The non-standard H-infinity control problem is the case where the direct feedthroughs from the input to the error and from the exogenous signal to the output are not necessarily of full rank. In this paper, this problem is reformulated based on the generalized chain-scattering representation (GCSR). The GCSR approach leads naturally to a generalization of the homographic transformation. The state-space realization for this generalized homographic transformation and a number of fundamental cascade structures of the H-infinity control systems are further studied in a unified framework of GCSR. Certain sufficient conditions for the solvability of the non-standard H-infinity control problem are therefore established via a (J, J')-lossless factorization of GCSR. These results present extensions to Kimura's results on the chain-scattering representation (CSR) approach to the H-infinity control in the standard case.