Projection-free approximate balanced truncation of large unstable systems

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.

Modal Analysis of Fluid Flows: An Overview

“…While Balanced POD has been successfully applied to model and control a number of fluids systems [90,[93][94][95][96], further research could validate and extend its use to a wider range of fluids applications. Work towards this goal currently includes application to the control of nonlinear systems [97], direct application to unstable systems [98], and use with harmonically forced data [99]. Further work could also seek algorithmic variants that improve efficiency.…”

Linearly Recurrent Autoencoder Networks for Learning Dynamics

“…We remark that in the infinite-horizon limit T → ∞, BPOD converges on a transformation which balances the generalized Gramians introduced in [38]. Application of BPOD to unstable systems is discussed in [6] which provides justification for the approach.…”

Model Reduction for Flow Analysis and Control

“…Applications, findings, and variants Ilak & Rowley (2008) Three-dimensional channel flow; BPOD models outperform POD models due to dynamically important, low-energy POD modes, and nonorthogonal projection in BPOD Barbagallo et al (2009) Control of cavity flow; balancing modes are more efficient than POD modes, but can be less robust Dergham et al (2011) Flow over a backward-facing step and square cavity; modified version of BPOD that uses harmonically forced data Semeraro et al (2011Semeraro et al ( , 2013 Modeling and control of Tollmien-Schlichting waves in a boundary layer to delay transition to turbulence, for both linearized and nonlinear systems Flinois et al (2015) Linearized flow past a cylinder; direct application of BPOD to unstable systems discrete time, we identify a discrete-time model of the form…”