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Mathematical and Computer Modelling of Dynamical Systems
, Volume 7 (2): 2 – Jun 1, 2001

/lp/taylor-francis/editorial-differential-algebraic-equations-and-descriptor-systems-uZWNiTlndI

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- Publisher
- Taylor & Francis
- Copyright
- Copyright Taylor & Francis Group, LLC
- ISSN
- 1744-5051
- eISSN
- 1387-3954
- DOI
- 10.1076/mcmd.7.2.131.3652
- Publisher site
- See Article on Publisher Site

Mathematical and Computer Modelling of Dynamical Systems 1387-3954/01/0702-131$16.00 2001, Vol. 7, No. 2, pp. 131±132 Swets & Zeitlinger Editorial Differential-Algebraic Equations and Descriptor Systems What is a differential-algebraic equation (DAE) and what is a descriptor system? Why does it make sense to model dynamical systems as DAEs? And how can we control DAEs and solve them numerically? Our special issue of Mathematical and Computer Modelling of Dynamical Systems addresses these questions. It also gives an overview on the application ®elds of computational engineering where DAEs are particularly important. We mention in this context mechanics and mechatronics, electrical engineering and chemical engineering. The general form of a DAE is F…t; x…t†; x _…t†† ˆ 0 with time t and vector valued functions x and F. A descriptor system, on the other hand, consists of a DAE with additional control. In contrast to this, an explicit or normal form of an ordinary differential equation (ODE) reads y _…t†ˆ f…t; y…t†† : If the DAE can be rewritten as ODE with the same state variables x y, then the DAE represents merely a system of implicit ODEs. In this special issue, however, we concentrate on those problems for which this rewriting is impossible, i.e., for which the Jacobian @F=@x _ is singular. Such differential- algebraic problems always impose constraints on the variables and show a behavior that strongly differs from standard ODE theory. As the authors in this issue demonstrate, there are good reasons to treat DAEs directly, rather than to try to transform them to an ODE. In many applications, the mathematical model takes quite naturally the form of a DAE, with state variables that have a physical signi®cance. Moreover, such models are generated automatically by simulation programs. The reduction to an ODE, on the other hand, may lead to less meaningful variables and is often time consuming or even not feasible. For instance, the widespread use of subsystem or network modelling leads in general to DAE models. Due to the efforts of the last decade, DAEs or descriptor systems have become popular in several application ®elds. Reliable and ef®cient numerical integration software is nowadays available, and the simulation process itself has become much easier. Nevertheless, there are still many open problems. 132 EDITORIAL Looking at analysis and design of descriptor systems, only linear problems can in general be solved. Major dif®culties arise for nonlinear problems. The growing need for re®ned models which include partial differential equations (PDEs) results also in challenging new questions. In this context, the new acronym PDAE for Partial Differential-Algebraic Equation has been created. In detail, the special issue is organised as follows. The ®rst part deals with modelling and analysis aspects. In the paper by P.C. Mu È ller, an overview on recent progress in analysis and synthesis of control systems in descriptor form is given. Next, Ch. Kraus, M. Winckler, and G. Bock discuss the use of certain natural coordinates for constrained mechanical systems, an application ®eld where DAEs and their appropriate modelling are particularly important. The third paper by K. Schlacher and A. Kugi concentrates on variational problems for descriptor systems and presents computer algebra algorithms in order to derive the necessary and Â Â suf®cient conditions of optimal control. The ®rst part closes with G. Fabian, D.A. van Beek, and J.E. Rooda who introduce an index reduction and discontinuity handling technique based on so-called substitute equations. Thereafter, several case studies show the importance of DAEs in application ®elds. M. Gu È nther investigates PDAE models for interconnected RLC networks in electrical circuit simulation. The transmission lines are here modelled as PDEs and then coupled with standard DAEs. In the following paper, G. Clau, P. Schwarz, B. Straube, and W. Vermeiren also focus on circuits and give an algorithm for symbolic index calculation. Moreover, C. Tischendorf and D.E. Schwarz draw attention to three essential problems in circuit simulation, namely the role of the index, consistent initial values and asymptotic stability. The second part ends with M. Schaub and B. Simeon. Their paper presents an analysis of models and simulation techniques for the system of pantograph and catenary, an application from mechanical systems. Finally, numerical methods and their properties when applied to DAEs are discussed. I. Higueras studies numerical methods for stiff DAEs of index 3 and extends the results to in®nite dimensional systems or PDAEs. Last but not least, J. Schropp shows how certain Runge±Kutta methods preserve geometric properties of nonautonomous DAEs of index 2. At this point, we would like to thank all the authors who contributed to the special issue. We hope that their work demonstrates the importance of the subject but also serves as starting point for future research. P.C. Muller and B. Simeon, Wuppertal and Karlsruhe, June 2001

Mathematical and Computer Modelling of Dynamical Systems – Taylor & Francis

**Published: ** Jun 1, 2001

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