Structural Methods in the Study of Complex Systems

290 Pages·2020·6.754 MB·English

by  Elena Zattoni, Anna Maria Perdon, Giuseppe Conte

Lecture Notes in Control and Information Sciences 482
Elena Zattoni
Anna Maria Perdon
Giuseppe Conte Editors
Structural
Methods in the
Study of Complex
Systems
Lecture Notes in Control and Information
Sciences
Volume 482
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Elena Zattoni Anna Maria Perdon
(cid:129) (cid:129)
Giuseppe Conte
Editors
Structural Methods
in the Study of Complex
Systems
123
Editors
ElenaZattoni Anna Maria Perdon
Department ofElectrical, Electronic and Dipartimento di Ingegneria
Information Engineering “G.Marconi” dell’Informazione
AlmaMater Studiorum Universitàdi UniversitàPolitecnica delle Marche
Bologna Ancona,Italy
Bologna, Italy
Giuseppe Conte
Dipartimento di Ingegneria
dell’Informazione
UniversitàPolitecnica delle Marche
Ancona,Italy
ISSN 0170-8643 ISSN 1610-7411 (electronic)
Lecture Notesin Control andInformation Sciences
ISBN978-3-030-18571-8 ISBN978-3-030-18572-5 (eBook)
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Preface
Complex dynamical systems emerge in a variety of disciplines and domains,
ranging from those that deal with physical processes (biology, genetics, environ-
mental sciences, etc.) to those that concern man-made systems (engineering,
energy, finance, etc.). Indeed, in these fields, it is becoming more and more com-
mon to refer to dynamical structures such as systems of systems, hybrid systems
and multimodal systems. In brief, the former ones consist of many interconnected
dynamical systems with various topological patterns and hierarchical relations; the
secondonesaredynamicalsystemsthatexhibitdynamicsofadifferentnature,both
continuous and discrete; the third ones are dynamical systems whose behaviour
may vary during their life cycle owing to different operating conditions or
depending on the occurrence of some events. The dynamical structures with these
characteristics are currently modelled as multi-agent systems, hybrid impulsive
systems, switching systems, implicit switching systems and so on.
Consequently, control design techniques have changed to adapt to the
ever-increasing system complexity. In this scenario, structural methodologies (i.e.,
those methods which have evolved from original graph theories, differential alge-
braictechniquesandgeometricapproaches)haveproventobeparticularlypowerful
for several reasons. Beforehand, the structural approaches privilege the essential
featuresofdynamicalsystemsandtheirinterconnections,thusyieldingabstractions
that can fit a wide variety of situations. Meanwhile, the geometric perspective,
whichisoftenatthebasisofthestructuralapproaches,introducesarelevantvisual
and intuitive component which fosters research advancements. Nevertheless, the
formalizationofstructuralandgeometricconceptsisrenderedwithalgebraictools,
which, in turn, have a direct correspondence with computational algorithms, thus
paving the way to actual implementation in engineering applications.
In the latest years, relevant theoretical achievements have been obtained within
the scope of each methodology encompassed in the sphere of the structural
approaches (i.e., graph-theoretic methods, differential algebraic methods and geo-
metricmethods)inrelationtofundamentalcontrolandobservationproblemsstated
forcomplexsystems(e.g.,multi-agentsystems,hybridimpulsivesystems,switching
systems,implicitswitchingsystems).Moreover,computationalalgorithmsandcase
v
vi Preface
studies have been developed together with the theoretical accomplishments. Thus,
thecorpusofconsolidatedresults(boththeoreticalandpractical/computationalones)
presentlyavailablemotivatesthisbook,whoseprimary aimistoillustratethestate
oftheartontheuseofmethodologicalapproaches,groundedonstructuralviews,to
investigate and solve paradigmatic analysis and synthesis problems formulated for
complexdynamicalsystems.Inparticular,thedifferentperspectivesemergingfrom
thevariouscontributions havethepurposeofdevelopingnewsensibilities towards
theselectionofthemostsuitabletoolstohandlethespecificproblems.Furthermore,
thethoroughdiscussionsofspecifictopicsareexpectedtooutlinenewdirectionsfor
solving open problems both in thetheory and intheapplications.
The book starts with a general description of complexity and structural
approaches to it, then it focuses on some fundamental problems and, finally, it
dwellsonapplications.Inmoredetail,anoverviewonthecomplexsystemsarising
inthevariousfields,onthenewchallengesofengineeringdesignandonhowthese
can be mastered by means of the structural approaches is provided first. A novel
geometric view, based on transformations which maintain the invariance of global
properties, such as stability or H norm, is described and shown to provide new
1
toolstoinvestigatestabilityandtoparameterizethesetofthestabilizingcontrollers.
Agraph-theoreticbasedapproachandtheoriginalnotionofzeroforcingsetarethe
toolsusedtoanalysecontrollability,faultdetectabilityandidentifiabilityofsystem
networks and, more generally, of systems defined over graphs. How solvability
oftheoutputregulationprobleminhybridlinearsystemswithperiodicstatejumps
can be investigated by structural methods is then illustrated. A mixed digraph
theory and geometric approach is exploited to introduce the novel concept of
subspace arrangement and solve the problem of right-inversion for over-actuated
linear switching systems. Furthermore, the synthesis of unknown-input state
observerswithminimumcomplexityistackledbystructuraltoolsinthecontextof
linearimpulsivesystems:necessaryandsufficientsolvabilityconditionsarederived
once a set of essential requirements has been disentangled. The disturbance
decouplingproblemisinvestigatedforaclassofimplicitswitchingsystemsthrough
geometric considerations inspired to the behavioural approach. In particular, the
theoretical results are applied to the synthesis of a Beard–Jones filter. Finally, a
structural perspective is adopted to analyse Huygens synchronization over dis-
tributed media and it reveals a complex, but structured behaviour behind a seem-
ingly chaotic one.
Thebookisintendedfor systems andcontrolscientistsinterested indeveloping
theoretical and computational tools to solve analysis and synthesis problems
involving complex dynamical systems. The different contributions aim at giving a
comprehensive picture of the available results together with a stimulating view of
possible new directions of investigation in the field. Since the presentations
emphasize methodologies supported by a solid computational background and
oftenbyspecificengineeringapplications,researcherseitherfocussedontheoretical
issues or mainly committed to applications may equally find interesting hints.
Preface vii
The idea of this book has stemmed from the workshop which the editors have
organized at the European Control Conference 2018 and its realization has been
madepossiblethankstothestrongandenthusiasticsupportoftheinvitedspeakers
and their co-authors, who have contributed their original work and latest achieve-
ments in the various chapters.
Bologna, Ancona Elena Zattoni
March 2019 Anna Maria Perdon
Giuseppe Conte
Contents
Part I Structure of Complex Dynamical Systems
1 Complex Systems and Control: The Paradigms of Structure
Evolving Systems and System of Systems . . . . . . . . . . . . . . . . . . . . . 3
Nicos Karcanias and Maria Livada
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 The Notion of the System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Integrated Design and Operations . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Integrated System Design and Model Complexity Evolution. . . . . 12
1.4.1 Integrated Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.2 Early–Late Design Models: The Family of Fixed-Order
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.3 Early–Late Design: Model Complexity Evolution . . . . . . . 16
1.5 Cascade Design System Evolution. . . . . . . . . . . . . . . . . . . . . . . . 21
1.5.1 Systems Composition and Complexity . . . . . . . . . . . . . . . 22
1.5.2 Systems Instrumentation and Forms of Evolution . . . . . . . 25
1.6 Integrated Operations and Emergent Properties. . . . . . . . . . . . . . . 32
1.6.1 The Multi-modelling and Hierarchical Structure
of Integrated Operations. . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.7 The Notion of System of Systems. . . . . . . . . . . . . . . . . . . . . . . . 40
1.7.1 The Empirical Definition of System of Systems . . . . . . . . 41
1.7.2 CompositeSystemsandSoS:TheIntegratedAutonomous
and Intelligent System . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.7.3 The Systemic Definition of System of Systems . . . . . . . . . 45
1.7.4 Methods for the Characterization of Systems Play. . . . . . . 47
1.8 Conclusions and Future Research . . . . . . . . . . . . . . . . . . . . . . . . 49
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
ix
x Contents
2 Stability and the Kleinian View of Geometry . . . . . . . . . . . . . . . . . . 57
Zoltán Szabó and József Bokor
2.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.1.1 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.1.2 A Projective View. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.2 A Glimpse on Modern Geometry—The Kleinian View. . . . . . . . . 62
2.2.1 Elements of Projective Geometry . . . . . . . . . . . . . . . . . . . 63
2.2.2 Projective Transformations. . . . . . . . . . . . . . . . . . . . . . . . 67
2.2.3 A Trapezoidal Addition . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.3 The Standard Feedback Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.3.1 Youla Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.4 Group of Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.4.1 Indirect Blending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.4.2 Direct Blending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.4.3 Strong Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.4.4 Example: State Feedback . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.5 A Geometry Based Controller Parametrization . . . . . . . . . . . . . . . 82
2.5.1 A Coordinate Free Parametrization . . . . . . . . . . . . . . . . . . 83
2.5.2 Geometric Description of the Parameters. . . . . . . . . . . . . . 85
2.6 From Geometry to Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3 Strong Structural Controllability and Zero Forcing . . . . . . . . . . . . . 91
Henk J. van Waarde, Nima Monshizadeh, Harry L. Trentelman
and M. Kanat Camlibel
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.2 Zero Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.3 Zero Forcing and Structural Controllability . . . . . . . . . . . . . . . . . 94
3.3.1 Strong Structural Controllability. . . . . . . . . . . . . . . . . . . . 95
3.3.2 Leader Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.3.3 Qualitative Subclasses . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.4 Targeted Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.4.1 Output Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.4.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.4.3 Targeted Controllability for QðGÞ. . . . . . . . . . . . . . . . . . . 101
3.4.4 Targeted Controllability for QdðGÞ . . . . . . . . . . . . . . . . . . 105
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112