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1
Content available remote

Descriptor fractional linear systems with regular pencils

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2013
|
tom 23
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nr 2
309-315
EN
Methods for finding solutions of the state equations of descriptor fractional discrete-time and continuous-time linear systems with regular pencils are proposed. The derivation of the solution formulas is based on the application of the Z transform, the Laplace transform and the convolution theorems. Procedures for computation of the transition matrices are proposed. The efficiency of the proposed methods is demonstrated on simple numerical examples.
2
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A realization problem for positive continuoustime systems with reduced numbers of delays

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2006
|
tom 16
|
nr 3
325-331
EN
A realization problem for positive, continuous-time linear systems with reduced numbers of delays in state and in control is formulated and solved. Sufficient conditions for the existence of positive realizations with reduced numbers of delays of a given proper transfer function are established. A procedure for the computation of positive realizations with reduced numbers of delays is presented and illustrated by an example.
3
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Pointwise completeness and pointwise degeneracy of positive fractional descriptor continuous-time linear systems with regular pencils

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2015
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tom 25
|
nr 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.
4
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Reduced-order fractional descriptor observers for a class of fractional descriptor continuous-time nonlinear systems

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2016
|
tom 26
|
nr 2
277-283
EN
Fractional descriptor reduced-order nonlinear observers for a class of fractional descriptor continuous-time nonlinear systems are proposed. Sufficient conditions for the existence of the observers are established. The design procedure for the observers is given and demonstrated on a numerical example.
5
Content available remote

Singular fractional linear systems and electrical circuits

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2011
|
tom 21
|
nr 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.
6
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Locally positive nonlinear systems

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2003
|
tom 13
|
nr 4
505-509
EN
The notion of locally positive nonlinear time-varying linear systems is introduced. Necessary and sufficient conditions for the local positiveness of nonlinear time-varying systems are established. The concept of local reachability in the direction of a cone is introduced, and sufficient conditions for local reachability in the direction of a cone of this class of nonlinear systems are presented.
7
Content available remote

Independence of asymptotic stability of positive 2D linear systems with delays of their delays

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2009
|
tom 19
|
nr 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.
8
Content available remote

Bilateral polynomial equations with unimodular right-hand-side matrices

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2003
|
tom 13
|
nr 4
511-514
EN
Necessary and sufficient conditions are established for the existence of a solution to some bilateral polynomial matrix equations with unimodular right-hand-side matrices. A procedure for the computation of the solution is derived and illustrated by a numerical example. Two examples of applications of bilateral polynomial matrix equations are presented.
9
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A relation between the concepts of transposition, similarity, and symmetry, as well as matrix equations

100%
Kaczorek T.
Mathematica Applicanda
|
2005
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tom 33
|
nr 47/06
EN
The author studies relationships among the notions of transpose, similarity, and symmetrization of matrices. It is shown that a square matrix is similar to its transpose, and that there exists a matrix that, simultaneously, carries this similarity transformation and symmetrizes the matrix. Furthermore, some equalities involving adjoint matrices are established as well. The proofs of the results are formulated for complex matrices, but they are valid also for other algebraically closed fields.
10
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Content available

Positivity and stability of fractional descriptor time-varying discrete-time linear systems

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2016
|
tom 26
|
nr 1
5-13
EN
The Weierstrass-Kronecker theorem on the decomposition of the regular pencil is extended to fractional descriptor timevarying 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.
11
Content available remote

Externally and internally positive time-varying linear systems

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2001
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tom 11
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nr 4
957-964
EN
The notions of externally and internally positive time-varying linear systems are introduced. Necessary and sufficient conditions for the external and internal positivities of time-varying linear systems are established. Moreover, sufficient conditions for the reachability of internally positive time-varying linear systems are presented.
12
Content available remote

Computation of realizations composed of dynamic and static parts of improper transfer matrices

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2007
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tom 17
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nr 1
23-25
EN
The problem of computing minimal realizations of a singular system decomposed into a standard dynamical system and a static system of a given improper transfer matrix is formulated and solved. A new notion of the minimal dynamical-static realization is introduced. It is shown that there always exists a minimal dynamical-static realization of a given improper transfer matrix. A procedure for the computation of a minimal dynamical-static realization for a given improper transfer matrix is proposed and illustrated by a numerical example.
13
Content available remote

Fractional positive continuous-time linear systems and their reachability

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2008
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tom 18
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nr 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.
14
Content available remote

Minimum energy control of positive continuous-time linear systems with bounded inputs

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2013
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tom 23
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nr 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
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Content available

Positivity and stabilization of 2D linear systems

100%
Kaczorek T.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2009
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tom 29
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nr 1
43-52
EN
The problem of finding a gain matrix of the state-feedback of 2D linear system such that the closed-loop system is positive and asymptotically stable is formulated and solved. Necessary and sufficient conditions for the solvability of the problem are established. It is shown that the problem can be reduced to suitable linear programming problem. The proposed approach can be extended to 2D linear system described by the 2D Roesser model.
16
Content available remote

Minimum energy control of fractional positive continuous-time linear systems with bounded inputs

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2014
|
tom 24
|
nr 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.
17
Content available remote

Similarity transformation of matrices to one common canonical form and its applications to 2D linear systems

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2010
|
tom 20
|
nr 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.
18
Content available remote

An extension of the Cayley-Hamilton theorem for nonlinear time-varying systems

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2006
|
tom 16
|
nr 1
141-145
EN
The classical Cayley-Hamilton theorem is extended to nonlinear time-varying systems with square and rectangular system matrices. It is shown that in both cases system matrices satisfy many equations with coefficients being the coefficients of characteristic polynomials of suitable square matrices. The proposed theorems are illustrated with numerical examples.
19
Content available remote

Infinite eigenvalue assignment by an output feedback for singular systems

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2004
|
tom 14
|
nr 1
19-23
EN
The problem of an infinite eigenvalue assignment by an output feedback is considered. Necessary and sufficient conditions for the existence of a solution are established. A procedure for the computation of the output-feedback gain matrix is given and illustrated with a numerical example.
20
Content available remote

Canonical forms of singular 1D and 2D linear systems

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2003
|
tom 13
|
nr 1
61-72
EN
The paper consists of two parts. In the first part, new canonical forms are defined for singular 1D linear systems and a procedure to determine nonsingular matrices transforming matrices of singular systems to their canonical forms is derived. In the second part new canonical forms of matrices of the singular 2D Roesser model are defined and a procedure for determining realisations in canonical forms for a given 2D transfer function is presented. Necessary and sufficient conditions for the existence of a pair of nonsingular block diagonal matrices transforming the matrices of the singular 2D Roesser model to their canonical forms are established. A procedure for computing the pair of nonsingular matrices is presented.
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