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1
Content available remote Performance improvement of aircraft pitch angle using the fractional order adaptive PID controller
Bensafia Yassine, Idir Abdelhakim, Zemmit Abderrahim, Khettab Khatir
Przegląd Elektrotechniczny
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2023
|
R. 99, nr 5
98--101
EN
Fractional calculus has been rediscovered by scientists and engineers in the last two decades, and applied in an increasing number of fields, namely control theory. The current research work presents the use of the fractional adaptive PID controller approach optimized by a genetic algorithm to improve the performances (rise time, setting time, overshoot, and mean absolute error) for aircraft by introducing a fractional order integrator and differentiator in the classical feedback adaptive PID controller. To validate the arguments, the effectiveness and performance analysis of the proposed fractional order adaptive PID controller optimized by a genetic algorithm have been studied in comparison to the classical adaptive PID controller. Numerical simulation and analysis are presented to verify the best controller. The fractional order adaptive PID gives the best results in terms of settling time, rise time, overshoot, and mean absolute error. This approach can also be generalized to other fractional and integer systems in order to improve their performances and noise rejection.
PL
Rachunek ułamkowy został na nowo odkryty przez naukowców i inżynierów w ciągu ostatnich dwóch dekad i stosowany w coraz większej liczbie dziedzin, a mianowicie w teorii sterowania. Obecna praca badawcza przedstawia zastosowanie podejścia adaptacyjnego regulatora PID ułamkowego zoptymalizowanego przez algorytm genetyczny w celu poprawy wydajności (czas narastania, czas ustawiania, przeregulowanie i średni błąd bezwzględny) dla samolotów poprzez wprowadzenie integratora i różniczkowania ułamkowego rzędu w klasycznym adaptacyjnym regulatorze PID ze sprzężeniem zwrotnym . Aby potwierdzić te argumenty, przeprowadzono analizę skuteczności i wydajności proponowanego adaptacyjnego regulatora PID ułamkowego rzędu zoptymalizowanego algorytmem genetycznym w porównaniu z klasycznym adaptacyjnym regulatorem PID. Przedstawiono symulację i analizę numeryczną w celu weryfikacji najlepszego sterownika. Adaptacyjny PID ułamkowego rzędu daje najlepsze wyniki pod względem czasu ustalania, czasu narastania, przeregulowania i średniego błędu bezwzględnego. To podejście można również uogólnić na inne systemy ułamkowe i całkowite w celu poprawy ich wydajności i tłumienia szumów.
2
Content available remote "Fractionalization" A new approach for comparing different approximation methods of fractional order systems and disturbances rejection in PID control
Dib Riad, Bensafia Yassine, Khettab Khatir
Przegląd Elektrotechniczny
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2023
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R. 99, nr 7
227-231
EN
Recently, many research works have focused on fractional order systems and their approximation methods. It has been shown to be a useful tool for enhancing plant dynamics in terms of time and frequency performance. In this paper we propose a new approach for comparing between the different approximations methods of fractional order systems and disturbance rejection in PID control of DC motor by fractionalizing an integer order derivative operator in the original integer system. The implementation of the fractionalized terms is realized by mean of the well established approximation methods and in order to determine the best method, the responses of original integer system are compared to those of fractionalized systems. Illustrative simulations examples show that the fractionalization approach give the best decision (selected method) ,a good tool for comparison between different approximation methods and it give the good rejection of disturbances in PID control of DC motor . This approach can also be generalized to others numerical approximation methods and it can also be used in the area of systems control.
PL
Ostatnio wiele prac badawczych koncentrowało się na systemach rzędu ułamkowego i metodach ich aproksymacji. Wykazano, że jest to przydatne narzędzie do zwiększania dynamiki instalacji pod względem wydajności czasowej i częstotliwościowej. W tym artykule proponujemy nowe podejście do porównywania różnych metod aproksymacji systemów ułamkowego rzędu i odrzucania zakłóceń w sterowaniu PID silnika prądu stałego poprzez frakcjonowanie operatora pochodnej rzędu całkowitego w oryginalnym układzie całkowitym. Implementacja wyrazów ułamkowych jest realizowana za pomocą dobrze znanych metod aproksymacyjnych iw celu wyznaczenia najlepszej metody porównuje się odpowiedzi oryginalnego układu całkowitoliczbowego z odpowiedziami układów ułamkowych. Ilustracyjne przykłady symulacyjne pokazują, że podejście frakcyjne daje najlepszą decyzję (wybrana metoda), jest dobrym narzędziem do porównywania różnych metod aproksymacyjnych i zapewnia dobre odrzucanie zakłóceń w regulacji PID silnika prądu stałego. Podejście to można również uogólnić na inne metody aproksymacji numerycznej, a także można je stosować w obszarze sterowania systemami.
3
Content available Pointwise completeness and pointwise degeneracy of fractional standard and descriptor linear continuous-time systems with different fractional orders
Kaczorek Tadeusz, Sajewski Łukasz
International Journal of Applied Mathematics and Computer Science
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2020
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Vol. 30, no. 4
641--647
EN
Descriptor and standard linear continuous-time systems with different fractional orders are investigated. Descriptor systems are analyzed making use of the Drazin matrix inverse. Necessary and sufficient conditions for the pointwise completeness and pointwise degeneracy of descriptor continuous-time linear systems with different fractional orders are derived. It is shown that (i) the descriptor linear continuous-time system with different fractional orders is pointwise complete if and only if the initial and final states belong to the same subspace, (ii) the descriptor linear continuous-time system with different fractional orders is not pointwise degenerated in any nonzero direction for all nonzero initial conditions. Results are reported for the case of two different fractional orders and can be extended to any number of orders.
4
Content available Application of the Drazin inverse to the analysis of pointwise completeness and pointwise degeneracy of descriptor fractional linear continuous-time systems
Kaczorek Tadeusz, Ruszewski Andrzej
International Journal of Applied Mathematics and Computer Science
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2020
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Vol. 30, no. 2
219--223
EN
The Drazin inverse of matrices is applied to the analysis of pointwise completeness and pointwise degeneracy of fractional descriptor linear continuous-time systems. It is shown that (i) descriptor linear continuous-time systems are pointwise complete if and only if the initial and final states belong to the same subspace, and (ii) fractional descriptor linear continuous-time systems are not pointwise degenerated in any nonzero direction for all nonzero initial conditions. The discussion is illustrated with examples of descriptor linear electrical circuits.
5
Content available The Caputo vs. Caputo-Fabrizio operators in modeling of heat transfer process
Oprzędkiewicz K., Mitkowski W., Gawin E., Dziedzic K.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2018
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Vol. 66, nr 4
501--507
EN
In the paper two non-integer order, state space models of heat transfer process are compared. The first uses a known Caputo operator and the second – a new operator proposed by Caputo and Fabrizio in 2015. Both discussed models are modifications of a known, integer order, state space, semigroup model of heat transfer process. Parameters of both models were identified by means of optimization of MSE cost function with the use of simplex method, available in MATLAB. Both proposed models have been compared in the aspect of accuracy and convergence. Analytical and numerical results show that the Caputo-Fabrizio model is faster convergent and easier to implement than the Caputo model. However, its accuracy in the sense of MSE cost function is worse.
6
Content available Fractional Signals and Systems
Ortigueira M. D., Machado J. T. M., Ostalczyk P.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2018
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Vol. 66, nr 4
385--387
EN
The special section in the current volume of the Bulletin of the Polish Academy of Sciences, entitled “Fractional Signals and Systems”, includes selected papers from the FSS17 International Conference, which was held in Łódź, Poland on October 9–11, 2017. The founder of the conference is Manuel Duarte Ortigueira from the New University of Lisbon, Portugal. The FSS17 is yet another in a series of conferences, which had previously taken place in: 1. Caparica, Portugal, 2009 2. Coimbra, Portugal, 2011 3. Ghent, Belgium, 2013 4. Cluj-Napoca, Romania, 2015. The FSS17 conference addressed a broad spectrum of the Fractional Calculus (FC) applications in technical sciences. Main topics included the fractional-order continuous-, and discrete-time linear or non-linear fractional-order control, dynamic system identification via fractional models, fractional order filtering, as well as image processing using fractional methods. The conference’s main organizers included the Institute of Applied Computer Science (Instytut Informatyki Stosowanej Politechniki Łódzkiej), the Lodz University of Technology (Politechnika Łódzka) and the Polish Information Processing Society – Łódź Branch (Polskie Towarzystwo Informatyczne – Oddział Łódzki).
7
Content available Different fractional order models for an experimental smart beam system
Markowski K. A., Birs I., Muresan C. I., Prodan O.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2018
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Vol. 66, nr 4
485--493
EN
The applicability of fractional calculus in system engineering outperforms classic identification techniques due to its ability to depict physical phenomena with increased accuracy. The present study explores the increased accuracy and flexibility of a fractional order model applied to an experimental smart beam depicting an airplane wing. The paper details the fractional order system identification of the beam and explores the possibility of realization of the model.
8
Content available Application of the Drazin inverse to the analysis of descriptor fractional discrete-time linear systems with regular pencils
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2013
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Vol. 23, no. 1
29--33
EN
The Drazin inverse of matrices is applied to find the solutions of the state equations of descriptor fractional discrete-time systems with regular pencils. An equality defining the set of admissible initial conditions for given inputs is derived. The proposed method is illustrated by a numerical example.
9
Content available remote Singular fractional linear systems and electrical circuits
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2011
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Vol. 21, no 2
379-384
EN
A new class of singular fractional linear systems and electrical circuits is introduced. Using the Caputo definition of the fractional derivative, the Weierstrass regular pencil decomposition and the Laplace transformation, the solution to the state equation of singular fractional linear systems is derived. It is shown that every electrical circuit is a singular fractional system if it contains at least one mesh consisting of branches only with an ideal supercapacitor and voltage sources or at least one node with branches with supercoils.
10
Content available remote Positivity and stabilization of fractional 2D linear systems described by the Roesser model
Kaczorek T., Rogowski K.
International Journal of Applied Mathematics and Computer Science
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2010
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Vol. 20, no 1
85-92
EN
A new class of fractional 2D linear discrete-time systems is introduced. The fractional difference definition is applied to each dimension of a 2D Roesser model. Solutions of these systems are derived using a 2D Z-transform. The classical Cayley-Hamilton theorem is extended to 2D fractional systems described by the Roesser model. Necessary and sufficient conditions for the positivity and stabilization by the state-feedback of fractional 2D linear systems are established. A procedure for the computation of a gain matrix is proposed and illustrated by a numerical example.
11
Content available remote Simple conditions for practical stability of positive fractional discrete-time linear systems
Busłowicz M., Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2009
|
Vol. 19, no 2
263-269
EN
In the paper the problem of practical stability of linear positive discrete-time systems of fractional order is addressed. New simple necessary and sufficient conditions for practical stability and for practical stability independent of the length of practical implementation are established. It is shown that practical stability of the system is equivalent to asymptotic stability of the corresponding standard positive discrete-time systems of the same order. The discussion is illustrated with numerical examples.
12
Content available remote Reachability of cone fractional continuous-time linear systems
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2009
|
Vol. 19, no 1
89-93
EN
A new class of cone fractional continuous-time linear systems is introduced. Necessary and sufficient conditions for a fractional linear system to be a cone fractional one are established. Sufficient conditions for the reachability of cone fractional systems are given. The discussion is illustrated with an example of linear cone fractional systems.
13
Content available remote Fractional positive continuous-time linear systems and their reachability
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2008
|
Vol. 18, no 2
223-228
EN
A new class of fractional linear continuous-time linear systems described by state equations is introduced. The solution to the state equations is derived using the Laplace transform. Necessary and sufficient conditions are established for the internal and external positivity of fractional systems. Sufficient conditions are given for the reachability of fractional positive systems.
14
Content available remote Optimal approximation, simulation and analog realization of the fundamental fractional order transfer function
Djouambi A., Charef A., Besançon A. V.
International Journal of Applied Mathematics and Computer Science
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2007
|
Vol. 17, no 4
455-462
EN
This paper provides an optimal approximation of the fundamental linear fractional order transfer function using a distribution of the relaxation time function. Simple methods, useful in systems and control theories, which can be used to approximate the irrational transfer function of a class of fractional systems for a given frequency band by a rational function are presented. The optimal parameters of the approximated model are obtained by minimizing simultaneously the gain and the phase error between the irrational transfer function and its rational approximation. A simple analog circuit which can serve as a fundamental analog fractional system is obtained. Illustrative examples are presented to show the quality and usefulness of the approximation method.
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