Linear fractional transformations and J-lossless factorization in H∞ control theory

The linear fractional transformation, the lossless (all-pass) system and the canonical (inner-outer) factorization are three key notions in the frequency domain approach to the H^∞ control problem. It is shown that these three notions are combined more naturally and systematically in the chain-scattering formalism to establish a conceptual framework of H^∞ control theory. In our framework, we can treat both the one-block and the four-block cases in a unified way, based on an extension of homographic transformation. It is shown that the problem is reduced to the J-lossless factorization of a certain matrix, which is a slight modification of a J-spectral factorization.