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1
Content available remote On spectrum of Metzler matrices
Domka Michał, Mitkowski Wojciech
Przegląd Elektrotechniczny
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2022
|
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 Global stability of discrete-time feedback nonlinear systems with descriptor positive linear parts and interval state matrices
Kaczorek Tadeusz, Ruszewski Andrzej
International Journal of Applied Mathematics and Computer Science
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2022
|
Vol. 32, no. 1
5--10
EN
The global stability of discrete-time nonlinear systems with descriptor positive linear parts, positive scalar feedbacks and interval state matrices is addressed. Sufficient conditions for the global stability of this class of nonlinear systems are established. The effectiveness of these conditions is illustrated using numerical examples.
3
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.
4
Content available Stability of interval positive fractional discrete-time linear systems
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2018
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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.
5
Content available Robust controlled positive delayed systems with interval parameter uncertainties: A delay uniform decomposition approach
Elloumi W., Mehdi D., Chaabane M.
International Journal of Applied Mathematics and Computer Science
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2018
|
Vol. 28, no. 3
441--450
EN
This paper is concerned with robust stabilization of continuous linear positive time-delay systems with parametric uncertainties. The delay considered in this work is a bounded time-varying function. Previously, we have demonstrated that the equidistant delay-decomposition technique is less conservative when it is applied to linear positive time-delay systems. Thus, we use simply a delay bi-decomposition in an appropriate Lyapunov–Krasovskii functional. By using classical and partitioned control gains, the state-feedback controllers developed in our work are formulated in terms of linear matrix inequalities. The efficiency of the proposed robust control laws is illustrated with via an example.
6
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.
7
Content available Decentralized stabilization of fractional positive descriptor continuous-time linear systems
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2018
|
Vol. 28, no. 1
135--140
EN
A method for decentralized stabilization of fractional positive descriptor linear systems is proposed. Necessary and sufficient conditions for decentralized stabilization of fractional positive descriptor linear systems are established. The efficiency of the proposed method is demonstrated on a numerical example.
8
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.
9
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
10
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
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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.
11
Content available Positivity of a class of fractional descriptor continuous-time nonlinear systems
Kaczorek T.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2015
|
Vol. 63, nr 2
561--565
EN
The positivity of a class of fractional descriptor continuous-time nonlinear systems is addressed by the use of the Weierstrass- Kronecker decomposition of the pencil of linear part of nonlinear system. Sufficient conditions for the positivity are established and illustrated by an example of fractional continuous-time descriptor nonlinear systems.
12
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
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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.
13
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.
14
Content available Minimum energy control of positive continuous-time linear systems with bounded inputs
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2013
|
Vol. 23, no. 4
725--730
EN
The minimum energy control problem for 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.
15
Content available Positive realizations on time scales
Bartosiewicz Z.
Control and Cybernetics
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2013
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Vol. 42, no. 2
315--327
EN
The problem of realization of a linear input-output map as a positive linear system on a time scale is studied. To state the criteria of existence of realization, modified Markov parameters corresponding to the input-output map are introduced. It is necessary for the existence of a positive realization that the modified Markov parameters be nonnegative. A necessary and sufficient condition for realizability is expressed in the language of positive cones in an infinite dimensional space. The sequence of modified Markov parameters generates one of the cones that appear in the criterion of realizability.
16
Content available On positive reachability of time-variant linear systems on time scales
Bartosiewicz Z.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2013
|
Vol. 61, nr 4
905--910
EN
Positive reachability of time-variant linear positive systems on arbitrary time scales is studied. It is shown that the system is positively reachable if and only if a modified Gram matrix corresponding to the system is monomial. The general criterion is then specified for particular cases of continuous-time systems and various classes of discrete-time systems. It is shown that in the case of continuous-time systems with analytic coefficients the conditions for positive reachability are very restrictive, similarly as for time-invariant systems.
17
Content available Minimum energy control of fractional positive continuous-time linear systems
Kaczorek T.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2013
|
Vol. 61, nr 4
803--807
EN
The minimum energy control problem for the fractional positive continuous-time linear systems is formulated and solved. Sufficient conditions for the existence of solution to the problem are established. A procedure for solving of the problem is proposed and illustrated by a numerical example.
18
Content available Comparison of approximation methods of positive stable continuous-time linear systems by positive stable discrete-time systems
Kaczorek T.
Archives of Electrical Engineering
|
2013
|
Vol. 62, nr 2
345--355
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.
19
Content available Improved stability and robust stability conditions for a general model of scalar continuous-discrete linear systems
Busłowicz M.
Pomiary Automatyka Kontrola
|
2011
|
R. 57, nr 2
188-189
EN
The paper improves the main result of the previous authors paper [1]. First it is shown that the conditions for asymptotic stability and for robust stability of a general model of scalar continuous-discrete linear systems given in this paper are only necessary. Next, the necessary and sufficient conditions are established. The conditions are expressed in terms of coefficients of the model.
PL
W pracy podano poprawione warunki stabilności oraz odpornej stabilności modelu ogólnego (1) skalarnych liniowych układów ciągło-dyskretnych, standardowych oraz dodatnich. Pokazano, że podane w pracy [1] warunki są tylko konieczne. Bazują one bowiem na warunku stabilności (7), który jest słuszny dla klasy (5) wielomianów dwóch zmiennych niezależnych. Wielomian charakterystyczny (4) rozpatrywanego układu nie należy do klasy (5), ale do klasy (8) wielomianów. Wobec tego do badania stabilności modelu (1) należy wykorzystać warunki (7) i (9), które są konieczne i wystarczające dla asymptotycznej stabilności klasy (8) wielomianów. Bazując na tych warunkach w twierdzeniu 1 sformułowano kryterium asymptotycznej stabilności analizowanej klasy układów. Warunki asymptotycznej stabilności oraz odpornej stabilności standardowego układu ciągło-dyskretnego podano w twierdzeniu 2 oraz w twierdzeniu 4, odpowiednio. Natomiast warunki asymptotycznej stabilności oraz odpornej stabilności dodatniego układu ciągło-dyskretnego podano w twierdzeniach 3 i 5. Wszystkie warunki są wyrażone w terminach współczynników modelu (1) (lub wartości krańcowych przedziałów (18), z których te współczynniki mogą przyjmować swoje wartości).
20
Content available Positivity and stability of fractional 2D Lyapunov systems described by the Roesser model
Rogowski K.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2011
|
Vol. 59, nr 2
195-200
EN
A new class of fractional 2D Lyapunov systems described by the Roesser models is introduced. Necessary and sufficient conditions for the positivity and asymptotic stability of the new class of systems are established. It is shown that the checking of the asymptotic stability of positive 2D fractional Lyapunov systems can be reduced to testing the asymptotic stability of corresponding positive standard 1D discretetime systems. The considerations are illustrated by a numerical example.
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