Table Of ContentSPRINGER BRIEFS IN ELECTRICAL AND COMPUTER
ENGINEERING CONTROL, AUTOMATION AND ROBOTICS
Andrew D. Lewis
Tautological
Control Systems
SpringerBriefs in Electrical and Computer
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Control, Automation and Robotics
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Andrew D. Lewis
Tautological Control
Systems
123
Andrew D.Lewis
Department of Mathematics andStatistics
Queen’s University
Kingston
Canada
ISSN 2192-6786 ISSN 2192-6794 (electronic)
ISBN 978-3-319-08637-8 ISBN 978-3-319-08638-5 (eBook)
DOI 10.1007/978-3-319-08638-5
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Preface
Thelackoffeedback-invarianceofmathematicalformulationsofnonlinearcontrol
theory has been a thorn in the side of understanding the basic structure of control
systems. Moreover, it is a thorn whose presence has largely come to be accepted,
andthishasprohibitedacompleteunderstandingofcertainfundamentalstructural
problemsfornonlinearsystems.Onewaytounderstandtheissueisasfollows:just
as an explicit parameterisation of system dynamics by state, i.e., a choice of
coordinates,canimpedetheidentificationofgeneralstructure,soitistoowithan
explicit parameterisation of system dynamics by control. However, such explicit
andfixedparameterisationbycontroliscommonplaceincontroltheory,leadingto
definitions, methodologies and results that depend in unexpected ways on control
parameterisation. This unexpected dependence makes it virtually impossible to
comprehensively address the fundamental structural problems in control theory,
such as controllability and stabilisability.
Inthismonograph,wepresentaframeworkformodellingsystemsingeometric
control theory in a manner that does not make any choice of parameterisation by
control;thesystemsarecalled‘TautologicalControlSystems’.Fortheframework
to be coherent, it relies in a fundamental way on topologies for spaces of vector
fields. As such, we take advantage of recent characterisations of topologies for
spaces of vector fields possessing a variety of degrees of regularity: finitely dif-
ferentiable;Lipschitz;smooth;realanalytic.Aspartofthepresentation,therefore,
locally convex topologies for spaces of vector fields are comprehensively
reviewed. It is these locally convex topologies that provide for the unified treat-
ment of time-varying vector fields that underpin the approach.
Thismonographpresentssimplythefoundationsoftheapproach,aswellasthe
basic results that indicate the structural attributes of ‘Tautological Control Sys-
tems’.Inparticular,weareabletoprovethefeedback-invarianceoftheapproach.
Future work will involve using this feedback-invariant approach to address the
basicproblemsofcontroltheory,e.g.,controllability,stabilisability,andoptimality.
The author was a Visiting Professor in the Department of Mathematics at
UniversityofHawaii,Manoa,whenthemonographwaswritten,andwouldliketo
acknowledgethehospitalityofthedepartment,particularlythatofMoniqueChyba
v
vi Preface
and George Wilkens. The author also thanks his departmental colleague at
Queen’s, Mike Roth, for numerous useful conversations over the years. While
conversationswithMikedidnotleaddirectlytoresultsinthismonograph,Mike’s
willingness tochat about complex geometry and to answer ill-informed questions
wasalwaysappreciatedand,ultimately,veryhelpful.Jointworkontopologiesfor
spaces of vector fields with the author’s Doctoral student, Saber Jafarpour, was
essential to the completion of this work.
Honolulu, May 2014 Andrew D. Lewis
Contents
1 Introduction, Motivation, and Background . . . . . . . . . . . . . . . . . . 1
1.1 Models for Geometric Control Systems: Pros and Cons. . . . . . . 2
1.1.1 Family of Vector Field Models. . . . . . . . . . . . . . . . . . . 2
1.1.2 Models with Control as a Parameter . . . . . . . . . . . . . . . 3
1.1.3 Fibred Manifold Models . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.4 Differential Inclusion Models. . . . . . . . . . . . . . . . . . . . 8
1.1.5 The ‘‘Behavioural’’ Approach. . . . . . . . . . . . . . . . . . . . 9
1.2 An Introduction to Tautological Control Systems . . . . . . . . . . . 9
1.2.1 Attributes of a Modelling Framework for Geometric
Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2 The ‘‘Essentials’’ of Tautological Control Theory. . . . . . 10
1.3 An Outline of the Monograph. . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Notation, Conventions, and Background. . . . . . . . . . . . . . . . . . 14
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Topologies for Spaces of Vector Fields . . . . . . . . . . . . . . . . . . . . . 21
2.1 An Overview of Locally Convex Topologies
for Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.1 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.2 Families of Seminorms and Topologies
Defined by These . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Seminorms for Locally Convex Spaces of Vector Fields . . . . . . 24
2.2.1 Fibre Norms for Jet Bundles . . . . . . . . . . . . . . . . . . . . 24
2.2.2 Seminorms for Spaces of Smooth Vector Fields. . . . . . . 26
2.2.3 Seminorms for Spaces of Finitely Differentiable
Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.4 Seminorms for Spaces of Lipschitz Vector Fields. . . . . . 27
2.2.5 Seminorms for Spaces of Holomorphic Vector Fields . . . 28
2.2.6 Seminorms for Spaces of Real Analytic Vector Fields. . . 28
2.2.7 Summary and Notation . . . . . . . . . . . . . . . . . . . . . . . . 29
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
vii
viii Contents
3 Time-Varying Vector Fields and Control Systems. . . . . . . . . . . . . 31
3.1 Time-Varying Vector Fields. . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.1 Time-Varying Smooth Vector Fields. . . . . . . . . . . . . . . 32
3.1.2 Time-Varying Finitely Differentiable and Lipschitz
Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.3 Time-Varying Holomorphic Vector Fields. . . . . . . . . . . 34
3.1.4 Time-Varying Real Analytic Vector Fields . . . . . . . . . . 35
3.1.5 Topological Characterisations of Spaces
of Time-Varying Vector Fields. . . . . . . . . . . . . . . . . . . 37
3.1.6 Mixing Regularity Hypotheses . . . . . . . . . . . . . . . . . . . 38
3.2 Parameterised Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1 The Smooth Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.2 The Finitely Differentiable or Lipschitz Case. . . . . . . . . 40
3.2.3 The Holomorphic Case . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.4 The Real Analytic Case. . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.5 Topological Characterisations of Parameterised
Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.6 Mixing Regularity Hypotheses . . . . . . . . . . . . . . . . . . . 43
3.3 Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.1 Control Systems with Locally Essentially
Bounded Controls. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.2 Control Systems with Locally Integrable Controls. . . . . . 44
3.3.3 Differential Inclusions. . . . . . . . . . . . . . . . . . . . . . . . . 46
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4 Presheaves and Sheaves of Sets of Vector Fields. . . . . . . . . . . . . . 49
4.1 Definitions and Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Sheafification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 The Étalé Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4 Stalk Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5 Tautological Control Systems: Definitions
and Fundamental Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.1 Tautological Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Open-Loop Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3 Trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Attributes that can be Given to Tautological
Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.5 Trajectory Correspondences with Other Sorts
of Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.6 The Category of Tautological Control Systems. . . . . . . . . . . . . 90
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Contents ix
6 Étalé Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.1 Sheaves of Time-Varying Vector Fields. . . . . . . . . . . . . . . . . . 98
6.2 An Alternative Description of Local Sections of Sheaves
of Time-Varying Vector Fields. . . . . . . . . . . . . . . . . . . . . . . . 100
6.3 Étalé Open-Loop Systems and Open-Loop Subfamilies . . . . . . . 104
6.4 Étalé Trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7 Ongoing and Future Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.1 Linearisation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.2 Optimal Control Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.3 Controllability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.4 Feedback and Stabilisation Theory . . . . . . . . . . . . . . . . . . . . . 115
7.5 The Category of Tautological Control Systems. . . . . . . . . . . . . 116
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
