A Physical experimental study of variable-order fractional integrator and differentiator

Sheng, Hu; Sun, HongGuang; Coopmans, Calvin; Chen, YangQuan; Bohannan, Gary W.

doi:10.1140/epjst/e2011-01384-4

Cited by 96 publications

(65 citation statements)

References 18 publications

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“…solution of system (3) will be calculated from (7) and (5b), time horizon T = 3 with sample time h = 1, and then k = h = 3. By (6), this yields α 1 = (α …”

Section: ) Hementioning

confidence: 93%

“…e-varying, begun to be ble order behavior can y when system's propactions. Experimental of physical fractional nted in [6]. The vario describe time evolurical implementations and differentiators can al variable order calriable order fractional r interpretation of the integrators, realized as pplications of variable in control [12,13,14].…”

Section: Definitions Of Variable Order Operatorsmentioning

confidence: 99%

“…The fractional variable order behavior can ntered for example in chemistry when system's propchanging due to chemical reactions. Experimental f an electrochemical example of physical fractional order system have been presented in [6]. The varir equations have been used to describe time evolurag expression in [7].…”

Section: ) Hementioning

confidence: 99%

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Numerical solution of fractional variable order linear control system in state-space form

Malesza

Macias

2017

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. The aim of this paper is to introduce a matrix approach for approximate solving of non-commensurate fractional variable order linear control systems in state-space form. The approach is based on switching schemes that realize variable order derivatives. The obtained numerical solution is compared with simulation and analog model results. In our paper, a method of finding a numerical solution of fractional variable order control system in a state-space form is introduced, both for time-invariant and time-variant case. Moreover, the obtained results are also valid for system of differential equations with different types of variable order derivatives. To validate our approach the fractional variable order state-space system was physically built and the experimental results were compared with numerical implementations.The paper is organized as follows. At the beginning, in Section 2, the few types of fractional variable order derivatives are recalled, together with their discrete approximations and matrix forms. In Section 3 the solution of linear control system in statespace form for time-variant and time-invariant non-commensurate fractional variable order system is presented. An analog model of particular type of fractional variable order state-space system is introduced in Section 4. The experimental and numerical results are presented in Section 5. Finally, Section 6 summarizes the main results. Fractional variable order operatorsBelow, we recall the already known different types of fractional constant and variable order derivatives and differences. In our paper, a method of finding a numerical solution of actional variable order control system in a state-space form introduced, both for time-invariant as well as time-variant der state-space system was physically build and the experimental results were compared with numerical implementations. The paper is organized as follows. At the beginning, in Sect. 2, the few types of fractional variable order derivatives are recalled, together with their discrete approximations and matrix forms. In Sect. 3 the solution of linear control system in state-space form for time-variant and time-invariant noncommensurate fractional variable order system is presented. An analog model of particular type of fractional variable order state-space system is introduced in Sect. 4. The experimental and numerical results are collected in Sect. 5. Finally, Sect. 6 summarizes the main results. Fractional variable order operatorsBelow, we recall the already known different types of fractional constant and variable order derivatives and differences. Definitions of variable order operatorsThe following fractional constant order difference of Grünwald-Letnikov type will be used as a base of generalization onto variable orderwhere α ∈ R, l = 0, . . . , k, and h > 0 is a sample time. We will consider the following four types of fractional variable order derivatives and their discrete approximations (differences). We admit the order is changing in time, i.e., α(t) ∈ R for t > 0; and in ...

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“…solution of system (3) will be calculated from (7) and (5b), time horizon T = 3 with sample time h = 1, and then k = h = 3. By (6), this yields α 1 = (α …”

Section: ) Hementioning

confidence: 93%

Section: Definitions Of Variable Order Operatorsmentioning

confidence: 99%

See 1 more Smart Citation

Numerical solution of fractional variable order linear control system in state-space form

Malesza

Macias

2017

Bulletin of the Polish Academy of Sciences Technical Sciences

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“…In [26], experimental studies of an electrochemical example of physical fractional variable order system are presented. In [21], the variable order equations were used to describe a history of drag expression.…”

Section: Variable Order Casementioning

confidence: 99%

“…In [21], the variable order equations were used to describe a history of drag expression. Papers [26,35] present methods for numerical realization of fractional variable order integrators or differentiators. The fractional variable order calculus can be also used to obtain variable order fractional noise [25], and to obtain new control algorithms [19].…”

Section: Variable Order Casementioning

confidence: 99%

Fractional Order Estimation Schemes for Fractional and Integer Order Systems with Constant and Variable Fractional Order Colored Noise

Sierociuk

Ziubinski

2014

Circuits Syst Signal Process

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The paper presents estimation schemes for discrete fractional and integer order state-space systems with fractional order colored noise. The fractional order colored noise is a generalization of the traditional colored noise (noise with dynamic dependency) for the case when the dynamics of noise is of fractional order. Proposed estimation algorithm additionally uses information about noise dynamics, which allows for obtaining better estimates of state vector. The numerical experiments of estimation integer order system with fractional noise are presented as well.

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Variable‐order Caputo derivative of LC and RC circuits system with numerical analysis

Naveen,

Parthiban

2024

Circuit Theory & Apps

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SummaryIn this paper, computational analysis of a Caputo fractional variable‐order system with inductor‐capacitor (LC) and resistor‐capacitor (RC) electrical circuit models is presented. The existence and uniqueness of solutions to the given problem are determined using Schaefer's fixed point theorem and the Banach contraction principle, respectively. The proposed problem's computational consequences are addressed and analyzed using modified Euler and Runge–Kutta fourth‐order techniques. Furthermore, the suggested model compares several orders, including integer, fractional, and variable orders. To demonstrate the utility of the proposed approach, computational simulations are carried out on LC and RC circuit models of various orders. Furthermore, a comparative analysis with previous investigations has been carried. For the given problem, the numerical solution results in high‐precision approximations.

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A Physical experimental study of variable-order fractional integrator and differentiator

Cited by 96 publications

References 18 publications

Numerical solution of fractional variable order linear control system in state-space form

Numerical solution of fractional variable order linear control system in state-space form

Fractional Order Estimation Schemes for Fractional and Integer Order Systems with Constant and Variable Fractional Order Colored Noise

Variable‐order Caputo derivative of LC and RC circuits system with numerical analysis

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