Mixed Potential Approach to Convergence of Nonlinear RLC Circuits with Memristors

Abstract

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.