具有通信时延的离散时间二阶多自主体系统.doc

精品论文具有通信时延的离散时间二阶多自主体系统的补偿一致性算法刘成林 轻工过程先进控制教育部重点实验室, 江南大学自动化研究所, 江苏 无锡 214122 摘要：针对具有通信时延的离散时间二阶多自主体系统系统的动态一致性问题，本文通过在常 规异步耦合一致性算法中引入邻居状态状态补偿来构造一致性算法。利用频域方法分析两个耦 合个体的一致性收敛，可得出补偿算法比常规异步耦合算法能容忍更大通信时延。根据线性分 式变换和小增益定理，得到一般有向拓扑结构下具有通信时延的离散时间二阶多自主体系统的充分条件。仿真结果证明了结论的正确性。 关键词：一致性，补偿算法，通信时延，二阶多自主体系统 中图分类号： tp242compensation-based consensus algorithm for discrete-time second-order multi-agent systems under communication delayliu cheng-linkey laboratory of advanced process control for light industry (ministry of education) institute of automationjiangnan university, wuxi 214122 chinaabstract: consensus algorithm, which is constructed by introducing compensation on the neighboring agents delayed states into the normal asynchronously-coupled consensus algorithm, is proposed to solve the dynamical consensus problem of discrete-time second-order multi-agent systems with communication delay. by analyzing the consensus convergence of two coupled-agents according to frequency-domain analysis, we come to the conclusion that the compensation-based algorithm can endure higher communication delay than the synchronously-coupled consensus algorithm. based on linear fractional transformation andthe small-gain theorem, sucient conditions are gained for discrete-time second-order multi-agent systems with communication delay under a general digraph. simulation illustrates the correctness of the results.key words: consensus, compensation-based algorithm, communication delay, second-ordermulti-agent systems,基金项目： specialized research fund for the doctoral program of higher education of china (20090093120006)作者简介： liu cheng-lin（1981-），male，associate professor，major research direction：coordination control ofmulti-agent systems.0 introductionin the last decade, coordination control of multi-agent systems based on communication network has attracted more and more attention for the rapid development of automatic control and communication technology. as one of the most important issues in coordination control of multi-agent systems, consensus problem, which requires the outputs of several distributed agents to reach a common value, has been extensively studied in various research elds.in a multi-agent network, communication delays corresponding to the information exchange between neighboring agents can not be neglected. under communication delay, consensus al- gorithms are usually divided into synchronously-coupled and asynchronously-coupled forms. in synchronously-coupled consensus algorithm, self-delays introduced for each agent in the coordination part equal the corresponding communication delays. in asynchronously-coupled consensus algorithm, each agent uses its delayed state with the delay dierent from the corre- sponding communication delay, or uses its current state to compare with its delayed neighboring agents states.in synchronously-coupled form, the convergence of consensus algorithms depends on the communication delay strictly for the multi-agent systems under xed 134 or switched topolo- gies 56789.the rst-order and second-order agents with stationary consensus algorithms in asynchronously- coupled form can achieve an asymptotic consensus without any relationship to the commu- nication delay 1011121314151617. for the dynamical consensus seeking problemof second-order multi-agent systems, asynchronously-coupled form can not drive the agents to achieve the desired dynamical consensus, e.g., the dynamical consensus algorithm com- posed of the position and velocity consensus coordination control parts makes the second-order agents just achieve the stationary consensus1819. thus, asynchronously-coupled consensus algorithms modied by introducing delayed state compensations are proposed to guarantee the second-order agents to converge the desired dynamical consensus 20212223. surpris- ingly, the compensation-based consensus algorithms in asynchronously-coupled form can toler- ate higher communication delay than that in synchronously-coupled form 20. furthermore, munz et al. 2425 investigated the homogeneous and heterogeneous multi-agent systems with agents dynamics described by strictly stable linear systems under diverse communication delays respectively. by analyzing the convex sets of the frequency-domain feedback matrix, set- valued consensus conditions have been obtained for the system under synchronously-coupled and asynchronously-coupled consensus algorithms respectively 2425.in this paper, asynchronously-coupled consensus algorithm, which is composed of the po- sition and velocity consensus coordination control parts, is accompanied with delayed state compensation in terms of neighboring agents delayed states so as to solve the dynamical con- sensus problem of discrete-time second-order multi-agent systems with communication delay.firstly, two coupled dynamic agents are taken as investigation object, and it is proved basedon frequency domain analysis that our proposed algorithm can tolerate higher communication delay than the synchronously-coupled consensus algorithm. then, sucient conditions are ob- tained for multiple agents with communication delay under general digraph based on linear fractional transformation and small-gain theorem.1 formulation1.1 discrete-time second-order agentsconsider the following discrete-time model with zero-order hold of the continuous-time second-order agentsxi (k + 1) = xi (k) + t vi (k) +t 2ui (k)2vi (k + 1) = vi (k) + t ui (k), (1)where xi r, vi r, and ui r are the position, velocity and acceleration, respectively, of the agent i, and t 0 is the sampling interval. under the frame of coordination control of multiple agents, the information ows between neighbors make all the agents form a network, of which interconnection topology is usually described as a digraph. agent refers to the nodes of the digraph, while the information ow between neighboring agents refers to a directed edge between the neighboring nodes.1.2 interconnection topologya weighted digraph g = (v, e, a) of order n is composed of a set of vertices v = 1, , n, a set of edges e v v and a weighted adjacency matrix a = aij rnn with aij 0. the node indexes belong to a nite index set i = 1, , n. a directed edge from the node i to j of the digraph g is denoted by eij = (i, j) e. we assume that the adjacency elements associated with the edges of the digraph are positive, i.e., aij 0 eij e. moreover, we assume aii = 0 for all i i. the set of neighbors of node i is denoted by ni = j v : (i, j) e. the laplacian matrix of the weighted digraph g is dened as l = d a = lij rnn , wherend = diagj=1 aij , i i is the degree matrix of g.in the digraph, if there is a path from one node i to another node j, then j is said to be reachable from i. if not, then j is said to be not reachable from i. if a node is reachable from every other node in the digraph, then we say it globally reachable.1.3 dynamical consensus seekingthe second-order dynamic agents (1) converge to a stationary consensus asymptotically, if the agents states under arbitrary initial values satisfylim xi (k) = c, lim vi (k) = 0, i i,kkwhere c is a constant. if the agents states under arbitrary initial values satisfylim xi (k) = k + , lim vi (k) = , i i,kkwhere and are two constants, we say that the second-order dynamic agents (1) achieve adynamical consensus asymptotically,in this paper, we just consider the dynamical consensus problem of the second-order multi- agent systems (1). to solve the dynamical consensus problem, the following consensus algorithm 2 is extensively adopted, ui (k) = d aij (xj (k) xi (k) + (vj (k) vi (k) (2)i jniwhere 0 and 0, di = jniaij , ni denotes the neighbors of the agent i, aij 0 is theadjacency element of a in the digraph g = (v, e, a).under communication delay, the consensus algorithm (2) in asynchronously-coupled form is given byui (k) = d aij (xj (k ) xi (k)i jni+(vj (k ) vi (k) (3)where 0 is the communication delay.remark 1. the second-order agents (1) under the consensus algorithm (2) can achieve a dynamical consensus if the control parameters satisfy some certain conditions 2. under asynchronously-coupled consensus algorithm (3), the dynamic agents (1) just can achieve a stationary consensus instead of a dynamical consensus, and the consensus convergence is de- pendent on the communication delay 1819.up to now, the results in 3 have proved that the following synchronously-coupled con-sensus algorithmui (k) = d aij (xj (k ) xi (k )i jni+(vj (k ) vi (k ) (4)can make the second-order agents (1) reach an asymptotic dynamical consensus under commu- nication delay.dierent from synchronously-coupled consensus algorithm (4), we modify the asynchronously-coupled consensus algorithm (3) by introducing the delayed state compensations as followsui (k) = d aij (xj (k ) xi (k)i jni(xj (k 2 ) xj (k )+(vj (k ) vi (k)(vj (k 2 ) vj (k ) (5)where (xj (k 2 ) xj (k ) and (vj (k 2 ) vj (k ) are compensations based on the delayed neighboring agents states. from the denition of dynamical consensus, the control input becomes zero when the second-order agents approach a dynamical consensus.2 consensus analysis for two coupled agentsto compare the delay robustness of the compensation-based consensus algorithm (5) to that of synchronously-coupled consensus algorithm (4), we just take two coupled agents as our investigation in this section.the closed-form of two coupled agents (1) with synchronously-coupled consensus algorithm(4) is given byxi (k + 1) = xi (k) + t vi (k)t 2+2jni(xj (k ) xi (k )+(vj (k ) vi (k )vi (k + 1) = vi (k) + t (xj (k ) xi (k )jni+(vj (k ) vi (k ). (6)dening x = x1 x2 and v = v1 v2 , we obtainx(k + 1) =x(k) + t v(k)t 2+ 2 (x(k ) x(k )v(k + 1) =+(v(k ) v(k ),v(k) + t (x(k ) x(k )+(v(k ) v(k ). (7)by taking the z transform of the system (7), the characteristic equation of the system (7)about x(k) is given by1 + ms (z) = 0,where ms (z) =2(t (z1)+ t (z+1)2(z1)22z .by computing, we obtain|ms (ej )| = t 2 sin2 ( ) + t 2 cos2 ( )2 4 2,and2arg(ms (ej ) = sin2 ( )22 + arctan( ttan(2t).tremark 2. from the lemma 2 in , when 2 + 1 1, ms (ej ) crossestthe real axis for the rst time at c (0, that satises c + c= arctan( 2 tan c ) and2t22 arg(m (ej ) 1 or 2 1,ttarg(m (ej ) holds for (0, , so we obtain that two coupled agents can not reach an asymptotic consensus.now, we pay attention to the closed-form of two coupled agents (1) with compensation- based consensus algorithm (5) as followsxi (k + 1) = xi (k) + t vi (k)t 22jni(xj (k ) xi (k)(xj (k 2 ) xj (k )+(vj (k ) vi (k)(vj (k 2 ) vj (k ),vi (k + 1) = vi (k) + (xj (k ) xi (k) (8)jni(xj (k 2 ) xj (k )+(vj (k ) vi (k)(vj (k 2 ) vj (k ).we can obtain the dynamics of x = x1 x2 and v = v1 v2 as followsx(k + 1) =x(k) + t v(k)t 2+ 2 (x(k ) x(k)v(k + 1) =+x(k 2 ) x(k )+(v(k ) v(k)+v(k 2 ) v(k ),v(k) + t (x(k ) x(k) (9)+x(k 2 ) x(k )+(v(k ) v(k)+v(k 2 ) v(k ).taking the z transform of the system (9), the characteristic equation of the system (9)about x(k) is given bywhere mac (z) =t (z1)+ t (z+1)22(z1)21 + mac (z) = 0,(1 + 2z z2 ). it is evident thatmac (ej ) = ms (ej )(1 + j sin( ).thus, we can easily get2 sin2 ( ) + t 2 cos2 ( ) 2|ms (ej )| = 2t 2 4 21 + sin2 ( ),andsin2 ( )2arg(m (ej ) = 2 + arctan( ttan( )2+ arctan(sin( ).tremark 3. obviously, arg(m (ej ) 1,calculating the derivation of arg(mac (ej ) on yieldsd arg(mac (ej )1= + t d21 + ( 42 2 cos( )+ 1 + sin2 ( )t 2 1) sin ( 2 ). (10)djthere must exist 0 (0, ) such that d arg(mac (e ) 0 holds for (0, 0 ). thus, thefrequencies at which mac (ej ) crosses the real axis are dened as ci , i = 1, .tremark 4. from remark 2 and remark 3, when 2 + 1 2 1 w, two coupled agents in the synchronously-coupled form (4) can not converge to a consensus asymptotically withany , but two coupled agents in the compensation-based form (5) can achieve an asymptoticconsensus by choosing proper . hence, we come to the conclusion that the the compensation- based consensus algorithm (5) can tolerate higher communication delay than the synchronously- coupled consensus algorithm (4) under the same control parameters.3 consensus criterion for multi-agent systems undergeneral digraphin this section, we will get the consensus condition for discrete-time second-order multi- agent systems with compensation-based algorithm under general digraph.xi (k + 1) = xi (k) + t vi (k)t 2 2 di aij (xj (k ) xi (k)jni(xj (k 2 ) xj (k )+(vj (k ) vi (k)(vj (k 2 ) vj (k ), vi (k + 1) = vi (k) + d aij (xj (k ) xi (k) (11)i jni(xj (k 2 ) xj (k )+(vj (k ) vi (k)(vj (k 2 ) vj (k ).taking z transforms of above system yields the characteristic equation about x = x1 , x2 , , xn tas followsdet(z 1)2 i + (t (z 1) +t 2(z + 1)2(d1 l(2z z2 ) + (1 z )2 i ) = 0. (12)take the elementary column and row transforms for the matrix d1 l by adding all the other columns to the rst column and subtracting the rst row from all the other rows as follows 10 0 1 0 0 0pt 1 1 0 1 1 1 0 . . . . . . d l = ,. . . . . . . . 0 l1 0 11 0 1where 0 = 0, 0, , 0t rn1 , p = l12 , l13 , , l1n t rn1 , and l is dened as l22d1l12d1l23d1l13.l2nl1n l = d2 d1.d2 d1 . d2 d1,ln2l12ln3l13lnnl1ndn d1dn d1 dn d1where lij is the element of the laplacian matrix l. therefore, the characteristic equation (12)equals(z 1)2 i + (t (z 1) +ort 2(z + 1)(1 z )2 i = 0(13)2t 2det(z 1)2 i + (t (z 1) +(z + 1)2(l(2z z2 ) + (1 z )2 i ) = 0(14)dene xi = xi x1 and vi = vi v1 , i = 2, , n, and the dynamics of x(k) = x2 (k), , xn (k)tand v(k) = v2 (k), , vn (k)t can be given byxi (k + 1) =xi (k) + t vi (k)t 2 +21 dijniaij (2xj (k ) xj (k 2 )(2xi (k ) xi (k 2 )+(2vj (k ) vj (k 2 )(2vi (k ) vi (k 2 )(xi (k) 2xi (k ) + xi (k 2 )(vi (k)