Implementation of Continuous‐Time Fractional‐Order Integrators, Derivatives, and Systems Using Takenaka‐Malmquist Functions

ABSTRACT The paper introduces a novel definition of the fractional‐order integrator, which is the counterpart to the Riemann‐Liouville one. The integrator is based on the generalized orthonormal functions approach, specifically the Takenaka‐Malmquist functions. The key feature of the introduced definition is its finite‐length approximation, which effectively models the continuous‐time integrator in the chosen frequency bound. Based on the above model, the approximation of the fractional‐order Riemann‐Liouville and Caputo derivatives and the continuous‐time non‐commensurate fractional‐order system are introduced. The paper also presents a poles' generating algorithm for the Takenaka‐Malmquist functions. Furthermore, through the number of simulation experiments, the paper demonstrates the high efficiency of the introduced approaches in modeling fractional‐order integrator, derivative, and non‐commensurate systems.