The practical applicability of control theoretic model reduction methods is still limited to linear middle-scale systems. This shows a clear contrast to the Proper Orthogonal Decomposition (POD), which is a simulation-based model reduction method that has been widely applied to nonlinear large-scale systems, but with no theoretical underpinnings for its application to controlled systems. In this paper, we show that these controllability-based and simulation-based methodologies are equivalent when the input port is open to a noisy environment.