In this paper, a distributed velocity-constrained consensus problem is studied for discrete-time multi-agent systems, where each agent's velocity is constrained to lie in a nonconvex set. A distributed constrained control algorithm is proposed to enable all agents to converge to a common point using only local information. {The gains of the algorithm for all agents need not to be the same or predesigned and can be adjusted by each agent itself based on its own and neighbors' information.}
It is shown that the algorithm is robust to arbitrarily bounded communication delays and arbitrarily switching communication graphs provided that the union of the graphs has directed spanning trees among each certain time interval. The analysis approach is based on multiple novel model transformations, proper control parameter selections, boundedness analysis of state-dependent stochastic matrices, exploitation of the convexity of stochastic matrices, and the joint connectivity of the communication graphs. Numerical examples are included to illustrate the theoretical results.
In this note, we discuss the problem of consensus finding in communication networks of agents with dynamically switching topologies. In particular, we consider the case of directed networks with unbalanced matrices of communication rates. We formulate sufficient conditions for consensus finding in terms of strong connectivity of the underlying directed graphs and prove that, given these conditions, consensus is found asymptotically. Moreover, we show that this consensus is an emergent property of the system, being encoded in its dynamics and not just an invariant of its initial configuration.
In this paper, we consider a robust consensus tracking problem of heterogeneous multi-agent systems with time-varying interconnection topologies. Based on common Lyapunov function and internal model techniques, both state and output feedback control laws are derived to solve this problem. The proposed design is robust by admitting some parameter uncertainties in the multi-agent system.
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…
Distributed consensus optimization has received considerable attention in recent years; several distributed consensus-based algorithms have been proposed for (nonsmooth) convex and (smooth) nonconvex objective functions. However, the behavior of these distributed algorithms on {\it nonconvex, nonsmooth and stochastic} objective functions is not understood. This class of functions and distributed setting are motivated by several applications, including problems in machine learning and signal processing. This paper presents the first convergence analysis of the decentralized stochastic subgradie…
The leader-following consensus of multiple linear time invariant (LTI) systems under switching topology is considered. The leader-following consensus problem consists of designing for each agent a distributed protocol to make all agents track a leader vehicle, which has the same LTI dynamics as the agents. The interaction topology describing the information exchange of these agents is time-varying. An averaging method is proposed. Unlike the existing results in the literatures which assume the LTI agents to be neutrally stable, we relax this condition, only making assumption that the LTI agent…
This paper presents a novel framework which combines a non-iterative solution of Real-Time Nonlinear Receding Horizon Control (NRHC) methodology to achieve consensus within complex network topologies with existing time-delays and in presence of switching topologies. In this formulation, we solve the distributed nonlinear optimization problem for multi-agent network systems directly, \emph{in real-time}, without any dependency on iterative processes, where the stability and convergence guarantees are provided for the solution. Three benchmark examples on non-linear chaotic systems provide valid…
In this paper, we study the discrete-time consensus problem over networks with antagonistic and cooperative interactions. Following the work by Altafini [IEEE Trans. Automatic Control, 58 (2013), pp. 935--946], by an antagonistic interaction between a pair of nodes updating their scalar states we mean one node receives the opposite of the state of the other and naturally by an cooperative interaction we mean the former receives the true state of the latter. Here the pairwise communication can be either unidirectional or bidirectional and the overall network topology graph may change with time.…
Convergence of Consensus-Based Particle Methods for Nonconvex Bi-Level Optimization
Department of Computer Science（计算机科学系）；Technical University of Munich（慕尼黑技术大学）；Munich Institute of Robotics and Machine Intelligence（慕尼黑机器人与智能机械研究所）；Mathematical Institute（数学研究所）；University of Oxford（牛津大学）
In this paper, we study finite-time state consensus problems for continuous nonlinear multi-agent systems. Building on the theory of finite-time Lyapunov stability, we propose sufficient criteria which guarantee the system to reach a consensus in finite time, provided that the underlying directed network contains a spanning tree. Novel finite-time consensus protocols are introduced as examples for applying the criteria. Simulations are also presented to illustrate our theoretical results.
Consensus optimization has received considerable attention in recent years. A number of decentralized algorithms have been proposed for {convex} consensus optimization. However, to the behaviors or consensus \emph{nonconvex} optimization, our understanding is more limited. When we lose convexity, we cannot hope our algorithms always return global solutions though they sometimes still do sometimes. Somewhat surprisingly, the decentralized consensus algorithms, DGD and Prox-DGD, retain most other properties that are known in the convex setting. In particular, when diminishing (or constant) step …
This paper studies the consensus problem of multi-agent systems with asymmetric and reducible topologies. Centralized event-triggered rules are provided so as to reduce the frequency of system's updating. The diffusion coupling feedbacks of each agent are based on the latest observations from its in-neighbors and the system's next observation time is triggered by a criterion based on all agents' information. The scenario of continuous monitoring is first considered, namely all agents' instantaneous states can be observed. It is proved that if the network topology has a spanning tree, then the …
This paper addresses the robust consensus problem under switching topologies. Contrary to existing methods, the proposed approach provides decentralized protocols that achieve consensus for networked multi-agent systems in a predefined time. Namely, the protocol design provides a tuning parameter that allows setting the convergence time of the agents to a consensus state. An appropriate Lyapunov analysis exposes the capability of the current proposal to achieve predefined-time consensus over switching topologies despite the presence of bounded perturbations. Finally, the paper presents a compa…
Connected and automated vehicles (CAVs) offer huge potential to improve the performance of automated vehicles (AVs) without communication capabilities, especially in situations when the vehicles (or agents) need to be cooperative to accomplish their maneuver. Lane change maneuvers in dense traffic, e.g., are very challenging for non-connected AVs. To alleviate this problem, we propose a holistic distributed lane change control scheme for CAVs which relies on vehicle-to-vehicle communication. The originally centralized optimal control problem is embedded into a consensus-based Alternating Direc…
In this paper, we study consensus problem in multi-agent system with directed topology by event-triggered feedback control. That is, at each agent, the diffusion coupling feedbacks are based on the information from its latest observations to its in-neighbours. We derive distributed criteria to determine the next observation time of each agent that are triggered by its in-neighbours' information and its own states respectively. We prove that if the network topology is irreducible, then under the event-triggered coupling principles, the multi-agent system reach consensus. Then, we extend these r…
Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays
Fudan University（复旦大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进部数理研究所）；Santa Fe Institute for the Sciences of Complexity（圣塔菲复杂性科学研究所）
In this preliminary paper we study the stability property of a consensus on the average algorithm in arbitrary directed graphs with respect to communication/sensing time-delays. The proposed algorithm adds a storage variable to the agents' states so that the information about the average of the states is preserved despite the algorithm iterations are performed in an arbitrary strongly connected directed graph. We prove that for any network topology and choice of design parameters the consensus on the average algorithm is stable for sufficiently small delays. We provide simulations and numerica…
Peking University（北京大学）；Shanghai Artificial Intelligence Laboratory（上海人工智能实验室）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Michigan State University（密歇根州立大学）；Fudan University（复旦大学）；Alibaba DAMO Academy（阿里巴巴达摩院）；Princeton University（普林斯顿大学）
Decentralized optimization is an emerging paradigm in distributed learning in which agents achieve network-wide solutions by peer-to-peer communication without the central server. Since communication tends to be slower than computation, when each agent communicates with only a few neighboring agents per iteration, they can complete iterations faster than with more agents or a central server. However, the total number of iterations to reach a network-wide solution is affected by the speed at which the agents' information is ``mixed'' by communication. We found that popular communication topolog…
This paper investigates the distributed event-based consensus problem of switching networks satisfying the jointly connected condition. Both the state consensus of homogeneous linear networks and output consensus of heterogeneous networks are studied. Two kinds of event-based protocols based on local sampled information are designed, without the need to solve any matrix equation or inequality. Theoretical analysis indicates that the proposed event-based protocols guarantee the achievement of consensus and the exclusion of Zeno behaviors for jointly connected undirected switching graphs. These …
Sichuan University（四川大学）；Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen Research Institute of Big Data（深圳市大数据研究院）；University of Florida（佛罗里达大学）
While many distributed optimization algorithms have been proposed for solving smooth or convex problems over the networks, few of them can handle non-convex and non-smooth problems. Based on a proximal primal-dual approach, this paper presents a new (stochastic) distributed algorithm with Nesterov momentum for accelerated optimization of non-convex and non-smooth problems. Theoretically, we show that the proposed algorithm can achieve an $ε$-stationary solution under a constant step size with $\mathcal{O}(1/ε^2)$ computation complexity and $\mathcal{O}(1/ε)$ communication complexity. When comp…
The alternating direction method of multipliers (ADMM) is widely used to solve large-scale linearly constrained optimization problems, convex or nonconvex, in many engineering fields. However there is a general lack of theoretical understanding of the algorithm when the objective function is nonconvex. In this paper we analyze the convergence of the ADMM for solving certain nonconvex consensus and sharing problems, and show that the classical ADMM converges to the set of stationary solutions, provided that the penalty parameter in the augmented Lagrangian is chosen to be sufficiently large. Fo…
We consider stationary consensus protocols for networks of dynamic agents with switching topologies. The measure of the neighbors' state is affected by Unknown But Bounded disturbances. Here the main contribution is the formulation and solution of what we call the $ε$-consensus problem, where the states are required to converge in a tube of ray $ε$ asymptotically or in finite time.
In this technical note we address the problem of achieving consensus in a network of homogeneous nonlinear systems. The communication network is supposed to be switching within a finite set of topologies which may be disconnected for finite time intervals. We prove that if the length of the time intervals in which connected topologies are active satisfy an average dwell-time condition consensus is achieved. Lyapunov arguments proposed in the field of hybrid nonlinear systems are adopted to prove the result.
This note is devoted to the distributed optimization problem of multi-agent systems with nonconvex velocity constraints, nonuniform position constraints and nonuniform stepsizes. Two distributed constrained algorithms with nonconvex velocity constraints and nonuniform stepsizes are proposed in the absence and the presence of nonuniform position constraints by introducing a switching mechanism to guarantee all agents' position states to remain in a bounded region. The algorithm gains need not to be predesigned and can be selected by each agent using its own and neighbours' information. By a mod…
In this paper, the fast consensus problem of high-order multi-agent systems under undirected topologies is considered. The direct link between the consensus convergence rate and the control gains is established. An accelerated consensus algorithm based on gradient descent is proposed to optimize the convergence rate. By applying the Routh-Hurwitz stability criterion, the lower bound on the convergence rate is derived, and explicit control gains are derived as the necessary condition to achieve the optimal convergence rate. Moreover, a protocol with time-varying control gains is designed to ach…
We establish average consensus on graphs with dynamic topologies prescribed by evolutionary games among strategic agents. Each agent possesses a private reward function and dynamically decides whether to create new links and/or whether to delete existing ones in a selfish and decentralized fashion, as indicated by a certain randomized mechanism. This model incurs a time-varying and state-dependent graph topology for which traditional consensus analysis is not applicable. We prove asymptotic average consensus almost surely and in mean square for any initial condition and graph topology. In addi…
This article investigates discrete-time matrix-weighted consensus of multi-agent networks over undirected and connected graphs. We first present consensus protocols for the agents in common networks of symmetric matrix weights with possibly different update rates and switching network topologies. A special type of matrix-weighted consensus with non-symmetric matrix-weights that can render several consensus control scenarios such as ones with scaled/rotated updates and affine motion constraints is also considered. We employ Lyapunov stability theory for discrete-time systems and occasionally ut…
This paper investigates the distributed continuous-time nonconvex optimization problem over unbalanced directed networks. The objective is to cooperatively drive all the agent states to an optimal solution that minimizes the sum of the local cost functions. Based on the topology balancing technique and adaptive control approach, a novel fully distributed algorithm is developed for each agent with neither prior global information concerning network connectivity nor convexity of local cost functions. By viewing the proposed algorithm as a perturbed system, its input-to-state stability with a van…
Second-Order KKT Guarantees for Bregman ADMM in Nonconvex and Non-Lipschitz Optimization
Department of Electrical and Computer Engineering, Iowa State University（电气与计算机工程系，爱荷华州立大学）；Department of Computer Science and Engineering, Ohio State University（计算机科学与工程系，俄亥俄州立大学）；DAMO Academy, Alibaba Group US（阿里巴巴集团美国DAMO研究院）
In this paper, the consensus problems of the continuous-time integrator systems under noisy measurements are considered. The measurement noises, which appear when agents measure their neighbors' states, are modeled to be multiplicative. By multiplication of the noises, here, the noise intensities are proportional to the absolute value of the relative states of agent and its neighbor. By using known distributed protocols for integrator agent systems, the closed-loop {system is} described in the vector form by a singular stochastic differential equation. For the fixed and switching network topol…
Distributed consensus in the Wasserstein metric space of probability measures on the real line is introduced in this work. Convergence of each agent's measure to a common measure is proven under a weak network connectivity condition. The common measure reached at each agent is one minimizing a weighted sum of its Wasserstein distance to all initial agent measures. This measure is known as the Wasserstein barycenter. Special cases involving Gaussian measures, empirical measures, and time-invariant network topologies are considered, where convergence rates and average-consensus results are given…
In this paper, we consider distributed optimization problems over a multi-agent network, where each agent can only partially evaluate the objective function, and it is allowed to exchange messages with its immediate neighbors. Differently from all existing works on distributed optimization, our focus is given to optimizing a class of non-convex problems, and under the challenging setting where each agent can only access the zeroth-order information (i.e., the functional values) of its local functions. For different types of network topologies such as undirected connected networks or star netwo…
It is desirable but challenging to fulfill system constraints and reach optimal performance in consensus protocol design for practical multi-agent systems (MASs). This paper investigates the optimal consensus problem for general linear MASs subject to control input constraints. Two classes of MASs including subsystems with semi-stable and unstable dynamics are considered. For both classes of MASs without input constraints, the results on designing optimal consensus protocols are first developed by inverse optimality approach. Utilizing the optimal consensus protocols, the receding horizon cont…
A new theorem on conditions for convergence to consensus of a multiagent time-dependent time-discrete dynamical system is presented. The theorem is build up on the notion of averaging maps. We compare this theorem to results by Moreau (IEEE Transactions on Automatic Control, vol. 50, no. 2, 2005) about set-valued Lyapunov theory and convergence under switching communication topologies. We give examples that point out differences of approaches including examples where Moreau's theorem is not applicable but ours is. Further on, we give examples that demonstrate that the theory of convergence to …
We study distributed stochastic nonconvex optimization in multi-agent networks. We introduce a novel algorithmic framework for the distributed minimization of the sum of the expected value of a smooth (possibly nonconvex) function (the agents' sum-utility) plus a convex (possibly nonsmooth) regularizer. The proposed method hinges on successive convex approximation (SCA) techniques, leveraging dynamic consensus as a mechanism to track the average gradient among the agents, and recursive averaging to recover the expected gradient of the sum-utility function. Almost sure convergence to (stationar…
We present convergence and error estimates of the time-discrete consensus-based optimization(CBO) algorithms proposed in [arXiv:1909.09249] for general nonconvex functions. In authors' recent work [arxiv: 1910.08239], rigorous error analysis of the first-order consensus-based optimization algorithm proposed in [arXiv:1909.09249] was studied at the particle level without resorting to the kinetic equation via a mean-field limit. However, the error analysis for the corresponding time-discrete algorithm was not done mainly due to lack of discrete analogue of Itô's stochastic calculus. In this pape…
This paper introduces a novel continuous-time dynamic average consensus algorithm for networks whose interaction is described by a strongly connected and weight-balanced directed graph. The proposed distributed algorithm allows agents to track the average of their dynamic inputs with some steady-state error whose size can be controlled using a design parameter. This steady-state error vanishes for special classes of input signals. We analyze the asymptotic correctness of the algorithm under time-varying interaction topologies and characterize the requirements on the stepsize for discrete-time …
