Imperial College London


Faculty of Natural SciencesDepartment of Mathematics

Chair in Applied Mathematics







753Huxley BuildingSouth Kensington Campus






BibTex format

author = {Flinois, TLB and Morgans, AS and Schmid, PJ},
doi = {10.1103/PhysRevE.92.023012},
journal = {Phys Rev E Stat Nonlin Soft Matter Phys},
title = {Projection-free approximate balanced truncation of large unstable systems.},
url = {},
volume = {92},
year = {2015}

RIS format (EndNote, RefMan)

AB - 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 therefore can 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.
AU - Flinois,TLB
AU - Morgans,AS
AU - Schmid,PJ
DO - 10.1103/PhysRevE.92.023012
PY - 2015///
TI - Projection-free approximate balanced truncation of large unstable systems.
T2 - Phys Rev E Stat Nonlin Soft Matter Phys
UR -
UR -
UR -
VL - 92
ER -