Aspects of Modeling Dynamical Systems by Differential-Algebraic Equations

Abstract

In recent years the analysis and synthesis of control systems in descriptor form has been established. The general description of dynamical systems by differential-algebraic equations (DAE) is important for many applications in mechanics and mechatronics, in electrical and electronic engineering, and in chemical engineering as well. In this contribution the pros and cons of system modelling by differential-algebraic equations are discussed and an actual state of the art of descriptor systems is presented. Firstly, the advantages of modelling are touched in general and illustrated in detail by Lagranges equations of ®rst kind, by subsystem modeling and by the statement of the tracking control problem. Secondly, the development of tools for numerical integration is discussed resulting in the comment that today stable and ef®cient DAE solvers exist and that the simulation of descriptor systems is not a problem any longer. Thirdly, the methods of analyzing and designing descriptor systems are considered. Here, linear and nonlinear systems have to be distinguished. For linear descriptor systems more or less the required methods to solve usual control tasks are available in principal. But actually a related program package for fast and reliable application of these methods is still missed. However, in the near future such a toolbox is expected. Main dif®culties arise for nonlinear problems. A few results on stability and optimal control are known only and still a lot of research work has to be effected. In spite of these de®ciencies, all over the descriptor system approach is very attractive for modeling and simulation, and will become attractive more and more for analysis and design.

Consistent‐inconsistent decomposition to initial value problem of descriptor linear systems

“…Descriptor systems (singular systems, differential-algebraic equations) are an interesting research topic in numerical mathematics, mechanics, and control theory recently [1][2][3][4][5][6][7][8][9]. Among many interesting phenomena, we point out the inconsistent initial value problem, which results in, for example, the need for reinitialization in numerical integration [6] and impulse elimination in control [2] for this type of systems.…”

Eigenvalue bounds for matrix pencils

“…In the last two decades the modelling of dynamical systems by descriptor systems (singular systems, differential-algebraic equations) became more and more familiar, cf. e.g [1,2,4]. For linear time-invariant uncontrolled systems the dynamic behavior is described by Eẋ(t) = Ax(t), rkE = l < n = dim x, where (λE − A) is regular, i.e.…”