351,852 research outputs found
Interpolatory methods for model reduction of multi-input/multi-output systems
We develop here a computationally effective approach for producing
high-quality -approximations to large scale linear
dynamical systems having multiple inputs and multiple outputs (MIMO). We extend
an approach for model reduction introduced by Flagg,
Beattie, and Gugercin for the single-input/single-output (SISO) setting, which
combined ideas originating in interpolatory -optimal model
reduction with complex Chebyshev approximation. Retaining this framework, our
approach to the MIMO problem has its principal computational cost dominated by
(sparse) linear solves, and so it can remain an effective strategy in many
large-scale settings. We are able to avoid computationally demanding
norm calculations that are normally required to monitor
progress within each optimization cycle through the use of "data-driven"
rational approximations that are built upon previously computed function
samples. Numerical examples are included that illustrate our approach. We
produce high fidelity reduced models having consistently better
performance than models produced via balanced truncation;
these models often are as good as (and occasionally better than) models
produced using optimal Hankel norm approximation as well. In all cases
considered, the method described here produces reduced models at far lower cost
than is possible with either balanced truncation or optimal Hankel norm
approximation
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
Empirical Model Reduction of Controlled Nonlinear Systems
In this paper we introduce a new method of model reduction for nonlinear systems
with inputs and outputs. The method requires only standard matrix computations, and
when applied to linear systems results in the usual balanced truncation. For nonlinear
systems, the method makes used of the Karhunen-Lo`eve decomposition of the state-space,
and is an extension of the method of empirical eigenfunctions used in fluid dynamics. We
show that the new method is equivalent to balanced-truncation in the linear case, and
perform an example reduction for a nonlinear mechanical system
Modeling of Transitional Channel Flow Using Balanced Proper Orthogonal Decomposition
We study reduced-order models of three-dimensional perturbations in
linearized channel flow using balanced proper orthogonal decomposition (BPOD).
The models are obtained from three-dimensional simulations in physical space as
opposed to the traditional single-wavenumber approach, and are therefore better
able to capture the effects of localized disturbances or localized actuators.
In order to assess the performance of the models, we consider the impulse
response and frequency response, and variation of the Reynolds number as a
model parameter. We show that the BPOD procedure yields models that capture the
transient growth well at a low order, whereas standard POD does not capture the
growth unless a considerably larger number of modes is included, and even then
can be inaccurate. In the case of a localized actuator, we show that POD modes
which are not energetically significant can be very important for capturing the
energy growth. In addition, a comparison of the subspaces resulting from the
two methods suggests that the use of a non-orthogonal projection with adjoint
modes is most likely the main reason for the superior performance of BPOD. We
also demonstrate that for single-wavenumber perturbations, low-order BPOD
models reproduce the dominant eigenvalues of the full system better than POD
models of the same order. These features indicate that the simple, yet accurate
BPOD models are a good candidate for developing model-based controllers for
channel flow.Comment: 35 pages, 20 figure
Multi-Robot Transfer Learning: A Dynamical System Perspective
Multi-robot transfer learning allows a robot to use data generated by a
second, similar robot to improve its own behavior. The potential advantages are
reducing the time of training and the unavoidable risks that exist during the
training phase. Transfer learning algorithms aim to find an optimal transfer
map between different robots. In this paper, we investigate, through a
theoretical study of single-input single-output (SISO) systems, the properties
of such optimal transfer maps. We first show that the optimal transfer learning
map is, in general, a dynamic system. The main contribution of the paper is to
provide an algorithm for determining the properties of this optimal dynamic map
including its order and regressors (i.e., the variables it depends on). The
proposed algorithm does not require detailed knowledge of the robots' dynamics,
but relies on basic system properties easily obtainable through simple
experimental tests. We validate the proposed algorithm experimentally through
an example of transfer learning between two different quadrotor platforms.
Experimental results show that an optimal dynamic map, with correct properties
obtained from our proposed algorithm, achieves 60-70% reduction of transfer
learning error compared to the cases when the data is directly transferred or
transferred using an optimal static map.Comment: 7 pages, 6 figures, accepted at the 2017 IEEE/RSJ International
Conference on Intelligent Robots and System
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