Analysis of Time-Delayed Neural Networks via Rightmost Eigenvalue Positions

Sun Yi,Jung H. Kim,Taher M. Abu-Lebdeh,Sangseok Yu

doi:10.3844/ajeassp.2015.1.10

Abstract

Neural networks have been frequently used in various areas. In the implementation of the networks, time delays and uncertainty are present and known to lead to complex behaviors, which are hard to predict using classical analysis methods. In this study, stability and robust stability of neural networks considering time delays and parametric uncertainty is studied. For stability analysis, the rightmost eigenvalues (or dominant characteristic roots) are obtained by using an approach based on the Lambert W function. The Lambert W function has already been embedded in various commercial software packages (e.g., MATLAB, Maple and Mathematica). In a way similar to non-delayed systems, stability is determined from the positions of the characteristic roots in the complex plane. Conditions for oscillation and robust stability are also given. Numerical examples are provided and the results are compared to existing approaches (e.g., bifurcation method) and discussed.

Highlights

During the last several decades, neural networks have received wide interest due to their applications in various areas, such as signal processing, image processing, power systems and optimization (Kim et al, 1996; Haque and Kashtiban, 2005)
Stability and robust stability of neural networks has been investigated through solving for the characteristic roots of delay differential equations
Delay Differential Equations (DDEs) render transcendental characteristic equations, the characteristic roots and dominant ones can be found by using the Lambert W function

Summary

Introduction

During the last several decades, neural networks have received wide interest due to their applications in various areas, such as signal processing, image processing, power systems and optimization (Kim et al, 1996; Haque and Kashtiban, 2005). Existing approaches for time-delay systems are limited in three critical ways: (1) They approximate time delays in modeling and, reduce accuracy (e.g., Padé approximation); (2) They rely on model-based prediction of future trajectories (e.g., Smith predictor), which is vulnerable to uncertainty; (3) Or, they are dependent upon Lyapunov functions, which induce conservativeness in the results. Those shortcomings mainly come from lack of analytical solutions for timedelay systems. The approach presented in this study will be of interest

Background

Conclusion

Funding Information