Calculation of Lyapunov exponents using Radial Basis Function networks for stability analysis of nonlinear control systems

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. Estimating Lyapunov exponents using a time series has the advantage in that no mathematical model is required. The downside lies in that the method is believed to be reliable only for estimating positive exponents, and to suffer from generating spurious exponents. In contrary, the model-based method is constructive and reliable for calculating both positive and non-positive exponents. The use of the system Jacobians is the key to the advantages of the model-based method. In this paper, a novel approach is proposed, where the system Jacobians are derived based on system approximation using the Radial Basis Function (RBF) network. The proposed method inherits the advantages of the model-based method, yet no mathematical model is required. Two case studies are presented to demonstrate the efficacy of the proposed method. We believe that the work can contribute to stability analysis of nonlinear systems of which the dynamics are either difficult to model due to complexities or unknown.