A new Solution of the Fractional-order Singular Linear Time-Invariant System

A new solution, the distributional solution of the fractional-order singular linear time-invariant control system (FOSLTIS) is studied in the paper. On the basis of proposing the definition of the Caputo fractional derivatives of Dirac function δ ( t ), the Laplace transform L [ 0 C D t α δ ( t ) ] = s α ( α ∈ R + ) was obtained, then L − 1 ( s α ) = 0 C D t α δ ( t ) ( α ∈ R + ) was derived. With regard to FOSLTIS, the system was decomposed into fast subsystem and slow subsystem by restricted equivalent transformation. Using Laplace transformation and taking L − 1 ( s α ) = 0 C D t α δ ( t ) ( α ∈ R + ) into account, distributional solution for the fast subsystem was obtained. Two examples illustrated the correctness of the distributional solution of the FOSLTIS.