Construction of Minimal Delay Steiner Tree Using Two-pole Delay Model

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GhostConv轻量级网络设计及故障诊断研究

第 37 卷第 1 期2024 年1 月振动工程学报Journal of Vibration EngineeringVol. 37 No. 1Jan. 2024GhostConv轻量级网络设计及故障诊断研究赵志宏1，李春秀2，杨绍普1（1.石家庄铁道大学省部共建交通工程结构力学行为与系统安全国家重点实验室，河北石家庄 050043；2.石家庄铁道大学信息科学与技术学院，河北石家庄 050043）摘要: 提出一种GhostConv轻量级网络模型并将其用于故障诊断。

GhostConv利用常规卷积生成一小部分特征图，然后在生成的特征图上进行多次特征提取来生成其余特征图，最大程度地节约了常规卷积中生成冗余特征图的成本，减少了模型参数，保证了模型的性能。

采用连续小波变换对振动信号进行时频变换生成二维时频图，之后利用设计的GhostConv搭建轻量级网络模型进行故障诊断。

采用凯斯西储大学轴承数据集进行验证，并与其他卷积结构网络模型进行参数量、计算量以及识别准确率的对比。

实验结果表明，与其他模型相比，所使用的网络模型在参数量和计算量较少的条件下依旧有较高的识别精度，且具有较好的鲁棒性和泛化能力，具有一定的工程应用价值。

关键词: 故障诊断；滚动轴承；轻量级网络； GhostConv；时频图中图分类号: TH165+.3；TH133.33 文献标志码: A 文章编号: 1004-4523（2024）01-0182-09DOI：10.16385/ki.issn.1004-4523.2024.01.018引言轴承作为旋转机械最重要的组成部分，在运行过程中出现故障会导致安全事故的发生，造成巨大的经济损失。

因此，对滚动轴承的故障诊断越来越受到研究人员的重视［1］。

目前，关于轴承故障诊断的研究已有多种方法，例如，Lu等［2］使用遗传算法和经验模式分解提取特征，然后使用支持向量机对故障进行分类和识别。

Mao等［3］提出了一种结合多孔排列熵和支持向量机的诊断方法，对轴承故障类型进行分类。

图论及其应用

图和子图图和简单图图 G = (V, E)，其中 V = {νv v v ,......,,21} V ---顶点集， ν---顶点数E = {e e e 12,,......,ε}E ---边集， ε---边数例。

左图中， V={a, b,......,f}, E={p,q, ae, af,......,ce, cf} 注意，左图仅仅是图G 的几何实现（代表），它们有无穷多个。

真正的图G 是上面所给出式子，它与顶点的位置、边的形状等无关。

不过今后对两者将经常不加以区别。

称边 ad 与顶点 a (及d) 相关联。

也称顶点 b(及 f) 与边 bf 相关联。

称顶点a 与e 相邻。

称有公共端点的一些边彼此相邻，例如p 与af 。

环（loop ，selfloop ）：如边 l 。

棱（link ）：如边ae 。

重边：如边p 及边q 。

简单图：（simple graph ）无环，无重边平凡图：仅有一个顶点的图（可有多条环）。

一条边的端点：它的两个顶点。

记号：νε()(),()().G V G G E G ==。

习题1.1.1 若G 为简单图，则εν≤⎛⎝ ⎫⎭⎪2 。

1.1.2 n ( ≥ 4 )个人中，若每4人中一定有一人认识其他3人，则一定有一人认识其他n-1人。

同构在下图中，图G 恒等于图H , 记为 G = H ⇔ V （G)=V(H), E(G)=E(H)。

图G 同构于图F ⇔ V(G)与V(F), E(G)与E(F)之间各存在一一对应关系，且这二对应关系保持关联关系。

记为 G ≅F 。

注往往将同构慨念引伸到非标号图中，以表达两个图在结构上是否相同。

de f G = (V, E)y z w cG =(V , E )w cyz H =(V ’, E ’)’a ’c ’y ’e ’z ’F =(V ’’, E ’’)注判定两个图是否同构是NP-hard 问题。

完全图(complete graph) Kn空图（empty g.） ⇔ E = ∅ 。

计算机网络(第四版)课后习题(英文)+习题答案(中英文)

ANDREW S. TANENBAUM 秒，约533 msec.----- COMPUTER NETWORKS FOURTH EDITION PROBLEM SOLUTIONS 8. A collection of five routers is to be conn ected in a poi nt-to-poi nt sub net.Collected and Modified By Yan Zhe nXing, Mail To: Betwee n each pair of routers, the desig ners may put a high-speed line, aClassify: E aEasy, M ^Middle, H Hard , DaDeleteGree n: Importa nt Red: Master Blue: VI Others:Know Grey:—Unnecessary ----------------------------------------------------------------------------------------------ML V Chapter 1 In troductio nProblems2. An alter native to a LAN is simply a big timeshari ng system with termi nals forall users. Give two adva ntages of a clie nt-server system using a LAN.(M)使用局域网模型可以容易地增加节点。

如果局域网只是一条长的电缆，且不会因个别的失效而崩溃(例如采用镜像服务-------------------------------------------器)的情况下，使用局域网模型会更便宜。

计算机网络第四版(课后练习+答案)

计算机网络第四版(课后练习+答案)第 1 章概述1.假设你已经将你的狗Berníe 训练成可以携带一箱3 盒8mm 的磁带，而不是一小瓶内哇地. (当你的磁盘满了的时候，你可能会认为这是一次紧急事件。

)每盒磁带的窑最为7GB 字节;无论你在哪里，狗跑向你的速度是18km/h 。

请问，在什么距离范围内Berníe的数据传输速率会超过一条数据速率为150Mbps的传输线?答：狗能携带21千兆字节或者168千兆位的数据。

18 公里/小时的速度等于0.005 公里/秒，走过x公里的时间为x / 0.005 = 200x秒，产生的数据传输速度为168/200x Gbps或者840 /x Mbps。

因此，与通信线路相比较，若x<5.6 公里，狗有更高的速度。

6. 一个客户·服务器系统使用了卫星网络，卫星的高度为40 000km. 在对一个请求进行响应的时候，最佳情形下的延迟是什么?答：由于请求和应答都必须通过卫星，因此传输总路径长度为160,000千米。

在空气和真空中的光速为300，000 公里/秒，因此最佳的传播延迟为160,000/300，000秒，约533 msec。

9. 在一个集中式的二叉树上，有2n -1 个路出器相互连接起来:每个树节点上都布一个路由器。

路由器i 为了与路由器j 进行通信，它要给树的根发送一条消息。

然后树根将消息送下来给j 。

假设所有的路由器对都是等概率出现的，请推导出当n很大时，每条消息的平均跳数的一个近似表达式。

答：这意味着，从路由器到路由器的路径长度相当于路由器到根的两倍。

若在树中，根深度为1，深度为n，从根到第n层需要n-1跳，在该层的路由器为0.50。

从根到n-1 层的路径有router的0.25和n-2跳步。

因此，路径长度l为：18.OSI 的哪一层分别处理以下问题？答：把传输的比特流划分为帧——数据链路层决定使用哪条路径通过子网——网络层.28. 一幅图像的分辨率为1024X 768 像素，每个像素用3 字节来表示。

第4章物流节点选址其他模型

指出的是本问题没有需求量和容量，故无需考虑约束(8-19)。
表候选点服务范围
居民点号
A( j)
B(i)
1
1， 2， 3， 4
1， 2， 3， 4
2
1，2，3
l，2，3
3
l，2，3，4，5
1， 2， 3， 4， 5
4
1， 3， 4， 5， 6， 7
1， 3， 4， 5， 7
5
3， 4， 5， 6
3，4，5
两个候选点作为仓库地址，使总运输成本最小。( p 2 )。
1
4
5
2
1
4
7 3
2
6
8
3
图超市及仓库候选点位置
1166
解答:
4 12 20 6
2
10
25 10
3 4 16 14
C ij
6
18
5 12
9 7
2
3
14 2 4 9
20 30 2 11
24 12 6 22
100
50
120
零售点应该在需求点 3 或者它下面的位置。结合 2 个方面的限制和图 8-7 的相对位置，在 y 方向，只能选择一个
有效的中值点： ys 3 km。
6 5
5
4
4
3
B3
2 1
A
2
0
1
y，千米
0
1
2
3
4
5
6
x，千米
图 8-7 可能的方案
综合考虑 x 、 y 方向的影响，于是最后可能的地址为 A、B 之间的一条线段(见图 8-7)。表 8-4 对 A、B 两个位

一种对流层散射通信斜延迟估计方法

一种对流层散射通信斜延迟估计方法胡邓华;刘继业;陈西宏;王洁【摘要】为实现对流层散射通信的实时性,针对散射通信延迟估计问题,提出了一种不事先进行信道测量的对流层通信延迟计算方法.首先利用全球压力和温度2(GPT2)模型计算气象数据,然后采用射线描迹法对大气层分层并积分求和,最后计算出对流层散射通信延迟.采用与射线描迹法相结合的方法,摆脱了射线描迹法对探空数据的依赖.最后选取我国三个典型测量站数据进行算例分析,计算结果与我国对流层延迟实际分布特征相吻合,为研究在不事先进行信道测量的情况下计算对流层散射通信延迟量提供了一种新思路.%In order to realize real time troposcatter communication,a method for calculating troposcatter communication delay without channel measurement is proposed. First,the meteorological data is calculated by Global Pressure and Temperature 2(GPT2)model.Then,the troposcatter communication delay is calcu-lated based on ray tracing method. The calculation result with the data of three typical measuring stations in China is coincided with the actual distribution characteristics of tropospheric delay in China. This meth-od is a hybrid method combining the GPT2 model and ray tracing which avoids the dependence on sounding data. This paper offers a new idea and method for calculating the troposcatter communication delay without channel measurement.【期刊名称】《电讯技术》【年(卷),期】2018(058)005【总页数】6页(P500-505)【关键词】散射通信;斜延迟;全球压力和温度2模型;射线描迹法【作者】胡邓华;刘继业;陈西宏;王洁【作者单位】空军工程大学防空反导学院,西安710051;空军工程大学防空反导学院,西安710051;空军工程大学防空反导学院,西安710051;空军工程大学防空反导学院,西安710051【正文语种】中文【中图分类】TN9261 引言当无线电波在对流层中传播时，除沿途遭受折射外，还被对流层散射体再次辐射，这种通信方式即对流层散射通信。

基于非hanan点的时延约束最小斯坦那树生成方法

收稿日期：０６１－２０ —１２４
当漏点ｉ的时延违反值为正时，际时延ｄｎ）实（；大于时延约束，能满足时延约束；时延违反值为不当很大负值时，能存在过设计的问题。因此，可时延违反值为正或者为很大的负值时，都不是最佳方案。在
为清晰描述时延约束最小Ｓｅｅ树，先引入ｔｎｒ首ｉ
“ 时延违反 ” 的概念。
单线网最小长度Ｓｅｅ树（ｔｉｒ最小Ｓｅｅ树）ｎｔｉｒ生成，ｎ是
时延驱动总体布线的重要研究内容。
定义１时延违反值）若漏点ｉ（的实际时延为ｄ
很多情况下，将漏点的违反值尽量向０靠，以大大可
Ｎｏ－ｎａｌｙＣｏｓｒｉｅｅｎｒＲｏｉｅｎｒｔｏｇｒｔｎ・ＨａｎＤｅａ－ｎｔａｎｔｄＳｔｉｅｕｔｎｇＴｒｅＧｅｅａｉｎＡｌｏｉｈｍ
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（ｉ，ｎ）ｔｎｉ＝ｄｎ）一Ｔ（）ｉａｏ（）（ｉｌｉ一ｉ（）１
文献［］出的Ｓｅｅｌｏｅ１提ｔｎｒｍｒ布线树（ＥＴ算法ｉＥＳＲ）是一种基于Ｅｍｒ时延模型的满足时延约束的最小斯ｌｏｅ
０引言
当超大规模集成电路特征尺寸达到深亚微米阶
段时，互连阻抗对布线性能的影响越来越大，因此时延驱动的总体布线方法很受关注。满足时延约束的

2012-57-9-TAC-采样一致+二阶积分器+非一致时变时延

A Sufﬁcient Condition for Convergence of Sampled-DataConsensus for Double-Integrator Dynamics With Nonuniform and Time-Varying Communication Delays Jiahu Qin,Student Member,IEEE,andHuijun Gao,Senior Member,IEEEAbstract—This technical note investigates a discrete-time second-order consensus algorithm for networks of agents with nonuniform and time-varying communication delays under dynamically changing communica-tion topologies in a sampled-data setting.Some new proof techniques are proposed to perform the convergence analysis.It isﬁnally shown that under certain assumptions upon the velocity damping gain and the sampling pe-riod,consensus is achieved for arbitrary bounded time-varying commu-nication delays if the union of the associated digraphs of the interaction matrices in the presence of delays has a directed spanning tree frequently enough.Index Terms—Double-integrator agents,sampled-data consensus,span-ning tree,time-varying communication delays.I.I NTRODUCTIONIn recent years,consensus problems for agents with single-integrator dynamics have been studied from various perspectives(see,e.g.,[4], [7],[10],[11],[14],[16],[17],[26]).Taking into account that double-integrator dynamics can be used to model more complicated systems in reality,cooperative control for multiple agents with double-integrator dynamics has been studied extensively recently,see[12],[18]–[20], [23],[28]for continuous algorithms and[1]–[3],[5],[6],[8],[13]for discrete-time algorithms.In[8],a sampled-data algorithm is studied for double-integrator dy-namics through a Lyapunov-based approach.The analysis in[8]is lim-ited to an undirected network topology and cannot be extended to deal with the directed case.However,the informationﬂow might be directed in practical applications.In a similar sampled-data setting,[1]studies two sampled-data consensus algorithms,i.e.,the case with an absolute velocity damping term and the case with a relative velocity damping term,in the context of a directed network topology by extensively using matrix spectral analysis.Reference[2]extends the algorithms in[1]to deal with a dynamic directed network topology.References[5]and[6] mainly investigate sampled-data consensus for the case with a relative velocity damping term under a dynamic network topology.In[5],the network topologies are required to be both balanced and strongly con-nected at each sampling instant.On the other hand,considering that it might be difﬁcult to measure the velocity information in practice,[6] Manuscript received November17,2009;revised September15,2010; August15,2011,and January24,2012;accepted January25,2012.Date of publication February17,2012;date of current version August24,2012.This work was supported in part by the National Natural Science Foundation of China under Grants60825303,60834003,and61021002,by the973Project (2009CB320600),and by the Foundation for the Author of National Excellent Doctoral Dissertation of China(2007B4).Recommended by Associate Editor H.Ito.J.Qin is with Harbin Institute of Technology,Harbin,China,and also with the Australian National University,Canberra,A.C.T.,Australia(e-mail:jiahu. qin@.au).H.Gao is with the Research Institute of Intelligent Control and Systems, Harbin Institute of Technology,Harbin150001,China(e-mail:hjgao@. cn).Color versions of one or more of theﬁgures in this paper are available online at .Digital Object Identiﬁer10.1109/TAC.2012.2188425proposes a consensus strategy using the measurements of the relative positions between neighboring agents to estimate the relative velocities. In[13],consensus problems of second-order multi-agent systems with nonuniform time delays and dynamically changing topologies is investigated.However,the paper considers a discrete-time model es-timated by using the forward difference approximation method rather than a sampled-data model.In general,a sampled-data model is more realistic.Also,in[13],the weighting factors must be chosen from a ﬁnite set.With this background,we study the convergence of sam-pled-data consensus for double-integrator dynamics under dynamically changing topologies and allow the communication delays to be not only different but also time varying.Here,considering the weighting factors of directed edges between neighboring agents usually represent conﬁ-dence or reliability of the transmitted information,it is more natural to consider choosing the weighting factors from an inﬁnite set,which is more general than theﬁnite set case in[2]and[13].Moreover,dif-ferent from that in[13],A(k),the interaction matrix in the presence of delays at time t=kT,is introduced in this technical note and the dif-ference between A(k)and A(k),the adjacency matrix at time t=kT, is further explored as well.The reason for introducing A(k)is that it is more relevant than A(k)to the strategies investigated in this technical note.It is worth pointing out that the method employed to perform the convergence analysis is totally different from most of the existing liter-ature which heavily relies on analyzing the system matrix by spectral analysis.By using the similar transformation as that used in[13],we can treat the sampled-data consensus for double-integrator dynamics as the consensus for multiple agents modeled byﬁrst-integrator dynamics. Then,in order to make the transformed system dynamics mathemati-cally tractable,a new graphic method is proposed to specify the rela-tions between0(A(k)),the associated digraph of the interaction matrix in the presence of delays,and the the associated digraph of the trans-formed system matrix.Finally,motivated by the work in[22,Theorem 2.33]and[27],by employing the product properties of row-stochastic matrices from an inﬁnite set,we present a sufﬁcient condition in terms of the associated digraph of the interaction matrix in the presence of delays for the agents to reach consensus.Note here that the proving techniques employed in this technical note can be extended directly to derive similar results by considering the discrete-time model in[13]. The rest of the technical note is organized as follows.In Section II, we formulate the problem to be investigated and also provide some graph theory notations,while the convergence analysis is given in Section III.In Section IV,a numerical example is provided to show the effectiveness of the new result.Finally,some concluding remarks are drawn in Section V.II.B ACKGROUND AND P RELIMINARIESA.NotationsLet I n2n2n and0n;n2n2n denote,respectively,the identity matrix and the zero matrix,and1m2m be the column vector of all ones.Letand+denote,respectively,the set of nonnegative and positive integers.Given any matrix A=[a ij]2n2n,let diag(A) denote the diagonal matrix associated with A with the ith diagonal element equal to a ii.Hereafter,matrices are assumed to be compatible for algebraic operations if their dimensions are not explicitly stated.A matrix M2n2n is nonnegative,denoted as M 0,if all its entries are nonnegative.Let N2n2n.We write M N if M0N 0.A nonnegative matrix M is said to be row stochastic if all its row sums are1.Let k i=1M i=M k M k01111M1denote the left product of the matrices M k;M k01;111;M1.A row-stochastic matrix M is ergodic0018-9286/$31.00©2012IEEE(or indecomposable and aperiodic )if there exists a column vector f2nsuch that lim k !1M k =1n f T .B.Graph Theory NotationsLet G =(V ;E ;A )be a weighted digraph of order n with a ﬁnite nonempty set of nodes V =f 1;2;...;n g ,a set of edges E V 2V ,and a weighted adjacency matrix A =[a ij ]2n 2n with nonnegative adjacency elements a ij .An edge of G is denoted by (i;j ),meaning that there is a communication channel from agent i to agent j .The adjacency elements associated with the edges are positive,i.e.,(j;i )2E ,a ij >0.Moreover,we assume a ii =0for all i 2V .The set of neighbors of node i is denoted by N i =f j 2V :(j;i )2Eg .Denote by L =[l ij ]the Laplacian matrix associated with G ,where l ij =0a ij ,i =j ,and l ii=n k =1;k =i a ik .A directed path is a sequence of edges in a digraph of the form (i 1;i 2);(i 2;i 3);....A digraph has a directed spanning tree if there exists at least one node,called the root node,having a directed path to all the other nodes.A spanning subgraph G s of a directed graph G is a directed graph such that the node set V (G s )=V (G )and the edge set E (G s ) E (G ).Given a nonnegative matrix S =[s ij ]2n 2n ,the associated di-graph of S ,denoted by 0(S ),is the directed graph with the node set V =f 1;2;...;n g such that there is an edge in 0(S )from j to i if and only if s ij >0.Note that for arbitrary nonnegative matrices M;N2p 2p satisfying M N ,where >0,if 0(N )has a di-rected spanning tree,then 0(M )also has a directed spanning tree.C.Sampled-Data Consensus Algorithm for Double-Integrator DynamicsEach agent is regarded as a node in a digraph G of order n .Let T >0denote the sampling period and k2denote the discrete-time index.For notational simplicity,the sampling period T will be dropped in the sequel when it is clear from the context.We consider the following sampled-data discrete-time system which has been investigated in [1],[2],and [8]asr i (k +1)0r i (k )=T v i (k )+12T 2u i (k )v i (k +1)0v i (k )=T u i (k )(1)where x i (k )2p ,v i (k )2p and u i (k )2p are,respectively,the position,velocity and control input of agent i at time t =kT .For simplicity,we assume p =1.However,all results still hold for any p2+by introducing the notation of Kronecker product.In this technical note,we mainly consider the following discrete-time second-order consensus algorithm which takes into account the nonuniform and time-varying communication delays as u i (k )=0 v i (k )+j 2N (k )ij (k )(r j (k 0 ij (k ))0r i (k ))(2)where >0denotes the absolute velocity damping gain,N i (k )de-notes the neighbor set of agent i at time t =kT that varies with G (k )(i.e.,the dynamic communication topology at time t =kT ), ij (k )>0if agent i can receive the delayed position r j (k 0 ij (k ))from agent j at time t =kT while ij (k )=0otherwise,and 0 ij (k ) max ,where ij (k )2,is the communication delay from agent j to agent i .Here,we assume ii (t ) 0,that is,the time delays affect only the in-formation that is transmitted from one agent to another.Moreover,we assume that all the nonzero and hence positive weighting factors areboth uniformly lower and upper bounded,i.e., ij (k )2[ ;],where 0< < ,if j 2N i (k ).Remark 1:In general,(j;i )2E (G (k ))or a ij (k )>0,which cor-responds to an available communication channel from agent j to agent i at time t =kT ,does not imply ij (k )>0even if the reverse is true.This is mainly because the communication topologies are dynamicallychanging and the communication delays are time varying,which may destroy the continuity of information.Note that ij (k )>0requires a ij >0for the whole time between k 0 ij (k )and k .DeﬁneA (k )= 11(k )111 1n (k )......... n 1(k )111 nn (k):To distinguish A (k )from the adjacency matrix A (k )at time t =kT ,we call A (k )the interaction matrix in the presence of delays to em-phasize that A (k )is closely related to not only the available commu-nication channel but also the information transmission in the presence of delays.Let L (k )be L (k )=D (k )0A (k ),where D (k )is a diag-onal matrix with the i th diagonal entrybeing n j =1;j =i ij (k ).In fact,0(A (k )),the associated digraph of A (k ),is a spanning subgraph of the communication topology G (k )at time t =kT .To illustrate,consider a team of n =3agents.The possible communication topologies are modeled by the digraph as shown in Fig.1.Assume the communica-tion delays 21(k )and 32(k ),k2,are all larger than 1T ,while the communication topology switches periodically between Ga and Gb at each sampling instant.Clearly,A (k )=03;3at each sampling instant.However,in the special case that there is no communication delay be-tween neighboring agents,0(A (k ))=G (k ).In the case that both the communication topology and the communication delays are time in-variant,0(A (k ))=G (k )after max time steps.We say that consensus is reached for algorithm (2)if for any initial position and velocity states,and any i;j 2Vlim k !1r i (k )=lim k !1r j (k )and lim k !1v i (k )=0:It is assumed that r i (k )=r i (0)and v i (k )=v i (0)for any k <0and i;j 2V .III.M AIN R ESULTSDenote G=f G 1;G 2;...;G m g as the ﬁnite set of all possible com-munication topologies for all the n agents.In the sequel,when we men-tion the union of a group of digraphs f G i ;...;G i g G,we mean a digraph with the node set V =f 1;2;...;n g and the edge set given by the union of the edge sets of G i ,j =1;...;k .Firstly,we perform the following model transformation,which helps us deal with the consensus problem for an equivalent trans-formed discrete-time system.Denote r (k )=[r 1(k );111;r n (k )]T ,v (k )=[v 1(k );111;v n (k )]T ,x (k )=(2= )v (k )+r (k ),andy (k )=[r (k )T x (k )T ]T.Then,applying algorithm (2)and by some manipulation,(1)can be written in a matrix form asy (k +1)=40(k )y (k )+`=14`(k )y (k 0`)(3)where we get the equation shown at the bottom of the next page,and 4`(k )=T2A `(k )0n;n2T +12T 2A `(k )0n;n;`=1;2;...; max :Here in 4p (k ),p =0;1;...; max ,the ij th element of A p (k )is either equal to ij (k )if ij (k )=p ,or equal to 0otherwise and L (k )is the Laplacian matrix of the digraph of A (k ).1ObviouslyA 0(k )+A 1(k )+111+A(k )=A (k ):The following lemma will allow us to perform the convergence anal-ysis by using the product properties of row-stochastic matrices.1NoteL (k )is different from the Laplacian matrix of the communicationtopology G(k).Fig.1.Two possible communication topologies for the three agents.Lemma 1:Let d (k )be the largest diagonal element of the Lapla-cian matrix L (k ),i.e.,d (k )=max if n j =1;j =i ij (k )g .If the ve-locity damping gain and the sampling period T satisfy the following condition:4 T 0 T >2and T 01 2T d (k )(4)then 4(k )=40(k )+41(k )+111+4(k );k2+,is a row-stochastic matrix with positive diagonal elements.Proof:It follows from A 0(k )+A 1(k )+111+A(k )=A (k )=diag L (k )0L (k )that4(k )=40(k )+41(k )+111+4(k )=411(k )412(k )421(k )422(k )(5)where 411(k )=(10( =2)T +( 2=4)T 2)I n 0(T 2=2)L (k ),412(k )=(( =2)T 0( 2=4)T 2)I n ,421(k )=(( =2)T +( 2=4)T 2)I n 0((2= )T +(1=2)T 2)L (k )422(k )=(10( =2)T 0( 2=4)T 2)I n .One can easily check from (4)that all the matrices 411(k ),412(k ),421(k ),and 422(k )are nonnegative with positive di-agonal elements.That is,4(k )is a nonnegative with positive diagonal elements.Finally,it follows straightforwardly from L (k )1n =1n that 4(k )is a row-stochastic matrix.Remark 2:By some manipulation,we can get that (4)is equivalent to the following condition:1+1+8T 2d (k )2T <p 501:(6)This is achieved by solving ( T )2+2 T 04<0and T 20 02T d (k ) 0,which can be considered the quadratic inequalities in T and ,respectively.In the sequel,4(k )will be used to denote the row-stochastic matrix as described in Lemma 1.In order to make the transformed system dynamics mathematically tractable in terms of 0(A (k )),the associated digraph of the interaction matrix in the presence of delays,we need to explore the relations be-tween 0(A (k ))and the associated digraph of the transformed system matrix 0(4(k )).To this end,a new graphic method is proposed as follows.Lemma 2:Given any digraph G (V ;E ).Let G 1(V 1;E 1)be a graph with n nodes and an empty edge set,that is,V 1=f n +1;n +2;...;2n g and E 1=.Let ~G(~V ;~E )be a digraph satisfying the fol-lowing conditions:(A)~V=V [V 1=f 1;...;n;n +1;...;2n g ;(B)there is an edge from node n +i to node i ,i.e.,(n +i;i )2~",for any i 2V ;(C)if (j;i )2E ,then (j;n +i )2~Efor any i;j 2V ;i =j .Then,G has a directed spanning tree if and only if ~Ghas a directed spanning tree.Proof:Necessity:Denote G s as a directed spanning tree of the digraph G .Assume,without loss of generality,`is the root node of G s .By rules (B )and (C ),split each edge (i;j )in G s into edges (i;n +j );(n +j;j )and add edge (n +`;`)for the root node `,then we canget a directed spanning tree for ~G.Sufﬁciency:Let ~Gs be a directed spanning tree of ~G .Note that by the deﬁnition of ~G,the digraph G can be obtained by contracting all the edges (n +i;i );i 2V in the digraph ~G.Thus,the operation of the edge contraction on ~Gs will result in a directed spanning tree,say G s ,of the digraph G .Based on the above lemma,now we have the following result.Lemma 3:Suppose that and T satisfy the inequality in (4).Let f z 1;z 2;...;z q g be any ﬁnite subsetof +.If the union of the digraphs 0(A (z 1));0(A (z 2));...;0(A (z q ))has a directed spanning tree,then the union of digraphs 0(4(z 1));0(4(z 2));...;0(4(z q ))also has a directed spanning tree.Proof:The union of the digraphs 0(4(z 1));0(4(z 2));...;0(4(z q ))hereby is exactly the digraph0(q l =14(z l )).Because and T satisfy (4),it follows that 4(z l ),l =1;2;...;q ,is a row-stochastic (and hence nonnegative)matrix with positive diagonal entries.Note that L (z l )=diag L (z l )0A (z l ).By observing the equation in (5),we get that there exists a positive number ,say =min f q (( =2)T 0( 2=4)T 2);(2= )T +(1=2)T 2g ,such that we get (7),as shown at the bottom of the page.It thus follows from ~M 12=I n that (n +i;i )20(q l =14(z l ))for any i 2V .On the other hand,~M 21=q l =1A (z l )implies that(j;i )20(q l =1A (z l ))if and only if (j;n +i )20(ql =14(z l ))for any i;j 2V ;i =j .Combining these arguments,we knowthat the digraphs0(q l =14(z l ))and0(ql =1A (z l ))correspondto the digraphs ~G and G ,respectively,as described in Lemma 2.Note that the digraph0(q l =1A (z l ))is just the union of digraphs 0(A (z 1));0(A (z 2));...;0(A (z q )).It then follows from Lemma 2that the digraph0(q l =14(z l ))has a directed spanning tree,which proves the Lemma.Let P be the set of all n by n row-stochastic matrices.Given any row-stochastic matrix P =[p ij ]2P ,deﬁne (P )=10mini;j k min f p ik ;p jk g [25].Lemma 4: (1)is continuous on P .40(k )=102T +4T2I n 0T2(diag L (k )0A 0(k))2T 04T2In2T +4T2I n 02T +12T 2(diag L (k )0A 0(k))102T 04T2I nql =14(z l )q2T 04T2I n2T +12T 2diag q l =1L (z l )0q l =1L (z l )0Inql =1A (z l )0= ~M 11~M12~M 21~M22:(7)Proof:2:P can be viewed as a subset of metricspace n .All the functions involved in the deﬁnition of (1)are continuous,and since the operations involved are sums and mins,it readily follows that (1)is continuouson n .The restriction of a continuous function is con-tinuous,so (1)is also continuous on P .Two nonnegative matrices M and N are said to be of the same type,denoted by M N ,if they have zero elements and positive elements in the same places.To derive the main result,we need the fol-lowing classical results regarding the inﬁnite product of row-stochastic matrices.Lemma 5:([25])Let M =f M 1;M 2;...;M q g be a ﬁnite set of n 2n ergodic matrices with the property that for each se-quence M i ;M i ;...;M i of positive length,the matrix productM i M i111M i is ergodic.Then,for each inﬁnite sequence M i ;M i ;...there exists a column vector c2n such thatlim j !1M i M i111M i =1c T :(8)In addition,when M is an inﬁnite set, (W )<1,where W =S k S k 111Sk,S k 2M ,j =1;2;...;N (n )+1,and N (n )(which may depend on n )is the number of different types of all n 2n ergodic matrices.Furthermore,if there exists a constant 0 d <1satisfying (W ) d ,then (8)still holds.Let d=(n 01) .Assume,in the sequel,that ;T satisfy (4= T )0 T >2and T 01 (2= )T d.Then,by Lemma 1,all possible 4(k )must be nonnegative with positive diagonal elements.In addition,since the set of all 2n ( max +1)22n ( max +1)matrices can be viewed as the metricspace [2n (+1)],for each ﬁxed pair ;T ,all possible 4(k )compose a compact set,denoted by 7( ;T ).This is because all the nonzero and hence positive entries of 4(k )are both uniformly lower and upper bounded,which can be seen by observing the form of 4(k )in (5).Let 3(A )=f B =[b ij ]22n 22n :b ij =a ij or b ij =0;i;j =1;2;...;2n g ,and denote by 5( ;T )the set of matricesM (40;41;...;4)=40411114014I 2n 0111000I 2n 11100 0111I 2nsuch that 40;41;...;423(4(k ))and 40+41+...+4=4(k ),where 4(k )27( ;T ).The set 5( ;T )is compact,since givenany 4(k )27( ;T ),all possible choices of 40;41;...;4are ﬁnite.Let (k )=[ 1(k ); 2(k );111; 2n (+1)(k )]T =[y T (k );y T (k 01);111;y T (k 0 max )]T22n (+1).Then,there exists a matrix M (40(k );41(k );...;4(k ))25( ;T )such that system (3)is rewritten as(k +1)=M (40(k );41(k );...;4(k )) (k ):(9)Clearly,the set 5( ;T )includes all possible system matrices of system (9).2Weare indebted to Associate Editor,Prof.Jorge Cortes,for his help with a simpler proof of this lemma.Given any positive integer K,deﬁne ~5(;T )=i =1M (4i 0;4i 1;...;4i):M (1)25( ;T )and there exists a integer ;1 K suchthat the union of digraphsj =04ij ;i =1;...; ;has a directed spanningtree :~5(;T )is also a compact set,which can be derived by noticing the following facts:1)5( ;T )is a compact set;2)all possible choices of are ﬁnite since is bounded by K;3)all possible choices of the directed spanning trees are ﬁnite;and 4)given the directed spanning tree and ,the followingset:i =1M (4i 0;4i 1;...;4i):M (1)25( ;T )and the union of the digraphsj =04ij;i =1;...; ;hasthe speci ed directed spanningtreeis compact (this can be proved by following the similar proof of [27,Lemma 10]).Note that the set ~5(;T )includes all possible products of ; K ,consecutive system matrices of system (9).The following lemma is presented to prove that all the possible prod-ucts of consecutive system matrices of system (9)satisfy the result as stated in Lemma 5,which in turn allow us to use the properties of in-ﬁnite products of row-stochastic matrices from an inﬁnite set to derive our main result.Lemma 6:If 81;...;8k 2~5(;T ),where k =N (2n ( max +1))+1,then there exists a constant 0 d <1such that(k i =18i ) d .Proof:We ﬁrst prove that for any 82~5(;T );8is an er-godic matrix.According to the deﬁnition of ~5(;T ),there exist pos-itive integer (1 K),M (4i 0;4i 1;...;4i )25( ;T ),i =1;...; ,such that 8= i =1M (4i 0;4i 1;...;4i)and the union of digraphs0(j =04ij ),i =1;...; ,has a directed span-ning tree.Since M (4i 0;4i 1;...;4i )25( ;T ),j =04ij must be nonnegative matrices with positive diagonal elements.Furthermore,there exists a positive number 1such that diag(j =04ij ) I 2n ,for any M (4i 0;4i 1;...;4i )25( ;T ).Speciﬁcally,by observing (5),we can choose as=min 1;10 2T + 24T20T 22(n 01) ;10 2T 0 24T2:Combining this with the condition that the union of digraphs0(j =04ij ),i =1;...; ,has a directed spanning tree,we can prove that matrix 8is ergodic by following the proof of [26,Lemma 7].Letd =max 82~5(;T )ki =18i :From Lemma 5,we know that(k i =18i )<1.This,together withthe fact that ~5( ;T )is a compact set and (1)is continuous (Lemma4),implies d must exist and 0 d <1,which therefore completing the proof.For notational simplicity,we shall denote M (40(k );41(k );...;4(k ))by M (k )if it is self-evident from the context.Based on the preceding work,now we can present our main result as follows.Theorem 1:Assume that and T satisfy (4= T )0 T >2andT 01 (2= )T d.Then,employing algorithm (2),consensus is reached for all the agents if there exists an inﬁnite sequence of con-tiguous,nonempty,uniformly bounded time intervals [k j ;k j +1),j =1;2;...,starting at k 1=0,with the property that the union of the di-graphs 0(A (k j ));0(A (k j +1));...;0(A (k j +101))has a directed spanning tree.Proof:We ﬁrst prove that consensus can be reached for system (9)using algorithm (2).Let 8(k;k )=I 2n (+1),k 0,and 8(k;l )=M (k 01)111M (l +1)M (l ),k >l 0.Assume,without loss of generality,that the lengths of all the time intervals [k j ;k j +1),j =1;2;...,are bounded by K.It follows from Lemma 3and the condition that the union of the digraphs 0(A (k j ));0(A (k j +1));...;0(A (k j +101))has a directed spanning tree that the union of the digraphs 0(4(k j ));0(4(k j +1));...;0(4(k j +101))also has a directed spanning tree for each j2+,which,together with the proof ofLemma 6,implies that 8(k j +1;k j )=k 01k =k M (k )2~5(;T ).Since 8(k j ;0)=8(k j ;k j 01)8(k j 01;k j 02)1118(k 2;k 1),it then follows from Lemma 5and Lemma 6thatlim j !18(k j ;0)=12n (+1)wT(10)where w22n (+1)and w 0.For each m >0,let k l be the largest nonnegative integer such that k l m .Note that matrix 8(m;k l )is row stochastic,thus we have8(m;0)012n w T =8(m;k l)8(k l ;0)012n wT :The matrix 8(m;k l )is bounded because it is the product of ﬁ-nite matrices which come from a bounded set ~5(;T ).By using (10),we immediately have lim m !18(m;0)=12n (+1)w T .Combining this with the fact that (m )=8(m;0) (0)yields lim m !1 (m )=(w T (0))12n (+1)which,in turn,implies lim m !1x (m )=(w T (0))1n and lim k !1v (m )=0,and there-fore completing the proof.Remark 3:Matrix A (k )is a somewhat complex object to study compared with the adjacency matrix A (k )(see Remark 1).It is worth noting that more general results in which the sufﬁcient conditions for guaranteeing the ﬁnal consensus are presented in terms of G (k )instead of the interaction matrix in the presence of delays can be provided if some additional conditions are imposed.For example,if in addition to the conditions on and T as that required in Theorem 1,it is further required that a certain communication topology which takes effect at some time will last for at least max +1time steps,then we can get that consensus can be reached if there exists an inﬁnite sequence of contiguous,nonempty,uniformly bounded time intervals [k j ;k j +1),j =1;2;...,starting at k 1=0,with the property that the union of the digraphs G (k j );G (k j +1);...;G (k j +101)has a directed spanning tree.This can be observed by reconstructing a new sequence of con-tiguous,nonempty and uniformly bounded time intervals which satis-ﬁes the condition in Theorem 1by using similar technique as that in in [26,Theor.3].IV .I LLUSTRATIVE E XAMPLEConsider a group of n =6agents interacting between the possible digraphs f Ga;Gb;Gc g (see Fig.2),all of which have 0–0.2weights.Fig.2.Digraphs which model all the possible communicationtopologies.Fig.3.Position and velocity trajectories for agents.Take and T as =2and T =0:6respectively.Assume that the communication delays ij (k )satisﬁes 21(k )= 32(k )= 43(k )=1T s , 52(k )= 54(k )=2T s ,while 65(k )= 61(k )=3T s ,for any k2+.Moreover,we assume the switching signal is periodically switched,every 3T s in a circular way from Ga to Gb ,from Gb to Gc ,and then from Gc to Ga .Obviously,the union of the digraphs 0(A (k ))across each time in-terval of 9T s is precisely the digraph G d in Fig.2,which therefore has a directed spanning tree.Fig.3shows that consensus is reached for algorithm (2),which is consistent with the result in Theorem 1.V .C ONCLUSIONS AND F UTURE W ORKIn this technical note,we have investigated a discrete-time second-order consensus algorithm for networks of agents with nonuniform and time-varying communication delays under dynamically changing com-munication topologies in a sampled-data setting.By employing graphic method,state argumentation technique as well as the product proper-ties of row-stochastic matrices from an inﬁnite set,we have presented a sufﬁcient condition in terms of the associated digraph of the interac-tion matrix in the presence of delays for the agents to reach consensus.Finally,we have shown the usefulness and advantages of the proposed result through simulation results.It is worth noting that the case with input delays is an interesting topic which deserves further investigation in our future work.。

一个有效的时延约束最小代价多播路由算法

时延抖动约束。另外，最短时延路径算法针对单一度量，ＤＣＤＶＡ在搜索路径时仅仅考虑了时延这个可加性参数，而没有考虑到链路的代价，这有可能导致算法为了满足时延和时延差异的限制而使用代价过大的路径，从而使得最终得到的多播树代价过大。
收稿日期：００—１２２１０—１基金项目：北京市自然科学基金资助项目（１２４）博士后专项基金资助项目（０９０６０４４００１；２０００１０１）１作者简介：陈月云（９６一）女，１６，河北景县人，副教授，主要从事无线和移动通信理论及无线通信新技术等研究
择具有最小Ａｅｇ（）的节点作为中心节点。本文认为这样缩小了中心节点的选择范围，ｖｒｅＲＤｅ值ａ可能会遗漏最佳的中心节点，使整棵多播树的代价未必优化到最小。所以，ＳＤＭＭＣ — ＣＣＲ算法采用从多播节点集
７０
空军工程大学学报（自然科学版）
时延约束的上限时尽量优化到最小。因此，只要选择合适的参数ａ就能找到一条最优的路径，，既能满足
时延的约束又能使代价最小且满足代价约束上限。参数ａ越大，ｗ的值越大，同时ｗ的值越小。这个关系：的正确性以及ａ的值取多大可以参看文献［］９。
ＤＭＡ的基本思想是：ＣＣ假设Ｗ和分别代表链路的代价和时延。首先采用Ｄｊｓａ，ｉｔ算法以代价函数ｋｒＣｅ为权值搜寻节点间的最小代价路径Ｐ并计算路径Ｐ上的端到端时延ｄｐ，（），（）验证ｄｐ是否满足时延约（）束上限ｃ。如果路径Ｐ上的时延满足时延约束，那么这条路径Ｐ就是所求路径，否则就采用Ｄｊｓａｉｔ算法以ｋｒ时延函数Ｄ（）ｅ为权值搜寻节点间的最小时延路径ｇ。同样计算路径ｇ的时延ｄｇ，上（）验证ｄｑ是否满足（）ｃ。如果满足，ｑ，则就是所求路径。否则ＤＭＡ程序就开始执行下面的循环，ＣＣ在每次循环中，路径Ｐ被一个拥有更低代价或更小时延的路径ｒ更新，路径ｒ是最短路径。循环进行直至找到更好的路径来更新。

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1020304050
#node

60
40
20
10

50
70
time(sec)

131