Stability of continuous-time distributed consensus algorithms

Moreau, Luc

doi:10.1109/cdc.2004.1429377

Cited by 573 publications

(512 citation statements)

References 30 publications

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“…The following preliminary result of [93] will be needed in Section 7.2. First, for any δ > 0 and any matrix B, the δ-digraph is defined as the digraph where edge (i, j) exists if B ij ≥ δ.…”

Section: Preliminary Resultsmentioning

confidence: 99%

Submodularity in Dynamics and Control of Networked Systems

Clark

Alomair

Bushnell

et al. 2016

Communications and Control Engineering

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Submodularity in Dynamics and Control of Networked Systems Andrew ClarkCo-Chairs of the Supervisory Committee:Radha Poovendran Electrical Engineering Linda Bushnell Electrical EngineeringControlling a networked dynamical system to reach a desired state is a fundamental challenge in applications including transportation, energy, social, and biological systems. One scalable approach to controlling such systems is to directly control the states of a subset of leader nodes, while relying on local interactions to steer the remaining nodes towards their desired states. The choice of leader nodes is known to affect system metrics including robustness to noise, rate of convergence to a desired state, and controllability of the system.Selecting an optimal subset of leader nodes, however, is inherently a combinatorial problem, making optimal leader node selection intractable in general.This thesis presents a submodular optimization framework to selecting leader nodes for control of networked systems. We investigate the problem of selecting a subset of leader nodes in order to minimize node state errors due to noise in the communication links between nodes. We prove that the error due to link noise is a supermodular function of the set of leader nodes, leading to the first polynomial-time algorithms for minimizing error due to link noise with provable optimality bounds. We develop our approach for networks with static topologies, as well as dynamic topologies due to random link failures, switching between predefined topologies, and arbitrary mobility.We study selecting leader nodes in order to minimize convergence error, defined as the error in the intermediate node states prior to reaching their desired values. We derive upper bounds for a class of convergence error metrics based on the hitting time of random walk on the network, which we prove to be a supermodular function of the set of input nodes.We present polynomial-time algorithms for minimizing convergence error with provable optimality bounds, for static as well as dynamic networks.Efficient algorithms have recently been proposed for selecting leader nodes to satisfy controllability, defined as the ability to drive the non-input nodes from any initial state to any desired state in finite time. These algorithms, however, do not incorporate performance criteria including robustness to noise and convergence rate. We study the problem of leader selection for joint performance and controllability, and prove that controllability can be formulated as a matroid constraint on the set of leader nodes. We propose efficient algorithms with provable optimality gap for selecting leader nodes for joint performance and controllability, and characterize the submodular structure of the largest controllable subgraph of a network.We also investigate selecting input nodes for guaranteeing synchronization in networked systems. Using the widely-studied Kuramoto model of nonlinear phase-coupled oscillators, we develop novel threshold-based conditions for a set of input nodes to ...

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Section: Preliminary Resultsmentioning

confidence: 99%

Submodularity in Dynamics and Control of Networked Systems

Clark

Alomair

Bushnell

et al. 2016

Communications and Control Engineering

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“…Therefore, from Theorem 1 in [14], we conclude that the variables ω a k , b a k and c a k asymptotically converge to the consensus valuesω,b andc respectively.…”

Section: Stabilization Of Relative Equilibria In the Presence Of Lmentioning

confidence: 69%

“…Since s j = s av , and x j is parallel to ω for every j ∈ G 2 , we conclude from (23) that this set does not correspond to global minima of (14). It can be shown that under suitable perturbations of x k , k ∈ G 2 (such that s av is constant) (23) is decreasing and therefore conditions (22) define an unstable set (the details are omitted due to space constraints).…”

Section: A Stabilization About An Axis Of Rotation With a Fixed Dirementioning

confidence: 94%

Stabilization of collective motion in three dimensions: A consensus approach

Scardovi

Leonard

Sepulchre

2007

2007 46th IEEE Conference on Decision and Control

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Abstract-This paper proposes a methodology to stabilize relative equilibria in a model of identical, steered particles moving in three-dimensional Euclidean space. Exploiting the Lie group structure of the resulting dynamical system, the stabilization problem is reduced to a consensus problem. We first derive the stabilizing control laws in the presence of allto-all communication. Providing each agent with a consensus estimator, we then extend the results to a general setting that allows for unidirectional and time-varying communication topologies.

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“…Flocking dynamics described by Reynolds' rule are known to be asymptotically stable under fairly weak conditions on the topology of the underlying graph [17], [18], [24], [25]. Stronger results, such as exponential stability, have been found for linear time-varying consensus protocol for single integrator systems [26], in the context of synchronization for second-order Euler-Lagrange systems [20]- [22], and using tools from dynamical systems theory for time-invariant, undirected graph topologies in second-order, two-dimensional (2-D) flocking dynamics [27]. Preliminary results on exponential stability of flocks under tree and star topology constraints are presented in [8].…”

Section: A Overview Of the Literaturementioning

confidence: 90%

Robotic Herding of a Flock of Birds Using an Unmanned Aerial Vehicle

et al. 2018

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Abstract-In this paper, we derive an algorithm for enabling a single robotic unmanned aerial vehicle to herd a flock of birds away from a designated volume of space, such as the air space around an airport. The herding algorithm, referred to as the mwaypoint algorithm, is designed using a dynamic model of bird flocking based on Reynolds' rules. We derive bounds on its performance using a combination of reduced-order modeling of the flock's motion, heuristics, and rigorous analysis. A unique contribution of the paper is the experimental demonstration of several facets of the herding algorithm on flocks of live birds reacting to a robotic pursuer. The experiments allow us to estimate several parameters of the flocking model, and especially the interaction between the pursuer and the flock. The herding algorithm is also demonstrated using numerical simulations.

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Stability of continuous-time distributed consensus algorithms

Cited by 573 publications

References 30 publications

Submodularity in Dynamics and Control of Networked Systems

Submodularity in Dynamics and Control of Networked Systems

Stabilization of collective motion in three dimensions: A consensus approach

Robotic Herding of a Flock of Birds Using an Unmanned Aerial Vehicle

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