Identification of fractional order systems using modulating functions method

TL;DR

This paper extends the modulating functions method for online identification of fractional order systems, enabling parameter estimation without initial value knowledge or derivative estimation of noisy outputs, and demonstrates robustness and efficiency through simulations.

Contribution

The paper introduces a generalized modulating functions approach for fractional systems that avoids initial value and derivative estimation, enhancing robustness and applicability.

Findings

Method is robust against high-frequency noise

No need for initial conditions or derivative estimation

Numerical simulations confirm efficiency and stability

Abstract

The modulating functions method has been used for the identification of linear and nonlinear systems. In this paper, we generalize this method to the on-line identification of fractional order systems based on the Riemann-Liouville fractional derivatives. First, a new fractional integration by parts formula involving the fractional derivative of a modulating function is given. Then, we apply this formula to a fractional order system, for which the fractional derivatives of the input and the output can be transferred into the ones of the modulating functions. By choosing a set of modulating functions, a linear system of algebraic equations is obtained. Hence, the unknown parameters of a fractional order system can be estimated by solving a linear system. Using this method, we do not need any initial values which are usually unknown and not equal to zero. Also we do not need to estimate the fractional derivatives of noisy output. Moreover, it is shown that the proposed estimators are robust against high frequency sinusoidal noises and the ones due to a class of stochastic processes. Finally, the efficiency and the stability of the proposed method is confirmed by some numerical simulations.