Tuning of fractional order proportional integral/proportional derivative controllers based on existence conditions

Fractional order proportional integral and proportional derivative controllers are nowadays quite often used in research studies regarding the control of various types of processes, with several papers demonstrating their advantage over the traditional proportional integral/proportional derivative controllers. The majority of the tuning techniques for these fractional order proportional integral/fractional order proportional derivative controllers are based on three frequency-domain specifications, such as the open-loop gain crossover frequency, phase margin and the iso-damping property. The tuning parameters of the controllers are determined as the solution of a system of three nonlinear equations resulting from the performance criteria. However, as with any system of nonlinear equations, it might occur that for a certain process and with some specific performance criteria, the computed parameters of the fractional order proportional integral/fractional order proportional derivative controllers do not fall into a range of values with correct physical meaning. In this article, a study regarding this limitation, as well as the existence conditions for the fractional order proportional integral/fractional order proportional derivative parameters are presented. The method could also be extended to the more complex fractional order proportional-integral-derivative controller. The aim of this research is directed toward demonstrating that when designing fractional order proportional integral/fractional order proportional derivative controllers, the choice of the performance specifications should be done based on some specific design constraints. The article shows that given a specific process and open-loop modulus and phase specifications, the gain crossover frequency (or in general, a certain test frequency used in the design), specified as a performance specification, must be selected such that the process phase fulfills an important condition (design constraint). Once this is met, the proposed approach ensures that the tuning parameters of the fractional order controller will have a physical meaning. Illustrative examples are included to validate the results.