Title: Convergence and Stability for Control-Affine Systems with Applications to Formation Control
This talk gives an overview of a new approach to distance-based formation control. We consider systems of autonomous agents, where each agent can only measure the distances to other members of the team according to an interaction graph. The goal is to find a distributed control law that steers the agents from any given initial state to a prescribed final formation. One of the most popular strategies to solve this kind of problem is the gradient control law, where each agent is driven into the direction of steepest descent of a suitable chosen local potential function. In recent years, it was shown that this control law steers the agent to a desired formation if, for example, certain rigidity conditions are satisfied. However, an implementation of the gradient law requires information about relative positions and is therefore not applicable if only distance measurements are available. To circumvent this problem, we approximate the trajectories of the gradient control law by a sequence of trajectories from a purely distance-based control law. This way it is possible to transfer exponential stability from the multi-agent system under the gradient control law to the system under the purely distance-based control law. The underlying mathematical theory, which was developed by H.J. Sussmann in the early 1990's, is explained in the talk. It is based on convergence of trajectories of control-affine systems to trajectories of so-called extended systems, which contain Lie brackets of the control vector fields.
Raik Suttner received the B.S. and M.S. degrees in mathematics from the University of Wuerzburg, Germany, in 2013 and 2015, respectively. Since April 2016 he is a PhD student in mathematics at the University of Wuerzburg, Germany.