Quadrature Based Approximations of Non-integer Order Integrator on Finite Integration Interval

Fractional calculus has found multiple applications around the world. It is especially prevalent in the domains of control and electronics. One of the key elements of fractional applications is the fractional integral (or integrator) which is a backbone of famous PIλD controller. It gives advantages of traditional PID with a limited phase lag. The are, however, issues with implementation, which will allow good low-frequency behavior. In this paper, we consider a diffusive realization of a fractional integrator with the use of quadratures. We implemented this method in numerical package SoftFrac, and we illustrate how different quadratures work for this purpose. We show superiority of bounded domain integration with logarithmic transformation and explain issues with behavior for extremely low frequencies.

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“…In this paper, we want to consider the following approaches (expanding on our earlier conclusions from [14,15]:…”

Section: Approximation Of Diffusive Realizationsmentioning

confidence: 99%

“…We have conducted initial research of this technique in [14,15]. Monteghetti et al conducted an in-depth analysis of quadrature properties (see in [16]).…”

Section: Introductionmentioning

confidence: 99%

Practical Applications of Diffusive Realization of Fractional Integrator with SoftFrac

2021

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“…where ẏ is the strong derivative and d 1 2 is the Caputo derivative defined in (6). The exact solution of (30) can be expressed using the Mittag-…”

Section: Fractional Differential Equationmentioning

confidence: 99%

“…Another approach consists in directly using a quadrature rule, without any split. To get back to a finite interval, one can either truncate the semi-infinite integration domain [2] or use a change of variable [46,12,4].…”

Section: Introductionmentioning

confidence: 99%

“…In [2], Gauss-Legendre and Curtis-Clenshaw quadrature rules are used on a truncated domain. A method proposed in [46], based on a Gauss-Laguerre quadrature rule with a change of variable, has been widely investigated and led to the definitions of methods based instead on the Gauss-Jacobi quadrature rule [12,4], see [4] for a comparison that favors [4,Eq.…”

Section: Introductionmentioning

confidence: 99%

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Time-local discretization of fractional and related diffusive operators using Gaussian quadrature with applications

Monteghetti

Matignon

Piot

2020

Applied Numerical Mathematics

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This paper investigates the time-local discretization, using Gaussian quadrature, of a class of diffusive operators that includes fractional operators, for application in fractional differential equations and related eigenvalue problems. A discretization based on the Gauss-Legendre quadrature rule is analyzed both theoretically and numerically. Numerical comparisons with both optimization-based and quadrature-based methods highlight its applicability. In addition, it is shown, on the example of a fractional delay differential equation, that quadrature-based discretization methods are spectrally correct, i.e. that they yield an unpolluted and convergent approximation of the essential spectrum linked to the fractional derivative, by contrast with optimization-based methods that can yield polluted spectra whose convergence is difficult to assess.

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SoftFRAC - Matlab Library for Realization of Fractional Order Dynamic Elements

Bauer¹,

Baranowski²,

Piątek³

et al. 2020

Advances in Intelligent Systems and Computing

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Quadrature Based Approximations of Non-integer Order Integrator on Finite Integration Interval

Cited by 4 publications

References 20 publications

Practical Applications of Diffusive Realization of Fractional Integrator with SoftFrac

Practical Applications of Diffusive Realization of Fractional Integrator with SoftFrac

Time-local discretization of fractional and related diffusive operators using Gaussian quadrature with applications

SoftFRAC - Matlab Library for Realization of Fractional Order Dynamic Elements

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