Mathematically Designing a Local Interaction Algorithm for Autonomous and Distributed Systems

In the near future, autonomous and distributed systems consisting of a huge number of sensor nodes are expected to play an important role. Due to their distributed nature, each node should control itself autonomously by means of a local interaction algorithm. Although such local interaction algorithms improve system scalability, how to design a local interaction algorithm has become an issue. In this paper, we describe a local interaction algorithm in a partial differential equation (or PDE) and propose a new design method whereby a PDE is derived from a pattern of control values. A pattern means a distribution of nodes´ control values over the network each of which is used to control the node. Nodes behave according to the pattern and as a result they collectively provide network functions such as collision and congestion avoidance. In this paper, we focus on a periodic pattern composed of sinusoidal waves and derive the PDE by exploiting the Fourier method.