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Results on the controllability of Caputo’s fractional descriptor systems with constant delays

Beata Sikora

Bulletin of the Polish Academy of Sciences. Technical Sciences

2023

Vol. 71, nr 4

art. no. e146287

The paper investigates the controllability of fractional descriptor linear systems with constant delays in control. The Caputo fractional derivative is considered. Using the Drazin inverse and the Laplace transform, a formula for solving of the matrix state equation is obtained. New criteria of relative controllability for Caputo’s fractional descriptor systems are formulated and proved. Both constrained and unconstrained controls are considered. To emphasize the importance of the theoretical studies, an application to electrical circuits is presented as a practical example.

Controllability of linear convex combination of linear discrete-time fractional systems

Kaczorek Tadeusz, Klamka Jerzy, Dzieliński Andrzej

Bulletin of the Polish Academy of Sciences. Technical Sciences

2022

Vol. 70, nr 5

art. no. e143102

In this paper the controllability properties of the convex linear combination of fractional, linear, discrete-time systems are characterized and investigated. The notions of linear convex combination and controllability in the context of fractional-order systems are recalled. Then, the controllability property of such a linear combination of discrete-time, linear fractional systems is proven. Further, the reduction of an infinite problem of transition matrix derivation is reduced to a finite one, which greatly simplifies the numerical burden of the controllability issue. Examples of controllable and uncontrollable, single-input, linear systems are presented. The possibility of extension of the considerations to multi-input systems is shown.

Fractional order tube model reference adaptive control for a class of fractional order linear systems

Balaska Hanane, Ladaci Samir, Djouambi Abdelbaki, Schulte Horst, Bourouba Bachir

International Journal of Applied Mathematics and Computer Science

2020

Vol. 30, no. 3

501--515

We introduce a novel fractional order adaptive control design based on the tube model reference adaptive control (TMRAC) scheme for a class of fractional order linear systems. By considering an adaptive state feedback control configuration, the main idea is to replace the classical reference model with a single predetermined trajectory by a fractional order performance tube guidance model allowing a set of admissible trajectories. Besides, an optimization problem is formulated to compute an on-line correction control signal within specified bounds in order to update the system performance while minimizing a control cost criterion. The asymptotic stability of the closed loop fractional order control system is demonstrated using an extension of the Lyapunov direct method. The dynamical performance of the fractional order tube model reference adaptive control (FOTMRAC) is compared with the standard fractional order model reference adaptive control (FOMRAC) strategy, and the simulation results show the effectiveness of the proposed control method.