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.     

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.     

Controllability and stabilization of Boolean control networks by the auxiliary function of flipping

In this article, a hybrid method combining flip matrix approach and an open (or closed) loop control is proposed to study global controllability and stabilization of Boolean control networks (BCNs). First, the necessary and sufficient condition for global controllability of BCNs, by flipping some members of a perturbation set and under free control sequences, is proposed. After that, using a search algorithm, the minimal perturbation sets for global controllability (MS‐GCs) of the BCN are obtained. Next, we propose a necessary and sufficient criterion for global stabilization of BCNs by flipping some members of a perturbation set and under a state feedback control. Similarly, an algorithm is given to search for the minimal perturbation sets for global stabilization (MS‐GSs) of the BCN. Moreover, the time‐optimal MS‐GSs of the BCN are also obtained by an algorithm. Some examples are given to illustrate the effectiveness of the results.     

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.     

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.     

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.     

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.     

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.     

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.     

An average optimal control approach to the set stabilization problem for boolean control networks

This paper addresses the set stabilization problem for deterministic Boolean control networks (BCNs). An optimal control approach is investigated to solve the problems by using the semi‐tensor product of matrices, where a policy iteration algorithm for the set stabilization problem is deduced. Finally, the intervention problem of a cAMP receptor protein is addressed in the framework of the set stabilization problem. The problem is solved to validate the effectiveness of the proposed policy iteration approach for a practical application.     

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.     

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.     

Set stabilization of boolean networks via sampled‐data control

The paper investigates the sampled‐data state feedback control (SDSFC) for set stabilization of Boolean control networks (BCNs). Set stabilization means that a system converges to a subset of the state space under certain controllers. Assume that the given subset is , where and sampling period is τ. We consider two conditions q ≤ τ;q > τ and for any given subset , calculate the corresponding largest control invariant subset (LCIS). Moreover, a design procedure to calculate all possible SDSFCs for set stabilization of BCNs is obtained. Ultimately, we provide an example to demonstrate the efficiency of the results.     

Stability and stabilization of evolutionary games with time delays via matrix method

This paper studies the stability and stabilization of a class of evolutionary games (EGs) with time delays via the semi‐tensor product (STP) method. First, the matrix expression of the EGs with time delays is given, and then it is converted into the form without delay equivalently. By designing Lyapunov function, a necessary and sufficient condition for the global stability of delayed EGs is obtained. To deal with the delayed EGs with multiple equilibriums, the controlled EGs with time delays is investigated. An intermittent state feedback control is proposed, in which the controller works only at certain strategy profiles. Based on the intermittent control, two necessary and sufficient conditions are provided to assure the stabilization of the controlled EGs with time delays. Some examples are worked out to illustrate the effectiveness of the theoretical results.     

Bayesian selection probability estimation for probabilistic Boolean networks

A Bayesian approach to estimate selection probabilities of probabilistic Boolean networks is developed in this study. The concepts of inverse Boolean function and updatable set are introduced to specify states which can be used to update a Bayesian posterior distribution. The analysis on convergence of the posteriors is carried out by exploiting the combination of semi‐tensor product technique and state decomposition algorithm for Markov chain. Finally, some numerical examples demonstrate the proposed estimation algorithm.     

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.     

状态翻转控制下布尔控制网络的可镇定性和Q学习算法

在给定一个子集的条件下, 本文研究了在状态翻转控制下布尔控制网络的全局镇定问题. 对于节点集的给定子 集, 状态翻转控制可以将某些节点的值从1 (或0)变成0 (或1). 将翻转控制作为控制之一, 本文研究了状态翻转控制下的 布尔控制网络. 将控制输入和状态翻转控制结合, 提出了联合控制对和状态翻转转移矩阵的概念. 接着给出了状态翻转 控制下布尔控制网络全局稳定的充要条件. 镇定核是最小基数的翻转集合, 本文提出了一种寻找镇定核的算法. 利用可 达集的概念, 给出了一种判断全局镇定和寻找联合控制对序列的方法. 此外, 如果系统是一个大型网络, 则可以利用一 种名为Q学习算法的无模型强化学习方法寻找联合控制对序列. 最后给出了一个数值例子来说明本文的理论结果.     

On controllability and stabilizability of probabilistic Boolean control networks

The controllability of probabilistic Boolean control networks(PBCNs)is first considered.Using the input-state incidence matrices of all models,we propose a reachability matrix to characterize the joint reachability.Then we prove that the joint reachability and the controllability of PBCNs are equivalent,which leads to a necessary and sufcient condition of the controllability.Then,the result of controllability is used to investigate the stability of probabilistic Boolean networks(PBNs)and the stabilization of PBCNs.A necessary and sufcient condition for the stability of PBNs is obtained first.By introducing the control-fixed point of Boolean control networks(BCNs),the stability condition has finally been developed into a necessary and sufcient condition of the stabilization of PBCNs.Both necessary and sufcient conditions for controllability and stabilizability are based on reachability matrix,which are easily computable.Hence the two necessary and sufcient conditions are straightforward verifiable.Numerical examples are provided from case to case to demonstrate the corresponding theoretical 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.     

Reconstructibility analysis and set-observer design for Boolean control networks

This paper deals with the issues of reconstructibility analysis and set-observer design for Boolean control networks (BCNs). A state estimation set is considered to find the state estimation of BCNs, and a reconstructibility judgment matrix (RJM) is obtained by compact vector of state estimation set. Then, a kind of state-reconstructible tree is constructed, and algorithms are also provided to construct it. The state-reconstructible tree can categorize BCNs to be globally reconstructible, locally reconstructible, and unreconstructible. The sufficient and necessary conditions for the reconstructibility of BCNs are given. Meanwhile, the relationship between input–output trajectory and estimated state is derived. Next, with the knowledge of input–output sequence, a set-observer is designed such that the state of BCNs for any global reconstructible BCNs is uniquely determined in a finite time. Some examples are also given to illustrate the effectiveness of the proposed methods.