Abstract
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.
Highlights
Many linear dynamical systems, such as the linearized Navier-Stokes equations, are composed of a large number of states O(105–108), but their behavior is dominated by a much smaller number of modes O(1–100)
induction hypothesis (In) order to check that the application of the approximate balanced truncation method to unstable systems results in satisfactory reduced-order model (ROM) for large-scale problems, we first apply the method to the linearized complex Ginzburg-Landau equation
We have shown that applying the snapshotbased, projection-free, approximate balanced truncation method to unstable systems theoretically yields a converged transformation that balances the Gramians for all t∞ → +∞, including the unstable subspace
Summary
Many linear dynamical systems, such as the (discretized) linearized Navier-Stokes equations, are composed of a large number of states O(105–108), but their behavior is dominated by a much smaller number of modes O(1–100). We focus instead on ROMs based on balanced modes These are ranked by their dynamical significance to the input-output relationship of the system and are better suited for feedback control purposes than POMs or global modes by design. The method was applied to the flow over a cavity [20], a flat plate at large incidence [26], and a cylinder [29] This extension requires the computation of the system’s unstable global modes and the projection of the system onto its stable subspace. For large systems, such as three-dimensional flows, the cost of this procedure may still be excessively large. V, the method is applied to a two-dimensional Navier-Stokes simulation of the flow over a cylinder to show that it performs well in this more realistic and computationally demanding setup
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