The Distributional Solution of the Fractional-order Descriptor Linear Time-Invariant System and Its Application in Fractional Circuits

The distributional solution of the Fractional-Order Descriptor Linear Time-Invariant System (FODLTIS) and its application in the fractional circuits are studied. Based on proposing the definition ofc0Dtαδ(t), the Laplace transform ofc0Dtαδ(t) is investigated. Next, with regard to FODLTIS, the system is decomposed into fast subsystem and slow subsystem by restricted equivalent transformation. Solution for the slow subsystem is known. Using Laplace transform ofc0Dtαδ(t), distributional solution for the fast subsystem is derived. Combining solutions of the slow subsystem and the fast subsystem, the distributional solution of FODLTIS is successfully obtained. The structure of the distributional solution of the system shows that the superposition principle for the system still holds for FODLTIS. Finally, the distributional solution of FODLTIS is applied in the fractional RC circuit, fractional RL circuit, and the fractional LC circuit with ideal operational amplifier. Numerical solution and figures are made to verify the correctness and stability of the solution.