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.

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 Research on Heat Transfer Dynamic Characteristics of Composite Layer Based on Laplace Transform

2019 ◽  
Author(s):  
Chunyu Xu ◽  
Junhua Lin ◽  
Wenhao Liu ◽  
Yuanbiao Zhang
Keyword(s):  
Heat Transfer ◽  
Composite Materials ◽  
Temperature Distribution ◽  
Transfer Function ◽  
Dynamic Characteristics ◽  
Transfer Functions ◽  
Composite Layer ◽  
Engineering Application ◽  
Second Order ◽  
Inverse Transformation

This paper predict and effectively control the temperature distribution of the steady-state and transient states of anisotropic four-layer composite materials online, knowing the density, specific heat, heat conductivity and thickness of the composite materials. Based on the transfer function, a mathematical model was established to study the dynamic characteristics of heat transfer of the composite materials. First of all, the Fourier heat transfer law was used to establish a one-dimensional Fourier heat conduction differential equation for each composite layer, and the Laplace transformation was carried out to obtain the system function. Then the approximate second-order transfer function of the system was obtained by Taylor expansion, and the Laplace inverse transformation was carried out to obtain the transfer function of the whole system in the time domain. Finally, the accuracy of the simplified analytical solutions of the first, second and third order approximate transfer functions was compared with computer simulation. The results showed that the second order approximate transfer functions can describe the dynamic process of heat transfer better than others. The research on the dynamic characteristics of heat transfer in the composite layer and the dynamic model of heat transfer in composite layer proposed in this paper have a reference value for practical engineering application. It can effectively predict the temperature distribution of composite layer material and reduce the cost of experimental measurement of heat transfer performance of materials.

Integrated and frequency-band magnitude, two alternative measures of the size of an earthquake

1970 ◽  
Vol 60 (3) ◽  
pp. 917-937 ◽  
Author(s):  
B. F. Howell ◽  
G. M. Lundquist ◽  
S. K. Yiu
Keyword(s):  
Transfer Function ◽  
Frequency Domain ◽  
Frequency Band ◽  
Transfer Functions ◽  
Body Wave ◽  
Average Amplitude ◽  
Equivalent Quantity ◽  
Short Period ◽  
Alternative Measures ◽  
Body Wave Magnitude

Abstract Integrated magnitude substitutes the r.m.s. average amplitude over a pre-selected interval for the peak amplitude in the conventional body-wave magnitude formula. Frequency-band magnitude uses an equivalent quantity in the frequency domain. Integrated magnitude exhibits less scatter than conventional body-wave magnitude for short-period seismograms. Frequency-band magnitude exhibits less scatter than body-wave magnitude or integrated magnitude for both long- and short-period seismograms. The scatter of frequency-band magnitude is probably due to real azimuthal effects, crustal-transfer-function variations, errors in compensation for seismograph response, microseismic moise and uncertainties in the compensation for attenuation with distance. To observe azimuthal variations clearly, the crustal-transfer functions and seismograph response need to be known more precisely than was the case in this experiment, because these two sources of scatter can be large enough to explain all of the observed variations.

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.

A Novel Method of Transfer-Function Identification for Unknown Systems

2013 ◽  
Vol 321-324 ◽  
pp. 1967-1970
Author(s):  
Chung Neng Huang ◽  
Chen Min Cheng
Keyword(s):  
Transfer Function ◽  
Electromechanical Coupling ◽  
High Efficiency ◽  
Transfer Functions ◽  
Second Order ◽  
System A ◽  
Derivative Free ◽  
Efficiency Control ◽  
Novel Method ◽  
Free Search

This study proposes a new modeling method for unknown systems. Through this method, the transfer functions can be identified. First, the input-output data pairs of the unidentified system should be collected. Then, the transfer function’s coefficients can be identified based on the errors via the derivative-free search methods such as GA etc. Here, a second-order transfer function is used in this study. For a second-order transfer function is difficult to approach each system, a plurality of transfer functions may be used depending on the precise requirement. Finally, following the previous steps, the other transfer function can be found in succession. In order to confirm the effectiveness of this proposed method, an electromagnetic flywheel (EF) system is used in this study. Such kinds of systems are always with many uncertainties as nonlinear electromechanical coupling and electromagnetic saturation, etc. They are difficult to modeling via traditional mathematic ways. In this study, the data pairs of EF system is collected by experiments. By assessing the results of the proposal and experimental data shows that this method is feasible to any unknown systems. system, achieve more saving energy and high efficiency control purposes.

scholarly journals Estimation of Transfer Function Coefficients for Second-Order Systems via Metaheuristic Algorithms

Sensors ◽  
2021 ◽  
Vol 21 (13) ◽  
pp. 4529
Author(s):  
Omar Rodríguez-Abreo ◽  
Juvenal Rodríguez-Reséndiz ◽  
Francisco Antonio Castillo Velásquez ◽  
Alondra Anahi Ortiz Verdin ◽  
Juan Manuel Garcia-Guendulain ◽  
...  
Keyword(s):  
Transfer Function ◽  
Transfer Functions ◽  
Standard Form ◽  
Estimation Method ◽  
Second Order ◽  
Parametric Estimation ◽  
Metaheuristic Algorithms ◽  
Step Response ◽  
Gray Wolf ◽  
Final Value

The present research develops the parametric estimation of a second-order transfer function in its standard form, employing metaheuristic algorithms. For the estimation, the step response with a known amplitude is used. The main contribution of this research is a general method for obtaining a second-order transfer function for any order stable systems via metaheuristic algorithms. Additionally, the Final Value Theorem is used as a restriction to improve the velocity search. The tests show three advantages in using the method proposed in this work concerning similar research and the exact estimation method. The first advantage is that using the Final Value Theorem accelerates the convergence of the metaheuristic algorithms, reducing the error by up to 10 times in the first iterations. The second advantage is that, unlike the analytical method, it is unnecessary to estimate the type of damping that the system has. Finally, the proposed method is adapted to systems of different orders, managing to calculate second-order transfer functions equivalent to higher and lower orders. Response signals to the step of systems of an electrical, mechanical and electromechanical nature were used. In addition, tests were carried out with simulated signals and real signals to observe the behavior of the proposed method. In all cases, transfer functions were obtained to estimate the behavior of the system in a precise way before changes in the input. In all tests, it was shown that the use of the Final Value Theorem presents advantages compared to the use of algorithms without restrictions. Finally, it was revealed that the Gray Wolf Algorithm has a better performance for parametric estimation compared to the Jaya algorithm with an error up to 50% lower.

Accuracy in LP electromagnetic seismograph calibration by least-squares inversion of the calibration pulse

1980 ◽  
Vol 70 (6) ◽  
pp. 2261-2273
Author(s):  
Robert W. McGonigle ◽  
Paul W. Burton
Keyword(s):  
Transfer Function ◽  
Least Squares ◽  
Gaussian Noise ◽  
Measurement Errors ◽  
Transfer Functions ◽  
Phase Delay ◽  
Noise Distribution ◽  
Dominant Source ◽  
Electromagnetic Seismograph ◽  
Source Of Error

abstract Uncertainties in the instrumental constants for the standard LP instruments of the WWSSN, arising from errors in amplitude measurements of the calibration pulse, are analytically quantified assuming that such measurement errors may be approximated by a perturbing Gaussian noise distribution. More importantly, for the practicing seismologist, uncertainties in magnification and phase delay of the entire instrumental transfer function are deduced from least-squares inversion of the seismogram calibration pulse. It is emphasized that reliable estimates of instrumental transfer functions are obtained by least-squares inversion of the calibration pulse, particularly for underdamped or near critically damped seismographs, where the Gaussian noise distribution is the dominant source of error.

A Numerical Evaluation of the Quadratic Transfer Function for a Floating Structure

2019 ◽  
Author(s):  
Zhitian Xie ◽  
Yujie Liu ◽  
Jeffrey Falzarano
Keyword(s):  
Transfer Function ◽  
Frequency Domain ◽  
Transfer Functions ◽  
Second Order ◽  
Wave Frequency ◽  
Wave Forces ◽  
Floating Structure ◽  
Transfer Function Matrix ◽  
Wave Induced ◽  
Function Matrix

Abstract The second order force of a floating structure can be expressed in terms of a time independent quadratic transfer functions along with the incident wave elevation, through which it is possible to evaluate the second order wave exciting forces in the frequency domain. Newman’s approximation has been widely applied in approximating the elements of the quadratic transfer function matrix while numerically evaluating the second order wave induced force. Through Newman’s approximation, the off-diagonal elements can be numerically approximated with the diagonal elements and thus the numerical calculation efficiency can be enhanced. Newman’s approximation assumes that the off-diagonal elements do not change significantly with the wave frequency and that hydrodynamic phenomenon regarding the low difference frequency are usually of interest. However, it is obviously less satisfying when an element that is close to the diagonal line in the quadratic transfer function matrix shows an extremum if the corresponding wave frequency is close to the natural frequency of the certain motion. In this paper, the full derivation and expression of the second order wave forces and moments applied to a floating structure have been presented, through which the numerical results of the quadratic transfer function matrix including the diagonal and the off-diagonal elements will be illustrated. This work will present the basis of numerically evaluating the second order forces in the frequency domain. The comparisons among various approximations regarding the second order forces in deep water will also be presented as a meaningful reference.

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