Stability of Dynamical Systems: On the Role of Monotonic and Non-Monotonic Lyapunov Functions

669 Pages·2015·8.456 MB·English

by  Anthony N. Michel, Ling Hou, Derong Liu (auth.)

Systems & Control: Foundations & Applications
Anthony N. Michel
Ling Hou
Derong Liu
Stability of
Dynamical
Systems
On the Role of Monotonic and
Non-Monotonic Lyapunov Functions
Second Edition

Systems & Control: Foundations & Applications
SeriesEditor
Tamer Bas¸ar, University of Illinois at Urbana-Champaign, Urbana, IL,
USA
EditorialBoard
KarlJohanÅström,LundUniversityofTechnology,Lund,Sweden
Han-FuChen,AcademiaSinica,Beijing,China
BillHelton,UniversityofCalifornia,SanDiego,CA,USA
AlbertoIsidori,SapienzaUniversityofRome,Rome,Italy
MiroslavKrstic,UniversityofCalifornia,SanDiego,CA,USA
H.VincentPoor,PrincetonUniversity,Princeton,NJ,USA
MeteSoner,ETHZürich,Zürich,Switzerland;
SwissFinanceInstitute,Zürich,Switzerland
RobertoTempo, CNR-IEIIT,PolitecnicodiTorino,Italy
Moreinformationaboutthisseriesathttp://www.springer.com/series/4895
Anthony N. Michel (cid:129) Ling Hou (cid:129) Derong Liu
Stability of Dynamical
Systems
On the Role of Monotonic
and Non-Monotonic Lyapunov
Functions
Second Edition
AnthonyN.Michel LingHou
DepartmentofElectrical DepartmentofElectricalandComputer
Engineering Engineering
UniversityofNotreDame St.CloudStateUniversity
NotreDame,IN,USA St.Cloud,MN,USA
DerongLiu
DepartmentofElectricalandComputer
Engineering
UniversityofIllinoisatChicago
Chicago,IL,USA
ISSN2324-9749 ISSN2324-9757 (electronic)
Systems&Control:Foundations&Applications
ISBN978-3-319-15274-5 ISBN978-3-319-15275-2 (eBook)
DOI10.1007/978-3-319-15275-2
LibraryofCongressControlNumber:2015931620
Mathematics Subject Classification (2010): 15-XX, 15A03, 15A04, 15A06, 15A09, 15A15, 15A18,
15A21,15A42,15A60,15A63,26-XX,26Axx,26A06,26A15,26A16,26A24,26A42,26A45,26A46,
26A48,26Bxx,26B05,26B10,26B12,26B20,26B30,26E05,26E10,26E25,34-XX,34-01,34Axx,
34A12,34A30,34A34,34A35,34A36,34A37,34A40,34A60,34Cxx,34C25,34C60,34Dxx,34D05,
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Preface
Intheanalysisandsynthesisofcontemporarysystems,engineersandscientistsare
frequentlyconfrontedwith increasinglycomplex models that may simultaneously
includecomponentswhosestatesevolvealongcontinuoustime(continuousdynam-
ics) and discrete instants (discrete dynamics); components whose descriptions
may exhibit hysteresis nonlinearities, time lags or transportation delays, lumped
parameters, spatially distributed parameters, uncertainties in the parameters, and
the like; and components that cannot be described by the usual classical equa-
tions (ordinary differential equations, difference equations, functional differential
equations,partialdifferentialequations,andVolterraintegrodifferentialequations),
as in the case of discrete-event systems, logic commands, Petri nets, and the
like. The qualitative analysis of systems of this type may require results for
finite-dimensional systems as well as infinite-dimensional systems, continuous-
timesystemsaswellasdiscrete-timesystems,continuouscontinuous-timesystems
as well as discontinuous continuous-time systems (DDS), and hybrid systems
involvingamixtureofcontinuousanddiscretedynamics.
Other than the first edition of this book, there are no texts on stability theory
that are suitable to serve as a single source for the analysis of system models
of the type described above. Most existing engineering texts on stability theory
address finite-dimensional systems described by ordinary differential equations,
and discrete-time systems are frequentlytreated as analogousafterthoughtsor are
relegatedtobooksonsampled-datacontrolsystems.Ontheotherhand,booksonthe
stabilitytheoryofinfinite-dimensionaldynamicalsystemsusuallyfocusonspecific
classes of systems (determined, e.g., by functional differential equations, partial
differentialequations,andsoforth).Finally,theliteratureonthestabilitytheoryof
discontinuousdynamicalsystems(DDS)ispresentlyscatteredthroughoutjournals
and conference proceedings. Consequently, to become reasonably proficient in
the stability analysis of contemporary dynamical systems of the type described
above may require considerable investment of time. As was the case for the
first edition, the present updated version of this book aims to fill this void. To
accomplish this, the book addresses again four general areas: the representation
and modeling of a variety of dynamical systems of the type described above, the
vii
viii Preface
presentationoftheLyapunovandLagrangestabilitytheoryfordynamicalsystems
definedongeneralmetricspaces,thespecializationofthisstabilitytheorytofinite-
dimensional dynamical systems, and the specialization of this stability theory to
infinite-dimensionaldynamicalsystems. Throughoutthebook,theapplicabilityof
thedevelopedtheoryisdemonstratedbymeansofnumerousspecificexamplesand
applicationstoimportantclassesofsystems.
In his groundbreaking result concerning the global asymptotic stability of an
equilibrium, Lyapunov makes use of positive definite, radially unbounded, and
decrescent scalar-valued functions of the system state and time (called Lyapunov
functions)which,whenevaluatedalongthemotionsofafinite-dimensionaldynam-
icalsystem(determinedbyordinarydifferentialequations),decreasemonotonically
with increasing time and approach zero as time approaches infinity. In this book,
suchfunctionsarecalledmonotonicLyapunovfunctions.
Lyapunov’sfamousresultforglobalasymptoticstabilityofanequilibriumyields
sufficient conditions. Subsequently, necessary conditions for uniform asymptotic
stability in the large were established as well. However, for general dynamical
systems, there are no results which constitute necessary and sufficient conditions
for the uniform asymptotic stability of an equilibrium. This points to limitations
inherent in the Lyapunov results which comprise the Direct Method of Lyapunov
(alsocalledtheSecondMethodofLyapunov).
Inrecentworks(concerningthequalitativeanalysisofdiscontinuousdynamical
systems,includingswitchedsystemsandhybridsystems),stabilityandboundedness
results were discovered which involve the existence of Lyapunov-like functions
which, when evaluated along the motions of a dynamical system, still need to
approachzeroastimeapproachesinfinity;however,thesefunctionsnolongerneed
to decrease monotonically with increasing time along the system motions. In this
book,suchfunctionsarecallednon-monotonicLyapunovfunctions.
As in the case of the classical Lyapunov stability and boundedness results
(involving monotonic Lyapunov functions), the stability and boundedness results
involving non-monotonic Lyapunov functions constitute sufficient conditions or
necessary conditions. For general dynamical systems, no results which constitute
necessaryandsufficientconditionsareknown.However,itturnsoutthat,ingeneral,
the principal stability and boundedness results involving monotonic Lyapunov
functions will always reduce to corresponding results involving non-monotonic
Lyapunov functions. Furthermore, for many cases, the classical Lyapunov results
involvingmonotonicLyapunovfunctionsturnouttobemoreconservativethanthe
correspondingresultsinvolvingnon-monotonicLyapunovfunctions.
The first edition of this book contains many stability and boundedness results
involving non-monotonicLyapunov functions. While these results were primarily
discoveredinthequalitativeanalysisofdiscontinuousdynamicalsystems(including
switched and hybrid systems), it was recognized that these results are applicable
to continuous dynamical systems as well. Nevertheless, the terms “monotonic
Lyapunov function” and “non-monotonic Lyapunov function” do not appear in
the first edition of this book. There are two reasons for this. First, at the time of
the publication of this book, the roles of the monotonic and the non-monotonic
Preface ix
behavior of Lyapunov functions along system motions were perhaps not fully
appreciated. Secondly, the development of the stability and boundedness theory
involvingnon-monotonicLyapunovfunctionswasincomplete.Forexample,when
the first edition of this book appeared, no invariance stability and boundedness
results involving non-monotonic Lyapunov functions had been discovered, the
stability and boundedness results for discrete-time dynamical systems involving
non-monotonic Lyapunov functions had not been established, results involving
multiple non-monotonic Lyapunov functions had not been developed yet, and so
forth.
Once the aforementionedobstacleshad beenremoved,a presentationof allthe
principalLyapunovstabilityandboundednessresultsinvolvingmonotonicandnon-
monotonicLyapunovfunctionswas madepossible. To accomplishthis, altogether
elevennewsectionshavebeenaddedtothefirsteditionofthisbook.Furthermore,
the entire text is replete with explanations of the roles of monotonic and non-
monotonicLyapunovfunctionsforthevariousresultsbeingpresented.
Indevelopingthesubjectonhand,wefirstestablishtheLyapunovandLagrange
stability results for general dynamical systems defined on metric spaces. Next,
we present corresponding results for finite-dimensional dynamical systems and
infinite-dimensionaldynamicalsystems.Ourpresentationisveryefficientbecause
thestabilityandboundednessresultsoffinite-dimensionalandinfinite-dimensional
dynamical systems are, in many cases, direct consequences of the corresponding
stability and boundednessresults of generaldynamicalsystems defined on metric
spaces.
In our presentation of the various stability and boundedness results, we use a
prescribed road map. First, for the case of continuous-time dynamical systems,
we establish results involving non-monotonic Lyapunov functions. These results
are applicable to discontinuous as well as continuous dynamical systems. Next,
wepresentthecorrespondingclassicalLyapunovstabilityandboundednessresults
for continuous-time dynamical systems. We prove these results by showing that
wheneverthehypothesesofagivenclassicalresultinvolvingmonotonicLyapunov
functions are satisfied, then the hypotheses of the corresponding result involving
non-monotonicLyapunovfunctionsarealsosatisfied.Weestablishthestabilityand
boundednessresultsfordiscrete-timedynamicalsystemsinvolvingnon-monotonic
and monotonic Lyapunov functions in an identical manner. Alternatively, since
every discrete-time dynamical system can be associated with a discontinuous
continuous-timedynamicalsystem with identical stability and boundednessprop-
erties, we also make use of the above stability and boundedness results for
continuous-timesystemsinvolvingnon-monotonicLyapunovfunctionstoestablish
the variousLyapunovstability and boundednessresults for discrete-time systems.
In addition to being very efficient in establishing the various assertions on hand,
theabovemethodofproofenablesusalsotoconcludethatthevariousstabilityand
boundednessresultsinvolvingmonotonicLyapunovfunctionswillalwaysreduceto
correspondingresultsinvolvingnon-monotonicLyapunovfunctions.Furthermore,
therelationshipbetweendiscontinuouscontinuous-timesystemsanddiscrete-time