
included in a thesis, dissertation or report submitted to this University or to any
otherinstitutionfor adegree, diplomaorotherqualifications.
Signed ....................................................
DongkunHan
i
Acknowledgements
Four years experience at the University of Hong Kong has been nothing short of
amazing. Since my first day at HKU on August 17th, 2010, I have been given
unique opportunities from a number of people without whom this thesis might not
havebeen writtenand to whomIamgreatly indebted.
First and foremost I wish to give my heartfelt gratitude to my supervisors Dr.
Graziano Chesi and Prof. Yeung Sam Hung, for their instruction, guidance and
support throughout my Ph. D. study. I am also thankful for the excellent example
that Dr. Graziano Chesi has provided as a successful young scientist and will al-
ways be a guiding light for my future career. I am indebted to his encouragements
when our paper was rejected, his enlightening ideas when my research sank in the
mire, his inspiring smile when I made any academic achievement and his all-out
recommendationsformyvisitingprogram lastyear.
My sincere thanks also go to Prof. Stephen Boyd and Dr. Sanjay Lall, for
offering me a position of visiting research student at Stanford University in their
groups and leading me to work on diverse interesting areas. I also appreciate the
helps and stimulating discussions from Pontus Giselsson, Madeleine Udell, Ernest
Ryu and EricChu at InformationSystemsLaboratory,Stanford University.
Iamindebtedtomanystudentcolleaguesforprovidingastimulatingandenjoy-
ableenvironmentinwhichtoendowandgrow. Particularly,Iamespeciallygrateful
toDr. PingLi andDr. Bo Shen fortheirsoundadviceandinsightfulideas. Iwould
liketo express my appreciation to other group members including Dr. Fei Mai, Dr.
Jian Zhang, Dr. Jiewei Li, Dr. Tiantian Shen, Dr. Xiongbo Zhang and Chuen Kit
ii
Lukfortheirhelps.
I wish to thank Alan Ng, the accommodation manager of HKU, who offered
me a position of Honorary Resident Tutor at Pokfield Road Residence during my
Ph. D. study. For the fun and jollities, I would like to thank the colleagues of PRR
including Ivy Lee, Jie Hing, Dr. Anson Zeng, Dr. Abby Ding, Dylan Zhang and
CarlaLung.
As long as I am writing names down, I cannot neglect these without whom my
lifeinPh. D.pursuitmaynotbealwaysfullofcolourandsunshine. Iamindebtedto
YiLong,GuolinNiu,XiongboZhang,ShizhouFu,ChunguoLi,FanSun,Liangyin
Zhang, Zhiguang Feng, Fei Wang, Yaxin Hu, Wenchao Hu, Xin He, Qinliang Su
and LinfengChen forthehappytimeweshared.
Last but certainly not least, my grateful thanks to my beloved family for their
endlessloveand boundlesssupport.
DongkunHan
August2014
CYC807, HKU
iii
Contents
Declaration i
Acknowledgements ii
TableofContents iv
ListofTables vii
ListofFigures viii
ListofPublications ix
ListofAbbreviations xi
ListofSymbols xii
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 LiteratureReview . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Consensusin NetworkswithFixedTopology . . . . . . . . 4
1.2.2 Consensusin NetworkswithChangingTopology . . . . . . 8
1.2.3 Consensusin NetworkswithNonlinearDynamics . . . . . 10
1.3 MathematicalPreliminaries . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 AlgebraicGraph Theory . . . . . . . . . . . . . . . . . . . 13
1.3.2 LyapunovStabilityTheory . . . . . . . . . . . . . . . . . . 14
1.3.3 Square MatrixRepresentation . . . . . . . . . . . . . . . . 18
iv
1.4 Problem Statements . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.6 ThesisOutline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Robust Consensus forUncertain andLinear Dynamics 27
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Continuous-timeDynamics . . . . . . . . . . . . . . . . . . . . . . 30
2.3.1 First-orderConsensus . . . . . . . . . . . . . . . . . . . . 31
2.3.2 Second-order Consensus . . . . . . . . . . . . . . . . . . . 36
2.4 Discrete-timeDynamics . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.1 First-orderConsensus . . . . . . . . . . . . . . . . . . . . 40
2.4.2 Second-order Consensus . . . . . . . . . . . . . . . . . . . 48
2.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.5.1 Example1 . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.5.2 Example2 . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.5.3 Example3 . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.5.4 Example4 . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3 Consensus forNonlinearDynamics 61
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 ConsensusConditionswithNonlinearDynamics . . . . . . . . . . 63
3.3.1 Local ConsensusConditions . . . . . . . . . . . . . . . . . 63
3.3.2 GlobalConsensusConditions . . . . . . . . . . . . . . . . 68
3.4 Robust LocalConsensuswith Time-varyingUncertainties . . . . . . 72
3.4.1 SystemApproximation . . . . . . . . . . . . . . . . . . . . 73
3.4.2 ConditionsviaUsingHPLF . . . . . . . . . . . . . . . . . 77
3.4.3 PolytopicConsensusMargin . . . . . . . . . . . . . . . . . 81
v
3.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.5.1 Example1 . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.5.2 Example2 . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.5.3 Example3 . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.5.4 Example4 . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4 Robust Consensus forUncertain andNonlinearDynamics 91
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.1.1 Basics ofContractionTheory . . . . . . . . . . . . . . . . 92
4.1.2 RobustConsensus Problems . . . . . . . . . . . . . . . . . 95
4.2 Robust ConsensusConditions . . . . . . . . . . . . . . . . . . . . 98
4.3 AnalysisviaHPD-PCM . . . . . . . . . . . . . . . . . . . . . . . . 104
4.4 Robust ConsensusPerformance . . . . . . . . . . . . . . . . . . . 110
4.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.5.1 Example1 . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.5.2 Example2 . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5 Conclusions andFuture Works 119
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2 FutureWorks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
A Runtime andNumerical Complexity 123
References 124
vi