An adaptive multiscale basis method for modelling discrete-time non-linear dynamical systems

Abstract

Based on locally applicable autoregressive moving average with exogenous (ARMAX) models this paper proposes two new adaptive modelling schemes for discrete-time non-linear dynamic systems based on the concept of a multiscale which originates in approximation theory. A general result on the convergence of modelling based on the multiscale basis functions with a least square estimated is derived. With the advantage of the multiresolution nature of the multiscale basis, the new modelling scheme can be used to capture both the global and local characteristics of a non-linear system over distinct scales. The algorithms also provide a trade-off between the model structure/model size and the modelling error, one of which emphasizes the modelling error with variable model structure whilst the other stresses a desired model size with better modelling error. Data-based modelling examples are used to demonstrate the effectiveness of the multiscale modelling schemes with different multiscale basis.