Contribution to the initialization of linear non-commensurate fractional-order systems for the joint estimation of parameters and fractional diﬀerentiation orders

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Contribution to the initialization of linear non-commensurate

fractional-order systems for the joint estimation of parameters and

fractional diﬀerentiation orders

Mohamed A. Bahloul1,2, Zehor Belkhatir3and Taous-Meriem Laleg Kirati2,4

1College of Engineering, Electrical Engineering Department at Alfaisal University,

Riyadh 11533, Saudi Arabia. E-mail: mbahloul@alfaisal.edu

2Electrical and Computer Engineering Department, KAUST, Saudi Arabia.

E-mail: mohamad.bahloul@kaust.edu.sa, taousmeriem.laleg@kaust.edu.sa

3School of Engineering and Sustainable Development, De Montfort University,

United Kingdom, E-mail: zehor.belkhatir@dmu.ac.uk

4National Institute for Research in Digital Science and Technology, Paris-Saclay, France.

October 19, 2022

Abstract

It has been recognized that using time-varying initialization functions to solve the

initial value problem of fractional-order systems (FOS) is both complex and essential

in deﬁning the dynamical behavior of the states of FOSs. In this paper, we investigate

the use of the initialization functions for the purpose of estimating unknown parame-

ters of linear non-commensurate FOSs. In particular, we propose a novel "pre-initial"

process that describes the dynamic characteristic of FOSs before the initial state and

arXiv:2210.10016v1 [stat.ME] 18 Oct 2022

consists of designing an appropriate time-varying initialization function that ensures

accurate convergence of the estimates of the unknown parameters. To do so, we

propose an estimation technique that consists of two steps: (i) to design of practical

initialization function that is output-dependent and which is employed; (ii) to solve

the joint estimation problem of both parameters and fractional diﬀerentiation orders

(FDOs). A convergence proof has been presented. The performance of the proposed

method is illustrated through diﬀerent numerical examples. Potential applications of

the algorithm to joint estimation of parameters and FDOs of the fractional arterial

Windkessel and neurovascular models are also presented using both synthetic and

real data. The added value of the proposed "pre-initial" process to solve the studied

estimation problem is shown through diﬀerent simulation tests that investigate the

sensitivity of estimation results using diﬀerent time-varying initialization functions.

1 Introduction

Over the past 325 years, fractional calculus (FC) has attracted the attention of mathematicians and

engineers working in various ﬁelds of science and engineering [1]. FC started to be used as a powerful

tool to describe and explore complicated dynamical systems of real-world applications, e.g., biology

[2], control [3], economic [4]. Indeed FC can describe systems with high-order dynamics and complex

nonlinear aspects employing fewer coeﬃcients than the integer-order calculus. The arbitrary order

of the derivatives provides an extra degree of freedom to concisely and precisely entail the hidden

dynamics having a diﬀerent origin of memory eﬀect. A distinctive feature of fractional-order calculus

is that contrary to integer-order calculus, the fractional-order one accounts for the system’s memory.

In fact, the fractional-order derivative depends not only on the local conditions of the evaluated

time but also on the entire history of the function. This peculiarity is usually valuable when the

studied system holds a long-term "memory," and any assessed point depends on the past values of the

function [5]. Accordingly, one essentially-crucial prerequisite to ensuring the fractional-order theory

functioning is the adequate initialization and the proper incorporation of the history of the system

[6].

Generally, the initialization of fractional-order systems is considered complicated and challenging.

This dilemma surfaced notably in the case of the non-zero initial value. In fact, it is challenging to

design appropriate initialization functions that guarantee the acquiring of exact states of the system

while accounting for its history, [7]. The pre-initialization process and the initialization functions

of non-zero initial value-based fractional-order systems remain an open and controversial question.

Various eﬀective methods have been developed to analyze the characteristic of fractional order deriva-

tives where initial values are insuﬃcient [8, 9, 10]. Lorenzo et al., [11] was the ﬁrst to demonstrate

that time-varying functions are more suitable for the initialization of fractional-order systems rather

than constant ones. The time-varying initialization has a profound eﬀect on the standard deﬁnitions

of fractional-order derivative and integral. Diﬀerent time-varying initialization functions may lead

to the same initial value from where the fractional-order operator starts; however, as the system’s

dynamic is related to the pre-initial process, diﬀerent initialization functions result in diﬀerent re-

sponses. This fact is known as the aberration phenomenon [12]. Accordingly, designing the proper

initialization function to acquire the desired convergence of the fractional diﬀerential system is deemed

very problematic.

For this reason, in this report, we introduce a novel pre-initialization process that warrants a fast and

precise convergence of the joint estimation of the parameters and diﬀerentiation orders of the frac-

tional diﬀerential system (FOS). The key hypothesis in the design is to consider an output-dependent

initialization function when estimating the unknowns. This will reduce the inﬁnite-dimensional space

of initialization functions into a ﬁnite parametric space where the remaining degree of freedom is

about the length of the history function to consider. In addition, we proposed a two-stage algorithm:

while the ﬁrst stage solves a system of linear equations to estimate the parameters of the FOS, the

second stage uses the iterative ﬁrst-order Newton’s method [13, 14, 15, 16] to estimate the fractional

diﬀerentiation orders. Our contributions are as follows: (i) we consider a time-varying function to

initialize the FOS; (ii) we design a pre-initialization process based on the output signal; and (iii) we

solve a simple linear equation system to estimate the parameters, and iteratively we apply Newton’s

method to estimate the fractional diﬀerentiation orders.

The performance of the proposed method is illustrated through diﬀerent numerical examples. Ad-

ditionally, potential applications of the algorithm are presented, which consists of estimating param-

eters and fractional diﬀerentiation orders of a fractional-order arterial Windkessel and neurovascular

models. To the best to the author’s knowledge, this is the ﬁrst study that accounts for the initializa-

tion function as part of the parameters estimation problem for fractional diﬀerential systems.

2 Notations

For a clear perception of the report, in this section, we present the adopted notation and the basic

deﬁnitions of the fractional-order integral and derivative. Through the following we consider a smooth

function f(t)such that f(t)is zero for t≤tabs and f(t)is fin(t)for tabs ≤t≤tin.

•tin Dα

tf(t)denotes the ’initialized’ αth order diﬀer-integration of f(t)from start point t0to t.

•tin dα

tf(t)represents the ’non-initialized’ generalized (or fractional) αth order diﬀer-integration

of f(t). It is equivalent to shifting the origin of function f(t)at the start of the point from

where diﬀer-integration starts.

dαf(t)

[d(t−tin)α]≡tin dα

tf(t).(1)

•Ψdenotes the time-varying initialization function or history function. It is also known as

the complementary function [17]. The expression between initialized diﬀer-integral and non-

initialized ones is:

tin Dα

tf(t) =tin dα

tf(t) + Ψ(f,α,tabs,tin,t)(2)

2.1 Deﬁnitions

There are many kinds of deﬁnitions for fractional-order diﬀer-integration. Here we present the more

commonly known ones, namely the Riemann-Liouville, Caputo and Grunwald-Letnikov deﬁnitions.

A detailed note on the diﬀerent deﬁnitions might be found in this reference, [11, 17].

Deﬁnition 1. The Riemann-Liouville (R-L) deﬁnition of fractional-order integration is drawn as:

tin d−α

tf(t) = 1

Γ(α)Zt

tin

f(τ)

(t−τ)αdτ, (3)

here tabs is noted as the terminal point as well. 0< α < 1and Γ(x)corresponds to so-called Gamma-

function, written as Γ(x) = R∞

0e−uux−1du.

Deﬁnition 2. The R-L deﬁnition of fractional-order derivative is based on the above deﬁnition (3)

and the standard integer-order derivative:

tin dα

tf(t) = d

dthtin d−(1−α)

tf(t)i.(4)

Deﬁnition 3. The Caputo deﬁnition of fractional-order diﬀerentiation takes the integer-order diﬀer-

entiation of the function ﬁrst and then takes a fractional-order integration:

tin dα

tf(t) = 1

Γ(n−α)Zt

tin

(1 −τ)−α−1+ndn

dtnf(τ)dτ. (5)

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Contributiontotheinitializationoflinearnon-commensuratefractional-ordersystemsforthejointestimationofparametersandfractionaldierentiationordersMohamedA.Bahloul1;2,ZehorBelkhatir3andTaous-MeriemLalegKirati2;41CollegeofEngineering,ElectricalEngineeringDepartmentatAlfaisalUniversity,Riyadh11533,SaudiA...

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Contribution to the initialization of linear non-commensurate fractional-order systems for the joint estimation of parameters and fractional diﬀerentiation orders

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