State Space Consistency and Differentiability

79 Pages·2015·0.832 MB·English

by  Serakos, Demetrios

SPRINGER BRIEFS IN OPTIMIZATION
Demetrios Serakos
State Space
Consistency and
Diff erentiability
123
SpringerBriefs in Optimization
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StephenM.Robinson
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MyT.Thai
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Demetrios Serakos
State Space Consistency
and Differentiability
123
DemetriosSerakos
Dahlgren,VA,USA
ISSN2190-8354 ISSN2191-575X (electronic)
SpringerBriefsinOptimization
ISBN978-3-319-14468-9 ISBN978-3-319-14469-6 (eBook)
DOI10.1007/978-3-319-14469-6
LibraryofCongressControlNumber:2014958663
MathematicsSubjectClassification(2010):93B28;47N70;47N20
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©DemetriosSerakos2014
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MemoryofGeorgia(Pantelopoulos)and
DemetriosSerakos

Foreword
This book presents an analysis of input–output systems with regard to the
mathematical concept of “state.” The state of a system condenses, in a usable
way,theeffectsofpastinputstothesystem.Thisbookistheresultofatheoretical
examinationofthestateconceptandtheconditionsunderwhichstateassumptions
arevalid.Theresultsdevelopedinthisbookareusefulaidsforunderstandingproper
modeling of physical systems, including the development of guidance and control
algorithms,andindevelopingsimulations.Anearlierversionofthisbookappeared
asaNavalSurfaceWarfareCenter,DahlgrenDivisiontechnicalreport.1
TheauthorisindebtedtoProfessorWilliamL.Root(1919–2007)forhelpwith
thetextandwithProposition20andforprovidingProposition15.Theauthorwishes
tothankanonymousreviewersforvariouscommentsandforsimplifyingExample8.
Mr.JohnE.Grayprovidedthefirstthreeparagraphsoftheintroduction.Also,the
authorwishestothankMs.RaziaAmzadfromSpringerandDr.GlennHeitman.
Dahlgren,VA,USA DemetriosSerakosPh.D.
1D. Serakos, “State Space Consistency and Differentiability Conditions for a Class of Causal
DynamicalInput-OutputSystems,”NSWCDD/TR08/3,NavalSurfaceWarfareCenter—Dahlgren
Division, Dahlgren, VA 22448. Posted on www.dtic.mil. This version was not sponsored by
NSWCDDanddoesnotnecessarilyreflecttheviewsorpoliciesoftheUSNavyorUSGovernment.
vii

Contents
1 Introduction ................................................................... 1
2 Preliminaries .................................................................. 5
3 SomeStateSpaceProperties ................................................ 13
4 StateDifferentiabilityPropertiesinInput–OutputSystems ............ 23
5 SummaryandConclusions .................................................. 39
A BoundednessandContinuityoftheNaturalState ........................ 41
B ProofofProposition20....................................................... 45
C OnShiftDifferentiabilityofTimeFunctions .............................. 63
D OnShiftDifferentiabilityofSystemTrajectories ......................... 67
References.......................................................................... 71
ix