678 research outputs found
A note on the cross Gramian for non-symmetric systems
The cross Gramian matrix is a tool for model reduction and system identification, but it is only applicable to square control systems. For symmetric systems, the cross Gramian possesses a useful relation to the system's associated Hankel singular values. Yet, many real-life models are neither square nor symmetric. In this work, concepts from decentralized control are used to approximate a cross Gramian for non-symmetric and non-square systems. To illustrate this new non-symmetric cross Gramian, it is applied in the context of model order reduction
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
N identical particles under quantum confinement: A many-body dimensional perturbation theory approach
Systems that involve N identical interacting particles under quantum
confinement appear throughout many areas of physics, including chemical,
condensed matter, and atomic physics. In this paper, we present the methods of
dimensional perturbation theory, a powerful set of tools that uses symmetry to
yield simple results for studying such many-body systems. We present a detailed
discussion of the dimensional continuation of the N-particle Schrodinger
equation, the spatial dimension D -> infinity equilibrium (D^0) structure, and
the normal-mode (D^{-1}) structure. We use the FG matrix method to derive
general, analytical expressions for the many-body normal-mode vibrational
frequencies, and we give specific analytical results for three confined N-body
quantum systems: the N-electron atom, N-electron quantum dot, and N-atom
inhomogeneous Bose-Einstein condensate with a repulsive hardcore potential
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