  In this article, we show that the projection-free, snapshot-based, balanced
truncation method can be applied directly to unstable systems. We prove that
even for unstable systems, the unmodified balanced proper orthogonal
decomposition algorithm theoretically yields a converged transformation that
balances the Gramians (including the unstable subspace). We then apply the
method to a spatially developing unstable system and show that it results in
reduced-order models of similar quality to the ones obtained with existing
methods. Due to the unbounded growth of unstable modes, a practical restriction
on the final impulse response simulation time appears, which can be adjusted
depending on the desired order of the reduced-order model. Recommendations are
given to further reduce the cost of the method if the system is large and to
improve the performance of the method if it does not yield acceptable results
in its unmodified form. Finally, the method is applied to the linearized flow
around a cylinder at Re = 100 to show that it actually is able to accurately
reproduce impulse responses for more realistic unstable large-scale systems in
practice. The well-established approximate balanced truncation numerical
framework can therefore be safely applied to unstable systems without any
modifications. Additionally, balanced reduced-order models can readily be
obtained even for large systems, where the computational cost of existing
methods is prohibitive.
