Average Consensus Distributed Algorithms for Multi-Agent Networks Instructor: K. Sinan YILDIRIM

Graphs and Matrices Several matrixes can be associated to graphs – Several graph properties can be deduced from the associated matrixes Graphs + Matrixes = Algebraic Graph Theory – Algebraic tools will be fundamental for linking Graph Theory to the study of multi-agent systems when seen as a collection of dynamical systems

Adjacency Matrix

Degree Matrix

Incidence Matrix

Laplacian Matrix

Consensus Protocol Consider agents with an internal state x i Consider an internal dynamics for the state evolution – in our case, single integrator x i =u i Consider an interaction graph G having the agents as vertexes Problem: design the control inputs so that – all the states x i agree on the same common value – by making use in u i of only relative information w.r.t. the neghbors’ state

Possible Applications rendezvous: meet at a common point (uniform the positions) alignment: point in the same direction (uniform the angles) distributed estimation: agree on the estimation of some distributed quantity (e.g., average temperature) synchronization: agree on the same time (regardless of phase shifts or different rates in the clocks)

Consensus Protocol - I Let u i be the sum of all the differences of the neighbors’ states w.r.t. the state of agent

Consensus Protocol - II Fact 1: Fact 2:

Consensus Protocol - III average of the initial state x 0

Example