Control of Singular Systems with Random Abrupt Changes

268 Pages·2008·2.769 MB·English

by  El-Kebir Boukas (auth.)

Communications and Control Engineering
SeriesEditors
E.D.Sontag·M.Thoma·A.Isidori·J.H.vanSchuppen
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El-K´ebir Boukas
Control of Singular
Systems with Random
Abrupt Changes
123
Prof.E.K.Boukas
MechanicalEngineeringDepartment
E´colePolytechniquedeMontr´eal
SuccirsaleCentre-VilleH3C
3A7Montreal,Quebec
Canada
email:[email protected]
ISBN:978-3-540-74344-6 e-ISBN:978-3-540-74345-3
DOI10.1007/978-3-540-74345-3
CommunicationsandControlEngineeringSeriesISSN:0178-5354
LibraryofCongressControlNumber:2007934281
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Dedicated tomywifeSaida andmy kidsImane,Ibtissamaand Anas
Preface
Singular systems also referred to as descriptor systems, implicit systems, general-
ized state-space systems, differential-algebraic systems or semi-state systems (see
[43,76])representsaninterestingclassofdynamicalsystemssinceitcombinesdif-
ferentialequationsandalgebraicequationsandgeneralizesthelineartimeinvariant
modelwhichis extensivelyused in linearcontroltheory.Thisclass of systemshas
been used to model varieties of systems like economics, chemical processes, me-
chanics,electricalsystems,etc.Singularsystemshaveattractedalotofresearchers
from the mathematicaland controlcommunitiesand a great numberof fundamen-
tal notionsand resultsin controland systemstheorybased onlinear time-invariant
systemswithstatespacerepresentationhavebeensuccessfullyextendedtotheclass
of singular systems. For more details on what it has been done on this subject we
referthereaderto[3,4,51,75,122,123,127,50,109,112,122,124,126],andthe
referencestherein.Morespecifically,wetackledthestabilityandthestabilizability
ofthe classoflinearcontinuous-timesingularsystemsandmanyresultshavebeen
reportedintheliteratureintheLMIsetting.Amongtheseresultswequotethoseof
[14].Therobuststabilityandtherobuststabilizationhavealsobeenstudied.Many
typesofuncertaintieshavebeenconsideredamongthemwequotethenormbounded
form,thelinearfractionaltransformation(LFT,whichgeneralizesthenormbounded
type)andpolytopicform.
TheH∞controlandthefilteringproblemshavealsobeentreatedandinteresting
resultswerereportedintheliterature.FortheH∞control,undertheassumptionsthat
theexternaldisturbanceshavefinitepowerorfiniteenergy,acontrollawisdesigned
to guarantee that the closed-loop system is regular, impulse-free and stable and at
the same time assures the disturbance rejection with a given level γ > 0. For the
filteringproblem,theobjectiveistodesignadynamicalestimatorthatestimatesthe
state vectorthat can be used in the controlto make the closed-loopof the singular
systemregular,impulse-freeandstable.
Someofthepracticalsystemsarestochasticormorespecificallywecanmodel
thembyaclassofstochasticsystemsdrivenbycontinuous-timeMarkovchainsthat
wewillrefertoastheclassofstochasticsystemswithabruptchanges.Thisclassof
systemsismoreappropriatetomodelmanypracticalsystems,whererandomfailures
VIII
andrepairsandsuddenenvironmentchangesmayoccur.Formoredetailonwhatit
has been done on the subject, we refer the reader to [27], [96] and the references
therein.Thisclassofsystemshasalsoattractedalotofresearchersfrombothmath-
ematicalandcontrolcommunity.Manyresultsonstochasticstabilityandstochastic
stabilization have been reported in the literature. For more details on these results
wereferthereaderto[29,26,49,60,94]andthereferencestherein,wheredifferent
approacheshavebeenused.TheH∞ controlproblemwasinvestigatedin[45,105],
wheresufficientconditionsforthe solvabilityofthisproblemwasproposed.When
timedelaysappearinaMarkovianjumpsystem,theresultsonstabilityanalysisand
H∞ controlwerealsoreportedin[31],[24]and[25]fordifferenttypesoftimede-
lays. For moredetail on Markovianjumpingsystems with time delay,we refer the
readerto[14,27]andthereferencestherein.
Ourgoalinthisvolumeistocombinetheclassofsingularsystemswiththeone
of systems with abrupt changes which will give the class of systems that we will
refer to as the class of stochastic singular systems with abrupt changes. Our main
objectiveinthisvolumeistotreatthestabilityandthestabilizationproblemsusing
differenttechniques.Wewillalsohandlethefilteringproblem.
Therestofthisbookisorganizedasfollows.InChap.1,thedifferentproblems
arestatedandthenecessaryassumptionsaregiven.Chapter2dealswiththestabil-
ity problemof the class of singularsystems with randomabruptchangesand LMI
conditionsaredevelopedtocheckifagivensystemofthisclassofsystemsispiece-
wise regular,impulse-freeandstochasticallystable.Therobuststabilityproblemis
alsoconsidered.Chapter3 treatsthe stabilizationproblemanditsrobustness.State
space controllersare consideredand LMI design approachesare developed.Chap-
ter 4 deals with the H∞ control for the class of singular piecewise deterministic
systems.InChap.5,thestaticoutputstabilizationistackledandLMIresultsarede-
veloped for the class of singular systems with random abrupt changes. The robust
caseisalsoconsidered.Chapter6dealswithobserver-basedoutputstabilizationfor
the class of Markovianjump singular systems. In Chapt. 7 the filtering problemis
consideredand design proceduresare proposedin the LMI formalism to solve the
theH∞filteringproblem.InChap.8,theguaranteedcostcontrolproblemistackled
andLMIresultsaredevelopedtosynthesizethestatefeedbackcontrollerthatmakes
the closed-looppiecewise regular,impulse-freeandstochastically stable andat the
sametimeguaranteedanupperboundforthecostforalladmissibleuncertainties.In
Chap.9,themixedH2/H∞ controlistackledanddesignprocedureisdevelopedto
synthesizeastatefeedbackcontroller.Finally,Chap.10providessometoolsthatcan
beusedtosolvealltheLMIconditions.Itgivesanideatothereaderonhowtowrite
hisprogramunderMathlabtosolvetheconsideredproblemofthisclassofsystems.
Contents
PartI Modelingandproblemstatements
1 Introduction .................................................. 3
1.1 ExamplesofSingularSystems................................ 4
1.2 ProblemStatements ........................................ 10
1.3 SolutionfortheDynamics ................................... 14
1.4 StochasticH2 andH∞ Norms................................ 16
1.5 Lemmas .................................................. 17
1.6 Notes..................................................... 18
PartII Stochasticstability
2 Stability....................................................... 23
2.1 ProblemStatement ......................................... 23
2.2 StabilityofSingularSystems................................. 25
2.3 RobustStability............................................ 32
2.4 NumericalExamples........................................ 39
2.5 Notes..................................................... 43
PartIII Stabilization
3 StateFeedbackStabilization..................................... 49
3.1 ProblemStatement ......................................... 49
3.2 DesignofStateFeedbackStabilization......................... 51
3.3 ConstantGainStateFeedbackStabilization..................... 67
3.4 NumericalExamples........................................ 77
3.5 Notes..................................................... 80
X Contents
4 H∞ Stabilization............................................... 81
4.1 ProblemStatement ......................................... 81
4.2 H∞ StateFeedbackStabilization ............................. 82
4.3 H∞ ConstantGainStabilization .............................. 113
4.4 NumericalExamples........................................ 127
4.5 Notes..................................................... 130
5 OutputFeedbackStabilization................................... 131
5.1 ProblemStatement ......................................... 131
5.2 StaticOutputFeedback...................................... 133
5.3 H∞ StaticOutputFeedbackControl........................... 143
5.4 NumericalExamples........................................ 151
5.5 Notes..................................................... 155
6 Observer-BasedFeedbackStabilization........................... 157
6.1 ProblemStatement ......................................... 158
6.2 Observer-BasedStabilizationofSingularSystems ............... 159
6.3 RobustObserver-BasedStabilizationofSingularSystems......... 168
6.4 NumericalExample......................................... 176
6.5 Notes..................................................... 179
PartIV Filtering
7 Filtering ...................................................... 185
7.1 ProblemStatement ......................................... 185
7.2 H∞ FilteringforNominalSystems............................ 187
7.3 RobustFiltering............................................ 196
7.4 NumericalExamples........................................ 209
7.5 Notes..................................................... 211
PartV SingularOptimalControl
8 GuaranteedCostControl ....................................... 217
8.1 ProblemStatement ......................................... 217
8.2 GuaranteedCostBound ..................................... 219
8.3 GuaranteedCostControlDesign.............................. 222
8.4 NumericalExample......................................... 228
8.5 Notes..................................................... 230
9 MixedH2/H∞ ControlProblem................................. 231
9.1 ProblemStatement ......................................... 232
9.2 MixedH2/H∞Control:NominalCase ........................ 235
9.3 MixedH2/H∞Control:UncertainCase ....................... 242
9.4 NumericalExample......................................... 248