Luo Chao, Wang Xingyuan, Liu Hong
1] School of Information Science and Engineering, Shandong Normal University, Jinan 250014, China [2] Shandong Provincial Key Laboratory for Novel Distributed Computer Software Technology, Jinan 250014, China.
Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, China.
Sci Rep. 2014 Dec 17;4:7522. doi: 10.1038/srep07522.
In this article, the controllability of asynchronous Boolean multiplex control networks (ABMCNs) with time delay is studied. Firstly, dynamical model of Boolean multiplex control networks is constructed, which is assumed to be under Harvey' asynchronous update and time delay is introduced both in states and controls. By using of semi-tensor product (STP) approach, the logical dynamics is converted into an equivalent algebraic form by obtaining the control-depending network transition matrices of delayed system. Secondly, a necessary and sufficient condition is proved that only control-depending fixed points of the studied dynamics can be controlled with probability one. Thirdly, respectively for two types of controls, the controllability of dynamical control system is investigated. When initial states and time delay are given, formulae are obtained to show a) the reachable set at time s under specified controls; b) the reachable set at time s under arbitrary controls; c) the reachable probabilities to different destination states. Furthermore, an approach is discussed to find a precise control sequence which can steer dynamical system into a specified target with the maximum reachable probability. Examples are shown to illustrate the feasibility of the proposed scheme.
本文研究了具有时滞的异步布尔多路控制网络（ABMCNs）的可控性。首先，构建了布尔多路控制网络的动力学模型，该模型假定采用哈维异步更新，且在状态和控制中均引入了时滞。通过使用半张量积（STP）方法，通过获取时滞系统的依赖控制的网络转移矩阵，将逻辑动力学转换为等效的代数形式。其次，证明了一个充要条件，即所研究动力学的依赖控制的不动点只有在概率为1时才能被控制。第三，分别针对两种类型的控制，研究了动态控制系统的可控性。当给定初始状态和时滞时，得到了公式以表明：a）在指定控制下时刻s的可达集；b）在任意控制下时刻s的可达集；c）到达不同目标状态的可达概率。此外，还讨论了一种方法来找到一个精确的控制序列，该序列能够以最大可达概率将动态系统引导到指定目标。通过实例说明了所提方案的可行性。
