Stability robust for fractional generalized multi-dimensional state–space models

Abstract

In this work, we consider a new class of generalized fractional linear multidimensional state–space systems described by the Roesser model. We discuss a novel technique for analysing robust stability, focusing specifically on the stability of the closed-loop system in terms of the ${\mathcal{H}}_2$ and ${\mathcal{H}}_{\mathrm{\infty }}$ norms. Both discrete-time and continuous-time cases are addressed across various regions of the complex plane. An extension of the bounded real lemma is proposed, dealing with both continuous and discrete cases. This lemma is used to provide sufficient conditions, in the form of linear matrix inequalities, to ensure stability margin for the perturbed system. Motivating examples are presented to demonstrate the effectiveness of our main results.

Keywords:

• Fractional dD systems
• stability radius
• stability region
• bounded real lemma
• linear matrix inequalities

Mathematics Subject Classifications:

• 39A30
• 34D23
• 34D20
• 34K37
• 35R11

Acknowledgments

This paper presents research results of the ACSY-Team (Analysis & Control systems team) and of the doctorial training on the Operational Research from the Pure and Applied mathematics Laboratory, UMAB, and Decision Support funded by the General Directorate for Scientific Research and Technological Development of Algeria (DGRSDT) and supported by National Higher School of Mathematics (NHSM), University of Mostaganem Abdelhamid Ibn Badis (UMAB) and initiated by the concerted research project on Control and Systems theory (PRFU Project Code C00L03UN270120200003).

Disclosure statement

No potential conflict of interest was reported by the author(s).