In response to increasing size and complexity of modern control systems, large-scale interconnected systems attract the interest of systems and control researchers since quite some time. The hope is that analysis and design problems can be solved based on the subsystems and thereby to obtain scalable solutions with decreased overall complexity while still maintaining guarantees for the performance of the overall system. Besides the individual subsystems, the constituent parts of interconnected systems are the interconnection topology and the individual links used to realize the interconnections. We thus have three dimensions of complexity in interconnected systems: System complexity, topological complexity and link complexity. It turns out that the three dimensions of complexity cannot be addressed independently. Elevated complexity along one dimension usually yields constraints along the other dimensions. In this talk, the tradeoff between system complexity and topological complexity in consensus and synchronization problems is explored. The key tool will be a novel internal model principle for synchronization, recently developed by our group.
Given two complementary distributions in the tangent bundle of a manifold, we find conditions to factorize an stochastic flow into a diffusion in the (infinite dimensional) Lie group of diffeomorphisms which preserve one distribution (horizontal), composed with a process in the Lie group of diffemorphisms which preserve the other distribution (vertical). This decomposition generalizes previous approach, e.g. using coordinate maps, by Ming Liao and others.
