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1
Content available Transfer matrices with positive coefficients of descriptor linear systems
Kaczorek Tadeusz, Sajewski Łukasz
Journal of Automation, Electronics and Electrical Engineering
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2024
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Vol. 6, No. 1
7--17
EN
Transfer matrices with positive coefficients of descriptor positive linear continuous-time systems are addressed. Two methods of checking of the positivity of descriptor linear systems are proposed. It is shown that if the positive descriptor system is asymptotically stable then all coefficients of its transfer matrix are positive.
2
Content available The positivity of the fractional order model of a two-dimensional temperature field
Oprzędkiewicz Krzysztof
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2023
|
Vol. 71, nr 4
art. no. e145675
EN
The paper presents analysis of the positivity for a two-dimensional temperature field. The process under consideration is described by the linear, infinite-dimensional, noninteger order state equation. It is derived from a two-dimensional parabolic equation with homogenous Neumann boundary conditions along all borders and homogenous initial condition. The form of control and observation operators is determined by the construction of a real system. The internal and external positivity of the model are associated to the localization of heater and measurement. It has been proven that the internal positivity of the considered system can be achieved by the proper selection of attachment of a heater and place of a measurement as well as the dimension of the finite-dimensional approximation of the considered model. Conditions of the internal positivity associated with construction of real experimental system are proposed. The postivity is analysed separately for control and output of the system. This allows one to analyse the positivity of thermal systems without explicit control. Theoretical considerations are numerically verified with the use of experimental data. The proposed results can be applied i.e. to point suitable places for measuring of a temperature using a thermal imaging camera.
3
Content available remote Unconditionally positive NSFD and classical finite difference schemes for biofilm formation on medical implant using Allen-Cahn equation
Tijani Yusuf O., Appadu Appanah Rao
Demonstratio Mathematica
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2022
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Vol. 55, nr 1
40--60
EN
The study of biofilm formation is becoming increasingly important. Microbes that produce biofilms have complicated impact on medical implants. In this paper, we construct an unconditionally positive non-standard finite difference scheme for a mathematical model of biofilm formation on a medical implant. The unknowns in many applications reflect values that cannot be negative, such as chemical component concentrations or population numbers. The model employed here uses the bistable Allen-Cahn partial differential equation, which is a generalization of Fisher’s equation. We study consistency and convergence of the scheme constructed. We compare the performance of our scheme with a classical finite difference scheme using four numerical experiments. The technique used in the construction of unconditionally positive method in this study can be applied to other areas of mathematical biology and sciences. The results here elaborate the benefits of the non-standard approximations over the classical approximations in practical applications.
4
Content available Positivity of fractional descriptor linear discrete-time systems
Kaczorek Tadeusz
International Journal of Applied Mathematics and Computer Science
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2019
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Vol. 29, no. 2
305--310
EN
The positivity of fractional descriptor linear discrete-time systems is investigated. The solution to the state equation of the systems is derived. Necessary and sufficient conditions for the positivity of fractional descriptor linear discrete-time systems are established. The discussion is illustrated with numerical examples.
5
Content available Positivity of fractional descriptor linear continuous-time systems
Kaczorek T.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2019
|
Vol. 67, nr 2
229--233
EN
The positivity of fractional descriptor linear continuous-time systems is investigated. The solution to the state equation of the systems is derived. Necessary and sufficient conditions for the positivity of fractional descriptor linear continuous-time systems are established. The considerations are illustrated by numerical examples.
6
Content available Analysis of the positivity and stability of linear electrical circuits with state-feedbacks
Kaczorek T.
Poznan University of Technology Academic Journals. Electrical Engineering
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2018
|
No. 93
21--36
EN
The analysis of the positivity and stability of linear electrical circuits by the use of state-feedbacks is addressed. Generalized Frobenius matrices are proposed and their properties are investigated. It is shown that if the state matrix of electrical circuit has generalized Frobenius form then the closed-loop system matrix is not positive and asymptotically stable. Different cases of modification of the positivity and stability of linear electrical circuits by state-feedbacks are discussed and necessary conditions for the existence of solutions to the problem are established.
7
Content available Nonnegativity, stability analysis of linear discrete-time positive descriptor systems: an optimization approach
Yang H., Zhou M., Zhao M., Pan P.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2018
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Vol. 66, nr 1
1--7
EN
This paper discusses an efficient approach to the analysis of positivity and stability of linear discrete-time positive descriptor system. Irs main objective is to convert the necessary and sufficient condition of characterizing positivity and stability of positive descriptor systems into an optimization problem, then propose a method to numerically check the positivity and stability of the positive linear discrete-time systems. Comparing with the other methods now available, the approach presented in this paper is less theoretical and easier to implement. Examples are provided in order to validate results.
8
Content available Positive and stable electrical circuits with state-feedbacks
Kaczorek T.
Archives of Electrical Engineering
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2018
|
Vol. 67, nr 3
563-–578
EN
The analysis of the positivity and stability of linear electrical circuits by the use of state-feedbacks is addressed. Generalized Frobenius matrices are proposed and their properties are investigated. It is shown that if the state matrix of an electrical circuit has generalized Frobenius form then the closed-loop system matrix is not positive and asymptotically stable. Different cases of modification of the positivity and stability of linear electrical circuits by state-feedbacks are discussed and necessary conditions for the existence of solutions to the problem are established.
9
Content available remote Modification of the stability and positivity of standard and descriptor linear electrical circuits by state feedbacks
Borawski K.
Przegląd Elektrotechniczny
|
2017
|
R. 93, nr 11
176--180
EN
The modification of the stability and positivity of standard and descriptor linear electrical circuits by state feedbacks is investigated. It is shown that: 1) There is a class of nonpositive and unstable R, L, e circuits that can be stabilized and modified to positive ones by state feedback; 2) There is a class of nonpositive and stable R, L, e circuits that can be modified by state feedback to positive ones without loss of stability. The modification of stability and positivity of linear descriptor electrical circuits is addressed. Considerations are illustrated by examples of linear electrical circuits.
PL
W pracy rozpatrzono problem modyfikacji stabilności i dodatniości standardowych i deskryptorowych liniowych obwodów elektrycznych poprzez sprzężenie zwrotne od wektora stanu. Pokazano, że: 1) Istnieje klasa niedodatnich i niestabilnych obwodów typu R, L, e, które mogą zostać ustabilizowane i zmodyfikowane do obwodów dodatnich; 2) Istnieje klasa niedodatnich i stabilnych obwodów typu R, L, e, które mogą zostać zmodyfikowane do obwodów dodatnich bez utraty stabilności. Rozpatrywany problem uogólniono dla klasy układów deskryptorowych. Rozważania zilustrowano przykładami obwodów elektrycznych.
10
Content available Positivity and stability of fractional discrete-time linear systems. The model without a time shift in the difference
Kaczorek T., Ostalczyk P.
Control and Cybernetics
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2017
|
Vol. 46, no. 1
27--36
EN
The positivity and stability of fractional discretetime linear systems described by a new model are addressed. Necessary and sufficient conditions for the positivity and asymptotic stability of the systems are established. New tests for checking of the stability are proposed.
11
Content available Analysis of positive and stable fractional continuous-time linear systems by the use of Caputo Fabrizio derivative
Kaczorek T.
Control and Cybernetics
|
2016
|
Vol. 45, no. 3
289--299
EN
Using the Caputo-Fabrizio definition of fractional order derivative, the positivity and asymptotic stability of the fractional continuous-time linear systems are investigated. The solution to the matrix fractional differential state equations is derived. Necessary and sufficient conditions for the positivity and asymptotic stability of the fractional linear systems are established. Tests for checking of the asymptotic stability of the systems are provided.
12
Content available Positivity and stability of time-varying fractional discrete-time linear systems
Kaczorek T., Borawski K.
Measurement Automation Monitoring
|
2015
|
Vol. 61, No. 3
84--87
EN
Necessary and sufficient conditions for the positivity of time-varying fractional discrete-time linear systems are established. The problem of asymptotic stability of the positive time-varying fractional discrete-time linear systems is analyzed and sufficient conditions are given. Considerations are illustrated by numerical examples.
13
Content available Fractional descriptor standard and positive discrete-time nonlinear systems
Kaczorek T.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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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.
14
Content available Descriptor standard and positive discrete-time nonlinear systems
Kaczorek T.
Archives of Control Sciences
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2015
|
Vol. 25, no. 2
227--235
EN
A method of analysis of 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 numerical examples.
15
Positivity and linearization of a class of nonlinear discrete-time systems by state-feedbacks
Kaczorek T.
Logistyka
|
2014
|
nr 6
5078-5083
EN
The positivity and linearization of a class of nonlinear discrete-time system by nonlinear state-feedbacks are addressed. Necessary and sufficient conditions for the positivity of the class of nonlinear systems are established. A method for linearization of the nonlinear systems by nonlinear state-feedbacks is presented. It is shown that by suitable choice of state-feedbacks it is possible to obtain asymptotically stable and controllable linear system.
PL
Praca jest poświęcona dodatniości i linearyzacji nieliniowych układów dyskretnych za pomocą nieliniowych sprzężeń zwrotnych od wektora stanu. Sformułowano warunki konieczne i wystarczające dodatniości tej klasy układów nieliniowych. Podano metodę linearyzacji tych układów za pomocą nieliniowych sprzężeń zwrotnych od wektora stanu. Wykazano, że przez odpowiedni dobór nieliniowych sprzężeń zwrotnych można uzyskać układ liniowy sterowalny i stabilny asymptotycznie.
16
Content available Positivity and asymptotic stability of descriptor linear systems with regular pencils
Kaczorek T.
Archives of Control Sciences
|
2014
|
Vol. 24, no. 2
193--205
EN
The positivity and asymptotic stability of the descriptor linear continuous-time and discrete-time systems with regular pencils are addressed. Necessary and sufficient conditions for the positivity and asymptotic stability of the systems are established using the Drazin inverse matrix approach. Effectiveness of the conditions are demonstrated on numerical examples.
17
Content available Design of regular positive and stable descriptor systems via state-feedbacks for descriptor continuous-time linear systems with singular pencils
Kaczorek T.
Archives of Control Sciences
|
2014
|
Vol. 24, no. 3
289--297
EN
A new method is proposed of design of regular positive and asymptotically stable descriptor systems by the use of state-feedbacks for descriptor continuous-time linear systems with singular pencils. The method is based on the reduction of the descriptor system by elementary row and column operations to special form. A procedure for the design of the state-feedbacks gain matrix is presented and illustrated by a numerical example.
18
Content available remote Constructability and observability of standard and positive electrical circuits
Kaczorek T.
Przegląd Elektrotechniczny
|
2013
|
R. 89, nr 7
132--136
EN
The constructability and observability of standard and positive electrical circuits composed of resistors, coils, condensators and voltage (current) sources are addressed. Necessary and sufficient conditions for the constructability and observability of the positive electrical circuits are established. Effectiveness of the conditions is demonstrated on examples of positive electrical circuits.
PL
W pracy rozpatruje się standardowe i dodatnie obwody elektryczne złożone z rezystorów, cewek i kondensatorów oraz źródeł napięcia i prądu. Podano warunki konieczne i wystarczające odtwarzalności i obserwowalności tych obwodów elektrycznych. Warunki te zostały zlustrowane na przykładach dodatnich obwodów elektrycznych.
19
Content available remote Positive unstable electrical circuits
Kaczorek T.
Przegląd Elektrotechniczny
|
2012
|
R. 88, nr 5a
187-192
EN
The instability for the positive linear electrical circuits composed of resistors, coils, condensators and voltage (current) sources are addressed. Three different classes of the positive unstable linear electrical circuits are proposed and analyzed. It is shown that positive electrical circuits are unstable for all values of their parameters if the electrical circuit has at least one mesh containing only inductances and source voltages.
PL
W artykule rozpatrywane są niestabilne dodatnie liniowe obwody elektryczne złożone z rezystorów, cewek, kondensatorów i źródeł napięcia lub prądu. Analizowane są trzy różne klasy dodatnich obwodów elektrycznych, które są niestabilne dla wszystkich wartości swoich parametrów tzn. rezystancji, indukcyjności i pojemności. Wykazano, że dodatnie obwody elektryczne są niestabilne dla wszystkich wartości swoich parametrów jeżeli zawierają one przynajmniej jedno oczko złożone tylko z cewek i źródeł napięcia.
20
Stability of descriptor positive linear systems
Kaczorek T.
Logistyka
|
2012
|
nr 3
EN
The purpose of the paper is to propose a new method for checking the positivity and asymptotic stability of descriptor continuous-time and discrete-time linear systems. The proposed method is based on the elementary operations applied to descriptor linear systems with regular pencils. The descriptor systems with regular pencils are transformed to equivalent standard systems which are used for checking of the asymptotic stability of descriptor systems.
PL
W pracy podano warunki konieczne i wystarczające dodatniości i stabilności asymptotycznej układów singularnych o pękach regularnych. Zaproponowano nową metodę sprawdzania dodatniości i asymptotycznej stabilności singularnych ciągłych i dyskretnych układów liniowych o pękach regularnych. Proponowana metoda jest oparta o działaniach elementarnych na pękach tych układów. W metodzie tej układy syngularne są redukowane do równoważnych postaci standardowych, które są wykorzystywane następnie do sprawdzania stabilności asymptotycznej tych układów singularnych.
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