Showing papers on "Consensus published in 2011"
TL;DR: This paper discusses the finite-time consensus problem for leaderless and leader-follower multi-agent systems with external disturbances, and proposes continuous distributed control algorithms designed for these agents described by double integrators.
816 citations
TL;DR: It is shown that the output consensus is reached if the (state) consensus is achieved within the internal models among the agent's controllers (even though the plant's outputs, rather than the internal model's outputs) are communicated.
Abstract: This technical note studies the output consensus problem for a class of heterogeneous uncertain linear multi-agent systems. All the agents can be of any order (which might widely differ among the agents) and possess parametric uncertainties that range over an arbitrarily large compact set. The controller uses only the output information of the plant; moreover, the delivered information throughout the communication network is also restricted to the output of each agent. Based on the output regulation theory, it is shown that the output consensus is reached if the (state) consensus is achieved within the internal models among the agent's controllers (even though the plant's outputs, rather than the internal model's outputs, are communicated). The internal models can be designed and embedded into the controller, which provides considerable flexibility to designers in terms of the type of signals that are agreed on among the agents.
629 citations
TL;DR: In this article, a necessary and sufficient condition for consensusability under a common control protocol is given, which explicitly reveals how the intrinsic entropy rate of the agent dynamic and the communication graph jointly affect consensusability.
Abstract: This paper investigates the joint effect of agent dynamic, network topology and communication data rate on consensusability of linear discrete-time multi-agent systems. Neglecting the finite communication data rate constraint and under undirected graphs, a necessary and sufficient condition for consensusability under a common control protocol is given, which explicitly reveals how the intrinsic entropy rate of the agent dynamic and the communication graph jointly affect consensusability. The result is established by solving a discrete-time simultaneous stabilization problem. A lower bound of the optimal convergence rate to consensus, which is shown to be tight for some special cases, is provided as well. Moreover, a necessary and sufficient condition for formationability of multi-agent systems is obtained. As a special case, the discrete-time second-order consensus is discussed where an optimal control gain is designed to achieve the fastest convergence. The effects of undirected graphs on consensability/formationability and optimal convergence rate are exactly quantified by the ratio of the second smallest to the largest eigenvalues of the graph Laplacian matrix. An extension to directed graphs is also made. The consensus problem under a finite communication data rate is finally investigated.
537 citations
TL;DR: It is proved that under the protocol designed, for a connected network, average consensus can be achieved with an exponential convergence rate based on merely one bit information exchange between each pair of adjacent agents at each time step.
Abstract: Communication data rate and energy constraints are important factors which have to be considered when investigating distributed coordination of multi-agent networks. Although many proposed average-consensus protocols are available, a fundamental theoretic problem remains open, namely, how many bits of information are necessary for each pair of adjacent agents to exchange at each time step to ensure average consensus? In this paper, we consider average-consensus control of undirected networks of discrete-time first-order agents under communication constraints. Each agent has a real-valued state but can only exchange symbolic data with its neighbors. A distributed protocol is proposed based on dynamic encoding and decoding. It is proved that under the protocol designed, for a connected network, average consensus can be achieved with an exponential convergence rate based on merely one bit information exchange between each pair of adjacent agents at each time step. An explicit form of the asymptotic convergence rate is given. It is shown that as the number of agents increases, the asymptotic convergence rate is related to the scale of the network, the number of quantization levels and the ratio of the second smallest eigenvalue to the largest eigenvalue of the Laplacian of the communication graph. We also give a performance index to characterize the total communication energy to achieve average consensus and show that the minimization of the communication energy leads to a tradeoff between the convergence rate and the number of quantization levels.
504 citations
TL;DR: This technical note studies the consensus problem for cooperative agents with nonlinear dynamics in a directed network through a combination of the tools of complex analysis, local consensus manifold approach, and Lyapunov methods.
Abstract: This technical note studies the consensus problem for cooperative agents with nonlinear dynamics in a directed network. Both local and global consensus are defined and investigated. Techniques for studying the synchronization in such complex networks are exploited to establish various sufficient conditions for reaching consensus. The local consensus problem is first studied via a combination of the tools of complex analysis, local consensus manifold approach, and Lyapunov methods. A generalized algebraic connectivity is then proposed to study the global consensus problem in strongly connected networks and also in a broad class of networks containing spanning trees, for which ideas from algebraic graph theory, matrix theory, and Lyapunov methods are utilized.
379 citations
TL;DR: This paper studies two problems which often occur in various applications arising in wireless sensor networks, and provides a diminishing step size algorithm which guarantees asymptotic convergence of the consensus problem and the problem of cooperative solution to a convex optimization problem.
Abstract: In this paper, we study two problems which often occur in various applications arising in wireless sensor networks. These are the problem of reaching an agreement on the value of local variables in a network of computational agents and the problem of cooperative solution to a convex optimization problem, where the objective function is the aggregate sum of local convex objective functions. We incorporate the presence of a random communication graph between the agents in our model as a more realistic abstraction of the gossip and broadcast communication protocols of a wireless network. An added ingredient is the presence of local constraint sets to which the local variables of each agent is constrained. Our model allows for the objective functions to be nondifferentiable and accommodates the presence of noisy communication links and subgradient errors. For the consensus problem we provide a diminishing step size algorithm which guarantees asymptotic convergence. The distributed optimization algorithm uses two diminishing step size sequences to account for communication noise and subgradient errors. We establish conditions on these step sizes under which we can achieve the dual task of reaching consensus and convergence to the optimal set with probability one. In both cases we consider the constant step size behavior of the algorithm and establish asymptotic error bounds.
366 citations
TL;DR: In this paper, the consensus problem of heterogeneous multi-agent systems is considered and sufficient conditions for consensus are established when the communication topologies are undirected connected graphs and leader-following networks.
Abstract: In this study, the consensus problem of heterogeneous multi-agent system is considered. First, the heterogeneous multi-agent system is proposed which is composed of first-order and second-order integrator agents in two aspects. Then, the consensus problem of heterogeneous multi-agent system is discussed with the linear consensus protocol and the saturated consensus protocol, respectively. By applying the graph theory and Lyapunov direct method, some sufficient conditions for consensus are established when the communication topologies are undirected connected graphs and leader-following networks. Finally, some examples are presented to illustrate the theoretical results.
293 citations
TL;DR: A leader-follower control problem in multiagent dynamical systems is considered, which reveals that to reach consensus the agents with very small degrees must be informed.
Abstract: This paper studies general higher order distributed consensus protocols in multiagent dynamical systems. First, network synchronization is investigated, with some necessary and sufficient conditions derived for higher order consensus. It is found that consensus can be reached if and only if all subsystems are asymptotically stable. Based on this result, consensus regions are characterized. It is proved that for the m th-order consensus, there are at most ⌊(m+1)/2⌋ disconnected stable and unstable consensus regions. It is shown that consensus can be achieved if and only if all the nonzero eigenvalues of the Laplacian matrix lie in the stable consensus regions. Moreover, the ratio of the largest to the smallest nonzero eigenvalues of the Laplacian matrix plays a key role in reaching consensus and a scheme for choosing the coupling strength is derived. Furthermore, a leader-follower control problem in multiagent dynamical systems is considered, which reveals that to reach consensus the agents with very small degrees must be informed. Finally, simulation examples are given to illustrate the theoretical analysis.
272 citations
TL;DR: The conditions that guarantee the finite-time consensus for the systems are identified and the signum protocol does not require explicit measurement of time signals from neighbors, and hence has the potential to significantly reduce the requirements for both computation and sensing.
248 citations
TL;DR: A novel technique is introduced to overcome the difficulties induced by the delays and noises in transmission channels and a consensus protocol with decaying gains satisfying persistence condition is adopted.
227 citations
TL;DR: A linear consensus protocol is proposed for solving such a consensus problem, which includes two parts: a feedback controller and interactions from the neighbours, and a sufficient and necessary condition for consensus in high-order systems is obtained.
Abstract: In this study, the lth order (l ≥ 2) consensus problem for multi-agent systems is considered, which generalises the existing second-order consensus algorithm. A linear consensus protocol is proposed for solving such a consensus problem, which includes two parts: a feedback controller and interactions from the neighbours. A sufficient and necessary condition for consensus in high-order systems is obtained. As special cases, criteria for second- and third-order systems are given, in which the exact relationship between feedback gain and system parameters is established. Finally, numerical simulations are reported to illustrate the effectiveness of this protocol.
TL;DR: It is shown that a simple adaptation of a consensus algorithm leads to an averaging algorithm, and lower bounds on the worst-case convergence time for various classes of linear, time-invariant, distributed consensus methods are proved.
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.
15 May 2011
TL;DR: The derivation of the optimal filter is based on the use of minimum principle of Pontryagin coupled with the Lagrange multiplier method and the results of generalized inverse of matrices for type-II sensors.
Abstract: This paper is concerned with the problem of filter design for target tracking over sensor networks. Different from most existing works on sensor networks, we consider the heterogeneous sensor networks with two types of sensors different on processing abilities (denoted as type-I and type-II sensors, respectively). However, questions of how to deal with the heterogeneity of sensors and how to design a filter for target tracking over such kind of networks remain largely unexplored. We propose in this paper a novel distributed consensus filter to solve the target tracking problem. Two criteria, namely, unbiasedness and optimality, are imposed for the filter design. The so-called sequential design scheme is then presented to tackle the heterogeneity of sensors. The minimum principle of Pontryagin is adopted for type-I sensors to optimize the estimation errors. As for type-II sensors, the Lagrange multiplier method coupled with the generalized inverse of matrices is then used for filter optimization. Furthermore, it is proven that convergence property is guaranteed for the proposed consensus filter in the presence of process and measurement noise. Simulation results have validated the performance of the proposed filter. It is also demonstrated that the heterogeneous sensor networks with the proposed filter outperform the homogenous counterparts in light of reduction in the network cost, with slight degradation of estimation performance.
TL;DR: It is proved that if the network is jointly connected, average-consensus can be asymptotically achieved, and the convergence rate is quantified, and if the duration of any link failure in thenetwork is bounded, then the control gain and the scaling function can be selected properly such that 5-level quantizers suffice for asymPTotic average- Consensus with an exponential convergence rate.
TL;DR: This investigation is carried out for both general consensus and average consensus; for each case, a class of algorithms is proposed, under which a necessary and sufficient graphical condition is derived to guarantee the corresponding consensus.
Abstract: We study distributed consensus problems of multi-agent systems on directed networks and subject to quantized information flow. For the communication among component agents, particular attention is given to the gossip type, which models their asynchronous behavior; for quantization effect, each agent's state is abstracted to be an integer. The central question investigated is how to design distributed algorithms and what connectivity of networks that together lead to consensus. This investigation is carried out for both general consensus and average consensus; for each case, a class of algorithms is proposed, under which a necessary and sufficient graphical condition is derived to guarantee the corresponding consensus. In particular, the obtained graphical condition ensuring average consensus is weaker than those in the literature for either real-valued or quantized states, in the sense that it does not require symmetric or balanced network topologies.
TL;DR: Pairwise Equalizing is developed, a gossip-style, distributed asynchronous iterative algorithm for achieving unconstrained, separable, convex consensus optimization over undirected networks with time-varying topologies, where each component function is strictly convex, continuously differentiable, and has a minimizer.
Abstract: In many applications, nodes in a network desire not only a consensus, but an optimal one. To date, a family of subgradient algorithms have been proposed to solve this problem under general convexity assumptions. This technical note shows that, for the scalar case and by assuming a bit more, novel non-gradient-based algorithms with appealing features can be constructed. Specifically, we develop Pairwise Equalizing (PE) and Pairwise Bisectioning (PB), two gossip algorithms that solve unconstrained, separable, convex consensus optimization problems over undirected networks with time-varying topologies, where each local function is strictly convex, continuously differentiable, and has a minimizer. We show that PE and PB are easy to implement, bypass limitations of the subgradient algorithms, and produce switched, nonlinear, networked dynamical systems that admit a common Lyapunov function and asymptotically converge. Moreover, PE generalizes the well-known Pairwise Averaging and Randomized Gossip Algorithm, while PB relaxes a requirement of PE, allowing nodes to never share their local functions.
TL;DR: This paper investigates the consensus problem in a multi-agent system with random delays governed by a Markov chain and converts the original system into a reduced-order one featuring the error dynamics, which is transformed into the stabilization of the error dynamic system.
TL;DR: Distributed adaptive/robust control laws are proposed such that the state of each system asymptotically converges to the desired trajectory with the aid of information interchange between systems.
TL;DR: Using tools from differential equations and stochastic calculus, together with results from matrix theory and algebraic graph theory, it is established that sufficient conditions under which the proposed consensus protocols lead to mean square average-consensus are established.
TL;DR: It is proved that under the proposed control protocol, the modified consensus problem can be solved if and only if the system matrices of the agent's dynamics are stabilizable and detectable, the input matrix is not a zero matrix, and the communication topology graph has a spanning tree.
04 Jun 2011
TL;DR: In this paper, the stabilizing consensus problem is studied and a simple randomized algorithm called median rule is proposed that, with high probability, just needs O(log m log log n + log n) time and work per process to arrive at an almost stable consensus for any set of m legal values as long as an adversary can corrupt the states of at most n processes at any time.
Abstract: In the standard consensus problem there are n processes with possibly different input values and the goal is to eventually reach a point at which all processes commit to exactly one of these values. We are studying a slight variant of the consensus problem called the stabilizing consensus problem [2]. In this problem, we do not require that each process commits to a final value at some point, but that eventually they arrive at a common, stable value without necessarily being aware of that. This should work irrespective of the states in which the processes are starting. Our main result is a simple randomized algorithm called median rule that, with high probability, just needs O(log m log log n + log n) time and work per process to arrive at an almost stable consensus for any set of m legal values as long as an adversary can corrupt the states of at most √n processes at any time. Without adversarial involvement, just O(log n) time and work is needed for a stable consensus, with high probability. As a by-product, we obtain a simple distributed algorithm for approximating the median of n numbers in time O(log m log log n + log n) under adversarial presence.
TL;DR: A distributed consensus protocol is proposed based on dynamic encoding and decoding and it is shown that for a connected network, as long as the time delays are bounded, the average consensus can be achieved with a finite-level quantizer.
Abstract: This paper considers the average consensus problem for multiagent networks with communication delays and limited data rate. On one hand, communication delays often exist in information acquisition and transmission; on the other hand, only limited state information of agents can be transmitted to their neighbors at each time step due to bandwidth constraints. The average consensus problem becomes much more complicated when both delays and data-rate constraints are to be considered. In this paper, a distributed consensus protocol is proposed based on dynamic encoding and decoding. It is shown that for a connected network, as long as the time delays are bounded, the average consensus can be achieved with a finite communication data rate. In particular, it is shown that merely a one-bit information exchange between each pair of adjacent agents at each time step suffices to guarantee the average consensus.
01 Sep 2011
TL;DR: Two ways to model delays are proposed, assuming each edge of a communication network has a fixed delay and a novel way to model random delays per edge is proposed.
Abstract: We study the effects of communication delays in distributed consensus and optimization algorithms. We propose two ways to model delays. First, assuming each edge of a communication network has a fixed delay, we characterize the consensus value exactly as a function of the delays and edge weights and obtain convergence rate bounds using results from non-reversible Markov chains. Second, we propose a novel way to model random delays per edge. Our model allows the reception of multiple delayed messages from the same sender in the same time slot, a situation that can happen in practice. Both models admit a description of the consensus updates in the presence of delays via linear equations. Finally, we briefly discuss how to apply our delay models to analyze distributed optimization algorithms in the presence of delayed information.
12 Apr 2011
TL;DR: The Adversarially Robust Consensus Protocol (ARC-P) is presented, which combines ideas from consensus algorithms that are resilient to Byzantine faults and from linear consensus protocols used for control and coordination of dynamic agents and solves the consensus problem in complete networks whenever there are more cooperative agents than adversaries.
Abstract: In the past decade, numerous consensus protocols for networked multi-agent systems have been proposed. Although some forms of robustness of these algorithms have been studied, reaching consensus securely in networked multi-agent systems, in spite of intrusions caused by malicious agents, or adversaries, has been largely underexplored. In this work, we consider a general model for adversaries in Euclidean space and introduce a consensus problem for networked multi-agent systems similar to the Byzantine consensus problem in distributed computing. We present the Adversarially Robust Consensus Protocol (ARC-P), which combines ideas from consensus algorithms that are resilient to Byzantine faults and from linear consensus protocols used for control and coordination of dynamic agents. We show that ARC-P solves the consensus problem in complete networks whenever there are more cooperative agents than adversaries. Finally, we illustrate the resilience of ARC-P to adversaries through simulations and compare ARC-P with a linear consensus protocol for networked multi-agent systems.
TL;DR: A distributed coordination algorithm based on sampled-data control is proposed to track the considered leader by employing M -matrix theory, and sufficient conditions on the sampling period and control parameters are derived to ensure that the tracking errors are bounded.
Abstract: In this article, the authors study a leader-following consensus problem for multi-agent systems in a sampled-data setting. A distributed coordination algorithm based on sampled-data control is proposed to track the considered leader. By employing M -matrix theory, the authors derive sufficient conditions on the sampling period and control parameters to ensure that the tracking errors are bounded. Numerical simulations are presented to illustrate the effectiveness of the theoretical results. Moreover, some previous results concerning the leader-following problem with switched coupling topology are improved.
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TL;DR: Zero-Gradient-Sum (ZGS) algorithms as discussed by the authors use a Lyapunov function candidate that exploits convexity to solve unconstrained, separable, convex optimization problems over undirected networks.
Abstract: This paper presents a set of continuous-time distributed algorithms that solve unconstrained, separable, convex optimization problems over undirected networks with fixed topologies. The algorithms are developed using a Lyapunov function candidate that exploits convexity, and are called Zero-Gradient-Sum (ZGS) algorithms as they yield nonlinear networked dynamical systems that evolve invariantly on a zero-gradient-sum manifold and converge asymptotically to the unknown optimizer. We also describe a systematic way to construct ZGS algorithms, show that a subset of them actually converge exponentially, and obtain lower and upper bounds on their convergence rates in terms of the network topologies, problem characteristics, and algorithm parameters, including the algebraic connectivity, Laplacian spectral radius, and function curvatures. The findings of this paper may be regarded as a natural generalization of several well-known algorithms and results for distributed consensus, to distributed convex optimization.
TL;DR: This network construction models the perceptual phenomenon of numerosity observed in animal groups exhibiting collective behavior and derives a closed form expression for the asymptotic convergence factor.
Abstract: We analyze the discrete-time consensus problem for a group of agents that communicate through a stochastic directed network with fixed out-degree. This network construction models the perceptual phenomenon of numerosity observed in animal groups exhibiting collective behavior. We find necessary and sufficient conditions for mean square consentability of the averaging protocol and we derive a closed form expression for the asymptotic convergence factor. Analytical results are illustrated through simulations.
TL;DR: This paper surveys Byzantine consensus in message-passing distributed systems by presenting the main theoretical results in the area, the main classes of algorithms and by discussing important issues like the performance and resilience of these algorithms.
Abstract: Consensus is a classical distributed systems problem with both theoretical and practical interest. Asynchronous Byzantine consensus is currently at the core of some solutions for the implementation of highly-resilient computing services. This paper surveys Byzantine consensus in message-passing distributed systems, by presenting the main theoretical results in the area, the main classes of algorithms and by discussing important issues like the performance and resilience of these algorithms.
TL;DR: Using the ordinary differential equation method, it is proved that the updates of a particular class of distributed consensus algorithms based on damped updates converge almost surely to the consensus average for various models of perturbation of data exchanged between nodes.
Abstract: This paper focuses on the consensus averaging problem on graphs under general imperfect communications. We study a particular class of distributed consensus algorithms based on damped updates, and using the ordinary differential equation method, we prove that the updates converge almost surely to the consensus average for various models of perturbation of data exchanged between nodes. The convergence is not asymptotic in the size of the network. Our analysis applies to various types of stochastic disturbances to the updates, including errors in update calculations, dithered quantization and imperfect data exchange among nodes. Under a suitable stability condition, we prove that the error between estimated and true averages is asymptotically Gaussian, and we show how the asymptotic covariance is specified by the graph Laplacian. For additive perturbations, we show how the scaling of the asymptotic MSE is controlled by the spectral gap of the Laplacian.
TL;DR: The sufficient conditions for consensus in both discrete-time and continuous-time networks in terms of conditional expectations of the underlying graph topology are derived and it is proved that if there exist $T>0$ and $delta>0 $ such that the conditional expectation of the union of the $\delta$-graph topologies across each $T$-length time interval has spanning trees, then the multiagent system reaches consensus.
Abstract: In this paper, we discuss the consensus problem in networks of multiagents with stochastically switching topologies. The switch of graph topology is modeled as an adapted stochastic process, which in principle can include any stochastic processes such as independent and identically distributed (i.i.d.) processes and Markov chains. We derive the sufficient conditions for consensus in both discrete-time and continuous-time networks in terms of conditional expectations of the underlying graph topology. We prove that if there exist $T>0$ and $\delta>0$ such that the conditional expectation of the union of the $\delta$-graph topologies across each $T$-length time interval has spanning trees, then the multiagent system reaches consensus. For comparison, we show that some previous results on this topic can be derived from our main theorem as corollaries. This includes important results when the switching topology can be modeled as the special and important stochastic models—the i.i.d. process and the Markov process—which implies that we generalize the previous results to some extent. As applications, we also give some corollaries concerning stochastic processes other than the i.i.d. process and Markov processes, such as independent but not necessarily identically distributed processes, hidden Markov models, and $\phi$-mixing processes.