Matrix Approach and Analog Modeling for Solving Fractional Variable Order Differential Equations

Malesza, Wiktor; Macias, Michał; Sierociuk, Dominik

doi:10.1007/978-3-319-09900-2_7

Cited by 9 publications

(8 citation statements)

References 10 publications

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“…riable order derivative definitions of variable 7, 18]. Numerical and ariable order differenly in [19,20] and [21]. numerical solution of in a state-space form s well as time-variant e also valid for system ypes of variable orders fractional variable order state-space system was physically build and the experimental results were compared with numerical implementations.…”

Section: Definitions Of Variable Order Operatorsmentioning

confidence: 99%

“…Alternative definitions of variable re introduced in [17,18]. Numerical and of linear fractional variable order differenresented, respectively in [19,20] and [21]. ethod of finding a numerical solution of rder control system in a state-space form for time-invariant as well as time-variant obtained results are also valid for system ons with different types of variable orders ate our approach the fractional variable ore.pw.edu.pl s@ee.pw.edu.pl der state-space system was physically build and the experimental results were compared with numerical implementations.…”

Section: Time-variant Control Systemmentioning

confidence: 99%

“…Solution of (5a), and thereby approximated solution of (3a), is COR by (7), of appr REM I n(k+1) i.e., n = in [19].…”

Section: Numerical Solution Of Variable Order Lmentioning

confidence: 99%

“…In a case of scalar system, i.e., n = 1, a condition under which a solution exists is given in [19].…”

Section: Numerical Solution Of Variable Order Linear Control Systemmentioning

confidence: 99%

“…Solution (7) admits singularities if marrix I n(k+1) −TW(−α)A is singular. In a case of scalar system, i.e., n = 1, a condition under which a solution exists is given in [19]. EXAMPLE 1.…”

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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Section: Definitions Of Variable Order Operatorsmentioning

confidence: 99%

Section: Time-variant Control Systemmentioning

confidence: 99%

“…Solution of (5a), and thereby approximated solution of (3a), is COR by (7), of appr REM I n(k+1) i.e., n = in [19].…”

Section: Numerical Solution Of Variable Order Lmentioning

confidence: 99%

“…In a case of scalar system, i.e., n = 1, a condition under which a solution exists is given in [19].…”

Section: Numerical Solution Of Variable Order Linear Control Systemmentioning

confidence: 99%

Section: ) Hementioning

confidence: 99%

See 3 more Smart Citations

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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Fractional Variable-Order Derivative and Difference Operators and Their Applications to Dynamical Systems Modelling

Dzieliński

Sierociuk

Malesza

et al. 2022

Studies in Systems, Decision and Control

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Variable-, Fractional-Order Linear System State-Space Description Transformation

Horla

2022

Studies in Systems, Decision and Control

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Matrix Approach and Analog Modeling for Solving Fractional Variable Order Differential Equations

Cited by 9 publications

References 10 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 Variable-Order Derivative and Difference Operators and Their Applications to Dynamical Systems Modelling

Variable-, Fractional-Order Linear System State-Space Description Transformation

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