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

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.     

Controllability of Boolean control networks with time delays in states

This paper investigates the controllability of Boolean networks with time-invariant delays in states. After a brief introduction on converting the logic dynamics to discrete time delay systems, the controllability via two kinds of controls is studied. One kind of control is generated by Boolean control networks, another kind of control is free Boolean sequences. In both cases, necessary and sufficient conditions of the controllability of Boolean control networks are proved. Finally, examples are given to illustrate the efficiency of the obtained results.     

Identification of Boolean control networks

In this paper the identification of Boolean control networks is addressed. First, necessary and sufficient conditions are obtained for the identification of state equation from input-state data. Then a necessary and sufficient condition for a controllable Boolean network to be observable is presented. Based on these two results, a necessary and sufficient condition for the identification from input–output data is achieved. To practically identify the model, a numerical algorithm is proposed. Two particular cases: (i) identification of systems with a known network graph; (ii) identification of a higher order Boolean network, are also investigated. Finally, the approximate identification for large size networks is explored.     

Feedback control and output feedback control for the stabilisation of switched Boolean networks

This paper presents the feedback control and output feedback control for the stabilisation of switched Boolean network. A necessary condition for the existence of a state feedback controller for the stabilisation of switched Boolean networks under arbitrary switching signal is derived first, and constructive procedures for feedback control and output feedback control design are provided. An example is introduced to show the effectiveness of this paper.     

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.     

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.     

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.     

An analysis and design for time-varying structures dynamical networks via state constraint impulsive control

This paper investigates the problems of exponential stabilisation criterion for time-varying structures dynamical networks via state constraint impulsive control. Two types of state constraint impulsive control schemes are proposed which include fully state constraint and partial state constraint. Combining with convex analysis, inductive method and matrix measure theory, some sufficient conditions are obtained to ensure exponential stabilisation for time-varying structures dynamical systems by using the constrains linear feedback impulsive controllers. Meanwhile, some simulation results are given to validate the effectiveness of the proposed criteria in the end.     

Dynamics and control of singular boolean networks

Consider a class of singular Boolean networks, which consist of two parts: difference part and algebraic part. Using the truth matrix of the algebraic part, the trajectories of the networks are obtained and certain properties are investigated. Then the results are extended to Boolean control networks and the controllability of singular Boolean control systems is investigated. Necessary and sufficient conditions are obtained.     

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.     

概率布尔控制网络的集可控

本文研究概率布尔控制网络的集可控性问题.首先,利用矩阵半张量积方法,得到概率布尔控制网络的代数表示.其次,借助一个新的算子构造不同的可控矩阵,进而通过可控矩阵考虑自由控制序列和网络输入控制下概率布尔控制网络的集可控性问题,得到了概率布尔控制网络集可控性的充要条件.最后,给出数值例子说明本文结果的有效性.     

Controllability and observability of Boolean control networks

The controllability and observability of Boolean control networks are investigated. After a brief review on converting a logic dynamics to a discrete-time linear dynamics with a transition matrix, some formulas are obtained for retrieving network and its logical dynamic equations from this network transition matrix. Based on the discrete-time dynamics, the controllability via two kinds of inputs is revealed by providing the corresponding reachable sets precisely. Then the problem of observability is also solved by giving necessary and sufficient conditions.     

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.     

具有切换概率分布的概率布尔网络的依分布稳定和镇定

布尔网络作为研究基因调控网络的一种重要模型,近年来引起了国内外很多学者的广泛关注.本文利用代数状态空间表示方法,研究具有切换概率分布的概率布尔网络的依分布稳定和镇定问题.首先,回顾针对切换布尔网络稳定性分析的现有的研究结果.其次,给出具有切换概率分布的概率布尔网络依分布稳定的定义,并利用矩阵的半张量积建立具有切换概率分布的概率布尔网络的代数表示.再次,基于该代数表示,建立具有切换概率分布的概率布尔网络的依分布稳定的充分必要条件.最后,给出具有切换概率分布的概率布尔控制网络镇定问题可解的充要条件,并给出相应的控制设计方法.     

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.     

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.     

Morgan's problem of Boolean control networks

This paper investigates the Morgan''s problem of Boolean control networks. Based on the matrix expression of logical functions, two key steps are proposed to solve the problem. First, the Boolean control network is converted into an output-decomposed form by constructing a set of consistent output-friendly subspaces, and a necessary and sufficient condition for the existence of the consistent output-friendly subspaces is obtained. Secondly, a type of state feedback controllers are designed to solve the Morgan''s problem if it is solvable. By solving a set of matrix equations, a necessary and sufficient condition for converting an output-decomposed form to an input-output decomposed form is given, and by verifying the output controllability matrix, the solvability of Morgan''s problem is obtained.     

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.     

Set reachability and set stability of Boolean networks with state-dependent asynchronous updating rule

Asynchronous behavior is significant for real biological systems. This paper explores the impact of state-dependent asynchronous updating rule (SDAUR) on the set reachability and set stability of Boolean networks (BNs). First, a set of state-dependent control variables is introduced to describe the SDAUR. Second, a new asynchronous set reachablity matrix is constructed based on an auxiliary input and semi-tensor product. Third, using the asynchronous set reachablity matrix, several necessary and sufficient conditions are obtained for the set reachablity and set stability of BNs with SDAUR. Finally, the new results proposed in this paper are verified through the immunomic network.