Balanced truncation model reduction of second-order systems

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On the use of modal derivatives for nonlinear model order reduction (2016, methodology): “Specifically in state-space formulations, balanced truncation is a favourite method [2, 3, 4, 5, 6], since it guarantees the observability and the stability of the system [7].”
Simulation process of flexible multibody systems with non-modal model order reduction techniques (2011, methodology): “In [39] the different approaches for balanced truncation of second-order systems are described.”
An improved numerical method for balanced truncation for symmetric second-order systems (2013, background): “Following the approach in [5], one can choose between four possible singular value decompositions”
Efficient balancing-based MOR for large-scale second-order systems (2011, unknown): “Unfortunately, the global balanced truncation error bound for the reduction is lost if the structure preserving balanced truncation is applied in the following [14, 16, 17].”