Robust Stability of Differential-Algebraic Equations
This paper presents a survey of recent results on the robust stability analysis and the distance to instability for linear time-invariant and time-varying differential-algebraic equations (DAEs). Different stability concepts such as exponential and asympt
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Abstract This paper presents a survey of recent results on the robust stability analysis and the distance to instability for linear time-invariant and time-varying differential-algebraic equations (DAEs). Different stability concepts such as exponential and asymptotic stability are studied and their robustness is analyzed under general as well as restricted sets of real or complex perturbations. Formulas for the distances are presented whenever these are available and the continuity of the distances in terms of the data is discussed. Some open problems and challenges are indicated. Keywords Differential-algebraic equation · Restricted perturbation · Robust stability · Stability radius · Spectrum · Index Mathematics Subject Classification 93B35 · 93D09 · 34A09 · 34D10
1 Introduction In many areas of science and engineering one uses mathematical models to simulate, control or optimize a system or process. These mathematical models, however, are typically inexact or contain uncertainties and thus, the following question is of major importance. How robust is a specific property of a given system described by differential or difference equations under perturbations to the data? N.H. Du · V.H. Linh (B) Faculty of Mathematics, Mechanics and Informatics, Vietnam National University, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam e-mail: [email protected] N.H. Du e-mail: [email protected] V. Mehrmann Institut für Mathematik, MA 4-5, Technische Universität Berlin, 10623 Berlin, Fed. Rep. Germany e-mail: [email protected] A. Ilchmann, T. Reis (eds.), Surveys in Differential-Algebraic Equations I, Differential-Algebraic Equations Forum, DOI 10.1007/978-3-642-34928-7_2, © Springer-Verlag Berlin Heidelberg 2013
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Here, we say that a certain property of a system is robust if it is preserved when an arbitrary (but sufficiently small) perturbation affects the system. An important quantity in this respect is then the distance (measured by an appropriate metric) between the nominal system and the closest perturbed system that does not possess the mentioned property, this is typically called the radius of the system property. In this paper, we deal with robustness and distance problems for differentialalgebraic equations (DAEs), with a focus on robust stability and stability radii. Systems of DAEs, which are also called descriptor systems in the control literature, are a very convenient modeling concept in various real-life applications such as mechanical multibody systems, electrical circuit simulation, chemical reactions, semidiscretized partial differential equations, and in general for automatically generated coupled systems, see [12, 39, 47, 63, 68, 84, 85] and the references therein. DAEs are generalizations of ordinary differential equations (ODEs) in that certain algebraic equations constrain the dynamical behavior. Since the dynamics of DAEs is constrained to a set which often is only given implicitly, many theoretical and numerical difficulties arise, which may lead to a sensitive behavior
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