Synthetic Biologists are increasingly interested in the idea of using synthetic feedback control circuits for the mitigation of perturbations to gene regulatory networks that may arise due to disease and/or environmental disturbances. Models employing Michaelis-Menten kinetics with Hill-type nonlinearities are typically used to represent the dynamics of gene regulatory networks. Here, we identify some fundamental problems with such models from the point of view of control system design, and argue that an alternative formalism, based on so-called S-System models, is more suitable. Using tools from system identification, we show how to build S-System models that capture the key dynamics of an example gene regulatory network, and design a genetic feedback controller with the objective of rejecting an external perturbation. Using a sine sweeping method, we show how the S-System model can be approximated by a linear transfer function and, based on this transfer function, we design our controller. Simulation results using the full nonlinear S-System model of the network show that the synthetic control circuit is able to mitigate the effect of external perturbations. Our study is the first to highlight the usefulness of the S-System modelling formalism for the design of synthetic control circuits for gene regulatory networks.