Matrix scaling for large-scale system decomposition

Many large-scale systems exhibit the structure of weakly connected components. In such cases, proper identification of weakly coupled subsystems will add insight into large-scale system behavior, and aid in related tasks such as the design of decentralized control. ε-decomposition is a well-known efficient graph theoretic algorithm for achieving a complete set of nested decompositions of a large-scale system. This paper shows how system matrix scaling can affect these decompositions, and determines that a system is properly scaled for ε-decomposition when it is max-balanced, a property associated with weighted directed graphs. Also, it is shown that an existing algorithm for max-balancing can be altered slightly to return the complete set of ε-decompositions, thus removing the need for two separate algorithms. Finally, the advantages of max-balancing before decomposition are shown explicitly for large-scale system stability tests and parallel solution of nonlinear equations