基于流转换理论的多自主体一致性及动力学行为研究

【作者】 孙晓辉；

【导师】 傅希林；

【作者基本信息】 山东师范大学 ， 管理决策理论与应用， 2015， 博士

【摘要】 复杂网络作为复杂系统中的新兴学科,一直以来受到了国内外许多不同领域研究者的广泛关注。多主体系统作为一类特殊的复杂网络模型,其一致性问题的研究成为了信息科学、管理科学和控制科学等众多学科所关注的热点,也成为了科技前沿的战略性课题之一。多自主体系统一致性问题的研究到目前已取得了很多的相关成果。但是现有的研究方法和研究成果还不全面,仍有很多方面没有涉及。根据该问题的研究现状,本文主要利用不连续动力系统中流转换理论对多自主体系统的一致性问题及离散一致性问题进行了研究和分析,并对一致性问题常用研究方法中的系统稳定性理论进行了探讨,给出了一种更有效、更便利的基于向量场的方法。本文主要有以下内容：1.利用不连续动力系统中的流转换理论研究了二阶连续多自主体系统在有限时间内的一致性问题,其中研究对象是具不同固有非线性特征的二阶自主体系统。本文在有领导者的前提下,通过一个形式简单、易于实现的一致性协议,研究了有限时间内系统中个体之间运动的一致性。在研究中,首先根据不连续动力系统理论将系统中每个个体的运动场按照一致性协议的限制划分为不同的子区域,在各个子区域上由于一致性协议的影响,个体的运动分别具有不同的动力学特征。在一致性协议的约束下,要实现系统的一致性,就要使得个体在子区域之间的边界上运动。因此,根据各子区域内系统的动力学特征,本文利用流转换理论对分隔边界附近各个子系统的运动情况进行了细致分析。通过定义相应的G函数,并通过G函数的符号及变化规律判断了子系统流的各种情况,以及各种可能出现的分支情形。通过子系统流的转换建立了判断系统一致性开始出现和一致性消失的切换条件,从而可以用来判断系统在非完全一致性时,部分时间段内的一致性状态,并进一步的建立了系统出现一致性的解析条件。基于本文的研究结论,在实际的应用中通过对结论中解析条件的分析,可以有针对性地为具体多自主体系统中参数的选择及一致性协议的选取提供理论上的决策依据。2.利用不连续动力系统理论中的映射动力学理论研究了一类非连续系统的离散一致性问题。研究对象是个体的动力学行为受到脉冲影响的二阶自主体系统。本文首先给出了个体运动在一次迭代下的伴随以及一致性定义,并进一步对离散一致性产生的条件进行了分析。根据一次迭代下一致性的定义,建立了系统中个体在相邻切换点处运动状态之间的合成映射,通过此合成映射给出了个体在一次迭代中能够形成一致性的充要条件,并且利用合成映射的分支情形给出了系统一致性开始和一致性消失的解析判定条件。从而能够从理论上为实际模型离散一致性研究提供可行的解决方案。3.利用不连续动力系统理论的流转换思想对一致性问题主要研究方法中的系统稳定性理论进行了研究。主要研究的对象是n维微分系统。我们提出了一种研究微分系统平衡点稳定性的向量场方法。首先利用系统的向量场和首次积分之间的关系得到了微分系统平衡点的一致稳定性。针对微分系统的首次积分难以确定的问题,本文将系统的向量场适当分解成两部分,使得其中一部分的首次积分可以相对容易地得到,而另一部分只要满足适当的条件,就可以得到判定系统平衡点稳定性的条件,并给出了微分系统平衡点一致稳定性的结论。然后,在此基础上进一步对系统稳定性的研究方法进行探讨,利用系统向量场和广义测度函数的关系研究了系统平衡点的一致稳定性,克服了首次积分对于问题的限制,并给出了微分系统平衡点一致稳定的结果。同时,本文利用上述方法对具有脉冲影响微分系统的稳定性也进行了研究,得到了相关结果。对上述结果分别通过数值模拟说明了结论的有效性。为系统稳定性理论在多自主体一致性问题研究中的应用提供了一种更便利、有效的方法。更多还原

【Abstract】 As the emerging discipline in the complex system, the complex networks have attracted extensive attentions of the domestic and international researchers in different fields. As a special kind of complex network model, the multi-agent system has become the focus in many disciplines, such as the information science, management science and control science, and the consensus of the multi-agent systems has been one of the strategic frontier subjects of science and technology. So far, there have been a lot of achievements on the consensus of the multi-agent systems. However, there are still many aspects which are not involved about the research methods and results. According to the research status quo of the problem, the theory for the discontinuous dynamical systems is used to analyze the consensus of the multi-agent systems in this thesis. At the same time, this thesis presents a method for judging the stability of the differential system. The main contributions are as follow:l.The consensus problems of the second-order continuous multi-agent systems within the limited time are investigated through the theory of flow switchability. Each equation in the systems has inherent characteristics of nonlinear dynamics. Through a simple consensus protocol, the consensus within limited time of the system with leader is discussed in the thesis. First of all, based on the theory of flow switchability, the direction field of each individual is divided into several different subdomains according to the restriction of the protocol. In each subdo-main, the individual’s motion has different dynamical characteristics respectively. In order to realize the consensus of the system under the protocol, the individu-al’s motion should be on the boundaries between the subdomains. Therefore, the motions of subsystems near the boundaries are analyzed in detail. By defining the corresponding G function, a variety of situations and possible bifurcations of the subsystems’flow are determined through the sign and variation of the G function. Through the flow switchability of the subsystems, the conditions for the onset and vanishing of the consensus are established. The consensus of the system in partial time intervals can be determined, when the full consensus does not exist. Furthermore, the analytical condition for consensus is presented.2. Using the mapping dynamical theory, the discrete consensus problem of a class of discontinuous systems is studied. The research is on the second order multi-agent system affected by the impulsive function. The definitions of com-panion and consensus under once iteration are established, and the consensus conditions are analyzed. According to the definitions, the composite mapping between the adjacent switch points is presented, and the necessary and sufficient conditions of consensus are discussed by using the mapping. The conditions for the onset and vanishing of the consensus are established.3. The stability of differential systems is investigated. The main research is on the n-dimensional differential systems. Based on the theory of flow switch-ability, we present a new method to study the stability of differential systems. The method takes the advantage of the relationship between the vector field and the first integral. In order to overcome the difficulty that the first integral of differential systems is hard to be determined, we decompose the vector field into two parts. The first integral of the first part can be got easily. When the other part meets the corresponding conditions, the stability of differential systems can be presented. Moreover, we discuss the stability of the impulsive differential sys-tems in this section. On this basis, the uniform stability of differential systems is discussed by using the relationship between the vector field and the generalized measure function. This method overcomes the restriction of the first integral, and presents results of uniform stability of equilibrium solution of the differential sys-tems. Also, we present the stability results for the impulsive differential systems. And the effectiveness of results is illustrated through numerical simulation.更多还原