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.
We develop a behavioural approach to linear, time-varying, differential algebraic systems.
The analysis is \almost everywhere" in the sense that the statements hold on R T, where
T is a discrete set. Controllability, observability and autonomy is introduced and related to
the behaviour of the system. Classical results on the behaviour of time-invariant systems are
studied in the context of time-varying systems.
The purpose of this paper is the analysis of relaxation methods for the numerical integration
of coupled systems of ODEs and DAEs. We will investigate convergence of relaxation methods and put
special emphasis on the Jacobi- and Gauss-Seidel methods. Furthermore, the fundamental difference
in the convergence behaviour of coupled ODEs and DAEs is pointed out. This difference is used to explain
why certain relaxation methods for coupled DAEs may fail. Finally, a remedy to this undesirable effect
is proposed that makes use of a so-called preconditioned dynamic iteration strategy. This regularization
also allows significant reduction of relaxation steps.
We consider a time-dependent optimal control problem, where the state
evolution is described by an ODE. There is a variety of methods for the treatment
of such problems. We prefer to view them as boundary value problems and apply to
them the Riccati approach for non-linear BVPs with separated boundary conditions.
There are many relationships between multiple shooting techniques, the Riccati
approach and the Pantoja method, which describes a computationally efficient
stage-wise construction of the Newton direction for the discrete-time optimal control
problem.
We present an efficient implementation of this approach. Furthermore, the wellknown
checkpointing approach is extended to a `nested checkpointing` for multiple
transversals. Some heuristics are introduced for an efficient construction of nested
reversal schedules. We discuss their benefits and compare their results to the optimal
schedules computed by exhaustive search techniques.
Systems of stiff ordinary differential equations (ODEs) can be integrated
properly only by implicit methods. For that purpose, one usually has to
solve a system of nonlinear equations at each time step. This system of equations
may be solved by variants of Newton's method. Here, the main computing effort lies
in forming and factoring the Jacobian or a suitable approximation to it.
In this paper, we examine a new approach of constructing an appropriate quasi-
Newton approximation for solving stiff ODEs. The method makes for the first time
explicit use of tangent and adjoint information that can be obtained using the forward
and the reverse mode of algorithmic differentiation (AD). We elaborate the
conditions for invariance with respect to linear transformations of the state space
and thus similarity transformations of the Jacobian. One new updating variant that
yields such an invariant method is presented. Numerical results for Runge-Kutta
methods and linear multi-step methods are discussed.
We present the mathematical theory of general over- and underdetermined
hybrid (switched) systems of differential-algebraic equations
(HDAEs). We give a systematic formulation of HDAEs and discuss existence
and uniqueness of solutions, the numerical computation of the switch
points and how to perform consistent initialization at switch points. We
show how numerical solution methods for DAEs can be adapted for HDAEs
and present a comparison of these methods for the real world example of
simulating an automatic gearbox.
The numerical simulation of very large scale integrated
circuits is an important tool in the development of new
industrial circuits. This topic has received increasing attention
within the last years. The main problem in circuit simulation is
that the model equations lead to differential algebraic equations
(DAEs). One known property of circuit DAEs is that they
may have an index larger than one, i.e., they may contain socalled
hidden constraints. The increased index has numerous
disadvantages on the numerical treatment of circuit DAEs.
The determination of these hidden constraints can be done
investigating the circuit topology. Until now, this information
has only been used for the consistent initialization of the circuit
equations. A recent approach has been to reduce the index of the
circuit DAE in order to improve their numerical behaviour. This
paper will give graph theoretical methods that lead to constraints
in a favorable formulation. Furthermore, the index reduction via
minimal extension will be performed for circuit DAEs, using these
constraints.
Element-based Topological Index Reduction for Differential-Algebraic Equations in Circuit Simulation
(2005)
The numerical simulation of very large scale integrated
circuit is an important tool in the development of new
industrial circuits. In the course of the last years, this topic has
received increasing attention. Common modeling approaches for
circuits lead to differential-algebraic systems (DAEs). In circuit
simulation, these DAEs are known to have index 2, given some
topological properties of the network. This higher index leads
to several undesirable effects in the numerical solution of the
DAEs. Recent approaches try to lower the index to improve
the numerical behaviour. These methods usually involve costly
algebraic transformations of the differential-agebraic equations.
Especially, for large scale circuit equations, these transformations
become too costly to be efficient.
We will present methods that change the topology of the network
itself, while replacing certain elements in oder to obtain a
network that leads to a DAE of index 1. This procedure can
be performed prior to the actual numerical simulation. The
decreasing of the index usually leads to significantly improved
numerical behaviour.
Classical results about the local existence and uniqueness of
DAE solutions are based on the derivative array [2] or on a geometrical
approach [13]. Thus these results can't be applied to equations with nonsmooth
coefficients. Also, sufficient conditions that guarantee solvability
are hard to check in general [6, 13]. In this paper a new approach to proving
local existence and uniqueness of DAE solutions is presented. The
main tool is a decoupling procedure that makes it possible to split DAE
solutions into their characteristic parts. Thus it is possible to weaken
the smoothness requirements considerably. In order for the decoupling
procedure to work we require a certain structural condition to hold. In
contrast to results already known, this condition can be easily verified.
A new MATLAB toolbox for computing eigenvalues
and invariant subspaces of Hamiltonian and skew-Hamiltonian
matrices is described. Based on orthogonal symplectic decompositions,
the implemented algorithms are both numerically backward
stable and structure-preserving. It will be demonstrated
how this toolbox can be used to address a number of tasks
in systems and control theory, including some model reduction
methods and the computation of the H? norm.