# Power Law Type Long Memory Behaviors Modeled with Distributed Time Delay Systems

*Fractal Fract’s*Editorial Board Members)

## Abstract

**:**

## 1. Introduction

- The use of fractional differentiation in the pseudo-state space description is not mandatory and only fractional integration is needed [16];
- Exact observability cannot be reached as all the system past must be known to predict its future [18];
- In modeling, several mathematical and interpretation problems can invalidate the models obtained [21].

## 2. A Class of Time Delay Systems That Exhibits a Power Law Long Memory Behavior

- Parameter $\nu $ affects the order of the power law behaviors;
- Parameter ${T}_{f}$ chosen such that ${T}_{f}=\text{}10/{\omega}_{l},$ controls the frequency band on which the power law behavior exists.

## 3. Power Law Long Memory Behavior Without Singular Kernel

## 4. Application

- -
- A system S
_{d}models the diffusion of lithium in the spherical particle and links the current $I\left(t\right)$ to the concentration of lithium ${C}_{s}\left(t\right)$ at the surface of the spherical particle; - -
- A nonlinear function $OCV\left(t\right)=f\left({C}_{S}\left(t\right)\right)$ links the concentration of lithium at the surface of the spherical particle ${C}_{S}\left(t\right)$ to the open circuit voltage $OCV\left(t\right)$;
- -
- A resistor R is used to model the cell internal resistance and contact resistance.

## 5. What Does the Proposed Approach Solve?

- In Equation (3), the variable $x\left(t\right)$ can be viewed as a real state and a physical meaning can be associated to it;
- There is no longer any ambiguity in the operator used for the definition of Equation (3) (in Equation (2), the Caputo, Riemann–Liouville, or other operators [12] can be chosen);
- The memory of Model (3) is of finite length;
- Initialization the Model (3) requires the knowledge of its state on a finite length and is well defined.

## 6. Conclusions

## Conflicts of Interest

## Appendix A

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**Figure 1.**Gain (

**a**) and phase (

**b**) diagrams of integral I(s) for various values of ν and T

_{f}= 10,000.

**Figure 5.**Frequency response (gain on the (

**a**), phase on the (

**b**)) of I(s) with the non-singular kernel η(t) for ν varying from 0.1 to 0.9 and with ${\omega}_{l}={10}^{-2}$ rd/s, ${\omega}_{m}={10}^{4}$ rd/s.

**Figure 6.**Frequency response H(s) for various values of ${\alpha}_{0}=1/{\omega}_{1}^{\nu}$, with ν = 0.5: gain (

**a**) and phase (

**b**).

**Figure 7.**Model of a lithium-ion cell proposed in [25].

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Sabatier, J.
Power Law Type Long Memory Behaviors Modeled with Distributed Time Delay Systems. *Fractal Fract.* **2020**, *4*, 1.
https://doi.org/10.3390/fractalfract4010001

**AMA Style**

Sabatier J.
Power Law Type Long Memory Behaviors Modeled with Distributed Time Delay Systems. *Fractal and Fractional*. 2020; 4(1):1.
https://doi.org/10.3390/fractalfract4010001

**Chicago/Turabian Style**

Sabatier, Jocelyn.
2020. "Power Law Type Long Memory Behaviors Modeled with Distributed Time Delay Systems" *Fractal and Fractional* 4, no. 1: 1.
https://doi.org/10.3390/fractalfract4010001