The authors propose a two sided moment matching method for model reduction of quadratic-bilinear descriptor systems. The goal is to approximate some of the generalised transfer functions that appear in the input–output representation of the non-linear system. Existing techniques achieve this by utilising moment matching for the first two generalised transfer functions. In this study, they derive an equivalent representation that simplifies the structure of the generalised transfer functions. This allows them to extend the idea of two sided moment matching to higher subsystems which was difficult in the previous approaches. Numerical results are given for some benchmark examples of quadratic-bilinear systems.