Nonlinear Dyn DOI 10.1007/s11071-017-3658-z ORIGINAL PAPER New alternatives for analog implementation of fractional-order integrators, differentiators and PID controllers based on integer-order integrators Carlos Muñiz-Montero · Luis V. García-Jiménez · Luis A. Sánchez-Gaspariano · Carlos Sánchez-López · Víctor R. González-Díaz · Esteban Tlelo-Cuautle Received: 20 April 2016 / Accepted: 5 July 2017 © Springer Science+Business Media B.V. 2017 Abstract In this work, we propose an alternative for the circuital realization of analog fractional-order dif- ferentiators and integrators without using ladder net- works. 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 fil- ters. The advantage of the proposed implementation is the requirement of only simple analog design blocks, C. Muñiz-Montero · L. V. García-Jiménez Electronics and Telecommunications Department, Universidad Politécnica de Puebla, 72640 Cuanalá, Puebla, Mexico e-mail: carlos.muniz@uppuebla.edu.mx L. V. García-Jiménez e-mail: luis.garcia@uppuebla.edu.mx L. A. Sánchez-Gaspariano · V. R. González-Díaz Faculty of Electronics, Benemérita Universidad Autónoma de Puebla, Jardines de San Manuel, 72592 Puebla, Mexico e-mail: luis.sanchezgas@correo.buap.mx V. R. González-Díaz e-mail: vicrodolfo.gonzales@correo.buap.mx C. Sánchez-López Department of Electronics, Universidad Autónoma de Tlaxcala, 90300 Apizaco, Tlaxcala, Mexico e-mail: carlsan@ieee.org E. Tlelo-Cuautle (B ) Department of Electronics, INAOE, Santa María Tonantzintla, 72840 Puebla, Mexico e-mail: etlelo@inaoep.mx E. Tlelo-Cuautle Computer Department, CINVESTAV, San Pedro Zacatenco, 07360 Mexico City, Mexico such as integrators (of integer order), differential ampli- fiers and two-input adder amplifiers. Most important, contrarily to other reported solutions, the proposed real- ization can be fulfilled using commercially available resistors and capacitors, with a reduced number of cal- culations, 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 val- idate the proposal of implementation, and as example of application, we present simulations (HSPICE, MAT- LAB) 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 pro- posed alternative of implementation presents advan- tages regarding a Cauer-network-based realization in terms of number of active and passive elements, num- ber of passive elements with non-commercial available values and design complexity. Keywords Fractional calculus · Fractional-order PID controllers · Field-programmable analog array 1 Introduction The fractances are circuit elements which are charac- terized by a magnitude response with roll-off ±20α decibels by decade and a constant-phase response at 123