The chain scattering-matrix description (CSD) of a two-port network, first introduced in Chap. 2, originated from the conventional electrical circuit theory. In contrast to the LFT, the CSD developed in the network circuits provides a straightforward interconnection in a cascaded way. The CSD can transform an LFT into a two-port network connection and vice versa. Thus, many known results which have been developed for a two-port network can then be used in control system analysis and synthesis. Due to its benefits of describing a linear system, the CSD was later extended to the design of robust control systems [1, 2], where different structures of the CSD were investigated. In this chapter, the CSD will be formally defined and explored for transformations and the descriptions of state-space realizations. The J-lossless and dual J-lossless systems are defined in this chapter. J-lossless and dual J-lossless both play an important role in the CSD control system manipulations, analysis, and synthesis. In particular, the properties of J-lossless and dual J-lossless are essential in synthesizing H∞ (sub)optimal controllers using the CSD approach.