On the maximal output set of fractional-order discrete-time linear systems

: pp. 262–277
Received: September 03, 2021
Revised: November 26, 2021
Accepted: November 27, 2021

Mathematical Modeling and Computing, Vol. 9, No. 2, pp. 262–277 (2022)

Laboratory of Analysis, Modeling and Simulation, Hassan II University Casablanca
Laboratory of Analysis, Modeling and Simulation, Hassan II University Casablanca
Laboratory of Analysis, Modeling and Simulation, Hassan II University Casablanca
Laboratory of Analysis, Modeling and Simulation, Hassan II University Casablanca

In this paper, we consider a linear discrete-time fractional-order system defined by \[\Delta ^{\alpha }x_ {k+1}=Ax_k+B u_k, \quad k \geq 0, \quad x_{0} \in \mathbb{R}^{n};\] \[y_{k}=Cx_k, \quad k \geq 0,\] where $A$, $B$ and $C$ are appropriate matrices, $x_{0}$ is the initial state, $\alpha$ is the order of the derivative, $y_k$ is the signal output and $u_k=K x_k$ is feedback control.  By defining the fractional derivative in the Grunwald–Letnikov sense, we investigate the characterization of the maximal output set, $\Gamma(\Omega)=\lbrace x_{0} \in \mathbb{R}^{n}/y_{i} \in \Omega,\forall i \geq 0 \rbrace$, where $\Omega\subset\mathbb{R}^{p}$ is a constraint set; and, by using some hypotheses of stability and observability, we prove that $\Gamma(\Omega)$ can be derived from a finite number of inequations.  A powerful algorithm approach is included to identify the maximal output set; also, some appropriate algorithms and numerical simulations are given to illustrate the theoretical results.

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