Table Of ContentEfim N. Rosenwasser
Torsten Jeinsch
Wolfgang Drewelow
Sampled-Data
Control
for Periodic
Objects
Sampled-Data Control for Periodic Objects
· ·
Efim N. Rosenwasser Torsten Jeinsch
Wolfgang Drewelow
Sampled-Data Control
for Periodic Objects
EfimN.Rosenwasser TorstenJeinsch
DepartmentofShipAutomation LehrstuhlRegelungstechnik
MarineTechnicalUniversitySt.Petersburg UniversitätRostock
SaintPetersburg,Russia Rostock,Mecklenburg-Vorpommern
Germany
WolfgangDrewelow
LehrstuhlRegelungstechnik
UniversitätRostock
Rostock,Mecklenburg-Vorpommern
Germany
ISBN 978-3-031-01955-5 ISBN 978-3-031-01956-2 (eBook)
https://doi.org/10.1007/978-3-031-01956-2
MathematicsSubjectClassification:93C05,93C35,93C57,93C80,93C83,93D15,93E20
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Wededicatethisbooktothememoryofour
goodfriendandcolleagueBernhardP.
Lampe,whomademajorcontributions tothe
underlyingtheoryofcomputer-controlled
systems.
Preface
Oneoftheactualproblemsinmoderncontroltheoryisthecontrolofobjectswith
periodicallychangingparameters.Manyoftheseobjectsaredescribedwithasuffi-
cientdegreeofadequacybymathematicalmodelsintheformofsystemsoflinear
differential equations with periodic coefficients. In the literature, such objects are
referredtoasFinite-DimensionalLinearContinuousPeriodic(FDLCP),alsosimply
referredtoasperiodicbelow.Anincompletelistanddescriptionofdifferenttypes
ofsuchFDLCPobjectscanbefoundin[1]–[10]andtheliteraturecitedthere.
This monograph deals with the problem of digital control of FDLCP objects.
The corresponding closed-loop system belongs to the class of sampled-data (SD)
systems,becauseitisbuiltofbothcontinuous-anddiscrete-timecomponents.
The main mathematical apparatus used in this book is a method based on the
concept of the parametric transfer function or Parametric Transfer Matrix (PTM),
respectively.Bydefinition,unliketheordinarytransferfunctionW(s)forlineartime-
invariantsystems,theparametrictransferfunctionW(s,t)dependsonthecomplex
variables andthetimet asparameters.
The concept of the parametric transfer function was introduced into control
theoryintheworkofL.A.Zadeh[11],[12].Theideaoftransferringthisconcept
tosampled-datasystemswithcontinuouslineartime-invariant (LTI)elementswas
putforwardintheworkofJ.R.RagazziniandL.A.Zadeh[13].Thepossibilityof
transferringtheparametrictransferfunctionmethodtoSDsystemswithaperiodic
non-stationarycontinuousobjectwassetin[14].
Asystematicdescriptionofthemethodsfortheanalysisandsynthesisofsingle-
input single-output sampled-data systems with continuous LTI elements based on
theparametrictransferfunctionisgiveninthemonographs[15]–[19].
This parametric transfer function approach has been generalized to multidi-
mensional computerized systems with continuous linear time-invariant objects by
introducingtheconceptofthePTM[20].
Inmonograph[21],thisapproachwasextendedtosystemswithdelay.
Themaincontentofthismonographisdividedintofiveparts.
vii
viii Preface
PartI,comprisingChaps.1–4,developstheresultsof[22]andisdevotedtothe
formulation of the frequency approach for the mathematical description and study
ofFDLCPobjects.
In Chap. 1, discrete operational transformations of functions of the continuous
argumentt intheinfiniteinterval−∞<t <∞,aswellasarepresentationofthis
transformationbythetwo-sidedLaplacetransformation,areintroduced.Thisleads
to the definition of the discrete Laplace transformation of functions of continuous
argumentsandthecorrespondingtransformationoftheimage.Besides,thefirstpart
containsachapterdevotedtotheoperatordescriptionofLTIobjects.
Chapter2providesnecessarymaterialfromthetheoryoflineardifferentialequa-
tionswithperiodiccoefficients,whichisanimportantbasisforthefurtherpresen-
tation. The methodology for calculating the response of LTI objects to periodic
andexponentialperiodicinputsignalsusingthediscreteLaplacetransformationis
introduced.
Chapter 3 defines the parametric transfer matrix of an FDLCP object based on
thecontentofthepreviouschaptersandgivesgeneralrelationsdefiningthisPTM.
Chapter4introducestheconceptoftheFloquet–Lyapunovdecomposition,which
makesitpossibletorepresentaFDLCPobjectintheformofaserialconnectionof
anLTIelementandmodulationelementsatitsinputandoutput,respectively.
PartIIconsistsofChaps.5–7.Thesechaptersdescribemethodsforconstructing
thePTMsofopenandclosedsampled-datasystemswithaperiodicobjectanddelay.
The discussion includes both synchronous systems, in which the period of
the FDLCP object is equal to the sampling period of the discrete controller, and
asynchronoussystems,wheretheseperiodsdifferbyanintegerfactor.
Chapter5dealswiththeopen-loopsampled-datasystemwithoutdelay.Chapter6
develops the parametric transfer functions for open-loop synchronous and asyn-
chronoussampled-datasystemsincaseofadelay.Chapter7providesgeneralexpres-
sionsthatdefinePTMsfordifferenttypesofclosed-loopSDsystemswithFDLCP
objectsanddelays.
PartIIIincludesChaps.8–12.Inthispart,thetechniqueofsolvingtheproblems
ofmodalcontrolandstabilizationofclosed-loopSDsystemsofconsideredclasses
is developed. Chapters 8–9 contain the information necessary for understanding
the subsequent presentation from the theory of polynomial and rational matrices.
Chapter10describesthemethodofsolvingproblemsofmodalcontrolandstabiliza-
tion of SD systems based on applying the mathematical apparatus of Determinant
PolynomialEquations(DPEs).Thepeculiarityoftheproposedapproachistheuse
of discrete models presented in polynomial form described by the backward oper-
ator(reverseshift)ζ.Usingthismodelformeliminatestheproblemofseparationof
thesubsetofcausalcontrollersfromthewholesetofstabilizingcontrollersbecause
alldiscretestabilizingcontrollersforastrictlycausaldiscretebackwardobjectare
causal.InChap.11,thetasksofsynchronouscontrolstabilizationofFDLCPobjects
areconsideredbasedonthedescribedapproach.Forthispurpose,adiscreteback-
wardmodelisbuiltforwhichthesetofallcausallystabilizingdiscretecontrollers
isgeneratedusingtheDPEmathematicalapparatus.InChap.12,asimilarapproach
Preface ix
is used to solve the problem of stabilizing asynchronous closed-loop SD systems,
includingdelayintheloop.
Part IV, formed by Chaps. 13–19, is devoted to the construction of a quality
functionalforsolvingthe H -optimizationproblemofthesynchronousclosed-loop
2
SD system Sτ with FDLCP object and delayed control. Chapters 13–15 establish
someadditionalrelationsforthePTMofthesystemSτ requiredfortheconstruction
ofthequalityfunctional.Chapters16and17showthatthesetoftransfermatricesof
allcausaldiscretecontrollersthatstabilizethesystemSτ canbeparameterizedbya
stablerationalmatrixθ(ζ),whichiscalledthesystemfunction,andarepresentation
ofthePTMofthesystemSτ bythesystemfunctionisconstructed.InChaps.18,19,
therelationsexpressingthe H2-normofthesystemSτ byitsPTMaregiven.Using
theserelations,anintegralqualityfunctionalisconstructedwhichdependsonmatrix
θ˜(s)=Δθ(e−sT)specifiedontheimaginaryaxisintheinterval−ω ≤Im(s)≤ ω with
2 2
ω= 2π.
T
PartV(Chaps.20–25)isdevotedtothemethodofH -optimizationofthesystem
2
Sτ, based on minimization of the quadratic quality functional, built in Chap. 19.
Chapter20providesauxiliaryinformationfromthetheoryofscalarandmatrixquasi-
polynomials, including an important factorization theorem from [23]. Chapter 21
deals with the algorithm of minimizing the quadratic functional on the unit circle,
based on the results of [24], [25]. Chapters 22 and 23 describe practical ways of
building the matrix η(s,t), which appears in the PTM representation through the
system function and matrix C˜ (s,t), which is included in the expression of the
T
constructedqualityfunctional.InChap.24,thegeneratedqualityfunctionalistrans-
lated into a functional specified on the circle |ζ| = 1 by replacing the integration
variable ζ = e−sT. It is shown that the coefficients in the integral expression of
thetransformedfunctionalarequasi-polynomials,wheretwoofthemonthecircle
|ζ| = 1 are non-negative. The final chapter (Chap. 25) is devoted to solving the
H2-optimization problem for the system Sτ. A general algorithm for minimizing
theconstructedtransformedqualityfunctionalisgiven.Asanapplicationexample
oftheproposedmethod,thesequentialsolutionofthe H -optimizationproblemfor
2
thesystemSτ withanFDLCPobjectoffirstorderisdemonstrated.
The book is intended for researchers and engineers, involved in research and
developmentofmoderncontrolsystems,aswellasforteachers,postgraduates,and
undergraduates.
TheauthorsverygratefullyacknowledgethefinancialsupportfromtheDeutsche
Forschungsgemeinschaft.Wewouldliketoexpressourgratitudeforthededicated
andcarefulworkofOliverJacksonofSpringer,whohelpedusgreatlyinovercoming
editorialproblems.WethankVladislavRybinskiiandRenateZieglerfortheirhelp
infinalizingthemanuscript.
SaintPetersburg,Russia EfimN.Rosenwasser
Rostock,Germany TorstenJeinsch
Rostock,Germany WolfgangDrewelow
Contents
PartI TheFrequencyApproachtotheMathematicalDescription
ofLinearPeriodicObjects
1 Discrete Operational Transformations of Continuous
ArgumentFunctionsandOperatorDescriptionofLinear
Time-InvariantSystems ........................................ 3
1.1 DiscreteLaplaceTransformationofContinuousArgument
Functions ............................................... 3
1.2 DiscreteLaplaceTransformoftheImage .................... 5
1.3 Operator Description of a Finite-Dimensional Linear
Time-InvariantSystem .................................... 9
1.4 TransferofanExponentiallyPeriodicSignalThrough
aLinearTime-InvariantSystem ............................ 11
2 State-Space Analysis of Finite-Dimensional Linear
ContinuousPeriodicObjects ................................... 17
2.1 State-SpaceDescriptionofPeriodicObjects .................. 17
2.2 TransferofPeriodicSignalsThroughPeriodicObjects ......... 20
2.3 Transfer of Exponentially Periodic Signals Through
PeriodicObjects ......................................... 23
2.4 Higher-OrderPeriodicObjects ............................. 27
3 TheFrequencyMethodintheTheoryofPeriodicObjects ......... 31
3.1 FrequencyDescriptionofLinearPeriodicOperators ........... 31
3.2 LinearPeriodicIntegralOperators .......................... 33
3.3 OperatorDescriptionofaBasicPeriodicObject .............. 39
3.4 ParametricTransferMatrixoftheBasicPeriodicObject ....... 43
3.5 ParametricTransferMatrixofaComplementedPeriodic
Object .................................................. 45
4 TheFloquet–LyapunovDecompositionandItsApplication ........ 51
4.1 Floquet–LyapunovTransformation .......................... 51
xi
xii Contents
4.2 Floquet–Lyapunov Decomposition and Its Parametric
TransferMatrix .......................................... 53
4.3 PeriodicObjectwithDelay ................................ 54
4.4 Low-Frequency Exponentially Periodic Excitation
oftheFloquet–LyapunovDecomposition .................... 58
PartII The Parametric Transfer Matrix Approach to
Sampled-DataSystemswithPeriodicObjects
5 Open-LoopSampled-DataSystemwithPeriodicObject ........... 63
5.1 MultivariableZero-OrderHold ............................. 63
5.2 LinearizedModeloftheDigitalController ................... 65
5.3 Open-LoopSystemwithTime-InvariantObject ............... 70
5.4 SynchronousOpen-LoopSystemwithPeriodicObject ......... 72
5.5 AsynchronousRisingOpen-LoopSystemwithPeriodic
Object .................................................. 75
5.6 Open-Loop System with Periodic Object
andHigh-FrequencyHold ................................. 78
6 Open-Loop Sampled-Data System with Periodic Object
andDelay .................................................... 83
6.1 Open-LoopSystemwithLinearTime-InvariantObject
andDelay ............................................... 83
6.2 SynchronousOpen-LoopSystemwithPeriodicObject
andDelay ............................................... 89
6.3 AsynchronousRisingOpen-LoopSDSystemwithPeriodic
ObjectandDelay ......................................... 94
6.4 Open-LoopSystemwithPeriodicObject,High-Frequency
HoldandDelay .......................................... 98
7 Closed-Loop Sampled-Data SystemwithPeriodic Object
andDelay .................................................... 101
7.1 SynchronousClosed-LoopSystem .......................... 101
7.2 AsynchronousRisingClosed-LoopSystem .................. 106
7.3 Closed-LoopSystemwithPeriodicObject,High-Frequency
HoldandDelay .......................................... 110
PartIII Determinant Polynomial Equations, Sampled-Data
ModalControlandStabilizationofPeriodicObjects
8 PolynomialMatrices ........................................... 115
8.1 GeneralDefinitionsandProperties .......................... 115
8.2 PolynomialMatricesEquivalence ........................... 117
8.3 LatentEquationandLatentNumbers ........................ 118
8.4 PairsofPolynomialMatrices ............................... 119
