Stability Analysis of Nonlinear Systems via Estimating Radial-Basis-Function-Network-Based Lyapunov Exponents From a Scalar Time Series

Abstract

The concept of Lyapunov exponents is a powerful tool for analyzing the stability of nonlinear dynamic systems especially when the mathematical models of the systems are available. However, for real world systems, such models are often unknown, and estimating these exponents reliably from experimental data is notoriously difficult. A novel method of estimating Lyapunov exponents from a time series is presented in this paper. The method combines the ideas of reconstructing the attractor of the system under study and approximating the embedded attractor through tuning a Radial-Basis-Function (RBF) network, which facilitates the derivation of the Jacobian matrices for applying the model-based algorithm. Simplified as a two-link inverted pendulum with one additional rigid foot-link, a standing biped with a Linear Quadrtic Regulator (LQR) is selected as a case study. The biped balance system has a spectrum including four negative Lyapunov exponents, of which the high numerical accuracy derived through the newly proposed method can be guaranteed even in presence of the measurement noise. We believe that the work can contribute to the stability analysis of nonlinear systems of which the dynamics are either unknown or difficult to model due to complexities.© 2012 ASME