Controllability and observability of conformable fractional finite dimensional linear systems

Abstract

In this research paper, we investigate the controllability and observability of continuous-time fractional dynamical systems. The characterisation of all fractional continuous one-parameter semi groups on a finite dimensional vector space as fractional matrix-valued exponential functions is given. This characterisation is used to express the solution of continuous-time fractional dynamical system. Furthermore, we analyse controllability and observability of linear conformable fractional systems in the mathematical context of linear operators on a finite dimensional vector space. By defining the controllability and observability maps, we establish necessary and sufficient conditions for controllability and observability of this type of systems. We also present the Hautus test for controllability and observability, and introduce controllability and observability Gramians, both for finite and infinite time interval. Finally, we present several illustrative applications and examples to validate all the obtained results.

Keywords:

• Fractional calculus
• conformable fractional systems
• controllability
• observability
• fractional semigroups

Disclosure statement

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

Additional information

Notes on contributors

M. Ouyadri

Mourad Ouyadri received his MASTER degree in mathematical engineering from Chouaib Doukkali University in 2015, followed by a Ph.D in control theory for positive systems from the same university in 2022. His primary research focus lies within the field of system and control theory, encompassing topics such as finite and infinite-dimensional system theory, controllability, observability, positive systems, stability analysis, optimal control theory, and fractional systems.

A. Binid

Abdellaziz Binid received his MASTER degree from Sultan Moulay Slimane University in Applied Mathematics in 2014, the Ph.D. degree in control theory for distributed systems from Chouaib Doukkali University in 2020. He was a Visiting Postdoc at Qatar University, Doha, Qatar in 2023. His main research interest is in the area of system and control theory include infinite dimensional system theory, observer design, positive systems, linear-quadratic optimal control, dynamical analysis and control of transport-reaction chemical processes and fractional systems.