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.
This paper aims at proposing a procedure to derive distributed algorithms for distributed consensus-based optimization by using distributed algorithms for network resource allocation and vice versa over switching networks with/without synchronous protocol. It is shown that first-order gradient distributed consensus-based optimization algorithms can be used for finding an optimal solution of distributed resource allocation with synchronous protocol under weaker assumptions than those given in the literature for non-switching (static) networks. It is shown that first-order gradient distributed r…
This paper investigates the distributed Kalman filtering (DKF) from distributed optimization viewpoint. Motivated by the fact that Kalman filtering is a maximum a posteriori estimation (MAP) problem, which is a quadratic optimization problem, we reformulate DKF problem as a consensus optimization problem, resulting in that it can be solved by many existing distributed optimization algorithms. A new DKF algorithm employing the dual ascent method is proposed, and its stability is proved under mild assumptions. The performance of the proposed algorithm is evaluated through numerical experiments.
Consensus of Multi-Agent Systems with General Linear and Lipschitz Nonlinear Dynamics Using Distributed Adaptive Protocols
School of Automation, Beijing Institute of Technology（北京理工大学自动化学院）；Department of Electrical and Computer Engineering, Utah State University（犹他州立大学电气与计算机工程系）
In this paper, we consider the problem of containment control of multi-agent systems with multiple stationary leaders, interacting over a directed network. While, containment control refers to just ensuring that the follower agents reach the convex hull of the leaders states, we focus on the problem where the followers achieve a consensus to the average values of the leaders states. We propose an algorithm that can be implemented in a distributed manner to achieve the above consensus among followers. Next we optimize the convergence rate of the followers to the average consensus by proper choi…
This paper focuses on the problem of distributed consensus control of multi-agent systems while considering two main practical concerns (i) stochastic noise in the agent dynamics and (ii) predefined performance constraints over evolutions of multi-agent systems. In particular, we consider that each agent is driven by a stochastic differential equation with state-dependent noise which makes the considered problem more challenging compared to non-stochastic agents. The work provides sufficient conditions under which the proposed timevarying distributed control laws ensure consensus in expectatio…
We revisit an algorithm for distributed consensus optimization proposed in 2010 by J. Wang and N. Elia. By means of a Lyapunov-based analysis, we prove input-to-state stability of the algorithm relative to a closed invariant set composed of optimal equilibria and with respect to perturbations affecting the algorithm's dynamics. In the absence of perturbations, this result implies linear convergence of the local estimates and Lyapunov stability of the optimal steady state. Moreover, we unveil fundamental connections with the well-known Gradient Tracking and with distributed integral control. Ov…
A Round-Robin Protocol for Distributed Estimation with $H_\infty$ Consensus
School of Engineering and Information Technology, UNSW Canberra at the Australian Defence Force Academy（新南威尔士大学堪培拉校区澳大利亚国防军学院工程与信息科技学院）；School of Electrical Engineering, Tel-Aviv University（特拉维夫大学电气工程学院）
Inspired and underpinned by the idea of integral feedback, a distributed constant gain algorithm is proposed for multi-agent networks to solve convex optimization problems with local linear constraints. Assuming agent interactions are modeled by an undirected graph, the algorithm is capable of achieving the optimum solution with an exponential convergence rate. Furthermore, inherited from the beneficial integral feedback, the proposed algorithm has attractive requirements on communication bandwidth and good robustness against disturbance. Both analytical proof and numerical simulations are pro…
The distributed optimal output consensus problem for high-order multi-agent systems has been studied recently. In this paper, we further focus on the same problem for high-order multi-agent systems subject to parametric uncertainties and aim at distributed robust controllers by measurement output feedback. We first develop a dynamic compensator to estimate the expected optimal consensus point and convert the problem into several decentralized robust tracking problems. Then, by combining the integral control technique and dirty derivative observer technique, we constructively propose a distribu…
Motivated by decentralized sensing and policy evaluation problems, we consider a particular type of distributed stochastic optimization problem over a network, called the online stochastic distributed averaging problem. We design a dual-based method for this distributed consensus problem with Polyak--Ruppert averaging and analyze its behavior. We show that the proposed algorithm attains an accelerated deterministic error depending optimally on the condition number of the network, and also that it has an order-optimal stochastic error. This improves on the guarantees of state-of-the-art distrib…
We consider the distributed leader-follower consensus problem with event-triggered communications. The system under consideration is a non-linear input-affine multi agent system. The agents are assumed to have identical dynamics structure with uncertain parameters and satisfying an incremental stabilisability condition. A distributed control law is proposed which achieves consensus based on two novel Communication Triggering Conditions (CTCs): the first one to achieve an asymptotic consensus but without any guarantees on Zeno behaviour and the second one to exclude Zeno behaviour but with prac…
We consider the global consensus problem for multi-agent systems with input saturation over digraphs. Under a mild connectivity condition that the underlying digraph has a directed spanning tree, we use Lyapunov methods to show that the widely used distributed consensus protocol, which solves the consensus problem for the case without input saturation constraints, also solves the global consensus problem for the case with input saturation constraints. In order to reduce the overall need of communication and system updates, we then propose a distributed event-triggered control law. Global conse…
This paper presents the formulation and analysis of a novel distributed maximum likelihood algorithm that utilizes a first-order optimization scheme. The proposed approach utilizes a static average consensus algorithm to reach agreement on the initial condition to the iterative optimization scheme and a dynamic average consensus algorithm to reach agreement on the gradient direction. The current distributed algorithm is guaranteed to exponentially recover the performance of the centralized algorithm. Though the current formulation focuses on maximum likelihood algorithm built on first-order me…
We investigate the use of distributed PID actions to achieve consensus in networks of homogeneous and heterogeneous linear systems. Convergence of the strategy is proved for both cases using appropriate state transformations and Lyapunov functions. The effectiveness of the theoretical results is illustrated via its application to a representative power grid model recently presented in the literature.
This paper studies the consensusability problem of multi-agent systems (MASs), where agents communicate with each other through Markovian packet loss channels. We try to determine conditions under which there exists a linear distributed consensus controller such that the MAS can achieve mean square consensus. We first provide a necessary and sufficient consensus condition for MASs with single input and i.i.d.\ channel losses, which complements existing results. Then we proceed to study the case with identical Markovian packet losses. A necessary and sufficient consensus condition is firstly de…
This paper considers the distributed consensus problem of linear multi-agent systems subject to different matching uncertainties for both the cases without and with a leader of bounded unknown control input. Due to the existence of nonidentical uncertainties, the multi-agent systems discussed in this paper are essentially heterogeneous. For the case where the communication graph is undirected and connected, a distributed continuous static consensus protocol based on the relative state information is first designed, under which the consensus error is uniformly ultimately bounded and exponential…
Results for estimating the convergence rate of non-stationary 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.
This paper presents the formulation and analysis of a fully distributed dynamic event-triggered communication based robust dynamic average consensus algorithm. Dynamic average consensus problem involves a networked set of agents estimating the time-varying average of dynamic reference signals locally available to individual agents. We propose an asymptotically stable solution to the dynamic average consensus problem that is robust to network disruptions. Since this robust algorithm requires continuous communication among agents, we introduce a novel dynamic event-triggered communication scheme…
We study a distributed allocation process where, repeatedly in time, every player renegotiates past allocations with neighbors and allocates new revenues. The average allocations evolve according to a doubly (over time and space) averaging algorithm. We study conditions under which the average allocations reach consensus to any point within a predefined target set even in the presence of adversarial disturbances. Motivations arise in the context of coalitional games with transferable utilities (TU) where the target set is any set of allocations that make the grand coalitions stable.
Consensus conditions and convergence speeds are crucial for distributed consensus algorithms of networked systems. Based on a basic first-order average-consensus protocol with time-varying topologies and additive noises, this paper first investigates its critical consensus condition on network topology by stochastic approximation frameworks. A new joint-connectivity condition called extensible joint-connectivity that contains a parameter $δ$ (termed the extensible exponent) is proposed. With this and a balanced topology condition, we show that a critical value of $δ$ for consensus is $1/2$. Op…
ADMM is a popular algorithm for solving convex optimization problems. Applying this algorithm to distributed consensus optimization problem results in a fully distributed iterative solution which relies on processing at the nodes and communication between neighbors. Local computations usually suffer from different types of errors, due to e.g., observation or quantization noise, which can degrade the performance of the algorithm. In this work, we focus on analyzing the convergence behavior of distributed ADMM for consensus optimization in presence of additive node error. We specifically show th…
Distributed optimization algorithms have been studied extensively in the literature; however, underlying most algorithms is a linear consensus scheme, i.e. averaging variables from neighbors via doubly stochastic matrices. We consider nonlinear consensus schemes with a set of time-varying and agent-dependent monotonic Lipschitz nonlinear transformations, where we note the operation is a subprojection onto the consensus plane, which has been identified via stochastic approximation theory. For the proposed nonlinear consensus schemes, we establish convergence when combined with the NEXT algorith…
We consider the problems of reaching average consensus and solving consensus-based optimization over unreliable communication networks wherein packets may be dropped accidentally during transmission. Existing work either assumes that the link failures affect the communication on both directions or that the message senders {\em know exactly}, in each iteration, how many of their outgoing links are functioning properly. In this paper, we consider directed links, and we {\em do not} require each node know its current outgoing degree. First, we propose and characterize the convergence rate of reac…
Linear consensus iterations guarantee asymptotic convergence, thereby, limiting their applicability in applications where consensus value needs to be used in real time to perform a system level task. It also leads to wastage of power and communication resources. In this article, an algorithm is proposed which enables each node to detect in a distributed manner and in finite number of iterations, when every agent in the network is within a user specified threshold of the consensus value (approximate consensus) and hence terminate further communications and computations associated with consensus…
Distributed averaging of agent initial conditions is a well-studied problem in context of networked systems where coordination amongst the agents is of paramount importance. The asymptotic nature of convergence of distributed averaging protocols and presence of communication delays, however, makes it challenging to implement in practical settings. It is important that agents develop the ability to detect on their own when average of the initial conditions of the agents is achieved within some pre-specified tolerance and stop further computations to avoid overhead expenses in the presence of de…
This paper presents the formulation of fault detection and accommodation schemes for a network of autonomous agents running internal model-based dynamic average consensus algorithms. We focus on two types of consensus algorithms, one that is internally stable but non-robust to initial conditions and one that is robust to initial conditions but not internally stable. For each consensus algorithm, a fault detection filter based on the unknown input observer scheme is developed for precisely estimating the communication faults that occur on the network edges. We then propose a fault remediation s…
This paper considers distributed average consensus using finite-bit bounded quantizer with possibly unbounded data. Under the framework of the alternating direction method of multipliers (ADMM), we develop distributed averaging algorithms where each node iteratively updates using only the local information and finitely quantized outputs from its neighbors. It is shown that all the agent variables either converge to the same quantization level or cycle around the data average after finite iterations. An error bound for the consensus value is established, which turns out to be the same as that o…
Distributed dynamic compensators, also known as distributed observer, play a key role in the output consensus problem of heterogeneous nonlinear multi-agent systems. However, most existing distributed dynamic compensators require either the compensators' information to be exchanged through communication networks, or that the controller for each subsystem satisfies a class of small gain conditions. In this note, we develop a novel distributed dynamic compensator to address the adaptive output consensus problem of heterogeneous nonlinear multi-agent systems with unknown parameters. The distribut…
The shift of power systems toward a smarter grid has brought devices such as distributed generators and smart loads with an increase of the operational challenges for the system operator. These challenges are related to the real-time implementation as well as control and stability issues. We present a distributed transactive control strategy, based on the projected consensus algorithm, to operate the distributed energy resources and smart loads of a power system toward optimal social welfare. We consider two types of agents: Generators, and smart loads. Each agent iteratively optimizes its loc…
This paper presents a novel distributed algorithm for tracking a maneuvering target using bearing or direction of arrival measurements collected by a networked sensor array. The proposed approach is built on the dynamic average-consensus algorithm, which allows a networked group of agents (nodes) to reach consensus on the global average of a set of local time-varying signals in a distributed fashion. Since the average-consensus error corresponding to the presented dynamic average-consensus algorithm converges to zero in finite time, the proposed distributed algorithm guarantees that the tracki…
We study the popular distributed consensus method over networks composed of a number of densely connected clusters with a sparse connection between them. In these cluster networks, the method often constitutes two-time-scale dynamics, where the internal nodes within each cluster reach consensus quickly relative to the aggregate nodes across clusters. Our main contribution is to provide the rate of the distributed consensus method, which characterize explicitly the impacts of the internal and external graphs on the performance of this method. Our main result shows that this rate converges expon…
We address the problem of distributed uncon- strained convex optimization under separability assumptions, i.e., the framework where each agent of a network is endowed with a local private multidimensional convex cost, is subject to communication constraints, and wants to collaborate to compute the minimizer of the sum of the local costs. We propose a design methodology that combines average consensus algorithms and separation of time-scales ideas. This strategy is proved, under suitable hypotheses, to be globally convergent to the true minimizer. Intuitively, the procedure lets the agents dist…
CADMM-Prox: A Bi-level Consensus ADMM for Non-smooth Non-convex Distributed Consensus Optimization
The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；The Hong Kong University of Science and Technology（香港科技大学）；North China University of Water Resources and Electric Power（华北水利水电大学）；KTH Royal Institute of Technology（皇家理工学院）
Distributed algorithms for solving additive or consensus optimization problems commonly rely on first-order or proximal splitting methods. These algorithms generally come with restrictive assumptions and at best enjoy a linear convergence rate. Hence, they can require many iterations or communications among agents to converge. In many cases, however, we do not seek a highly accurate solution for consensus problems. Based on this we propose a controlled relaxation of the coupling in the problem which allows us to compute an approximate solution, where the accuracy of the approximation can be co…
Average consensus protocols emerge with a central role in distributed systems and decision-making such as distributed information fusion, distributed optimization, distributed estimation, and control. A key advantage of these protocols is that agents exchange and reveal their state information only to their neighbors. Yet, it can raise privacy concerns in situations where the agents' states contain sensitive information. In this paper, we propose a novel (noiseless) privacy preserving distributed algorithms for multi-agent systems to reach an average consensus. The main idea of the algorithms …
In this paper, we propose a distributed Newton method for consensus optimization. Our approach outperforms state-of-the-art methods, including ADMM. The key idea is to exploit the sparsity of the dual Hessian and recast the computation of the Newton step as one of efficiently solving symmetric diagonally dominant linear equations. We validate our algorithm both theoretically and empirically. On the theory side, we demonstrate that our algorithm exhibits superlinear convergence within a neighborhood of optimality. Empirically, we show the superiority of this new method on a variety of machine l…
Distributed quantized weight-balancing and average consensus over fixed digraphs are considered. A digraph with non-negative weights associated to its edges is weight-balanced if, for each node, the sum of the weights of its out-going edges is equal to that of its incoming edges. This paper proposes and analyzes the first distributed algorithm that solves the weight-balancing problem using only finite rate and simplex communications among nodes (compliant to the directed nature of the graph edges). Asymptotic convergence of the scheme is proved and a convergence rate analysis is provided. Buil…
We propose a weight design method to increase the convergence rate of distributed consensus. Prior work has focused on symmetric weight design due to computational tractability. We show that with proper choice of asymmetric weights, the convergence rate can be improved significantly over even the symmetric optimal design. In particular, we prove that the convergence rate in a lattice graph can be made independent of the size of the graph with asymmetric weights. We then use a Sturm-Liouville operator to approximate the graph Laplacian of more general graphs. A general weight design method is p…
This paper is concerned with the leader-following output consensus problem in the framework of distributed nonlinear observers. In stead of certain hypotheses on the leader system, a group of geometric conditions is put forward to develop a novel distributed observer strategy with less conservatism, thereby definitely improving the applicability of the existing results. To be more specific, the improved distributed observer can precisely handle consensus problems for some nonlinear leader systems which are invalid for the traditional strategies with the certain assumption, such as Elastic Shaf…
We study the stability properties of linear time-varying systems in continuous time whose system matrix is Metzler with zero row sums. This class of systems arises naturally in the context of distributed decision problems, coordination and rendezvous tasks and synchronization problems. The equilibrium set contains all states with identical state components. We present sufficient conditions guaranteeing uniform exponential stability of this equilibrium set, implying that all state components converge to a common value as time grows unbounded. Furthermore it is shown that this convergence result…
