Revisiting four approximation methods for fractional order transfer function implementations: Stability preservation, time and frequency response matching analyses

Deniz, Furkan Nur; Alagöz, Barış Baykant; Tan, Nusret; Köseoğlu, Murat

doi:10.1016/j.arcontrol.2020.03.003

Cited by 57 publications

(29 citation statements)

References 45 publications

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“…The zeros and poles are computed by using ( 16) and (17). Note that in [5], a slight modification was reported compared with (19), where the bottom index of the product starts from -N instead of 1, as in the original Outstaloup's notation. Nevertheless, both approximations, reported by Equation ( 23) in [5] and (19) are equivalent.…”

Section: Refined Oustaloup's Approximationmentioning

confidence: 99%

“…Note that in [5], a slight modification was reported compared with (19), where the bottom index of the product starts from -N instead of 1, as in the original Outstaloup's notation. Nevertheless, both approximations, reported by Equation ( 23) in [5] and (19) are equivalent. According to Table 6, ( 19) is easily coded in the Maple 18 environment as shown in Table 7.…”

Section: Refined Oustaloup's Approximationmentioning

confidence: 99%

“…This approximation method was reported in [19,32], which is an improved version of the stability boundaries locus fitting approximation method [33]. According to [19], the method uses a set of logarithmically spaced frequency sampling points w k = [w l , w 2 , w 3 . .…”

Section: Modified Stability Boundary Locus (Msbl) Fitting Approximationmentioning

confidence: 99%

“…Therefore, this survey not only describes and analyzes the main approximation methods reported to date, but two model order reduction methods are also reviewed in order to obtain rational transfer functions of reduced order and with the target of simplifying the electronic synthesis process. Unlike [18,19] and to help beginners get started quickly and learn arbitrary-order calculus, the corresponding Maple 18 code of each approximation method is herein reported.…”

Section: Introductionmentioning

confidence: 99%

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Rational Approximations of Arbitrary Order: A Survey

et al. 2021

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This paper deals with the study and analysis of several rational approximations to approach the behavior of arbitrary-order differentiators and integrators in the frequency domain. From the Riemann–Liouville, Grünwald–Letnikov and Caputo basic definitions of arbitrary-order calculus until the reviewed approximation methods, each of them is coded in a Maple 18 environment and their behaviors are compared. For each approximation method, an application example is explained in detail. The advantages and disadvantages of each approximation method are discussed. Afterwards, two model order reduction methods are applied to each rational approximation and assist a posteriori during the synthesis process using analog electronic design or reconfigurable hardware. Examples for each reduction method are discussed, showing the drawbacks and benefits. To wrap up, this survey is very useful for beginners to get started quickly and learn arbitrary-order calculus and then to select and tune the best approximation method for a specific application in the frequency domain. Once the approximation method is selected and the rational transfer function is generated, the order can be reduced by applying a model order reduction method, with the target of facilitating the electronic synthesis.

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Section: Refined Oustaloup's Approximationmentioning

confidence: 99%

Section: Refined Oustaloup's Approximationmentioning

confidence: 99%

Section: Modified Stability Boundary Locus (Msbl) Fitting Approximationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rational Approximations of Arbitrary Order: A Survey

et al. 2021

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“…For this reason, ideal digital realization of fractional-order elements uses growing computational resources in time as a result of the long memory effect. Therefore, for the practical realization of this controller, approximate fractional-order models are used to implement fractional-order derivative and integral elements [43][44][45][46]. In practice, FOPID controllers are commonly implemented by using these approximate fractional-order models.…”

Section: Fractional Calculus and Fractional-order Systemsmentioning

confidence: 99%

A NARX Model Reference Adaptive Control Scheme: Improved Disturbance Rejection Fractional-Order PID Control of an Experimental Magnetic Levitation System

et al. 2020

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Real control systems require robust control performance to deal with unpredictable and altering operating conditions of real-world systems. Improvement of disturbance rejection control performance should be considered as one of the essential control objectives in practical control system design tasks. This study presents a multi-loop Model Reference Adaptive Control (MRAC) scheme that leverages a nonlinear autoregressive neural network with external inputs (NARX) model in as the reference model. Authors observed that the performance of multi-loop MRAC-fractional-order proportional integral derivative (FOPID) control with MIT rule largely depends on the capability of the reference model to represent leading closed-loop dynamics of the experimental ML system. As such, the NARX model is used to represent disturbance-free dynamical behavior of PID control loop. It is remarkable that the obtained reference model is independent of the tuning of other control loops in the control system. The multi-loop MRAC-FOPID control structure detects impacts of disturbance incidents on control performance of the closed-loop FOPID control system and adapts the response of the FOPID control system to reduce the negative effects of the additive input disturbance. This multi-loop control structure deploys two specialized control loops: an inner loop, which is the closed-loop FOPID control system for stability and set-point control, and an outer loop, which involves a NARX reference model and an MIT rule to increase the adaptation ability of the system. Thus, the two-loop MRAC structure allows improvement of disturbance rejection performance without deteriorating precise set-point control and stability characteristics of the FOPID control loop. This is an important benefit of this control structure. To demonstrate disturbance rejection performance improvements of the proposed multi-loop MRAC-FOPID control with NARX model, an experimental study is conducted for disturbance rejection control of magnetic levitation test setup in the laboratory. Simulation and experimental results indicate an improvement of disturbance rejection performance.

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Direct Approach for Modelling a Class of Fractional-Order System Using Two Generating Functions

Sungoh,

Swarnakar

2024

Lecture Notes in Electrical Engineering

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Revisiting four approximation methods for fractional order transfer function implementations: Stability preservation, time and frequency response matching analyses

Cited by 57 publications

References 45 publications

Rational Approximations of Arbitrary Order: A Survey

Rational Approximations of Arbitrary Order: A Survey

A NARX Model Reference Adaptive Control Scheme: Improved Disturbance Rejection Fractional-Order PID Control of an Experimental Magnetic Levitation System

Direct Approach for Modelling a Class of Fractional-Order System Using Two Generating Functions

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