Stability Analysis of Markovian Jump Systems

200 Pages·2018·4.205 MB·English

by  Yu Kang,Yun-Bo Zhao,Ping Zhao (auth.)

Yu Kang
Yun-Bo Zhao
Ping Zhao
Stability
Analysis of
Markovian
Jump Systems
Stability Analysis of Markovian Jump Systems
Yu Kang Yun-Bo Zhao
(cid:129)
Ping Zhao
Stability Analysis
of Markovian Jump Systems
123
YuKang PingZhao
University of Science Jinan University
andTechnologyof China Jinan
Hefei China
China
Yun-Bo Zhao
ZhejiangUniversity of Technology
Hangzhou
China
ISBN978-981-10-3859-4 ISBN978-981-10-3860-0 (eBook)
DOI 10.1007/978-981-10-3860-0
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Preface
Markovian jump systems typically consist of a finite number of subsystems and a
jumping law governing the active/deactivate mode switches among these subsys-
tems. The subsystems are usually modeled as differential/difference equations, and
thejumpinglawisacontinuous-time/discrete-timeMarkovchain.Markovianjump
systems are a powerful modeling tool in many engineering areas. For instance,
abruptchangesareoftenseeninpracticalsystems,duetotheabruptenvironmental
disturbances,thecomponentandinterconnectionfailures,theabruptchangesofthe
operation point for the nonlinear plant, etc. The system can be modeled as having
differentdynamicsbeforeandaftertheabruptchanges,andthechangesareusually
memoryless and thus Markovian, hence resulting in a Markovian jump system.
Indeed, Markovian jump systems can often be seen in the study of networked
controlsystems,circuitandpowersystems,flightcontrolsystems,roboticsystems,
and so on, where the stability analysis, tracking, fault-tolerant control, etc., have
been extensively discussed. However, the theoretical development of Markovian
jump systems has its own challenges, mainly due to the exclusive Markovian
jumpinglaw.Itiswell-knownthatthewholesystemcanstillbeunstableevenifall
the subsystems are stable, while the whole system can be stable even if all the
subsystems are unstable. Furthermore, the existence of random noises, delays,
nonlinearity, modeling error and disturbance, robust stability, H control and fil-
1
tering, adaptive control, practical stability and optimal control, etc. are also
important topics in Markovian jump systems.
This book discusses the stability analysis of different Markovian jump systems
as well as some applications. With multiple stability definitions, we analyze and
design Markovian jump systems in a systematic manner. This book is written
primarilyforpostgraduatestudentsincontroltheoryandapplications,andcanalso
beusefulfortheresearchersandengineersinthisarea.Inordertousethisbook,the
reader should have the basic knowledge on linear control theory, matrix analysis,
optimization techniques, probability and stochastic processes.
This book contains seven chapters. A brief description of each chapter goes as
follows. Chapter 1 introduces the related history and background of Markovian
jump systems as well as the necessary definitions and notations. Chapter 2 deals
v
vi Preface
with the robust stability and H control issues for a class of uncertain Markovian
1
jumpsystemswithdelays.Chapter3investigatesvariousstochasticstabilitycriteria
for nonlinear Markovian jump systems with asynchronous switching and extended
asynchronousswitching.Chapter4discussesarobustadaptivecontrolschemefora
class of nonlinear uncertain Markovian jump systems with nonlinear
state-dependent uncertainty. Chapter 5 studies the practical stability in probability,
practical stability in the pth mean, and the practical controllability for stochastic
nonlinear Markovian jump systems. Chapter 6 considers the Markovian jump
systemmodelfornetworkedcontrolsystems.Chapter7discussestwoapplications
basedontheMarkovjumptheory,i.e.,thefault-tolerantcontrolforwheeledmobile
manipulators and the jump linear quadratic regulator problem.
We hope the reader will find this book useful.
Hefei, China Yu Kang
May 2017
Acknowledgements
ThanksgotoProf.GuopingLiuattheUniversityofSouthWales,toProf.Ji-Feng
Zhang at the Chinese Academy of Sciences, to Prof. Hongsheng Xi at the
University of Science and Technology of China, to Prof. Shuzhi Sam Ge at the
National University of Singapore (NUS), for their valuable contribution to some
part of the book, and to our students, Mr. Pengfei Li, Mr. Binglin Wang, and Mr.
Jian Zhou, for their time and effort in preparing the materials of the book. Also,
thanks go to the editors at Springer for their reliable help and patience in dealing
with this project.
ThisworkwassupportedinpartbytheNationalNaturalScienceFoundationof
China under grants 61422307, 61673350, and 61673361. The authors also grate-
fullyacknowledgethesupportsfromtheYouthTop-notchTalentSupportProgram,
the 1000-talent Program of China and Zhejiang, and the Youth Yangtze River
Scholarship.
vii
Contents
1 Introduction to Markovian Jump Systems .... .... .... ..... .... 1
1.1 Background .... .... ..... .... .... .... .... .... ..... .... 1
1.1.1 Robust Stochastic Stability .... .... .... .... ..... .... 2
1.1.2 Imprecise Jumping Parameters . .... .... .... ..... .... 3
1.1.3 Nonlinear Markovian Jump Systems. .... .... ..... .... 4
1.1.4 Practical Stability ... .... .... .... .... .... ..... .... 5
1.1.5 Networked Control Systems ... .... .... .... ..... .... 5
1.2 Model Description and Preliminaries .. .... .... .... ..... .... 6
References.. .... .... .... ..... .... .... .... .... .... ..... .... 10
2 Robust Stochastic Stability..... .... .... .... .... .... ..... .... 15
2.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 15
2.2 Uncertain Markovian Jump Linear Systems with Time Delays.... 16
2.3 Robust Control.. .... ..... .... .... .... .... .... ..... .... 20
2.4 Robust H Disturbance Attenuation... .... .... .... ..... .... 31
1
2.5 Numerical Simulation. ..... .... .... .... .... .... ..... .... 37
2.6 Summary .. .... .... ..... .... .... .... .... .... ..... .... 40
References.. .... .... .... ..... .... .... .... .... .... ..... .... 40
3 System with Imprecise Jumping Parameters... .... .... ..... .... 43
3.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 43
3.2 Asynchronous and Extended Asynchronous Switching. ..... .... 45
3.2.1 Asynchronous Switching.. .... .... .... .... ..... .... 47
3.2.2 Extended Asynchronous Switching.. .... .... ..... .... 49
3.3 Stability Analysis Under the Two Switchings.... .... ..... .... 51
3.3.1 Stability Analysis Under Asynchronous Switching... .... 51
3.3.2 Stability Analysis Under Extended Asynchronous
Switching .... ..... .... .... .... .... .... ..... .... 62
ix
x Contents
3.4 Numerical Simulation. ..... .... .... .... .... .... ..... .... 74
3.4.1 Asynchronous Switching.. .... .... .... .... ..... .... 74
3.4.2 Extended Asynchronous Switching.. .... .... ..... .... 83
3.5 Summary .. .... .... ..... .... .... .... .... .... ..... .... 93
References.. .... .... .... ..... .... .... .... .... .... ..... .... 94
4 Nonlinear Markovian Jump Systems. .... .... .... .... ..... .... 97
4.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 97
4.2 Description of Nonlinear Uncertain Jump System .... ..... .... 98
4.3 Robust Adaptive Control for Nonlinear
Uncertain Jump Systems.... .... .... .... .... .... ..... .... 100
4.4 Numerical Simulation. ..... .... .... .... .... .... ..... .... 108
4.5 Summary .. .... .... ..... .... .... .... .... .... ..... .... 111
References.. .... .... .... ..... .... .... .... .... .... ..... .... 112
5 Practical Stability ... .... ..... .... .... .... .... .... ..... .... 113
5.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 113
5.2 Markovian Jump Nonlinear Systems with Time Delays..... .... 114
5.3 Comparison Principle. ..... .... .... .... .... .... ..... .... 115
5.4 Practical Stability .... ..... .... .... .... .... .... ..... .... 116
5.5 Practical Controllability .... .... .... .... .... .... ..... .... 119
5.6 Optimal Control . .... ..... .... .... .... .... .... ..... .... 121
5.7 Summary .. .... .... ..... .... .... .... .... .... ..... .... 128
References.. .... .... .... ..... .... .... .... .... .... ..... .... 128
6 Networked Control System: A Markovian
Jump System Approach .. ..... .... .... .... .... .... ..... .... 131
6.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 131
6.2 Description of Networked Control Systems . .... .... ..... .... 132
6.3 Packet-Based Control for NCSs .. .... .... .... .... ..... .... 134
6.4 Stochastic Modeling and Stabilization . .... .... .... ..... .... 137
6.4.1 The MJS Model of the Packet-Based Control Approach
for NCSs. .... ..... .... .... .... .... .... ..... .... 138
6.4.2 Stochastic Stability and Stabilization. .... .... ..... .... 140
6.5 Numerical Simulation. ..... .... .... .... .... .... ..... .... 143
6.6 Summary .. .... .... ..... .... .... .... .... .... ..... .... 146
References.. .... .... .... ..... .... .... .... .... .... ..... .... 146
7 Applications Based on the Markov Jump Theory... .... ..... .... 149
7.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 149
7.2 Robotic Manipulator System. .... .... .... .... .... ..... .... 152
7.2.1 Introduction to the System .... .... .... .... ..... .... 152
7.2.2 Output Feedback Controller Based on High-Gain
Observer. .... ..... .... .... .... .... .... ..... .... 155
7.2.3 Markovian Model and Problem Statement .... ..... .... 156
Contents xi
7.2.4 Stability Analysis ... .... .... .... .... .... ..... .... 159
7.2.5 Numerical Simulation .... .... .... .... .... ..... .... 165
7.3 Optimal Control Problem of MJLS.... .... .... .... ..... .... 169
7.3.1 An Description of Optimal Control Problem... ..... .... 169
7.3.2 Two-Level Regulating Method . .... .... .... ..... .... 171
7.3.3 Two Special Cases .. .... .... .... .... .... ..... .... 178
7.3.4 Numerical Simulation .... .... .... .... .... ..... .... 184
7.4 Summary .. .... .... ..... .... .... .... .... .... ..... .... 186
References.. .... .... .... ..... .... .... .... .... .... ..... .... 187
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 191