Inverting fractional order transfer functions through Laguerre approximation

2004 ◽  
Vol 52 (5) ◽  
pp. 387-393 ◽  
Author(s):  
Guido Maione
Keyword(s):  
Fractional Order ◽  
Transfer Functions

scholarly journals Comparative Study of Discrete Component Realizations of Fractional-Order Capacitor and Inductor Active Emulators

2018 ◽  
Vol 27 (11) ◽  
pp. 1850170 ◽  
Author(s):  
Georgia Tsirimokou ◽  
Aslihan Kartci ◽  
Jaroslav Koton ◽  
Norbert Herencsar ◽  
Costas Psychalinos
Keyword(s):  
Comparative Study ◽  
Fractional Order ◽  
High Performance ◽  
Continued Fraction ◽  
Transfer Functions ◽  
Operational Amplifiers ◽  
Current Conveyors ◽  
Continued Fraction Expansion ◽  
Current Feedback ◽  
Discrete Component

Due to the absence of commercially available fractional-order capacitors and inductors, their implementation can be performed using fractional-order differentiators and integrators, respectively, combined with a voltage-to-current conversion stage. The transfer function of fractional-order differentiators and integrators can be approximated through the utilization of appropriate integer-order transfer functions. In order to achieve that, the Continued Fraction Expansion as well as the Oustaloup’s approximations can be utilized. The accuracy, in terms of magnitude and phase response, of transfer functions of differentiators/integrators derived through the employment of the aforementioned approximations, is very important factor for achieving high performance approximation of the fractional-order elements. A comparative study of the accuracy offered by the Continued Fraction Expansion and the Oustaloup’s approximation is performed in this paper. As a next step, the corresponding implementations of the emulators of the fractional-order elements, derived using fundamental active cells such as operational amplifiers, operational transconductance amplifiers, current conveyors, and current feedback operational amplifiers realized in commercially available discrete-component IC form, are compared in terms of the most important performance characteristics. The most suitable of them are further compared using the OrCAD PSpice software.

Asymptotic magnitude Bode plots of fractional-order transfer functions

2019 ◽  
Vol 6 (4) ◽  
pp. 1019-1026 ◽  
Author(s):  
Ameya Anil Kesarkar ◽  
Selvaganesan Narayanasamy
Keyword(s):  
Fractional Order ◽  
Transfer Functions ◽  
Bode Plots

On the approximate inverse Laplace transform of the transfer function with a single fractional order

2020 ◽  
pp. 014233122097766
Author(s):  
Ali Yüce ◽  
Nusret Tan
Keyword(s):  
Transfer Function ◽  
Laplace Transform ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Transfer Functions ◽  
Inverse Laplace Transform ◽  
Fractional Order Systems ◽  
Approximate Inverse ◽  
Classical Mathematics ◽  
Curve Fitting Method

The history of fractional calculus dates back to 1600s and it is almost as old as classical mathematics. Although many studies have been published on fractional-order control systems in recent years, there is still a lack of analytical solutions. The focus of this study is to obtain analytical solutions for fractional order transfer functions with a single fractional element and unity coefficient. Approximate inverse Laplace transformation, that is, time response of the basic transfer function, is obtained analytically for the fractional order transfer functions with single-fractional-element by curve fitting method. Obtained analytical equations are tabulated for some fractional orders of [Formula: see text]. Moreover, a single function depending on fractional order alpha has been introduced for the first time using a table for [Formula: see text]. By using this table, approximate inverse Laplace transform function is obtained in terms of any fractional order of [Formula: see text] for [Formula: see text]. Obtained analytic equations offer accurate results in computing inverse Laplace transforms. The accuracy of the method is supported by numerical examples in this study. Also, the study sets the basis for the higher fractional-order systems that can be decomposed into a single (simpler) fractional order systems.

Fractional-Order Modeling and Simulation of Magnetic Coupled Boost Converter in Continuous Conduction Mode

2018 ◽  
Vol 28 (05) ◽  
pp. 1850061 ◽  
Author(s):  
Zirui Jia ◽  
Chongxin Liu
Keyword(s):  
Mathematical Model ◽  
Mathematical Models ◽  
Fractional Order ◽  
Transfer Functions ◽  
Circuit Simulation ◽  
Boost Converter ◽  
Averaged Model ◽  
Conduction Mode ◽  
State Average ◽  
Continuous Conduction Mode

By using fractional-order calculus theory and considering the condition that capacitor and inductor are naturally fractional, we construct the fractional mathematical model of the magnetic coupled boost converter with tapped-inductor in the operation of continuous conduction mode (CCM). The fractional state average model of the magnetic coupled boost converter in CCM operation is built by exploiting state average modeling method. In these models, the effects of coupling factor, which is viewed as one generally, are directly pointed out. The DC component, the AC component, the transfer functions and the requirements of the magnetic coupled boost converter in CCM operation are obtained and investigated on the basis of the state averaged model as well as its fractional mathematical model. Using the modified Oustaloup’s method for filter approximation algorithm, the derived models are simulated and compared using Matlab/Simulink. In order to further verify the fractional model, circuit simulation is implemented. Furthermore, the differences between the fractional-order mathematical models and the corresponding integer-order mathematical models are researched. Results of the model and circuit simulations validate the effectiveness of theoretical analysis.

scholarly journals Minimum MOS Transistor Count Fractional-Order Voltage-Mode and Current-Mode Filters

Technologies ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 85
Author(s):  
Panagiotis Bertsias ◽  
Costas Psychalinos ◽  
Ahmed S. Elwakil ◽  
Brent Maundy
Keyword(s):  
Fractional Order ◽  
Transfer Functions ◽  
Circuit Complexity ◽  
Current Mode ◽  
Cmos Process ◽  
Mos Transistors ◽  
Correct Operation ◽  
Mos Transistor ◽  
Voltage Mode ◽  
Filter Structures

Voltage-mode and current-mode fractional-order filter topologies, which are capable of realizing various types of transfer functions, are introduced in this paper. Thanks to the employment of the transconductance parameter of the MOS transistors, the derived filter structures offer the benefit of the electronic adjustment of their frequency characteristics. With regards to the literature, the number of MOS transisitors is minimized leading to significant reduction of the circuit complexity and power dissipation. Simulation results, derived using the Design Kit of the 0.35 μm Austria Mikro Systeme CMOS process and the Cadence IC design suite, confirm the correct operation of the presented filter structures.

Stability and resonance analysis of a general non-commensurate elementary fractional-order system

2020 ◽  
Vol 23 (1) ◽  
pp. 183-210 ◽  
Author(s):  
Shuo Zhang ◽  
Lu Liu ◽  
Dingyu Xue ◽  
YangQuan Chen
Keyword(s):  
Fractional Order ◽  
Transfer Functions ◽  
Order Parameters ◽  
Fractional Order System ◽  
Order System ◽  
Stability Conditions ◽  
Order Restriction ◽  
Resonance Analysis ◽  
General Kind ◽  
The Stability

AbstractThe elementary fractional-order models are the extension of first and second order models which have been widely used in various engineering fields. Some important properties of commensurate or a few particular kinds of non-commensurate elementary fractional-order transfer functions have already been discussed in the existing studies. However, most of them are only available for one particular kind elementary fractional-order system. In this paper, the stability and resonance analysis of a general kind non-commensurate elementary fractional-order system is presented. The commensurate-order restriction is fully released. Firstly, based on Nyquist’s Theorem, the stability conditions are explored in details under different conditions, namely different combinations of pseudo-damping (ζ) factor values and order parameters. Then, resonance conditions are established in terms of frequency behaviors. At last, an example is given to show the stable and resonant regions of the studied systems.

scholarly journals Modeling and Analysis of the Fractional-Order Flyback Converter in Continuous Conduction Mode by Caputo Fractional Calculus

Electronics ◽  
2020 ◽  
Vol 9 (9) ◽  
pp. 1544
Author(s):  
Chen Yang ◽  
Fan Xie ◽  
Yanfeng Chen ◽  
Wenxun Xiao ◽  
Bo Zhang
Keyword(s):  
Theoretical Analysis ◽  
Fractional Order ◽  
Transfer Functions ◽  
Circuit Simulation ◽  
Mutual Inductance ◽  
Power Electronic ◽  
Power Electronic Converters ◽  
Flyback Converter ◽  
Conduction Mode ◽  
Continuous Conduction Mode

In order to obtain more realistic characteristics of the converter, a fractional-order inductor and capacitor are used in the modeling of power electronic converters. However, few researches focus on power electronic converters with a fractional-order mutual inductance. This paper introduces a fractional-order flyback converter with a fractional-order mutual inductance and a fractional-order capacitor. The equivalent circuit model of the fractional-order mutual inductance is derived. Then, the state-space average model of the fractional-order flyback converter in continuous conduction mode (CCM) are established. Moreover, direct current (DC) analysis and alternating current (AC) analysis are performed under the Caputo fractional definition. Theoretical analysis shows that the orders have an important influence on the ripple, the CCM operating condition and transfer functions. Finally, the results of circuit simulation and numerical calculation are compared to verify the correctness of the theoretical analysis and the validity of the model. The simulation results show that the fractional-order flyback converter exhibits smaller overshoot, shorter setting time and higher design freedom compared with the integer-order flyback converter.

Analytical Design of Fractional Order Proportional Integral Controller for Spherical Tank

2014 ◽  
Vol 573 ◽  
pp. 279-284 ◽  
Author(s):  
Neenu Elizabeth Cherian ◽  
K. Sundaravadivu
Keyword(s):  
Fractional Order ◽  
Dead Time ◽  
Transfer Functions ◽  
Design Method ◽  
Analytical Design ◽  
Proportional Integral ◽  
First Order ◽  
Spherical Tank ◽  
Proportional Integral Controller ◽  
Fopi Controller

This paper presents an analytical design method for fractional order proportional integral (FOPI) controller for the spherical tank which is modelled as a first order plus dead time (FOPDT) process. The design is based on the Bode’s ideal transfer function and fractional calculus. By using frequency domain, the proposed FOPI tuning rules are directly derived for a generalized first order plus dead time process and then applied to the transfer functions obtained at various operating points of the spherical tank. The performance of the designed FOPI controller is compared with the conventional integer order proportional integral derivative (IOPID) controller in simulation.

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