Final Consensus State Research Articles

Distributed optimal consensus of multi-agent systems: A randomized parallel approach

In this paper, a randomized parallel algorithm is proposed to solve the distributed optimal consensus problem of multi-agent systems. Involving both the transient response and the final consensus state, the problem is described as a constrained non-separable optimization problem. Inspired by the randomized Jacobi proximal alternating direction method of multipliers, the proposed algorithm makes it possible for only a fraction of agents to solve their private subproblems in parallel at each iteration, which greatly saves computational resources and enhances running efficiency. The convergence analysis of the algorithm gives fully distributed convergence conditions. A trade-off between the convergence speed and resource savings is then obtained, where the convergence rate is estimated to be at least O1t. Furthermore, the algorithm can be accelerated to enjoy a convergence rate of O1t2 by adaptively adjusting the auxiliary parameters properly. Numerical simulations demonstrate the effectiveness of the theoretical results.

Fast Consensus of High-Order Multiagent Systems

In this paper, the fast consensus problem of high-order multi-agent systems (MASs) under undirected topologies is considered. The direct link between the consensus convergence rate and the control gains is established. A gradient descent based algorithm is proposed to optimize the convergence rate. By applying the Routh-Hurwitz stability criterion, the lower bound on the convergence rate is derived, and explicit control gains are derived as conditions to achieve the optimal convergence rate. Moreover, a protocol with time-varying control gains is designed to achieve the finite-time consensus. Explicit formulas for the time-varying control gains and the final consensus state are given. Numerical examples and simulation results are presented to illustrate the obtained theoretical results.

Interval consensus of switched multiagent systems

This paper studies interval consensus for switched multiagent systems over directed networks, which consist of a continuous-time subsystem and a discrete-time subsystem regulated by a switching rule. Interval consensus refers to a state constrained consensus, where each agent is allowed to propose an acceptable interval to saturate their expressed states and the final consensus state lies in the nonempty intersection of all these intervals. We establish conditions guaranteeing interval consensus for switched multiagent system under arbitrary switching rules. Furthermore, we introduce the scaled interval consensus notion, which allows both the convergence of states to a pre-assigned proportion and the scaled consensus values lying in the desired range. Simulation results are provided to verify the effectiveness of the theoretical results.

Bi-layer voter model: modeling intolerant/tolerant positions and bots in opinion dynamics

The diffusion of opinions in social networks is a relevant process for adopting positions and attracting potential voters in political campaigns. Opinion polarization, bias, targeted diffusion, and the radicalization of postures are key elements for understanding the voting dynamics’ challenges. In particular, social bots are currently a new element that can have a pronounced effect on the formation of opinions during electoral processes by, for instance, creating fake accounts in social networks to manipulate elections. Here, we propose a voter model incorporating bots and radical or intolerant individuals in the decision-making process. The dynamics of the system occur in a multiplex network of interacting agents composed of two layers, one for the dynamics of opinions where agents choose between two possible alternatives, and the other for the tolerance dynamics, in which agents adopt one of the two tolerance levels. The tolerance accounts for the likelihood to change opinion in an interaction, with tolerant (intolerant) agents switching opinion with probability 1.0 (\(\gamma \le 1\)). We find that intolerance leads to a consensus of tolerant agents during an initial stage that scales as \(\tau ^+ \sim \gamma ^{-1} \ln N\), who then reach an opinion consensus during the second stage in a time that scales as \(\tau \sim N\), where N is the number of agents. Therefore, very intolerant agents (\(\gamma \ll 1\)) could considerably slow down dynamics towards the final consensus state. We also find that the inclusion of a fraction \(\sigma _{{\mathbb {B}}}^-\) of bots breaks the symmetry between both opinions, driving the system to a consensus of intolerant agents with the bots’ opinion. Thus, bots eventually impose their opinion to the entire population, in a time that scales as \(\tau _B^- \sim \gamma ^{-1}\) for \(\gamma \ll \sigma _{{\mathbb {B}}}^-\) and \(\tau _B^- \sim 1/\sigma _{{\mathbb {B}}}^-\) for \(\sigma _{{\mathbb {B}}}^- \ll \gamma \).

An Accelerated Algorithm for Linear Quadratic Optimal Consensus of Heterogeneous Multiagent Systems

An accelerated algorithm is proposed in this article for solving the linear quadratic optimal consensus problem of multiagent systems. To optimize the linear quadratic response and the final consensus state simultaneously, a nonseparable multiobjective optimization problem with coupled constraints on decision variables is formulated. The main difficulty in solving the optimization problem lies in the nonlinear coupling of objectives, which is overcome by separating the problem into two independent and solvable single-objective optimization subproblems using the alternating direction method of multipliers. The proximal gradient decent scheme is then introduced to approximate the precise optimal solutions of the subproblems so as to improve the computing efficiency. Convergence analysis is performed to estimate the convergence rate and derive the convergence condition, which is independent of any global information of the system and, therefore, is fully distributed. Furthermore, the solution of each subproblem is obtained in a distributed form, allowing the multiagent system to achieve optimal consensus. Numerical examples show the effectiveness of the accelerated algorithm.

Distributed Consensus Algorithms for a Class of High-Order Multi-Agent Systems on Directed Graphs

We consider the consensus problem of high-order multi-agent systems, described by multiple integrator dynamics, under general directed graphs. In contrast to the existing results, we propose a fully-distributed consensus algorithm, albeit employing static state feedback. Specifically, it is shown that, with a suitably designed similarity transformation, consensus is reached under only the necessary and sufficient condition of an interconnection graph having a spanning tree. The proposed approach relaxes some restrictive assumptions commonly considered in the available literature, such as imposing global gains in the network and/or exploiting additional information from neighboring agents other than their position-like states. In addition, the proposed approach enables the explicit determination of the final consensus state even in the presence of constant communication delays. Simulation results are provided to illustrate the effectiveness of the proposed approach.

Finite-time scaled consensus through parametric linear iterations

ABSTRACTThis paper deals with finite-time scaled consensus problems over undirected and directed topologies, wherein agents’ states reach prescribed ratios in a finite time. We develop distributed linear iterations as a function of a linear operator on the underlying network and present necessary and sufficient conditions guaranteeing scaled consensus in a fixed number of steps equal to the number of distinct eigenvalues of a related linear operator. We identify the dependence of the final consensus states on the initial state condition, which can be conveniently and freely tuned by designing suitable parameters. Our results extend the recently developed approach on successive nulling of eigenvalues from complete consensus to scaled consensus, and from undirected topologies to directed topologies. Numerical examples and comparison studies are provided to illustrate the effectiveness of our theoretical results.

Finite-Time Weighted Average Consensus and Generalized Consensus Over a Subset

In this paper, the finite-time consensus for arbitrary undirected graphs is discussed. We develop a parametric distributed algorithm as a function of a linear operator defined on the underlying graph and provide a necessary and sufficient condition guaranteeing weighted average consensus in $K$ steps, where $K$ is the number of distinct eigenvalues of the underlying operator. Based on the novel framework of generalized consensus meaning that consensus is reached only by a subset of nodes, we show that the finite-time weighted average consensus can always be reached by a subset corresponding to the non-zero variables of the eigenvector associated with a simple eigenvalue of the operator. It is interesting that the final consensus state is shown to be freely adjustable if a smaller subset of consensus is admitted. Numerical examples, including synthetic and real-world networks, are presented to illustrate the theoretical results. Our approach is inspired by the recent method of successive nulling of eigenvalues by Safavi and Khan.

Scaled cluster consensus of discrete-time multi-agent systems with general directed topologies

ABSTRACTThis note proposes a notion of scaled cluster consensus, wherein the final consensus states within different clusters converge to prescribed ratios. Unlike most results in existing literature on cluster consensus, no constraints are imposed on the system topologies under the designed protocol, i.e. the agents are not required to possess any cluster affiliation information of others. For the delay-free case, an explicit scaled cluster consensus function is provided by exploring the characteristics of stochastic matrices. Diverse input delays and asymmetric communication delays are both considered, and sufficient condition for scaled cluster consensus is derived based on frequency domain analysis. Finally, numerical examples are given to illustrate the effectiveness of the presented results.

Robust finite-time synchronization of coupled harmonic oscillations with external disturbance

This paper considers the problem of finite-time synchronization for multiple coupled harmonic oscillators with a leader–follower architecture. By using the techniques of finite-time control and saturation control, a class of bounded finite-time state feedback controllers are first proposed. Then to address the case in the presence of external disturbance and lack of velocity measurement, a finite-time convergent observer is constructed to estimate both the unknown velocity information and the disturbance in a finite time. Finally, a disturbance observer-based bounded finite-time output feedback controller is developed. Rigorous proof shows that the systems output can reach synchronization in a finite time and the final consensus states are the leader׳s states.

Nonlinear finite-time bipartite consensus protocol for multi-agent systems associated with signed graphs

In this paper, finite-time multi-agent consensus problems are considered under networks associated with signed graphs whose edge weights can be not only positive but also negative. A nonlinear consensus protocol is proposed to guarantee the states of all agents to converge in a finite time. If the signed graph is structurally balanced, then the final consensus states of all agents are the same in modulus but not in sign. Otherwise, if the signed graph is structurally unbalanced, then the states of all agents converge to zero. Moreover, the final consensus states of agents can be provided uniformly regarding a signed-average quantity that depends on both the initial states of agents and the topology structure of the whole multi-agent network. Numerical simulations illustrate that the protocol is effective in achieving the finite-time consensus of agents under signed graphs and can particularly solve the finite-time average consensus problem of agents when their associated graph has all positive edge weights.

Consensus analysis for a class of mixed-order multi-agent systems with nonlinear consensus protocols

The consensus problem for a class of mixed-order multi-agent systems is investigated in this paper, where the multi-agent system is composed of first- and second-order dynamic agents. Firstly, consensus protocols are proposed for solving the finite-time consensus problem for mixed-order multi-agent systems. By employing the finite-time stability theory and LaSalle’s invariance principle, some sufficient conditions for finite-time consensus are given for multi-agent systems with directed communication topologies. Then, a class of nonlinear consensus protocols are given to solve the asymptotic consensus problem for mixed-order multi-agent systems. In addition, an invariant quantity is introduced to specify the expressions of the final consensus states when asymptotic consensus is reached. Finally, a simulation example is provided to demonstrate the effectiveness of the theoretical results.

Finite-Time Synchronization of a Class of Second-Order Nonlinear Multi-Agent Systems Using Output Feedback Control

This paper considers the problem of finite-time synchronization for a class of second-order nonlinear multi-agent systems with a leader-follower architecture. By using the finite-time control technique and homogenous systems theory, a finite-time state feedback controller is first proposed. Then to address the lack of velocity measurement, a finite-time convergent observer is constructed to estimate the unknown velocity information in a finite time. Finally, an observer-based finite-time output feedback controller is developed. Rigorous proof shows that the systems output can reach synchronization in a finite time and the final consensus states are the leader's states. In addition, for some special second-order multi-agent systems, a bounded finite-time output feedback controller can also be designed.

Multi-state voter model on weighted social networks with committed agents

There exist scaling correlations between the edge weights and the nodes' degrees in weighted social networks. Based on the empirical findings, we study a multi-state voter model on weighted social networks where the weight is given by the product of agents' degrees raised to a power θ and there exist persistent individuals whose opinions are independent of those of their friends. We find that the fraction of each opinion will converge to a value which only relates to the degrees of initial committed agents and the scaling exponent θ. The analytical predictions are verified by numerical simulations. The model indicates that agents' degrees and scaling exponent can significantly influence the final coexistence or consensus state of opinions. We also study the influence of degree mixing characteristics on the dynamics model by numerical simulations and discuss the relation between the model and the other related opinion dynamics models on social networks with different topological structures and initial configurations.

Evolutionary Origin of Asymptotically Stable Consensus

Consensus is widely observed in nature as well as in society. Up to now, many works have focused on what kind of (and how) isolated single structures lead to consensus, while the dynamics of consensus in interdependent populations remains unclear, although interactive structures are everywhere. For such consensus in interdependent populations, we refer that the fraction of population adopting a specified strategy is the same across different interactive structures. A two-strategy game as a conflict is adopted to explore how natural selection affects the consensus in such interdependent populations. It is shown that when selection is absent, all the consensus states are stable, but none are evolutionarily stable. In other words, the final consensus state can go back and forth from one to another. When selection is present, there is only a small number of stable consensus state which are evolutionarily stable. Our study highlights the importance of evolution on stabilizing consensus in interdependent populations.

Asymmetric coevolutionary voter dynamics

We consider a modification of the adaptive contact process that, interpreted in the context of opinion dynamics, breaks the symmetry of the coevolutionary voter model by assigning to each node type a different strategy to promote consensus: Orthodox opinion holders spread their opinion via social pressure and rewire their connections following a segregationist strategy; heterodox opinion holders adopt a proselytic strategy, converting their neighbors through personal interactions, and relax to the orthodox opinion according to its representation in the population. We give a full description of the phase diagram of this asymmetric model, using the standard pair approximation equations and assessing their performance by comparison with stochastic simulations. We find that although global consensus is favored with regard to the symmetric case, the asymmetric model also features an active phase. We study the stochastic properties of the corresponding metastable state in finite-size networks, discussing the applicability of the analytic approximations developed for the coevolutionary voter model. We find that, in contrast to the symmetric case, the final consensus state is predetermined by the system's parameters and independent of initial conditions for sufficiently large system sizes. We also find that rewiring always favors consensus, both by significantly reducing convergence times and by changing their scaling with system size.

Consensus control for a class of heterogeneous multi-Agent systems

The consensus problem for a class of heterogeneous multi-Agent systems was investigated in this paper,where the multi-Agent systems were composed of first-order Agents and second-order Agents.First,consensus protocols were proposed for first-order Agents and second-order Agents,respectively.Then,necessary and sufficient condition was presented for heterogeneous multi-Agent systems with fixed and directed communication topology to reach consensus by using tools from graph theory and matrix theory.Besides,the final consensus states were specified if consensus is achieved to the case of fixed communication topology.Furthermore,sufficient condition was derived for heterogeneous multi-Agent systems with switching and directed communication topologies to reach consensus.Finally,a simulation example was given to demonstrate the effectiveness of theoretical results.

Admissible Consensus of Multi‐Agent Singular Systems

ABSTRACTConsensus problems are studied for both continuous‐time and discrete‐time multi‐agent singular systems with time‐invariant and directed communication topologies. Under restricted system equivalence of singular agents, sufficient and necessary conditions are obtained for admissible consensus ability with static protocols, which are based on both the relative information of the dynamic states and the absolute information of the static states. For a network of continuous‐time singular systems, the existence of admissible consensualization can be cast into strong stabilizability of the agent dynamics. Once discrete‐time multi‐agent singular systems satisfy the condition of reaching nontrivial final consensus states, strong stabilizability is a sufficient condition to achieve admissible consensualization. Two algorithms are proposed to construct two protocols, which are based on a linear matrix inequality and a modified Riccati equation, respectively. Finally, the algorithms are illustrated by two simulation examples.

Second-order consensus for directed multi-agent systems with sampled data

This paper deals with the consensus problem of second-order multi-agent systems with sampled data. Because of the unavailable velocity information, consensus problem is studied only by using the sampled position information. The final consensus states of multi-agent system are given. And a necessary and sufficient consensus condition is provided, which depends on the parameters of sampling interval, eigenvalues of Laplacian matrix, and coupling strengths. Then, the case that both the sampled position and velocity information can be obtained is discussed. On the basis of introducing a time-varying piecewise-continuous delay and proposing a novel time-dependent Lyapunov functional, the sufficient consensus condition is presented, and the upper bound of sampling interval can be estimated. Simulation examples are provided finally to demonstrate the effectiveness of the proposed design methods. Copyright © 2013 John Wiley & Sons, Ltd.

Universal and nonuniversal features of the generalized voter class for ordering dynamics in two dimensions

By considering three different spin models belonging to the generalized voter class for ordering dynamics in two dimensions [Dornic et al., Phys. Rev. Lett. 87, 045701 (2001)], we show that they behave differently from the linear voter model when the initial configuration is an unbalanced mixture of up and down spins. In particular, we show that for nonlinear voter models the exit probability (probability to end with all spins up when starting with an initial fraction x of them) assumes a nontrivial shape. This is the first time a nontrivial exit probability is observed in two-dimensional systems. The change is traced back to the strong nonconservation of the average magnetization during the early stages of dynamics. Also the time needed to reach the final consensus state T(N)(x) has an anomalous nonuniversal dependence on x.