H ∞ optimal control, which minimizes the H ∞-norm of a closed-loop system, has been developed in the last 30 years and been applied in various domains. The original H ∞ optimal control problem involves an equivalent model matching problem, which can be transformed into a four-block distance problem. By applying spectral factorizations, the four-block distance problem can be reduced to a Nehari problem, and Hankel norm approximation can be considered [2–4, 6]. Operator theory approach is very mathematics involved, and numerical solution procedures are difficult to be developed for general form problems. Notable progress was made in finding suboptimal solutions of a general control synthesis problem by solving two algebraic Riccati equations (AREs) [2, 3, 10, 13]. However, even with such solution procedures, questions of “why?” and “how?” often arise from students and engineers who want to understand and use them. An alternative development based on the framework of J-lossless coprime factorizations was proposed by Green in which the solutions can be characterized in terms of transfer function matrices . A similar framework based on a single chain scattering description (CSD) was initially proposed by Kimura . As described in Chap. 8, the general four-block problem can be solved by augmenting with some fictitious signals. Furthermore, since the transformation from LFT to CSD does not guarantee stability of the resulting CSD matrix, the J-lossless factorization with an outer matrix cannot be found directly by the coprime factorization-based method. In this book, the proposed H ∞ CSD solution framework involves constructing two coupled (right and left) CSD matrices by solving two J-lossless coprime factorizations and is fairly straightforward. The method is generally valid and does not need to introduce any fictitious signals for matrix augmentation. Based on Green’s approach of J-lossless coprime factorizations, the proposed CSDs framework is significantly different from Kimura’s approach.