Matrix approach to simplification of finite state machines using semi‐tensor product of matrices

In this paper, we use a new mathematical tool, semi‐tensor product of matrices, to investigate the problem of simplification of finite state machines (FSMs) in a mathematical manner. First, based on the dynamic equations of state transition and output behavior which are developed recently, an algebraic criterion of k‐difference states is established. Second, using the criterion, a scheme is designed to construct the incompatible graphs of FSMs. Third, with the incompatible graphs and the method of searching internally stable sets of graphs proposed by the authors, a solution is proposed to obtain all of the compatible state set (CSS) of FSMs. Then, with the aid of the CSS, we investigate three kinds of structures of state space of FSMs, including compatible cover of state set (CCSS), representative set of state set (RSSS), and minimum representative set of state set (MRSSS); necessary and sufficient conditions are proposed to formulate the three kinds of structures. Finally, examples are given to exemplify minimum realizations of FSMs by these conditions.     

Language acceptability of finite automata based on theory of semi‐tensor product of matrices

Using the theories of many‐valued logic and semi‐tensor product of matrices (STP), this paper investigates how to mathematically determine whether or not a regular language is recognized by finite automata (FA). To this end, the dynamic behaviour of FA is first formulated as bilinear dynamic equations, which provides a uniform model for deterministic and non‐deterministic FA. Based on the bilinear model, the recognition power of FA understanding of regular languages is investigated and several algebraic criteria are obtained. With the algebraic criteria, to judge whether a regular sentence is accepted by a FA or not, one only needs to calculate an STP of some vectors, rather than making the sentence run over the machine as traditional manners. Further, the inverse problem of recognition is considered, an algorithm is developed that can mathematically construct all the accepted sentences for a given FA. The algebraic approach of this paper may be a new angle and means to understand and analyse the dynamics of FA.     

On disturbance decoupling problem of Boolean control networks

In this paper, the disturbance decoupling problem for Boolean control networks is investigated in a new viewpoint. A new concept called original disturbance decoupling is proposed and necessary and sufficient conditions for the original disturbance decoupling are given. Similarly, an equivalent condition for the s‐step original disturbance decoupling is obtained. Finally, an algorithm and some simulations for an example are provided to validate the obtained theoretical results.     

Stability and stabilization of Boolean networks

The stability of Boolean networks and the stabilization of Boolean control networks are investigated. Using semi‐tensor product of matrices and the matrix expression of logic, the dynamics of a Boolean (control) network can be converted to a discrete time linear (bilinear) dynamics, called the algebraic form of the Boolean (control) network. Then the stability can be revealed by analyzing the transition matrix of the corresponding discrete time system. Main results consist of two parts: (i) Using logic coordinate transformation, the known sufficient condition based on incidence matrix has been improved. It can also be used in stabilizer design. (ii) Based on algebraic form, necessary and sufficient conditions for stability and stabilization, respectively, are obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

Topologies on quotient space of matrices via semi‐tensor product

An equivalence of matrices via semi‐tensor product (STP) is proposed. Using this equivalence, the quotient space is obtained. Parallel and sequential arrangements of the natural projection on different shapes of matrices lead to the product topology and quotient topology respectively. Then the Frobenious inner product of matrices is extended to equivalence classes, which produces a metric on the quotient space. This metric leads to a metric topology. A comparison for these three topologies is presented. Some topological properties are revealed.     

Subspace controllability of Boolean networks

Many Boolean control networks contain independent uncontrollable subnetworks, which may affect other nodes; and the rest of the system is called subspace of sub‐controllable states. This paper investigates the problem of subspace controllability under free input sequences, while the presumption that the initial states of those independent subnetworks are designable is canceled. An algorithm based on the common asymptotic periodic properties of the states is developed to find the reachable sets. Accordingly, the existing controllability criteria for subspaces when initial states of subnetworks are designable is improved, and a necessary and sufficient condition of subspace controllability via subnetworks and free inputs is derived. A design technique involving a kind of newly defined addition is presented to construct desired controls.     

Observability conditions of Boolean control networks

Observability is a basic, yet challenging, issue when studying Boolean control networks (BCNs). Recently, a criterion for observability of controllable BCNs was proposed by using the algebraic representation of logical dynamics based on the technique of the semi‐tensor product of matrices. In this paper, we present new necessary and sufficient conditions guaranteeing observability of a BCN without preassuming its controllability. The conditions are hence more general. Some examples are worked out to illustrate the obtained results. Copyright © 2013 John Wiley & Sons, Ltd.     

Minimum‐Time State Feedback Stabilization of Constrained Boolean Control Networks

Ministry in 2005, and the National Prize of Natural Science of China in 2008. Currently, he is Associate Editor IMA Journal of Math Control and Inform., and a Technical Committee member of IFAC (TC2.3). This paper investigates the state feedback stabilization of Boolean control networks (BCNs) with state and input constraints by using the semi‐tensor product of matrices. Firstly, a kind of constrained input‐state incidence matrix is proposed for constrained BCNs, which contains all the reachability information of the constrained systems. Secondly, based on the constrained input‐state incidence matrix, a necessary and sufficient condition is presented for the stabilization of constrained BCNs via state feedback. Thirdly, a general procedure is proposed to design all possible minimum‐time state feedback stabilizers. The study of an illustrative example shows that the new results obtained are effective in analyzing the stabilization of constrained BCNs.     

切换奇异布尔网络的稳定性分析

基因调控网络的稳定性分析是系统生物学的研究热点问题之一.本文利用矩阵半张量积方法研究了切换奇异布尔网络的稳定性问题.首先给出了切换奇异布尔网络的代数表示,基于该代数表示,建立了系统解存在唯一的充要条件.然后通过将切换奇异布尔网络转化为等价的切换布尔网络,分别得到了系统在任意切换下稳定以及切换可稳的充要条件.最后给出例子验证所得结果的有效性.     

Controllability and observability of state‐dependent switched Boolean control networks with input constraints

This paper investigates the controllability and observability of state‐dependent switched Boolean control networks (SDSBCNs) with input constraints. First, the algebraic form of SDSBCNs is derived via the semi‐tensor product of matrices. Then constrained model‐input‐state (CMIS) matrices are introduced for SDSBCNs with input constraints. On the basis of CMIS matrices, two necessary and sufficient conditions are provided for the controllability and observability. Finally, two examples are given to show the effectiveness of the main results.     

State feedback controller design for anti‐synchronization of Boolean control networks: An event‐based idea

In this paper, we investigate the event‐based state feedback control for the anti‐synchronization of Boolean control networks (BCNs) under the configuration of drive‐response coupling. Two equivalence properties for the anti‐synchronization of BCNs are obtained by the algebraic representations of the logical dynamics. Based on the analysis of the event conditions, an algorithm is established to design the event‐based state feedback controller, and a necessary and sufficient condition guaranteeing the anti‐synchronization of the drive‐response coupled BCNs is formulated. An example is finally given to demonstrate the validity of the obtained results.     

Controllability of time-variant Boolean control networks and its application to Boolean control networks with finite memories

This paper investigates the controllability of time-variant Boolean control networks(BCNs).For the time-variant BCNs,a necessary and sufcient condition for the controllability is given,and a control design algorithm is presented.For a BCN with fnite memories,an equivalent transformation to a time-variant BCN is constructed.Then a necessary and sufcient condition for the controllability and a control design algorithm are obtained.     

A Semi‐Tensor Product Approach to Pseudo‐Boolean Functions with Application to Boolean Control Networks

Using the semi‐tensor product method, this paper investigates several fundamental problems of general pseudo‐Boolean functions with application to the optimal control of Boolean control networks, and establishes a new framework to deal with pseudo‐Boolean inequalities, the optimization problem and the best linear approximation of pseudo‐Boolean functions. First, the pseudo‐Boolean function is expressed in the algebraic form via constructing its unique structural matrix. Second, based on the matrix expression, solving pseudo‐Boolean inequalities is converted into finding solutions to algebraic inequalities, and a set of new formulas are presented. Third, the optimization problem and the linear approximation of the pseudo‐Boolean function are considered and several new results are established. Finally, as an application, we investigate the optimal control of Boolean control networks, and present a new optimal control design procedure. It is shown through the study of illustrative examples that the new results proposed in this paper work very well.     

L2 stability,H∞ control of switched homogeneous nonlinear systems and their semi‐tensor product of matrices representation

This paper investigates the problems of L₂ stability and H_∞ control of switched homogeneous nonlinear systems. This paper gives that if a switched homogeneous nonlinear system with disturbance is internally homogeneously asymptotically stable, then it has a finite L₂ gain, and if a switched homogeneous nonlinear system with disturbance and controls is homogeneously stabilizable under the zero disturbance condition, its H_∞ control problem is solvable under some mild conditions, and a homogeneous solution is given. Then, via the semi‐tensor product of matrices method, the aforementioned obtained results are transformed to linear‐like forms, and the aforementioned obtained Hamilton–Jacobi–Isaacs inequality is transformed to a linear‐like matrix inequality, which makes it feasible to compute the solutions by computer. Copyright © 2012 John Wiley & Sons, Ltd.     

布尔控制网络的输出稳定与镇定

矩阵的半张量积是将逻辑变量转化为向量研究的主要工具.本文利用半张量积把逻辑控制系统表示为离散时间仿射线性系统,在逻辑系统的状态空间框架下研究了以布尔控制网络为代表的逻辑动态系统的输出稳定与镇定.首先给出布尔网络输出稳定的定义,研究了布尔网络输出稳定的充要条件;其次讨论了布尔控制网络的输出镇定,分别得到了布尔控制网络由常值输入变量、自由控制序列、状态反馈控制序列输出镇定的条件.本文讨论的系统输出稳定与镇定是(部分)变量稳定与镇定的推广.     

Simultaneous stabilization for a set of Boolean control networks

This paper investigates simultaneous stabilization of a collection of Boolean control networks (BCNs) by using the semi-tensor product method, and presents a number of new results. First, an algebraic expression of the BCNs is obtained by the semi-tensor product, based on which some necessary and sufficient conditions are presented to solve the simultaneous stabilization problem by a free control sequence, a state-feedback control, and an output-feedback control, respectively. Second, using the column stacking form of matrices, a new procedure is established to design both state-feedback and output-feedback controllers for the simultaneous stabilization problem. The study of two illustrative examples shows that the new results obtained in this paper are very effective in solving simultaneous stabilization of a collection of BCNs.     

Partial stability and stabilisation of Boolean networks

In this paper, we investigate the stability of Boolean networks and the stabilisation of Boolean control networks with respect to part of the system's states. First, an algebraic expression of the Boolean (control) network is derived by the semi-tensor product of matrices. Then, some necessary and sufficient conditions for partial stability of Boolean networks are given. Finally, the stabilisation of Boolean control networks by a free control sequence and a state-feedback control is investigated and the respective necessary and sufficient conditions are obtained. Examples are provided to illustrate the efficiency of the obtained results.     

Robust stability and stabilisation of Boolean networks with disturbance inputs

This paper deals with the problems of robust stability analysis and robust stabilisation control design for Boolean networks with disturbance inputs. The main tool used in this paper is the semi-tensor product of matrices. First, a necessary and sufficient condition is presented for the robust stability of Boolean networks based on the algebraic representation of Boolean networks with disturbance inputs. Second, the concept of one-step robust reachability is defined for Boolean control networks (BCNs) with disturbance inputs, based on which a constructive procedure is proposed for the robust stabilisation control design of BCNs. The study of an illustrative example demonstrates the effectiveness of the obtained new results.     

Stability and stabilization of Boolean networks with impulsive effects

This paper investigates the stability and stabilization of Boolean networks with impulsive effects. After giving a survey on semi-tensor product of matrices, we convert a Boolean network with impulsive effects into impulsive discrete-time dynamics. Then, some necessary and sufficient conditions are given for the stability and stabilization of Boolean networks with impulsive effects. Finally, examples are provided to illustrate the efficiency of the obtained results.     

基于牵制控制的切换布尔网络的分布式集合镇定

切换布尔网络是一种典型的网络化控制系统, 在基因调控、信息安全、人工智能、电路设计等领域具有重 要应用. 本文基于牵制控制方法, 研究切换布尔网络在任意切换下的分布式集合镇定问题. 首先, 利用矩阵半张量积 方法,得到切换布尔网络的代数形式. 其次, 基于代数形式, 提出构造性的算法来实现切换布尔网络在牵制控制的 作用下任意切换集合镇定, 并设计出状态反馈牵制控制器. 再次, 利用逻辑矩阵分解技术和分布式控制方法, 设计任 意切换下切换布尔网络的分布式集合镇定控制器, 并提出分布式控制器存在的充分条件. 文中给出3个例子来说明 所获得结果的有效性.