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Containment for double-integrator multi-agent systems in heterogeneous networks

Research Center for Complex Systems and Network Sciences

Southeast University, Nanjing

Saturday 13 December 2014

Jiahu Qin1, Wei Xing Zheng2, Huijun Gao3, and Qichao Ma1

1. University of Science and Technology of China2. University of Western Sydney, Australia3. Harbin Institute of Technology, China

Reach agreement about some variable of interests. E.g. position, velocity, voltage, phase, frequency

Distributed algorithm: using local information

Leader-following/Leaderless framework

Consensus/Synchronization

Background

More than one leaders

Follower agents move into the convexhull spanned by the leaders

Containment

Background

Social animal groups: herding of large groups

Social insect groups: silk-worm moth

Civilian & Military Application: mobile robots,

obstacle avoidance

G. Notarstefano 2011, AutomaticaHummel & Miller 1984 , Springer Verlag

Containment for single-integrator agents

4

Background

• Double-integrator agents

• Leaders of constant velocity

• Leaders of time-varying velocity

• Simulations

• Conclusion

Outline

A broader class of vehicles are modeled by double integrator dynamics: mobile robots, underwater vehicles, etc.

· ·

System dynamics ,

Double-integrator agents

d'Andrea-Novel, Bastin & Campion 1991, ICRARen 2008, IEEE-TAC

7

ii vx ii uv

· ·

General setup: agents communicate over heterogeneous network:Gp≠Gv


Position and velocity may be measured in different ways, e.g. using different sensors

Information loss may also lead to heterogeneity of position and velocity interaction topology

Goldin & Raisch 2013, AJCQin & Yu 2013, IEEE-TNNLS

8

9


System model

Leaders

Followers

• xi: position state of agent i

• vi: velocity state of agent i

• ui: control input (containment algorithm) for agent i

• Leaders are assumed to move with the same velocity vr(t)




• Simulations

• Conclusion

Outline

10

Leaders of constant velocity

Case I: velocity topology for follower agents is balanced

Containment algorithm:

• ：set of agent i’s neighbours inposition interaction topology

• ：set of agent i’s neighbours in velocity interaction topology

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piN

viN

T h e g r a p h G i s s a i d t o h a v ea united spanning tree if for each of thefollower agents, there exists at least oneleader that has a directed path to thefollower agents in graph G.

Graph theoretic basic

f1

l2

f5

f4

l1

f3

f2

Example of having a united spanning tree:


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Main result I ( is balanced):

Theorem 1: Assume that is undirected. If both Gp and Gv have a unitedspanning tree, then employing algorithm (1), all the follower agents moveasymptotically into the convex hull spanned by the leaders. In particular,

and

• ( )：position(velocity) interaction topology for follower agent

• : a row stochastic matrix causing ‘containment’p21p1LL-

pfG

pfG

vfG

vfG

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Case II: general velocity interaction topology for follower agents

Containment algorithm:

● Existence of can be guaranteed if Gv (velocity interaction topology for leader and follower agents) has a united spanning tree.

Λ

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Main result II:

Theorem 2: Assume that is undirected. If both Gp and Gv have a unitedspanning tree, then employing algorithm (3), all the followers moveasymptotically with velocity vr into the convex hull spanned by the leaders.In particular,

and

pfG

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• Simulations

• Conclusion

Outline

16

Leaders of time-varying velocity

Containment algorithm

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Main result III:

where λ1,…, λn2 are the n2 eigenvalues(repetition may exist) of matrix Lv1Lp1,then employing algorithm (4), the follower agents move asymptotically intothe convex hull spanned by the leaders with velocity vr(t). In particular,

Theorem 3: Assume that both Gp and Gv have a united spanning tree. If allthe eigenvalues of matrix Lv1Lp1 have positive real parts and further

-1

-1

and

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where λ1,…, λn2 are the n2 eigenvalues(repetition may exist) of matrix Lv1Lp1,then employing algorithm (4), the follower agents move asymptotically intothe convex hull spanned by the leaders with velocity vr(t). In particular,

Theorem 3: Assume that both Gp and Gv have a united spanning tree. If allthe eigenvalues of matrix Lv1Lp1 have positive real parts and further

-1

-1

and

Two special cases:• and are undirected•

pfG v

fGvp GG

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Main result III:

Outline




• Simulations

• Conclusion

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Simulation results

Leaders of constant velocity (general )vfG

• Leaders: 1,2,3

• Followers:4,5,6,7,8

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Simulation results

Containment is reached

Leaders of constant velocity (general )vfG

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Position trajectories using algorithm (3) Velocity trajectories using algorithm (3)

Simulation results

Leaders of the same time-varying velocity

))cos(( t

1010

r1x

))cos(( t

1010

-xr2

)sin()( tt 10rv

• Leader 1:

• Leader 2:

)sin()( tt 10rv

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Simulation results

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Leaders of the same time-varying velocity

Position and velocity trajectories using algorithm (4):Line formation is achieved

Outline




• Simulations

• Conclusion

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Containment control for double-integrator agents over heterogeneous networks

Different containment algorithms are proposed respectively for



Sufficient conditions guaranteeing containment control are provided re provided

Conclusion

Contribution

Future work

Extension to directed position interaction topology

Dynamic interaction topologies

Thanks! Thanks!

Questions? Questions?

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