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.     

状态和输入受限的切换奇异布尔控制网络的最优控制

本文研究了状态和输入均受限的切换奇异布尔控制网络的最优控制问题.利用矩阵半张量积方法获得受限切换奇异布尔控制网络的等价代数形式.然后通过类似针变化得到了存在最优控制的必要条件,并且提出了一个算法设计切换序列和控制策略使收益函数最大化.最后给出例子验证所得结果的有效性.     

Topological structure and optimal control of singular mix-valued logical networks

This paper introduces singular mix-valued logical networks and singular mix-valued logical control networks. Via semi-tensor product, a singular mix-valued logical network can be converted to an algebraic form. On this basis, the normalization problem and solvability are discussed. Then, fixed points and cycles of singular mix-valued logical networks are also studied. Furthermore, the optimal control problem of singular mix-valued logical control networks is presented and a necessary condition for the existence of the optimal control is supplied, based on which, an algorithm is provided. Illustrative examples are given to show the feasibility of the results.     

A temporal context model for boosting video annotation

In this paper, we propose a new method to model the temporal context for boosting video annotation accuracy. The motivation of our idea mainly comes from the fact that temporally continuous shots in video are generally with relevant content, so that the performance of video annotation could be comparably boosted by mining the temporal dependency between shots in video. Based on this consideration, we propose a temporal context model to mine the redundant information between shots. By connecting our model with conditional random field and borrowing the learning and inference approaches from it, we could obtain the refined probability of a concept occurring in the shot, which is the leverage of temporal context information and initial output of video annotation. Comparing with existing methods for temporal context mining of video annotation, our model could capture different kinds of shot dependency more accurately to improve the video annotation performance. Furthermore, our model is relatively simple and efficient, which is important for the applications which have large scale data to process. Extensive experimental results on the widely used TRECVID datasets exhibit the effectiveness of our method for improving video annotation accuracy.     

Deterministic and stochastic control of discrete-time bilinear systems

Optimal control of a class of time invariant single-input, discrete bilinear systems is investigated in this paper. Both deterministic and stochastic problems are considered.

In the deterministic problem, for the initial state in a certain set ∑₀, the solution is the same as the solution to the associated linear system. The optimal path may be a regular path or a singular path.

The stochastic control problem is considered with perfect state observation, and additive and multiplicative noise in the state equation. It is demonstrated that the presence of noise simplifies the analysis compared to that in the determinstic case.     

An integer programming approach to optimal control problems in context-sensitive probabilistic Boolean networks

A Boolean network is one of the models of biological networks such as gene regulatory networks, and has been extensively studied. In particular, a probabilistic Boolean network (PBN) is well known as an extension of Boolean networks, but in the existing methods to solve the optimal control problem of PBNs, it is necessary to compute the state transition diagram with 2ⁿ nodes for a given PBN with n states. To avoid this computation, an integer programming-based approach is proposed for a context-sensitive PBN (CS-PBN), which is a general form of PBNs. In the proposed method, a CS-PBN is transformed into a linear system with binary variables, and the optimal control problem is reduced to an integer linear programming problem. By a numerical example, the effectiveness of the proposed method is shown.     

Stationary LQG control of singular systems

The stationary linear-quadratic-Gaussian control problem is formulated and solved for single-input single-output singular systems. The control system is required to be internally proper and stable in order to avoid both impulsive and unstable exponential behavior. The set of all controllers resulting in such a control system is specified in parametric form. All controllers that yield finite cost are identified, once again in parametric form, within this set. Necessary and sufficient conditions are then established for an optimal controller to exist. All optimal controllers are shown to possess the same transfer function. The problem is analyzed in the complex domain. The transfer functions are expressed as quotients of proper, strict-Hurwitz rational functions. By means of this maneuver, the powerful tools of algebra are made available. The synthesis of the optimal controller is reduced to the solution of two linear Diophantine equations whose coefficients are obtained by spectral factorization.     

Minimum time control of large‐scale boolean control networks with constraints

The minimum time control that steers a large‐scale Boolean control network with constraints from an initial state to a given state is investigated. This problem is studied under the partition of the large‐scale Boolean control networks. Based on that, the problem is converted to finding the minimum time control for each subnetwork. Hence, the minimum time control for small size Boolean control networks with constraints is considered firstly. Then, a method to solve the problem for large‐scale Boolean control networks with constraints is presented. Finally, a numerical example on T‐cell receptor kinetics is given to show the effectiveness of the main results.     

Realization of Boolean control networks

Based on the linear expression of the dynamics of Boolean networks, the coordinate transformation of Boolean variables is defined. It follows that the state space coordinate transformation for the dynamics of Boolean networks is revealed. Using it, the invariant subspace for a Boolean control network is defined. Then the structure of a Boolean control network is analyzed, and the controllable and observable normal forms and the Kalman decomposition form are presented. Finally the realization problem, including minimum realization, of Boolean control networks is investigated.     

使用模型检测解决概率布尔网络优化控制

系统生物学期望对复杂生物系统建立一个真实的、可计算的模型,以便于以系统的角度去理解生物系统的演变过程。在系统生物学中,一个重要的主题是通过外部的干预控制发展关于基因调控网络的控制理论,以作为未来基因治疗技术。目前,布尔网络及其扩展的概率布尔网络已经被广泛用于对基因调控网络进行建模。在控制问题的研究中,概率布尔控制网络的状态迁移本质上构成一条有限状态空间的离散时间马尔科夫决策过程。依据马尔科夫决策过程的理论,通过概率模型检测方法解决网络中有限范围优化控制问题和无限范围优化控制问题。针对带有随机干扰且上下文相关的概率布尔控制网络,使用概率模型检测器PRISM对其进行形式化建模,然后将两类优化控制问题描述为相应的时序逻辑公式,最后通过模型检测寻找出最优解。实验结果表明,提出的方法可以有效地用于生物网络的分析和优化控制。     

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.     

布尔网络的分析与控制——矩阵半张量积方法

布尔网络是描述基因调控网络的一个有力工具. 由于系统生物学的发展, 布尔网络的分析与控制成为生物学与系统控制学科的交叉热点. 本文综述作者用其原创的矩阵半张量积方法在布尔网络的分析与控制中得到的一系列结果. 内容包括: 布尔网络的拓扑结构, 布尔控制网络的能控、能观性与实现, 布尔网络的稳定性和布尔控制网络的镇定, 布尔控制网络的干扰解耦, 布尔 (控制) 网络的辨识,以及布尔网络的最优控制等.     

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.     

Controllability of probabilistic Boolean control networks

This paper deals with the controllability of probabilistic Boolean control networks. First, a survey on the semi-tensor product approach to probabilistic Boolean networks is given. Second, the controllability of probabilistic Boolean control networks via two kinds inputs is studied. Finally, examples are given to show the efficiency of the obtained results.     

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.     

The effects of redundant control inputs in optimal control

For a stabilizable system, the extension of the control inputs has no use for stabilizability, but it is important for optimal control. In this paper, a necessary and sufficient condition is presented to strictly decrease the quadratic optimal performance index after control input extensions. A similar result is also provided for H ₂ optimal control problem. These results show an essential difference between single-input and multi-input control systems. Several examples are taken to illustrate related problems.     

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.     

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.     

Neighbouring-extremal control for singular dynamic optimisation problems. Part II: multiple-input systems

Dynamic optimisation provides a unified framework for improving process operations while taking operational constraints into account. In the presence of uncertainty, measurements can be incorporated into the optimisation framework for tracking the optimum. For non-singular control problems, neighbouring-extremal (NE) control can be used to force the first-order variation of the necessary conditions of optimality (NCO) to zero along interior arcs. An extension of NE control to singular control problems has been proposed in the companion paper for single-input problems. In this article, a generalisation to multiple-input systems is presented. In order for these controllers to be tractable from a real-time optimisation perspective, an approximate NE feedback law is proposed, whose application guarantees, under mild assumptions, that the first-order variation of the NCO converges to zero exponentially. The performance of multi-input NE control is illustrated by the case study of a steered car.     

Neighbouring-extremal control for singular dynamic optimisation problems. Part I: single-input systems

A powerful approach for dynamic optimisation in the presence of uncertainty is to incorporate measurements into the optimisation framework so as to track the optimum. For non-singular control problems, this can be done by tracking active constraints along boundary arcs and using neighbouring-extremal (NE) control along interior arcs. Essentially, NE control forces the first-order variation of the necessary conditions of optimality (NCO) to zero. In this article, an extension of NE control to singular control problems is proposed. This article focuses on single-input systems, while the extension to multiple-input systems is investigated in the companion paper. The idea is to design NE controllers from successive time differentiations of the first-order variation of the NCO. Approximate NE feedback laws are also proposed, which are both easily implementable and tractable from a real-time optimisation perspective. These developments are illustrated by the case study of a semi-batch chemical reactor.