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Adaptive Consensus of Discrete-time Heterogeneous
Multi-agent Systems
Takashi Okajima (The University of Tokyo)Koji Tsumura (The University of Tokyo)Tomohisa Hayakawa (Tokyo Institute of Technology)Hideaki Ishii (Tokyo Institute of Technology)
○
Multi-Agent Dynamical Systems (MADSs)
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Active research field [Fax & Murray 2004] [Olfati-Saber et al. 2007]
Application for engineering Cooperation control of robots Vehicle formation
Unmanned Aerial Vehicles (UAVs) Automated Highway Systems (AHSs)
Control
J.A. Fax and R.M. Murray, “Information flow and cooperative control of vehicle formations”, 2004.
R. Olfati- Saber et al. “Consensus and cooperation in networked multi-agent systems”, 2007.
Multi-Agent Dynamical Systems (MADSs)
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Agent = Dynamical system (with controller) communicates with neighbor agents. autonomously acts on local information.
The system as a whole achieves some objective.
Local information
Communication
Global objectiveControl
Consensus problem of MADSs
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Consensus problem To reach an agreement on a certain quantity
In this presentation,we consider state consensus problem.
: state vector of ’th agent (at time step )
Research objective
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Homogeneous MADSs have been mainly researched.
Large number of the agentsThe dynamics of them are
heterogeneous & uncertain.
?
?
?
?
state consensus control of uncertain heterogeneous MADSs
[Wieland et al. 2011] :heterogeneous
[Fax & Murray 2004] [Olfati-Saber et al. 2007]
P. Wieland et al. “An internal model principle is necessary and sufficient for linear output synchronization,” 2011.
H. Kim el al. “Output consensus of heterogeneous uncertain linear multi-agent systems”, 2011.
[Kim et al. 2011] :heterogeneous & uncertain
Contents
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Introduction Adaptive control approach Framework Numerical example Conclusion
Adaptive control approach (1)
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A standard adaptive control can be regarded as a consensus problem of 2 agents.
Extension to more general communication topology.
Adjustmentmechanism
Feedback Controller Plant
Model
Agent 1
Agent 2
2
1
uncertain dynamics
known (desired) response
A standard adaptive control can be regarded as a consensus problem of 2 agents.
Extension to more general communication topology.
Adaptive control approach (2)
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4
12
1
3
62
5 7
Previous study (Adaptive control approach) Kaizuka & Tsumura, 2010. continuous-time systems
Corresponding result for discrete-time systems is not straightforward.
Adaptive control approach (3)
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4
12
1
3
62
5 7
Contents
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Introduction Adaptive control approach Framework Problem formulation Distributed adaptive controller Main result
Numerical example Conclusion
Communication topology
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Edge⇔ Agent receives information of Agent .
A special agent is called leader. (⇔known model )
The other agent are called followers.(⇔uncertain plants)
4
l=1 3
2
N
There are no edges to the leader. Edges between followers are bidirectional. There are directed spanning trees.
Assumption about Communication topology
Heterogeneous dynamics of agents
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Leader :Follower :
Agent Dynamics state: input:
:known, asymptotically stable
:unknown, heterogeneous, (possibly unstable):known, full column rank
:reference input, bounded:control input to Agent
Assumption (Matching condition)
:unknown, heterogeneous‘the true value of gain’
++
=
State consensus problem
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Design control inputs . Achieve state consensus.
Use only the states of the neighbors and its own.
Leader :Follower :
Agent Dynamics state: input:
3
6
5 7
4
1
2
Adaptive Controller of Follower
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++
:estimation (at time ) of .Feedback control law:
Gain update law:
Controller of Follower
:Neighbors of agent
3
6
5 7
4
1
2
( )
Gain update law
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Continuous-Time
[Hayakawa et al. 2004]
Discrete-Time
2l
[Kaizuka & Tsumura 2010]
summation
standard
multi-agent summation
Main result
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Theorem
is non-increasing.
The system of {state errors & gain errors} is globally stable.
(state consensus)
⇒ The gain update law is a globally stable estimator.
The controlled MADSs satisfies following.
(Details are given in the proceedings.)
( : Gain error at agent )
Contents
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Introduction Adaptive control approach Framework Numerical example Conclusion
Numerical example
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l=12 3
45
Leader :Follower :
Agent Dynamics
Communication topology
Poles(stable)
(unstable)
Consensus achieved
2 3
45
Conclusions
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Conclusions We proposed a state consensus control framework
for discrete-time uncertain heterogeneous MADSs. We showed the adaptive control based framework
is also valid for discrete-time systems.
Future works Output feedback case The case that none of the agent dynamics is known.
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Thank you for your attention!
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Continuous-time vs. discrete-time
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Continuous-time case
Discrete-time case
Lyapunov function:
2
l=1
Update law:Linear!
⇒
(Positive definite) cross term!Lyapunov function:
Update law:
: large enough.⇒