Average Consensus Problem Research Articles

Adaptive distributed dynamic average consensus with prescribed performance

AbstractA novel adaptive distributed estimation algorithm is developed for dynamic average consensus problem. The algorithm enables a group of agents to accurately track the average value of signals that are available locally to each agent. Unlike most of existing approaches that rely on error transformation techniques, our algorithm employs an adaptive‐gain based barrier function to simultaneously meet prescribed performance specifications of transient and steady state. It is both simple and easy to implement and is not dependent on bounded reference signals and their derivatives. Additionally, it can operate under less strict assumptions than those that assume absolute bounds on the signals. Simulation results are presented to demonstrate the effectiveness of the developed algorithm.

Invariant properties of linear-iterative distributed averaging algorithms and application to error detection

We consider the problem of average consensus in a distributed system comprising a set of nodes that can exchange information among themselves. We focus on a class of algorithms for solving such a problem whereby each node maintains a state and updates it iteratively as a linear combination of the states maintained by its in-neighbors, i.e., nodes from which it receives information directly. Averaging algorithms within this class can be thought of as discrete-time linear time-varying systems without external driving inputs and whose state matrix is column stochastic. As a result, the algorithms exhibit a global invariance property in that the sum of the state variables remains constant at all times. In this paper, we report on another invariance property for the aforementioned class of averaging algorithms. This property is local to each node and reflects the conservation of certain quantities capturing an aggregate of all the values received by a node from its in-neighbors and all the values sent by said node to its out-neighbors (i.e., nodes to which it sends information directly) throughout the execution of the averaging algorithm. We show how this newly-discovered invariant can be leveraged for detecting errors while executing the averaging algorithm.

Differentially private average consensus with improved accuracy-privacy trade-off

This paper studies the average consensus problem with differential privacy of initial states, for which it is widely recognized that there is a trade-off between the mean-square computation accuracy and privacy level. Considering the trade-off gap between the average consensus algorithm and the centralized averaging approach with differential privacy, we propose a distributed shuffling mechanism based on the Paillier cryptosystem to generate correlated zero-sum randomness. By randomizing each local privacy-sensitive initial state with an i.i.d. Gaussian noise and the output of the mechanism using Gaussian noises, it is shown that the resulting average consensus algorithm can eliminate the gap in the sense that the accuracy-privacy trade-off of the centralized averaging approach with differential privacy can be almost recovered by appropriately designing the variances of the added noises. We also extend such a design framework with Gaussian noises to the one using Laplace noises, and show that the improved privacy-accuracy trade-off is preserved.

Privacy-preserving finite-time average consensus and its application to power allocation of battery energy storage systems

In this paper, the privacy-preserving problem of finite-time discrete-time average consensus is studied by adding random numbers. The privacy leakage conditions are given for systems with curious but not malicious nodes and eavesdroppers, respectively. By introducing the random numbers and node decomposition mechanism, the privacy information of the nodes cannot be estimated by the adversaries. The proposed algorithm is applied to the power allocation problem of distributed battery energy storage systems. Based on the previous proposed multi-rate sampling method, the privacy-preserving finite-time discrete-time average consensus is utilized to estimate the accurate average value within finite steps. The accurate power allocation based on the same relative SoC variation rate can be achieved with the privacy preservation of the transmission information. Finally, based on a networked battery energy storage system, simulation studies are presented to verify the effectiveness of the proposed method.

Dynamic event‐triggered fixed‐time average consensus for general linear multi‐agent systems under switching topologies

AbstractThis article studies the practical fixed‐time average consensus problem of general linear multi‐agent systems under switching topologies. First, a distributed fixed‐time consensus control protocol based on dynamic event‐triggered mechanism is designed by using the local information exchange between neighboring agents. The protocol introduces an internal dynamic variable and the triggering condition changes in real time based on the dynamic variable obtained online, which can significantly reduce the number of triggering events and effectively decrease the energy consumption of the multi‐agent system. Then, the sufficient conditions for the multi‐agent system to solve the practical fixed‐time average consensus problem are given, and it is proved that there is no Zeno behavior in the system. Furthermore, a self‐triggered control scheme is proposed such that the neighboring agents communicate with each other only when one of them is triggered. In other words, this scheme does not require continuous communication between neighboring agents and thus reduces communication cost of the system. Finally, the simulation results verify the effectiveness of the proposed method.

A novel dynamic event-triggered mechanism for dynamic average consensus

This paper studies a challenging issue introduced in a recent survey, namely designing a distributed event-based scheme to solve the dynamic average consensus (DAC) problem. First, a robust adaptive distributed event-based DAC algorithm is designed without imposing specific initialization criteria to perform estimation task under intermittent communication. Second, a novel adaptive distributed dynamic event-triggered mechanism is proposed to determine the triggering time when neighboring agents broadcast information to each other. Compared to the existing event-triggered mechanisms, the novelty of the proposed dynamic event-triggered mechanism lies in that it guarantees the existence of a positive and uniform minimum inter-event interval without sacrificing any accuracy of the estimation, which is much more practical than only ensuring the exclusion of the Zeno behavior or the boundedness of the estimation error. Third, a composite adaptive law is developed to update the adaptive gain employed in the distributed event-based DAC algorithm and dynamic event-triggered mechanism. Using the composite adaptive update law, the distributed event-based solution proposed in our work is implemented without requiring any global information. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.

Differentially Private Average Consensus With Logarithmic Dynamic Encoding-Decoding Scheme.

This article is concerned with the differentially private average consensus (DPAC) problem for a class of multiagent systems with quantized communication. By constructing a pair of auxiliary dynamic equations, a logarithmic dynamic encoding-decoding (LDED) scheme is developed and then utilized during the process of data transmission, thereby eliminating the effect of quantization errors on the consensus accuracy. The primary purpose of this article is to establish a unified framework that integrates the convergence analysis, the accuracy evaluation, and the privacy level for the developed DPAC algorithm under the LDED communication scheme. By means of the matrix eigenvalue analysis method, the Jury stability criterion, and the probability theory, a sufficient condition (with respect to the quantization accuracy, the coupling strength, and the communication topology) is first derived to ensure the almost sure convergence of the proposed DPAC algorithm, and the convergence accuracy and privacy level are thoroughly investigated by resorting to the Chebyshev inequality and ϵ -differential privacy index. Finally, simulation results are provided to illustrate the correctness and validity of the developed algorithm.

Model-Based Distributed Asynchronous Sampling for Multiagent Consensus With Event-Detecting Delays and Network Delays

In this paper, the practical average consensus problem of linear multi-agent systems with model-based distributed asynchronous sampling is investigated. Both internal event-detecting delays and external network delays are allowed in systems. Data-sampling instants are generated asynchronously by local event-triggering conditions. Particularly, data-sampling instants and event-triggering instants of each agent are also asynchronous due to the existence of event-detecting delays. Prediction models are utilized in detecting events and updating control inputs. Artificial interpolation instants are introduced in each sampling interval to address theoretical challenges brought by internal and external delays. To deal with this kind of complex asynchronous setup with multiple time delays, based on Barbalat's Lemma, a new procedure for deriving conditions for average consensus and a theoretical analysis approach are presented.

Distributed fixed-time and prescribed-time average consensus for multi-agent systems with energy constraints

This paper is devoted to the investigation of the fixed-time and prescribed-time average consensus problem for multi-agent systems (MASs) with energy constraints. For the purpose of the fixed-time and prescribed-time average consensus of MASs, two control algorithms are firstly developed in this paper on the basis of the fixed-time technique and the prescribed-time technique. Then, the time integral of the quadratic function is constructed to calculate the energy consumption by the error between each agent and the reference state. And the sufficient conditions for the stability of the Lyapunov function are provided by designing the parameters and the gain matrix. Ultimately, the theoretical results are effectively and correctly verified by utilizing simulations.

Average Consensus Problem in Multi-Agent System in an Environment with Obstacles

Multi-agent systems (MASs) are composed of multiple autonomous agents, and the overall behavior of an MAS is determined by the local interactions between its agents. In recent years, research on MASs aimed at their application in real-world environments has garnered considerable interest. In particular, consensus control in environments with constrained communication range of agents and obstacles is being extensively investigated. In this study, the consensus problem of MASs is examined by considering the effects of obstacles. In particular, an obstacle avoidance algorithm is developed as a control method for real-world autonomous machines such as autonomous mobile robots. In conventional control methods, obstacles are provided as coordinate points; however, in this study, they are introduced as variously shaped polygons. Furthermore, the graph topology is dynamic, which enables more realistic situations. Finally, a simulation study is performed, and the effectiveness of the developed algorithm for applying to the MAS consensus problem is demonstrated.

Appointed-Time Dynamic Average Consensus: A Holistic Planning and Multi-Step Control Framework

Most existing solutions to the dynamic average consensus (DAC) problem assume access to the full dynamic of reference signals or their derivatives. Such constraints are more stringent and impose restrictions on use. Hence, we revisit the DAC problem in this paper and present its solution by developing a holistic planning and multi-step control (HPMSC) framework. Here, the HPMSC allows the agents to achieve DAC in an appointed settling time without involving any other parameters, which is called appointed-time dynamic average consensus (AT-DAC). Under the HPMSC framework, a sampling-data-based AT-DAC controller is designed, which guarantees that agents can track the average value of dynamic reference signals at a pre-appointed time. Compared with the existing results on DAC, a nice highlight of this paper is the implementation of zero-error DAC with weaker conditions imposed on the reference signals, without the need to know the upper bounds on their derivatives and without additional common function conditions. Finally, the performance validation of theoretical results is demonstrated by a numerical simulation.

Faster Supervised Average Consensus in Adversarial and Stochastic Anonymous Dynamic Networks

How do we reach consensus on an average value in a dynamic crowd without revealing identity? In this work, we study the problem of average network consensus in Anonymous Dynamic Networks (ADN). Network dynamicity is specified by the sequence of topology-graph isoperimetric numbers occurring over time, which we call the isoperimetric dynamicity of the network. The consensus variable is the average of values initially held by nodes, which is customary in the network-consensus literature. Given that having an algorithm to compute the average one can compute the network size (i.e., the counting problem) and vice versa, we focus on the latter. We present a deterministic distributed average network consensus algorithm for ADNs that we call isoperimetric Scalable Coordinated Anonymous Local Aggregation, and we analyze its performance for different scenarios, including worst-case (adversarial) and stochastic dynamic topologies. Our solution utilizes supervisor nodes, which have been shown to be necessary for computations in ADNs. The algorithm uses the isoperimetric dynamicity of the network as an input, meaning that only the isoperimetric number parameters (or their lower bound) must be given, but topologies may occur arbitrarily or stochastically as long as they comply with those parameters. Previous work for adversarial ADNs overestimates the running time to deal with worst-case scenarios. For ADNs with given isoperimetric dynamicity, our analysis shows improved performance for some practical dynamic topologies, with cubic time or better for stochastic ADNs, and our experimental evaluation indicates that our theoretical bounds could not be substantially improved for some models of dynamic networks.

Privacy-Preserving Average Consensus in Multiagent Systems via Partial Information Transmission

This article investigates the privacy-preserving average consensus problem in multiagent systems. A new approach is proposed to achieve the accurate average consensus value while protecting the initial state information of agents. The key idea to achieve these goals is based on partial information transmission. Each agent decomposes its initial state information into several different subinformation. The values of these subinformation are chosen randomly but with their mean value fixed to the original initial value. Then, based on multiple communication channels, agents share all their subinformation, but ensure that each neighbor can only obtain partial subinformation. We prove that the privacy can be protected under our method if each agent has at least two neighbors, and one of the neighbors is neutral. The generalization on the directed graph about this method is also considered. Finally, two examples are provided to illustrate the design process and practical applications of the proposed approach.

Predefined Time Synchronization of Multi-Agent Systems: A Passivity Based Analysis.

This paper deals with the predefined-time synchronization for a class of nonlinear multi-agent systems. The notion of passivity is exploited to design the controller for predefined-time synchronization of a nonlinear multi-agent system, where the time of synchronization can be preassigned. Developed control can be used to synchronize large-scale, higher-order multi-agent systems as passivity is an important property in designing control for complex control systems, where the control inputs and outputs are considered in determining the stability of the system in contrast to other approaches, such as state-based Control We introduced the notion of predefined-time passivity and as an application of the exposed stability analysis, static and adaptive predefined-time control algorithms are designed to study the average consensus problem for nonlinear leaderless multiagent systems in predefined-time. We provide a detailed mathematical analysis of the proposed protocol, including convergence proof and stability analysis. We discussed the tracking problem for a single agent, and designed state feedback and adaptive state feedback control scheme to make tracking error predefined-time passive and then showed that in the absence of external input, tracking error reduces to zero in predefined-time. Furthermore, we extended this concept for a nonlinear multi-agent system and designed state feedback and adaptive state feedback control scheme which ensure synchronization of all the agents in predefined-time. To further strengthen the idea, we applied our control scheme to a nonlinear multi-agent system by taking the example of Chua's circuit. Finally, we compared the result of our developed predefined-time synchronization framework with finite-time synchronization scheme available in literature for the Kuramoto model.

An Operator-Theoretic Approach to Robust Event-Triggered Control of Network Systems With Frequency-Domain Uncertainties

In this paper, we study the robustness of the event-triggered consensus algorithms against frequency- domain uncertainties. It is revealed that the sampling errors resulted by event triggering are essentially images of linear finite-gain L2-stable operators acting on the consensus errors of the sampled states and the event-triggered mechanism is equivalent to a negative feedback loop introduced additionally to the feedback system. In virtue of this, the robust consensus problem of the event-triggered network systems subject to additive dynamic uncertainties and network multiplicative uncertainties are considered, respectively. In both cases, quantitative relationships among the parameters of the controllers, the Laplacian matrix of the network topology, and the robustness against aperiodic event triggering and frequency-domain uncertainties are unveiled. Furthermore, the event-triggered dynamic average consensus (DAC) problem is also investigated, wherein the sampling errors are shown to be images of nonlinear finite-gain operators. The robust performance of the proposed DAC algorithm is analyzed, which indicates that the robustness and the performance are negative

Distributed Event-Triggered Algorithms for Finite-Time Privacy-Preserving Quantized Average Consensus

In this article, we consider the problem of privacy preservation in the average consensus problem when communication among nodes is quantized. More specifically, we consider a setting where nodes in the network can be curious, while certain nodes in the network want to ensure that their initial states cannot be inferred exactly by these curious nodes. Curious nodes are not malicious, i.e., they try to identify the initial states of other nodes based on the data they receive during their operation (and some of them might even collude) but do not interfere in the computation in any other way. Each node in the network (including curious nodes) can opt to execute a privacy-preserving algorithm (so as not to reveal the initial state it contributes to the average calculation) or its underlying (plain) average consensus algorithm (if privacy is not a concern). We propose two privacy-preserving event-triggered quantized average consensus algorithms. Under certain topological conditions, both the proposed algorithms allow the nodes that adopt privacy-preserving protocols to preserve their privacy and at the same time to obtain, after a finite number of steps, the exact average of the initial states while processing and transmitting <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">quantized</i> information. We also present illustrative examples and comparisons of our algorithms against other algorithms in the existing literature and discuss a motivating application in which smart meters in a smart grid collect real-time demands in a privacy-preserving manner.

Fundamental Performance Limitations for Average Consensus in Open Multi-Agent Systems

We derive <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fundamental performance limitations</i> for intrinsic average consensus problems in open multi-agent systems, which are systems subject to frequent arrivals and departures of agents. Each agent holds a value, and the objective of the agents is to collaboratively estimate the average of the values of the agents presently in the system. Algorithms solving such problems in open systems are poised to never converge because of the permanent variations in the composition, size, and objective pursued by the agents of the system. We provide lower bounds on the expected mean squared error achievable by any averaging algorithms in open systems of fixed size. Our derivation is based on the analysis of a conceptual algorithm that would achieve optimal performance for a given model of replacements. We obtain a general bound that depends on the properties of the model defining the interactions between the agents, and instantiate that result for all-to-one and one-to-one interaction models. A comparison between those bounds and algorithms implementable with those models is then provided to highlight their validity.

Predefined-time stability for a class of dynamical systems and its application on the consensus control for nonlinear multi-agent systems

This paper focuses primarily on the predefined-time stability for a class of fixed-time stable systems, with the least upper bound of the settling time as an explicit parameter. It is mathematically proved that the systems under consideration herein can reach stability within any preassigned time, rather than the overestimation of convergence time found in previous fixed-time stability. Comparisons show that our studied system is more general than some classical ones, and that many existing results can be derived. As an application of the exposed stability analysis, static and adaptive predefined-time control algorithms are designed to study the average consensus problem for nonlinear leaderless multi-agent systems. Particularly, the adaptive control protocol guarantees predefined-time consensus without requiring the global topology information of the systems under consideration. Numerical simulations are provided throughout to support our theoretical results.

Identification of Malicious Activity in Distributed Average Consensus via Non-Concurrent Checking

We consider the problem of average consensus in a network system under a fixed, undirected communication topology, when there are malicious nodes present that may try to influence the average calculation. In the setting considered, the average consensus is performed by the nodes in a distributed fashion using a linear iterative algorithm. We assume malicious nodes can manipulate, in an arbitrary manner, the value of their state in the aforementioned algorithm; the problem is then to check whether or not each node is correctly performing the updates of its state. To address this problem, we propose a distributed algorithm whereby each node is in charge of checking the updates performed by its neighboring nodes based on information that it receives from them and also from the neighbors of its neighbors. The algorithm leverages ideas from non-concurrent error detection schemes and its main advantage is that information from two-hop neighbors is only needed infrequently—a relaxation that significantly reduces the communication overhead associated with the requirement to make such information available

Differentiated Output-Based Privacy-Preserving Average Consensus

This letter investigates the differentiated output-based privacy-preserving average consensus problem over digraphs. A new stochastic obfuscation algorithm is proposed to achieve better privacy-preserving effect. When the output messages for at least one out-neighbour are not leaked, the algorithm can be designed to achieve any pre-given consensus accuracy and privacy-preserving level simultaneously by properly selecting the private weights. Even if all the output messages are leaked, the algorithm can still ensure that each agent’s initial state is protected to a certain extent. The mean square convergence of the algorithm is proved. The efficiency of the algorithm is verified by a numerical example.