Average Consensus Problem Research Articles

Multi-stage discrete time and randomized dynamic average consensus

In this paper we propose a novel local interaction protocol which solves the discrete time dynamic average consensus problem, i.e., the consensus problem on the average value of a set of time-varying input signals in an undirected graph. The proposed interaction protocol is based on a multi-stage cascade of dynamic consensus filters which tracks the average value of the inputs with small and bounded error. We characterize its convergence properties for time-varying discrete-time inputs with bounded variations. The main novelty of the proposed algorithm is that, with respect to other dynamic average consensus protocols, we obtain the next unique set of advantages: i) The protocol, inspired by proportional dynamic consensus, does not exploit integral control actions or input derivatives, thus exhibits robustness to re-initialization errors, changes in the network size and noise in the input signals; ii) The proposed design allows to trade-off the quantity of information locally exchanged by the agents, i.e., the number of stages, with steady-state error, tracking error and convergence time; iii) The protocol can be implemented with randomized and asynchronous local state updates and keep in expectation the performance of the discrete-time version. Numerical examples are given to corroborate the theoretical findings, including the case where a new agent joins and leaves the network during the algorithm execution to show robustness to re-initialization errors during runtime.

New Internal Model Average Consensus Estimators with Light Communication Load

The dynamic average consensus problem for a group of agents is considered. Each agent in the group is supposed to estimate the average of inputs applied to all agents and the estimation should be done in a distributed way. By reinterpreting the proportional integral type estimator, a new structure for the average estimator which can embed the internal model of inputs is proposed and conditions which result in the zero estimation error in the steady state are derived. We present constructive design procedures for the cases of constant inputs and time-varying inputs employing the root locus for the former and LQR-based design for the latter. The theory is validated through numerical simulations.

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

AbstractThis 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.

Tutorial on Dynamic Average Consensus: The Problem, Its Applications, and the Algorithms

Technological advances in ad hoc networking and the availability of low-cost reliable computing, data storage, and sensing devices have made scenarios possible where the coordination of many subsystems extends the range of human capabilities. Smart grid operations, smart transportation, smart health care, and sensing networks for environmental monitoring and exploration in hazardous situations are just a few examples of such network operations. In these applications, the ability of a network system to (in a decentralized fashion) fuse information, compute common estimates of unknown quantities, and agree on a common view of the world is critical. These problems can be formulated as agreement problems on linear combinations of dynamically changing reference signals or local parameters. This dynamic agreement problem corresponds to dynamic average consensus, which, as discussed in Summary, is the problem of interest of this article. The dynamic average consensus problem is for a group of agents to cooperate to track the average of locally available time-varying reference signals, where each agent is capable only of local computations and communicating with local neighbors.

Consensus Driven by the Geometric Mean

Consensus networks are usually understood as arithmetic mean driven dynamical averaging systems. In applications, however, network dynamics often describe inherently non-arithmetic and nonlinear consensus processes. In this paper, we propose and study three novel consensus protocols driven by geometric mean averaging: a polynomial, an entropic, and a scaling-invariant protocol, where terminology characterizes the particular nonlinearity appearing in the respective differential protocol equation. We prove exponential convergence to consensus for positive initial conditions. For the novel protocols, we highlight connections to applied network problems: The polynomial consensus system is structured like a system of chemical kinetics on a graph. The entropic consensus system converges to the weighted geometric mean of the initial condition, which is an immediate extension of the (weighted) average consensus problem. We find that all three protocols generate gradient flows of free energy on the simplex of constant mass distribution vectors albeit in different metrics. On this basis, we propose a novel variational characterization of the geometric mean as the solution of a nonlinear constrained optimization problem involving free energy as cost function. We illustrate our findings in numerical simulations.

Optimal Filter Design for Signal Processing on Random Graphs: Accelerated Consensus

In graph signal processing, filters arise from polynomials in shift matrices that respect the graph structure, such as the graph adjacency matrix or the graph Laplacian matrix. Hence, filter design for graph signal processing benefits from knowledge of the spectral decomposition of these matrices. Often, stochastic influences affect the network structure and, consequently, the shift matrix empirical spectral distribution. Although the joint distribution of the shift matrix eigenvalues is typically inaccessible, deterministic functions that asymptotically approximate the matrix empirical spectral distribution can be found for suitable random graph models using tools from random matrix theory. We employ this information regarding the density of eigenvalues to develop criteria for optimal graph filter design. In particular, we consider filter design for distributed average consensus and related problems, leading to improvements in short-term error minimization or in asymptotic convergence rate.

Adaptive output stationary average consensus for heterogeneous unknown linear multi‐agent systems

This study deals with the output stationary average consensus problem for heterogeneous linear multi-agent systems with unknown parameters under undirected or balanced directed networks. The output stationary average consensus means that the outputs of agents reach an agreement on the average of their initial outputs. Each agent is modelled as an unknown linear system of any order (which might extensively vary between the agents) with the relative degree one or two. The agent in the network has only access to its own output and the relative outputs of its neighbours. Based on a direct model reference adaptive control scheme, a novel decentralised adaptive output consensus protocol is proposed to ensure that the output of each agent converges to the output of its reference model, as well as reaching output stationary average consensus. Finally, an illustrative example is presented to better highlight the merit of the proposed protocol.

Average Consensus of Multi-Agent Systems Under Directed Topologies and Binary-Valued Communications

This paper is concerned with the average consensus problem of multi-agent systems with binary-valued communications under directed topologies. The information that each agent communicates with others is binary-valued with a fixed threshold and the control is a quantity of transportation from one agent to another, which causes that a decrease of the transmitter’s state leads to an increase of the same amount of the receiver’s. Due to the limitation of the information obtained, each agent needs to wait for a period of time to collect enough binary-valued information to estimate. Hence, we construct a two-scale control algorithm. At the small-time scale, each agent estimates its neighbors’ states based on the accumulation of binary-valued information for a period of time, during which the control is zero. At the large-time scale, each agent designs the control signal based on the estimation, which results in state updating of the multi-agent system. Then, each agent estimates the new states of its neighbors and the process of alternating estimation and control will be repeated. Finally, we prove that the estimation is convergent and give the mean square convergence rate as the reciprocal of the estimation time. More importantly, the multi-agent system is proved to achieve average consensus and the consensus speed is given. Simulation results are given to validate the algorithm.

Cartesian product-based hierarchical scheme for multi-agent systems

In this paper, we solve the average-consensus problem using a hierarchical scheme based on Cartesian product of strongly connected balanced graphs — an algebraic approach to design complex networks. We show that the Cartesian product based hierarchical scheme for multi-agent systems outperforms the single-layer control strategies for average-consensus problem in terms of convergence rate, and also the system matrix produced by Cartesian product-based hierarchical (CPH) scheme do not necessarily exhibit block circulant symmetry. We analyze that if the factors graphs in Cartesian product are cyclic pursuit graphs, then the CPH scheme provides the same convergence rate while requiring the same communication links as in the hierarchical cyclic pursuit (HCP). We provide simulation results to demonstrate the key theoretical results.

Stationary average consensus protocol for a class of heterogeneous high-order multi-agent systems with application for aircraft

ABSTRACTThis paper investigates the stationary average consensus problem for a class of heterogeneous-order multi-agent systems. The goal is to bring the positions of agents to the average of their initial positions while letting the other states converge to zero. To this end, three different consensus protocols are proposed. First, based on the auxiliary variables information among the agents under switching directed networks and state-feedback control, a protocol is proposed whereby all the agents achieve stationary average consensus. In the second and third protocols, by resorting to only measurements of relative positions of neighbouring agents under fixed balanced directed networks, two control frameworks are presented with two strategies based on state-feedback and output-feedback control. Finally, simulation results are given to illustrate the effectiveness of the proposed protocols.

Distributed Control of Active–Passive Networked Multiagent Systems

An active–passive networked multiagent system framework is introduced and analyzed, which consists of agents subject to exogenous inputs (active agents) and agents without any inputs (passive agents). Specifically, we propose an integral action-based distributed control approach and establish its transient time and steady-state performance characteristics using results from graph theory, matrix mathematics, Lyapunov stability, and ${\mathcal L}$ stability. Apart from the existing relevant literature, where either none of the agents are subject to exogenous inputs (i.e., average consensus problem) or all agents are subject to these inputs (i.e., dynamic average consensus problem), the key feature of our approach is that the states of all agents converge to the average of the exogenous inputs applied only to the active agents. We further discuss the conditions when the performance of the proposed distributed controller specializes to the performance of standard distributed controllers used for average consensus and dynamic average consensus of leaderless networks, and draw connections between pinning control and containment control of leader–follower networks. Several illustrative numerical examples are provided to demonstrate the efficacy of the proposed distributed control approach.

A new finite-time average consensus protocol with boundedness of convergence time for multi-robot systems

Multi-robot consensus has been extensively applied in robotics. In this study, a new protocol is proposed to solve the finite-time average consensus problem. The protocol can improve the convergence rate. The upper bound of the convergence time is obtained. Analysis shows that there exists a limit value of the convergence time when the disagreement of initial states tends to be infinitely large, and the value is irrelevant to the initial states. The relationship between convergence time and initial states, communication topology, parameter is analysed. Lastly, the effectiveness of the results is verified by simulations.

A local average broadcast gossip algorithm for fast global consensus over graphs

Motivated by applications to wireless sensor, peer-to-peer, and social networks, the canonical average consensus problem is considered in random and regular graphs in this paper. A local average information exchange (LAIE) algorithm is developed to compute the global consensus of the initial measurements of the nodes at every node in the network. In the proposed algorithm, each node interacts with all of its neighboring nodes in each round of the diffusion process to compute and exchange the local average value, such that all nodes can asymptotically reach a global consensus in a distributed manner very quickly. This is in contrast to the conventional random gossip scheme, where each node only interacts with one of its neighboring nodes, leading to very long convergence time. Results show that in a random graph with n nodes, the convergence time of the LAIE algorithm is bounded below by Ω(n−1)lognΔ,1 where the parameter Δ denotes the largest degree of the graphs. When a network has n nodes represented by d-regular topology graphs (d>2), where each node has the same number of neighbors d, the convergence time of the LAIE algorithm is bounded below by Θn(d+1)logn(2+d+2d−1)(d−2d−1). This shows that the proposed algorithms can achieve quicker convergence to the global consensus than other schemes based on the classic random gossip algorithm. Finally, we assess and compare the communication cost of the local average algorithm to achieve consensus through numerical results.

Output average consensus over heterogeneous multi-agent systems via two-level approach

In this paper, a novel two-level framework was proposed and applied to solve the output average consensus problem over heterogeneous multi-agent systems. This approach is mainly based on the recent technique of system abstraction. For given multi-agent systems, we first constructed their abstractions as the upper level and solved their average consensus problem by leveraging well-known results for single integrators. Then the control protocols for physical agents in the lower level were synthesized in a hierarchical way by embedding the designed law for abstractions into an interface between two levels. In this way, the complexity coming from heterogeneous dynamics of agents is totally decoupled from that of the coordination task and the communication topologies. An example was given to show its effectiveness.

Differentially private average consensus: Obstructions, trade-offs, and optimal algorithm design

This paper studies the multi-agent average consensus problem under the requirement of differential privacy of the agents’ initial states against an adversary that has access to all the messages. We first establish that a differentially private consensus algorithm cannot guarantee convergence of the agents’ states to the exact average in distribution, which in turn implies the same impossibility for other stronger notions of convergence. This result motivates our design of a novel differentially private Laplacian consensus algorithm in which agents linearly perturb their state-transition and message-generating functions with exponentially decaying Laplace noise. We prove that our algorithm converges almost surely to an unbiased estimate of the average of agents’ initial states, compute the exponential mean-square rate of convergence, and formally characterize its differential privacy properties. We show that the optimal choice of our design parameters (with respect to the variance of the convergence point around the exact average) corresponds to a one-shot perturbation of initial states and compare our design with various counterparts from the literature. Simulations illustrate our results.

Non-fragile consensus algorithms for a network of diffusion PDEs with boundary local interaction

ABSTRACTIn this study, non-fragile consensus algorithm is proposed to solve the average consensus problem of a network of diffusion PDEs, modelled by boundary controlled heat equations. The problem deals with the case where the Neumann-type boundary controllers are corrupted by additive persistent disturbances. To achieve consensus between agents, a linear local interaction rule addressing this requirement is given. The proposed local interaction rules are analysed by applying a Lyapunov-based approach. The multiplicative and additive non-fragile feedback control algorithms are designed and sufficient conditions for the consensus of the multi-agent systems are presented in terms of linear matrix inequalities, respectively. Simulation results are presented to support the effectiveness of the proposed algorithms.

Distributed Source Identification by Two-Hop Consensus Dynamics with Uniform Time-Varying Communication Time-Delays

This paper considers the average consensus problem by the 2-hop communication with time-varying time-delays. Each agent updates its state by a discrete-time consensus dynamics with fixed and undire...

Stationary average consensus for high‐order multi‐agent systems

The stationary average consensus problem is studied for high-order multi-agent systems (MASs) under balanced directed networks. The objective is to bring the positions of agents to the average of their initial positions while allowing all the remaining states (e.g., velocity, acceleration, and higher-order states) to converge to zero. To this end, the authors propose two consensus protocols for high-order MASs in two cases: (i) state-feedback control, which assumes that each agent has access to its own states as well as the relative positions of its neighbours; and (ii) output-feedback control, where each agent measures only its own position and the relative positions of its neighbours. Two case studies are given to illustrate the advantages and effectiveness of the proposed protocols. In the first case study, a state-feedback controller is designed for consensus-based formation control of a team of vertical take-off and landing aircraft. In the second case study, an output-feedback consensus protocol is designed for a third-order MAS and compare it with a recent technique from the literature.

Linear Time Average Consensus and Distributed Optimization on Fixed Graphs

We describe a protocol for the average consensus problem on any fixed undirected graph whose convergence time scales linearly in the total number nodes $n$. The protocol relies only on nearest-neig...

Event-triggered consensus for multi-agent networks with switching topology under quantized communication

The main purpose of this paper is to study average consensus problems in multi-agent networks with switching topologies under digital communication channels. A novel communication framework that can smoothly combine the quantization scheme and event-triggered strategy is developed, which apparently improves the communication efficiency of networks. In pursuit of saving the network resource and reducing the computational load of microprocessors embedded in all agents, a specifically designed distributed event-triggered strategy is proposed for well treatment with consensus problems in digital networks with switching topologies. Furthermore, we also prove that the average consensus of networks is asymptotically achieved, and the states evolution of all agents is well. Finally, some numerical simulations are given to illustrate the correctness of the theoretical results and the feasibility of presented protocols.