【作者】 王振华；

【导师】 张焕水；

【作者基本信息】 山东大学 ， 控制理论与控制工程， 2015， 博士

【摘要】 本文研究了具有通信时滞的线性多智能体系统的趋同问题,基于无向的网络拓扑研究了两种情形：第一,系统临界不稳定、通信时滞为时变情形的趋同问题；第二,系统不稳定(至少临界不稳定)、通信时滞为定常情形的趋同问题.主要学术贡献：对高阶临界不稳定连续多智能体系统,给出了时滞为时变情形的趋同条件,本质上改进了之前系统为低阶或者要求时滞为定常的趋同结果；当系统不稳定时,将智能体自身的历史输入信息用在协议的设计中,减弱了传统结果中时滞对趋同结果的限制.具体研究内容、研究结果和创新按照章节顺序叙述如下：1.研究了具有时变时滞的高阶临界不稳定连续多智能体系统的趋同问题.用参数化的代数Riccati方程的唯一正定解设计趋同增益,分别研究了相对状态信息延迟与邻居状态信息延迟两种情形的趋同控制问题,基于Razumikhin稳定性定理给出了趋同的条件.创新之处：与已有的文献相比,本文研究的系统是高阶系统,通信时滞是时变且可以是未知的.2.研究了具有定常时滞的一阶不稳定连续多智能体系统的趋同问题.当时滞只影响相对状态信息时,为了减弱时滞对趋同结果的限制,将智能体的历史输入信息引入到协议的设计中,得到如下结果：·当智能体自身的历史输入信息用在协议的设计中时,得到了与通信时滞、网络拓扑及智能体结构三者相关的趋同条件.在智能体系统结构给定时,可以提高网络拓扑的一致性来增大时滞的容许范围；另一方面,通信时滞固定时,可以提高网络拓扑的一致性来容许更不稳定的系统达到趋同.特别地,当连通图为完全图时,任意有界的时滞及不稳定的智能体结构都可容许趋同.·当智能体与邻居间的相对历史输入信息引入到协议的设计中时,得到的趋同条件只与网络拓扑有关.当无向图连通时,任意有界的时滞都可容许趋同,此时的结果与无时滞时的结果类似.通信时滞只延迟邻居状态信息时,给出了只与网络拓扑有关的趋同条件.当条件满足时,任意有界的时滞都可容许趋同,此时的时滞还可以证明是未知且时变的.创新之处：与基于一阶积分器或者临界不稳定(极点在闭左平面)的高阶系统的趋同结果相比,本文研究了一阶不稳定(极点在闭右平面)的情形.此外,通过在趋同协议中引入智能体的历史输入信息,减弱了时滞对趋同结果的影响.当时滞只影响邻居状态信息时,将之前的结果改进为时变的情形.3.研究了具有定常时滞的一阶不稳定离散多智能体系统的趋同问题.相比连续情形,这方面的结果不是很多.与第2种情形类似,本文分别研究了时滞影响相对状态信息与邻居状态信息的趋同问题,并给出相应的趋同条件.特别地,当通信时滞为一步延迟时,包含智能体自身历史输入信息的协议的趋同条件不仅是充分的而且是必要的.创新之处：现存的结果往往针对一阶积分器或临界不稳定(极点在闭单位圆内)的高阶系统,与此相比,本文研究的是一阶不稳定(极点在单位圆上或之外)的系统.利用包含智能体历史输入信息的协议减弱了时滞对趋同结果的限制.如果时滞只影响邻居状态的信息,将之前的定常时滞改进为时变情形.4.研究了具有定常时滞的高阶不稳定连续多智能体系统的趋同问题.用时滞信息与参数化代数Riccati方程的唯一正定解设计增益,基于常微分方程的常数变异法、Razumikhin稳定性定理等给出了趋同条件.·当协议只包含相对状态信息时,通过分析一凹函数的极值给出了容许趋同的时滞上界,时滞上界与网络拓扑的一致性有关,在完全图时达到最大.·当协议包含相对状态信息与智能体自身的历史输入信息时,时滞上界仍基于一凹函数的极值给出.当网络拓扑为完全图时,任意有界时滞和任意高阶不稳定系统都可容许趋同.创新之处：与以往的结果相比,研究了至少临界不稳定的高阶连续多智能体系统的趋同问题,问题更具有一般性；在协议中引入智能体的历史输入信息,减弱了时滞对趋同结果的限制.更多还原

【Abstract】 This thesis investigates the consensus problem for linear multi-agent systems with communication delay. Under the assumption that the network topology is undi-rected, two issues are considered. Firstly, we study the consensus of critically un-stable high-order multi-agent systems that are subjected to time-varying commu-nication delay; Secondly, the consensus problem of unstable agents with constant communication delay is concerned.The main contribution are as:For the high-order continuous multi-agent sys-tems that are critically unstable, consensus conditions are provided for time-varying communication delay, which improves the existing results essentially; when the agent dynamic is unstable (at least critically unstable), agent’s own historical in-put information is used in the protocol design, and the results weaken the effect of delay on consensus.In the order of chapters, the main contents and innovations are listed as follows:1. Suppose that the systems are critically unstable, we study the consensus problem of high-order continuous multi-agent systems with time-varying communi-cation delay. The consensus gain is designed with the unique positive-definite so-lution of a parametric algebra Riccati equation. Then, by applying the Razumikhin stability Theorem, consensus conditions are provided respectively based on the de-layed relative information and the delayed neighbors’ informationThe main innovation is that consensus conditions are obtained for high-order multi-agent systems with time-varying and unknown communication delay, which extends the existing researches.2. For the first-order unstable continuous-time multi-agent systems, the con-sensus problem are addressed in case of constant communication delay. When the delayed relative information is available, agent’s own historical input information is used in the protocol to recede the effect of delay on consensus, and obtain the following results:● If the protocol contains agent’s own historical input information, consensus conditions related to communication delay, network topology and agent dy-namics are acquired. For any fixed agent dynamics, the better synchronizabil-ity corresponds to larger delay bound for consensus. On the other hand, if the communication delay is fixed, more unstable agent dynamic is allowed for consensus if the synchronizability of the network is better. Especially, when the undirected graph is completed, any bounded communication delay and un-stable agents are tolerant for consensus.●If the relative historical input information among agent and neighbors are used in the protocol, any bounded delay is allowed for consensus just as the undi-rected graph is connected, which is similar to the delay-free case.If the delay only influences the neighbors’ information, consensus condition related to network topology is provided for any bounded delay, and the delay in this case is allowed to be time-varying and unknown.The main innovations are that we study the unstable (the poles of the system matrix are on the closed right-half plane) agent dynamics, which are different from the results in the literature that are mainly on single-integrator or the systems that are critically unstable (the poles of the system matrix are on the closed left-half plane). Furthermore, the effect of delay on consensus is significantly weakened by introducing agent’s historical input information in the protocol design. In addition, we extend the constant delay to time-varying case on condition that the delay only affects the neighbors’information.3. The consensus problem for unstable first-order discrete multi-agent systems that are subjected to constant communication delay is studied. To the best of our knowledge, the relevant results are rare. Similarly as case 2, we respectively study the consensus problem on account of the delayed relative information and the de- layed neighbors’information, and the corresponding consensus conditions are fol-lows. Especially, when the protocol includes agent’s own historical input informa-tion, necessary and sufficient condition is obtained in case of one step delay.The main innovations are extending the existing results which are aimed at single-integrator or critically unstable(the poles of the system matrix are inside or on the closed unit circle) systems to the unstable(the poles of the system matrix are on or outside the unit circle) case. The historical input information in protocol definitely weakens the influences of delay on consensus. Furthermore, when the delay only affects neighbors’information, we also improve the constant delay to the time-varying case.4. In case of constant communication delay, the consensus problem of unstable high-order continuous multi-agent systems is discussed. We use the delay informa-tion and the unique positive-definite solution of a parametric algebra Riccati equation to design the control gain, and the consensus conditions are given by means of the variation of constants formula and Razumikhin stability Theorem.● If the protocol only contains the relative information, an allowable delay bound related to the maximum of a concave function is obtained, and the biggest delay bound is achieved in case of complete graph.● If the protocol includes the relative information and agent’s own historical input information, the allowable delay bound is also related to the maximum of a concave function. Especially, any bounded delay and high-order unstable agent dynamics are tolerant for consensus if the graph is complete.Comparing with the existing work, the system investigated are of high-order and at least critically unstable. The adoption of agent’s own historical input infor-mation in the protocol weakens the constraint of communication delay on consensus.更多还原