Random consensus in nonlinear systems under fixed topology

This paper investigates the consensus problem in almost sure sense for uncertain multi-agent systems with noises and fixed topology. By combining the tools of stochastic analysis, algebraic graph theory, and matrix theory, we analyze the convergence of a class of distributed stochastic type non-linear protocols. Numerical examples are given to illustrate the results.Comment: 9 pages, 5 figures. arXiv admin note: text overlap with arXiv:0909.316

Leader-following Consensus Problems with a Time-varying Leader under Measurement Noises

In this paper, we consider a leader-following consensus problem for networks of continuous-time integrator agents with a time-varying leader under measurement noises. We propose a neighbor-based state-estimation protocol for every agent to track the leader, and time-varying consensus gains are introduced to attenuate the noises. By combining the tools of stochastic analysis and algebraic graph theory, we study mean square convergence of this multi-agent system under directed fixed as well as switching interconnection topologies. Sufficient conditions are given for mean square consensus in both cases. Finally, a numerical example is given to illustrate our theoretical results.Comment: 12 pages 3 figure

Containment control of multi-agent systems with measurement noises

In this paper, containment control of multi-agent systems with measurement noises is studied under directed networks. When the leaders are stationary, a stochastic approximation type protocol is employed to solve the containment control of multi-agent systems. By using stochastic analysis tools and algebraic graph theory, some necessary and sufficient criteria are established to ensure the followers converge to the convex hull spanned by the leaders in the sense of mean square and probability 1. When the leasers are dynamic, a stochastic approximation type protocol with distributed estimators is developed and necessary and sufficient conditions are also obtained for solving the containment control problem. Simulations are provided to illustrate the effectiveness of the theoretical results.Comment: 8 page

Synchronization in Networks of Coupled Harmonic Oscillators with Stochastic Perturbation and Time Delays

In this paper, we investigate synchronization of coupled second-order linear harmonic oscillators with random noises and time delays. The interaction topology is modeled by a weighted directed graph and the weights are perturbed by white noise. On the basis of stability theory of stochastic differential delay equations, algebraic graph theory and matrix theory, we show that the coupled harmonic oscillators can be synchronized almost surely with perturbation and time delays. Numerical examples are presented to illustrate our theoretical results.Comment: 9 pages 5 figure

On Non-Consensus Motions of Dynamical Linear Multi-Agent Systems

The non-consensus problems of high order linear time-invariant dynamical homogeneous multi-agent systems are concerned. Based on the conditions of consensus achievement, the mechanisms that lead to non-consensus motions are analyzed. Besides, a comprehensive classification for diverse types of non-consensus phases in accordance to the different conditions is conducted, which is jointly depending on the self-dynamics of agents, the interactive protocol and the graph topology. A series of numerical examples are demonstrated to illustrate the theoretical analysis

Consensus Seeking in Multi-Agent Systems with Multiplicative Measurement Noises

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 topologies cases, constant consensus gains are properly selected, such that mean square consensus and strong consensus can be achieved. Especially, exponential mean square convergence of agents' states to the common value is derived for the fixed topology case. In addition, asymptotic unbiased mean square average consensus and asymptotic unbiased strong average consensus are also studied. Simulations shed light on the effectiveness of the proposed theoretical results

Consensus of switched multi-agent systems

In this paper, we consider the consensus problem of switched multi-agent system composed of continuous-time and discrete-time subsystems. By combining the classical consensus protocols of continuous-time and discrete-time multi-agent systems, we propose a linear consensus protocol for switched multi-agent system. Based on the graph theory and Lyapunov theory, we prove that the consensus of switched multi-agent system is solvable under arbitrary switching with undirected connected graph, directed graph and switching topologies, respectively. Simulation examples are also provided to demonstrate the effectiveness of the theoretical results.Comment: 16 pages, 4 figure

Approximate Consensus Multi-Agent Control Under Stochastic Environment with Application to Load Balancing

The paper is devoted to the approximate consensus problem for networks of nonlinear agents with switching topology, noisy and delayed measurements. In contrast to the existing stochastic approximation-based control algorithms (protocols), a local voting protocol with nonvanishing step size is proposed. Nonvanishing (e.g., constant) step size protocols give the opportunity to achieve better convergence rate (by choosing proper step sizes) in coping with time-varying loads and agent states. The price to pay is replacement of the mean square convergence with an approximate one. To analyze dynamics of the closed loop system, the so-called method of averaged models is used. It allows to reduce analysis complexity of the closed loop system. In this paper the upper bounds for mean square distance between the initial system and its approximate averaged model are proposed. The proposed upper bounds are used to obtain conditions for approximate consensus achievement. The method is applied to the load balancing problem in stochastic dynamic networks with incomplete information about the current states of agents and with changing set of communication links. The load balancing problem is formulated as consensus problem in noisy model with switched topology. The conditions to achieve the optimal level of load balancing (in the sense that if no new task arrives, all agents will finish at the same time) are obtained. The performance of the system is evaluated analytically and by simulation. It is shown that the performance of the adaptive multi-agent strategy with the redistribution of tasks among "connected" neighbors is significantly better than the performance without redistribution. The obtained results are important for control of production networks, multiprocessor, sensor or multicomputer networks, etc.Comment: 10 pages, 8 figure

Robust Consensus for Multi-Agent Systems Communicating over Stochastic Uncertain Networks

In this paper, we study the robust consensus problem for a set of discrete-time linear agents to coordinate over an uncertain communication network, which is to achieve consensus against the transmission errors and noises resulted from the information exchange between the agents. We model the network by means of communication links subject to multiplicative stochastic uncertainties, which are susceptible to describing packet dropout, random delay, and fading phenomena. Different communication topologies, such as undirected graphs and leader-follower graphs, are considered. We derive sufficient conditions for robust consensus in the mean square sense. This results unveil intrinsic constraints on consensus attainment imposed by the network synchronizability, the unstable agent dynamics, and the channel uncertainty variances. Consensus protocols are designed based on the state information transmitted over the uncertain channels, by solving a modified algebraic Riccati equation.Comment: 9 pages and 3 figures. Submitted to Automatic

Critical Connectivity and Fastest Convergence Rates of Distributed Consensus with Switching Topologies and Additive Noises

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 $\delta$ (termed the extensible exponent) is proposed. With this and a balanced topology condition, we show that a critical value of $\delta$ for consensus is $1/2$. Optimization on convergence rate of this protocol is further investigated. It is proved that the fastest convergence rate, which is the theoretic optimal rate among all controls, is of the order $1/t$ for the best topologies, and is of the order $1/t^{1-2\delta}$ for the worst topologies which are balanced and satisfy the extensible joint-connectivity condition. For practical implementation, certain open-loop control strategies are introduced to achieve consensus with a convergence rate of the same order as the fastest convergence rate. Furthermore, a consensus condition is derived for non stationary and strongly correlated random topologies. The algorithms and consensus conditions are applied to distributed consensus computation of mobile ad-hoc networks; and their related critical exponents are derived from relative velocities of mobile agents for guaranteeing consensus.Comment: 36 pages, 0 figur