Variable-, fractional-order discrete PID controllers

The fractional calculus is the area of mathematics that handles derivatives and integrals of any arbitrary order (fractional or integer, real or complex order) [1,2,3,4]. Nowadays it is applied in almost all areas of science and engineering. Here one can mention its numerous and successful applications in dynamical systems modeling and control with increasing number of studies related to the theory and application of fractional-order controllers, specially ones. In such controllers μ_k <;0 and v_k >; 0 denote the integration and differentiation order, respectively. Now research activities are focused on developing new analysis and closed-loop system synthesis methods for fractional-order controllers being an extension of classical control theory. In the fractional-order controller tuning there are two additional parameters μ_k <; 0 and v_k >; 0. This impedes the controller tuning procedure but leads to new (unattainable in classical PID control [5]) closed-loop system transient responses. The closed-loop system with fractional controller must satisfy typical requirements among which one can mention the system robustness due to the plant model uncertainties.