Cluster Consensus Problem Research Articles

Cluster consensus of finite-field networks with stochastic communication topology

Most of the multi-agent systems (MASs) in real life are with limited storage and communication capabilities; studies on such systems, including consensus problem, however have been mainly based on real number field models. In this article, the cluster consensus problem of MASs with stochastic communication topology over finite-field known as the cluster consensus of finite-field networks (FFNs), is considered. Using the semi-tensor product of matrices, the dynamics of FFNs with stochastic communication topology are equivalently converted into a logical algebraic form, which facilitates further studies. Then a permutation system is introduced. The relationship between cluster consensus of FFNs with stochastic communication topology and the set stability with probability one (SSPO) of the permutation system is revealed. In such an approach, some verifiable criteria can be derived for the cluster consensus of FFNs with stochastic communication topology, without requesting the connectivity of the whole network. Finally, an example is presented to validate the main results.

Cluster consensus for nonlinear multi‐agent systems restricted with input saturation by event triggered control

AbstractThis article studies the cluster consensus problem for the nonlinear multi‐agent systems subject to input saturation under the context of event triggered control. Taking ordinary consensus object as a special case, this article considers the problem of cluster consensus. The event triggered mechanism is designed and the Zeno phenomenon is ruled out. The sufficient conditions of control protocol and range estimation of domain of attraction are formulated by the linear matrix inequalities for the constraints of input saturation and nonlinear terms existed in the system dynamics. Finally, the proposed results are validated by one example in the background of spring damping system.

Adaptive cluster consensus control of nonlinear multi‐agent system via the dynamic event‐triggered strategy

SummaryIn this article, the cluster consensus problems of two classes of nonlinear multi‐agent systems (NMASs) are studied under the dynamic event‐triggered control (ETC) strategy, where cooperative‐competitive interactions among agents and high‐order dynamics are considered. First, two dynamic ETC protocols are developed to realize the cluster synchronization of NMASs. Different from the static ETC, a dynamic parameter regulated by an adaptive law is used in place of a constant which can improve the trigger time intervals and reduce the energy consumption. Subsequently, an adaptive control mechanism based on the proposed ETC strategy is designed to adjust the controller parameter for agents. Some sufficient conditions for achieving the cluster consensus of NMASs are derived by using the Lyapunov stability theory and algebraic graph theory. Moreover, it is proved in theory that Zeno phenomenon can be excluded with these control schemes. Finally, two simulation examples validate the effectiveness of the theoretical results.

Adaptive dynamic event-triggered cluster consensus for second-order delayed MASs with input saturation

In this paper, the cluster consensus problem of second-order multi-agent systems (SOMASs) is investigated in which cooperative-competitive interactions among agents, input time delay, and input saturation are involved simultaneously. A novel adaptive dynamic event-triggered control (ETC) protocol is first designed to realize cluster consensus. Since the constant in static trigger condition is replaced by a dynamic parameter, the new control scheme can reduce the trigger times and save energy consumption. Then, switching topology problem of heterogeneous multi-agent systems (MASs) is also discussed based on the dynamic ETC strategy. Some sufficient conditions for achieving the cluster consensus of SOMASs are proposed by utilizing the Lyapunov stability theory and algebraic graph theory. Especially, to solve the saturation problem, Lagrange mean value theorem is adopted. Moreover, it is also proved that no Zeno behavior occurs under the proposed control protocol. Finally, two examples justify the significance of our theoretical results.

Cluster consensus for inherently nonlinear cooperative networks

SummaryThis paper investigates the cluster consensus problem for inherently nonlinear cooperative networks with first and second‐order system dynamics. For a first order multi‐agent network evolving over a directed graph, not necessarily containing a spanning tree, a sufficient condition based on Lyapunov theory is derived on the single controller parameter so that the system achieves cluster consensus. These stability results are extended to second‐order nonlinear systems that utilize two parameters in the distributed control law. Furthermore, the total number of clusters and members of each cluster are explicitly computed from the primary and secondary layer subgraphs of the underlying directed graph. The results are the first in the literature to address the cluster consensus problem for inherently nonlinear cooperative networks evolving over directed graphs that do not contain a spanning tree.

K-filter-based distributed adaptive output cluster consensus under cooperative-competitive networks

This paper investigates the K-filter-based output cluster consensus problem under cooperative-competitive networks. The agents in the network are divided into several clusters where the relationships among agents belonging to the same cluster are only cooperative while agents in different clusters have both cooperative and competitive interactions with each other. Firstly, K-filters are introduced to estimate the state variables of the system with unknown control coefficients. Then new neighborhood errors are constructed to make the couplings among agents belonging to the same cluster strong enough. Based on the K-filters and the new neighborhood errors, a distributed output-feedback control protocol is designed to achieve output cluster consensus. It is proved that all signals in the closed-loop systems are bounded and the output of each agent can track the reference signal asymptotically if the network is in-degree balanced and each cluster contains a directed spanning tree. Finally, a numerical example is provided to verify the effectiveness of the proposed control protocol.

Cluster consensus with first and higher-order antagonistic interaction dynamics

This paper studies the cluster consensus problem for networks with antagonistic interactions modelled by adjacency matrices with negative weights. By introducing an extended digraph representation consisting of purely cooperative interactions with lifting approach, we relate the trajectories of the system to those of an extended system with a positive digraph. The behaviours of agents in a signed network are extracted from its extended digraph. Consequently, the number of clusters and the cluster members are explicitly determined for any signed digraph by using primary and secondary layer subgraph concepts. The relation between the dynamics of a signed system and its extended representation is investigated for first and higher-order systems. The conditions that make both systems stable are stated. Additionally, the control parameters to achieve cluster consensus are derived explicitly for first, second and third order systems. The obtained results for continuous-time networks are subsequently extended to discrete-time networks. Finally, theoretical results are illustrated via numerical examples.

Cluster consensus for leader-following high-order linear multiagent systems

<p indent="0mm">Cluster consensus for leader-following high-order linear multiagent systems (MASs) is investigated. Based on the information of an agent and its neighbors, state feedback and output feedback control protocols are designed for each follower, respectively. Sufficient conditions for leader-following cluster consensus are given using tools from the algebraic Riccati equation and matrix analysis. It is shown that if the communication topology of an MAS composed of the leader and followers is interactively balanced and contains a spanning tree, the MAS can achieve leader-following cluster consensus under the proposed control protocols by properly selecting control gains. Particularly, it is found that the leader-following cluster consensus problem based on the output feedback protocol can be transformed into a static output feedback problem using an error system. If the input and output matrices of a follower satisfy a certain rank condition, the MAS can achieve leader-following cluster consensus under the designed output feedback control protocol.

Event-triggered cluster consensus for two classes of nonlinear multi-agent systems

This paper investigates the cluster consensus problems of two classes of leader-following nonlinear multi-agent systems (LFNMASs). The event-triggered strategies are applied to solve the problems, which can significantly reduce energy consumption and the frequency of the controller updates. The sufficient conditions are presented to realize the cluster consensus for two classes of LFNMASs. Then, it is proved that the Zeno behavior can be avoided under the proposed event-triggered control strategies. Finally, two numerical examples are given to demonstrate the effectiveness of the theoretical results.

Distributed Cluster Consensus of Higher-Order Linear Continuous-Time Systems with Input Delay

This paper studies the higher-order cluster consensus problem for linear multi-agent systems with fixed input delay. It is proved that there is always a non-empty set of control parameters that makes a delayed third-order system stable. In light of this result, a control parameter selection procedure is proposed for higher-order systems under fixed input delay. Theoretical results are verified by numerical examples.

Leader-Following Cluster Consensus as a Graphical Differential Game With a Nash Equilibrium Solution

This paper formulates the leader-following cluster consensus problem of a homogeneous linear time-invariant multi-agent system over a directed graph as an N-player graphical differential game. The agents utilize distributed state feedback control laws based on the available information to calculate their best response. In general, the absence of centralized information prevents the best responses of agents from constituting a Nash equilibrium solution. We show that the best responses depending on the unique solution of an algebraic Riccati equation constitute a Nash equilibrium for the proposed graphical differential game with a modified cost function. A numerical example illustrates the Nash equilibrium solution by showing that the agents can not improve their performance by altering their responses while solving the leader-following cluster consensus problem.

Bipartite multi-tracking in MASs with intermittent communication

In this paper, a novel cluster consensus problem related with the bipartition of the graph of multi-agent systems (MASs) is studied. To track the virtual leaders and reach the expected consensus, a new type of pinning consensus protocol with aperiodic intermittent effects is designed according to the graph structure, and a new kind of aperiodic intermittent communication is defined. Moreover, the protocol is applied to construct networked systems with intermittent communications. Lyapunov functional is applied to get sufficient conditions for solving the multi-tracking problem under a dual subsystem framework. Finally, some numerical simulations are given to illustrate the effectiveness of the theoretical results.

Resilient Cluster Consensus of Multiagent Systems

We investigate the problems of resilient cluster consensus in directed networks under three types of multiagent dynamics, namely, continuous-time multiagent systems, discrete-time multiagent systems, and switched multiagent systems composed of both continuous-time and discrete-time components. Resilient cluster censoring strategies are proposed to ensure cluster consensus against locally bounded Byzantine nodes in a purely distributed manner, where neither the number/identity of Byzantine nodes nor the division of clusters is assumed. We do not require complicated algebraic conditions or any balance conditions over intercluster structures, distinguishing the current work from previous results on cluster consensus problems besides <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a fortiori</i> the attack-tolerant feature. Sufficient conditions are established in all the three scenarios based on the graph robustness. Furthermore, we solve the heterogenous cluster robustness problems and resilient scaled cluster consensus problems as extensions. The theoretical results are illustrated through numerical examples including the Santa Fe collaboration network.

Cluster consensus of fractional-order non-linear multi-agent systems with switching topology and time-delays via impulsive control

In this paper, a cluster consensus problem for fractional-order nonlinear multi-agent systems with fixed and switching topologies and time delays is considered. The problem is solved in three cases: (i) fixed topology and time delays, (ii) switching topology, and (iii) switching topology and time delays. Cluster consensus is achieved for fractional-order multi-agent systems by using the impulsive control method. Cluster consensus algorithms are studied by using Mittag–Leffler stability and the fractional Lyapunov direct method and some sufficient conditions are obtained to achieve cluster consensus of fractional-order multi-agent systems with nonlinear dynamics under switching topologies and time delays. Finally, simulation results are presented to show the validity of the proposed control strategies.

Semiglobal Cluster Consensus for Heterogeneous Systems With Input Saturation.

In this article, the semiglobal cluster consensus problem is investigated for heterogeneous generic linear systems with input saturation. A general case in a leaderless framework is studied first, and then in order to broaden the scope of application, we consider a special case in which the leader nodes are pinned intermittently. To tackle the above problems, we propose a linear control scheme by using the low-gain feedback technique under the assumptions that each node is asymptotically null controllable and the underlying topology of each cluster (the extended cluster under the intermittent pinning control) has a directed spanning tree. The Lyapunov-based method and the low-gain feedback technique are developed for convergence analysis. It is shown that for both cases, the convergence rate is explicitly specified, which depends on the low-gain parameter and system matrices. Finally, two numerical examples are provided to verify the effectiveness of the theoretical findings.

Cluster consensus in first and second-order continuous-time networks with input and communication delays

In this paper, we investigate the cluster consensus problem for first and second-order multi-agent systems with or without delay in continuous-time. Contrary to previous studies on cluster consensus in the literature, the clusters are assumed not to be pre-determined. One of the main contributions of the paper is to compute the number and members of the clusters for a first-order multi-agent network evolving over an arbitrary directed graph. Another novel result is its extension to systems with second-order dynamics. The effect of input and communication delay on cluster consensus is also studied in detail and it is shown that there is an upper bound of input delay whereas communication delay does not adversely alter the cluster consensus properties. Theoretical results are illustrated via several simulations.

New methods to realize the cluster consensus for multi‐agent networks

AbstractThis paper investigates the cluster consensus problem for multi‐agent networks via a new method. Firstly, a distributed cluster consensus protocol is proposed by constructing a new Laplacian matrix, which is not required to be row‐zero‐sum. It is shown that the cluster consensus of directed multi‐agent networks can be achieved under the presented protocol, and the final cluster consensus values of the considered system are also given in this paper, which only depend on the initial states of all agents. Next, a pinning cluster consensus law is provided to force the dynamics of the agents to the desired trajectory. Furthermore, it is found that the cluster consensus of directed networks can be reached even in presence of arbitrary finite communication delays. Finally, effectiveness of the obtained results is demonstrated by numerical examples.

Cluster output regulation of heterogeneous multi-agent systems

This paper studies the cluster output regulation problem of heterogeneous linear multi-agent systems under fixed directed topology, where dynamics of all agents are allowed to be different. Compared with the standard consensus problem, the interactions among clusters, which can be negatively weighted, impose a challenge on dealing with the issue. Based on the output regulation theory and the small-gain theorem, distributed state feedback and output feedback controllers are proposed to solve the problem. In each case, a small-gain criterion is given to guarantee cluster output regulation. Furthermore, a unified design procedure is given for both cases. Our results provide a new way to understand the relationship between the standard consensus problem and the cluster consensus problem. Numerical simulations are provided to verify the effectiveness of theoretical results.

Analysis of Second Order Cluster Consensus for Directed Networks with Time Delay

Abstract In this paper, we consider the cluster consensus problem in delayed directed networks where agents are described by second-order discrete-time dynamics. Unlike first order systems, delay is shown to lead to instability in second order systems if the controller parameters are not chosen properly. Furthermore, the numerical results reveal that as the amount of delay increases, the stability region in the controller parameter space shrinks.

Weighted Consensus Problem for Multiagent Systems with Edge‐ and Node‐Weighted Directed Graphs

In this paper, the concept of consensus is generalized to weighted consensus, by which the conventional consensus, the bipartite consensus, and the cluster consensus problems can be unified in the proposed weighted consensus frame. The dynamics of agents are modeled by the general linear time‐invariant systems. The interaction topology is modeled by edge‐ and node‐weighted directed graphs. Under both state feedback and output feedback control strategies, the weighted consensus problems are transformed into the equivalent conventional consensus problems. Then, some distributed state feedback and output feedback protocols are proposed to solve the weighted consensus problems. For output feedback case, a unified frame to construct the state‐observer‐based weighted consensus protocols is proposed, and different design approaches are discussed. As special cases, some related results for bipartite consensus and cluster consensus can be obtained directly. Finally, a simple example is given to illustrate the effectiveness of our proposed approaches.