On a new definition of fractional variable-order derivative

In this paper, a switching strategy for recursive fractional variable-order derivative is proposed. This strategy can be interpreted as an explanation of order switching mechanism for this particular type of derivative. Additionally, important properties of variable fractional order derivatives, required for prove the main result, are introduced both in a difference equation and a matrix form. Duality between the recursive and standard variable-order derivative is detailed derived. Based on the switching scheme, an analog realization of the recursive variable-order derivative definition is presented. Experimental results obtained for the analog realization are compared to the numerical results.

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“…The fractional constant-order difference definition (that can be obtained from Definition 1 for α(t) = const) can be rewritten as follows [27]:…”

Section: Recursive Definition For Variable-order Differ-integralmentioning

confidence: 99%

On the Recursive Fractional Variable-Order Derivative: Equivalent Switching Strategy, Duality, and Analog Modeling

Sierociuk

Malesza

Macias

2014

Circuits Syst Signal Process

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“…general types of variable order derivative n given. Alternative definitions of variable re introduced in [17,18]. Numerical and of linear fractional variable order differenresented, respectively in [19,20] and [21].…”

Section: Time-variant Control Systemmentioning

confidence: 99%

“…Applications of variable lso in control [12,13,14]. f variable order derivative tive definitions of variable [17,18]. Numerical and al variable order differentively in [19,20] and [21].…”

Section: ) Hementioning

confidence: 99%

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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“…A very good example of the fractional noise is a temperature noise. As it was presented in [30], the dynamics between the source heat flux and temperature in beam heating process is of fractional order type. In case when source heat flux is a white noise without correlated dynamics, the temperature will be a noise with fractional order dynamics.…”

Section: Introductionmentioning

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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On a new definition of fractional variable-order derivative

Cited by 40 publications

References 15 publications

On the Recursive Fractional Variable-Order Derivative: Equivalent Switching Strategy, Duality, and Analog Modeling

On the Recursive Fractional Variable-Order Derivative: Equivalent Switching Strategy, Duality, and Analog Modeling

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

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