A comparison of identification techniques for fractional order systems

Abstract

This thesis compares the performances of various identification methods of deterministic and linear systems described by fractional order models. A detailed introduction to fractional calculus and fractional differential equations is presented. In this respect, the definitions of fractional calculus by Cauchy, Gr¨unwald-Letnikov, Riemann- Liouville and Caputo as well as their properties and integral transforms are covered. Both analytical and numerical solutions of fractional differential equations as well as the initial condition problem are given in this thesis. Nonparametric and parametric system identification techniques for integer order systems are reviewed. The investigated fractional order identification methods are parametric techniques based on minimizing the prediction error. The modeling is done in black-box approach where the structure of the fractional order differential equation is selected at the start of the identification procedure. The estimation of the parameter vector can be performed in time and frequency domain. Time domain identification is carried out by using linear regression form and Gr¨unwald-Letnikov’s definition while the investigated frequency domain methods are Levy’s method and Levy’s method with Vinagre’s weights. As benchmark systems, semi-integrating electrical circuits and Bagley-Torvik’s viscoelastic system are used. Identification results have revealed that in general the proposed fractional order models are more successful at predicting the system output than the proposed integer order models. The persistency of excitation from integer order system identification has to be redefined for fractional order system identification. Time domain methods can be applied directly while in frequency domain system’s frequency response must first be estimated by nonparametric methods. Original contribution of this thesis is the comparison of integer and fractional order models for the chosen benchmark systems.