Linear Index-1 DAEs: Regular and Singular Problems

Several features and interrelations of projector methods and reduction techniques for the analysis of linear time-varying differential-algebraic equations (DAEs) are addressed in this work. The application of both procedures to regular index-1 problems is reviewed, leading to some new results which extend the scope of reduction techniques through a projector approach. Certain singular points are well accommodated by reduction methods; the projector framework is adapted in this paper to handle (not necessarily isolated) singularities in an index-1 context. The inherent problem can be described in terms of a scalarly implicit ODE with continuous operators, in which the leading coefficient function does not depend on the choice of projectors. The nice properties of projectors concerning smoothness assumptions are carried over to the singular setting. In analytic problems, the kind of singularity arising in the scalarly implicit inherent ODE is also proved independent of the choice of projectors. The discussion is driven by a simple example coming from electrical circuit theory. Higher index cases and index transitions are the scope of future research.     

Averaging for switched DAEs

Switched differential-algebraic equations (switched DAEs) are suitable for modeling many practical systems, e.g. electrical circuits. When the switching is periodic and of high frequency, the question arises whether the solutions of switched DAEs can be approximated by an average non-switching system. It is well known that for a quite general class of switched ordinary differential equations (ODEs) this is the case. For switched DAEs, due the presence of the so-called consistency projectors, it is possible that the limit of trajectories for faster and faster switching does not exist. Under certain assumptions on the consistency projectors a result concerning the averaging for switched DAEs is presented. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Distributional solution theory for linear DAEs

A solution theory for switched linear differential–algebraic equations (DAEs) is developed. To allow for non–smooth coordinate transformation, the coefficients matrices may have distributional entries. Since also distributional solutions are considered it is necessary to define a suitable multiplication for distribution. This is achieved by restricting the space of distributions to the smaller space of piecewise–smooth distributions. Solution formulae for two special DAEs, distributional ordinary differential equations (ODEs) and pure distributional DAEs, are given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Simple Structural Analysis Method for DAEs

We describe a straightforward method for analysing the structure of a differential-algebraic system. It generalizes the method of Pantelides, but is more directly informative and applies to DAEs with derivatives of any order. It naturally leads to a numerical method for the initial value problem that combines projection and index reduction. We illustrate the method by examples, and justify it with proofs. We prove that it succeeds on a fairly wide class of systems encountered in practice, and show its relation to the Pantelides method and to the Campbell-Gear derivative-array equations.This revised version was published online in October 2005 with corrections to the Cover Date.     

Decoupling of Abstract DAEs

We consider linear and time-invariant abstract differential-algebraic equations (DAEs) which are equations of the form E (t) = Ax (t) + f (t), x (0) = x₀. x (·) and f (·) are functions with values in Hilbert spaces X and Z and the operator E: X → Z is assumed to be bounded, whereas A is closed and defined on some dense subspace D (A). Motivated by the Kronecker normal form for the finite dimensional case, the decoupling of abstract DAEs is investigated. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical stroboscopic averaging for ODEs and DAEs

The stroboscopic averaging method (SAM) is a technique for the integration of highly oscillatory differential systems dy/dt=f(y,t) with a single high frequency. The method may be seen as a purely numerical way of implementing the analytical technique of stroboscopic averaging which constructs an averaged differential system dY/dt=F(Y) whose solutions Y interpolate the sought highly oscillatory solutions y. SAM integrates numerically the averaged system without using the analytic expression of F; all information on F required by the algorithm is gathered on the fly by numerically integrating the originally given system in small time windows. SAM may be easily implemented in combination with standard software and may be applied with variable step-sizes. Furthermore it may also be used successfully to integrate oscillatory DAEs. The paper provides an analytic and experimental study of SAM and two related techniques: the LIPS algorithm of Kirchgraber and multirevolution methods. An error analysis is provided that indicates that the efficiency of all these techniques increases even further when combined with splitting integrators.     

Numerical Methods for Stiff Index-3 DAEs

Space semidiscretization of PDAEs, i.e. coupled systems of PDEs and algebraic equations, give raise to stiff DAEs and thus the standard theory of numerical methods for DAEs is not valid. As the study of numerical methods for stiff ODEs is done in terms of logarithmic norms, it seems natural to use also logarithmic norms for stiff DAEs. In this paper we show how the standard conditions imposed on the PDAE and the semidiscretized problem are formally the same if they are expressed in terms of logarithmic norms. To study the mathematical problem and their numerical approximations, this link between the standard conditions and logarithmic norms allow us to use for stiff DAEs techniques similar to the ones used for stiff ODEs. The analysis is done for problems which appear in the context of elastic multibody systems, but once the tools, i.e., logarithmic norms, are developed, they can also be used for the analysis of other PDAEs/DAEs.     

Funnel control for electrical circuits

We study the zero dynamics and funnel control for linear passive electrical circuits. We show that asymptotic stability of the zero dynamics can be characterized by criteria on the circuit topology. Thereafter we consider the output regulation problem for electrical circuits by funnel control. We show that for circuits with asymptotically stable zero dynamics, the funnel controller achieves tracking of a class of reference signals within a pre-specified funnel. This result can be relaxed to the case of non-autonomous zero dynamics by requiring the reference trajectory to evolve within a certain subspace. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stabilized multistep methods for index 2 Euler-Lagrange DAEs

We consider multistep discretizations, stabilized by -blocking, for Euler-Lagrange DAEs of index 2. Thus we may use nonstiff multistep methods with an appropriate stabilizing difference correction applied to the Lagrangian multiplier term. We show that orderp =k + 1 can be achieved for the differential variables with orderp =k for the Lagrangian multiplier fork-step difference corrected BDF methods as well as for low orderk-step Adams-Moulton methods. This approach is related to the recently proposed half-explicit Runge-Kutta methods.     

Singularity analysis of quasi-linear DAEs

A reduction method is introduced to explore the quasi-linear DAE. On the basis of reduction, three kinds of singularities of quasi-linear DAE are discussed.     

Consistent initialization for DAEs in Hessenberg form

For nonlinear DAEs, we can hardly make a reasonable statement unless structural assumptions are given. Many results are restricted to explicit DAEs, often in Hessenberg form of order up to three. For the DAEs resulting from circuit simulation, different beneficial structures have been found and exploited for the computation of consistent initial values. In this paper, a class of DAEs in nonlinear Hessenberg form of arbitrary high order is defined and analyzed with regard to consistent initialization. For this class of DAEs, the hidden constraints can be systematically described and the consistent initialization can be determined step-by-step solving linear subproblems, an approach hitherto used for the DAEs resulting from circuit simulation. Finally, it is shown that the DAEs resulting from mechanical systems fulfill the defined structural assumptions. The algorithm is illustrated by several examples.     

Controllability characterization of switched DAEs

We study controllability of switched differential algebraic equations (switched DAEs) with fixed switching signal. Based on a behavioral definition of controllability we are able to establish a controllability characterization that takes into account possible jumps and impulses induced by the switches. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Index 2 DAEs in Plasticity Theory

The equations of rate‐independent elastoplasticity form a differential‐algebraic equation (DAE) with discontinuities. For the numerical solution, implicit Runge‐Kutta methods are applied and combined with the return mapping strategy of computational plasticity. It turns out that the convergence order depends crucially on the switching point detection. Further, it is shown that algebraically stable Runge‐Kutta methods preserve the contractivity of the elastoplastic flow.     

Reversible methods of Runge-Kutta type for Index-2 DAEs

Summary. A new interpretation of Runge-Kutta methods for differential algebraic equations (DAEs) of index 2 is presented, where a step of the method is described in terms of a smooth map (smooth also with respect to the stepsize). This leads to a better understanding of the convergence behavior of Runge-Kutta methods that are not stiffly accurate. In particular, our new framework allows for the unified study of two order-improving techniques for symmetric Runge-Kutta methods (namely post-projection and symmetric projection) specially suited for solving reversible index-2 DAEs.Mathematics Subject Classification (1991): 65L05, 65L06     

Observer Design based on Constant-Input Observability for DAEs

For differential-algebraic equations (DAEs) an observability notion is considered which assumes the input to be unknown and constant. Based on this, an observer design is proposed. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

多项式微分代数系统的奇点性质

本讨论多项式微分代数方程的奇点性质，证明了经用吴方法整序后的系统的奇点与原系统的相应奇点有相同的鞍点性质。