scholarly journals Fractional hold circuits versus positive realness of discrete transfer functions

2005 ◽  
Vol 2005 (3) ◽  
pp. 373-378 ◽  
Author(s):  
M. de la Sen ◽  
A. Bilbao-Guillerna
Keyword(s):  
Transfer Function ◽  
Fractional Order ◽  
Transfer Functions ◽  
Input Output ◽  
Appropriate Use ◽  
First Order ◽  
Positive Realness ◽  
Direct Input

The appropriate use of fractional-order holds (β-FROH) of correcting gainsβ∈[−1,1]as an alternative to the classical zero-and first-order holds (ZOHs, FOHs) is discussed related to the positive realness of the associate discrete transfer functions obtained from a given continuous transfer function. It is proved that the minimum direct input/output gain (i.e., the quotient of the leading coefficients of the numerator and denominator of the transfer function) needed for discrete positive realness may be reduced by the choice ofβcompared to that required for discretization via ZOH.

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.

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.

scholarly journals On the Essential Instabilities Caused by Fractional-Order Transfer Functions

2008 ◽  
Vol 2008 ◽  
pp. 1-13 ◽  
Author(s):  
Farshad Merrikh-Bayat ◽  
Masoud Karimi-Ghartemani
Keyword(s):  
Transfer Function ◽  
Fractional Order ◽  
Characteristic Equation ◽  
Stability Condition ◽  
Transfer Functions ◽  
Branch Points ◽  
The Stability ◽  
Unstable Branch

The exact stability condition for certain class of fractional-order (multivalued) transfer functions is presented. Unlike the conventional case that the stability is directly studied by investigating the poles of the transfer function, in the systems under consideration, the branch points must also come into account as another kind of singularities. It is shown that a multivalued transfer function can behave unstably because of the numerator term while it has no unstable poles. So, in this case, not only the characteristic equation but the numerator term is of significant importance. In this manner, a family of unstable fractional-order transfer functions is introduced which exhibit essential instabilities, that is, those which cannot be removed by feedback. Two illustrative examples are presented; the transfer function of which has no unstable poles but the instability occurred because of the unstable branch points of the numerator term. The effect of unstable branch points is studied and simulations are presented.

scholarly journals Approximation method for a fractional order transfer function with zero and pole

2014 ◽  
Vol 24 (4) ◽  
pp. 447-463 ◽  
Author(s):  
Krzysztof Oprzędkiewicz
Keyword(s):  
Transfer Function ◽  
Fractional Order ◽  
Approximation Method ◽  
Transfer Functions ◽  
Control Algorithms ◽  
Model Based ◽  
Interval Approach ◽  
Special Control

Abstract The paper presents an approximation method for elementary fractional order transfer function containing both pole and zero. This class of transfer functions can be applied for example to build model - based special control algorithms. The proposed method bases on Charef approximation. The problem of cancelation pole by zero with useful conditions was considered, the accuracy discussion with the use of interval approach was done also. Results were depicted by examples.

scholarly journals Validation of Fractional-Order Lowpass Elliptic Responses of (1 + α)-Order Analog Filters

2018 ◽  
Vol 8 (12) ◽  
pp. 2603 ◽  
Author(s):  
David Kubanek ◽  
Todd Freeborn ◽  
Jaroslav Koton ◽  
Jan Dvorak
Keyword(s):  
Transfer Function ◽  
Least Squares ◽  
Fractional Order ◽  
Frequency Band ◽  
Operational Amplifier ◽  
Transfer Functions ◽  
Second Order ◽  
Experimental Results ◽  
Analog Filters ◽  
Lowpass Filter

In this paper, fractional-order transfer functions to approximate the passband and stopband ripple characteristics of a second-order elliptic lowpass filter are designed and validated. The necessary coefficients for these transfer functions are determined through the application of a least squares fitting process. These fittings are applied to symmetrical and asymmetrical frequency ranges to evaluate how the selected approximated frequency band impacts the determined coefficients using this process and the transfer function magnitude characteristics. MATLAB simulations of ( 1 + α ) order lowpass magnitude responses are given as examples with fractional steps from α = 0.1 to α = 0.9 and compared to the second-order elliptic response. Further, MATLAB simulations of the ( 1 + α ) = 1.25 and 1.75 using all sets of coefficients are given as examples to highlight their differences. Finally, the fractional-order filter responses were validated using both SPICE simulations and experimental results using two operational amplifier topologies realized with approximated fractional-order capacitors for ( 1 + α ) = 1.2 and 1.8 order filters.

Algebraic bound for the phase–frequency response of the commande robuste d'ordre non-entier approximation of fractional differentiators and its applications in control systems analysis

2021 ◽  
pp. 107754632098776
Author(s):  
Khashayar Neshat ◽  
Mohammad Saleh Tavazoei
Keyword(s):  
Transfer Function ◽  
Frequency Response ◽  
Control Systems ◽  
Fractional Order ◽  
Upper Bound ◽  
Systems Analysis ◽  
Sufficient Conditions ◽  
Approximation Process ◽  
Positive Realness

This article deals with analyzing the phase–frequency response of commande robuste d'ordre non-entier approximations of fractional-order differentiators. More precisely, an algebraic tight upper bound is derived for the phase of the approximations obtained from the commande robuste d'ordre non-entier method. Then, some applications for this achievement are discussed in the viewpoint of control systems analysis. These applications include usefulness of the obtained upper bound in stability preservation analysis during the commande robuste d'ordre non-entier–based approximation process and applicability of such a bound in finding necessary or sufficient conditions for test of positive realness/negative imaginariness of a fractional-order transfer function.

scholarly journals Control device synthesis by polynomial matrix fractional descriptions method with limitation on controller structure

2021 ◽  
Vol 2131 (3) ◽  
pp. 032021
Author(s):  
A Voevoda ◽  
V Shipagin ◽  
K Bobobekov
Keyword(s):  
State Space ◽  
Linear Model ◽  
Polynomial Matrix ◽  
Control Device ◽  
Output Channel ◽  
Input Output ◽  
Output Vector ◽  
First Order ◽  
Direct Input ◽  
Proposed Modification

Abstract Modification of the algorithm for the polynomial synthesis of a multi-channel controller was proposed to preserve all control channels in this article. In order to test the functionality of the proposed modification, an example of a linear model of an unstable multi-channel plant is considered. The choice of the plant was determined by the possibility of a visual algorithm demonstration for polynomial synthesis of the controller, taking into account the proposed modifications.The plant was represented as three series-connected standard links: an aperiodic link of the first order, an unstable link, and an integrator, and has three input and two output channels. The control in the system is carried out in the feedback of the system and is summed up with the input impact. The feature of the plant is to limit the task to the second output, since it is essentially a derivative of the first output. In addition, the plant has a direct input–output channel. That is, the traversal matrix of the system is nonzero (when described through the state space). The synthesis task was set as follows: it is necessary to achieve certain quality indicators of the output vector value while maintaining all three control channels of the plant.

scholarly journals (N + α)-Order low-pass and high-pass filter transfer functions for non-cascade implementations approximating butterworth response

2021 ◽  
Vol 24 (3) ◽  
pp. 689-714
Author(s):  
David Kubanek ◽  
Jaroslav Koton ◽  
Jan Jerabek ◽  
Darius Andriukaitis
Keyword(s):  
Transfer Function ◽  
Fractional Order ◽  
Transfer Functions ◽  
Function Optimization ◽  
Pass Filter ◽  
Operational Transconductance Amplifiers ◽  
High Pass Filter ◽  
Approximation Errors ◽  
Low Pass ◽  
High Pass

Abstract The formula of the all-pole low-pass frequency filter transfer function of the fractional order (N + α) designated for implementation by non-cascade multiple-feedback analogue structures is presented. The aim is to determine the coefficients of this transfer function and its possible variants depending on the filter order and the distribution of the fractional-order terms in the transfer function. Optimization algorithm is used to approximate the target Butterworth low-pass magnitude response, whereas the approximation errors are evaluated. The interpolated equations for computing the transfer function coefficients are provided. An example of the transformation of the fractional-order low-pass to the high-pass filter is also presented. The results are verified by simulation of multiple-feedback filter with operational transconductance amplifiers and fractional-order element.

scholarly journals Geomagnetic coast effect at two Croatian repeat stations

2017 ◽  
Vol 59 (6) ◽  
Author(s):  
Eugen Vujić ◽  
Mario Brkić
Keyword(s):  
Transfer Function ◽  
Function Method ◽  
Transfer Functions ◽  
First Order ◽  
Transfer Function Method ◽  
Coast Effect ◽  
Inductive Effects ◽  
Induction Arrows ◽  
Coastal Effect ◽  
Induced Currents

<p>Knowledge of inductive effects is important for the reliability of geomagnetic surveys as well as reduction of measurements, and hence for the accuracy of models and maps of the Earth’s magnetic field. Detection of anomalous induced fields, due to the geomagnetic coast effect, was carried out by the transfer function method to estimate the induction arrows indicating areas of anomalous induced currents. To determine the transfer function at the two coastal Croatian repeat stations used in this study, the so-called geomagnetic plane-wave events from July 2010 were used. Analysis of transfer functions for Krbavsko polje and Sinjsko polje first order repeat stations, using observatories Grocka and Tihany as references, revealed the existence of the Adriatic coastal effect on periods of 10-65 minutes.</p>

scholarly journals An Effective Approach of Approximation of Fractional Order System using Real Interpolation Method

2017 ◽  
Vol 1 (1) ◽  
pp. 39
Author(s):  
Dung Quang Nguyen
Keyword(s):  
Transfer Function ◽  
Fractional Order ◽  
Transfer Functions ◽  
Interpolation Method ◽  
Fractional Order System ◽  
Real Interpolation ◽  
Order System ◽  
Frequency Method ◽  
Time Frequency ◽  
Real Interpolation Method

Fractional-order controllers are recognized to guarantee better closed-loop performance and robustness than conventional integer-order controllers. However, fractional-order transfer functions make time, frequency domain analysis and simulation significantly difficult. In practice, the popular way to overcome these difficulties is linearization of the fractional-order system. Here, a systematic approach is proposed for linearizing the transfer function of fractional-order systems. This approach is based on the real interpolation method (RIM) to approximate fractional-order transfer function (FOTF) by rational-order transfer function. The proposed method is implemented and compared to CFE high-frequency method; Carlson’s method; Matsuda’s method; Chare ’s method; Oustaloup’s method; least-squares, frequency interpolation method (FIM). The results of comparison show that, the method is simple, computationally efficient, flexible, and more accurate in time domain than the above considered methods.  This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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