Indirect Approach for Closed-loop System Identification with Fractional ...

ICFDA'14 Catania, 23 - 25 June 2014 Copyright © 2014 IEEE

ISBN 978-1-4799-2590-2

Indirect Approach for Closed-loop System Identification with Fractional Models Z. YAKOUB∗ , M. CHETOUI∗† , M. AMAIRI∗ and M. AOUN∗ ∗ University

of Gabes, National Research Unit MACS 06/UR/11-12, Rue Omar Ibn El Khattab, Gabes 6029, Tunisia. † Universit´e de Bordeaux, Laboratoire IMS, CNRS-UMR 5218, 351 Cours de Lib´eration-33405, Talence Cedex, France.

Abstract—This paper deals with fractional closed-loop system identification using the indirect approach. Firstly, all differentiation orders are supposed known and only the coefficients of the closed-loop fractional transfer function are estimated using two methods based on least squares techniques. Then, the fractional open-loop process is determined by the knowledge of the regulator. A numerical example is presented to show the effectiveness of the proposed scheme. Keywords—Closed-loop, fractional calculus, identification, least squares, state variable filter.

I.

I NTRODUCTION

The fractional calculus has become, recently, an active research field due to his capacity of complex behavior modelling (see [1] and references therein). This field has been applied in many domains such as modelling, identification and control [2]. Concerning the fractional order system identification, much research has reported on the identifying the fractional order model in the time domain [3][4][5] or in the frequency domain[6][7]. The fractional order system identification requires estimation of the model coefficients and/or fractional orders. Many approaches have been proposed in literature and two classes have been distinguished: the first one is based on an equation error [8][3][9] [4][5] and the second is based on an output error [10][11][6]. Almost all the proposed methods in the literature are for open-loop fractional system identification which is the classical case. However, in practice there are many situations when the open-loop system identification can not be possible. For example, the case of systems being unstable in the open-loop, the case of security or when production recommendation do not permit regulators to be removed during an identification experiment [12], [13]. Hence, in these situations, the identification of the process can be only obtained on the closed-loop conditions. Three approaches of the closed-loop identification are presented in literature: the direct approach, the indirect approach and the joint input-output approach [12]. The proposed method is an extension of the classical indirect approach for closed-loop system identification with rational models which has also received a considerable attention in the literature [14][15]. This identification approach is based on equation error, such as the fractional models is obtained by the discretization of continuous fractional order

differentiation equation and then the algorithm based on a least squares method is used to identify the closed-loop fractional system. This paper is organized as follows: Section II mathematical background presents a brief reviews of fractional systems. The three approaches of the closed-loop identification are detailed in section III. In Section IV the indirect closed-loop identification of fractional systems using the least squares method is developed. The performance of the proposed approach are evaluated by a numerical example presented in Section V. Finally, Section VI concludes the paper. II.

M ATHEMATICAL BACKGROUND

A SISO linear fractional order system is governed by a fractional differential equation: y(t) +

N X

αn

an D

y (t) =

n=1

M X

bm Dβm yc (t)

(1)

m=0

where (an , bm ) ∈ R2 are the linear coefficients of the differential equation and where yc (t) and y (t)  are  respectively d the input and the output signals and D = is the time dt domain differential operator . The fractional orders αn and βm are allowed to be non integer positive numbers such as: α1 < α2 < · · · < αn ;

β0 < β1 < · · · < βm

The ν−order fractional derivative of a continuous-time function f (t), when f (t) = 0 for t ≤ 0, is numerically evaluated using the Gr¨unwald definition [16]:   ∞ 1 X ν k ν D f (t) = ν (−1) f (t − kh), ∀ t ∈ R∗+ (2) k h k=0   ν where h is the sampling period and is the Newton’s k binomial generalized to fractional order:   ( 1 if k = 0 ν = ν (ν − 1) (ν − 2) . . . (ν − k + 1) k if k > 0 k! (3) The Laplace transform of the ν−order fractional derivative of a function f (t) relaxed at t = 0, i.e. f (t) = 0 for t ≤ 0,