Research Article H Consensus for Discrete-Time Multiagent Systems

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1 Discrete Dynamics in Nature and Society Volume 05, Article ID 8084, 6 pages Research Article H Consensus for Discrete- Multiagent Systems Xiaoping Wang and Jinliang Shao School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 67, China Correspondence should be addressed to Xiaoping Wang; xiaopingwang@uestc.edu.cn Received May 05; Accepted 4 July 05 Academic Editor: Driss Boutat Copyright 05 X. Wang and J. Shao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An H consensus problem of multiagent systems is studied by introducing disturbances into the systems. Based on H control theory and consensus theory, a condition is derived to guarantee the systems both reach consensus and have a certain H property. Finally, an example is worked out to demonstrate the effectiveness of the theoretical results.. Introduction The research booms for consensus problems of multiagent systems following DeGroot s literature [. Consensus problem of multiagent systems has attracted considerable attention in recent years. Among various studies of linearconsensus algorithms, a noticeable phenomenon is the fact that algebraic graph theory and linear matrix inequalities (LMIs) play an important role in dealing with consensus problems. There has been a tremendous amount of interest in consensus problems of multiagent systems. In 00, Jadbabaie et al. explained the phenomenon mentioned in [ by theoretical techniques like undirected graph theory, algebraic theory, andthespecialpropertiesofstochasticmatricesin[.the distributed complete consensus problems for continuoustime have been intensively investigated [4 5, since the theoretical framework of consensus problems for first-order multiagent networks was proposed and solved by Olfati- Saber and Murray in [6. They presented the conditions on consensus in terms of graphs for three cases. Furthermore, Ren and Beard presented looser conditions to guarantee the complete consensus based on the graph theory in [7 than the results in [6. The consensus problem for discrete-time multiagent systems has attracted numerous researchers from mathematics, physics, biology, sociology, control science, and computer science. Xiao and Wang investigated the consensus problems for discrete-time multiagent system with timevarying delays; it is worth mentioning that the augmentation system was first proposed to deal with the consensus problem in [8. In [9, the cluster consensus problem for firstorder discrete-time multiagent system based on the stochastic matrix theory was solved. In recent years, many researchers analysed the external disturbances effects on stability and the convergence performance of networks, which is called H consensus. In [0, the H consensus problems were investigated for both discrete-time and continuous-time multiagent systems. In [, the sufficient conditions to guarantee all agents achieve H consensus were obtained by algebraic graph theory firstly. After that, Guo et al. extended general H consensus problemtoclustersynchronizationin[andobtained some sufficient conditions to realize cluster synchronization of the Lurie dynamical networks both without time delay and with time delay. In [, Liu and Chen addressed H consensus control for second-order multiagent systems; a sufficient condition is derived to guarantee H consensus for the systems by Lyapunov-Krasovkii functional. Inspiredbytheaboveanalysis,fordiscrete-timemultiagent systems, we discussed a distributed H consensus problem of which the objective is to design appropriate protocols to reach consensus while satisfying the described H performance for multiagent system with external disturbances. The paper is organized as follows. Section presents the mathematical model as well as relevant graph theory. Some analysis results on H are derived in Section. A simulation result is presented in Section 4. The conclusion is given in Section 5. Notations. Throughout this paper, the following notations are used: I denotes the identity matrix with an appropriate

2 Discrete Dynamics in Nature and Society dimension; denotes the space of square integrable vector functions over [0,); represents an ellipsis for the term introduced by symmetry.. Preliminaries.. Graph Theory. Denote by G =(V, E, A) an undirected graph with n nodes, where V ={,,...,n}and E V V indicate the set of vertices and edges, respectively. Here, (i, j) E if agent j can communicate with agent i. Inthis paper, we assume that there is no self-loop in the graph; that is, (i, i) E. LetN i = {j V (j, i) E} be the neighborhood set of vertex i. For any edge of the undirected graph G, (i, j) E if and only if edge (j, i) E. Theterm of path refers to a sequence of distinct vertices. A path P between two vertices V 0 and V k is the sequence {V 0,...,V k }, where (V i, V i ) E for i =,,...,k.graphg is called connected if there exists a path between any two vertices in G.LetA =[a ij R n n be the adjacency matrix of G,where a ij 0ifandonlyif(j, i) E; otherwisea ij = 0. Define L = D A as the Laplacian matrix of the undirected graph G,whereD =[d ii R n n is diagonal with d ii = n j= a ij.the degree of node i is defined as d i = n j= a ij... Mathematical Models. We suppose that a multiagent system under consideration consists of n agents with discretetime dynamics, and the dynamics of ith agent is given by x i (k+) =x i (k) +u i (k) +b i w i (k), z i (k) =c i x i (k) +d i w i (k),,,..., where x i (k) is the state, u i (k) is the control input to be designed, and w i (k) and z i (k) are the external disturbance and the output, respectively. Here, we assume parameters b i, c i,andd i are all known constants. Here, for (), we design the following consensus protocol: u i (k) = n j= () a ij (x j (k) x i (k)), () where a ij 0 is the entry of weighted matrix A. To end this section we give two definitions. Definition. Given the dynamic system (), one says that protocol () asymptotically solves an H consensus problem if the states of agents satisfy lim k x i (k) x j (k) = 0, i, j N. Definition. Let x i (0) betheinitialstateofagentv i.then protocol () is said to solve an average consensus problem asymptotically if the states of agents satisfy lim t x i (k) n j= x j(0)/n = 0, i N.. Analysis Results In this section, we discuss an H consensus problem of the proposed protocol for the discrete-time multiagent system. We first transfer system () into one in the form of matrix for completing theory analysis conveniently. Let x i (k) =x i (k) n j= x j (0), () n anditiseasytoknowthat x i (k) 0whenk +. Based on () and (), one gets x i (k+) = x i (k) + u i (k) +B w i (k). (4) Define x(k) = [ x (k),..., x n (k) T, z(k) = [z (k),..., z n (k) T,andw(k) = [w (k),...,w n (k) T,andthen()canbe rewrittenintoavectorformas where x (k+) = A x (k) + Bw (k), z (k) = C x (k) + Dw (k), a ij, { i =j; A=(a ij )= n { a ij, i = j. { j=,j=i And B =diag{b,b,...,b n }, C =diag{c,c,...,c n },and D = diag{d,d,...,d n }. Let H=... d [ [ (n ) n (5) (6). (7) It is obvious that H is a full row rank matrix. We introduce a linear transformation x(k) = H x(k) for(5)and note that x(k) = H T (HH T ) x(k).thus,(5)istransformedto x (k+) =H x (k) =H A x (k) +H f (k) Furthermore, (4) can be written as =H AH T (HH T ) x+h f (k). x (k+) =Ax (k) +Bw(k), z (k) =Cx (k) +Dw(k), where A=H AH T (HH T ), B=H B, C= CH T (HH T ), and D= D. Definition. System (9) is said to realize consensus with H performance index γ>0, if (8) (9) () system (9) with zero disturbance can reach consensus;

3 Discrete Dynamics in Nature and Society () system (9) satisfies z(k) γ w(k), where is the -norm of a vector in R n and γ is aconstant. Lemma 4 (Schur Complement). S=[ S S R n n is S S a symmetric matrix, where S R r r with r<n; then the following three conditions are equivalent: (a) S<0; (b) S < 0, S S S S < 0; (c) S < 0, S S S S < 0. Basedontheaboveanalysis,wecanderivethefollowing main result. Theorem 5. For given constants γ>0,system()canachieve H consensus if there exists a symmetric positive definite matrix X,suchthat X AX B 0 X 0 C T < 0. (0) [ γ I D T [ I Proof. We first consider the system without disturbances as follows: x (k+) =Ax (k), () and we prove that system (9) without disturbances is stable. Define a Lyapunov function as V (x (k),k) =x(k) T Px (k), () where P is a symmetric positive definite matrix. The difference of Lyapunov function V(x(k), k) yields V (x (k),k) =V(x (k+),k+) V(x (k),k) =x(k) T (A T PA P) x (k). The difference is negative if and only if () A T PA P < 0, (4) whichisequivalentbyschurcomplementto [ P AP P A P <0. (5) Under(0),itiseasytoknowthattheinequalityaboveholds by denoting X=P. Hence, system () is stable; that is, system (9) without disturbances is stable. In order to prove that the system has the H consensus property, that is, satisfying the performance z(k) γ w(k), we define J= [z T (k) z (k) γ w T (k) w (k). (6) Agent Agent Agent Figure :. Since the system is stable and lim k + V(x(k), k) = 0, for w(k) L [0,+),wehave J [z T (k) z (k) γ w T (k) w (k) + [V (x (), ) V(x (0), 0) + [V (x (), ) V(x (), ) + + [V (x (k+),k+) V(x (k),k) +. Since V(x(k), k) < 0, we have J [z T (k) z (k) γ w T (k) w (k) + V(x (k),k). Furthermore, letting ξ(k) = [x T (k), w T (k) T,wecanobtain J (7) (8) ξ T (k) Sξ (k), (9) where S=[ S S and S =A T PA P+C T C, S = S C T D+A T PB, ands = γ I+D T D+B T PB. Itiseasy to know that J 0ifS<0. That indicates z(k) γ w(k). By Schur Complement Lemma, we know that S<0ifand only if P A B 0 P 0 C T < 0. (0) [ γ I D T [ I

4 4 Discrete Dynamics in Nature and Society Agent Agent Agent (a) Agent Agent Agent (b) Figure : (a) when γ=0.00. (b) when γ= Furthermore,byusingSchurComplementLemmaagain,we can derive equivalently P AP B 0 P 0 C T < 0 () [ γ I D T [ I which exactly is (0) by letting P = X. According to Definition, we obtain that system () can achieve H consensus. The proof is completed. 4. Simulation In this section, we provide some simulations of two protocols for system () under the effect of different positive parameter γ with agents. The state trajectories of all agents are described in all the figures. In particular, the solid curve and thedottedcurvedescribethestatetrajectoriesofagentand agent, respectively; the remaining curve describes the state trajectory of agent. The horizontal axis and vertical axis describe time and state trajectories of all agents, respectively. In order to verify the effect of positive parameter γ,wepresent some simulations for Example 7 under different γ. Example 6. Consider a multiagent system with agents and the weighted adjacency matrix is A= [ () [ We suppose the disturbance is w(k) = [sin(k) sin(k), sin(k) T ; the positive parameter γ = 0.00 which is defined in Definition. Let the initial values be x (0) =, x (0) =, and x (0) =. The state trajectories of all agents are described in Figure. By simple matrix computation, we know that the model satisfies the condition in Theorem 5. It is easy to see that the system reaches consensus (three curves reach coincident) at about time 6 in Figure. Example 7. Consider a multiagent system with agents and the weighted adjacency matrix is A= [ () [ We suppose the disturbance is w(k) = [sin(k), sin(k), sin(k) T ; we provide some simulations with different γ. Let theinitialstatevaluesbex(0) =[,, T. In Figure (a), the H consensus for system () is under topology () and γ = 0.00; in particular, all the agents converge to a constant consensus state at about time 7. In Figure (b), the H consensus for system () is under topology () and γ=0.005; in particular, all the agents converge to a constant consensus state at about time 7. In Figure (a), the H consensus for system () is under topology () and γ= 0.0; in particular, all the agents converge to a time-varying consensus state at about time 8. In Figure (b), the H consensus for system () is under topology () and γ=0.08; in particular, all the agents converge to a time-varying consensus state at about time 7. In Figure 4(a), the H consensus for system () is under topology () and γ=0.; in particular,

5 Discrete Dynamics in Nature and Society Agent Agent Agent (a) Agent Agent Agent (b) Figure : (a) when γ=0.0. (b) when γ= Agent Agent Agent (a) Agent Agent Agent (b) Figure 4: (a) when γ=0.. (b) when γ=. all the agents converge to a time-varying consensus state at about time 7. In Figure 4(b), the H consensus for system () is under topology () and γ=; in particular, all the agents converge to a time-varying consensus state at about time Conclusions An H consensus problem for discrete-time multiagent system with external disturbances is investigated. Based on system transformation as well as Lyapunov stability theory, the sufficient condition in form of LMIs is obtained to guarantee H consensus of discrete-time multiagent with disturbances. A simulation result is finally provided to verify the effectiveness of our theoretical results. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

6 6 Discrete Dynamics in Nature and Society Acknowledgments The authors would like to thank the reviewers for their valuable comments and suggestions. This work was supported by NSFC (640064) and the Fundamental Research Funds for the Central Universities (ZYGX0Z005). References [ M. H. DeGroot, Reaching a consensus, the American Statistical Association,vol.69,no.45,pp.8,974. [ T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Novel type of phase transition in a system of self-driven particles, Physical Review Letters,vol.75,no.6-7,pp.6 9, 995. [ A. Jadbabaie, J. Lin, and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Transactions on Automatic Control, vol. 48, no. 6, pp , 00. [4 Y. Hong, J. Hu, and L. Gao, Tracking control for multiagent consensus with an active leader and variable topology, Automatica,vol.4,no.7,pp.77 8,006. [5 V. Gazi, Stability of an asynchronous swarm with timedependent communication links, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol.8,no., pp , 008. [6 Y. Cao and W. Ren, Distributed coordinated tracking with reduced interaction via a variable structure approach, IEEE Transactions on Automatic Control,vol.57,no.,pp. 48,0. [7 Z.-J. Tang, T.-Z. Huang, J.-L. Shao, and J.-P. Hu, Leaderfollowing consensus for multi-agent systems via sampled-data control, IET Control Theory & Applications, vol. 5, no. 4, pp , 0. [8G.Shi,Y.Hong,andK.H.Johansson, Connectivityandset tracking of multi-agent systems guided by multiple moving leaders, IEEE Transactions on Automatic Control, vol. 57, no., pp , 0. [9X.Dong,J.Yao,andJ.Wang, H Formation control and obstacle avoidance for hybrid multi-agent systems, Applied Mathematics,vol.0,ArticleID07,pages,0. [0 J. Yu, M. Yu, J. Hu, and B. Liu, Group consensus in multi-agent systems with sampled data, in Proceedings of the nd Chinese Control Conference,pp ,July0. [ B. Liu, G. Xie, Y. Gao, J. Wu, J. Zhang, and W. Luo, Consensus analysis of second-order multiagent systems with general topology and time delay, Applied Mathematics, vol. 0, Article ID 59750, 8 pages, 0. [ H. Xia, T.-Z. Huang, J.-L. Shao, and J.-Y. Yu, Second-order leader-following consensus of multiagent systems with time delays, Mathematical Problems in Engineering,vol.0,Article ID 50544, 8 pages, 0. [ D. Meng, Y. Jia, J. Du, and J. Zhang, On iterative learning algorithms for the formation control of nonlinear multi-agent systems, Automatica, vol. 50, no., pp. 9 95, 04. [4J.Y.Yu,L.Wang,andJ.Y.Yu, Groupconsensusofmultiagent systems with undirected communication graphs, in Proceedings of the 7th Asian Control Conference (ASCC 09),pp. 05 0, IEEE, Hong Kong, China, August 009. [5 J. Y. Yu and L. Wang, Group consensus in multi-agent systems with switching topologies and communication delays, Systems and Control Letters,vol.59,no.6,pp.40 48,00. [6 R. Olfati-Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on Automatic Control,vol.49,no.9,pp.50 5, 004. [7 W. Ren and R. W. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Transactions on Automatic Control,vol.50,no.5,pp , 005. [8 F. Xiao and L. Wang, Consensus protocols for discrete-time multi-agent systems with time-varying delays, Automatica,vol. 44,no.0,pp ,008. [9 Y.Chen,J.Lü, F. Han, and X. Yu, On the cluster consensus of discrete-time multi-agent systems, Systems & Control Letters, vol.60,no.7,pp.57 5,0. [0 J. H. Kim and H. B. Park, H state feedback control for generalized continuous/discrete time-delay system, Automatica, vol. 5,no.8,pp.44 45,999. [ P. Lin, Y. Jia, and L. Li, Distributed robust H consensus control in directed networks of agents with time-delay, Systems and Control Letters,vol.57,no.8,pp.64 65,008. [ L. Guo, X.-H. Nian, H. Pan, and Z.-T. Bing, Robust H cluster synchronization analysis of Lurie dynamical networks, Chinese Physics B,vol.,no.4,ArticleID04050,04. [ J. Liu and Z. Q. Chen, Robust H consensus control concerning second-order multi-agent systems with time-varying delays on dynamic topologies, International Computer Mathematics,vol.88,no.,pp ,0.

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