布尔控制网络的系统分解

发布时间：2018-09-02 09:42

【摘要】：随着系统生物学的快速发展,布尔动态模型作为分析基因间相互作用的有效工具,迅速成为系统生物学家们研究的热点问题.由于布尔控制网络可以较好的模拟基因调控网络,因而布尔控制网络的研究在系统生物学领域和控制科学领域引起了广泛的关注.近年来,程代展及其团队提出了矩阵半张量积运算,这为布尔控制网络的研究提供了新的理论工具.运用矩阵半张量积,布尔控制网络的逻辑动态方程转化为双线性方程,这大大推动了布尔控制网络的发展.本文在矩阵半张量积的理论框架下,运用代数及图方法讨论了布尔控制网络的系统分解问题,提出并完整地解决了布尔控制网络关于输入的分解问题、关于输出的分解问题以及Kalman分解问题.在已有的研究布尔控制网络系统分解的文献中,作者在布尔控制网络的最大不能控子空间和最大不能观子空间是正则的条件下,得到了布尔控制网络可系统分解的充分条件.与已有的结果相比,本文没有沿用布尔控制网络状态空间及子空间的相关内容,而是运用代数和图的知识对三种分解问题分别进行了讨论,并得到了各分解问题可解的充分必要条件.本文的研究内容是作者在攻读博士学位期间所做工作的总结,具体如下：1.第2章首先在布尔控制网络的代数框架下,定义了布尔控制网络的状态转移图.随后运用集合论中等价类关系的相关知识,分析了状态转移图中顶点集和有向边之间的关系,并将布尔控制网络的结构与点集划分的相关知识有效地结合起来了.2.第3章讨论了布尔控制网络关于输入的分解问题,即系统的状态是否可分解成受外界输入控制和不受外界输入控制的两部分.首先,基于逻辑坐标变换,给出了布尔控制网络关于输入的分解的定义.利用有向图中顶点集和有向边之间的关系,得到了布尔控制网络关于输入可分解的充分必要条件,并进一步得到了构造逻辑坐标变换的方法.随后设计了算法来寻找布尔控制网络关于输入的分解形式.3.第4章研究了布尔控制网络关于输出的分解问题,即系统的状态是否可分解成影响输出和不影响输出的两部分.首先,给出了布尔控制网络关于输出的分解的定义.利用有向图中顶点集和有向边的关系以及各顶点的输出信息,得到了布尔控制网络关于输出可分解的充分必要条件,并进一步得到了构造逻辑坐标变换的方法.随后,设计了算法来寻找布尔控制网络关于输出的分解形式.最后,分析了布尔控制网络关于输出的可分解性与布尔控制网络的不能观测性之间的联系.4.第5章在布尔控制网络关于输入的分解和关于输出的分解二者的基础上,讨论了布尔控制网络的Kalman分解问题,即依据系统的状态是否受外界输入控制以及是否影响输出,能否将系统分解为四个子系统.利用已有的系统关于输入的分解和关于输出的分解的相关结论,给出了布尔控制网络可Kalman分解的充分必要条件,并设计了一个算法来寻找布尔控制网络的Kalman分解形式.基于得到的充要条件,得到了布尔控制网络异于传统线性控制系统的一个本质特征：并非所有的布尔控制网络都可Kalman分解.
[Abstract]:With the rapid development of system biology, Boolean dynamic model, as an effective tool to analyze the interaction between genes, has become a hot issue for system biologists. Because Boolean control network can better simulate gene regulatory network, the research of Boolean control network is in the field of system biology and control science. In recent years, Cheng and his team have proposed matrix semi-tensor product operation, which provides a new theoretical tool for the study of Boolean control networks. By using matrix Semi-tensor product, the logical dynamic equations of Boolean control networks are transformed into bilinear equations, which greatly promotes the development of Boolean control networks. In the framework of matrix semi-tensor product theory, the decomposition of Boolean control networks is discussed by algebraic and graph methods. The decomposition of input, output and Kalman decomposition of Boolean control networks are proposed and solved completely. Under the condition that the maximum uncontrollable subspace and the maximum uncontrollable subspace of a Boolean control network are regular, sufficient conditions are obtained for the system decomposition of a Boolean control network. This paper is a summary of the work done by the author during his Ph.D. study. It is as follows: 1. Chapter 2 defines the state transition diagram of Boolean control networks under the algebraic framework of Boolean control networks. In set theory, the relation between vertex set and directed edge in state transition graph is analyzed, and the structure of Boolean control network and the knowledge of point set partition are effectively combined. 2. Chapter 3 discusses the decomposition of input in Boolean control network, that is, whether the state of the system can be decomposed into acceptance. Firstly, based on the logical coordinate transformation, the definition of decomposition of Boolean control network is given. By using the relationship between vertex set and directed edge in directed graph, the necessary and sufficient conditions for decomposition of input in Boolean control network are obtained, and the structure is further obtained. Then an algorithm is designed to find the decomposition form of the Boolean control network with respect to the input. Chapter 4 studies the decomposition of the Boolean control network with respect to the output, i.e. whether the state of the system can be decomposed into two parts that affect the output and do not affect the output. By using the relation between vertex set and directed edge in a directed graph and the output information of each vertex, the necessary and sufficient conditions for decomposable output of Boolean control network are obtained, and the method of constructing logical coordinate transformation is further obtained. Finally, the relationship between the decomposability of Boolean control network with respect to output and the observability of Boolean control network is analyzed. 4. Chapter 5 discusses the Kalman decomposition of Boolean control network based on the decomposition of input and output, that is, whether the state of the system is subject to external conditions. Input control and whether it affects the output can be decomposed into four subsystems. Based on the existing results of decomposition of input and output, the necessary and sufficient conditions for Kalman decomposition of Boolean control networks are given, and an algorithm is designed to find the Kalman decomposition basis of Boolean control networks. Based on the necessary and sufficient conditions, an essential characteristic of Boolean control networks different from traditional linear control systems is obtained: not all Boolean control networks can be decomposed by Kalman.
【学位授予单位】：南京师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O231

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