Optimizing the Diffusion System Based on Continuous-Time Consensus Algorithm

Abstract

Traditionally, systems governed by linear Partial Differential Equations (PDEs) are spatially discretized to exploit their algebraic structure and reduce the computational effort for controlling them. Due to beneficial insights of the PDEs, recently, the reverse of this approach is implemented where a spatially-discrete system is approximated by a spatially-continuous one, governed by linear PDEs forming diffusion equations. In the case of distributed consensus algorithms, this approach is adapted to enhance its convergence rate to the equilibrium. In previous studies within this context, constant diffusion parameter is considered for obtaining the diffusion equations. This is equivalent to assigning a constant weight to all edges of the underlying graph in the consensus algorithm. Here, by relaxing this restricting assumption, a spatially-variable diffusion parameter is considered and by optimizing the obtained system, it is shown that significant improvements are achievable in terms of the convergence rate of the obtained spatially-continuous system. As a result of approximation, the system is divided into two sections, namely, the spatially-continuous path branches and the lattice core which connects these branches at one end. The optimized weights and diffusion parameter for each of these sections are optimal individually but considering the whole system, they are suboptimal. It is shown that the symmetric star topology is an exception and the obtained results for this topology are globally optimal. Furthermore, through variational method, the results obtained for the symmetric star topology are validated and it is shown that the variable diffusion parameter improves the robustness of the system too.