Analysis And Synthesis Of Discrete-Time And Discrete-State Dynamical Systems

Nature depends on a diversity of large-scale complex networks for its various existence and evolution.Compared with continuous systems,the discrete-time and discrete-state dynamical systems(DTDSs)have some prominent advantages while qualitatively describing the mechanisms of some realistic phenomenon.For instance,Boolean networks(BNs),as a classical model to characterize gene regulatory networks,can macroscopically emphasize the generic principles rather than the quantitative parameter details,thus it overcomes the low availability due to the insufficient dynamics parameters.Besides,it is more realistic to model the wireless sensor networks with limited storage capacities and constrained channel bandwidths by finite-field networks(FFNs).In this article,concerning on the existing limitations and problems in the areas of DTDSs,we provide some solution strategies from several perspectives.The main research work can be divided into the following aspects:Since the existing results and methods heavily depend on the node dynamics,the concepts of structural controllable graphs and structural observable graphs are proposed for the first time;they respectively guarantee the controllability and observability of DTDSs from the viewpoint of network topology.Moreover,a feasible type of structural controllable graphs and structural observable graphs are respectively presented,and we prove that they are tight for BNs.Subsequently,we consider how to control the minimal number of vertices and modify their adjacency relation or impose the sensors such that an arbitrary digraph becomes a structural controllable or structural observable one.To this end,we prove that the minimum realization of structural controllable graphs is NP-hard;and two classes of minimum realization of structural observable graphs can be both realized via two polynomial-time algorithms.As usual,the identification of node dynamics is different for large-scale networks,but it is easy for network topology.The defined structural controllable graphs and structural observable graphs can analyze and control the DTDSs without available node dynamics.Concerning that the obtained data is always noisy,two types of stochastic BNsprobabilistic BNs(PBNs)and Markovian jump BNs(MJBNs)are respectively studied.Based on the algebraic state space representation(ASSR)method,we present a necessary and sufficient condition for the asymptotic stability of PBNs from the viewpoint of steady distribution of Markov chain.Based on the topological sorting idea,two algorithms are designed to check the finite-time stability of PBNs.Due to the forward compactness of time evolution,we prove that the bounded and coincident delays have no effect on the stability,even if they are time varying.It dramatically reduces the time complexity from O((?+1)2n)to O(2n).In order to save the time consumption or control cost,the minimum-time control and minimum-triggering control are respectively investigated for MJBNs.A maximum principle is established to derive a necessary condition for the minimal observable time,and the corresponding control sequence is also designed;With regard to event-triggered output feedback observability,an efficient procedure is developed to minimize the number of triggering events.These results can ensure the system observability with minimum time consumption or control cost.Additionally,with the help of independence between switching signal and system state,the time complexity to analyze and control MJBNs can be reduced to a certain extent.Since the traditional ASSR method involves in utilizing 2n × 2n-dimensional network transition matrix,it is usually unacceptable for large-scale BNs.To this end,based on the network topology,a novel pinning control framework is proposed for observability,controllability,set stabilization and stabilization in probability.Compared with the traditional pinning framework,the pinning controllers designed here only utilize the n × n-dimensional dependency graph instead of 2n × 2n-dimensional network transition matrix.Thus,the time complexity is reduced from O(2n)to O(n23d*),where d*is the largest in-degree of network vertices.The designed controllers only depend on the local information,so it is simpler in the form than the traditional one.Additionally,the pinning nodes can be searched within time O(n2),and only impose on a small fraction of state nodes.In the traditional framework,selecting pinning nodes needs to reversely solve the 2n×2n-dimensional logical matrix,thus the control inputs may be imposed on all nodes.Noting that the existing analysis methods depend on the precise communication of state values between nodes,it sometimes violates the individual privacy.Motivated by the node decomposition and weight mechanism,the privacy-preserving strategy is proposed for the first time:every node state into two substates,where the first one involves in the same computation as the original one does,and the other one only interacts with first substate.Such scheme not only guarantees the individual privacy,but also makes the system convergence to the exact value without any error.The techniques that is proposed in this article overcomes some limitations and problems in the field of DTDSs.They respectively provide the feasible strategies for the situations of unavailable node dynamics,limited time or control cost,large-scale networks,as well as privacy preserving.