Search

This paper proposes scalable, distributed algorithms for solving linear equations by integrating two mechanisms, termed consensus and conservation, in double-layered multiagent networks. The multiagent network considered in this paper is composed of clusters and each cluster consists of an aggregator and a subnetwork of agents. By achieving consensus and conservation through agent–agent communications in the same cluster and aggregator–aggregator communications among different clusters, respectively, distributed algorithms are devised for agents to cooperatively achieve a solution to the overall linear equation. These algorithms outperform existing algorithms, including but not limited to the following aspects—first, each agent does not have to know as much as a complete row or column of the overall equation; second, each agent only needs to control as few as two scalar states when the number of clusters and the number of agents are sufficiently large; third, the dimensions of agents’ states in the proposed algorithms do not have to be the same (while in contrast, algorithms based on the idea of standard consensus inherently require all agents’ states to be of the same dimension). Both analytical proof and simulation results are provided to validate exponential convergence of the proposed distributed algorithms in solving linear equations. [ABSTRACT FROM AUTHOR]

Two asynchronous distributed algorithms are presented for solving a linear equation of the form $Ax=b$ with at least one solution. The equation is simultaneously and asynchronously solved by $m$ agents assuming that each agent knows only a subset of the rows of the partitioned matrix $[A\ \ b]$, the estimates of the equation's solution generated by its neighbors, and nothing more. Neighbor relationships are characterized by a time-dependent directed graph whose vertices correspond to agents and whose arcs depict neighbor relationships. Each agent recursively updates its estimate of a solution at its own event times by utilizing estimates generated by its neighbors which are transmitted with delays. The event time sequences of different agents are not assumed to be synchronized. It is shown that for any matrix-vector pair $(A, b)$ for which the equation has a solution and any repeatedly jointly strongly connected sequence of neighbor graphs defined on the merged sequence of all agents’ event times, the algorithms cause all agents’ estimates to converge exponentially fast to the same solution to $Ax=b$ . The first algorithm requires a specific initialization step at each agent, and the second algorithm works for arbitrary initializations. Explicit expressions for convergence rates are provided, and a relation between local initializations and limiting consensus solutions is established, which is used to solve the least 2-norm solution. [ABSTRACT FROM PUBLISHER]

Use of a gradient descent control law has been a popular method to effectively stabilize undirected rigid formations, by assuming that interagent distances between a certain set of neighboring agent pairs can be accurately specified and measured. This paper examines the collective motion behavior for an infinitesimally rigid formation in a three-dimensional ambient space, in the case that neighboring agent pairs have slightly differing views or measurements of the desired interagent distances they are tasked to maintain. It is shown that the formation shape will converge exponentially fast to a rigid one, while additional rigid helical motions of the final formation will occur. We further discuss the convergence to the equilibrium motions, and derive certain motion parameter formulas to describe the rigid formation movements by employing the angular momentum concept from classical mechanics. Finally, we explain how the idea can be used for steering a rigid formation to move as a rigid body. [ABSTRACT FROM AUTHOR]

A distributed algorithm is described for solving a linear algebraic equation of the form Ax = b assuming the equation has at least one solution. The equation is simultaneously solved by m agents assuming each agent knows only a subset of the rows of the partitioned matrix [\matrixA & b], the current estimates of the equation's solution generated by its neighbors, and nothing more. Each agent recursively updates its estimate by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a time-dependent directed graph \BBN(t) whose vertices correspond to agents and whose arcs depict neighbor relations. It is shown that for any matrix $A$ for which the equation has a solution and any sequence of “repeatedly jointly strongly connected graphs” $\BBN(t)$, $t = 1, 2, \ldots$, the algorithm causes all agents' estimates to converge exponentially fast to the same solution to $Ax = b$. It is also shown that, under mild assumptions, the neighbor graph sequence must actually be repeatedly jointly strongly connected if exponential convergence is to be assured. A worst case convergence rate bound is derived for the case when $Ax = b$ has a unique solution. It is demonstrated that with minor modification, the algorithm can track the solution to $Ax = b$, even if $A$ and $b$ are changing with time, provided the rates of change of $A$ and $b$ are sufficiently small. It is also shown that in the absence of communication delays, exponential convergence to a solution occurs even if the times at which each agent updates its estimates are not synchronized with the update times of its neighbors. A modification of the algorithm is outlined which enables it to obtain a least squares solution to $Ax = b$ in a distributed manner, even if $Ax = b$ does not have a solution. [ABSTRACT FROM AUTHOR]

This paper proposes a continuous-time distributed algorithm for multiagent networks to achieve a solution with the minimum $l_1$ -norm to underdetermined linear equations. The proposed algorithm comes from a combination of the Filippov set-valued map with the projection-consensus flow. Given the underlying network is undirected and fixed, it is shown that the proposed algorithm drives all agents’ individual states to converge in finite time to a common value, which is the minimum $l_1$ -norm solution. [ABSTRACT FROM AUTHOR]

In this technical note, a distributed algorithm is proposed for multiagent networks to achieve a least squares solution of a system of linear equations, in which each agent only knows part of the overall equations and communicates only with its nearby neighbors. The proposed algorithm is discrete time but does not involve small or time-varying step sizes. Given that the network is fixed, connected, and undirected, the proposed algorithm enables all agents in the network to achieve exponentially fast the same least squares solution; this is validated by simulations. [ABSTRACT FROM AUTHOR]

The distributed averaging problem is a consensus problem whose objective is to devise a protocol which will enable all the members of a group of autonomous agents to compute the average of the initial values of their individual consensus variables in a distributed manner. Periodic gossiping is a deterministic method for solving the distributed averaging problem by stipulating that each pair of agents which are allowed to gossip, do so repeatedly in accordance with a prespecified periodic schedule. Agent pairs which are allowed to gossip correspond to edges on a given connected, undirected graph. In general, the rate at which the agents' consensus variables converge to the desired average value depends on the order in which the gossips occur over a period. The main contributions of this paper are first to characterize the classes of periodic gossip sequences which have the same convergence rate and second to prove that if the graph of allowable gossips is a tree with each edge restricted to gossiping once per period, the convergence rate is quite surprisingly, fixed and invariant over all possible periodic gossip sequences the graph allows. To arrive at these results, a new and unusual graph theoretic concept, namely the transfer function of a node of an undirected graph, is used. Among all the trees with the same number of edges, optimal tree structures, which yield the fastest convergence rate, can then be sought. [ABSTRACT FROM AUTHOR]