Self-Triggered Set Stabilization of Boolean Control Networks and Its Applications

Set stabilization is one of the essential problems in engineering systems, and self-triggered control(STC) can save the storage space for interactive information, and can be successfully applied in networked control systems with limited communication resources. In this study, the set stabilization problem and STC design of Boolean control networks are investigated via the semi-tensor product technique. On the one hand, the largest control invariant subset is calculated in terms of the strongly connected components of the state transition graph, by which a graph-theoretical condition for set stabilization is derived. On the other hand, a characteristic function is exploited to determine the triggering mechanism and feasible controls. Based on this, the minimum-time and minimum-triggering open-loop, state-feedback and output-feedback STCs for set stabilization are designed,respectively. As classic applications of self-triggered set stabilization, self-triggered synchronization, self-triggered output tracking and self-triggered output regulation are discussed as well. Additionally, several practical examples are given to illustrate the effectiveness of theoretical results.

Semi-tensor product approach for partially symmetric games

In this paper,a criterion for the partially symmetric game(PSG)is derived by using the semitensor product approach.The dimension and the basis of the linear subspace composed of all the PSGs with respect to a given set of partial players are calculated.The testing equations with the minimum number are concretely determined,and the computational complexity is analysed.Finally,two examples are displayed to show the theoretical results.

Semi-tensor product of matrices and its application to Morgen's problem

This paper proposes a new matrix product, namely, semi-tensor product. It is a general-ization of the conventional matrix product. Meanwhile, it is also closely related to Kronecker (tensor) product of matrices. The purpose of introducing this product is twofold: (i) treat multi-dimensional da-ta; (ii) treat nonlinear problems in a linear way. Then the computer and numerical methods can be easily used for solving nonlinear problems. Properties and formulas are deduced. As an application, the Morgan's problem for control systems is formulated as a numerically solvable problem.

Matching Algorithms of Minimum Input Selection for Structural Controllability Based on Semi-Tensor Product of Matrices

In 2011,Liu,et al.investigated the structural controllability of directed networks.They proved that the minimum number of input signals,driver nodes,can be determined by seeking a maximum matching in the directed network.Thus,the algorithm for seeking a maximum matching is the key to solving the structural controllability problem of directed networks.In this study,the authors provide algebraic expressions for matchings and maximum matchings proposed by Liu,et al.(2011)via a new matrix product called semi-tensor product,based on which the corresponding algorithms are established to seek matchings and maximum matchings in digraphs,which make determining the number of driver nodes tractable in computer.In addition,according to the proposed algorithm,the authors also construct an algorithm to distinguish critical arcs,redundant arcs and ordinary arcs of the directed network,which plays an important role in studying the robust control problem.An example of a small network from Liu’s paper is used for algorithm verification.

Modeling and reachability analysis of synchronizing transitions bounded Petri net systems based upon semi-tensor product of matrices

The reachability problem of synchronizing transitions bounded Petri net systems （BPNSs） is investigated in this paper by constructing a mathematical model for dynamics of BPNS. Using the semi-tensor product （STP） of matrices, the dynamics of BPNSs, which can be viewed as a combination of several small bounded subnets via synchronizing transitions, are described by an algebraic equation. When the algebraic form for its dynamics is established, we can present a necessary and sufficient condition for the reachability between any marking （or state） and initial marking. Also, we give a corresponding algorithm to calculate all of the transition paths between initial marking and any target marking. Finally, an example is shown to illustrate proposed results. The key advantage of our approach, in which the set of reachable markings of BPNSs can be expressed by the set of reachable markings of subnets such that the big reachability set of BPNSs do not need generate, is partly avoid the state explosion problem of Petri nets （PNs）.

A Semi-Tensor Product Based All Solutions Boolean Satisfiability Solver

Boolean satisfiability (SAT) is widely used as a solver engine in electronic design automation (EDA). Typically, SAT is used to determine whether one or more groups of variables can be combined to form a true formula. All solutions SAT (AllSAT) is a variant of the SAT problem. In the fields of formal verification and pattern generation, AllSAT is particularly useful because it efficiently enumerates all possible solutions. In this paper, a semi-tensor product (STP) based AllSAT solver is proposed. The solver can solve instances described in both the conjunctive normal form (CNF) and circuit form. The implementation of our method differs from incremental enumeration because we do not add blocking conditions for existing solutions, but rather compute the matrices to obtain all the solutions in one pass. Additionally, the logical matrices support a variety of logic operations. Results from experiments with MCNC benchmarks using CNF-based and circuit-based forms show that our method can accelerate CPU time by 8.1x (238x maximum) and 19.9x (72x maximum), respectively.

矩阵左半张量积的正定性

对两个实矩阵的左半张量积为正定矩阵的情况进行了研究,从特征值的角度给出了某些实矩阵的左半张量积为正定矩阵的一系列充要条件,并得到了一些相关结论.

关于矩阵左半张量积迹的几个不等式

矩阵的迹是矩阵理论中的重要课题,有很重要的理论和实际应用价值,本文主要研究了左半张量积迹的一些性质,推广了矩阵乘积迹的一些不等式,进一步丰富了半张量积的理论知识。

矩阵左半张量积的特征值不等式

首先给出了任意两个复矩阵做左半张量积的特征值的不等式,然后给定两个(半)正定矩阵A、B以及它们的特征值,给出了矩阵A、B的左半张量积的特征值不等式以及一个精确估计,得到了一个不断缩小A×lB特征值的下、上限间距离的方法.

多重线性映射问题的左半张量积方法

讨论了矩阵左半张量积乘法新的运算性质,并将所取得的研究成果应用于多重线性映射,为矩阵方法解决多重线性映射问题提供了更简单直接的方法.

左半张量积的一些性质和定理

对左半张量积,矩阵的一种新的运算,进行了研究,获得了一些新的性质,得出了一些重要的结论.

Semi-tensor product approach to networked evolutionary games

In this paper a comprehensive introduction for modeling and control of networked evolutionary games （NEGs） via semi-tensor product （STP） approach is presented. First, we review the mathematical model of an NEG, which consists of three ingredients： network graph, fundamental network game, and strategy updating rule. Three kinds of network graphs are considered, which are i） undirected graph for symmetric games; ii） directed graph for asymmetric games, and iii） d-directed graph for symmetric games with partial neighborhood information. Three kinds of fundamental evolutionary games （FEGs） are discussed, which are i） two strategies and symmetric （S-2）; ii） two strategies and asymmetric （A-2）; and iii） three strategies and symmetric （S-3）. Three strategy updating rules （SUR） are introduced, which are i） Unconditional Imitation （UI）; ii） Fermi Rule（FR）; iii） Myopic Best Response Adjustment Rule （MBRA）. First, we review the fundamental evolutionary equation （FEE） and use it to construct network profile dynamics （NPD）of NEGs. To show how the dynamics of an NEG can be modeled as a discrete time dynamics within an algebraic state space, the fundamental evolutionary equation （FEE） of each player is discussed. Using FEEs, the network strategy profile dynamics （NSPD） is built by providing efficient algorithms. Finally, we consider three more complicated NEGs： i） NEG with different length historical information, ii） NEG with multi-species, and iii） NEG with time-varying payoffs. In all the cases, formulas are provided to construct the corresponding NSPDs. Using these NSPDs, certain properties are explored. Examples are presented to demonstrate the model constructing method, analysis and control design technique, and to reveal certain dynamic behaviors of NEGs.

A Matrix Approach to the Modeling and Analysis of Networked Evolutionary Games With Time Delays

Using the semi-tensor product method, this paper investigates the modeling and analysis of networked evolutionary games(NEGs) with finite memories, and presents a number of new results. Firstly, a kind of algebraic expression is formulated for the networked evolutionary games with finite memories, based on which the behavior of the corresponding evolutionary game is analyzed. Secondly, under a proper assumption, the existence of Nash equilibrium of the given networked evolutionary games is proved and a free-type strategy sequence is designed for the convergence to the Nash equilibrium. Finally, an illustrative example is worked out to support the obtained new results.

ON THE OBSERVABILITY OF FREE BOOLEAN NETWORKS VIA THE SEMI-TENSOR PRODUCT METHOD

This paper investigates the observabihty of free Boolean networks by using the semi-tensor product method,and presents some new results.First,the concept of observability for free Boolean networks is proposed,based on which and the algebraic form of Boolean networks,a kind of observabihty matrix is constructed.Second,by the observability matrix,a new necessary and sufficient condition is given for the observability of Boolean networks.Third,the concept of observabihty index for observable Boolean networks is defined,and an algorithm is established to calculate the observability index.Finally,a practical example of D.Melanogaster segmentation polarity gene networks is studied to support our new results.The study of the illustrative example shows that the new results obtained in this paper are very effective in investigating the observability of free Boolean networks.

The Computation of Nash Equilibrium in Fashion Games via Semi-Tensor Product Method

Using the semi-tensor product of matrices, this paper investigates the computation of purestrategy Nash equilibrium （PNE） for fashion games, and presents several new results. First, a formal fashion game model on a social network is given. Second, the utility function of each player is converted into an Mgebraic form via the semi-tensor product of matrices, based on which the case of two-strategy fashion game is studied and two methods are obtained for the case to verify the existence of PNE. Third, the multi-strategy fashion game model is investigated and an algorithm is established to find all the PNEs for the general case. Finally, two kinds of optimization problems, that is, the so-called social welfare and normalized satisfaction degree optimization problems are investigated and two useful results are given. The study of several illustrative examples shows that the new results obtained in this paper are effective.

矩阵左半张量积的{1,2,3}与{1,2,4}-逆的反序律

利用矩阵的广义Schur补的最大秩及最小秩的表达式,给出了两矩阵左半张量积的{1,2,3}-逆的反序律,即给出了B{1,2,3}■A{1,2,3}■(A■B){1,2,3}成立的一个充要条件;然后按照同样的方法,给出了两矩阵左半张量积的{1,2,4}-逆的反序律,即B{1,2,4}■A{1,2,4}■(A■B){1,2,4}成立的一个充要条件,并且给出了一些相关的结论.

多边矩阵的剖面广义交叉乘法和半张量积及其数据挖掘

介绍了多边矩阵的剖面广义交叉乘法概念,给出了多边矩阵的剖面广义交叉乘法的一些性质,证明了多边矩阵剖面广义交叉乘法满足结合律和分配律。作为应用,研究了多边矩阵剖面广义交叉乘法和一般矩阵半张量积、数据挖掘之间的关系。

Stability Analysis of Networked Evolutionary Games with Profile-Dependent Delays

This paper investigates the networked evolutionary games(NEGs)with profile-dependent delays,including modeling and stability analysis.Profile-dependent delay,which varies with the game profiles,slows the information transmission between participants.Firstly,the dynamics model is proposed for the profile-dependent delayed NEG,then the algebraic formulation is established using the algebraic state space approach.Secondly,the dynamic behavior of the game is discussed,involving general stability and evolutionarily stable profile analysis.Necessary and sufficient criteria are derived using the matrices,which can be easily verified by mathematical software.Finally,a numerical example is carried out to demonstrate the validity of the theoretical results.

Controllability of Boolean control networks with multiple time delays in both states and controls

In this paper,the problem of controllability of Boolean control networks(BCNs)with multiple time delays in both states and controls is investigated.First,the controllability problem of BCNs with multiple time delays in controls is considered.For this controllability problem,a controllability matrix is constructed by defining a new product of matrices,based on which a necessary and sufficient controllability condition is obtained.Then,the controllability of BCNs with multiple time delays in states is studied by giving a necessary and sufficient condition.Subsequently,based on these results,a controllability matrix for BCNs with multiple time delays in both states and controls is proposed that provides a concise controllability condition.Finally,two examples are given to illustrate the main results.

Algebraic form and analysis of SIR epidemic dynamics over probabilistic dynamic networks

The outbreak of corona virus disease 2019 has profoundly affected people’s way of life.It is increasingly necessary to investigate epidemics over social networks.This paper studies susceptible-infected-removed(SIR)epidemics via the semi-tensor product.First,a formal susceptible-infected-removed epidemic dynamic model over probabilistic dynamic networks(SIRED-PDN)is given.Based on an evolutionary rule,the algebraic form for the dynamics of individual states and network topologies is given,respectively.Second,the SIRED-PDN can be described by a probabilistic mix-valued logical network.After providing an algorithm,all possible final spreading equilibria can be obtained for any given initial epidemic state and network topology by seeking attractors of the network.And the shortest time for all possible initial epidemic state and network topology profiles to evolve to the final spreading equilibria can be obtained by seeking the transient time of the network.Finally,an illustrative example is given to show the effectiveness of our model.