Table Of ContentLecture Notes in Control and Information Sciences 492
Xiaoli Luan
Shuping He
Fei Liu
Robust Control
for Discrete-Time
Markovian Jump
Systems in
the Finite-Time
Domain
Lecture Notes in Control and Information
Sciences
Volume 492
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· ·
Xiaoli Luan Shuping He Fei Liu
Robust Control
for Discrete-Time Markovian
Jump Systems
in the Finite-Time Domain
XiaoliLuan ShupingHe
KeyLaboratoryofAdvancedProcess KeyLaboratoryofIntelligentComputing
ControlforLightIndustry(Ministry andSignalProcessing(Ministry
ofEducation) ofEducation)
InstituteofAutomation SchoolofElectricalEngineering
JiangnanUniversity andAutomation
Wuxi,Jiangsu,China AnhuiUniversity
Hefei,Anhui,China
FeiLiu
KeyLaboratoryofAdvancedProcess
ControlforLightIndustry(Ministry
ofEducation)
InstituteofAutomation
JiangnanUniversity
Wuxi,Jiangsu,China
ISSN 0170-8643 ISSN 1610-7411 (electronic)
LectureNotesinControlandInformationSciences
ISBN 978-3-031-22181-1 ISBN 978-3-031-22182-8 (eBook)
https://doi.org/10.1007/978-3-031-22182-8
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Preface
Inthefieldofmodernindustry,therearemanyhybridsystemsinvolvingbothcontin-
uousstateevolutionanddiscreteevent-driven,suchasbiochemicalsystems,commu-
nicationnetworks,aerospacesystems,manufacturingprocesses,economicsystems.
These systems often encounter component failure, external environment change,
andsubsystemcorrelationchange,whichwillcauserandomjumpingorswitching
of system structure and parameters. That is, the switching between each mode is
random but may conform to certain statistical laws. If it conforms to Markovian
characteristics,itiscalledstochasticMarkovianjumpsystems(MJSs).Thedynamic
behaviorofMJSsconsistsoftwoforms:oneisthediscretemode,whichisdescribed
byasetofMarkovianchainsvaluedinafiniteintegerset.Theotherisacontinuously
changingstate,characterizedbyadifferential(ordifference)equationforeachmode.
In this sense, the MJSs belong to a category of hybrid systems, and their particu-
laritylyeinthatthediscreteeventsandcontinuousvariablescanbeexpressedbya
stochasticdifferentialequationordifferenceequation.Thisprovidesideasforpeople
toapplythestatespacemethodinmoderncontroltheorytostudysomeproblemsof
MJSs.
Ontheotherhand,thecontroltheoryhasfocusedonthesteady-statecharacter-
isticsofthesystemsintheinfinite-timedomainforalongtime.However,formost
engineeringsystems,thetransientcharacteristicsoverafinite-timeintervalaremore
practical. On the one hand, an asymptotically stable system does not imply good
transitioncharacteristics.Sometimes,thesystemevenappearsviolentshocks,thus
cannotmeettheproductionrequirements;ontheotherhand,manypracticalproduc-
tionprocessesareshorttimerunningsystems,suchasbiochemicalreactionsystems,
economic systems, and people are more interested in their transient performance
in a given time domain. Therefore, this book introduces the theory of finite-time
controlintostochasticdiscrete-timeMJSs,considersthetransientcharacteristicsof
thediscrete-timeMJSsoverafinite-timeinterval,establishesitsstability,bounded-
ness,robustness,andotherperformancesinagiventimedomain,andensuresthat
the state trajectory of the system is limited within a certain range of the equilib-
riumpoint.Inthisway,theengineeringconservativenessofasymptoticstabilityof
conventionalcontroltheoryisreducedfromthetimedimension.
v
vi Preface
Thisbookaimsatdevelopinglessconservativeanalysisanddesignmethodology
for discrete-time MJSs via finite-time control theory. It can be used for final year
undergraduates, postgraduates, and academic researchers. Prerequisite knowledge
includes linear algebra, linear system theory, theory of matrix, stochastic systems,
etc.Itshouldbedescribedasanadvancedbook.
Wuxi,Jiangsu,China XiaoliLuan
[email protected]
Hefei,Anhui,China ShupingHe
[email protected]
Wuxi,Jiangsu,China FeiLiu
fl[email protected]
Acknowledgements
Theauthorswouldliketoexpressoursincereappreciationtothosedirectparticipation
in various aspects of the research leading to this book. Special thanks go to Prof.
Pedro Albertos from the Universidad Politécnica de Valencia in Spain, Prof. Peng
Shi from the University of Adelaide in Australia, Prof. Shuping He from Anhui
University in China, Profs. Fei Liu, Jiwei Wen, and Shunyi Zhao from Jiangnan
University in China for their helpful suggestions, valuable comments, and great
support.Theauthorsalsothankmanycolleaguesandstudentswhohavecontributed
technical support and assistance throughout this research. In particular, we would
liketoacknowledgethecontributionsofWeiXue,HaiyingWan,PengHe,Ziheng
Zhou, Chang’an Han, Chengcheng Ren, Xiang Zhang, and Shuang Gao. Finally,
weareincrediblygratefultoourfamiliesfortheirnever-endingencouragementand
supportwhenevernecessary.
This book was supported in part by the National Natural Science Foundation
of China (Nos. 61991402, 61991400, 61833007, 62073154, 62073001), Scien-
tificResearchCooperationandHigh-levelPersonnelTrainingProgramswithNew
Zealand(No.1252011004200040), theUniversitySynergy Innovation Programof
AnhuiProvince(No.GXXT-2021-010),AnhuiProvincialKeyResearchandDevel-
opment Project (No. 2022i01020013), and Anhui University Quality Engineering
Project(No.2022i01020013,2020jyxm0102,021jxtd017).
vii
Contents
1 Introduction .................................................. 1
1.1 MarkovianJumpSystems(MJSs) ........................... 1
1.1.1 NonlinearMJSs ................................... 2
1.1.2 SwitchingMJSs ................................... 3
1.1.3 Non-homogenousMJSs ............................. 4
1.2 Finite-TimeStabilityandControl ........................... 4
1.2.1 FTSforDeterministicSystems ...................... 6
1.2.2 FTSforStochasticMJSs ............................ 8
1.3 Outline ................................................. 9
References .................................................... 12
2 Finite-TimeStabilityand Stabilization for Discrete-Time
MarkovianJumpSystems ...................................... 21
2.1 Introduction ............................................. 21
2.2 PreliminariesandProblemFormulation ...................... 22
2.3 StochasticFinite-TimeStabilizationforLinearMJSs .......... 24
2.4 StochasticFinite-TimeStabilizationforNonlinearMJSs ....... 26
2.5 SimulationAnalysis ...................................... 33
2.6 Conclusion .............................................. 36
References .................................................... 36
3 Finite-Time Stability and Stabilization for Switching
MarkovianJumpSystems ...................................... 39
3.1 Introduction ............................................. 39
3.2 PreliminariesandProblemFormulation ...................... 40
3.3 StochasticFinite-Time H∞Control ......................... 42
3.4 Observer-BasedFinite-Time H∞Control .................... 51
3.5 SimulationAnalysis ...................................... 60
3.6 Conclusion .............................................. 66
References .................................................... 66
ix
x Contents
4 Finite-TimeStabilityandStabilizationforNon-homegeneous
MarkovianJumpSystems ...................................... 69
4.1 Introduction ............................................. 69
4.2 PreliminariesandProblemFormulation ...................... 70
4.3 StochasticFinite-TimeStabilization ......................... 72
4.4 StochasticFinite-Time H∞Control ......................... 75
4.5 Observer-BasedFinite-TimeControl ........................ 79
4.6 SimulationAnalysis ...................................... 85
4.7 Conclusion .............................................. 90
References .................................................... 90
5 AsynchronousFinite-TimePassiveControlforDiscrete-Time
MarkovianJumpSystems ...................................... 93
5.1 Introduction ............................................. 93
5.2 Finite-TimePassiveControl ............................... 94
5.3 AsynchronousFinite-TimePassiveControl .................. 99
5.4 SimulationAnalysis ...................................... 103
5.5 Conclusions ............................................. 106
References .................................................... 106
6 Finite-Time Sliding Mode Control for Discrete-Time
MarkovianJumpSystems ...................................... 109
6.1 Introduction ............................................. 109
6.2 Finite-TimeSlidingModeControl .......................... 110
6.3 AsynchronousFinite-TimeSlidingModeControl ............. 115
6.4 SimulationAnalysis ...................................... 125
6.5 Conclusion .............................................. 128
References .................................................... 128
7 Finite-Frequency Control with Finite-Time Performance
forMarkovianJumpSystems .................................. 131
7.1 Introduction ............................................. 131
7.2 Finite-Time Stabilization with Finite-Frequency
Performance ............................................. 132
7.3 Finite-Time Multiple-Frequency Control Based
onDerandomization ...................................... 137
7.4 SimulationAnalysis ...................................... 142
7.5 Conclusion .............................................. 147
References .................................................... 150
8 Stochastic Finite-Time Consensualization for Markovian
JumpNetworkswithDisturbances .............................. 151
8.1 Introduction ............................................. 151
8.2 PreliminariesandProblemFormulation ...................... 152
8.3 Finite-TimeConsensualizationwithStateFeedback ........... 154
8.4 Finite-TimeConsensualizationwithOutputFeedback ......... 157
8.5 SimulationAnalysis ...................................... 160
