Introduction aux Systèmes Collaboratifs Multi-Agentsfabio/Eng/documenti/Teaching/DR15-16/... ·...

Institute for Design and Control of Mechatronical Systems

Introduction aux Systèmes Collaboratifs Multi-Agents

UPJV, Département EEA

Fabio MORBIDI

Laboratoire MIS Équipe Perception et Robotique

E-mail: [email protected]

Année Universitaire 2015/2016

Jeudi 13h30-16h30, Salle 8

M1 EEAII - Découverte de la Recherche (ViRob)


Harald Kirchsteiger 2011/04

Consensus protocol

Directed weighted networks

2 Fabio Morbidi



Reaching consensus: directed weighted networks

3

Consider the weighted digraph in the figure, which corresponds to the first-order dynamics:

1

2

4 3

Fabio Morbidi




4

We can compactly rewrite the previous system as: 1

2

4 3 where

Recalling the definition of in-degree Laplacian for digraphs, we can rewrite the dynamics of the networked system as:

where is the underlying directed interconnection between the vertices

Fabio Morbidi



Another example:

5

• Three robots coordinate their speeds according to the chain of command in the figure

1

2 3 1

1/2 1/2 1

Fabio Morbidi


• The dynamics of the resulting system can be written as:


Harald Kirchsteiger 2011/04 6

The previous equations can be rewritten as:

where

1

2 3 1

1/2 1/2 1

The matrix in the system above corresponds to the negated in-degree Laplacian of the network, thus:

where is the weighted digraph of the network

Fabio Morbidi


Another example:




7

Are there necessary and sufficient conditions on the graph that lead to the convergence of system to the consensus set ?

As in the case of undirected networks, the rank of the Laplacian matrix and how this relates to the structure of the graph, plays a critical role

The following notion parallels that of spanning tree for undirected graphs

A digraph is a rooted out-branching if:

a) It does not contain a directed cycle

b) It has a vertex (root) such that for every other vertex there is a directed path from to

Definition (Rooted out-branching or directed rooted tree)

Fabio Morbidi




8

Example of rooted out-branching

Fabio Morbidi




9

A digraph on n vertices contains a rooted out-branching as a subgraph if and only if

Proposition

Theorem (Main result) For a digraph containing a rooted out-branching, the state trajectory

generated by initialized with , satisfies

where and , are respectively, the right and left eigenvectors associated with the zero eigenvalue of , normalized such that

As a result, one has for all initial conditions if and only if contains a rooted out-branching

Fabio Morbidi



Reaching consensus: remark I

10

Let be the left eigenvector of the digraph in-degree Laplacian associated with its zero eigenvalue. Then the quantity:

remains invariant under the consensus dynamics:

Proposition (Constant of motion)

Fabio Morbidi

Remark: The vector is a left eigenvector of the matrix with associated eigenvalue if:



Reaching consensus: remark II, balanced digraphs

11

A digraph is called balanced if, for every vertex, the in-degree and out-degree are equal

Definition (Balanced digraph)

Balanced digraph Unbalanced digraph in-degree = out-degree in-degree ≠ out-degree

Fabio Morbidi



Reaching consensus: remark II, balanced digraphs

12

When the digraph is balanced, in addition to having , one has

The consensus protocol over a digraph reaches the average consensus for every initial condition if and only if the digraph is weakly connected and balanced

Theorem

Thus, if the digraph contains a rooted out-branching and is balanced, the common value reached by the consensus protocol is the average value of the initial states, i.e. the average consensus, since

i.e.

Fabio Morbidi

Launch the Matlab file: “Rendezvous_directed.m”



Consensus protocol Some extensions for undirected networks

13 Fabio Morbidi



Extensions of the consensus protocol

1. Consensus protocol with uniform communication time-delays

“Consensus problems in networks of agents with switching topology and time-delays”, R. Olfati-Saber, R.M. Murray, IEEE Trans. Automat. Contr., 49(9):1520-1533, 2004

Consider the uniformly delayed consensus dynamics over a connected, weighted undirected graph, specified by:

for some This delayed protocol achieves average consensus if and only if

where is the largest eigenvalue of the weighted Laplacian

• Trade-off between faster convergence rate and tolerance to uniform delays on the information exchange links

Fabio Morbidi




2. Consensus protocol for double-integrator agents

Consider the following second-order dynamics for agent :

where and are, respectively, the position and velocity of agent w.r.t. an inertial frame, and is the control input (an acceleration), with

Inspired by the consensus protocol, we can define the control for agent as

“On Consensus Algorithms for Double-Integrator Dynamics”, W. Ren, IEEE Trans. Automat. Contr., 53(6):1503-1509, 2008

where is a positive gain

Fabio Morbidi




3. Discrete-time consensus protocol

An iterative form of the consensus protocol can be stated as follows in discrete time:

where is the step size, and is the maximum degree of the network

The collective dynamics of the network can be written in compact form as

where and

Fabio Morbidi




3. Discrete-time consensus protocol

• is referred to as the Perron matrix of the graph with parameter • is a stochastic matrix, i.e. the row-sum is equal to 1:

• The conditions for achieving consensus in discrete-time are the same as in continuous-time • The convergence speed to the consensus set is dictated by:

the second largest eigenvalue of

For more details on discrete-time consensus protocols and their connection to the theory of Markov chains, see “Consensus and Cooperation in Networked Multi-Agent Systems”, R. Olfati-Saber, J. A. Fax, R. M. Murray, Proc. IEEE ,Trans. Automat. Contr., 95(1):215-233, 2007

Fabio Morbidi



Sujets de projet

18 Fabio Morbidi



Sujets de projet pour les trinômes

19 Fabio Morbidi Fabio Morbidi

• Pour la recherche d’articles scientifiques liés au projet: Google Scholar

1. Contrôle de formations de robots mobiles

2. ‘‘Flocking’’ ou consensus sur l’orientation d’un groupe de robots

3. Contrôle de cohortes de micro UAVs (Unmanned Aerial Vehicles)

4. Graphes de proximité et cartes spatialement distribuées

5. Problème de la poursuite cyclique (« n-bug »)

6. Écran robotique

7. Surveillance aérienne coopérative et décentralisée

8. Contrôle de couverture avec un réseau de caméras mobiles

9. L‘algorithme de consensus et les réseaux sociaux

10. L’algorithme de consensus et l‘algorithme PageRank de Google

11. Synchronisation d‘un réseau de générateurs électriques (parcours ES)



Publications scientifiques


On peut distinguer les publications scientifiques selon leur support de parution :

1. Les revues à comité de lecture ou ‘peer-reviewed’ (jusqu‘à 20 pages)

2. Les comptes-rendus de congrès scientifique à comité de lecture (6 pages)

3. Les ouvrages collectifs rassemblant des articles de revue ou de recherche autour d'un thème donné, coordonnés par un ou plusieurs chercheurs appelés éditeurs

Maisons d‘édition les plus importantes pour l‘ingénierie: IEEE (Institute of Electrical and Electronics Engineers), Elsevier, Springer, John Wiley & Sons, Sage, ACM (Association for Computing Machinery). Visitez par example: www.ieeexplore.ieee.org et www.sciencedirect.com

Structure d‘un article scientifique Titre (10-15 mots clés) Abstract ou résumé (10-20 lignes) Introduction (état de l’art et contributions originales par rapport à la littérature) Formulation et résolution du problème Validation (simulations numériques et/ou expérimentations) Conclusions et travaux futurs Bibliographie



La recherche scientifique ...


« Doing research is a hard process, involving many erroneous assumptions, disappointments, and dead ends. Being positive, imaginative, and keeping good judgment are always helpful. Research is all about generating many, many ideas - and just making sure to keep the good ones. »

Panagiotis Tsiotras (IEEE Control Systems Magazine, april 2016)

« A researcher must accept to slog away at a problem for a hour, a day, or all his life. Rather, he uses up his energy excessively with respect to the results, he asks himself several questions, he gropes in the dark, he moves forward step by step. This is a hard task; then, at a certain point, the illumination comes. It is often unexpected but it is the result of a huge amount of unsuccessful reflections. » Laurent Schwartz (1915-2002)

Introduction aux Systèmes Collaboratifs Multi-Agentsfabio/Eng/documenti/Teaching/DR15-16/... ·...

Documents

Transcript of Introduction aux Systèmes Collaboratifs Multi-Agentsfabio/Eng/documenti/Teaching/DR15-16/... ·...