Analog Circuit Implementation of Fractional Order Damped Sine and Cosine Functions

Nezzari, Hassene; Charef, Abdelfatah; Boucherma, Djamel

doi:10.1109/jetcas.2013.2273854

Cited by 13 publications

(13 citation statements)

References 4 publications

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“…For 0 < m < 1, the rational functions approximating the fundamental functions Ff 3 (s) and Ff 4 (s) of (25), in the frequency band of interest [0, ω H ], are given as [24,25]:…”

Section: Fundamental Functions For a Pair Of Complex Eigenvalues Withmentioning

confidence: 99%

The solution of state space linear fractional system of commensurate order with complex eigenvalues using regular exponential and trigonometric functions

Boucherma¹,

Charef

Nezzari³

2015

Int. J. Dynam. Control

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In a previous work, we have derived the general solution of the state space linear fractional system of commensurate order for real simple and multiple eigenvalues of the state space matrix. The obtained solutions of the homogeneous and non-homogeneous cases have been expressed as a linear combination of introduced fundamental functions. In this paper, the above work has been extended to solve the state space linear fractional system of commensurate order for complex eigenvalues of the state space matrix. First, suitable fundamental functions corresponding to the different types of complex eigenvalues of the state space matrix are introduced. Then, the derived formulations of the resolution approach are presented for the homogeneous and the nonhomogeneous cases. The solutions are expressed in terms of a linear combination of the proposed fundamental functions which are in the form of exponentials, sine, cosine, damped sine and damped cosine functions depending on the commensurate fractional order. The results are validated by solving an illustrative example to demonstrate the effectiveness of the proposed analytical tool for the solution of the state space linear fractional system of commensurate order.

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“…For 0 < m < 1, the rational functions approximating the fundamental functions Ff 3 (s) and Ff 4 (s) of (25), in the frequency band of interest [0, ω H ], are given as [24,25]:…”

Section: Fundamental Functions For a Pair Of Complex Eigenvalues Withmentioning

confidence: 99%

The solution of state space linear fractional system of commensurate order with complex eigenvalues using regular exponential and trigonometric functions

Boucherma¹,

Charef

Nezzari³

2015

Int. J. Dynam. Control

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“…In this first example, the numerical values of the observation samples y(k) which will be used in the parameters identification are generated by using the fractional order systems implementation methods developed in [31][32][33]. This means that we do not use the linear fractional order differential equation of (41) to generate the numerical values…”

Section: Examplementioning

confidence: 99%

“…of the observation samples y(k) as it is done for the identification algorithms in the literature. The transfer function of (42) can be rewritten as From [31][32][33], the three fractional order fundamental functions of (47) can be easily approximated by a rational function, in a given frequency band [0, ω H = 10000 rad/s] and ω max = 100 000ω H , as follows…”

Section: Examplementioning

confidence: 99%

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Linear fractional order system identification using adjustable fractional order differentiator

Idiou

Charef

Djouambi

2014

IET signal process.

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In previous decades, it has been observed that many physical systems are well characterised by fractional order models. Hence, their identification is attracting more and more interest of the scientific community. However, they pose a more difficult identification problem, which requires not only the estimation of model coefficients but also the determination of fractional orders with the tedious calculation of fractional order derivatives. This study focuses on an identification scheme, in the time domain, of dynamic systems described by linear fractional order differential equations. The proposed identification method is based on the recursive least squares algorithm applied to an ARX structure derived from the linear fractional order differential equation using adjustable fractional order differentiators. The basic ideas and the derived formulations of the identification scheme are presented. Illustrative examples are presented to validate the proposed linear fractional order system identification approach.

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“…Utilization only proportional-integral (PI) controllers is sufficient in wide range of industrial control problems. Although digital controllers are used more often, the role of controllers designed via analogue technique should not be underestimated [8] [14]. Our brief literature survey indicates various implementation techniques for the fractional-order integral operator s l , where -1 < l < 0 realization in the Laplace domain [15], [16].…”

Section: Introductionmentioning

confidence: 99%

Analogue Implementation of a Fractional-PI^λ Controller for DC Motor Speed Control

Herencsár

Kartci

Koton

et al. 2019

2019 IEEE 28th International Symposium on Industrial Electronics (ISIE)

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In this paper, an approach to design a fractionalorder integral operator s  where-1 < λ< 0, using an analogue technique, is presented. The integrator with a constant phase angle-80.1 degree (i.e. order l =-0.89), bandwidth greater than 3 decades, and maximum relative phase error 1.38% is designed by cascade connection of first-order bilinear transfer segments and first-order low-pass filter. The performance of suggested realization is demonstrated in a fractional-order proportionalintegral (FOPI ) controller described with proportional constant 1.37 and integration constant 2.28. The design specification corresponds to a speed control system of an armature controlled DC motor, which is often used in mechatronic and other fields of control theory. The behavior of both proposed analogue circuits employing two-stage Op-Amps is confirmed by SPICE simulations using TSMC 0.18 μm level-7 LO EPI SCN018 CMOS process parameters with ±0.9 V supply voltages.

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Analog Circuit Implementation of Fractional Order Damped Sine and Cosine Functions

Cited by 13 publications

References 4 publications

The solution of state space linear fractional system of commensurate order with complex eigenvalues using regular exponential and trigonometric functions

The solution of state space linear fractional system of commensurate order with complex eigenvalues using regular exponential and trigonometric functions

Linear fractional order system identification using adjustable fractional order differentiator

Analogue Implementation of a Fractional-PI^λ Controller for DC Motor Speed Control

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Analog Circuit Implementation of Fractional Order Damped Sine and Cosine Functions

Cited by 13 publications

References 4 publications

The solution of state space linear fractional system of commensurate order with complex eigenvalues using regular exponential and trigonometric functions

The solution of state space linear fractional system of commensurate order with complex eigenvalues using regular exponential and trigonometric functions

Linear fractional order system identification using adjustable fractional order differentiator

Analogue Implementation of a Fractional-PIλ Controller for DC Motor Speed Control

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Product

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Analogue Implementation of a Fractional-PI^λ Controller for DC Motor Speed Control