The necessity of providing reduced models of dynamical systems is growing continuously. Model-based control and model-based design are exemplary fields of applications. In this contribution, the reduction of a controlled drivetrain of a rolling mill using the method of Proper Orthogonal Decomposition is investigated, where the specific choice of the control law leads to equations of motion with time-periodic coefficients. Depending on amplitudes and frequency parameters of the time-periodic coefficients, artificial damping is introduced, referred to as parametric control. The maximum damping effect depends on these parameters in a nonlinear manner, as it is shown by means of a stability-parameter from Floquet theory. The reduced model set-up approximates the stability-parameter of the full model in an appropriate way within a wide range of the parameters. A measure based on the linear time-invariant system is developed that gives insight into the effect of the simulated timeseries on the properties of the reduced model.