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1
Content available remote On spectrum of Metzler matrices
Domka Michał, Mitkowski Wojciech
Przegląd Elektrotechniczny
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2022
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R. 98, nr 12
121--123
EN
In the paper it was proven that spectrum of Metzler matrix must belong to a certain cone in the complex plane. The result is derived from the analytical characterization of spectra of positive matrices obtained by Karpelevich. Furthermore, it was shown that in case of 3x3 matrices this property yields also a sufficient condition that a set of numbers must satisfy in order to be spectrum of some Metzler matrix.
PL
W pracy wykazano, że widmo macierzy każdej Metzlera należy do pewnego stożka na płaszczyźnie zespolonej. Wykorzystano w tym celu analityczną charakteryzację widma macierzy dodatniej wyznaczoną przez Karpielewicza. Ponadto wykazano, że w przypadku macierzy 3x3 własność ta pozwala wyznaczyć również warunek wystarczający, który spełniać musi zbiór liczb zespolonych, aby był widmem pewnej macierzy Metzlera.
2
Content available Positivity and cyclicity of descriptor electrical circuits with chain structure
Kaczorek Tadeusz, Borawski Kamil
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2022
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Vol. 70, nr 1
art. no. e139955
EN
The positivity and cyclicity of descriptor linear electrical circuits with chain structure is considered. Two classes of descriptor linear electrical circuits are analyzed. Some new properties of these classes of electrical circuits are established. The results are extended to fractional descriptor linear electrical circuits.
3
Content available Positive electrical circuits with the chain structure and cyclic Metzler state matrices
Kaczorek Tadeusz
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2021
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Vol. 69, nr 4
art. no. e137731
EN
The cyclicity of the state matrices of positive linear electrical circuits with the chain structure is considered. Two classes of positive linear electrical circuits with the chain structure and cyclic Metzler state matrices are analyzed. Some new properties of these classes of positive electrical circuits are established. The results are extended to fractional linear electrical circuits.
4
Content available On controllability of fractional positive continuous-time linear systems with delay
Sikora Beata, Matlok Natalia
Archives of Control Sciences
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2021
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Vol. 31, No. 1
29--51
EN
In the paper positive fractional continuous-time linear systems are considered. Positive fractional systems without delays and positive fractional systems with a single delay in control are studied. New criteria for approximate and exact controllability of systems without delays as well as a relative controllability criterion of systems with delay are established and proved. Numerical examples are presented for different controllability criteria. A practical application is proposed.
5
Content available Realizacje dodatnie stabilne liniowych układów ciągłych niecałkowitego rzędu z macierzą systemową symetryczną Metzlera
Kaczorek T., Borawski K.
Pomiary Automatyka Kontrola
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2014
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R. 60, nr 10
822--825
PL
Podano warunki dodatniości i stabilności liniowych układów ciągłych niecałkowitego rzędu. Sformułowano problem realizacji dodatnich stabilnych liniowych układów ciągłych niecałkowitego rzędu z macierzą systemową symetryczną Metzlera. Zaproponowano metodę sprowadzania macierzy stanu w postaci kanonicznej Frobeniusa do postaci symetrycznej stabilnej Metzlera. Metodę zobrazowano przykładem numerycznym.
EN
A dynamical system is called a fractional-order system if its state equations are given by fractional-order derivative of the state vector. Using that theory, more precise mathematical models of systems can be obtained. A dynamical system is called positive if its all inputs, outputs, state variables and initial conditions are nonnegative. Variety of models having positive behavior can be found in engineering, biology, economics etc. Conditions for positivity and stability of linear continuous-time fractional-order systems are presented in the paper. A positive stable realization problem for linear continuous-time fractional-order systems with symmetric system Metzler matrix is formulated. The method for finding the realization is given. The problem is solved and conditions for the existence of the realization are established. The paper is organized as follows. In Section 2 the conditions for internal positivity and stability of linear continuous-time fractional-order systems are given. This section also contains the formulation of the positive stable realization problem for linear continuous-time fractional-order systems with symmetric system Metzler matrix. In Section 3 the procedure for computation of the realization is given. An example illustrating the method proposed is presented in Section 4. Section 5 contains the concluding remarks.
6
Content available Determination of the set of Metzler matrices for given stable polynomials
Kaczorek T.
Pomiary Automatyka Kontrola
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2012
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R. 58, nr 5
407-412
EN
The problem of determination of the set of Metzler matrices for given stable polynomials is formulated and partly solved. For stable polynomial of the second degree there exists a set of Metzler matrices if and only if the polynomial has only real negatives zeros. If the stable polynomial has only real negative zeros then the set of corresponding Metzler matrices is given by the set of lower or upper triangular matrices with diagonal entries equal to the negative real zeros and any nonnegative off-diagonal entries. Sufficient condition are establish for the existence of the set of Metzler matrices for stable polynomials with a real negative zeros and the complex conjugate zeros).
PL
W artykule sformułowani i częściowo rozwiązano problem wyznaczania zbioru macierzy Metzlera dla danych stabilnych wielomianów. Wykazano, ze dla stabilnych wielomianów stopnia drugiego istnieje zbiór macierzy Metzlera wtedy i tylko wtedy, gdy wielomian ten ma tylko ujemne pierwiastki rzeczywiste. Jeżeli stabilny wielomian dowolnego stopnia ma tylko pierwiastki rzeczywiste, to odpowiadający jemu zbiór macierzy Metzlera jest dany zbiorem macierzy dolno lub górno-trójkątnych z elementami na głównej przekątnej równych ujemnym zerom tego wielomianu oraz nieujemnymi elementami poza główną przekątną. Warunkiem koniecznym na to, aby dla danego stabilnego wielomianu istniał zbiór macierzy Metzlera jest posiadanie przez ten wielomian co najmniej dwóch zer przeczystych. Podano warunki dostateczne na istnienie zbioru macierzy Metzlera dla danych stabilnych wielomianów z ujemnymi zerami rzeczywistymi i zespolonymi parami sprzężonymi.
7
Content available Existence and determination of the set of Metzler matrices for given stable polynomials
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2012
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Vol. 22, no. 2
389-399
EN
The problem of the existence and determination of the set of Metzler matrices for given stable polynomials is formulated and solved. Necessary and sufficient conditions are established for the existence of the set of Metzler matrices for given stable polynomials. A procedure for finding the set of Metzler matrices for given stable polynomials is proposed and illustrated with numerical examples.
8
Content available Selected classes of positive electrical circuits and their reachability
Kaczorek T.
Computer Applications in Electrical Engineering
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2012
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Vol. 10
1--28
EN
Conditions for the positivity of linear electrical circuits composed of resistors, coils, condensators and voltage (current) sources are established. It is shown that: 1) the electrical circuit composed of resistors, coils and voltage source is positive for almost all 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 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, 3) the positive n-meshes electrical circuit with only one inductance in each linearly independent mesh is reachable if all resistances of branches belonging to two linearly independent meshes are zero 4) the electrical circuits of the structure shown on Fig.5.2 is reachable if and only if the condition (5.11) is satisfied.
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