Nonparametric Identification of Nonlinear Dynamics of Systems Based on the Active Experiment

Abstract

Identification is a process (measured experiment and numerical procedure) aiming at determining a quantitative model of the examined object’s behaviour. Identification of dynamics is a process which tries to define a quantitative model of variation of system state with time. The goal in the experiment is to measure inputs and outputs of the examined object, the system excitations and reactions. In special cases the model can be treated as a “black box” but it always has to be connected with physical laws and can not be inconsistent with them. The most commonly used models of system dynamics are differential equations – general nonlinear, partial, often nonlinear ordinary, rarely linear ordinary, additionally nonstationary and with deviated arguments. Sometimes one considers discrete-time models presented in a form of difference equations, which are simplified models of a certain kind. Integral equations, functional equations etc. are models of a different kind. If a model structure is a priori known or if it can be assumed, the identification consists in determination of model parameters and it is defined as parametric identification. If the full model structure or its part is not known, nonparametric identification has to be used. In domain of linear models an equivalence of linear ordinary differential equations is transfer function, transient response or frequency response. They can be obtained using experiments of various types: passive – observation of inputs or outputs without interaction upon object or active – excitation of the examined object by special signals (determined: impulse, leap, periodic, a periodic, lottery: white noise, coloured noise, noise with determined spectrum...). Many possibilities lead to a variety of identification methods. In the last decades various identification methods have been developed. Rich bibliography connected with this thematic includes Uhl’s work (Uhl, 1997) which describes computer methods of identification of linear and nonlinear system dynamics, in time domain and also frequency, with short characteristic and a range of their applications. There are many methods of parametric and nonparametric identifications of linear dynamics of systems (Eykhoff, 1980), (Iserman, 1982), (Soderstrom & Stoica, 1989). There are fewer useful methods applied for systems with nonlinear dynamics (Billings & Tsang, 1992), (Greblicki & Pawlak, 1994), (Haber & Keviczky, 1999), thereby a presented simple solution