Consensus seeking in multi-agent systems with noisy and delayed communication in digraphs having spanning tree

This paper studies the consensus problem of continuous-time single-integrator multi-agent systems with measurement noises and time delays under directed fixed topologies. Each agent in the team receives imprecise and delayed information from its neighbours. The noises are considered white noises, and time delays are assumed to be uniform for all the received information states. An analysis framework based on graph theory and stochastic tools is followed to derive conditions under which the asymptotic unbiased mean square linear χ-consensus is achieved in directed fixed topologies having a spanning tree. Then, conditions to achieve asymptotic unbiased mean square average consensus are deduced for fixed balanced digraphs having a spanning tree. The effectiveness of the proposed algorithms is proved through some simulations.     

Distributed consensus of discrete‐time multi‐agent systems with multiplicative noises

In this paper, we consider the consensus problem of discrete‐time multi‐agent systems with multiplicative communication noises. Each agent can only receive information corrupted by noises from its neighbors and/or a reference node. The intensities of these noises are dependent on the relative states of agents. Under some mild assumptions of the noises and the structure of network, consensus is analyzed under a fixed topology, dynamically switching topologies and randomly switching topologies, respectively. By combining algebraic graph theory and martingale convergence theorem, sufficient conditions for mean square and almost sure consensus are given. Further, when the consensus is achieved without a reference, it is shown that the consensus point is a random variable with its expectation being the average of the initial states of the agents and its variance being bounded. If the multi‐agent system has access to the state of the reference, the state of each agent can asymptotically converge to the reference. Numerical examples are given to illustrate the effectiveness of our results. Copyright © 2014 John Wiley & Sons, Ltd.     

Mean square average-consensus under measurement noises and fixed topologies: Necessary and sufficient conditions

In this paper, average-consensus control is considered for networks of continuous-time integrator agents under fixed and directed topologies. The control input of each agent can only use its local state and the states of its neighbors corrupted by white noises. To attenuate the measurement noises, time-varying consensus gains are introduced in the consensus protocol. By combining the tools of algebraic graph theory and stochastic analysis, the convergence of these kinds of protocols is analyzed. Firstly, for noise-free cases, necessary and sufficient conditions are given on the network topology and consensus gains to achieve average-consensus. Secondly, for the cases with measurement noises, necessary and sufficient conditions are given on the consensus gains to achieve asymptotic unbiased mean square average-consensus. It is shown that under the protocol designed, all agents’ states converge to a common Gaussian random variable, whose mathematical expectation is just the average of the initial states.     

Leader-following consensus for single-integrator multi-agent systems with multiplicative noises in directed topologies

This paper proposes a leader-following consensus control for continuous-time single-integrator multi-agent systems with multiplicative measurement noises under directed fixed and switching topologies. The consensus controller is developed by combining the graph theory and stochastic tools. The control input for each agent relies on its own state and its neighbours’ states corrupted by noises, the noises are considered proportional to the relative distance between agents, both of the noisy case and the noise-free case are studied, and conditions to achieve mean square convergence under noisy measurement and asymptotic convergence in absence of noises are derived. Finally, in order to prove the validity of the consensus control, some simulations were carried out.     

Sampled-data based average consensus with measurement noises: convergence analysis and uncertainty principle

In this paper, sampled-data based average-consensus control is considered for networks consisting of continuous-time first-order integrator agents in a noisy distributed communication environment. The impact of the sampling size and the number of network nodes on the system performances is analyzed. The control input of each agent can only use information measured at the sampling instants from its neighborhood rather than the complete continuous process, and the measurements of its neighbors’ states are corrupted by random noises. By probability limit theory and the property of graph Laplacian matrix, it is shown that for a connected network, the static mean square error between the individual state and the average of the initial states of all agents can be made arbitrarily small, provided the sampling size is sufficiently small. Furthermore, by properly choosing the consensus gains, almost sure consensus can be achieved. It is worth pointing out that an uncertainty principle of Gaussian networks is obtained, which implies that in the case of white Gaussian noises, no matter what the sampling size is, the product of the steady-state and transient performance indices is always equal to or larger than a constant depending on the noise intensity, network topology and the number of network nodes.     

Stochastic consensus of single‐integrator multi‐agent systems under relative state‐dependent measurement noises and time delays

This paper proposes a consensus algorithm for continuous‐time single‐integrator multi‐agent systems with relative state‐dependent measurement noises and time delays in directed fixed and switching topologies. Each agent's control input relies on its own information state and its neighbors' information states, which are delayed and corrupted by measurement noises whose intensities are considered a function of the agents' relative states. The time delays are considered time‐varying and uniform. For directed fixed topologies, condition to ensure mean square linear χ‐consensus (average consensus, respectively) are derived for digraphs having spanning tree (balanced digraphs having spanning tree, respectively). For directed switching topologies, conditions on both time delays and dwell time have been given to extend the mean square linear χ‐consensus (average consensus, respectively) of fixed topologies to switching topologies. Copyright © 2016 John Wiley & Sons, Ltd.     

Stochastic consensus of double-integrator leader-following multi-agent systems with measurement noises and time delays

This paper studies a leader-following consensus problem of continuous-time double-integrator multi-agent systems with measurement noises and time-varying communication delays under directed topology. By utilising the neighbour position and velocity information, which are delayed and disturbed by measurement noises whose intensities are considered a function related to the neighbour position and velocity of agents, a distributed consensus protocol is presented, sufficient conditions of the tracking consensus in the sense of mean square are derived. Finally, the effectiveness of the proposed consensus protocol is proved by some simulations.     

Consensus of linear multi‐agent systems with stochastic noises and binary‐valued communications

This article studies consensus problems of discrete‐time linear multi‐agent systems with stochastic noises and binary‐valued communications. Different from quantized consensus of first‐order systems with binary‐valued observations, the quantized consensus of linear multi‐agent systems requires that each agent observes its neighbors' states dynamically. Unlike the existing quantized consensus of linear multi‐agent systems, the information that each agent in this article gets from its neighbors is only binary‐valued. To estimate its neighbors' states dynamically by using the binary‐valued observations, we construct a two‐step estimation algorithm. Based on the estimates, a stochastic approximation‐based distributed control is proposed. The estimation and control are analyzed together in the closed‐loop system, since they are strongly coupled. Finally, it is proved that the estimates can converge to the true states in mean square sense and the states can achieve consensus at the same time by properly selecting the coefficient in the estimation algorithm. Moreover, the convergence rate of the estimation and the consensus speed are both given by O(1/t). The theoretical results are illustrated by simulations.     

Second-order consensus of multi-agent systems with noises via intermittent control

Second-order consensus of multi-agent systems with noises via intermittent control is investigated in this paper. First, we study the mean-square consensus problem with communication noises by intermittent control. In order to reach consensus, under the strong directed interacted topology, by using the tools of graph theory and Lyapunov method, a distributed control protocol is proposed based on the noises and periodical intermittent information. The upper bound of noise strength in the sense of matrix norm and the lower bound of communication time duration are obtained. Second, a class of coupled system models which include delay-terms in their nonlinearities in the noisy environment is discussed. Under the balanced strongly connected topology, the sufficient conditions to achieve the mean-square average-consensus are obtained. Finally, simulations are given to illustrate the effectiveness of our results.     

一种基于切换拓扑结构的一致性协议及其稳定性

本文研究了一种新的基于最近邻法则的分布式离散形式的一致性协议,该协议只参考两个邻域成员状态进行状态演变、系统网络拓扑为无向连接切换拓扑图。文中还研究了邻域部分信息选取的策略,给出了策略的具体算法,并设计了公共Lyapunov函数,进行了系统的稳定性分析,对影响协议收敛的调整因子进行分析和证明。最后给出了对应的仿真实验,仿真实验以11个移动智能体的汇聚为例,实验结果表明,在新的协议下,智能体的移动能够保持时时连通,该多智能体系统的汇聚过程能表现出全局渐近稳定的结果,同时系统演化过程当中所需的存储量、计算量大为减少。     

Consensus of double-integrator multi-agent systems without relative states derivative under relative-state dependent measurement noises*

This paper proposes a consensus protocol for continuous-time double-integrator multi-agent systems under noisy communication in directed topologies. Each agent’s control input relies on its own velocity and the relative positions with neighbours; it does not require the relative velocities. The agent receives its neighbours’ positions information corrupted by time-varying measurement noises whose intensities are proportional to the absolute relative distance that separates the agent from the neighbours. The consensus protocol is mainly based on the velocity damping gain to derive conditions under which the unbiased mean square χ-consensus is achieved in directed fixed topologies, and the unbiased mean square average consensus is achieved in directed switching topologies. The mean square state errors are quantified for both the positions and velocities. Finally, to illustrate the approach presented, some numerical simulations are performed.     

Synchronization of General Linear Multi‐Agent Systems With Measurement Noises

This paper studies the synchronization of general linear multi‐agent systems with measurement noises in mean square. It shows that the conventional consensus protocol is efficient and robust to the additive and multiplicative measurement noises in mean square. For the additive measurement noises which are independent of the relative‐states, it shows that the multi‐agent systems can achieve synchronization in practical mean square. For the multiplicative measurement noises which are dependent of the relative‐states, it shows that the multi‐agent systems can achieve synchronization in (strict) mean square. Furthermore, the new consensus protocol is better than the conventional one at some specific situations, i.e., the multi‐agent systems with additive measurement noises can also achieve synchronization in (strict) mean square. Numerical simulations are also provided and the results show highly consistent with the theoretical results.     

Quantised consensus by using the PD-like protocols in directed networks

This paper studies the distributed consensus problem in sampled-data multi-agent systems with directed network topologies subject to a quantisation constraint. Different from the widely used consensus protocol which exploits current information, we adopt a distributed proportional-differential (PD)-like protocol. First, we provide a necessary and sufficient condition of control gains which guarantee the consensus with the assumption that real-valued communication information can be obtained. Next, we analyse the quantisation effects in system performances under the proposed protocol; it is proved that the quantised consensus can be achieved. Finally, by providing numerical examples, we show that with appropriate parameters, the consensus can be achieved and the quantisation noises can be attenuated effectively.     

Multi-vehicle consensus with a time-varying reference state

In this paper, we study the consensus problem in multi-vehicle systems, where the information states of all vehicles approach a time-varying reference state under the condition that only a portion of the vehicles (e.g., the unique team leader) have access to the reference state and the portion of the vehicles might not have a directed path to all of the other vehicles in the team. We first analyze a consensus algorithm with a constant reference state using graph theoretical tools. We then propose consensus algorithms with a time-varying reference state and show necessary and sufficient conditions under which consensus is reached on the time-varying reference state. The time-varying reference state can be an exogenous signal or evolve according to a nonlinear model. These consensus algorithms are also extended to achieve relative state deviations among the vehicles. An application example to multi-vehicle formation control is given as a proof of concept.     

Distributed consensus for multi-agent systems with delays and noises in transmission channels

This paper studies the distributed consensus problem for linear discrete-time multi-agent systems with delays and noises in transmission channels. Due to the presence of noises and delays, existing techniques such as the lifting technique and the stochastic Lyapunov theory are no longer applicable to the analysis of consensus. In this paper, a novel technique is introduced to overcome the difficulties induced by the delays and noises. A consensus protocol with decaying gains satisfying persistence condition is adopted. Necessary and sufficient conditions for strong consensus and mean square consensus are respectively given for non-leader–follower and leader–follower cases under a fixed topology. Under dynamically switching topologies and randomly switching topologies, sufficient conditions for strong consensus and mean square consensus are also obtained. Numerical examples are given to demonstrate the effectiveness of the proposed protocols.     

Consensus control of second-order stochastic discrete-time multi-agent systems without velocity transmission

This article studies the almost-sure and the mean-square consensus control problems of second-order stochastic discretetime multi-agent systems with multiplicative noises. First, a control law based on the absolute velocity and relative position information is designed. Second, considering the existence of multiplicative noises and nonlinear terms with Lipschitz constants, the consensus control problem is solved through the use of a degenerated Lyapunov function. Then, for the linear second-order multi-agent systems, some explicit consensus conditions are provided. Finally, two sets of numerical simulations are performed.     

Leader-following consensus of double-integrator multi-agent systems with noisy measurements

This paper proposes a leader-following consensus control for continuous-time double-integrator multi-agent systems in noisy communication environment with a constant velocity reference state. Each follower in the team inaccurately measures its neighbors’ positions and the leader’s position if this follower has access to the leader, that the measured positions are corrupted by noises. The constant velocity of the leader is a priori well known. The consensus protocol is constructed based on algebraic graph theory and some stochastic tools. Conditions to ensure the tracking consensus in mean square are derived for both fixed and switching directed topologies. Finally, to illustrate the approach presented, some numerical simulations are carried out.     

Stochastic bounded consensus tracking of second-order multi-agent systems with measurement noises based on sampled-data with general sampling delay

This paper investigates the stochastic bounded consensus tracking problems of second-order multi-agent systems, where the control input of an agent can only use the information measured at the sampling instants from its neighbours or the virtual leader with a time-varying reference state, the measurements are corrupted by random noises and the signal sampling process induces the general sampling delay. First, the stochastic bounded consensus tracking protocol based on sampled-data with the general sampling delay is presented by using the delay decomposition technique. Second, the augmented matrix method, the probability limit theory and some other techniques are employed to derive the necessary and sufficient conditions guaranteeing the mean square bounded consensus tracking. The theoretical results show that the convergence of the proposed protocol simultaneously depends on the constant feedback gains, the network topology, the sampled period and the sampling delay, and that the static consensus tracking error depends on not only the above-mentioned factors, but also the noise intensity and the upper bound of the velocity and the acceleration of the virtual leader. The obtained results cover no sampling delay and the small sampling delay as two special cases. Simulations are provided to demonstrate the effectiveness of the theoretical results.     

Average Consensus for Multi‐Agent System with Measurement Noise and Binary‐Valued Communication

This paper considers the average consensus problems of a class of multi‐agent systems (MAS) with binary‐valued communication. Each agent can only obtain its neighbor's binary‐valued information under measurement noise because of limited bandwidth in communication channels. To seek consensus, we propose a two‐scale multi‐agent consensus algorithm with distributed strategy by combining state estimation and control design alternately. An exponential step size is chosen in the state estimation process and the estimation method can be proved to be asymptotically efficient. Additionally, by utilizing a distributed control law designed based on the estimates of the neighbors' states with a constant gain, we further prove that the proposed average consensus algorithm is convergent. Furthermore, the proposed average consensus algorithm is given and proved. Finally, some simulation results, which illustrate the effectiveness of the obtained results, are also given in the paper.     

Stochastic Bounded Consensus Tracking of Second-Order Multi-Agent Systems with Measurement Noises and Sampled-Data

This paper investigates the stochastic bounded consensus tracking problems of second-order multi-agent systems, where the control input of an agent can only use the information measured at the sampling instants from its neighbors or the virtual leader with a time-varying reference state, and the measurements are corrupted by random noises. The probability limit theory, the algebra graph theory, and some other techniques are employed to derive the necessary and sufficient condition guaranteeing the mean square bounded consensus tracking. It turns out that the maximum allowable sampling period depends on not only the network topology but also the constant feedback gains. Furthermore, the effects of the sampling period on tracking performance, including the tracking speed and the static tracking error, are also analyzed. The results show that reducing the sampling period can accelerate the tracking speed and decrease the static tracking error. Simulations are provided to demonstrate the effectiveness of the theoretical results.