On the Electronic Realizations of Fractional-Order Phase-Lead-Lag Compensators with OpAmps and FPAAs

It is well known that the fractional-order phase-lead-lag compensators can achieve control objectives that are not always possible by using their integer-order counterparts. However, up to now one can find only a few of publications discussing the strategies for parameters’ tuning of these compensators, with only simulation results reported. This is due in part to the implicit difficulties on the implementation of circuit elements with frequency responses of the form \(s^{\pm \lambda }\) that are named “fractances”. In this regard, there exist approximations with rational functions, but the drawback is the difficulty to approximate the required values with the ones of the commercially-available resistances and capacitors. Consequently, fractional compensators have not been appreciated by the industry as it is in the academia. Therefore, motivated by the lack of reported implementations, this chapter is structured as a tutorial that deals with the key factors to perform, with the frequency-domain approach, the design, simulation and implementation of integer-order and fractional-order phase-lead-lag compensators. The circuit implementations are performed with Operational Amplifiers (OpAmps) and with Field Programmable Analog Arrays (FPAA). Emphasis is focused in the obtaining of commercially-available values of resistances and capacitors. Therefore, the design procedure starts with the use of equations that provide the exact and unique solution for each parameter of the compensator, avoiding conventional trial-and-error procedures. Then, five OpAmp-based configurations for integer-order and fractional-order realizations are described in terms of basic analog building blocks, such as integrators or differential amplifiers, among others. The corresponding design equations are also provided. Then, six examples are presented for both, OpAmp-based and FPAA-based implementations with the simulation and experimental results discussed regarding other results reported in the literature.