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1
Content available Transfer matrices with positive coefficients of descriptor linear systems
Kaczorek Tadeusz, Sajewski Łukasz
Journal of Automation, Electronics and Electrical Engineering
|
2024
|
Vol. 6, No. 1
7--17
EN
Transfer matrices with positive coefficients of descriptor positive linear continuous-time systems are addressed. Two methods of checking of the positivity of descriptor linear systems are proposed. It is shown that if the positive descriptor system is asymptotically stable then all coefficients of its transfer matrix are positive.
2
Content available Stability of convex linear combinations of continuous-time and discrete-time linear systems
Kaczorek Tadeusz
Archives of Control Sciences
|
2023
|
Vol. 33, No. 4
789--799
EN
The asymptotic stability of the convex linear combination of continuous-time and discrete-time linear systems is considered. Using the Gershgorin theorem it is shown that the convex linear combination of the linear asymptotically stable continuous-time and discretetime linear systems is also asymptotically stable. It is shown that the above thesis is also valid (even simpler) for positive linear systems.
3
Content available New stability tests for fractional positive descriptor linear systems
Kaczorek Tadeusz, Ruszewski Andrzej
Archives of Control Sciences
|
2023
|
Vol. 33, No. 1
71--81
EN
The asymptotic stability of fractional positive descriptor continuous-time and discrete-time linear systems is considered. New sufficient conditions for stability of fractional positive descriptor linear systems are established. The efficiency of the new stability conditions are demonstrated on numerical examples of fractional continuous-time and discrete-time linear systems.
4
Content available New stability tests for positive descriptor linear systems
Kaczorek Tadeusz
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2022
|
Vol. 70, nr 3
art. no. e140688
EN
The asymptotic stability of positive descriptor continuous-time and discrete-time linear systems is considered. New sufficient conditions for stability of positive descriptor linear systems are established. The efficiency of the new stability conditions are demonstrated on numerical examples of continuous-time and discrete-time linear systems.
5
Content available Observers of fractional linear continuous-time systems
Kaczorek Tadeusz
Archives of Control Sciences
|
2022
|
Vol. 32, No. 1
73--84
EN
The concepts of full-order and reduced-order observers are extended to the fractional linear continuous-time systems. Necessary and sufficient conditions for the existence of the observers for fractional linear systems are established. Procedures for designing of the observers are given and illustrated by examples.
6
Content available Application of the Lagrange-Sylvester formula to computation of the solution to state equations of fractional linear systems
Kaczorek Tadeusz
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2021
|
Vol. 69, nr 2
art. no. e136729
EN
The Lagrange-Sylvester formula is applied to the computation of the solutions of state equations of fractional continuous-time and discrete-time linear systems. The solutions are given as finite sums with their numbers of components equal to the degrees of the minimal characteristics polynomials of state matrices of the systems. Procedures for computations of the solutions are given and illustrated by numerical examples of continuous-time and discrete-time fractional linear systems.
7
Content available Positivity of fractional descriptor linear continuous-time systems
Kaczorek T.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2019
|
Vol. 67, nr 2
229--233
EN
The positivity of fractional descriptor linear continuous-time systems is investigated. The solution to the state equation of the systems is derived. Necessary and sufficient conditions for the positivity of fractional descriptor linear continuous-time systems are established. The considerations are illustrated by numerical examples.
8
Content available Absolute stability of a class of positive nonlinear continuous-time and discrete-time systems
Kaczorek Tadeusz
Archives of Control Sciences
|
2019
|
Vol. 29, No. 1
157--167
EN
The positivity and absolute stability of a class of 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 nonlinear systems are also given.
9
Content available Absolute stability of a class of nonlinear systems with nonpositive linear parts
Kaczorek Tadeusz
Archives of Control Sciences
|
2019
|
Vol. 29, No. 2
247--258
EN
The positivity and absolute stability of a class of nonlinear continuous-time and discrete-time systems with nonpositive linear part 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 nonlinear systems are also given.
10
Content available Stability of interval positive continuous-time linear systems
Kaczorek T.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2018
|
Vol. 66, nr 1
31--35
EN
It is shown that the convex linear combination of the Hurwitz polynomials of positive linear systems is also the Hurwitz polynomial. The Kharitonov theorem is extended to the positive interval linear systems. It is also shown that the interval positive linear system described by state equation x ̇ = Ax, A ϵ ℜn×n, A1 ≥ A ≤ A2 is asymptotically stable if and only if the matrices Ak = 1, 2 are Hurwitz Metzler matrices.
11
Content available Responses of positive standard and fractional linear systems and electrical circuits with derivatives of their inputs
Kaczorek T.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2018
|
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.
12
Content available A new method for determination of positive realizations of linear continuous-time systems
Kaczorek T.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2018
|
Vol. 66, nr 5
605--611
EN
A new method for determination of positive realizations of given transfer matrices of linear continuous-time linear systems is proposed. Necessary and sufficient conditions for the existence of positive realizations of transfer matrices are presented. A procedure for computation of the positive realizations is proposed and illustrated by an example.
13
Content available A new method for computation of positive realizations of fractional linear continuous-time systems
Kaczorek T.
Archives of Control Sciences
|
2018
|
Vol. 28, No. 4
511--525
EN
A new method for computation of positive realizations of given transfer matrices of fractional linear continuous-time linear systems is proposed. Necessary and sufficient conditions for the existence of positive realizations of transfer matrices are given. A procedure for computation of the positive realizations is proposed and illustrated by examples.
14
Content available Relationship between the observability of standard and fractional linear systems
Kaczorek T.
Archives of Control Sciences
|
2017
|
Vol. 27, no. 3
441--451
EN
The relationship between the observability of standard and fractional discrete-time and continuous-time linear systems are addressed. It is shown that the fractional discrete-time and continuous-time linear systems are observable if and only if the standard discrete-time and continuous-time linear systems are observable.
15
Content available Stability conditions for linear continuous-time fractional-order state-delayed systems
Busłowicz M.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2016
|
Vol. 64, nr 1
3--7
EN
The stability problem of continuous-time linear fractional order systems with state delay is considered. New simple necessary and sufficient conditions for the asymptotic stability are established. The conditions are given in terms of eigenvalues of the state matrix and time delay. It is shown that in the complex plane there exists such a region that location in this region of all eigenvalues of the state matrix multiplied by delay in power equal to the fractional order is necessary and sufficient for the asymptotic stability. Parametric description of boundary of this region is derived and simple new analytic necessary and sufficient conditions for the stability are given. Moreover, it is shown that the stability of the fractional order system without delay is necessary for the stability of this system with delay. The considerations are illustrated by a numerical example.
16
Content available Analysis and comparison of the stability of discrete-time and continuous-time linear systems
Kaczorek T.
Archives of Control Sciences
|
2016
|
Vol. 26, no. 4
551--563
EN
The asymptotic stability of discrete-time and continuous-time linear systems described by the equations xi+1 = Ākxi and x(t) = Akx(t) for k being integers and rational numbers is addressed. Necessary and sufficient conditions for the asymptotic stability of the systems are established. It is shown that: 1) the asymptotic stability of discrete-time systems depends only on the modules of the eigenvalues of matrix Āk and of the continuous-time systems depends only on phases of the eigenvalues of the matrix Ak, 2) the discrete-time systems are asymptotically stable for all admissible values of the discretization step if and only if the continuous-time systems are asymptotically stable, 3) the upper bound of the discretization step depends on the eigenvalues of the matrix A.
17
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.
18
Content available Pointwise completeness and pointwise degeneracy of fractional descriptor continuous-time linear systems with regular pencils
Kaczorek T.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2015
|
Vol. 63, nr 1
169-–172
EN
Pointwise completeness and pointwise degeneracy of the fractional descriptor continuous-time linear systems with regular pencils are addressed. Conditions for the pointwise completeness and pointwise degeneracy of the systems are established and illustrated by an example.
19
Content available Robust stability of a class of uncertain fractional order linear systems with pure delay
Busłowicz M.
Archives of Control Sciences
|
2015
|
Vol. 25, no. 2
177--187
EN
The paper considers the robust stability problem of uncertain continuous-time fractional order linear systems with pure delay in the following two cases: a) the state matrix is a linear convex combination of two known constant matrices, b) the state matrix is an interval matrix. It is shown that the system is robustly stable if and only if all the eigenvalues of the state matrix multiplied by delay in power equal to fractional order are located in the open stability region in the complex plane. Parametric description of boundary of this region is derived. In the case a) the necessary and sufficient computational condition for robust stability is established. This condition is given in terms of eigenvalue-loci of the state matrix, fractional order and time delay. In the case b) the method for determining the rectangle with sides parallel to the axes of the complex plane in which all the eigenvalues of interval matrix are located is given and the sufficient condition for robust stability is proposed. This condition is satisfied if the rectangle multiplied by delay in power equal to fractional order lie in the stability region. The considerations are illustrated by numerical examples.
20
Content available Minimum energy control of positive time-varying linear systems
Kaczorek T.
Acta Mechanica et Automatica
|
2015
|
Vol. 9, no. 4
225--228
EN
The minimum energy control problem for the positive time-varying 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.
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