New alternatives for analog implementation of fractional-order integrators, differentiators and PID controllers based on integer-order integrators

In this work, we propose an alternative for the circuital realization of analog fractional-order differentiators and integrators without using ladder networks. This alternative is obtained by a mathematical manipulation of a rational function in a similar way to the reported for the synthesis of the variable-state filters. The advantage of the proposed implementation is the requirement of only simple analog design blocks, such as integrators (of integer order), differential amplifiers and two-input adder amplifiers. Most important, contrarily to other reported solutions, the proposed realization can be fulfilled using commercially available resistors and capacitors, with a reduced number of calculations, and without negative impedance converters or inductors. In addition, the orders of the fractional derivative and integral can be modified just varying the gain of the differential amplifiers and adders. To validate the proposal of implementation, and as example of application, we present simulations (HSPICE, MATLAB) and experimental results of a first-order plus dead time plant controlled by fractional-order PI and PID controllers. The experimental results were obtained from a realization using field-programmable analog arrays. A comparison analysis highlights that the proposed alternative of implementation presents advantages regarding a Cauer-network-based realization in terms of number of active and passive elements, number of passive elements with non-commercial available values and design complexity.