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t
;
s
ij
= 0 if otherwise. Over time, binary changes of
S
(
t
) occur at an infinite
sequence of time instants, denoted by
{
t
k
:
k
∈
Ω
}
,where
Ω
{
1
,
2
,...,
∞}
,and
S
(
t
) is piecewise constant as
S
(
t
)=
S
(
t
k
) for all
t
∈
[
t
k
,t
k
+1
).
Problem 1
: Design the local control law for agent
i
to designate how the agent
processes available information at time
t
−
1 in order to formulate a decision
v
i
at time
t
as
v
i
(
t
)=
U
i
s
i
1
(
t
)
v
1
(
t
1)
,where
−
1)
,s
i
2
(
t
)
v
2
(
t
−
1)
,...,s
in
(
t
)
v
n
(
t
−
i
=
)is the local control law for agent
i
at time
t
.
Intuitively, it would be sucient for all the agents to be controlled properly if
each of them can receive enough information from its neighboring units. However,
it is not practical. So the minimum requirement on communication needs to be
specified properly to ensure the system level objective is satisfied. So it will be
the problem as below.
Problem 2
{
1
,
2
,...,n
}
,
U
i
(
·
: Determine the sensing/communication matrix in order to ensure the
global objective is desirable while minimizing the communication costs.
Through the design of local communication topology, the candidate sequences
of
should be chosen appropriately to ensure model developed
by using game theory converge to the desired equilibrium.
{
S
(
t
0
,S
(
t
1
,...
))
}
3 Rule of Communication Topology Design
In order to ensure the global objective can be achieved, local information needs to
be shared among the agents. Heuristically, the more information channels there
are, the faster the convergence to the desired global behavior. In order to guaran-
tee the validness of the proposed control strategy with the minimal information
requirement, we will give the rule of the communication topology in this section.
Rule
: The sequence of sensing/communication matrices
S
∞
:0
=
{
S
(
t
0
)
,S
(
t
1
)
,...
}
should be sequentially complete [15].
The sequentially completeness condition is a very precise method to schedule
local communication among agents. Especially, it gives the cumulated effects in
an interval of time and shows that the cumulated communication network can
be connected even if the network may not be connected at some time instants.
4 State Based Ordinal Potential Game Design
4.1 State Based Ordinal Potential Game
Different from but complementary to the result in Marden [6]and Li [13], we
focus on a more general potential game, termed as state based ordinal potential
games, to get the desirable solutions for the optimization problem in 1.
Definition 1
(state based ordinal potential game): A state based ordinal potential
game denoted by
G
=
, consists of a player set
N
and an underlying finite state space
X
. Each agent
i
has a state invariant action
set
A
i
, and a state dependent payoff function
U
i
:
X
{
N,
{
A
i
}
i∈N
,
{
U
i
}
i∈N
,X,f,ϕ
}
×
A
→
R
, also a both state
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