Consensus under Communication Delays Alexandre Seuret* Dimos V. - - PowerPoint PPT Presentation

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Consensus under Communication Delays

Alexandre Seuret* Dimos V. Dimarogonas** Karl H. Johansson***

* ** *** NeCS-team MIT

• Dep. of Automatic Control

GIPSA-lab CNRS/INRIA RA Laboratory for Information and ACCESS Linnaeus Centre Grenoble, France Decision Systems KTH, Stockholm, Sweden Cambridge, MA, U.S.A

31 Mars 2009

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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

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Introduction of communication delays…Where?

A: Adjacency matrix

Δ: Diagonal matrix

L=-(Δ-A) : Laplacian

1 2 3 4 5

τ

τ

• 1. Problem formulation

1 2 3 4 5

τ

+ Conserve averaging properties

• Not realistic in case of unknown delays,

packet losses, samplings,...) + Realistic setup

• Does not (always?) conserve aver. prop.

[Olfati-Saber et al, 07] [Olfati-Saber et al ,04 07], … [Olfati-Saber et al ,07], [Moreau, 04,05],...

Laplacian matrix Disturbance due to the delay

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• 1. Problem formulation
• Time-delay system or hereditary system:

Systems where the evolution depends not only on the current state but also

• n a part of its past.
• Fonctionnal differential equations

Delay

t x

h

t x Infinite dimension

Initial conditions are taken

• ver an interval

Infinite number of roots

Delays & Time-delay systems

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Assumptions on the multi-agent set:

• A1. Arbitrary connected graph
• A2. All com. delays are equal and constant
• A3. The diagonal components of L are equal (arbitrary network)

1 2 3 4 5

τ

• 1. Problem formulation:

Problems to solve:

• P1. Analytic expression of the agreement
• P2. Convergence

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An appropriate representation: Lemma 1: Model transformation Proof:

1) Eigenvalue decomposition of the laplacian matrix 2) 3)

• 2. Model transformation

(in the case of symmetric graphs, B can be diagonal)

(1) (2)

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• 3. Stability analysis

a) Existence of a consensus:

Lemma 2: Proof: Stability and limit of

s solution and (2) has stable solution (2)

• 1. Consider the Laplace transform of
• 2. Characteristic equation :
• 3. Final limit theorem.

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Theorem 1: Consensus stability:

• 3. Stability analysis

b) Arbitrary networks:

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• 3. Stability analysis

b) Arbitrary networks:

Proof:

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) (1) (2)

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• 3. Stability analysis

c) symmetric networks: Theorem 2: Proof:

s solution and (1) has asymp. stable solution

• 2. Characteristic equation of (1):
• 1. Symmetric communication graph L is symmetric B is diagonal

Horn and Johnson, 1987

(1) (2)

• 3. Application of Lemma 2 as previously

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Depends on the delay and

• n the initial conditions over the delay interval:

Two corollaries:

Cor 2: Consider initial conditions: Then: Non delayed case Cor.1: Consider initial conditions: Then : Attenuation

• 4. Discussions on the consensus equilibrium

The consensus value is given by:

Event Time

• τ

τ

Transmission

• Dist. Control

Reception

• τ

τ Event Time

Transmission

• Dist. Control

Reception

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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

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• 5. Example

Convergence and attenuation

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 Event Time

• τ

τ

Communication

• Dist. Control

Reception

Event Time

• τ

τ

Communication

• Dist. Control

Reception τ=0.6, Cor. 2

• τ

τ Event Time

Communication

• Dist. Control

Reception

• τ

τ Event Time

Communication

• Dist. Control

Reception

Event Time

Communication

• Dist. Control

Reception

15 τ=0,1 τ=0.6 τ= 1.2 τ=0.6

• 5. Example

Convergence and attenuation

Event Time

• τ

τ

Communication

• Dist. Control

Reception

Event Time

• τ

τ

Communication

• Dist. Control

Reception τ=0.6, Cor. 2

• τ

τ Event Time

Communication

• Dist. Control

Reception

• τ

τ Event Time

Communication

• Dist. Control

Reception

Event Time

Communication

• Dist. Control

Reception

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

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2 4 6 8 10 12 14 16 18 20

• 5

5 10 15 x y

τ=0 τ=0.1 τ=0.6

• 5. Example

Motion in a 2D plan

1 2 3 4

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τ=0,1 τ=0.6 τ= 2 τ=0.6

, Cor. 1 , Cor. 1 , Cor. 1 , Cor. 2

• 4. Example

Convergence rate

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• 6. Conclusion and perspectives

Summary:

• Analysis of consensus stability under constant communication delay

Model transformation 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…

Relaxation of initial restrictions Heterogenous communication delays Time-varying delays (including PL, Samplings,…)

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• 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.

Some references

R.A. Horn and C.R. Johnson, Matrix analysis, Cambridge, UK Cambridge Univ. Press, 1987.