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1
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.
2
Content available Fractional discrete-continuous model of heat transfer process
Oprzędkiewicz Krzysztof, Dziedzic Klaudia
Archives of Control Sciences
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2021
|
Vol. 31, No. 2
287--306
EN
The paper proposes a new, state space, finite dimensional, fractional order model of a heat transfer in one dimensional body. The time derivative is described by Caputo operator. The second order central difference describes the derivative along the length. The analytical formulae of the model responses are proved. The stability, convergence, and positivity of the model are also discussed. Theoretical results are verified by experiments.
3
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.
4
Content available Metzler cyclic electric systems
Mitkowski W.
Zeszyty Naukowe Akademii Marynarki Wojennej
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2017
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R. 58 nr 2 (209)
109--118
EN
In applications we meet with positive (nonnegative) systems. A system generated by a linear differential equation with a Metzler-matrix is called a nonnegative system. The paper shows the basic dynamic properties of nonnegative continuous-time systems. Considered was the example of an oscillating electrical circuit with a cyclic Metzler-matrix, enabling the correct physical interpretations.
PL
Układy dodatnie (nieujemne) można spotkać w wielu zastosowaniach. Układem nieujemnym jest układ generowany równaniem różniczkowym liniowym z macierzą Metzlera. W artykule przedstawiono podstawowe własności dynamiczne układów nieujemnych z czasem ciągłym. Rozważono przykład elektrycznego obwodu oscylacyjnego z cykliczną macierzą Metzlera umożliwiający odpowiednie interpretacje fizyczne.
5
Content available remote New stability conditions for positive continuous-discrete 2D linear systems
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2011
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Vol. 21, no 3
521-524
EN
New necessary and sufficient conditions for asymptotic stability of positive continuous-discrete 2D linear systems are established. Necessary conditions for the stability are also given. The stability tests are demonstrated on numerical examples.
6
On stabilizability-holdability problem for linear discrete time systems
Rumchev V., G., Higashiyama Y.
Systems Science
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2010
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Vol. 36, no 2
33-38
EN
Consider the following problem. Given a linear discrete-time system, find if possible a linear state-feedback control law such that under this law all system trajectories originating in the non-negative orthant remain non-negative while asymptotically converging to the origin. This problem is called feedback stabilizability-holdabiltiy problem (FSH). If, in addition, the requirement of non-negativity is imposed on the controls, the problem is a positive feedback stabilizability-holdabiltiy problem (PFSH). It is shown that the set of all linear state feedback controllers that make the open-loop system holdable and asymptotically stable is a polyhedron and the external representation of this polyhedron is obtained. Necessary and sufficient conditions for identifying when the open-loop system is not positive feedback R+n-invariant (and therefore there is no solution to the PFSH problem) are obtained in terms of the system parameters. A constructive linear programming based approach to the solution of FSH and PFSH problems is developed in the paper. This approach provides not only a simple computational procedure to find out whether the FSF, respectively the PFSH problem, has a solution or not but also to determine a linear state feedback controller (respectively, a non-negative linear state feedback controller) that endows the closed-loop (positive) system with a maximum stability margin and guarantees the fastest possible convergence to the origin.
7
Content available Względna obserwowalność dodatnich układów ciągło-dyskretnych
Trzasko W.
Pomiary Automatyka Kontrola
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2010
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R. 56, nr 5
396-399
PL
W pracy rozpatrzono problem obserwowalności dodatnich liniowych dwuwymiarowych układów ciągło-dyskretnych. Sformułowano definicje oraz podano warunki konieczne i wystarczające obserwowalności oraz względnej obserwowalności. Podano też metodę wyznaczania nieujemnego stanu początkowego dla dowolnej zadanej odpowiedzi układu. Rozważania zilustrowano przykładem.
EN
In the paper the observability problem for linear 2D positive continuous-discrete time systems described by the differential-difference state equations (formulas (1a) and (1b)) and output equation (1c) with boundary conditions (3) is considered. It should be noted that such a model is called a hybrid system in literature [1, 3, 4]. A 2D hybrid system is a dynamic system that includes both continuous- and discrete-time dynamics. It means that the 2D hybrid system state vector contains continuous-time and discrete-time state variables; its input and output vectors depend on continuous time t and discrete step i. The results presented in this work are based on solving the state equations (formula (4)) given in [1]. For simplicity it is assumed that the vector x2(t):=x2 of boundary conditions (3) is constant in the whole interval The definition and necessary and sufficient conditions for the continuous -discrete system positivity are formulated. There is also introduced the definition of observability at the point and a simple condition of observability (Theorem 3) is given. Moreover, based on the left-inverse of the observability matrix (the formula (16)), a simple method for computing the initial state (10) when assuming the knowledge of the output sequence (15) at the points is proposed. In Section 4 the relative observability with respect to state at the point is investigated. This is the special case of observability, but more useful in practice. The considerations are illustrated by a numerical example. Numerical calculations were performed in the Matlab program environment.
8
Content available remote Algorytm wyznaczania realizacji dodatniej układów dwuwymiarowych z opóźnieniami
Markowski K. A.
Pomiary Automatyka Robotyka
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2009
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R. 13, nr 2
485-495
PL
W artykule przedstawiono metodę wyznaczania realizacji dodatniej układu dwuwymiarowego z opóźnieniami opisanego za pomocą modelu ogólnego. Do wyznaczania realizacji użyto teorii wielowymiarowych grafów skierowanych oddziaływań. Zaproponowaną metodę zilustrowano prostymi przykładami numerycznymi.
EN
In this paper a method for determination positive realization of two dimensional systems described by general model with delays using multidimensional digraphs theory is presented. The proposed method is illustrated by numerical examples.
9
Content available remote Independence of asymptotic stability of positive 2D linear systems with delays of their delays
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2009
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Vol. 19, no 2
255-261
EN
It is shown that the asymptotic stability of positive 2D linear systems with delays is independent of the number and values of the delays and it depends only on the sum of the system matrices, and that the checking of the asymptotic stability of positive 2D linear systems with delays can be reduced to testing that of the corresponding positive 1D systems without delays. The effectiveness of the proposed approaches is demonstrated on numerical examples.
10
Content available remote Asymptotic stability of positive 2D linear systems with delays
Kaczorek T.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2009
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Vol. 57, nr 2
133-138
EN
New necessary and sufficient conditions for the asymptotic stability of positive 2D linear systems with delays described by the general model, Fornasini-Marchesini models and Roesser model are established. It is shown that checking of the asymptotic stability of positive 2D linear systems with delays can be reduced to the checking of the asymptotic stability of corresponding positive ID linear systems without delays. The efficiency of the new criterions is demonstrated on numerical examples.
11
Content available remote Fractional positive continuous-time linear systems and their reachability
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2008
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Vol. 18, no 2
223-228
EN
A new class of fractional linear continuous-time linear systems described by state equations is introduced. The solution to the state equations is derived using the Laplace transform. Necessary and sufficient conditions are established for the internal and external positivity of fractional systems. Sufficient conditions are given for the reachability of fractional positive systems.
12
Maintainability of positive linear discrete-time systems
James G., Rumchev V.
Systems Science
|
2007
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Vol. 33, no 3
21-30
EN
The concept of maintainability for (time-invariant) positive linear discrete-time systems (PLDS) is introduced and studied in detail. A state x(t) of a PLDS is said to be maintainable if there exists an admissible control such that x(t + 1) = x(t) for t = 0, 1, 2, ... For time-invariant systems, if a given state is maintainable it is maintainable at all times. The set of all maintainable states is called a maintainable set. Maintainability and stability are different concepts - while stability is an asymptotic ("long-term") notion, maintainability is a "short-term" concept. Moreover, stability always implies maintainability but maintainability does not necessarily imply stability. If no additional constraints are imposed on the states and controls except the standard non-negativity restrictions, the maintainable sets are polyhedral cones. Their geometry is determined completely by the structural and spectral properties of nonnegative system pair (A, B) ≥ 0. Different cases are studied in the paper and relevant numerical examples are presented. PLDS with two-side bounded controls are also discussed and an interesting result is obtained namely the maintainable set of an asymptotically stable PLDS coincides with its asymptotic reachable set.
13
Content available remote Positive partial realization problem for linear discrete-time systems
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2007
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Vol. 17, no 2
165-171
EN
A partial realization problem for positive linear discrete-time systems is addressed. Sufficient conditions for the existence of its solution are established. A procedure for the computation of a positive partial realization for a given finite sequence of the values of the impulse response is proposed. The procedure is illustrated by four numerical examples.
14
Content available remote Determination of reachability subspace of two-dimensional positive linear systems described by Roesser model
Markowski K.A.
Przegląd Elektrotechniczny
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2005
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R. 81, nr 3
89-92
EN
A method of determinination reachability subspace of the positive two dimension systems described by Roesser model using digraph theory is proposed. A procedure for computation of the reachability subspace is also proposed. The procedure illustrated by a numerical example.
PL
Przedstawiono metodę wyznaczania obszaru osiągalności dodatnich układów dwuwymiarowych opisanych za pomocą modelu Roessera. Zaproponowaną procedurę zilustrowano prostym przykładem numerycznym.
15
Content available remote Stability of positive linear discrete-time systems
James G., Rumchev V.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2005
|
Vol. 53, nr 1
1-8
EN
The main focus of the paper is on the asymptotic behaviour of linear discrete-time positive systems. Emphasis is on highlighting the relationship between asymptotic stability and the structure of the system, and to expose the relationship between null-controllability and asymptotic stability. Results are presented for both time-invariant and time-variant systems.
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