Fractional Proportional Linear Control Systems: A Geometric Perspective on Controllability and Observability

The paper presents a detailed analysis of control and observation of generalized Caputo proportional fractional time-invariant linear systems. The focus is on identifying controllable states and observable systems within the controllable subspace, null space, and unobservable subspace of the proposed system. The necessary conditions for the controllable subspace and the necessary and sufficient conditions for observability criteria are firmly established. The controllable subspace is treated geometrically as the set of controllable states, while the observable system is characterized by a zero unobservable subspace. The results are reinforced by examples and will immensely benefit future studies on fractional-order control systems.