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2962 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 11, NOVEMBER 2013 REFERENCES [1] C. M. Agulhari, R. C. L. F. Oliveira, and P. L. D. Peres, “Static output feedback control of polytopic systems using polynomial Lyapunov functions,” in Proc. 49th IEEE Conf. Decision Control, 2010, pp. 6894–6901. [2] Z. Artstein, “Stabilization with relaxed control,” Nonlin. Anal., pp. 1163–1173, 1983, TMA 7. [3] S. Battilotti, “Robust stabilization of nonlinear systems with pointwise norm-bounded uncertainties: A control Lyapunov function approach,” IEEE Trans. Autom. Control, vol. 44, pp. 3–17, 1999. [4] J. Bernussou, J. C. Geromel, and R. H. Korogui, “On robust output feedback control for polytopic systems,” in Proc. 44th IEEE Conf. Decision Control, Eur. Control Conf., Seville, Spain, 2005, pp. 5018–5023. [5] S. Boyd, L. EL Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [6] E. Feron, P. Apkarian, and P. 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Available: http://researcher. nsc.gov.tw/public/Jenq_Lang_Wu/Attachment/31131651071.pdf [28] R. Yang, P. Shi, and G. P. Liu, “Filtering for discrete-time networked nonlinear systems with mixed random delays and packet dropout,” IEEE Trans. Autom. Control, vol. 56, no. 11, pp. 2655–2660, Nov. 2011. Consensus of Discrete-Time Linear NetworkedMulti-Agent Systems With Communication Delays Chong Tan and Guo-Ping Liu, Fellow, IEEE Abstract—This technical note investigates the problem of consensus for discrete-time networked multi-agent systems (NMASs), where information is exchanged through the network with a communication delay. Based on the networked predictive control scheme and the dynamic output feed- back, a novel distributed protocol is proposed to compensate for the net- work delay actively. For NMASs with a directed topology and non-uniform agents, sufficient conditions of the consensus are obtained. A numerical simulation demonstrates effectiveness of the proposed theoretical results. Index Terms—Consensus, discrete-time linear systems, networkedmulti- agent systems (NMASs), networked predictive control (NPC). I. INTRODUCTION The problem of consensus for multiple autonomous agents has re- ceived extensive attention due to its broad applications in cooperative control, formation control, design of sensor networks, flocking of social insects and so on [1], [2]. One critical aspect of the consensus problem is to design protocols such that the group of agents can agree on certain quantities of interest based on the local information. Olfati–Saber and Murray [3] first posed and solved the theoret- ical framework of the consensus problem. For multi-agent systems with first-order dynamics and stochastic communication noises, the mean square average-consensus was discussed in [4], based on the probability limit theory. For multiple agents with double-integrator dynamics, convergence of the consensus strategies is studied in [5]. Because the states of all agents can be unavailable, consensus proto- cols via an observer or a dynamic output feedback (DOF) have been Manuscript received July 17, 2012; revised December 05, 2012; accepted April 23, 2013. Date of publication May 01, 2013; date of current version Oc- tober 21, 2013. This work was supported in part by the National Natural Science Foundation of China under Grants 61273104 and 61021002. Recommended by Associate Editor Y. Hong. C. Tan is with the CTGT Center, School of Astronautics, Harbin Institute of Technology, Harbin 150001, China and is also with the School of Automation, Harbin University of Science and Technology, Harbin 150080, China (e-mail: tc20021671@126.com). G.-P. Liu is with the Faculty of Advanced Technology, University of Glam- organ, Pontypridd CF37 1DL, UK and is also with the CTGT Center, Harbin Institute of Technology, Harbin 150080, China (e-mail: gpliu@glam.ac.uk). Color versions of one or more of the figures in this technical note are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2013.2261177 0018-9286 © 2013 IEEE IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 11, NOVEMBER 2013 2963 designed in [6]–[9]. Besides, the existence of consensus protocols (i.e., consensusability) has been studied in [10]–[12]. It is noted that most of the existing works neglect the prediction in- telligence of each individual. Inspired by numerous results on the pre- dictive intelligence of natural bio-groups [13], [14], Zhang et al. [15] have designed a small-world predictive protocol for the A/R andVicsek models, and proposed centralized and decentralized model predictive control protocols for linear dynamic networks without leaders, which shows that the predictive protocols can accelerate consensus speeds and reduce sampling frequencies. For the input saturation constraints case, a decentralized predictive mechanism is proposed to achieve the consensus under some mild assumptions in [16], [17], which applies predictive pinning control to improve consensus performance. Differ- ently, the eigen-spectrum of the state matrix is optimized to accelerate the consensus convergence speed in [18]. However, network communication delays are not considered in [15]–[18]. In networked control systems, it is inevitable that the network-induced delays will occur while exchanging data among multiple agents through a network, due to the limited bandwidth of communication channels and the finite transmission speed. Time delay can degrade the performance of control systems and even destabilize systems. With this background, this technical note addresses the consensus problem of discrete-time networked multi-agent systems (NMASs) with network transmission delays, based on the networked predictive control scheme (NPCS) proposed in [19]. A novel dis- tributed protocol is put forward to compensate for the communication delay actively rather than passively. For NMASs with a directed topology and non-uniform agents described by linear discrete-time time-invariant systems, sufficient conditions of the consensus are presented under mild assumptions. The main idea of [15]–[18] is that the decision on the next-step behavior of each individual is not only based on the currently available state information, but also on the pre- diction of future states. By setting and minimizing the moving horizon optimization index, the model predictive control laws are calculated and implemented. Nevertheless, the objective of this technical note is that, based on available historical local information and the NPCS, the current states of agents are predicted efficiently, the network delay is compensated actively, and the consensus protocol is designed via the DOF and predictive data at current time. The rest of the technical note is organized as follows. Necessary notations and concepts are described in Section II. In Section III, the protocol based on the NPCS and the DOF is designed, and the con- sensus analysis of NMASs with non-uniform agents and a constant communication delay is discussed. A numerical example is provided in Section IV to illustrate the feasibility of the theoretical results. Fi- nally, some concluding remarks are drawn in Section V. II. PRELIMINARIES The set of nonnegative integers is denoted by . Let be the set of all -by- matrices over a field , and is ab- breviated to , where is the real number field or complex number field . For , -inverse and Moore-Penrose inverse of are denoted by and [20], respectively; Left null space and (right) null space of are denoted by and , respectively, i.e., and . A vector valued function of a matrix is defined as , where is the k-th column of , . A matrix is said to be Schur if is square and , where is the spec- trum of , and is an open unit disk centered at the origin. Given two sets and , expresses the subtraction of and . Empty set is denoted as . The dimension of a finite-dimensional linear space is denoted as . Let denote a -dimension column vector with all ones, 0 and represent zero matrix and identity matrix with an appropriate dimension, respec- tively. The Kronecker product of matrices is denoted by . norm on vectors or its induced norm on matrices is represented by . A block diagonal matrix is denoted by , where is block diagonal, . Since the communication among interacting agents in NMASs is achieved by a network, the communication topology of agents can be modeled by a digraph with the set of nodes denoting the agents, the set of edges , and a nonnegative weighted adjacency matrix . The directed edge from node to node means that agent can receive information from agent . Adjacency element as- sociated with edge is positive. No self-cycle is allowed, hence , . The set of neighbors of node is denoted by . A directed path is a sequence of edges in a digraph of the form , where , and . If there exists a di- rected path from node to node , then node is said to be reachable from node . The set of all reachable nodes to node is denoted by . The Laplacian matrix of digraph is defined as and , . Obviously, all the row-sums of are zero. The Laplacian matrix of digraph has ex- actly one zero eigenvalue if and only if has a spanning tree [21]. III. CONSENSUS OF NMASS WITH NON-UNIFORM AGENTS A. Design of the Protocol Based on NPCS and DOF Consider an NMAS composed of agents, where the dynamics of agent are described by a linear discrete-time system as follows: (1) where , and are the state, control input and measured output of agent , respectively; , and are constant matrices; is a transmission delay of the network, by which information exchanged among all agents is achieved. It implies that agents are compelled to re- ceive data with -step lag. , and represent the ini- tial state, initial control input and initial output, respectively. Naturally, from (1), the dynamic structure of each individual can be different. For the simplicity of the consensus analysis, the following assumption can reasonably be made: Assumption 1: I) Network delay is a constant and known positive integer. II) The states of all agents can not be available but their outputs can be measured. III) Each agent can receive information from agent . Due to the network delay, old information rather than current infor- mation is received by agents. Obviously, it is not accurate using delayed data to design controllers and regulate systems. Therefore, NPCS pro- posed in [19] is exploited to compensate for the network delay actively and predict the current states of agents efficiently. Because agent receives information from agent with time delay , in order to overcome the effect of the network delay, 2964 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 11, NOVEMBER 2013 based on the output data of agent up to time , the state predictions of agent from time to are constructed as (2a) (2b) where and are the one-step ahead state prediction and the input of the observer at time , respectively; and can be designed using observer design approaches; is a state prediction of agent at time on the basis of the information up to time , and is the input at time , ; , , is the initial state of observer (2a), . It follows from (2) that, based on the data up to time , the output prediction of agent at time can be constructed as , . Thus, the following protocol based on the DOF is designed for agent : (3) where is the protocol state, and denotes output prediction difference between agent and agent ; is the initial state of the protocol; is the weighted adjacency matrix of digraph , , , , and are matrices to be designed. Remark 1: The routine consensus protocol is , where , , which exploits the state difference between agent and agent at time . It is implicitly assumed that all states are available for measurement. However, in many practical problems, it may be either impossible or impractical to measure the states directly. Hence, in industrial applications, the output feedback is more realistic and more useful. So more practical consensus protocol (3) based on the DOF is designed in this technical note. When , DOF (3) is a more general situation, which can increase free degree of the design and be more flexible. When , DOF (3) is simplified into a static output feedback , . In order to fulfill the proposed NPCS and compensate for the net- work transmission delay, it is necessary that every agent maintains both predictor (2) and controller (3) for agent . The net- worked predictive control scheme is explained in detail as follows. At time , agent can only receive data from agent . An observer (2a) is first used to provide an one-step ahead state prediction , based on . Then, using available data , and at time , input and protocol state can be obtained by (2) and (3). Thus, can be constructed by (2b). Sequentially, from data , and , input and protocol state can be also obtained in a similar manner. Hence, input data and , , , may be iteratively implemented. Of course, the proposed NPCS will inevitably increase the computational burden. However, the predictions of current states can be still obtained according to (2) and (3), which will be di- rectly used to compensate for the network transmission delay actively. Hence, at least in theory, the proposed NPCS is always feasible. Spe- cially, when , only one-step ahead state prediction is necessary and adequate based on observer (2a), but predictive process (2b) is not needed, i.e., , . When the network has a bounded and time-varying delay at time , agent receives information from agent with time delay , where and are known positive integers. The dwell-time approach can be used to handle the bounded and time-varying delay [22], [23]. When , data in the net- work are compelled to dwell such that the time delay achieves the upper bound . Then, the time-varying delay is transformed into a constant delay. Therefore, it is assumed that the network delay is constant in this technical note. Although the dwell-time approach is slightly conserva- tive, it provides a method of investigating the time-varying delay when it is difficult to deal with it directly. Definition 1: For NMAS (1) with a communication delay , pro- tocol (3) is said to solve the consensus problem if the following condi- tions hold: (c1) , ; (c2) , ; (c3) , ; where is the one-step ahead estimate error satisfying that , , and , , is the initial condition, . Definition 1 indicates that protocol (3) solves the consensus problem if and only if the difference of the states of any two agents asymp- totically converges to zero, dynamic output feedback controller (3) is asymptotically stable, and tracking errors of observers asymptotically converge to zero. B. Consensus Analysis Let , , , , , , . Protocol (3) solves the consensus problem if and only if , and hold simultaneously. For convenience of pro- tocol designs, the following condition is given. Condition 1: (F1) , where and (4) (F2) There exists such that, for every (5) Lemma 1: The following three conditions are equivalent: i) . ii) . IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 11, NOVEMBER 2013 2965 iii) There exist a positive integer and a non-zero ma- trix , such that . Proof: is equivalent to . So the equivalence of (i) and (ii) is obvious. If (ii) is true, then there exists and , which implies that for every , and for some . Then . Taking derives that , i.e., (iii) holds. On the other hand, if (iii) is true, which implies that there exists a non-zero matrix such that . Then , which implies that . Let , where , . Then there exists for some , such that and , i.e., . Then (ii) holds. So (ii) and (iii) are equivalent. If (F1) in Condition 1 holds, it follows from Lemma 1 that there exist non-zero matrices and such that (6) Besides, the processes of proof of Lemma 1 propose a method of con- structing and . Furthermore, if (F2) in Condition 1 holds too, then the following results hold: (F3) The matrix equation has a solution and the general solution is , where is arbitrary. (F4) For every , thematrix equation has a solution and the general solution is , where is arbitrary. For simplicity, denote , , , , , , , . For NMAS (1)