Table Of ContentYang Liu · Jianquan Lu
Liangjie Sun
Sampled-data
Control of Logical
Networks
Sampled-data Control of Logical Networks
Yang Liu • Jianquan Lu (cid:129) Liangjie Sun
Sampled-data Control
of Logical Networks
YangLiu JianquanLu
CollegeofMathematicsandComputer SchoolofMathematics
Science SoutheastUniversity
ZhejiangNormalUniversity Nanjing,Jiangsu,China
Jinhua,Zhejiang,China
LiangjieSun
DepartmentofMathematics
UniversityofHongKong
HongKong,China
ISBN978-981-19-8260-6 ISBN978-981-19-8261-3 (eBook)
https://doi.org/10.1007/978-981-19-8261-3
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Foreword
The last two decades have witnessed unprecedented progress in the research of
Boolean networks as well as general logical (control) networks. The motivation
forthisdevelopmentismainlyfromthedemandofvariousapplications,including
biology, game theory, coding, finite automata, graph theory, etc. Meanwhile, the
semi-tensorproduct(STP)ofmatrices,proposedrecently,providesausefultoolin
formulating,analyzing,anddesigningcontrolsforlogicalnetworks.STPenhances
significantlytheinvestigationsonlogical(control)networks.
ThoughtherearealreadyseveralbooksontheSTPapproachtoBooleancontrol
networks, this book is the first to introduce sampled-data control into the study
of Boolean control networks. The book consists of four parts with a total of six
chapters. It mainly discusses three research problems: Boolean control network
under periodic sampled-data control, Boolean control network under aperiodic
sampled-datacontrol,andlogicalcontrolnetworkwithevent-triggeredcontrol.
The contents of this book cover the most updated results on sampled-data
control of Boolean networks, including the authors’ contributions as well as other
researchers’. The materials are well selected and organized. The book is carefully
written.Allthetheoreticalresultsareclearlypresentedandproperlyproved.Many
numericalexamplesarepreparedtohelpreaderstodigestthemainresults.
The authors of this book are leading researchers in this field. I am confident
that this book will be useful for graduate students and young scientists, who are
interested in finite value dynamic (control) systems, such as Boolean networks.
Readerswillbebenefitedalotfromthisbook.
May17,2022 DaizhanCheng
v
Preface
Inrecentyears,logicalnetworks,suchasBooleannetworks,havebeenextensively
studiedduetotheirwiderangeofapplications,includingsystemsbiology,cryptog-
raphy,etc.Designingcontrollerstoachieve thedesiredgoalisalwaysanessential
workinlogicalnetworks.
Mostliteratureonlogicalcontrolnetworks(Booleancontrolnetworks)consider
state feedback control, whose state is updated at each discrete time. However, if
the working principle of the controller is intermittently sampled to the control
system, the loss of the controller can be greatly reduced and its advantages can
be reflected. Moreover, in practical applications, the controller is often difficult
to update at each time. Therefore, in this book, we focus on the study of logical
controlnetworksundersampled-datacontrol.Thefollowingissueswillbepresented
indetail:(1)Booleancontrolnetworkunderperiodicsampled-datacontrol,whose
samplingperiodisconstant.(2)Booleancontrolnetworkunderaperiodicsampled-
datacontrol,whosesamplingintervalisuncertainand/ortimevarying.(3)Logical
controlnetworkunderevent-triggeredcontrol.
This book aims to introduce some recent research work on the logical control
networksundersampled-datacontrol.Thebookisorganizedasfollows:
Chapter 1: This chapter begins with the background of logical networks.
Subsequently,theresearchproblemsofthisbook,someimportantdefinitions,useful
lemmas, and some basic knowledge about semi-tensor product of matrices are
introduced.
Chapter 2: Stabilization of sampled-data Boolean control networks is inves-
tigated. Here, we first derive necessary and sufficient conditions for global sta-
bilization by sampled-data state feedback control and present two algorithms to
constructsampled-datastatefeedbackcontrol.Somedifferencesbetweensampled-
data feedback control and state feedback control for Boolean control networks
are noted. Then, we study the set stabilization of Boolean control networks under
sampled-data state feedback control. We find that sampled cycles allow elements
to be repeated and not every element can be regarded as an initial state, and this
is quite different from conventional cycles of Boolean control networks. Based on
the sampled point set and the sampled point control invariant set, a necessary and
vii
viii Preface
sufficientconditionisderivedfortheglobalsetstabilizationandsampled-datastate
feedbackcontrollersarealsodesigned.Wealsofindaninterestingthingthatifastate
entersthesampledpointcontrolinvariantsetasanunsampledpoint,thenitmayrun
outofthegivensetagain,whichisinsharpcontrasttoconventionalBooleancontrol
networks.
Chapter 3: Controllability, observability, and synchronization of sampled-data
Boolean control networks are investigated. First, we observe some surprising
phenomenathatsampled-dataBooleancontrolnetworkcanlosethecontrollability
orobservabilitycomparedtoasystemwiththesamplingperiodequalingone(which
is a conventional Boolean control network). Necessary and sufficient conditions
forthecontrollabilityandobservabilityofsampled-dataBooleancontrolnetworks
are presented. We propose several algorithms for determining the observability of
sampled-data Boolean control networks. Especially, an algorithm based on graph
theory is designed, and its computational complexity is independent of the length
of the constant sampling periods. Then, we investigate the sampled-data state
feedback control for the synchronization of Boolean control networks under the
configuration of drive-response coupling. Necessary and sufficient conditions for
the complete synchronization are obtained. Unlike studying the synchronization
problem of Boolean control networks under state feedback control, the sampling
periodaffectsthecyclesofthesystem,sochoosinganappropriatesamplingperiod
iscriticaltoachievingsynchronization.
Chapter 4: Stabilization of probabilistic Boolean control networks under
sampled-data control is investigated. We first address the stabilization of
probabilistic Boolean control networks under sampled-data state feedback control
basedonthealgebraicrepresentationoflogicfunctions.Anecessaryandsufficient
condition is derived for the existence of sampled-data state feedback controls
for the global stabilization of probabilistic Boolean control networks. Since the
considered system is a probabilistic Boolean control network, which is more
general and complicated than a Boolean control network, the analysis is more
difficultandchallengingthanthatinChap.2.Moreover,comparedwiththemethod
ofdesigningsampled-datacontrollermentionedinChap.2,weproposeadifferent
and simpler method of designing sampled-data control. Then, we investigate the
partial stabilization problem of probabilistic Boolean control networks under
sample-data state-feedback control with a control Lyapunov function approach.
Here,theclassicalcontrolLyapunovfunctionapproachofthedynamicalsystemsis
extended to the sampled-data state feedback stabilization problem of probabilistic
Boolean control networks. It is indicated that the existence of the sampled-data
state feedback control is equivalent to that of a control Lyapunov function, then
a necessary and sufficient condition is obtained for the existence of control
Lyapunov function. We also propose two different methods to solve all possible
sampled-datastatefeedbackcontrollersandcorrespondingstructuralmatricesofa
controlLyapunovfunction.Twoexamplesaregiventomakeacomparisonwiththe
methods.Comparedwiththeexistingresults,themainadvantagesofourresultsare
we consider the stabilization problem under sampled-data state feedback control,
and we can design all possible sampled-data state feedback controllers as well as
Preface ix
the corresponding control Lyapunov function using the Lyapunov theory. Finally,
we investigate the finite-time and infinite-time set stabilization of probabilistic
Boolean control networks under sampled-data state-feedback control. The finite-
time and infinite-time set stabilization of probabilistic Boolean control networks
are first studied via semi-tensor product approach. In addition, some criteria for
checkingeachkindofsetstabilizationareproposed.Twoalgorithmsareproposed
to find the sampled point set and the largest sampled point control invariant set
of probabilistic Boolean control networks by sampled-data state-feedback control.
Then, a criterion is given for the finite-time set stabilization, and the time-optimal
sampled-data state-feedback controller is designed. A necessary and sufficient
criterionforinfinite-timesetstabilizationofprobabilisticBooleancontrolnetworks
by sampled-data state-feedback control is obtained, and all possible sampled-data
state-feedback controllers are designed. The association and distinction between
finite-timeandinfinite-timesetstabilizationareillustratedbyanumericalexample.
Chapter 5: Stabilization of aperiodic sampled-data Boolean control networks is
investigated. We first study the global stability of Boolean control networks under
aperiodic sampled-data control. By converting the Boolean control network under
aperiodicsampled-datacontrolintoaswitchedBooleannetwork,theglobalstability
of Boolean control networks under aperiodic sampled-data control is first studied.
Here, the switched Boolean network can only switch at sampling instants, which
does not mean that the switches occur at each sampling instant. For the switched
Boolean network containing both stable subsystems and unstable subsystems, we
denotetheactivationfrequenciesofthestablesubsystemsandunstablesubsystems,
respectively. A sufficient condition for global stability Boolean control networks
under aperiodic sampled-data control is derived and an upper bound of the cost
function is determined. An algorithm is presented to construct aperiodic sampled-
data controls for the global stabilization of Boolean control networks. Then, we
consider the case that all subsystems of switched Boolean networks are unstable.
By means of the discretized Lyapunov function and dwell time, a sufficient
conditionforglobalstabilityisobtained.Finally,weinvestigatetheglobalstochastic
stability of Boolean control networks under aperiodic sampled-data controls by a
delay approach . A sufficient condition for global stochastic stability of Boolean
control networks under aperiodic sampled-data control is obtained by using the
Lyapunov function and augmented method. Here, when the sampling instants are
uncertain and only the activation frequencies of the sampling interval are known,
bytransformingtheaperiodicsampled-datacontrolintodelayedcontrol,theglobal
stochastic stability of the Boolean control network under this aperiodic sampled-
data control is first considered; for sampled-data state feedback control (constant
samplinginterval),wecanalsoconvertitintodelayedcontrol,andthen,theglobal
stabilityoftheBooleancontrolnetworkundersampled-datastatefeedbackcontrol
canbestudiedbyadelayapproach.
Chapter 6: Event-triggered control for logical control networks is investigated.
We first investigate the global stabilization problem of k-valued logical control
networksviaevent-triggeredcontrol,wherethecontrolinputsonlyworkatseveral
certain individual states. A necessary and sufficient criterion is derived for the
x Preface
event-triggered stabilization. And a constructive procedure is developed to design
all time-optimal event-triggered stabilizers. Moreover, the switching-cost-optimal
stabilizer, which is event-triggered and has a minimal number of controller exe-
cutions, is designed. The labeled digraph is constructed to describe the dynamic
behavioroftheevent-triggeredcontrolledk-valuedlogicalcontrolnetwork.Based
on knowledge of graph theory, the number of controller executions is minimized
throughauniversalprocedurecalledminimalspanningin-treealgorithm.Itdeserves
formulating that this algorithm can handle all circumstances and overcome the
constraintofthetraditionalmethod.Then,weinvestigatethedisturbancedecoupling
problemofBooleancontrolnetworksbyevent-triggeredcontrol.Ontheonehand,
when the triggering mechanism is satisfied, controllers are updated such that the
disturbance decoupling problem of Boolean control networks is solvable. On the
other hand, in complex systems, the whole system may be controlled well by
controlling only a small subset of variables directly. This strategy is referred to as
the partial control. Motivated by this, event-triggered control is used to study the
disturbance decoupling problem of Boolean partial control networks as well. The
presentedevent-triggercontrolofBooleanpartialcontrolnetworksnotonlyreduces
the control time and cost but also cut down the number of controllers. Finally, we
investigatetheoutputregulationproblemofprobabilistick-valuedlogicalsystems
withdelaysbyanintermittentcontrolscheme.Asufficientandnecessarycondition
for the output regulation problem of probabilistic k-valued logical systems with
delaysisobtained.Twotypesofapproachesaregiventodesigntheevent-triggered
controllaws.
Jinhua,China YangLiu
Nanjing,China JianquanLu
HongKong,China LiangjieSun
April2022
Acknowledgments
Insomesense,wehavebeenworkingonthisbookfor6years,andwehavelotsof
peopletothank.First,wewouldliketothankDr.DaizhanChengandhisteam.The
semi-tensorproducttheyproposedisthemaintoolforstudyinglogicalnetworksin
thisbook.Itisworthmentioningthatthesemi-tensorproductisveryusefulinthe
expressionandanalysisoflogicalnetworks.
Then we would like to express our appreciation to Dr. Jinde Cao at Southeast
University,China;Dr.JungangLouatHuzhouUniversity,China;Dr.DanielW.C.
HoatCityUniversityofHongKong,China;Dr.Wai-KiChingattheUniversityof
Hong Kong, China; Dr. Yuqian Guo and Dr. Weihua Gui both at Central South
University, China; Dr. Zheng-Guang Wu at Zhejiang University, China; Dr. Jie
ZhongatZhejiangNormalUniversity,China;Dr.LeszekRutkowskiatCzestochowa
UniversityofTechnology,Poland;Dr.BowenLiatNanjingUniversityofPostsand
Telecommunications, China; Dr. Kit Ian Kou at the University of Macau, China;
andDr.LiYuatZhejiangUniversityofTechnology,China;Mr.ShiyongZhu;Mr.
QunxiZhu;Miss.LiyunTong;Miss.LiqingWang;Miss.JingyiHe;Miss.Mengxia
Xu;andMr.JiayangLiuwhohaveallcoauthoredwithusafewpapers,whichhave
been included in this book. We would like to thank Mr. Ziyu Xuan, who helped a
lotinproofreadingthemanuscripts.
Moreover, this work was supported by the National Natural Science Foun-
dation of China under Grants 62173308, 61573102 and 11671361; the Natural
Science Foundation of Zhejiang Province of China under Grants LD19A010001,
LR20F030001, and D19A010003; the Natural Science Foundation of Jiangsu
Province of China under Grant BK20170019; the China Postdoctoral Science
Foundation under Grants 2015M580378 and 2016T90406; the National Train-
ing Programs of Innovation and Entrepreneurship under Grants 201610345020,
201710345009, 201810345005, and 201910345013; and the Jinhua Science and
TechnologyProjectundergrant2022-1-042.
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