bg from scalar to matrix weighted consensus overview applications and open issues 1518

Ongoing studies and Conclusions 6/4/2022 Hanoi University of Science and Technology 2... Introduction to consensus algorithm6/4/2022 Hanoi University of Science and Technology 3... Model

From Scalar to Matrix-Weighted Consensus: Overview, Applications, and Open Issues

Dr Minh Hoang Trinh (Trịnh Hoàng Minh)

Faculty of Automation Engineering School of Electrical and Electronic Engineering Hanoi University of Science and Technology

WPEC 2021

1 Introduction to consensus algorithm

2 Matrix-weighted consensus and Matrix-scaled consensus

3 Ongoing studies and Conclusions

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Introduction to consensus algorithm

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Multi-agent systems (MASs)

• MASs: a system consisting of multiple subsystems

interacting with each other

• In nature: bird flocking, fish schooling, firefly synchronization

• Engineering systems: formations of robots/autonomous

vehicles, sensor network, smart-grid, traffic systems,…

• Human: social networks, opinion dynamics, collaboration

Modelling of multi-agent systems

• Agent’s model: simple dynamical models

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Modelling of multi-agent systems

• Graph: 𝐺 = (𝑉, 𝐸, 𝐴)

• Vertex ↔ agent (𝑉 = 1, … , 𝑛 , 𝑉 = 𝑛)

• Edge ↔ interaction (uni-directional, bi-directional) (𝐸 ⊂ 𝑉 × 𝑉, 𝐸 = 𝑚, 𝑒𝑖𝑗 = 𝑖, 𝑗 : edge)

• 𝐴 = 𝑎𝑖𝑗 ∈ ℝ+ 𝑖,𝑗 ∈𝐸: scalar edge weights

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• Graph 𝐺: 𝑉 = 1,2,3,41,2 : unidirectional2,3 , 3,4 : bidirectional

1 2

3

4

Information flow between robots

Modelling of multi-agent systems

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Room 4, Temperature 𝑇4

1 2

Modelling of multi-agent systems

• Connected graph: there exists a path connecting

every two vertices in the graph

Relative state of agents 𝑖

and 𝑗

ሶ𝒙 𝑡 = −𝑳𝒙 𝑡

Laplacian matrix

R Olfati-Saber, and R M Murray (2004) "Consensus problems in networks of agents with switching topology

and time-delays." IEEE Transactions on Automatic Control 49(9): 1520–1533-1520–1533.

Formation control

• Formation control:

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• 𝐺, 𝒑 : formation

• 𝒑𝑖 ∈ ℝ𝑑 𝑑 = 2,3 : position of agent 𝑖 in the space

• 𝒑 𝑡 = vec 𝒑1, … , 𝒑𝑛 : configuration at time 𝑡 ≥ 0

• 𝒑 ∗ = vec 𝒑1∗, … , 𝒑𝑛∗ : desired configuration

Target configuration

Agent

Neighbor agents

Initial configuration

K.-K Oh, M.-C Park, H.-S Ahn, “A survey of multi-agent formation control” Automatica, 53, 2015, 424 − 440.

𝒑𝑖∗ − 𝒑𝑗∗: desired displacement

• Local coordinate systems are aligned

• Each agent senses the displacement

𝒑𝑖 − 𝒑𝑗 and knows the desired displacement 𝒑𝑖∗ − 𝒑𝑗∗

W Ren, R.W Beard, “Distributed Consensus in Multi-vehicle Cooperative Control:

Theory and Applications”, Springer, 2008

• 𝒑 𝑡 = vec 𝒑1, … , 𝒑𝑛 : configuration at time 𝑡 ≥ 0

• 𝒑∗ = vec 𝒑1∗, … , 𝒑𝑛∗ : desired configuration

Formation control

• Displacement-based formation control:

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𝒑𝑖𝑖 − 𝒑𝑗𝑖: local displacement measured

in agent 𝑖’s coordinate system

𝑎𝑖𝑗 𝒑𝑗 − 𝒑𝑖 = 𝒑𝑗 − 𝒑𝑖 2 − 𝒑𝑗∗ − 𝒑𝑖∗ 2

• The scalar weights 𝑎𝑖𝑗 are dependent on states (can be positive, zero, or negative)

• 𝒑𝑖∗ − 𝒑𝑗∗ : the desired distance

B.D.O Anderson, C Yu, B Fidan & J.M Hendrickx Rigid graph control architectures for autonomous formations.

IEEE Control Systems Magazine, 28(6): 2008, 48 − 63.

Điều khiển đội hình

• Distance-based formation control:

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ሶ𝒑𝑖𝑖 𝑡 = − ෍

𝑗∈𝑁𝑖

𝑎𝑖𝑗 𝒑𝑗 − 𝒑𝑖 𝒑𝑖𝑖 𝑡 − 𝒑𝑗𝑖 𝑡

The system converges to two different configurations

with two initial configurations

𝑎𝑖𝑗 𝒑𝑗 − 𝒑𝑖 = 𝒑𝑗 − 𝒑𝑖 2 − 𝒑𝑗∗ − 𝒑𝑖∗ 2

M.C Park, Z Sun, M H Trinh, B D O Anderson, and H.-S Ahn, 2016, “Distance-based control of K4 formation

with almost globalconvergence” 2016 IEEE 55th Conference on Decision and Control (CDC), 904-909

• Matrix form:

ሶ𝒑 𝑡 = −𝑹 𝒑 ⊤𝒆

Analysis involves rigidity theory The problem becomes

complicated due to existence of undesired equilibria

Altafini’s model explains the bipartite

consensus in a network with antagonistic

interactions

Agents’ states converge to one of two values having the same magnitude

but opposite sign.

The signed graph satisfies the structured sign balance condition:

+ + = + +−= −

−+= −

−−= +

C Altafini (2013) "Consensus problems on networks with antagonistic interactions."

IEEE Transaction on Automatic Control 58(4): 935-946.

Matrix-weighted consensus

Matrix-scaled consensus

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Positive semidefinite edge

Positive definite edge

Consensus condition: rank 𝑳 = 𝑑𝑛 − 𝑑

Clustering phenomenon happens even if 𝐺 is connected

M H Trinh, C V Nguyen, Y H Lim and H S Ahn (2018) "Matrix-weighted consensus and its applications."

Automatica 89: 415-419.

ker 𝑳 ⊇ im 𝟏𝑛 ⊗ 𝑰𝑑

Graph conditions for reaching a consensus

• Sufficient condition: existence of a positive spanning tree

• Expanding the positive spanning tree

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Think of each matrix weight as a bridge between positive trees ker 𝑨𝑖𝑗1 ∩ ⋯ ∩ ker 𝑨𝑖𝑗𝑘 = ∅ ⟹ equiv with a positive definite edge

Bearing-based network localization

Bearing vector from 𝒑𝑖 to 𝒑𝑗

S Zhao, and D Zelazo, 2016 Localizability and distributed protocols for bearing-based network localization in

arbitrary dimensions Automatica, 69, 334-341.

𝒈𝑖𝑗

𝒙

𝑷𝒈𝑖𝑗𝒙

Bearing-based network localization

• Bearing-based network localization:

• Each node has an estimate: ෝ𝒑𝑖 ∈ ℝ𝑑, sense the bearing vectors

𝒈𝑖𝑗

𝑖,𝑗 ∈𝐸 and want to determine the real position 𝒑𝑖 ∈ ℝ𝑑

• Network localization law: ሶො𝒑𝑖 = − σ𝑗∈𝑁𝑖 𝑷𝒈𝑖𝑗 ෝ 𝒑𝑖 − ෝ 𝒑𝑗 , 𝑖 = 1, … , 𝑛

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• Matrix form: ෝ𝒑 = vec ෝ𝒑1, … , ෝ𝒑𝑛 ∈ ℝ𝑑𝑛

ሶෝ𝒑 𝑡 = −𝑳𝑏𝒑 𝑡ෝ

• The network is localizable if there are 2 reference nodes

and the bearing Laplacian satisfies:

rank 𝑳𝑏 = 𝑑𝑛 − 𝑑 − 1 (cứng hướng vi phân)

p

4

M.H Trinh, T.T Nguyen, N.H Nguyen, H.-S Ahn, "Fixed-time network localization based on bearing measurements,"

American Control Conference (ACC), Denver, CO, USA, 2020.

Bearing-based formation control

• Bearing-based formation control:

• Leaderless formation: the final formation differs from the target formation by a

translation and a scaling

𝒑 𝑡 → 𝒑∗ ∈ ker 𝑳𝑏 = im 𝟏𝑛 ⊗ 𝑰𝑑, 𝒑∗ , 𝑡 → +∞

Zhao, S and Zelazo, D., 2015 Translational and scaling formation maneuver control via a bearing-based approach.

IEEE Transactions on Control of Network Systems, 4(3), pp.429-438.

Bearing-based formation maneuver

• Formation maneuver/tracking:

• Leaders: move with a reference velocity

• Followers: acquires a target formation and move with the leaders’ velocity

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𝒖𝑖 = formation acquisition + formation tracking terms, 𝑖 = 1, … , 𝑛

Design based on assumption of the

leaders’ reference velocity:

• PID, sliding-mode, back-stepping

• Observer-based

D V Vu, and M H Trinh "Decentralized sliding-mode control laws for the bearing-based formation tracking

problem." International Conference on Control, Automation and Information Sciences (ICCAIS) IEEE, 2021

H M Nguyen, M H Trinh, "Leader-follower matrix-weighted consensus: a sliding-mode control

approach", accepted to VCCA 2021.

Multi-dimensional opinion dynamics

• Continuous-time multi-dimensional Friedkin-Johnsen model:

Opinion changes due to interaction between agents process of individual 𝑖The inner cognitive

M Ye, M H Trinh, Y H Lim, B D O Anderson and H S Ahn (2020) "Continuous-time opinion dynamics

on multiple interdependent topics." Automatica 115(108884).

Topic 1: Mentally challenging tasks are just as exhausting as physically challenging tasks Topic 2: E-sport (games) should be considered a sport in the Olympics

𝑪 = 1 0

0.7 0.3

Multi-dimensional opinion dynamics

• Opinion dynamics model:

• 𝒙𝑖 ∈ ℝ𝑑: opinion of 𝑖 about 𝑑 interdependent topics

H.-S Ahn, Q V Tran, M H Trinh, M Ye, J Liu and K L Moore (2020) "Opinion dynamics with cross-coupling topics:

Modeling and analysis." IEEE Transactions on Computational Social Systems 7(3): 632-647

⟹ Each agent 𝑖 reach a different level of consensus, final value inverse proportional to 𝑠𝑖

⟹ The final vales of all agents lie on a straight-line going through 𝟎

S Roy (2015) Scaled consensus Automatica, 51, 259-262.

• sign 𝑺𝑖 = 1 (resp., −1) if 𝑺𝑖 is p.d (resp., n.d.)

• Matrix form: 𝒙 = vec 𝒙1, … , 𝒙𝑛 ∈ ℝ𝑑𝑛

ሶ𝒙 𝑡 = − diag sign 𝑺𝑖 𝑳 ⊗ 𝑰𝑑 𝑺𝒙 𝑡

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Q V Tran, M H Trinh, and H.-S Ahn (2018) "Surrounding formation of star frameworks using

bearing-only measurements." European Control Conference (ECC) IEEE, 2018.

The signum function guarantees the agents to reach matrix-scaled consensus 𝑺of agent 𝑖 with a global coordinate (maybe non-𝑖: a matrix relates the local coordinate system

Interpretation of the model?

• MSC allows individuals to have different final opinions (cluster consensus):

• 𝒙𝑖 ∈ ℝ𝑑: opinion of individual 𝑖 on 𝑑 inter-logical topic

• 𝑺𝑖 ∈ ℝ𝑑×𝑑: Individual 𝑖’s belief system

• Virtual consensus point: 𝒙𝑎 = σ𝑗∈𝑁𝑖 𝑺𝑖 −1 −1 σ𝑗∈𝑁

𝑖 sign 𝑺𝑖 𝒙𝑖 0 - jointly determined by initial states and 𝑺𝑖

• 𝒙𝑖 𝑡 → 𝑺𝑖−1𝒙𝑎: final opinion of agent 𝑖 is biased by the scaling matrix 𝑺𝑖−1 (magnitude and direction) from the virtual consensus point.

• Individuals with the same 𝑺𝑖 converges to the same point

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Ongoing studies

• Matrix-weighted consensus:

• Directed graphs

• Delays, normalized matrix-weighted Laplacian (applications in network localization)

• Disturbance rejection, tracking consensus, robustness issues (applications in based formation)

bearing-• Agent’s model and connectivity

• Develop the matrix-scaled consensus model

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