Qualitative features of matrix pencils and DAEs arising in circuit dynamics

Abstract

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.

Saddle-Node Bifurcations in Classical and Memristive Circuits

“…The absence of purely imaginary eigenvalues amounts, essentially, to a result proved in [Riaza and Tischendorf, 2007], because of the fact that the non-passive resistor may be understood to yield an opencircuit at equilibrium owing to the assumption ('(v^) = 0; note however that now there is a zero eigenvalue in the dynamics (contrary to the setting considered in [Riaza and Tischendorf, 2007]). The eigenvalueeigenvector equations for the pencil XA -B* can be easily checked to read as (33a) (33b) of (33b), namely…”

“…From a qualitative point of view, it is well-known and can be traced back at least to [Haggman and Bryant, 1986] the property that VL-loops and/or IC-cutsets are responsible for the presence of null eigenvalues in the linearized circuit dynamics. In the absence of these configurations, asymptotic stability of equilibria follows from a strict passivity assumption [Riaza and Tischendorf, 2007].…”

“…in [Chua, 1980;Haggman and Bryant, 1986;Riaza, 2008;Riaza and Tischendorf, 2007]. Specifically, a graph-theoretic index analysis of DAE circuit models under passivity assumptions was developed in [Estévez-Schwarz and Tischendorf, 2000;Tischendorf, 1999Tischendorf, , 2003.…”

Structural characterization of classical and memristive circuits with purely imaginary eigenvalues

“…In this context, we extend in the present paper our previous research concerning qualitative properties of electrical circuits [52,53] by focusing on the hyperbolicity problem. A linear time-invariant VIRLC circuit (that is, a circuit composed of independent voltage and current sources, and linear time-invariant resistors, inductors, and capacitors) is said to be hyperbolic if all the eigenvalues arising in the state-space description of the dynamics are away from the imaginary axis.…”

Purely Imaginary Eigenvalues (PIE) in Extended Circuits with Mem-Devices

“…Recent approaches are based on the use of semistate (differential-algebraic) circuit models, accommodating both nodal and hybrid techniques for setting up the circuit equations [24,25,26,27,28,29,30,31,32]. In the previous research there is qualitative properties of electrical circuits [33,34] by focusing on the hyperbolicity problem. A linear time-invariant VIRLC circuit (that is, a circuit composed of independent voltage and current sources, and linear time-invariant resistors, inductors, and capacitors) is said to be hyperbolic if all the eigenvalues arising in the state-space description of the dynamics are away from the imaginary axis.…”