Decomposition and Projection Methods for Distributed Robustness Analysis of Interconnected Uncertain Systems

We consider a class of convex feasibility problems where the constraints that describe the feasible set are loosely coupled. These problems arise in robust stability analysis of large, weakly interconnected uncertain systems. To facilitate distributed implementation of robust stability analysis of such systems, we describe two algorithms based on decomposition and simultaneous projections. The first algorithm is a nonlinear variant of Cimmino's mean projection algorithm, but by taking the structure of the constraints into account, we can obtain a faster rate of convergence. The second algorithm is devised by applying the alternating direction method of multipliers to a convex minimization reformulation of the convex feasibility problem. Numerical results are then used to show that both algorithms require far less iterations than the accelerated nonlinear Cimmino algorithm.