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1
Content available Pointwise completeness and pointwise degeneracy of fractional standard and descriptor linear continuous-time systems with different fractional orders
Kaczorek Tadeusz, Sajewski Łukasz
International Journal of Applied Mathematics and Computer Science
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2020
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Vol. 30, no. 4
641--647
EN
Descriptor and standard linear continuous-time systems with different fractional orders are investigated. Descriptor systems are analyzed making use of the Drazin matrix inverse. Necessary and sufficient conditions for the pointwise completeness and pointwise degeneracy of descriptor continuous-time linear systems with different fractional orders are derived. It is shown that (i) the descriptor linear continuous-time system with different fractional orders is pointwise complete if and only if the initial and final states belong to the same subspace, (ii) the descriptor linear continuous-time system with different fractional orders is not pointwise degenerated in any nonzero direction for all nonzero initial conditions. Results are reported for the case of two different fractional orders and can be extended to any number of orders.
2
Content available Application of the Drazin inverse to the analysis of pointwise completeness and pointwise degeneracy of descriptor fractional linear continuous-time systems
Kaczorek Tadeusz, Ruszewski Andrzej
International Journal of Applied Mathematics and Computer Science
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2020
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Vol. 30, no. 2
219--223
EN
The Drazin inverse of matrices is applied to the analysis of pointwise completeness and pointwise degeneracy of fractional descriptor linear continuous-time systems. It is shown that (i) descriptor linear continuous-time systems are pointwise complete if and only if the initial and final states belong to the same subspace, and (ii) fractional descriptor linear continuous-time systems are not pointwise degenerated in any nonzero direction for all nonzero initial conditions. The discussion is illustrated with examples of descriptor linear electrical circuits.
3
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
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2018
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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.
4
Content available Decentralized stabilization of fractional positive descriptor continuous-time linear systems
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2018
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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.
5
Content available Minimum energy control of descriptor fractional discrete-time linear systems with two different fractional orders
Sajewski Ł.
International Journal of Applied Mathematics and Computer Science
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2017
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Vol. 27, no. 1
33--41
EN
Reachability and minimum energy control of descriptor fractional discrete-time linear systems with different fractional orders are addressed. Using the Weierstrass–Kronecker decomposition theorem of the regular pencil, a solution to the state equation of descriptor fractional discrete-time linear systems with different fractional orders is given. The reachability condition of this class of systems is presented and used for solving the minimum energy control problem. The discussion is illustrated with numerical examples.
6
Content available Reduced-order fractional descriptor observers for a class of fractional descriptor continuous-time nonlinear systems
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2016
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Vol. 26, no. 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.
7
Content available Positivity and stability of fractional descriptor time-varying discrete-time linear systems
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2016
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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.
8
Content available Fractional descriptor continuous-time linear systems described by the Caputo–Fabrizio derivative
Kaczorek T., Borawski K.
International Journal of Applied Mathematics and Computer Science
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2016
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Vol. 26, no. 3
533--541
EN
The Weierstrass–Kronecker theorem on the decomposition of the regular pencil is extended to fractional descriptor continuous-time linear systems described by the Caputo–Fabrizio derivative. A method for computing solutions of continuous-time systems is presented. Necessary and sufficient conditions for the positivity and stability of these systems are established. The discussion is illustrated with a numerical example.
9
Content available Robust sensor fault estimation for descriptor-LPV systems with unmeasurable gain scheduling functions: Application to an anaerobic bioreactor
López-Estrada F. R., Ponsart J. C., Theilliol D., Astorga-Zaragoza C. M., Camas-Anzueto J. L.
International Journal of Applied Mathematics and Computer Science
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2015
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Vol. 25, no. 2
233--244
EN
This paper addresses the design of a state estimation and sensor fault detection, isolation and fault estimation observer for descriptor-linear parameter varying (D-LPV) systems. In contrast to where the scheduling functions depend on some measurable time varying state, the proposed method considers the scheduling function depending on an unmeasurable state vector. In order to isolate, detect and estimate sensor faults, an augmented system is constructed by considering faults to be auxiliary state vectors. An unknown input LPV observer is designed to estimate simultaneously system states and faults. Sufficient conditions to guarantee stability and robustness against the uncertainty provided by the unmeasurable scheduling functions and the influence of disturbances are synthesized via a linear matrix inequality (LMI) formulation by considering H∞ and Lyapunov approaches. The performances of the proposed method are illustrated through the application to an anaerobic bioreactor model.
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
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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 Minimum energy control of fractional descriptor positive discrete-time linear systems
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2014
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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.
12
Content available Discretization of singular systems and error estimation
Karampetakis N. P., Karamichalis R.
International Journal of Applied Mathematics and Computer Science
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2014
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Vol. 24, no. 1
65--73
EN
This paper proposes a discretization technique for a descriptor differential system. The methodology used is both triangular first order hold discretization and zero order hold for the input function. Upper bounds for the error between the continuous and the discrete time solution are produced for both discretization methods and are shown to be better than any other existing method in the literature.
13
Content available Solution of the state equations of descriptor fractional discrete-time linear systems with regular pencils
Kaczorek T.
TTS Technika Transportu Szynowego
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2013
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R. 20, nr 10
415--422, CD
EN
A method for finding of the solutions of the state equations of descriptor fractional discrete-time linear systems with regular pencils is proposed. The derivation of the solution formula is based on the application of the Z transform and the convolution theorem. A procedure for computation of the transition matrix is proposed. The effectiveness of the proposed method is demonstrated on a simple numerical example.
PL
Podano metodę wyznaczania rozwiązań równań stanu deskryptowych układów dyskretnych rzędów niecałkowitych o pękach regularnych. Rozwiązanie to zostało wyprowadzone korzystając z przekształcenia zet i twierdzenia o transformacie splotu. Zaproponowano procedurę wyznaczania macierzy tranzycji tych układów. Proponowaną metodę zilustrowano przykładem numerycznym.
14
Content available Descriptor fractional linear systems with regular pencils
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2013
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Vol. 23, no. 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.
15
Content available D-stability for a class of discrete descriptor systems with multiple time delays
Pan S., Chen C., Hsieh J.
Control and Cybernetics
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2002
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Vol. 31, no 1
79-89
EN
In this paper, the research on discrete descriptor systems is extended to include discrete multiple time-delay descriptor systems. The impulse-free and D-stability problem for a class of discrete descriptor systems with multiple time delays is investigated. A delay-dependent criterion is first derived to guarantee that the system is proper. A delay-dependent stability criterion in terms of spectral radius is then presented to ensure the D-stability of the system. Furthermore, a delay-dependent criterion is proposed to guarantee that the system is nebular, impulse-free, and D-stable. Finally, a numerical example is provided to illustrate our main results.
16
Content available On descriptor systems and related linear quadratic problem
Gessing R., Czornik A.
Control and Cybernetics
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2000
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Vol. 29, no 1
79-89
EN
It is shown that a descriptor system under the condition of impulse controllability, Cobb (1984), may be converted, by means of linear transformations, to a system described in a state space and composed of state and output equations. The transfomations determine one to one correspondence between the solutions of both the systems. It is noted that the control in a feedback form may not determine a unique solution of the descriptor system what is often overlooked in many previous papers. It is also shown that the LCD problem formulated in a descriptor space for the impulse observable system, Cobb (1984); may be converted by means of linear transformations to the usual LQ problem formulated in the state space. It is stressed that the second problem may be regular even then, when the weighting matrix of the control, in the cost functional of the first problem, is singular. The proposed approach simplifies the calculations related to the LQ problem solution significantly.
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