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1
Content available remote Performances improvement of fractional order systems using the fractional order adaptive PID controller
Asradj Zahir, Bensafia Bouira, Merzouk Imad
Przegląd Elektrotechniczny
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2023
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R. 99, nr 8
257--260
EN
Fractional order systems are widely used in industrial application for its different advantage such us high efficiency, and flexibilities. The applications of fractional order systems in a range of scientific fields have caught the attention of researchers especially in control strategy. The current research work presents the use the fractional adaptive PID controller approach, optimized by genetic algorithm, to improve the performances (rise time, setting time and overshoot) for fractional systems by introducing fractional order integrator and differentiator in the classical feedback adaptive PID controller. To validate the arguments, effectiveness and performances analysis of the proposed approach optimized by 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 PID gives the best result in terms of settling time, rise time, overshoot and mean absolute error.
PL
Systemy ułamkowego rzędu są szeroko stosowane w zastosowaniach przemysłowych ze względu na różne zalety, takie jak wysoka wydajność i elastyczność. Zastosowania systemów rzędu ułamkowego w wielu dziedzinach nauki przykuły uwagę badaczy, zwłaszcza w dziedzinie strategii sterowania. Obecna praca badawcza przedstawia wykorzystanie podejścia ułamkowego regulatora adaptacyjnego PID, zoptymalizowanego przez algorytm genetyczny, do poprawy osiągów (czas narastania, czas ustawiania i przeregulowanie) układów ułamkowych poprzez wprowadzenie integratora i układu różniczkowego ułamkowego rzędu do klasycznego regulatora PID z adaptacyjnym sprzężeniem zwrotnym. Aby zweryfikować argumenty, przeprowadzono analizę skuteczności i wydajności proponowanego podejścia zoptymalizowanego za pomocą algorytmu genetycznego w porównaniu z klasycznym adaptacyjnym regulatorem PID. Przedstawiono symulację i analizę numeryczną w celu weryfikacji najlepszego sterownika. PID rzędu ułamkowego daje najlepsze wyniki pod względem czasu ustalania, czasu narastania, przeregulowania i średniego błędu bezwzględnego.
2
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.
3
Content available Fractional time-invariant compartmental linear systems
Kaczorek Tadeusz
International Journal of Applied Mathematics and Computer Science
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2023
|
Vol. 33, no. 1
97--102
EN
Fractional time-invariant compartmental linear systems are introduced. Controllability and observability of these systems are analyzed. The eigenvalue assignment problem of compartmental linear systems is considered and illustrated with a numerical example.
4
Content available Two classes of infinitely many solutions for fractional Schrödinger-Maxwell system with concave-convex power nonlinearities
Khiddi Mustapha
Journal of Mathematics and Applications
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2022
|
Vol. 45
87--105
EN
Employing critical theory and concentration estimates, we establish the existence of two classes of infinitely many weak solutions fractional Schrödinger-Poisson system involving critical Sobolev growth. The first classe of solutions with negative energy is found by using of notion genus while the second one contains infinitely many weak solutions with positive energy via Fountain theorem.
5
Content available Divisibility of the second-order minors of the nominators by minimal denominators of transfer matrices of cyclic fractional linear systems
Kaczorek Tadeusz
International Journal of Applied Mathematics and Computer Science
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2021
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Vol. 31, no. 4
627--633
EN
The divisibility of the second-order minors of the numerators of transfer matrices by their minimal denominators for cyclic fractional linear systems is analyzed. It is shown that all nonzero second-order minors of the numerators of the transfer matrices are divisible by their minimal denominators if and only if the system matrices of fractional standard and descriptor linear systems are cyclic. The theorems are illustrated by examples of fractional standard and descriptor linear systems.
6
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.
7
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.
8
Content available Absolute stability of a class of fractional positive nonlinear systems
Kaczorek Tadeusz
International Journal of Applied Mathematics and Computer Science
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2019
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Vol. 29, no. 1
93--98
EN
The positivity and absolute stability of a class of fractional nonlinear continuous-time and discrete-time systems are addressed. Necessary and sufficient conditions for the positivity of this class of nonlinear systems are established. Sufficient conditions for the absolute stability of this class of fractional positive nonlinear systems are also given.
9
Content available Stability of interval positive fractional discrete-time linear systems
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2018
|
Vol. 28, no. 3
451--456
EN
The aim of this work is to show that interval positive fractional discrete-time linear systems are asymptotically stable if and only if the respective lower and upper bound systems are asymptotically stable. The classical Kharitonov theorem is extended to interval positive fractional linear systems.
10
Content available Minimal positive realizations of linear continuous-time fractional descriptor systems: Two cases of an input-output digraph structure
Markowski K. A.
International Journal of Applied Mathematics and Computer Science
|
2018
|
Vol. 28, no. 1
9--24
EN
In the last two decades, fractional calculus has become a subject of great interest in various areas of physics, biology, economics and other sciences. The idea of such a generalization was mentioned by Leibniz and L'Hospital. Fractional calculus has been found to be a very useful tool for modeling linear systems. In this paper, a method for computation of a set of a minimal positive realization of a given transfer function of linear fractional continuous-time descriptor systems has been presented. The proposed method is based on digraph theory. Also, two cases of a possible input-output digraph structure are investigated and discussed. It should be noted that a digraph mask is introduced and used for the first time to solve a minimal positive realization problem. For the presented method, an algorithm was also constructed. The proposed solution allows minimal digraph construction for any one-dimensional fractional positive system. The proposed method is discussed and illustrated in detail with some numerical examples.
11
Content available A new method for computation of positive realizations of fractional linear continuous-time systems
Kaczorek T.
Archives of Control Sciences
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2018
|
Vol. 28, No. 4
511--525
EN
A new method for computation of positive realizations of given transfer matrices of fractional linear continuous-time linear systems is proposed. Necessary and sufficient conditions for the existence of positive realizations of transfer matrices are given. A procedure for computation of the positive realizations is proposed and illustrated by examples.
12
Content available Minimum energy control of descriptor fractional discrete-time linear systems with two different fractional orders
Sajewski Ł.
International Journal of Applied Mathematics and Computer Science
|
2017
|
Vol. 27, no. 1
33--41
EN
Reachability and minimum energy control of descriptor fractional discrete-time linear systems with different fractional orders are addressed. Using the Weierstrass–Kronecker decomposition theorem of the regular pencil, a solution to the state equation of descriptor fractional discrete-time linear systems with different fractional orders is given. The reachability condition of this class of systems is presented and used for solving the minimum energy control problem. The discussion is illustrated with numerical examples.
13
Content available Reduced-order fractional descriptor observers for a class of fractional descriptor continuous-time nonlinear systems
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2016
|
Vol. 26, no. 2
277--283
EN
Fractional descriptor reduced-order nonlinear observers for a class of fractional descriptor continuous-time nonlinear systems are proposed. Sufficient conditions for the existence of the observers are established. The design procedure for the observers is given and demonstrated on a numerical example.
14
Content available Positivity and stability of fractional descriptor time-varying discrete-time linear systems
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2016
|
Vol. 26, no. 1
5--13
EN
The Weierstrass–Kronecker theorem on the decomposition of the regular pencil is extended to fractional descriptor time-varying discrete-time linear systems. A method for computing solutions of fractional systems is proposed. Necessary and sufficient conditions for the positivity of these systems are established.
15
Content available Fractional descriptor continuous-time linear systems described by the Caputo–Fabrizio derivative
Kaczorek T., Borawski K.
International Journal of Applied Mathematics and Computer Science
|
2016
|
Vol. 26, no. 3
533--541
EN
The Weierstrass–Kronecker theorem on the decomposition of the regular pencil is extended to fractional descriptor continuous-time linear systems described by the Caputo–Fabrizio derivative. A method for computing solutions of continuous-time systems is presented. Necessary and sufficient conditions for the positivity and stability of these systems are established. The discussion is illustrated with a numerical example.
16
Content available A new approach to the realization problem for fractional discrete-time linear systems
Kaczorek T.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2016
|
Vol. 64, nr 1
9--14
EN
A new approach to the realization problem for fractional discrete-time linear systems is proposed. A procedure for computation of fractional realizations of given transfer matrices is presented and illustrated by numerical examples.
17
Content available Reachability of standard and fractional continuous-time systems with constant inputs
Rogowski K.
Archives of Control Sciences
|
2016
|
Vol. 26, no. 2
147--159
EN
The reachability of standard and fractional-order continuous-time systems with constant inputs is addressed. Positive and non-positive continuous-time linear systems are considered. Necessary and sufficient conditions for the existence of such constant inputs that steers the system from zero initial conditions to the given final state in desired time are derived and proved. As an example of such systems the electrical circuits with DC voltage sources are presented
18
Content available Pointwise completeness and pointwise degeneracy of positive fractional descriptor continuous-time linear systems with regular pencils
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2015
|
Vol. 25, no. 2
217--221
EN
Pointwise completeness and pointwise degeneracy of positive fractional descriptor continuous-time linear systems with regular pencils are addressed. Conditions for pointwise completeness and pointwise degeneracy of the systems are established and illustrated by an example.
19
Content available Minimum energy control of fractional positive continuous-time linear systems with bounded inputs
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2014
|
Vol. 24, no. 2
335--340
EN
A minimum energy control problem for fractional positive continuous-time linear systems with bounded inputs is formulated and solved. Sufficient conditions for the existence of a solution to the problem are established. A procedure for solving the problem is proposed and illustrated with a numerical example.
20
Content available Minimum energy control of fractional descriptor positive discrete-time linear systems
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2014
|
Vol. 24, no. 4
735--743
EN
Necessary and sufficient conditions for the positivity and reachability of fractional descriptor positive discrete-time linear systems are established. The minimum energy control problem for descriptor positive systems is formulated and solved. Sufficient conditions for the existence of a solution to the minimum energy control problem are given. A procedure for computation of optimal input sequences and a minimal value of the performance index is proposed and illustrated by a numerical example.
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