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1
Content available Necessary and Sufficient Stability Conditions of Fractional Positive Continuous-Time Linear Systems
100%
Kaczorek T.
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tom Vol. 5, no. 2
52-54
EN
Necessary and sufficient conditions for the asymptotic stability of fractional positive continuous-time linear systems are established. It is shown that the matrix A of the stable fractional positive system has not eigenvalues in the part of stability region located in the right half of the complex plane.
2
Content available Positivity and Reachability of Fractional Electrical Circuits
100%
Kaczorek T.
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tom Vol. 5, no. 2
42-51
EN
Conditions for the positivity of fractional linear electrical circuits composed of resistors, coils, condensators and voltage (current) sources are established. It is shown that: 1) the fractional electrical circuit composed of resistors, coils and voltage source is positive for any values of their resistances, inductances and source voltages if and only if the number of coils is less or equal to the number of its linearly independent meshes, 2) the fractional electrical circuit is not positive for any values of its resistances, capacitances and source voltages if each its branch contains resistor, capacitor and voltage source, It is also shown that the fractional positive electrical circuits of R, C, e type are reachable if and only if the conductances between their nodes are zero and the fractional positive electrical circuits of R, L, e type are reachable if and only if the resistances belonging to two meshes are zero.
3
Content available Stability of fractional positive nonlinear systems
100%
Kaczorek T.
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tom Vol. 25, no. 4
491--496
EN
The conditions for positivity and stability of a class of fractional nonlinear continuous-time systems are established. It is assumed that the nonlinear vector function is continuous, satisfies the Lipschitz condition and the linear part is described by a Metzler matrix. The stability conditions are established by the use of an extension of the Lyapunov method to fractional positive nonlinear systems.
4
Content available Comparison of approximation methods of positive stable continuous-time linear systems by positive stable discrete-time systems
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Kaczorek T.
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tom Vol. 11
1--10
EN
The positive asymptotically stable continuous-time linear systems are approximated by corresponding asymptotically stable discrete-time linear systems. Two methods of the approximation are presented and the comparison of the methods is addressed. The considerations are illustrated by three numerical examples and an example of positive electrical circuit.
5
Content available Responses of positive standard and fractional linear systems and electrical circuits with derivatives of their inputs
100%
Kaczorek T.
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tom 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.
6
Content available Positive realizations with reduced numbers of delays for 2D continuous-discrete linear systems
100%
Kaczorek T.
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tom Vol. 60, nr 4
835-840
EN
A new method is proposed for determination of positive realizations with reduced numbers of delays of linear 2D continuousdiscrete systems. Sufficient conditions for the existence of the positive realizations of a given proper transfer function are established. It is shown that there exists a positive realization with reduced numbers of delays if there exists a positive realization without delays but with a greater dimension. The proposed method is demonstrated on a numerical example.
7
Content available Positive stable realizations of discrete-time linear systems
100%
Kaczorek T.
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tom Vol. 60, nr 3
605-616
EN
The problem of existence and determination of the set of positive asymptotically stable realizations of a proper transfer function of linear discrete-time systems is formulated and solved. Necessary and sufficient conditions for existence of the set of the realizations are established. A procedure for computation of the set of realizations are proposed and illustrated by numerical examples.
8
Content available Positive time-varying continuous-time linear systems and electrical circuits
100%
Kaczorek T.
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tom Vol. 63, nr 4
837--842
EN
The positivity of time-varying continuous-time linear systems and electrical circuits are addressed. Necessary and sufficient conditions for the positivity of the systems and electrical circuits are established. It is shown that there exists a large class of positive electrical circuits with time-varying parameters. Examples of positive electrical circuits are presented.
9
Content available Electrical circuits with state-feedbacks and desired dynamical properties
100%
Kaczorek T.
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tom R. 56, nr 5
371-376
EN
Linear electrical circuits composed of resistors, inductances, capacitances and voltage (current) sources with state-feedbacks are addressed. It is shown that for large class of nonpositive electrical circuits it is possible to choose gain matrices of the state-feedbacks so that the closed-loop systems are positive and have some desired dynamical properties. Sufficient conditions for nonnegativity of B matrices of linear electrical circuits are established. Considerations are illustrated by three examples of linear electrical circuits.
PL
W pracy są rozpatrywane liniowe obwody elektryczne złożone z rezystancji, pojemności, indukcyjności i źródeł napięcia (prądu). Wykazano, że dla szerokiej klasy niedodatnich obwodów elektrycznych można dobrać macierz wzmocnień statycznych sprzężeń zwrotnych od wektora stanu tak, aby układ zamknięty był dodatni i miał pożądane właściwości dynamiczne. Podano warunki wystarczające nieujemności elementów macierzy B liniowych obwodów elektrycznych. Rozważania ogólne zostały zilustrowane przykładami obwodów elektrycznych.
10
Content available remote Positivity and stability of discrete-time and continuous-time nonlinear systems
100%
Kaczorek T.
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tom R. 91, nr 8
127--130
PL
Przedstawione zostaną dodatnie i stabilne asymptotycznie nieliniowe układy dyskretne i ciągłe. Podane zostaną warunki wystarczające dodatniości i stabilności asymptotycznej układów nieliniowych. Proponowane metody badania stabilności zostaną oparte na uogólnieniu metody Lyapunova. Efektywność testów zostanie zademonstrowana na przykładach numerycznych.
EN
The positivity and asymptotic stability of the discrete-time and continuous-time nonlinear systems are addressed. Sufficient conditions for the positivity and asymptotic stability of the nonlinear systems are established. The proposed stability tests are based on an extension of the Lyapunov method to the positive nonlinear systems. The effectiveness of the tests are demonstrated on examples.
11
Content available Stabilization of positive linear systems by state-feedbacks
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Kaczorek T.
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tom R. 45, nr 3 (z. 1)
EN
A method based on Gersgorinás theorem of stabilization of postive linear continuous-time and discrete-time systems by state feedbacks is presented. It is shown that the stabilization problems can be reducted to suitable quadratic programming problems with constraints. The method is illustrated by two numerical examples.
PL
Przedstawiono metodę stabilizacji dodatnich, liniowych układółw ciągłych i dyskretnych, za pomocą sprzężeń zwrotnych od wektora stanu, opartą na twierdzeniu Gersgorina. Wykazano, że problemy stabilizacji mogą być sprowadzone do odpowiednich problemów programowania kwadratowego z ograniczeniami liniowymi. Metodę zilustrowano dwoma przykładami liczbowymi.
12
Content available remote Similarity transformation of matrices to one common canonical form and its applications to 2D linear systems
100%
Kaczorek T.
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2010
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tom Vol. 20, no 3
507-512
EN
The notion of a common canonical form for a sequence of square matrices is introduced. Necessary and sufficient conditions for the existence of a similarity transformation reducing the sequence of matrices to the common canonical form are established. It is shown that (i) using a suitable state vector linear transformation it is possible to decompose a linear 2D system into two linear 2D subsystems such that the dynamics of the second subsystem are independent of those of the first one, (ii) the reduced 2D system is positive if and only if the linear transformation matrix is monomial. Necessary and sufficient conditions are established for the existence of a gain matrix such that the matrices of the closed-loop system can be reduced to the common canonical form.
13
Content available Global stability of nonlinear feedback systemswith positive descriptor linear parts
100%
Kaczorek T.
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tom Vol. 67, nr 1
45--51
EN
The global (absolute) stability of nonlinear systems with negative feedbacks and positive descriptor linear parts is addressed. Transfer matrices of positive descriptor linear systems are analyzed. The characteristics u = f (e) of the nonlinear parts satisfy the condition k1e ≤ f (e) ≤ k2e for some positive k1, k2. It is shown that the nonlinear feedback systems are globally asymptotically stable if the Nyquist plots of the positive descriptor linear parts are located in the right-hand side of the circles (– 1/k1, – 1/k2).
14
Content available Nilpotency indices of state matrices and linear dependence on time of state variable of electrical circuits with state feedbacks
100%
Kaczorek T.
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tom Vol. 59, nr 1-2
21-34
EN
Electrical circuits with state-feedbacks are addressed. It is shown that by suitable choice of the gain matrices of state-feedbacks it is possible to obtain the closed-loop system matrices with nilpotency indices equal to two and their state variables are linear functions of time. The considerations are illustrated by linear electrical circuits.
15
Positivity and stability of time-varying linear electrical circuits
100%
Kaczorek T.
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tom nr 3
2060--2071, CD 1
EN
The positivity and asymptotic stability of time-varying linear electrical circuits are addressed. Necessary and sufficient conditions for the positivity of the time-varying linear systems and electrical circuits are established. Using the Lyapunov method the asymptotic stability of a large class of positive electrical circuits is shown. Examples of positive and asymptotically stable electrical circuits are presented.
PL
Praca poświęcona jest badaniu dodatniości i stabilności asymptotycznej liniowych układów i obwodów elektrycznych o zmiennych w czasie parametrach. Podano warunki konieczne i wystarczające dodatniości liniowych układów i obwodów elektrycznych zmiennych w czasie parametrach. Pokazano, że istnieje obszerna klasa liniowych obwodów elektrycznych dodatnich o zmiennych w czasie parametrach. Podano również warunki dostateczne oraz proste kryteria badania stabilności tej klasy liniowych układów i obwodów elektrycznych. Rozważania zilustrowano przykładami dodatnich stabilnych asymptotycznie liniowych obwodów elektrycznych o zmiennych w czasie parametrach.
16
Content available Singular Fractional Continuous-Time and Discrete-Time Linear Systems
100%
Kaczorek T.
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tom Vol. 7, no. 1
26-33
EN
New classes of singular fractional continuous-time and discrete-time linear systems are introduced. Electrical circuits are example of singular fractional continuous-time systems. Using the Caputo definition of the fractional derivative, the Weierstrass regular pencil decomposition and 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 systems if it contains at least one mesh consisting of branches with only ideal supercondensators and voltage sources or at least one node with branches with supercoils. Using the Weierstrass regular pencil decomposition the solution to the state equation of singular fractional discrete-time linear systems is derived. The considerations are illustrated by numerical examples.
17
Standard and positive electrical circuits with zero transfer matrices
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Kaczorek T.
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tom No. 85
11--28
EN
Standard and positive electrical circuits with zero transfer matrices are addressed. It is shown that there exists a large class of electrical circuits composed of resistances, inductances, capacitances and voltage (current) sources with zero transfer matrices. The electrical circuits are unreachable, unobservable and unstable for all values of the resistances, inductances and capacitances. An extension of these considerations to fractional electrical circuits is given.
18
Content available remote Characteristic polynomials of positive and minimal-phase electrical circuits
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Kaczorek T.
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tom R. 92, nr 6
79--85
EN
Characteristic polynomials of positive and minimal-phase electrical circuits are addressed. It is shown that the characteristic polynomials of the electrical circuits are independent of the choice of their reference mesh and of their reference node. Conditions are established under which the positive electrical circuits have real eigenvalues and are minimal-phase linear systems. Sufficient conditions for cancelation of zeros and poles of minimal-phase electrical circuits are given.
PL
W pracy wykazano, że wielomiany charakterystyczne obwodów elektrycznych są niezależne od wyboru oczka odniesienia w metodzie oczkowej i węzła odniesienia w metodzie węzłowej analizy tych obwodów. Podano warunki przy spełnieniu których dodatnie obwody elektryczne mają rzeczywiste wartości własne i są minimalnofazowymi obwodami elektrycznymi. Podano również warunki wystarczające skracania zer i biegunów w minimalnofazowych obwodach elektrycznych. Rozważania zostały zilustrowane przykładami dodatnich i minimalnofazowych obwodów elektrycznych.
19
Content available remote Singular fractional linear systems and electrical circuits
100%
Kaczorek T.
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tom 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.
20
Content available remote Pointwise completeness and pointwise degeneracy of standard and positive fractional linear systems with state-feedbacks
100%
Kaczorek T.
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tom Vol. 19, no. 3
295-306
EN
The pointwise completeness and pointwise degeneracy of standard and positive fractional linear discrete-time and continuous-time systems with state-feedbacks are addressed. It is shown that the pointwise completeness and pointwise degeneracy of the fractional positive continuous-time systems are invariant under the state and output feedbacks. Necessary and sufficient conditions are established for the existence of gain matrices of state-feedbacks for standard and positive linear systems such that the closed-loop systems are pointwise complete. Considerations are illustrated by numerical examples.
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