Distributed Robustness Analysis of Interconnected Uncertain Systems Using Chordal Decomposition

Large-scale interconnected uncertain systems commonly have large state and uncertainty dimensions. Aside from the heavy computational cost of solving centralized robust stability analysis techniques, privacy requirements in the network can also introduce further issues. In this paper, we utilize IQC analysis for analyzing large-scale interconnected uncertain systems and we evade these issues by describing a decomposition scheme that is based on the interconnection structure of the system. This scheme is based on the so-called chordal decomposition and does not add any conservativeness to the analysis approach. The decomposed problem can be solved using distributed computational algorithms without the need for a centralized computational unit. We further discuss the merits of the proposed analysis approach using a numerical experiment.Comment: 3 figures. Submitted to the 19th IFAC world congres

Exploiting chordal structure in polynomial ideals: a Gr\"obner bases approach

Chordal structure and bounded treewidth allow for efficient computation in numerical linear algebra, graphical models, constraint satisfaction and many other areas. In this paper, we begin the study of how to exploit chordal structure in computational algebraic geometry, and in particular, for solving polynomial systems. The structure of a system of polynomial equations can be described in terms of a graph. By carefully exploiting the properties of this graph (in particular, its chordal completions), more efficient algorithms can be developed. To this end, we develop a new technique, which we refer to as chordal elimination, that relies on elimination theory and Gr\"obner bases. By maintaining graph structure throughout the process, chordal elimination can outperform standard Gr\"obner basis algorithms in many cases. The reason is that all computations are done on "smaller" rings, of size equal to the treewidth of the graph. In particular, for a restricted class of ideals, the computational complexity is linear in the number of variables. Chordal structure arises in many relevant applications. We demonstrate the suitability of our methods in examples from graph colorings, cryptography, sensor localization and differential equations.Comment: 40 pages, 5 figure