State-space approach to factorization of lossless transfer functions and structured matrices☆

Abstract The concept of J-lossless triangular state-space models is reviewed, and its relation to Lyapunov equations and to matrices with a displacement structure is characterized in detail. A new recursive procedure for cascade synthesis of such state-space models is introduced. In contrast to previous state-space-based techniques for cascade decomposition (factorization) of J-lossless transfer functions, which require conversion of a given state-space representation into an equivalent balanced form, our new procedure recursively determines a sequence of unbalancedJ-lossless state-space models. The cascade decomposition of a given J-lossless transfer function obtained by the new procedure is the same as the one obtained by the previous techniques, but the attendant computational requirements are significantly reduced. Furthermore, the final computational formulation of our procedure subsumes many previous methods for efficient triangular factorization of structured matrices—the so-called “generalized Schur” or “generalized fast Cholesky” algorithms.