Adaptive Consensus Control of Multi-Agent Systems with Large Uncertainty and Time Delays *

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1 Journal of Robotcs, etworkng and Artfcal Lfe, Vol., o. (September 04), 5-9 Adaptve Consensus Control of Mult-Agent Systems wth Large Uncertanty and me Delays * L Lu School of Mechancal Engneerng Unversty of Scence and echnology Bejng, Bejng 00083, Chna Yu Meng** School of Mechancal Engneerng Unversty of Scence and echnology Bejng, Bejng 00083, Chna Dongxu Zou School of Automaton and Electrcal Engneerng Unversty of Scence and echnology Bejng, Bejng 00083, Chna Wecun Zhang School of Automaton and Electrcal Engneerng Unversty of Scence and echnology Bejng, Bejng 00083, Chna ** Correspondng author. E-mal: myu@ustb.edu.cn Abstract A weghted mult-model adaptve control (WMMAC) method s proposed to acheve consensus of mult-agent system wth large parameter uncertanty and communcaton delays, n whch H control s adopted as local control strategy to deal wth smaller parameter uncertanty. Moreover a smple and effectve weghtng algorthm s adopted to assgn weghts for each local controller, based on whch the global control law s obtaned as a weghted sum of all the local control at each tme nstant. he smulaton results demonstrate the effectveness of the proposed method. Keywords: Adaptve consensus; Mult-agent system; WMMAC; H control. * hs work was supported by Major State Basc Research Development Program (973 Program) (o. 0CB800), atonal Hgh-ech Research and Development Program of Chna (863 Program) (o. 0AA060408), Bejng Hgher Educaton Young Elte eacher Project (o. YEP036), atonal Key echnology Research and Development Program of Chna (o. 03BAB0B07) Publshed by Atlants Press Copyrght: the authors 5

2 . Introducton Durng the past decades, the study of decentralzed control has been focused on mult-agent systems, such as flockng or swarmng behavors [-3], mult-robot formaton control [4,5], and path plannng [6] and so on. he consensus problem, the most mportant and fundamental ssue n the cooperaton control of multagent system, s of theoretcal value and practcal sgnfcance. For a mult-agent system, consensus means that the states of all agents tend to be dentcal asymptotcally under gven protocols (control law) based on the communcaton networks. Recently, some researchers have solved the consensus problem of mult-agent systems wth tmevaryng external dsturbances and random communcaton delays [7]. In [8], model uncertanty,.e., the small parameter uncertanty, was taken nto consderaton, and a robust H controller was desgned accordngly. In ths paper, the consensus problem s consdered for mult-agent systems wth large parameter uncertanty, whch wll be dvded nto smaller ntervals that can be dealt wth by each local H controller. External dsturbances and communcaton delays are also taken nto account. WMMAC strategy showed ts prorty of dealng wth large parameter uncertantes [9], whch conssts manly of two components,.e. the local controller set and the weghtng algorthm. he local controller set was desgned accordng to a local model set whch was determned off-lne to cover the large uncertanty of system parameters. he weghtng algorthm was desgned to get weghts for each local controller, based on whch the global control law s obtaned as a weghted sum of all the local control at each tme nstant. Fnally, smulaton results ndcate that under the proposed protocol,.e. the WMMAC strategy, multagent systems wth large parameter uncertanty can reach the desred consensus performance n the presence of communcaton delays.. Uncertan Mult-Agent Systems Consder a second-order mult-agent system consstng of n dentcal agents wth the th one modeled by as the form of (). () x () t = Ax () t + Bω () t + B u (), t x () t = [ x (), t x ()] t, () t L (0, ] ω x () t s the state of agent, whle ω () t represents approprate external dsturbance that belongs to L (0, ]. u () t denotes the control protocol. If system matrces AB,, B are uncertan, then they are supposed to follow the forms of (). In (), A0, B0, B0 are constant matrces and A(), t B(), t B() t are tme-varyng matrces. E and F, =,, 3 are constant matrces wth approprate dmensons and Σ() t s tme-varyng uncertan matrx satsfyng Σ() t Σ() t I. It s also assumed that ( A0, B0 ) s stablzable. ( ) A= A0 + At, B = B0 + B() t B = B + B () t 0 [ A( t) B( t) B () t ] = E Σ( t)[ F F F3] he consensus of mult-agent system s that the states of agents satsfy (3) under control protocol u () t. lm x t x t = 0,, j n (3) t ( ( ) j ( )) 3. Weghted Mult-model Adaptve Control Weghted mult-model adaptve controller s made up of the followng components. 3.. Model set Consderng the large uncertanty of plant parameters cannot be dealt wth by a sngle H controller, then we need to use multple models,.e. a model set, to cover the uncertanty of the plant. he model set s desgned as Ω= { M =,,, }. Obvously, the performance of mult-agent control system depends on the model set and the local control strategy, as well as the rght convergence of the weghtng algorthm. 3.. Controller set he controller set C s shown as the form of C = { C =,,, } n whch the controller C s desgned accordng to model M accordng to H control strategy or other feasble control strategy. In ths paper, robust H method s selected as control strategy. Consderng the mult-agent system (), the output s supposed to follow (4). We can reformulate the system () as the form of (5) that satsfes (6) wth the help of Kronecker product. () Publshed by Atlants Press Copyrght: the authors 6

3 n z ( t) = x ( t) x ( t), =,..., n (4) j n j = xt () = ( In Axt ) () + ( In B) ω() t + ( In B) ut () (5) zt () = ( L I ) xt () c m mn xt () = [ x() t, x() t,..., xn mn ω() t = [ ω() t, ω() t,..., ωn mn ut () = [ u() t, u() t,..., un mn zt () = [ z() t, z() t,..., zn ( n ) n, j n n = Lc = L R, L = cj c j n, j Accordng to the neghbors states, the protocol of agent can be desgned as (7) to acheve the above condtons. ( ) ( ) ( ( t) ) ( ( t) ) u t = K a t x t d x t d j j j ( t) (7) where 0 dt () dd, > 0 s the tme-varyng communcaton delay, () t 0s the neghbor set of agent at tme nstant t, a () t are adjacency elements j of the correspondng nteracton graph, and K s an undetermned feedback matrx. Substtutng control protocol (7) nto system (5), we obtan ( ) = ( n ) ( ) + ( σ ) ( ( )) ( I B ) ω ( t), z( t) ( L I ) x( t) xt I Axt L BK xt dt + = n c m (8) Fnally, the consensus problem of mult-agent system s converted to calculate the feedback matrx K. Under control protocol (7), system () acheves consensus wth a gven H ndex γ, f there are a scalar m m α>0, a postve defne matrx P R and a matrx m m Q R that satsfes Ψ σ +Ψσ αλσbq B dψ P σ αλσq B α P 0 dαλσq B 0 B 0 0 γ I db 0 < d Ψσ dαλσbq db α P 0 P I Ψ = AP+ λ BQ σ 0 σ (6) (9) where λ σ, =,, n s the postve egenvalues of L σ, for detals please be referred to Reference [8]. If LMI (9) s feasble for the maxmum and mnmum egenvalues of all the connected topologcal graphs, the feedback matrx can be defned by K = QP. If LMI (9) s not feasble, then we need to dvde the uncertanty of parameters,.e. A(), t B(), t B() t nto lmted number of smaller ntervals, such that for each smaller nterval, LMI(9) s feasble. And then, we have the modfed protocol of agent ( ) ( ) ( ( t) ) ( ( t) ) u t = p K a t x t d x t d j j = j ( t) where p ( k ) can be calculated accordng to the weghtng algorthm descrbed n the next subsecton Weghtng Algorthm he method based on probablty-weghted s that the weghted sum of controller set s taken as the global control nput. he weght of controller can be calculated by partton theorem. Suppose the control output of controller u k, the weght s ( ) C s ( ) p k, the global control nput s ( ) u k, the system output s y( k ), the output of th model s y ( k ). Let the output dfference between th model and plant be e = y y. hen the weghtng algorthm can be desgned as n [9]. Frst, weghts of all models are ntalzed to, f there are models n the model set,.e. ( ) ( ) p 0 = r 0 =, =,,..., (0) hen, t s conductve to mprove the ant-jammng capablty by selectng the cumulatve mean squared error as the performance ndex shown n (). k l k = + e k () ( ) ( ) k j = Fnally, the computatonal process of weghts s denoted by (). hs method s much smpler than probablty-weghted method. he performance ndexes are forced to compete at every step n order to make the weghts converge. Publshed by Atlants Press Copyrght: the authors 7

4 () mn ( ) = mn ( ) l k l k ( ) r k p 4. Smulaton results l =,..., r( k ) r r mn = l = = In ths secton, we wll llustrate the consensus of multagent system under the method of WMMAC. Consderng a mult-agent system of four agents, the parameter matrces are desgned as (3) A0 =, 0, 0, B = B = At () =, B ( t), B () t t = 0 = 0 6 ( e ) 0 (3) he communcaton delay s assumed to be constant he external dsturbances are supposed to be band-lmted whte noses shown as (4). ( t) ( t) ( t) ( t) ( t) ω = ω ω ω3 ω4 (4) = ω( t).5ω( t).ω( t).8ω( t) For smplcty, the H performance ndex s chosen as γ =, the constant scalar α =0., λ = and all the nonzero weghtng factors of adjacency matrx are assumed to be. he smplfed undrected nteracton graph s shown n Fg A =, A, A3, A4 = 4 = = 6 8 (5) Accordng to the LMI (9), the feedback matrces K can be determned as (6) wth the help of Matlab LMI toolbox K =, K = K3 =, K4 = (6) he smulaton results are shown n fgures -3, n whch Fg. represents the trajectores of states x and x, Fg. 3 represents the energy relatonshp between the system output zt () and the external dsturbance ω () t Fg. (a) State curve of x Fg. smplfed undrected graph We can get the maxmum and mnmum nonzero egenvalues λσ = , λ 4 σ = by calculatng the Laplace matrces correspondng to the network graph as shown n Fg.. It s assumed that there are four models n the model set,.e. Fg. (b) State curve of x Publshed by Atlants Press Copyrght: the authors 8

5 References Fg. 3 Energy of zt () and ω () t If the system states are asymptotcally stable, the consensus of mult-agent system s all up to the steadystate values of each agent. herefore, the states of agents are desgned to be dvergent n order to dsplay the consensus of mult-agent system sgnfcantly. It s easy to say that the mult-agent system acheves consensus accordng to Fg. When the state matrx A changes to 0 A = 7.5 the weghts of all models are dsplayed n Fg.4.. R. Olfat-Saber Flockng for mult-agent dynamc systems: algorthms and theory. IEEE ransactons on Automatc Control, 006, 5(3): H. G. anner, A. Jadbabae, G. J. Pappas, Flockng n fxed and swtchng networks. IEEE ransactons on Automatc Control, 007, 5(5): H. S. Su, X. F. Wang, W.Yang, Flockng n mult-agent systems wth multple vrtual leaders. Asan Journal of Control, 008, 0(): M. Egerstedt, X. Hu, Formaton control wth vrtual leaders and reduced communcatons. IEEE ransactons on Robotcs and Automaton, 00, 7(6): R. Olfat-Saber, R. M. Murray, Dstrbuted structural stablzaton and trackng for formatons of dynamc mult-agents. In Proceedngs of the 4th IEEE Conference on Decson and Control, 00, pp F. Lan, Cooperatve path plannng of dynamcal multagent systems usng dfferental flatness approach, Internatonal Journal of Control, Automaton, and Systems, 008, 6(3): W. Ren, R. W. Beard, D. B. Kngston, Mult-agent Kalman consensus wth relatve uncertanty, Proc. of the Amercan Control Conference, 005. pp Y. Lu, Y. Ja, Robust H consensus control of uncertan mult-agent systems wth tme delays, Internatonal Journal of Control, Automaton, and Systems, 0, 9(6): W. Zhang, Stable Weghted Multple Model Adaptve Control: Dscrete-me Stochastc Plant, Internatonal Journal of Adaptve Control and Sgnal Process, 03, 7(7): Fg.4 weghts of models In Fg.4, the weght of the 4th model s convergent to. hen the controlled quantty of the 4th model s chosen to be the global control nput of the mult-agent system. 5. Conclusons By applyng the weghted MMAC methodology, we have addressed the adaptve consensus problem of mult-agent system wth large parameter uncertanty and communcaton delays. Accordng to the states dfferences between the plant and each model, we can obtan the weght of each local controller wth the weghtng algorthm mentoned. Fnally, the controller correspondng to the closest model to the plant wll be chosen as the rght controller. Publshed by Atlants Press Copyrght: the authors 9