Parameter estimation of nonlinear systems with stable, chaotic and periodic behaviours at different initial conditions - a new approach

Abstract

Real-life systems are generally nonlinear in nature. Thus, accurate nonlinear system modelling is very important to analyse and control of the same system. Nonlinear systems are highly complex and exhibit different behaviours with the variation of their initial conditions and parameters. The class of chaotic systems is nonlinear systems with high sensitivity to initial conditions. In this paper, parameters estimation of the well known Lorenz system, as an example, is considered with the variation of its parameters and initial conditions. The physics informed neural networks, a recent advance in machine learning techniques, is used for this purpose. The variation of parameters leads to stable, periodic and chaotic behaviours of the Lorenz system, and these parameters are estimated with different initial conditions. The simulation results reveal the accuracy of the estimation as (0.03 to 0.39)% for stable behaviour, (0.08 to 0.43)% for periodic behaviour and (1 to 6)% for chaotic behaviour. The transient time for the parameters estimation is checked by estimating the parameters in stable - chaotic - stable sequences. The transient time is less when the physics informed neural networks are in use for some time. The short transient time is suitable for real-life applications using the used approach. Most of the available literature for the estimation of parameters of a chaotic system considers only the set of parameters for chaotic behaviour and with only one initial condition; and thus, signify the contributions of this paper.