Solutions of singular constrained differential equations: A generalization of circuits containing capacitor-only loops and inductor-only cutsets

Häggman, Börje; Bryant, P.R.

doi:10.1109/tcs.1984.1085466

Cited by 30 publications

(14 citation statements)

References 7 publications

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“…Additionally, the book [44] presents a systematic qualitative analysis of non-reciprocal circuits. A different, geometric approach stems from the work of Brayton and Moser [45,46]; later results in this direction can be found in [47][48][49][50][51][52][53]. Other developments in the last two decades have concerned singularities in DAE models [10,11,29,30,33], general circuit modelling [14, 15, 17-19, 31, 39, 52-56], and qualitative results for certain circuit families [27,28,57,58], besides of course numerical aspects.…”

Section: Qualitative Results Based On State-space Modelsmentioning

confidence: 99%

Qualitative features of matrix pencils and DAEs arising in circuit dynamics

Riaza

Tischendorf

2007

Dynamical Systems

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The dynamical behaviour of nonlinear electrical circuits is usually modelled in the time domain by differential-algebraic equations (DAEs). The differentialalgebraic formalism drives qualitative analyses based on linearization to a matrix pencil setting. In this context, the present paper performs a spectral analysis of matrix pencils and DAEs arising in nonlinear circuit theory. Specifically, the non-singularity, hyperbolicity and asymptotic stability of equilibria are addressed in terms of circuit topology. The differential-algebraic framework puts the results beyond those already known for state-space models, unfeasible in many actual problems. The topological conditions arising in this qualitative study are proved independent of those supporting the index, and therefore they apply to both index-1 and index-2 configurations. The approach illustrates how graph theory, matrix analysis and DAE theory interact in the dynamical study of nonlinear circuits.

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Section: Qualitative Results Based On State-space Modelsmentioning

confidence: 99%

Qualitative features of matrix pencils and DAEs arising in circuit dynamics

Riaza

Tischendorf

2007

Dynamical Systems

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“…The relation between certain pathological conÿgurations (V -C and V -L loops, I -L and I -C cutsets) arising in di erent approaches to circuit modelling [10,[12][13][14]31] have been addressed in this paper as a limit case of the spectral transformation following from a reactive dual construction. This approach has led, from the already-known property that the index of strictly passive circuits without controlled sources does not exceed two, to the discussion of a dual property stating that the index of a null eigenvalue in a strictly passive circuit cannot exceed one.…”

Section: Discussionmentioning

confidence: 99%

“…¿0) for any x ∈ R n − {0}; we do not assume W to be symmetric. The properties stated below are addressed within a geometric framework in References [12][13][14]. Explicit proofs for both claims in the present setting can be found in References [9] and [31], respectively.…”

Section: Pathological Conÿgurations In the Semistate Contextmentioning

confidence: 99%

“…On the other hand, so-called degenerate conÿgurations such as V -L loops and I -C cutsets are known to yield di culties concerning operating points and isolation of equilibria (see Reference [11] for details). Both types of conÿgurations were extensively discussed, within a geometric framework, by Haggman and Bryant in References [12] and [13,14], respectively.…”

Section: Introductionmentioning

confidence: 99%

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Time‐domain properties of reactive dual circuits

Riaza

2006

Circuit Theory & Apps

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SUMMARYThe present work explores some e ects 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 → −1 in the system spectra. In the semistate context, this covers in particular extremal cases in which null eigenvalues or inÿnite ones with higher index appear in the matrix pencil associated with the model; these cases describe certain pathological circuit conÿgurations. This approach leads to a discussion of new properties of strictly passive circuits; speciÿcally, 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, deÿned by an index-two doublezero 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.

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“…Practical examples have been studied by Campbell and Rose, Haggman and Bryant, Luenberger, Petzold, Singh and Liu, Wang and Dai, among many other authors (see [2], [7], [9], [10], [11], [13]). Many of classical systems results have been generalized to the case of singular linear systems.…”

Section: Introductionmentioning

confidence: 99%