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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 Fractional time-invariant compartmental linear systems
Kaczorek Tadeusz
International Journal of Applied Mathematics and Computer Science
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2023
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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.
3
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.
4
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
|
2021
|
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.
5
Content available Global stability of nonlinear feedback systems with fractional positive linear parts
Kaczorek Tadeusz
International Journal of Applied Mathematics and Computer Science
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2020
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Vol. 30, no. 3
493--499
EN
The global (absolute) stability of nonlinear systems with fractional positive and not necessarily asymptotically stable linear parts and feedbacks is addressed. The characteristics u = f(e) of the nonlinear parts satisfy the condition k1e ≤ f(e) ≤ k2e for some positive k1 and k2. It is shown that the fractional nonlinear systems are globally asymptotically stable if the Nyquist plots of the fractional positive linear parts are located on the right-hand side of the circles (−1/k1,−1/k2).
6
Content available Absolute stability of a class of fractional positive nonlinear systems
Kaczorek Tadeusz
International Journal of Applied Mathematics and Computer Science
|
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.
7
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.
8
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
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2018
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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.
9
Content available Boundary controllability of nonlinear stochastic fractional systems in Hilbert spaces
Mabel Lizzy R., Balachandran K.
International Journal of Applied Mathematics and Computer Science
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2018
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Vol. 28, no. 1
123--133
EN
Sufficient conditions for the controllability of nonlinear stochastic fractional boundary control systems are established. The equivalent integral equations are derived for both linear and nonlinear systems, and the control function is given in terms of the pseudoinverse operator. The Banach contraction mapping theorem is used to obtain the result. A controllability result for nonlinear stochastic fractional integrodifferential systems is also attained. Examples are included to illustrate the theory.
10
Content available Responses of positive standard and fractional linear systems and electrical circuits with derivatives of their inputs
Kaczorek T.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2018
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Vol. 66, nr 4
419--425
EN
Responses of positive standard and fractional continuous-time and discrete-time linear systems with derivatives of their inputs are presented herein. It is shown that the formulae for state vectors and outputs are also valid for their derivatives if the inputs and outputs and their derivatives of suitable order are zero for t = 0. Similar results are also shown for positive standard and fractional discrete-time linear systems.
11
Content available Reduced-order perfect nonlinear observers of fractional descriptor discrete-time nonlinear systems
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2017
|
Vol. 27, no. 2
245--251
EN
The purpose of this work is to propose and characterize fractional descriptor reduced-order perfect nonlinear observers for a class of fractional descriptor discrete-time nonlinear systems. Sufficient conditions for the existence of these observers are established. The design procedure of the observers is given and demonstrated on a numerical example.
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
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2017
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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 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.
16
Content available Fractional descriptor standard and positive discrete-time nonlinear systems
Kaczorek T.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2015
|
Vol. 63, nr 3
651--655
EN
A method of analysis of the fractional descriptor nonlinear discrete-time systems with regular pencils of linear part is proposed. The method is based on the Weierstrass-Kronecker decomposition of the pencils. Necessary and sufficient conditions for the positivity of the nonlinear systems are established. A procedure for computing the solution to the equations describing the nonlinear systems are proposed and demonstrated on a numerical example.
17
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.
18
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.
19
Content available Descriptor fractional linear systems with regular pencils
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2013
|
Vol. 23, no. 2
309--315
EN
Methods for finding solutions of the state equations of descriptor fractional discrete-time and continuous-time linear systems with regular pencils are proposed. The derivation of the solution formulas is based on the application of the Z transform, the Laplace transform and the convolution theorems. Procedures for computation of the transition matrices are proposed. The efficiency of the proposed methods is demonstrated on simple numerical examples.
20
Content available Approximation of fractional positive stable continuous-time linear systems by fractional positive stable discrete-time systems
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2013
|
Vol. 23, no. 3
501--506
EN
Fractional positive asymptotically stable continuous-time linear systems are approximated by fractional positive asymptotically stable discrete-time systems using a linear Padé-type approximation. It is shown that the approximation preserves the positivity and asymptotic stability of the systems. An optional system approximation is also discussed.
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