A non integer order, state space model for one dimensional heat transfer process

Oprzędkiewicz, Krzysztof; Gawin, Edyta

doi:10.1515/acsc-2016-0015

Cited by 25 publications

(28 citation statements)

References 12 publications

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“…The usefulness of the nonintegerer approach has been presented by many authors, see e.g. , [5], [6], [9], [12], [18]. Many real applications, to mention model-based conl, model-based fault detection, require to implement a ninteger-order model at a digital platform like PLC or GA.…”

Section: Definition 2 the Gamma Function Is Defined Asmentioning

confidence: 99%

“…e RL or C definitions are considered, the Laplace transan also be given (see for example [11]) as a generalizaf the Laplace transform for the integer-order case: FINITION 6. The Laplace transform for the Riemannille operator is as follows:…”

Section: Discrete-time Approximations Of Fo Operatormentioning

confidence: 99%

“…The usefulness of the noninteger-order approach has been presented by many authors, see e.g. [1][2][3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

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A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equation

Oprzędkiewicz¹,

Stanisławski²,

Gawin³

et al. 2017

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. In the paper, a new method for solution of linear discrete-time fractional-order state equation is presented. The proposed method is simpler than other methods using directly discrete-time version of the Grünwald-Letnikov operator. The method is dedicated to use with any approximator to the operator expressed by a discrete transfer function, e.g. CFE-based Al-Alaoui approximation. A simulation example confirms the usefulness of the method. A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equationbstract. In the paper, a new method for solution of linear discrete-time fractional-order state equation is presented. The proposed method simpler than other methods using directly discrete-time version of the Grünwald-Letnikov operator. The method is dedicated to use with ny approximator to the operator expressed by a discrete transfer function, e.g. CFE-based Al-Alaoui approximation. A simulation example onfirms the usefulness of the method. Many real applications, to mention model-based conrol, model-based fault detection, require to implement a oninteger-order model at a digital platform like PLC or PGA. Known discrete-time state space models of nonintegerrder systems are typically based on the Grünvald-Letnikov GL) definition. An accurate implementation of this model, in articular at the bounded resource platforms, requires a (very) ong-length approximation of the GL-based system [33].The purpose of this paper is to propose a new, discrete-time tate space model of a noninteger-order system, constructed ith the use of the continuous fraction expansion (CFE) imlemented for the Al-Alaoui operator. The use of such an aproximant enables to obtain a much more effective model in hat the memory length is quite low. This recommends its use t industrial digital platforms.The paper is organized as follows. Heaving recalled the ackground of the paper in Section 1, Section 2 outlines the undamentals of fractional-order calculus and introduces a operator is defined aswhere a and t denote time limits for calculation of the operator, α ∈ R denotes the noninteger order of the operation.Next, an idea of the Gamma (Euler) function (see for example [12]) can be given: DEFINITION 2. The Gamma function is defined asThe fractional-order, integro-differential operator (1) can be described by different definitions, given by Grünvald and Letnikov (GL definition), Riemann and Liouville (RL definition) and Caputo (C definition). All these definitions are given below. With respect to particular additional assumptions these definitions can be considered equivalent. A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equationAbstract. In the paper, a new method for solution of linear discrete-time fractional-order state equation is presented. The proposed method is simpler than other methods using directly discrete-time version of the Grünwald-Letnikov operator. The method is dedicated to use with any approximator to the operator expressed by a discrete tra...

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Section: Definition 2 the Gamma Function Is Defined Asmentioning

confidence: 99%

Section: Discrete-time Approximations Of Fo Operatormentioning

confidence: 99%

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A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equation

Oprzędkiewicz¹,

Stanisławski²,

Gawin³

et al. 2017

Bulletin of the Polish Academy of Sciences Technical Sciences

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“…The model considered follows directly from the semigroup model given by Oprzędkiewicz and Gawin (2016) as well as Oprzędkiewicz et al (2016a). It employs a new, continuous fraction expansion (CFE) based solution method proposed by Oprzędkiewicz et al (2017b).…”

Section: Introductionmentioning

confidence: 99%

A Memory–Efficient Noninteger–Order Discrete–Time State–Space Model of a Heat Transfer Process

Oprzędkiewicz

Mitkowski

2018

International Journal of Applied Mathematics and Computer Science

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A new, state space, discrete-time, and memory-efficient model of a one-dimensional heat transfer process is proposed. The model is derived directly from a time-continuous, state-space semigroup one. Its discrete version is obtained via a continuous fraction expansion method applied to the solution of the state equation. Fundamental properties of the proposed model, such as decomposition, stability, accuracy and convergence, are also discussed. Results of experiments show that the model yields good accuracy in the sense of the mean square error, and its size is significantly smaller than that of the model employing the well-known power series expansion approximation.

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“…The model considered in this paper can be applied not only for scalar systems, but also for multidimensional systems with diagonal state operator. The example of such a class of real, physical processes are heat transfer processes described by a semigroup model [16].…”

Section: Introductionmentioning

confidence: 99%

Practical and asymptotic stability of fractional discrete-time scalar systems described by a new model

Ruszewski

2016

Archives of Control Sciences

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Practical and asymptotic stability of fractional discrete-time scalar systems described by a new model ANDRZEJ RUSZEWSKIThe stability problems of fractional discrete-time linear scalar systems described by the new model are considered. Using the classical D-partition method, the necessary and sufficient conditions for practical stability and asymptotic stability are given. The considerations are il-lustrated by numerical examples.

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A non integer order, state space model for one dimensional heat transfer process

Cited by 25 publications

References 12 publications

A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equation

A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equation

A Memory–Efficient Noninteger–Order Discrete–Time State–Space Model of a Heat Transfer Process

Practical and asymptotic stability of fractional discrete-time scalar systems described by a new model

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