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5 International Journal of Applied Mathematics and Computer Science

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Positivity of fractional descriptor linear discrete-time systems

Kaczorek Tadeusz

International Journal of Applied Mathematics and Computer Science

2019

Vol. 29, no. 2

305--310

The positivity of fractional descriptor linear discrete-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 discrete-time systems are established. The discussion is illustrated with numerical examples.

A reference trajectory based discrete time sliding mode control strategy

Bartoszewicz Andrzej, Adamiak Katarzyna

International Journal of Applied Mathematics and Computer Science

2019

Vol. 29, no. 3

517--525

This study presents a new, reference trajectory based sliding mode control strategy for disturbed discrete time dynamical systems. The desired trajectory, which is generated externally according to an existing switching type reaching law, determines the properties of the emerging sliding motion of the system. It is proved that an appropriate choice of the trajectory generator parameters ensures the existence of the quasi-sliding motion of the system according to the definition by Gao et al. (1995) in spite of the influence of disturbances. Moreover, the paper shows that the application of the desired trajectory based reaching law results in a significant reduction in the quasi-sliding mode band width and errors of all state variables. Therefore, in comparison with Gao’s control method, the system’s robustness is increased. The paper also presents an additional modification of the reaching law, which guarantees a further reduction in the quasi-sliding mode band in the case of slowly varying disturbances. The results are confirmed with a simulation example.

Stability of interval positive fractional discrete-time linear systems

Kaczorek T.

International Journal of Applied Mathematics and Computer Science

2018

Vol. 28, no. 3

451--456

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.

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

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.

Realization problem for positive multivariable discrete-time linear systems with delays in the state vector and inputs

Kaczorek T.

International Journal of Applied Mathematics and Computer Science

2006

Vol. 16, no 2

169-174

The realization problem for positive multivariable discrete-time systems with delays in the state and inputs is formulated and solved. Conditions for its solvability and the existence of a minimal positive realization are established. A procedure for the computation of a positive realization of a proper rational matrix is presented and illustrated with examples.