This paper investigates control laws allowing mobile, autonomous agents to optimally position themselves on the line for distributed sensing in a nonuniform field. We show that a simple static control law, based only on local measurements of the field by each agent, drives the agents close to the optimal positions after the agents execute in parallel a number of sensing movement computation rounds that is essentially quadratic in the number of agents. Further, we exhibit a dynamic control law which, under slightly stronger assumptions on the capabilities and knowledge of each agent, drives the agents close to the optimal positions after the agents execute in parallel a number of sensing communication computation movement rounds that is essentially linear in the number of agents. Crucially, both algorithms are fully distributed and robust to unpredictable loss and addition of agents.
In @cite_0 , uniform coverage algorithms are derived using Voronoi cells and gradient laws for distributed dynamical systems. Uniform constrained coverage control is addressed in @cite_39 where the constraint is a minimum limit on node degree. Virtual potentials enable repulsion between agents to maximize coverage and attraction between agents to enforce the constraint. In @cite_10 , gradient control laws are proposed to move sensors to a configuration that maximizes expected event detection frequency. Local rules are enforced by defining a sensing radius for each agent, which also makes computations simpler. The approach is demonstrated for a nonuniform but symmetric density field with and without communication constraints. Further results for distributed coverage control are presented in @cite_37 for a coverage metric defined in terms of the Euclidean metric with a weighting factor that allows for nonuniformity. As in @cite_37 , the methodology makes use of Voronoi cells and Lloyd descent algorithms. The papers @cite_15 @cite_25 identified a class of non-convex regions for which the coverage problem may be solved by reduction to the convex case through a well-chosen transformation of the region. The papers @cite_36 @cite_16 explored an optimization-based approach to some complex variations of the covering problem.
@cite_37 @cite_36 @cite_39 @cite_0 @cite_15 @cite_16 @cite_10 @cite_25
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In this paper we address the problem of dynamic coverage control of a convex polygonal region in the plane using N agents with bounded velocities and finite instantaneous area of coverage. The proposed coverage algorithm guarantees finite- time search of the region, does not depend on gradient-based methods, and can be carried out by the agents in a distributed fashion. We provide an upper bound on the completion time as well as the number of messages that need to be exchanged by the agents. A simulation is provided to illustrate the algorithm. In this paper, we investigate nonuniform coverage of a planar region by a network of autonomous, mobile agents. We derive centralized nonuniform coverage control laws from uniform coverage algorithms using cartograms, transformations that map nonuniform metrics to a near Euclidean metric. We also investigate time-varying coverage metrics and the design of control algorithms to cover regions with slowly varying, nonuniform metrics. Our results are applicable to the design of mobile sensor networks, notably when the coverage metric varies as data is collected such as in the case of an information metric. The results apply also to the study of animal groups foraging for food that is nonuniformly distributed and possibly changing. In Part I of this paper, we proposed and analyzed a novel algorithmic framework for the minimization of a nonconvex (smooth) objective function, subject to nonconvex constraints, based on inner convex approximations. This Part II is devoted to the application of the framework to some resource allocation problems in communication networks. In particular, we consider two non-trivial case-study applications, namely: (generalizations of) i) the rate profile maximization in MIMO interference broadcast networks; and the ii) the max-min fair multicast multigroup beamforming problem in a multi-cell environment. We develop a new class of algorithms enjoying the following distinctive features: i) they are across the base stations (with limited signaling) and lead to subproblems whose solutions are computable in closed form; and ii) differently from current relaxation-based schemes (e.g., semidefinite relaxation), they are proved to always converge to d-stationary solutions of the aforementioned class of nonconvex problems. Numerical results show that the proposed (distributed) schemes achieve larger worst-case rates (resp. signal-to-noise interference ratios) than state-of-the-art centralized ones while having comparable computational complexity. We define and analyze a random coverage process of the @math -dimensional Euclidian space which allows one to describe a continuous spectrum that ranges from the Boolean model to the Poisson-Voronoi tessellation to the Johnson-Mehl model. Like for the Boolean model, the minimal stochastic setting consists of a Poisson point process on this Euclidian space and a sequence of real valued random variables considered as marks of this point process. In this coverage process, the cell attached to a point is defined as the region of the space where the effect of the mark of this point exceeds an affine function of the cumulated effect of all marks. This cumulated effect is defined as the shot noise process associated with the marked point process. In addition to analyzing and visualizing this continuum, we study various basic properties of the coverage process such as the probability that a point or a pair of points be covered by a typical cell. We also determine the distribution of the number of cells which cover a given point, and show how to provide deterministic bounds on this number. Finally, we also analyze convergence properties of the coverage process using the framework of closed sets, and its differentiability properties using perturbation analysis. Our results require a pathwise continuity property for the shot noise process for which we provide sufficient conditions. The model in question stems from wireless communications where several antennas share the same (or different but interfering) channel(s). In this case, the area where the signal of a given antenna can be received is the area where the signal to interference ratio is large enough. We describe this class of problems in detail in the paper. The obtained results allow one to compute quantities of practical interest within this setting: for instance the outage probability is obtained as the complement of the volume fraction; the law of the number of cells covering a point allows one to characterize handover strategies etc.