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of the process. In Li [13], an extension of potential games, termed as state based
potential game, was presented to cope with the design challenges by introducing
an additional state variable into the game environment.
The main contribution of this paper is to extend the results of Li [13] to a more
general game framework based on matrix theory. The matrix theory [3] are used
in this paper to develop a new framework for analyzing the interaction behavior
and the minimal information requirement among agents are provided. Using
the canonical form of matrix theory, the results in Li [13] are extended to the
general form of state based ordinal potential games. We will show that the game
designed in Li [13]is a special case of the general game model proposed in this
paper. Furthermore, this general game model provides us with much more degree
of freedom to design local control laws and both cases with connected and time-
invariant interaction topology in Li [13]are extended to the practical conditions
including time-variant and not always connected communication topology.
2 Problem Setup
2.1 System Model
We are interested in optimization algorithm that can be distributed across the
decision-makers. Suppose there is a multi-agent system consisting of
n
≥
2 agents
denoted by the set
N
=
N
is endowed with a set of
possible decisions (or values) denoted by
A
i
which is a nonempty convex subset
of
{
1
,
2
,...,n
}
. Each agent
i
⊆
R
. We denote a specific joint decision profile by the vector
v
{
v
1
,v
2
,...,v
n
}
i∈N
V
i
,where
V
is the closed, convex and non-empty set consist-
ing of all possible joint decisions. Suppose the global objective
φ
:
V
and
v
⊆
V
R
that
system designer seeks to minimize is differentiable and convex. More specifically,
the distributed optimization problem takes on the general form:
→
min
v
i
φ
(
v
1
,v
2
,...,v
n
)
s.t.
v
i
∈
V
i
,
∀
i
∈
N
(1)
2.2 Problems to Be Solved
The sensing and communication among agents is described mathematically by
a time-varying and piecewise-constant matrix whose dimension is equal to the
number of dynamical agents and the elements assume binary values. The matrix
can be defined without loss of any generality [3]:
⎛
⎞
s
11
s
12
···
s
1
n
⎝
⎠
s
21
s
22
···
s
2
n
...............
s
n
1
s
n
2
···
S
(
t
)=
s
nn
where
s
ii
= 1 because agent can always acquire its own information. In general,
s
ij
= 1 if the agent
i
can get the information of agent
j
for any
j
=
i
at time
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