Source: Dissertation Abstracts International, Volume: 76-07(E), Section: B.

Adviser: A. Stephen Morse.

Access restricted by licensing agreement.

This item must not be sold to any third party vendors.

This thesis investigates two problems in distributed control of a network of autonomous agents: the formation control problem and the problem of solving linear equations in a distributed way..

Gradient control is perhaps the most comprehensive currently in existence for controlling undirected formations based on graph rigidity. One aim of this thesis is to explain what happens to such formations if neighboring agents have slightly different understandings of what the desired distance between them is supposed to be or equivalently if neighboring agents have differing estimates of what the actual distance between them is. In either case, what one expects would be a gradual distortion of the formation from its target shape as discrepancies in desired or sensed distances increase. While this is observed for the gradient laws in question, something else quite unexpected happens at the same time. It is shown that for any rigidity-based, undirected formation which is comprised of three or more agents, that if some neighboring agents have slightly different understandings of what the desired distances between them are suppose to be, then almost for certain, the trajectory of the resulting distorted but rigid formation will converge exponentially fast to a closed circular orbit in two-dimensional space which is traversed periodically at a constant angular speed. In addition this thesis also contributes to globally stabilizing a class of minimally rigid, acyclic directed formations.

A distributed algorithm is proposed for solving a linear algebraic equation of the form Ax = b where A is a matrix for which 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 [A b], the current estimates of the equation's solution generated by its neighbors, and nothing more. Each agent recursively updates its estimate of a solution by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a time-dependent directed graph N( 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" N(t), t = 1, 2, ..., the algorithm causes all agents' estimates to converge exponentially fast to the same solution to Ax = b..