Characterizing maximum output sets in fractional-order discrete-time linear systems utilizing fractional feedback control

Document Type : Research Article

Author

AMINE El Bhih ^{1, 2}

¹ Multidisciplinary Research and Innovation Laboratory (LPRI), Moroccan School of Engineering Sciences (EMSI), Honories University. Casablanca, Morocco

² Laboratory of Analysis, Modeling and Simulation, Department of Mathematics and Computer Science, Faculty of Sciences Ben M’Sik, Hassan II University, Casablanca, BP 7955, Sidi Othman, Casablanca, Morocco

10.22067/ijnao.2025.94705.1693

Abstract

This paper examines a linear controlled discrete-time fractional-order system, as defined by 
$$
\left\{

\begin{array}{ccc}

\Delta ^{\gamma }x_{k+1} & = & Ax_{k}+B \Delta ^{\beta }u_{k} \\

x_{0} & = & x_{0}%

\end{array}%

\right. 
$$
The corresponding output, denoted as $y_{k} = Cx_{k}$, is intended to be stable, i.e., $\underset{k\rightarrow \infty }{\lim} y_k=0 $. More precisely, we suppose the existence of a fractional feedback control denoted as  $\Delta ^{\beta }u_{k}=\Delta^{\beta }Lx_{k}$, where the matrices $A$, $B$, $C$, and $L$ are suitably chosen. The characterization of the maximal output set $\Upsilon \left( \mathcal{R} \right)$ is investigated by defining the fractional derivative in the Grunwald-Letnikov sense. Specifically, $ \Upsilon \left( \mathcal{R} \right) =\left\{ x_{0}\in \mathbb{R} ^{n}/ y_{k} \in \mathcal{R},\text{ }\forall k\geq 0 \right\} $ where $\mathcal{R}$ $ \subset $ $\mathbb{R}^n$ represents a constraint set. It can be shown, through the use of stability and observability hypotheses, that a finite number of inequations can derive $ \Upsilon \left( \mathcal{R} \right)$. The maximal output set is determined using an advanced algorithmic approach. In order to clarify the theoretical results, relevant algorithms and numerical simulations have been included.

Keywords

Fractional Order Systems

Stability

Observability

asymptotically stable

Discrete-time System

Subjects

• Control and Optimization

---

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