Two optimal control problems for instationary magnetization
processes are considered in 3D spatial domains that
include electrically conducting and nonconducting regions. The magnetic
fields are generated by induction coils. In the first model, the induction coil
is considered as part of the conducting region and the electrical current is taken
as control. In the second, the coil is viewed as part of the nonconducting region and the
electrical voltage is the control. Here, an integro-differential equation accounts
for the magnetic induction law that couples the given
electrical voltage with the induced electrical current in the
induction coil.
We derive first-order necessary
optimality condition for the optimal controls of both problems. Based on them,
numerical methods of gradient type are applied. Moreover, we report on the application
of model reduction by POD that lead to tremendous savings. Numerical tests are
presented for academic 3D geometries but also for a real-world application.
We propose a framework for deformation-based surface modeling that is interactive, robust and intuitive to use. The deformations are described by a non-linear optimization problem that models static states of elastic shapes under external forces which implement the user input. Interactive response is achieved by a combination of model reduction, a robust energy approximation, and an efficient quasi-Newton solver. Motivated by the observation that a typical modeling session requires only a fraction of the full shape
space of the underlying model, we use second and third derivatives of a deformation energy to construct a low-dimensional shape space that forms the feasible set for the optimization. Based on mesh coarsening, we propose an
energy approximation scheme with adjustable approximation quality. The quasi-Newton solver guarantees superlinear convergence without the need of costly Hessian evaluations during modeling. We demonstrate the effectiveness of the approach on different examples including the test suite introduced in [Botsch and Sorkine 2008].
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 propose a model reduction method for positive systems that ensures the positivity of the reduced-order model. In the standard as well as in the descriptor case, for continuous-time and discrete-time systems, our approach is based on constructing diagonal solutions of Lyapunov inequalities. These are linear matrix inequalities (LMIs), which are shown to be feasible. Positivity and stability are preserved and an error bound in the $\mathcal{H}_\infty$-norm is provided.
In this paper we consider structure-preserving model reduction of
second-order systems using a~ba\-lan\-ced truncation approach.
Several sets of singular values are introduced for such systems,
which lead to different concepts of balancing and different
second-order balanced truncation methods. We compare the
properties of these methods on numerical examples.
We review some known results for POD model reduction applied to ODEs. Then, these
results are generalized to several types of DAEs. We provide algorithms for the
model reduction and error bounds for the reduced order models. Some limits of
the approach are pointed out and alternative methods for reduced order subspace approximation
are presented. The POD approach is tested and evaluated for a medium sized DAE example
from multibody dynamics.
In this paper we give an overview of model
order reduction techniques for coupled
systems. We consider linear time-invariant
control systems that are coupled through
input-output relations and discuss model
reduction of such systems using moment
matching and balanced truncation.
Structure-preserving approaches to model
order reduction of coupled systems are also
presented. Numerical examples are given.
Descriptor systems present a general
mathematical framework for the modelling, simulation and control of complex dynamical systems arising in many areas of mechanical,
electrical and chemical engineering. This
paper presents a survey of the current theory of descriptor systems,concerning
solvability, stability, model reduction, controllability, observability and optimal control.