Theory and Implementation of Distributed-Order Element Networks

Li, Yan; Chen, YangQuan

doi:10.1115/detc2011-48063

Cited by 5 publications

(8 citation statements)

References 32 publications

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“…A similar example consists of modeling the response of an electrical circuit with a distributed network of capacitors exhibiting the well-known fractional-order Curie’s law. According to this law, current through a capacitor varies with time t as

, where

is a constant voltage and

[ 210 ]. These simple examples suggest that there exists a class of physical problems that can be better described by DO operators.…”

Section: Relevance Of Distributed-order Operatorsmentioning

confidence: 99%

“…The impulse response and asymptotic behavior of the DO controllers have been derived in [ 335 , 340 ]. Additionally, a physical realization of the DO integrator using a series of capacitors has been developed in [ 210 , 340 ]. The DO controllers have been applied to control motors [ 338 ] and robots [ 331 ] among many other applications [ 36 ].…”

Section: Applications To Control Theorymentioning

confidence: 99%

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Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

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Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date.

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, where

is a constant voltage and

[ 210 ]. These simple examples suggest that there exists a class of physical problems that can be better described by DO operators.…”

Section: Relevance Of Distributed-order Operatorsmentioning

confidence: 99%

Section: Applications To Control Theorymentioning

confidence: 99%

Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

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“…Impulse response of fractional distributed order integrators in the Laplace domain is simply obtained by replacing the Laplace operator “ s ” with

W ((), - ln s) = \int_{0}^{1} w ((), α) s^{α} italicdα

, where w ( α ), α ∈ [0, 1] is called the weight function. For instance, in case the weight function is in the pulse form w ( α ) = H ( α − a ) − H ( α − b ), 0 < a < b < 1, it is expressed in time domain by Li and Chen as follows

L_{s \to t}^{- 1} ({}, ln s / ((), s^{b} - s^{a})) = \frac{1}{π} \int_{0}^{+ \infty} \frac{exp ((), - italicxt) \int_{a}^{b} x^{τ} sin ((), πτ) italicdτ}{{(\int_{a}^{b} x^{τ} cos (italicπτ) dτ)}^{2} + {(\int_{a}^{b} x^{τ} sin (italicπτ) dτ)}^{2}} italicdx .

Hoping to obtain a simpler expression in the next theorem, an alternative representation of the impulse response of distributed order integrator is obtained based on Theorem for the general case of weight functions.Theorem Let F ( s ) = 1/ W ( s ) where W ( s ) = L α → s { w ( α )}. Then, the impulse response of distributed order integral operator with a non‐negative weight function w ( α ), 0 < α < 1 satisfying

\int_{0}^{1} w ((), α) italicdα \neq 0

is given by

L_{s \to t}^{- 1} ({}, 1 / W ((), - ln s)) = \frac{1}{t} \sum_{r = 1}^{+ \infty} γ_{r} F^{(r)} ((), ln italict), t > 0

…”

Section: Applicationsmentioning

confidence: 99%

“…Distributed order operators have recently found important applications in signal processing, control systems, electrical circuits, etc. Therefore, the exact impulse response of distributed order operators is of utmost importance when designing circuits containing distributed order elements for instance.…”

Section: Applicationsmentioning

confidence: 99%

“…Applications of the results are therefore presented subsequently, including analytic solutions of fractional distributed order relaxation processes and time domain impulse response of fractional distributed order operators in new simple series expansions. Due to the physical implications and importance of these two problems as well as omnipresence of fractional (distributed order) differential equations, numerical schemes for these kinds of equations and analytical exact and approximate expressions of their solutions that are simple and easy to compute are interesting to scientists . In addition to that, new alternative series expansions for some special functions are obtained in this paper.…”

Section: Introductionmentioning

confidence: 99%

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The use of partition polynomial series in Laplace inversion of composite functions with applications in fractional calculus

Taghavian

2019

Math Methods in App Sciences

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This paper presents an analytical method towards Laplace transform inversion of composite functions with the aid of Bell polynomial series. The presented results are used to derive the exact solution of fractional distributed order relaxation processes as well as time‐domain impulse response of fractional distributed order operators in new series forms. Evaluation of the obtained series expansions through computer simulations is also given. The results are then used to present novel series expansions for some special functions, including the one‐parameter Mittag‐Leffler function. It is shown that truncating these series expansions when combined with using potential partition polynomials provides efficient approximations for these functions. At the end, the results are shown to be also useful in studying asymptotical behavior of partial Bell polynomials. Numerical simulations as well as analytical examples are provided to verify the results of this paper.

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Robust stabilisation of distributed‐order systems

Muñoz‐Vázquez

Fernández‐Anaya

Sánchez‐Torres

et al. 2022

Math Methods in App Sciences

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Funding informationThere are no funders to report for this submission.This paper presents the design of a novel methodology for robust stabilisation of distributed-order systems, which are subject to both matched and mismatched disturbances. Matched disturbances are coped through a nonlinear controller, while mismatched disturbances are rejected by a pseudo-state feedback, whose gain is adjusted by solving alinear matrix inequality. Since nonsmooth techniques induce not-necessarily integer-order differentiable solutions, the dynamical systems under consideration are defined through an operator that extends the distributed-order differentiation of integer-order differentiable functions to the case of not necessarily integer-order differentiable ones. Numerical simulations highlight the reliability of the proposed scheme.

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Theory and Implementation of Distributed-Order Element Networks

Cited by 5 publications

References 32 publications

Applications of Distributed-Order Fractional Operators: A Review

Applications of Distributed-Order Fractional Operators: A Review

The use of partition polynomial series in Laplace inversion of composite functions with applications in fractional calculus

Robust stabilisation of distributed‐order systems

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