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Project
Model Reduction for a Class of Nonlinear Electrical Circuits by Reduction of Linear Subcircuits
(2010)
We analyze a model reduction approach for a class of electrical circuits containing nonlinear resistance.
In our approach
the linear subcircuits are extracted and replaced with linear passive reduced-order models. The resulting nonlinear
reduced-order model preserves admissibility and passivity. Moreover, we derive a-priori bounds for the error between
the input-output map of the original circuit equations and that of our reduced-order model. These bounds
are valid for all inputs. Since only linear subcircuits are reduced, our approach is effective when the number of nonlinear resistances is relatively small. The performance of our approach is illustrated numerically.
We study the class of linear differential-algebraic m-input m-output systems
which have a transfer function with proper inverse.
A sufficient condition for the transfer function to have proper inverse
it that the system has 'strict and non-positive relative degree'.
We present two main results:
First, a so called 'zero dynamics form' is derived: this form is - within the class of system equivalence - a simple "almost normal" form of the DAE; it is a counterpart to the well-known Byrnes-Isidori form for
ODE systems with strictly proper transfer function.
The 'zero dynamics form' is exploited to characterize structural properties such as
asymptotically stable zero dynamics,
minimum phase, and high-gain stabilizability.
The zero dynamics are characterized by (A,E,B)-invariant subspaces.
Secondly, it is shown that the 'funnel controller' (that is a static nonlinear output error feedback) achieves, for all DAE systems with asymptotically stable zero dynamics and transfer function with proper inverse, tracking of a reference signal by the output signal within a pre-specified funnel. This funnel determines the transient behaviour.
We consider the behavior of a modulated wave solution to
an $\mathbb{S}^1$-equivariant autonomous system of differential equations under an external
forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the
modulation frequency of the modulated wave solution, while the wave frequency of the forcing is supposed to be far
from that of the modulated wave solution. We describe the domain in the three-dimensional
control parameter space (of frequencies and amplitude of the forcing)
where stable locking of the modulation frequencies of the forcing and the modulated wave solution
occurs.
Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external
periodically modulated
optical signals.
Functional Magnetic Resonance Imaging inherently involves noisy measurements and a severe multiple test
problem. Smoothing is usually used to reduce the effective number of multiple
comparisons and to locally integrate the signal and hence increase the
signal-to-noise ratio. Here, we provide a new structural adaptive segmentation
algorithm (AS)
that naturally combines the signal detection with noise reduction in one procedure.
Moreover, the new method
is closely related to a recently proposed structural adaptive smoothing
algorithm and preserves shape and spatial extent of activation areas without
blurring the borders.
Diffusion weighted imaging is a magnetic resonance based method to investigate
tissue micro-structure especially in the human brain via water diffusion.
Since the standard diffusion tensor model for the acquired data failes in
large portion of the brain voxel more sophisticated models have bee developed.
Here, we report on the package dti and how some of these models
can be used with the package.
Given a system of closed convex domains (inclusions) in n-dimensional Euclidean space, new computational meshes are introduced which partition the convex hull of the inclusion set into simple geometric objects. These partitions generalize the concept of Delaunay triangulations by interpreting the inclusions as generalized vertices while the remaining elements of the partition serve as connections between generalized vertices and therefore assume the classical role of edges, faces, etc. The proposed partitions are derived in two different ways: by exploiting duality with respect to certain generalized Voronoi partitions and by generalizing the well known Delaunay (empty circumcircle) criterion.
Generalized Delaunay partitions are of practical importance for the modeling of particle- and fiber-reinforced composite materials
since they enable an efficient conforming resolution of the highly complicated component geometries. The number of elements in the partitions is proportional to the number of inclusions which is minimal.
We describe the appearance and stability of spatio-temporal periodic
patterns (rotating waves) in unidirectional rings of coupled oscillators
with delayed couplings. We show how delays in the coupling lead
to a splitting of each rotating wave into several new ones. The appearance
of rotating waves is mediated by Hopf bifurcations of the symmetric
equilibrium.
We also conclude that the coupling delays can be effectively
replaced by increasing the number of oscillators in the chain.
The phenomena are shown for Stuart-Landau
oscillators as well as for coupled FitzHugh-Nagumo systems interacting
via excitatory chemical synapses.
The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6
(2010)
Let $B$ be a fractional Brownian motion with Hurst parameter
$H=1/6$. It is known that the symmetric Stratonovich-style Riemann sums
for $\int g(B(s))\,dB(s)$ do not, in general, converge in probability.
We show, however, that they do converge in law in the Skorohod space of
c\`adl\`ag functions. Moreover, we show that the resulting stochastic
integral satisfies a change of variable formula with a correction term
that is an ordinary It\^o integral with respect to a Brownian motion
that is independent of $B$.
Classical surface parameterization algorithms often place singularities
in order to enhance the quality of the resulting parameter map. Unfortunately, singularities of positive integral index (as the north pole of a sphere) were not handled since they cannot be described with piecewise linear parameter functions on a triangle mesh. Preprocessing is needed to adapt the mesh connectivity. We present an extension to the QuadCover parameterization algorithm [KNP07], which allows to handle those singularities. A singularity of positive integral index can be resolved using bilinear parameter functions on quadrilateral elements. This generalization
of piecewise linear functions for quadrilaterals enriches the space of parameterizations. The resulting parameter map can be visualized by textures using a rendering system which supports quadrilateral elements, or it can be used for remeshing into a pure quad mesh.