Solution of Time-Variant Fractional Differential Equations With a Generalized Peano–Baker Series

Abstract

Time-variant fractional systems have many applications. For example, they can be used for system identification of lithium-ion batteries. However, the analytical solution of the time-variant fractional pseudo state space equation is missing so far. To overcome this limitation, this letter introduces a novel matrix approach, namely the generalized Peano–Baker series, which is comparable to the transition matrix in the case of ordinary systems. Using this matrix, the solution of the time-variant fractional pseudo state space equation is derived. The initialization process is taken into account, which has been proven to be a crucial part for fractional operator calculus. Following this initialization, a modified definition of a fractional pseudo state is presented.