Sam Safavi

Revisiting Finite-Time Distributed Algorithms via Successive Nulling of Eigenvalues

In this letter, we characterize the finite-time behavior on arbitrary undirected graphs. In particular, we derive distributed iterations that are a function of a linear operator on the underlying graph and show that any arbitrary initial condition can be forced to lie on a particular subspace in a finite time. This subspace can be chosen to have the same dimension as the algebraic multiplicity of any (arbitrarily chosen) eigenvalue of the underlying linear operator and is spanned by the eigenvectors corresponding to the chosen eigenvalue. In other words, finite-time behavior is completely characterized by the algebraic multiplicity of the eigenvalues and the corresponding eigenvectors of the underlying linear operator. We show that finite-time average-consensus can be cast naturally in this setup for which we further develop the necessary and sufficient conditions.

Leader-Follower Consensus in Mobile Sensor Networks

In this letter, we study the consensus-based leader-follower algorithm in mobile sensor networks, where the goal for the entire network is to converge to the state of a leader. We capture the mobility in the leader-follower algorithm by abstracting it as a Linear Time-Varying (LTV) system with random system matrices. In particular, a mobile node, moving randomly in a bounded region, updates its state when it finds neighbors; and does not update when it is not in the communication range of any other node. In this context, we develop certain regularity conditions on the system and input matrices such that each follower converges to the leader state. To analyze the corresponding LTV system, we partition the entire chain of system matrices into non-overlapping slices, and relate the convergence of the sensor network to the lengths of these slices. In contrast to the existing results, we show that a bounded length on the slices, capturing the dissemination of information from the leader to the followers, is not required; as long as the slice-lengths are finite and do not grow faster than a certain exponential rate.

Unbounded connectivity: Asymptotic stability criteria for stochastic LTV systems

In this paper, we study the asymptotic stability of linear time-varying systems with (sub-) stochastic system matrices. Motivated by applications in distributed dynamic fusion (DDF), we impose some mild regularity conditions on the elements of time-varying system matrices, and provide sufficient conditions under which the asymptotic stability of the underlying LTV system is guaranteed. We partition the sequence of system matrices into non-overlapping slices, and by introducing the notion of unbounded connectivity, we obtain stability conditions in terms of the slice lengths and some network parameters. In particular, we show that the system is asymptotically stable if the unbounded lengths of an infinite subset of slices grow slower than an explicit exponential rate.

Asymptotic Stability of LTV Systems With Applications to Distributed Dynamic Fusion

We investigate the behavior of Linear Time-Varying (LTV) systems with randomly appearing, sub-stochastic system matrices. Motivated by dynamic fusion over mobile networks, we develop conditions on the system matrices that lead to asymptotic stability of the underlying LTV system. By partitioning the sequence of system matrices into slices, we obtain the stability conditions in terms of slice lengths and introduce the notion of unbounded connectivity , i.e., the time-intervals, over which the multi-agent network is connected, do not have to be bounded as long as they do not grow faster than a certain exponential rate. We further apply the above analysis to derive the asymptotic behavior of a dynamic leader-follower algorithm.

On the convergence of time-varying fusion algorithms: Application to localization in dynamic networks

In this paper, we study the convergence of dynamic fusion algorithms that can be modeled as Linear Time-Varying (LTV) systems with (sub-) stochastic system matrices. Instead of computing the joint spectral radius, we partition the entire set of system matrices into slices, whose lengths characterize the stability (convergence) of the underlying LTV system. We relate the lengths of the slices to the rate of the information flow within the network, and show that fusion is achieved if the unbounded slice lengths grow slower than an explicit exponential rate. As a motivating application, we provide a distributed algorithm to track the positions of an arbitrary number of agents moving in a bounded region of interest. At each iteration, agents update their position estimates as a convex combination of the states of the neighbors, if they are able to find enough neighbors; and do not update otherwise. We abstract the corresponding position tracking algorithm as an LTV system, and introduce the notion of slices to provide sufficient conditions under which the system asymptotically converges to the true positions of the agents. We demonstrate the effectiveness of our approach through simulations.

A distributed range-based algorithm for localization in mobile networks

In this paper, we provide a distributed algorithm to locate an arbitrary number of agents moving in a bounded region. Assuming that each agent can estimate a noisy version of its motion and the distances to the nodes in its communication radius, we provide a simple linear update to find the locations of an arbitrary number of mobile agents when they follow some convexity in their deployment and motion, given at least one anchor, agent with known location, is present in Rm. At each iteration, agents update their location estimates as a convex combination of the states of the neighbors, if they lie inside their convex hull, and do not update otherwise. We abstract the corresponding localization algorithm as a Linear Time-Varying (LTV) system, and using slice notation we show that it asymptotically converges to the true locations of the agents. We study the effects of noise on our localization algorithm, and provide simulations to verify our analytical results.

On the convergence rate of swap-collide algorithm for simple task assignment

This paper provides a convergence rate analysis of the swap-collide algorithm for simple assignment problems. Swap-collide is a distributed algorithm that assigns a unique task to each agent assuming that the cost of each assignment is identical and has applications in resource-constrained multiagent systems; prior work has shown that this assignment procedure converges in finite-time. In this paper, we provide an analytical framework to establish the convergence rate of swap-collide, and show that for a network of size N, the lower and upper bounds for the convergence rate are O(N3).

Distributed Localization: A Linear Theory

Fifth-generation (5G) networks providing much higher bandwidth and faster data rates will allow connecting vast number of stationary and mobile devices, sensors, agents, users, machines, and vehicles, supporting Internet-of-Things (IoT), real-time dynamic networks of mobile things. Positioning and location awareness will become increasingly important, enabling deployment of new services and contributing to significantly improving the overall performance of the 5G system. Many of the currently talked about solutions to positioning in 5G are centralized, mostly requiring direct communication to the access nodes (or anchors, i.e., nodes with known locations), which in turn requires a high density of anchors. But such centralized positioning solutions may become unwieldy as the number of users and devices continues to grow without limit in sight. As an alternative to the centralized solutions, this paper discusses distributed localization in a 5G-enabled IoT environment where many low power devices, users, or agents are to locate themselves without a direct access to anchors. Even though positioning is essentially a nonlinear problem (solving circle equations by trilateration or triangulation), we discuss a cooperative linear distributed iterative solution with only local measurements, local communication, and local computation needed at each agent. Linearity is obtained by reparametrization of the agent location through barycentric coordinate representations based on local neighborhood geometry that may be computed in terms of certain Cayley–Menger determinants involving relative local inter-agent distance measurements. After a brief introduction to the localization problem, and other available distributed solutions primarily based on directly addressing the nonlinear formulation, we present the distributed linear solution for stationary agent networks and study its convergence, its robustness to noise, and extensions to mobile scenarios, in which agents, users, and (possibly) anchors are dynamic.

Dynamic leader-follower algorithms in mobile multi-agent networks

In this paper, we consider a Linear Time-Varying (LTV) model to describe the dynamics of a leader-follower algorithms with mobile agents. We first develop regularity conditions on the LTV system matrices, according to a random motion of the agents and the underlying communication protocol. We then study the convergence of all agents to the state of the leader, and show that this requires the underlying LTV system to be asymptotically stable. We introduce the notion of slices as non-overlapping partitions of the LTV system matrices, and relate the convergence of the multi-agent network to the length of these slices. Finally, we demonstrate the convergence and steady-state of a dynamic leader-follower network through simulations.