101 research outputs found
Linear Control Theory with an ℋ∞ Optimality Criterion
This expository paper sets out the principal results in ℋ∞ control theory in the context of continuous-time linear systems. The focus is on the mathematical theory rather than computational methods
Data-driven Balanced Truncation for Predictive Model Order Reduction of Aeroacoustic Response
Rapid prediction of the aeroacoustic response is a key component in the
design of aircraft and turbomachinery. While it is possible to achieve accurate
predictions using direct solution of the compressible Navier-Stokes equations,
applications of such solvers is not feasible in design optimization due to the
high cost of resolving wave phenomena in an Eulerian setting. In this work, we
propose a technique for highly accelerated predictions of aeroacoustic response
using a data-driven model reduction approach based on the eigensystem
realization algorithm (ERA), as a non-intrusive balanced truncation method.
Specifically, we create and compare ERA ROMs based on the training data
generated by solving the linearized and nonlinear Euler equations with Gaussian
pulse inputs, and use them for prediction of the aeroacoustic response of an
airfoil subject to different types of gust loading. The results show that both
ROMs are in good agreement with the full-order model (FOM) solution in a purely
predictive setting, while achieving orders of magnitude reduction in the online
computation time. Using ERA for prediction of the acoustic response requires
activating each input channel separately in the FOM for training ROMs, and
operating on a large Hankel matrix, that can become computationally infeasible.
We address this bottleneck in two steps: first, we propose a multi-fidelity
gappy POD method to identify the most impactful input channels based on a
coarser grid. Therefore, we reduce the computation cost on the FOM and ROM
levels as we build the Markov sequence by querying the high-resolution FOM only
for the input channels identified by gappy POD. Second, we use tangential
interpolation at the ROM level to reduce the size of the Hankel matrix. The
proposed methods enable application of ERA for highly accurate online acoustic
response prediction and reduce the offline computation cost of ROMs
emgr - The Empirical Gramian Framework
System Gramian matrices are a well-known encoding for properties of
input-output systems such as controllability, observability or minimality.
These so-called system Gramians were developed in linear system theory for
applications such as model order reduction of control systems. Empirical
Gramian are an extension to the system Gramians for parametric and nonlinear
systems as well as a data-driven method of computation. The empirical Gramian
framework - emgr - implements the empirical Gramians in a uniform and
configurable manner, with applications such as Gramian-based (nonlinear) model
reduction, decentralized control, sensitivity analysis, parameter
identification and combined state and parameter reduction
Non-intrusive Balancing Transformation of Highly Stiff Systems with Lightly-damped Impulse Response
Balanced truncation (BT) is a model reduction method that utilizes a
coordinate transformation to retain eigen-directions that are highly observable
and reachable. To address realizability and scalability of BT applied to highly
stiff and lightly-damped systems, a non-intrusive data-driven method is
developed for balancing discrete-time systems via the eigensystem realization
algorithm (ERA). The advantage of ERA for balancing transformation makes
full-state outputs tractable. Further, ERA enables balancing despite stiffness,
by eliminating computation of balancing modes and adjoint simulations. As a
demonstrative example, we create balanced ROMs for a one-dimensional reactive
flow with pressure forcing, where the stiffness introduced by the chemical
source term is extreme (condition number ), preventing analytical
implementation of BT. We investigate the performance of ROMs in prediction of
dynamics with unseen forcing inputs and demonstrate stability and accuracy of
balanced ROMs in truly predictive scenarios whereas without ERA, POD-Galerkin
and Least-squares Petrov-Galerkin projections fail to represent the true
dynamics. We show that after the initial transients under unit impulse forcing,
the system undergoes lightly-damped oscillations, which magnifies the influence
of sampling properties on predictive performance of the balanced ROMs. We
propose an output domain decomposition approach and couple it with tangential
interpolation to resolve sharp gradients at reduced computational costs
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