Local Stability Analysis of Discrete-Time, Continuous-State, Complex-Valued Recurrent Neural Networks With Inner State Feedback

Abstract

Recurrent neural networks (RNNs) are well known for their capability to minimize suitable cost functions without the need for a training phase. This is possible because they can be Lyapunov stable. Although the global stability analysis has attracted a lot of interest, local stability is desirable for specific applications. In this brief, we investigate the local asymptotical stability of two classes of discrete-time, continuous-state, complex-valued RNNs with parallel update and inner state feedback. We show that many already known results are special cases of the results obtained here. We also generalize some known results from the real-valued case to the complex-valued one. Finally, we investigate the stability in the presence of time-variant activation functions. Complex-valued activation functions in this brief are separable with respect to the real and imaginary parts.

A Continuous-Time Recurrent Neural Network for Joint Equalization and Decoding – Analog Hardware Implementation Aspects

“…11. If we consider only the equalization loop in Figure 19, we notice that it describes exactly the dynamical behavior of discrete-time recurrent neural networks [19,25,33,[54][55][56] u½l þ 1¼½ I−…”

Approximation of activation functions for vector equalization based on recurrent neural networks

“…3 shows a continuous-time RNN. The dynamical behavior is given by the following pair of state space equations time cases [9].…”

“…Reviewing the conditions, which an activation function in an RNN must fulfill to guarantee the Lyapunov stability [8], [9] and comparing them with the above listed properties, we notice that the optimum activation function for vector equalization based on RNNs fulfills these conditions. The importance of this approximation for analog implementation lies in the fact that the tanh can be easily realized in analog by differential amplifiers.…”

Dissipativity and passivity analysis for discrete-time complex-valued neural networks with time-varying delay

“…Recently, neural networks have been electronically implemented and they have been used in real-time applications. However in electronic implementation of neural networks, some essential parameters of neural networks such as release rate of neurons, connection weights between the neurons and transmission delays might be subject to some deviations due to the tolerances of electronic components employed in the design of neural networks (Aizenberg, Paliy, Zurada, & Astola, 2008;Hu & Wang, 2012;Mostafa, Teich, & Lindner, 2013;Wang, Xue, Fei, & Li, 2013;Wu, Shi, Su, & Chu, 2011). As we know, time delays commonly exist in the neural networks because of the network traffic congestions and the finite speed of information transmission in networks.…”

“…Over the past decades, some work has been done to analyze the dynamic behavior of the equilibrium points of the various CVNNs. In Mostafa et al (2013), local stability analysis of discrete-time, continuous-state, complex-valued recurrent neural networks with inner state feedback was presented. In Zhou and Song (2013), the authors studied boundedness and complete stability of complex-valued neural networks with time delay by using free weighting matrices.…”