Finite-time consensus for switching network topologies with disturbances

In this paper we investigate the properties of a decentralized consensus algorithm for a network of continuous-time integrators subject to unknown-but-bounded time-varying disturbances. The proposed consensus algorithm is based on a discontinuous local interaction rule. Under certain restrictions on the switching topology, it is proven that after a finite transient time the agents achieve an approximated consensus condition by attenuating the destabilizing effect of the disturbances. This main result is complemented by an additional result establishing the achievement of consensus under different requirements on the switching communication topology. In particular, we provide a convergence result that encompasses situations in which the time varying graph is always disconnected. Lyapunov analyses are carried out to support the suggested algorithms and results. Simulative tests considering, as case study, the synchronization problem for a network of clocks are illustrated and commented on to validate the developed analysis.     

Discrete leader‐following consensus

In this paper, a leader‐following consensus of discrete‐time multi‐agent systems with nonlinear intrinsic dynamics is investigated. We propose and prove conditions ensuring a leader‐following consensus. Numerical examples are given to illustrate our results.     

Variable impulsive consensus of nonlinear multi-agent systems

In this paper, the consensus problem for nonlinear multi-agent systems with variable impulsive control method is studied. In order to decrease the communication wastage, a novel distributed impulsive protocol is designed to achieve consensus. Compared with the common impulsive consensus method with fixed impulsive instants, the variable impulsive consensus method proposed in this paper is more flexible and reliable in practical application. Based on Lyapunov stability theory and some inequality techniques, several novel impulsive consensus conditions are obtained to realize the consensus of multi-agent systems. Finally, some necessary simulations are performed to validate the effectiveness of theoretical results.     

Convergence rate for consensus with delays

We study the problem of reaching a consensus in the values of a distributed system of agents with time-varying connectivity in the presence of delays. We consider a widely studied consensus algorithm, in which at each time step, every agent forms a weighted average of its own value with values received from the neighboring agents. We study an asynchronous operation of this algorithm using delayed agent values. Our focus is on establishing convergence rate results for this algorithm. In particular, we first show convergence to consensus under a bounded delay condition and some connectivity and intercommunication conditions imposed on the multi-agent system. We then provide a bound on the time required to reach the consensus. Our bound is given as an explicit function of the system parameters including the delay bound and the bound on agents’ intercommunication intervals.     

Dynamic output feedback consensus of continuous‐time networked multiagent systems

This article addresses the dynamic output feedback consensus problem of continuous‐time networked multiagent systems. Both a fixed topology and Markovian switching topologies are considered. The consensus algorithms are on the base of the output information of each agent's itself and its neighbors. Some sufficient conditions for consensus of multiagent systems are obtained in forms of bilinear matrix inequalities. The algorithm based on the homotopy continuation method is given to compute the feasible controller gains. Numerical simulations are given to show the effectiveness of the proposed results. © 2014 Wiley Periodicals, Inc. Complexity 20: 35–42, 2015     

Practical scaled consensus for nonlinear multiagent systems with input time delay via a new distributed integral-type event-triggered scheme

This paper studies scaled-based practical consensus issue for multiagent systems with input time delay by a fully continuous communication-free integral-type event-triggered scheme. By choosing the proper scales, scaled consensus can be induced to synchronization, bipartite consensus or cluster consensus. By defining a continuous communication-free measurement error for the integral-type event-triggered mechanism, the new integral-type event-triggered condition is proposed which can not only reduce the energy consumption but also prolong the interevent time. Then, with time domain analysis method, the distributed integral-type event-triggered control problem for nonlinear general multiagent systems involving input time delay is investigated and then the second-order counterpart, with a calculated upper-bound for time-delay. Moreover, it is concluded that with such event-triggered protocols, practical scaled consensus can be achieved without the exhibition of Zeno behavior. At last, simulations are shown to support the results.     

Impulsive consensus of multi-agent systems with stochastically switching topologies

In this paper, the impulsive consensus problem for multi-agent systems is investigated. The purpose of this paper is to provide a valid consensus protocol that overcomes the difficulty caused by stochastically switching structures via impulsive control. Some sufficient conditions of almost sure consensus are proposed when the switching structures are the independent process or the Markov process. It is shown that the sum-zero rows of matrix play a key role in achieving group consensus. Furthermore, simulation examples are provided to illustrate and visualize the effectiveness of these results.     

On consensus speed of multi-agent systems with double-integrator dynamics

The performance of multi-agent systems is an important issue. In this paper, it is focused on consensus speed for multi-agent systems with double-integrator dynamics and fixed undirected graphes under a kind of consensus protocols. It is revealed that, under some conditions, the maximum consensus speed is determined by the largest and the smallest nonzero eigenvalues of the Laplacian matrix of the undirected connected graph. Based on the mentioned results, arbitrary desired consensus speed can be achieved by choosing suitable feedback gains. Numerical simulations are given to illustrate the main results.     

Tracking consensus of nonlinear MASs with asymmetric communication delays in noisy environments

This paper investigates the tracking consensus problem of nonlinear multi-agent systems (MASs) with asymmetric time-varying communication delays in noisy environments under the conditions of fixed and switching directed topologies. A novel stochastic analysis approach is proposed, which guarantees that the designed two distributed tracking protocols can guide the controlled systems to achieve tracking consensus in the sense of mean square. In order to further reveal the influence of asymmetric communication delays on the tracking consensus ability for MASs, some new delay-dependent sufficient conditions for mean-square consensus are also developed. A simple example is finally given to illustrate the effectiveness of the proposed theoretical results.     

Voter models and external influence

ABSTRACT

In this paper, we extend the voter model (VM) and the threshold voter model (TVM) to include external influences modeled as a jump process. We study the newly-formulated models both analytically and computationally, employing diffusion approximations and mean field approximations. We derive results pertaining to the probability of reaching consensus on a particular opinion and also the expected consensus time. We find that although including an external influence leads to a faster consensus in general, this effect is more pronounced in the VM as compared to the TVM. Our findings suggest the potential importance of external influences in addition to local interactions.     

Consensus seeking in multi-agent systems via hybrid protocols with impulse delays

This paper studies the consensus problem of multi-agent systems with both fixed and switching topologies. A hybrid consensus protocol is proposed to take into consideration of continuous-time communications among agents and delayed instant information exchanges on a sequence of discrete times. Based on the proposed algorithms, the multi-agent systems with the hybrid consensus protocols are described in the form of impulsive systems or impulsive switching systems. By employing results from matrix theory and algebraic graph theory, some sufficient conditions for the consensus of multi-agent systems with fixed and switching topologies are established, respectively. Our results show that, for small impulse delays, the hybrid consensus protocols can solve the consensus problem if the union of continuous-time and impulsive-time interaction digraphs contains a spanning tree frequently enough. Simulations are provided to demonstrate the effectiveness of the proposed consensus protocols.     

How long does it take to consensus in the Hegselmann-Krause model?

Hegselmann and Krause introduced a discrete-time model of opinion dynamics with agents having limit confidence. It is well known that the dynamics reaches a stable state in a polynomial number of time steps. However, the gap between the known lower and upper bounds for the worst case is still immense. In this paper exact values for the maximum time, needed to reach consensus or to discover that consensus is impossible, are determined using an integer linear programming approach. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Robust fixed-time consensus protocols for multi-agent systems with nonlinear state measurements

This paper solves the robust fixed-time consensus problem for multi-agent systems with nonlinear state measurements. Sufficient conditions are established for the proposed protocol to reach fixed-time consensus under time-varying undirected and fixed directed topology with the aid of Lyapunov functions. It is proved that the finite settling time of the presented protocol for robust consensus is uniformly bounded for any initial condition, which makes it possible for people to design and estimate the convergence time off-line. Numerical simulations are preformed to show the effectiveness of our proposed protocol.     

A two-phase algorithm for consensus building in AHP-group decision making

Group decision making through the AHP has received significant attention in contemporary research, the primary focus of which has been on the issues of consistency and consensus building. In this paper, we concentrate on the latter and present a two-phase algorithm based on the optimal clustering of decision makers (members of a group) into sub groups followed by consensus building both within sub groups and between sub groups. Two-dimensional Sammon’s mapping is proposed as a tool for generating an approximate visualization of sub groups identified in multidimensional vector space, while the consensus convergence model is suggested for reaching agreement amongst individuals in and between sub groups. As a given, all decision makers evaluate the same decision elements within the AHP framework and produce individual scores of these decision elements. The consensual scores are obtained through the iterative procedure and the final scores are declared as the group decision. The results of two selected numerical examples are compared with two sets of results: the results obtained by the commonly used geometric mean aggregation method and also the results obtained if the consensus convergence model is applied directly without the prior clustering of the decision makers. The comparisons indicated the expected differences among the aggregation schemes and the final group scores. The matrices of respect values in the consensus convergence model, obtained for cases when the decision makers are optimally clustered and when they are not, show that in the latter case the decision makers receive lower weights of respect from other members in the group. Various tests showed that our approach is efficient in cases when no clusters can be visually and undoubtedly identified, especially if the number of group members is high.     

On the consensus protocol of conspecific agents

We present a class of consensus protocols over groups of agents with stochastically switching, directed, and weighted communication topologies. In this protocol, an agent’s traits, that is, the cardinality of its neighbor set and the weight assigned to its neighbors in the updating process, are given by two jointly distributed random variables and the neighbors of an agent are selected with equal probability. We provide closed form results for the asymptotic convergence rate and for the steady state mean square deviation in the presence of additive noise. These results are specialized to consensus protocols based on Erd?s–Rényi and numerosity-constrained networks.     

Impulsive control for one-side Lipschitz nonlinear MASs under semi-Markovian switching topologies with partially unknown transition probabilities

This article addresses the consensus problem of impulsive control for the multi-agent systems under uncertain semi-Markovian switching topologies. Considering the control and information exchanging cost in the implementation of multi-agent systems, an impulsive control protocol is developed not only to relieve the network burden but address the consensus problem. In addition, globally Lipschitz condition, as required in many existing literatures, is not needed in this article, so we introduce one-side Lipschitz condition to loosen the constraint of Lipschitz constant and widen the range of nonlinear application. According to cumulative distribution functions and Lyapunov functional, sufficient criteria are derived for the mean square consensus of multi-agent systems. It is shown that the impulsive sequence is not only inconsistent with switching sequence but also mode-dependent. Finally, simulation results are given to validate the superiority of the theoretical results.     

Design and analysis of quantizer for multi-agent systems with a limited rate of communication data

This paper concerns the problem of consensus for multi-agent systems with a limited rate of communication data. The effect of the quantization error on consensus is analyzed for the case that the nodes’ dynamics is described by the first order integrator with a quantized message input. In order to describe how exactly the consensus is achieved, the notion of consensus level is introduced. Based on these, a suboptimal method of resource allocation to reduce such effect of quantization error is proposed, and a dynamic quantizer is designed to meet the requirement of the consensus level, under the constraints that the total quantization rate of overall system is limited. At last, a numerical example is included for demonstration, and the outcome of the simulations suggests the validity of the presented theoretic results.     

A new approach using multiple Lyapunov functions for bipartite consensus of multi-agents over directed switching signed graphs

This paper investigates a new approach for the bipartite (cooperative–competitive) consensus control design for a class of nonlinear agents with Lipschitz dynamics under directed switching topologies. The design technique utilizes multiple Lyapunov functions (MLFs), inequality-based criteria, and average dwell-time (ADT) for switching instances to develop relaxed and less conservative constraints. The results are derived for structurally balanced signed graphs which switch among different configurations with persistent or frequent directed spanning tree (DST), rooted at the leader node. Further, results are also investigated for the dynamic leader agent with non-zero norm-bounded control input. To the best of our knowledge, bipartite consensus of a generic form of Lipschitz nonlinear agents under directed switching topologies has been addressed for the first time. In addition, advanced concepts of MLFs and ADT are used for dealing with switching among signed communication topologies. Numerical simulations to validate the proposed theoretical analysis are provided for different conditions.     

An axiomatic study of Majority-rule (+ ) and associated consensus functions on hierarchies

Consensus trees have been used routinely in systematic biology for over 30 years with the majority-rule consensus tree among the most popular. In this paper we give axioms that characterize a new consensus function, called majority-rule (+), that is based on a generalization of majority-rule to the supertree case. In addition, two other related consensus functions are characterized.     

Consensus of the second-order multi-agent systems with an active leader and coupling time delay

This article investigates the consensus problem of the second-order multi-agent systems with an active leader and coupling time delay in direct graph. One decentralized state control rule is constructed for each agent to track the active leader and it is proved that the proposed control scheme enables the consensus to be obtained when the adjacency topology is fixed/switched. Simulation results show effectiveness of the proposed theoretical analysis.