Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte

Structured penalties for functional linear models---partially empirical eigenvectors for regression

One of the challenges with functional data is incorporating spatial structure, or local correlation, into the analysis. This structure is inherent in the output from an increasing number of biomedical technologies, and a functional linear model is often used to estimate the relationship between the predictor functions and scalar responses. Common approaches to the ill-posed problem of estimating a coefficient function typically involve two stages: regularization and estimation. Regularization is usually done via dimension reduction, projecting onto a predefined span of basis functions or a reduced set of eigenvectors (principal components). In contrast, we present a unified approach that directly incorporates spatial structure into the estimation process by exploiting the joint eigenproperties of the predictors and a linear penalty operator. In this sense, the components in the regression are `partially empirical' and the framework is provided by the generalized singular value decomposition (GSVD). The GSVD clarifies the penalized estimation process and informs the choice of penalty by making explicit the joint influence of the penalty and predictors on the bias, variance, and performance of the estimated coefficient function. Laboratory spectroscopy data and simulations are used to illustrate the concepts.Comment: 29 pages, 3 figures, 5 tables; typo/notational errors edited and intro revised per journal review proces

A Matlab toolbox for the regularization of descriptor systems arising from generalized realization procedures

In this report we introduce a Matlab toolbox for the regularization of descriptor systems. We apply it, in particular, for systems resulting from the generalized realization procedure of [16], which generates, via rational interpolation techniques, a linear descriptor system from interpolation data. The resulting system needs to be regularized to make it feasible for the use in simulation, optimization, and control. This process is called regularization.DFG, SFB 1029, Substantial efficiency increase in gas turbines through direct use of coupled unsteady combustion and flow dynamic

Applications of the local state-space form of constrained mechanical systems in multibody dynamics and robotics

This thesis explores several areas in dynamics which can be viewed as applications of the local state-space form of a mechanical system. The simulation of mechanical systems often involves the solution of differential algebraic equations (DAEs). DAEs occur in every mechanism containing kinematic loops. Such systems can be found in a wide range of areas including the aerospace, automotive, construction, and farm equipment industries. The numerical treatment of DAEs is a topic which is relatively recent and continues to be studied. One can regard DAEs as ordinary differential equations (ODEs) on certain invariant manifolds after index reduction. Thus, the numerical solutions of the DAEs can be obtained through integration of their underlying ODEs. In certain circumstances, difficulties may occur since the numerical solutions of the underlying ODE can drift away from the invariant manifold. In this thesis, the underlying ODEs are locally transformed into ODEs of minimal dimension via local parameterizations of the invariant manifold. By their nature, such ODEs are local and implicit, but their solutions do not suffer from the drift phenomenon. Since the states of these minimal ODEs are independent, they are known as a local state-space form of the equations of motion. This work focuses on generalizing the application of the local state-space form and applying it towards problem areas in multibody dynamics and robotics. The first application of the local state-space form is in deriving a formulation of dynamics called the Singularity Robust Null Space Formulation. This formulation utilizes several aspects of the singular value decomposition for an approach which is efficient, does not fail at singularities, and is better suited than most near singularities. The second application area in this work is the study of the linearized mechanical system. Since the linearized model is also useful in optimization and implicit integration problems, an efficient recursive algorithm for its construction is derived. The algorithm appeals to a formulation of the dynamics found in robotics to ease in a coherent derivation

Index preserving polynomial representation of nonlinear differential-algebraic systems

Recently in (9) a procedure was presented that allows to reformulate nonlinear ordinary differential equations in a way that all the nonlinearities become polynomial on the cost of increasing the dimension of the system. We generalize this procedure (called `polynomialization') to systems of differential-algebraic equations (DAEs). In particular, we show that if the original nonlinear DAE is regular and strangeness-free (i.e., it has differentiation index one) then this property is preserved by the polynomial representation. For systems which are not strangeness-free, i.e., where the solution depends on derivatives of the coefficients and inhomogeneities, we also show that the index is preserved for arbitrary strangeness index. However, to avoid ill-conditioning in the representation one should first perform an index reduction on the nonlinear system and then construct the polynomial representations. Although the analytical properties of the polynomial reformulation are very appealing, care has to be given to the numerical integration of the reformulated system due to additional errors. We illustrate our findings with several examples