# M21 â€ ST PETERBURG 02/05/2016â€06/05/2016

## Distributed control and computation

**A. Stephen Morse**

Department of Electrical Engineering

Yale University, USA

http://www.eng.yale.edu/controls/

morse@sysc.eng.yale.edu

### Abstract of the course Â

Over the past decade there has been growing in interest in distributed control problems of all types. Among these are consensus problems including flocking and distributed averaging, the multiagent rendezvous problem, and the distributed control of multi-agent formations. The aim of these lectures is to explain what these problems are and to discuss their solutions. Related concepts from spectral graph theory, rigid graph theory, nonhomogeneous Markov chain theory, stability theory, and linear system theory will be covered. Among the topics discussed are the following.**Flocking:**We will present graph-theoretic results appropriate to the analysis of a variety of consensus problems cast in dynamically changing environments. The concepts of rooted, strongly rooted, and neighbor -shared graphs will be defined, and conditions will be derived for compositions of sequences of directed graphs to be of these types. As an illustration of the use of the concepts covered, graph theoretic conditions will be derived which address the convergence question for the widely studied flocking problem in which there are measurement delays, asynchronous events, or a group leader.

**Distributed Averaging:**By the distributed averaging problem is meant the problem of computing the average value of a set of numbers possessed by the agents in a distributed network using only communication between neighboring agents. We will discuss a variety of double linear iterations and deadlock-free, deterministic gossiping protocols for doing distributed averaging.

**Formation Control:**We will review recent results concerned with the maintenance of formations of mobile autonomous agents {eg robots} based on the idea of a rigid framework. We will talk briefly about certain classes of â€œdirectedâ€ rigid formations for which there is a moderately complete methodology. We will describe recently devised potential function based gradient laws for asymptotically stabilizing â€œundirectedâ€ rigid formations and we will illustrate and explain what happens when neighboring agents using such gradient laws have slightly different understandings of what the desired distance between them is suppose to be.

### Topics will include:

1. Flocking and consensus2. Distributed averaging via broadcasting

3. Gossiping and double linear iterations

4. Multi-agent rendezvous

5. Control of formations

6. Convergence rates

7. Asynchronous behavior

8. Consensus-based approach to solving a linear equation

9. Stochastic matrices, graph composition, rigid graphsperformance recovery via