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.

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.

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 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.

HEART SOUND MODEL BASED ON CASCADED AND LOSSLESS ACOUSTIC TUBES

2019 ◽  
Vol 19 (05) ◽  
pp. 1950031
Author(s):  
KAI WANG ◽  
XIEFENG CHENG ◽  
YAMIN CHEN ◽  
CHENJUN SHE ◽  
KEXUE SUN ◽  
...  
Keyword(s):  
Transfer Function ◽  
Transmission Line ◽  
Approximation Method ◽  
Heart Sound ◽  
Generation Mechanism ◽  
Traditional Model ◽  
Normal Heart ◽  
Tube Radius ◽  
Formant Frequency ◽  
Model Based

In order to further understand the generation mechanism of Heart Sound, we introduce a new method for simulating Heart Sound by using cascaded and lossless acoustic tubes. Based on the theory of acoustics, we abstract the ventricles and arteries inside of the heart as multistage tubes with equal length and different radii. By controlling the radii of tubes, we simulate the process of relaxation and constriction of the ventricles and arteries. Then, we calculate the transfer function of the tubes based on the theory of reflective transmission line. To gain the tubes’ radii, we use formant frequency as the model target parameters and put forward an approximation method. Finally, the experimental results show that compared with traditional model, the model based on cascaded and lossless acoustic tubes could better reflect the state of the ventricles and arteries. Meanwhile, by comparing the model tube radius of normal Heart Sound and pathological Heart Sound, we can give a better explanation to the cause of pathological Heart Sound.

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.

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 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.

Limit cycles in relay systems with fractional order plants

2019 ◽  
Vol 41 (15) ◽  
pp. 4424-4435
Author(s):  
Ali Yüce ◽  
Nusret Tan ◽  
Derek P Atherton
Keyword(s):  
Time Delay ◽  
Transfer Function ◽  
Limit Cycle ◽  
Fractional Order ◽  
Limit Cycles ◽  
Transfer Functions ◽  
Feedback Systems ◽  
Relay Feedback ◽  
Cycle Frequency ◽  
Plant Transfer

In this paper, limit cycle frequency, pulse width and stability analysis are examined using different methods for relay feedback nonlinear control systems with integer or fractional order plant transfer functions. The describing function (DF), A loci, a time domain method formulated in state space notation and Matlab/Simulink simulations are used for the analysis. Comparisons of the results of using these methods are given in several examples. In addition, the work has been extended to fractional order systems with time delay. Programs have been developed in the Matlab environment for all the theoretical methods. In particular, Matlab programs have been written to obtain a graphical solution for the A loci method, which can precisely calculate the limit cycle frequency. The developed solution methods are shown in various examples. The major contribution is to look at finding limit cycles for relay feedback systems having plants with a fractional order transfer function (FOTF). However, en route to this goal new assessments of limit cycle stability have been done for a rational plant transfer function plus a time delay.

scholarly journals FPAA-Based Realization of Filters with Fractional Laplace Operators of Different Orders

2021 ◽  
Vol 5 (4) ◽  
pp. 218
Author(s):  
Stavroula Kapoulea ◽  
Costas Psychalinos ◽  
Ahmed S. Elwakil
Keyword(s):  
Transfer Function ◽  
Fractional Order ◽  
Transfer Functions ◽  
Pass Band ◽  
Laplace Operators ◽  
Conventional Procedure ◽  
Programmable Analog ◽  
Field Programmable ◽  
Low Pass ◽  
Band Pass

A simple and direct procedure for implementing fractional-order filters with transfer functions that contain Laplace operators of different fractional orders is presented in this work. Based on a general fractional-order transfer function that describes fractional-order low-pass, high-pass, band-pass, band-stop and all-pass filters, the introduced concept deals with the consideration of this function as a whole, with its approximation being performed using a curve-fitting-based technique. Compared to the conventional procedure, where each fractional-order Laplace operator of the transfer function is individually approximated, the main offered benefit is the significant reduction in the order of the resulting rational function. Experimental results, obtained using a field-programmable analog array device, verify the validity of this concept.

Routh Approximations for Reducing Order of Discrete Systems

1982 ◽  
Vol 104 (1) ◽  
pp. 107-109 ◽  
Author(s):  
Chyi Hwang ◽  
Yen-Ping Shin
Keyword(s):  
Transfer Function ◽  
Approximation Method ◽  
Transfer Functions ◽  
Discrete Systems ◽  
Higher Order ◽  
Order System ◽  
High Order System ◽  
Numerical Example ◽  
Higher Order Terms ◽  
Z Domain

Routh approximation method is extended to the simplification of z-transfer functions. The procedure includes (1) transformation of the z-transfer function into the w-domain, (2) γ-δ expansion of the w-transfer function, (3) truncation the higher-order terms in the γ-δ expansions, and (4) transformation of the reduced w-transfer function into the z-domain. The reduced model is always stable if the original high-order system is stable. A numerical example is inclinded to illustrate the procedure.

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