PrefaceDifferential-algebraic equations (DAEs) have been the object of increasing attention in the last three decades. Nowadays, they provide a valuable tool for system modeling and analysis within different fields, including nonlinear electric and electronic circuit theory, constrained mechanics, and control theory, among others.The first part of this book addresses analytical properties of such differential-algebraic systems. The few existing monographs on DAEs are mainly focused on numerical aspects and, in most cases, they are restricted to a specific approach, structured around the differentiation, geometric, tractability, perturbation or strangeness indices, respectively. By contrast, the present book attempts to discuss a variety of analytical frameworks for the study of DAEs. The emphasis will be on projector methods based upon the tractability index, and also on reduction techniques supported on the geometric index. The differentiation index will also be briefly examined; note that it has received comparatively more attention in the DAE literature than the projector-based and reduction frameworks, in spite of the many benefits displayed by these approaches.Projector-based methods, introduced for linear DAEs in Chapter 2, allow for precise input-output functional descriptions and explicit solution characterizations in terms of the original problem variables: this holds for linear time-varying DAEs with arbitrary index, under mild smoothness requirements. These methods have been mainly developed by Roswitha März and, accordingly, the material in Chapter 2 is crucially based on her work. Nevertheless, some recent or new contributions can also be found in this Chapter, concerning e.g. the so-called Π-projectors, a simplification of the decoupling of DAEs with properly stated leading term, or a detailed characterization of standard form linear problems. Reduction methods, based on the research of Rabier, Rheinboldt and other authors, define a powerful framework for the analysis of nonlinear DAEs. Chapter 3 is mainly focused on these techniques, introducing in particular a local approach for quasilinear DAEs in settings where the global assumptions of Rabier and Rheinboldt do not hold, and paving the way for subsequent analyses of singular problems. The differentiation index is also discussed in this Chapter; cf. Sections 3.1, 3.2 and 3.7.From a dynamical point of view, the essential differences between DAEs and explicit ordinary differential equations (ODEs) arise in so-called singular problems, which lead to new dynamic phenomena such as those displayed at impasse points or singularity-induced bifurcations. Recent results on the classification and analysis of singularities are extensively discussed in Chapter 4. The topics covered on singularities of linear time-varying problems are the result of recent research, whereas the material on singular points of quasilinear DAEs is completely new.The second part of the present monograph is focused on the analysis of DAEs arising in electrical circuit theory, emph...
PrefaceDifferential-algebraic equations (DAEs) have been the object of increasing attention in the last three decades. Nowadays, they provide a valuable tool for system modeling and analysis within different fields, including nonlinear electric and electronic circuit theory, constrained mechanics, and control theory, among others.The first part of this book addresses analytical properties of such differential-algebraic systems. The few existing monographs on DAEs are mainly focused on numerical aspects and, in most cases, they are restricted to a specific approach, structured around the differentiation, geometric, tractability, perturbation or strangeness indices, respectively. By contrast, the present book attempts to discuss a variety of analytical frameworks for the study of DAEs. The emphasis will be on projector methods based upon the tractability index, and also on reduction techniques supported on the geometric index. The differentiation index will also be briefly examined; note that it has received comparatively more attention in the DAE literature than the projector-based and reduction frameworks, in spite of the many benefits displayed by these approaches.Projector-based methods, introduced for linear DAEs in Chapter 2, allow for precise input-output functional descriptions and explicit solution characterizations in terms of the original problem variables: this holds for linear time-varying DAEs with arbitrary index, under mild smoothness requirements. These methods have been mainly developed by Roswitha März and, accordingly, the material in Chapter 2 is crucially based on her work. Nevertheless, some recent or new contributions can also be found in this Chapter, concerning e.g. the so-called Π-projectors, a simplification of the decoupling of DAEs with properly stated leading term, or a detailed characterization of standard form linear problems. Reduction methods, based on the research of Rabier, Rheinboldt and other authors, define a powerful framework for the analysis of nonlinear DAEs. Chapter 3 is mainly focused on these techniques, introducing in particular a local approach for quasilinear DAEs in settings where the global assumptions of Rabier and Rheinboldt do not hold, and paving the way for subsequent analyses of singular problems. The differentiation index is also discussed in this Chapter; cf. Sections 3.1, 3.2 and 3.7.From a dynamical point of view, the essential differences between DAEs and explicit ordinary differential equations (ODEs) arise in so-called singular problems, which lead to new dynamic phenomena such as those displayed at impasse points or singularity-induced bifurcations. Recent results on the classification and analysis of singularities are extensively discussed in Chapter 4. The topics covered on singularities of linear time-varying problems are the result of recent research, whereas the material on singular points of quasilinear DAEs is completely new.The second part of the present monograph is focused on the analysis of DAEs arising in electrical circuit theory, empha...
SUMMARYWe discuss in this paper several interrelated nodal methods for setting up the equations of non-linear, lumped electrical circuits. A rather exhaustive framework is presented, aimed at surveying di erent approaches and terminologies in a comprehensive manner. This framework includes charge-oriented, conventional, and hybrid systems. Special attention is paid to so-called augmented node analysis (ANA) models, which somehow articulate the tableau and modiÿed node analysis (MNA) approaches to nonlinear circuit modelling. We use a di erential-algebraic formalism and, extending previous results proved in the MNA context, we provide index-1 conditions for augmented systems, which are shown to be transferred to tableau models. This approach gives, in particular, precise conditions for the feasibility of certain state-space reductions. We work with very general assumptions on device characteristics; in particular, our approach comprises a wide range of resistive devices, going beyond voltage-controlled ones.
The memory-resistor or memristor is a new electrical device governed by a nonlinear flux-charge relation. Its existence was predicted by Leon Chua in 1971, and the report in 2008 of a physical device with such a constitutive relation has driven a lot of attention to this circuit element. The memristor and related devices are expected to play a very relevant role in electronics in the near future, specially at the nanometer scale. The special form of the voltage-current characteristic, which reads as either v = M (q)i or i = W (ϕ)v, implies that any equilibrium point is embedded into a center manifold of equilibria whose dimension is defined by the total number of memristors in the circuit. We characterize the normal hyperbolicity of these manifolds of equilibria in graph-theoretic terms. Moreover, when the assumptions supporting the normal hyperbolicity of such manifolds fail, the differential-algebraic nature of circuit models is shown to lead to certain bifurcations without parameters not exhibited by explicit ODEs. The results are illustrated by several circuit examples, some of which arise in the design of superconducting quantum bits based on the Josephson junction.
SUMMARYThe present paper addresses index characterizations in differential-algebraic models of electrical circuits without the need for passivity assumptions. Positive definiteness conditions on the conductance, capacitance and inductance matrices are replaced by certain algebraic assumptions on the so-called proper trees for augmented node analysis and normal trees for modified node analysis. The current discussion is restricted to index-0 and index-1 systems; for the latter, the analysis is based upon certain matrix factorizations which split the topological information from the electrical features of the devices. Several examples illustrate the scope of our framework.
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