Showing papers on "Consensus published in 2009"

Distributed Consensus Algorithms in Sensor Networks With Imperfect Communication: Link Failures and Channel Noise

Abstract: The paper studies average consensus with random topologies (intermittent links) and noisy channels. Consensus with noise in the network links leads to the bias-variance dilemma-running consensus for long reduces the bias of the final average estimate but increases its variance. We present two different compromises to this tradeoff: the A-ND algorithm modifies conventional consensus by forcing the weights to satisfy a persistence condition (slowly decaying to zero;) and the A-NC algorithm where the weights are constant but consensus is run for a fixed number of iterations [^(iota)], then it is restarted and rerun for a total of [^(p)] runs, and at the end averages the final states of the [^(p)] runs (Monte Carlo averaging). We use controlled Markov processes and stochastic approximation arguments to prove almost sure convergence of A-ND to a finite consensus limit and compute explicitly the mean square error (mse) (variance) of the consensus limit. We show that A-ND represents the best of both worlds-zero bias and low variance-at the cost of a slow convergence rate; rescaling the weights balances the variance versus the rate of bias reduction (convergence rate). In contrast, A-NC, because of its constant weights, converges fast but presents a different bias-variance tradeoff. For the same number of iterations [^(iota)][^(p)] , shorter runs (smaller [^(iota)] ) lead to high bias but smaller variance (larger number [^(p)] of runs to average over.) For a static nonrandom network with Gaussian noise, we compute the optimal gain for A-NC to reach in the shortest number of iterations [^(iota)][^(p)] , with high probability (1-delta), (epsiv, delta)-consensus (epsiv residual bias). Our results hold under fairly general assumptions on the random link failures and communication noise.

Decentralized Robust Adaptive Control for the Multiagent System Consensus Problem Using Neural Networks

Abstract: A robust adaptive control approach is proposed to solve the consensus problem of multiagent systems. Compared with the previous work, the agent's dynamics includes the uncertainties and external disturbances, which is more practical in real-world applications. Due to the approximation capability of neural networks, the uncertain dynamics is compensated by the adaptive neural network scheme. The effects of the approximation error and external disturbances are counteracted by employing the robustness signal. The proposed algorithm is decentralized because the controller for each agent only utilizes the information of its neighbor agents. By the theoretical analysis, it is proved that the consensus error can be reduced as small as desired. The proposed method is then extended to two cases: agents form a prescribed formation, and agents have the higher order dynamics. Finally, simulation examples are given to demonstrate the satisfactory performance of the proposed method.

Convergence Speed in Distributed Consensus and Averaging

Abstract: We study the convergence speed of distributed iterative algorithms for the consensus and averaging problems, with emphasis on the latter. We first consider the case of a fixed communication topology. We show that a simple adaptation of a consensus algorithm leads to an averaging algorithm. We prove lower bounds on the worst-case convergence time for various classes of linear, time-invariant, distributed consensus methods, and provide an algorithm that essentially matches those lower bounds. We then consider the case of a time-varying topology, and provide a polynomial-time averaging algorithm.

Distributed Consensus Filtering in Sensor Networks

Abstract: In this paper, a new filtering problem for sensor networks is investigated. A new type of distributed consensus filters is designed, where each sensor can communicate with the neighboring sensors, and filtering can be performed in a distributed way. In the pinning control approach, only a small fraction of sensors need to measure the target information, with which the whole network can be controlled. Furthermore, pinning observers are designed in the case that the sensor can only observe partial target information. Simulation results are given to verify the designed distributed consensus filters.

Convergence speed in distributed consensus and averaging

Abstract: We study the convergence speed of distributed iterative algorithms for the consensus and averaging problems, with emphasis on the latter We first consider the case of a fixed communication topology We show that a simple adaptation of a consensus algorithm leads to an averaging algorithm We prove lower bounds on the worst-case convergence time for various classes of linear, time-invariant, distributed consensus methods, and provide an algorithm that essentially matches those lower bounds We then consider the case of a time-varying topology, and provide a polynomial-time averaging algorithm

Leader-following consensus problem with a varying-velocity leader and time-varying delays

Abstract: In this paper, we study a leader-following consensus problem for a multi-agent system with a varying-velocity leader and time-varying delays. Here, the interaction graph among the followers is switching and balanced. At first, we propose a neighbor-based rule for every agent to track a leader whose states may not be measured. In addition, we consider the convergence analysis of this multi-agent system under two different conditions: the connection between the followers and the leader is time-invariant and time-varying. For the first case, a novel decomposition method is introduced to facilitate the convergence analysis. By utilizing a Lyapunov–Krasovskii functional, we obtain sufficient conditions for uniformly ultimately boundedness of the tracking errors. Finally, two simulations are also presented to illustrate our theoretical results.

The Heard-Of model: computing in distributed systems with benign faults

Abstract: Problems in fault-tolerant distributed computing have been studied in a variety of models. These models are structured around two central ideas: (1) degree of synchrony and failure model are two independent parameters that determine a particular type of system, (2) the notion of faulty component is helpful and even necessary for the analysis of distributed computations when faults occur. In this work, we question these two basic principles of fault-tolerant distributed computing, and show that it is both possible and worthy to renounce them in the context of benign faults: we present a computational model based only on the notion of transmission faults. In this model, computations evolve in rounds, and messages missed in a round are lost. Only information transmission is represented: for each round r and each process p, our model provides the set of processes that p “hears of” at round r (heard-of set), namely the processes from which p receives some message at round r. The features of a specific system are thus captured as a whole, just by a predicate over the collection of heard-of sets. We show that our model handles benign failures, be they static or dynamic, permanent or transient, in a unified framework. We demonstrate how this approach leads to shorter and simpler proofs of important results (non-solvability, lower bounds). In particular, we prove that the Consensus problem cannot be generally solved without an implicit and permanent consensus on heard-of sets. We also examine Consensus algorithms in our model. In light of this specific agreement problem, we show how our approach allows us to devise new interesting solutions.

Consensus Optimization on Manifolds

Abstract: The present paper considers distributed consensus algorithms that involve $N$ agents evolving on a connected compact homogeneous manifold. The agents track no external reference and communicate their relative state according to a communication graph. The consensus problem is formulated in terms of the extrema of a cost function. This leads to efficient gradient algorithms to synchronize (i.e., maximizing the consensus) or balance (i.e., minimizing the consensus) the agents; a convenient adaptation of the gradient algorithms is used when the communication graph is directed and time-varying. The cost function is linked to a specific centroid definition on manifolds, introduced here as the induced arithmetic mean, that is easily computable in closed form and may be of independent interest for a number of manifolds. The special orthogonal group $SO(n)$ and the Grassmann manifold $\text{{\it Grass\/}}(p,n)$ are treated as original examples. A link is also drawn with the many existing results on the circle.

Consensus over directed static networks with arbitrary finite communication delays

Abstract: We study the consensus problem in directed static networks with arbitrary finite communication delays and consider both linear and nonlinear coupling. For the considered networked system, only locally delayed information is available for each node and also the information flow is directed. We find that consensus can be realized whatever the communications delays are. In fact, we do not even need to know the explicit values of the communication delays. One well-informed leader is proved to be enough for the regulation of all nodes' final states, even when the external signal is very weak. Numerical simulations for opinion formation in small-world and scale-free networks are given to demonstrate the potentials of our analytic results.

Consensus of multi-agent systems based on sampled-data control

Abstract: This article studies the consensus problem in directed networks, assuming that each agent is with double-integrator dynamics and only obtains the measurements of its positions relative to its neighbours at sampling instants. We propose a protocol based on sampled-data control and derive an equivalent characterisation of the solvability of the consensus problem under this protocol. In virtue of this equivalent characterisation, we further consider two cases: fixed topology and switching topology. For the first case, we present a set of sampling periods and feedback coefficients which ensure that the protocol can solve a consensus problem. For the second case, we derive sufficient conditions for the protocol to solve a consensus problem under arbitrary switching signals and under a class of switching signals, respectively. Finally, simulations are provided to illustrate the effectiveness of the theoretical results.

Using Three States for Binary Consensus on Complete Graphs

Abstract: We consider the binary consensus problem where each node in the network initially observes one of two states and the goal for each node is to eventually decide which one of the two states was initially held by the majority of the nodes. Each node contacts other nodes and updates its current state based on the state communicated by the last contacted node. We assume that both signaling (the information exchanged at node contacts) and memory (computation state at each node) are limited and restrict our attention to systems where each node can contact any other node (i.e., complete graphs). It is well known that for systems with binary signaling and memory, the probability of reaching incorrect consensus is equal to the fraction of nodes that initially held the minority state. We show that extending both the signaling and memory by just one state dramatically improves the reliability and speed of reaching the correct consensus. Specifically, we show that the probability of error decays exponentially with the number of nodes N and the convergence time is logarithmic in N for large N. We also examine the case when the state is ternary and signaling is binary. The convergence of this system to consensus is again shown to be logarithmic in N for large N, and is therefore faster than purely binary systems. The type of distributed consensus problems that we study arises in the context of decentralized peer-to-peer networks, e.g. sensor networks and opinion formation in social networks - our results suggest that robust and efficient protocols can be built with rather limited signaling and memory.

Group consensus of multi-agent systems with undirected communication graphs

Abstract: We investigate a new consensus problem in networks of dynamic agents, where the agents in a network can reach more than one consistent values asymptotically. It contains such consensus problem as a special case that all agents in a network reach a consistent value asymptotically. When information exchange is undirected, a novel consensus protocol is designed to solve the group consensus problem. The convergence analysis is discussed and several criterions are established based on graph theories and matrix theories. Simulation results are presented to demonstrate the effectiveness of the theoretical results.

Consensus Problems with Distributed Delays, with Application to Traffic Flow Models

Abstract: This paper focuses on consensus problems for a class of linear systems with distributed delay that are encountered in modeling traffic flow dynamics. In the application problems the distributed delay, whose kernel is a $\gamma$-distribution with a gap, represents the human drivers' behavior in the average. The aim of the paper is to give a characterization of the regions in the corresponding delay parameter space, where a consensus is reached for all initial conditions. The structure and properties of the system are fully exploited, which leads to explicit and computationally tractable expressions. As a by-product a stability theory for distributed delay systems with a $\gamma$-distribution kernel is developed. Also explicit expressions for the consensus function(al) of time-delay systems with constant and distributed delays that solve a consensus problem are provided. Several illustrative examples complete the presentation.

Polynomial Filtering for Fast Convergence in Distributed Consensus

Abstract: In the past few years, the problem of distributed consensus has received a lot of attention, particularly in the framework of ad hoc sensor networks. Most methods proposed in the literature address the consensus averaging problem by distributed linear iterative algorithms, with asymptotic convergence of the consensus solution. The convergence rate of such distributed algorithms typically depends on the network topology and the weights given to the edges between neighboring sensors, as described by the network matrix. In this paper, we propose to accelerate the convergence rate for given network matrices by the use of polynomial filtering algorithms. The main idea of the proposed methodology is to apply a polynomial filter on the network matrix that will shape its spectrum in order to increase the convergence rate. Such an algorithm is equivalent to periodic updates in each of the sensors by aggregating a few of its previous estimates. We formulate the computation of the coefficients of the optimal polynomial as a semidefinite program that can be efficiently and globally solved for both static and dynamic network topologies. We finally provide simulation results that demonstrate the effectiveness of the proposed solutions in accelerating the convergence of distributed consensus averaging problems.

Distributed linear estimation over sensor networks

Abstract: We consider a network of sensors in which each node may collect noisy linear measurements of some unknown parameter. In this context, we study a distributed consensus diffusion scheme that relies only on bidirectional communication among neighbour nodes (nodes that can communicate and exchange data), and allows every node to compute an estimate of the unknown parameter that asymptotically converges to the true parameter. At each time iteration, a measurement update and a spatial diffusion phase are performed across the network, and a local least-squares estimate is computed at each node. The proposed scheme allows one to consider networks with dynamically changing communication topology, and it is robust to unreliable communication links and failures in measuring nodes. We show that under suitable hypotheses all the local estimates converge to the true parameter value.

Further results on decentralised coordination in networks of agents with second-order dynamics

Abstract: This study investigates consensus problems for networks of second-order agents, where each agent can only access the relative position information from its neighbours. We first introduce two new protocols with and without time-delay. Then we provide a convergence analysis in three cases: (a) networks with fixed topology; (b) networks with switching topology; (c) networks with switching topology and time-delays. Several conditions are presented to make all agents asymptotically reach consensus while accomplishing some tasks such as moving to a common value and moving together with a constant velocity or with a constant acceleration. Finally, simulation results are provided to demonstrate the effectiveness of our theoretical results.

A Cooperative Spectrum Sensing Consensus Scheme in Cognitive Radios

Abstract: Cooperative spectrum sensing is attracting more attention in cognitive radio networks. This paper proposes a fully distributed consensus-based cooperative spectrum sensing scheme to cope with both fixed and random bidirectional connections among secondary users. In the proposed scheme, secondary users can maintain coordination based on only local interactions without a centralized common receiver. Moreover, we prove that even inter-connected with unreliable bidirectional communication links, secondary users can still make an average consensus. Simulation results show that the proposed scheme can have significantly lower missing detection probability and false alarm probability.

Leader-follower consensus of multi-agent systems

Abstract: This paper addresses the leader-follower consensus problem of multi-agent systems with a time-invariant communication topology consisting of general linear node dynamics. A distributed observer-type consensus protocol based on relative output measurements is proposed. It is shown that the leader-follower consensus of multi-agent systems can be cast equivalently into the stability of a set of matrices of the same dimension as a single agent. The notion of consensus region is then introduced and analyzed by using tools from the stability of matrix pencils. It is further demonstrated that there exists an observer-type protocol that solves the leader-follower consensus problem, and meanwhile yields an unbounded consensus region, if and only if the agent dynamics are stabilizable and detectable. A multi-step consensus protocol design procedure is finally presented. The effectiveness of the theoretical results is demonstrated through numerical simulations.

Convergence Speed of Unsteady Distributed Consensus: Decay Estimate Along the Settling Spanning-Trees

Abstract: Results for estimating the convergence rate of nonstationary distributed consensus algorithms are provided, on the basis of qualitative (mainly topological) as well as basic quantitative information (lower-bounds on the matrix entries). The results appear to be tight in a number of instances and are illustrated through simple as well as more sophisticated examples. The main idea is to follow propagation of information along certain spanning-trees which arise in the communication graph.

Self-adapting modular robotics: A generalized distributed consensus framework

Abstract: Biological systems achieve amazing adaptive behavior with local agents performing simple sensing and actions. Modular robots with similar properties can potentially achieve self-adaptation tasks robustly. Inspired by this principle, we present a generalized distributed consensus framework for self-adaptation tasks in modular robotics. We demonstrate that a variety of modular robotic systems and tasks can be formulated within such a framework, including (1) an adaptive column that can adapt to external force, (2) a modular gripper that can manipulate fragile objects, and (3) a modular tetrahedral robot that can locomote towards a light source. We also show that control algorithms derived from this framework are provably correct. In real robot experiments, we demonstrate that such a control scheme is robust towards real world sensing and actuation noise. This framework can potentially be applied to a wide range of distributed robotics applications.

Consensus in leaderless networks of high-order-integrator agents

Abstract: This paper investigates the consensus problem for a group of high-order-integrator agents with fixed topology. A linear distributed consensus protocol is proposed, which only depends on the agent's own information and its neighbors' partial information. A necessary and sufficient condition for convergence to consensus is established. It is proved that the topology having a spanning tree is a necessary condition for convergence to consensus. Based on the consensus protocol for networks of high-order-integrator agents, a consensus controller is provided for a group of identical agents with dynamics described by a completely controllable single-input linear time-invariant (LTI) system. It is shown that the consensus of this kind of networks is equivalent to that of networks of high-order-integrator agents. Finally, the parameter design of the protocol is discussed.

Distributed consensus over network with noisy links

Abstract: We consider a distributed consensus problem where a set of agents want to agree on a common value through local computations and communications. We assume that agents communicate over a network with time-varying topology and noisy communication links. We are interested in the case when the link noise is independent in time, and it has zero mean and bounded variance. We present and study an iterative algorithm with a diminishing stepsize. We show that the algorithm converges in expectation and almost surely to a “random” consensus, and we characterize the statistics of the consensus. In particular, we give the expected value of the consensus and provide an upper bound on its variance.

Consensus of high order linear multi-agent systems using output error feedback

Abstract: In this paper, we study the consensus problem of multi-agent systems in which each agent adopts the same linear model that can be of any order. We consider the case where only the relative output error between the neighboring agents can be measured. In order to solve the consensus problem, two kinds of decentralized control laws are designed. We first show that a static output error feedback control can solve the consensus problem if some further constraints on the system model is imposed. Then we use an observer based approach to design a dynamic output error feedback consensus control. We note that with the observer based approach, some information exchange between the agents is needed unless the associated adjacent graph is completely connected.