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.
We introduce a transformation between the generalized symplectic
pencils and the skew-Hermitian/Hermitian pencils. Under the transformation
the regularity of the matrix pencils is preserved, and the
equivalence relations about their eigenvalues and deflating subspaces
are established. The eigenvalue problems of the generalized symplectic
pencils and skew-Hermitian/Hermitian pencils are strongly related to
the discrete-time and continuous-time robust control problems, respectively.
With the transformation a simple connection between these two
types of robust control problems is made. The connection may help
to develop unified methods for solving the robust control problems.
Abstract. We consider a mathematical model (the so-called traveling-wave system) which describes longitudinal
dynamical effects in semiconductor lasers. This model consists of a linear hyperbolic system
of PDEs, which is nonlinearly coupled with a slow subsystem of ODEs. We prove that a corresponding
initial-boundary value problem is well posed and that it generates a smooth infinite-dimensional dynamical
system. Exploiting the particular slow–fast structure, we derive conditions under which there exists a lowdimensional
attracting invariant manifold. The flow on this invariant manifold is described by a system
of ODEs. Mode approximations of that system are studied by means of bifurcation theory and numerical
tools.
We simulate and analyse a 1D-PDE model describing the dynamics of multisection semiconductor lasers. We demonstrate how a semi-analytical computation of the spectrum and the corresponding eigenfunction expansion of the computed solutions provides a useful information allowing to achieve a better understanding of the laser dynamics. Basic algorithms implemented into a corresponding software tool are described.
The Modified Nodal Analysis leads to differential algebraic equations
with properly stated leading terms. In this article a special structure of the DAEs
modelling electrical circuits is exploited in order to derive a new decoupling for
nonlinear index-2 DAEs. This decoupling procedure leads to a solvability result and
is also used to study general linear methods, a class of numerical schemes that covers
both Runge-Kutta and linear multistep methods. Convergence for index-2 DAEs is
proved.
In this paper we give a survey on balanced truncation model order
reduction for linear time-invariant continuous-time systems in descriptor form. We
first give a brief overview of the basis concepts from linear system theory and then
present balanced truncation model reduction methods for descriptor systems and
discuss their algorithmic aspects. The efficiency of these methods is demonstrated
by numerical experiments.
Transient analysis in industrial chip design leads to very large systems
of differential-algebraic equations (DAEs). The numerical solution of these
DAEs strongly depends on the so called index of the DAE. In general,
the higher the index of the DAE is, the more sensitive the numerical
solution will be to errors in the computation. So, it is advisable to use
mathematical models with small index or to reduce the index.
This paper presents an index reduction method that uses information
based on the topology of the circuit. In addition, we show that the
presented method retains structural properties of the DAE.
We study linear, possibly over- or under-determined, differentialalgebraic
equations that have the same solution behavior as linear
differential-algebraic equations with well-dened strangeness index. In
particular, we give three different characterizations for differentialalgebraic
equations, namely by means of solution spaces, canonical
forms, and derivative arrays. We distinguish two levels of generalization,
where the more restrictive case contains an additional assumption
on the structure of the set of consistent inhomogeneities.
In this paper we discuss the time integration of multibody system model equations
following a novel approach that has originally been developed for the time integration of linear
differential-algebraic equations of arbitrary high index. We do not restrict ourselves to classical
constrained mechanical systems but consider the more complex model equations that are actually
used in state-of-the-art multibody system simulation packages. The equations of motion form a
system of differential-algebraic equations of differentiation index 3 with a special structure that we
will exploit in the numerical solution. We replace the equations of motion by a so-called projected
differentiation index-one differential-algebraic equation with the same solution set.
Traveling wave equations are used to model the dynamics of multisection semiconductor lasers. To perform a bifurcation analysis of this system of 1-D partial differential equations its low dimensional approximations are constructed and considered. Along this paper this analysis is used for the extensive study of the pulsations in a three section distributed feedback laser. Namely, stability of pulsations, different bifurcation scenaria, tunability of the pulsation frequency and its locking by the frequency of electrical modulation are considered. All these pulsation qualities are highly important when applying lasers in optical communication systems.