Nonsymmetric Interconnection Research Articles

Convergence and Multistability of Nonsymmetric Cellular Neural Networks With Memristors.

Recent work has considered a class of cellular neural networks (CNNs) where each cell contains an ideal capacitor and an ideal flux-controlled memristor. One main feature is that during the analog computation the memristor is assumed to be a dynamic element, hence each cell is second-order with state variables given by the capacitor voltage and the memristor flux. Such CNNs, named dynamic memristor (DM)-CNNs, were proved to be convergent when a symmetry condition for the cell interconnections is satisfied. The goal of this paper is to investigate convergence and multistability of DM-CNNs in the general case of nonsymmetric interconnections. The main result is that convergence holds when there are (possibly) nonsymmetric, non-negative interconnections between cells and an irreducibility assumption is satisfied. This result appears to be similar to the classic convergence result for standard (S)-CNNs with positive cell-linking templates. Yet, due to the presence of DMs, a DM-CNN displays some basically different and peculiar dynamical properties with respect to S-CNNs. One key difference is that the DM-CNN processing is based on the time evolution of memristor fluxes instead of capacitor voltages as it happens for S-CNNs. Moreover, when a steady state is reached, all voltages and currents, and hence power consumption of a DM-CNN vanish. This notwithstanding the memristors are able to store in a nonvolatile way the result of the processing. Voltages, currents and power instead do not vanish when an S-CNN reaches a steady state.

Consensus of second-order multi-agent systems with nonsymmetric interconnection and heterogeneous delays

In this paper, a consensus algorithm of multi-agent second-order dynamical systems with nonsymmetric interconnection and heterogeneous delays is studied. With the hypothesis of directed weighted topology graph with a globally reachable node, decentralized consensus condition is obtained by applying generalized Nyquist criterion. For the systems with both communication and input delays, it is shown that the consensus condition is dependent on input delays but independent of communication delays.

Scalable robust stability for nonsymmetric heterogeneous networks

Scalable decentralized stability certificates in networks i.e. decentralized stability guarantees for an arbitrary interconnection of heterogeneous dynamical systems, are often based on certain symmetry assumptions in the way the systems are interconnected. Such structure simplifies the mathematical analysis significantly, nevertheless a potential pitfall needs to be addressed: the stability proof might offer no robustness guarantees to deviations from protocol symmetry. This is, for example, the case for Internet congestion control stability results for arbitrary networks i.e. the stability proofs break down with an arbitrarily small deviation from protocol symmetry. We propose in the paper conditions that can guarantee scalable robust stability in a nonsymmetric interconnection setting for a class of networks that includes Internet congestion control models and consensus protocols. The certificates derived are decentralized and scale with the degree of nonsymmetry.

SCALABLE ROBUSTNESS FOR CONSENSUS PROTOCOLS WITH HETEROGENEOUS DYNAMICS

It is shown in the paper how robustness can be guaranteed for consensus protocols with heterogeneous dynamics in a scalable and decentralized way i.e. by each agent satisfying a test that does not require knowledge of the entire network. Random graph examples illustrate that the proposed certificates are not conservative for classes of large scale networks, despite the heterogeneity of the dynamics, which is a distinctive feature of this work. The conditions hold for symmetric protocols and more conservative stability conditions are given for general nonsymmetric interconnections. Nonlinear extensions in an IQC framework are finally discussed.

Global stability analysis in delayed cellular neural networks

In this paper, without assuming the boundedness, monotonicity, and differentiability of the activation functions, we present new conditions ensuring existence, uniqueness, and global asymptotical stability of the equilibrium point of cellular neural network models with fixed time delays. The results are applicable to both symmetric and nonsymmetric interconnection matrices, and all continuous nonmonotonic neuron activation functions.

Global stability analysis in delayed Hopfield neural network models

In this paper, without assuming the boundedness, monotonicity and differentiability of the activation functions, we present new conditions ensuring existence, uniqueness, and global asymptotical stability of the equilibrium point of Hopfield neural network models with fixed time delays or distributed time delays. The results are applicable to both symmetric and nonsymmetric interconnection matrices, and all continuous nonmonotonic neuron activation functions.

New conditions for global stability of neural networks with application to linear and quadratic programming problems

In this paper, we present new conditions ensuring existence, uniqueness, and Global Asymptotic Stability (GAS) of the equilibrium point for a large class of neural networks. The results are applicable to both symmetric and nonsymmetric interconnection matrices and allow for the consideration of all continuous nondecreasing neuron activation functions. Such functions may be unbounded (but not necessarily surjective), may have infinite intervals with zero slope as in a piece-wise-linear model, or both. The conditions on GAS rely on the concept of Lyapunov Diagonally Stable (or Lyapunov Diagonally Semi-Stable) matrices and are proved by employing a class of Lyapunov functions of the generalized Lur'e-Postnikov type. Several classes of interconnection matrices of applicative interest are shown to satisfy our conditions for GAS. In particular, the results are applied to analyze GAS for the class of neural circuits introduced for solving linear and quadratic programming problems. In this application, the principal result here obtained is that these networks are GAS also when the constraint amplifiers are dynamical, as it happens in any practical implementation. >