Consensus protocol

Institute for Design and Control of Mechatronical Systems

Introduction aux Systèmes Collaboratifs Multi-Agents

UPJV, Département EEA

Fabio MORBIDI

Laboratoire MIS

Équipe Perception Robotique E-mail: [email protected]

Jeudi 13h30-16h30, Salle 8

M1 EEAII - Découverte de la Recherche (ViRob)

Introduction

  Agreement is one of the fundamental problems in multi-agent coordination

–  A collection of agents are to agree on a joint state value

  We will study the dynamics of the so-called consensus protocol over static undirected and directed networks

  We will see the interdependency between the convergence properties of the consensus protocol and the structural attributes of the underlying network

Consensus protocol

Undirected networks

Consensus protocol: a motivating example

  A group of sensors are to measure the temperature in Picardie

20^o 17^o

19^o 20^o

21^o 22^o

19^o

22^o

Consensus protocol: a motivating example

  Using an information-sharing network, the sensor group is required to agree on a single value (e.g. 20^o) which represents the temperature of the region

  For this, the sensor group needs a protocol over the network, allowing it to reach consensus on what the common sensor measurement value should be

20^o 20^o

20^o

20^o 20^o

20^o 20^o

20^o

Consensus protocol

  The consensus protocol involves dynamic units, interconnected via relative information-exchange links

  The rate of change of the state (e.g. the temperature) of each unit is

assumed to be governed by the sum of the relative states with respect to the neighboring units

Example:

1 2

3 4

Consensus protocol

  Denoting the scalar state of node as one has then:

where is the neighborhood of node in the network (i.e. set of nodes adjacent to ).

  The overall system can be rewritten in vector form as:

•  is the Laplacian of the agents‘ interaction network

• 

Consensus dynamics

Consensus protocol

  Remark (Case of nonscalar states)

If (e.g. the 2D position of a robot) one can still obtain a

compact description of the consensus dynamics using the Kronecker product The Kronecker product of two matrices and , is the block matrix:

We then have that:

where is the identity matrix and

Consensus protocol: electrical example

  Example (Resistor-capacitor circuit)

•  Let us suppose that R = 1 Ω, C = 1 F.

•  Kirchhoff‘s current and voltage laws lead to:

C

C C

R

R

which describes the dynamics

R

of the capacitors‘ voltages

represents the set of nodes in the circuit that are connected to node via a resistor

Consensus set

  From the previous examples we note that the state of each vertex in the network is ‘‘pulled“ toward the states of the neighboring vertices

  Will all vertices reach some weighted average of their initial states, which also corresponds to the fixed point of their collective dynamics?

The consensus set is the subspace i.e.

Definition (Consensus set)

Example:

Reaching consensus: undirected networks

The spectrum of the Laplacian for a connected undirected graph is:

and is the eigenvector corresponding to the zero eigenvalue Consider the spectral factorization of the Laplacian ( is a symmetric matrix):

where

is the

n

n

and

  The solution of initialized with is

Using the spectral factorization of the Laplacian, we get:

Reaching consensus: undirected networks

Let be a connected graph. Then the protocol converges to the consensus set with a rate of convergence that is dictated by

Proof:

Theorem

The proof follows directy from the equation

by observing that for a connected graph for as always Thus,

and hence As is the smallest positive eigenvalue of the Laplacian, it dictates the slowest mode of convergence in the limit above

(Average Consensus)

Reaching consensus: undirected networks

Example

The Laplacian spectrum of is:

Complete graph

Cycle

The Laplacian spectrum of is:

Then Then

Reaching consensus: undirected networks

Reaching consensus: remark I

Therefore, the quantity

i.e. the centroid of the network states, is a constant of motion for the consensus dynamics

Note that:

  The state trajectory generated by the consensus protocol converges to the projection of its initial state onto the consensus subspace, since

Reaching consensus: remark II

A spanning tree of a connected graph is a tree composed of all the vertices and some (or perhaps all) of the edges of

Informally, a spanning tree of is a selection of edges of that form a tree spanning every vertex

Definition (Spanning tree) Definition (Tree)

A tree is connected graph without cycles

Reaching consensus: remark III

Proposition

A necessary and sufficient condition for the consensus protocol

to converge to the consensus set from an arbitrary initial condition , is that the underlying graph containts a spanning tree

Example:

Two possible spanning trees

of (blue)

Reaching consensus: remark III

Example 1: rendezvous problem

  A collection of mobile agents with single integrator dynamics

  This location is not given in advance and the agents do not have access to their global positions (no GPS)

  The agents can only measure the relative displacement w.r.t. their neighbors are to meet at a single location. Here, denotes the position of agent

Consensus protocol ‘‘in action“

Consider the following control input for agent : Solution:

Example 1: rendezvous problem

Rendezvous point

  Simulation (Network topology: )

Launch the Matlab file:

“Rendezvous.m”

  Consider different network topologies and study the rate of convergence to the rendezvous point

  Study the behavior of the system when all the agents initially lie on the same line (see the figure below)

Exercise

a₁

a₂

a₃

a

Example 1: rendezvous problem

Remark Wheel graph Petersen graph

with 10 nodes

Example 2: cyclic pursuit (‘‘ n -bug problem‘‘)

  Consider

n

and suppose that we want agent to pursue agent modulo

n

  We can then select the following control input

for agent :

1

2

3

4

where is a positive constant

  This results in the following system:

where and

Circulant matrix

1

2

3

Example 2: cyclic pursuit (‘‘ n -bug problem‘‘)

4