Power Law Type Long Memory Behaviors Modeled with Distributed Time Delay Systems
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 affects the order of the power law behaviors;
- Parameter chosen such that controls the frequency band on which the power law behavior exists.
3. Power Law Long Memory Behavior Without Singular Kernel
4. Application
- -
- A system Sd models the diffusion of lithium in the spherical particle and links the current to the concentration of lithium at the surface of the spherical particle;
- -
- A nonlinear function links the concentration of lithium at the surface of the spherical particle to the open circuit voltage ;
- -
- 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 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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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
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 StyleSabatier, 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