Complex Dynamic Behavior And Applications Of Logical Dynamic Systems

Logical dynamic systems(LDSs)are a special kind of nonlinear systems,whose states,inputs and outputs belong to finite sets.The dynamic equations of LDSs can be represented by multi-valued or mixed-valued logical functions.The main characteristic of LDSs is that the dynamics are parameter-free,which can be used to model large-scale systems and facilitate the computer simulation verification.With the rapid development of systems biology,network security,artificial intelligence and game theory,LDSs have attracted increasing attention from scholars at home and abroad.During the study of practical LDSs,time delays,switching phenomena,asynchronization and random disturbances are ubiquitous.Thus,the study of dynamic behaviors for LDSs with the above factors is indispensable.the complex dynamic behaviors of LDSs are studied,and the results of this thesis are applied to classical problems such as output tracking and observer design.The main contents of this thesis are listed as follows:1.The stability of logical networks with time delays is considered.In order to reduce the computational complexity of the existing methods for constructing the augmented system,a new method is proposed by using the parallel technique,and a necessary and sufficient condition is given for the existence and uniqueness of the solution to the augmented system.Then,the topological structure of the augmented system is analyzed.After defining the concepts of fixed point and cycle of the augmented system,a necessary and sufficient condition is presented for verifying the stability of the augmented system.Finally,using the parallel technique,the logical networks with time delays are converted into an augmented system,and the stability of the logical networks with time delays is verified.2.The stabilization of deterministic asynchronous LCNs is considered.Firstly,using the algebraic state space representation of periodic switched LCNs,a kind of periodic switching point controllability matrix is constructed,based on which,a necessary and sufficient condition is presented for the periodic switching point reachability and controllability of periodic switched LCNs.Secondly,by constructing a series of reachable sets,a constructive procedure is proposed to design time-variant state feedback controllers for the periodic switching point stabilization of periodic switched LCNs.In addition,by converting the dynamics of deterministic asynchronous LCNs into the form of periodic switched LCNs,an effective procedure is proposed to design the time-variant state feedback stabilizers for a deterministic asynchronous LCNs.3.The observability and reconstructibility of probabilistic logical control networks(PLCNs)are considered.Based on the partition of the state pair space,an observable verification system is established.The equivalence between the stabilization of observable verification system and the observability of PLCNs is revealed,and a necessary and sufficient condition is established for solving the observability of PLCNs.In addition,the relationship between observability and reconstructibility of PLCNs is unveiled.By defining the ring on the space of state pair,some new criteria are established to solve two kinds of reconstructibility problems for PLCNs.Finally,an example of a biological network,apoptosis network,is given to show the feasibility and superiority of the obtained results.4.The optimal state estimation problem of LCNs with stochastic disturbances is considered.First,the problem of optimal state estimation for logical networks is considered.Based on the equivalent algebraic form of logical networks with stochastic disturbances,the prediction matrix and update matrix are defined,based on which,an effective procedure is given to design the optimal state estimator by using the semi-tensor product of matrices.Under the framework,an effective method is proposed to design the Luenberger-like observer based on the equivalent algebraic form of LCNs with stochastic disturbances.Moreover,by resorting to the Luenberger-like observer,an effective approach is established to design the optimal state estimator in the sense of minimum mean square error.In addition,the multi-valued Kalman filter is established,including prediction step and update step,based on which,both optimal state estimator and minimum mean square error can be calculated effectively.Finally,the obtained results are applied to the finite-horizon output tracking problem and the optimal control problem of LCNs with stochastic disturbances.The main contributions of this thesis contain three folds.(1)Reducing computational complexity: This thesis proposes a new method for constructing the augmented system.The introduction of the augmented system is useful for promoting the development of LDSs such as the stability of logical networks with time delays.Compared with the existing augmented system method of dealing with time delays in LDSs,the dimension of the augmented system developed in this thesis is much lower.In addition,in order to reduce the computational complexity of studying the observability of LDSs using the general augmented system method,this thesis presents a new kind of observable verification system for PLCNs,which provides a new framework for the observability analysis of PLCNs.(2)Provide a new research method for deterministic asynchronous LDSs: This thesis presents a new criterion for the periodic switching point controllability of periodic switched LCNs,based on which,a constructive procedure is proposed to design time-variant state feedback stabilizers of periodic switched LCNs.By converting the dynamics of deterministic asynchronous LCNs into the form of periodic switched LCNs,the time-variant state feedback stabilization problem of deterministic asynchronous LCNs is solved,and a new idea is provided for the study of deterministic asynchronous LCNs.(3)Improve the estimation accuracy and optimize the estimation algorithm: This thesis proposes an approach for LCNs with stochastic disturbance to design a Luenberger-like observer,which provides a new research idea for the reconstructibility of PLCNs.The Luenberger-like observer is applied to solve the optimal state estimation problem of LCNs in the sense of minimum mean square error and a new approach is presented to find the optimal state estimator.Compared with the existing results,the estimation accuracy of the results obtained in this thesis is better.