Luca Pancioni

Mauro Di Marco, Mauro Forti, Luca Pancioni et al.

The paper considers a large class of nonlinear circuits, termed RLCM, containing all four basic circuit elements, i.e., resistors, inductors, capacitors and memristors. A companion paper [1] has introduced a mixed potential for RLCM circuits generalizing that found by Brayton and Moser for circuits without memristors. In this paper, systematic Lyapunov-like results on convergence of RLCM circuits are proved by means of the mixed potential. These hold under the basic assumption that an RLCM circuit has a complete set of variables in the flux-charge domain and they require, roughly speaking, that there is a balance, which is quantitatively estimated, between capacitors and inductors. The convergence results are robust with respect to circuit parameter variations and they include cases where the memristor circuits possess multiple stable equilibrium points, which is of importance for instance to implement content addressable memories (CAMs). The results extend to circuits possessing all four basic circuit elements previous results that pertain to circuits without memristors or memristor circuits without inductors. The main proofs are conducted by using the flux-charge analysis method (FCAM) to analyze RLCM circuits in the flux-charge domain.

Mauro Di Marco, Mauro Forti, Luca Pancioni et al.

In two seminal articles published in 1964, Brayton and Moser introduced the concept of a mixed potential as a fundamental theoretic tool to describe and analyze a class RLC of nonlinear circuits containing resistors, capacitors and inductors. In this paper, it is shown for the first time that a mixed potential can be introduced for a class RLCM of RLC circuits containing also memristors. This is possible provided a memristor circuit is analyzed not in the traditional voltage-current domain but rather in the flux-charge domain. The flux-charge analysis method (FCAM) plays a crucial role in the extension, in particular, a key step is an equivalence principle established via FCAM between an RLCM circuit in the flux-charge domain and a nonlinear RLC circuit in the voltage-current domain. Several examples are discussed where the mixed potential is explicitly found. These include basic circuits with memristors, such as Chua's circuit with a memristor and also large-scale memristor arrays with a neural architecture. This paper is mainly devoted to the introduction of a mixed potential for memristor circuits and the study of its main theoretic properties, as the possibility to write the circuit state equations in the flux-charge domain in an effective and compact form via the mixed potential. In a companion paper [1], the mixed potential is used to obtain in a systematic way Lyapunov-like results on convergence of RLCM circuits. Those results will extend existing results on convergence that do not cover the important case where there is the simultaneous presence of capacitors and inductors in a memristor circuit.