Ricardo Riaza

Differential-Algebraic Systems: Analytical Aspects and Circuit Applications

Differential-algebraic equations (DAEs) provide an essential tool for system modeling and analysis within different fields of applied sciences and engineering. This book addresses modeling issues and analytical properties of DAEs, together with some applications in electrical circuit theory. Beginning with elementary aspects, the author succeeds in providing a self-contained and comprehensive presentation of several advanced topics in DAE theory, such as the full characterization of linear time-varying equations via projector methods or the geometric reduction of nonlinear systems.Recent results on singularities are extensively discussed. The book also addresses in detail differential-algebraic models of electrical and electronic circuits, including index characterizations and qualitative aspects of circuit dynamics. In particular, the reader will find a thorough discussion of the state/semistate dichotomy in circuit modeling. The state formulation problem, which has attracted much attention in the engineering literature, is cleverly tackled here as a reduction problem on semistate models.

On singular equilibria of index-1 DAEs

Some qualitative properties of singular equilibria arising in semiexplicit index-1 differential-algebraic equations are discussed in this paper. We extend a taxonomy of singularities from index-0 problems to address situations not covered by the singularity induced bifurcation, singular flow, and associated theorems. Although some related phenomena may occur, different behavior can be expected when the assumptions supporting these results are not satisfied, as is illustrated by several examples. Special attention is paid to the effect that the singular nature of the underlying ordinary differential equation and, eventually, of the solution manifold, has on the behavior of the original singular index-1 problem.

On the singularity-induced bifurcation theorem

The singularity-induced bifurcation theorem (SIBT) is extended in this note to quasi-linear singular ordinary differential equations. The hypotheses supporting this result are simplified and rewritten in terms of matrix pencils. This approach shows that the SIB follows from a minimal index change at the singularity. The use of a quasi-linear reduction leads to a simple statement of the SIBT for semiexplicit index-1 differential algebraic equations.

Manifolds of equilibria and bifurcations without parameters in memristive circuits

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

Time‐domain properties of reactive dual circuits

v=M(q)i

Linear Index-1 DAEs: Regular and Singular Problems

or

A matrix pencil approach to the local stability analysis of non-linear circuits

i=W(\varphi)v

Non-linear circuit modelling via nodal methods

, 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 ...

Stability Loss in Quasilinear DAEs by Divergence of a Pencil Eigenvalue

The present work explores some effects of the replacement of capacitors by inductors and vice versa in state and semistate models of lumped circuits. Such a replacement, when performed together with an inversion of the capacitance and inductance matrices, yields a transformation of the form λλ−1 in the system spectra. In the semistate context, this covers in particular extremal cases in which null eigenvalues or infinite ones with higher index appear in the matrix pencil associated with the model; these cases describe certain pathological circuit configurations. This approach leads to a discussion of new properties of strictly passive circuits; specifically, from the known fact that the index of strictly passive circuits does not exceed two, we derive that the index of null eigenvalues in this setting cannot exceed one. This precludes in particular Takens-Bogdanov degeneracies, defined by an index-two double-zero eigenvalue, in strictly passive circuits. Although the results are addressed in a linear context, they can be extended via linearization to non-linear problems, as it is the case in the transformation of singularity-induced bifurcation phenomena into steady bifurcations discussed at the end of the paper. Copyright

Linear differential-algebraic equations with properly stated leading term: A-critical points

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.