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]
M1 EEAII - Découverte de la Recherche (ViRob)
Consensus protocol
Directed weighted networks
Consider the weighted digraph in the figure, which corresponds to the first-order dynamics :
1
2
Reaching consensus: directed weighted networks
We can compactly rewrite the previous system as:
1
2
3 4
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
Reaching consensus: directed weighted networks
Another example:
• Three robots coordinate their speeds
according to the chain of command in the figure
1
3 2
1 1/2
1/2 1
Reaching consensus: directed weighted networks
• The dynamics of the resulting system can be written as:
The previous equations can be rewritten as:
where
1
3 2
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
Reaching consensus: directed weighted networks
Reaching consensus: directed weighted networks
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)
Reaching consensus: directed weighted networks
Example of rooted out-branching
root
Reaching consensus: directed weighted networks
A digraph on 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
Reaching consensus: directed weighted networks
Example:
root
Blue: a rooted out-branching of (not unique)
Reaching consensus: remark I
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)Remark:
The vector is a left eigenvector of the squre matrix with associated
eigenvalue if:
Reaching consensus: remark II, balanced digraphs
A digraph is called balanced if, for every vertex, the in-degree and out-degree are equal
Definition (Balanced digraph)
Balanced digraph (unitary weights) Unbalanced digraph (unitary weights)
in-degree = out-degree in-degree ≠ out-degree
Reaching consensus: remark II, balanced digraphs
Balanced digraph Unbalanced digraph
in-degree = out-degree in-degree ≠ out-degree
2
3 5
2
5
5
1
2 4
3
1/2
2
Reaching consensus: remark II, balanced digraphs
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.
Launch the Matlab file:
“Rendezvous_directed.m”
Consensus protocol
Some extensions for undirected networks
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., vol. 49, n. 9, pp. 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 linksExtensions of the consensus protocol
τ < π
2 λ
n(L(G ))
λ
n(L(G ))
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 (acceleration), with
Inspired by the consensus protocol, we can define the control for agent as:
where is a positive gain
Extensions of the consensus protocol
or equivalently
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 graph
The collective dynamics of the network can be written in compact form as
where and
Extensions of the 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, in Proc. IEEE, vol. 95, n. 1, pp. 215-233, 2007
Extensions of the consensus protocol
3. Discrete-time consensus protocol
• Antagonistic interactions (e.g. friends vs. adversaries in a social network), can be modeled as negative
weights on the communication graph
• On signed networks all agents can converge to a consensus value which is the same for all agents except for the sign (“bipartite consensus”)
• Bipartite consensus can be achieved if the graph is structurally balanced (this means that all cycles of the graph are positive, i.e. they contain an even
number of negative edge weights)
“Consensus Problems on Networks With Antagonistic Interactions”, C. Altafini, in IEEE Trans. Automat. Contr., vol. 58, n. 4, pp. 935-946, 2013
"Predictable Dynamics of Opinion Forming for Networks with Antagonistic Interactions", C. Altafini, G. Lini, in IEEE Trans. Automat. Contr., vol. 60, n. 2, pp. 342-357, 2015
Extensions of the consensus protocol
4. Consensus protocol with antagonistic interactions
Bipartite consensus
time [s]
Group 1
Group 2
Extensions of the consensus protocol
Example (signed undirected graphs, ):
-2 -4
1
-2 4
1
2 4
1
Structurally balanced Structurally unbalanced Structurally balanced 4. Consensus protocol with antagonistic interactions
Sujets de projet
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1. Les revues à comité de lecture ou ‘peer-reviewed’ (jusqu‘à 20 pages) 2. Les comptes-rendus (‘Proceedings’) de congrès scientifique à comité de
lecture (typiquement 6 pages)
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Introduction (état de l’art et contributions par rapport à la littérature) Formulation et résolution du problème
Validation (simulations numériques et/ou expérimentations)
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Auteurs et affiliations
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