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1

Frequency response based curve fitting approximation of fractional-order PID controllers

Bingi Kishore, Ibrahim Rosdiazli, Karsiti Mohd Noh, Hassam Sabo Miya, Harindran Vivekananda Rajah

International Journal of Applied Mathematics and Computer Science

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2019

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Vol. 29, no. 2

311--326

EN

Fractional-order PID (FOPID) controllers have been used extensively in many control applications to achieve robust control performance. To implement these controllers, curve fitting approximation techniques are widely employed to obtain integer-order approximation of FOPID. The most popular and widely used approximation techniques include the Oustaloup, Matsuda and Cheraff approaches. However, these methods are unable to achieve the best approximation due to the limitation in the desired frequency range. Thus, this paper proposes a simple curve fitting based integer-order approximation method for a fractional-order integrator/differentiator using frequency response. The advantage of this technique is that it is simple and can fit the entire desired frequency range. Simulation results in the frequency domain show that the proposed approach produces better parameter approximation for the desired frequency range compared with the Oustaloup, refined Oustaloup and Matsuda techniques. Furthermore, time domain and stability analyses also validate the frequency domain results.

2

35 mm ammunition’s trajectory model identification based on firing tables

Baranowski L., Gadomski B., Majewski P., Szymonik J.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2018

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Vol. 66, nr 5

635--643

EN

The article presents a procedure designed for identification of projectile’s trajectory model through aerodynamic coefficients estimation. The identification process is based on firing tables artificially prepared (firing tables prepared using mathematical flight model for the projectile instead of trajectories recorded on field tests) with the use of modified point–mass and rigid body trajectory models. All the necessary data, including physical parameters of the projectile and its aerodynamic characteristics are provided. The detailed results of estimation of chosen aerodynamic coefficients are presented in both visual and tabular form. The main purpose of this paper is to establish the minimum number of trajectories (as characterized in firing tables), and the permissible error of initial parameters being passed to the mathematical model that would allow the correct identification of projectile’s trajectory model.

3

Interim fuel consumption estimation based on ship service parameters in real weather conditions with the use of ndCurveMaster curve fitting software

Cepowski T., Drozd A.

Zeszyty Naukowe Akademii Morskiej w Szczecinie

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2017

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nr 50 (122)

30--35

EN

This paper deals with fuel consumption estimations relating to container ships on the basis of ship service and wave parameters. Data, on which to base estimations, was measured and recorded from a container ship during 96 months at sea. Approximating functions were calculated by the use of curve fitting techniques and regression methods, utilizing newly developed software named ndCurveMaster. The approximation function presented in this paper could have practical application for the estimation of container ship fuel consumption, while considering weather routing. In addition the study clearly shows the relationship between the fuel consumption of a container ship and the number of months since its last docking. These results may form the basis for further research in this direction.

4

Application of a multicriterial optimization to the resolution of X-ray diffraction curves of semicrystalline polymers

Rabiej M.

Polimery

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2017

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T. 62, nr 11-12

821--833

EN

The analysis of wide angle X-ray diffraction (WAXD) curves of semicrystalline polymers is connected with their decomposition into crystalline peaks and amorphous components. To this aim a theoretical curve is constructed which is a best fitted, mathematical model of the experimental one. All parameters of the theoretical curve are found using an optimization procedure. As it has been already proved, a reliable decomposition can be performed only by means of a procedure which effectively performs a multicriterial optimization. It consists in minimization of the sum of squared deviations between the theoretical and experimental curves and simultaneous maximization of the area of the amorphous component. So, the objective function in the optimization procedure is constructed of two criterial functions which represent the two requirements. The proportions between the criterial functions and their significance at different stages of the procedure must be determined by suitable weights. A proper choice of the weights is an important part of the procedure. In this paper a new solution of this problem is presented: the weights are changed dynamically in subsequent steps of the optimization procedure. A few different algorithms of the weights determination are presented and evaluated by means of several statistical method. The optimization procedures equipped with these algorithms are tested using WAXD patterns of popular polymers: Cellulose I, Cellulose II and PET. It is shown that the optimization procedures equipped with the dynamic algorithms of weights determination are much more effective than the procedures using some constant, arbitrarily chosen weights.

PL

Dokonanie analizy krzywych dyfrakcyjnych WAXD polimerów semikrystalicznych, związanej z ich dekompozycją na składowe krystaliczne i amorficzne, wymaga zbudowania funkcji stanowiącej matematyczny model zarejestrowanej krzywej dyfrakcyjnej. Do wyznaczenia parametrów tej funkcji stosuje się nieliniową optymalizację wielokryterialną, polegającą na minimalizacji sumy kwadratów odchyleń krzywej teoretycznej od krzywej eksperymentalnej oraz maksymalizacji pola pod składową amorficzną. Funkcja celu jest więc skonstruowana z dwóch funkcji kryterialnych, które reprezentują te dwa warunki. Proporcje między tymi funkcjami i ich znaczenie na poszczególnych etapach procedury optymalizacyjnej muszą być określane przez odpowiednie wagi. Właściwy dobór wag dla funkcji kryterialnych jest jednym z trudniejszych zadań. Zaprezentowano nowe rozwiązanie problemu doboru wag dla funkcji kryterialnych: wagi zmieniano dynamicznie w kolejnych krokach procedury. Zaproponowano kilka różnych algorytmów wyznaczania wag. Algorytmy porównano i oceniono za pomocą testów statystycznych. Procedury wyposażone w te algorytmy zastosowano do rozkładu krzywych dyfrakcyjnych celulozy I, celulozy II i poliestru. Wykazano, że procedury wyposażone w dynamiczne algorytmy wyznaczania wag są znacznie bardziej skuteczne niż procedury wykorzystujące wagi stałe, ustalone arbitralnie.

5

A new approach for the classification of event related potentials for valid and paradox reasoning

Papaodysseus C., Zannos S., Giannopoulos F., Arabadjis D., Rousopoulos P., Stachtea X., Papageorgiou C.

Biocybernetics and Biomedical Engineering

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2016

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Vol. 36, no. 1

292--301

EN

In the present paper, a novel approach is introduced for comparing and classifying recorded ERP signals from subjects applying valid (Aristotle's) and paradox (Zeno's) syllogisms. In fact, the authors conceived and realized a corresponding experiment, as well as a new method for processing, fitting and classifying the corresponding captured ERP signals into groups according to their similarity. Subsequently, for each such group, an ideal curve that represents all signals of the group has been evaluated for valid and paradox reasoning separately. These ‘‘ideal representatives’’ manifest essential statistical differences per subject for a considerable number of electrodes (5 electrodes with 99% level of confidence, 14 electrodes with 95% level of confidence, 17 electrodes with 90% level of confidence). These results support the assumption that the obtained ideal representatives may indeed reflect essential differences in the underlying brain functions which generated the obtained ERPs. Equivalently, one may claim that the performed experiment and the associated results manifest statistically essential differences between the mental functions during valid and paradox reasoning.

6

Analysis of cells elasticity based on force-distance curves obtained from atomic force microscopy

Płaczek B.

Journal of Medical Informatics & Technologies

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2015

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Vol. 24

173--179

EN

Automated techniques for measuring elasticity parameters of cells enable development of new diagnosis methods. An important elasticity parameter is the Young’s modulus (YM), which has been effectively used to characterize different cell properties, e. g., platelet activation, locomotion, differentiation, and aging. This paper deals with the problem of automated determination of cells YM based on the force-distance curves obtained from atomic force microscope. During experiments, the YM of cells was determined by using contact point detection and curve fitting algorithms. Experimental results were compared for two theoretical models of indentation: Hertz model, and Sneddon model. The results show that single indentation model allows a satisfactory accuracy to be obtained only for a subset of the force-distance curves. The most appropriate model for a given curve can be selected based on the fitting error analysis.

7

Hyper-elastic modelling of intervertebral disc polyurethane implant

Pawlikowski M., Skalski K., Sowiński T.

Acta of Bioengineering and Biomechanics

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2013

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Vol. 15, no. 2

43--50

EN

Artificial materials including various kinds of polymers like polyurethanes are more and more widely used in different branches of science and also in biomedical engineering. The paper presents the process of creating a constitutive equation for a polyurethane nanocomposite which is considered to be hyper-elastic. The constitutive modelling was conducted within the range of application of the material as one of the components of lumbar intervertebral disc prosthesis. In the paper, the biomechanics of the lumbar spine and the most frequently applied intervertebral disc prostheses are described. Also a polyurethane nanocomposite as a new material to be applied in prostheses is presented. The way of formulating a constitutive equation by means of mathematical formulae is described. Four various hyper-elastic potential functions are considered, i.e., Ogden, Neo-Hookean, Yeoh and Mooney–Rivlin. On the basis of monotonic compression tests the best hyper-elastic model for the material considered was chosen and hyper-elastic constants were calibrated. Finally, the constitutive model was validated on the basis of FE analysis. The paper ends with a conclusion and presentation of further plans of research directed towards the development of a constitutive equation and its application in computer simulations by means of the finite element method.

8

ANFIS Approach for Noise Reduction of Lightning Current Online Monitoring System

Yan N., Fu Z., Li J.

Przegląd Elektrotechniczny

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2012

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R. 88, nr 7a

108-112

EN

A novel de-noising algorithm, based on adaptive neural-fuzzy inference system (ANFIS) is proposed for noise reduction of the lightning current online monitoring system. The paper presents the theory and the implement procedure of the fuzzy neural system. Comparisons among the traditional strategies, such as curve fitting (CF), wavelet transform (WT) methods and the proposed ANFIS strategy are carried out. The simulation results demonstrate the superiority of the proposed method. Moreover, the employed approach has been tested on the practical measured current of lightning current online monitoring system. The testing results validate the proposed approach.

PL

Zaproponowano nowy algorytm odszumiania bazujący na adaptacyjnym neuro-fuzzy systemie interferencji ANFIS. System zastosowano przy monitorowaniu prądu wyładowań. System porównano z innymi dotychczas stosowanymi – dopasowanie krzywej czy transformata falkowa.

9

Ship’s horizontal plane determination by means of three laser range sensors in pilot navigation and docking system

Zalewski P.

Journal of KONBiN

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2012

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No. 4 (24)

161--169

EN

Modern ship’s navigation support systems relay mainly on GNSS technology, as prime source of position. The advantages of such solution are obvious and undeniable. On the other hand in critical situations the dependency of the one system presents a great deal of risk. Taking above into consideration the idea of building GNSS independent Pilot Navigation and Docking System (PNDS) was put into practice. PNDS utilizes the range measurements between laser head and ship’s side to determine the ship’s horizontal plane presented on the screen. The study comprises the case in which three networked rangefinders were deployed on the berth side in a line. Such location of sensors causes uncertainty in determining of ship’s outline contour and speed in relation to assigned coordinate reference system. The final algorithm presented in the article, taking into account all underdetermined conditions, has been tested in PNDS system constructed in Maritime University in Szczecin.

PL

Współczesne systemy wspomagania nawigacji opierają się głównie na technologii GNSS, jako podstawowym i często jedynym źródle informacji pozycyjnej. Zalety tego rozwiązania są ewidentne, jednakże w sytuacjach krytycznych pod względem bezpieczeństwa zależność od jednego systemu powoduje znaczny wzrost ryzyka. Nawigacyjny System Pilotowo-Dokujący (PNDS) został zaprojektowany z wykorzystaniem alternatywnej metody wyznaczenia położenia kadłuba podchodzącej do nabrzeża jednostki. Zastosowano w nim pomiary odległości pomiędzy głowicami laserowymi a burtą statku w celu wyliczenia położenia umownej obwiedni (konturu) kadłuba. Badania objęły przypadek trzech dalmierzy laserowych ustawionych w linii prostej na nabrzeżu. Takie usytuowanie czujników powoduje niepewność wyznaczenia położenia kadłuba i prędkości w przyjętym układzie współrzędnych. Zaprezentowany w artykule algorytm, biorąc pod uwagę przypadki niedookreślone, został przetestowany w systemie PNDS zbudowanym w Akademii Morskiej w Szczecinie.

10

Udoskonalona metoda dopasowywania krzywych do danych niezawodnościowych uwzględniająca różnice między próbkami

Sun R., Peng W. W., Huang H. Z., Ling D. L., Yang J.

Eksploatacja i Niezawodność

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2012

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Vol. 14, No. 1

62-71

PL

W praktyce inżynieryjnej stykamy się z problemem wykorzystania zgromadzonych danych do oceny maszyn lub sprzętu. Dopasowywanie krzywych to metoda powszechnie używana do rozwiązywania tego typu problemów. W procedurze tej szeroko stosuje się metodę najmniejszych kwadratów. Jeżeli dane wejściowe dopasowywane krzywą można pogrupować tak by tworzyły oddzielne próbki, a różnice między próbkami w sposób oczywisty odzwierciedlają pewną właściwość dotyczącą gromadzenia danych, której nie można pominąć, to konwencjonalna metoda dopasowywania krzywych nie pozwala na analizę takich danych wejściowych. Aby przezwyciężyć to ograniczenie, przedstawiamy udoskonaloną metodę dopasowywania krzywych. Poprzez analizę danych wejściowych, możemy określić związek pomiędzy położeniem próbki a czynnikiem ważonym w dopasowaniu krzywej oraz wykorzystać czynniki ważone przy dopasowywaniu krzywej. Aby osiągnąć jak najdokładniejsze przybliżenie do krzywej rzeczywistej wprowadziliśmy procedurę iteratywną modyfikującą czynniki ważone. Zastosowanie zaproponowanej metody zilustrowano na przykładzie danych z badań niezawodnościowych.

EN

In engineering practice, we face a problem of using some collected data to evaluate a kind of machine or equipment. Curve fitting is a common method to solve this problem. Least square method is wildly applied in this procedure. If the source data of curve fitting can be grouped in samples and the distinction of samples obviously express some character in source data collecting which cannot be ignored. Conventional curve fitting method cannot handle these source data. To deal with this disadvantage, we introduce an improved curve fitting method. Through source data analysis, we can find out the relationship between sample location and weight factor in curve fitting, and use these weight factors for curve fitting. To approach the true curve, we introduce an iterative procedure to modify the weight factors. An engineering example is given to illustrate this proposed method.

11

A new approximation method for the trend hypersurface analysis for elevation of drilhole data

Karakus D., Safak S., Onur A. H.

Archives of Mining Sciences

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2011

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Vol. 56, no 1

47-58

EN

This paper presents a new mathematical approach to a well known interpolation technique of the trend surface analysis method that allows for fast and accurate estimation of the spatial distribution of mineral resources. The most popular and accurate method for estimating the quality of an ore body spatial distribution uses geostatistics built upon complex variogram modeling. To simplify this application and maintain sensitivity, a trend hypersurface analysis method was reformulated, and a novel computer algorithm was established to analyze data collected from a limestone quarry in Turkey. However 4 dimensional variables obtained from drill hole data and data type limit the surface analysis studies. This study compares the measured CaCO3 contents obtained from 31 drill holes with data derived from mathematical modeling.

PL

Praca opisuje nowe podejście matematyczne do znanej techniki interpolacji stosowanej w analizie powierzchni trendu, umożliwiające szybką i dokładną estymację rozkładu przestrzennego złoża surowców mineralnych. Najbardziej znaną i najdokładniejszą metodą estymowania jakości rozkładu przestrzennego złoża rudy wykorzystuje metody geostatystyki zbudowane w oparciu o modelowanie z użyciem złożonych wariogramów. Aby uprościć tę aplikację jednocześnie zachowując wrażliwość metody, przeformułowano metodę analizy hyper-powierzchni trendu i stworzono nowatorski algorytm umożliwiający analizę danych zebranych w kopalni wapienia w Turcji. Jednak że 4-wymiarowe zmienne otrzymane na podstawie danych otworowych ograniczają zasięg analizy powierzchni. W pracy tej porównano zmierzone zawartości CaCO3 otrzymane z 31 otworów z wynikami badań modelowych.

12

Modelowanie matematyczne i badania fizykochemiczne równowag kwasowo-zasadowych w układach binarnych rozpuszczalników

Michałowski T., Pilarski B., Dobkowska A., Młodzianowski J.

Wiadomości Chemiczne

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2010

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[Z] 64, 1-2

123-154

EN

Conductometric titration (CT) is used as a complementary, non-specific technique against potentiometric (pH) titration (PT). The CT + PT procedure is applied to binary-solvent systems: HmL + Z/A + B (or HmL + Z/W + B) formed of a weak acid HmL and a pH-modifying agent Z (e.g. NaOH, NH3 or HCl), and totally miscible organic solvents A and B (or W = H2O and B) chosen as co-solvents, where acid-base equilibria are considered. The solvent composition in such systems is expressed by the mole fraction x of B. Concentrations of solutes: HmL and Z are the same in the solution titrated (titrand, D) and in titrant (T) and changes in pH value resulted only from differences in acid-base properties of solvents applied in D and T. On the basis of results obtained from PT and CT, the functional relationships related to the solutions of weak acids HmL (m = 1, 2, ...) were formulated, namely: (1) pKi = pKi(x) for acidity parameters; (2) Lp = Lp(x) for molar conductivities, related to the species formed by di- and triprotic acids. For this purpose, the Kohlrausch law on the independence of migrating ions has been applied. The pKi = pKi(x) relationships were formulated for diprotic and triprotic acids on the basis of enderson.Hasselbalch functions and modeled with use of polynomial functions. The pKi(1) and Lp(1) values were obtained by extrapolation. The approach suggested is the essential novelty in contemporary literature related to the subject in question. This way, the paper provides a new methodology, a new dimension (x-scale) to the problems known hitherto. In the experimental part, the pK1 = pK1(x) relationships were formulated for results of mutual pH titrations: HmL/B . HmL/W or HmL/B . HmL/A, obtained for the systems not involving Z. The systems were modeled with the use of Redlich.Kister (R.K) and orthogonal (normal, shifted) Legendre polynomials. Asymmetric functions by Myers.Scott (M.S) and by Michałowski were also suggested for this purpose. It was stated that all approximating functions fit the experimental data very well. The asymmetric functions are applicable for modeling of more bended relationships pK1 = pK1(x), particularly when referred to the limiting x-values in the interval < 0, 1 >. A software that enables to formulate the relationships pKi = pKi(x) according to different approximating functions after introducing the experimental data {(Vj, pHj) | j = 1, ..., N} obtained from PT, was also prepared, together with statistical evaluation of pKi(0) and pKi(1) values, obtained by extrapolation. The results thus obtained are presented in tables and figures. Some criteria of validity of the results obtained according to PT for pK1(0) and pK1(1) are also presented. The first criterion testifying about reversibility of indications of the measuring system is a position of the point of intersection of the plots obtained in the mutual titrations. The second criterion stems from comparison of results obtained from three pH titrations: (1) HmL/B Ť HmL/W, (2) HmL/B Ť HmL/A and (3) HmL/A Ť HmL/W. The convergence of the related plots at x = 0 and x = 1, where x = xB in (1) and (2) and x = xA in (3) testifies on account of the calibration procedure applied for glass electrode.

13

Problem doboru wspólnej krzywej dla dwóch replikacji na przykładzie procesu kiełkowania ziaren zbóż stymulowanych polem magnetycznym

Bochniak A., Wesołowska-Janczarek M.

Inżynieria Rolnicza

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2006

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R. 10, nr 5(80)

39-47

PL

W pracy przedstawiony jest problem doboru wspólnej krzywej obrazującej przebieg badanego procesu, gdy mamy do dyspozycji dwie replikacje. Wybrana krzywa powinna być dobrym dopasowaniem dla obu replikacji. Do analizy, na podstawie której zostanie przedstawiony problem dopasowania wspólnej krzywej, wybrano krzywą logistyczną, przy pomocy której daje się oszacować przebieg procesu kiełkowania ziaren zbóż stymulowanych polem magnetycznym. Jako kryterium dobroci dopasowania krzywej zastosowano, często w tym celu wykorzystywany, współczynnik determinacji R2. Podano sposób wstępnej oceny, czy do otrzymanych replikacji daje się dopasować wspólną krzywą logistyczną, która może być oszacowaniem zarówno dla pierwszej, jak i drugiej replikacji. Praca ma na celu przedstawienie różnych możliwości doboru takiej krzywej z wykorzystaniem współczynnika R2 jako kryterium dobroci dopasowania krzywej i zwrócenie uwagi na zaobserwowane własności tych metod.

EN

A problem of fitting a common curve representing examined process, when there are two replications, is presented in the paper. The chosen curve should be good fitted for both replications. Logistic curve, which can be used to estimate sprouting process of corn seeds stimulated with magnetic field, was chosen to illustrate this problem. A determinant coefficient R2 was used as a criterion of goodness of fit. A method of initial estimation, when such a common curve can be fitted, is given. This paper presents different methods of choosing a common curve with coefficient R2 as a criterion and their observed properties.