Analysis and control of Boolean networks: a semi-tensor product approach PDF
Preview Analysis and control of Boolean networks: a semi-tensor product approach
Communications and Control Engineering Forothertitlespublishedinthisseries,goto www.springer.com/series/61 SeriesEditors (cid:2) (cid:2) (cid:2) (cid:2) A.Isidori J.H.vanSchuppen E.D.Sontag M.Thoma M.Krstic´ Publishedtitlesinclude: StabilityandStabilizationofInfiniteDimensional SwitchedLinearSystems SystemswithApplications ZhendongSunandShuzhiS.Ge Zheng-HuaLuo,Bao-ZhuGuoandOmerMorgul SubspaceMethodsforSystemIdentification NonsmoothMechanics(Secondedition) TohruKatayama BernardBrogliato DigitalControlSystems NonlinearControlSystemsII IoanD.LandauandGianlucaZito AlbertoIsidori MultivariableComputer-controlledSystems L2-GainandPassivityTechniquesinNonlinearControl EfimN.RosenwasserandBernhardP.Lampe ArjanvanderSchaft DissipativeSystemsAnalysisandControl ControlofLinearSystemswithRegulationandInput (Secondedition) Constraints BernardBrogliato,RogelioLozano,BernhardMaschke AliSaberi,AntonA.StoorvogelandPeddapullaiah andOlavEgeland Sannuti AlgebraicMethodsforNonlinearControlSystems RobustandH∞Control GiuseppeConte,ClaudeH.MoogandAnnaM.Perdon BenM.Chen PolynomialandRationalMatrices ComputerControlledSystems TadeuszKaczorek EfimN.RosenwasserandBernhardP.Lampe Simulation-basedAlgorithmsforMarkovDecision ControlofComplexandUncertainSystems Processes StanislavV.EmelyanovandSergeyK.Korovin HyeongSooChang,MichaelC.Fu,JiaqiaoHuand StevenI.Marcus RobustControlDesignUsingH∞Methods IanR.Petersen,ValeryA.Ugrinovskiand IterativeLearningControl AndreyV.Savkin Hyo-SungAhn,KevinL.MooreandYangQuanChen ModelReductionforControlSystemDesign DistributedConsensusinMulti-vehicleCooperative GoroObinataandBrianD.O.Anderson Control WeiRenandRandalW.Beard ControlTheoryforLinearSystems HarryL.Trentelman,AntonStoorvogelandMaloHautus ControlofSingularSystemswithRandomAbrupt Changes FunctionalAdaptiveControl El-KébirBoukas SimonG.FabriandVisakanKadirkamanathan NonlinearandAdaptiveControlwithApplications Positive1Dand2DSystems AlessandroAstolfi,DimitriosKaragiannisandRomeo TadeuszKaczorek Ortega IdentificationandControlUsingVolterraModels Stabilization,OptimalandRobustControl FrancisJ.DoyleIII,RonaldK.PearsonandBabatunde AzizBelmiloudi A.Ogunnaike ControlofNonlinearDynamicalSystems Non-linearControlforUnderactuatedMechanical FelixL.Chernous’ko,IgorM.AnanievskiandSergey Systems A.Reshmin IsabelleFantoniandRogelioLozano PeriodicSystems RobustControl(Secondedition) SergioBittantiandPatrizioColaneri JürgenAckermann DiscontinuousSystems FlowControlbyFeedback Orlov OleMortenAamoandMiroslavKrstic´ ConstructionsofStrictLyapunovFunctions LearningandGeneralization(Secondedition) MalisoffandMazenc MathukumalliVidyasagar ControllingChaos ConstrainedControlandEstimation Zhangetal. GrahamC.Goodwin,MariaM.Seronand JoséA.DeDoná ControlofComplexSystems Zecˇevic´andŠiljak RandomizedAlgorithmsforAnalysisandControl ofUncertainSystems RobertoTempo,GiuseppeCalafioreandFabrizio Dabbene Daizhan Cheng (cid:2) Hongsheng Qi (cid:2) Zhiqiang Li Analysis and Control of Boolean Networks A Semi-tensor Product Approach Dr.DaizhanCheng ZhiqiangLi AcademyofMathematicsandSystems AcademyofMathematicsandSystems Science(AMSS),InstituteofSystems Science(AMSS),InstituteofSystems Science Science ChineseAcademyofSciences ChineseAcademyofSciences 100190Beijing 100190Beijing China,People’sRepublic China,People’sRepublic [email protected] HongshengQi AcademyofMathematicsandSystems Science(AMSS),InstituteofSystems Science ChineseAcademyofSciences 100190Beijing China,People’sRepublic ISSN0178-5354 ISBN978-0-85729-096-0 e-ISBN978-0-85729-097-7 DOI10.1007/978-0-85729-097-7 SpringerLondonDordrechtHeidelbergNewYork BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ©Springer-VerlagLondonLimited2011 MATLAB® and Simulink® are registered trademarks of The MathWorks, Inc., 3 Apple Hill Drive, Natick,MA01760-2098,USA.http://www.mathworks.com Apartfromanyfairdealingforthepurposesofresearchorprivatestudy,orcriticismorreview,asper- mittedundertheCopyright,DesignsandPatentsAct1988,thispublicationmayonlybereproduced, storedortransmitted,inanyformorbyanymeans,withthepriorpermissioninwritingofthepublish- ers,orinthecaseofreprographicreproductioninaccordancewiththetermsoflicensesissuedbythe CopyrightLicensingAgency.Enquiriesconcerningreproductionoutsidethosetermsshouldbesentto thepublishers. Theuseofregisterednames,trademarks,etc.,inthispublicationdoesnotimply,evenintheabsenceofa specificstatement,thatsuchnamesareexemptfromtherelevantlawsandregulationsandthereforefree forgeneraluse. Thepublishermakesnorepresentation,expressorimplied,withregardtotheaccuracyoftheinformation containedinthisbookandcannotacceptanylegalresponsibilityorliabilityforanyerrorsoromissions thatmaybemade. Coverdesign:eStudioCalamarS.L. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Motivated by the Human Genome Project, a new view of biology, called systems biology, is emerging [5]. Systems biology does not investigate individual genes, proteins or cells in isolation. Rather, it studies the behavior and relationships of all of the cells, proteins, DNA and RNA in a biological system called a cellular network.Themostactivenetworksmaybethoseassociatedwithgeneticregulation, whichregulatethegrowth,replication,anddeathofcellsinresponsetochangesin theenvironment. How do these genetic regulatory networks function? In the early 1960s Jacob and Monod showed that any cell contains a number of “regulatory” genes that act asswitchesandwhichcanturneachanotheronandoff.Thisshowsthatagenetic networkisof“on–off”type[7]. Boolean networks, first introduced by Kauffman, have become powerful tools fordescribing,analyzing,andsimulatingcellularnetworks[2,3].Hence,theyhave received much attention, not only from the biology community, but also from re- searcherswithbackgroundsinphysics,systemsscience,etc. The purpose of this book is to present a new approach to the investigation of Boolean(control)networks.Inthisnewapproach,alogicalrelationisexpressedas analgebraicequation,andalogicaldynamicalsystem,suchasaBooleannetwork, is converted into a standard discrete-time linear system. Similarly, a Boolean con- trol network is converted into a discrete-time bilinear system. In this way, various tools for solving conventional algebraic equations and dealing with difference or differentialequationscanbeusedtosolvelogic-basedproblems.Underthisframe- work,thetopologicalstructuresofBooleannetworksarerevealedviathestructures of their network transition matrices. The state space, subspaces, etc., are then de- fined as sets of logical functions. This framework makes the state-space approach to dynamical (control) systems applicable to Boolean (control) networks. Using this new technique, we investigate the properties and control design of Boolean networks. Many basic problems in control theory are studied, such as controlla- bility, observability, realization, stabilization, disturbance decoupling and optimal control. Thefundamentaltoolinthisapproachisanewmatrixproduct,calledthesemi- tensor product (STP). The STP of matrices is a generalization of the conventional v vi Preface matrixproducttothecasewherethedimension-matchingconditionisnotsatisfied. Thatis,weextendthematrixproductAB tothecasewherethecolumnnumberof AandtherownumberofBaredifferent.Thisgeneralizationpreservesallthemajor propertiesoftheconventionalmatrixproduct. Using the STP, a logical function can be converted into a multilinear mapping, called the matrix expression of logical relations. Under this construction, the dy- namicsofaBooleannetworkcanbeexpressedasaconventionaldiscrete-timelinear system.Inthelightofthislinearexpression,certainmajorfeaturesofthetopology of a Boolean network, such as fixed points, cycles, transient time, and basins of attractors,canbeeasilyrevealedviaasetofformulas. WhenthecontrolofaBooleannetworkisconsidered,thebilinearsystemrepre- sentationofaBooleancontrolnetworkmakesitpossibletoapplymosttechniques developedinmoderncontroltheorytotheanalysisandsynthesisofaBooleancon- trolnetwork. Themaincontentsofthisbookareasfollows. Chapter 1 consists of a brief introduction to propositional logic. This is very elementaryandinvolvesonlythepropositionallogicrequiredinthisbook.Areader whoisfamiliarwithmathematicallogiccanskipit. InChap.2weintroducesomebasicconceptsandpropertiesoftheSTP,whichis theprincipaltoolusedinthisbook.TheSTPisageneralizationoftheconventional matrix product in cases where the dimension-matching requirement for the factor matricesfails.Thisgeneralizationpreservesthemajorpropertiesoftheconventional matrixproduct. In Chap. 3 we consider the matrix expression of logical relations. Identifying T (true)and F (false)withvectors [1,0]T and [0,1]T,respectively,alogicalvari- able becomes a 2-dimensional vector variable. Using the STP, a logical function canbeexpressedasamultilinearmappingwithrespecttoitslogicalargumentsso that each logical function is uniquely determined by a matrix, called its structure matrix. Chapter4isdevotedtosolvinglogicalequations.Usingthematrixexpressionof logicasystemoflogicalequationscanbeconvertedintoalinearalgebraicequation. Ignoringthecomplexityofcomputation,thesolutionofsystemsoflogicalequations becomes theoreticallyequivalentto the solution of algebraic equations, which can beachievedwithstraightforwardcomputation. Chapter5considersthelinearexpressionofBooleannetworks.Usingthetech- nique developed in previous chapters, the dynamics of a Boolean network is con- verted into a conventional discrete-time linear system. In the light of this linear expression,thetopologicalstructuresofBooleannetworksareinvestigatedviatheir transition matrices. Formulas are obtained to calculate the fixed points, cycles of differentlengths,transientperiod,andthebasinofeachattractor. The input-state structures of Boolean control networks are studied in Chap. 6. Thecompoundedstructureofcyclesininput-statespaceisobtained.Thisapproach isappliedtotheanalysisofBooleannetworkswithcascadingstructure.The“rolling gear” structure of cycles is revealed, which explains the phenomenon that tiny at- tractorscandeterminethevastorderofthenetwork[4]. Preface vii Chapter7presentsatechniquetobuildthedynamicmodelofaBooleannetwork viaobserveddata.Insteadof buildingthe logicaldynamicsof a Booleannetwork, we first identify its algebraic form, so the conversion of the algebraic form of a Booleannetworkbacktoitslogicalformisfirstinvestigated.Afterageneralmodel constructiontechniqueisintroduced,severalspecialcasesarestudied,includingthe knownnetworkgraphcase,theleastin-degreemodel,theuniformmodel,etc.The problemofdealingwithdatacontainingerrorsisalsodiscussed. In Chap. 8 a systematic state-space description is developed. The state space (and its subspaces) of a Boolean (control) network are defined in a dual way, i.e., theyaredefinedassetsoflogicalfunctions.Itisshownthatthisdescriptionisvery convenientinrevealingthepropertiesofBooleannetworksandinthecontroldesign ofBooleancontrolnetworks. Chapter9isdevotedtoBooleancontrolnetworks.Usinglinearexpressions,itis shownthatBooleancontrolnetworkscanbeconvertedintolinearcontrolsystems. Some basic control problems such as controllability and observability of Boolean controlnetworksaretheninvestigatedviatheirequivalentforms forlinearcontrol systems. Chapter10considerstherealizationproblemofBooleancontrolnetworks.First, coordinatetransformationsareconsidered,andthenthe Kalmandecompositionof Booleaninput–outputnetworksisproposed.UsingtheKalmandecomposition,the minimumrealizationofaBooleaninput–outputmappingisobtained. ThestabilityandstabilizationproblemisdiscussedinChap.11.Theapplicable setfrommetric-basedconvergenceanalysis[6]isenlargedbytheuseofcoordinate transformations. Based on the analysis of the network transition matrix, necessary and sufficient conditions are then obtained for stability and stabilization by either open-loopcontrolorclosed-loopcontrol.Severalexamplesareincluded. Chapter 12 considers the disturbance decoupling problem. First, the output- friendlysubspaceisintroduced.Formulasandalgorithmsareprovidedtoconstruct a minimum regular subspace, which is called the “friend” of output y. The de- signtechniqueforconstructingthefeedbackandsolvingthedisturbancedecoupling problemispresented.Toconstructaconstantstabilizingcontrol,thecanalizingmap- ping,whichisageneralizationofthecanalizingfunction,isproposedanditsmain propertiesarerevealed. In Chap. 13 we consider the coordinate-independent geometric structure of Boolean (control) networks. Based on this structure, the feedback decomposition of Boolean control networks is studied. The input-state decomposition, including cascadingandparalleldecompositions,andinput–outputdecompositionofBoolean control networks are investigated, and necessary and sufficient conditions are pre- sented. Chapter14dealswiththemultivaluedlogicwhichcouldprovideamoreprecise description for real networks such as gene regulation networks, etc. The structure of k-valued logical networks is first investigated. Controllability and observabil- ity of k-valued logical networks are then considered. In fact, almost all the argu- ments and results about Boolean networks can be extended to the k-valued logic setting. viii Preface Chapter 15 considers the optimal control of Boolean control networks. To deal withBoolean (or k-valued)games with s-memory, higher-order Boolean(control) networksareintroduced,andtheiralgebraicformsarealsopresented.Theone-to- onecorrespondencebetweenthecyclesoftheoriginalnetworkandthecyclesofits algebraicformisestablished.Theoptimalcontrolproblemistheninvestigatedand theoptimalcontrolisdesigned. Chapter16introducesausefultool,calledtheinput-stateincidencematrix,which isanalgebraicdescriptionoftheinput-statetransfergraph.Controllabilityandob- servability of Boolean control networks are revisited and some further results are presented. The topological structures of Boolean control networks with free con- trols are also investigated. Finally, the results are extended to mix-valued logical dynamicalsystems. Chapter 17 investigates the identification of Boolean control networks. First, a newobservabilityconditionisobtainedwhichprovidesawaytoconstructtheinitial stateofatrajectoryfromitsinput–outputdata.Anecessaryandsufficientcondition foridentifiabilityisthenpresented.Anumericalalgorithmisproposedforpractical application. Chapter 18 considers an application to game theory. We consider a game with finitely many players and where each player has finitely many possible actions. When the game is infinitely repeated, a strategy using finite memory becomes a logical dynamical system. Hence, the results obtained for Boolean or logical net- works are applicable to finding Nash or sub-Nash solutions for the infinitely re- peatedgames. The primary objects of this book are deterministic Boolean networks, but in Chap.19weprovideabriefintroductiontorandomBooleannetworks.Basiccon- ceptsarepresentedandthenthesteady-statedistributionofarandomBooleannet- workisinvestigated.Finally,thestabilizationofarandomBooleannetworkisstud- ied. Recently, random Boolean networks have been the subject of much research, andsoadetaileddiscussionisbeyondthescopeofthiswork. AppendixAexplainsrelevantnumericalcalculations.Asoftwaretoolboxforthe algorithmsisavailableathttp://lsc.amss.ac.cn/~dcheng/. AppendixBcontainsproofsofsomekeypropertiesofthesemi-tensorproduct, whicharetranslatedfrom[1],withthepermissionofSciencePress. This book is self-contained. The prerequisites for its use are linear algebra and somebasicknowledgeofthecontroltheoryoflinearsystems.Themanuscriptwas originally prepared when the first author was visiting Kyoto University. The first authorwouldliketoexpresshisheartythankstoProfessorYutakaTakahashiforhis proof-reading and useful suggestions for parts of the manuscript. The manuscript has been used as lecture notes in a series of seminars organized jointly by the Academy of Mathematics and Systems Science, Tsinghua University, and Peking University.Manycolleaguesandstudentsattendingtheseseminarshavecontributed to this book via useful discussions, suggestions, and corrections. Particularly, Dr. Yin Zhao helped in the preparation of Chaps. 15–17. Dr. Yifen Mu, Dr. Zhenning Zhang,Dr.YinZhao,Dr.XiangruXu,andDr.JiangboZhanghelpedwiththefinal galleyproofofthemanuscript.TheauthorsarealsoindebtedtoMr.OliverJackson forhiswarmheartedsupport. References ix TheresearchpresentedinthisbookwaspartlysupportedbytheChineseNational NaturalScienceFoundationundergrantnumberG60736022. Beijing DaizhanCheng HongshengQi ZhiqiangLi References 1. Cheng,D.,Qi,H.:Semi-tensorProductofMatrices—TheoryandApplications.SciencePress, Beijing(2007)(inChinese) 2. Kauffman,S.:Metabolicstabilityandepigenesisinrandomlyconstructedgeneticnets.J.Theor. Biol.22(3),437(1969) 3. Kauffman,S.:TheOriginsofOrder:Self-organizationandSelectioninEvolution.OxfordUni- versityPress,London(1993) 4. Kauffman,S.:AtHomeintheUniverse.OxfordUniversityPress,London(1995) 5. Kitano,H.:Systemsbiology:abriefoverview.Science259,1662–1664(2002) 6. Robert,F.:DiscreteIterations:AMetricStudy.Springer,Berlin(1986).TranslatedbyJ.Rolne 7. Waldrop,M.:Complexity:TheEmergingScienceattheEdgeofOrderandChaos.Touchstone, NewYork(1992)
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