Consensus under Communication Delays Alexandre Seuret* Dimos V. Dimarogonas** Karl H. Johansson*** * NeCS-team GIPSA-lab CNRS/INRIA RA Grenoble, France
** MIT Laboratory for Information and Decision Systems Cambridge, MA, U.S.A
31 Mars 2009
*** Dep. of Automatic Control ACCESS Linnaeus Centre KTH, Stockholm, Sweden
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Introduction •
Decentralized consensus control of multi-agent systems: agents aim at attaining a common value of some value function with limited information on the other agents’s goals/states.
•
Applications: multi-robot systems, distributed estimation and filtering in networked systems.
Influence of the communication network on the consensus control: Communication delays
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Outline 1) Problem formulation 2) Model transformation 3) Stability analysis a) Existence of a consensus b) Arbitrary networks c) Symmetric networks
4) Discussions on the consensus equilibrium 5) Example 6) Conclusion & Perspectives 3
1. Problem formulation
Introduction of communication delays…Where? τ
τ
2
4
τ
[Olfati-Saber et al ,04 07], …
1
3
0
5
2
[Olfati-Saber et al ,07], [Moreau, 04,05],...
1
3
4
5
A: Adjacency matrix
Δ: Diagonal matrix L=-(Δ-A) : Laplacian + Conserve averaging properties - Not realistic in case of unknown delays, packet losses, samplings,...)
Laplacian matrix Disturbance due to the delay + Realistic setup - Does not (always?) conserve aver. prop. [Olfati-Saber et al, 07]
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1. Problem formulation
Delays & Time-delay systems x
Delay t
•
x
h
t
Time-delay system or hereditary system: Systems where the evolution depends not only on the current state but also on a part of its past.
•
Fonctionnal differential equations Infinite dimension ¾ Initial conditions are taken over an interval ¾ Infinite number of roots 5
1. Problem formulation:
τ
Assumptions on the multi-agent set:
0 2
1
A1. Arbitrary connected graph A2. All com. delays are equal and constant
3
4
5
A3. The diagonal components of L are equal (arbitrary network)
Problems to solve: P1. Analytic expression of the agreement P2. Convergence
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2. Model transformation An appropriate representation: Lemma 1:
Model transformation
(1) (2)
(in the case of symmetric graphs, B can be diagonal) Proof:
1) Eigenvalue decomposition of the laplacian matrix 2)
3) 7
3. Stability analysis
a) Existence of a consensus:
Stability and limit of
(2)
Lemma 2:
Proof: 1. Consider the Laplace transform of
2. Characteristic equation :
s solution
and (2) has stable solution
3. Final limit theorem. 8
3. Stability analysis
Theorem 1:
b) Arbitrary networks:
Consensus stability:
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3. Stability analysis
b) Arbitrary networks:
Proof:
(1) (2)
1)
2) Consider the LKF : (Stability of TDS [Niculescu, 2003], Corollary 5.5, pp222)
If the conditions of theorem 1 are satisfied, then 3) Lemma 2:
4)
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3. Stability analysis
c) symmetric networks:
Theorem 2:
Proof: 1. Symmetric communication graph
L is symmetric
B is diagonal (1) (2)
2. Characteristic equation of (1):
s solution
and (1) has asymp. stable solution
Horn and Johnson, 1987 3. Application of Lemma 2 as previously 11
4. Discussions on the consensus equilibrium
The consensus value is given by: Depends on the delay and on the initial conditions over the delay interval:
Two corollaries: Cor.1: Consider initial conditions:
Cor 2: Consider initial conditions:
Then:
Then :
Non delayed case
Attenuation
Event
Event Dist. Control
Dist. Control Transmission
Transmission Reception
Reception
Time
Time -τ
0
τ
-τ
0
τ
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3. Convergence rate : c) Main result
Theorem 2:
Exponential convergence of the set multi-agent:
Proof: Based on L-K theory and exponential stability [Seuret et al, 04] Precision on the case of symmetric communication graph 13
5. Example Convergence and attenuation 1
4
2
3
1
4
2
3
1
4
2
3
1
4
2
3
Event Dist. Control Communication Reception
Time 0
-τ
τ
Event τ=0.6 , Cor. 2
Dist. Control Communication Reception
Time -τ
0
τ 14
5. Example Convergence and attenuation 1
4
2
3
1
4
2
3
1
4
2
3
1
4
2
3
τ=0,1 Event Dist. Control
τ=0.6
Communication Reception
Time 0
-τ
τ
τ= 1.2
Event τ=0.6 , Cor. 2
Dist. Control
τ=0.6
Communication Reception
Time -τ
0
τ 15
5. Example Motion in a 2D plan
4
2
15
10
3
y
1
τ=0 τ=0.1 τ=0.6
5
0
-5
0
2
4
6
8
10 x
12
14
16
18
20
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4. Example Convergence rate
τ=0,1 , Cor. 1 τ=0.6 , Cor. 1
τ= 2 , Cor. 1
τ=0.6 , Cor. 2 17
6. Conclusion and perspectives
Summary: • Analysis of consensus stability under constant communication delay ¾ Model transformation ¾ Time-delay systems theory
• Influence of the initial conditions and the delay • Study of the convegence rate of a consenus
On going work and possible extensions: • Delay-independent stability criteriafor arbitrary networks • To a more realistic Setup… ¾ Relaxation of initial restrictions ¾ Heterogenous communication delays ¾ Time-varying delays (including PL, Samplings,…)
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Some references
R.A. Horn and C.R. Johnson, Matrix analysis, Cambridge, UK Cambridge Univ. Press, 1987. L. Moreau, Stability of continuous-time distributed consensus algorithms, 43rd IEEE Conference on Decision and Control,2004. L. Moreau, , Stability of multi-agent systems with time-dependent communication links, IEEE Trans. on Automatic Control 50 (2005), no. 2, 169–182. R. Olfati-Saber, A. Fax, and R.M. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE 95 (2007), no. 1, 215–233. R. Olfati-Saber and R.M. Murray, Consensus problems in network of agents with switching topology and time delays, IEEE Trans. on Automatic Control 49 (2004), no. 9.
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