Controllability, observability, and stability of φ-conformable fractional linear dynamical systems

This paper focuses on two main aspects. Firstly, we introduce a generalized form of conformable fractional derivatives called phi-conformable fractional derivatives C-delta,C-phi along with their corresponding integrals I-delta,I-phi. We also present several theorems related to these derivatives. Secondly, we investigate the observability, controllability, and stability of a fractional dynamical system known as the phi-conformable fractional dynamical system (phi-CF-DS). We establish the connection between controllability, the controllability matrix, and the fractional phi-conformable fractional differential Lyapunov equation (phi-CF-DLE). We prove that the phi-CF-DS is stable if and only if all eigenvalues of the matrix M have negative real parts. Moreover, we provide theorems regarding the observability of an phi-CF-DS. To demonstrate the efficacy of our results, we include illustrative examples.