Distributed Optimization and State Based Ordinal Potential Games - Intelligent Computing for Sustainable Energy and Environment - page 120

observed unless many of the communication channels stop working at the same
time. This shows that the local communication network can be designed to be
robust to the time varying and intermittent conditions according to the rule in
section (3).
Consequently, the consensus problem can be formalized as the following op-
timization problem: min v∈V φ ( v )= Σ i∈N,s ij =1 ( v i − v j ) 2 .

We simulated this average consensus problem by applying the state based or-
dinal potential games with the parameter α =0 . 2 and the gradient play learning
process. The results are presented in Fig. 2. In the top of Fig. 2, it illustrates
the dynamics of the values of the agents and shows that all the values con-
verge to the common scalar value after about 1 , 000 iterations by applying the
model we design and the gradient play learning algorithm for potential games.
In the bottom of Fig. 2, it demonstrates the evolution of the potential function
ϕ ( x,a )= ϕ φ ( x,a )+ αϕ e ( x,a ). And it is obvious to notice that at the beginning,
the value of the potential game will be very large since the agents are initially
dispersed. After about 500 iterations, the value of potential game will rapidly
converge to zero. This plot demonstrates the potential function designed in our
game model will tolerate the error caused by the introduction of estimate items
and achieve the desired global objective rapidly.

Fig. 2. Top figure: convergence of the dynamics of the 12 agents. Bottom figure: evo-
lution of the potential function.

6 Conclusions

In this paper, a new theoretical framework for analysis and design of distributed
optimization problem is developed based on the cooperative control methodol-
ogy and game theory. The matrix theoretical approach provides the basis for
state based ordinal potential game design, which gives the system designer addi-
tional freedom to design local control laws with the locality of information and
the eciency of the resulting equilibrium. Also the minimal requirement on the
interaction topology among the agents is provided. Further direction includes