Protocol specification using parameterized communicating extended finite stte machines - a
TDC-GP22用户手册
模拟输入电路
斩波稳定低漂移比较器,可编程offset,±35 mV 第一个波检测: 在第一波检测后比较器自动调整
offset到0,可以选择相对于第一个波的测量回波 第一个波脉冲宽度测量用于信号质量检测以及气
泡的检测 用于输入选择的内部集成模拟开关 外部电路仅需要2个电阻和2个电容
acam-messelectronic gmbh - Friedrich-List-Str. 4 - D-76297 Stutensee-Blankenloch - Germany - www.acam.de
2-1
时间数字转换器
TDC-GP22
2
特性和规格
2.1
电气特性
绝对最大额定值
供电电压
Vcc vs. GND
时间数字转换器
临时数据手册
TDC-GP22
超声波热表水表特定双通道时间数字转换器
2012年1月26日 版本号码.: DB_GP22_cn V0.1
时间数字转换器
Published by acam-messelectronic gmbh © acam-messelectronic gmbh 2011
2
acam-messelectronic gmbh - Friedrich-List-Str. 4 - D-76297 Stutensee-Blankenloch - Germany - www.acam.de
TDC-GP22
目录
1 概述
2 特性和规格
3 配置 & 读寄存器
4 前端转换器
5 详细描述和特殊功能 6 应用 7 其它
- 0.3 到 4.0
V
Vio vs. GND
计算机网络原理习题讲解精编版
计算机网络原理习题讲解精编版MQS system office room 【MQS16H-TTMS2A-MQSS8Q8-MQSH16898】ChapterI1. Whatisthedifferencebetweenahostandanendsystem??2. Whatisaclientprogram?Whatisaserverprogram?Doesaserverprogramrequesta ndreceiveservicesfromaclientprogram?3. ,companyaccess,ormobileaccess.4. Dial-upmodems,HFC,,providearangeoftransmissionratesandcommentonwhetherthe transmissionrateissharedordedicated.5. .6. Whatadvantagedoesacircuit-switchednetworkhaveoverapacket-switchednetwork?WhatadvantagesdoesTDMhaveoverFDMinacircuit-switchednetwork?7. 2,000km 8102⨯prop d trans d trans d t =prop d trans d trans d prop d trans d trans d 8105.2⨯=s prop d trans d 6108⨯下列说法中,正确的是()。
A.在较小范围内布置的一定是局域网,币在较大范围内布置的一定是广域网B.城域网是连接广域网而覆盖园区的网络C.城域网是为淘汰局域网和广域网而提出的一种网络技术D.局域网是基于广播技术发展起来的网络,广域网是基于交换技术发展起来的向络 解答:D 。
通常而言,局域网的覆盖范围较小,而广域网的覆盖范围较大,但这并不绝对。
有时候在一个不大的范围内采用广域网,这取决于应用的需要和是否采用单一网络等多种因素。
中国商用车发动机CAN通讯规范
SRR-代用远程请求 SA# - SAE J1939目标地址#n
RTR-远程传输请求位 DP-SAE J1939数据页
IDE-标识符扩展位 PF# - SAE J1939 PDU格式位#n
r # - CAN保留位 PS# - SAE J1939 特定PDU位#n
DLC# - 数据长度码位#n
（d）- 显性位
EEC
Electronic Engine Controller
转速表
TCO
Tachograph
车辆智能中心
VIC
Vehicle Intelligence Centre
排气缓速器
XR
Exhaust Retarder
高字节
MSB
Most Significant Byte
低字节
LSB
Least Significant Byte
参数组编号（PGN）
在CAN数据帧的数据域中需要指明参数组时，PGN是表示成24位。PGN是一个24位的值，包括以下要素：保留位、数据页位、PDU格式域（8位）和群扩展域（8位）。各个位转化到PGN的过程如下。若PF值小于240(F016)，PGN的低字节置0。注意：用本文档规定的范例，并非全部131,071种组合都可用于分配(计算如下：2页*[240+(16*256)]=8,670)。见SAE J1939附录A，可查现行的最新分配。参见PGN表，表6，位和其相应的十进制转化。
1376.1-2013版规约与376.1-2009版规约差异分析v1.1
1376.1-2013版规约与376.1-2009版规约差异分析V1.12014年1月目录目录I第1章概述 (2)1.1背景 (2)1.2目的 (2)第2章规约变更内容 (3)2.1帧格式变更 (3)2.1.1功能码 (3)2.1.2帧序列域SEQ (3)2.1.3数据单元标识 (4)2.1.4附加信息域AUX (4)2.2报文应用及数据结构变更 (5)2.2.1确认/否认(AFN=00H) (5)2.2.2链路接口检测(AFN=02H) (5)2.2.3设置参数(AFN=04H) (6)2.2.4控制命令(AFN=05H) (11)2.2.5身份认证及密钥协商(AFN=06H) (12)2.2.6请求终端配置及信息(AFN=09H) (19)2.2.7请求1类数据(AFN=0CH) (21)2.2.8请求2类数据（AFN=0DH） (29)2.2.9请求3类数据(AFN=0EH) (38)2.2.10文件传输(AFN=0FH) (42)2.2.11数据转发(AFN=10H) (44)第3章与重庆规约扩展冲突 ..................................................... 错误!未定义书签。
3.1控制命令(AFN=05H)...................................................... 错误!未定义书签。
3.1.1下行报文..................................................................... 错误!未定义书签。
第1章概述1.1背景国家电网公司在2013年颁布了Q/GDW1376—2013《电力用户用电信息采集系统通信协议》。
该协议是根据《关于下达2012年度国家电网公司技术标准制（修）订计划的通知》（国家电网科[2012]66号）的要求，对Q/GDW 376—2009《电力用户用电信息采集系统通信协议》的修订。
ccp标定协议和uds协议
ccp标定协议和uds协议CCP标定协议（CCP Calibration Protocol）和UDS协议（Unified Diagnostic Services）是汽车行业中常用的两种标准协议，用于对车辆进行诊断和校准。
本文将分别介绍CCP标定协议和UDS协议的特点和应用。
一、CCP标定协议CCP标定协议是汽车行业中广泛使用的一种通信协议，用于对车辆的控制单元进行标定和调试。
在车辆制造过程中，各个控制单元需要进行标定，以确保其正常工作。
CCP标定协议提供了一种可靠的通信机制，使得工程师能够通过诊断工具与车辆的控制单元进行通信，并进行参数的读取和写入。
CCP标定协议通过CAN总线进行数据传输，其中包括两个主要部分：CCP DAQ（Data Acquisition）和CCP Calibration。
CCP DAQ用于实时获取车辆控制单元的数据，包括传感器数据和控制单元输出数据。
CCP Calibration用于对控制单元的参数进行标定和调整，使得其能够满足设计要求。
CCP标定协议具有以下特点：1. 实时性：CCP标定协议能够实时获取车辆控制单元的数据，使得工程师可以在车辆运行状态下进行标定和调试。
2. 灵活性：CCP标定协议支持多种标定和调试方式，可以根据具体需求选择合适的方式进行操作。
3. 可靠性：CCP标定协议通过CAN总线进行通信，具有较高的信号传输可靠性和抗干扰性。
二、UDS协议UDS协议是汽车行业中一种广泛应用的车辆诊断协议，用于对车辆的各个控制单元进行故障诊断和维修。
UDS协议提供了一系列的诊断服务和功能，使得技术人员能够准确地定位车辆故障并进行修复。
UDS协议定义了基本的诊断服务、会话控制、安全访问和通信管理等功能，包括以下几个主要部分：诊断会话控制、ECU诊断和ECU编程。
诊断会话控制用于建立和管理诊断会话，包括会话的建立、切换和结束。
ECU诊断用于对车辆的各个控制单元进行故障诊断，包括故障码读取、数据流读取和清除故障码等功能。
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.。