Some Topics Research On Observability Of Boolean Networks Based Via STP Method

Learning and modeling genetic regulatory networks are an important problem in systems biology.Boolean network(BNs)was first proposed by Kauffman,which are proposed to simulate and analyze gene regulatory networks.BNs has attached considerable attention from many experts and scholars because of its simplicity and potential applications.In BNs,there are two ways to represent each gene node at discrete time:1 or 0(ON or OFF),and it is updated according to logical functions at each discrete time.So the BN is a discrete of the time logic system.Despite the conceptual simplicity,BNs seem suitable for genetic regulation and provide much useful information about various real systems.Most of the results on BNs can be extended to multi-valued logical dynamic systems.As the mathematics tool for BNs is more suitable for semi-tensor product(STP),so we use STP to transform the logical systems in BNs into algebraic form,which facilitates the analysis of BNs.To be concrete,the contributions of this dissertation are as followsChapter 1 describes the research background of this paper.First,the development of BNs and switched Boolean networks(SBNs),and the definition and properties of semi-tensor products are introduced.Then the matrix representation of the logical function in BNs and its algebraic expression are also introducedChapter 2 studies the observability and minimal observability of BNs.First,a necessary and sufficient condition has been obtained for determining the observability of BNs.Moreover,the minimal observability problem is defined.In BNs,with the increases of nodes are directly measured,the observability of the system will be im-proved,that is to say an already observable state will not be affected by a new observer.A procedure for designing an observer for unobservable BNs has been proposed,and a sufficient and necessary condition for determining the minimal number of nodes has been obtained,which needs to be directly measurable.Finally,examples have been provided to show the efficiency of main results of this paperChapter 3 studies the observability of a switched Boolean control network(SBC-N).Using semi-tensor product of matrices,the dynamics of a SBCN can be transformed into an algebraic form.First,the observability of the SBCN is determined under desir-able switching signals and arbitrary switching signals by encoding the switching signal as a Boolean input.Then an algorithm is designed for determining the observability.Furthermore,feedback control laws are obtained to guarantee the observability of S-BCNs.Finally,a biological example is provided to illustrate the effectiveness of the results.In Chapter 4,the observability of Boolean networks(BNs)is studied,using semi-tensor product of matrices.First,unobservable states can be divided into two types,and the first type of unobservable states can be easily determined by blocking idea.Second,it is found that all states reaching to observable states are observable.Based on subgraph of transition matrix and blocking idea,the second type of unobservable states can be also determined.Approaches are obtained to directly determine some observable or unobservable states.An algorithm is designed for determining the ob-servability of BNs as well.Examples are given to illustrate the effectiveness of the given results.