Generalization of the modulating functions method into the fractional differential equations

Drapała

International Journal of Applied Mathematics and Computer Science

2019

The paper presents new concepts of the identification method based on modulating functions and exact state observers with its application for identification of a real continuous-time industrial process. The method enables transformation of a system of differential equations into an algebraic one with the same parameters. Then, these parameters can be estimated using the least-squares approach. The main problem is the nonlinearity of the MISO process and its noticeable transport delays. It requires specific modifications to be introduced into the basic identification algorithm. The main goal of the method is to obtain on-line a temporary linear model of the process around the selected operating point, because fast methods for tuning PID controller parameters for such a model are well known. Hence, a special adaptive identification approach with a moving window is proposed, which involves using on-line registered input and output process data. An optimal identification method for a MISO model assuming decomposition to many inner SISO systems is presented. Additionally, a special version of the modulating functions method, in which both model parameters and unknown delays are identified, is tested on real data sets collected from a glass melting installation.

Section: Basics Of the Mfmmentioning

confidence: 99%

An Adaptive Identification Method Based on the Modulating Functions Technique and Exact State Observers for Modeling and Simulation of a Nonlinear Miso Glass Melting Process

Drapała

International Journal of Applied Mathematics and Computer Science

2019

The 27th Chinese Control and Decision Conference (2015 CCDC)

“…An improved Luus-Jaakola algorithm has been proposed in [12] basing on the discretization of fractional order systems. Liu et al [13] extended the method of modulating functions to the estimation of fractional order systems. On the other hand, the identification methods in frequency domain were shown in [14] [15].…”

Section: Introductionmentioning

confidence: 99%

Memory identification of fractional order systems: Background and theory

Zhao

2015

This paper presents a novel work that how to determine the memory (initialization function) of fractional order systems by using the recent sampled input-output data. The background and basic theories of initialized fractional order systems are introduced. A practical algorithm is proposed to estimate the initial value of initialization function, which is adaptive to all system parameters. A P-type learning law is applied so that the initialization function can be computed accordingly. The whole process is optimized by using finite system information. The above strategy is available for both Caputo and Riemann-Liouville fractional order systems, where the initial values are applied instead of the initial conditions.

2013 American Control Conference

“…Hence, instead of solving a fractional differential equation where the initial values are often unknown, the problem of identification is transformed into solving a linear system where the initial conditions are not required. Generalization of modulating functions method to fractional systems has been already proposed, for example in [13]. However, the authors of this paper proposed only to reduce the orders of the derivatives in a fractional differential equation.…”

Section: Introductionmentioning

confidence: 99%

Identification of fractional order systems using modulating functions method

Liu

Laleg‐Kirati

Gibaru

et al. 2013

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.