Application of the Volterra Series Paradigm to Memristive Systems
The advent of the memristor in the panorama of fundamental passive circuit elements and the recent development of physical nano-devices with memory resistance opens new opportunities in IC electronics. However, considerable progress in the design of novel
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Application of the Volterra Series Paradigm to Memristive Systems Alon Ascoli, Torsten Schmidt, Ronald Tetzlaff, and Fernando Corinto
5.1 Introduction The modeling [1] and investigation [2] of the nonlinear dynamics of memristive systems [3, 4] is one of the hottest topics of current research. The nonlinearity of these unique systems calls for the need to employ techniques from nonlinear system theory [5] to investigate and model their behavior and the dynamics occurring in memristor circuits. One of the most well-known theories for modeling dynamical systems is the Volterra series paradigm [6]. Let us consider a dynamical system with input x(t) and output y(t). Its block diagram is shown in Fig. 5.1. The Volterra series representation of the output y(t) to the system is given by y(t) =
∞
∑ yn (t),
(5.1)
n=1
whose nth-order component yn (t) is described by yn (t) =
+∞ −∞
...
+∞ −∞
hn (τ1 , . . . τn )x(t − τ1 ) . . . x(t − τn )d τ1 . . . d τn .
(5.2)
A. Ascoli () • R. Tetzlaff Technische Universität Dresden, Mommsenstraße 12, 01062 Dresden, Germany e-mail: alon.ascoli@tu-dresden; ronald.tetzlaff@tu-dresden T. Schmidt Hochschule Ansbach, Residenzstraße 8, 91522 Ansbach, Germany e-mail: [email protected] F. Corinto Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy e-mail: [email protected] R. Tetzlaff (ed.), Memristors and Memristive Systems, DOI 10.1007/978-1-4614-9068-5__5, © Springer Science+Business Media New York 2014
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Fig. 5.1 Block diagram of a Volterra system x(t)
{hn(τ1,...,τn)}
y(t)
Here hn (τ1 , . . . τn ) is a real valued function of τ1 , . . . , τn [7] called nth-order Volterra kernel in the time domain. In Fig. 5.1 the whole set of Volterra kernels for n ∈ [1, ∞] is denoted as {hn (τ1 , . . . τn )}. The nth-order Volterra kernel in the s-domain is defined as [8] Hn (s1 , . . . , sn ) =
+∞ −∞
...
+∞ −∞
hn (τ1 , . . . τn )e−(s1 τ1 +···+sn τn ) d τ1 . . . d τn .
(5.3)
Letting sk = jωk (k ∈ {1, . . . , n}) (5.3) may be recast as Hn ( jω1 , . . . , jωn ) =
+∞ −∞
...
+∞ −∞
hn (τ1 , . . . τn )e−( jω1 τ1 +···+ jωn τn ) d τ1 . . . d τn , (5.4)
denoting the nth-order Volterra kernel in the frequency domain. Sect. 5.2 presents a brief historical overview of the Volterra series theory including some of the most recent works and gives some insight on its application for the modeling and investigation of memristive systems. Sects. 5.3 and 5.4, respectively, introduce a systematic approach for modeling the dynamics of a class of single- and two-memristor circuits. Finally conclusions are drawn in Sect. 5.5.
5.2 Application of the Volterra Series Paradigm to Memristive Systems The determination of solutions of differential equations by applying Volterra series representations has been addressed by many authors. An approach based on a power series expansion of the solution of a differential equation is addressed in the pioneering work of Barrett [9] by separating the linear and nonlinear parts of the considered
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