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This paper addresses the problem of consensus in networks structured in several clusters consisting of general linear node dynamics. The communication of each cluster is represented by fixed, directed spanning tree. Most of the agents interact only with some agents belonging to the same cluster in a continuous way. The states of each cluster are continuously evolved and achieved a local agreement by an observer type consensus protocol. To enforce a global consensus, each cluster has an agent, called leader, which updates its value periodically by using states from agents outside its cluster. Due to practical limitations in some circumstances, the states of the agents are usually unavailable for designing distributed consensus protocols. Therefore, distributed observers based on relative output measurements of neighbor agents are utilized to reconstruct the states of nodes, which will be first designed. Secondly, sufficient Linear Matrix Inequality (LMI) conditions for the global uniform exponential stability of the linear reset system are given. The consensus value and its characterization are also analysis and determination. Finally, a transformation of the network to a system, composing of two independent subsystems, which has lower dimension. Furthermore, an algorithm to properly select the feedback gain matrices and coupling strengths is presented which allows reaching a global consensus value. Simulation results are demonstrated to show the effectiveness of the theoretical results.

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egin{IEEEkeywords}

Multi agent system, consensus, reset system, LMI

end{IEEEkeywords}

section{Introduction}

label{sec:introduction}

IEEEPARstart{T}{he} problem of consensus or synchronization has attracted increasing attention from researchers in a variety of fields such as mechanisms cite{Chen2013}, engineering cite{Khazaei2017}, biology cite{Pavlopoulos2011}, sociology cite{Lorenz2005} and so on. Generally speaking, the primary task in the consensus problems is to design relevant protocols and algorithms such that agents reach an agreement asymptotically under exchanged information among agents. The issue of consensus for first-order dynamics is discussed in cite{Olfati-Saber2004} with different frameworks: directed or undirected networks, fixed or switching topology, and delayed communication or without time-delays. The consensus problem of the continuous and discrete first-order of the multi-agent system is also studied in cite{Ren2005}. It proved that the first-order multi-agent system could achieve asymptotically consensus if the directed interaction topology contains a directed spanning tree. Recently, a variety of remarkable contributions, for instant see cite{Ren2008, Xiao2009}, have been primarily originated from the papers as mentioned above. Extensions of consensus algorithms to second-order with linear or nonlinear dynamics under the fixed or the time-varying interaction graph, sampling information are investigated in cite{Ren2008, Yu2013} respectively. One limitation problem in most previous publications is that the agent dynamics are often restricted to be single or double integrators. Another worth problem is that most of the proposed distributed consensus protocols are based on the state feedback control approach. However, in many cases, the state information of agents may not be available. To overcome these problems, some distributed consensus protocols are thus proposed in cite{ZhongkuiLi2010, Li2015, Feng2017}, just to name a few. Most of the previously mentioned works are concerned with the complete consensus, i.e., all the agents converge to the common state.

In a realistic world, to deal with the unanticipated situations or changes, the network topology is partitioned into several subgraphs, called clusters/communities. The connections inside each cluster happen continuously. Due to the energy constraints occurring in long-distance or long time interactions, and communication constraints (harsh environment, energy optimization or opinion preferences for instance), the inter-communities interactions are only allowed to communicate outside their clusters at discrete time instants. Examples include team hunting of predators, obstacle avoidance of animal herds, cooperative searching of autonomous vehicles for multiple objects, and task allocation over the network between groups cite{Hou2015, Qin2013a, Wen2015}. In social networks, the opinion of each individual continuously evolves with respect to the views of the members belonging to its community in order to achieve a common agreement. Nevertheless, at specific instants, one individual (called leader) can change its opinion by exchanging with leaders outside its community. They will reset their opinion taking into account the ones of other leaders. These inter-cluster interactions can be considered as resets of the opinions. Consequently, a network dynamics is described in term of reset systems. In cite{Bragagnolo2016a}, in order to enforce a global agreement over the whole network, the paper proposed the discrete control action in a decentralized way. The sufficient LMI conditions are given to guarantee the global uniform exponential stability of the consensus with an assumption that network of clusters is directed and strongly connected graphs. In the case, the reset instants are event-triggered, i.e., defined by the occurrence of specific events cite{Mor2016}. The sufficient conditions for consensus in networks of linear reset systems are proposed to impulses corresponded to point-wise activation of some interconnections, with an assumption that the communication structure is directed spanning tree.

The above-mentioned problem mainly motivates our work in this paper. Instead of investigating the cluster consensus problem for multi-agent systems with integrator dynamics, we focus on a more general model. That is consensus problem of clusters for general linear multi-agent systems, in which each agent is modeled by a general linear system dynamics. Our main contributions of this paper are summarized as follows: Compared to the integrator case, in which each agent, in general, has no dynamics, see cite{Bragagnolo2016a, Mor2016}. When there are only interactions among clusters at the specific instants, the collective behavior of a generic linear multi-agent system is defined by not only the dynamical control protocols with respect to the isolated clusters, but also the discrete interactions among the leaders. This makes consensus/coordination problem for general linear agents much more challenging than that of the integrator case. Furthermore, most proposed cluster consensus based on the relative states between neighboring agents, see cite{Bragagnolo2016a, Yu2010, Qin2013a}. However, in many scenarios, the state information of agents is not available. Thus, the distributed full order observer type consensus based on relative output measurements is proposed in this paper. With an assumption that clusters have a containing of directed spanning tree, and all the leaders will quasi-periodically reset their state corresponding to the states of some their neighboring leaders. There are some properties of topology, corresponding to left and right eigenvalues, are given. Then, by the disagreement vector, the distributed consensus problem is equivalent to the stability problem of the disagreement dynamics. Furthermore, in order to analyze the global uniform exponential stability of the equilibrium point, we also propose LMI conditions that can be adapted for further goals of the paper including the design an algorithm allows to reach consensus. The next contribution is related to the characterization of the consensus value in the framework under consideration. The consensus value depends on not only the initial conditions, the topology of each community, and the network that associated with the leaders, but also the system dynamic of each agent. Besides, the reset sequence of leader’s states does not affect the consensus value. It is important to notice that the consensus value in this paper is more comprehensive than that in cite{Bragagnolo2016a}, which is as a particular case. Another contribution of the article is related to a transformation of the network to a system, composing of two independent subsystems, which has lower dimension. Furthermore, an efficient algorithm for selecting the feedback gain matrices and coupling strengths for the controllers and observers. To guarantee the consensus problem can be achieved, sufficient conditions with parameters solved by optimization problem is derived. It is noteworthy that we can reach consensus the inter-cluster interactions in a distributed way instead of the decentralized methodology mentioned in cite{Bragagnolo2016a}.

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The paper is organized as follows. In Section 2 we formulate the problem under consideration. The agreement behavior and the possible consensus value are determined in Section 3. The new linear reset system is given based on disagreement vector, and the conditions for the global uniform exponential stability of the consensus are provided in Section 4. These conditions are given in the form of a parametric LMI concerning in Section 5. Section 6 is assigned to numerical simulations which illustrate the results.

extit{Notation}. The following standard notation used in the paper. The sets of nonnegative integers, real and non-negative real numbers are denoted by $mathbf{N}$, $mathbf{R}$ and $mathbf{R}^+$, respectively. $A^T$ denotes the transpose of a matrix $A$. Vector $ extbf{1}_N$ and $0_N$ are the column vectors of size $N$ having all the components equal $1$ and $0$, respectively. We also use $x(t_k^- ) = lim_{t

ightarrow t_k, t leq t_k} x(t)$. $ A otimes B$ is denoted by the Kronecker product of matrices and satisfies the following property $(A otimes B)(Cotimes D)=(AC)otimes(CD)$.