Controllability and Observability of Non-homogeneous Granular Descriptor Fractional Dynamical Systems Applied in Electrical Circuit

This paper deals with the study of controllability and observability of fuzzy fractional descriptor dynamical system in terms of granular differentiability. The granular fuzzy solution for granular descriptor fractional dynamical system (GrDFDS) is obtained by using the Mittag-Leffler function and granular Laplace transform and the solution is represented in terms of the state transfer matrix. The controllability and observability of GrDFDS is analysed with the help of theorems using the controllability Gramian matrix and observability Gramian matrix. In order to illustrate the efficacy of our findings, we hereby give the controllability analysis of an engineering problem pertaining to the compatibility of a descriptor fractional electric circuit. The graph visually represents the outcome, indicating that the granular descriptor dynamical system of the electric circuit can be effectively controlled, leading to the successful transition of its state from the granular initial to the final state.